The Econometrics of Financial Markets
John Y. CamPgeU AndrewW.Lo I
A. Craig MacKinJay
Princeton University Press Princeton, New Jersey
C"I'yri\l,ht © 1',)97 hy I'rillc elOll Ulliw "ily I'n'" ('fI''' . 41 Willi ,,," SII ,'('1.
Publisht·" hy Princeton Univt"rsily Plince\()II. New Jersey \lilr)~\l
In the linite tl Kingdonl; prinCt.'tlln Unil' t'"ity
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Contents
List of Figures List of Tables Preface
xiii
xv xvii
Introduction 3 1.1 Organization of the Book 4 1.2 Useful Background. . . . {) 1.2.1 Mathematics Background {) 1.2.2 Probability and Statistics Background (j 1.2.3 Finance Theory Background 7 1.3 Notation................ H 1.1 Prices, Returns, and Compollnding . 9 9 1.1.1 Defmitions and Conventions. 1.4.2 The Marginal, Conditional, anclJoint Distribution of Returns. . . . . . . . . . . . . . . . . . . . . . .. 13 I.:) Market Efliciency . . . . . . . . . . . . . . . . . . . . . .. 20 1.5.1 Efficient MarkrL~ and the l.aw of Iterated . . . . . . 22 Expectations .. . . . . . . . Is Market EffIciency Testable? 24 TIle Predictability of Asset Returns 2.1 The Random Walk Ilypotheses . . . . . . . . . . . . . 2.1.1 The Random Walk I: lID Incremenl~ . . . . . . 2.1.2 The Random Walk 2: Independent Increment!! 2.1.3 The Random Walk 3: Uncorrclated Increments Tests of Random Walk I: lID Increlllents . 2.2.1 Traditional Statistical Tests . . . . . 2.2.2 SCf)uences and Reversals, and Runs
27 2R 31 32 33
viii COII/m!.1 2.~
Test s of Ran doll l Wal k 2: Inde pcn den t Incr eme nts 2.3.1 Filte r Rule s . . . . . . . . . ........ . 2.3.2 Tech nica l Analysi~ . . . . . ....
..... . Test s of Ran doll l Wal k :\: Ul\c ond .\led Incr emc nts 2.4.1 Aut ocor rela tion Coe flici ents 2.4.2 Port man teau Stat istic s 2.'1.3 Vari ance Rali m . . . . . . . Lon g-Il oriz on Rell lrBs . . . . . . . 2::>.1 Prob lem s with LOllg-1 [oriz on Infc renn :s Test s For Lon g-Ra nge Dep end encc . . . . . . . 2.ti.l Exal llple s of I.on g-Ra ngc lkpc /lclc ncc . 2.6.2 The Hur sl-M ancl clbr ot Resc aled ·Ran gc Stat istic Uni l ROOl Test s . . . . . . . Rec cnt EmpiriC
2,4
2.!:i
2.G
2.7 2.H
2.n
3
Mar ket Mic rost ruct ure 3.1 { ~
I
3.2 I I I
3.3
! I
'j I
3.4
I
'} r, ••.• 1
\
I
Non sync hron olls Trad ing . . ....... . 3.1.1 A Mod el of NOl lSyn c!tro llous Trad ing 3.1.2 Exte nsio ns and Gen eral izat ions The Bid-Ask Spr e'ld . . . . . . . . . . . . . 3.2.1 Rid-Ask Bou ncr . . . . ...... . 3.2.2 CompOnel\L~ of the \)id-A~k Spre ad Mod elin g Tran sact ions Data . ... . 3.3.1 Mot ivati on . . . . . . . . . . . 3.3.2 Rou ndin g
4 Eve JStu dy Analysis 4.\ ; Out line of an Evel lt Stud y ......... . 'I.~ All Exa mpl e of an Evcn t Stud y . ...... . 4.:-1 Mo
mal I'crfOnll'
41
·IH :):) :>7 !/l
!",!l li2 li4 Ii:> titi ti8 74 7H KO
83 \H K:> ~)H !)~)
W\ lin 107 10K 114 122 12H 12K I:H 1:I!i 144
ix
CCJllifll 1.1
4..1
4.:> Hi 4.7 4.K 4 .~l
4.10
'1.:~.3 Other Statistical Mlldds ,1.::1.'1 Ecollomic Modds . . . . Me;\sul'illg and AnalYl.illg Abl\ormal Retlll"lls . 4.4.1 Estimation of the Market Model . . . . '1.4.2 Statistical Properties of Ahllonnal Rellll'lls 4.4.:~ Aggregation of AIlllormal Returns . . . . . 4.4.4 Sensitivity to Nor11lal Rl'llll'l\ Modd . . . . 4.4.5 CARs for the Eamiligs-Allllolllll'l'ment Example 4A.G Inferences with Cilisterillg Moclifying the NullllYl'othesis Allalysis of Power . . . Nonparametric Tests . Cross-Sectional Models Further Issues . . . . . 4.9.1 Role of the Sampling lntel'val 4.9.2 Inferences with Evelll-Date Uncertainty 4.9.3 Possible Biases. Conclusion
5 The Capital Asset Pricing Model
:1. I S.~
:>.:~
:>.4 5.:> S.li :>.7
?i.H :).~l
6
Review or the CAI'M .. . Results from Efficient-Set Mathematics . . . . . . Statistical Framework for Estimatioll and Testing. 5.3.1 Sharpe-Lintner Version :l.:t2 nlack Versioll Size of Tests . . . . . . . . . . Power of Tests . . . . . . . . . Nonnormal and Non-lID Returns Implementation of Tests . . . . . :>.7.1 Summary of Empirical Evidence 5.7.2 Illustrative Implementation :l.7.~ Unobservability of the Market Portfolio Cross-Sectional Regressions Conclusion . . . . .
Multifactor Pricing Models Theoretical Background . . . . . . . . . . . . . Estimation and Testing . . . . . . . . . . . . . . 1;.2.1 Portfolios as Factors with a Riskfrcc A.sset li.2.2 Portfolios as FaClors without a Riskf'ree Asset ti.~.:~ Macroeconomic Variables as Factors . . . . . (i.!!A Factor i'ortfillios Spallllilll-\" the l\kall-V;triance Frolllicr .. ' . . . . . .
G.I (;.2
I:)~
15ti [57 15H
159 IGO IG~
1U3 ltiG
107 IGH
172 173
175 175 176 177 178
I
181
IHI 184 IH8 IH9 1%
203 204 208 211 211 21~
213
215 217 219
219 222 223
224 226 .22H
•
IU li,4
1i5 li.li
1i.7 7
Estimation of Risk PrellIia and Expcelcd Returns Sdcelion of FaclOrs . . . . . . li.'1.1 Statistical Approaches .. li..l.~ NlImher of FaCiors . . . {i.'l.:\ TllI'oretical Approach('s Empirical R('sults . . . . . . . . Interpreting Deviations from Exact Factor Pricing {i.{i.1 Exael Factor Pricing Models, Mean-Variance Analysis, and the Optimal Orthogonal Portfillio li.li.2 Squared Sharpc' Ratios . . . . . . . . . . . . . . (i.fi.:~ Implications fi)J' Sc'parating Altcrnativc Th('mil's Conclusion .................... .
Present-Value Relations
253
7.1
~:I·1
The Relation I)('twec'n Prices, Dividends, and Returns 7.1.1 'I'lli'I .inear Pn'sc'nt-Value- Relation with Constant
Expected Relllnls . . . . . . . . . . . . . . . . . . . ~:);) Rational Blibbles . . . . . . . . . . . . . . . . .. :!:.~ 7.1.:1 All Approxilllalc'l'n'sc'nl-Vahlc Relation wilh TillleVarying Exp('cled R('lurns . . . . . . . . . . . . . . . <'(ill 7.1.4 Prices and Retul'lls in a Simple Example ~(i·1 Present-ValliI' Relations and US Stock Pricc Ikhavior ~(i7 7.~.1 I.ollg-iioriwil Regressions . . . ~(;7 7.~.'2 Volatility 'Ii'sts . . . . . . . . . . n:; 7,'23 Vector AlilOregn'ssive Mc,tllOds Conclilsion 7.1.~
7.'2
8
Intertemporal Equilibrium Models H.I H.'2
HA
The- Stochastic Discounl FaCIOI' . . . . . . . . . . . . H.I.I Volatility BOllncls . . . . . . . . . . . . . . . . Consumptioll-Basl'd Asset Pricing with Pown Utility. H.:!. I Powc'r Utility ill a I.ogllol'lnal Model. . . . H.'2.'2 Power lItility and (;('ll\'rali'.cll Method of MOlllents . . . . . . . . . . . . . . . . . . . . . . . . Market Friel ions H.:I.l Market FriClion~ and Il;ulsclI:)agalinathan Boullds . . . . . . . . . . . . . . . . . . . . . H3.'2 Markc,t Frictions and Aggre-gatc Consllmption Data . . . . . . . . . . . . . . . . . . . . . . . More (;('nnall 1tility FunC'lions . . . . . . . H..t.1 Iiallit Forillation . . . . . . . . . . . H.·I.'2 Psychological Mockls or Pr('fucnc('s ( :ondllsion
291 2~l:\
2% :10,\ :1(Hi
:q 4
:11:. :1 I Ii
9
Derivative Pricing Models 339 9.1 Brownian Motion . 341 341 I 9.1.1 Constructinr; Brownian Motion . 9.1.2 Stochastic Differential Equations 346 [ 349 . 9.2 A Brief Review of Derivative Pricinr; Methods. 9.2.1 The Black-Scholes and Merton Approach . 350 9.2.2 The Martingale Approach . . . . . . . . . 354 !1.3 Implementing Parametric Option Pricing Models 355 9.3.1 Parameter Estimation of Asset Price Dynamics 356 9.3.2 Estimating (j in the Black-Scholes Model . . . 361 9.3.3 Quantifying the Precision of Option Price Estimators . . . . . . . . . . . . . . . . . . . . . . . . 367 9.3.4 The Effects of Asset Return Predictability. 369 9.3.5 Implied Volatility Estimators . . . . . . . . . . .. 377 9.3.G Stochastic Volatility Models. . . . . . . . . . . .. 379 9.4 Pricing Path-Dependent Derivatives Via Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 9.1.1 Discrete Versus Continuous Time . . . . . 383 9.1.2 How Many Simulations to Perform . . . . 384 9.4.3 Comparisons with a Closed-Form Solution 384 386 9.4.4 Computational Efficiency . 390 9.1.5 Extensions and Limitations. 391 Conclusion '"
10 Fixed-Income Securities 10.1 Basic Concepts . 10.1.1 Discount Bonds 10.1.2 Coupon Bonds 10.1.3 Estimating the Zero-Coupon Term Structure 10.2 Interpreting the Term Structure of Interest Rates 10.2.1 The Expectations Hypothesis . . . . . . 10.2.2 Yield Spreads and Interest Rate Forecasts 10.3 Conclusion . . . . . . . . . . . . . . . . . . . .
395 396 397 401 409 413 413 418 423
11 Term-Structure Models 11.1 Affine-Yield Models . 11.1.1 A Homoskedastic Single-Factor Model 11.1.2 A Square-Root Single-Factor Model 11.1.3 A Two-Factor Model . . . . . . . . . 11.1.4 Beyond Affine-Yield Models . . . . 11.2 Fitting Term-Structure Models to the Data 11.2.1 Real Bonds, Nominal Bonds, and Innation 11.2.2 Empirical Evidence on Affine-Yield Models
427
428 429 435 438 441
442 442 445
xii
(.'U/I 11'11 1.1
11.3
11.4
Pricing Fixed-Incomc Dcrivativc Sccuritics .. 11.3.1 Filling the Currcnt Terlll Structure Exactly 11.3.2 Forwards and Futures . . . . . . . . . . . . 11.3.3 Optioll Pricing in a Tcrlll-Structure Modc1 Conclusion . . . . . . . . . . . . . . . . . . . . .
12 Nonlinearities in Financial Data Nonlinear Structure in Univari
12.1
467 11iK
470 47:) 479 4HI 490
494 49K !",oo :)02 504
!i07 :)12 512 :) I Ii 51H
!ilK 51\1 :)~:{
:,24
1
Appendix A Linear Instrumelllal Variablcs . . . . . . . . . . Generalized Mcthod of MOlllcnts . . . . . . . . A.2 A.3 Serially Correlated and f Icteroskeoastic Errors. A.4 GMM and MaximulII Likelihood . . . . . . . .
I
References
541
Author Index
587
Subject Index
597
List of Figures
1.1 I.~
<~.l
:t2
:).:) ~.-1
'1.1 4.2
-1.3
4.4
:),1 :i.~
(i.l
Dividend Payment Timing Convention . . . . . . . . Comparison 0(' Stable and Normal Density Functions
12 II:!
Nontrading-Induced AutocolTelations . . . . . . . . Histogram of Daily Price Fractions and Price Changes for Five NYSE Stocks from January 2, I~)~)O to December 31. 1992 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2-Histories of Daily Stock Returns luI' Five NYSE Stocks from .Ianuary2, 1990 to Deccmber:ll, 1992 .. . . . . . . . . . . TIll: Ordered Probit Model .
96
Tinlt.' l.ine for an Event Study (a) Plot of Cumulative Market-Model Abnormal Return for Earning Announcements; (Il) Plot of Cumulative Conslant-Mean-Return-Modcl Abnormal Return for Earning Announcements . . . . . . . . . . . . . . . . . . . . . . .. Power of Event-Study Test Statistic Jl to Reject the Null Hypothesis that the Abnormal Retllrn Is Zero. When the Square Root of the Average Variance of the Abnormal Return Across Firms is (a) 2% and (b) 4% . . . . . . . . . . . . . . . . . , Power of Evelll-Study Test Statistic Jl to Reject the Null Hypothesis that thc Abnormal Return is Zero. for Differelll Sampling Intervals, When the Square Root of the Average Variance of the Abnormal Return Across Finns Is 1% for the Daily Interval . . . . . . . . . . . . . . . . . . . . . . MinimulIl-Variance POrll'olios Without RisHree Asset Minimulll-Variance Portfolios With Riskfree Asset . . Di~trihlllions
III 113 125
157
165
171
176
187 189
for the CAPM Zero-I II tercept Tesl Statistic fur
FOIII' Ilv\lollH'~(,s
'lUI
7.1 7.'2 7.:1
H.I
H.2
H.~
!I. I !I.'2
I.og Real Slock 1'1 ice and Dividend Series, An""al US Data, lH72 to I ~)~ltl . . . . . . . . . . . . . . . . . . . . . . . . . . Log Real Stock Pric(' and Estimated Dividend Compo"e"t, Annual US Dala, I H7/i 10 I !'!H . . . . . . . . . . . . . . . .. I.og Dividend-Pric(, Ratio and ESlimaled Dividend COlllPO1I('nl, Annual US d;lIa, IH7G 10 Iq!H . . . . . . . . . . . . . (a) Mean-Standard Deviation Diagram for Asset Returns; (h) Implied Sla"c\arc\ Deviatio,,-Mean Diagram for Stochaslic Discount Factors . . . . . . . . . . . . . . . . . . . . . . (a) Mean-Standard Dcviatio" Diagram for a Single Excess A~ set Return; (h) Implied Standard Deviation-Mean Diagram for Stochastic Discount Factors . . . . . . . . . . . . . . .. Feasible Region for Slochastic Discounl Faclors Implied hy AnllllallJS Dala, I H~J\ to I ~'~J1 . . . . . . . . . . . . . . Salllple Path of a Discret('-Tillle Random Walk. . . . . Salllple Path and Conditional Expectation of a l\rownian Motion with Drilt . . . . . . . . . . . . . . . . . . . . . . ..
~H~
'21':\ '2~·1
2~'9
:\02 ~O.'l ~·12
:\4:.
10.2 10.:1 lOA I lUI
Zero-Collpon Yidd a"d Forwanl-Rate Cllrves in Jallllal), I !IH7 . . . . . . . . . . . . . . . . . . . . . . . . Cash Flows in a Forward Trallsactioll . . . . Calculatioll of llmation (1I' a Coupon Hond . . . The Prire-Yil'ld Relationship. . . . . . . . . . . . Short- alld I.ong-Terrn I n(erest Rates 1952 to 19!11
11.1 11.2
Change ill Short Rate I )ivided hy Short Rate to the Power y Sample and Theoretical Average Forward-Rate Curves
4;,() 4!'H
The Tent Map. . . . . . . . . . . . . . . . . . . . . . Monthly Ex('('ss I.og liS Siock Returns, 1926 to 1!J!J4 Shifled ,\lHI Tilll'd Ahsoillte-Valll(, FIIllctioll 12.4 Simulation of}', = Sin(X,) + O,:'f, . . . . . . . . . . . 12.;. K('(IIeI Estimalor . . . . . . . . . . . . . . . . . . . . 12./i Bullish Venicil Spl'l';ld 1':lyolf Fllnclioll and Silloothed Vnsion .. . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Biliary Thr('shold t-.I"dcl . . . . . . . . . . . . . . . . . I :!.H COlliparisoll of I kavisidc and I.ogistic Activation FIIIICliolls 12.!) MIIJtila)'('I' I'CI'Cl'I"l'On wilh a Single IIidden l.ayer . 12.10 MI.!'( \.:.) Model of l', = Sin( X,) -+ O':'E, ....... 12.11 Typir:1l Silllillated Trainill)!; !'ath . . . . . . . . . . . . 12.12 Typicallkha\'ior or FOIII'-Nonlill<'ar-Tl'I'lIl RBF Model
47·1 41'2 'IH() r.ol
10.1
12.1 1'2.2 12.:\
:\~)H
'100 402 40/ 41 Ii
:.(n o. \ () :d:1 :; 1:1 :d 1 !llfi :.~tl
:.21
List of Tables
l.l
Stock market returns, 1962 to 1994 . . . . . . . . . . . . . .
2.1 2.2 2.3 2.4
Classification of random walk and martingale hypotheses. Expected runs for a random walk with drift J.I.. • • • • • • • Autocorrelation function for fractionally dilTerenced process. Autocorrelation in daily, weekly, and monthly stock index returns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . " Variance ratios for weekly stock index returns . . . . . . '. Variance ratios for weekly size-sorted portfolio returns. . . .. Variance ratios for weekly individual security returns. . . . . , Cross-autocorrelation matrices for size-sorted portfolio returns. Asymmetry of cross-autocorrelation matrices.
29
Summary statistics for daily returns of five NYSE stocks. Relative frequencies of price changes for tick dara of five stocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. :t::la Expected upper bounds for discreteness bias: daily returns. ::I.::Ih Expect.ed upper hounds for discreteness bias: monthly returns. '\.::1(' Expected upper bounds for discreteness bias: annual returns. :t4 Autocorrelation matrices for size-sorted portfolio returns. :t:J Estimates of daily nontrading probabilities. :tli NOlltrading-implied weekly index autocorrelations. . ::1.7 SUJllmary statistics for transactions data of six stocks. :~.Ha Estimates of ordered probit partition boundaries. :tHh Estimates of ordered probit "slope" coefficients. . . .
109
25 2.fi 2.7 2.H 2.9 3.1 3.2
4.1 4.~
Abnormal returns for an event study of the information contelll of earnings
~ 67 69 71 73 75 77
112 118 119 120 131 1::12 13:~
13H 141 142
164 170
xvi I.isl oj '['abil's
5.1
Fini te-sa mpl e size of tests of tile Sha rpe- l.int ner C'AI'M !Ising large-salllple test statistics . . . . ............... . Pow er of f'-test of Sha rpe- Lint ner CAPM usin g statistic Jl .. Emp irica l results for tcsts of thc Sha rpe- I.ill tncr versio/l of theC APM . . . . . . . . . . . . . .............. .
5.2 5.3 6.1 7.1
7.2 8.1 H.:.! 9.1 9.2a 9.2h
9.3 9.4 9.5 9.li
10.1 10.2 10.3
1SUJIlmary of results lilr tests of exaCl
blCIO !' pric ing lIsing 'zcro -illl erce pt F-tes!. . . . . . . . . . . . . . . . : Lon g-ho rizo n regr essio lls of' log stoc\:. retu rns on the log ,div iden d-pr ice ratio . . . . . . . . . .•. . . . . . . . II.on g-ho rizo n regr cssi ons oflo g stoc k relll rns on the stoc has\tically detr end ed shor t-ter lll inte rest rate. . . . . . . .
I
2·11
I
no
\Molllents of cons ulIll lliol l grow th and asset retu rns'. . . . . \1~lstrumental variables rq~rcssiollS for retll ms and COIlSlllllP,tlOlI grow th . . . . . . . . . . . . ............... .
~lIltiplic~tion rulcs for slOchast~c diffe rent ials. {\symptol1c stan dard erro rs for a . ...... . Asymptotic stan dard erro rs for 0 2 • ••••• ~uto{rva!lIes for colllparativc statics of VI' • • ... .;\sYlllptotic variances of' Black-Sc holes call pric c sensitivity estim ator s. . . . . . . . . . . . ............ . Opt ion prices on asseL~ with nega tively aillo corr elat eu retu rns. . . . . . . . . . . . . . . . . . . . . . . . . Mon tc Carl o estim atio ll of look back opti on pric e. . . . Macaulay's and mod ificd dura tion for sele cted hon ds. Mea ns and stan dard devi atio ns of terll l-str uctu re vari able s. R.egression coef licie nls fill and YII' ... ... ... ... . .
30l:\
:no
Pr efa ce
Thc secd s of lhis hoo k wcr e plan led 01'('1 ' fifteell year s ag-o. al lhc very slar l of 0111' prok ssio ll;t1 care ers. Whi le sllidyillg- fillallcial ('con ollli cs, and as wc 1)('g-;1I1 to lcac h iI, we disc over cd seve ral exce llcn l ICXlbooks for (ina nc;:tl Iheo ry-D uffi e (1 U92 ), I'lua llg alld l.ilzcnbcrg-l'I' (19HH), and Ing-ersoll ( 1~)K7), for cxa lllp lc-b ul no equi vale llt ICXlbook for elllp irira lmc lhod s. Dur ing lhe sam e pcri od, we parl icip aled in resc arch con fcrc nces 011 Fill;lllcial M:lrkels and MOllelary Eco noll lics. held und er lhe allsp ices of lhe Nati ollal BlIr eau orEc onu lIlic Resc arch ill Call1ilridge, Massaciluscl lS. Man y or Ihe pape rs lhal cap lure d our allel liiol l al lhcs c Illceling-s invo lved new ccol lolll eiric IIwlhoc!s or new elllp irica llind illgs ill filla ncia l econ ollli cs. We felli hat this was SOIl l(' of the Illos l exci lillg rese arch hein g don e in !ina nce • •lIlC! lhal sllld cIIls shou ld be exp osed 10 lhis Illal crial al an earl y slag e. III 1!)H!) we bega n to disc uss lhe idea OrWriling- a hoo k lhal wou ld cove r ecol lolll elric IlIClhoc!s as app lied 10 filla llce. alol lg wilh som e of lhe mor e prOlllill('Il1 elllp irica l resu lts ill lhis arca . We hcga ll wl'iling- in earn esl in I ~)91, com plel ing lhis ardu ous pr<~jccI five ycar s and almO SI six hUl ldre d page s latcr. This boo k is consi
c l, lkll lknt allk e. Sln' (' CC('ChClli ../oh ll Cox . AlIgll~ Dea loll. (;cll e Faill a. Ihllc e (;rlll lcl),. Jerr y llauSIlI
Man)' individuals ha\'(' also I'rovid('d us with invaluahle COIIIIIH'nts alld discussions n·ganling the COIIIl'nis and I'XPOSilioll of this hook. WI' thauk David Backus. Nil-k Barhnis. D,wid Barr. David Hates. KI'I\ Ikdllllallil. Dilllitris I\l'rtsilllas. Tilll IlolIl'lsll'\'. Peter ChriSloffl'rsl'lI. Susall KelT Christo/~ krsl'n. (;I'orgl' Conslantinidl's . .John Cox. Xavier Gahaix. Lorenzo (:iorgianlli •.Il'rl'lny (;0111. I.ars I Iotusl'n. Camphdlllarvl'Y•.Iohn JIeaton. I.udgl'r Ilenlsdll'l, Rogl'r 111I_lIlg. R_\\'i.lagannathan. Shmllel Kandel. (:aulam K.III!. .Iung-Wook Killl. 'I')dcl f\lillon. Dan Ndson. Amlan Roy. Boh Shiller. Marc Shivers. Rohert SI;lIllhaugh. Tom Stoker• .Iean-Lllc Vila • .Iiallg Wang. and Ihl' ph.n. SllIdl'nls at IIaITard. MI'I: I'rincetoll. alld Wharton Oil wholll Ihis lIIaterial was "test-marketed" alld rdinl'd. \"'1' have rdied heavily on Ihe ahle research assislance o/'Pl'lr Adalllek. Sangjoon Kim. f\brtill 1.(·l\all. Terenn' Lim. Conslanlin Pelrov. Chllllshellg Zhou. and parlicllbrly Mall Van Vlack and Luis Vin·ira. who undertook the diffinllt lasks 01' prool'n'ading Ihe llI;l\lllsnipt and prq>aring the ilHlex. WI' an' grall'ful 10 Sll'phani(' Ilogu\' fill' h<'l' greal skill and carl' ill prl'parillg Ihl' l'Il'Clmllic version of this llIalluscripl. alldlhe Iypcselll'rs ,It Archl'l)'pe fill' producing Ihl' final \,I'rsion oflhc hook. WI' Ihank 1'('ln \)oughl'rly. 0\11' ('(Iilo)' al Princelon UniversilY Press. fur his pati('IHT, ('II .. "urag(·IlH'nl, alld support throughout this project. Se\,l'ral orgalli/atiolls provid('d IlS wilh generous suppon during \',11ious S\;lges of Ihis hook's geslalion; in paniclllar, we thank Ballcryman h Financial Managellll'nl, the National I\lIreall of E.conomic Rl'search. till' Nalional Science Foundalion, Ihe John M. Olin Foundation, Ihe Alfred 1', Sloan FOllndalion, alld research cl'nlers at I larvaI'd, MIT, Prin('('toll. and Whanoll. i\lltllillal\y, we OWl' 11101'1' thall WI' ('all say 10 the SlIpport alld lo\'l' 1lJ' "Ill' /;lIl1ilics .
.lye AWl.
,\(:M
The Econometrics of Financial Markets
•
1 Introduction
is a highly empirical discipline, perhaps the most cmpirical among the branches of economics and even alllong the social sciences in general. This should come as no surprise, for financial markets arc not mere figments of theoretical abstraction; they thrive in practice :llId playa crucial role in the stability and growth of the global economy. Therefore, although some aspects of the academic finance literature may seenl abstract at first, there is a practical relevance delllanded of financial lllodeis that is often waived for the ll10dels of other comparable disciplines. l Despite the empirical nature of financial economics, like the other soci:.il sciences it is almost entirely noncxperilllentai. Therefore, the primary method of inference for the financial economist is model-based statistical inference-financial cconolJleu·ics. While ecollometrics is also essential in other branches of economics, what distinguishes financial economics is the central role that uncertainty plays in both linanrial theory amI its empirical implementation. The starting point for every financialmodcl is the uncertainty facing investors, and the substance of every financial model involves the impact of uncertainty on the behavior of investors and, ultimately, on market prices. Indeed, in the absence of ullcertainty, the problems of financial ecollomics reduce to exert"ises in basic microeconomics. The very cxistence of financial economics as a discipline is predicated on uncertainty. This has important cOllsequences fi)!" finaJlcial ecollometrics. The randOIH fluctuatiolls that require the use of statistical theory to estimate and test tiluncial models are intimately related to the ullcertainty on which those Ill()(lcls
FINANCIAL ECONOMICS
i
B<", ml .. in (I ~J~:l) provides a highly, c.ulab,," '''"("'H'''I "I Ih .. inl<·rpl •• y h.·lween theory and
pr;l("tiu~ ill tilt" dt·."c!OpJIIClll of Illodt'nl fiualu"iai econolllics.
social sciences, although it has been the hallmark of the natural sciences I()r finite some tillie, Il is ont' of the mosl rewarding aspens of Iin
1,1
Or~ani7.ation
of the Book
In orgallizillg this hook. we 11;1\,(' ",lIowed two gelleral prillciples, First, tIl<' ('ady chaptns nlll('('lItrat(' ('xdllsivdy Oil stock markets, Although many 01 the IIl('thO(\s discllsse" c
;5
I
ties (Chaptcrs 10 and II). Thc last chaptcr of thc book prescnts nonlin~ar methods, with applications to both stocks and derivatives. \ Second, we start by presenting statistical models of assct returns, and then discuss more highly structured ecollomic models. In Chapter 2, for example, we discuss mcthods for predicting stock returns from their past history, without much atlention to institlllional detail; in Chaptcr 3 we show how the microstructure of stock markets affecL~ thc short-run behavior of returns. Similarly, in Chapter 4 we discuss simplc statistical models of the cross-section of individual stock returns, and thc application of these models to evcnt studies; ill Chaptcrs 5 and G we show how the Capital Asset Pricing Model and multifactor models such as the Arbitrage Pricing Theory restrict the parameters of the statistical models. In Chapter 7 we discuss longer-run evidence on thc predictability of stock returns from variables oth~r than past stock returns; in Chaptcr 8 wc explore dynamic equilibrium models which can gcncratc persistent timc-variation in expcctcd returns. We use the samc principle to divide a basic treatmcnt of fixed-income securitics in Chapter 10 from a discussion of equilibrium term-structure models in Chapter II. We havc tricd to makc each chaptcr as scll~o[\tained as possiblc. While SOIllI' chapters naturally go together (c.g., Chapters 5 and G, and Chapters 10 .mll II), there is certainly no need to read this book straight through frolll heginning to cnd. For classroom usc, most teachcrs will find that there is too much matcrial hcrc to he covcrcd in onc semcstcr. Therc are scvcral ways to usc thc hook in a onc-semcster coursc. For cxamplc onc teachcr might start by discllssing short-run time-serics bchavior of stock priccs using Chaptcrs 2 and 3, then covcr cross-scctional models in Chaptcrs 4, 5, and 6, then discuss intcrtcmporal cquilibrium models using Chaptcr 8, and finally (over dcrivative securitics and nonlinear methods as advanced topics using Chapters 9 and 12. Anothcr tcachcr might first prcscnt the cvidence on short· and long-run prcdictability of stock returns using Chapters 2 and 7, then discuss static and intcrtcrnporal equilibrium thcory using Chaptcrs 5, {i, and R, and finally covcr flxcd-incomc securities using Chapters 10 and II. Therc arc somc important topics that wc havc not been able to include in this texl. Most obviously, our foclls is almost cxclusively on US domestic asset markcts. Wc say vcry littlc about asset markets in othcr countries, and we do not try to covcr intcrnational topics such as exchange-ratc bchavior or the homc-bias punic (the tcndency for each country's investors tei hold a disproportionatc sharc of their OWJl country's assets in their portfolios). We also omitslIch important economctric subjects as Bayesian analysis! and frequency-domain methods of time-serics analysis. In many cases our choice of topics has becn influcnccd by the dual objectives of the book:l 10 ('xplain the methods of financial cconoIllctrics, and to review the ("IIl-' piric .. 1 literature in finance. We havc tended to conccntrate on topics that
0"11
I I
I
i
I
' ,.' • , .,:: "" • ~ t; •
1
inv~lve
cconometric issucs, somctilllcs at (hc expensc ofo(her equally ill('/'IHuch rcccHt work ill hchavioral !inan('t'-that is ccbllollletrically more straightforwanl.
cstill~ l1Iatcrial-indudill~
1.2 Useful Background The lIIany rewards of financial cconometrics come at a price. A solicll>ackground in m.athematics. prohahility and statistics, and finance theory is nc('essary for the practicing financial economctrician. for precisely the reasolls that make financial cconometrics such an en~a~ing endeavor. To assist readers in obtaining this background (since only the most focused and directed of st\l(ients will have it already), we outline in this scction the IOpics in mathematics, probability, statistics, and finance theory that havc becollle indispensahle to financial ecollomctrics. We hope that this outline call scrve as a self-study guidc for the lIIore enterprising rcaders and that it will bc a partial substitute for includin~ backgruund material ill this houk. 1.2.1 Atllt/wlllltirs B(!ckg'HlUUd
The mathcmatics background most useful for !inaneial ecollonletri('s is not unlike the background necessary for econollletrics in general: lIlultiv'\I·iate calculus, linear algebra, and matrix analysis. References for each or these topics arc Lang (1973), Strang (197!i), and Magnus and Neudecker (I ~IHH), respectively. Key concepts include • • • •
multiple intcgration multivariate constrained optimization matrix algeura basic rules of matrix difti.'rentiation.
In addition, optiun- and other dcrivativ{:-pricin~ models, and contillllollStime asset pricing models, require somc passing familiarity with the Illi or .I/or/untie ((l/eu/us. A lucid and thorough treatment is provided hy Merion (1!190), who pioneered the application of stochastic calculus to lin.uH"ial economics. More mathematically inclillcd readers fIIay also '''ish to {"O!lSuit Chllll'fand Williams (1990). 1.2.2 "lOllIlhilitv
111111
SllItisliLl i!tlfi
Basic ill"ohahility theory is a prerequisite f(,r any disci pi inc in which u!l{"{'rt
illluilion alld suhllclies or deillclltar), I'rohahilistic rcasollin~. An amazillgly dmable t'lassic that ta"es just this appn><\ch is Fe\ler (I ~I(;H). nrielllan (I ~)~)~) provides similar intuitioll hut al ;1 IIlGlslllc-theoretic level. Key COIIc('pts incillde • • • • • • •
ddillilioll of a ralldom variable illdepelldellce di,trii>ulioll and density fUllctiolls c()nditional prob~lbility modes of cOllvcrgenct: laws of large numbers centrallilllit theorems.
Sialistics is, of course, the prilllal)' engine which drives the illferences that fillallcial ecollolllt:tricians draw from the data. As with probability theory, statistics can be taught at various levels of mathematical sophistication. Moreovcr,unli"e the narrower (and sOllie would say "purer") focusofprobahilit)' theory, statistics has increased its breadth as it has matured, giving birth 1(> nWlly well-delined subdisciplines such as lIIultivariate analysis, nonparalIlelrics, tillie-series allalysis, order statistics, allalysis of variance, decisioll Ihcor)" Ha)'esian statistics, etc. Each of these subdisciplines has been drawn upon by financial econometricians at on(' tilllc or another, ma"ing it rather difficult to provide a single rderellct: for all or these topics. Amazingly, sllch ;1 rdercllce does exist: Stuart alld Ord's (I ~)H7) three-volullle lour de jil/ce. A lIIorc cOlllpacl reference that contains most of the relevant material lor 0\,1' purposes is the elegant IIlollog,-aph by Silwy (197:'). For topics ill ti,"e-series analysis. Hamilton (\ !l94) is an excellenl comprehensive text. Key concepts include Neyman-Pearson hypothesis testin~ lint'ar n:gression IIl
• • • •
F()r ("(JlIi ill1lOm-lilIlc financiallllodds, an additional dose of stochastic pron'ss('s is
J.2.3 Filllll/u' 'filml)' HllrkgmHlld
Since lhe mi.l()11 drlrf of financial c("onOllll'trics is the t'mpirical ill1plelllen1~lIioll and cvaluation of financial Illl)dds, a solid \);ld.grolllld ill finance III('()'), i, Ill(' most in,portant of all. Scvcral texis pro\'id(' exccllent coveragc
or this material: Duffie (I!l!l~), /-luang and Litzenberger (19HH), Ingersoll .14·'- L"~~I.~ t,1·· .... ; ," J_I" !~"'" .'~I:~·:)L" iIJCJudt" 1
• ,hI. ,.I<·ISIIIJI .tlllI ''''IWI 11'11-,<111111' IIJ('ory • s!
1.3 Notation We have fimnd that it is 1;lr rmm silllple to devise a consistcnt notational schellle lilr a hook or Ihis scope. Thc dirricllity cOllies rmlll thc fan thai linancial econometrics spans s('\'('I'al \'ery different strands of the financ(' literature, ('ach replc-h' with its own lirlllly estahlished sct of potational ('OIl\'('lltiollS. I\llt till' COllVl'lltiollS in Ollt' litcrallll'C orten conflict wilh Ihe convcntions ill another. Ilna\'oidahly, tll('ll, WI' IIIl1sl sanifice either ill\('Illal notational (,(lIlsistl'ncy across dilli'n'llt chapters of Ihis tcxt or cxtcrIlal cOllsisll'lH'Y with the 1I0tatioll used ill th(' professional lit('ralun~. Wc hav!, chos('n Ihl' «mll('r as Ih(' kS'lT ('\'il, hUI we do mainlain Ih(' «)\Iowing COII1'('lliions throughout lilt' hook: • We IISC holell;\("(' for \'cctors alld matriccs, and rq.(ular race for scalars.
•
•
• • • •
Where possihle, W(' US(' hold IIppercase for lIIatrices and hold lowcrcase filr vcctors. Thlls x is a \'('ctor while X is a matrix. Where possihle., we lise IIppercase letters ror thc levds or variahles aJ:<1 lowercase ktters for thc lIatllrallogarithllls (logs) ofthc sallie variabks. Thlls ir I' is an assct price,/! is th(' log asset price. Our stalldard notation filr all innovation is thc Creek Ic-lI('1' L When' we nced to defille se\'eral dilfl'l'I'lIt illnovations, we IIS(' the alternalivc (;rl'l'k 1I'1I1'rs 'I, l;, alld (. Wher(' possihle, WI' IIS(' (;n'l'k 1e1l('rS to dellote paranll'ters or parallll'tl'l' \'('('\ors. WI' uS(' 1iJ(' (;n'l'k II'th'r" to
j for asset subscripts; k, m, and n for lead and lag subscripls; and j as a {!c.-Dc.-riC subscripl .. M--~' J.tr.. JlDllt;!' GIIIW!':llllU'1 u.:u.':...;, ;ur.mitt": *' cia!rri.1 lifru'-I~ the end of period t. DllIs RI dcnotes a rcturn on an asset held from the end of pcriod /-1 to the end of pcriod t. • In writing variance-covariance matrices, we use n for the variancecovariance matrix of asset returns, L for the variance-covariance matrix of residuals from a time-serics or cross-sectional model, and V for the variance-covariance matrix of parameter estimators. • We usc scriptlellers sparingly. N denotes the normal distribution, and L denotes a log likelihood function. • We usc Pr(·) to denote the probability (lfan event.
The professional literature uses Illany specialized terms. Inevitably we also use these frequently, and we italicize them when they first appear in the book.
1.4 Prices, Returns, and Compounding Virtually every aspect of financial economics involves returns, and there are at leasltwo rcasops for focusing our attcntion all returns rather than on pri~es. Firsl, for the average investor. financial markeL~ may be considered c1os
1.4.1 Definitions and Conventions Denote by 1', the price of an assct at date I and assume for now that this asset pays no dividends. The Jim/lie nfl rl'lum, on the asset between dates I - 1 and I is defined as
n,
H,
1', --I. P'-I
(1.4.1 )
Tht" silll/Ill' {!!OH rp/um on the asset is just one pillS the net return, 1 + R,. Frolll this definition it is apparent that the asset's gross return over the most recent k periods from date / - k to date t, written I + ~(k), is simply
p<~'~":': ;~Jt .
,: 10 ..
I. Ill/mt/ue/illll
cC]ualto thc product of the k single-period returns frolll 1- k + I to I, i.('.,
1+ R,(k)
+ I?d . (I + /{,-I)'"
+ I?,-k+d
-
(I
==
--.--.--."---
(l
I',
1',-1
I',_.~
I',-H I
1',-1
I',_~
P'-:I
I',-k
I',
-J',-k
( I.·U!)
and its net rcturn over the 1I1osl recent k periods, written /{,(k), is simply cCjualto iLS k-pcriod gross return minus olle. These llluitiperiod returns are callcd com/Jound rcturns. Although returns arc scale-free, it shollid he emphasized that they are 1I0t IIl1illess,'uut are always defined with respect to some tilllc intcrval, e.g., onc "pcriod." In fact, R, is lIIore properly called a rale of retlll'll, which is mon "jcullluersomc tcrminology butlllore accurate in refcrring to HI as a rate or, in! cconomic jargon, a flow variable. Thereforc, a return of 20% is not a coniplete dcscription of the invcstmcnt opportunity without specification of the retllrn horizon. In the academic literature, the relllrn horizon is generally given explicitly, oftell as part of the data description, e.g., "The CRSP monthly returns file was used." 14owever, among pr~ctitioners and in the limlIlcial press, a returnhoriz{lII of one ycar is usually assumed implicitly; hencc, unless stated othcrwis
\ \
Annualized[RI(k)) ==
\
[Hn(l +
R'_j)
Jl/k
-
I.
(1.'1.3)
j:O
l~ingle-period
Sincc rcturns are gencrally slllall in magnitudc, the following a~proxilllation bascd on a first-ordcr Taylor expansion is often used to annu~lize multiyear returns: Anllualizl'd[U,(k)] ~
I
A-I
k L Ilt-r
(1.'1.'1)
j=1l
Whether such an approximation is ade<)l1ale depends on the particlliar application at hand; it may suffice for a <)ui<"k and coarse comparisoll or invcstmcnt pcrformance across many assets, bllt for finer calculations in which thc volatility of returns plays all important role, i.e., whell the higher.. order terllls in thc Taylor expansion arc not negligiblc, the approxill1ation t (1.4.4) lIlay hreak down. The only advantage orslich an approximation is •. (OIlVcnience-it is easicr to calculate all arithmetic rather than a geolllet• ric avcrage-howevcr, this advantage has diminished considerably with the .. advclll of cheap and convenienl computing power.
,.~ j:.
1.-1.
}'ril'f.\', 1II'lUnt.l, !luti
C0mfruUIlfLillg
11
(:oll/ill Will.\' (:olll/mulldilll{
'! '", diflicully or manipulating g\~ol\wtrir ,Iverages surh as (1.4.:~) motivates ,t .. · .. <'1' approach to compound returns, (In(' which is not approximate and
,dso has important implications for modeling asset returns; this is the notion or con tin uous COlli pounding. The (Oll/jlluowly COIII/lOlllllil'd rflum or {ogrelum r, ofan asset is dclined to be the naturallogaritlilll o('its gross return (I + H,): ( 1.4.5) \"here h == log 1',. When we wish 10 ('lllphasiJ.e the dbtinl'lion between R, and rl, we shall refer to R, as a sim/)ifo return. Our notation here deviates slighLly from our convention that lowercase letters denote Ihe logs of uppercase lettel>, since here we have rl == log(l + lll) rather than log(/l,); we do this to maintain consistency with standard conventions. The advantages of continuously compounded returns become clear "hen we consider multi period I'eturns, since 1',(11)
==
logO
+ R,(h)) ==
logO
+ H,) + logO + H,_I) + ... + logO + RH+I)
r,
log«l
+ T,_I + ... + r,-ktt,
+ HI)' (1 + RI _ I )··· (1 + U'-H1) ( 1.4.ti)
and hence the continuously compounded llIultiperiod retum is simply the sum of continuously compounded single-period returns. Compounding, a multiplicative operation, is converted to an additive operation by taking logarithms. However, the simplification is not merely in reducing multiplicatioll to additioll (since we argued above that with modern calculators and computers, this is trivial), but 1II0re in the modeling of the statistical behavior of asset returns over time-it is far easier to derive the lime-series properties of additive processes thall of multiplicative processes, as we shall ~('e in Chapter 2. Continuously compounded returns do have one disadvantage. The siJnpie relU!'ll Oil a portfolio ofassels is a weighted average of the simple rellIrns 011 the assets thelllselves, where the weight 011 each assel is the share of the portrolio'~ value invested in that asset. ]j' portfolio /) places weight W,p in asSel i, thell lhe return 011 the portfolio at litlle /, HI'" is related to the returns Oil individual assets, Rjlo i == I ... N, hy R/" = L:':'d W,/,Hil . UnforlUnatrly conlinllously compounded returlls do lIot share this conveniellt property. Since Ihe log or a SUIII i~ 1I0t the sallie as the stun or logs, 'i" docs not equal \~.v
L...,=1 1l'I/ lrli'
In empirical applicalions this problem is IIsllally minor. When returns ,Ire lIwasured over short intervals of lime, alld are therefore dose to zero, Ill(' cOJltinllously (,OIllPOIIlHlcd /'(>Iurll Oil ,I port!,.!io is close to Ihl' weightl'd
Ii,
Ii",
J', 1+ I I Jil'idl'lld I'mll/PIII "'ill/il/~ CIIIIl'rIIlill/!
Fi}.,"In' 1,1,
average of the continllollsl\' COlJlpoIIIHled returns
'i,' ""
L:t lfIll"i"~
Oil
lhe individual
a~scts:
We lise this approxiliialioll in Chapter :1, Nonl,thelc,s
it is conllnon to USI' ~illlpk returlls when a (Toss-section of assets is heillg stlldied, as in ChaplCls ·I-(i, ;IIId cOlllilluously cOIlIpounded relUnls whl'1I Ihe lelllporal behavior of relllnlS is the lilcus of interesl, as in Chapll'!s '2 and 7, I )ir/it/nlll I'ayllll'll/.\
For ;\.sSI'ls which IIIake periociic ciivicielld paYl11l'nlS, we I11I1SI IlIociil)' our ddinilions or 1'('1 II rns anci compounciillg, lknole hy /), the assel's divicielld pa)'IIIelll al dale 1 alld aSS"IlIe, »lIlrly as a malleI' or COnV('IIlioll, Ihat Ihis di\'idcnd is paidjllst herore Ihe ciale-I price 1', is recorded; hence 1', is lakell 10 be Ihl' (',\'·dir,it/I'IIII price ;11 dall' I, AlIel'llalivcly, one Illight dl'snihl' I', ;" all elld-ol~period assct price, as showII ill Figure 1,1, Then Ill!' IIet silllpk l'('UII'Il at dall' 1 1ll;1)' 1)(' dc/illl'd as f{,
1'1 + /), ---I. 1',-1
(J.oI,7l
Multipl'riod and cOlllillllollsly COIIIIHHllldcti relllrns 1\Iay hc oi>uilll'ci ill Ihe sallie way as in Ihe lIo-dividellds case. Nute that the cOlltillllOllSI~' compollllckd rellll'll Oil a di\'idl'lld-payillg assel, r, = log(/'I + /),) -log(l',_1 ), is a nonlincar rllll('(ioll or log priccs alld log divicknds. Whell thc ratio or prices to tlidclcnds is nol ICHI \'ariahle, however, tbis rlllll'liOll elll hc approxilllaled hy a lilH'ar rllllClioll or log prices alld di\'idelld~. as discllssed in detail in Chaptci 7, I':,\H',\.\
~I
U,'/un/.\
II is oftcn COII\'I'lIit'lIl 10 work willI ;111 assel's exccss return, c1dincd as III<' dini.'n'lIl,(, hl'IW(,I'!I Ihl' ;lss('I\ rl'llIr!l and the retllrll Oil S01\le refl'I"'IlC(' aSSl'!. Thl' n'f(on'!I((' ;ISSl'1 is ofl('!1 ;Issulllcci to Ill' riskless alld ill pranicc i, IIsllallya shorl-l
I :.',,, \1", hmt1 \d"'"'1i''H'l''' I
,lIIlw
1I"i.t'd 10 Id,lIf' ,ilJll'k
IIHtlllIHHI",ItO\
,11111
{.(·BH\\~,.(H"'4."u"i.,,(.'
,1\ S('( titlll ~I. t .'1(11 (:h.l\ltt"1 I,.
'''lIliIIlIOIl,h (PlIIl'0lltult·clIC·IUI n~"
silllple excess return on asset i is 0.4.8) where I~Jt is the reference return. Alternatively one can define a log excess return as ( 1.4.9) The excess return can also be thought of as the payoff on an amitmgt /Jllrtjo/io that goes long in asset i and short in the reference asset, with no net investment at the initial date. Since the initial net investment is zero, the return on the arbitrage portfolio is undefined bllt its dollar payoff is proportional to the excess return as dc!inecl above.
1.4.2 The Marginal, Conditional, and joint Distribution of Rrtums I"laving defined asset returns carefully, we can now begin to study their behavior across assets and over time. Perhaps the most important characteristic of asset returns is their randomness. The return of IBM stock over the next month is unknown today, and it is largely the explicit modeling of the sourres and nature of this uncertainty that distinguishes financial economics from other social sciences. Although other branches of economies and sociology do have models of stochastic phenomena, in none of them does uncertainty play so central a role as in the pricing of finanrial assets-without uncertainty, much of the financial economics literature, both theoretical and empirical, would be superfluous. Therefore, we must articulatl' at the very start the types of uncertainty that asset returns mi~ht exhibit. "
! Thr joiJl ( lJistn"blltion
\ COllsider a collection of N assets at date t, each with return R,/ at dat9 t, where I = 1•... , T. Perhaps the most general model of the collection lor returns I a,ll is iL~ joint distribution function: I ! ( 1.4.10) , where x is a vector of slale variables, variables that summarize the economic {,llvirol1ment in which asset returns arc determined, and (J is a vector of fixed parameters that uniquely determines C. For notational convenience, we shall suppress the dependence of G Oil the parameters (J unless it is Ill'l'ded. , The probability law G governs the stochastic bebavior of asset returns alld x. and represents the SUIll total of all knowable information about tbem. WI' lIIay then view financial econolJletrics as the statistical inference of giv{,Jl (; and realizations of IR"l. Of course, (1.4.10) is far too general to
e,
I. Illlroliuriio/l
he of any lise for statistical inference, and we shall have to place further restrictions on G in the cOllling sections and chapters. Ilowev(~r, (1.'1.10) docs serve as a cOllvenient way to organize the many models of asset returns to be developed here and ill latcr chapters. For exalllple. Chapters ~ t11}ough 6 deal exclusively with the joint distribution of (ll'll, leaving additi9nal state variahles x 10 be considered in Chapters 7 and H. We write this joim distribution as Gil. I Many asset pricing models, sllch as the Capital Asset Pricing Model (cArM) of Sharpe (1964), Lintner (1965a,b), and Mossin (l%li) considcred in Chapter 5, dcscribe the joint distrihution of thc cross section of r('tIIfns lUll' ... , ){Ntl at a single date I. To n~duce (1.4.10) to this essentially stalic structure, we shall have to assert that returns arc statistically indepcn
FUlil' ... ,Rrr)
=
/';1 (R.d . /';~Ul,t I n,d· /'i:l(fl,:1 ... /';'r\R,'f
I
lilT-I •. ··
.fl,d.
R,2. R,d
(1.'1.11)
Frolll (1.4.1 I). the tcmporal depeudencies implicit in {R'II are apparent. Issues of predictability in asset rcturns involve aspects of their fOlUliliOIl(l{ distributions and. in particular, how the conditional distributions evolve Ihrough time. By plac-ing further restrictiollS on the conditional distributions 1';,(·), we shall be able to estimate thc paramcters 0 implicit in (1.4.11) and examinc (he predictability of asset retuflls explicitly. For example, one versioll of the random-walk hypothesis is obtained by the restriction that thc conditional distribution of rcturtl /lit is equal to iL~ marginal distributioll, Le .• l'il(1l/l I .) = Jo;I(1~iI)' If this is the case, thcn returns are temporally indqll'ndent and thcrcfore unpredictahle using past r('(urns. Weaker versions oftlH' randolll walk are obtained by illlposing weaker restrictions Oil I';,(ll/l I . ). The UllfOllliilioll(l{ Distributioll In cases where an assct f(~tllfll'S conditional distribution differs from its llIarginal or IIlIconditional distriillttioll. it is dearly'the {"(mdilional distrihlt-
1.". Prias, Returns, and Compoundillfi
15
tion that is relevant for issues involving predictahility. However, the properties of III(' unconditional distribution of returns may still be of sOllie interest, l:~pc('i;dll' ill cases where we expect pn'dictability to be minimal. One or Ihc most cOlllmon models for asset returns is the temporally inde[Jelldel1tly and identically distrii>uteo (liD) normal 1Il0del, in which returns arc assumed to be indepellden I (lvel' tillle (although perhaps crosssectionally correlated), identically distributed over time, and normally disIributed. The original formulation of tlw CAPM employed this assumption of' normality, although returns were only implicitly assulllcd to be temporall), liD (since it was a static "two-period" model). More recently, models of asymllletric information such as Grossman (19H9) and Grossman and Stiglilz (1980) also use normality. While the temporally lID normal modclmay be tractable, it suffers from al leasltwo important drawbacks. First, most financial assets exhibit limited liability, so that the largest loss an investor can realize is his total investment and no more. This implies that the smallest net return achievable is -1 (,r -100%. But since the normal distributioll's support is the entire real lill':, this lower bound of -1 is clearly violated by normality. Of course, it Illay be argued that by choosing the mean and variance appropriately, the probability of realizations below -I can be made arbitrarily small; however il will never be lero, as limited liability requires. Second, if single-period returns arc assumed to he normal, then multiperiod returns cannot also be normal since they arc the produrtsoflhe singleperind returns. Now the sums of norlllal single-period returns arc indeed normal, hut the slim of single-period silllple rellll'IIS does not have any economically meaningful interpretation. However, as we saw in Section 1.4.1, the slim of single-period continuously COlllpOlllHlcd rclllrns docs have a meaningful interpretation as a llIultiperiod continuously compounded return.
'Jile 1.og71oTlIlal Distribution A sensible alternative is to assume that cOlllinllollsly compounded singleperiod returns Til are lID norlllal, which implies that single-period gl'Oss silllpll~ returns arc distributed as liD lO!-,'1wnllal variates, since Til == log( I + RII ). We lIIay express the lognormal model then as (1.4.12)
l: ndt:r the lognormal model, if' tltt· 11I(,~\l1 and V~II'i;\llCC or r,l arc 111 alld a/, respectively, tilen the mean and variance of simple returns are given by (1.4.13)
(1.4.14 )
AIII'matin-h', if \\'1' ;1.~~II/lJ(' that Ihl' /lIl'an and variance orsimpk retllJ'll' a'l arc III, and I;, rl'sp('ctin'II-, thl'lI IIlHkr the IO):(lIorlllallllodt'llill' Ill!'all ,llld \'"ri;l/l('c of 1;1 arl' gin'II bl'
111,+ I
log --;:=====
I -I- ( ~ " )~
log [I
\\111',11
+
(_S' )~]. 111,+ I
(J..l.1 til
'1'111' 10):(nonll;1I Illodl'l has till' added adl'antage of IIot violatillg lilllil!'d liability, sincl' limilt'd liability ddds ;1 low('\' hound of /.no on (I + I?'II. which is satisficc! h)' (I + No) = 1,1" whell I;, is assllnwc\ to 1)(' normal. The IO):(lIormalllllldd has a long alld illustrions history. hegillnill):( with the dissl'rtation ort hI' Frl'nch III;\thl'llIatician l.otlis nachdin (I ~lO(), wili(h contained the lIIathematics or Brownian lI\otion and hcat conduction, Ii\"(' years prior to Eillstl'i" 's (I !l()!"l) ElIllOIIS paper. For other reasolls that will 1)('1'0111(' appalTnt in later chapters (SI'(', I'specially, Chapter !l), the lognonllal model has bl'('ollw tIll' workhorse or the fillancial asset pricing litnaturc. nllt as attractivl' as thl' IO):(lIortllallllodcl is, it is not consistent with al\ the properties or histol"ical stock returns. At short horizons, historical retllrns show weak evidence of skewness "nd strong evidence or exn'ss kurtosis. The Jkntlnr.f.\, or norlllali/l'c1 third IllOIll('nt, or a randolll variahl(' ( with IlIl'al1 /1 alld \'arialKI' (J ~ is ddilwt\ by ( 1.4. Ii)
The
lil/r/llli.I,
or lIortn;tii/.l'd ',l\lItll I\lOllll'lll, of £ is defined hy
Thl' \lonl\al disirilllitillll 1,,1' S\....'WIII·S' "'111011 10 l.l'J"(I, as do all othn ': IllIIll'tric distribllli""s. Th .. "Orlll,1i dislriblltioll has kurtosis l'ljl\al 10 :1. IlIll .!',I-Iaifl'/{ dist rihlltiolls wit h <,xll;\ Iliohahility IIlass ill t 11(' tail art'as lIaw highn or 1'\'('11 illli"it!' kurlosis. Skcwllt'ss alld kurlosis I ;111 hI' t'SI illlatni ill ;\ salllpk of (bt;l hI' COllstl1 /( till!!: till' ,,11\·ioll' sallll'l,' ;I\ ... ra~,·': IIll' ,a III ph- 1\1 ('a II
/1
( 1.1. I (I)
I .. /.
1.7
l'nuI, I!I'LrII7l.I, 1I1It! COII//IIJlllldillJ;
Ill(' sample variance T
a~ - TI '\'(( ,.)2 L-'-'" , 1:1
Ihe salllple skewnf'Ss .
r
I",
-3
.Ii -- 'fo-:I L-(f, - Ii) ,
(1.4.21)
1=1
\
alld the sample kurtosis
L T
l\." == -I-4
Ta
-
(E, - Ii) 1 .
(1.4.22)
1=1
III lar~e samples of normally distrihuted data, the estimators ,~and k are lIormally distributed with means 0 and 3 and variances 6/ T and 24/ T' respectively (sec Stuart and Ord [ 1987, Vol. I]). Since 3 is the kurtosis ofth~ normal distribution, sample ex(f'SS kurtosis is defined to be sample kurtosis less 3. Sample estimates of skewness for daily US stock returns tend to be IIq!;ative for stock indexes hut close to zero or positive for individual stocks. Sample estimates of excess kurtosis for daily US stock returns are large and positive for hoth indexes and individual stocks, indicating lhat returns have more lIIass in the tail areas than would be predicted by a normal distribution. Stable Distributions Early studies of stock market returns attempted to capture this excess kurtosis by modeling the distribution of continuously compounded returns as a memher of the stable class (also called the stable Parf'icrLivy or stable Parefilm), of which the normal is a special case. 3 The stable distributions arc a natural generalization of the norlllal in that, as their name suggests, they are slahlc under addition, i.e., a sum of stable random variables is also a stable random variable. However, nOllllorl11al stable distributions have more probability mass in the tail areas than the normal. In fact, the nonnormal stable distributions arc so fat-tailed thaI their variance and all higher moments are infinile. Sample estimates of variance or kurtosis for random V<\riablcs with "The Frenrh probabilist raul Levy (1924) was porhap' the Ii ... t to initiate a genel4l inve,ti· gation "("table di.tributions and proVided a complete characterization of them Ihrough their log~ haranrristic functions (,e~ Iwlow). l.evy (192:» .Iso ,howed Ihal the tail probabilities 01 st"hlt· db{r,hut~on~ approximat{" those of the Part'to distribution, hence the t~nn "stable }';II"('lo-l.(·\,y" or "s[ahl(, rarrlil't ~ ,tnd (;onedes (1974); Failla (196:'); Fam. and Roll (1971); Fieliu (1976); Fielitz and HOII'll (1~1H:1); (;ran~rr atld Mor~e"'l.rn (1970); IbRennan (197H); Ihn, Miller, and Wichern (I <17·\); ~bt"klhrot (I%~,); Mandell>rol3l,,1 Taylor (1%7); OfTicer (1972); Samuelson (1967, I '17Ii); Silflklll,'ill and Ik.,II,'., (I')HO); and Tttrker (11)92).
J~
1. Introductioll
, \
,,"
\ \
"
I
n.lo
"
Cauchy '.
I II
-3 Figure 1.2.
-2
-I
0
3
CompariiOll o/Slab'" (wd Nomlfll Dl'Ilsily FIUlcliolU
these distributions will not converge as the sample size increases. but will tend to increase indefinitely. Closed-form expressions for the densilY functions of stable randolll variables are available for only three special cases: the normal, the Cauchy, ami the Bernoulli cases. 4 Figure 1.2 illustrates the Cauchy distrihution, wilh density function j(x) = T(
y2
y
+ (x -
<5)~
In Figure 1.2, (1.4.23) is graphed with parameters <5 == 0 and y \, and il is apparent from the comparison with the normal density fUllction (dashed lines) .that the Cauchy has faller tails than the normal. Although sc.able distributions were popular in the 1960's and early 1970's, they are less commonly used today, They have fallen out of favor partly because they make theorelicallllodelling so difficult; standard finance theory 4I!uwever;I..e"'! (1925) deriveilihe \ol\owillK ~xplicil exp ....,sioll for Ih" 10Ka";III", of Ihl' ch'lra 0 (or fI < 0) Ihe distribution is skewed to the riKht (or lefl). Wh"11 fI = () ")'d a = I we have the Cauchy di,lIihulion. ,lIId when" 1/:l, fl I, & ll, allli r I we han! Ihe Bernoulli diMriblltion,
,len."",
=
=
=
=
1.4. Pl'ires, Uellmls, IHld CornpoullliillK
19
almost always rcquires finite second IIlOnJents of returns, and often fmitc highcr moments as well. Stablc distriblllions also have sOllie counterfacIllal implications. First, thcy illlply thaI sample estimates of the variance ,md higher lIlomcnts of returns will tend to incrcasc as the sample sizc increascs, whereas in practice thcsc cstimalcs sccm to convergc . .sccond, they imply th,\t long-horizon returns will he just as non-normal as short-horizon returns (s'nce long-horizon returns are SUIllS of shon-horizon returns, and these distributions arc stable undcr addition). III practice the evidence for non-normality is much weaker for long-horizon returns than for shorthorizon returns. Recent rcsearch tends instead to llIodel returns as drawn from a fattailcd distribution with finite higher moments, such as the I distribution, or as drawn from a mixture of distributions. For cxample the return might bc conditionally normal, conditional on a variance parameter which is itself random; thcn the unconditional distribution of rcturns is a mixture of normal distributions, somc with small conditional variances that concentrate mass around the mean and others with large conditional variances that put mass in thc tails of the distribution. The result is a fat-tailed unconditional distribution with a flllite variance and finite higher moments. Since all momcnls are (initc, the Central Limit Theorem applies and long-horizon rcturns will tend to bc closer to the norlllal distributioll than short-horizon rcturns. It is natural to model the conditional variance as a time-series process, and we discuss this in detail in Chapter 12. All
J~'IIlIJl'riCllllllllslmliuTl
Table 1.1 contains some samplc statistics for individual and ap;gregate stock rcturns from thc Ccnter for Research in Sec\IJ'ities Priccs (CR.SP) for 1962 to 1994 which illustrate some of the issues discussed in the previous sections. Sam pic moments, calculated in the straightforward way described ill (1.4.19)-( 1.4.22}, arc reported for value- and equal-weighted indexes of stocks listed on the New York Stock Exchange (NYSE) and American Stock hchange (AMEX), and for ten individual stocks. The individual stocks wcre selccted from market-capitalization deciles using 1979 end-ofyear markct capitalizations for all stocks in the CRSP NYSE/ AMEX universe, whcre in[em'llional Business Machines is the largest decile's representative and Contincntal Materials Corp. is the smallcst dccilc's rcpresentative. Panel A reports statistics for daily rcturns. The daily index returns have ('xlrelllcl)' high sample exccss kurtosis, :11.!) and 2(i.O respcctively, a clear sign of fat tails. Although the exccss kurtosis estimates for daily individual stock returns arc p;cncrally less than those for thc indcxes, they are still large, r'-lngillg 1'1'01\1 3.35 to 59.4. Sincc thcrc are H179 obscrvations, the standard error for thc kurtosis estimatc undcr thc lIull hypothesis of normality is J24/H 179 = 0.0;)4, so these estimalcs of excess kurtosis arc overwhelmingly
·
1o
JltjlfJtl/l11l1J1I
statistically si!!;nificant. TIl(' skcwncss ('stilllates are ne!!;,lliV(' lill' lhe daily index retllrns, -I.:B ;lnd -O.!I:~ respectively, but gCII!'rally positive 1'01 Ihe individual stock retUrIlS, rangillg from -O.IH to 2.25. Many of Ihe ske\\'n('~s estimates art' also statistically sign ifirall t as the standard error ullder lhe null hypothcsis of nonllalit)' is /ti/H 17!) = 0.027. Panel B reports salllplt' statistics f(lI' lIlonthly returns. These arc COIlsiderably less leptoklll'tic than daily rcturns-the vallle- and eqllal-weighled CRSP monthly indt'X returns haw' excess kurtosis of only 2.12 and 4.11, ITspeclively, all orcin of magnilude smaller than the excess kurtosis of daily n~tllrns. As Ihert' are ollly :~90 ohs('l'valions the slandard error for Ihe kurtosis estimate is also Illllch larger, O.21H. This is olle piece of evidence Ihat h,ls led researchers to usc hll-Iailcd distrihlllillns with linile higher moments, f()r which Ihe Centrall.illlil Theorelll applics and drivcs longn-horil.oll rt'lmIlS lowards normality.
1.5 Market Efficiency
I
\
I
I I I
\
II \
!
Tht' origins or Ih(' EHici('nt Markt'ts II),polhesis (EMH) can he tra('(~d hack at least as far as tht' pion('('I'in!!; thcort'tical cOlltrihlltioll ofBachelit'r (I!JO()) and the empirical research of Cowles (I!J:\:\). The modern literature in economics hegins with Samuelsoll (I!Hi!',), whose contrihution is neatly Sllllllll
-11\("IH"'.\t'tl1 ( I ~t~I'.!\ di,( U"t"' Ill(" (01111 ihu1iuH' (ll Itl("hc:li<"I. (:o\\"1t:Ir\. S,\llltu:l~on •• uul",.",,· o,I\\"1" ... ;.nt\" a\l1hol'_ I"IH" .ntH k, H'P'\Hl\od \1\ loU (lq~H,) indHd(' somt' of ,itt" mos.\ impolt.ulI
p.'I)t'rs
ill
1his
httOLHUI to.
I.).
M(lTkt'1
21
Jll'rinley 1abk 1.1.
S/ork marl,,/ "/lIm.<, 196210 J 994.
Standard Deviation
Exce",
St'rurit)'
M~an
Valllt'-Wt'i!(htt'd Index
0.044
O.R2
-1.11
14.92
-111.\0
1I.1I7
Eqllal-W~i!(hted
(um
0.76
-0.'1:\
26.03
-14.19
9.83
Om!)
1.42 I.f,f,
-0.11l
12.48
-22.96
11.72
0.054
(Ull
3.3:'
-13.46
9.43
0.072
1.45
-f).OO
11.03
-18.67
11.89
Interiak .. Corp.
0.041
2.16
0.72
12.35
R"ylrrh Corp. AlIlpco-l'ill,bllrgh Corp.
0.050 0.053 0.054 0.070
B9 2.41 1.41 2.79 2.35 5.24
2.2:' 0.66 0.27 0.74
59.40 5.02 5.91 6.18 7.13 6.49
-17.24 -57.90 -19.05 -12.82 -23.53 -16.67 -26.92
7:'.00 19.18 11.11 22.92 19.07 50.00
SkCWIIt'\S
Kllrtosis Minimum
Maximum
Pond A: Daily Rrlurru
Index
Intefnati(Hlal Bw;inc5..o;
Machines Crr1l"r,,1 Si!(n,,1 Corp; Wri!(lt'yCo.
Encrgt"n Corp.
Ceneral 11"'1 Corp. Caran Inc.
0.079 COlitint"nt,,1 Materials Corp. 0.143
0.72 0.93
23.08
Pan.' B: Monthly Returm Valll"-W"i~hled
Index Eqllal-Weighted Index Intt"rnational Business Machillt"s (;"n"lal Si~n;\1 Corp. Wrigley Co. Intc-ridke Corp. R;lylech Corp.
0.96
4.33
-0.29
2.42
-21.81
1651
1.25
:'.77
0.07
4.14
-26.80
33.17
0.81
6.IR
-0.14
0.83
-26.19
18.95
1.17
R.I'1 6.68 9.311 14.IIR
-0.02
1.87
-36.77
29.73
1.31 4.09 22.70 2.04 12.47 1.11
-20.26 -30.28 -45.65 -36.08 -24.61
29.72 54.84 142.11 46.94
-38.05
42.8fi
ETlt"Jgell Corp.
1.10
~'.7:'
Cent'ral Ilost Corp.
I.:n
II.fi7
0.30 0.67 2.7:l 0.77 1.47 OY,
Gar-all Inc.
1.64
11.30
0.76
2.30
-35.48
51.f>O
CmHinelllal Maleriah Corp. 1.64
17.76
1.13
3.33
-58.09
84.78
Al1\p("n-rilt.c:..hllr~h
Corp.
I." I 0.H6 O.R:! 1.06
IO.M
,
i
•
48.~fi
SllInllJary statistics for daily ami monthly returns (in percent) of CRSP equal- and valueweighted SlOck indexes and ten individual securities continuously lisled over the entire sample p<,riod frolll.!"ly~, 19fi2 to Derember 30,1994. Individual securities are selected to represent 'torks in each SilO decile. Statistics are defined in (1.4.1~)-(L4.22) .
. . . implies that it is impossible to make economic profits by trading on the basis of [tbat information setl. Malkil'l's first sentence repeats Failla's definition. His second and third sentellCl'S expand the deflJlition in two alternative ways. The second sentence SltJ!:J.!;('.~ts that Illilrket efficiency can he tested hy revealing information to
22l
I. Ill/mi/w/iol/
rna kel pilrticipanlS and mcasuring thc rcaction of sccurity prin's. If prices do \101 rnove when information is revealed. thell the market is dlicil'llt with reS~)CctlO thaI information. Although this is clear conceptually. it is hard to carty out such a tcst in practice (except perhaps ill a lahoratory). Malkicl's third sentence suggesL~ an alternative way to judge the dliciellcy of a market, by measuring the profits that can bc made by tradillg Oil information. This idca is thc foundation of almost all thc cmpirical work 011 market efficieucy. It has heen lIsed ill two main ways. First, lIIallY 1'1'searchers have tried to measure the proliL~ carncd by markct proiCssionals such as mutual fund managers. II' thcse managers achicve supcri(Jr rcturns (after a
i
Weak-Corm Efficiency: The information set includcs only the history of
prices or returns themselves. Semistrong-Form Efficiency: The information set includcs all informatioJl known to all market participaJlts (publicly available information). Strong-Form Efficiency: The information set includes all inforlllatioll known to any market participant (private information). The next step is to specify a 1II0dei of "no rill a I" returns. Here the dassil' assllmption is that the normal returns on a security arc constant over time, hut in recenl ycars there has hecn increased intt'l'est in equilibriulIl models Wilh time-varying normal security returns. Finally, abnormal security returns are computed as the dill'erclllT betwecllrthe return on a security and iL~ normal return, and forecasts of the ahllor/,Ial returns are constructed lIsing the chosen infilfluution set. If till' abnormal security return is unforec<1stahlc, and in this sense "randolll," th('n the h)~pothesis of market effici('ncy is not rc.:iected.
1.5.1 Ffficif'll/
\
MtlTkrl.1
(/Ilil/hr i.alll o/lIrlil/l'd EX/ll'r/a/ioll.\
The idea that efficient security returns should be random has oftcn caused confll~ion. Many people scem to think that ;lnefficicllt security price shollid
I
J.5. Mil/lid J~jli(il'l/ly' he SlllOOlh ralhn thall randOln. Black (I !)71) h;ls ;Illa('k(~d this idea rather
t'ffl-nil't'ly: :\ perren lIlarket for a stor\;. is olll' ill whidl then' arc Ill) proliL~ to he made by people who haw no special inforlllation about thc company, and in which it is dilIicult eVl'1I for people who do havc spccial inlill'mation to make profits, beGIllS(~ the pritT a(ljusts so rapidly as the information becomes available .... TilliS we would like to see randomness ill the prices of successive transactions, rather than great continuity .... Randomness means that a series of slllall upward lIIovelllellts (or slllall downward movcments) is very unlikely. Ir the price is going to move up, il should move lip all at OIlCC, rathcr than in a scries or small steps .... I .;lrge price movcments arc desirable, so long as they arc not cOllsistently i(lllowed by price movemcnts in the opposite direction. Underlying this confusion lIIay be a bclief that retunts cannot be randolll if security prices are dctermincd by discounting future cash 1I0ws. Smith (1968), for examplc, writes: "I suspect that even if the random walkers announced a pcrfcct11lathematic proor ofrandolllncss, I would go on believing tha, in the long run future earnings influencc prcsent value." III fan, the discounted present-valuc model of a sccurity price is entirely consistent with randomness ill security returns. The key to understanding lhis is the so-called Law oj Iterated Hxpe(/aliufIJ. To state this result we define information scts I, and j" where I, C J, so all the information in I, is also in J, bUl JI is superior bccausc it contains some extra information. We consider cxpcn,\lions of a random variable X conditional 011 Ilws(' illrormatioll S{·ts, wrillcn E[ X I I,] or E[ X I J,j. The l.aw of Iteratcd Expectations says that E(X I ILl = E(E[X I jLl I I,]. In words, if one has limited information II> the best forecast one can make or a random variable X is the forecast of the forccast one would make of X if one had superior information J" This can be rewriuen as ~:[X - E[X I J,] I I,J = 0, which has an illtuitive interprctation: Onc cannot lise limited information lito predict thc forecast crror one would make if one had superior information J,. Samuelsou (1965) was the first to show the relevance of the Law of Iterated Expectations ror sccurity market analysis; I.e Roy (l9H9) givcs a lucid review of thc argulIlent. We discuss the point ill detail in Chaptcr 7, hill a brief sllllllllary may be helpful here. Suppose that a security price at lime I, /'" can be wrillen as the rational expectation of some "fundamental value" 1", conditional on information I, available at tillle t. Then we have P,
:=
1':[ V' I I,] = I':, V'.
(\.5.1)
The s;lIne equation holds one period ahead, so 1',+ I = E( V' I I"
I
I =
E'l I \,'.
( 1.5.2)
filiI Ihen Ihe ex(>eclalioll of III(' chan).!;e in the price over the next period is
1-:,1/'1+1-1',1 = E,IE,tIIV'j-F./[V'))
=
0,
Iwcallse I, C l,t I. so E,I 1-:/ + II \"11 = E,I V' J hy Ihe I.aw of Iter;\!ec\ ExpectaliollS. Thlls rl';lli/ell ch;lIIgl's in plin's arc IInlilrcr;lslable ).!;iven information ill the sci 1/.
1.5.2 !., lI/(///U'I/':llifil'llry '/r.\lab!r?
Althollgh the l'mpirirallllt,thodoloh'Y slIllImari7.ed here is wdl-('SI;lhlishcd, Ihere arc sOllie seriolls C\if'firultics ill inllTpreting ils resllJrs, Firsl, allY lest of efficiency nnlst asslln\(' an eqllilibrilllll model thai c\ermes nOflllal secoritv relurns. If dIicicncy is rejeclec\, Ihis could be because the markel is Iruh· incl'lkienl or bccallse all illcorrcci cqllilibrium model has hcen asslIl\lcd, This joillllr:v/ltlllrl'.li,1 pl'OlIl('\1I \\leans 11\;11 lIIarkel crrlciency as slIch em newr he n:je('\I'c\, Second. perli'('\ I'fficiency is an lin realistic benl'hlllark th;1I is uillikely 10 hold in pranicl'. Evell in thcory. as Grossman alld Sliglitz (I !lHO) h;l\c shown. allllonnal \'('tllI'llS will exist iftherl' are costs of ).(.lIherillg and pmcessill)!; ill(III'lIIalioll. Thcse rellirns are necessary 10 compensate illv('siors fill' their infonnalion-gathnill).!; alld informatioll-processing eX)lenses, and are 110 lon).!;er ahllol'mal whell Ihese expellSes arc properly accollnlcd (i,l'. III a large and liqllid mark('I. inlilrlllalion cosls are likely tojllstify (111)' snd} allllOrmall'l'llInlS, hili il is dimcllit 10 say hnw small. evell if sllch costs could he measllred precisc\}', The nolioll or rrialil'f efficicncy-Ihe efficiency of one marketmc;lSlll('d agaillst allothcr. e,g,. tlie N('w York Stock Exchange vs, Ihe Paris BOllrsc. 1',,lures markets vs, SpOl markets, or all('\ion vs, dealer markels-llIay he a mort' IIscrlll rOllccpl Ihall Ihc all-Ol'-llothill).( view lakcn by milch or Ihe tradilional lIIarkel-eflicit'IKY lilnatllre, The adv;llIla).!;es of rclaliw elliciellcy over absolule efliciellcy arc easy 10 SC(' hy way of all ;lIIal0!n'. Physical syslelIIs arc ofIe II ).!;iv(,11 all ('nici('III"}, rat ill).!; has('d Oil the rclalivc propol'.tioll of cllergy or flld ('ollwr!ed to IIsd'lIl work. Therdc)J'(\ a pis Ion ellgille lIIay he rated al n()'){, eflici('lIcy, 1I\('allillg Ihal Oil avcra).!;c 1;0% of tilt" ellnh,)' cOlllailled ill the ell).!;illc'S fllel is IIseli 10 tllm lhe nallkshafr, wilh the remaillill).!; 40'7" lost \0 olhel' forms of work slich as hcal.lighl. or noise. Few ell).!;illeers wOllld e\'eI' t'ollsilln pCfrOrtllill).!; a stalislicaltesllo (11:1('1'lIIille whclhn or 1101 a givclI ('lIgillC i.~ perfcnly d'lirienl-sllch all cllgill!' exists ollly ill 11\l' idt',11 i/.ed frid iOllll'ss world of Ihe imaginatioll, I\lIlnwasll rill).!; I'elalivc eflkiclI('\'-rclalivc 10 Ihe friniolliess idcal-is conHllonplan', hHh-I'I\. wc ha\'\' fllllH' \1) ,"xpcn SlIl'h IIH'aSlln'lI\ellts (i)\' lIIallY hOllsl'!lold prodll('I~: ail' fOlldilioll('JS, hoi wal('J healers. rcfri).(('raIOl's, ('tf. Similarly,
25 market efficiency is an ideali7.ation that is economically unrealizable, but that serves as a useful benchmark for measuring relative efficiency. For these reasons, in this book we do not take a stand on market efficiency iL~e)f, but focus instead on the statistical methods that can be used to test the joint hypothesis of market efllciency and market equilibrium. Although many of the techniques covered ill these pages are central to the market-dflciency debate-tests of variance bounds, Euler equations, th¢ ('APM and the APT-we feel that they can be more profitably applied to measuring efficiency rather than to testing it. And if some markets tuI"it oUI to he particularly inefficient, the diligent reader of this text will be wel~ prepared to take advantage of the opportunity. ! I
1
f I
I "
2 The Predictability of Asset Returns
and most enduring questiollS of financial econometrics is whether linancial asset prices are forecastable. Perhaps because of the obviolls analogy between financial investments and games of chance, Ill
the notion of IOIlv;-rallgl' dCpCII(/CI\(T, and a test ()r this phenolllenon is prest'lIt('(1 ill SCrlion ~.(i. For complewlll'ss, we provil\e a hricr discussion or tests for IInit roots, which an' son\('lin\('s confused wilh lesls of 11((' randolll walk, III S('CliOIl ~.H WI' prescnt s('\'l'I'al cmpirical illustratiolls that docull1ellt illlportantdepartur('s from thc random walk hypothesis for r(,Cl'nl US stork markel dala.
2.1 TIle Random Walk Hypotheses A uscfnl way 10 organizl" Ihl' various versions or Ihe ralldolll walk alld mar· lillv;ale IIwdds Ihal 11'(' shall ]In'st'llt 1)('1011' is 10 consider the varions killd, or depl'llIl('ncl' Ihal can exisl helwl'I'1I an asset's returns T, and T'H all\\'11 dales t alld (+ k. To do Ihis. dl'lIlIl' Ihe random variahles !(r,) alld ~(7i+d where I(') and g-(.) are Iwo arhilrary functions, and consider thl' situatioll' in which
(:01'1 I( r, l. g( r, \ ~)I
o
CU.!)
ror all I and 1'01' ko;fO. For appropriately chosen !(.) and K(·). virtually all versions or the random walk and lI1artingale hypotheses an.' captured bv (2. I ,I), which lIlay he interpreted as an orthogo7lality condition. For ('Xalllple, if I(') alld g(.) an' ('('striucd to he arhitrary /illPllI' fULl" lions,then (2.1.1) implies th.ll returns arc serially uncorrc1atec1. corresponding to the Ufl/II/om Walk J lIlodel described in Section 2.1.3 h .. low. Alternatively, ir !(-) is unrestricted but g(-) is restricted to be linear, then (2.1.1) i, equivalent to the marling'lle hypothesis Ilescribed in Sectioll ~.I. FilJ;llly. if (2.1.1) holds I()!' all functions!(.) and ~(.), this implies that returns art' 11111wally independellt, COIT('SI)(l\\(iillg to the /lam/om Walk I ;lIld /law/om Hfdh 2 l1lodels discllssl'I\ in St't·tiolls 2.1.1 and ~.1.~. respectively. This dassilicatioll is sUIIIIll;lrized ill Tahle 2.1, AhhouV;h there an' several other ways to characteri7.e the various ralldOI1l walk alld marlingal(' lIlodels. COllditioll (2.1.1) and "nIhil' 2.1 are pankIIlarly rdevallt 1(11' ('('ononli!' hypotheses sinct' almost all eqllilibriulll asselpricing lIIodels can he reduced to a sct or orthogonality cOllditiolls. This interpretation is explored ('xtl'lIsively in Chaptl'l's Hand 12. Thl' MtlTlillKillt' M(I(it-l }'erhaps the ear/il'st III1Hkl oflillallrial asset prices was the mrlTlil1Ka/r 1ll00kl. whose oriv;in lies in tIll' hi~tory or gallles or chance and till' hirlh of prohahility theor\,. The pr
Table 2.1.
Co\'[j(r,), g(T,+.»)
=0
Cuusificalion of random
'l'f1{p.
and martingalR hypo/hPsrs.
g(T,+.).
g(r, •• ),
Yg(.) Linear
'V g(.)
Uncorrelated Increments. Random Walk 3:
f( r,). VfO Linear Proj[r,+.lr,) = J.L
Martingale/Fair Game:
fer,). Vf(·)
Independent Increments, Random Walks I and 2:
E[r'HIT,] = J.L pdf(r'Hir,) = pdf(r, •• )
"Proj[.y I xl" denote, the linear projection of J onto x, and "pdfC)" denotes the probabit:t)' densit), function of its argument.
30
2. 77le Predictubili/.v oj A.U ft
Uptuml
Lud o Ak ae (Th e Boo k oJ GmllfS 0JC hal lrf) , in wh ich he wro te: 1
Th c JIlost fun dam ent al pri nci ple of all in gal llb lin g is silllply eql lal ('111 dit ion s, e.g., ofo pp on 1 cn ts, ofh yst and ers , of mo ney , ofs itll atio ll.· >1 dic c box , an d of the die Ihe itself. To the ext ell l to wh ich YOI l dep;1I1 1111111 tha t cqu alit y, if it is in yo ur op po ne nt' s favollr , YOI I arc a fool, all d if ill yo ur ow n, you arc IIn jllst. Th is pas sag e clc arl y con tai ns the no tio n ofa Jai rgt lllle , a gam e wh ich in yo ur favor no r you is ne ith er r op po nen t's, and thi s is the ess enc c of a lIla sto cha stic pro ces s {I',} rtil/ jial l', a wh ich satisfies the fol low ing con dit ion : E[ /"I I I I'/, ["- I, ... J =:= 1'/, (2. 1.2 ) or, equivalently,
E[ I', tl- I'/ 11 ',,1 '/-1 , ... 1 = o. If P, rep rcs cnt s on e's rtll llli lati ve win nin gs or we alth at dat e I fro sOllie galliC of cha nce lll playiflg eac h pcr iod , the ll a fai r gal lic is oll e (l>l' wh exr }'c ted wc alth nex t irh the per iod is sim ply eql lal to thi s pcr iod 's (2.1\.2)), con dit ion ed we alth (se e 011 the his tor y of the gam e. Alt ern ativ ely if tli,c exp ect cd inc rcm , a gal lic is (~Iir ent al win nin gs at any sta ge is zer o wh en con on the his tor y of the gam dit ion ed e (sc c ('2 .1. 3». If 1', is tak cn to be an ass et's pri ce at dat e I, the n the ma rtin gal e hyp oth esis sta tes tha t tom orr ow 's pri ce is exp ect ed to be equ al to tod ay' givch thc ass et's ent ire s pri tT, pri ce history. Alternativ ely, the ass et's exp ect cha lig c is zer o wh en con ed pri ce dit ion ed on the ass et's pri ce his tor y; hen ce iL~ is ju~t as likely to rise pl'i re as it is to fall. Fro m a illr cca stin g per spe cti mar~ingale hyp oth ve, the esi s im pli cs tha t the "be st" for eca st of tom orr sim~y tod ay' s pri ow 's pri ce is cc, wh ere "bc st" me ans mi nim all llc an- sql lar ed Ch al> ter 7). err or (se e An oth er asp cct of the ma rtin gal e hyp oth esi s is tha t no no ver iap pri ed cha ng cs arc un cor pin g rcl ate d at all lea ds and lags, wh ich im pli es the efTccliveness of all line inar for eca stin g rul es for fut ure pri ce cha ng on hIstorical pri ces alo es bas cd ne. Th e hlct tha t so sw eep ing an im pli cat ion cOlin! fro m as sim ple a cou ld mo del as (2.1.2) for esh ado ws the im po rta nt the nla rtin gal e hyp oth rol e Iha l esi s will play in the lIIo del ing of ass et pri ce dyn (se c thc dis cus sio n bel alll ics ow and Ch apt er 7). In fact, the ma rtin gal e was lon g con sid ere d to be a nec ess ary con dit for an 1firienl assct ma ion rke t, on e in wh ich the in( llfI lla tio n con tai ned pri ces is instantly, fully, in pas l an d per pet ual ly ref l(,c ted ill the ass et's cll rre If the ma rke t is eff icie nt pric(':~ nt, th( 'n il slio lll< lno t be pos sib le to pro llt by tra dil lg Oil 'Se e \laI d (1990, Cha pte r 4) III' \",.,
ll<'r ,\('t;,i\" 1Se~ Sam llet ."" (I !Iii,>, I !In , \'IT \) , R"h t·rt, (1\)1,7) form market effIciency.
}1("
.,lso ,\d' lu's an
~'s~('1
,;oil, Iht· martjll~;ol .. hyp "IIo,·,j,
malKet to he .\,mi.um ll):-fontl
Il'I'fl h'
.HH t "IOU!["{mm
2. I. The R(l//(~{)1Il Walk Hypolhe.lf .l
the illfo rma tioll cont aille d ill the asse t's pric (' histo ry; hCl ln'lh e cond ition al expc n;lIi oll offl ltllr e pric e chan ges, cond itiol lal olltl ae pric e histo ry, cann ot ill' eith er p"'l ii,,' or nega live (ifsh or\s alcs arc I.... asih le) and ther efor e nllls t he 1.('1"0. Thi~ lIoti on of enic iellc y has a won derf ully cOll lller intll itive and secm ingl y cOll trad icto ry flavo r to it: The llIor e dlk icnt the mar ket, the lIIor e r;tlHlolll is thc sC'l uell ce of pric e chan ges gell erat ed hy Ih(' mar ket, alld the IllOSt eflic ielll mar ket of all is olle ill whic h pric e chan ges arc com plet ely r.lndOIll anel ullp redi ctah le. How cver , one of the cell irall enei s ofln odc m fina llcia l econ omi cs is Ihe ncce ssity of sOlnc trad c-of fbet wee ll risk and exp ecte d retll rn, and alth oug h the lIlar ting ale hypo thes is plac es a restr ictio n on exp ecte d retu rns, it doc s Ilot ,\CCOUllt for risk in any way. In part icul ar, if an asse t's exp ecte d pric e chal lge is posi tive, it may be the rewa rd nece ssar y 10 attra ct inve stor s to hold the assel and bear its asso ciate d risks. The refo re, d('sp ite the intu itive iq)j> calth at the fair-galliC inte rpre tatio ll lIIight have , it has been show n that th" Illar ting ale prop erty is neit her a ncce ssal )' nor a suff icien t cOll ditio n fOI· riltionally dete rilli ned assc l pric es (sec , for exal ilple . I.em y [197 3]. Luc as [ : ~)7H]. and Cha pter 8). Nev crth eles s, the mar ting ale has beco me a pow erfu l 1001 in prob abil ity anel .,tatistics and also has illlp orta nt appl icat ions ill mod em theo ries of assct pric es. For cxal llplc . onc e
2.1.1 The RandulIl Wal/( I: /lj) !1l0f llle/l1 5 Perh aps the simp lesl vers ion of the rand olll walk hyp othe sis is the inde pend entl y and iden tical ly dist ribu ted (liD ) incr ellle nis case in whic h the d!'na lllic s of {I'd arc give n by the follo wing equa tion :
(2.1.4) wile re J1 i~ t he exp ecte d pric e chan ge or drill, and 11»( 0, 17~) dcn otes thai f, is illde pend elltl y and iden tical ly dislr iilut ed Wilh IIICi lll () an
indt'pcnd(,IHT orlht' in('Jt'III('nls {f,} implics Ihatthc randolll walk is also a /;Iirgamt', hUI in a IIl1l<"h sll"OlIgt'rst'lIsc thalllhc martingale: Indq)('ncll'll("(' illlplit's Ilot ollly Ih;11 illn('IIH'nls an' IlIlcolTclaled, hilt Ihal allY Ilolllilll'ar fUllclions of Ihl' ill<TI'IIH'IIIS ;11(' abo IUlcolTl'ialcd. WI' shall call Ihis Ihl' NII/llillm I\(tih I model or RW I. 'Iil dl'l'ciop SIIIlII' inluilioll 1m RWI, consider ils coudiliollallllcall ;11111 \'ariallcc al dall' I, cOIHlilioll;ti Oil SOIll(' inilial valu(' al dall' 0:
n
1-:1/',
/'01 (:1.1.1; )
which /illlows frolll r('cursin' suhSlilulion oflagged 1', in (2.1"1) ;lIlIllh(' IIll ilHT('m('nls assumplion. From (2.1.:1) and (2.1.0) il is apparenl Ihal Ihe ralldom walk is lIollslalionary alld Ihal ils conditional IIIcall and variance art' hOlh lint'ar inlimt'. Tht'st' implicalions also hold for Ihc IWo olher lill'lllS or lilt' random walk 1I~'P0IIH'sis (RW2 and RW~) dcsnihcd helow. I'l'rilaps 111(' moSI CIIIIIIIIOII dislribulional assumplion for III(' inllo\';Itions or in('J'(,llIl'nls f, is normality. If t hc f /s arc liD N (0, a ~), Ihcn (2.1.'1) is l'lJllivaknl III all flrilh/llt'/i,. Hmwl/;'lII /II 01 i'lI/, sampled at rq~ularly SP;ll'l'd unil inlervals (Sl'l' Sl'('liou ~1.1 in ChapIn 9). This dislrihuliollal assumption silllplifies lIIany of Ihe calculations slIIToulHling the randolll walk, but suffers frolll Ihl' same problclII Ihal afIlicts norlllally distrihuted returns: \'iolalioll or\imiled liahilily. Irlhe condilional distrihution of 1', is normal, Ihcn therc will always he a posilil'l' prohabililY thai 1', <0. To avoid viola ling limiled liabilily, we may use Ihe salllc device a, in Section 1.4.2, namely. 10 assert Ihal Ihe nalural logarithm of prices /It == loj.{ 1', follows a random walk wilh llorlllallY distrihuled inCl'cmcllls; hence
I', ==
/1
+ /It_I +f"
(2.1.7)
This itllplies Illal conlilluously COIllPOIlIH\c(IITlllrtlS arc liD normal \,~lrialc, wilh 1I11'
2.1.2
nil' NIII/I/OIII
II'II/Ii 2: ftlth'l"'lIt/pl/lln(/'PIIII'I//I'
Ill'spile 1IIl' degalHT alld silllplilil)' of RWI, Ihe assumplion of'id('nlil'all\' dislrihlllni illlTl'lIll'lIls is 1101 plallsihle iiII' financial assel priCl'S OWl' IOllg Ii IIIl' spa liS. For l'xalll pie, ovcr Illc Iwo-IIIII HII'l~d-ycar h iSlory of Ih c New York SllIck Exdl;lIIgc, Ilwre 11;1\'(' b('('11 counlless changes ill Ihe (,CIlIlOlllic, soci;d. Ic.-iliiological. ill'lillllillll;d. alld rcglllaloryellvironllH'll1 ill wllicll siock prin's arc dClnlllilll'd, 'I'll(' assl'l'lioll 111;\1 IlIl' proh;,hililv!oJ", of' daily ""Hk
simpl~
relllrns has rem-ained the same over this two-hundred-year period is implausible. Therefore, we relax the assumptions of RWI to include processes with independent but not identically distributed (INID) increments., and we shall call this the Random Walk 2 model or RW2. RW2 clearly contains RWI as a special case, but also contains considerably more general price pr; ('('sses. For example, RW2 allows for unconditional heteroskedasticity in the (/'s, a particularly useful feature given the time-variation in volatility of man)' financial asset return series (see Section 12.2 in Chapter 12). • Although RW2 is weaker than RWI (sec Table 2.1), it still retains the most interesting economic property of the lID random walk: Ally arbitrary transformation of future price increments is unforecastable using any arbitrary transformation of past price incremenL~. 2.1.3 The Random Walk 3: Uncomiall'd incremmts
• An even more general version of the random walk hypothesis-the one most often tested in the recent empirical literature-may be obtained by relaxing the independence assumption of RW2 to include processes with dependent but un correlated increments. This is the weakest form of the random walk hypothesis, which we shall refer to as the Random Walk J model or RW3, and contains RWI and RW2 as special cases. A simple example of a process that satisfies the assumptions of RW3 but not of RWI or RW2 is any process for which Cov[( to Et-kl = 0 for all k '" 0, but where COV[E;. E;_kl -I 0 for some k j O. Such a process has uncorrelated increments, but is clearly not independent since its squared incremenl~ are correlated (see Section 12.2 in Chapter 12 for specific examples).
2.2 Tests of Random Walk 1: lID Increments Despite the fact that RWI is implausible from a priori theoretical considerations, nevertheless tests of RWI provide a great deal of intuition about the behavior of the random walk. For example, we shall see in Section 2.2.2 that tbe drift of a random walk can sometimes be misinterpreted as predictability if not properly accounted for. Before turning to tbose issues, we begin with a brief review of traditional statistical tests for tbe llD assumptions in Section 2.2.1. 2.2.1 Traditional Statistical Tests
Since the assumptions ofIID are so central to classical statistical inference, it' should come as no surprise that tests for these two assumptions have a long ': and illustrious history in statistics, with considerably broader applications i than to the random walk. Because of their breadth and ubiquity, it is virtually \
34
2. 'f'lte Predictability oJ A55el Uelu71/.1
impossible to catalog all tests of 110 in any systematic fashion. and we shall mention only a few of the most well-known tcsts. Since liD are propcrties of random variables that arc not specific I', .1 particular parametric family of distributions. many of thcse tests fall uillin the rubric of nonparamelrir tcsts. Some examples arc the Spearmall rallk correlation tcst, Spearman's footrule tcst, the Kendall r correlation test, and other tests based on linear combinatiolls of ranks or R-statistics (5('(' Randles and Wolfe [1979] and Serflin~ [1980]). ny using information contailled solely in the ranks of the observations. it is possible to develop tests or ~ID that are robust across parametric familics and invariant to changes ill Un\L~ of measurcment. Exact sampling theories for such statistics arc generally available but cllmbersome. involving transformations of the (discrete) uniform distrihution over the set of permutations of the ranks. llowever, fill mo~t of these statistics, normal asymptotic approximations to the samplillg distributiolls have been developed (sc(~ Serfling ( 1980]). ; More recent techniqul's based on the empirical distributioll fllllnioll of the data havc also been used to construct tests of lID. These tests of~ ten :require slightly stronger assumptions on the joint and lIlar~inal distribut\on functions of the data-gcncrating' proccss; hellcc they fall illto the clas~ of umiparametric tesL~. Typically, such tcsts form a direct rDlllpariSOli (between the joint and marginal empirical distribution functions or an indirect comparison using the (juantiles of the two. For these test statistics,!(~xact sampling thcories are generally unavailable, amt we must rely on i asymptotic approximations to perfilflll the test.~ (see Shorack and Wellner [ 19H6]). ~nder paramctric assumptions, tests of lID arc gencrally easier to COIIstruCt. for example, to test fiJr indepcndcncc among k vectors which Me jointly normally distributed. several st;lIistics may be used: the likelihood ratio statistic, the canonical correlation, eigellvalues of the covariance matrices, etc. (see Muirhead 11 !)83]). Of course, the tractability of sitch [csts /IIust be traded ofT against their dependence 011 specific paramc[ric assumptions. Although these tests an~ oftcn more pownful than their nonparametric counterparts, evell small departures frOI\1 the hypothcsized parametric family can read to large difTcrl'lIces hetwecn the actual alld nominal sizes of the t('st.~ in finite samples.
2.2.2 SI'f{III'1/(fJ IInti Unwr.\(/Is, (/1/(1 Hu7ts Thc early tests of the randolll walk hypothesis were largely tesL~ of I{W I and RW2. Although they arc now primarily of historical interest, nevcrtheless we can learn a great deal about the propcrtics of thc random walk from slIch tests. Moreover, several recelllly developcd econometric tools rely heavily on RWI (sec. for example, Sections 2.5 and 2.(i), hence a discIISsion oftl\('se
2.2.
Tflls
of R(/lldom Walk ): 11)
lests also provides us with an opportunity we shall 1't'
35
)IlC1I'IIlflll.1
to
develop
SOIllC
lIIachillery that
flIul Unwna!.s
V','e begill with the logarithmic vClsion or RW I or geol1letric Brownian 1110lion ill which the log pricc proccss PI is asslllllcd to li)lIow an lID random walk wil/LUul drift:
(2.2.1 ) and denote by I, the following r;III
I,
g
==
if
1',
/1, -
/'t-I > 0
if
1'1 -
/'t -
P,--I
<
(2.2.2)
O.
~'!Il('h like the classical Bernoulli coin-lOss, I, indicates whether the
N, -
2:" Y
I,
1'1 -
II I(j-j
+ (I
- / / )(1 - IIt-d
(2.2.3)
'~I
N, ==
11-
(2.2.4)
N,.
If log prices follow a driftlcss lID random walk (2.2.1), and if we add the further restriction that the distributioll of the increlllclll.S ( I is symmetric, then whether rl is positive or negative sho\lld be equally likely, a fair coin-toss with probability one-half of cither outcome. This implies that for any pair of consecutive returns, a sequence and a reversal arc equally probable; hence the C:ow\cs:Joncs ratio N,I N, should be ;lpproxilllatdy equal to one. More formally, this ratio m siuce:
q ;::
(J _
N, N,
N,fn
N,/n
Jr,
I, -
n,
..
-I"
Jr,
I - Jr,
j
q
2 1 q
== I.
/"
where "--" d('lIo\('s cOllvngclKl' ill prohahility. The fart that this ratio (·xn·(·(h·i\ Olll' Ii)r mall)' hislOriral stock returns scries kd Cowlcs andJolH'S (1!1:l7) to cOllrl\J('" Ihallhis "reprCsl'lIls conclusivc evidellcc orstrllCiIllC ill stock pritTs ... :1 II0wcVl'r. Ihe assumption of a I,ero drift is critical in dctermining the vallll' of q. In particular. q will l'xITcd Olle for an liD randolll walk with drift. sincc a drift-cil her positive or negative-c1carly makes sefll I l'Jl( cs lJ\ore likely than n·\'l'Isals. To sce this. suppose that log pritTS f()lIo\\' a normal random walk with dril't: /', = /1
+ 11,_1 + ~,.
Theil the illdicator variahle I, is uo IOllger a fair coin-toss hut is hiased ill Ihe direclion orlhe dril't. i.c .. I,
=
{Io
with prohahility
7T
wilh prohahility I -
(2.2,:i) 7T.
where rr
=:
I'r(r, >
0)
=
¢l (;;').
(2.2.1i)
If the drift JA is posilive Ihen Jr > ~. and ifil is ncgative Ihcn Jr < ~. Under Ihis 11101'1' gt'n('ral spccification. Ih .. I'alio of Jr, to I - Jr. is given hy Jr~+(I-Jr)2
q
=
2Jr (I _ Jr)
::: 1.
A~
long as the drifr is 1I0nl.ero. it will nlwllYS be the case that sequcllc('s are more likely thall reversals. simply because a nonzero drift induces a trend in the process. It is ollly for Ihe "fair-game" case of 7'( == ~ that CJ achi<. ves ils lower hound of OIlC. To sec how large all cflen a 1I0llzero drift might have Oil q. suppose thai /1 == O.OH alld f1 == 0.21. values which correspolld ronghly to anllual US stock ITltll'llS indexes OWl' Ih(' 1;lsl hal f-l'Cn tllry. This yieJ!ls the folJowill~ estimate of'rr:
IT
¢l
(
O,OH) == O.fi1R4
-(1.~1
Jr,
ir~+(I-IT)~
q
J.I!J.
0.5440
:\111 a late· .. siudy. (:owl('s (I!Uifl) (OIIC'C" tor hi;\."i('s ill linH'~aV('r;lg('<1 price.' ".ata ;111<1 . . [I!) (;J lali,,, ill c'Xc C'" of 01iC'. I h"'."'\'c'I, hi, (1I1l( hl,jnn i.Ii sOlJlt'what mOl"(, H".II(\(''': .... ,. whIle
fillfJ..
ollr \';11 iuu, ;1II.1Jy~C·.' h;l\"(' (li~( )u,,,d ~I Ictldc'll( ill Ilei (a"iC' i~ C"lJ.\h.
y loward."
p(·rsi."i.I('lIct' ill stork 1)1 in'
mO\'(,IIH'lIh.,
Ihi.Ii ~lIlIici('1l1 10 pJurick 11100e' Ih.tII II('gligihlt" I)lolif~ aher p.IY"U'1I1 orhrnJ..c'I.lgc·
which is close to the value or 1.17 that Cowles and Jones (1937, Table II) report ror the annual returns or an index or railroad stock prices rrom 1~35 to 1935. Is the difference statistically significant? . i To perform a formal comparison of the two values 1.19 and 1.17, ~e reqllire a sampling theory ror the estimator Such a theory may be ~h. tained by noting rrom (2.23) that the estimator N, is a binomial random variable, i.c., the sum or n Ikrnoulli randolll variables YI where
CJ.
y = {I I
0
with probability IT, = rr2 with probability 1 -
+ (I
- rr)2;
Jf,
hcnce we may approximate the distribution or N, ror large n by a normal distribution wi~h mean E[N,] == nJf, and variance VarIN,). I\ecause each pair ofa
Var[N,)
+ 2nCov[Yf, YHd rr,) + 2 (rr 3 + (1 - rr)~ - Jf;).
mr,(1 - rr,) nrr,(1 -
(2.2.7)
Applying a first-order Taylor approximation or the delta method (see Section AA of the Appendix) to == N,/( n - N,l using the normal asymptotic approximation for the distribution or N, then yields
q
(2.2.8)
where ,,~" indicates that the distributional relation is asymptotic. Since the Cowles andJones (1937) estimate or 1.17 yields JT, == 0.5392 and JT == 0.6399, with a sample size n or 99 returns, (2.2.8) implies that the approximate standard error of the 1.17 estimate is 0.2537. Thererore, the estimate 1.17 is not statistically significantly different from L 19. Moreover, under the null hypothesis rr has a mean of one and a standard deviation or 0.20 10; hence neither L 17 or L 19 is statistically distinguishable rrom one. This provides little evidence against the random walk hypothesis. On the other hand, suppose the random walk hypothesis were ralsewOllld this he detectable by the CJ statistic? To see how departures from the randoJll walk might affect the ratio CJ. let the indicator II be the following
= 4, q
'III bn. 1', is a tw(~stat~ Markov chain with prohahilities Pr( Y, /,1"1/1', ,md Pr( Y, == 0 I Y,_I = 0) = 1/2.
=1I
Y'-1
= I) =
2. TIll' I'mli(/abilil)' 1I/A.I.II'IIMum.\ Iwo-slate Markov chain:
\ I,
I (I .-
o
() (j
Ii
\\"11('1:(' (j denutes thc fU1II/ili(J1I1I1 prolJ'lhililY Ih,lI 1",+ I is negalivl', condilioll.1I a \msilivc Ii. and fl elCIlOIt'S Ih(' mJldili(JllIli proh"hility 11I.1I1111 is POSiliv<', n"l(~itiol\al on a IIcgative I;. If a = 1- fJ.this rcd\lces 10 thc rasc exanlinl'd aho\'(' (setl! = I - a): lhc Ill) r,llldol\l walk wilh drift. As IOllg as a t- I - fl. I, (hl'nce ,.,) will he serially corrdatl'd. \'iolating RWI. In this cas('. th(' thl'ol\etiral \'aille of lhe ratio (:J is givl'1l hy
Oil
\
,
(.). ==
\
(I -
whidi ran take tahk.\
Oil
O'}fl + (I
,lilY nOlIlIl'g.llin' I ('al
n.:!\)
u
0.:10 0.'10 O.!)O lI.tiO 0.70 II.HO
0.·10
!l.OO
:),:!~)
li':'O :),(;7 r).~:)
5.00 -I XI 4,71 4.1;~
O.!lO
'U)(i
1.00
4.:,0
!Ui7 :1.17 2.:1:1 J.!1'2 1.1;7 '2.:n 1.:)0 '2.'21 1.:iH '2.1 :1 1.'29 '2'()(; 1.'2'2 '2.00 1.17
as illlistratcd hy Ill!' ")lIowiJlg
Ii
n.1O o.:m n.:lO fi.:'O ·1.00 :1.17 '2.7:) '!.511
(:!,~.l 0)
'
\',JlII("
I
Il.I 0
- fJ)a
'J II _0'/,
I
1.~1'2
0.:)0 :).00 '!..:)O l.ti7
1':'0
1.~:}'
1.~:}
1.00 OXI 0.71 0.1;:1
'!..7:)
I.OH o.'IIi 0.H7 O.HI 0.7:,
IU)li
0.:)0
0.70
O.HO
11.~IO
1.llI)
·1.71 '2.21 1.:IH 0.% 0.71 O.!):) 0.·1:1 0.:1·\ II.:I~I 1I.'!.7 O':\:I 0.'21
·!.I,:I
·151; '!..(lIi I.'!.'!. O.HI 0.:)1; O.:I!I 0.'27 O.IH n.11
·150 '!..oo 1.17 0.7:) 0':'0 O.TI O.'!.I O.I'!.
D.I)('
0.00
0.1i0 ·IXI '!..:n 1':'0 I.OH OXI 0.1i7 0.:,:) 0.·\1;
2.1 :~ 1.2~)
O.H7
O.li:1 OAIi 0.:"1 0.'2:) O.IH Il.I '2
lUll;
A~ 0: and fJ hoth approach Oill'. tile lik('lihood of revn~als increases alld hencc q approaches O. A~ either Ci or fJ approadlt's I.no. th(' likelihood of sequl'nces increases and q ill('l'l'as('s wilhout bOllnd. In such CISl'S. ( ] i~ d('arly a rcasonablc illclicator of deparlurt's from RW I. [Iowt'\'('r. lIot(' that there exisl combillatious of (a. {I) for which at-l-{i alld q= I. ('.g., (o:.fi)=(~. ~); hence the (;1 statistic canllot distinguish illt'sl' ca.~(·.~ froJlI RWI (s('t' I'robl('1I1 2.:~ for hlrtlH'r discussioJJ). /{un\
Allot her cOllllllon test for RW I is th(' fllII.1 11'.1/, ill II'hidl tht' llullliln 01 Sl'qlll'tH,(,S ofcol\s('clltin' posili\'(' ,tilt! n('g,lIi\'(' n·tlllll~, or runs, i~ tahulated and t'O\l\IMITd against its sa\llplill~ distrilllJliOIl ulldn tht' ralldom II'dlk hypothesis. For cxalllpk, \lsillg tht' inclicalor \'ariahlt' I, ddill('d ill (~.~.~). a partir\llar s('qucnce or 10 l('tunlS Illa\' he rqll('~('llt('tI b\' 10011 \0\00. Clllltaillill!-,: thrl'(' rllIIS or 1~ «(JrI(,Il~1 It I. :\. alld I. \'("IH'r\ ill'II') ;IIHllhn'(' 1I11l'
2.2.
'1'1'.1[.1
o/fill/lli(J/Il'Walk I: /If)
39
/111'11'1111'111.1
of Os (oflcnl-\th~, I, and 2, respectively), thus six runs intota!' III contrast, the seqll('Jl(T 0000011111 l'OlllaillS thl' Ser ofnllls N,,,m in a saillplc of Ii. Mood (1940) was the (irst to provide a cOlllpn:hellsivc allalysis of rllllS, .111<\ we shall provide a brkf slllllillary of his \IIost general resulL~ here. Suppose that each of 11 liD observations takes 011 olle of q possiblc v;t1ues with probability Jr" i = I, ... , I{ (hence Li Jri = I). In Ihe case or the indicator variable I, defined in (2.2.2), q is equal to ~; wc shall return to this special case below. Dellote by Nrun,(i) the total Ilumber or runs of type i (of allY lellgth), i I, ... , q; hence the tOlal IIl1mber of nllls N,,,.,, ;;:: N,,",,(i). Using combinatorial argulIlenls and Ihe properlies of the Illullinolllial distribution, Mood (1940) derives the discrete distribution or N,,,,,,(i) frolll which he calculates the f()lIowing 1Il01lle11lS:
or
=
Li
E[N,,",,(i)]
Val' [N""" (i) ]
1IJr,( I - Jr,) IIlf,(l -
·Ilf,
+ If/
(2.2.11 )
+ (;If,~
- :~lf;l)
+ If,~ C~ -- Hlf,l- !)Jf /) Cov(N,,,m(i), Nru",(j)]
-Illfi If,( 1 - ~lf, - 2lf, - lfi If,(2Jri
+ 2lf,
(2.2.12)
+ 37(i Jrj)
- 5Jri Jr,).
Mor('( 'vcr, Mood (I D40) shows thal the distribution oj' the lIumber of nlIIs converges to a normal distributioll aSYlliptotkally whell properly lIormaliLccI. III particular, we have N,,,,,,(i) - Illfi( I - If,) -
x,
If /
fit N(O,
lfi(l - If,) - :'If/( I '- If,)l)
(2.~.14)
(2.2.15) N,,,,,, - 11(1- L,lf,~)
x
==
fit (2.2.16)
where .. ~ .. indicates that the eCJllalil), holds aWlllplolicallr. Tests of RWI b(' pl'I'j'ol'llll'd using- IIze aSYlllptOlic appl'O~ill1~tli()lls C!.~.14) or
1lI;1)' (itcll
Tahle 2.2.
1:\lJnIPfIIllIH!"r" /(/Ildom walk with fiJl!/11. 1/
1,000 1,000 1,000 I,OOIl 1,000 I ,DOll 1,000 1,000 1,000 1,000 1.000
/1
0 '2 ·1 Ii H 10 I:?
rr
0':'00 O.:,?tH 0.:>7Ii IUil:? 0.li4H O.liH:\ 0.7lfi J.I O.7·1H IIi 0.777 IH O.HO·\ 20 O.H:\O
ErN",,,,] SOO.:> 497.ti 'IH~).I
47:>.'2 4:>C>':'
4:n.ti 407.2 :nH.I :\47.:'> :~ I :>.:, 2H:t:>
EXPt'{"I('d lotallillmht'f of III"" il1.I ... ;tlHP!c· oln ind(·I)t·,)(It."nl8~rnoulii (rials rt'I)J('s('lIting po,,· iti\'('/lIc'gati\'(' fOlllinlloll.lril\' fOIU}lOlllHlt'd r('lurn" fCu' a GillI.~"iian gromerri(' I\rowllial1 mOlioJl wilh II .. in/1 = Of:k .... ~O'.)'r, allli !'Il.lIUi;lId d('viation n == ~1 %.
(2.2. Hi), anrr of rullS in lhl' sample of 1/ lhal an' Iht' ilh l)'p('; Ihlls 1/ = II,. To dn'dop SOI1l(' S(,IlS(, of Ih(' h('havior or Ih(' tolal nlllllb('r or rullS, consider Ih(' B('rnolilli caS(' /( = 2 corr('sponding 10 lhe indicalor variable I, dcfined in (2.2.2) or S('nioll 2.2.2 wh('rc rr dCIIOICS lh(' prohahilit:· tllal I, = 1. In Ihis caSt', Ill(' ('XIW(·\('(IIOI.tllllllnher of nllls is
L,
ErN,I/,,,] == 211JT(1 _rr)+n 2 +(I_n)2.
(2.2.1'7)
()hserV<' Ihal for an)' II :::: I, (2.2.17) is a glob'llly concave quadralic rUllllioll illlT Oil 10, I J whirh ,lIlaills a maximlllll value of (11 + 1)/2 allT :::: ~. Th('1'('/1m', a driftkss ralldolll walk maximil(,s Ih(' exp('cled lotalnlllllbcr or rllm for all)' (ix('d s~u\lpk sill' /I or, ait('ntaliVl'ly, Ihl' pres(,llce or a drift o( (,ither sigl1 will (kcH'as(' Ih(' ('''I"'rll'd 10lal \lumber of runs. To S('(' Ih(' Sl'lIsilivill' or El NIII/,'] wilh r('sp('('( 10 Ih(' clrift, ill Tahle 2.~ we r('port 11)(' ('"1'('('((''' IOI,t\ 11111111)('1' or runs for a sarnpl!' or II == I.(JO(J ohs('rvaliolls fil!' a J.:l'oll)('lric ralldolll walk wilh lIormally dislrihul('d ill(11/0). Frolll Tahk ~.~ WI' S(T Ihal ;tS Ihl' drift in('l'('ases, Ihl' ('X 1)('('\ ('{I lolal ntlllllH'r or I tillS dnlil\('s (ollsidnahly, (rolll :)()().:) 1'01' l('ro-ilrirl 10 21-\:1.:) ror ;t 21l(.~. drirt. Ilo\l'('n'I, all or Ih('s(' \'ah\('~ an' slill COJlsistt'nl wilh 11)(' ralldOiIl \\;dk hvp()1 h('sis.
41 To perform a test for the random walk in the Bernoulli case, we may calculate the following statistic:
z ==
Nun, - 2n7r(1 - rr) ~ N(O, 1) 2Jmr(l - rr)[ 1- 3rr(l - rr»)
.
and perform the usual test of significance. I\. slight adjustment to this statistic is often made to account for the fact that while the normal approximation yields different probabilities for realizations in the interval [Nun.. Nrum + I), the exact probabilities are constant over this interval since Nruns is integer. valued. Therefore, a continuity (orrection is made in which the z-statistic is eval· uated
z ==
NJlln1
+~ -
2nrr(l - rr)
2.Jnrr(l-rr)[I-3rr(l-rr»)
•
~
N(O, I).
Other aspects of nms have also been used to test the lID random walk, such as the distribution of runs by length and by sign. Indeed, Mood's (I !140) seminal paper provides an exhaustive catalog of the properties of runs, including exact marginal and joint distributions, factorial moments, centered moments, and asymptotic approximations. An excellent summary of these resulL~, along with a collection of related combinatorial problems in probability and statistics is contained in David and Barton (1962). Fama ( I !I(5) presen ts an extensive empirical analysis of runs for US daily, four-
.
2.3 Tests of Random Walk 2: Independent Increments The restriction of identical distributions is clearly implausible. especially when applied to financial data that span several decades. However. testing for independence without assuming identical distributions is quite difficult; . particularly for time series data. If we place no restrictions on how the. marginal distributiolls of the data can vary through time. it becom~s virtually illlpossible to conduct statistical inference since the sampling distribution, of l"VCll the most elementary statistics cannot be derived.
2. Tht' Prt'diclabilil.y of Awl
/{dllnl.1
I
: SOllie oflhe lIonparamclric lIIethods melltioned in Section ~.~.I such ;IS correia lions do lesl for independencc wilhoUI also rC!Juiring identical dislhhlliio/ls. bUI the number of distinrl marginal distriblitions is typically a lillite and slIIall lIullIber. For t·x~!lllpk. ~I tcst of independt'\lIT ('l'tWl'l'lI IQ tcores and academic performance involves two distinct margin;ll distrii>lllions: one for IQ scores and the other f()(" academic perf()J"JIl'IIHT. Mul~iple observations are drawn frOIll ealh lIIarginal dislributioll alld variOilS \Ilonparametric tesls can he designed to check whether the prodllrt of Ihe ~Ilarginal distribulions equals Ihe joint distribution of the paired ohserv\ltions, Such an approach ubviously GlIIllot slJcceed if we hypothesi!.(' a l1Jiifllle marginal dislri1>ltIion fplied to thcse modes of analysis, hut rather that the standards of ('vidence ill this literature have evolved along very different paths. Therefore, we shall present only a cursory review of thl'se techniques. ran~
2.3.1 Fillt'r Rules To test RW2, Alexander (1961, 19(1) applied a filter rule ill which all asset is purchased when its price increases by x%, and (short)sold when its price drops by x%. Such a rule is said to be an x% filter. and was proposed by Alexander (1961) for lhe following reasons: Suppose we tentatively assume the existence of trends in stock market pdccs but believe them to he masked by the jiggling of the market. We might filter out allmovelllellls smaller than a specified sile and examine the remaining movements. The total return of this dynafllil portfolio strateb'Y is then taken to be a mcasllre of the predktability in assel returns. A comparison of the tOlal retllrn to the retllrn from a huy-and-hold stratq..'Y for thc Dow Jones and Standard ,lIld Poor's industrial averages led Alexander to conclude that " ... Ihere liTe trends in siock lIl
2. J. TI'Jls (if /{1/111101ll Walk 2: hllll'{'I'III/1'II1
/lIrrl'llll'IIi.1
43
more frequellt trading, Fallla and IIlullle (I !)(j(j) show that evcn a 0.1 % roulldtrip transaction cost is enough 10 eliminale the prolils from such 1iII(T rules.
2. J. 2 -Ife/wiml Ibwly.,i.l :\s ;\ measure of predictability, til(' (iller rule has lhl" ;I
variables.
_.
~ ' . . . IfHII
HIIIIIII)'
f~1
J"',Uf'/ J{t'/lIrll.\
COlltrast this with the statclII('lIt: The pn'S('IHT or dearl}' id(,11 tifled,~lIpp()rt and resistallcc levels, couplc:d \vith a olH,-thinl retra("('IIH'lIt parametcr whclI prices li(, betwcen thelll, slIggests the pn'selH"(, of strong buying alld scllillg opportullities in the ncaHenll. l\oth statclllellts ha\'e !Ill' sallie IIIcaning: Using historical priccs, one Gill predict rlltllre prices to sOllie extent in the short ntn. Hilt becausc the tW
2,4 Tcsts of Random Walk 3: Uncorrclated Increments
1 1 I
()nl' or the 1II0st direct alld illlllilivl' tests or the ralldolll walk ,lIl(illlanillgale hypotheses iill' all individual lilll(, series is to check ror .Inial (on"rlnlioll, correlatioll hctw('cn two ohservations of the S,III1C series at difkrelll dates. Under the weakest version of the /'
==
(:ov[x, ),J JVar!xl.jVar[y!·
(2.4.1 )
, 45
2.4. Tfjls oj ilandom Walk 3: UI/(orre!ated Increments
Given a covariance-stationary time series {T,}, the hth order autocovariance and autocorrelation coefficients, y(k) and p(k), respectively, are definedtts6
COV[T,.
y(k)
T'H1
Cov[r,.
p(k)
(2.4.2)
T,Hl
Cov[
==
r,. r'Hl
Var[ rtl
=
y(h)
yeO) •
(2.4.3)
where the second equality in (2.4.3) follows from the covariance-stationarity of {T,I. For a given sample {T,I;"'I' aULOcovariance and autocorrelation coefficients may be estimated in the natural way by replacing population moments with sample counterparts: 1 T-A
y(k)
T 2:)r, - fr)(Tt+A -
==
TT).
0 ::; k < T
(2.4.4)
,=1
p(k)
==
TT
-
y(k)
(2.4.5 )
yeO) 1 T TLT,.
(2.4.6)
,=1
The sampling theory for y(k) and p(k) depends. of course. on the datagenerating process for I rtl. For example. if r, is a finite-<>rder moving average, M T,
== LakE,-b .=0
where {E,) is an independent sequence with mean 0, variance 0'2, fourth moment 1')(14, and finite sixth moment, then Fuller (1976, Theorem 6.3.5) shows that the vector of aULOcovariance coefficient estimators is asymptotically multivariate normal:
JT[ y(O)-y(O)
y(l)-y(l) ... y(m)-y(m)]' :.- N(O, V).
(2.4.7)
where
v ==
[Vi]
v,)
(1] - ?»y(i) y(j) +
1 00
-
L
[y(i)
y(f-i+j)
(:;:-00
+ y (l+ j)
y (l- i) ] .
(2.4.8
IIThr- rrC]uirement of (~uvarian(e-stationariry i:o. pr)Jn;nHy for notational conv~nien[e (llht'",i, .. y(k) and p(k) may be functions of I a, well as k. and may nOI even ~ well-defined i s(,fond moments are not finite.
2. The Predictability oj A.\sft Hfturn.! Under the same assumptions, Fuller (1976, Corollary 6.3.5.1) shows Ihal the asymptotic distribution of the vector of autocorrelation coerlicient estimators is also multivariate normal:
v'T[ p(O)-p(O)
p(m)-p(m)], ~ N(O, G).
p(l)-p(l)
(VUI)
where
00
g'j
-
'I
L
[p(e) p(l'-i+ j)
+ p(l'+ j) p(l'-i)
- 2p(j) p(f) p(f-i)
(=-00
rI
- 2p(i) p(l') p(/'- j)
+ 2p(i) p(j) p2(e) ].
(2.4.10)
!
For purposes of testing the random walk hypotheses in which all thc population autocovariances are l.ero, these asymptotic approximations reduce to simpler forms and more can be said of their finite-sample means and varia~ces. In particular, if hI satisfies RWI and has variance o~ and sixth mo~ent proporlionalto 0 6 , then I T-k . (2.4.1 I) E[p(k) 1 ---+ O('r- 2 ) T(T-l) \
=
Cov[p(k), p(m
I;;J. + O( r2) ,. { O(T-2)
=
\
if k
I'
otherwise.
i: 0
(2.4.12)
From\ (2.4.11) we see that undcr RWl, where p(k)==O for all k>O, the sample autocprrelation coefficients i>(k) are negatively biased. This negative bias comer from the fact that the autocorrelation coefficient is a scaled sum of cross-p,roducts of deviations of T/ from i\.~ mean, and if the mean is unknown it must be estimated, most commonly by the sample mean (2.4.6). But deviations from the sample mean sum to zero by construction; therefore positive deviations must eventually be followed by negative deviations on average and vice versa, and hence the expected value of cross-produc\.~ of deviations is negative. for smaller samples this eITecl can be significant: The expected value or p(l) ror a sample size of 10 observations is -)0%. Under RWI, fuller (1976) proposes the following bias-corrected estimator p(k):7 -
p(k)
==
_
p(k)
+ --,'{-It --" ( I ( f-I)<
-2)
- p (I,) .
(2.4.1:1)
7 NIlI Ihal pIA) i~ nllt IInbia",,": 11ll' I<'nll "hias
2. ·1. ·I;'.I/s 0/ Url/II/O/ll IVlIlI, J:
47
Ullmnl'/a/I'I/IIIOI'1Il1'll1.1
Wilh IIllililr!lIf)' houtlded sixth 1II01lletltS, he shows Ihat the s.\llIl'le autororrelatioll coefficicnts arc asymploticall), independcnt and normally distribilled wilh distribution:
Nw.
(2..1.14)
I)
(2.4.15)
N(O, I).
These r('stllls yidel a variel), or atllocorrdalioll-lJasnl lesls or the ralldolll ""Ilk hypothesis RWI. 1.0 and M.ICKillhty (1~18H), Richarclsoll and SllIith (1!1~14), all(l Romano and Thombs (f9!l(i) derive asymptotic approxilllatioll~ for salllple aUlOcorrelatioll codliciellis under evell weaker cOliditioIiS-lIl1corrdatcc\ weakly dq>ellc\ellt observatiollS-and Ihesl' results lllay he used 10 COllstruct tests or RW~ and RW:~ (sec Sectioll 2.4.3 helow).
2.4.2 Portmall/Pal! .'i/a/istirs Sillce RW I implies that all aUlOcorrclatiolis arc zero, a simple test statistic of RWI that has power against mallY alternative hypothescs is Ihe Q-statislic due to ~ox and Pierce (1970):
~II
'"
-
TLp~(lC).
(2.4.16)
h=1
Ullder the RWI nlill hypothesis, and using (2.4.14), it is ea~y to sec that c1.. = 'I"'£;~I P(lc) is asymptotically distrihuled as X~,. qUllg and Box (197H) provicle the (ollowing finite-sample correctiotl whirh yidds a helll'r (it lO the X~, for slIlall sample sizes: '"
fl' "'-'II
2(k)
== '1'('1'+2) "" -p-
b
T-k
.
(2.4.17)
By summing the squared autocorrclations, the Box-Pierce Q-statistic is designed to detect departures from zero alllOl'orrclatiom in either direction anel al all lags. Therefore, it has power' against a broad r'angt> or alternative hypotheses to the random walk. However, selecting the \l\lInber of autucorrelations III re'l"irc~ SOlllC care-if too /CW arc used, the prcsencc of highcr-orcler autocorrelation lIlay hc IIIbsccl; if too lllallY arc IIsed, the test Illa), lIol have lllllCh power due to insignificant higher-order aUlOcorrelalions. Therefore, while sllth a portlllanteall st,l\islir does have sOllle appeal, IWlLcr tesL~ of thc random walk hypothcscs lIlay he availahle when spedne allcl'Il;lIiVl' h)'I)otheses can be identified. We shall lui'll to slIcll examples ill III<' /ll'xi sCl'lions.
L.,
2, -I, }
\{II1I1//(/'
1111' 1'll'Il/dli/II/I/,\' "/ 11.1.11'/ UI'/IIII1,\
IIn/io,1
An important pl'Opnl\' of allthn'(' randolll walk hypotheses is th,1\ thl' \'alianft' of ral1dom walk iIHT('I111'nIS ""ISI Ill' a Iin('ar rUllclioll or Ihc lilllC il1tt'rvaJ. H For ('"ample. Ulldl'l' RW I 1<11' lo~ prices wher(' conlillll()IISI~' COIII»otllllied r('tlll'\lS I,"" lo)!; I',-Io)!; I', I alT liD, the varialllT of 1",+1"" I mllst he twin' the \'ariaIHT or 1'" Thneiorc, thc I'Jausihilit)' or til(' ralldolll \1';r1k modcllllay I)l' r\1('('k('d In' l'IlIIlparill)!; III(' variallcc of 1,+Ii_1 10 tll'ice (he varianc(, or r,,!' Ofcoltrsc, ill practicl' thl'Sl' will not he 1l1l111l'ril"ally idclltical ('\'I'n if RW I were trill', hut their rat io shonld be statistically indistinguishahll' from OIH', Tl1l'rl'fon', (0 rOllslnlcl a slalislical I('SI or IIH' ralldoll1 walk 11\'pOlhcsis usin~ I'arianrc ratios, liT n'quirc Ihcir samplillg distrihulioll under tIll' ralldolllwalk nlill h\'jlotill'sis,
l'o/l11/a/ioll I'm/lI'l/il'l 0/ \ II/iiII/O' NII/illl Ikillfl' (Icriving slich s;lIliplilig dislrihlltiolls, we develop sOllie intuitioll for the pOjlllhllioll valll('s of t!rl' varianrl' ratio statistic Ilnlln variolls scenarios, COl1Sic\l'I' again IIIl' ralio or Ihl' variancc or a two-period (onlilluol"ly compoundcd ITlurn I,I:!) == 'I + '1,,1 (0 Iwirl' the variance or a olle-period H'tllnl 1',. and 1111' the 11101111'111 kt liS aSSIIIII(, lIothing "hollt the time scri('S of n'turlls othn th;\1l st;lIionarity, Thcn this varianre r;ltio, whirh WI' Wrill' as \'R(~), I'('cluccs 10:
VOId I, + Ii· I I ~ V;ld 1',1 ~
\'011'1 It 1+ 2 Cov! Tt • Ii-II 2Var[r,J
VR(2)
C~.4,IH)
+p(l).
Wlll'IT p( I) is thc Iit-st-onln alltocorn'lation ('od'llcil'l1t of \'('turns {I,\, For any slationary lillie series, Ihl' populatioll value or thc variallcc ralio stali~lic VR(2) is simply OUI' plus Ihl' first-md('!' aUlocorrelation co('fficiellt. III particular, IIlIdl'r RW I a!lthl' ,lIlIOCOlTl'btiollS an' 7.1'1'0, helll'(' VR(2)= I in tltis rase, as ('''periI'd, IlIth(' pn'sl'IH,(, ofposilil'l' lirsl-orcln autocorrelatioll, VRe!) will ('xl'l'l'd Ollt', If rl'tllrllS arl' positi\'d\' ;lll\o('olTclal('d, Ihl' variallce of 11t(' SIIIII of 111'0 "Thi\ Iilu-arilr
propPllV j, iliOn'
difli, lilt
rase.' of R\\'~ and R\\,:\ hl'r;IIIS(' du' 11ow("\'(" 1", (,\'("11 in Iht-,(, ('''''I'S .li(' \'ar-jallfT and Ihis b Ihc.' lilU'aril), IJlopc'ny \dlirh lilt'
10 ,';IIt' ill Iht"
\';uiann's 01 iIlIT('IIU'III."; ilia\" ";11\' 1IIIIIIIgla lilllt'.
oltlu' .";11111 11111,1 f·fjll.tllhe· '11111 (If lh(' \';lIi;lIl(('S, \;11 i'lllt t' lalio ,,"~I (·xploih. \\'f' .. h.tli ('Oll"nll I h· ... b of;,lIllIn'c' hypfllh(,~f''; helo\\,. ')~I;III\ .. llIdic', 1t,1\"t' ("1'10111'" liIi .. I" CIlu', IV 01 lilt' lalUlo11l walk hypolh,"" ill dc" j .. illg ,'IIII,i, ic.II 1(""«,' 1)1 ('flu fahilil\,: I ('f CIII ",.11111 ,Ie' ill( Illcle (~HHplU'11 ;lIIcI ~Ltlll..i,\' ( I ~IH7), ( :( If 1lr.1l1t' (I!IHH), F.III,1 (lqlJ'.!l. 1.41 ,IIICI i\1.h
(I!I!I:\). .11111
Ri( h.tld",1t ;11111 SIOf I..
Killl;1\
(PIXX).
'J:I~~II.
I'ott'rha
;\1111 SUIIHIIC'I' (1~'HN),
Ric
h,llcI"'fllI
2.4. "I'r.I!'1 of Ralldolll H'tilk J: Ullron'f'llIlrrllllnrlllflllJ
49
one-period returns will he larger dian the sum of the one-period return's variances; bel.ICe variances \ViII grow faster than linearly. Alternatively, in lhl" presence of negative first-order autocorrelation, the variance of the slim 01 two one-period returns will he smaller than the sum of the one-period r('IIII'Il's variances; hence variances will grow slower lhan linearly. For comparisons beyond one- and lw(}-period returns, higher-order autocorrclatio!lS come into play. In p,lrticuiar, a similar calculation shows that the gener;'!1 q-period variance ratio statistic VR(q) satisfies the relation:
VR(q)
==
Var[r,(q)]
q. Varlr,l
== 1 + 2
L.
'II ( LI
k)
I - - p(k). q
(2.4.1\)
+... + r'-H I ,md p(k) is the kth order autocorreiatiQ,n coefficient of (rrl. This shows that VR(q) is a particular lineaT combination of the first k-\ autocorrelation coefficients of I Ttl, with linearly declinil1g weights. : Under RWI, (2.4.19) shows that for all q, VR(q)=I since in this Ca.'le p(k)==() for all k~ 1. Moreover, even lInder RW2 and RW3, VR(q) must still eqllal one
",lIn(' r,(k) '"' r,+ r'_1
VR(q)
Relations sllch as this arc critical for constructing alternative hypotheses for which the variance ratio tesl has high and low power, and we shall return lO I II is isslle below. Srull/dinK lJislribulioll of VD(q) (Il1d W(q) lmdrr HWI To cOllStrllct a statistical test for RWI we follow the exposition of 1,0 and MacK.illla), (19HH) and begin by stating the null hypothesis Ho tinder which Ih(' samplillg distribution (lfthe test statistics will be derived.'" Let p, denote lile log pric(' pr()Ct:~s and r, == /Ir- /J,_I ,ol1till\lOIlSly [ompounded returns. '''For ,"'1'111,11;,,(' c·,po,itiolls ,,·c· Call1pll<'l1 allci ~1;,lIkiw (I). l'otC'. Ita .111(1 StllIlI"''''', (I !IHK). Rirhanholl (I!I(I:I) .•\I,d RirhA«j"on A",I Stuck (19K!I).
50
2. The PTedirtability of A.I.II'I
/(1'1/1 m.1
The n the ilUli hypo thes is we cons ider ill this sect ion is II
Ho : Let our data cons ist of 211+ I obse rvat ions of log pric es {f~). 1'1 . .... I'l" \. 'IIU\ cons ider the follo wing estim ator s for p and a~: '2'1
[J.
cr; '2 °b
-
~II
L (I'. - 1'.-1 )
I
"i""II (/'2"
"'-I
- 1~1l
(2.
'l.u
-
-
L (I'. - I'H - II) 211 .=
211
,~
(2.4.21 )
I
L" ({'!. -
.
{'l.- ~ - 2/~) 2 .
.~t
(2.4.~~)
Eqll ation s (2.4 .20) alf(l (2.'1.~ I) Me the IIslIal sam ple mea n and vari ancc estil llato rs. The y are also the maxi mum-likeWwod estim ator s of f.1 and a ~ (sct" Sect ion 9.3.2 in Cha pter 9). Thc seco nd estim ator of a ~ mak es usc of tIlt· rand om walk natu re of PI: Und er RWI the mea n and vari ance of incr cme nts arc line ar in the incr eme nt inter val. henc e the a ~ can be estim ated hy onchalf the sam ple vari ance of the incr cmc ills of even -num bere d obse rv;lI ions I/~)./). 1'4 •...• pln)' Lind er stan dard asym ptot ic thco ry. all thre e estim ator s are stro ngly COIlsisten~: Hol ding all othe r para met ers cons tallt . as the tota lnul llbe r of ohse rvatioris 2n incr ease s with out hou nd the estim ator s conv erge almo st slIrely to thei r j>opulation values. In addi tion . it is well know n that cr'/.' and a,7 possess the following norm al limi ting distr ibut ions (see , for exam ple. Stua rt and Ord [~987»:
a/,
\
I' i
I I
,j'j;;(a/,-a~J
:.:.- N«J ,4a 1 ).
(2.'1.24)
lIow erer . we seek the limi tillg dislI 'ihut ioll of the ratio of the vari ancc s. Alth ol/g h itllIa y read ily be show n that the ratio is also asym ptot icall y Ilorm al with IIlrit mea n und er RW I. the vari ance of the limi ting dist ribu tion is not appar~nt siricc the two v;lriallce cstil llato rs arc dear ly /lot asym ptot icall y IInc orfe lated . Bilt sinc e the estim ator is asym ptot icall y crtic ient und er the nllll hypo tlles is RW I. we may \lse llall sma n's (1~7H) insig ht that the asym ptot ic
cr,;
II \\'t' a\.lIjume nonn ality only tor t·xpo . . itioll.lI ("onn 'lIicn rt'-th l' resl1l t~ in this s('nio ll "pply 1I111ch 1II0re gelle rally to log price I'lOces,,"s with liD illrn"II1<"lIt' th;\l posse ss fillite I(lIIllh Iltnm eilis.
2, ·1, 'li',I/.1 4UflllI/OII/ Walk J: l/lIomdfllnllllrll'll/l'/Il.l
",Iriallce of tite differenct" of a consistent estilllator and an asymptotically dlicil'llt estilllator is simply the C\iI"i"el"clln' of II\{' aSYIll(>lOtic varian(es,l~ II IV(' ddill(' tlw vari'lllce di(fc.'ITIIlT estimator as VI)('2) == tl\{'n C!"I.~:-\), (~.'I.~.j). and I'!ausman's result implies:
n;; - n,;.
TIll' Illll1 hypothesis" can then he t('stnlusillg (~ .. I.~:» alld allY cOllsistcnt ('stilll;llOr :!a'i of 20'1 (for example. :2(a~)~): COllstrud the standardized
sl;ltistic \'1)(:2)1 ~ which has a lilllitill~' standard lIoml;d distrihutioll 1111dn RW I. and reject the null hypothcsis at th(' :if;;, hoyd if it lies outside the illt(,ly;dl-I.%,I,%j, The as),lllptotic distrihutioll of the t,,'o-lwriod v;lriaIK(' ratio statistic \"71\(2) == a,~ now follows directl), fmlll (:2A,2!i) usillg a (irst-onierlil),lor applOxilllatioJl or the delta Illethod (se(' SeClion/\..J of the Appendix):I:1
In,;
'I'll< 111111 !!y(>othesis 110 can he t('stnl Il)' ("ollllllilillg the standardized statistic ~(VR(2)-I)/~ which is asynl()tolic;dly st,lIHttrd Jlormal-if it lies ()llIside the illtnval 1-1.~l(i. I.~HiJ. RWI Ilia), 1)(' rl'jnwd at thl' :)'}{, kvd of si~Jlili(allce,
AlillOugh litl' vari,III('C ratio is ,,1"1 ell pn'krrl'd to Ihe \',lIi,III("(' dint'n'll(T 1)('("'\\ls(' Ill(' ralio is scalc-fret", ohserv(' Iltat ir~(a,;)~ is IIsed 10 eSlilllale 2a I. lhell lite sl,llllbrd sigllilicalHT ll:Sl ofVD=() lor tilL' dilkrelllT will yidd lhl: saille inferellces as the correspollding test ofVR-I=O (l!' lhe ratio SiIlCl':
~VT)(2)
ff,i(a,; - a,;)
JET,}
~a,;
Tltnt'i"ure, ill Illis silllple cOlltexl th(' IWo tesl statistics arl: ('ljuivaklli. Ilow('\"l'r, 111('1'(' ;11"(' othl:l" reasolls that Illake tll(' I'ariallcc ratio Illore appealing \'.!
HI il'll}, 11.111'1111.111 (I ~)7H) c.'xploib III(' LH t 11,,11 ,IllY .I:'I~ IIIIHllllf.dly dlidc.'11i
I..':.lilll.lIor
01
Ii,., 1II11:-.t I't):-.~c.·s., the.' p"oP('II~' Ih.11 it i~ 'I'~ IIIpIOlit';dly unfOI n'lo"t:d with Iill' 11111"1 ('lit"(' (i" -- (i,. where: (itt i~ au)' olher (·~lilll.ltor 0111. II lIot, dlt:1I tllC.'''''' t"xi!oots a liue.n ("Ollibill.tlillll (II (i,. ,lIltl (ill -fir IhOit is iliOn.' c.'Hi("it'lll IIi.lllli, (CJlllr.uli< lillg Illl' ''-,.'IIIH«,'d l'Ilirielu"y
.1 p.II'IlIl('Ic.'1 (I,
~.I\"
I
0111,
"111(' I ("'1111 1t)lIo\\'~
dirt'nly. thcll. sinn';
,'\',III,i,,1 '"
,,\';11
=>
Iii, + Ii" -,i,1
,'\',11 I Ii,
,I\'a'IO" -
,'\';111';" I
Ii,
I
",hefe ;I\',III·J d('IIIJI('~ thl' a~>'llIptolic \',uian("('
I
,'\',III,i" - ,i, I - ,1\';11 \Ii, I,
"pCI ;11111'.
\'~111 p.lIlirlll.tr, apply Ihl" ddt.llIlethod to I({il.,i'.!l:::;(i,/fi',! \\liel"(' fjl=n/~-n,;. fi,;!:=(j,;. aud oh,cl \"(" [h.1I r1f~ ,11111 n-,; ,II (' a.,YIIIIHCJ!i( all~· 1111( 011 (,l.lIc·d IIt'( .111'(' n,! j,;111 dfifi('1I1 {".,lim:.tlol".
-r.-:;
and tll<'s(' arc disnlss{'d ill (;ochr,lIIe (l~lHH). Falls\ (19~1~), ',1I1d 1.0 and MacKillla), (I!IHH, I!lH!I). The I'aliall(,(' elilklt'llIT anel lalio sialislics elll })(' easil)' gcnl'r;di/l'd Itl IIllihipnioC\ ITllirIlS. I.t'! 0111' salllpll' cOllsist of IIq+ I ohsl'I"l'aliollS 1/~1o /1\, ... ,/1",,1. wh('l'(' '/ is "III' illtl').(('J' J.(lc"tcr than one "lid define Ihl' eslim,,lors: 1/11
.-
II
}
L /1'1
(Ilk -
11. - I )
(I'k -
/1.·-1 -
-
'=1
L
(II,,,,
-/~I)
(2.4.:2H)
1/({
III/
-" (1,;
-
11,/ k,-I
L"
°/~(q) //11
\'1)(1/)
-
(/'"k -
,~ II)
/1"*_,, -
(:~. ·1.:2!I)
'
qll) ~
C~.·I.:W)
k~1
0/; (1/)
-
G,;,
VR(q)
Cr/;(q)
-
0'1.
(2.4.:\1 )
"
llsillJ.( silllilar arglllll('lIlS, Ihc aSYlIlplolic distributiolls of VOl'll alld \/R(,/) IItHlcr till' RW I 111111 h)'JJotlwsis are
JiIii (1)( 'I)
N(O,'2(q-I)IT' I )
JIif/(Vlt(q) - I)
N(O,'l(q-I)).
(2.4:1:\)
Two illlportalli ll'iin('nH'nts of these statistics can improve their finitesample propntics slIhstantially. Th(' first is to lISC OlIIT/aNlinK q-peric,d rcIlIrns ill estimaling Ih(' variances h)' dellning Ihe following alternative l'slimalor for IT ~: .. "
I'~
IT,-Utl = -'7 /Ill
L (I'k k~"
.... J
Ilk-" - 'Ill)-.
This eslimalor conlaills /1,/- q+ I t('J'IIlS, whereas the l'SlimalOro;; ('I) (ontains only 1/ tl'l'Ins. Using overlappillg 'I-(l('l'iod rCllll'lls yields a lIlore effici('nt cSlimalor and hell('e a Illore pownflll Il'SI. The s('('olld rdill<'m('1I1 im'Olves (·(lIT(·(·ting lh(' bias ill Ihe valiant'{· eslimators 0,; alld h,~ herOIC dividing OIl(' hy Ihe olher. Denole the IInbiased estimators as i1~ and (II), whnl'
n;
1
,"/
""
- - L (II. -/1.-1 '''/ - 1 k~ I
",
-
' ~ Jl)
•. ,
J
• ..... UV
53
...
(2.4.36)
m
and define the statistics: (2.4.37)
This yields an unbiased vari;lIlce difference estimator, however, the variance ratio estimator is still biased (due to jensen's Ine<]uality). Nevertheless, simulation experiments reported in Lo and MacKinlay (1989) show that the fi~te-samplc properties of VR(q) arc closer to their asymptotic limits thall VR(q). Ullcler the null hypothesis II, the asymptotic distributions of the varialice difference and variance ratio arc given by . a
VD(q)
~
J1lii (VR(q) - I)
~
a
( 2(2q-l)(q-l) N 0, a 3q
N
(0,
1)
2(2 q-l)(q-l)).
3q
(2.4.38)
(2.4.39)
These statistics can then be standardized ill the usual way to yield asymptOlically standard normal test statistics. A5 before, if 0 4 is estimated by ~in standardizing the variance difference statistic, the result is the same as the stanclarclized variance ratio statistic:
Vr (q)
J1lii(VR(q) - 1)
(2(2 Q-,I)(q-I))-1/2 $q
J'ilfjVf5(q) (2(2 Q-1)('1- I
r;;4
ya!
»)-1/2
(2.4.4P) I
~
N(O. 1).
3q
Sam/}/ing Distribution of VR(q) under RW3 Since there is a growing consensus among financial economists that volatilities change over time (sec Section 12.2 in Chapter 12), a rejection of the random walk hypothesis because of heteroskedasticity would not be of much interest. Therdore. we seek a test for RW3. A5 long as returns are lIncorrelated, even in the presence of heteroskedasticity the variance ratio must sti.1I approach unity as the lIumber of observations increases without bound, ft' the variance of the SUIll of un correlated increments must still equal the sum of the variances. Howeyer, the asymptotic variance of the variance ratios will clearly depend on the type and degree of heteroskedasticity present. One approach is to model the heteroskedasticity explicitly as in Section 12.2 of Chapter 12, and thell calculate the asymptotic variance ofVR(q) uncler this specific Ilull hypothesis. However, to allow for more general forms
54
2. The Predictability oj A.I.Ip ( HI'll/m.1
of hete rosk edas ticil y, we follo w Ihe app roac h take n by 1.0 and Mac Kin lay (198 8) \'{hidJ relie s on the hete rosk edas ticit y-{: onsi sten l met hod s of Whi te (198 0) and Whi te and ))om owit l. (1 ~184). This app roac h appl ies 10 a IIlttc h broa der class of log pric e proc esse s !tIll than the liD 1I0r mai incr ellll 'nls proc ess of the prev ious seCl ioll, .\ part icul arly rele vant cOll cern for US stoc k retu rns as Tab le 1.1 illus trate s. I 1 • Spec ifica lly, let r Jl + ii, l alld defi ne the follo wing com pou nd nlill hypo thes is H~:
=
(Ill) For ailt, E[ir l
== O,ll luIE [i l il_ r ) = OJura71yr '" O.
(112) Ii rl is CP-lIiixillg wilh curfJicieu ts cP (lit) vJ siu r/ (2r- l) vr is a-mixillg witli (oifficimts a(m ) oj size r/(r -I), wher e r > I, such that Jor all t (lwl Jur (lilY r ~ 0, iI!l're exists s07l/eli > 0Jor whi rhE lklil _rl'l (,H) ) < D. < 00. I nq , ., lim E[i; J = a- < 00. "'1_ 00
nq
L
1:::01
j'llr rlllt, E{il il_; fl (,-.1 = 0 jor (lil)'
/Wl/Z. I'I'O j and k will'l l' j l' k. Con ditio n (Il I) is the unco rrcla ted incr eme nts prop erly of the rand om walk Jhat we wish to test. Con ditio ns (112) and (H3 ) are restr iClio ns on the max illlu m degr ee of dep end ence and hete roge ncit y allow able whil e still perll~illing sO/lle form of the Law of Larg e Num bers and thc Cen trall .illli t TheO rellJ to obta in (sec Whi te [I mH) for the defi nitio ns of cp- and a-m ixin g rand+1IJ sequ ence s). Con ditio n (1!4 ) imp lies that the sam ple aUlO corr elation spf' l are asym ptot icall y IIJlc orrel ate< i; this cond ition may be w('a kene c\ cons iller ably at the expe nse of cOll lput ation al silllp licit y (see !lole I:». 'I~Jis com pou nd null hypo thes is assu mes that jJ, poss esse s lInc one lalc d incr etne nlS but allow s for 'Illit e gene ral form s of hetc rosk edas ticit y, incll ldillg dtte rmin istic chan ges in the vari allce (due . for exam ple, to seas onal factor~) and Eng le's (1~)H 2) ARC II proc esse s (ill whic h the cond ition al variance (~epe~~ on past info nnat ioll) . Sihc e VR( q) still appr oach es one IInd er lit" we need only com pute it~. asym p'tot ic vari ance [call it U(Ij) J to perf orm the stan dard infe renc es. 1.0 and Mac Kin lay (198 8) do this ill two step s. First , reca ll that the follo wing equa lity hold s asym ptot icall y und er 'Illit e gene ral rond ition s: 1
VR(I" j) ==
I
+28 L '~I
(I - ;k) p(h).
(VI Al)
I
rOIlI~(". second 11I0ll lt'ub an' ",1111 .t....!'IIIIBe d 10 hc' fillite ; other\\,ls(', Iht" \'ari~lIln' lougt"r \\'e11 defin ed. This rull'., r.1l1t1 out (1i:"'l Iibulio lls with IIIlilll tc \·"lri~lIlre. such as thoM ' Ihe st.thlt' P,lrt" lo.. Le\y f.uuil), (with ill flt.n,1 ( It'n:"l. lif eXpO llt'Ill. \ th.1l are Ic~s tll.tn 2) prop ~1.11l(1t·lhr(}t (19tj3 ) al1d o\('d In Failla (I~)I;:)). J1{)WC H'r, 1l1;IIIY olll('r fC)lIII!'o ofl('p tokur t()\is ~,rt· .111(Jwc'd. "I< h 'I' Illal ~enerdled hy ElIgl e', (1 \IH~) ""tol q~ .. ""in · (olldi lioll,l lIv 1t~lel ",~,'d'''li( (AI{( 11IOf~" \'ee ScClill1l 12.2 :111 ill CIt"PI"\' 1'2) l.Jor
j, 110
1,{JI/~-lllJriwll
2, 5,
1&llLm.l
Secolld, II' ,It' Ihal undl'J' 1I~ (conditioll (114» the autocorrelation coelEel,':11 estilll(k) are asym(ltotkally uncorreiatedY' II' the aSYllJptotic \\lri.1J1Ce O. ( " of the p(k)'s call he obtai lied Ul\dn II;,. the asymptotic \'~Iriallre (}(q) oj \'R('1) lIlay he calnilated as the weil-(hted SUIII 01' the Ilk's, whne Ihe weigh Is arc simply the weighL~ ill relalion (~.4.41) squared. Denote by Il. and U(q) the asymptotic variances of Ii(k) and VR(q), respectively. Then ullder Ihe lIull hypothesis II~ Lo and MacKinlay ( I UHH) show that I, The slatistics VD(q), and VR(q)-1 converge almost surely to zero for all q as 11 increases without bound. ') The following is a heleroskedastirity-n)Jlsistenl eSlimator of Il.: (2.4.42)
:t The followillg is a heleroskedastit:ily-rollsisll'nl estimator of O(q): 1- 1
8(q) -
,.
4L (I - ~). 8•. ~~t
(2.4.43)
q
Despite the presence of general heteroskl'dasticity, the standardized test st:11 i,)tic if' (if) if;' (q)
CIII 1)(' used
10
lesl
il,~
J1zij(VR('/) - I)
fii
N(O,i)
(2.4.44)
ill lile usual way.
2.5 Long-Horizon Returns St'\'nal recent sludies have focused on the properties of long-horizon returns to test the random walk hypotheses, in some cases using 5- to 10yt'ar ITtUJ'IlS OWl' a 65-ycar salllple. There are fewer nonovcrlapping longhori/ol1 rcturns for a given tillle span, so samplillg errors are generally l\lthollgb Ihl., 1("~lriClion 011 the fOllrth ('r()!\!\'lIlOlIIl'l1l~ 01 f;, III .. }' ~t'l"Jll sOIllt'whdt lIlIiJlIIl~ it i", :-.ati:-.f"ll'd tor .~ny process with itHkpelHlent illcrelllelits (legaJ(lIess of heterogeneity) .IIHI ~Ibt) luI' Ijllt'~lr (;all:-.~iaJl ARCII processes. This a . . sumptioll !Hay he relaxt'd entirely, req\lir~ IlIg tilt' l"lilll~llioll of the asymptotic rov.uiuHTS of tht' ~IHI()n)IT(·l.ttioll (,!'ttiIllClIOr:'i in order In l'l
ill\(',
\",illl"'" lilt' lilllitillg ';tri"Jl(e f) of VR(q) vi" (~,~,'II), Allhollgh thl' r,'slIilillg estimator of II \\(lIdd he 1110)"(' c.:olllplicatcd than equdliou ('2AA.:-\). ill~ coun'pln.lIly Mlaigilllorw.... d dud lIIay I(,adily IH' forlll('d .d()l1~ the lines of N,·w('y .uHI \t\'(.'~t (I ~JH7). All ('\"('11 III()((~ Kt'llC'lal (and pos~ :-.ihly won.' t'xart) sampling thew), ftH"lht' v,lriillln' r~llio~ m;IY he olHailJ('d IIsing- the result'" uf nll"",r (I ~IH I) alld 1)1I1<,"r ""d Roy (IYW,), Again, lhis wo"ld ",nili",' IlIlIrh of till' ,implicily of (lur ;1.'Ylllptotir u'.'HIL,.
larger fllr statistil'S hased Oil long-horizon returns. Hut for some a\lertlati\'es to the randolll walk. long-horil.on returns can he 1II0re informative than their shorter-horizon ('Ollllterparts (sec Scction 7.2.1 in Chapter 7 and 1.0 ami MacKinlay I 1~IH9j), One motivation flu' using long-horizoll returns is thc permancnt/transitory componcnts alternativc hypothesis. lirst proposed hy Muth (1!)(iO) in a macroC(,(l11omic context. In this model. log prices arc c0111posed of 1\\'0 componcnls: a rando111 walk and a stationary process.
W,
)i
)',
+ 11',_.1 + f,.
any l.{'ro-l11ean stationary proccss.
and (w,1 and \),,1 ;\1'1' 1111\111ally indcpendcllt. Thc common interpretalion h)r (2.:1.1) as a nlOdc! of stock prices is that II', is the "fundamental" COIllponent that r('lIe('\s the efficient markets pricc. anc! _v, is a ZCfO-l11ean ~ta tiouary COI11POllCIH Ihal rdkrls a short-Ierm or transitory (\eviatioll from the e('fi('ienl-lnark('ls price II'" il11pl)'ing the prescnce or "fads" or othef market inefficiencies. Sin('c )', is slationary. it is mcan-rcvcrting by delinilioll and revcrts to its 111ean of zcro in the 10llg rUII. Although there are sevcral difficulties with such an illtnprctatioll of (2.5.1 )-lIot the least of which is the 1;I('t that market efficiellcy is talltological without additional rt'IlII(ill1i( strlll:tllrt·-ncvcrthdess. sllch an altcrnative provides a good bhorat()ry for studying the variance ratio's perilll'mance. While VR(q) can behavc in l11any ways under (25.1) for small If (<1<:pending on the cOlTdatiol1 structure of)',). as q geL~ larger the hehavio, of VR( ,,) becol11es less arhitrary, ) II partirular. observe that
1',
Ji,-Ji,
1
= II+f,+.v,-,~'-1
,/-1 1',( II)
L'
q-I
t -,
If)L
+
L f,-k +
,y, - .V'-'I
(::!,:,,:I)
k=()
,...=:/I
Val'l 1',(1/) I
(~,c),~)
'P~ 1- ::!)'I(O) - '2y\(q),
(:!,:l'I )
wh('1'C y,(q)= COVi.v, . .v1l ,/1 is IIH' aulol'ovariance runctioll of .y,. Thncrol('. ill this case the poplllatiol1 valll(, of the variance ratio he('()n](',~
VR(If)
l/a 1
+ 2y\.(O) -
If (a~
+ '2y,(O)
'2y\(q)
- '2y\( I»)
.,
0-
a~
+ 2y\(O) -
2y\(I)
as
q -+ 00
(2.5.6)
2y\(O) - ~y,(I)
1-------o~
+ 2y,(O)
- 2y,(I)
Var[ l'.y] I - ----'----Var[l'.y] + Var[l'.w] VR(q)
-+
1-
Var[l'.y] Var[l'./J]
(2.5.7)
.
whl'l'e (2.:l.6) requires the additional assumption that y,(q)-+O as q-+oo, ,11\ asymptotic independence condition that is a plausible assumption for most economic time series. II; This shows that for a sufficiently long horizon q. the permanent/transitory components model must yield a variance ratio less than one. Moreover, the magnitude of the difference between the long-horizon variance ratio and one is the ratio of the variance of l'.YI to the variance of l'.PIo a kind of "signal/ (signal+noise)" ratio, where the "signal" is the transitory component and the "noise" is the permanent markets cOlllponent. In fact, one might consider extracting the "signal/noise" ratio frolll VR(q) in the obvious way:
1 ---I VR(I/)
-+
Varl.6.yJ Var[l'.tuJ
2.5. J Problems with Long-/lorizon Inferences There are, however, several difficulties with long-horizon returns that stem frolll the f;\ct that when the horizon q is large relative to the total time span 'J'= "'1. the asymptotic approx:mations that are typically lIsed to perform infercnces break down. For exalllple, consic\er the test statistic VR(q)-l which is asymptotically nOrlnal with mean 0 and variance: 2('2q- 1)( q-I) ?>l1q~
(2.5.8)
IIn<\('I" the RW I nnll hypothesis. Observe lhal for all q>2, the bracketed term in (25.R) is hounded hetween ~ and I and is monotonically increasing in q. Therefore, for fixed n, this implies upper and lower bounds for V are ,i,; and respectively. Now since variances cannot be negative, the lower
t"
HI'I his j... ilnpliecl hy ergociirit}'. {lJld ('\'('11 Iht' St·rtinn '2.t; sati . . ty this nmditioll.
1()lIg~r..tTl~e . .depencit"lIt time ~rie~ di~{lI~\('d!in
58
2. The Predictability oj A.w'l J{d/lrm
hound for VR(q)-1 is -I. Buttilcn the smallest algebraic vallie thaI the test statistic (VR(q)-l)/.fV can tak(' on is: .
Mill
VR(I/) - I
"'V
"V
=
-I
r.;-
== -,,'211 =
Min.fV
_
~)'I'I
'11"'11 11,
~.
suprlOSC that If is set at two-thirds oj" the salllple sill' 'f' so that TI '1= This implies that the lIormalized lest statistic VR(q)/,fV can never he less thall -1.73; hence the test will nnWT n~ectthc lIull hypothesis at the 9:>% level or significance, regardless or the data! or course, the test statistic can still rt~jen the lIull hypothesis by drawing from the right tail, but against alternative hypdtheses that imply variance ratios less than one for large II-such as the pernbanent/transitory compOnenL'i model (2.5.1 )-the variance ratio test will II ave very lillIe power whclI If / T is not close to zero. ~ \\lore explkil illustration oj' the prohlems that arise when III T is largc may ~)e obtai lied by perrorllling an alternal(~ asymptotic analysis, olle in which q grows with 'j'so that q( T)I Tapproachessome limitS strictly between zero ~nd one. In this case, Hilder RWI Richardson and Stock (1989) show that ~he un normalized variance ratio VR(q) converges in distributioll to the following: I \
-.
!
"
Xs(r)
111
-
o
'J X;(r) tir
(25.10)
J
!J(r) - H(r-o) - SU(!).
(2.:>.11 )
where no is standard Brownian motion ddillcd on the unit interval (see Section 9.1 in Chapter 9). Unlike the standard ufixed-q" asymptotics, ill this case VR(q) does not converge in probability to one. Instead, it converges in distribution to a random variable that is a runctional or Brownian llIotioll. The ex'pecled value of this limiting distribution ill (2.5.10) is
In our ('arlier example where Ifl T = ~,the alternative asymptotic approxiIllation (2.5.10) implies that EIVR(q) I ~'onvtTges to ~. considerably less than Ollt' despite the ract that RW I holds. . These biases arc not unexpected in light of the daunting demands we are pbcing on long-horizon returns-withont more specific economic strllc\\Ire, it is extremely difficult to infer llIuch abollt phenomena that SP;lI\S a signiflrallt portioll of the entire dataset. This problem is closely related to olle ill spectr;ll analysis: estilll~lling thl' ~pcnral density function n('ar frcqUl'llCY 1 ('-ro. Fr('qu(,IH'il'~ Ileal' len) COlTcspmHI !O ('xtrnl1cly IOllg pnio(h..
59
2,0, 'If.!t,\ N,r /,ollg-U(Wgl' DI'/JI'lu/l'Il(f
.11)(\ it i~ llotoriou~ly difticult to draw illkrCflce~ ahout periodicities that excCl'd :), ": ',111 uCthe data. 17 We ~ha!l M'e explicit l'vil\eflce orsudl difficultics ill lhl' l'11'i,11 ir.tI resulL~ orSectioll ~,H, llowcvcr. ill ~OJll{' ca~es lon~-horizon 1t'llll'IlS call yield importallt in~ighls. <,specially Wlll'll other ecollomic vari'lhks slid I as the dividend-price ratio COIlIC illto play-sc(' Scctioll 7.2.1 III CllaptlT 7 for further discussiofl,
2.6 Tests For Long-Range Dependence There is olle departurc f.-om thc r.Illt!OI11 walk hypothesis that is ullL~ide the slalistical framework we have developed so I;tr, and that is the phenomenon of long-rangf de/Jelllimce. LO\lg-rall~{'-dcpelHlcllt limc series exhibit an unu~Llally high (lcgree ofpersistellce-ill a seme to be made precise beluw-su that observatiom in thc rcmute past arc llulltrivially correlated with uhservatioIlS in the distant future, even as the time span between the two observatiuns increases. Nature's predilection towards lon~-range dependcnce hJS becn well-documented in the natural sciences such as hydrology, meteorolu6,)" and geophysics, and some have argued that cconomic time series are also long-range dependent. In the frequency domain, such time serics cxhibit power at thc lowest frequencies, and this was thought to he so cOIlllllonplace a phenumenon that (;rallger (J ~(ili) dubbed it the "typical spectral shape of an economic variable." MandcIbrot and Wallis (1968) used the more colorful term "Joseph Effect," a rdl'\'cnce to the passagc in the Book of' Genesis (Chapter 41) ill which Juseph rOl'elold the sevcn years 0(' plenty followed by the seven ycars of lamine that Egypt was to experience. 1H
2.6.1
examples oj Long-Range De/Wilder/a
A typical example of long-range dependence is given by the fractionally differenced time series models ofGran~er (19t:lO), Granger andJoyeux (1980), and Hosking (!9t:ll), in which PI satislies the following dificrence equation:
(2.6.1)
I"here I, is the lag operator, i.e., I./'t = /It-l. Cl-angcr and Joycux (1980) and I1usking (19H I) show that when the quantity ([ - L)d is extended to nOllil\tt'~cr powers of Ii in the mathematically natural way, the result is a 17 ~t't· tilt' di~r\l~~iull ;,uHl allJlysis ill Se( 11011 L.h lUI further clr.[.lib. ll'iThi:"l biblir.ll aualoK)' is Hot (Omplt'ldy 'I i\'O!OIl."I, _.,inn' IOl1g-l~lllge depend.ence h.lS been
doculIlenled in variolls hydrological Silldi"., nol th .. It'd" of whirh was lIurst's (1951) seminal OIl lIlt:a~lII ill~ the IOJlg-lcnn sloraKt' cap.H:ity of n.,~t~l"'\'o~rs. hull-eel. much of )furst's rt:'~earch Wi.l." motivated by his erupiricdl ohs.t'rvations oj the Nile. Ihe \'{'ry saIBt' river Ihat played:\o prolllill('lIt a rolt' in.Jo~t·ph\ J)J'opiln it,\.
~(Udy
·'0'
J"
well-defined lillie seri('s that is said to he frartionally dif/fTl'1lm/ of order Ii (or, r()llival('Jllly, fi'flrliolllll~y illlrgrlltn/ ordcr -Ii), Briefly, this involvcs expanding th(' ('xpn'ssion (I-I,)" via the hinomial thcorelll fi)r nOllilltq~er powers:
or
(I - /.)"
d(d-l)(d-2)---(d-k+ I)
Ii!
and Ihell applying Ihe expansioll
10
(2.1;_2)
III: 00
LAk/ll-k ==
flo
(2,(i,3)
k~tJ
wh('J'(' t\le autoregressive C()('flicicnts Ak arc often re-expressed in t('l"IlIS or the gamma funt'lion:
Ak =
(-I)
k(d)It
\'(11 -
d)
/11 may also he viewed as an infinite-order MA process since /J
k
=
nli + d} J'(d} ['(Ii
+ I)
It is not ohviou's that such a definitioll orfractional dirrcrcllcing might ):cld a tlscrul stochastic process, bllt Granger (1980), Granger andJoyeux (I !IHO), and I-Iosking ( 19H I) show that the characteristics or fractionally clirrerenccd time series arc interesting indeed_ For example, they show thaI/II is statiollary and invertihle fill' dE (- ~, ~) (scc Ilosking (19H I) and exhihits a unique kind of c1qwnd('Il('(' that is positive or negative depending on whether d is positive or ncgat iv(', i,c" t he autocorrelation coefficients of /Jr arc oft\tc sallle sign as d, So slowly c\o the alltocoJ'lclatiollS decay that when dis posili\'e thdr slim
d,'I'("""'"1,
\11
Table 2.3.
A utororrriatioll Jlmrtioll Jor Jrartionally diJJtrrnud pmass. pp(k)
PI,(k)
Pp(lt)
[d==~ ]
[<1== - ;]
[ARO)'.p == .5]
-(l.::~:)()
:)
0.500 0.4()0 0.350 O.3IH O.29!i
-0.015
0.500 0.250 0.125 0.0[,3 0.031
10 2:) :)0 100
0.235 0.173 0.137 0.109
-0.00:; -0.001 -3.21 x 10- 4 -1.0~ x 10- 4
0.001 2.98 x 10-" R.RR X 10- 1• 7.89 x 10;-31
l.at: k
~
3 1
-0.071 -(l.03e)
-tum
Comparison ofautocorrelatioll functions of fractionally differenced time series (1- L)d p, = (, fo, d ~. -~, with that of an AR(l) p, "'1>,-1 to yield" unit V'anance for p, in all Ihree ca..... ,.
=
=
+ f"
'"
= .5. The variance of (, wa. chosen '
series have first-{)rder autocorrelations of 0.500, at lag 25 the AR( 1) lcorrelation is 0.000 whereas the fractionally differenced series has correl~tion 0.173, declining only to 0.109 at lag 100. In fact, the defining characteristic of long-range dependent processes has been taken by many to be this slow decay of the autocovariance function. More generally, long-range dependent processes (1',1 may be definc:d to be those processes with a~l\ocovariance functions Yp(k) such that .
(k)
Yp •
~
kV JI (k)
for
v
E (-1.0)
{ _kv Ji (k)
for
v
E (-2, -I)
or,
as
k
-4
00,
(2:6.6)
where 11 (k) -is any slowly varying function at infinity.20 Alternatively, I~ng range dependence has also been defined as processes with spectral density functions seA) such that IX
E
(-1,1),
(2.6.7)
where h(k) is a slowly varying function. For example, the autocovariance
~"t\ function
I fl.
(0).
Il> be slowly varyin); 'II 00 if lim._ oo f(lx)l/(x) = 1 for a\\ I E x is an rxample ofa slowly varying function at infinity.
fix) is sait!
The (unction
lo~
2. The Predictability 11 AI.lrl
{(tlllm.1
I
function and spectr,t1 density near frequency zero of the fractionally dillerenced process (2.6.1) is Yp(k)
r(d) r(l-d) r(1I
+
I - ell (Vi.H)
\I \
A
-+
n.
\ I~here dE (-~, ~). Dqll'llding on whether Ii is nq:;ative or positive, th(' spectral demity of (2.ti.l) at frequt'Jlcy lero will either he I.ero or infinite.
2.6.2 The JIllnl-AlllIHlrlbmllll'.I((Ilrd
/{lIllge
Slali.llil'
The importance of long-range ,kpl'ndl'llct' ill asset llIart.:ets was tirsl sllldied hy Mandelbrot (1971), who proposl'd \I~ing tltl' range over standard deviation, or R/S, statistic, also called tlte rr.\CIlled range, to deten long-rangt' dependence in economic time sl'ries. The R/S statistic was originally dl'\'doped by the English hydrologist Harold Edwin lIurst (l 951) ill his studies of river discharges. The R/S statistic is the range of partial SlIllIS of deviations of a time series rrolll iL~ mean, rescaled by its standard deviation. Specifically, consider a sample of continuously compounded asset rl'turns {Tl, T2 • •••• Tn} and let Til denote the sample Jnt'an ~ L, ']. Then tlte classical rescaled-range statistic, which w(~ shall call ~" is given by (2.(j.lll) whc~c
s" is the usual (maximullJ likelihood) stanclard deviatioll estilllator, S"
~
'-;-(1) - _r,,) ~]l(t
I " [
-;;
(~.().ll
)
The first term in brackets ill (~.(j.1 0) is the maximulIl (over k) of Ihe panial sums of the first k deviations or ri from the salllple mean. Since the slim of all n deViations of T, 's frolll their Illean is zero, this maximulIl is always nonnegative. The second lerlll in (2.li.IO) is the minillllllll (over k) or this same sequence of partial SIIIllS, ami hence it is always nOllposilive. Tht' differellce of the two quantities, calkd the rallgr for obviollS reasollS, is always nonnegative and helHT <1,:::(J.~1
Q,. m~\y hl' 1>l'I1("1" tIlUit"f!
11The- hehavior or ~t\\dil'S
) I
03
III sncral sCJllillal papers Mandcl lm)(, Taqqll, alld Wallis demons trate lite sllJ>criorily of R/S analysis to lIIore nJllvclll iollal JIlelhod s of detenni llillg IOllg-rallge depend ence, such as allalyzil lg alllOcor relatioll s, variance Lllios, and spectral decomp ositiolls . For exampl e, Malldcl hrot and Wallis (I ~)(j~)I» show by Monte Carlo simlliali oll llial Ihe R/S statistic call delen IOllg-range depelld ence ill highly Iloll-Gaussiall lillie series with large s!--ewlless alld/or kllrtosis . III f~ICl, Malldd brot (I!)7:!, 1!175) reports the allllOSI-SlIre converg ellce of the R/S statistic fi)!· stochasl ic process es with infinite variance s, a distinct advanta ge over Iltested to sOllie degree, it is a well,·slablis hcd fact tliatlon g-range depelld ence ell I indel"d he detecte d by the "cLlssical" R/S statistic. Howeve r, perhaps the lIIost illlporta nt shortco llling 01· tile rescaled range is its sensitivi ty to short-ra llge depend ence, implyin g tlLlt allY illcoIllp atibility betweel l the data and the predicte d behavio r of tIle R/S statistic under the null hypothe sis need not COIIIC from long-ra nge depeIld cIIce, but Illay merely he a symptol ll or short-te rIIl Illeillory . III particul ar 1.0 (199 I) shows that under RWI the asympto tic distribution of (I/.[ii) Q,. is given by the randoIll variable V, the range of a firownia ll bridge, but under a stationa ry AR( I) specific ation with autoregressive coeffici ent ¢ the Ilormali Led R/S statistic converg es to ~ V where ~ == .j( I +¢) /( I-¢). For weekly retums of some portfoli os of com IlIon stork, i> is as large as 50%, implyin g that the !Ilcan of C6./.[ii may be biased uI}-
III11S1 he eli",en 10 allow ti)r tlllcillalio ll.' ill Iii .. supply of IVal .. r abm·.. IIie dalll while .Iill Ill~lillt;ljllillg a relatively cunstant flow ofwatc.'r below the dalll. Siufr dam rOJlsrfuctiun costs al to illllllt.'ll~e. IIH~ ililportdn ce of e~[ill1aliIIK Ihe reservoir rapacity Ilt"cessary to meet long-term slOra,;" Ilted., is apparellt. The nUlKe is all eslilllale of ,his quanlity. If JS i. the riverflow (per ,,"it lillle) a/Jove Ihe dam and X n i. Ihe d,,,ired rivedlow below Ihe dalll, Ihe brackeled 'I' .... ltily ill (\!.6.1 0) is Ihe capacilY oflhe reservoir n("~ded 10 tm ....e lhis .Illoolh How Kiven Ihe p.lllnll of Ilows ill p .... iuds I IhrouKh n. For exalllple, Sllpp"S" '"lllllal river How. are ""'"ll1ed to I,,· 100, ;,0, 100. alld :,0 ill years I IhrollKh 4. If a cu,,-,WlIl allllllal flow of 75 below the dam i~ d(,~1I"l'd c.'arli yc:ar, a reservoir must have a minimuJI l total CiJMrity of~:, since it must store 25 111111., ill y.. ars 1 alld:{ 10 provide for Ihe rt"/alivt'ly chy y.. ars:l ,,,"1~. Now slIl'l'ose inslead Ihal Ill .. "'llllrall" 'll'·fIl of riverflow is 100,100, :,0, :,0 ill Y",I" I tilr
!'olwll.," ill E1-.'Ypt Ihott sparkt"d J IlIrst's lifelong Lis<"ill.llioll wilh Iht' Nilt', Ic',ulillg t"v("IHllall y to III . . 11I['·lt· .... 1 III dl(' It'.'i(';ilt'd ra I1gt' ,
.,"
•
"1'''111""".,
" / ,1'111.\111111/.
ward II)' 7:\'X,! Silln' tilt' IIlt'all 01 V is Jrr72~ 1.2:), the "wall or !Ill' classical n's('ah'd rangl' would Ill' '2. Hi ror slll'h an AR( I) process. 1.0 (I!I!II) dl'\'l'Iops a Illodilicatioll or the R/S statistic to ;1I'1'otllll lilt' the clfl'l'ls or short-rallgl' dqll'lHletllT, ckrivl's an asymplolic s;lI11pling lhl'or\' under sl'\'I'l'al nllil anei altl'rtJatin' hypotheses, and demonslr;ttl's yi;t !\lonte Carlo silllllbtions alld I'lllpirical examples drawn rlOm rl'Cl'lIt hi""rical stock llIarkl't data that the Inoeiilil'd R/S statistic is I'onsidcrahh' nlOl'(' accurate, often I'idding infi'rl'lIct's that contradicllhose of its classical cotlntl'rparl. In particllbr, II'hal Ihc I'arli(,r lilcrature had asslIllIed was 1'\'id('IIIT of 10llg-rangl' dq)(,lIeil'lICI' ill ltS stock retllrns Illay well he Ihl' 1'1"1111 of quickly decal'illg sholl-r;lIIge dq)('ndl'nce inste;lIl.
2.7 Unit Root Test.. t\ IIIOl'e I'C(,('lIt alld nHm' spcciali/l'd class or lesls thaI art' orten conruscd wilh lests or lhl' random walk hypotheses is Ihe colleclion or Iwit rolll lests in which Ihl' lIuli hl'pollit'sis is .\',
~,
/1
1.\',
I
+ f"
('27.1 )
often with til<' followillg ,tlt('JII,tti\'(' h"l'0thesis:
X, -/il =
f,
I -. /1(/-·1))
-t
f"
E (-1.1),
is all)' Il'n)-IlH'an st;\lioll;t1T process, stich Ihat
() <
" (J,~
=
lim E
I ., '-
[~(tf')~] I
<
(Xl,
,=1
I klll'istil'alil', condition (:2.7.:\) r(,quirl's thaI variance of the partial stlln 2:.:=1 f, inneasl' at approxim;ttl'ly thl' sanlt' rate as T, so Ihal each ne\\' f, added to thl' palli;Ii sum has a nontrivial contrihulion 10 the partial SIIIlI's \'ariancl'.~~ This condition enSllrl'S th;\I Ihe usual limit Iheorl'ms are applicahle to the f,'S, alld it is satisfied hy virtllally all of the stational')' pron"sl's th;\I wc sh;tli han' " .... asion to sttldl' (l'xCCpl for those in Set'lion '2'(i). '!'!1f
0, IIH'
lilt,
r.
\\'e'I!' In glll\..· ,10\\,('" Ih;1I1 ';'0 that tilt, lilllil ill ('l. 7.:\ I \\('1(' the' ,«,cJlU'lIe (. 01 ( I \ \\'lIl1ld he "clIICt'l1il1g 011'" o\'(,r lilll(, ;11,,1 \\'ollid 1101
p;lIli,tI ,11111 \ \".11 i.111I t'
IIIH,{,lt;lilll\' ill
IH'.I \1'1'\' lI,dllllllCUld 01 1.111(1(1111 I" ic (' d~II.lIlIi('..;,. All ('xall1plt' 01 !'i1tch ;\ pr()('('~' j, all f\1,\( I) Wilh,llIlIitloell.i.t' .. (/;;-· ,], ··'1, I.WIH·IC'II,i,whil(·noisc. II tile' p.III1.&I '11111\ \·.III.llle c' \\'('11' 10 glO\\, 1.1'"'' th.1II ~o that till' lilllil ill (~.7.:q \\'('1(' "X.." ,hi ... \\'0'11,\ hi' .111 ,·"U\lph· 01 IIIII.I.! HIli,!!" d"',,·IlIt,·Hft'. ill whirh tlu: ~\\lh)( 01 .. \'I,uio" hUH ti(lll (II IIII' , / \ clcCI\· ... \C., \ ,Ie 1\\ h .. \11 (·".Hllpll' I I' '"1'11 ,I plon's!'i i~ a fr.u·lioll.II" dill"I(,lIn'd I" ell c'" II 'I,. \\ IH'II' 'I, I .... \\ Iliit' lie li . . e·. \cc' SCTIIIJlI ~.ci ;1IIc1I.o (I~,q I) 101 1IIIIh('1 eli" 11"11111
or,
I,", /
The unit root test is designed to reveal whether X, is diff"mc~stationary (the null hypothesis) or trend-stationary (the alternative hypothesis); this distinction reSL~ on whether rp is unity, hence the term unit root hypothesis. The test iL~elf is formed by comparing the ordinary least squares estimator ¢ to unity via its (nonstandard) sampling distribution under the null hypothesis (2.7.1), which was first derived hy Dickey and Fuller (1979).2' Under the null hypothesis, any shock to X, is said to be permanmt since E[ XI+k I XI) = /-lk + XI for all k>O, and a shock to X, will appear in the conditional expectation of all future XII... In this case X, is often called a slochfl.ltic trend since iL~ conditional expectation depends explicitly on the stochastic variable X" In contrast, under the alternative (2.7.2). a shock to XI is said to he tem!lOrary, since E[X'+k I X,J = /l(t+k) + rpl(X,-/lt), and the influence of X, on the conditional expectation offuture X,+k diminishes as k increases. Because the (,'S are allowed to be an arbitrary zero-mean stationary process under both the unit root null (2.7.1) and alternative hypothesis (2.7.2), the focus of the unit root test is not Oil the predictability of X" as it is under the random walk hypotheses. Even under the null hypothesis (2.7.1), the incremenl~ of XI may be predictable. Despite the fact that the r,lIldOIll walk hypotheses arc contained in the unit root null hypothesis, it is the permanent/temporary nature of shocks to X, that concerns such tesL~. Indeed, since there are also nonrandom walk alternatives in the unit root llull hypothesis, tests of unit roots are clearly not designed to detect predictability, but are ;,\ fact insensitive to it by construction.
2.8 Recent Empirical Evidence
pre(~ictability in asset returns is a very broad and active research topic, and jt
is illlpossiple to provide a complete survey of this vast literature;n just a fe~ pages. Therefore, in this section we focus exclusively on the recent empiric,\1 literature. 24 We hope to give readers a sense for the empirical relevance ~f predictability in recen t equity markets by applying the tests developed in the earlier sections to stock indexes and individual stock returns using daily and weekly data from 1962 to 1994 and monthly data from 1926 to 1994. Despite ! ~'Since then, advance, in econometric method, have yielded many extensions and generaiintio", to thi, 'imple framework: te,ts for multiple unit roots in multivariate ARIMA systems, tt'st, for rointrgration, consistent estimation of mode" with unit roots cointegration, etc. (~e Cal~lphell and Perron [19911 for a thorough survey of this literature). i H Ilowrver, we would be remiss if we did not cite the rich rmpiricaltradition on which th( reft'nt literature i, built, which includes: Alexander (1961. 1964), Cootner (1964), CowIe,' (19W), Cowles and Jone' (19~7), Fama (1965), Fama and Blume (1966) Kt,ndall (1953), C;r"n~('f and Morgenstern (1963), Mandelbrot (1963), Osborne (1959, 1962), Roberts (1959), ;llld Workill~ (191;0).
2. '/'lIe Predictability of Auet ReinDl,\
~pecific.ity
the of these examples, the empirical resullS illustrate many of the issufs that have arisen in the broader search for predictability alllOIlf.: assel retu\rtls.
,
2.8. I 111l/ocorrela/iulO
I
:
Table 2.4 reporlS the means, standard deviations, alltocorrelations, and Hox· Pierce Q-statistics for daily, weekly, and llIonthly CRSI' ~lOck retlll'lls indexes fl'OmJuly 3,1962 to Deccmher 31, 1~\l4.~~ During this period, panel A of Table 2.4 reporlS that thc daily equal-weighted CRSP index has a first-ordcr autocorrelation p(l) of 3~.O%. Recall rrolll Section 2.4.1 that under lhe I[l) random walk nu1\ hypothesi~ RW I, the asymptotic samplinf.: distribution of .0(1) is normal with mcan 0 and standard deviation I/JT (sce (2.4.14». This implics that a sample siJ.e or 8,179 observations yields a standard error of 1.11 % for p( I); hcnce an .0% implies thaI 12.3% of the variation ill the daily CRSP equal-weighted index return is predictable using the preceding day's index return.
X;
,~
'l')UIl?t'~ stated othenYist'. we rakt' returns [0 he rOlllillllollsly compounded.
POlltoliu
returns ~re raldtlated first from sirupit' returlls and tht"n are converted to a (outilllluu.;,ly
compo"nded relllm, The weekly H'I"rn of ""ch St'cmilY i, colllpllletl a5 Ilw retllrn III>Ill Tuesday'I' closing price \0 Ihe followill~ TII""lay's dosill); price. If Ihe t()llowillg TIIl'M!;,Y's price is Illi"ing. Ihell Wednesday's pri('e (01 MOllddy'S if Wednesday's is also Illissill~) i, \lSl,d, If hOlh ~Ionday's and Wednesday's prices are missing. Ihe relurn for Ihal week is "'1',,,1,,,1 ..., mi",iol!!; llois occurs ollly ,,,,ely. To com pUle weekly relurns Oil sile-SOrled pOrliotios. tor each week all slOcks wilh nonmissinK relllrn, Ihal week are "-'Signed 10 ponh)lios hased Oil Ihe lJeKillllirlK of year market value. If Ih< \>C)lilllliIlK "fY"ar lII.lr\c.c1 value is missing.lhell Ihe .
,,' 2.4.
S.IIUple Period
A ullJrurrelatiml ill dllily, wrri
Sample M Sil.e call
SI)
(t,
A. [hill' RCllll'llS
CRSI' V'lllIt'-Wci~hl('d Index ti:!:07:tn-94:12::lO H,I79 0.041 O.H24 17.6 -0.7 li~:07:0:~-7H: 10:27 4,090 0.02H O.nH 27.H 1.2 71-1: 1O::{O-~)1: 12:30 1,OH9 O.!)!)4 0.901 IO.H -2.2
0.1 -O.H 4.ti ~.~ -2.\l -:\5
21i:D
2695
:~2\1.4
~435
695
72.1
CRSI' Eqllal-Weighted Index 1i:!:07:0:{-94: 12::\() H,179 0.070 0.761 :~;,.() ii:!:07:0:\-7H: 10:27 4,090 0.Ofi3 0.771 4:1.1 7t'.: 1O::\O-\J4: 12::\0 'I,OK\! 0.07H 0.75ti ~(j.~
~J.:\
1:\.0 4.\)
H5 1,>.3 2.0
9.9 1,301.'J 1,36'J5 1:>.2 1,062.2 1,110.2 4.\) :HH.'J 379.5
1\. W('(')"I), Retllrns
CRSI'
Vahll~-Wt'igh\ed
In(h',.
h:!:07: 10-\)·1: I '2:'27 I,W:> 0.1% VI\l3 15 -2.:) ii:!:07:IO-7H:IO:(l:\ H4H 0.141 1.9~)4 :>.Ii -:n 7H:10:IO-94:1'2:'27 H47 0.248 2.IHH -2.0 -1.5
CRSP
:-15 -0.7 I.li :>.H I.t; -:>.:\
H.H 'J.O !l.3
36.7 215 2!l.2
·I.H ti.1 2.2
!14.:{ tiO.4 33.7
IO!I.3 tiH.5 !ll.3
4.:\ -5.:1 -I.:{ -0.4 7.:{ ti.4 -:\.8 li.2 U -6.3 -H.3 -7.7
ti.H 3.'J 7.5
12.5 'J.7 14.0
12.H 75
21.3 12.t\ 14.2
Eqllal-Wci~hled
ti:!:07: 10-\14: I '2:~n l,fi9.'i 0.339 ~D2I 20.3 h'2:07: 10-78: 10:0:\ H·18 0.324 2.4liO '2I.H 7li:J 0: I O-'J4:12:27 H47 0.351 2.171 IHA
Iudex ti.1 75 4.:~
\l.I 11.\) :).:)
C. Monthly RcLUms CRSP
Valllc-W"i~hl .. d
ludex
ti:!:07::{1-94: 1:!::lO li'2:07::{ 1-7H:m):29 7H: I (I::{ 1-\14: 12::\0
:-190 0.861 4.336 19!1 O.64ti 4.21\1 1% 1.076 4.4:,()
li~:07::\I-\H: I ~::{()
:1'J0 1.077 5.749 17.1 -3.-\ --:\.:\ -1.6 195 1.049 fi.I4H 18.4 -2.:) 204 '1.4 1'J5 1.105 5.:\3li 1!i.0 -I.ti - 12.4 -7.4
CRSI' Eqllal-W .. iglllcc\ huh',. (i~:1)7 ::\ 1-7H:0\):2\)
7!-l: I ()::\ 1-!I4: I :!::{{)
H,9
l\'II>I'I>'"'\'\\i,," H)('fti. i., lib (ill p('rrelll) ;,,1<) II .. ,-Pi ......· (l-,t.lli,lir, 101 t:/{SI' dJily, weeki)', alld IlIollllrly val 11('- alld (,<)lIJI-w('ighl('cI 1('1\ II'll indexes I()r tire ,.11111'1,' period frolllJllly :{, IYI;~ 10 ))"«'II1IJ<'1' :1O. I!)!H and slIbperiods.
i.
j hI' j'mlidllhility
oj AUft
/{I"tll/I/.\
The weekly alld lIlollthly retllrn aUlOcorrelations reponed ill pallels B and C of Table ~.'1, respectively, exhibit pallerns similar to those of the daily autocorrelatiolls: positiv(, alld statistically si~nificallt at the first lag over the clltire samplc and Ii)!' all sllhsamplcs, with smaller and sometimes lIegative higher-orcin an[('("(lIre\ati(IIIS,
2,8.2 \'rllil/llrl' /lotios The fact that the auto(orreiations of daily, weekly, and mOllthly index retllrns in Tahle ~A an' positive and orten significantly different frolll 7,no has illlplications for the hehavior or the variance ratios of Senioll 2.'1 alld we explore these illlplicatiolls ill this sectioll for the returns of ill de xes, portfolios, and individllal sl'rllritit's.
CIlSP hllll'Xf.1 The alltocorreiations ill Tahle 2.4 sllggest variance ratios greatn thall Olle, and this is confirllled ill 'nlhk 2.:' which reporL~ variallce ratios VR defilled in (2.4.:\7) alld, in parenth('ses, hewroskedasticity-nl!1sistellt aSYlllptoticallv standard normal test statistics 1/I'(If} defined in (2.4.44), for weekly CRSP eljual-and va It Ie-wei gh ted market retllrn indexes. 2t\ Panel A (olltains results lilr tl\(' ('qllal-weighted indl'x and panel n contains results !i)r Ihl' \"011111'wd~hted index. Withifl each pand, the first row presents the variall(e ratios afld test statistics fi,r the elltire l,fi!I!i-week sampk and the lIext two rows presl'nt silllilar results lin' the two suhsampks of H4H and H47 w('eks. rand A shows that the r;llldolll walk null hypothesis RW3 is rejected at all the usual si~nifkal1("e levels for th(' entire time period and all ,ubperiods fOl' the equal-weight(,d index. Moreover, the rejections are not !III(" to changing varianc(,s sinc(' til(' ",'('1)\ arc heteroskedasticity-collsistenf. Th(' estimates ofth(, variance r,ltio arc {((rgrr thau olle for all cases. For example, the entries ill thl' Iirst cohunll of panel A corresponoto varianc(' ratios with an aggrl'g;lIioll valu(' 'I of~. Invi('w of (2.4.18), ratios with '1=2 are approximately ('(]lIal to I plus th(' lirst-or!ln autocorrelation coefficient estim,llor of weekly retunts; hellc(', [he elltry ill the first row, ) .20, implies that th(' first-order autocorrelation fi,r we(,kly returns is approximately 20 'f.! , which is consistent wilh Ih(' \'alue rqHll\ct\ in l;lh\c 2.1. With a corresponding 1/1' ('I) stat ist ic of '15:1, the randolll walk hypot hesis is resoundingly rejected. The suhsamplc r('sults show that although RW3 is easily n:inlet! over Imth halves of 111t' sa III pit' period, Ihe varian('c ratios arc slightly brgcr ,\Ild the rejections sli~hllv strollger ovn th(' first haIL This pallerll is re)leated in Tahle 2.li and in other eUlpirical studit's of predictahility in l;S siock :!liSinc c' ill 0111 ~alllplt'
;II\\'ays \pan'
W(' I('POII
olily rhc'
1/,' (1/)--( CllIlplllt'd under Ilu' 111111 hrpoliu'sis R\\':~-·-.II(· ,h;tll rhe' \".tllle' ... of 1/1(//) (';dclllau'
till" \";1111(,~ III
sl~tli~li("lIy Ic~~ ,igllific.lIIl mOl('
((lIi.\("
\';ui\"c :"ot;lIi"lic~,
69
2.8. R.ecenl r:mpirical Evidence
Table 2.5.
Sample period
Vminnu rat ins Jor wakl) stock intkx Yl'turm.
Number q of base observations aggregated Number 10 form variance ratio nq of base -,_._._II obserJations 2 4 16 8
A. CRSP Equal-Weighted Index 62:07: I 0-91: 12:27
1.695
848
78: I 0: I 0-94: 12:27
847
1.20 (4.53)* I.n (:'.17)*
i
1.19 (2.%)*
1.42 (5.30)· 1.47 (4,44)· 1.35 (2.96)"
1.65 (5.84)· 1.74 • (4.87)· 1.48 (3.00)·
I. 74 (4.85)* 1.90 I (4.24)+ 154 " '(255)~
1.02 (O.!'>I) 1.06 (1.11 ) 0.98 (-0.45)
1.02 (0.30) 1.08 (0.89) 0.97 (..(J.40)
1.04 (0.41 ) 1.14 (1.05) 0.93 (..(J.50)
1.02 (0.14) 'i 1.19 : (0.95), 0.88 (-0.64)
n. CRSI' Value-Weighted Index 62:07: I 0-94:12:27
1,69:)
62:07: 1()... 78: I 0:03
848
78: I 0: I ()"'94: 12:27
847
VJri:lIlrl··ratio test of the random walk hypothesis for CR5P equal- and value-weighted indexe•• for the sample period from July 10. 19fi~ to December 27.1994 and subperiod •. The variance ratios VR(q) are reported in the main rows, v..ith heteroskeuasriciry
returns: the degree of predictability seems to be declining through time. To the extent that such predictability has been a source of "excess" profits, iLS decline is consistent with the fact that financial markets have become increasingly competitive over the sample period. The variance ratios for the equal-weighted index generally increase with q: the variance ratio climbs from 1.20 (for q=2) to 1.74 (for q 16), and the subsample results show a similar pattern. To interpret this pallern, observe that an analog of (2.4.18) can be derived for ratios of variance ratios:
=
VR(2q)
- = VR(q)
1+ fJq(l)
(2.8.1 )
where (lq( 1) is the first-order autocorrelation coefficient for q-period retums rt + rt-i + ... +rt - q+ t. Therefore, the fart that the variance ratios in panel A ofTahle 2.5 are increasing implies positive serial correlation in multiperiod
70
2. 'f'/u' I'II'ILir/abilily of 11.\\('1/(1'111111.1
retu rns. For cxam plc, VR( 4)/VR(!1 )== 1.42 / 1.20== I.IH , whic h illlp lies thai 2wcck rctu rns havc a lirst -ord er aUlO corr elati oll coeH icien t of approxiJ II;III'I,
IH%.
Pane l B of Tab le 2.5 show s tllat the valu e-we ight ed inde x beha v(', c1iITerclllly. Ove r the ellli re sanl plc peri od, the vari allce ratio s arc ,III gl ('.11" 1 thal l onc, but not by muc h. rang ing from 1.02 for q=2 to 1.04 for q=H. Mor eove r, the test stati stics "" (,,) arc all stati stica lly insig llilic allt, hen ce RW:-I cann ot he reje cted for any ". The subs amp le resulL~ show that duri ng Ihe firSI half ofth c sam ple peri od, the vari allce ratio s for the valu e-we ight ed illde x do incr ease with q (imp lyin g posi tive seria l corr elat ion for lIlul tipe riod retllJ 'Jls), but duri ng thc scco nd half of the sam ple, the vari ance ratio s decl ine with q (imp lyin g nega tive scria l corr elat ion for Illlli tipcr iod retu rns) . The se two opp osin g patt erns are resp onsi ble for Ihe relat ively stab le beha vior of Ihe vari ance ratio s over the ellli re sanl ple peri od. Alth ough the test stati stio ill Tab le 2.:) arc base d Oil nom illal stoc k retu rns, it is appa rent Ihat vil'lu ally Ihe saJlle rcsult.~ wou ld obla ill wilh real or exce ss retu rns. Sillc e the \'olat ililY of weekly Ilom inal retu rns is so Illuc h larg er than that of the infla lioll alld Trea sury -bill rates , the use of nOll linal , real, or exce ss retu rns in vola tility -bas ed tesls will yield prac tical ly idcn tical infc rcnc cs.
Size-Sorled Pori/olio.! Thc fact that RW3 is rcjc ctcd by Ihe eCJual-weighled inde x but IlOI by Ihe valu e-we ight cd illde x SllggesL~ Ihat mar ket capi taliz atio n or size lIlay play a role in the bcha vior of the val'i ance ratio s. To obta in a bell er sellSc or Ihis intu itiol l, Tahl e 2.6 presenL~ vari ance I'alios fol' Ihe relul 'lls of sil.e-SO/'l('(l porl foVo s. We com pute wcekly retu rns for live size-SOrice! pOrl f()lio s from Ihe Cl~)P NYSE-AMEX daily relU /'llS file. Stoc ks wilh reili ms f()J' any give n wcek ;ll'e assig lled to port folio s base d on whic h <[llilftile thei r begi nllil fgof~year"/Ilarket capi taliz atiol l belo lfgs to. The pOr lfoli m are equa l-we ight ed and ha.\.e a chan ging com posi tion Y Pane l A ofT ahle 2.6 repo rts the resu lts for the, port folio of Sill all firm s (Iil'sl quin tilc) , pane l B repo rts the rcw lts for the;p ortfo lio of med ium -size finm (thir d Cfuinlile), alfd pane l C r('po lls the resu lts for the port folio ofla rgc linn s (fift h quin lile) . EI'(dcllC(' agai nst the rand olll walk hypo thes is for Ihe port folio of COIUpani cs ~n the slllall('st <]Uilllile is siro llg f()!' the ('nli re sam ple and for b011t sllhs,UI~l'lcs: ill pane l A alltl te '1,'('1) Slalistics arc well abov e the :)% niti ral I'allll' or 1.!J6, rang ing frolll ·1.li7 10 10.7·1. The l'
hTlgh lt·d!lI I.lI kt'l illde·x . '.
I
I.,,').~(· .. t \.lhH· ·\\Tig hlt·d «jllili lilt' I~ quilt' ~illlil;lr 10
tilC'
\'.1 II It'·
71
Table 2.6.
Vllrillltf"P mlio.\ jor lII("'dy .liU-.lollnl flllrijo/io Idllm.l.
NliJIlh"r of has" ohservations
NlIIlIl)('r
I{
of has" ohser\';lIions aggrq;att'd to
1111
rOJ III \,~triallc(,
ralio
I(i
H
A. P()rtfolio of linns with markct vallics ill sJIlalkst CRSI' quintile
(i:!:07: 10-~H: I 2:27
l,ti9:1
(;:!:07: I 0-7H: 10:0:-1
H4H
7S: 10: I 0-94: I 2:27
H47
I.:{:I (7.1:1)' 1.:14 (:1.47)* U7 (4.67)*
1.77 (9.42)* 1.7(; (7.:t1)* I. 7~J (f).\J 1)*
2.24 ( 10.74)* 2.'!.'!. (H.O::I)* 2.22 «(Ul!I) *
2Ati (9.:-1:\)* 2.4fi
(G.Y7)* 2.49 (6.60)*
B. Ponf(,lio of firms with market vallics ill central CRSI' qllintilc (i~:07: I 0-~J4:
I 2:27
1,695
I 0-7H: 10:0:'1
H4H
7S: I 0: I 0-\14: I 2:27
H47
li~:07:
1.20 (4.25)* 1.21 (:1.2:,)* 1.19 (2.7~)*
I.:{!J (4.H5)* 1.4:1 (4.0:{)* 1.:-1:'1 ('2.71)*
1.59 (5.16)* 1.66 (4.27)* 1.41 (2.6:-1)*
1.65 (4.17)* 1.79 (3.67)* 1.47 (2.14)*
C. 1'()J:folio of firms with markct valllt"s in largest CRSI' qllintile (;:!:07: I 0-\/·1: I '2:27
1#):,
li:!:07: I 0-7H: I 0:0:1
H4H
70: 10: 10-\/·': I '2:'27
H47
I.(H; (1.71 ) 1.11 ('2.0:)* 1.01 (0.'2')*
1.10 ( 1.4ti) 1.'21 ('2.1 :,) * 1.00 «(!.Of, )
VIII.III(l'-r;Hio Il· . . ' of tlie ralldDIlI walk h}'P()lhl'~i:) for :o.i/l·-~ortl'd
pl"riodlroIll.l"ly 10. lV!i2 ~\l·l· H.'ported in tht"
10
Decelllher '27,1\1\14.
1.14 (UH) 1.:10 ('2.1'2)* O.'/H (-0.1:1)
pOI,rolin,.
1.I1 (0.76) l.3'2 ( 1.59) O.\J'!. (-{HI)
tor tht" !'lam pIc:
TIo," variallce ralios VR('11
Blain rows, with ht"tt"ro~kl"d.l:-.licily-("Ollsislt'nt te:-.( statist irs "'.(q) Kiven ill
illlllU.:didtdy ht'luw cach lnalll l"tn\'. Ullder lht' f.llltioill \\r~llk Hull hYPOlh~sis. the \";1111(' 01 Iht" \',II-j;IIlCt' ratio is one and the.' le.'M st~ltiMin, h~\\-l' ~\ ~\~\1Hl"lI"tl lloJ'lllal ciiMrihutioll p.lrcllIlil'st':o.
a_'Ylllpl(Jlic;dly_ T(':\1 :\t"ti~ti("s lII.uked with ;.1Mt"rbks indicate tllat tilt, (orrt'spondinH varid.nre r;ttiu:\
,1I"e
st.lll;\tic;tlly diJkrt"llt from on~ at the :)IYcJ 1t"\'l'I oj :\igllilirall("c.
(h;lIl OIlC, illlplying a firsl-order autocorrelation of :1:J'X, lill· weekly returns tl;e elltirc s;ullplc pniod. Fur lite purtfolios of Illcdilllll-sil.l' ("OIIlJ!~lllic~, lltt" of!' (If) statistics in p~llid 1) shows thal there is also strong t'vic\eIlCl' agaillst RW::I, although til<" variance ratios arc slllaller 110W, illlplying lower serial correlation. For tit,· port!i.lio or the largest flrllls, pallel C shows th;l( evidcllcc against RW:~ i, 'par,,·, lilllil .. d !lilly III the first ~lalr or th .. salllpk period.
ll\'lT
TIll' rl'snlls lill' sitl'-hased pOr\lillios art' gem'rally ('ollsislelll ",illl those fi/!' Ihl' Illarkel indexes: varian('e ralios an' gcnerally grealer than onl' and increasing in ", implying posilil'e serial corrdalion in IIIl1lriperiod r('llIms, sial iSI il'ally sign i fici III Ii n' pori Ii II ios all 1>111 IhI' largl'sl ('ompa \l it'S, .\IId 1l10re sigllili('anl during Ihe lirsl Iialf of Ihe sampll' period Ihan IiiI' S(,(,OIlc! half.
or
Illditlit/llul SI'I'//lili",1 Ilavillg shown that Ihe rallc\olll walk hypothesis is in('onsistellt wilh th(' behavior of the equal-weighted index and portlillios of Sill. tll- .l1\d IlIcdiulll-si/,t' ('ompanil's, 1\'1' 1I0W IIIJ'IIIO Iht, cast' orin
Table 2.7.
Variance ratios for weeki) individual security returns. !\'umber q of base observations aggregated to form variance ratio
:--'-umbcr Sample
nq of base observations
2
4
8
16
A. Averages of variance ratios m'er individual securities All stocks (411 stocks)
1.695
0.96 (0.04)
0.92 (0.07)
0.89 (0.11 )
0.85 (0. (4)
Small stocks (100 stocks) Medium stocks (100 stoc ks) Large stocks (l00 stocks)
1,695
0.95 (0.06) 0.96
0.90 (0.09) 0.93 (0.07) 0.91 (0.06)
0.88 (0.12) 0.90 (0.09) 0.89 (0.11)
0.85 (0.15) 0.B5 0.86 (015)
1.29 (1.99)-
1.695
(0.04)
l.fi95
0.95 (0.03)
(0.13)
B. Variance ratios of equal- and '"
1,695
/.11 (2.75)*
1.20 (2.83)"
1.30 (2.88)'
Value-weighted portfolio (411 stocks)
1,695
0.99 (-0.26)
0.97 (-0.43)
0.96 (-0.12)
0.93 (-0.53)
Means of variance ratios over all individual securities with complete reCum histories during the sample period from July 10. 1962 to December 27, 1994 (411 stocks). Means of variance ratios for the smallest 100 stocks. the intermediate 100 MOCks. and the largest 100 stocks are also reported. For purposes of comparison. panel B reporu the ,,,riance ratios for equal- and value-weighted portfolios, respectivdy. of the ~ II stocks. Parenthetical entries for average, of indi\idual securities (panel A) ar~ ~tandard dniauons of the cros.s section of variance ratios. &cau-'t' lhe "-:Jriance ratios are not cro~~ctionall}' independent. the sI.1ndard deviation cannot be used to perform the usual si>:nificance tests; they are reported onl)' to pro\ide an indication of Ihe ,,,riance ratios' cro.'>Hectional di.'persion. Parenth~ticaJ entries for portfolio \"riance ratio, (panel B) are the heteroskedasticil)'
74
2.
'J'III'
p,.n!i,t(/bility of AI.lft Uptllnn
by forming portfoli os, we wOlild expect to IIIICOVCl ' the predicta hle .IJltl'/lU/t i, compon ellt more readily wh('l1 securitie s arc cOlllbin ed. Neverth eless. tht' weak negative autocor rclatioll s ofthe individu al securitie s arc an intnesti ng contrast to the stronge r positive ".\lIt()(.:orrd;ltion of the por'th)li o returns. 2.8.3 Cros5-A lltO(ondl ltiullJ
llltt! I.mtf-I.ll g ReilltiuH.1
Despite the fact that ill(\ividu;d security relUrm arc weakly negative ly autocorrcl ated, portfoli o returns -which ',\re essclHia lly average s of individual security returns -are strongly positivel y autocor rclated. This sOlllewh at paradoX ical result can lIlean only o/le thing: large positive crOSS-
[It I /12 .. , /1N]' rmti autocouarirwu lIlatrices E[(R _ - ,.,.)(R t k t ,.,.)'] ,; r(k) wherl',withnoloHojgl'llerality. welakek?:.Osinc er(k) ::: r'(-k) . ::: ,.,. 1=
.
If Lis (Icfincd to bc a vcctor of olles [1 .. , 1]'. we can express the e<\II;IIweigh\c d markel index as L'Rr/ N. The lirst-Qrd er alitocovari
not ""
l
I Cov[ Hmt - I • II mt J
'
= Cov
rL--;::;- 'tv L'Rt] L'R t _ 1
=:
LT(I)L N~
(~.H.2)
I
alld th~rcforc the first-ord er ',lIllo('o rrcblioll of U"., call be express ed ,IS I COy
,Umt -
I,
U,.II
\, ar! fl,.,]
LT(I)
L
LT(O)
L
LT(!)
tr(1'(I)) ------- + -- . LT(O) L LT(O) L L -
11'(1'(1»
('nu)
where 11'(') is the trrt(l'opc ralo)' which SIIIIlS the diagoll;t1 entries ofils s<ju;I)'('malrix argulJle nt. The first 11'1111 of Ihe ri~hl side of (2.H.3) (olliain s only (A 1) illi IlIJ(h' 101 IIl1l.lIlI)lI.11 ~illll'lH Ily. ~ill( (' jOint ("O\'~II i.1I1n·-~t~ul olI.1I ily .11. to t"iimill.Ht' til1le:IIH'ext:~ 110m popul.ltio IlIllOI1ll'Il l\ such ~lS}i. awl r(kL the qll~\hl.tli\'t· tt'·dt\lrl'~ of our rt."sU'L\ win not (hange ulHlt'r the weakt"f a.'i~uJllplion~ of weak.ly dqH.'n
lnw.,
\I,
\ \ ')'IOr) lor 1111 Ihf,.
1~'/11I,i1i((jl
2.8. Rnm/
Tablr 2.8.
(:w\.\-Ilu/c u ont'iulioll IIlft/rir t'\ for ';1.1'-'>01 it'd
II"
(I."''
H"
O.!l:1H O.H!)':.! O.H:1\) 0.7'2H
H:lI
111
11."
nil II,.,
N~I_'
11:11 -
\
Il;i-\ I~,,-\
l
/(~,-~ n,,_~
/{1I .• l Ilr.,_~
U: I/ . , 1l;'.4 I~.,_,
fI. "
0.171 O.IK':.! 0.1\17 0.201 0.IK7
[{"
0.11:; O.I':.!\) 0.147 0.1:;3 0.147
H.W,li H.\) 14 H.\lti I 1.000 Ilr"
00")
o.o:n
0.0:,:1 0.059 0.057
11:"
Il;,
I~"
0.141 0.1:15 0.121 n.OK4
0.tl:1':.! O.O':.!\) 0.0:12 0.02K 0.111 ':.!
-0.010 '-0.005 -0.006 -0.016
C""
0.141 n.14:1 0.1:17 O.I':.!O
[I,~,-,
1l.~)H
on")
0.0:,7 O.O!) I 0.0:,1 O.lHli 0.02:;
C flit
14
tl.W)t;
0.9-\4 0.\17\) I.oo(l o. \Hi I
Nli
N~,_,\
11,,-4
0.\).\·1
1.(lOO 0.97 1 )
O.OK\) 0.07K 0.07\) 0.071 0.04:,
I55
11."_,,
1l.~)7ti
0.~)7ti
1/,.,
/III
/(,,_:1
Il;,_, //'.,-:1
[{II O.H:\~)
0.3:10 O.3':.!4 0.310 0.':.!ti5
/("
Y"
f(.\, O.HI)~
O.':.!':.!li 0.':.!:1':.! 0.':.!44 0.'242 O.':.!2:1
N,,_~
Y
U::!I
II~,
C'"
/IO,.(/oli(l rf'/unu.
1).lnH 1.\lOIl
/("
11,,- \
Y\
75
Hl'itimfr
rIO'
OJl97 U.UY!'! 0.100 0.094
lIu
11.1,
[1,4,
O.IOli I>.lOO 0.105 0.\04 O.0\)3
0.o7·l 0.071 0.077 0.07\) 0.074
Il.W,O 1),0:,0 0.0:,8 O.Olll 0.061
flll
/(,,,
0.063 0.062 II.I)(iO O.Oti7 0.Oli4
0.0:16 O.O:lti 0.033 O.O:l\) O.03H
n"
0.016 0.017 lUll :, 0.023 0.025
-Iun
127 ) <1' O.\l:\I 0.03Y 0.044 0.047 Ilr.,
-<10<>7 -0.006 ) -11.011 -11.004 -0.001
II{., J' whe,.e Il" is Ihe wee". '\IIl()[o,.n ·blioll 11I.llri('c. of lhe veclor X, '" I /(" /(~, H" nil the I,iI '1Uilililc.'. i:;:: I ....• 5 (quintile I return Ull dw equal-wci ghted portfoliu of Sl()rk~ ill tUJlIl.Ju1y 10. 1962 tu the ~1I1~IIlt"st ~locks). fur the s~lIl1plt.: of ,P\;YSl-:·AMEX stocks
I cOllt;lius U·t/~EI(X'_'-I'J(X'-I')')D-t/~ llC('l·IJlht 'J':n. I!J!H (1.(i!J:.ob, nV'dlioJlS ). Notelh'd ll(k) '" correlatio ll hetweenll ll_1t dnd \\'I1t"l"e D == cliag(o: . .... 0::); thus tht· (i. j)th element is tht:" Jatiolls under all Ill) lluB hypothesi~ are given /(,1. A";YllIptotic stalldanl errurs for the alHo("orre
h\'
I/./'T
= O.O:!4.
anccs. If cross·au toco\,ar iances and the secolld term ollly the OIvll-au tocovari ariance autocov index and , ncgative y generall are ances ocovari the own-aut er. the Moreov . is positive, thell the cr05s-au tocovari ances must bc positive of the sum the exceed to as large so cross'
'Clhle ~,H reports autocor relatioll matrices T(k) of the vcctor of weekly rctunIs of/i\'(' sil.e-sor ted portfolio s, formed from the sample of stocks u~inl{ wcekly reluI"IIs fromJ Illy 10, I!Hi2, 10 Dccemb er 27, I ~1~11 (1,(i~I:) obsl'l"vatiollS), 1.('1 XI dello((' Ihe \'('((01' I UtI U~I /l:ll HII U',I J', ",h('l"e U,I is Ihe r('(lInt ollihe ('<Jllal-weil{hled porlfoli o of stocks inthc ith qllilllil(' , Thclllh c klh order alliocor reialioll /lwflixo fX , is I{ivcll by I(k) == D-I/~EI (X'_k Jl)(X I -ltnD-I/~, wh('l"e D == dial{(rr(, '" and It == EIX/J, By Ihis cOllven lion, the i, jlh elclllcnt of I(k) is the correlat ion of U,I - k wilh Il/" The cstimato r T(k) is Ihe IIsual sample alltocor relation matrix, An interesli lll{ pallel'll emerge s from Table 2,H: Thc enlrics helow thc diagollals ofT(k) arc almosl always larl{er th;1II thosc ahove the dial{olla ls, For exampl e, the firsl-ord er alliocor relation helwecn Iasl week's rei II I'll Oil 1;lrl{(' slorks (/l,.,_t ) wil h th is \\'eek 's ret til'll on small stocks (Ill I ) is 2(i5';;" wher(,'ls Ihe first-ord er aUlocor relalioll l)('t\\'e(,11 last week's retllrll Oil small stocks (Ili/-I) with this week's relurn Oil larl{c stocks (I?r./) is only 2.1%, Similar pallenls mOl)' he se(,11 illihe hil{her- order autocor relation malrice s, allhong h the magllitl ldes a 1'(' sm;1I I1'1' sillce Ihe higher-o rder cross-au loforrcla liollS dcray, '1'11<' as)'lIl11H'lrV of Ihe TUl) malrice s implies thai Ihe aUlo('ov ari;III(,(' malrix eSlimato rs h/l) ;11'<' also as)'lnnH 'trir. This intriguin l{ 11'fIr/,lfI,i!, pallel'll, whcrc largcr capitaliz alion slocks lead alld small('l" rapilali/ ,alioll slOcks );Ig, is morc apparcn t ill 'Llble 2,9 which reports Ihe diff('rell(,(' of the aUlocor relation malrices and II1<'ir transpos es, Ever)' lower-d iagonal entr)' is posilive (hencc cver), upper-d iagonal eiliry is lIegalive ), implyin g Ihal Ihe correlal ioll between Cllrrcllt reltlrns of smaller stocks alld pasl rctllrns of largcr stocks is always largcr than the cOlTe:atioll hetw('('11 currl'nl retllrllS of larger slocks and past retllrns of' smaller stocks, Of COII/'Sl', the nOlltnH lillg llIodel of Chapter 3 also yields all as:'IlllIIell'ic alltocor rdalio/l matrix, Iloweve r, we shall sec ill that chapter that IInr('alis lically high probabi lities of' nonlrad ing are reqllired to g('ll('rat<' Cl'Oss-a lllocorrd aliolls of 111(' magnill lde reported ill 'Elhlc ~,H, The resulls ill'l;lhle s 2,H and ~,~I point to the cOlllplt' x palll'l'ns of CI II~S effi'cts alllollg sennilie s as significa lll sourfl'S of positive illdex alllocOITl'lalion, luclcl'd, 1.0 alld MacKill lav (1~1!)(lc) show that OVlT half of Ihe positivI' iudex ,llllocol Tclaliol l is atlrihut ahk to positive {'f'oss-{'fT('cts, They also ohsC'l'vl' Ihal positive noss-dJ' e('(s CUI explaiu the apparen t profitah ilil\' of f/ll/lmrif/ 1/ ill\'esllIl<'lIl slral('gic s, slralegi( 's Ihal are coulrary 10 III<' gellnal lllarkel direnio ll, Thesc siralegic s, predica ted ou the Ilotioll th,lI ill\'('.'lllrs lelld 10 O\'('IH'OIn 10 illforlllOllillll, ('ollSisl of sellillg "wilIlH'r s" alld IHll'ilig "Ios('rs," Sellillg Ihe willll('rs ;1I1e1 bllyillg til(' losers will ('anI posilive eXpC('ieel prolils ill Ihe pn's('II(,(' "fllegal il'(' scrial cOiTclal ioll h,'('alls(' ('111'1'('111 losers an'likel \' 10 IH'(,OIlI<' 1'111111'1' willllcrs alld CIIITellt willllers are likeh' 10 I)('colll(' hlllln'lo sers,
,rr,;)
Table 2.9.
AIYIn1Wtl) of rmIHlIl/or()rrriation matrir,s. III III R~
Y(I) - Y'(I)
II, HI fir,
0.104 0.153 0.19:) 0.241
C""" HI
Y (2) - Y' (2)
III R2 R, f4 f~
(""00 0.052 0.079 0.OH9 (1.094 HI
YCI) - Y'(3)
HI U2 R, U4
1Ir,
(00"0
(l.03S 0.069 0.087 (l.093 HI
II, 112 Y(4) - Y'(4)
U, f4 fir,
COOO 0.033 0.059 0.084 0.102
/I~
I~
-0.104 0.000 0'{)61 0.113 O.IHI
II, -0.153 -0.061 0.000 0.0:,4 0.134
R.. -0.195 -0.113 -0.054 0.000 0.088
/12 -0.052 0.000 (l.029 0.042 0.0:)5
H, -0.079 -0.029 0.000 0.014 0.029
R.. -0.089 -0.042 -(1.014 (1.0(10 0.018
-0.035 0.000 O.O:H 0.054 0.062
H, ·-0.069 -0.024 0.000 0.022 0.035
-0.OH7 -0.054 -0.022 (),(l00 0.018
/12 -0.033 0.000 0.024 0.050 0.070
H, -0.059 -0.024 0.000 0.023 0.049
R..
llo
-0.084 -0.050 -0.023 0.000 0.030
-OJ02)
/I~
R..
-02<1 ) -0.11l1 -0.134 -0.088 0.000
•
llo
-om,)
-(l.O55 -0.029 -O.UIH 0.000
Il"
-O@')
-0.062 -0.035 -0.018 0.000
-0.070 -0.049 -0.030 0.000
Diflcrenccs between autocorrelation matrices and their transposes for the v("Clor of !izeX, - I III, II~, II" 11." /lo, J' where Fl., is Ihe week·, retUrn on Ihe e'l"al-weighled portfolio of .rocks in Ihe ilh qllintile, i= I, ... ,5 (quintile 1 contains Ihe smallcst 'lOeb), for Ihe sample of NYSE·AMEX stocb from July 10, 1962 10 December 27, 1994 (I,W!l' ohservalions). NOle Ihal Y(k) = O-1/ 2 EI(X,_. - jl)(X, - I,)')D- I/', where D 0= di'I~la~, ... ,a~;J.
sorted portfolio returns
But the presence of positive cross-efTecl~ provides another channel through which contrarian strategies can be profitable. If, for'example, a high return for security A today implies that security B's return will probably be high tomorrow, then a contrarian investment strategy will be profitable even if each security's returns are unforecastable using past returns of th~t security alone. To see how, suppose the market consists of only the two stocks, A and B; if A's return is higher than the market today, a contrarian sells it and buys 13, But if A and 13 are positively cross-autocorrelated, higher return for A today implies a higher return for n tomorrow on average, and thus the contrarian will ha\'e profitcd from his long position in I} Oil average.
a
7M 2. Till' Pre dirl abi lily 'if An l'i
lll'i llrl ll
No wh ere is it reCJuired th~tt the sto rk ma rke t ove rre act s, i.e., vid ual ret urn s arc neg (ha t ind iativ elv aut oco rr( 'lat ed. Th ere for e, the fan tha t cO lltr ari all str ate gie s son ic h;I\"(' jlositive exp ect ed pro Jits lIe ed not ma rke t ove rre act iol l. ilil ply sto ck In f~lIl, for the par tic ula r con tra ria n strat('h and Ma cK in lay (19!Hk ,)' tha t 1.0 ) eX~lInine, ov er hal f of the exp ect ed pro to cro ss- eff ect s and no lils is du e t 10 lIe gal ive aut oco rre lat ion in ind ivi dua ret urn s. l s'T llri ty Th ese cro ss- eff ect s In,l ), ;1\sO exp lai n til(' app are llt pro fita bil ity ofs oth er tra din g str ate gie l'v('f;11 s tha t hav e rec ent ly bec oil le po pu lar in the (om lnu nit )'. Fo r exa fin anc ial mp le, IUII!!,/slwrl or IIltL rkl' l-l/l 'lllm l str ate lon'f, pos itio ns arc off gie s in wh ich set dol lar -fo Hlo lla r by sho rt pos itio ns GU I ear reu frn s in exa ctly the n snp eri or bsh ion des cri bed abo ve, des pit e the f~tClt des ign ed to tak e adv ant hal llley arc age of ow n-c llc us, i.e" pos itiv e an d neg ;lli ve ofi ,)d ivi dua l sec uri tie fon 'l'a sh s' ('x p(' dn l ret urn s. Th e per for ma nce of or pai n tra din g str ate /lIIIII'II/'d·/mok gie s '~1I1 als o 1)(' alt l'ib llle d 10 cro ss- cll ecb own-cfrecls. as well as . i\lt ho ug h sev era l slu di( ,s ha\'(' all l'lI Ipt ed to exp lai n the se slr iki ng lag qft e(\ s (se e, for exa Ica dmp le, Ha dri nal h, Ka le, and No e [1! '!': Ij, Rich1anlson, am i Wh Bo ud ou ldl , itel aw lI9!141. Jeg ade esh and Swamin~ttll
\
2.8.4 '1i-.llJ (hil l!!, f.U II/iIIU liw lI IIrlurn.1 sev el\t l rec cnt slll
79 slIiJslalltiaIIlH'all-revlies that with a return horil,on of % 1Il0nths alld a sample period o/" (iO years, o=Hj(iO =O.133 hence the expeeled variance ratio is (I -8 )2=0.7:> I , despite IIIC fact thal RW I is assulJle d to nolr exallipl e, Richard soll alld Stock (19WJ, 'Elhk!» ), which are typically so large dS to yield z-statistics close to l('l"O regardle ss of the point estimate s. Richard son (199:1) and Richard son and Stock (19H~J) show that properly adjustin g 1'01' the Slllall salllple sizes, alld for otl)('~' statistic al issues associat ed Wilh 10ng-horiLol1 returns, reverses many of the inferenc es of Fama and French (I9HHb) and I'oterba and SUlIlme rs (I YHH). Moreov er, the point estimate s of autocor relation coeffici enlS and other time series parallle ters tend to exhibit conside rable samplin g variatio n for long-ho rizon returns. For exampl e, silllple hi:ls a<\justm enlS can change the signs of the autocor relation s, as Killl, Nelson, and Startz (19HH) and Richard son and Stock (I9H9) demons trate. This is not surprisi ng given the extreme ly slIlall sample sizes that long-ho rilOn returns produce (sec, for example , the lIlagnitu de of the bias adjllsln) ('nls in Section 2.4.1). Finally, Kin), Nelson, and Stanl (PIHH) sholl' Ihallhe IIcg:llive serial cor~ Icbtion ill long-ho rizon returns is ('xtrelllc ly sensitivc to the saJllple period and lIIay hc largely dlle to the first tell years of lhe I ~J~(; to 19H5 salllple. Althoug h ten years is a very signific ant portion of the data anci cannot be exclude d withollt careful considc ration, Ileverth eless il is trollhlin g thaI the
sig ll of the ser ial co nd ali oll coe ffic iell t hil lge s Oil clata (i'o m prc ssi oll . Th is CO IIII the (;r ea tlk IHl rlll l1- wh (·th er to om it dat a iIlIlIlCI1CC cat acl ysm ic ('\'1'111. or t\ Ill' a sill gle 10 illc ilid c it all d arg ll(' tha I sllc h all 1'1'(,111 .~cllta Iil'c of II J(' C(O is rcp reIlOIIl ic sVSI e m- III lt\l 'rsc ore s tl J(' fra gil il)' sta tist ica l illf ('lc ll(T . of sllIa II-sa III I'll ' {)l 'n,I II. lit nt' is litt le cvi dcl lcc f(lI' II1Call IOl lg- hot i/o ll rct llrn rev ers iol l ill s. Iho ug h I his lIIay be lIIo re of ,I sym Jllo m pie sil.es ral lw r Iha ll of SIII;III sall lCOllclllSivc C"icll-IUT aga ills tlll cal l rel 'er sio cal l1l ot lell . ll-w (' sill lply Th esc ' cOllsicic-ralioll s poi lll 10 sito rl-i tor il.o ll reI urn s ,IS the ilio C\iale .~OIllTC rro lll wh n' illl llle ich C'\"ic\CI1("(' of prc c\ic tab ilit y mi ghl be clil Ilo t to say IIial a can led . TIi is is ·fll l ill\ "('s liga liol l of ret url ls ovc r IOllgc'r bc Ull ill( )fJl lali \"(· . lllc tilll(' spa lls will \cl' d. ilm ay hc oll ly att hc sc low er freCjllellci im pac l or ITO Il() mir cs IIia llli c LtctO)S sllc h as thc bus illc ss cyclc- is dClc·CI'liJ 01'(' 1", to the cxt k. Mo rcell l Iha llra lls ani oll cos ts arc grc ate r (1I' slr alc gie s exp loi lill g sllOrI-llOrilOlI prl 'cli cl,l bili ty. IOl lg- hor iwl I pre e!i nah ilil Y lIlay uill (' for lll ofl lllc xpl oil bc ;1 11101"1' gC'lICc \ (llOlil opp ort ull ity . Nc vcr the les s, Ihe C'C cha llC 'llg cs POSI'c\ b)' OllOlIlc'lric IOl lg- hor i/o ll rC'llIrns arc' ('ol lsic lna blc -. all for acl (lit ioll al C'conO d Ihe llC'cd lllic slr un llr c is par tic ula rly gre al ill sllCh cas cs.
2.9 Co nc lus ion Rc ('cl lt C'CollOlIll'tric ac\ val lcc s ane! I'm pir ica l evi dcl lcc sec III 10 fin anc ial ass ct rct url sug gc" 1 Ill
Pr ob lem s-C ha pte r 2 2.1
If 11',1 is a IIlarliligaic-. sho w tha t: (I) the llli nim ull llll ('an -sc llia rnl nr fill Tca st of I'H I. con or dit iol lcd oll thc en tin ' his tor y 11',.1',_1 1',; (2) lIo ll(} \'l'r iap pil ig ... . 1. is sim pl) ' hlh ciif fer cnc cs arc ull cor rci at( 'c\ at atl lea ds lor all /; > O. ,lI1 dla gs 2.2 Ilo w all ' till ' RW I. RW~, R\'\':l, .lIl Clm art ing ak Iiy pot hes ('s clu de' a \'(, lIn di;l gr; rda ted (in Jm 10 illl lslr ;llc III<' rcl ati ons am on g lill ' fou r Pro vid c spl 'cif ic c'x am lIIod('ls)~ p!c s of I'ac lt.
lH 2.3 Characterize the set of all two-state Markov chains (2.2.9) that do not satisfy RWI and for which the CJ statistic is one. What are the general properties of such Markov chains, e.g .. no they generate sequences, reversals. etc.? 2.4 Derive (2.4. I 9) for processes with stationary increments. Why do the weights decline linearly? Using this expression. construct examples;ofnonrandom-walk processes for which the variance ratio test has very low power. 2.5 Using daily ami monthly returns data for ten individual stocks imd the equal- and value-weighted CR.',!' market indexes (EWRETD and VW1lliTD). perform the following statistical analysis using any statistical package;ofyour choice. Note that some of the stocks do not have complete return histories. so be sure to lise only valid observations. Also. for subsample analys~s, split , the available observations into equal subsamples.
a.
2.5.1 Compute the sample mean i<. standard deviation and first-orcier autocorrelation coefficient p(l) for daily simple returns over the entire 1962 to 1994 sample period for the ten stocks and the two innexes. Split the sample into four equal subperiods and compute the same statistics in each subperiod-are they stable over time?
a.
2.5.2 Compute the sample mean il, standard deviation and first·()rder autocorrelation c"efficient p(l) for continuously compounded daily returns over the entire 1962 to 1991 period, and for each of the four equal subperiods. Compare these to the results for simple returns-
82
2. The Predictability oj AUft Uft /117/J
subperio ds. Which of thc skewncss, kurtosis, and sllldent ized ran~e estiIljates are statistically differen t from the skcwness, kurtosis, and s!tll!entifecl range ofa norma! random variable at thc 5% level? For these twelve s(/rics, perform thc samc calculat ions using monthly data. What do YOll c911c1ude about thc normali ty of these rClllrn serics, alld why?
3 Market Microstructure
\\'1 III.E IT l~ AI.WAY~
the case that sOllie features of the data will bc lost in the process of modeling economic phenomena, determining which features to focus 011 requires some care and judgmcnt. III exploring the dynamic pmpcrties of financial asset prices ill Chapter 2, we have taken prices and returns as the principal objects of interest without explicit reference to the institutional structures in which they arc determined. We have ignored thc fan that security prices arc generally denominated ill fixed increments, typically eighths of a dollar or tirks for stock prices. Also, securitics do not tra(ic at evenly spaced intervals throughout the day, and on sOllle days they dOllol trade at all. Indeed, the very process of Ira ding can have an important ;llIpact 011 the statistical properties of financial asset priccs: In markets with designated lIlarketmakers, the cxistence of a J!nmd betwcclI the price at which the marketmaker is willing to buy (the bid price) and the price at which the markctmaker is willing to sell (the oJJ/'( or as" price) can have a llontrivial impact on the serial correlation of price changes. For some purposcs, such aspects of thc market's micTOstructure can be safely ignorcd, particularly when longcr investmcnt horizons arc involved. For example, it is ulllikely that bid-ask bounce (to he de filled in Section :~.~) is responsiblc for the negative autocorrelation in the five-year returns of US stock indcxes such as the Standard and Poor's ;)00, t even though the existellce of a bid-ask spread c10es induce negative autocorrelation in returns (see Section :{.2.1). Ilowever, for other purposes-the measurement of execution costs and Ilurket liquidity, thc comparison of allernative lIIarkelmaking mechanisms, the impact of competition and thl' potelltial li'r collusioll among marketmakcrs-markct microstructure is central. Indeed, market Illicrostructure is IlOW olle of the most active research areas in {'COllOlllics and finance, spanISt:~ ~c((iOIl ~.!) ill Ch"pl~r:1 "nd Section 7.2.1 ill Ch"PI'" 7 lor ""lher
Ilillg- lIlall)' 11\;11 \..I'IS al\(I lIIallY lIIod els,~ To lesl sOllie of Ihes e mod els, alld dele rllli llc IIII' illip onal lce of lIIar ket lIlic roslr llctl ire effe cts for otlll 'r resear ch area s, WI' rcqu ire SOI\lI' clllp irica lllle aslir es of lIlar kctl llirr ostr llctl ire effec ls, We shal l l'onslJ'llct sllch lIlea sure s ill Ihis chap ler, In Sl'l'I ion :\,1, Ive pres enl a silll ple mod el of Ille Iradillg- proc ess 10 cap1111'1' IIII' cfkl 'ls of nons ),nc llrol lous Iraclill~, III Scct ioll :~,~, WI' cons ider tile eff(ocis of the hid-ask spre ad on IIII' tillie-series prop ertie s of pric e dlan ges, and in Sect ioll :1.:1 W(' (,xp lore scve ral Icch lliqu l's for lIIo ddin g Iran s
3,1 NOn.';ynchronous Tra ding Till' 1I00H )'II(li mllll/ /1 flll/li llK or lIollf mdil lg effe ci arisc s whe n lillie serie s, IISUally assel pric es, are lakc n 10 he reco rded at time inler vals of olle leng th whc n in bl'l Illey an' rc('or (J'IHI i),Il., vi" allcliiolo (I~I~I:\). !'''"lk (;,IIIi, ,11111 l:i"I', 1 ''',
".1(
( I II~I,I) ,
lIlIlil li 11
".10;),
K.lf!d ,111<1 RII'Io (I ~1~'r)J. I...
(1~1~,r»). (l'II .. r..
(I ~'~I:»), a,,<1 SH:
J.1.
HUII'YIIUIIUIWIO I/llUlIlg
also induce spurious own-autocorrelation in the daily returns of A: During periods of nontrading, A's observed return is zero and when A does trade, its observed return reverts to the cumulated mean return, apd this meanreversion creates negative serial correlation in A's returns. These effects have obvious implications for tests of predictability and nonlinearity in asset returns (see Chapters 2 and 12), as well as for quantifying the trade-offs between risk and expected return (see Chapters 4-6). Perhaps the first to recognize the importance of non synchronous prices was Fisher (1966). More recently, explicit models of nontrading have been developed by Atchison, Butler, and Simonds (1987), Cohen, Maier, Schwartz, and Whitcomb (1978,1979), Cohen, Hawawini, Maier, Schwartz, and Whitcomb (1983b), Dimson (1979), Lo and MacKinlay (1988, 1990a, 1990c), and Scholes and Williams (1977). Whereas earlier studies c~msidered the effects of nontrading on empirical applications of the Capital Asset Pricing Model and the Arbitrage Pricing Theory,3 more recent attention has been focused on spurious autocorrelations induced by nonsynchronous trading.4 Although the various models of nontrading may differ in their specifics, they all have the common theme of modeling the behavior of asset returns that are mistakenly assumed lO be measured at evenly spaced time intervals when in fact they are nolo
3.1.1 A Model of Nonsynchrvnous Trading Since most e;npirical investigations of stock price behavior focus on ,returns or price changes, we take as primitive the (unobservable) return-gerlerating process of a collection of N securities. To capture the effects of nontrading, we shall follow the nonsynchronous trading model ofLo and MacKinlay ([990a) ~hich associates with each security i in each period tan uno?served or virtual continuously compounded return ril. These virtual returns represent changes in the underlying value of the security in the absence of any trading frictions or other institutional rigidities. They reflect both companyspecific information and economy-wide effects, and in a frictionless market these returns would be identical to the observed relUrns of the security. To model the nontrading phenomenon as a purely spurious statistical artifact-not an economic phenomenon motivated by private information and strategic considerations-suppose in each period t there is some probability Jri that security i does not trade and whether the security trades or not is independent of the virtual returns {rjl J (and all other random variables
~S.... , for .. xample, ('.ohell, Haw'"wini, Maier, Schwaru, and Whitcomb (1983a, b), Dimson (1~)7~1),
Schol .. s and Williams (1977), and Shanken (19R7b). ·S.." Atchison, Butler, and Simonds (1987), Cohen, Maier. Schwartz. and Whitcomb (1979.
19Hfi), and 1.0 and MacKinlay (I 98H, 19H!!b, 1990a, 1990c).
q'~:' .,
B6i
3. Markel Mirroslmrlu/'r
~hiS
Therefore~
in model).5 this nontrading process can be viewed as all IlQ sequence of coin tosses," with different nontrading probabilities across sec~rities. By allowing cross-sectional dificrcnces ill thc random IIOIlII ingi processes, we shall bc able to capture the effects of 1l01ltratiil1).! " ret~lrns of portfolios of securities. iThe observed return of security i, fj~' depends 011 whether s('cmil), i trades in ~eriQ(l/: Ifsecurity i does nottradc in pcriod t, let its obsCl'ved retllrll be zcnr,-ifno trades occur, then the closing price is set to thc prcvious period's el0tin g price, and hence ri~ = log(jJI///Jjt_l) = log 1 = 0, II', on the other han ,security i docs trade in period I, let its observed return be the SIIlIl of the I irtual returns in period I and in all prior r01l5eculiue periods in which i did ;not trade, 'For example, consider a sequence of five consecutivc periods in which security j trades in periods I, 2, and 5, and docs not trade ill periods 3 and 4, The above nontrading mechanism implies that: the observed return in period 2 issimply the virtual return (T,2 = Tj2); the observed returns ill period 3 and 4 arc both zero (Ti~~ = Tj1 = 0); and the observed return ill period !i is the sum of the virtual returns from periods 3 to 5 (f,~, = T,~ + f,4 + f ,).7 " This captures the essential feature of nontrading as a source of spurious autocorrelation: News affects those stocks that trade more freqnently first and influences the returns of more thinly traded securities with a lag, III this framework the impact of news 011 returns is captured by the vi"tllal retllrtJ~ process and the impact of the lag induced by lIontrading is captured hy the ohserved returns process f,~, To complete the specification of this nontrading model, slIppose that virtual returns are governed hy a one-factor linear model: rll
=
/1,
+ fJ,/t + E,t
i
=
I, .. ,' N
CU.I)
where f, is some lero-mean COllllllon factor and Ellis zcro-mean idiosyncratic noise that is temporally and cross-sectionally independent at all leads and lags, Sillce we wish to foclls on nontrading as the sole source of alltot'l>rrebtion, we also assume that the (01ll1110n factor It is lID and is independellt or ~JThr ca~e when' Irildil1~ is nllT('(at('d with viltHal IT\ltrIlS i:o. lIot without iIH('U'.,t, htll it j,
with the spirit or the n01111"1uling- a:\ it killd of 1I1(';IStll"t'UlerH error. In tlU" IJlt· ... (·lIt (' of priv4ltc informalion ant' Mratt"~ic h{'havior, trading anivily does typically dqu:nd on \'i(I\1,\\ l('l\ln\~ (~uit",bly defincd>. ;,uld ~trat{'gi(" Iradill~ COlli indllt:e ~crial correlation ill 01,'('1 \('<1 H'turm, hUl ,uch corr('lalioll fall hardly hl' ,li~1I\i",'d as -'pminus. Set' St'rlioll :I.I.~ lor furl I..... di'Cll",ioll, ';Thi, "''llmpli
,\
,
3, I, No/t.\Yll rItlVllull s'fmrliug
H7
(",., for all i, I, and k.x Each period's virtual retlllll is ralldom alld capture s m()Vl'lll ents c;lIlsed by informa tioll arriv;t1 ;IS well as idimync ratic noise. The p
X,,(II)
I (no trade) { o (trade)
with probahi lity n, with probahi lity I -
(I-8i1)8i'_18/,.~
.. ·8,,_.,
If,
(:\.I.:!)
I: > ()
with prohahi lity (I-lf/)lf ,' with probabi lity I -- (I -If,)lf,·
(:1. 1.3)
where X,,(O) == I - 8 i " {8,d is assllmed to be illdepcl ldent of {8jt} for i 1= j a:1d tempora lly lID for each i = I, ~, . " . N, The indicato r'v;lriab le 8" lakes on the vallie (Jlle whell seclirity i docs not trade in period I and is zero olherwi se, X,,(ld is also all illdicato r variahle and takes on the value one when sccurity i tr"dcs ill period f but has nol traded ill any of the Ii previou s (,OIlSl'!'IltiVl' I)('riods, alld is 1,('10 otherwi se, Si'ICC If, is withill the Ullil illlerval , for large II the variahle X,,(Ii) will he zero with high probabil ity, This is 1I0t slII'prisillf.( sillce it is highly unlikely that s(,curity i shollid lrade loday but lIever ill the past. I Iavillg defilled lhe X,,(k) 's it is 1I0W a simple 111.11 t ('I to d(')'iv(' all explicil expressi oll for observe d returns 1;';: 00
r/~ =
L X"Ud 'i,-.
I.,,,,N ,
(:U .4)
.~()
If security idol's 1101 trade ill period I. thCIIO,, = I which implies lhal X,,(k)=O liJr all Ii. alld lhus /';;=0, If i docs trade ill period I, thell its observe d return is equ.d to tlie sum ortoday 's virtual retllrll I',/ ;IIHI its past Ii, virtual r('turns, ",Iinc titl' ralldolll variable h, is tlie IlIlml)('r or past ((I11,lfnll ill!' periods that i Ii
==
~ { ~J tI"_1 }
n,
Althoug h I .'1) will prove to be Ilion' (,Ollvl'lIiellt IiII' suiJsequ ent ('aleulaliollS, Ii, 111<1)' be IIs('d to give a sOllu'wIi;1I 1110)'(' illtlliti\'( ' defillilio ll of the XTh('~(' ... Irullg ;t~."iIlIllPIi()Il~ ~ll'(' IIl.Hlc' plilll.llil}' f"l ("oll,idt'rahl),. S('(' Scnioll :\.I.~ for hlllhl'l III"
1l'1.1~(·d
t'XI'Il'lllltl l,d I fllI\"II"'II!
11\'11111.
C' .lIlff III.I}'
he
........
I. .\I(/tllI'I.\li"' ...I/nIl1,m·
ohs('rv('d
1I'llIrll~
procl'ss: I,
,."
L'"
"
CU.!)
I •.... N.
J.:~II
When'as (:\.1"1) shows Ihal ill Ii II' presenct' of nonlrading Iht' ohservt'd n'llInlS procl'ss is a (siochaslic) fllllclion of till pasl ITIIII'lIS. Iht' t'C)lIi,'al('lll rdation C\.I.li) rl'wals Ihal /';; Ilia), also ht' vit'wt'd as a ralldolll SIIIII wilh a randolll nllllll){'r of 1I'I'lllsY A Ihinl alld pnhaps llIost naillral way to view ohserved rt'llIrns is the followillg: 0
wilh prohabililY Jr,
rtf
with prohabililY (\-Jr,)~
1'"
'~t
,,"
wilh probability (I-JrI)~Jr,
+ 1'", I I
1",1_ I
+ r,,_~
wilh prohabililY (I-JrI)~Jr;
(~.1.7)
"
1- "", k
Exprl'sst'd in Ihis wa)" II IS appan'lIl Ihal 1I01llradill/{ Gill illdllcc spmiolls st'ri;1l corrdalion in ohst'rvcd rctllnts hecause each r;; conlains withill il Ihc SIIIlI of pasl Ii ('oIlS('('Uli\'(' virlllal rClllrns for t'very k wilh sOllie pusilive prohahilil), ! I - Jr,)~Jr,k. 'Ie) st'c hoI\' Iht' nOlllrading prohahility Jr; is rdaled 10 Iht' duralioll of nOlllrading. ('ollsidl'!' Ihe llwan and variance of h,: Jr ,
1-:1 ",I
==
I - Jr ,
Varlk,l ==
Jr,
C~,I,H)
(I - Jr,)~
If Jr,== ~ IhI'li S('(,II ri Iy i gOt'S wi Ih01l1 Iradin/{ lill' olle pniod al a Iilll(, on
;1\'('1"
ag(,; if Jr , == ~ Ih('11 Ih(' ;\\'('rag(' IlIlInhn of ('onst'flllivt' pniods of lIonlradillg "Th,~ b
"mH.u \" ~p;r'\
\1)
,Ill' S, holt-, .\1U\ \\'miam~ ('~)77) :\uhmtlill.l1C·" :\10('h;l~ti(' I"on'"
rq)l"('st'litalion 01 oh:\('I"\'('" 1I'llInl'. although \\'t' do 1101 rt':\trin tilt' trading lime's
in
1101.
lix('cI
lilill" illl('n,ll.
\\'itll
~tllf,lhlt, nonl1~lli/~"ioIlS il
(0
Iilkc' \';dll("~
may Iw :\11O\v11 thai 0111' IHllllradillg IIlCukl c'olln'lge''i wc·;tI... h IOliae'("(llIliIIlIOlh'lillH' POi."-MIil pron':\s ofSdlOlt·., ,uHI \Villi.IIII.' (1~177), 10'10111 (:\'1,,1) III(' OhM'1 nod 11'luI"II' III 01"('" molY abo 1)(' ("on~i,lc'r('d 'lIt illlilliu' ..ultl,·r lIIonllg ~'n'lag" ell \"illllall('l1l1l1\ \\'h('n' du' \1:\« 111'11 It il'lll~ an" ~lorh"~liL This i~ in ("olllra.'1111 (:011('11, ~1.liC'l. S, h\\';tl"ll, .IIHI \\'hil( IlIlIh (1 qHli. (:h;'pll'r f.) ill whirlt oh~('n'('d n'lIl1l1'i an' a'i'lIl11c·d Ie, hc' ;, lillih' ..uult'l ~L\ pIC II I'" with 1l01l .. llIch;I'lic nlC'lIil"i"lIh, /\!though 0111' nOlltl.uling Ilion· .... i.. 11101(' g(·III'r,lI. Ihc'il 0),"'1 \'('d 1"1'111111' 1)lIIIT:\~ ill( IlHlt,:\ a hid-;Isk .. prt"ad ("01111)(1111'111; 0111" dot'S .1
3.1.
.vul/.\.\"lIdIlUIIUII.\
nat/iug
is three. As expected, if the security trades every period so that1T; the mean anel variance of kl are zero.
=0, both
1IIIIIliraiions Jar individual Securily Rrlums To see how nontrading can afTect the time-series properties of the observed returns or individual securities, consider the moments of r;~ which, in turn. depend on the mOlJlents of X'I(k).11I For the nontrading process (3.1.2)(.~.l.:{), the observed returns processes {r,~1 (i = I, ... , N) are covariancestationary with the following first anel second moments:
(3.1.9)
/1,
t 21T; t 0+--/1
Var[r;~l
I ,
=
I -
11;
(3.1.10)
,
-~;rr:
11-".)11-"1) I ".",
= j,
n>O
for i =1= j,
n~O
for i
fJ fJ
~
(3.1.11) 1/
' J Of 1Tj
(3.1.12)
n > 0,
0/
where o,t == Var[r;ll and == Var[/tl. From (3.1.9) and !3.1.10) it is clear that nonrradingdoes notafTectthe mean of ohserved returns bill docs increase their variance ifLhe security has a nonzero expected return. Moreover, (3.1.12) shows that having a nonzero expected return induces negative serial correlation in individual security returns at all leads and lags which decays geometrically. The intuition for this phenomenon follows frolll the faClthat during nontrading periods the oliserved return is zero and during trading periods the observed return reverts back to its cumulated mean return, and this mean reversion yields negative serial correlation. When Il;=O, there is no mean reversion h<;nce no llt'galive serial correlation in this case.
Mllximal S/mrious AuloforTelalion
I
These momenL~ also allow liS to calculate the maximal negative autocorre1'lliOll attributable to nontrading in individual security returns. Sincei the autocorrelatioil of observed retllrns (3.1.12) is a nonpositive continuous function of 11; that is zero at IT;=O and approaches zero as IT; approaches unity, itlJlustattain a minimum for some IT; in [0,1). Determining (his Iqwer bound is a straightforward exercise in calculus, and hence we calculate it only for the first-order alllocorrelatioll and leave the higher-order cases to the reader. IIITo ("OIlM'I"\'(" span'. w(" sllllIl1Iaril.t'
( I !)!)O". I !)!)O("j I.... fllrtl ... r d.·I;liI •.
lht' rt'MIIt~ ht'rt· and n:ft'T
rt"adt"n 10
l.u and MacKinlay
90
3. Market Miovslmrlurr
Under (3.1.2)-(3.1.3) the minimulII first-<Jrder autocorrelation of the observed returns process Irj~l with respect to nontrading probahilities IT, is given by
M ·In Corr l· Tit' ,n,1 where ~j
• 1=
T,I+I
- (I~il) rn I + v21~;1
t
== J1.;!ai. and the minimum is attained at I IT i = : : '
I
Over all values of 11"; E [0, I) and .~
+ ./21~,1
~i E (-00, +(0),
Inf Corr[r,~, r~+IJ
I",,~,I
we have I
= -;-. 2
(~.I.I:)
whic I is lhe limit of (3.1.13) as 1~,1 [nneases without bound, bllt is never allail{led by finite ~;. Although the lower bound of - ~ seems quite significant, it is virtually It' unattainable for any empirically plausible parameter values. For exalllple, if we\ consider a period to he one trading day, typical values for II; and ~ (1j ar1.05% and 25%. respectively, implying a typical value or 0.02 for ~,. t Acco 'ding to (3.1.13), this would induce. a spurious autocorrehnioll or at It most 0.037% in individual security returns and would require a nontradillg prob bility of97.2% to allain, which cOITesponds to an average nontradillg ~ durat on of 35.4 days! ~. lese results also imply that nontrading-induced autocorrelation is ~ magn fied by taking longer sampling intervals since under the hypothesized irtual returns process, doubling the holding period doubles J1.; bllt i" only ultiplies (1; by a factor of ./2. Therefore more extreme negative alltocor elations are feasible for longer-horizon individual returns. However, this iSlnot of direct empirical relevance since the effects of time aggregation hflve been ignored. To see how, observe that the nontrading process (3.1.2)-(3.1.3) is not independent of the sampling interval but changes in a nonlinear fashion. For example, if a period is taken to be olle week, Ihe possibility of daily nontrading and all iL~ cOllcomitant cffecL~ all weekly observed returns is eliminated by assumptioll. A proper comparison or ohserved returns across distinct sampling intervals must allow for lion tra
l
i
AS)'lIlmrlr1C CroH-A UIOfOlIar1anrrJ Se\'eral olher important empirical illlplications of this nontrading lIlodd are captured by (3.1.11). In particular. the sign of the cross-autoc()\'ariallct's
J. l.
!11
NUIIS.yndlHJ/UIll.I 'J/1UliIlK
is deterJllined by the sign of /1,/1,. Also, the expression is nol symmctric with respcct to i and j: If 7r; 0 and 7r} of 0, thcn there is spuriolls crossall tocovariance betwcen r;~ alld 1';; III but 110 lTOSS-;llItO("llVariance hetweell
=
Ii';
alld
r;;+11
for allY
H>
0. 11 The intuitioll for this reslilt is simple: Whcn
j exhibils nOlllrading, the relurns to a constantly trading securilY i call forecasl j due to Ihe common f;tctor it prcsent in hoth I·eturns. That j exhibits nontrading implics that future obs('J"ved retllrns 1';;+11 will he a
st~curily
weighted average ofal! past virtllal retllrns r,I+/I_' (with the Xil~lI(k)'s as random weights), of which one lerm will he the l"llncnt virtual returll 1',1' Since the cOlltemporaneous virtual returlls 1"" and 'l ' ;Ire UIITl'iatl'ti (beGl\lse of the common factor). 1':; can forecast ';;4 /I' IloIVevcl", ,.;; is itself unforecastablc bccallse 1';; = Ti' for all t (since 7r, = 0) ,Illt! r,l is II!) by assumption. thus Ij'; is uncolTdatcd with ";;+11 for any 11 > n. The aSyllllllcll)' of (3.1.11) yields all empirically testahle restriction on the crosS-;l\llOcovariallces of returns. Since the only source of asymmetry ill (:\.1.11) is cross-scClional differences in the probabilities of Ilolllrading, :nforlll
IL "= E[ r;'I. Dellotillg !l,e (i,j)th clelllellt of 1'/1 hy Y,,(ll) =
Y'j(II),
(I - lli)( I - ll,) I-ll,ll,
we
h,IVl'
OU.Hi)
hy defillition
Ii, Ii (J~/ Jr"
(:1.1.17)
"
Ir Ihe lion trading probabilities ll, differ across securities, 1'/1 is asymmetric. From (3.1.17) il is cvidel1llhat
(3.1.18)
Therefore relalivc nontrading probabilities lIlay be eSlimatcd directly using sam pic autocovariances n' To derive estimates uf lhe prohahililics 7ri thelllselves we Ileed only estimalc olle SIIl"h probability, say lli.
r
II :\11 ;thCIII.lliv(' illtcrprct;lIioll or Ihis ;ISYIlIIHt'll y 111;1)' he 101ilid ill IIII' ,illH"~s("l if'S literaluu' cOlln"rlling Cr~lIlgt'r. <:allsality (!\ee (;rangl'r II!Jh~)j}. ill whirl, ,;; IS :-."id 10 (;ranK"-((IIL" r;; if Ihe relllrn to i predicts Ihe retllrn to j. III lilt, ,,,hove.' example, s{'("urily i (;'flnK"-<m.Hr.~ ~cllriry j when j is s\l\~jc-(t to nontrading but i is noL Sillce our HOlllTddillg prO(Ts..~ fUlly he vi{"w{"d a.1Ii. (\ form of \lll'a:-o\l1Tllwllt error, Ihe taCi Ihal Ih(' I"('11II1I."i 10 ollc "i('('111 i1r III;')' he l'xogellotl.,\ wirh l'e:-opeC't to 1~)77).
the
ret\lrns of allother ha~ lW('1I ))IOpo.... (·(l ulldel'
;I
diflt'n'nl glli.'\(' ill Sims
(I!J71,
/1Il/,limlio/H jill' I'orlfolio Ul'llIT/I.I Suppose securities are ~rouped hy their Ilontradillg prohahilities and eqllalweighted portll)lios are limlled hased on this grouping so that portfolio A contai/ls N" set'llrities with idt'/lti("al /lolllrading prohahility TC", a/ld similarly lill' pOrlli)lio Il. Iknott' hy ';~I and I~'I the ohserved time-t returns on these t\l"1 pOrl/illios rt'SIIt't'ti\'t'i)" whit'h art' approximately averages of the individll; I n'turlls: K
{I,
h.
where the summatio/l is over all securities i in the set of indices I, whirl, t'omprist' pmtli)lio 1<. Tht' rl';lson (:\.I,19) is 1l0lexaCI is th;\! hoth o\)seryt'c\ and virtual retllms art' assumed to he continuously compounded, and tht' logarithm of a sum is 1101 the sum of the logarithms. 12 Ilowever, if r:; t;lkt's Oil slIIall vahlt,s allll is 1I0t too volatile-plausihle ,lssumptiollS for the shm I returt! int!'l'vals thaI /lollsYllchrollous trading models typically focus 011-' the approximatioll !'I'ror ill (:\,I,19) is IIl'gligible, The timl'-seril's properlies of (:{.1.19) may be derived from a Sill'l,l,' asymptotic approximation that exploits the cross-sectiollal ill
LlT:fr-k,
I', -\- (I -IT,)fl.
('I.I~()
k=O
where CU~I)
i,,'
I:!.\ PIC"
IIlh'llHt'l,lIioll 01 ':', j, tilt, 1('lIl1"n 10
i'
pC)lllolio whOM' \,;,111(' is (".1)( 111;11('(\ .1'
all 11I1'\'C'lghh'd g"UIIH'lIil' ;I\'('lagt' 411 tht' ("(HIlPOllt'JH St.'flilitit·s' prin's, TIH' t'Xp('r!I'cI n'lllIll ~\\fh .., pontuhu \\,m ht· 10\\'\'1" 1h.m ,h . " t)f .\\\ t'(}\1/,\'~wt',ghtt't\ pontnho \... hn~t' u'\urn' ~\1t' ""kulah',l ;1.' IIH' arililllu'lil tIIe'HI.' of lilt' "implt' r('turns 01 Iht· (ompont'lll ~(·nlrili('s. This i",,~ is ,·x;""i ..... t ill ~n';""1 d",,,il hv Mod.,s' ;"HI Sll,,,\;ort's,," (1'lH:I) ;"I
01"
IIlllii I~'HH.
(I ~IH:\;t), (:halllhc'rI.,ill ;uHIH.Olh.'1 hilcl (I~IH:\). alld \\';lIlg III tht,,,, wC';&k('r (1IlIcliliolh j" ~jmply lu alltl\Y tI I..IW of I.argt· (\;tllllh('", 10
i"SSt'l". lUI ('X,I1I1I'''', (:h,tlllhc',I.,ill
(I!II.I:\). Th('
ht, applic'd
In
'" 'ht, nn,~
('."C'II( t'
IIII'
;1\'c'l;tgl' III
"'11'1111
gun\·,.
II",
di'llIl 11.11 liT.", ~o lia;1I .. itlio."ylH· ..."ir .. j,~" \',lIIbht,S
almo",
\111 (,1\'
for K = a. b. 'n.e first and second moments of the' poruqlio!s'feturns are then given by
E[r:/ ] Var[r:/l Cov[r:1' r: lh 1 Corr[r:l • r:l+ n)
Cov [r:1' r:l+ n)
= E[r./]
11.
2C-]'(')a~
a
ft. a
(3.1.22)
1 +]'(.
(3.1.23)
I
C-]'(.)
2 fJ.2 - - ]'(.n aI' 1+]'(•
n
~
.!!.
7r,," ,
a
(l-]'(a)(I-]'(b) 2 n fJafJbal ]'(b' 1- ]'(a]'(b
n ~ 0
0
(3.1.24) (3.1.25)
(~.1.26) I
!
where the symbol ~;;" indicates that the equality obtains only asymptotically. From (3.1.22) we see that observed portfolio returns have the same mean as the corresponding virtual returns. In contrast to observed i?dividual returns. the variance of 1 is lower asymptotically than the variarce of iL~ virtual counterpart ral since !
r:
ral
:<::: n
1 -Lril Na lEI, 11. +
1
= l1a + {Jalt + N Lfit d
(J.It,
ie/.
(~.1.27)
II (3.1.28)
where (3.1.28) follows from the law of large numbers applieq to tte last term in (3.1.27). Thus Var[ rad ~ fJ;a which is greater than or equal to Var[r: I ]· Since the nontrading-induced autocorrelation (3.1.25) declines geometrically. observed portfolio returns follow a first-order autoregressive process with autoregressive coefficient equal to the nontrading probability. In contrast to expression (3.1.11) for individual securities. the autocorrelations of observed portfolio returns do not depend explicitly on the expected return of the portfolio. yielding a much simpler estimator for ]'(.: the nth root of the nth order autocorrelation coefficient. Therefore. we may easily estimate all nontrading probabilities by using only the sample first-order own-autocorrelation coefficients for the portfolio returns. • Comparing (3.1.26) to (3.U1) shows that the cross-autocovariance between observed portfolio returns takes the same form as that of observed individual returns. If there are differences across portfolios in the nontrading probabilities. the autocovariance matrix for observed portfolio returns will be asymmetric. This may give rise to the types of lead-lag relations empirically documented by Lo and MacKinlay (1988) in size-sorted portfo-
J,
!J4
J.
M(lJ"krt M;n'U.IIl'llrlll/l'
lios. Ratios of the cross-autocovariances /IIay be forllled to estimate relativc nontrading proba~ities for portfolios, since rllft 'i,l.. + 11 I C~ov I" Cov[ ri:J't 1'::'+11)
III
~
(rr/~ )"
n.I.:l~l)
If"
~ddition, for purposes of testing the ovcrall spedfication of the
I~a{iing 1lI0del,
11011-
these ratios give rise to many over-idelllifying restricliollS.
slll~'e
I I
Y•• ,(n) y., •• (lI) Y•• K,(II)·" Y" ,.,(11) y.,1.(lI) Y.,a( n) Y••• , (11) Y.,K, (II) ... YK,., __ , (II) Yb., (1/)
==
(lflo)" -
n.I.:)()
If.
for 6ny arbitrary sequence of distinct indices KI. K2 • ••.• K,. !l -I b. r ::: N,,, whdre N" is the number or distinct )lOrti()lios and YK<• I (11) == Cov[r"/. ,.u1<" ," n I. \ II{,
Th9refore. although there arc N,; distinct autocovariances in r ll • the restrictiolls implied by the nOlllrading process yield far fewer degrees of freedom.
TinT Aggregation . Th~discrete-tillle framework we have adopted so far docs not require the specIfication of the calendar length of a "period." This advantage is more apP'Irent than real since any empirical implementatioll of the nOll trading mod~1 (3.1.2)-(3.1.3) mllst either implicitly or explicitly define a period 10 be a panicular fixed Gllentlar lime interval. Once the calendar tillle interval has been chosen. the stochastic behavior of coarser-sampled data is rcslrict('d by the parameters of' the most finely sampled process. For example. if the length of a period is taken to be one day, then the rnomen L~ of observed monthly returns may be expressed as fUllctions of the parameters or the daily observed returns process. We derive such restrictions in this sectioll. To do tflis, denote by li~(f{) the observed return of security i at time r where one unit of T-time is equivalent to q units of I-lime. tlIlIS: 'q
L
/=(,-llq+1
Then under the nontrading process (3.1.2)-(3.1.3), it can he shown Illal the time-aggregated observed retllrns )ll'Occsses (r;;(q)} (i = I ..... N) arc covariance-stationary with the following first and second lIloments (s('e I.\l and MacKiIl!ay [1990a]):
E[r;,(q)]
=
q/l;
=
.) qa;
'2 IT ;(\
-
IT ()
.)
+ (\ _ rrY 11;
n·I.:\:\)
3. 1. NOIlJy"dmmuu.\·. 'li"fUlillg
= -J1~ ITII/ I
Cord r;; ('I), r:; ll/(q) 1
II'ill
,
(I I -
" > (I
(3.1.34)
~;( I - IT?)~lT,I/'i 'il I
I -
==
IT,
- ------------
X(I-lT?)~ where~,
rr?)~
i of
IT,
j,
1/
> 0,
I(;/Oi as hefore.
Altho\lf!;h expected relllrns tilll<'-af!;f!;regate linearly, (:1.1.:13) shows that \';Iriances do nol. As a result of the negative snial correlatioll in r;;. the \'
=
or~lO%.
Although the alltocorrelatioll of coarser-sampled retllrns sllch as IlJolllllly or quarterly have lIIore extrelllc minima, they ;\IT ;lltailled ollly at highcr nOlltrading probabilities. Also, tillle-aggregalioll Ilced 1I0t always yidd a lIlore lIegative autocorrc\atioll, as is apparclll frolll the portioll of Ihe graphs to the left of, say, IT = .HI); in that I cgiolJ, all increase ill thc aggr('g;ltiulI \'.lIl1e '/ leads to all alltllcorrcl;lIioll dosn to l(,I'O. IlIde(~d as If ill(Tt';lSl'S without hOlilld the .1l11ocond;ttioll CI.I.:I;I) approaches I.cm for fixcd IT,, alld thus Ilolltradillf!; has lill1c illlpatt Oil IOIlf!;('I'-hOlilOIl returns. 1"Vaill(" lor ~ well' ohl.!illl'li hy takillg lIlt' I'.,lio "lllIe ',lIl1pie IIl('alllo Ihe ,alllple .\[;)1111.11<1 d('\'ialioll for d"ily, w('l'kly, alld monlhly ('qnal-w('I~IIIl'd I'eltll,,, i"dex .... lor Iltl' .ample period h'lll" 1~1t;:L 10 1\IK7 '" rl'pOI'Il't\ in 1.11 a"d ~1."·Kinl,,y (I\IKK, ·L,hl.·, L,-i·). Allhongh
"0"
tIH',\(' \,;llll(,~ 111.1)' he ilion' l('pn'S('n,ali\'(' of .,Ior)..
w'n'\ tlwl{'~~ lor
"H'
~;,k('
illClt·x('.\ I;lill("
of illu~trtuiul1 "u'Y ~honld :-oullic (',
11t.11I IIlfli\'idll.t1 .... l'("Ilfili(·.~.
I
)
" /'/ V.
/.
~ "~
I;
I
I
-
In
'.If
'"
1I••III·p' ...... '"\·
(
:~
~
.
-
,,'
'"
Nil
III
'III
;'11
)
"~
)'i)
~
/
'"
:f
+:
~
~
~
/;
'~
, I
..
r: "-
iI
.'-
I" < j
~
-
I
I
!
-
III'
;.tI
1.11
1II,'lq.III"OI'III\-
'"
'.11
111-
l.u-
1.1r-
1I",wl·,II.'M·III\'
t u-
'.11'
.
The effects of increasing ~ are traced out in Figures 3.1 band 3.1c. Even if we assume ~ == 0.21 for daily data, a most extreme value, the nontradinginduced autocorrelation in weekly returns is at most -8% and requires a daily nontrading probability of over 90%. From (3.1.8) we see that when = .90 the average duration of nontrading is nine daysl Since no security listed on the New York or American Stock Exchanges is ina€live for two weeks (unless it has been delisted), we infer from Figure 3.1 that the impact of nontrading for individual short-horizon stock returns is negligible.
rrj
Time Aggregation For Porlfolios Similar time-aggregated analytical results can be derived for observed portfolio returns. Denote by r;, (q) the observed return of portfolio A at time r where one unit of I-time is equivalent to q units of I-time; thus 'q
L
r:r(q) -
r:,t
(3.1.37)
1=(r-l1q+l
r:,
where is given by (3.1.19). Then under (3.1.2)-(3.1.3) the obselVed portfolio returns processes (r;, (q)) and (rb' (q)) are covariance-5tationary with the following first and second moments as N. and Nb increase without bound: i a
E[ r:, (q)]
Var[r:, (q)]
Cov[r:,(q), r:,+n(q)]
a
a
(3.1:.38)
qf-LK
[ 2rrK I-rr!] I_rr K 13 22 [I-rr.] [I -rr!r l+rr.l-rr. q-
K al
nq-q+ 1 f32
xrr • Corr[ r:,(q), r:r+ .. (q)]
[I-
Cov[ T,:r(q),
2
(l - rr!)2 rr;q-q+1
a
q(l -
rr;) - 2rrK
(l - rr!) ,
b
T ,+II(q)]
".(1-":)(1-",)%+,,,(1-":)(1-"01']
n
q
(I-"e)(I-Jf,) I-If.",
for
K
==
fl,
n > 0
• al ,
(I-If.)(I-",)
[~] 1-",
2
rr
nq - q+1
b
P.P'''t I-".If,
f3 f3 a 2 n
b
I
for
n == 0
for
n > 0
(3.1.42)
Ii, q > 1, and arbitrary portfolios a, h, and time r.
98
3. Mmlifl Minollrurllln'
Equation (3.1.40) shows thattillle a~~re~ation also affecls Ihe '1II\oCO!"relation of observed portfolio relurns in a highly nonlinear f;lshion. 111 contrast to the alltocorrelation for ti/lle-a!{~reg-ated individual securities, (3.1.40) approaches unity for any fixed I{ as 1(. approaches unity; IIIen,rol"(' the maxim,11 autocorrelation is one. To investig.lle the behavior of the portfolio autocorrelation we plot il as a function of the portlolio nontradin~ probability 1( in Fig-me :t Id Itll' q = !i. 22. G6. and 244. Besides differing in sign. portfolio and individual autocorrclations also differ in absolute magnitude. the former heing much larger than the bner for a givell nOll trading probability. If the nontrading phenomenon is extant. it will be most evident in portfolio returns. Also, portfolio autocorrelations arc 1JI00IOtonically decreasing in if so titat lillie aggregation always decreases nontrading-induced serial dependence ill portfolio relurns. This implies thaI we .Ire lIIostlikely to lind evidcllfc of non trading in short-horizon returns. We exploit both these illlplicatiolls ill Ihe empirical analysis of SeCiioli :1.4.1. 3.1. 2 l~xlensions and GPIleraliwlions
Despite the simplicity of the model of Ilonsynchronous lradill~ in Se(!ion 3.1.1. its implications luI' ohs('I"vcd tillle series arc surprisingly rkh. The framework can bc cxtcnded and gencralized in many directions with lillie dilllculty. It is a simple mailer to relax the assulllption lhat individual virllt.tI rclurn;lare lID by allowing Ihe COIllIltOIl faclOt· 10 be attLOcolTclaled allel the diSH! "bances to be cross-sectionally correlaled. For example, allowill~ ji to he a stationary AR( 1) is conceptll,tlly strai~htr()rward. although the t'alclllatilOns become somcwhat lIIore involved. This specification will yield a dcco61position of observcd <\utocorrclations into two componenls: one due 10 thJ COlllmon factor and anothcr dlle to 1I01l1l·adill~. Mlowing cross-section'll dependence in Ihe disturbanc'es also nllllplicates lthc momenl calculations hilt docs not crcate allY intrartahiliti('s.I~. Indeed, generalizations 10 lIIuhiplc l;tctors. lime-series depcndence of the diSIlIlI> 'pelldcncc Gin he built into Ihe nontrading process itsclfhy asslllltin~ thattl e O,l'S arc Markov dlains, so II1<1ttl1(' conditional probability of Iradillg
I
tli.'nl~.,,(·d (';U1it'l, ,OIlU" Itll III (II t rO~.'·~l"t licltl.tl wc.".,}.;, dCpt·l1th-lIet· 11111."1 he illlJ)('~('cl so thai (tht' as\"tuptotic ~1I"~lltnl·tlt~ of the port(oliu r('suh~ s.till ubtain. of ("UUt"S(', 'l1rh ,\t\ .\X\n1l1ppUI\ In.W nn1 alw.w'\ \w ,\pplilpri.\\t, .l~. tor t',,\.ul1ph-.ln the ra~p of n))npallit'~ "ilhil11lie ~.IIIW il1(lll~tly. whOM' n·~id\l.tl ri~k~ W(' l1lig.ll1 ('Spt'ft 10 Ilt' pn~ilin'l~ rnrn·ble(t. TIlC'It·ltu t·, till' .,'\1\1\,\(:\\\\' ~'ppn.,\.in\.\ti\ln ,,'ill bt' 1\\n~t .\r(\\L\~\' hn \\TH--i;\\\\·\· ... 1tlt·l\ punt'oHo .... '\\"iWt'
3,2, '/'Ill' Hid-fbI! Sf/mid
!I!I
tOll\orrow dqwllds 011 whellwl' 01' lIot ~I trad~' o<"nll'S tod"y, Ahl\()ugh this specificatioll docs adlllit cOlllpact and cI('g~1I1i ('xpressions Ill!' the lIlonJents of the obsel'ved r('tUI'IIS proccss, w(' shaillcave their derivalion to the rl'ael('1' (Sl'(' Problelll :t:I), However, a bl'i('f slIlllm.II'Y of thc illlplications for the lillle-snies properties or observed Idlll"\lS lIIay lit' worthwhile: (I) Individual SCCIII'ity retums lIIay he positively '1l1t()cond;ltcd and ponfolio IClllrns ilia), be nq~alively aUlocol'l'c\alcd, hUl these possibilities arc unlikely given empirically relevant paf'alllctcl' valucs; (2) It is possible, but IInlikely, 1(,1' autocorrelation matrices to be sYlllllletric; alltl C\) Spurious index autocorrelation illtluced by nontrading is higlll'r (or lown) whcn there is pIJsitivl' (01' negative) persistence in uOfltrading, In principle, propeny (:{) might be sufficiellt to explaill tlte lllagnitu(1c or index aUIO('olTdatiolls in f'Ccenl stock market dala, However, sevcl'al calibralioll experilJlcllls illdicatc the dq.;rce or persislellce in nontradill~ required (0 yield weekly aUlocorrclatiollS or :{O% is empirically impbllsihlc (sec \.0 alld MacKill]ay [1990c] 1'01' details), Olle (illal directiofl for fllnher invcstigation is the possibility of depelldence betweefl the nontrading and virtual retuf'flS processes, If virtual rctUf'flS are taken to he new inforfllation thell the exteflt to which traders exploit this information in deterflliniflg whell (and wh'lt) to trade will show itsc\f as correlation between r,( anti ,5 ,(, i'vLllly stralt'gic cOllsidn;ltions are illYolwcl ill l1\odels of ifl(orfllalioll-hased trascd llol11r.ulillg. ill what ,,'liS" is this .1ll\o('oIT('latioll SPIlriolls? The prclllis~~ or the extensive literature Ofl flUfls),flcilrouolfs lradiflg is that fI()fltrading is an outCOfllC or institutional features sllch as lagged adjllslIlll'llts anel nonsynchl'()flously reported prices, I\ut if nOJlsYllrhrollicilY is purposeful alld illlorlll
3.2 The Bid-Ask Spread 011(' or Ihe mosl important charac\el istics th;1I ill\'('stors look for in all 0)'g;llli,,'(\ lill;\nri.d nlarket is li(plitiity. till' ;dJilitv to hili' \)\' ,ell significlIIt
IIoSOIlH'
good iltll~tr.Hi()m. of tl\t· kind til
Il,Hhng hell.I\I(1I 111.lI (.111 .lIi\"
helm ""lIall'gi."
("oll\idt'l ;llioll~ .If"(' (oillaillt'd ill Adlllali ;11111 PI1t'id(,f(,1" , I~.HH. I~.H~)). Ikll\llII;l.\ ;111(\ 1.0 (1~~lh).
b"I",' ""d 0'11.",. (1!IH7, I!J!JO), K)it, (I!JH.'>l. ""d \I'",,~ (I'I!U, I!I!IIJ,
qualllilil'S oj a sl'nu ill I(lIiddr, '1I101l),IIHlllsly, and wilh rehllivl'l), lillie pricl' illlpan. 'Ii) mailllaill liqllidity, lIIallY oq~allized exchanges USI' marketlllakl'I'S, individuals who stand rl'ady to lilly or sell whenl'ver the pllhlic wishes Itl sl'll or hllY. III rl'11I1"II for providing liqllidity, markctlllakt'rs art' grallter! mOllopoly righls hy Ihl' I'xchallgl' 10 post dilTerent prices for purchases alld. sail'S: They IIuy OIl Ihl' hit! pricl' "" alld sl'lI al a higher fll/( price I~/ This ahility to huy low alld sl'lI high is Ihl' Illarkeimaker's primary SOlll'Cl' of rOIll)ll'nsatioll /(11' providing lilluidil)', alld although the hid-ask spread I'" - Ph is rardy larger thall (HIl' or two ticks-the N}'SI'; Fad /look: lCJIJ.I /)ala reports Ihat the slHl'ad was $0.'2:) or Il'ss in 90.Kt;:, of the NYSE hid-ask qllott's frolll I~)\H-o\'er a large nlllllhl'l' or trades Illarketlllakers can earn ellollgh to comp('IISall' Ihl'lII for Iheir sl'I'vicl's, Thl' dilllillluivl' Sill' or typical spreads also helil's thl'ir pOll'lItial illlportallcl' ill dell'J'lllillillg Ihl' lillie-series propl'rties of asset relurns. For l'xaIII pll', Phillips alld SlIIith (I !'KO) show that lIIost of the ah II llI'lII a I r('turns associated with particlliar options lrading slrategies art' dilllin
(pn'~I) .11 .. 41 dIU I1I1H'II1'- 1\1(' Id.llil III h"1\\("I'IIII,hcl f,l}C-IU1.11 .llIolllali('.\ (tilt" \\'('(')"'('111\ huht\.'y dk, \ .... l'le) ."u' . . ". . h·II\.Hi, m"\\'lIu'ub ht'\wt'c.'l\ lht· hill ~\1H' ;\:'\k prin·~.
17 ""jm
dlt'\ \,
.• '. ..."!i~~
J. Q..l Bid-Ask Bounce
To account for the impact of the bid-ask spread on the time-series prop~rties of asset returns, Roll (1984) proposes the following simple model. Denote by the time-/ fundamental value of a security in a frictionless econ~my, and denote by s the bid-ask spread (see Glosten and Milgrom [1985~, for example). Theil the observed market price PI may be written as '
r;
== II
•
P,
5
(3~2.l)
+ 11-2
lID {+I -I
with probability ~ (buyer-initiated) with probability ~ (seller-initiated)
tra~sac.
where II is an order-type indicator variable, indicating whether the tion at time t is at the ask (buyer-initiated) or at the bid (seller-initiated) price. The assumption that is the fundamental value of the security implies that E(Itl == 0, hence Pr(/I=l) == Pr(/I== - 1) ~. Assume for the lIIoment that there are no changes in the fundamentals oftl}e security; hence P; = P' is fixed through time. Then the process for price changes' t. PI is given hy
P;
=
(3.2.3)
• and under the assumption that II is IID the variance, covariance, and autocorrelation of t.PI may be readily computed Var[ t.PI 1 Cov( t.PI -
1 ,
s2 l
t.Ptl
Cov[ t.PI _ k
,
t.Ptl
Corr[ t.PI _ 1
,
t.PI J ==
(3.2.4)
2
(3.2.5)
4 0,
k > 1
2
(3.2.6)
(3.2.7)
Despite the fact that fundamental value P; is fixed, 6.PI exhibits volatility and negative serial correlation as the result of bid-ask bounce. The intuition is clear: If P' is fixed so that prices take on only two values, the bid and the ask, and if the current price is the ask, then the price change between the current price and the previous price must be either 0 or s and the price change between the next price and the current price must be either 0 or -So The sallie argument applies if the current price is the bid, hence the serial correlation between adjacent price changes is non positive. This intuition
102
J. Markd ,HirTU.s/rurturr
applies more generally to cases where the order-type indicator I, is not IID,IH hence the model is considerably Illore general than it may seeJJJ. The larger the spread s, the higher the volatility and the lirsHmll'l' autocovariance, oOlh increasing proportionally so that the first-onkr autocorrelation remains constant at Observe from (3.2.0) that the bid-ask spread docs not induce any higher-order serial correlation. Now let the fundamental value 1',. change through tillie, 11111 slIppose that its increments are serially uncorrclated and independent of 1,.19 Theil (3.2.5) still applies, but the first-order autocorrelation (3.2.7) is no lonp;er - ~ because of the additional variance of 6.P,. in thc denominator. Specifically if a 2 (6.I'.) is the variance of 6.1>,', then
b.
< O.
Although (3.25) shows that a given spread.l implies a first-order alltocovari)nce of _s2 /4, the logic may be reversed so that a givcn autocovariance codlicient and value of /1 imply a particular value for s. Solving for J in (3.2~5) yields
i
s
= '2)-
Cov[M',_I, 6.1',] ,
n·'2·~)
hen(e s may he easily estilllated frolll the sample autocovariances of price cha~ges (see the discussion in Section 3.4.2 regarding the empiric;!1 illlplemcn'lation of (3.2.9) for further details). l:stimating the bid-ask spread lIIay seelll superfluous given the 1;l('t th;lt bid-+k quotes are observable. Ilowever, Roll (1984) argucs that tile (I'loted spre~d may often differ from the 1Jerliue spread, i.e., the spread between the lual market prices of a sell order and a buy order, In many installces, trans clions occur at prices wi/hin the bid-ask spread, perhaps hecause Illarketm kers do nol always update their quotes in a timely fashion, or hecause they ish to rebalance their own inventory and are willing to "beller" their quot s momentarily to achieve this goal, or because they
hOllon' indllces nc~alive ,('rial (onebtioll ill price chall)(e" altholl)(h il do(" .dlnt the nlOlRni\mle. See Choi, Salandro, ;lI1d Shaslri (19HH) for an explicit allalrsi, of Ihi' C''''. 19Roll (19t\4) argues Ihal pricc chanf(c, IIIlI,t h .. serially lIlIcorrelated ill all illfollnatiollally elliden! market. Ilowt"ver, I.eroy (1 117:\), \.\lcas ( 1~17H), ami otirers have shown thai Ihi.' 1I("'d not be the {ase. NeverthelcS""" for ~hnn-hnril.()11 It'turns, f".g-., daily or inlradaily H'1ur1iS. il i~ diflicuh to p,,,e all {'mpirirally pbllsihk "'1"ilihrilun 111",,,"1 of ."'('1 "'\lilli' Ihat ('xhihits ~iKnilic;lI1t serial correiation.
103
3.2. Thl' fJ/d-A\k ,\/Jrl'a{/
accountillg fill' the or asset returns.
effecL~
3.2.2
of the hid-ask sprcad on the tillie-series properties
COIII/m/lt'llil
o!tlu·/Jid-A,/i S/I/md
Although Roll's lIIodel of the bid-ask spread captllrcs OIlC illlportant aspect of iL~ crfCu on transaction prices, it is by no lIIeallS a cOlllplete theol")' or the ecollomic detenninanLS and the dynamics or the spn.'ad, In parlicul;lI; Roll (I ~l81) takes .\ as given, hut ill practice the sit.l' of the spread is the single most important quantity that marketm'lkCls cOlltrol in their strategic interactions with other market participanL~. In bet, (;Iostl'n alld Milgnlln (19R5) argue convincingly that .\ is d('lermi ned clldogl'nously and is unlikely to be independent or P' as we have assumed in Sn:tion :1.~.I. Other theories or the markcll\laking process have decolllposed the spread into more fundamental UJIl1pOllellt5, .lIId thcse componcnts often behave in din'erent ways through lillie and across securities. Estimating thc separatc componcnts of the hid-ask spread is critical for properly implcIlIcnting these theorics with transactiolls data. III this sectioll wc shall turn to somc or the econometric issues surrounding this task. There arc three primary economic sources for the hid-'lsk spread: ordcrpr()ce~~ing costs, inventory costs, and adv('rsc-~elcctioll costs. Thc first two consist of the basic setup and operating COSL~ or trading and rccordkeeping, alld the carrying or undesired inventory subject to risk, Although these CoSL~ have been the main fucus or earlicr Iiterature,~() it is the adverse-selcction compollcnl that has receivcd lIIuch recent atlention.~1 Adverse selection cosb arise be calise somc investors arc beller informcd ahollt a sccurity's valuc [han the markclfllakcr, and trading with such investors will, on avcrage, be a losing proposition ror the lIlarketfllakeL Since IIl<1rketlllakers have no way to distinguish thc inforllled from the uninfi>nlled, they are rorced to engage in these losing trades and must be rewarded accordingly. Therefore, a portion of thc marketlllaker's bid-ask spread Illay bc vicwcd as cOllJpensalion for taking the otller side of potcnti;il inforlllation-based tr;\{!cs. Bccause this information COlllp'1I1enl can have very different stalistical properties from the order-processing ;\Ild invelltor)' conlponclIL~, it is critical to distinguish betwcen them in empirical applications. To do so, Glosll'n (I 9H7) provides a simple as)'Il11llcU'ic-ill ['onllatiol\ Illodel that captures the saliellt fe'lIllres or adverse sdertion fiJI' the COlllj)()nCllts or [he bid-ask spread, ,Illd wc shall present an ablJre\'ia)cd version or his elegallt analysis hl.'IT (scc, also, (;losl('11 and I larris [ I ~lHH I alld Stoll [ I ~IH~1 J). "'Sec, 1( ... ""11111'\'" Alllilll"\ alld M(,II(lchOIl (I'IHO). tl.lgd",l (t~171l. Ikln""1 (l'lhll), I I" '"1(1 Stoll (1~1I1).S',,11 (19711), '"1(1 Tilli\' (197'2). tt See l\a~,.ltot (1971), (;"1',,\;11111 <111<1 (;,ll"i (I (IKI). Fo,,',,"y "nd 0'1 I.n,\ (I ~11I7), (;\"",'1\ (I ~I!\7), Clm),," "IHI I brri, ( I~IIIII), (;I",,,'n ,11111 Mllgmlll \ I ~III', l. .lIld S,oll \ I !III!I).
J.
,l/tII/{'"
AII/'/wlll/l'llI/1'
(;!t1l11'1I \ JIt'/mll/iII,1 1/1" " Ikllott' Ill' /'" alld /'" 11)(" hid allc\ ask prices. respe('(ivdy. allc\ It,t /' Ill' the "11'111''' or 1'/111/ 11/11/1·111/0/,11//11 iOIl lll;lI"kl't price. t he price t hat all invest ors ,\'i t hout private intilllll;lIioll (1IIIill/(m/ll'l/ investors) ap;rel' "pOIl. Under risknl'lItrality. till' (,Olllllloll-int'lrlllatioll price is p;ivclI by J' == E[/"ISl] ",here n c\I'noll's Ihl' ('0111111011 or pllhli(' illtill'lllatioll set alld I"~ tIl'lIot('S tlle price thaI wOllld reslilt if l'VI'I)'Olle had access to all informalioll. TIll' hid ami ask prices llIay thl'1I Ill' e"pressed ;IS the fol\owillp; sums: C~·~.IO)
[' - 1\" - Ct.
[' + ;1" -I-
1)"
I
(:-\.~.!
(:.,
/'" - ['" = (..I"
+ Ad + (COl + C,,),
!)
C).:1.I~)
",line A"+A,, is the ac\verse-selection component or thl' spreatl. til Ill' 11t-le!'lnilll'd !wlo\\', allli ('>1-(;" illtllllil's Ihe onlcr-procl'ssillp; and illvt'lltory COlllPIHlI'lIts whil'h (;toStl'l\ c;lIls the gmu /m1il compOIH'nt ;\IItI takes as 1')(qp;I'ntlUs.~~ !f II IIi II forml'd ill\'l'stors observe a pllrchase at Ihe ask, thell they will revise their valliatioll of the ;Issel from /' to /'+A" to aCCOlIlI1 for thl' possihilitv thaI Ihe tratll' was illlill'lllatioll-I\\olivaled. alld Sillli!;lrI:', if;\ sail' at the hid is ohsl'J'\'('d, tlH'1I /' will hI' revised to /'-/1", BUI how are II" alit! AI. delt'l'IlIillnl; ClostI'll aSSllIlIeS lhat ;111 pOlt'ntia! lIIarketlllakers ha\'(' alTl'SS to (,Olll111(111 iidill'lllalioll 01111', alld he defilles Iheir IIplblinp; rllk ill respolIst' 10 Irallsaniolls at \';lIioIlS possibll' hid alld ask pricl's as
I I I
II(X)
1':[ I"~
Sl U I illveslor hllys ;It
r[ I"~
Q U {
xl ]
inwstor st'lls at )'}
J.
'\11 alld ",. art' Ihl'lI ).\ivl'lI hy lilt' follolVillp; rt'laliolls: il., '"
aU',,) - /',
lllllll'l' sllitahle n'slricli<\lIS li,r Ill,) alld f,(.), allt'ljllilihriulll alllollp; (,OIllJll'tJIIal'k"llIIak!'l,s will d('ll'IlIIillt' hid alld ask pric('s so Ihat the t'''pl'l'ted protils frolll ilia I kt'lllIakill~ aniviti(" will cover all coslS, inchuling (.',,+(.,. alld tI,,+ . h: 111'11('('
ill~
['"
//( [',,) + (:"
I',. ==
['(/'J,) ...
:.''!S~'(· ,\lIIilllui
,II
lit
[' + (,,(/',,)- I') + COl
= /'
+ '\" + (.~,
CI, -- /·_·([· .. f,(I'J,))-Ci. = /'-A"-C,,. !\kllci('I,tlll ( P);-J.O); (:lIh(,lI. T\.LIit" r, Sdlh'.lrt/. ,11111 \\'1111('01111.
aucl Sloll (I!IHI); ,lfld ."iloll 'I~J7H) Itll lIuldl'" 01 these co .. I.\.
(:\.~.
I Ii)
(:\.:1.17) (I ~)H 11; 110
105
J.2. The Bid-Ask Spread
An immediate implication of (3.2.16) and (3.2.17) is that only a portion of the total spread, Cn+Ch, covers the basic costs of marketmaking, so that the quoted spread An+Ab+Ca+Cb can be larger than Stoll's (1985) Mef_ fective" spread-the spread between purchase and sale prices that ~cur strictly within the quoted bid-ask spread-the dilTerence being the adverseselection component A,,+Ab. This accords well with the common practice of marketmakers giving certain customers a beller price than the q\loted bid or ask on certain occasions, presumably because these customers are perceived to be trading for reasons other than private infoonation, e.g., liquidity needs, index-portfolio rebalancing, etc.
Im/liiwlio1l5 Jor Transaction Prices i To derive the impact of these two components on transaction prices, denote hy the price at which the 11th transaction is consummated, and let i
1\
I>'.
I
= Pal" + Pbh.
(3.2.18)
where I" (lb) is an indicator function that takes on the value one if the transaction occurs at the ask (hid) and zero otherwise. Substituting (3.2;16)(3.2.17) into (3.2.IR) then yields
I>,.
fl"
= -
E[I'·Ir2 U All" + E[P"Ir2 U B1h + Cala - Cbh
(3.~.19)
1'.. + C.. Q,.
(3.2.20)
E[p·lr2 U AlIa + E[P"Ir2 U B1h
(3.2.21 )
ell -
{c,.
if buyer-initiated trade
Cb
if seller-initiated trade
Q,.
{ +1 -I
-
(3.2.22)
if buyer-initiated trade
(3.2.23)
if seller-initiated trade
where A is the event in which the transaction occurs at the ask and B is the event in which the transaction occurs at the bid. Observe that PI! is the common information price lifter the nth transaction. Although (3.2.20) is a decomposition that is frequently used in this literature, Glosten's model adds an important new feature: correlation between 1'" and Q,•. If P is the common information price before the nth transaction and I'" is the common information price afterwards, Glosten shows that
Cov[!'".
Q"IPl
= E[AIPl
where
Aa A:; { Ab
if Q,.
== +1
if Q,.=-l.
(3.2.24)
That I'" ,\Ild Q,. mllst be correlated follows from the existence of adverse selectioll. If Q,.= + I, the possibility that the buyer-initiated trade is informalioll-hased will cause an upward revision in P, and for the same reason,
106
3. Market Microstrurlurr
Q,.=-I will cause a downward revision ill P. There is only one case in which Pn and Q,. are uncorrelated: when the adverse-selection componellt of the spread is zero. Implications fOT Transaction Price Dynamics To derive implications for the dynamics of transactions prices, denote hy f" the revisions in 1'.-1 due to the arrival of new public information between tra(y~s n-I and n. Then the nth transactioll price may be wrillell as
1'" == 1',,_1
\
I
rt~.~:)
+ t" + A"Q".
Taki/lg the first difference of (:~.~.~O) then yields
==
I \
1\"<2,,
!
+
('II
+ ((;"Q" -
(;,,_1 Q,,-I),
whiejl shows thattrallsaction price changes are comprised of a gross-profils com mnent which, like Roll's (1984) model of the bid-ask spread, exhihilS reverlsals, and an adverse-selection componelll that tends to be permanent. The;fore, Glosten's allrii>utioll or the effective spread to the gross-profits com onerit is not coincidental, hilt well-JIlotivated by the fact that it is this omponent that induces negative serial correlation in returns, nol the advc se-sclection component. Accordingly, Glosten (1987) provides alternativf relations between spreads and return covariances which incorporate this d,stinction between the adverse-selection and gross-profiL~ compollents. In pa~ticular, under certain simplifying assulJlptions Glostell shows thal~:l
E(i41
= U(I + yfJ),
Cov( lit-I,
Tk
_
1 == -
2
ys"
4
(3.~.2H)
where Pn
-
Ph
y -
C C+A '
fJ -
and where [4, Il, arc the per-period market and true returns, respectively, and 7. is the continuously cOlllpoullded pcr-pcrio(l market return. These relations show that the presellce of adverse selection (y < 1) has an additional impact on \l1eam and covariallces of returns that is Ilot raptured hy other models of the bid-ask spread. Whether or not the adverse-selectioll 2"Specificallr, he ."'llilles Ihac (I) Tnt" n'\lIll" an' ill(\<")I'IH\I'Il\ of all 1''''1 hi.'lor)": (:!) The ~preact i~ synull("tric ahout (Ill' trut' }It"in'~ ,nul C~) The gr()S~prOlil COlUpOIlt'nl dot's nol catl~e conditional drift in pri(C~.
J. J. Mudrlill/i '1l"tlllSartiulls Data
107
("()lllpOllell! i~ ('collomically important is largely all t'mpirical issue that has yet to be detCl"mined decisively/ I neverthelcss (;Iostcil 's (I !IH7) model shows that advcrse seicrlion call have very dilli-n'llt ililplicatiolls lill' the statistical properties of trallsauions data than other COmpOII('IJls of the hid-ask spread.
3,3 Modeling Transactions Datil Olle of Ihe most exLiting recent developments ill clilpirical linance is tlw avaibhility oj" low-cost tm/l.\tlctioll.l datab. 1\19:», and K"im ali(I Madhd'.I!J\!t;). 27S<"e llolli. 'lillOIlS, and 'l,cho{'~1 (I\IW»): ehri.,li<". Il.llri,. ,11111 Sdillitl (I!J\H); Chr;"i,' alld Scltllitl (1!J\11); (;oodhan and Cllrdo (1\1\10); f ian i., (1\1\11); Nied{'lholkr (IV!;:), IV!;(;);
Nied('rholll'1" illld (hhonu.' ( I ~JlHi); and ()~hor rH' ( I ~Hi'l). '!HS Cl' <:oluol1, M;li("r, Schwart/.. and Whilnuuh (1!IH{i), ILIII i.\. Sofiallo" .1IIe1 Slhlpilo (1!'c),O,
11,lSl>rolirk (I \)\)1,1. h). Madhavan alld Sillidl (I \1\)1). ,11111 SlolI .tIId II'h,lh'y (1'1\10).
11/0
i. i\ltII/(1'I Mir/Os/nU"/III"t'
Ih('s(' qlll'SliollS an' IH'W 10 Ihl' n'(,(,111 lileralllre, the killd of allswers WI' call pWl'id,' hal'(' challgl'd dramatically, thallks to transaniolls dala, En'lI the C\,('lll stlldy, \\'hich traditiollally employs daily rcturtls data, has heclI applied recl'lIlly to Irallsat'liolls dala 10 sift Ollt Ihl' impact of nl'ws allllOllllCl'mcllts lI,i/hilllhe dOlI' (s('(', fill' I'sampl(', Barday alldl.itzclI!>ngcr 11!IHH]), The ric!lIl1'ss of Ihl'sl' d:llascls dol'S 1101 COIlH' withollt a pricl'-tr:lIlsat'liollS datascls al'l' cOllsid('raill), 11101'1' diflicull 10 malliplliate alit! all:dv!.(' heCIIIS(' of Iheir ,111'('1' sill'. For cxalllple, ill 1!l!14 the NYSE COllslIllllllat('(1 0\'1'1' ,1!1 IIlillioll lI'alls:lI'liolls, alld li.r 1':1c!1 Irallsaction, Ihe NYSE's Trad('s alld QIIO((,S (TI\Q) d:llab:lsc )TCOllis sl'wr:l! )licn's of illforlllalioll: Ir'llisaclioll price, lillie of Iradl', VO!t III 11', alld variolls COlldilioll codes dl'scriilillg Ihl' Irad('. Bid-ask qlloll'S alld dl'plhs a/'(' also rl'corded, EvclI fi.r illdividual sl't'llrili('s, a S'lIlIplt- si/.(' or 100,000 ohst'rvaliolls (i)r a sillglc ycar of Iralls:lI'liolls dala is 1101 1IIIIIsIIai.
1.
>. I
fIIo/iIlII/illll
Trallsat'liollS dal;t POSI' a 11I1I1I1H'r of IIlIic]!l!' ('COIIOIlIt'lric challl'lIgl's thaI do 1101 ('asil), Iii illio IIII' frolllH'work w(' have d('l'cloped so 1;11'. For 1')(:1111ph', t .. ans;tl'lioll' (latl,\,lIr-alid Ihis pn'sl'lIls a nllml)('r of prohlems for stalld;tnl ('CllIlOltll'lric Iltockls: oilsl'rvations are IllIlikt'ly to 1)(' idl'lIlically disIrihllll'c\ (sillcl' SOItH' obs( 'rva Iiolts arc vcr)' doselyspaced i II Iillll' wh ilc 01 It crs lIIay Ill' sql:uat('(1 hI' ltoUI SOl' davs), il is dil'flcIIlt to caplllH' scasollal !'Ifl'!'ls (sllch as liItH'-ol:;I'I\· rl')!;lIlarili('s) wilh silllple indicator fllildiolts, all.! «'/'('caslill)!; is 110 loltger a ~Irai)!;hll(.r\\'anl exercise Iwealls(' lite Iransaction t;il\l'S an' ranclolll. I\lso, Ir'lIIs:lI'lioll prin's are always qlloled in dis\Tell' units or lidonIITCn"" $0. I:!:. Ii,,· I'qllilil's, $()'<)(;~:' fi,,· eqllil)' oplions, $().W. for fllllll'CS conlracts 011 Ihl' SI.lIId.llIl alldl'oor'.; :.00 index, $O.O:\I~:, fi.r liS Trea,nrv hOllds alld I\OIt'S, al\tI sc> 1))1. \I\'hilt- t\wr" an' no II/Iliori 1\II'oJ'\,ticalIT:lsons 10 I'll'" 0111 COlllilllle,," pricl's, tIll' tralls;tclions coSIS associall'd wilh qllolillg and p\l)n'~sing slIl'h prin's 1Il
.It
I"l( CO,"
1.11 ClIII\\'('igh 1111' IH1!I'IIII,aI
I
t' tli'in (·I('IH"~!'o. Ih('rt, '('C'III' 10 lit' gellt'I.,) .dil(· dial tht· dlie it'll( \' g.lill~ hU1I1 H'h'
I Olllp.tll\ III it
.lgll'{'UWHI ,lIliCllIg c" 1111111111,1, .1I1t1 1'1.t( Iii II IlIc'1.'
n\h 01 illdid~ihlc.' Iradillg lOb.
ii,,,
IIOht·\'C'r. ,Ill 1IIII"{',,,I\('(\
i'~lIt· j, tlu' oIl/m/ft I ""~"'I' til di"'I('I('IW", ,dlie II 1J.ll.tlln·~ IIil' ("osts of ilHll\·t~ihihtit·' ag;tin:-t Iht·
lu'udih lit «ii" 1("11'11(''''''. hll ("",lIlIpk. 1l1I11i(' NYSE. tlu- minimuJIII)!in' 1Il0\'('IIIC"1I1 flf~lorl, ","h Pl\t\" gU',\h'l Ih;\1\ IH t'l1't.I\ 'f.1 i" niH' ,if\.., hUllhi., minimulII jilin' \';nialioll W;\' SC" \'t',lI~ .'go hdoll' lilt, .l«h(,111 III higil-'I"'c'" diglt.II ('ollllllllc'l~ ;lIld Ctli n'~p(lildillg «,I"flll !lIil" Iractillg IIU" h.llli'III'. II j, IlIIt 1",11 ',",WIlIl'1 til IItll ;\11 t'iglllh 01 a doll.ll" i, tilt, oplilll,ll d("~It'c' ul cli~c u'!t'w'" 10.1,1\ Illckcd, II" "III dl" ""IUII' 1."1\\'1'('11 Ih,' NYSF ;IIHI lIu' lIS . . . ,.( III Iii", .uul F\f 1I.1IIgl' (:fllHlIlI"itlll """111 II) lIuli,.IIt'.\ 1110\'" IIJ\\';lIlh dn;mlllr:alul1IlIlIdt'r "'''iell 1'1 in', anc.l '\HOh,'S ,In' (h-l\om'l\,\\\'~' HI ~\"\I'. Sn' 1\.\Il. T",uus, ;uHI "~("hot'g' t l~nFl): nll'llIl.HI .lIU' (:upl'l,lIul ( jl,IHH): 11.1111' t (I)!II L .11111 lilt' "F(:\ ( I!I!I I) l\lmNd .?()(HhlllCh·11I1 1111 tllt'l iii" ""1011.
,I)
Motlt'/irl~
J.J.
Transartioll.l /JII/a
Tabid. I.
'lOY
SU71l71lmy j/lI/i~li(j Jor daily If/urns offive NY.'iE s/{)(/u.
KAB
CHS
CCB
86.750
216500
629.750
7.250
1.375
40.fi25
I 29.()()()
3fiO.25()
2.!l7S
I'
3.353
!i.'i.!l78
173.924
467.H44
4.665
a(/')
O.HII
II.:{HO
I8.H77
53.251
O.Hlfi
21.4:{
fiAH
6.58
4.94
16.13
-14.29
-5.49
-7.H3
-9.43
-12.50 0.00 3.48
Statistic
MC
APD
/JIIl •tX
5.250
J~1II11
1'"", ('Yo) '"111111
Ii
((X»
('Yo)
0.12
0.11
0.02
-0.00
a(l/) (%)
4.88
I.!i I
1.45
1.46
SUllImary sialislics for daily relUrns clala from January 2, 1990, to December 31, 1992. for live NYSE ,Iock>: Me"" Anacom,,; API) = Ah Proc.!uc!.O and Chemic.i.; CBS = Columhia Broa(k"~lillg Sy~tl·m:
CC\\
=
C;lpi\al Citic...o;, Al\C; KAB = Kaneb St"rv1(es.
discreten('ss is less problematic for coarser-sampled data, which may be wellapproximated by a continuous-state process. But it becomes more relevant for transaction price changes, since such finely sampled price changes typically take on only a few distinct values. For example, the NYSE Fact Book: J99.J nata reporL~ that in 1994, 97.4% of all transactions on the NYSE occurred with no change or a one-tick price change. Moreover, price changes greater than 4 licks are extremely rare, as documented in Hausman, \.0, and MacKinlay (1992). j)jlrrfif1lfJs and Prias
I
DislTetcnt'ss affects bOlh prices and returns, but in somewhat different ~ays. With respect to prices, several studies have documented the phenome~on of /nla rlus/nin{;, the tendency for prices to fall more frequently on cerp.in valucs than on others.:\\' For example, Figure 3.2a displays the histograms of the fractional part of the daily closing prices of the following five NYSE slo('ks during the three-year period from January 2, 1990, to December,31. 1!l92 (see Tahle 3.1 for sUllImary statistics); Anacomp (AAC), Air Prod~cts and Chcmicals (AP!)), C,oltunbi,l firoa
(I~I~I\);
Iy)
3.
Mar/1ft Mi(/IJ.I/Illrllllf
Nk
(CCll), and Kaneb Services (KAll). The histogram for CBS is a paltiluljlrly good illustration of the classic price-clustering pallel'll: Prices lend to'fall more frequently 011 whole-dollar multiples than on half-dollar IIlllltijlles, more frequently on half-dollars than on qllaner-dollars, and lIIore fr quently on even eighths than on odd eighths. Price-cilistering is ('veil III Ire pronounced for transactions data. The importancc of these pallerns of discretencss has been highlighted by the recent controversy. and litigation surrounding the puhlicatioll of tW) empirical studies by Christie alld Schultz (1 !l(1) and Christie, IlalTis, al~1 Schultz (1991). They argue that the tendency for bid-ask quotes Oil N SDAQ stocks to cluster 1Il0re frequelltly Oil even eighths thall Oil odd ci hths is an indication of tacit collusion among NASDAQ dealers to maillt
I:!o lou
-
Hn
.;:
In
tit)
'211
:\1)11 . , - - - - - - - - -_ _- ' -_ _ _ _-,
~r-
r-
r - r-
0-
r-,-r-
j
c
~ :!O!)
t
1;,1)-
L.L.
1,,0
\I
.;:
l ~l(1
·1
1~\1
~,-
_.,
-1
II
h
I'..!(J
I
I I
11HI
,--
In
.1
I
:~n
-II
-K
,--
~ III
Ii"
~.~Inn
/I
.~;,II .:\7;1 .!,IIH) .II:!;) .7!'IH ,M7;, l'lll \. fl,\( ~I~II\ I\lr A1\<:
.1',6
r-
r-
rr-
ooo~
·10t,lI ~o
.J:!:•. :!~,II .:\7;1 .;100 J'IUI'I-"I.lClitlllltll
.li~;1
-
n·
.7:,0 .M7!',
t,LL,ll,.ll,.u.,u.,.u.,.Il,J:t,.J •H
A"I)
:'Utl ~:,D
~ ~
-h
- 1 -~ It -I Ii I',il" (.h,lIIgt· tllt'k ... ) Itll ,\I'n
---.---- ..- - -_____
.
'LIIO
~ 1;10-
~
1(10:ICI-
II
.1 'l.r, .:!!',,, .:i7!', . .r,Ot) 1'1111' 1-"1.\(111111
'1Il!)
:\~I{)
._-
.j;~:1
.7!,j) .K/.r,
Q91.-kr41.;u:;ll,lCjk,.y
n ._)0\
-Ii
-1
-'l
n
h
iI'l <:BS
- - - . - - - •• _ _ __
..
:wo~ ~r,()
~ l~11I
~ 'LOO
;. :Wo
[I!'I"
~ '~,Il 100
I(/fl
50
n..lL.,-.l-L,.-LL,-..w:::;=O..L.,-L.c;:::W-,-LL-,-JJ (I
.1:6
.:!~,O
1'11(('
I'.!II llll!
-
-
5-
...t
K\)
.:n;, .:,1111 .Ii:!!', .7;,1) .to!',
() -'Y-r-''r-.-''?-r9-r"Y-''''''''"T'9-,.-''i'-r"T' _H
·1,
-I
-'1.
FI.ltli'Iuiul <:(;1\
I ,--r-,--r-
r-
.'.!~ln .:n~,
.X7!',
,-
till .1Il
:!o .l '.!:,
5UU
.,;'.!~, .7~IH
1'111 (. h.lc·U\)lIlol iV\B
(a)
(ill
Figure 3.2. Ili.IIII/-''1"IIIIIII/LJllily P,lIP I';wlim/J 1I1/f11'riu (.'/11/111'/' /111 h"f NrSf SllKkJ /111111 January 2. 1991110 f)rmlll"'r ) I. 1 'J'J2
).
~
Tcll,/., 1.2.
,ll{///wl,UUW.I/IIIl"/ur"
lidJ/(i""/"'I/'IJ'IIU,'" "/lll'il'( 1'/11,"/-:(.1 finlirk dlllll
"f/i"f .• ,,,,in.
1
Slllck
I
NlIIllht'1
·1
.,1"1'",\11<-,
-:\
--:!
·-1
0
+1
+:!
+:1
?: +4
,\,\( :
IH,W,li
I),I)~
o.ln
1).17
I:!.·J.I
74.:\-1
I~':'H
O.IH
n.!).)
O.IH
API)
~t}.~'o:,
o.:\:! H.'II
:\.~:!
1:IAH
(i'1.·10
I '1.~:1
;1.1·'
0..11
H.:N
( :IIS
~
'2.~
I
7.V.
7.:!
:)~..1~
7.'1:1
7.·1:!
li.:11
~.,t:1
(:( :1\
:!:\,I:!H
1:1.7:!
0.70
I.li!I
:\.!lO
:1:'.11
451i
1.1i!} 05H
I:d~:.
"An
~t,OOH
0.01
(l.(lO
(1.1 Ii
11.77
7:1.7!1
IVJ.I
0.00
fl.07
1,:11:,
O.t "
\
\
\ \
ltt-Luin' Irl'(f1H'Jlf~' COIIIII, ill P(,H"t'JlI, lor ,,11 1~)~)1 tr;Ulsactioll prin' challg('.'i in ricks lor Ih'e
NYSE slOt''': AM :~'\II'\I'''ml': ,\I'l1=Air I'rod."", "lid (:hemjc .. I,: (:IIS~( :.. 11 ,,"hi .. llro:ldc,,'ill~ SI'''''''': 1:( :11" (:"l'il •• 1 I :i'\l" AliI:: K.\I1~ KoII\l'" S.. n·ice,.
I "n' :I.~ "sill~:l1I 011111' slocks' 1""IIS:lCliolls dllrill)!; Ihe I!l!ll call',"I",. Yl'a,.. Th .. low<'I'!" il .. d ,It I( ks-K,\l\ .1\111 AA( :-h,\V(' V.' I)' f.. w l,.allSal'lioll pricl' I'hall)!;l's "<'\'wld IIII' - I lick 10 + I lick rail)!;"; these thr .. e vallil's aC('Olillt fCl' !1!1./i';', a"d !I!I.:II;', IIf :111 thl' tr.ldl's (CII' KAn alld AAC, r"spl'clil'dl'. III (·01I1"I.~t, fill' a hi).(hl'l-I',icl'd sllIl'k likl' CCH, with all aVlTa)!;(' price or $4(iH
+
or
or
o
.rt /
i ./." , ' !ffiTIJ" ,-'--.. I; I ", - ",.. '\
.......
! .: ",
.~ "'~
,;< 1;-'--
l:-----~ 1
:
..
J
...
'.-- :~ t ~:.':"':';
~..-/ '"
; .~.-.-.---~~-----,~.~-~- , .......
_..... : '._--'. ,"
I
",
..
. .
.
2-Hi.tory of AAC Remrn., .
~;.
,",,"
.
....... --
P = $~B5!1
!
.:
... .... -...... i
.
,'.,,:(.F ,
,
;--:.~-:.~~~
..
...... -- :
f> = $55,878
2-History of APD Return.,
.
. ,
,
I.
J.
. ....... - .....
f> = $17S.924
2-Hi.tory ofCSS Returns,
I; ,·.::f5~~s~~;::·> ','.'
...,_\0
,,:~,
;.,"
...... -- :
2-Hi.tory of CCB Return.,
.....
. l"
,/
"I
"
,",__
1 ~~-"'£"-':.A"'~~.-;'-u.-
.......
.'.,....'.
.. '-....
,',
.'
P= $467,844
'
Figure 3.3_ III J)frnllbrr
,
'
--~"""'...:,;;-r-""""~"If
2-Hhtory of KAB Return., (a)
..
",'
(b)
........ -- :
f> = $4.665 (c)
2-J/is/or1es of Daily Siock llelurns for Fiv, NYSE Stocks from january 2, 1990 3/, 1992
ture than the lower.priced stocks but more than the higher-priced stocks. Figures S.;!h and S.Sc show that changing the scale of the plots can often reduct, .mel, in the case of APD, completely ohscure the regularities associated wilh discreteness. For further discussion of these 2-histories, see Crack ancll.ecloit (1996).
114
3. Markrl Miov,llrul'lwl'
These empirical observations have motivated several explicit models or rice discreteness, and we shall discnss the strengths ;md weaknesses of each of these models in the followinl{ sections. 3.3.2 Uoundillg and /Jamer Models
ever;il models of price discreteness begin with a "truc" uut unobserved cplltinuous-slatc price process PI' and obtain the observed price process 1';' by discretizing 1'1 ill some fashion (sec, for example, Ball [1988), Cho and Frees [1988J, and Gottlieb and Kalay [1985 J). Allhough this may be a convenient starling point, the usc ofthe term "true" price for the continuous-stale price process in this literalllre is an unfortunate choice of terminology-it implies that the discrete observed price is an approximation to the tme price whell, in fact, the reverse is true: continllous-state models are approximations to actual market prices which arc discrete, When the approximation errors inherent ill (,OlltillllOlIS-SI
LxJ
greatest integer < x
rxl
least integer
~
x
(!Ioor functioll) (ceiling function),
(:L1.1) (:-l,:U!)
for any real number x. J ) Using (:l.3.1) ,lIId (3.:'\,2), we call express lire three !lIost common methods of discretizillg 1'1 compactly as .
,nThc question of which prirc is Ihe "In\('" p. in' III"Y lIolh .. (n.ci •• 1for Ihe ~lalisllr,,1 ~"I)('(ts (If 't()(II,t~ of lli~rcl(,Il('.s-"hcr all, whelher 0111' is .u. "pproxim,'lion 10 lhe olht·) III vi. ('-\'(" ',I 31th- t~ uilly the sign of the approximation ('lTor, nut its f\hsolutt!' magnitude-hut it i~ c(,Htral to t 1e motivation and illtcrprcto.ttiu1\ uf thr- Te:mlt'\ (Sl'l' the discllssion dt the l'lId 01 Sl'niott :I.:t¥ for examples), Therefore. although we ~hall atlopl .111' lerlllinology of this lill'r",,,," luI' lhc ,noment, the reader is a.c:.ked 10 kt·(·p thi." amhiguity ill tHind whlle rcadi .. ~ \hi!'i :,\('( 1iol1. ~'\Fur rurther propf'rties and app1ications of tlu.-sc intl:gl:f functioJls, see Graham, Knuth, 3nd'l'alashhik (1989. Chapter 3),
I i
3.3. ModrlinK '/ImllafliollJ iJatll
(3.3.4 )
p", where thc first method rounds dlllol/. the se(,(>IId n>llnds IIfl. alld the third rounds to the lIearest multiple of d. For simplicity, we shall consider only (3.3.3), although our analysis easily cxtcnds to the othcr two methods. Atthc hcart of the discreteness issuc is thc difli:rence betwecn the retllrn X/ based on continuouS-Slate prices and the return XI" based on discretile<1 prices. To develop a sense of just how different these two returns can he, we shall construct an upper bound for the Ijuantity IXI" - Xtl = Ill; - 141. where H/ and R;' denote the simple net return of the continuous-state and discretil.ed pritT processes, respectively. I.et x and y he allY two ,lrhitrary nonlH~gative real nUlnbers such that .v > I, alld observc that x- I
--- <
y
lxJ lyJ
x
-- <
(3.3.6)
---
y-
Subtracting x/y from (3.3.6) then yields
lxJ ly J
x Y
--<----<
Y
x
y(v --
1)
•
CD.7)
which implies the inequality <
-I Max
Y
[x - - - , I ]. y-I
(3.3.H)
Assuming that 1'/ > d for all t, we lIlay sct x '" 1',1 tI,.r == 1',_11 d and substitute these expressions into (3.3.H) to ohtain the following upper bound: (3.3.!1) where 0,_1 == til P,_I is defined to be the K'rid .Iizr at tillle t-I. Although the upper bound Ct3.!J) is a strict ineCJuality, it is in fact thc least upper /JO II nd, i.e., for any fixcd d and any f > O. there always exists sOllie combination of 1',-1 and X/ for which III;' - Uti exceeds 1.(0. X/o I',-tl - L Therefore, (:t3.!l) measures the worst-rase deviation of II;' from N,. and it is the tightest of all such measures. Note tilal (:t:t!l) docs not yield a unif(lI"IlI upper hound in r/, since I. depends on 'i: CU. 10)
J, /lilli/Wi MiI7'IJ,I'Il'Ilrllll'l' Nevel'llwlt,ss, it still provides
Oil
(If /', " II is illlporl~11I1 10 "('('P ill nlind thaI (3.:t!l) is ollly an IIpp!'r hO\llld, ,\111\ while il dol'S I" oviel(' a II lI'aSIlrl' of Iht' Ill(Jn/-flI\e disl'l't'palll'y h('I\\'I'(,11 Il, alld U;', il is 1101 a IIwasur!' of lite discf('PI'I'l<'d ltpP('" boltlld S,"'IllS to Ilt' relatively illsl'lIs;I;\,(' 10 cIiallg(" ;11 Ill<' lIIeal\ ;1Ilt! vari'lIln' or x,. so lIial whl'lI measllred as a IH'rn'l\lage or till' ('XIIl'CIl'd 1'1'1111'111-:1 X,I. Ihl' n:pl'I'lt'd IIp(lt·\' hOllnd dol'S declinl' for IOllge\,horizon retllrns. I\y sp!'firyillg a pallicl1iar proCI'SS fi,r /'" \\Ie call ('valllate the expl'q~llillll of /.(.) 10 dl'velop sonll' Sl'nSl' lill' the lIIagnitlldes of !'xpeflec\ diSlTl'l!'IICSS hias E[ln;' - Udllhal an' possihk, For example, kt /', filllo\\l a gl'olw'lric randol\l walk wilh drift /1 alld diffllsion fO!'Ili!'i!'''1 a so lhat log 1',/ /'(-1 arc III> nonnal randoll\ variahles wilh IIl('all /1 ant! variance n~. III litis fas!', W!' hal'!'
1':[ /.( rI, .\"
I',
I)
I /'(
_8_ {
I ]
(JI I
'4 (I> (
,)
log( 1-8) - /1 n
) 1,
CD.II)
wit!'r!' cfJ(·) is Ihl' 1I01illai CDF,'I"
lid ,Ii ....... ')'. hll\ il is '"1'1" isill~lv dimr"l, ,.. tin ','c'lIl(' cli,,'I1~!'\ioll heluw n.·~."
:111''''011''', w,· \..ollitt lik .. '0,11.11 ... I'" ill' III;' -, ~ow11h .\H~' ''''gu'''
01
gC'lIt'1 .ili1)',
110\\'1'\'1'1,
',H'C IIi,' p.1I ;1111('11 it ;t .... 'IIIIIIHion..; for XIt mun' ptt·t+."" fh;u;tncri/.lIioll!'l III;', an' :lv,lil.,hl,', ·'t:'NolI' Ih" ,illlil.1I j". IWIn-tTI! rs.:~.II) ;11111 11i(' Hlack-Srholt·s ('all-uplioll pfidll~ 101"l1l1l1a.
h.1I1 in lIIo,ld'-IIIUI('I 01 fht· tlisnc'h'IH'!'\'
117
3. J. Moddillg TransurtionJ Data
Tahles 3.3a-c report numerical values of (3.3.9) for price levels Pr-I = $1, $5, $10, $50, $100, and $200, and for values of IJ. and (J corresponding to anllual means and standard deviations for simple returns fanging from 10% to 50% each, respectively, and then rescaled to represent daily returns in Table 3.3a, monthly returns in Table 3.3b, and annual returns in Table
:-\':-k. Table 3.3a shows that for stocks priced at $1, the expected upper bound for the discreteness bias is approximately 14 percentage points, a substantial hias indeed. However, this expected upper bound declines to approximately O.2:l percentage points for a $50 stock and is a negligible 0.06 percentage points for a $200 stock. These upper bounds provide the rationale for the empirical examples of Figures 3.3a-c and the common intuition tliat discreteness has less of an impact on higher-priced stocks. Table 3.3a also shows that for daily returns, changes in the mean and standard deviation of returns have relatively little impact on the magnitudes of the upper bounds. Tahles 3.3b and 3.3c indicate that the potential magnitudes of discreteness bias are relatively stable, increasing only slighlly as the return-horizon increases. Whereas the expected upper bound is about 2.5 percentage points for daily returns whcn PI_ 1 = $5, it ranges from 2.8% to 3.9% for annual rctllrns. This implics that as a fraction of the typical holding period relurn, discreteness bias is much less important as the return horizop incrcases. Not surprisingly, changes in the mean and standard deviation of returns havc.more impact with an annual rcturn-horizon. ROllI/ding Models Evcn if E[jR," - Rtl] is small, the statistical properties of P," can still differ in subtlc but important ways from tbose of PI' If discreteness is an unavoid~ble aspect of the data at hand, it ITIay be necessary to consider a more explicit statistical model of the discrete price process. As we suggested above, a rounding model can allow liS to infer the parameters of the continuous-State process from observations of the rounded process. In particular, in much of II\(' roundinv; literature it is assuITIed that PI follows a geometric BroWJ;lian Illotion ell' = J.l.Pdt + a PdW, and the goal is to estimate J.I. and a fiom thc obscrved price process P,". Clearly, the standard volatility estimator iJ based on con tinuously compounded observed returns will be an inconsisten t
(·stimator of a, converging in probability to /E[(Iog ~+I
-
log P,')2] rather
a
than to jE((Iog PI +1 -log PI )2] • Moreover, it can be shown that will be an oVfTestimate of a in thc prcscnce of pricc-discreteness (see Ball [1988, Tablc I] and Gottlieb and Kalay [1985, Table I] for approximate magnitudes of this upward bias). nail (1988), eho and Frees (1988), Gottlieb and Kalay Thi, is 110 acridc"l. sillce Max (X,. I-Ii) ",ay be rewrillen as Max( XI - (J -Ii), 01 + J -Ii; hence Ih,' "pP"r houlld Illay h.....•.. 'L'I as Ih,·payolf of a call oJllion on XI wilh slrike price J.
Table J.Ja.
1'1-1
=:
.1=20%
.. = ~O%
,= 40%
14.2895 14.2930 14.2961 14.2991 14.3018
14.2H95 14.2930 14.2961 14.2991 14.301H
14.2H95 14.2930 14.29fi1 14.2991 14.301H
14.2H9:1 14.2930 14.29(i\ 14.2991
2.5648 2.56:14 2.5660 2.5665 2.5670
2.5650 2.565:, 2.5660 2.5t;()5 2.:,fi70
2.5li76 2.5672 2.5671 25(i72 2.5fi74
2.5721 2.570\1 2.5701 2.:)t)9!) 2.:>692
25772
0.2511 0.2511 0.2511 0.2511 0.2511
0.2:,16 O.251!', O.2!l15 0.2:,15 0.2:/11
0.2520 0.2520 0.2519 O.251H 0.2:>lH
O.252!l 0.2524 0.2523 0.2522 O.2!'121
O.2:12!) 0.252H 0.2:127 O.2!'12ti O.2!'12!'1
0.1254 0.1254 0.1254 0.1254 0.1254
0.1256 0.1256 0.1256 0.1256 O.12:,(i
0.1259 O.12:lll O.12511 O.125H (J.1257
0.1261 O.12tiO 0.12fiO O.12(i0 0.1259
O. 12{j~ O.12ti2 0.1262 0.12(;1 O.12til
O.Ofi27 0.0627 0.0627 O.Ofi27 0.0627
O.O(j2H 0.Ofi2H (l.Ofi2H O.()(i2H O.1l62H
0.0629 0.0629 (1.0629 O.t){WI O.O(j211
O.O(i30 0.0630 0.Ofi30 0.01;29 O.OG29
O.0/i31 O.()(i:\\ 0.O(j3\
$1
10% 20% 30% 40% 50%
1',-1
=:
= r,O'Yt,
10%
.<=:
In
F.xl'frifd 1I/11'fT bOlilUt. fllr tii.mrlf//(ss him: daily l'flllll"
.1
._"_.. -._--
14.~018
14.2H9~,
14.2!)30 J.).2!Hll 14.2!)!1I l'UOIH
$5
10% 20% 30% 40% 50%
2.:,7!)!"1
25711 2.57:\0 2.5721
PI-I = $50 10%' 20% 30% 40% 50%
1"_1 = $100 10% 20% 30% 40% 50%
I~_I = $200
1% 0
I!O%
~O%
~O%
~lO% 1 I
O.O(i:\O
O.Oti:\O
l-:xpeclrd upper bound~ for tlisrf(·(l'nc.·~s hi;L' ill siu\I,h.' n.·turn~ Ill;' - ntl x 100 lindt,,, a gc.·\\11lt'tric. r;1I~ltlm w-dlk fnr prices 1', wilh drill <1",1 lIill'lI,;olll''Ir'lIlIl'I'''-' I' anll" c"hh.-al(·d 10 .1111.",,1111<'''11 all(\ standard devialioll of simp I!' relmn~ In ,11111.1. "·'I"·l'li\'(·ly. ""ch r""Hill!: frolll IH% 10,.0'7<,. "11(\ lhen rescaled to malch daily 1I,lla, i .••.. I'/:\r~). 17/J:liiii. LJiscrcli/.('.t prin" I';' E lI',ldJd. d ~ f). I:!!'" ... c IIscd III fakll!;"!' n'III' ' " U;' : (/';'11';'_ I) - I.
\
1
1,.,,11' J. Jb. III
!',. I
.1
I,'x/lfflnl 1I/'I1t',./1II1I1II1.,
= IWX,
.\ =
~Wy"
1=
== $1
:~O%
·10% !iO%
:\()'J"
14AOI,7 14.:.0M 14.1iOl9
1·1.'17HH
.1 ~.,
14.(i!I~O
14.7011 14.7HO·j
- .. __ .
14.li~I!)
I',. I ':"_$_:,____
....
~
·III'Y.,
.1
__ ._ ..
..
.------ - -
=!,O%
_-_._14.7(j~I'
I·Lld 17 I·LIi·I·!') I·LW07 1·!.7·!m 1·1.HOHI
J.I.:,.J('~
14.7767
h;m.' /I/lII,lhly,I'II1I71.I.
_." .. _-_.... -----
-_.._----I,U9\Hi 1·1.!i011 14.I'OI!i 1·I.I'!!1 !I 14.771i7
10% 20%
1m ";11.,,-11'//1'.11
14.77~:~
14.7!)44 I4.H~n
JoI.HIiHH ..
_ - ..- - -
2.li:,O I 2.lifi1:,
~.I'7!i!)
~.7004
:1.1i7H~
:1.71110
2.(1:)~J~J
~.liHII'
~.70~7
~.lil'I'1
~.IiH:.')
~.7O:.3
Vi!iH!!
2.(,73H
~.mll
~.70HH
O.~:.44
{).~:.ti!l
(1.~:.!)01
II.~I'I!)
().~(i4~
~O%
0.:6:11
0.~:.71;
O.~I'~I
O.~(;·13
30% 10%
O.~:;(j(i
O.~!iH1
O.:I(j~·1
(1.~(j1:.
O.~:.HO
O.~!i!I:\
0.:l1;~'1
0.~Ci17
:.O'Yc,
O.~:.93
O.~IiO~
o.:!,.!)!) II.:I(;(H 0.2(,10 O.21i17
O.:lli:\1
O.~Ii'"
10%
O.I~70
0.I~H3
~O%
0.1~7Ii
O.I~Hli
O. I 2% 0.12!IH
30% 10% !i0%
O.I~H:I
O.I~!lO
O.I:~OO
O.I~HH
O.I~!I!i
0.1 :m!i
O.I:~OO
O.I:IIH 0.I:m7
0.llI,11 0.1)(,4:1 0.01,1:. 0.01,17 0.I)(i1!1
0.111,,17 O.llIioIH 0.111;:.0 0.111;:,1 1I.III;r,:1
10%
~':'!I4!i
Vi~~H
~O%
:1.li07!i
:~()%
2.(i~~~
Vi300 Vi3H:.
·10'7<, !iO%
2.(,:171
Vi4H~
2.lifi~3
!{)'X,
P"I
--_._------_.._-
= $:.0
1',.1 = $100
._---
1".1 = $~~12. ___ ~O%
:10% 40% !i0%
0.I:H9 O.I:I~O 0.13~1
II.I:~U
0.13~~
0.1 :{I:.
0.1:{2·!
-------- _._--- -------
O.OW:. 0.01,:17 0.llIi10 0.01,14 0.111;,17
10%
._--------
11.1 :~IIH 11.1 :\0<) 11.1 :111
0.01;:.:1 O.llIi:I'I 11.111;:.:, II. Oi;r,(j 0.01;:,7
0,01,:;9 O.llIi"\) O.OI,IiO 1).0til,1 1).0til;1
Expected uppel" hound., fOI'c1i.~crt·tl"lIt·s .., hia.'\ ill o.;illlj)lc- n·(IIII1 ... tU;' ... /(11 )( 100 Hlult'r il gl'olHl'lI if randolil walk for pricco'\ 1'( with drift and dillllSilIU p.1I ;11111'1('1.\ II ;11,,1 (1 (,llilu.II«,d to 0I1I1H1.111l1t·;1II alld !Iotandard d('viatioll of~illlpk r('turll' m alld \, J(· ... ,H·(·li\(·k. (',If Ii r;lIlgIlig Irolll 1orf.., 10 ;)(J'}f" and tliell
1"<: .... (.II('cI (0
d = O,I~:)"lI(·
tI .... l'd
JIIatch mOl/lh~)·d.If;t.
i.t,.,
Il/I~. n/JI"":!. 1>1'.( I (·II,,·d IHi(c . . I};' == If'(/dJd.
loralrulalc r('tlInl\ U;':, (/';'/I',"' 1) - I
'Ii/MI' J. k I=-
11/
1~.\/.,.tll'd 11/1/11'1 Ilfl/lllt/Ijill t1iv/,rlr//I·.1I
III'Y,. I'" :.'1l'Y" .------.
1. == $1
}',
!!Of}{,
:10% .IO 'Y" :1(1(.~)
}".I .::'. $:.
lillY" 20% :\0% ·10%
0"
2.'11 W. :U07li :1.:1·\07 :15!lO!I :1.H·lti:1
10',1"
11.:.'77:.
~Uf,v.)
O.:IOO~I
'IH'X,
1I.:12'IH H.:V,Ol) 1l.:I7',"
1I.:.'X·lIi 0.:10:1'1 n.:12Iili o.:l:d H IUllin
r)()(,~.
I
r)O(){,
I~'_I_==- S~ UU I (}(){. tI.I:IHli !!t)(}{J
:\0%
0.1'.0:.' H.lfi:.'7
!)f){Yt,
1I.lx77
.«.lYc,
(1.1,12 I 0.1:,) 7 1I.lli:11 0.17'.:1 H.IH77
O.17'.~
/' 1 = $21111
.'
111%
:\0'1" ·111'.'(.
:,0';:,
I
\
17.1).):1!)
17.!i:~:!0
17.!I~HH
IH.7':tn :!1J.()4Ii.J 21.·!:i!lIi
IH.!IHH!I 211.1!157
2.!)!)!'.H :1.1677 '1.:{7:ili :i,li04li :1.H:lOli
::l.OHI:. :i.2:lW. :1.424H
1I.2!)~!)
1I.:IO!I7 n.:i2!IH
0.:1111:1 O.:lltili O.:I:I·IH
O.:I:.~·I
II.:F.:.:.
11.:I71i·1
1l.:l7HIl
~150HII
--
.. ---
li.:.:!·17 IH.:H'iH I!U:!o:1 :!().42!1!I 21./;:.·11
::l.li:~Ii·1
:Ulti7:1
:1. I li·1-! :1.:lIIH :1.·II\'Hi :I.tiH07 :1.H
- - - - -------O.I1Ii:l 0.1:.·17 H.lli"7 11.1760 II.IHHO
0.:10"·1 0.:12:1:-\ O.:H07 IU:ml IUHO!)
- - - - - - - - ---H.I !l'l"l O.I:.or, II.I:,HI O.lIi I 7 11.1701 0.1 Ii?:! 0.177:. 0.17!17 o.I!)()2 o.IHHH
_.- --
1I.07 r ,x II.HK I:, II.HK71i O.II!):IK
0.07:11 0.077:1 O.OH2:1 H.OK7!) O.O!I:I!)
0.07'.2 0.07!)0 O.OH:lli 0.IIHH7 O.O!)·I:\
0.11772 O.OHOH O.OW,II 1I.0K!IH O.O!I:, I
I'
htllllul," eli" 1('!c'IU',:\ hi.I' 11I.,illll'''· n.-turliS IU;'- Uti x 100 lIIult· .... gt'Ollle'lriC" h'I pile \, .. /', w'lh ,1I ,h ~'1H1 ,liU" .. ",,, par;Ulwtt'I"SII ;\lul n ('~,lihrah·tI10a1\I\\I;t1 \\It'~'1\ and., •. lIu1.ln) ct"\i.llic III 01 ,11111'1" Ic'IUl"l1' lII.lIul " n'~p("'fliv('Jy. t'ad. rangillg' from 10'::, 10 ~IO%, Hi" n'ti,,'" 1'1 Ie C" I';' ,::. t /',1 did. d .-.' 0.1 :1:-.. ;tf{' ",c'd (0 (aiculalc n'Uull' U;' =: (/J;'/ I';'. 1.
Fxl'(" Ic'd
\
Ili54:.'4
:)()I~,
-------.--- .".-------"----lI.mlll
H.llii1 II.IIHI:I H.IIH7Ii lI.o'nK
~t)(X,
\
\ ==
= $:.11
'I(II:{,
!
== .1f)'Yc,
.1
_____ - - - - -
2.H:17:! :1.077H :1.:1:1:1:1 :15H!I7 :I.H-Iti!
/'/
== :10%
/11/1111111 Iflllrl/.I.
-------~-.-------.--.-.-.
Ili.IH!)H 17.:!:."'7 IH.:lH:17 :.'0.001·1 :.' 1.·I:.'Hli
1:•. 7:.'W. 17.1·1:111 IK':'71·1 :.'0.0000 21.I:.'Hti
10%
'rI
.1
hill.l:
"PP'"
I~UH'''''' \,·.,l~
I' -
121
3.3. Modfiin~ TranJ(l(/iu7Is /)a/a
( I !lR5), and Harris (1990) all provide methods for estimating 0 consistently from the observed price process P," .:11;
Ill/nlY)" Mot/fis
A slightly diITerent but closely related set of models of price discreteness has h('('n proposed by Cho and Frees (I9HR) and Marsh and Rosenfeld (l9RIi) which we shall call barrier models. In these models, the continuous-state "true" price process 1', is also a continuous-time process, and trades are observed whenever P, reaches cerlain levels or barriers. Marsh and Rosenfeld (I9Rfi) place these barriers at multiples of an eighth, so that conditional OIl the most recent trade at, say 40~, the waiti.ng time until the next trade is the first-passage time of P, to two barriers, one al 40~ and the other at 40~ (assuming that P, has positive dfin). Cho and Frees (19R8) focus on gross returns instead of pric.esand define stopping times Til as
f/. (_1 ,I+d ) } . I+d
(3.3.12)
Therefore, according to their model a stock which has just traded at time Til_I at $10.000 a share will tracle next at time Tn when the unohserved cOlltinuolls-state gross returns process Pt/$IO.OOO reaches either 1.125 or 1/1.125, or when P, reaches either $10.125 or $8.888. If P, reaches $8.88R, the stock will trade next when P, reaches either $10.000 or $7.90 I, and so OIL
This process captures price-rliscreteness of a very diITerent nature since the price increments defined by the stopping times arc not integer multiples of any fixed quantity (for example, the lower barrier 1/1.125 does not correspond to a one-eighth price decline). However, such an unnatural definitiol\ of discreteness docs greatly simplify the characterization of SlOpping limes and the estimation of the parameters of P" since the first-rliITerence T" is lID. i I Under thc more natural specificltioll of price discreteness, not COl\sidcred hy Cho and Frees (19HH). the stopping time hecomes
or
{
P
T,; = inf I > T,,_I: - - ' ,
I'(T,,_I)
rt
(1-d d- )} - - - ,I + - 1'(1,,-1) P(T.-Il
(3.3.13)
which reduces to lhe Marsh and Rosenfeld (19Rfi) model in which the incrcments of stopping times are 110t lID. i
,
:\11110\'0'('\'(''', ~t"f: lh.: diSCII.It'\ioll ,at tlw I:lul of Section :t:t2 for ~om(" C3\'eab about (he mod\\uiUIl for tht's(" mudds.
,,. ...
]22
3. Market Microstructure
Limitations Although all of thc prcvious rounding and barricr lIlodcls do capturc pricc discrctcncss and admit cOllsistent cstimators of thc paramctcrs of the IInol)scrvcd continuous-statc price proccss, they suffer from at least three illlportantlimitations. First, ror unobscrved price processes other than geomctric Browniall motion, these models and their correspollding parameter estimators becOllie intractable. Second, the rounding and barrier models focus exclusively 011 prices alld allow no role for other economic variables thatlllight influcnce price behavior, e.g., bid-ask sprcads, volatility, trading volumc, etc. Third, and most importantly, thc distinction between the "true" and obscrved price is artificial at best. and the econoJllic interpretation of the two quantitics is unclear. For example, Ball (lUSH), Cho and Frees (I!IHH), Gottlicb and Kalay (1985), alld lIarris (I U90) all provide methods rO/" estimating the volatility of a continuOUS-lime prke process frpm discrele' 01>selved prices, never questioning the motivation of this arduous task. If lhe continuous-time price process is an approximation to actual market prices, why is the volatility of the approximating process of interest? One lIIight arguc that derivativc pricing models such as the mack-Scholes/Merton formulas depend on thc parameters of stich continuous-time processes, hut thost'i models arc also approximations to market prices, prices which exhibit ~iscreteness as well. Thcrefore, a case must he made for lhe ecollomic ~ rc\cv,!nec of the parameters of continuoUs-slate price processes to properly ~' motiVate the statistical models or discreteness in Section 3.3.2. ,;~ h~ the absencc of a wcll-articulated model of "truc~ pricc. it secms U\Inatur~lto argue that thc "truc" pricc is continuous, implying that ohserved discre~e market priccs are somchow less genuine. After all, the economic dcfini~ion orprice is that quantity oflllllllerairc at which two mutually COliscntif~economiC agents are willing to consummate a tradc. Despite thc f;ICt that if stitutional restrictions llIay rcquirc prices to fall on discrete values, as lon' as both buyers and sellers are aware of this discreteness ill advance and af still willing to engagc in trade, thcn discrete prices correspondi\lg to ma tct trades arc "true" prices ill every sense.
I
J.J.J The OI1/rrl'd l'mbit Modrl
To ad1rcss the limitations of the rOllllding and barrier lIlodels. Hausmall, Lo, anf'! MacKinlay (1992) propose ,11\ altern'ltive ill which price rlulIIgr.\ arc 1lI0dclkd directly using a statisticallllodel known as ordered !)Tobit. a technique used niost frequently in empirical studies ofdepe\ldent variables that take on only a finite mnllbcr of values possessing a \IaluL,1 ()nlering.~7 Heuristically, '7For cxal1\pl~. Ih~ dqwml"111 \'ariahlt- lIlif(hl h,' Ih .. kwl Ill' ,-,Iucllion. as 111<"""1<'" h\' ,Int·(· ('t\l(·goric.·~: I(·~.'\ Ihan hif.!,h !'(huot, hi~h ~fhool. aTIC) ("(llIq~t' (,duration, Tht· ,It'IU'lUlt'Ut
3.3. Modeling '/hlllJ(l(liiJlls Dala
12~
ordered prohil analysis is a gellerali /,alion or lile linc;lr regressi on lIludei 10 cases where lhe dependc lll variable is discrelc . As such, ,11110111{ lhc exisling models or slock pricc dis('lTI( 'I\('SS-(' .g., lIall ( I !)HH), (:ho and Frces ( 198H), COlllieb and Kalay (198:)), I Iarris ( I !)!)()), ;lIId Marsh and Rosenld d (19H(i) -orderc d prohil is Ihe only specifica lioll lh;11 elll easily Capll\l"e Ihe illlP;I('( of "explan alory" variahle s on pri(T eh;III),((" whik abo accounl ing f'II' pritt' dis(I'cl(: ness alld irregula r lransacl ion inll'n'als , niP
/Jrl.\ir S/)f'('ijiwlion
Specific ally, conside r a scquenc e of Iransat'l ion pi ices 1'(1,,), 1'(11), ... ,1'(1 11 ) sampled al limcs 10, II, .,., III' and dellOI(' hy VI, }'~, ... , I'll Ill!' corresp onding price changes , where Y. == 1'(1.) - 1'(lk .. l) is ;\S,ullled 10 be an inlcgn llIulliple or SOllie divisor, e.g., a lick. I ,1'1 r ' d('1I01l' ;11\ ullOhsl' lvabk conk linnous random variable sllch Ihal
r;
= X~rj
+
0,
CI.:U4)
",h"n' lhe ('I x I) vector X. == [ X.. ,'\'" J' is a veClor of explana lory variable s lhal delermi nes Ihc condilio nal mcan of and "INID" indicate s thai thc E. 's arc indcpcn dclllly hUI nol identica lly dislrihu led, an imJ>ortall! diffcren ce from standard econom clric models whit'h we shall relurn 10 shortly. NOle Ihat subscrip ls arc used to dcnote Imll,\(/rli '!Il lillle, whereas time argullle nts I. dcnotc calenda r or r1urh tilile. a COnV('lllion we shall follow through ollt Sectioll :1.33. The hearl of Ihe ordered prohil lllodel is the assllillpl ioll that ohserve d pricc changes Y. arc related to the cOlllillUOUS v;uiabks ill Ihe 1()lIowing llIanner : if )" E ;11 .II
1';
1';
,\~
y.
J",
if
• •E
)"
/I~ (~.:\.15)
if }"k c ;\",>
wherc the sets AI furm a /Hlrliliull of the statc spale S' of 1'; . i.e., S' = U;'~ I Aj and Ai n Aj = 11 for i 'I j, and the ~/s arc the diMTl'le valul's that compris e the state space S of J'., The motivati on for the ordered probil specific alioll is to ullcover thc mappin g lJetween S' and Sand relale il 10 a sel of ('Con 0 lIlic variable s. In Hausma n, I.o, and MacKill lay (I~1~12), Ihl' ,\,\ arc defilled as: 0,
-k, +k.
variable is uisncw awl is 1I<11ur
"II"
-~. +~. and so OIl. For sinl(llicilY, Ihl' slale-span.' partilion ofS' is usually ddil)('d to 1)(' illft'l va).;:
:1 I
-
(-00 ,
(:L\.Ifi)
;\~
-
{!XI '
atl U~ I
(:13.17)
A,
-
(U,_I , 0', )
CU.IK)
1\ '"
'r!
!
,
I
(0' 111-'
,
(0).
(:t:~. I ~))
Althou~h the ohser\'('d price chall~1' call be allY nUlllber of ticks, posilive or n<'l!;alivl'. WI' aSSllllle Ihal //I in C~,:~,I:)) is finile 10 keep Ihl' Illlmller of unknClwn parallH'I('I'S Iillil(', This (loses 110 diffICulties sinct.' we Illay always leI SOIl\(' slall's ill S r"l)J'('s('1l1 a lIIultiple (anc! possihly counlably illlillile) nlllnlll'r ofvallll's for lIlt' o!ls!'fvl'd prin' change, For example, in Ihe empirical applicalion of lIauslllan, Lo, ;\Ild MacKinby (19~12), .II is deflnec! to be a price dlangl' of -,I licks orlt'.I.I, ."1 to hI' ;1 price change of +4 ticks or /II0rf, anc! .I'.! 10 .IX 10 he pricl' changes or -:~ ticks to +:~ ticb. respectively. This parsimollY is ohlaillt'd al Ill<' ('osloI' losillg t,ria 11'.\Olul;OIl. ThaI is, lIlldcr this spl'('ifil'atioll JIll' onll'n't\ prohil model docs not distinguish hetwel'1l priCI' changes or +,1 alld price changes grealer thatl +4, sinCl' the +4-tick oUlcOllle anc! Ihe gl'ealer Ihan +4-li('k oUlcome have Ileen groupec! logether inlo a cOlllmon eVt'nl. The same is IrIIC for price changes of -4 licks ;lI1d price Changes less Ihan -4, This partilioning is illustrated in Figure ~,4 whirh superimposes thl' parlition boundaries lail on the c!ensity funClipn of and till' sill'S of the regions enclosed hy the partitions detl'rmine the prohahililies 71, oflhe discrele ('\'l'nts. Moreover, ill prillcipll' Ihe resollliion lIlay he lIlade arbilrarily flllCJ' hy simply introC\willJ.( ilIOn' Siall's, i.e" by illcreasing' 111, A~ IOllg as (~3.14) is cOITl'nly spt'dlil'll, illtTl'asillJ.( prin' resoilltion wiJlllot allen the ('stilllal(~(1 {-J asymptolicallv (allllou).!;h lillill'-s;\llIpk pmperti\~s may lIilfl'r). Ilo\\'e\'<'r, ill practicl' 1111' clal;1 will impos(' a lilllil Oil Ihe fllleness of price resolulioll simplY hecause Ihen' will he 110 ohS<'rv;tliolls in IIII' I'xln'lIl(' siall's \\'111'11 111 is tllO larg(,. ill whirh CISI' a sllhsl'l ollh(' parallH'll'rS is 1101 idclltilkc! alld call1lOI III' "SliU\;\ll'd.
1';.
'fill' COl/di/iol/lIl ni'/libll/iol/ 0/1'1';11' (;},(/I/,I;I'.I Ohsl'r\'l' that Ihl' 's ill (:\.:-1.1·1) arc asstlllll'd 10 be lIolli(\(,lIlicaIlv clisIrihlllt'll, I1l1Hli1i1l1l1'l1 ollthl' Xl's. The 111'('(1 for Ihis sllnll'WhalllOllslalllLlId as~umplioll COIIII'~ Irolll IIIl' irrl')?;lIlar anc! r.ltldolll sp'King of Ir"t1~',\"lioIlS (\;lIa. Ir. Ill!' 1',(;lIl1plt-, Irall'aniol! prin" wcn' (it-tcntlilll'd hy thl' Illodd ill Marsh alld RO'l'llldd (](IHli) whl'n' 1111' l't\ an' illCn'lIH'1l1s of arilhllll'lic
f.
125
J.J. Modtling 1ransactions Data
y'k Figure 3.4.
'f7te Ordered ProW Malhi
a;
BmwJlian motion with variance proportional to !'>.I. == I. - I.-I, must he a lineal' function of !'>.Ik which varies from one transaction lO the next. More generally, to allow for more general forms of conditional heteroskedasticity, let us assume that is a linear function of a vector of predetermined variables W. == [ WI •... Wu ]' so that
a;
Ek
2
Yo
2
+ YI
INID N(O, aJ)
Wlk
2
+ ... + YL Wu,
(3.3.20)
(3.3.21)
where (3.3.20) replaces the corresponding hypothesis in (3.3.14) and the conditional volatility coefficients IYJ} are squared in (3.3.21) to ensure t~at the conditional volatility is nonnegative. In this more general framework. the arithmetic Brownian motion model of Marsh and Rosenfeld (1986) he easily accommodated by setting
can
(3.3.22) (3.3.23)
•
III this case, W k contains only one variable, Ill. (which is also the only variable contained in X k ). The fact that the same variable is included in hOlh X k and W k does not create perfect multicollinearity since one vector while the other affects the conditional affl'l'ls the conditional mean of variance.
Y;
126
J. Mmkf'/ AlilTo.l/rttrlw('
The dependence structure of the observed process Yk is clearly ilHluced by that of Y; and the definitions of the A) 's, since
W k arc temporally independelll, the observed process YA is also temporally independent. Of course, these are fairly ITstriclive assulllptions and are certainly not necessary for any of the statistical inferences that follow. We I'equire only that the Ek'S be rOluJiliol/a'(v independent, so that all serial dependence is captured by the Xk'S and the W. 'so Consequently, the independence of the <. 's docs not imply that the V;'s arc independently distributed because no restrictions have been p\an'd Oil the temporal dependence of the Xk'S or W. 's. '~ The conditional distribution of observed price changes Y., clllI
\
1'(
r.
.I,/X., W.)
\
\
\ =
I
r(X~f3+E.
P( 0',_1 1'(
a,._1
::::
I
0'1
< X~f3 < X~f3
+
Xk,W k :::: 0',
if i
)
I X k , Wd
+ f A X k , W. )
=I
if I < i < if i =
11/
C\.:I.~(;)
III
I\~ :((:~:;))_¢(U"-X;f3) :::i~i<m ",(W , )
",(W,)
\
I
....
-'>'
(u •. ,-x;f3) ",
aq~umellt
.(. , 1 1=
Ill,
whe\·e aA(W.) is written as all ofWk to show how the conditioning varilbles enter the' conditional distribution, and (.) is the standard normal cUlllulative distribution functioll. To develop some inlllilion for the onlcn~d probit lIlodel, ohserve lhat the probability of any particular observed prin~ change is d{'tCl'mined hy where the condition
J, J, /'v/{)(iPlillg TrllIBllrlivl/.l' DIIIII
127
condilional heteroskedasticilY ill the ordered logil lIlodel, we have chosell the normal distribution, Civell the partition bOllndaries, a higher condition.II 1I1ean X~r~ implies a higher probahility of observing a InOIT extreme pmitiw st'lte, Of course, the labeling of states is arbitrary, but the {)llirmi prohit model makes usc 0(' the natlll'al ordering of the states, The regressors allow us to separate the dl'ccts of various economic bums that influl'llcl' the likclihooy a\lowing the data to oetet'minc the partitioll bOlllldaries 0, thc cocnicicnts (3 the conditional JIIl'an, and the cOllditiollal variall(,c ak~' the oJ(kred probit Illodd captures the elllpirical rl'latioll Iwtwl'cn the unobservable continuous st;lle sp;lce S' ;uHllh(' observ('d di"'J'('Il' state space S as a i'lIl1C1ioll of' the economil' variahles X k ;Illd W",
or
M,:xilllllllll.ihrlilwod I~Sli1/llllioll Let hCi) he an indicator variable whirh takes Oil the value one if the ITalizalioll of' the hth observatioll Vk is the ith state ,Ii, and zero otherwise, Theil the log-likelihood function C (il!' the vel'lOl of' price changes Y I 1', y~ l'" j', conditional Oil Ihe expl
L(YIX, W)
(3,:t28)
0;
Althmlgh is "llowed to vary lillcarly wiill Wk. lit('l'(' ar(' SOllie constraints that 11I1lst be placed 011 the parameters to ;ll'liieV(' idelltilication since, for example, doubling the o's, the {3\, ;\l1l1 Ok kavcs the likelihood unchanged, A typical idelltilil'atioll ;lssulIlptioll is to S('t Yo = I, Wc are Ih(,11 1I'l't with tilrcc issllcs tltat llIlist bc resolvcd hl'll)\(' l'sti\ll~ltioll is possihle: (i) tlie lIumber or states III; (ii) lhe specifiratioll of' the rcgll'ssors X k ; and (iii) the spcci licatioll of' Ihe cOllditional varialll'C In choosillg Ill, we 11I11st ha\an('l' prire rI'SOlllli,," ag~lillst the practit'al cOlIstraillttilatlOo large all //I will yield 110 observatiolls ill thc I'xltTIllC states 'I alld s"" For ('xample, ir we sct /1/ to I () I alld ddille the stalcs ~I alld 5101
a;,
J. J\!lIrkl'l J\!inmlrwllllf sYIIIIllI·trically to Ill' price dI
a;
3.4 Recent Empirical Findings Tlrc clllpirical 111.11\..('1 IlIicroSll'lll'llIrt' lilcratllre is an cXlI'nsivc Olll', slraddling hOlh ac"dl'mic "lid indllstr), p"hlications, and it is difficlllt if nOI impossihl(' 10 provide ('V('II " supl'rfici"l rcvil'W in a li~w pa),!;cs. IlIslead, we shall p\'t'S1'1I1 Ihn'l' spl't'ilk III"rkl'l lIIicrosll'llctllre "pplicaliolls ill this seclion. cach in SOIlH' dqllh, 10 ),!;ivc rcadcrs a more COII('I'('IC illllstration of empirical n's('"rdl ill Ihis l'XCilillg- "lid rapidly growill),!; lill'ratllre. S('ctioll :1.'1,1 provid('s all ('lIlpirical '1I1OII),sis of lIonsYllrhronolls tradill),!; ill which the lIIa)!;lIitudt, of lht' nOlllratiin),!; hias is mcasllreli llsin),!; daily, weeki)" ·md nHlIllhly slock rl'tlll'l\s. SI'('\ioll :t4.~ reviews the empirical all;\lysi~ of d~ fl'Clivl' hid-ask sJl('('ads hased Oil Ihl' modd ill Roll (19H4). Alief St'uioll :1.4.:1 prl'SelllS all applicllioll or Ihl' ordered prohil modd 10 IrallsaCliollS dala.
1. ,I. I
NI>II\y"rllIllI/lIll.1
TTl/ding
Bt'li l\"(' \'(\I\~idl'l'i 11),\ I hI' \'III pi rica I I'vidl'lIl'I' 1'01' !lOll I raeli IIg clTI'('\s WI' SIIIl \ I\larill' Ihl' qualitativ(, illlplic;Iliolls or thl' lIo11lradillg model or Seclioll :t 1.1, Althollgh 111;111\' th('s(' illlplicatiolls are cOllsisl('1I1 wilh olher models of IIOIlSYlldlrollOllS Iraelillg, Ih(' sharp comparaliv(' slatic r(,sllll.~ alld \'xposi-
or
'1"'1"01" (·"tllllph-. I Lu',IIl:lIl. 1.11, ;11111 i\l.lfl\illl.l)' (I!I~)!!) ~('I III == !J lor Ihe 1;lrgl'r .,Iorks, implying- ('xII 1'111(' SI;t1,'" 01 -·1 lid•., 01 It"", .lIId f··1 tid.s or 11101'(', and ,"'" III = :) f(,r 111(' ~I11;1IIt'r sllll 1..'" illll'lvillg .'\In'III(' '1.lIe', 01 :! Ii. ,,-, III Ic'" ;lIul +:! lid;..'i or mon', NOI(' (11;11 ;dlhutlgh till' fldilllllllil 01 ~1.11c, IIl'"d IItll be '~IIIJlH'lIil ( .. I.IIt' 'I I ;111 I", -Ii Ii, k!ll til Ic~!\. illll'l~'iTlg Ihal Mah' "I j .. -f '.! IiI ,,-, 01 IIIfll('), lilt, !\yllllllt'!I\' 01 Ihl' hi'logralll 01 prin' (·h.,"g('~ ill their d.lla'i('1 ~1I~~t'sts;, \\'111111('11 it ddlllilioll 411 III"
\, \.
3.4. UfI"fllt l~mJJirj((l1 Fiuding!
•
129
tional simplicity are unique to this framework. Under the assumptions of Section ~.I.I, the presence of nonsynchronous trading
). c10es not alTect the mean of either observed individual or portfolio returns. 2. increases the variance of observed individual security returns that have nonzero means. The smaller the mean, the smaller the increase in the variance of observed returns. 3. decreases the variance of observed portfolio returns when portfolios are well-diversified and consist of securities with common nontrading probability. 4. induces geometrically declining negative serial correlation in observed individual·security returns that have nonzero means. The smaller the absolute value of the mean, the closer is the autocorrelation to zero. 5. induces geometrically declining positive serial correlation in observed portfolio returns when portfolios are well-diversified and consist of securities with a common nontrading probability, yielding an AR( I) for the observed returns process. Ii. induces geometrically declining cross-autocorrelation between observed returns of securities i and j which is of the same sign as fJ,flj' This cross-autocorrelation is generally asymmetric: The covariance of current observed returns to i with future observed returns to j need not be the same as the covariance of current observed returns to j with future observed returns to i. The asymmetry arises from the fact that different securities may have different nontrading probabilities. 7. induces geometrically declining positive cross-autocorrelation between observed returns of portfolios A and B when ponfolios are well-diversifted and consist of securities with common nontrading probabilities. This cross-autocorrelation is also asymmetric and arises from the fact that securities in different portfolios may have different nontrading probabilities. H. induces positive serial dependence in an equal-weighted index if the betas of the securities are generally of the same sign, and if individual returns have small means. 9. and time aggregation increases the maximal nontrading-induced neg: ative autocorrelation in observed individual security returns. bUl lhi~ maximal negative autocorrelation is attained at nontrading probabili~ ties increasingly closer to unity as the degree of aggregation increases.; 10. and time aggregation decreases the nontrading-induced autocorrela-l \ tion in observed portfolio returns for all nontrading probabilities. Since tile effects of nOllsYllchrollollS trading are more apparent in securities grouped by nontrading probabilities than in individual stocks, our empirical application uses the returns of ten size-sorted portfolios for daily,
)30
3, MllIlut MirllJ.ltm(/lIr,.
weekly, and mOllthly data from 1962 to 1994, We use market Glpitali/.alion to group securities because the relative thinness of the market for any given stock is highly correlatcd with the stock's total J\lark{~t vahl{'; h 'lIce stocks with similar market values are likely to have similar nontraclilll-: p 'obabilities,39 We choose to form tell portfolios to maximi/.e the hOll)og neity of nontrading probabilities within each portfolio while still maintaIning reasonable diversification so that the asymptotic approximation ur d.1.20) might still obtain, ~o
l i
Df.ily Nontrading Probabilitifs Implicit ill A u/ocorre/ations 3.4 reports first-ordcr autocorrelation matrices rl for thc vector of fopr of the ten size-sorted portfolio returns using daily, weekly, and monthly d*a taken from the Cemer for Research in Security Prices (CRSP) database, P9rtfolio'l contains stocks with the smallest market values and portfolio 10 cO~llains those with the larges1. 41 From casual inspeClion it is apparent thfl these autocorrelation matrices arc not symmetric. The second colUl1111 of\matrices is the autocorrelation matrices minus their transposes, and it is ~videnlthat elements below the diagonal dominate those above it. This cO\lfirms the lead-lag pattern reported in 1.0 and MacKinlay (l990c), \ The fact thaI the returns of large stocks tend to lead those of sl11aller stolks does suggest that Ilonsynchronous trading may be a source of corrcl4tion, However, the magnitudes of the autocorrelations for weekly ;\lId monthly returns imply an implausible level of nontrading, This is lIIost evi(lcilt in Table 3.5, which reports estimates of daily nontrading probabilities implicit in the weekly and monthly own-alllOcorrclatiolls of Table 3.4, For example, using (3, lAO) the daily nontrading probability implied by all estimated weekly autocorrelatioll or 37% for portfolio I is estilllated to be 7I.7%,~2 Using (3.1.8) we estimate the average time between tracks to
T~ble
190nly ordinary cummon shares arc included in this analysis, Exchukd an' Alllni",," Depository Receip15 (AORs) and other specialized secllrities where lIsing market val\l~ to d,,"acteri/.e nontrading is Ie.., meaningful. ""The returns 10 these portfoliusarl' n)lltinuOIlslycOIlII)(Hllldcu relurns ofitldividtl.,1 ~il1\I)It· returns arithmetically averaged. Wt' have repeated tht, curr("l;tlion analysi:t. for ('oUtiIl1l0US)Y con'pounded rf'turn~ of p()rtfolio~ Wh05C value.ra. 4lre calculateLl as unweightC"d RCOlllt.'trir av.. {'rdg{'s of included securities' prices. The result, for these portfolio returns arc pr"rtically identical to those for the continuously compolllHkd returns of c(l"al.wciglll(·d pol"l1011O\. "We report only a ,ullset of four pnru"lios for the sake of hrevity, 41SIandard error. fm alllocorrl'l,uioll,h,l,.. d probability aud lIotl\r"dinH duration ("ti",,"("
r("turn~}.
J.".
NUI'IlI 1~'IIII)itiral Findings
7able 3.4.
131
A 1/11)(()n~I(lljl/1l 1II111";l'r.1 JilT .1;1.I'-.11/I"lr" f"'I"'folio
J\ I Daily
<1
7
.~~)
.~I
10
,I 7
,
(""
.11 AO
.:H
.:1!)
.~H
.~~
10
.:H
.:\ti
.:14
.11 .I!> . I ~)
T'
4
7
10
.19 .:11 .:1:1 .19
.1:1 .15 .17 .15
1
<1
7
.10 .16 .IV .IH
.Oti
'>7)
I
,I
Wcekly
Monthly
4
.:'11
7 10
.:1:1 .:11
.1:1 .I!) .~7
.:.!:.
7 10
'("
4 7 10
.:1H .:10 .:17
4 7 10
.15 .~o
.~ti
1', .11 .14
.14
(1)
1
7
-.1:) .00 .OH . I ~)
-.~O
1'1
10
.0,1 .05 .0:1
-.1 ~) -.m) .00 .1'1
10
-") -.~:)
- 1'1 .00
i'l - f'1 I
_II') , ("" -.00 .0:1 .01
7
, (""
1'1 I
1'1 -i"1
-.1:1 .00 .O!)
<1
I/"I/nll.
I
,
4 7 10
(""
.IH .:14 .~ti
-.OH .00 .14
-") -.IV
-.14
.00
-, -I',
4
7
-,,~O
-.:1!> -./IH .00
.00 .OH .101
10
.O~I
10
-~') -.14 -.09 .00
lir~I~)nll'r;lIl1ororrcLlIi(lIlIJl.HJ"ix i",'ol"tlw (-I x I) slIln'('( lor I ,;' '.~' ,; 1;'111' of ohs('I"\'('" fl'ltlfll:-' lD tell eqllOlI-weig-htcd sile-~orted pOrlfolios w .. illg d~'il)'. week.ly. alld monthly NYSEAMEX (0111111011 stock returns data from th(· CRSP lilt·s for Iht' lime pt:riod .lilly :l. 1962 In Den'mllei" :~(). I!)~H. Storks ;uc assiglled to portfolios 'lT1ll1lall>' lIsing th(~ IH01rket value.1t the cnd urlhe prior }'(·;If. If this market value is missil1K Ihl" cud 01 year markt,t \'.dut' is used. IflxHh mark('1 val lit's ,II"(' IlIis..... ill~ lil(' stork is 1I0t illt. Itlflt'd. ()lIly .\t'nll ilu':-, Wllit nllllpl('I(' daily 1('1(11 II hi!'lloril':-' withiu a gi\'('IJ mOllth arc inrilHh-d ill Ihe d'lily U'WIIiS f.lklll.tlio ll ..... ,;' i. . th(' U'tlil It 10 the portfolio containin g securities with tilt.' smelliest menkel ",,,III('s
ponfolio ofs('curiti{'s wilh thc lar~csl. Th('I(~ ,lit' approxilll
is abo r{'pont·d.
'L~ymlJletry
.uHI';; .
in IlleSt' CllltocOIIl" latiulI mati i(t'.o;" the- dillt'n'lIfC (.... 1-
t'l
be 2.5 days! The corresp onding daily nontrad ing prohahil ity is HG.(i% using monthly returns. implyin g an aver;lge lIolltrad illg duratiol l ofG,:, days. For compar ison Tahle 3.5 also reporls estimate s of th\' lIolltrad illg prohabilities using daily dat,l and using tr;1(1e informa tion from th(' CRSP files. In the abSCtll'C or timc aggrq~atioll OWIl-.lutocolTdatioIlS of porlf(,li o returns are consiste llt estimato rs ofllontr ading probabil ities; thus the clltries ill thc columll orTable ::I.5lahe lled "rr.(q = I)" arc sill'l)ly takclI rrom the diagolla l of the autocov ariallce matrix ill Table :t4. For the smallcr securitie s, the poilll cstim.lle s yield plausibl e Ilolltrad illg duration s. hut the estimate d duratiol ls declille (111)' 'lI;lrgill;l Ily for larger-
.l.
1·.~\/iIl/1I1,·\
'1lIhle J.5.
fr,
K
ir,('1=
.~~!")
·1 7 III
(11.111:1 )
. :I!I·I (O.O:!li)
(IHI7)
.o;.:! (O.OIH)
.:\.1:1 (0.02:1)
(O.W.)
.01!1 (O.OO:!)
.:I:!H IO.Oilil
IH~I
.Oll:! (0.11111 )
.IHH 10.111'1)
I':\lim~'h'~
"/ tlllil\' lItH/lrmli,,/{ Im,l",bilili,.I.
f.:1 ii, I
I)
ir,(f!=
1~lk,J
5)
ir,(f!
= :!2)
LI h, J
2.:.-1 (0.·12)
.Hlili «1.0:!!l)
li.!7 (Ui·!)
(O.W.!I)
I.:!H (0.:11)
.1-\:17 (11.0·1:1)
:.. 12 (1.1i I )
(fI.o·1)
.·I!I7 (O.Oli:!)
O.!I!I (O.:!:.)
.HI!I (O.O·IH)
1.:-.2 (I.·lli)
II.:!:I (0.0:1)
.I:I!I (O.i:!li)
O.lli (11.17)
.!") I:) (ll.·lti I )
1.IIIi (I.'IIi)
(IN•
.717 (O.O:H)
.:.1iI)
O.:,!!
of(.bily uoutLHhng pruh.dlilitit·:o, impli('it in
Inliu n'unn ,\\lh.)( unt'I.H\tH\S-.
II/a/Ii,,/ MU'/m/m(/w"l'
(c.'1\
wt'c:ldy ~'nd monthly \i/c.'·\ortc:d port-
r""i,'\ '" lin- ('ulumn l~\I)C.'tlc.·(t "rr ... ~Ul' ~\\·(·r~,~(·\ofthc.· harti(ltl of
,to, \U','it'~ '" pUllluhu /rt, lh.u ,1ill lin' tLH\t· uU \'~U h 'ra"in~ ,t,y. Wht'H' tht· ~\"t'ri\At' ,\ rnmp\1h't\ U\Tr "~II lr.uhng Il.,y, h om.}u\\-:t EU)~ to' h'n'mlu'r :\U. 14J94. Enll't'~ in tht· "rr .. ,,/ = 1)" fplumll ~'n' 1ht· Iii \H lICit·, aHtonJlIt'1.lIion fodlidt'llb of (lail), ponfoHn f('\nrl\!'\, \"hit h "ft' fOJ)!'oisu'ut ("!\lim;lIu"~
of 'bil\' IIOllll,ulill).,{ proh;thililit,."i, EIII ..it-S ill Ihe.' '"if" (,/
1"1111111""
,It" ('~lilll,IIt-!\.
monthly
pUlllulio 1"1'111111 ,11110("01 Id.lIiOIi ('udlid"UIS,
~k('lioli :\,~
('1
~
:1 101
= :,)" ;lIul "n,. (q
==
~~) ..
01 ,tlil\' lIolllr;uling I)luh.lhililit's ohtailU'«\ Irolll li""I-ol(lt-r wn'kly ;IIHI wc',,}"'1\' It'tunl' .IIHI
'I =
~~
"!'Iillg Iht'
lilll(,
ag-gu'g'lIioli rd.llion'
III
for IllOlUhly n'IIII"II." ;o.i 1I«."t' the.'f(' an':1 ~1I1(\:!~
fr.ulillg 1I;t\'s ill a \\'(·(·k awl,lllIullth, n"IH'( li\'t'ly), Enlrit';o. ill (·Ohllllll."i l;tlwlled
Mf:f k/l" alt' ('sri-
male.'s of Ih,' e.'Xp('fu'd IIl1mh,'" of nHt",'nlli\"(' days withour Ifading" implic.'d hy lilt' prohahility ('Mim;,"';o. ill rUIIlIllIl!\ to lIu' illlllu'di;III' It·fl. Standard ('nors an- r('porle'lI ill pal(·11111(·;o.(·": all ;an' hC.·h'I·'Js"(·(Ia.,IH·ily- ;'1111 illlltH', In(·l;lticIIH·(HI.'ii.... I(·,11.
sil(' portli.lios. A dllral iOIl of lIt'ad)" Ollt' lillirth of a day is IIIlIch 100 Iarl!;c «II' s('cllriti('s ill til(" lall!;t'sl portJ(,li, •. More direct evid(,lIc(' is provided ill th(' ("{)hllllll lahdlt-d n" which rqlllr\s the awrage !i'actioll ofs('cllritics ill a giv('n portfolio Ihal (10 not Irade dllring ('acll trading day.n This ;Iv('l";Ige is COllllllllt'd OV('I" alllra
nTh" iulium.Hlon" IBU\'"l,'" HI '\11' (J{Sl' ",Illy hh·!\tn ",hit'll "11' (luo;,tng pdcT nl;, ,('nully i, "'pol1t'C' 1o ht' Ih,' ,H·g.II"".,l IIII' .I\·C'J ;Igc' ot Ih,' hill alltl ask prin'!'i (lJI (I.ly~ wht'll th,11 .'C'C IIril)' Ilid 1101 II.U"', SLuul.lld "1101 ... 1111 proh.lhilil\' ,',lim.lit'S .tlt' h,l."i('(\ olllht· d.lily lilll{' Iht' iI~lnitlll
nlll,hlC'lIl.
ni
110-11;111(-,.
1"1,,'
stalld.lnl ('nul's an' h(·I(-ro!;,k(·daSlkil~'·
,nul
'(Til"
01
;1Il10fOIT('Lllillll-
J.4. Ilfcmt
Table 1.6.
Estimator of If i Nq~ative
133
I~",piri((ll Findillg.~
N(Jl/lmt1il/~-illll,lif(/
wffkly i,ll/ex flulocomlnlions.
Implied Index PI ('Yo) (Ill == 1. /ll" == 1)
Implied Indt;x PI ('Yo)
1.4
I.H
4.H
5.9
Share Price
Daily AUlOcorrelatio"
(Ill
= 1.5.13111 = ()5)
IlIIpli .." firsH>rtier autocondation PI ofw"ekly relurns of an equal-weighted portfolio o(ten portfolios (which approximales all equal-weighted portfolio of all ~curitie.). u~iJlg two difkr~nt t'stilllators of daily lIontrading prohahilities for the portfolios: the aver.tge frdnion of n~~ati\'(' sl"lrt' pric .. s reportt'd hy CRSI'. and daily nontr.tding prohahilities implied hy lirsiordl'r aUlororrdalion, of d;lily returns. Sinet' the index autocorrt'lation depends on the IX-tas of till' tt'n portt(>lio,. it is colllputed tilr two set' of betas. one in which all beta.~ are set w I.U ;lIId anoth"r in which tht' hetas decline linearly fmm III = 1.5 to Pili = 0.5. The sample weekly ""t"r""r~hlti()n for an "'lual-weil\htecl portfolio of the I('n portfolios is 0.21. Results are ha",," on ,1"la frl)IlI.1nly :>. I\l\i:.! III nec.ember :>0. 1\1\14. .
si/t·-"'rt~"
M!l!5ynfitron()1L5
1rading and Index Autocorrelation
Dellote by r,~,( the ohserved return in period 110 an equal-weighted portfolio of all N s(·curities. Its autocovariance and autocorrelation are readily shown to he (' r(n) (
~
(3.4.\1)
•
where r" is the contemporaneous covariance matrix of T," and ( is an (N xl) vector of ones. If the betas ofthe securities are generally of the same sign and if th(' Illean return of each secnrity is small, thell r~, is likely to be positively alltocorrelated_ Alternatively, if the cross-al1l0Covariances are positive and dOlllinate the negative own-au(Ocovariances, the equal-weighted index will exhibit positive serial dependence. Can this explain Lo and MacKinlay's ( 19RRh) strong rejection of the random walk hypothesis for the CRSP weekly (,qual-weighted index, which exhibits a first-{)rder autocorrelation over 20%? Wilh little loss in generality we let N = 10 and consider the equalweighted portfolio of the ten size-sorted portfolios, which is an approximalely equal-weighted portfolio of all securities. Using (3.1.36) we may calculate the weekly autocorrelation of r;;,( indnced by particular daily nontracling probabilities 7f; and beta coefficients {3;. To do this, we need 10 sel(,ct ('JlIpirically plausible values for IT; and 13;, i :::: 1.2 •...• 10. This is don(' in Table 3.6 \Ising two differ(,nt methods of estimating the 7T;'S and IWo dill't'J'enl aSSlllllptioll5 for the 13,'5. The first row corresponds 10 weekly autocorrelatiolls computed with lhe nontrading prohabilities obtained from the fractions of negative share prin's J'<'portet\ by CRSP (sec T~lbie 3.5). The first entry. 1.4%. is the first-
3. Markel MimH/l"Ilrlurr
qrdcr autocorrelation of thc wcckly cqual-weighted indcx aSSlIllling tklt all twcnty portfolio bctas arc 1.0, and the second entry, 1.8%, is cOlllputed UII- . dcr thc altcrnativc assumption thatthc bctas declinc linearly frorn III = 1.:> f~)r thc portfolio of smallest stocks to IlIU = 0.5 for tl!c ponlc>lio of the 11rgcst. The sccond row rcports similar autocorrclatiolls implicd hy 11011l~ading probabilitics cstimated fmlll daily autocorrclations using n.IAI). The largcst implicd !irst-order autocorrelation for the weekly equalw ightcd returns indcx reported in Table 3.6 is only 5.9%. Usillg direct c. timatcsofnontrading via lIegativc sharc priccs yields an autocorrelation of I~' s than 2%. Thcsc magnitudcs arc still considcrably smallcr than the '21 'Yv s· IIlplc autocorrelation of thc e<Jual-wcigllled indcx retUnI. In SIlIllIll,II")', tl e rcccnt cmpirical cvidence provides lillie support for nOlllrad'lIlg as all important source of spllrious correlation in the rcturns of common stock o~er daily and longcr frcquencies. 41
f.
3.4.2 1~,\tiIlUllil/li till' I:J/rrlivc
Bid·A~" .'>/))"('(1/[
In implementing thc model of Scction 3.2.1, Roll (1981) argucs that thc percentage bid-ask spread s, may bc more easily intcrpreted than thc al)solute hid-ask spread s, and he shows that thc Ilrst·urdcr 'lIl1ocovari'lllcc of simple returns is relatcd to .I, in the following way: Cov[ R - l '
s,
.\~1
.\;
--4 Hi
H, I
-
JJ>1I 1',.
~
s,~
4
(:H.'2) (:H.:\)
'
where 5, is defincd as a pcrccntage of thc geometric average of the avcragc bid and ask priccs Po and Ph' Using the approximation in (3.4.2), thc pcrcentage spread may be recovercd as (:H.4)
Notc that (3.4.4) and (3.'2.9) arc only wcll-deflllcd whcn thc returtl al1tocovariance is negalive, sincc by conslruction the hid-ask bOllllce elll only inducc negative !irsl-order serial correlation. Ilowever, in praclice, po~i tive scrial correlation in returns is lIot IInCOllllllon, and in thcse cases, Roll simply defines thc spread to he (sc(' footnotcs (l ,lIld b of his Tallk I):
Hnuudollkh, Richa,.dsoll, alld Whit,·law 1199:», M"('h (I!I!I:I) alld Sias alld Sta,b (I!I!I-!) prt·~ClIl
additional empiricfll results on l1onlra
p"pf'r~ do nut a~re(" on the I('\'l"i of auto("(lITf.'laliun indu('('d
rO"rltHl" thatllolllradilll: ,';,"lol rompl("I<"I), "e<'<>II1l1 Ii,,· Ill(" 01>"'1'1'("<1 ;\lU()('o"rl"li"m.
1,
3,4, Recmll:III/,irim/ Findings This conv('ntion seems dirticuh to .iustilY 011 ('('onomic ~rolll\ds-ne~ative spreads arc typically associated with lI1arkdll1;",i\l~ aClivity, i.e., the provision ol'liquia) propose a [\\or(' gCllerallllodcl th,ll cOlltains these (lthlT specilications as special cases and estimate the ('()lllpOllellts or tile spread to be \!l 'X, adverse-selection costs, 14% iIlVl'lIlOry-holdin~ n)sls, alld (i:)'X) ()nln-plO('('ssi\l~ costs usin~ 199:ltrall~
10 a \':lIil'll' (II d:llaS('ls 10 ~allgl' tlte I'xplallalOry pow('r alit! st:lhilit\, 01 ('ach lIIodd.
,. -1.1 nt/I/.lt/dill//.\' nolll III I tlllSlll:lIl, 1.0, :11111 /'bcKin!;I\' (I~)q~), Ihrl'l' spl'cilic aspl'ns of Ir:lllsa('lions data arl' I'xaillilll'd \Isillg thl' ordl'rl'd prohil modd of Sl'niOll :\.:\.:\: (I) Do('s thl' parliclIl:tr II',{III'IIO' ollratll's affl'cI till' cOlldiliollal dislriblliioll of prin' changl's, I'.g., dOL's Ihl' Sl'ljlH'llCl' of three pricl' challgl's -I- I, - I, -I- I ha\'l' II Ie S:\1 III' dIi'('( Oil the cOlltiitiollal distrihution ofthl' III'XI price change as Ihl' S('qtll'IIII' -I, + I, + I? (~) DOL'S Iradl' sizl' alTecI priet, chall!-\cs, alld if so, wh:11 is thl' pritT illlP:I('\ pn Ullil volullll' of Irade frolll Olle IrallSaclioll 10 Ihl' III'XI? e\) Dol'S pricl' disITell'Il1'SS l\Ialln? III particular, ell \ Ihl' cOlldiliollal di~lrihlllioll or pricl' chall!{l's he 1II0deled as a simple lillcar rl')~rt'ssioll of pricl' r11:lIlgl'S onl'Xplallalory variables withotll accolllltillg for dislTl'll'llI'SS al "II? Til acldn'ss 1IlI'sl' Ihn'l' qllesliolls, Ilatlsmall, 1,0, alld MacKillLi), (I !)92) ('stimall' Ihl' o,dl'l;'t\ plllhil II10clt'I for 19HH IrallsaCliolls dala of o\'er a hUlldrl'd slod.. s. 'Iii ('{IIISI'r\'I' span', wc rOnts (111)' Oil Iheir sllIalln ane! IIlIIr(' d('I:lilnl s:lIl1pll' of six ~1()cb-lnll'rllali()llal IIl1silH'ss Machilu'" Corpmat i()n (\HI\!) , (.2"antllll (:lu'lllical (:orporation «:U E), Foster Whcl'kr Corporation (F\\'( :), llalle!v alit! Ilannan COlllpany (I INl \), N:l\'isl:lr 111Il'I'llaliollal Corporalioll (NAVj, aile! American TI'I('plloll(' and Tc'lcgraplt IlIl'orpor:llt'd ('1'), For !iIl'SI' six slocks, Ihey foctls ollly Oil ililmtit/y Ir:lItsaclion prin' dlan~l's since illt:IS IWl'lI wdl-docIIIIII'llled Iltal overnight relurns di/kr sllhslallliall), /'rolll illlrad:l), J'('llInts (SCI', for c'xalllpk, Alllihlld atlel Ml'llclt'lson II!lH7I, Sloll and Whaky [19!1()], alld Wood, Mdllish, ancl Orel [ I!)H!, I), Thl'v also impose sl'vl'ral oliter filters 10 l'IilJlin:ttl' "prohklll" lransanions and qllOlt'S, which ~'idclt-cl salllpll' sizes rangillg frolll :1,17,1 Iracles «II' IINlllo ''!Ot;,7~)'(lralks ror IBM, Thl')' ;dSIlIISI' hid alld ask pric('s i II Iheir :llIalysis, alld since hid-:lsk qllOlcs arl' I'I'Jlortl'd olll\' ",hl'lI IIII'\' an' r('vis('cl, SCllIIl' efrorl is J'I'qllirl'cl 10 Ilialch qllOll'S 10 Ir:\lI~:ll'liolls, :\ natural algorilhm is 10 lIlall'h each Ir:lllsaniol\ pric(' 10 Ihe 1I10~1 1'1'1'1'11111' reported qllOI(' /willr 10 Ihl' Ir:lIIsauion; howevcr, Brollfrllan (Ill!)!) :11111 1,('(' alld Rl'ad)' (l!l!ll) have shown llial prices or Iradl's lIial prl'cipilalt, qlloll' r(,visiolls art' SOllll'lilll(,S report cd wilh a lag, so Ihal Ihl' onlt-,' or qllOl1' r(,vision and IrallSauiol\ price is reversed in ol'licial r('conls sllch as Ihl' (:ollsolid:lIl'd '1:11'('. To addl'('ss Ihis issll(" J lallsillall, 1.0, alld M:ld"illl:1\' (I'lc)~) 1I1:,td, trallsaclioll prices \0 qllotes Ihal arc set til Imlt/i1,/' \/'/II/ld, {"i",. 10 IhI' Irallsacl iOIl-Ihe c'vidl'ncl' in 1.1'(' and Ready (1!1'1I) sllggC'SIS Ihal Ihis will accolllII for 1II0s1 01'1\1(' Illissl'C(II('IICilll!;. This is 01111' OIl<' \'x:lIl1pk of till' killd of IIl1ill'll' challengc's lIi:11 Iransaclions dala po.~(',
J.".
/It'ct'lll
Empirical Filldings
137
To provide some intuition for [his enormous dataset, we report a few slIllllllary statistics in Tahle 3.7. Our sample contains considerable price dispersion. with the low stock price ranging from $3.125 for NAVto $104.250 for \HM. and the high ranging from $7.875 for NAV to $129.500 for IBM. 1\[ $219 million, HNH has the smallest market capitalization in ollr sample. and mM has the largest with a market value of $69.8 hillion. The empirical analysis also requires some indicator of whether a transaction was buyer-initiated or seller-initiated, othenvise the notion of price impact is ilI-defll1ecl-a 100.000-share block-purchase has quite a different price impact from a 100.000-share block-sale. Obviously. this is a difficult task because for every trade there is always a buyer and a seller. What we hope [0 capture is which of the two parties is more anxious to consummate the trade and is therefore willing to pay for it by being closer to the bid price or the ask price. Perhaps the most obviolls indicator is whether the transaction occurs at the ask price or at the bid price; if it is the former then the transaction is most likely a "buy" and if it is the latter then the transaction is most likely a "sell." Unfortunately. a large number of transactions occur at prices strictly withirl the bid-ask spread. so that this method for signing trades will leave the majority of trades indeterminate. Hausman. Lo, and MacKinlay (1992) use the well-known algorithm of signing a transaction as a buy if the transaction price is higher than the mean or the prevailing hid-ask quote (the most recent quote that is set atlea~t five seconds prior to the trade); they classify it as a sell if the price is lower. If the price equals the mean of the prevailing bid-ask quote, they classify the trade as an indeterminate trade. This method yields far fewer indeterminate trades than classifying according to transactions at the bid or at the; ask. Unfortunately. little is known about the relative merits of this methOd of c\;\ssific;ltion versus others such as the tick lesl (which classifies a transaction as a buy, a sell, or indeterminate ifits price is greater than,less lhan, orequaJ to the previous transaction's price. respectively). simply because it is virtually impossible to obtain the data necessary to evaluate these alternatives.
Thf J~ml)iriral Specification To estimate the parameters of the ordered probit model via maximum likelihood. three specification decisions must be made: (i) the number of stales 111, (ii) the explanatory variables Xk • and (iii) the parametrization of the . 2 vartance G k • In choosing In, we must balance price resolution against the practical constraint that too large an In will yield no observations in the extreme states ·\1 and .1 m • For example. if we set m to 10 I and define the states SI and SIOI syml\lctrically to be price changes of -50 ticks and +50 ticks, respectively, WI' would find no Y. 's among ollr six stocks falling into these two states. Usin~ the empirical distribution of the data as a guide, Hausman, Lo, and
138
3. Markel Mirm.IITllrlllTf' TableJ.7.
Summary stnti.,tirJ fOT tmll.lflrtiollJ Ilntn cif.lix .• /orh.,.
Variable \Low Price ,lIigh Price 'Markel Value (SBmiolls) \
IBM
CUE
FWC
IINII
NAV
T
104.250 129':'()0 ('9.HI"
6:,':'()() IOH.2:,O 2.1t17
11.500 17.250 0.479
14.2:,0 HI.500 0.219
~.12:'
24.12:,
7.H7:, H.9!J1-\
:\0.:\7:, 211.\I\lO
4:1.81 12.ijij 43.5:\
4~.19
:'17.13 2~.',8
22.:,3 26.2K 51.20
40.110 11-\.11 41.0!l
:12.:\7
18.67 :lK.14
-0.0010 0.7530
O.OOJ(' 1.23:>3
-().0017 O.l'~!JO
-0.0021l O.74!l2
-O.1l002 0.644"
(1.000 I 1l.!i:,40
27.21
20:1.,.2 :l1l2.lij
2\1(,.:.4 416.49
1129.37 1497.44
:1~L\(;
:11.00
:14.I~
7(i5:1
:\·1.:\\1
1.9470 1.462:,
3.2909 \.(,203
2.01'30 1.1682
2.4707 0.8994
l.46lti 0.(,713
l.Ii,.1;4 0.79%
-0.0000 -(LOOO4 0.0716 l1.l3!l7
-0.0017 0.147,.
-0.0064 0.1903
1).0001 O.103K
-(WOOl 0'<171;:.
0.00211 0.9:146
O.!I:,W, -0.(2)(; -0.2!llj7 0.8739 0.8095 H.900'.
-O.I)02H 0.9(Wl
-0.0\1:1:1 O.K:1:'(;
O.IO',!) 6.1474
0.:\:,74 -1l.O:,23 6.27911 ".6643
-1.%43 6.0I:l!lO
57,37,.
40.900
(;,1:,0
",:\(;3
.% Trades at Prices:
Mid'luote
;>
= Mid'luole < Mid'l"ole
39.29
2:J.~)2
41.71
Price Change. }. Mean: SId. Dey.: Time Belweell li-ade••
"It
Mean: Std. Dey.:
Bid-A.
0.0332 -0.42,.(; 6.\)70:, 75H4!i 3,O()O
7.\1,.0
Summary statistics for lr~ll.\a(tiOl) priet· . . alul «)rn'sIHlIHii1Jg '>rdercd probj( ('xpl'lIldlc)I y \'~II i·
able. oflnlernationaillusinc.., Marhincs Corporation (IUM. 20(;,794 trades). Qllantutll <:l>t'lIIkal C.orpor~\ion (CUE, 26,n7 lraclesl. Foster Whet'ler Corporation (FWC, IH.I\''I tradt's). Handy and Harman Company (IINII, ~.174 t,-adt'sl, Nayi
r
!/f),11' I I
MacKin lay (1992) set m =: 9 for the larger stocks, implying extreme Slates of ~4 ticks or less and +4 ticks or more, and set m = 5 for the two smaller slocfs, FWC and HNI-I, implying extreme stales of -2 ticks or lcss and +2 lick1 or more. 1fhe explanatory variables Xk arc selected to capture several aspccls of traJlrction price (hanges: dock-time cfTeCls (such as the arithmetic IkoIV-
3.4. NI'("('nll~lIIfiiriflll FindinK-'
nian mol ion model). the cfleCls or hid-ask houn("(' (silln' many transact iolls are mCl'el)' II10VenH.'lIls rromthc hid price 10 III(' ask pritT or vice versa).l he si/.e or Ihc Irallsacl ion (so prin' illll'al'l (";111 lit' delcl"llIill('(1 as a function or Ihe fjllalllilY Iraded). alld Ihe illll'acl or "SYSlclllalic" or Illarkelw ide 1II0vemenL~ olllhc cOllditio nal dislrihll iion of an individll;1 1 slock's pritT challges . These aspens call for the followin g expiallalOlY val iahlcs: 61k: The lillie clapsed betweell transaCl iolls Ii-I alld It. in seconds . An k _ l : The hid-ask spread prevailin g al lillie Ik __ I. in licks. Yk -,: Three lags [l = I. 2. :~J orthc dqH:lld ellt v;uiahk r • J{ccalith atl(lIh III = !l. price changes less than --4 licks arc sel eqllal 10 -·1 licks (slalc .\1). and price changes greater thall +., licks are scI e«lIal to +4 ticks (Slate -~l). and similarly for 111 = [,. V k _,: Thrce lags [I = 1. 2. :q of Ihe dollar volllllle or Ihe (It-I)th transanion. ddinc:d as Ihe price of the (il-/)Ih Iransact ion (in dollars. nol tic ks) lillles the nlllllber of shares Iraded (dellom illated in hundred s of shares); hcnce dollar volume is ckllollli naled in hundred s of llollars. To reduce the influenc e of oUlliers. if the share volume or a trade exceeds the (19.5 percenti le of the empirica l dislrihu lioll or share volume for that stock. it is set equllllo Ihc !I(l':' perccilli le. SP500 k __ ,: Three lags [I = I. 2, :1 I of fivc-lIlinule continu ously COIIIpounde d rctllrllS of the Siandar d alld Poor's (S&l') [,00 index futures price, for Ihe conlrac t maturin g in the c10scsll llonlh beyolld the lIIonlh in which transact ion It - I occurre d. where Ihc: [('Iurn is compul ed with Ihe: flllures price recorde d OIl(' minule hC'fore IIle nearcsl roulld millule firiur 10 Ik-' alld the price recorde d five millules hcf()Jc this. mS k _ / : Three lags [I = I. 2. :Ij of all illdicato r variahle: that takes Ihe vaillc + I if the (It - /Jth transact ion price is grcaler Ihan the average or Ihe qlloted bid and ask prices al tilll!' Ik _,. Ihe vallie -I if the (It-I)th transact ion price is less th,1I\ the average of Ihe bid and ask prices at lime I.-I. and zero otherwis e. i.e.,
IBS h _ 1
!
II -I
if
I'. I > ~ (/'k'-,
+ I'~'_I)
if
1'._1
~ U'h'_1
+ 1':'_1)
if
I'k I < W'k'-I -I- 1';_/)'
The spccific ation of X~fJ is then givell by the followillg expressi on: X~fi
=
+ fi~ Yh- I + fi:1 Yh--~ + fi·1 Yk-:I + /I,.SI':,OO._I + fib SI'500.- 2 + fi7SI'500h_:1 + fiXIlISk-1 + fi!,IIIS k- 2 + fiIOIBSh-~ + filiI '/i..(Vk-!l· IBS H I + fil~ 1·I!.(\'k_~) .IIIS._ 2 ) + fil:11 T), (Vk_:l) . IIIS k -:1 ) . fil 61h
The v,lriablt- 61h is illcillde d in .\.
10
allow
fill
clock-lilll!' df(-cls
Oil
Ihe
J. All11kl'l
MiOI lJ/III r·/III l'
rOlu lilio llal lIIeall of I·;. II plic es an' slah le ill Irall saCl ioll lillie ralh er Ihall doc k lillII', Ihis (odf icin ll shou ld 1)(' zel'O. l.a~~l'd prir e chall ~(,s arc illcl ilded 10 aC("OIlIlI lor sni;t 1 dq)( ,lIc1 enci es, and la~~ed reI lints 01 Ill(' SJ(-I':)"/) illd( 'x fllll lln prin ' an' illci llded 10 acco unl for mark ('I-w ide dkC ls 011 prir e challg-(·s. 'Ii) m('asU((' Ihe prir e ililp acl or a Irad e p('r unil \'olll llle, Iht' 1('I"In '1;,(V~_tl is indl lded , whi dl is doll al volll llle Irall sllll' l\l('d ac("ordill~ 10 the Box alld Cox (1!lfi·l) Ilalls ll'I"I lIatio ll '1;,(·); '1;.(\ )
x" I'
whe re I' E 10, II is also ;1 para lll('l er 10 he eSlil llale d. The Box -Cox IrallSf()l"Jllalion allow s doll ,lr voltlll\(' 10 ('n(( 'r into Ihe cond ilion al lIIea ll lIoll lin('arl y,;, parl intla rl), imp orta nt inllo valio n sinc e cOll llllo n intll ilion sugg esls Ihal plic( ' illlp arl ilia), exhi hil ('CO IIOllli('s of scal e with resp eC! to doll ar \'0111111('; i.('., alth oug h IOla ll'ric e imp arl is likely to incr ease with volu me, the lIlar~inal pric( ' illlp;'!'1 proh ahly do('s no!. The Box -Cox trall slim nati capIlIn' s the lil)(' ar sp('c ifica lioll (I' = I) and conc ave spec ifica tion s lipoll to alld illrludin~ till' lo~arilhlllic rlillClioll (I' = 0). The eSli mate d cllrv atur e or Ihis IraIl Sf(.n llalio ll will pl;,y all illlpol"lalll rolt- ill Ihe 1II('''SIl)"{'I I)(,1I1 of pric e illlp act. The Irall sforl ll('d doll ar \'Ollllll(, vari ahk is inle ract ed with IBS k _ I , an indi calo r of ",hel hl'l" Ih(' Irad (' was hll)TI~illitiated (IBS = I), selle r-ini h tiat( ,d IIBS~= - I \. or il)(I( 'I('fl llina ll' (IBS k =/)) . A posi live fill wou ld illlply thaI hlly( 'r-in ilial ed Ilad( 's tl'lId 10 pllsh pri('(~s up and selle r-ini tiate d tlad es t(,lId 10 driv e prir es dow n. Such a rela lion is pred icte d hy seve ral info rm"l .ionhas( 'd lIIod eis of Iradill~, e.g-., Easl ey and O'H ara (I9R 7), Mor eove r, Ihe lIIag llilll de of /ill is II ... p('r-IIl1il \'0111111(' illlp act Oil Ihe cOll ditio llal lIIeall of r~', whic h lila), he f(~adily Ir.lII slale d into the imp act Oil the ('oll dilio llal proh ahil ilics of ohs( 'rv('d pric( ' chan~('s, The sig-n and lIIa~nit\l
a;
III ""'"" ;Ir\" , Ih .. 'hl"I (" S;lI'l iIi, ,lIioll 11''I "ires rill' ('slil llali' "1 of:!· f l'"r;I I"I'tns: Ih(' p;lrl ili .. II """,, cla,· il" UI • ... , (rH, Ih(' vari ance p;lra lllel crs YI alld Y~,
I ~
J.4. UI'I'l'nl Empirical Findin~
141
if ordn"fd !J/1Jbit partition boulIdarUs.
Table}.Ba.
l~.'limfllf.\
IBM
CUE
FWC
HNH
NAV
-4.670 (-145.65)
-6.213 (-IIl.92)
-O71l (-25.24)
-4.456 (-5.91l)
-7.263 (-39.23)
(-56.~5)
-4.157 (-157.75)
-5.447 (-11l.99)
- 1.712 (-25.96)
-1.801 (-5.92)
-7.010 (-36.53)
-7.270 (-62.40)
-3.109 ( -171.59)
-2.795 (-19.14)
1.679 (26.32)
1.923 (5.97)
-6.251 (-37.22)
a.
-1.344 (-155.47)
-1.764 (-11l.95)
4.334 (25.26)
4.477 (5.1l5)
-1.972 (-34.59)
a~
1.326 (154.91)
1.605 (11l.1l1)
1.938 (34.66)
1.977 (62.1l2)
ali
3.126 (l67.IlI)
2.774 (19.1 I)
6.301 (36.36)
5.371l (62.43)
a7
4.205 (152.17)
5.502 (19.10)
7.742 (31.63)
7.294 (57.63)
aH
4.732 (131l.75)
6.150 (11l.!14)
1l.631l (30.26)
8.156 (50.23)
i
Paramcter al O':!
u:\
TI -IUJ73
-5.472 (-63.~) -I. .0
(-61.4H
Maximulll likelihood estimates of the partition boundaries of the ordered probit model for transaction price changes of International Business Machines Corporation (IBM, 206.794 trades). Quantum Chemical Corporation (CUE, 26,927 trades). Foster Wheeler Corporation (~WC. IH,I99 trades), Handy and Hannan Company (HNH. 3.174 tr..des). Navi'Ulr International Corporation (NAV. 96.127 trades). and American Telephone and Telegraph Company (1'. I HO,7'.!ti tr.. des). for the period from January 4. 1988 to December 30.1988.
the coefficients of the explanatory variables fl •. _... fl.3, and "the Box-Cox parameter v. The 5-state specification requires the estimation of only 20 parameters. 'J'hf Maximum Likelihood Rslimales
Tables 3.Ha and 3.1 Ob report the maximum likelihood estimates of the ordered prohit model for the six stocks. Table 3.8a contains the estimates of the houndary partitions a, and Table 3.8b contains the estimates of the ·slope" coelTIcients /3. Entries in each of the columns labeled with ticker symbols are the parameter estimates for that stock; z-statistics, which are asymptotically standard normal under the null hypothesis that the corresponding cocfficient is zero, are cOn\ained in parentheses below each estimate. Tahlc 3.Ra shows that the partition boundaries are estimated with high prl'cision for all stocks ancl, as expected, the %-Statistics are much larger for thost· slOr.:ks with many more observations. Note that the partition bound-
i
."
Table3.8b.
H,lill/l//i·., o/mdrmllJlobil .... ,01" .. mrijlt'iml.,.
mM
CUE
~wc
HNII
NAV
T
YI : AI/100
0.3\19 ( 1:>.:>7)
0.'199 ( 1l.G2)
o:n!i ( 11.2(i)
O.Il37 (1.07)
O.42H (HUll )
O.:IH7 (H.H!I)
Yt :AIL 1
0.:>15 (71.0H)
I.\IU (l5.3!1)
0.72:1 (145'1)
\.109 (4.4H)
O.HW (\\1.\1:1)
(l.HliH elH. IIi)
-0.11[, ( -11.12)
-0.014 (--2.H)
-(l.OI:\ -IU)IO (-:1.:,0) ( -Vi!))
-O.O:I~
( -:tH2)
-0.127 (-!151)
Paramclcr
III : A/flOO
111
: Y.. 1
-1.012 -0.:1:13 -l.:tl:, -0.740 -2.liO\I (-13!i57) (-13.41i) (-24.49) (-[>.I H) (-:\(i.:{2)
fl:, : >'-t
-0.:,:12 (-HS.OO)
fll : 1'.:.
-0.211 (-17.1:,)
fl:. : 51'500_ 1 {Iii : S1'500 .. t
fJ7
: S1'500_ 3
fJK : illS_I fJ'
: l
mS~2
fll :
ms_~
I
\
fJlj
:
't~(V _I )IBS_ I
i tJl~
:
·t;, (V -2)[85_ 2
IIJ : ·t;,(V .. ,)IB5_ 1
"'T
-(l.OOO -O.G:{H -0.40G -1521 -1..112 (-tI.03) (-IGAr,) (-4.O(i) (-31.1:1) (-:,(;.:,21 -(l.III; -OSH; -0.:.01 -0.0211 -O.~~:I (-!I.~:{) (-U14) ( -31.1;3) (- -I7.!1I ) ( -1.42) ~.2!1~
1.120 (54.22)
(1:1.:.4)
-0.:2:,7 (-12.!Hi)
1.:173
0.302
(!l.(iI )
(~.!13)
1I.(i77 (5.1:.)
( 1.!J7)
O.OO!; «(l.~(i)
-2.:I·lli (-ti~.7'1)
1.359 (I :~.49)
O.~04
0.472 ( I.:~{j)
(li.Wl)
tUi2 c, (17.12)
O.14H ( I.~O)
0.1:.0 (2.H7)
0.177 (·I.!/(i)
O.:lHH ( 1.13)
0.1 r.!J
(/.1.1\
(:\.(l~)
n.!/:\)
0.41!1
-l.I:n -UH:> (-('3.1i4) (- " •. 31i) -O.:l1i!/ -0.~79 (-~I.:,:,) (-:1.:\7)
-0.7·1\) -0501 -0.7!JI -O.HO:l (-7.HI) (-2.H!J) (-17.:{H) (-2:1.01)
-0.174 (-1O.2!J)
tU)79 (1I.!lH)
0.122 (47.37)
(12.97)
-().~!l!l -0.177 -0.022 -0.3111 ( -:l.Ii'l) (-0.17) (-I :,.:17) (-I!/.7HI (l.W,() O.03H tUII.1 0.0:12 ( I.HO) (U.5:,) (~5ti) (·I.rll I
U.!H7 (IH57)
(l.O:lt; (2.H:'»
O.III!/ (7.70)
(l,007 (0.5!1)
0.~17
-0.:\70 -0.:1-10 -O.IH1 -O.IH4 (-:\.fiG) (-0.75) (-I:>.:\H) (-IH.II)
(1.03(i (O.!iS) tUlI " -o.()(/(; (15Ii) (-0.34)
0.01:, (1.:,4 )
0.011
0,1/1·1
(~,:,'I)
('U!1)
11.000, (\1.()9)
n.m!)
II.OW,
MOlX;IHll1U likdBu)(){l l·~timat('!\of the ":-.Iopc,''' ("odJiril'Ht!O.ol thl' oH.h.'ICd I)lohit moth'llu .. lIall!'o~
pO". II "~'" ':' """'~"h'...~, ~'.""'~'~ M.,,,,, ...., 0"',, .... ,,," II "". "",."" ....,
"e '.
Ch<'IIIII',,1 (AlI'l'oralloll (CUI'., ~h,~1~7 lI;uk,), ~"'I"r Wh""\pr (.01'1'01;\1'0" (n\( .. lIandy ""d II,Hl""" C""'P;\I,)' (I INII, :1,174 II'"des). N",;sl,,1' Inl<'I'II",i"II"\ Co,! oralion (NAV, 96,127 ,r;u\~,). a"d AlIIl'I'ir;", 'li'I"I,hOl\(' ;\lId TcI<')~""ph Co",I'''''\' (I'. IIlO,7:lC, Ira~\e'), for 11\('I"'l'i",1 trolll.l;llllla ..\, ·1, 19KK 10 llt-r'·lIIb.. 1' :111, l!IKII.
QII" Hum
111.1 19
Irade~).
arie, arc not cvl'nly spaced, e.g., ja:\ -a.a! = 1.7(i:/. whereas ja'l-a,.j = ~.(i7() (it can hc shown that these two values arc statistically
a;
Y;.
Y;
r;.
On]/,/, Now, /)i;(I'plplIl'.B, awl I'ri((' /111/Hlfl More striking is the significance and sign or the lagged price change coefficients fit, fi:\, and E'I, which arc negative f(lI' all stocks, implying a tendency towards price reversals. For example, ir the past three pricc changes were each one tick, the conditionallllean or r; change~ hy IJ~+~:d'~I' lIowevCl', ir thc sequence of price changes was 1/-1/ I, thell the dli:l'l on the conditional mean is ~~-fiJ+fi" a quantity closer to I.!TO fill' cach orthl' security's parameter estimates. :-.Iote that these coefficients llleasure reversal tendencies bryolld that induced by the presellce of a cOllstant hid-ask spread as in Roll (\9H4). The effect of bid-ask bounce on the conditional mean should be captured by the indicator variables mS A_ I , ms •. ~, alld IBS • .:\. III thc abscllce of all other information (such as market 1Il0VCIlIellts or past pricc changcs), these vari,lbles pick up any price dkcts that bu)'S alld sells might have Oil the ronditioll,d mean. A~ expel'led, the l'stim,llc(l coe/lidents arc gener
. ' " " .• " . , , I U / t
illlparl-IIII' dkn 01 a Iralk Oil IIII' lIIark('1 prire-fall hI' 'Illantilied wilh rdalivdy high prl'ci~ioll. il dol's illITl'ase wilh tralll' sizl' althollgh lIot linI'ady, and it dilfns 1'111111 slock to ~tork. The ll10re liquid stocks sUfh as IBM t('ntlto 11;1\'" rd,lIi\'t"l\" 11,11 prin'-illlpan fllnctions, whereas less liquid ~torks sllch as IINII .111' Ilion' s('nsitivl' to tradl' sizl' (SCI', in particu!ar. I {allsmall, 1.0, alld IIla(Kill!a\" II!I!I~, Fi!-\lIn' ,11). Also, disnt'«'IIt'Ss dOl's lIIaUl'r, ill Ihl' SCIlSI' that Ih,' cOllditiollal di~lIihll lioll orpricl' challgl"s illlpli('d by IIH' ol"ll('n~d prohil SllI'cilicalioll call caplure ("('I"laill 1I01lIillcalilies-prin'-t"!lIsl('I"illg Oll,~vell eighlhs Vl'I"SIIS odd eighlhs, fill' ,'xampll'-Ihal oill<'r 1I'("lIl1iqll(,s Sllt'h as ordillary least sflllares call not. While il is slill 100 t'arlv 10 say whelher Ihl' ordered pmhit mOlk! will hOI\'!' hroadl'r applicatiolls ill llIarkl't microstructllre sltlliil's, it is cllrn'lItl), th,' ollly II10dd Ih'lI call capilli"!' t1isnett'ness, irrcglll
:·\.5 Conclusion
\
Then' an' III.UI\" oUlslandillg c("ollolllic aud l'COIIOIlIt'trit" issues Ihat call now hI' n'sol\'l'cI ill III .. "1"1",'1 lIIi.-rosl IIII'I 1111" lilt'rallll"l' thallks 10 Iht' plelhora or I),'wly a\'ail,IIII<' Ir'1I1sa ... ioIlS ,LII"hasl's. III Ihis chapter WI' have 10IH'h('(1 Oil ollly Ihn'" of th,' issul's Ihat an' pari of the hurg"(lIIing markt,t microstructure Iiter.Hure: 1I01lS),llt'itrollOtlS Irading, the hid-ask spread, alld modeling transactiolls dala. IlolI't'ver, lIlt' t'olllhillalion of transactions dalal>ast's and ever-increasing t'OIlIPUtillg )lower is surc to Cfeate many new direCliolls of research, For ,'xalllple. th,' nlt'asllrt'menl anti cOlltrol of trading costs has hel'lI of prilllar)' COIII','rn 10 largt· institlltional inveslOrs, hili thefe has 1,"1'11 rdalil'dy !illle acad,'lIli(' rest'arch dl'votl'd to Ihis importalll topi,' b"elus,' Ihe lH't't'SS'II'1' dala wt'n' 1I11.1\·"ilahlc IIlIlil rccelllly, Similarly, meastlres of markel trallspan'IH'Y, liqllidit\', alld t'oIllJl('titiveness all liguI"I' promillelltl), ill 1'1'(,('111 IIH'on'lit'al lIIodds of senility prices, hilI it has het'll \'irtllally impossihlt' 10 illlpll'III1'1I1 allY of Ihl'sC Ilworks IInlil rl'cemly I){'causl' of a lack of dala. Tilt' t'x!lt'rilllt'llIal markt'ls lilerature has also cOlllrii>lIll'd mallY illsights illiO lIIad,el IlIitToSlrllctlln' issllt's hilI ils ('1I0r1llOIlS P(lIt'lIlial is ollly I){'~illllillg 10 Ill' It·ali,,·d. (;il't'li tiH' growillg illltTl'st ill mar"t't IlIin()~lntC Illn' hy aCldt'lllics, iIlV"SIIIH'1I1 pl'oit'ssiollals alld, 1I10st n'tTully, policYIIl;lkt'I'S ill\'olved ill n'\\'/ ilill),!; secllrilics lIIalkcls regulalions, Ihl' lIext I'cw n',lrs art' surt' 10 I){' ,til t'xllellH'lv t'x!'ilillg and fertile period for Ihis art'a.
\
/lmMI'IIIS-C!lUpter J
\
I
\ \
\ I
I
:~.l
l>tolin' lilt' 1IH'
" luvi"CIIU
145
3.2 Under the nontrading process defined by (3.1.2)-(3.1.3), and assuming that virtual returns have a linear one-factor structure (3.1.1), show how nontrading affects the estimated beta ofa typical security. Recall that a security's beta is defined as the slope coefficient of a regression of [he security's n'lurns on the return of the market portfolio. . 3.3 Suppose that the trading process {8 il l defined in (3.1.2) were not 110, but followed a two-state Markov chain instead, with transition probabil\ties 8il ! given by o !
(3.~.1 ) I
3.3.1 Derive the unconditional mean, variance, first-<>rder autocovari'lIlce, and s~eady-state distribution of Oil as functions of 1rj and 1r;. i
3.3.2 Calculate the mean, variance, and autocorrelation function of the observed returns process Ti~ under (3.5.1). How does serial correlatio'1 in 8il affect the moments of observed returns? I Using daily returns ror any individual security, estimate the paJm(·ters rri and rr; assuming that the virtual returns process is 110. Are the estimates empirically plausible?
3.3.3
3.4 Extend the Roll (1984) model to allow for a serially correlated ordertype indicator variable. In particular, let I, be a two-state Markov with -1 and I as the two states, and derive expressions for the moments of 6P, in terms of s and the transition probabilities of I" How do these results differ from the llD case? How would you reinterpret Roll's (1984) findings in light of this more general model of bid-ask bounce? 3.5 How docs price discreteness affect the sampling properties of the mean, standard deviation, and first-Qrder autocorrelation estimators, ifat all? Hint: Simulate continuous-state prices with various starting price levels, round to the nearest eighth, calculate the statistics of imerest, and tabulate the ("('Ievant sampling distributions.
3.6 The following questions refer to an extract of the NYSE's TAQDalabllJe which consists of all transactions for IBM stock that occurred on January 4th and 5th, 1988 (2,748 trades). 3.6.1 Construct a histogram for IBM's stock price. Do you see any evjc\t-nce of price clustering? Construct a histogram for IBM's stock price rJUlIlj;I'J. Is there any price-change clustering? Construct the following 1\\1(, histograms and compare and contrast: the histogram of price changes conditional on prices falling on an even eighth, and the histogram of price changes conditional on prices falling on an odd eighth. Using these his-
146
J. Markel Miomtrurtllrp
tograms, comment on the importance or unimportance of discretl~ prices for statistical inference. 3.6.2 What is the average lillie hetween trade$ for InM? Cunstruct a 95% conCidence intel"al about this average. Using these quantities and the central limit theurem, what is the probability that IBM docs 1101 trade in aflY given one-minute interval? Divide the trading day into olle-lIIinule intervals, and estimate directly the unconditiunal and conditiol/al probabilities of nontrading, where the conditiunal probabilities arc conditioned on whether a trade occurred during the previous minute (hilll: think abollt Markov chains). Is the nontrading process independent? 3.6.3 Plot price and volullle on the same graph, with tillle-of·day as the horizontal axis. Are there any discernible patterns? Propose and perlonn statistical tests of stich patterns and other patterns that might not be vi$ihle to the naked eye but are motivated by economic considerations; e.g., block trades are followed by larger price changes than nonblock trades, etc-Y' 3.6.4 Devise and estimate a model that measures price impact, i.e., the actual cost of trading n shares of IBM. Feci free to use any statistical methods at your disposal-there is no single right answer On particular, ordered prohit is not necessarily the best way to do this). Think carefully : ahoutthe underlying economic motivation for measuring pric.:e impact.
1 3:7
The following questions refer to an extract of the NYS['s '//\Q /)1I/alm,11' W\ich consists of bid-ask quote revisions and depths for IBM stock that were d splayed duringJanuary 4th and 5th, 19HH (1,327 quote revisions), 3.7.1 Construct a histogram for IBM's bid-ask spread. Can you conclude from this that the dynamics of the bid-ask spread are unimportant! Why or why not? You may wish to constrnct various conditional histognllns to roperly answer this Cjuestion .
~
•7.2 Are there any discernible rclatioll$ between revisiollS ill the bid-ask uotes and transactions? That is. do revisions in hid-ask quotes "cause" rades to occur, or do trades motivate revisiolls in the quotes? Propose estimate a modclto answer this question.
tnd
*.7.3 How are changes in the hid alld ask prices related to vulume, ir at all? For example, do quote revisiolls c.:allse trades to occur, or do trades IllOtivate revisiuns in the quu(es? Propose alld estimate a modclto answ('\' this Cjucstion. 3.7.4 Consider an asset allocation rule ill which an investor invests fully ill stocks until experiencing a scquence of three consecutive dedilles, .tfter "'The NYSE deline> a block trade a, any tr;lde (O",;,tillK of 10,000 ~har("' or mor,·,
147 which hc will switch complet ely into honds lin til cxpcricncin~ a scquenc e of six rlJlI.ll'(ul iw advance s, Implelll ent this rille for .111 inilial investm ent of SI 00,000 wilh the transacl ions data, hill do il two ways: (I) lise the avera~e of the bid-ask spread fi,,· pllrchas es or saks; (:!) lise the ask prilT for pllrchas es ancl the bid price (('I" sales. Ilow IIlllch do YOIl have left at the end of two days of trading? YOII may a"I1I1I<' a I.ero riskfrcc rate ((, .. this exercise ,
•
4
Event-Study Analysis
ECONOMISTS ARE FREQUENTLY ASKED to measure the effect of an economic event on the value of a firm. On the surface this seems like a difficult task, but a measure can be constructed easily using financial market data ill an event study. The usefulness of such a study comes [rom the fact that, given rationality in the marketplace, the effect of an event will be reflected immediately in asset prices. Thus the event's economic impact ("all he measured using asset prices ohserved over a relatively short time pcriod. In .contrast, direct measures may require many months or even years or observation. The general applicability of the event-study methodology has led to its whle lise. In the academic accounting and finance field, event-5ludy IIlcthodology has been applied to a variety of firm-specific and economywide events. Some examples include mergers and acquisitions, earnings announcemen!S, issues ornew debt or equity. and announcements of mac roe[onomic variables such as the trade deficit. I However. applications in other fields are also abundant. For example. event studies are used in the field of law and economics to measure the impact on the value ofa finn ofa change in the regulatory environment,2 and in legal-liability cases event studies are used to assess damages.~ In most applications, the focus is the effecqof an event on the price of a particular class of securities of the firm, most often common equity. In this chapter the methodology will be discussed in \erms of common stock applications. However, the methodology can be applied to debt securities with little modification. Event studies have a long history. Perhaps the first published study is Dolley (1933). Dolley examined the price effects of stock splits. studying nominal price changes at the time of the split. Using a sample of 95 splits
IW.. will rurther discuss Ih,' firsllhr~.· "xamples laler in Ihe chapler. McQueen anrl,Rolry ( I ~I~):~) provide an ilIusfration using macrocconomic new.IIi annnuncement5. ~S~~Schwnl (19HI). "See Milchell .1Ilt! Neller (1994).
149
15U
frolll 1921 to 1931, hc found th
~
4,1 Outline of an Event Study AI Ihe outset it is nsefnl to gi\"(~ a brief ollllinc of the structure of an ('\"('111 study. While thcr!' is no uniqut' strtl('lun~, the ,lIIalysis ("an he \"inwd
4.1. O/ltlill~o/alll~vf1lt Stw/.V
l!il
as h,lvin!{ seven steps: I. 1:'wllt t/llil/iti()l l. The initial task of cOllductill!{ an ('velll stlldy is to define the l'vcnt orintere st and identify the period over whirh the security prices of the firms involved in this I'\'l~nt will hI' l'x,lInin l'd-the I'llf'l/t willlimll. For example , if one is looking at the illf<)rJnation content of an earnill!{S announ cement with daily data, the event will he the earnillgs announ cemelll and the l'Wlll window might he the onc day or the annOllll celllenl. In pr.lcticc , thl' event window is ofkn expand ed to two days, the day or the anllOlln Centent and the d,IY after the ,1IInOUIK eIllt'nl. This is donc to captllre the price errects ofannou nlTml'n ts which occllr after the stock lIIarket closes on the announ cement day. The period prior to or after the evcnt lIlay also he or interest and included Sl'par'llely in the analysis, For example , in the l'arnin!{ s-annou neemen t case, the market may aCfJuire informa tion about the ('arnin!{s prior to Ihe actual announ cement and one call illv('stigatl' this possihili ty hy l'xaillinin!{ pre-even t returns. ~ . .)l'lrrliol1 rrilrria. After identify ing the evellt or illtCn.:st, it is Ilcccssar y to determi ne the selectio n criteria for thc inclusio n or a given linn in tltl' study. The criteria lIIay involve restriCliolls imposed by data avail,lhility such as listing Oil the NYSE or AMEX or ma), involve restricti ons such as llIembe rship in a spccific induslly At this stage it is Ilseflll 10 sllllllll'lrir.e some characte ristics of the data sample (e.!{., finn market capitaliz ation, industry represel ltation, distribu tion of events through time) ,Ind lIote allY potentia l biases which m'l}' hav!' 1)('('11 illtrodu ced through the sample selectiol l. :1. NOn/wi al/d almo17l1lli 7l~tllnlS. To appraise the I'\,('nt's impact we n'lltlire a measure of the abnorm al relllrll. The abnorm al return is the actual "X /lost return of the security over the event window minus the normal return orthe firm over the event window. The normal return is defined as the return that would be expecte d if the e\'l'nt did not take place. For each firm i and event date r WI' hav('
<, <,,
= il" - E[lI"
I X,].
('1.U)
whl're I!", and E(il,,) arc the abnorm al, anllal, and normal returns, rcsplTti vdy, for tillle period t. X, is the l'IlIlditionin!{ inl()J'Itla tion for the norlllal perform ance lIIodel. There al'l' two COlllnlOU choices for lIlodeliu g the nonn;d rt'llIl'11 -the rOl/.lt{/I//-III1'III/·I('/lIrJ1 lIIodl'! whnl' X, j, a constan t, and the lIl(l1krl lIwdl'l where X, is the IIlarket return, The COl1stanl-IIII';tn-return model, as the n,lIne inlplil's, as,ulII('s that the IIlcan return 01'.1 given secllrity is constan t through tillll'. Th(' mark('t nlodel assumes a stahle linear relation betwcl'n t h!' IIlarkcl return and till' secllrit)' rl'turn.
4, /:1'1'111-.'11111/,1' :\//(//)'1;1
'I. /':,1;111111;"/1
/W/(/'dllli', (JIIl'1' a lIonnal pI'I'lill'lllallCl' lIlodl'l has hl'l'lI SI'"'I"II'd, Ihl' p;ILlnll'lns ol'lhl' nllHlcllllllSI Ill' \'Slilllatt'd \lsillg a suhSl'1 01'1111' I\;(LI \..11011'11 ;IS IIII' 1'.\1;lIIl1lill// W;IIt!Ol/l, Tht' mosl CtlllllllOlI r\wict', WII<'III",I,iblt-, is lOllS,' IIII' 1'1'1 iod prior 10 IIII' 1"'l'III willdow I'ollhl' I'sliIlialiollll'illdOl", For I'X;lIl1plt-, ill all 1"'1'111 SIllIly IIsilig Ilaily lIala alld Ihl' lIIarkl'l IIH1,kl, iiII' 11I;II'kl'l-llilHII'I param{'ll'ls cOllld 1)(' ('slimall'd oVl'r IIII' I:!() davs prior 10 Ihl' ""1'111. Cl'lIl'rall: Ihl' l'I'I'1I1 pl'l'iod ilsdf is 1101 illdlllll'li illdll' I'slilllalioll plTiod 10 prl'VCllllhl' l'V1'1i1 I'rolll illflllelicillg iiII' 1101'111;11 111'1'101'111;1111'1' 1I11.d'" l'arallll'll'l' l'slilllall'S, :1, '/i'llillg/",/(,'dllli', \\'ilh Ihl' P;II;IIIII'II'I' l'slilllall's IiII' Ihl' lIonll;a! pnli)l'IIlalln' IHodel, IIII' ;lilllormal rl'lurllS I'all hI' calculall'd, NexI, \I'{' IIlTd 10 dl'sigll Ihl' (l"lillg I'rallll'\\'oll. lill' Ihe alllloflllal rl'llIl'IIs, ilnporlalll l'ollsidnaliollS an' ddillillg 11ll' 111111 hYjlotlll'sis alld dl'll'rlllillillg Ihl' 1('dllliqlll'S lill' aggl'l'galing IIII' allllol'lIIal relllrns Orilldividllallinlls, Ii, /':I11/Jiriml li'IIIII" Thl' PI'I'S"III:!lioll of Ihl' t'mpil'ical I'{,SlIlts rolloll's Ih{' formilialioll 01'1111' ('COIIOIIII'll'ic dl'sigll, III addilioll 10 PI't'St'lIlillg IIII' basir I'mpil it"al n'sillts, III<' Pl'l'Sl'lIlalioll of diaglloslics call hI' I'I'Uilflll. (kclsiollalh-. ,'sl)('('iallv ill sludil's wilh a limill'd III IIII her 01"'1'1'111 obsl'l'";lIiollS, IIII' ('llIpirie:!1 I'('SlIlts call be hl'oll'ily illlhll'lHTd by OIH' or Iwo linns, Kll1lldl'dg" or Ihi' is illlportalil fi.1' J!;auJ!;illJ!; Ihl' illlport;III
4.2 An Example of an Event Study '1'111' Fill
4.3. M()(/,lJ Jar Mrasunng Nanna/ Prrjonnanrr each firm and quarter, three pieces of information are compiled: the pate of the announcement, the actual announced earnings, and a measure of the expected earnings. The source of the date of the announcement is Datastream, and the source of the actual earnings is Compustat. . If earnings announcemenl~ convey information to investors, one wiuld expect the announcement impact on the market's valuation of the firm's eC]uity to depend on the magnitude of the unexpected component of the announcement. Thus a measure of the deviation of the actual announced earnings from the market's prior expectation is required. We use the mean quarterly earnings forecast from the Institutional Brokers Estimate System (I/R/E/5) to proxy for the market's expectation of earnings. I/8/E/S cO[llpiles forecas15 from analysts for a large number of companies and reports sumlllary statistics each month. The mean forecast is taken from the last month of the quarter. For example, the mean third-<Juarter forecast from September 1990 is used as the measure of expected earnings for the third C]uarter of 1990. In order to examine the impact of the earnings announcement on the value of the firm's equity, we assign each announcement to one of three categories: good news, no news, or bad news. We categorize each announcement using the deviation of the actual earnings from the expected earnings. If the actual exceeds expected by more than 2.5% the announcement is designated as good news, and if the actual is more than 2.5% less than expected the announcement is designated as bad news. Those announcements where the actual earnings is in the 5% range centered about the expected earnings are designated as no news. Of the 600 announcements, 189 are good news, 173 are no news, and the remaining 238 are bad news. With the announcemen15 categorized, the next step is to specify the sampling interval, event window, and estimation window that will be used (0 analyze the behavior of firms' equity returns. For this example we set the sampling inten'allo one day; thus daily stock returns are used. We choose a 4 I-day event window, comprised of 20 pre-event days, the event day, and 20 post-cvent days. For each announcement we use the 250-trading-day period prior to the event window as the estimation window. After we present the methodology of an event study, we use this example as an illustration.
4.3 Models for Measuring Normal Performance A Illllnbcr of approaches are available to calculate the normal return of a givcn security. The approaches can be loosely grouped into two categories-;statistical and economic. Models in the first category follow from statistic~1 assumptions c.oncerning the behavior of asset returns and do not depend on
.154
4. Event-Study Allfllysi.l
any 1conomic arguments. In contrast, models in the second category rely on a~sumptions concerning investors' behavior and are not based solely on stati$tical assumptions. It should, however, be noted that to use economic models in practice it is necessary 10 add statistical assumptions. Thus the poteintial advantage of economic models is not the absence of statistical aSsUiPtions, but the opportunity to calculate more precise measures of the nor al return using economic restrictions. or the statistical models, it is conventional to assume that asset return arejointly multivariate norlllal and independently and identically distrib ted through time. Formally, we have:
(All
Let R, be an (N x I) vector of aJset relllrns fur calendar time Ileriuti t. R, i.l inde endl'1ltly multivariate )!anllally distributed with mean /1. and COIJarianfe matlix
n
fot flll t.
l
This distributional assumption is sul'ficient for the con~tant-lIIean-relllrn modrl and the market model to be correctly specified and permits the development of exact finite-sample distributional results for the estimators and statistics. Inferences using the Ilormal return lIIodels are robust to deviations from the assumption. Further, we can explicitly accol1lmodate deviations using a generalized lIlethod oflllOlllenL~ framework.
4.3.1 CUll.llrmt·Mrnll-Ul'tunl Model Let /1.;, the ith element of /1., be the mean return for asset i. Then the constant-mean-return model is (4.3.1)
El slI ] = ()
Var[s,tl
where Il;" the ith clement ofR" is the period-t retun! on security i, ~" is the disturba/lce term, and a~~ is Ihe (i. i) clement of n. Although the constant-meall-return model is perhaps the simplest model, Brown and Warner (I9HO, 1985) lind it often yields resu1L~ similar to those of more sophisticated models. This lack of sensitivity to the model c1lOice can he attributed to the faCl that the variance of the abnormal rel\lrn is frequently not reduced much by choosing a more sophisticated model. When using daily data the model is typically applied to nominal returns. With monthly data the model can be applied to rca! returns or excess returns (the return in excess of the nominal riskfree return generally measured using the US Treasury bill) as well as nominal returns.
".3. lVIIJ!lrLI.!;J/· MPtl.llllillg NIJnnall'njll/lI/f/I/i/'
l!i!i
4.3.2 Mil/hI'! MIJ"d
The market model is a statistical modd which rdatcs the retlll'll or allY givell security to the return of the lIIarket portfolio. Thl' lIIodel's linear Sl)('cificatioll f()lluws from the assllmed joillt Ilormality of asset rcturns:1 For any security j we have
H"
=
(X,
+ {l,U"" + E "
where Ji alld N"" are the period-t rc!lIl'11S Oil scclII'ity i alld the market " portrolio, respectively. and f;, is the zero meall disllIrhallcc term. (X" fi;. alld (1f~ are the parameters or the market model. III applirations a broadhased stork illdex is used ror the market ponf()lio, with the S&P!iOO index, the CRSP valllc-weighted index, alld the CRSI' eqll;li-weiglitcd index heilll-: popular choices. The market model represellts a potclltial illlJ)rovemellt over the CUIlstant·mean-return mudel. By remuvinl-: the portiun uf the return thal is related to variatiull ill the market's returll, the variance of the abnormal rCllIl'Il is reduced. This can lead to increased ahility to dcteCl evellt effects. The h(,lIdit from using the market model will depelld IIpOIl the Ji2 of thc lIlarket-lllo(kl regressioll. The higher the Ji~, the greater is thl' variallce reduction of' the abnormal returll, and Ihe la, gcr is the gain. Sel' Sel'lion 4.'!A for nlore discllssion of this poillt.
".3.3 ()thrr S/Illi.ltira/ Mo"dl A number uf other statistical mudds have hc('n proposed Ii)r mudeling the nurmal return. A general type of statistical model is the Jar/or model Factor models putentially provide the henefit of reducing the variance ur the abnormal return by explaining mure of the variation in the norl1lal return. Typically the ractors are portfolios of traded securities. The market model is an example of a onc-f;\ctor l\Iodel, but in a \\Iultifactur model one mi~ht include industry indexes in addition to the market. Sharpe (1970) and Sharpe. Alexander, and Bailey ( I \195) discuss index models with factors based un industry classification. Anuther variant uf a bctor model is a procedure which ca!culates the abno\'\\Ial retul'II hy taking the difference between the a(\ual retul'll and a ponfolio of [inns of silllilar sit.e, where si/.c is mcasllrcd hy market value of <,<)uity. III Ihis approach typically tell Sill' groups arc cunsidered and the loadill~ Oil the sil.e portfolios is restricted "The spt·rilircttion OIctl1
they h"ve lillie dkct
Oil
('lllpirical work.
to Hllit)'. This pmcetllll'l' implicitl\' aSSllmes that expected r('tlll'lI is t1irl'nl\ rdaled 10 Ih(' III;uk!'1 \';1111(, of eqllity. III praclic(' Ih(' gaills from cmployillg milltifactor models for ('I'CIII silldit'S arc limit!,(\. The n'ason t(lI'lhis is Ihal the llIar~illal explanalory power of ad(litiollallil('\ors 11I'yolld Ihl' market factor is small, and hcnce there is littlt' r('duClioll ill Ihl' 1':1Ii;IIHT of III<' ahllormal I'l'l Ill'll. The variallce r('dunioll will Iypically h(' g;rl'alt'SI ill cases 1\'111'1'1' Ih(' sample linns hal'<' a COlllmOlI charat'lnislic, for example Ihey aI'(' all 1I1('mhcrs of one indllslry or Ihey art' all linns con('('lIlrall'(1 ill OJle market capitalization grollP, 111 Ihese casn Ihc lise or a mllltif;\('lor mod .. 1 walTallls cOllsideralioll. SOllll'lilll(,s lilllil('d dala al'ailahilily lIlay diclalc Ihc IIS(' of a reslricled llIodd sllch as Ihl' I//m!w/-llIljIH/nl-rrllll'll IIIIIt/rl. For som(' ('vellts it is lIot f(:asihk to haw a pn'-('I'('nl ('stimatioll pniod for the normalmodd par:lllletns. alld a market-adjllste(1 almorm:ll retlll'll is IIsed. The markehl<\jllslcd-rCI\II'1I modd can hc I'iew('d as a reslricled lIIarkcl model with ex . cOllslraillt'd 10 he () and fl, conslrained 10 hI' I. Since Ihe llIodel cocfficients arc prespedfied. all cstimatioll pniod is lit)! required to oht,Iill parameter estimates. This modd is oftcn IIsl'd 10 stllt!y Ihe IIlIderpricing of initial pllhli(' offerillgs.'· A g('lIt'1'al I'('comnH'IHlation is to tlS(' stich reslricled models only as a last resort. alld to keep ill lIlilid that hiases lIlay arise if the restriclions arc false.
Economic Illodcls reslrict Ihe parameters of stalistical models \0 provide \\lore conslrained norlllall't'tlll'li IIH)(lds. Two common cconomic models which provide restrictions are the Capital A~scl Pridn~ Model (CAPM) and exacl vcrsions of Ihe Arhitrage Pricing Theory (APT). The CAI'M, (hie to Sharpe (191;1) alld I.illll\('r (I!}(i;lh), is all e(JllilihriulII thcory whne Ihe expccled retlll'll of a gil'clI assel is a lillear functioll of its covariall('e wilh Ihe 1'('1111'11 oftl\(' \\Iarkel port/illio. The AI'T. dill' to Ross (197G). is all asset pricing' Iheory whl'I'(' in Iht' ahsellcl' of asymptotic arhitra~e the cxpl'ctl'd r('tllrn oC'a gil'l'n assl'l is dl·tel'lnined hr its nll'ariances wilh multiple bClOrs, Chapt<'l's !'i and Ii plOvidl' ('xll'nsiv(' treatnwlI\S of these IWO Iheories. Tht' Capital Asst't I'riring Mot\1'l was ('()(Jllnonly used in evellt studies durillg the I !170s. Durillg Ihe last 11'11 years, however, deviations from lhe CAI'M have he('11 discOl'I'll'd. alld Ihis casts doubt on the validity or Ih(' restrit'liollS imposed hy lIlt' CAI'M 011 tht' markel mode\. Sillcc Ihest' ICstriniolls CIII 1)(' rdax('(1 at lillie ('osl hI' IIsillg Ihl' IIIarkel IIIodd, the liSt' or the CAI'M ill e\'(,111 stlltlil's has allllOst c('ased. SOIlI(' silldi('s haw IIsed IlIlIltirat'lor norlllal performallce 1II011cis 1110tivated hI' the Arhilrag(' I'l'it'illg Th 1'0 II', Thl' APT call he lIIade 10 fil lilt'
-1.-1. Mealunllg and Analyzing Abnonnal Returns
Tillie Line:
, \
(
est~malion ] wmdow
event ] ( window
po~t~venl ] ( Window
o r • Figure 4.1.
Ti= l.ine Ja,. an Eumt Study
cross-section of mean returns. as shown by Fama and French (1996a) and others. so a properly chosen APT model does not impose false restrictions on mean returns. On the other hand the use of the APT complicates the implementation of an event study and has little practical advantage relative to the llnrestricted market model. See, for example. Brown and Weinstein ( 19R!i). There seems to be no good reason to use an economic model rather than a statistical model in an event study.
4."4 Measuring and Analyzing Abnormal Returns In this section we consider the problem of measuring and analyzing abnormal returns. We use the market model as the normal perfonnance return model. bllt the analysis is virtually identical for the constant-mean-return model. We first define some notation. We index returns in event time using r. Defining r := 0 as the event date. r = 1'1 + 1 to r = 1'2 represents the event window. and r == To + 1 to r = 1'1 constitutes the estimation window. Let LI = T t - To and l.Jl == 71. - 1'[ be the length of the estimation window and the event window. respectively. If the event being considered is an announcement on a given date then 1'2 == 1'1 + 1 and L;. = 1. If applicable. the post-event window will be from r == 1'2 + 1 to r = 1', and its : leni!;th is L:\ = 1~ - 71.. The timing sequence is illustrated on the time line . ill Fi~\lrt· 4.1. We interpret the abnormal return over the event window as a measure Oflh(" impact of the event on the value of the firm (or its equity). Thus, the Illcthodology implicitly assumes that the event is exogenous with respect to lhe change in market value of the security. In other words. the revision in V
. dAi ri .' J..~-~;f. .. ,,':··~~.;t' .;;~~.
r;~' .~:.
4. HlIflll-SIIll!y illI(/!y.\i~
158
the event is endogenous. For these cases, Ihe usual inlerprelalion will be incorrect. It is typical/or the estimation window and the event window lIotlo overap. This design provides estimators for the parameters of the normal return odd which are not inlluenced by the event-related returns. Including the vent window in the estimation of the normal model parameters cOllld lead o the event returns having a large influence 011 the norlllal return Ille
'1',hr I'vIflr/{1'l Mudd
Recall that the market model for security i ami observalion r in event time is (4..1,1) U" = u, + Ii,RIIIT + (", The estimation-window ohservalions call he expressed as a regn'ssioll system, R, = X;O, + {" where Ri ::::: [R;7i,+ I ... Ri'li l' is an (1'1 xl) vector of estimation-window returns, Xi::::: [L R.,] is an (1'1 x2) matrix with a vector of ones in the lirst colulHn and the vector of market return observations R,. [U,.7i,+I·' ,Il"'/i J' in thesecondcolulllll, allll (), =:: [u,fi,l' isthe (2x 1) parameterveclOr. Xhas a subscript because the estimation window lIlay have timing that is specific to firm i. Under general conditions urdinary least squan:s (01.5) is a collsistent estimation procedure for the market-Illudel parameters. Funher, /-(ivelJ the assumptions uf Section 4.3, 01.5 is efficient. The 01.5 estimalors of Ihe lIlarket-model parameters usill/-( all estimatioll window or LI oilserv
=
0,
=
II
---ti.
1-1 -~ , '
R, -
Vade,]
X,B,
I \V~ next show how to use these OI.S estimators I
i
(1.4.4)
(4.4,ti) to l1leasur(' Ihe slatistical
'1. ·1. Mf'{/.\IIl'ill~ (II/(I AI/(I~l'Zill~ tlil/Ilml/( I/
/{('/l/m.1
1!i!1
propcrti es of ahnol"ln al rdurns. First w(' consid(, r till' "hllornm l return proJlcrti es of a given security "nd tlll'n we aggrq;a tl' "lTIIliS senlrilic s. 4.4.2 S[ali.llim ll'm/wr/il 'J I{ A/IIIII/I/lft! N,'IIIIII.I
(;i\'('ll Ihl' lll
€;
== wh('l'('
R;
=
R'I -X'O I /.
(4.4.7)
I U,r,+1
... UI/'1' is all (i,!x I) V('("(OI' of ('I'l'lIt-w illdow returllS, is
X; == It R;"I
E[R; -
X;O,
I
X; 1
I':[(R; - X;O,) - X;(O, - 0,)
o.
I
X;
1
(4.4.8)
I is Ih(' (I . ! x I ..J identity matrix. From (·I.·l.l'l) we sec Ihatth(' ahllol'lllIIoJ'lI Ial return vector frolll (-I.·U)) has two parts. 'I'll(' lirsl I('r11 I ill th(' Slllll is Ihe variallc( ' due to lhl' futllre disturha llces alld th(' '('C() III I tnlll i~ lIte addilion al varianc(' dll(' 10 IIt(' salnplin g l'rror ill (),. This .'
fill' allthc dCIIH'IIIS olthe ahllonllalrctllrn vector, wililcad to s('rial correlalion of Ihe ahllonll.d reI urns dcspile lhe bet lhal lhe lrue dislurbances arc indqlCndellt Ihrollgll lillie. As Ihl' length of the cstilllatioll willdo\\' 1,1 Iwcolllcs large, IiiI' second lerlll will approach zero as till' salllpling clTor of lhl' paralllell'l's vallishes, alld Ihe abllorlllal relurns arross lime periods will hecolllc illliq)(,lIdclIl ;tSl'lIIplOlicalll'. 1I11dl'l' tlie lIull hl'polhesis. 11 11 , Ihat the givcn ewnl has 110 impaci Oil IIH' nlt'an or variancc of 1I'lurns, wc call lise (4.4.H) and (4.4.9) and Ihcjoilll normalily of Ihe ahunrmal rei urns 10 draw inferl'lIces. Ullder Ilu, for the \'cflol' of e\'l'lll·window salllpll' ahllol'lllal rt"lIlnlS Wl' 'have ('1.4.10)
F(luation (,1..1.10) gil'es us IIJ(' dislrihution for any singk ahnormal rl'tul'll ohsl'I'l'alioll. We ncxl hllild on Ihis resul! and cOllsider Ihe aggregalion of ahnOrlllall'l'turlls.
·1. '1.1 :\g.t:'fl'gillillll II/ Abl/lInllll[ Urlllnls
The almonnal rt'llIrn ohsn\'aliolls mllsl he aggregated in order to draw o\'erall inli-n'lln'o; ItH' III<' ('\'elll or ill\<·I'l's\. The aggregalion is alollg two diml'nsiolls-through time and across securities. We will (irst considcr aggregalioll Ihrough lilll!' for all iuliil'idual secllrity alld Ihell will consi(\er aggregalioll hOlh across S('('urilies and Ihrough lillie. We inlroduCl' Ihl' l'Iulllllali\'c ahnormal reI urn 10 accolllllHHial1' Illultipk salliplilig illll'l'\'als withill the eve II I window. Defille CAR/(rl, r~) a~ the 1'llllllllaliVl' ahnol'mal relllm fill' security i from rl 10 r2 where '1'1 < rl :::: r~ :::: ·/~. l.l'I, he all (I 'J X I) v('clor with Olil'S ill posilions rl - '1'1 10 r~ - '1'1 ;lllI\1.l'rm's ds('wlwfl'. Th<'11 Wl' ha\'e
,-.
('1.4.11 )
1"/ V"r\(:A\{'(TI.
T~)\ = O/~(TI' r~)
= ,'ViI'
(4.4.12)
I! ii,\Iows frolll (·1.·1.10) Ihal ulHlcr II",
VI'., (';111 cousln, .. 1 a It'sl (.1' II" 1'01 se .. ulily i from (4.'1.1 :') lIsing th., SI,lIld,ll'di/l'd nl'lnillali\'(' alHlol'llial 1<'111111. SO\R,(rl. r~)
CAR,(rl'
T~)
(T/(r\, r~)
(.1. .!.l.! )
",herea/(rl, re) iSl'OlkuLlled wilh o,~ frolll (4.4.4) SlIhSlilUll'd li)r(1,~. L'nder lhl' nlll1 hVl'ollll'si'; IIII' dislrihulioll ofS{:AR;(rl' r:!) is Sludelil I wilh 1.\ - 2
4.4. Mfll.'iuring and Analyting Abnonnal Returns
161
degrees of freedom. From the properties of the Student t di~tribution. the expectation ofSCAR;(rl. r2) is 0 and the variance is (~). For a large estimation window (for example, 1.1 > 30), the distribution ofSOO;(rl. r2) will he well approximated by the standard normal. The above result applies to a sample of one event and must be extended for Ih(' uSIIal case where a sample of JIlany event observations is aggregated. To "ggregate across securities and through time, we assume that there is not any correlation across the abnormal returns of different securities. This will generally he the case if there is not any clustering, that is, there is not any overlap in the event windows of the included securities. The absence of any overlap and the maintained distributional assumptions imply that the abnormal returns and the cumulative abnormal returns will be independFnt' across seCllrities. Inferences with clustering will be discussed later. ! The individual securities' abnormal returns can be averaged using frolll (4.4.7). Given a sample of N even ts, defining f· as the sample averitge of the N abnormal return vectors, we have
i;
(4.4.15) \
! (4.4.16) I
We can aggregate the elements of this average abnormal returns vectpr through time using the same approach as we did for an individual security's vector. Define CAR(, •• '2) as the cumulative average abnormal return fr<\m 'I to ,~ where TI < " :5 '2 :5 1~ and 'Y again represents an (/1), xl) vect~r with ones in positions " - T. to '2 - T. and zeroes elsewhere. For the cUlIIulative average abnormal return we have (4.4.17) (4.4.18) Equivalently, to obtain CARCr •. r~), we can aggregate using the sample cUlllulative abnormal return for each security i. For N events we have (4.4.19)
(4.4.20)
162
4.
J~III'11I.SllUly
'\I/(/IY.I;'I
In (4.4.16), (4.4.18), and (4.4.~O) we use the assumption that the event windows of the N securities do not overlap to set the covariallce terllIs t<) zero. Inferences about the cumulative abnormal returns can be dra\"llllsill~ (4.4.~
1)
since under the null hypothesis the expectation of the ahnormal returns _2 is zerD. In practice, since a- 2(TI, T2) is unknowll, we can lise a (TI, r~) = ;& L::I u;(rl, r2) as a consistent estimator and proceed to test II" using CAR(rl r2)
11 ==
2
[J
•
I
a
~
N(O,I).
(rl' r2)p
This distributional resuit is for large samples of events and is lIot exact because an estimator of the variance appears in the denominator. A second method of aggregation is to ~ive equal weighting 10 the indio vidual SCAR;'s. Defining SCAR( rl. r2) as the average over N secnrities frol\l event time II to T2, we have
*
N
SCAR(Tt. T2)
=
I)fu,(TIo r2)'
(4.4.~3)
,=1
Assuming that the evellt windows of the N securities do not overlap in calendar time, under 11o, SCAR(r,. r2) will be normally distributed in large samples with a mean of zero and variance (N;C:1)' We can tesl the null hypothesis lIsing
12
= (
NU., 1.1 -
4»); __ I
,SCAR(r,. r2) .::. N(O. I). ~
When doing an eventsllldy one will have to choose betweellllsing 11 or 12 for the test statistic. One would like to choose the statistic with higher power, and this will depend on thc alternative hypothesis. If the tflle ahnormal return is constant across securities then the better choice will give more weight to the securities with the lower abnormal return variance, which is what Jl docs. On the other hand if the true abnormal return is larger for \ securities with higher variance, then the beller choice will give e(l"al weight i to the realized cumubtive abnormal retllrn of each security, which is what 11 Idocs. In most sllldies. the result.~ ,Ire notlikcly to be sensitive to the choice of 11 versus /~ because thc variance of the CAR is of a similar magnitude across SeCllnl1es.
l I \
4.4.4 .\'cIHililll(V 10 Normal Ul'lllni I'vlodel
I\we have developed reslllL~ using the market model as the Ilormal retLlnt model. As previollsly noted, Ilsing the market Illodrl as oppmcd to the
-I. ,1. Mm.\111711K mul AII(llyzi"K AIJlltlnll{/llIl'lllru,1
IIi:!
,I
constant-mean-return model will Ie,HI to redlldioll in Ihe ahnormal ret\lrn variance. This point call he shown hy C()nlp'lrin~ the ahnormal rclul"\l variances. For this illustr,ltion we take the llormal returll model parameters ,IS gIven. The variance 0(' the alJllortnal rellirn l(lI' the market model is
a,~
Yar[ [(" Yarl RII I
(x, -
fl, Um ,\
- Ii; Y,III Umll
(I - U;lYar[UIIJ.
(4.4.25 )
U;
where is the /{2 of the markel-lIIodel regressioll for security i. For the cOllstant-mean-rellirn model, the variance of the abnormal retUI'll ~" is the variance of the uncolHlitional retul'll, Yar[ U" J. that is,
o~~
:=
YarlU" -II,!
:=
Yar[Uill.
(4.4.26)
Combining (4.4.25) and (4.4.2li) we have
(4.4.27)
H;
Since !;es betwccn zero and one, tbe variance or the abnormal return using the market model will be less than or eqllal to the abnormal retllrn variance using the constant-mean-return model. This lower variance for the market model will Gln'Y OVCI' into all the av;v;n-v;atc abnormal retlll'll lIleaSllres. As a result, usinv; the ",arkel ",,,dd .-all lead 10 llIore precise infl~rellres. The v;ains will he v;reatl'st for" '''ll\plc of "'<:Ilriti,'s with hiV;h lIlarket-model U2 statistics. III principle further illcreases in U 2 cOllld h(' achieved by using a Illultiractor model. In practice, howevCJ', the g.lins in /{2 fwm adding additional ractors are usually small.
4.4.5 C1R.I for the 1~'(/rnil/g\-AII1/oll/l(flllfllt Exam/lie The earllin~s-anno\lllcement example illustrale, the use of sample abJlorrnal returns and sample cumulative ahnormal retUrlls. Table 4.1 presents the abnormal returns averav;el! across the :~() lirIllS as well as the a"cra~ed cIIllItdative almormal relUnl I()I" each of Ihe IhnT earnings news categories. Two norlllal return models are considered: the Illarket model and, for comparison, the constant-Illean-rctlll'n llIodel. 1'1015 of the ctllnulative al)normal retul'llS arc also included, wilh the CAR" hUIIl the market lIlodel in Figure 1.201 and the CARs frolll the cnmlallt-IllI',lI\-1('tUl1l model ill Figure 4.2h. The results of this example ar(' lar~ely nJllsisl(,lI1 wilh the exisling- literature on the inforlllation ronll'nl of' earllin~s, The ('vidcnce slrongly
Table 4.1.
..lhI/lJUI/I/III'IIIu/'
/111" (/1/
(1'1'111 ,111111'
"lIlit' ili/iJUI//lliulI
rolllfill "l('(lft/illg' /111-
11011111""""'11/\, \1;1It...'1
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i'
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.Will
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--,177 - "HI
(lIH
,(I!IH ,11(1 - .1).11
- IHII
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-.'!.:\r)
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-- 11111
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"II -1:\ ··1!! -II -1(1
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n()~
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:! FI:\ - II'.!!) .OX) '.!.:!'\j .017 .\7'! '.! .. \("' .W"
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-.ti:I'.! -,li77
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'1'1"
.:H'.! --.:111 .:U I .:\17 .~h:~
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-,I""
,U:II
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.fi!",·' -.170
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-,~12 -L~i:1
.or...
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I.~IX:,
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~,:I~(i
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I :~II - 1.07'\ ...
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- ,II:IH - 1.11'1 ,1171 --I.IIIH .uP) -I.II'!." -,111:1 • 1.117~
-.OMI; - I Jrl!' .0:,0 1.'.!II!'i
,."'!
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-,liH7 ,X71
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~.:n·1
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-.1171 -1 I/.-,Ii ,'.!Iii - 7XQ .Ollti
. IX:\
.1117
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'!..o~lt;
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-.ilili -,ti:I'.! - .~IIH
-.II:!:!
-.:,WI
-,IIHI
·-.(jiB
-,11',,1
- i:!·1
-,11,1 .O:!1i -,II'•
-,~x'i
-(\(i7 -,,\:.1 -.O'!.I ·... 17:, -.017 ~.OtiM -.U:I~I -':1:\·1 ,111:1 ~,171 -,II'lii -,',HII
.lI(it; !!.:!:n -.OHM ·-.ti77 ,I HI ~,:\,17 ,(\21 --,''''(i
-'.11-1:\
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.1":'1) . -.fJX!'i .. I.'!.·W
-.O:,~,
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---------- ---- -----
-.:q~J
-.lIi:1
-.'.!:!i -I.IIIH
PI
·.WIO
-.W,i
. I'.!·'
1.1l:1-1
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'.I
-, III
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. - ---"----,,------------
,I",
Tlw :-""uph· ,u""", HI ~, t\l\.,1 01 I~UI '1'1.\\ \\" Iy ;\1HHIUUn',u,'u\:\ fur ,In- Ihi1 'Y rump.""", iu IlUII'.IUIII" 11I
, 165
-1.-1. MI'llSWillg mul A1wlyz.ing A/manllal Returns
0.03
."
n.o:?
\
0.01
6
Nn-NC'WI Fir"1I
....... .
0
-0.01
-0.02 ~.03~~~~ww~~WW~~WW~LUWW~WU~~
-20
o
-10
10
20
Event Time
Figure 4.20.
Plot of Cumulative Market·Model Abnormal Return for Earning Announce-
111m I.'
0.03 0.02 Couci·Nt.,W!\o
.,. (j ~
()
"
Firm~
\.--
0.01
No-Ncws Firml
..... .
-0.01 8ad·Nr~
-0.02
-20
-10
Firms
o
10
20
Event Time
I I
Figure4.2b. Piol of Cumulatillf COll5lanl·Mffln·Rdurn·Model Abnormal Return for Earn· ill~ AIlIWWICfllU'1ltS i
SlIpports the hypothesis that earnings announcements do indeed convey i~. fill'lllation IIseful for the valuation of firms. Focusing on the announcement clay (clay zero) the sample average abnormal relUrn for the good-news fir~ I
•
166
4.
Ellellt-Slud.~
A 1I1/(~sis
using the market model is 0.965%. Since·the standard error Ilf the one-day good-news average abnormal return is O. I 04%, the value of 11 is !1.~H and the null hypothesis that the evellt has 110 impact is strongly n:iecled. The story is thc same for the had-news finns. The event day sample ahnorJnal return is -0.679%, with a standard error of 0.098%, leading to 11 eqllal to -6.93 and again strong evidence against the null hypothesis. /\s wOllld he expected, the abnormal re\Urn of the no-news firms is slllall at -o.ml1 % and, with a standard error of 0.098%. is less than one st'UHlanl error frolll zero. There is also some evidence of the announcement e1lect on day OlW. The average abnormal returns arc 0.25 I % and -0.204% for the good-news and the bad-news firms respectively. Both these values are more than two standard errors from zero. The source of these day-one effects is likely 10 be that some of the carnings announccmcnts arc made on cvent day zero after thc c10sc of thc stock markct. In thesc cases the cffects will he caplured in the return on day onc. Thc conclusions IIsing the abnormal returns from the constant-Illcanrcturn model arc consistent with those from the market modcl. Ilowever. there is some loss of precision Ilsing the constant-Illean-return lIlodel, as Ihc variancc'of the avcragc abnormal rcturn increascs for a)) thrcc categorics. When measuring abnormal rcturns with the constant-mean-relllrll model the standard crrors increasc from 0.104% to 0.1300/0 for good·news finns, from 0.098% to O. I ~4% for no-ncws firms, and from 0.098% to O.I:H % for~)ad-news finns. Thesc increascs arc 10 bc cxpcctcd whcn consideril1g a s~mplc of large IInns sllch as those ill the Dow Index since Ihese slOl'ks ten~1 to have an important markct compunenl whosc variability is elimillated using the market modcl. !The CAR plots show that to SOIllC cxtent thc markct gradually Icams abo~1\ the forthcoming announcemcnt. The avcrage CAR of thc good-news fil"Ofls gradually drifts up in days -20 to -I, and the avcragc CAR of thc bad ncws firms gradllally drifts down over this period. In thc days after the am unccmcnt the CAR is relatively stahlc, as would be expcctcd, although the c docs tcnd to hc a slight (hut statistically insignificant) increase for the bad news firms ill days two through eight.
4.4.61I1jl'll'II(f.\ with
Clll~/l'rillg
In a lalylillg aggrcgatcd abnormal returns, we have thus far assullll'd Ihat thc hnorrnal rcturns on individual securities arc uncorrc!ated ill titl' cross sect 011. This will gellcrally he a re.lsonahlc 'ISSlllllption irthe evcnt windows of tHe includcd securities do not overlap in calcndar tilllc. Thc assulllption allows us to calculatc thc variancc or the aggregated salllple clllnulativc abnormal returns wilho\lt concern .Ibotlt covarianccs betwcell individual sample CARs, sincc they arc I.cm. Ilowcver, when the cvcnt windows do
4.5. Mor/ijrillJ.: 1111' NlIllll)'jJIIlhrJi.1
l\i7
OIlTtip, the covarianccs betwc('n th(' abnorlllal r('lurns lIlay dilfer 1'1'0111 Z(TU, and th(' distributional results preselll('d Ii) .. Ih(' a~~rq~al('d almormal r('lums art' uot applicable. lkmanl (1~IK7) dis(lIss('s SDIll(' "I' tl\(' prohlems relaled 10 c1ustcrin!{. \"'hen thcre is onc evcllt date in calendar tillll', dusterinv; call he accom1ll()(\atl'd ill two dilkrcnt wa),s. First, thl' abnormal returns Gill he aggr('g;llcd into a portfulio dated using (,\'(,lIt tillll', and thc sccurity ieI'd analysis ofScnion 4.4 can be applied to the porth)lio. This ;Ipproach allows for cross correlation of thc abllormal retllrns. A sccond way to handle clustering is to analyzc th(' almurmal relllrns wilhout
4.5 Modifying the Null Hypothesis Thus Ell' we have focused on a singl(, null hypolhesis-Ihat Ihe given event has no imp,'Cl on the hdlavior or secllrity relurns. With Ihis 111111 hypothesis either a mcan effect or a variance elfel:t rq>n'sl'lllS a violation. Ilowcva. ill sOIl](, applications we lIlay he inl('resled in testing only for a lIlean effect. In these cases, W(' lleeci to expand the llullhypolhesis to ,Illow for challv;ing (usually increasin!{) variances. To accomplish Ihis, we need to elilllinate all}' rcliallc(' Oil pasl returns in {'slimalill!{ the variance ur the a!{!{rcg,lted clIlIIltiative ,tilllonnal returns. Instcad, we lise the cross section of flnlllllative almonn,tl returns to form all eSlilllator of Ih(' variance. Boehmer, J'vlllsllilleci, ;\1111 PoulsclI (I!I!II) discuss lhis Illcthotiolov;y, which is bes\ appli('d (Ising tite const
-I, J:'1'1'II1-SllIIly A/I(//y,li,
cross senitlll 10 1<11'111 eSlimalors of lilt' vi\I'hll":t's wc havI'
(-1,:1,1 )
I
v
'
---
'"
N~ I)SCARj(TI. T2) - SCAR(TI' T~»", (4,:,,2) I":"')
For Ihe,~e I'slilllalOr~ of Ihe v;lriant'l's 10 he consistelll WI' require the ahllormal relll!'llS III hI' IIl1coITl'I;lll'c1 ill the cross sectioll, An absellce of clllstering i~ sunicil'lIl for Ihis reqllirclIlcnt. NOlc thai cross-scctiollal holIIoskedaslicil), is 1101 n'qllirl'd iill' (ollsislt'ncy, CiVl'1I Ihese variallcc eslilll;!(or~, Ihe !llIlI hypolhesis Illal till' nllllul"liw allllol'lllal n'lurn~; arc I.I~ro Gill Ihl'lI hI' It'stl'd usiug lar)!;I' salllpll' thl'ory giVl'1I thc l'Ollsislellll'SlimalOrs Dr 11ll' variancl's ill (.I.:).~) alld (·1.:),1), 0111' ilIa), also hI' illll'l'l'stl'c1 ill IIIl' impart of all ('VI'1I1 on Ihl' risk of a finJl, Thl' rdevalll llIeaSlIre or risk 1II11St hI' ddilled hdi)rt, this iss\lc (all hI' addressl'd, 011(' dlOil'!' as a risk IIlCaSllre is the llIarkl,t-lIlodcl bela as implil'd hy Ihl' Capilal Assel Priring Model. Givcn this choict', the markel lJIollel rail Ill' fOrllllll;lIl'd 10 allow Ihl' hela 10 change over Ihe cwnl window alld Ihe siahilil)' of Ihe hela call he I'xaluillt'ti, Set' Kalil' alld Unal (I \lHH) (ill' all applicalioll til' Ihis idea,
~I
I I t
I
4,6 Analysis of Power 'Ii) illll'rprl'l all
\ I
\
I) I
I
1'1'1'111 silldy, 11'1' 1I1"l'd III kllow whal is our ahililY 10 ([('ItTI Ihl' pn'sl'lIn' of a noulero ahIlOI'lIl;\I relurll, III Ihis seuioll WI' ask ",hal is IIII' likdihood Ih"l ;llll'VI'IlI-SllId)' It's I rcjcl'ts the 111111 hypolh('~is for a gin'lI level or aillloflllal 1'('1111'11 associaled Ivilll all 1'1'1'111, thaI is, "'I' ('vaillal(' i11t' POIl't'!' or Ihl' It'S!. '\'1' cOllsidl'r a two-sided lesl "('Ihl' 111111 hypothesis \Ising Ihl' l'\Il\1ulalil'l''1IlIIOl'lllal-n'lllrll-h"~I'd slalistil./I ('rolll ('1.'1.~2), WI' aSS\lIlll' Ihalthc ;Ihll!)rlHall'l't\lrns arl' \lllrlllTdall'c! a I\} I ( I - u /~) 11'111'1'1' 'I) (,) is I he st;lIllianl IIormal l'IlIlllllal ivl' tiisl rihlll ion IIIIII'Iioll (CI>F),
169
4.6. AllalY.fiso/Power
we
Given an alternative hypothesis HA and the CDF of 11 for this h~pothesis. tahulate the power of a test of size a lIsing I
Gill
Pea, H A)
Pr(jl < -I (~)
+
I
H A)
Pr(jl > -I (1-~)
I HA).
(4.6.1 )
With this framework in place. we need to posit specific alternative hypotheses. Alternatives are constructed to be consistent with event studies using data sampled at a daily interval. We build eight alternative hypotheses using four levels of abnormal returns. 0.5%. 1.0%. 1.5%. and 2.0%. and two lcvels for the average variance of the cumulative abnormal return o~ a given sccurity over the sampling interval. 0.0004 and 0.0016. These varia~ces correspond to standard deviations of2% and 4%. respectively. The sample size, that is the number of securities for which the event occurs, is varied from I to 200. We document the power for a test with a size of 5% (a 0.05) giving values of -1.96 and 1.96 for <1>-1 (a/2) and <1>-I(I-a/2), respectively. In applications. of course, the power of the test should be considered when selecting the size. The power results·are presented in Table 4.2 and are plotted in Figures 4.3a and 4.3b. The results in the left panel of Table 4.2 and in Figure 4.3a are for the case where the average variance is 0.0004, corresponding to a standard deviation of 2%. This is an appropriate value for an event which does not lead to increased variance and can be examined using a one-day event window. Such a case is likely to give the event-study methodology its highest power. The results illustrate that when the abnormal relUrn is only 0.5% the power can be low. For example. with a sample size of20 the power of a !l% test is only 0.20. One needs a sample of over 60 firms before the power reaches 0.50. However, for a given sample size, increases in power arc suhstantial when the abnormal return is larger. For example, when the abnormal return is 2.0% the power of a 5% test with 20 firms is almost 1.00 with a value of 0.99. The general results for a variance of 0.0004 is that whcn the ahnormal return is larger than 1% the power is quite high even for slllall sample sizes. When the abnormal return is small a larger sample size is necessary to achieve high power. In the right panel of Table 4.2 and in Figure 4.3b the power results are presented for the case where the average variance of the cumulative abnormal return is 0.0016, corresponding to a standard deviation of 4%. This casc corresponds roughly to either a multi-day event window or to a one-day event window with the event leading to increased variance which is ;ICcol\1l1loc!ated as part of the nllll hypothesis. Here we see a dramatic decline in the power of a 5% test. When the CAR is 0.5% the power is only 0.09 with 20 firms and only 0.42 with a sample of200 firms. This magnit\ude
=
II
4. Event-Stut/.v All11~VJiJ
Tat1e 4.2. "film is
Power of roml-Jlml)' Ir.<1 .1/f/1;.
10 rrjr<1I11,
11111/ ")'/loil".I;.1 111,,1111,' "/,,/(,, 11/,,/
ZLm.
i
Sample ISize
1.0%
0.06 0.06 0.07 0.08 0.09 0.09 0.10 0.11 0.12 0.12 0.13 0.14 0.15 0.15 0.16 0.17 lUll 0.19 0.19 0.20 0.24 0.211 0.32 0.35 0.39 0.42 0.49 0.55 0.61 0.66 0.71 0.78 0.84 0.S9 n.92 0.94
0.08 0.11 0.14 0.17 0.20 0.23 0.26 0.29 0.32 U.35 0.311 0.41 0.44 0.46 0.49 0.52 0.54 0.56 0.59 O.fil 0.71 O.7B H.1I4 O.II!! 0.92 0.94 0.97 0.99 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.01l
I
(I
I
2 3 4 5 6 7 8 9 IU II 12 13
14 15 16 17
III 19 20
25 30
:lr.
40 45 50 60 70 80 90 100 120 140 100 1110 200
Abnormal Return
Abnormal Return 0.5%
15%
2.0%
05%
1.0%
0.17 0.2!1 0.41 0.52 0.61 0.69 0.75 OJ\! IU\;, 0.119 Il.!ll 0.93 0.95 0.96 0.97 0.98 0.98 0.99 n.99 0.99 LOO LOO 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.110 1.00 1.00 1.00 1.01l 1.00
0.05 0.05 0.06 0.06 O.Ofi 0.n6 n.06 0.06 0.07 0.07
0.06 0.06 0.07 O.OS O.ll9 0.09 0.10 0.11 0.12 0.12 0.13 0.14 0.15 0.15 0.16 0.17
=2% 0.12 0.19 0.25 0.32 0.39 0.4!> 0.51 0.:)1)
O.f,1 O.fil; D.70 0.74 0.77 O.!!O 0.83
ON, 0.117 0.119 0.90 0.92 O.9\i
O.!!II O.!l!l 1.HO 1.0H 1.00 I.on 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
(I
o.m 0.07 0.07 0.08 0.08 0.08 (I.OH (1.011 (l.OIl O.O!) (1.1 0 0.1 I 0.11 0.12 0.13 0.14 O. ](; 0.18 0.20 0.22 0.24 0.211 0.32 0.35 1l.:19 042
15%
2.0''',
= 4%
(l.~H
0.19 0.19 0.20 0.24 n.211 0.:12 0.3:. 0.39 0.42 0.4!)
0.:):) 0.61 0.6ti 0.71 0.711 0.84 O.S!I 0.92 0.94
0.07 0.011 O. ((l 0.12 0.1:1 0.1!> 0.17 O.I!! 0.20 D.22
1l.24 n.2:, 0.27 0.29 0.31 0.32 0.:14
n.:\(; 0.:\7 n.:I!1 0.47 0.:,4
n.w CUili 0.71 1I.7h 0.8:1 0.1111 1I.!)2 O.!)4 O.!)I; O.!!II O.')!) 1.00 LOt) I.Ot)
0.011 0.11 0.14 0.17 0.20 0.2:1 O.2ti O.2!1 IU2 0.3:, 0.38 nAI 0.44 0.41; 0.49 0,!',2 0.;"1 0.:,1; ()59
IUil lUI 0.711 CUI·l CUI!1 H.92 II.!H 0.!)7 n.!'!) II.!)!) LOll 1.00 1.00 1.00 I.I~I
1.00 1.0\1
pow~r
is reportc(1 for a tcst with a !'lilt" of' 5%. Th(' sample sil.c is the IHlIl1h('1 of t'\'('rH included in the study. Olnd rr is the sqllal't' root of the average \'~tri"I\('(' of the abnormal return ",cro~'" f\nu~.
The
uh~crv"tions
of abno~mal relurn is difficult to delect with the larger variance of 0.00 IIi. In conlraSl, when the CAR is as lar~e as 15% or 2.0% the 5% lCst slill has reasonable power. For example. when the abnormal return is 1.!'J'Yo and
4.6. Analysis ,,/Powl'r
171
0
'" 0
<; 0
" .,. 0
c..
0 0'1
6
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40
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:."c
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0
.,. ::0
-
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.\1'11'01'11..1 H. 10,," I II'" -~--------
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()
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I0
~I)
:111
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;,1)
W
70
HI)
~II)
100
Sallll'i,' Sill' (h)
Figure 4.3. 1'0"''''' "l ElII'II/-S/wly '/i'.11 SllIliM;"./1 (II Ufjrrl tllf' NlIlI IIY/I(II/t,.li.\ l/tal tiff AbllOnluzlllPillfll i.I /.I'IlJ. ~\?lfl! Iii, S'/IUII' illlllt "J rI,,' '~!"''''r;r \'II;f/I/,r IIf Illr Abllormal liflum Arml.' i'ima i.\ la) 2% f/Tlff IIJ) 4%
there is a sample sizc of :'0. thc power is 0.:>'1, Generally ir lhe abnormal return is large one will havc little dif'/iculty n~ectillg th(' lIull hypothesis of IlO abnormal return, We have calculated power analytically using distributional assumptions, If these distributional assumptions are inappropriatc thcII ollr power calculations IIIay be inaccuratc_ llowcver. Brown and Wamcr (1 9H:)) cxplore this
./.
J:lII'lI/-S/III/y .. II/(//pil
issu(' ;lIl1llilld Ih'll III(' ;III,III'li(al (OIIlIlIlI,lIiO\lS a\ld Ih(' "lllpil'icII }l()\\'(T ,\I'l' do,,'.
\'('1'1'
" i\ dillin!ll 10 \(',uh gnl('lal (ollclllSiollS rOIHTl'lIillg till' Ihl' ,Ibilill'
Ill' 1'\'I'llI-SIIIII, IlH'lhogl' 10 lh(,I'\\'is(' 0111' ,hollid sl'arch ICII' ways of illlTl'asillg' III(' PO\I'I'r, This clIll)(' dOli I' 1)\' illlT(';Jsillg Ihl' sa III ph- sizl', shorll'llillg' Ihl' ('1'1'111 willdo\,', or hI' dl'l'l'lopillg l1Iol'e spl'cilir prl'din iOlls of I h e nil II IlI'poll H'sis,
4.7 NOllparametric Tests TII(' IIIl'I IIn(h disllI"I'd 10 til is pOill1 ;111' parallll'l ric ill lIallln', ill tklt spccilic aSsllIllptiollS hOI\'(' h('('I1 Illadl' aholll lIlt' distrihulioll of ahllol'lllal 1'1'\ II \'Il S, ,\\tnllalil'l' lIollparallll'lri(' approa('hl's an' availahle whi('h an' rn'l' or SPI'dliras'lIll1pliolls ('OIIlTlllillg thl' distrihulioll 0(' 1"('llIl'IIs. III lhis s('('(ioll W(, disCII's 1\\'0 1't1l\\I\\t)\II\tlllpal'alllt'lri.- lI'sts for ('\'I'llt slIulil'S, Ihl' sigllll'si alld lhl' I'allk II'S\. Th .. sigll 1.. ,1. ,dlidl i, hOlS('" Oil Ih(' sigll of lilt' allllClI'Ill;tI \'('1111"11, n'l(uin's lhal IIII' ahllol"l\lal rl'llIrllS (or IllOn~ gl'lll'rally (,1I11l1l!alil'l' abnormal n'III\'IIs) 011'1' illdq)('IHll'1I1 ;tlTOSS sl'Cllrili('s and Ihal IIII' n:pl'('ll'd proportioll of positiw allllOlIIl.tll'l'llII'IIS 1I11dl'\" I he \lull hypolhesis is 05. The basis of Ih(' II'SI is Ih;1I IIl1d .. r 111(' 111111 hl'pOlh('sis it is equally prohahll' tlnl 111(' CAR will hI' (losilil'l' or IIq~ati\'l'. IL ((II' l'xample, Ihe altl'rnalil'l' !t"pothl'~is is Ihal Ihen' is;t pmitil'" ~Ihllonllal r('turn associated witll a gin'lI ('\'l'lIt, Ihe 111111 hl'(lollll'sis is Ilu: II ::: 0,:-. alld Ihl' altt'rnativl' is 11.\: II > lUi IdH'rc /, == I'r(C,\\{, ::: (1.11). 'Ii. calnll'III' IIII' II'SI sialisli .. WI' Ill'l'd Illl' 1l1l1ll1H'1' oj" (';lSI'S 11'11('1'1' IIII' ahllolillal l'l'tlll"ll is POSilil'I', N'\ , ane! Ihl' 101011 1IIIIIIhl'r of casl'S, N. I.cltillg ,It hI' Ihl' tl'st slatislic, 1111'11 asymptolically as N illlTl'aSl'S 11'1' h,ll'!'
/1
,
=
] N'/~ Vf -'- - 0.:-. - - ~ [ N 0.:1
N(o.
I) ,
For ;III'sl "I"sil(' (I .. tr), Ilu i, n:j('("\('(1 if/:! > 4)-1(0'). t\ II'l'akll(,ss of Ih .. sil-:II \('SI is thai it llIay 1101 he well specilit,d if Ilw dislrihlliioll ot' allllol"lllal I"I'IIII"IIS is skl'wec!, as call he till' case wilh claily dala. With sknl'l,d al)lHllllIOIi \('IIII"IIS, IIH' ('xIH'C!e!l proportion 01" positive ahllllnnal I"I'IIIIIIS 1.111 dill('\" hllill lllll' h,llt' l'l'l'll IIllder thl' 111111 hl'(lOlhl'sis, III I(,SPOIIS(, 10 IlIi, I'0s\ihl(' SIlOlI('OIl1illg, (:orrado (I~)H~I) proPOSI'S 01 110111'00Iallll'II"ic 1"'lId, 11,,11"1" ;11'"111"111.\1 1"'1 II)nl1.IIKt' i111'V1'1I1 sllIdil'S, 1;\'1' III idly d('snilll' hi, 1t',1 "I IIIl' 1I111111l'pOI\wsi, Ihal Ih('n' is 110 OIlIlIol'lllal 1'('111111 Oil
173
4.H. Cross-Sectional Modtls
event day zero. The framework can he easily altered for events occurring over lIlultiple days. Drawing on notation previollsly introduced, consider a sample of LJ abnormal returns for each of N securities. To implement the rank test it is necessary for each security to rank the abnormal returns from I to loIJ. Ddine KIT as the rank of the abnormal return of security j for evenl time period r. Recall that r ranges from Tl + 1 to T:z and T = 0 is the event day. The rank test uses the fact that the expected rank under the null hypothesis is The test statistic for the null hypothesis of no abnonnal return on event day zefo is:
¥.
14 = s(lJl)
=
I~
N
N
~
(
IJl
+
I)
K~l - ~ /s(ld
t
Ll r='1i+1
(~t (K N
;=1
_ loIJ IT
(4.7.1)
+ 2
I)Y
(4~ 7.2) :
Tests of the null hypothesis can be implemented using the result that the asymptotic null distribution of J~ is standard normal. Corrado (1989) gives further details. . I Typically, these nonparametric tests are not used in isolation bllt in conjunction with their parametric counterparts. The non parametric tests enable one to check the robustness of conclusions based on parametric tests. Such a check can be worthwhile as illustrated by the work of Campbell ami Wasley (1993). They find that for daily returns on NASDAQ stocks the non parametric rank test provides more reliable inferences than do: the standard parametric tests. I
4.8 Cross-Sectional Models Theoretical models often suggest that there should be an association~be tween the magnitude of abnormal returns and characteristics specific to the event observation. To investigate this association, an appropriate tool is a cross-sectional regression of abnormal returns on the characteristics of interest. To set up the model, define y as an (Nx 1) vector of cumulative abnormal return observations and X as an (NxK) matrix of characteristics. The first column of X is a vector of ones and each of the remaining (K - I) columns is a vector consisting of the characteristic for each event ohservation. Then, for the model, we have the regression equation
(4.8.1 ) when' 0 is the (Kxi) coefficient vector and fl is the (Nxl) disturbance E[X'fll = 0, we can consistently estimate (1 using OUi.
wC\or. A~suming
174
4. El/t'1ll-Sllli(V 1I11f11.\',li,1
For thc OLS estimator wc have
Assuming thc c\clllcnlS of 1/ arc cross·sectionally uncorrclatetl allli hOllloskcdastic, infcrcnccs can be derived using the usual OLS standard errors. Dcfining (J~ as thc variancc of thc clcmcnts of 1/ we have (4.H.3)
Using thc unbiascd cstimator for a~,
.~ a 1/ --
I.,. (N - K)
(4.HA)
1/1/ ,
wherc r, == y - xiJ, wc call construct I-statistics to asscss the statistical si!{lIiliGIlIce or the clelllel\l~ or 0. Alternatively, without assuming hOl\loskcdastkity, wc can construct hctcroskcdasticity-colISisten t 2-statistics usin!{
\ \
I
Var[9]
= .!. (X'X)-I N
[t XIX;r,;]
(X'X)-I,
i=1
wherc x; is thc ith row of X and i'i is th<; ith element of r,. This cxpression for \he standard errors can be dcrivcd using thc Gencralizcd Method of Mome~lS framework in Scction A.2 of the p.ppem\ix and also follows rrOIll the rcs Its or Whitc (1980). The usc of heteroskedasticity-consistent standard err rs is adviscd since therc is no rcason to expcct thc residuals or (4.H.I) to b homoskcdastic. quith and Mullins (19HG) provide an examplc of this approach. The tw lay cumulative abnormal rctufll for thc announcemcnt of ,III equity offe ing is regrcsscd on thc sizc of the offering as a perccntagc of the value of tl e lotal cquity of thc firm and Oil the cUlllulative abnormal relul'II ill thc leven months prior to the annollnccmcnt month. They lind that the mag itudc ofthc (ncgative) abnormal rcturn associatcd with the allllOllllcemcn~ or cquity offcrings is relaled to both thest' variables. Llrger pre'Cvcllt cuml.lalivc abnormal rcturns arc associatcd with less negative abnormal rend-ns, and largcr offcrings arc associated Wilh lIIorc ncgativ(' ahnormal retll~ns. Thcse findings are consistent with theoretical prcdiclioJ1S which thcy discuss. Onc lIIust be carel'ul ill interpreting the results of the Cfoss-senional rcgression approach. In many situations, the event-window abnormal r('llIm will he relatcd to firm characteristics not only through the valuatioll effects of thc cvcnt but also through a relation between thc linn characteristics and thc cxtcnt to which the cvent is anticipated. This call hapJl('1l wht'll
4.9.
FIII"//I/'I" lUIIt'.!"
illve,tor s ratiollall y IIS(~ firlll characte ristics to f'II'('clst the likeliho od of the evcllt oeCllnil lg. III these cases, a lillear rl'l"tion IwtW(,{,1I til(' linn characteristics ;11)(1 the valuatio n e1kct of the ('vellt C;llI 1)(' hiddell. M"latest a alld Thulll PSOll (I ~JH:J) alld I.anell and Thomps oll (I !)HH) provide example s of this situation . Technic ally, the relation betweel lthe linu ch;ll
4.9 Further Issues A IlIunllCr of further issues OftCII aris(, whell cOllduct illg an ('\'Cllt study. We di~cuss sOllie of these in this section. 4.9./ Rolf o/lllr Slllllj}/ill f; Ill/i'/HI!
If the timing of an evellt is known precisely , thell tht' ability to statistica lly identify the effect of the event will be highn ((Jr a shOrln samplin g interval. The increase results frOIll reducin g the varianec of the aiJnonn al relllrn willH)lll changin g the mean. We cvaluate the enlpiric d importa nce of this issue by cUlllpar ing the analytic al formula for the power of the test statistic JI with a daily salllplin g interval to thc pO\\,l'I' with a w('ekly and a monthly interval. We assUllle that a week cOllsists of liv(' days and a month is 22 days. The varianc( , of the almorm al return for an individu al evellt observa tion is assumed LO bc (4%)~ on a daily hasis and lillear ill tilll(,. [II Figu,!: 4.1, we plot the power of the test 01 110 ('vellt-e llen agaillst the alternat ivc of all abnorm al rcturn of I % fiJI' I to 200 securitie s. As olle \\'ould expect givell the allalysis of S('( tioll 4.(;, the d('(Tcas e in power going frolll a daily interv~d to a mOllthly illterval is se\'('I('. For example , with :JO sentritie s lhe (lower for a !i% test using d;lily data is O.!J4, whereas the power using weekly alld IIIonthly data is only O.:FJ ;llld 0.12, respcetiv ely. The clear llIessage is that there is a substant ial payolf ill terms of increase d power from reducill g the lellgth of til!' ('VI' II I willdow. 1'"lors(' ( I !JH1) prescnts detailed analysis olthe choi(:l' of daily V('l'SUS IllOllthly data alld draws the same COII( lusioll.
-I.
1:IJrI/I-Sll/fly Al/aly.I;.!
'~.
t
c:.-
-.- •••• " - O, ... ·W,·,·k III'm.,1
-
-
-
-
-
-
----~
OJl('-MlIlllh Inh'l\,11
!
(I
:!II
.(0
fill
---
HO
,
1110
1:!O
1-10
1fill
1HO
:!OO
S;lIIII'''" Sill'
Figllre 4.4.
1',,/1 1,"1
"/
1-."1'1'II1-SllIIly 'li'll Slal;I/;I".h III Jlpjrd Ihp NIIII "Y{i(llhl'.lil Ihal Ihp
I,·m . ./111" /Ji//i-mll Sfllll/J/iIlK IIIIPlVoll. II'lim Ihp S,/lIf11P /llIot AI'rllIK" I inial/II' II/thl' ..\11/1111"11/111 HI'IIII"II .. lfllI.\I Finll.1 1.1 ,,'X. fill" Ihp /Jail\' 1IIII'nllli
:lhllllnllal JlpllIlIl il
lit IIII'
A sampling illll'II';ti of OIl(' d;I\' is nol Ihl' shorll'sl ililer"al possible, Wilh Ihl' inITl'aSl'd al'aibhilill' of 11;lnsa('lion dala, re('elll sllIdies hal'l' used ohselvalion illll'll'als of dUlalion slioller Ihan Olll' day. The USl' of' inlradaily Iiaia in\'oll'l's SOIlll' ('oJllplicalions, however, or Ihe sor! dis(wsecl in Chapin :\, and so lli{' 111'1 \ll'ndil of wry shon inlervals is IIndear. H"l'rlay and [.ilt"II\wlgl'l' (Il)HH) discllss Ihl' IISl' of in Ira-daily dala inl'l'elll sllIdies .
./, f). 2 III/m'IIn'1 willi FIIf'III-f)(Ilf' lIlIl"rrlaillly
Thlls br Wl' kll'l' ;\SSII\\l('(llklllhl' ('\,(,111 d'lIl' call he idl'lIliliec! wilh n'naillil'. Ilowever, ill sonll' sllllli,'s il Ill;\\, he difficult to idelllify Ihl' exan dale. 1\ l'OIllIllOIlI'X;lIl1pll' is when ('olkclinJ,!; "I'l'nl dates fromlinanrial puhlicaliolls sHch as Ihl' Ilidl SIIi'I'I.!m//'IIal. When IIJ(' I'Vl'llt anllOlllll'l'lllent appears ill Ihe III'WSpa!H'!' onl' ,all 1101 !)(' l'l'l'laill if the markl'l was inlilrllled Iwji)l'" IIII' dose 01 III(' 1II;lIk"1 IIII' priol Irading day. If this is Ihl' ('ase Ihell Ihe prior d;w is II\(' ('WIlt ,\;tv; if lilli, Ihl'lI Ihl' (,UITl'nl day is Ihl' eVl'nl day. The uSllal nll'lhod of handling 111i\ 1'1'0111"11\ is 10 expa\\d Ih .. 1'\Tnl wi Ild 0\\" 10 111'0 days-dal' 0 and ,b)' + I. Whil .. Ihel'l' is a ('osl 10 ('xp;llldillJ,!; th .. ('\'('111 windo\\', IIll' Inll!" ill S'Tlion 'I.ti illdicall' Ihallhe pOlI'I'r propl'rties or 111'0day ('Will willdows ;11,' still gOlld, ~lIgg,'slillg Ihal il i~lI'orlll h('aring 1111' ('oSI 10 ;lvoiclllll' risk Ilr Illissillg Ihl' 1'1'1'111.
-I. '.J. Furtiu-r Issues
flail and Torous (1988) investigate this issue. They develop a maximumlikelihood estimation procedure which accommodates event-date uncertainty and examine resull5 of their explicit procedure versus the informal procedure of expanding the event window. The resull5 indicate that the informal procedure works well and there is little to gain from the more elahorate estimation framework.
4.9.3 Possible Biases
Event studies arc subject to a number of possible biases. Nonsynchronous trading can introduce a bias. The nontrading or nonsynchr~nous trading effect arises when prices are taken to be recorded at time intervals of one length when in fact they are recorded at time intervals of other possibly irregular lengths. For example, the daily prices of securities usually employed in event studies are generally "closing" prices, prices at which the last transaction in each of those securities occurred during the trading day, These closing prices generally do not occur at the same time each day, but by calling them "daily" prices, we have implicitly and incorrectly assumed that they arc equally spaced at 24-hour intervals. As we showed in Section 3.1 of Chapter 3, this nontrading effect induces biases in the momenl5 and co-momenl5'of returns. The influence of the nontrading effect on the variances and covariances of individual stocks and portfolios naturally feeds into a bias for the marketmodel beta. Scholes and Williams (1977) present a consistent estimator of beta in the presence of nontrading based on the assumption that the true return process is uncorrelated through time. They also present some empirical evidence showing the nontrading-adjusted beta estimates of thinly traded securities to be approximately 10 to 20% larger than the unadjusted estimates. However, for actively traded securities, the adjustmenl5 are gencrally small and unimportant. i .Jain (1986) considers the influence of thin trading on the distribution of the abnormal returns from the market model with the beta estimated IIsing the Scholes-Williams approach. He compares the distribution of these ahnormal returns to the distribution of the abnormal returns using the'usual 01.') betas and finds Ihat the differences are minimal. This suggesl5 that in gcneral the adjustment for thin trading is not important. The statistical analysis of Sections 4.3, 4.4, and 4.5 is based on the assumption that returns are jointly normal and temporally liD. Departures from this assumption can lead to biases. The normality assumption is important for the exact finite-sample resull5. Without assuming normal!\)', all results would be asymptotic However, this is generally not a problem for event studies since the test statistics converge to their asymptotic distributiolls rather quickly. Brown and Warner (1985) discuss this issue.
4. Euenl-Sllu(V tll/f/ly.\is
There can also be all upward bias ill cumulative ahnormal returns wh('11 these are calculated in the llslIal way. The bias arisdi frolll the ohservalionby-obselVation rebalancing to equal wcighL~ implicit in the calculation or the aggregate cumulative abnormal retum combined with the lise or transaction prices which can represent hoth the birl alld the ask side or the market.. Blume and Stambaugh (1983) analyze this bias and show that it can be important for studies using low-market-capitalization finns which have, ill percentage terms, wide bid-ask spre;lds. III these cases the bias Gill be cjminated by considering cUllIulative ahnormal returns that represent bUY-ind-hold strategies.
I
1
I
4.10 Conclusion
In c1jsin g, we briefly discuss examples of evelll-study successes and limitations. Perhaps the most successful applications have been in the are,1 of corp rate finance. Event studies dominate the empirical research ill this area. lmportant examples include the wealth effects of mergers and acquisition! and the price elTects of financing decisions by firms. Studies of these evellllltYPiCall Y [ocus on the ahllonnal return around the date of the lirst anllOl ncelllent. II the 1960s there was a paucity of e!npirical evidence 011 the wealth elTecL<;! of mergers and acquisitions. For example, Manne (196:,) discllsses the vaJious arguments for and againstlllergers. Al that time lhe d(,hate ("('11tered ~n the eXlenllO which mergers should be regulated in order to foster comP9lilion in the product markets. Manne argiles that mergers represent a natural oUlcome in an efficiently operating market for corporale control and consequently provide protection for shareholders. He downphlys the importance of the argumenl thal mergers reduce compelition. At the conclusion of his article Manne suggests that the two competing hypotheses for mergers cOllld be separated by studying the price eflects of the involved corporalions. He hypothesizes thal if mergers created markel power one would obselVe price increases for both the target and acqnirer. In contrast if the merger represented the acquiring corporation paying for control of the target, one would obsen'e a price increase for the target only and 1I0t for the acquirer. However, at thal time Manlle cOllcludes ill refen'I\C<~ to the price effects of lIlergers that " ... no data are presently availahle on this subjecl." Since thaI time an enormous hodyofempirical evidence on mergers and acquisitions has developed which is dominated hy the usc of evelll studi(·s. The gellera! result is that, given a successful takeover. the abn,onnal retllrns ufthe targets are large and positive aud thc ahnormal returns of the aClJllircr ue close to zero. Jarrell and Poulsen (19H~) lind that the average ahnorlll;ll
4.10. COllrill.li oll
179
retul'll li>r larf\el shareho lders excenls ~()(X, li>r a s'"l1pk of titi3 successf ul takeover s frolll I~Hi() 10 19H5. III cOlllras llhc "lmOl"III,1I relum (ill' acquirer s is close to zero OIl 1.14%, and evell nef\,lliv(' al -1.10% in Ihe 19HO·s. Eckbo (I ~IH:1) explicitl y addresse s Ihe role of increase d markel power ill explaini ng nlergcr- related ahnorm al r('lurIlS. I Ie separate s nH~lw~rs of competi nf\ lirms rrolll other merf\ers and linds no evid"n'T that thc wealth circus for cOlllpeti nf\ lirms arc differen t. Furtllt'r. he linds no cvidenc c that l'iV,lls of' linns IIIt~rf\in!{ horizont ally "xperit'I HT lIegativc abnorlll al I ctllms. FrOIll this hc concllld cs that redllced cOlllpct itioll ill the prodllrt markct is not an importa nt explana tion ror III'T!{Cr gains. Thi.~ leaves (t)lnpcti tion for corpora lc control a more likely cxplana tion. Much addition al empiric al work in thc arca or mergers anti acqllisit ions has Iwen conduct cd . .Jensen alld Ruback (I !lH3) and.larr ell, Brickley. alld Netter (I ~IHH) provide detailed survcys or this work. A number of robust results havc bcclI dcvelop cd from cvent studies or financil lg dccisioll s by corpora tiolls. Whcn a corpora tion announ ces tI.'lt it will raise capital in external lIIarkcts there is on average a negative abnorm al retul'll. The magnitu de of the abnorm al rcturn depends on the sourcc or cxtcrnal financin g. Asquith and Mullins (l~lH()) study a sample of 2GG firms ,1I11l011llCillg an equity issue in the pniod I~)W to 19HI and filld that tile two-day average ahnorm al rt'l1ll'1I is -~.7%, wllil,' on a sam pIc of 80 finns for the period 197~ to 19H~ Mikkels on and Pal tch (I!lHfj) find that the two-day avcra!{c ahnorm al relurn is -:{,;>!i'X,. In contrast . when lirms decide to usc straight dcbt linancin! {. Ihe averag'~ ahllorlll al rcturn is doser to I.ero. Mikkels oll alld Partch (1!IH(i) lilld tht' ;IWLlg" ;\I>11orlllal return for debt issues to be -O.~3% for a sample of 171 issues. Finding s such OIS tllesc provide thc fud for the dcvelop ment of new theories . For example . these external financin g results Illotivatc thc pecking order theory of capital Siructur e develop ed by Myers and Majluf (1~IH'I). A major success related to thosc in the corpora le fillancc area is the implicil accepta nce of event-st udy methodo logy hy Ihe lJ .S. Suprem e Court for dctCl'm ining material ity in imider trading cases and li>r determi ning appropr iate disgorge lllent amollnt s in cases of fralld. This implicit aCtTptance in lhc I ~IHH Basic, Incorpo rated v. J .cvinsoll case alld its importa nce for sccuritic s law is discusse d in Mitchell and Neller (1~194). TIH'rc havc also been less slltTt'ssful applicat iolls of cvent-st udy IIlcthod oIOh'Y' An importa nt characte ristic of a slIcccssful e\'t'nt stud), is the ability to idellliry precisely Ihc date of the cvelli. In cascs \\'hne the date is difficult to idelllify or the evcnt is partially antirip'i lcd. eVCIII studics have been less useful. For cxample . thc wealth dferls of n'!{uIator), changes j(>r alkned entilics can hc diHicult to dClect usillg t'\'t'nt-st udy IIlelhodolol-.'Y- The problem is that regulato ry challgc.~ arc often dcbated in Iht' polilic" ,IITlla ovt'J' time alld any aC(,(>lllpanying wealth cf(('((S will he ineol porated graduall y inlO
Ihe vahlt, of a IOIPOl;lIioll as Ih(' probahilily of Ihe change heillg adopled illcreascs. \);11111 alld .lallH's (1~IH:!) discllss Ihis isslle ill Iheir sludy of Ihc illlpaci of t\cposil illll'n'sl ralc t"\'ilillgs Oil Ihrili illSlilllliolls. They look OIl challges ill ralc ('(·jlillgs, hili dccidc 1101 10 cOllsider a changc ill I!173 hecallsc il was lhlt' 10 Icgislalivc aClioli aud 11('11('(' was likely 10 have Iwcll anlicipalcd hy Ihe market. Schipper alld TIIOIllPSOII ( I !IH:{, I !IH5) also ('nCO\llller Ihis prohll'lIl ill a sllidy or lIIergl'l'-I"I'lall'lI rcgulaliolls. They allclIlJlI 10 cirnllllvc:nl Ihe problelll of allticipated reglllatory changes by idenlifying dalf's when the I'rohahilil)' of a IcgUiatol'l' challgc illcrcascs or decreases. Ii owevc 1', lhcy find largely illsigllilicalll n·slllts.lcaving 0JlCJI Ihe possihilily Ihallhc absellce of dislinn evelll datt's accounls {ill' I h(' \;tck or wealth effeclS. Much has I)('clilcal'll('d frolll Ih(' hody ofrf'sf'an:h thaI uses ('V('llt-sludy lJlt'llloC\ology. Mosl gCJlnally. ('venl sllldi('s have showll Ihat. as we would (')(.)1('('( ill a r;lliOlI'.llllIarkt·lp!a('(·. prin's do r('sJlond 10 new inrOI'III;lIioll. We ('''1)('('\ Ihal CV('llI~llIdi('s will (,Olililllll' 10 Ill' a vaillable allel widely IIsed 1001 ill I'UllIOlllics alld lillaIH ....
4,1 Show Ihal whellllsillg till' 1\I;1I·1;.('IIIIOd('IIO nu'aSlllT allllol"lllal retlll"\lS, Ihe salllple ahllol'lllal rellll"JlS frolll eqllalioJl (4.4.7) an' aSYlllplolically illc\ept'llIl('1I1 as Ihe knglh 01 lIlt' ('slilllalioll ",inllow (1. 1) ill('J'('as('s 10 inlinily.
4,2
\ I
\
\
YOII are giv(,11 Ihe li,lIowillg illlimnalioll for an eW1I1. AhllorJllal I'l'IIlI'llS an' salllpkd al all illl("l"val of 0111' day. Th(' (,vcnl-window Ienglh is Ihn'(' days, Th(' IIll'all ahnol'ln,tI relllrn owr Ihe f'V{'nl window is (I.;\'}{, pn da)'. '1'011 ha\'(' a salll)llc oi" :,0 ('VI' II I ohservations. Tht' ahnorJllal r('lllrns an' int!('p(,lld('11l across thl' ('\'('nl observalions as well as across ('Vl'llI days for a giv(,1l CVI'III ohs('I"valion. For :?:, o/Ihe ('\'('nl ohservalions IIII' daily slandalCl d('vialion of Ihl' alllHlllllal 1'('1111'11 is :~'y,) and lilr llle rl'mainillg 2:, OhSI'IT;Iliolls Iht' flail), siandard dl'vi;lIioll is li%. Civclllhis infilrmatioll, whal would 1)(' 111(' pow('\" of Ihe leSI for ;111 ('v('nl sllldy IIsing Ille clIlIllllaliVl' ahllormal rl'tllrn test statistic ill (''1l1alioll (·1.·1.2~)? What would hI' Ihl' pOWl'l' IIsillg lhl' stalldanii/('d C1mlllialiv(' ahllol'lllal n'llIrII lest slalislic in eqllation (4.4. ~·I)) For Ihl' pow('r (,;licubliollS, aSSIlIl\(' lilt' st;IIHbrd dl'viation Ilf II,ll' abnol'lllal rt'llIrllS is knolVlI. 4.3 Whal wOllldl)!' Ih(' answl'I's III qlll'Slioll '1.2 il'lhe IIll'an ahnllrmal r('llIrn is O.li'Y., pl'r dav f(,r Ihe ~:, linll.s willi Ihe larger siandard dcvi;llion?
The Capital Asset Pricing M0ge1
ONE OF TilE IMPORTANT PROBLEMS of modern financial economics is Ihe CluantifIcation of the tradeoff between risk and expected return. A1tho~gh common sense suggests that risky investments such as the stock market will generally yield higher returns than investments free of risk, it was only 'tith the development of the C~pital Asset Pricing Model (CAPM) that economists were able to Cluantify risk and the reward for hearing it. The CAPM implies lhatlhe expected return of an asset Illust be linearly related to the covariance of ils return with the return of the market portfolio. In this chapter we discuss the econometric analysis of this model. The chapter is organized as follows. In Section 5.1 we briefly review the CArM. Section 5.2 presents some results from efficient-set mathematics, including those that are important for understanding the intuition of econometric tests of the CAPM. The methodology for estimation and testing is presenled in Section 5.3. Some tests are based on large-sample statistical theory making the size of the test an issue, as we discuss in Section 5.4. Section 5.5 considers the power of the tests, and Section 5.6 considers testing with weaker distributional assumptions. Implementation issues are covered in Section 5.7, and Section 5.8 considers alternative approaches to testing hased on cross:sectional regressions.
5.1 Review of the CAPM Markowitz (1959) laid the groundwork for the CAPM. In this seminal research, he cast the investor's portfolio selection problem in terms of expeeled return and variance of return. He argued that investors would optimally hold a mean-variance efficient portfolio, that is, a portfolio with the highest expected return for a given level of variance. Sharpe (1964) and l.intner (I !l65b) built on Markowitz's work to develop economy-wide implicatiolls. They showed that if investors have homogeneous expectations IRI
182
5. The Capital And 1'lirillK M{)(M
a~d optimally hold mean-variance ellicient portfolios then, in the ahsclKl' o market frictions, the portfolio of all invested wealth, or the market port~ lio, will itself be a mean-variance ef/lcient portfolio. The IIsllal CAI'M e~uation is a direct implication of the mean-variance efficiency or the mark~t portfolio. . The Sharpe and l.intner derivations of the CAl'M aSSllllle the eXisH~IIlT o~ lending and borrowing at a riskfree rate of interest. for this version or tlie CAPM we have for the expected return of asset i, •
I
E[R;]
~
=:
Ilj
+ fiilll(E[R",]
{Jim
=:
Covlll" Nm ] , Var[Ntn ]
-- IV)
(:;.1.1) (:).I.~)
wi erC Il m is thc relltrn on the market portfolio, and Ilj is the return 011 the ris free asset. The Sharpe-Lintner version can be most compactly expressed in ~erms of returns in excess of this riskfree rate or ill terllls of ,xrru n·lum.l. Le\ lj represent the reUlrn 011 the ilh asset in excess of the riskfree r,lle, Z; H. - Hj . Then for the Sharpe-Lintner CAPM we have
:=
E[l,]
Plln
iJimml",]
([d.:1)
Covl1.j , 1.111 J Var(l,n)
(:>.1.4 )
where l,. is the excess return 011 the market portfolio of assets. Because the riskfree rate is treated as being nonstochastic, equations (5.1.2) alld (!d.4) are equivalent. In empirical implementations, proxies for the riskfree rate arc stochastic and thus the betas can differ. Most empirical work rebting to the Sharpe-Lintner version employs excess returns and thus uses (5.1.4). Empirical tests of the Sharpe-Lintner CAPM have focused on three implications of (5.1.3); (1) The intercept is zero; (2) Beta completely captures the cross-sectional variation of expected excess returns; and (3) The market risk premium, E[z..J is positive. In much of this chapter we will foclls on the first implication; the last two implications will be considered later, in Section S.B. In the absence of a riskfree asset, mack (I ~172) derived a more general version of the CArM. In this version, known as the Black version, tht' expected return of asset i in excess of the l.era-het'l return is linearly related to ito; beta. Specifically, for the expected return of asset i, E[ R,l. wt' havc
E[N,)
=:
Elll"",]
+ fl,,,,(Elll,,,1
- E[ll. .. ).
ll,. is the return on the market portfolio, and It .. is the return on thl' um· portfolio associated with III. This portfolio is defined to he the portfolio that has the minimullI variance of all portfolios nncorrelated with 1/1. (Ally
brla
5.1.
1I1'l/jl'w
oflhl' CAI'M
18;\
other 11l1('olTdatcd ponfolio wOllld have till: same expcctl'd retlll'll, hilt a highl'l'variancl'.) Sincc it is wealth in n~.d tnms th"t is relevant, lill' the Black Illodcl, retllrns arc gener.lily stated on ,III inll.lliOll-"djusted hasis and fI"" is ddilll'(t in tt'nns of real retllrllS, fi,m =
Cov I N,. N", I -----.-. V.u"[l{", I
Econoilletric analysis of the Black version or the CAI'M treats the I.('IO-h('ta ponfolio rellll'll as an u1Iobserved qu.,ntit)', making the 'lll'll),sis more con.plicatcd than that of the Sharpe-Lilll1l('" vl'l'sioll. The Black vlTsion Gill he tested as a restrictioll on the re;!I-retul'll market III(HIl'I. For the real-retltrn m'lrkct model we have E[N;] = u""+/i""E[N,,,I,
"Ild tht' ill'lllicati()11 or the Black vt'l'sioll is
ex"" = E[N"",]
(I -
fI",,)
Vi.
Ud.Hl
III \I'ol'ds, the Black model restricts thl' ;I~set-spt'cilit illtlTt't'j>l of the realretul'll market modcl to be equal to the ('xlH't'tt'd l('lo-Ill'ta portli,lio retul'll tilll('s olle minlls the asset's beta. The CAI'M is a single-period Iliodd; hence (:',.1.:\) and (:,.1.:1) do lIot h'll'c .1 tilllc di mellsion. For ecollometric analysis of till' Illot kl, it is nefessary to add an asslllnption concernin!{ the time-series heh
\\'11('1"(' i d"lIoles llie ;!sSC'1 and I d,'lIo«'S Ihe lilllC period, I = I, ... , 'f. /." alld /."" are II,,' rl';,li/l'd ex\'css n'IIII'IlS ill lilll!' period I for asst'l i alld lile lIlarkel portt(,li", /l·'I)('clil'dy. 'I)'pically Ihe Sialldard alld Poor's !iOO Illd,'x s('rves as a prox\' 1(.1 11f(' Illalkel portlellio, alld tlie US Tr('asury hill r;lIe proxi,'s li.1' III(' riskIn'" n'llIl"/I. 'I'll,' eqllalioll is lIlosl cOlllIIIOllly ,'stilllOltl'd IIsillg :. ye;lrs "rlllotllltl" dala (.,. ::::: (iD). (;iwll all eSlilllale of Ihe beta, the
cost of capital is caielllalt'd IIsillg a liistoric;.1 average lell' til(' excess I'l'l Ill'll 011 the S&'-P :.00 0\,('1' 'licaslll"Y hills. This sort ofapplicatioll is olllyjllstilied if Ihe (:AI'1\1 II/m'id,'s a good desniplioll of tlic dala.
5.2 Results from Efficient-Set Mathematics III this secliotl lI'e 1('I'ie\\' Ihe IlIath('fllatics of lIleall-variallcc cfficielll s('h, The illln,'sll'd Il'adn is I'l'fiolTcd 10 MlTlolI (I \J7~) alld Roll (l \In) lill' d('tailed 111';11 II 1('11 Is. :\11 IIt1dclslalldillg of this topic is IIselitl for illierpletillg IlIlIrl! of th,' 1'lllpiric;t\ITsl';lrch rdatillg 10 the CAI'M, hecallse Ihe key tcslahle ill'l,lical iOIl of lite (:1\ I'M is 11t;,t Ihl' markel porlldio of risky assels is a lIH'all-l'ari;IIII'1' <'IIi, i"111 portfolio. Effici"IlI-set malhemalicsalsoplal.s a mit- ill lltl' ,tl\;tlysis "f Illltllibclor pridllg mOllels ill CIt;lpln 'i. \\'e slarl wilh SOIllI' 1I0laiioli. 1.1'1 there he N risky assels wilh JIlean VCI'lol' It ;\1111 cO\';lIiallll' IIlalrix 0. t\ ss III Ill' lhat II\(' l'xlll'ned n'III\"I\S of al It'asl 1\\'0 assl'lS dilkl ;l1l1llhal Ihe covariance malrix is of filII rallk. Define w" as the (Nx I) \'ector ofporti()lio weights for an arbitrary portfolio II \I'ith w,,'J.', and weip;hls sllInlllinp; to \1\1 it)'. Portlellio a has llH'an return /1" I'ariance w"'Ow,,. The covariance hctwcen any two portfolios II 'Illti b is w,,'f!w/,. (:i\'{'11 till' poplIl;ltioll of assets wc IICXI ('ollsitin minimulllvariance I'0rtli.lios ill Ihl' ah"'IHT of a riskfn'e asset.
=
(1,; =
Defillitioll. ['fIIl/i,lill /, i.1 till' illilli))/u))/-T'llfitlllff /)()rljc,/io of a/l/)()rlf()/ios wilh
'11,'11t!
/1'11/1'11 /11' il ill /J(I"I/iJ/ill 1I 11'ij;ltl Tl('f/or i.1 Ihl' .m/ll/ioll 10 Iltl' f()lIl1lllillj; {Ol/.I/milll'li Il/i/illliwlilll/:
lIlillw'nw
...,
(!•. ~.I
)
.whil'l"ll()
w',.
1.
·Ii. sol\'(' thi~ 1'1"11111,'111, WI' ".nullll' I.agl'all(!;iall functioll I., dil"!i-relliiale with rl'spl'ct 10 w, Sl'1 IIIl' r(,~\Ilti/l).( Cq\l;It'IO/lS 10 zero, a/lCilhl'1I solve for w. For Ihe I.ap;rall),(iall fllllClioll \\,t· h;\I't'
185
5.2. Ur.lIIlI.I /IT!1It i-Jfirifllt-.'ifl MalitfllUllirs
whnt' L is a conforming vector of ones and 0I and lit are l.agrange m\lltipliers. Differentiating L with respeC! to wand selling the result equal 10 zero, we have (5.2.5) 2nw - Oill. - 02L = O. COlllbining (:).2.5) with (:).2.2) ane! (5.23) we find the solution (5.2.6) whnt' g and hare (N xl) vectors.
g
=
(5.2.7) ,
. h
=
(~.2.R)
\
! anel.A == L'n-IJ-L, n == /l.'n-I/l., C == L'n-1L, anel D::::: Be - A2, Next we summari7.e a. number of resU!IS from efficient-sel malhen1atics for minimum-variance portfolios. These results follow from the form of the sohllion for thc ~ininHim-variance portfolio weights in (5.2.6). .
i
Result 1: The minimum-variance frontier can be gcnerated from any two C\islinct minimum-variance porlfolios. Result I': Any portfolio of minimum-variance portfolios is also a minimumvariance portfolio. Result 2: Let p and r be any two minimum-variance portfolios. The covariance of the return of p with the return of r is ' Cov[H,,, H,]
C ( /J.p - C A') (/J.r - CA) + C'1 = D
(5.2.9)
Result 3: Define portfolio g as the global minimum-variance portfolio. For portfolio g, we have wi:
f..l x 2
Ox
-In-I L
(5.2.10)
A C
(5.2.11 )
C
1
-C
(5.2.) 2)
Result 4: For e,tch minill1l1l1l-vari,\llce portfolio /1, except the global minililt I Ill-variance portfolio g. lhere exists a unique minimum-variance porllil!io that has zero covariance with /1. This portfolio is called the zero1I("la portfolio wilh respe<.:t 10/1.
5. The Capital A~set l'rid111i Modrl
B6 t
esult 4': The covariance of the return of the global miniIlIlIlJl-Variallc(' portfolio g with any asset or portfolio of asseL~ a is Cov[llg'I1,,} =
I
C.
Figure 5.1 illustrates the set of minimum-variance portfolios ill the a sence of a riskfree asset in mean-standard deviation space. Minimumv riance portfolios with an expected return greater than or eqll(' of Ihl' millimulll-variance frontier 011/) limes Ihe standard deviation of ponli)lio I). Result 5: Consider a multiple regression of the return 011 any asset or [Jollfolio ll" on the return of any minimum-variance portfolio l?p (exrept for the globaiminilllllm-varianfe portfolio) alld the return or its associat{'([ zero-beta ponlolio R'1" (!i.2.14)
For the regression
coefliciellt.~
Ih {JI
we have
Cov[ Un. RI,I 2
{Ja/,
Op
COV! Un' I~,) 2
(:).2.17)
I - {J"p
Oof·
fin
(!>.2.1H)
== 0
where l1al' is the beta of assel (j with respect to portfolio Result 5': For the expecled return of a we have ,
'1wc J1~xt
Ii" = (I - fia/,)Il,,/,
+ /3,,/,/J./,.
I). (!).2.1!1)
introduce a riskf"ree asset into the analysis alld consider portfi>lios Fomposed of a combinatioll of the N risky assets and the riskfree assel. Witl, a riskfrce asset the ponlolio weights of the dsky assets are not COIIslraj~led 10 slim to I, since (I - 'w' L) Gill be invested in the riskfree asst't. I
187
5.2. UI'S II lis from FJjirielll-Sfl Malhl'/lwlirJ I'
I'
" _. - - -' - - _ •• - - - - - -'"
If} (J
Figure 5.1.
MilliJlllllll·\'rllilll/(f'I'orl/oliol lVii/will fli.lk/i,·e A.ufl
Given ;\ riskfree asset with return UJ tilt" minillllllll-\'ariance portfc"fio with expected return ill' will be the solution to the constrained optimization mill w'Dw w
sul~iect to
(5.2.21)
As in the prior problem, we form the Lagrangian fUllctioil I., differentiate it with respect to w, set the resultillg equations to zero, alld thcll solvc for w. For the Lagrallgian function we have I.
==
w'Dw
+ 8 (PI' -
<-
W'IL - (I - w'L)IIJ ).
(5.2.22)
Differentiatillg I. with respect to wand settillg the resllit equal to I.em, we have (5.2.23) 2 nw - 8(IL - II,L) = O. Combinillg (5.2.23) wilh (5.2.21) we have (J.l.p - 'V) ----'--;-"---n -I (IL {JL -
IVL),n-1 (IL
-
Il/ i)
... 11/,.).
NOh' that WI' elll I'xprl'ss Wt, as ;) scalar whirh depends on Ihe lllt'all of lillH'S a portfolio w('ighl v('nor which does nol depend on Ii,
/1
where (:I,2,~t;)
alld Thlls wilh;) riskfnT aSse'1 alllllilliIlHIIIl-I'ariall('e ponli,lios an' a COlllhillalioll ofa J..:il'l'll risk" assel pOll ",Iio wil h weighls propOrlionallo W ;lIlcllhe riskrr('c assel. This pOI'lf()lio of risky assels is callecllhc tanJ..:t'llCY pOI'lI()lio anel has wl'iJ..:hl 1'1'('(01' I w'l = I rrl(/t - HIt},
d!
(/1 -
Uti)
WI' IIS(' Ihe sllhsnil'l 1[10 id('lllil~' till' lallJ..:I')I('Y portfolio, Eqllalioll (!" .. ~,2K) divides Ihe d('llH'IlIS or W hI' tllI'ir slim 10 gel a veclor whose elelllellls SIIIll III 0))(', Ih;11 is, a pO 1'1 fill ill wl'iglll I'(,(,(OJ'. FiJ..:llre :),2 illllstrales 1111' sel of lIlillillllllll-variann' pOllllllim ill IIII' presellce or ;1 riskl'n'(' assel. Wilh;\ riskrn'(' assel all dlil'i('111 pOrlfi,lios lit' alollJ..: tlw lille frolll Ihe riskfre!' assel thrtlllJ..:h portti,lio I" The I'XIIl'C\('d l'Xcess )'('1111'11 pn IIllil risk is IIsl'lirllO prol'id(' a hasi~ ror ecollomic inlerprelalioll of lesls or 11ll' CAPM. The Shmj'P m/in ml'aSI')'('S Ihis qllalllilY. For ally assel or pOrlli,lio II, Ihe Sharpe ralio is IIdilled as Ihe 1111'0111 exn'ss 1'1'1111'11 dil'ided hy til!' slalldard (\t:viatioll of relllrn,
\ : :;: '~j
I'" - lit
(:I,2,2~l)
III Figure :I,:! Ihe Sh;ll"!lC r;\lio is Ihl'slopl' orlh(' lillt: frollllh(' riskfrt,(, rl'lurll (Ill' 0) 10 IIII' portfolio !II,,. n,,), Tlw lallgelJcy portfolio q can he characterized as Ihl' portfolio lI'ilh Ihe Illaxi 11111 III Sharp!' ralio orallportfolios or risk\, assels, '11'sl illg Ihe Illt'arr-I';lrialll'c cfficicllcy of a J..:ivl'I1 pori folio is eqlliva\err I 10 ICSlillJ..: II'lwlI\('r Ih(' Sharpl' ralio of Ihal portfolio is IIH' IIIaxinllllll of Iltc sci of Sharpl' ralios of all possihk pOll fol ios,
5.:) Statistical Framework for Estimation and Testing Illilially 1\'1' IISC till' ;1\SIII"l'lillll Ilral ill\'cslllrs call horrow alit! !clld al a riskfn'c rail' or 11'111111, alld \\'c l'ollSit\n lire Sharpc-Lillillcr v('lsioll of'llrl' CAI'M, 'I'lli'll, 1\'(' c1illlill;IIC Ilris a~Slllllpli()" ;1Ilc\ analyzt' 11r!' Black versioll,
. 5.3.
St(lti~tim{ Fmmt!luorhfor Estimation
189
and Jesting
q
"
-'
"
" "
.
"
,
" ,,'"
R /", I o
Figure 5.2.
Minimum·Variance Portfolios With RUkjne Ami
5.3.1 Slw1'/Je-Lintner Version Define Z, as an (Nx 1) vector of excess returns for N assets (or portfolios of assets). For these N assets, the excess returns can be described using the excess·return market model:
z,
=
0:
+ (3 lm, + f/
(5.3.1) (5.3.2)
E[ftl = 0
(5.3.3)
E[f/f/l =:E E[l.. ,]
=
Jl .. ,
E[(l,., - Jlm)2]
Cov[l.. "
f,l
= O.
=
..
0'2
(5.3.4) (5.3.5)
{3 is the (Nx 1) vector of betas, lin' is the time period t market portfolip excess return, and 0: and "', are (N x I) vectors of asset return intercepts and
disturbances, respectively. As will be the case throughout this chapter wb have suppressed the dependence of 0:, (3, ancl "" on the market portfolio or its proxy. For cOllvenience, with the Sharpe·Lintner version. we redefine I! 10 refer to the expected excess return. .
90
5. The Capital Asset Priring Melilri
The imp licat ion of the Sha rpe- I.int ner vers ion of the CAI'M fi)r (:J.:t I) that all of the elem ellts of the vect or are zero . This imp lkat ion /iJllows from com pari ng the unco ll we have i~
Ct
and sillc e exce ss retu rns are temp oral ly liD, give n T obse rvat ions , the join t prob abili ty dens ity func tioll is J(ZI , Z2, ... , Z-r
I 7.",1, Z",2 •... , Z",T)
-r
==
n
tJ(Z,
I 7.,.,)
,=1 T
== n(27T)-~I~I-~ 1=1
~
Given (!i.3.M) alld the exce ss-re turn ohsel-vations, the para lllet ers o/" th' exce ss-re turn mar ket mod el call be estil llate d usin g maximum li/{r/ilwori. Th's appr oach is desi rabl e beca use, given cerla ill regu larit y cond ition s, 1II1xirlllllJllikclihood cstil llato rs are cons istel ll, asym ptot icall y c/lic ielll . alld asym ptoti cally norm al. To defi ne the lIIaxilJlulIl like liho od estim ator , we form the log-likelihood JUllctioll, that is, the loga rithm of the join t prob ;lhil ity del1sity func tion viewed as a func tioll of the unkn owll par; ullet ers, n, /3, ;Ind ~'IDenotillg £. as the log- likel ihoo d fUllction we h;i'vc,;:
I
r(Ct.f3. E)
I
==
NT
-T log(27T) - -;zT logl EI I T .
- ~ L(Z, - (} - (:J7.,,")'E- I(Z, ~I
.
Q
-
f3%ml)'
(!i.:t !l)
5.3. Slllli.lliml Fm7llflllorkjor l~lilllllli(}11 a/l(l 'jfJlil'K
HII
The maxilllulll likelihoud estimators are the values of the parameters which maximize L To find lhese estimators. wc differcntiate L with respeClto 0:. (3. alld :E. and set the resulting equatiolls to I.ero. The pallial derivalives arc
ilL. iJo:
ilL iJ{3
t(Z, -
1;-1 [
0:
-/;z",,)]
U·,.:t 10)
,=1
ta, -
:E- 1 [
0: -
r:11.",,)Zm,]
([/.3.11 )
'=1
ilL iJ:E (5.3.12) Selling (5.:t 10). (5.3.11). alld (!i.:t 12) IIIUIll likelihood estimators. These arc
ii. - ihi. m 'L;=I(Z, -
l.crO, we
(',Ill
solve fur the maxi-
(5.:~.13)
,,1' (. L...,,=I
to
M)(Z"" - {i m ) A) .•
Z",I -
Jim ....
(5.3.15)
where T
JL = A
TI "L..,Z,
alld
/Ill/
,=1
As already Iloted. the~e are just the /()\'JIlUlas fill' OLS estimaturs of the parameters. The distrihutiuns of the maxillllllnlikdihood estimators conditional on the excess return of the market. 1.",1, I",~, ... , Im/.lilllow from the assumed joint normality or excess n:ltIrns and the II D assumptiun. The variances and covariances uf the estimaturs can he derived using the illverse of the Fishrr infonnatioTl lIlatrix. As discussed in the Appendix, the Fisher information matrix is minus the expectation of the secolld old('l' derivative of the 101{likelihood fUllction with respect to the wetOI or the paramcters.
.1.
II/t' (,1/1'1/1/11\.1.\1'1 I'Hfll//!.
/1//1111'/
Tht' (,(lIl1liliollal dislribuliolls arc
.tV
(0'r~ [I + Ii;',] I:) G,;,
(:l.:t Hi)
(:d.IH)
where /i", is as pn'vious\1' .!dill(·d all.! - '. 11.;. =
r
" . TI 'L.."(Z",, -
_ ,.
/1",)-.
,~I
r-
The nOlalion W.v( ~. ~) illdicalt's Ihal Ihe (N x N) malrix TY;, has a ,,,'ishan dislrihlllioll wilh (T - ::!) degrees of freed 0 III and covariance maIrix This disirilllllion is a 1I11tllivarialt' generalizalion of Ihe chi-square dislrihillioll. Andl'l'soll (1~IH·l) allel Muirheael (\9R::\) provide discussions of its propeni('s.
I:.
'I'll(' ctl\'ari;lIICl'
tI(
c"\ alld ii is
_~ [ii.III]~. 'f
II
a~
E is inekpt'lldelli .,rholh c"\ alld j-J. Using Ih(' lIJ\colIslraillt'd cSlimalors, we call form a Wald It'SI slatisti': of Iht' nllll h),polhcsis, II,,: n = 0 against the ;Iltnllat in' hypot hesis,
The Walcll('sl slalislic is
/i;,,]-I (\-'",,-I n,
'J' [I -I- -
n:!II.
L,.,
WIl('ITW(' kl\'('slIhslillllnl (rolll (r.. :I.I(i) forVar!c\!. Ulltler Ihe lIuli hypolhesis./" will have ;1 chi-s'l";"(' dislrihlliion wilh N degrecs of frcedolll. Sillce ~ is ullknowlI, 10 lise ), «II" leslill),.; II". W(' SUhslilule a ("oll\islelll eSlilllalor Ill!" ~ ill (:l,:\.~~) alld tllI'lI asvlllpiolically Ih(' lIull dislrihulioll will 1)(' chisquare wilh N elegn'('s of Ircceloll\. Till' llIaXillllllll likelihood eslilllator of fall S.TV., as a fonsis\('111 ,'slilllalor.
I:
5. J. St(ltiltira{ Framnllork Jor /,!itilll(ltio/l Ilmi Tflting
Ilowever, in this case we need 1I0t resort (0 large-sample distribution theto draw inferences using a Wald-type test. The finite-sample distribution. which i~ developed in MacKinlay (I9R7) and Gibbons, Ross, and Shanken (19WI), can be determined hy applying the ()Uowing theorem presented in ' Muirhead (19R3):
01)'
Theorem. 1.1'1
"If
I
m-lIl'clor di.\IJ-ivu/ni W",(n, 0) with
x VI' di51riiJull'li N(O, D), il't thl' (rnx m) matrix A hI' (71 ~ 111), and lrl x (lnd Ave indepmdrot. Thm: I
(n-m+l)
I
_I
-'------xA x
~
'
1-", ... _ •• +1.
111
a
/1/
To apply this theorem we ~et x = JT[I + Ji;./a;1-1/2 ,A :: 7r'E, = N, and n :: (T - 2). Then defining JI as the test statistic we havel
J1
=
I
(T-N-l)
N
[
Jl.'2m +'2
am
J-
1
"i-- I
•
QL.JQ.
(5.3.23) I
I Under the null hypothesis, JI is unconditionally distributed central F with N degrees of freedom in the numerator and (T - N - I) degrees of freedom in the denominator. We fan construct the Wahl test Jo and the finite-sample F-test JI usi:ng only the estimators from the unconstrained model, that is, the excess-retll-n market Jllodel: To consider a third test, the likelihood ratio test, we need Ihe estimators of the constrained model. For the constrained model, the Sharpe-I .intner CAPM, the estimators follow from solving for (J and E from (:).:'\.11) and (5.3.12) with Q constrained to be zero. The constrained estim;ttors arc
13' ,
E
(5.3.24)
::
.
T
::
.. .• TI " ~(Z/ - f3 Zm/)(Z, - f3 Z.. /)'.
(5.3.25)
'~I
The distributions of the constrained estimators under the null hypothesis arc
N({3'+['21111l+I GJE) 2
(5.3.26)
III
T'E'
(5.3.27)
(:in'lI bOlh the unconstrained and constrained maximum likelihood esti1\l,llors, we can test the restrictions implied by the Sharpe-Lintner version
...
•.
; : ~~!~
,.~.tl'"
5. The Capital Asset lJriejll/{ /Hodel
194
using the likelihood ratio test. This test is based 011 the lo'garithm of the likelihood ratio, which is the value of the collstrained log-likelihood function minus the ullconstrained log-likelihood fUllction evaluated at the maximum likelihood eSlimators. Denoting CR as the log-likelihood ratio. we have
c-£
CR
T -. -~(logIE I-IogIEI].
(5.:t~H)
whcrc C reprcscnts the constrained log-likelihood function. To derive (5.3.28) we havc used the (act that summation in the last terlll in hoth thc unconstrained and constrained likelihood function evaluated at the maximum likelihood estimators simplifies to NT. We now show this for the unconstrained function. For the summation of'the last term in (5.3.9), evaluated at the maximum likelihood estimators, wc have T
})Z/ -
0: - j3z,.,)'t-
1
(Z, -
/37./n')
0: -
(5.3.2~)
1=1
T
= L trace[t- I (Z, -
6 - j3ZIII')(Z, - it -
137... ,)']
(:•. 3.30)
1=1
\ I;
--1"" L,.(Z, -
trace E [
=
T
'
,
Q - (3Zm')(Z, - Q - (3Z",,)' ]
(5.3.31)
,=1 • -1
tracc[E
'
(TE)] = Ttracc[l] = NT.
1
The step from (5.3.29) to (5.3.30) uses the rcsultthat trace All == trace EA. a~d the step to (5.3.31) uses the result that the trace of a SUIIl is equal to the sl\m of a trace. In (5.3.32) we lise the result that the trace of the iC\clltity mttrix is equal to its dimension. The test is based 011 the asymptotic result that, under the null hypothesis, times the logarithm of the likelihood ratio is distributed chi-square with d grees of freedom equal to lh(' IIlllJlher of restrictions untler II". That is, w call test 110 using
-r
1
J~
-Z£R
I
I
I. Interestingly. here we need not reson to large-sample theory to conduct a likelihood ratio test. JI in (5.3.~3) is .iL~e1f a likelihood ralio test statistic. This result, which we nexi develop. follows from the fact that JI is a monotonic transformation of J'1' The constrained maximum likelihood
5.3. S/(//islim/ FramnuorkJor /,:I/i/llu/ioll
tlllt!
I!I5
Tr.l/iIlK
('stimators call be expressed ill terms of the UIHollstnlillCd estimators. For WI' have
;1'
...
It",
...
!-J + ~--.-~ n. J.l",
. • ~
+ ° 111
T
'""' .• .• TI L(ZI -/~ £",,)(Z, -('1 £",,)' 1=1'
I ~
~ [ (ZI- n.- 'In",,) . L -I- ( I .
I ,=t
X Notin~
. [ (ZI
-
.
0: -
' . +(I fJ1,ntl
Ii '"
£",1 ) -:-~--':! u. ]
ILm
+(1",
fi '" Z"'I ) .] I ~--'i a lim
-t- (fm
(5.3.35)
that 1 T '""'(Z - 0:. -fJ' Z-..1 )' ( 1- ' 1 ", 1.-ml• ~ ) L 1 _~
'=1
0,
• 0:
(5.3.36)
111/1+0,.
we have
-. = ~• + ( _~ a;' . ~ ),_' au.
~
J1. m
(5.3.37)
+ am
Taking the determinant of both sides we have (:>.3.313)
where to go rrom (5.3.37) to (:).3.313) we factorize "E and lise the result that II + xx'I = (I + x'x) for the identity matrix I and a vector x. Substituting (5.3.38) into (:).3.28) ~ives
T log [( ~i 17 /;' ) n...1:,-1.0 CR = -:2
11,.+01/1
+ I]
.
(5.3.39)
[t]·i - I )
(5.:UO)
and for./I we have
JI
=
(T - N - I) (
N
exp
which is a mOllotonic transformation of J~. This shows that JI can be interpreted as a likelihood ratio test. Since the finite-sample distribntion of./I is known, equation (5.3.40) can also be used to derive the finite-sample distriblltion of./l. A~ we shall
/.
"It'
I.
see, ulldl'r IIII' lIull hvpollll'sis IIII' fillill'-sample dislrihulion of)~ rail diffl'r frolll ils laq~I'-salllpl(' dislrilHllioll, JohsoJl alld Korkic (I !IH:!) suggesl all ,UljllSlllH'JlI 10 .h whirll has ht'lIl'1" fillile-sample properties, Dl'iilling JI as Ihe llIodified slalislic, \\'1' 11;1\'1' (J' _. ~ - ~)
.h
---.,-.-.h (r--;¥-~)llogrt'I-logltl]:' X~,
\Vt' will visil IIII' isslI!' of Ihl' fillilc-sample propcrtics of .h alld JI III SI'CliOIl rIA. A II st' I'll I !'rolloillic illlnprl'lalioll can he made of Ihl' II'SI stalistic Jt IISing results fronl I'flid('nl-sc( ma(h('malics, Gihbons, Ross, and Shankl'n (1!1H!») show IhOlI
_ CJ'- N -
.
N
,i; ;;: -
I)
/1 - - - - -
(
'.~) n!
,."
I+~ n;;,
Wh('H' Ihl' portfolio dl'lI1111'd II\' If rl'prl'S('lltS Ihl' t'x /10.11 l;\ngl'lll"y portfolio cOllslnrrll'l1 as ill (!·I.~.~H) rrOlIl IiiI' tV illrlllded assets /1/111 tlie markel portfillio. RI'r;11l frolll Sl'l"Iioll :I.~ (hal IIII' pllr(fillio wilh (Iw maxillllllll sqllared Sharpe ralio of all pOrl/lllios is thl' \;Illgency portfolio, TilliS whell 1'.': /lu11 (he llIarkl'( (lort/illill is Ihe langl'nry ponfillio JI williII' l'qllallo 1.('ro, and as Ihl' sqllOlrl'd SharpI' ralio of Ih(' lIIarkl'l dccreases, .It will increas!', illdicating slrongl'l l"\·id('lllT "gainsl Ihl' I'fficiellcy of the lIIark!'1 jloll/illio. III S('l"liOIl !i.7.~ W(' pn's('lIt ;111 I'lIlpirical l'XOlJllpic using ./1 afll'r cOllsidl'ring IIIl' Black versioll or IIII' CAPM ill IIII' IIl'xl sl'ction,
'i.
,.2 1111/111 Vt'nillll
III Ihl' allS('lllT of a riskl"nT asset WI' cOllsicll'r Ihe Black version of Ihe CAI'1'\'1 ill (:1. I.!"! ). TIll' ('Xp('("\1'I1 1'1'1111"11 1111 Ihe I('nl-ill'''l porllillio EI /{,,'" I is In'ated as all 1.1II0hsl'l \"ahl(' alld hl'lIn' h(,("(III1I'S an unknown lIIodd paralllcll'J". Dl'Iilling IIII' Ino-Ill'\;1 )lortlillio ('xI)('cll'd 1'1'1111'11 as y, Ihe I\brk vl'\"sioll is FIR,I
I.y
+ (~(El a,",l -
(I-fi)y
y)
+,I3EII("'II,
Wilh Iht' mack Illodd, lire 11l1("ollstraill('c11ll0l1l'l is thl' ("('al-relurn markl't
197
5.3. SIIIIi51irlli Framework/or E51imation lind Te51ing
lIIodel. Deline R, as an (N x I) vector of real returns for N assets (or portf()li().~ of asseL~). For these N assets, the real-return market model is
R,
=:
a:
+ {3R mt + ft
(5.3.44) (5.3.45) (5.3.46)
E[Rill' 1
(5.3.47)
/lin'
=:
Cov[Rmt> Ed
=:
(5.3.48)
O.
{3 is the (N x I) vector of asset betas, Rot is the time period t market portfolio relllrn, and a: and f, are (N x 1) vectors of asset return intetcepts a,nd disturbances, respectively. ' The testable implication of the Black version is apparent from comparing the unconditional expectation of (5.3.44) with (5.3.43). The implicatipn IS Q
=
(~- (3)y.
(5.3,49)
This implication is more complicated to test than the zero-intercept restriction of the Sharpe-Lintner version because the parameters {3 and y enter ill a nonlinear fashion. (;iven the lID assumption and the joint normality of returns. the Black version of the CAPM can be estimated and tested using the maximum likelihood approach. The maximum likelihood estimators of the unresuicted Illodel, that is, the real-return market model in (5.3.44), are identical to the estimators of the excess-return market model except that real returns ar~ slIbstituted for excess returns. Thus jJ., for example. is now the vector of sample mean real returns. For the maximum likelihood estimators of ttt parameters we have
=
=
{l- {3[J.m
'L.:I (R
t -
(5.3.50) jJ.)(R ml - ft •• )
(5.3.51 )
'L.~I(Ulnl - ftm)2
+
T
=
L(Rt -
ex - j3R,.t)(R, - ex - {In,,.,)',
1=1
where 1 T
-"-R T~ 1=1
mI'
(5.3.52)
5. The Capital Asset Priring Me}(M
198
Conditional on the real relllrll of the markel, R"", butions are
U",~,
.. , , U,. /", the distri-
where T
•2
am :::
~
Tle covariance of it and I I,,
T1 \.""""""' L(I~.I -
•
JIm
~I
)2
•
jJ is
• ., It .. ] . Cov[a,i3J = - [ ~ E.
am
\ For the conslrained modd, that is, the Black version of the CAI'M, the log-likelihood function is • C(y. i3. E)
NT
= -T
T
"2
log(2JT) -
T
-21 'L" (R , -
y(t -
log lEI
m - i3R
ml ) '
E- ,
1=1
)( (R, - y(t -
m - i3R",I)'
(5.3.57)
Oil crcntiating with respect to y. i3. and E. we have (5.3.!iH)
t
:~
==
iJL
== -- E- 1 + - E- 1
(}E
E-
1
T 2
[
(R , -
I 2
Y{L -
[
m - i3U
"1' '"
S (R, -
X
ml )
(Hm,
yU -
(R, - y (t
-
y)]
(J}--- (JJ{ml)
~ m- i3Rmt)'] E- 1• (5.3JiO)
5.3. Slatistirtll Fmllll'work Jor /::ltilll(lfiOIl lIwl ·/"Ifill!!.
19!1
Selling (5.:!.!)!)), (:).:1.59), alld (~).:\'(i()) to {
(~
i/ t'
+. t(R
1-
-.
(it -
y'
......
~.
- li)'1:
L:~I(RI
-
(j Ii.,,)
I
Y~L)(/?""
L,'~tU{,,,(
-
(r,.:Uil)
" ..
(I, -,
0 ) -
y')
y')~
Y' (~- b') - /3' Uml)(R , - Y' (L -
i~') -- /3' U>nl)'.
e',.:t()3)
I;J
Equations (5.:I.GI), (:).3.()2), alld (!1.3.G:\) do 1I0t allow liS to solve explicitly for the maximum likelihood estimators. The nl
t
(~
- {j)y
(!).:H4) (:1.:\.65)
A likelihood ratio lest can be cOllstructed ill a lIlallll\T analogolls to the test constructed 1'01' the Sharpe-Lintner versioll in (:).:1.:tl), Ddillillg Jl as the test statistic, we have
(5.3.G6) Notice that the degrees of freedoJll of tbe null distributioll is N - I. Relative to Ihe Sh'lrpc-l.illtller vcrsioll of the model, the Black versioll loscs OIlC dcgree of freedom hecausc the zCJ'(>-heta expected return is a frcc paramctcr. In addition to the N(N - I )/2 parameters in tile residual covarialH:e matrix, the IIIH"Ollstrailled model has 2 N paramcters, N parameters comprising the venor Q and N comprising the vector p. The constrained model has, in additioll to the sallle lIulllber of covariance llIatrix par;uIlClcrs, N paralllcters cOlliprising the veri or f3 antlthc paranH'\n lor 11](' cxpeclnl/,cro-hctIIleXI (If Ihl' HIOlrk \'t'rsion 01"111" (:APM, (;,1,1.011\ (I q~:!l fit \1 Sh'"IKl'1I (t\IKClh) pm,"ill," d"I,liI"1I '"I;(\y,i,.
dC'veloped lhi~ ('!
).
WI' rail also
.f;. as thl'
1 ht' (;II/JlIIII ,Iul'/ I',-irillg ,1/lIdt'1
~\lli",t
;lIli"~tl'd
./1 to improve IIH·lillitt·-samplt· propt,rtit's. Ddillillg tl'St ,tatistic I\'C han' "
X.V-.I·
lillillite salllplt-s, till' 111111 dislrihlltiollor). willillore dosdv lIIalch the chisqllarl'distrihlltioll. (S('(' Senioll :•.. , 1'01' a Clllllparisoll ill tl)(' COli text or tl)(' Sharpl'-I.itllIH'I \'('rsioll.) There are t\\'o drawhacks to tht' IIIt'tltods wt' Itavejllst discllssed. First, the t'stilllatioll is sOIlH'\\,hattcdiollS sillcc olle mllst iterate ovcr the Iirst-ordn fllllditions. St'cond, tht' t('st is hased 011 largt'-sample theory and call haw wry poor flllite-salltpk properties. We 1'.\11 \lse tite res\llts of Kandd (1~IH4) alld Shallkell ( I!)Hli) 1(. O\'l'rCOllle Ihes(' drawharks, Thes(' alit hoI's show how to ('akllialt' cxact tllaxilllllm likelihood estimators alld how to impl('lI\clt! all approxilllate tcst with good lillite-salltple perli>rtnanCt', For tht, \lllcollstrailled mode\. cOllsidt'r the market lIIodel cxpressl'd ill tt'flllS of r,'lItrlls ill I'xct'ss of Ihl' \'Xlll'(·t,'d 1.,'1'0-1)('1;\ relllrll y:
R, .-
)'1.
""
It·/
(J(ll"" -
y)
+ 10,.
(!i.:UiHl
ASSlIllle y is kIlOWIl. Theil Iht' IItaxilllllJII likelihood estimators fi.n the IlItcOllstraitll'(IIlIOdcl an'
j1. - YI. - /3(it", - y), r • • L,=I(R, - p.)(RIII , -/1",) L:~I(/l.,.,
-
(5.:t!i!l) (:i.:DO)
il",)~
alld ./
E=
+:LIR,-it-/J(/(m,-ilmlIIR,-iL-{3{1l""-(I,,,)J'. (:).:DI) '~I
Thc lIlIcomtrailled I'slilllators "I' {J alld E do 1I0t dept'lld 011 tht' vaillt' or y hili, as indirated, tht' estilllator or n dot'S, The vallie or tht, IIncollstraillt'd log-likelihood rllllctioll ('\,alll;lIl'
II'hirh dOl's Ilot dl'pl'lld Oil )'.
I"g(~rrl-
'J' ~
• NT IOlfIEI- h
~
5. J.
Slfllisli{(l[
Frmnt'l/Jork for r.Jli/llflliOIl (lwl
Constraining
0:
201
'IrSlillg
to be zero, the constrained estimators are
~.
i
=
(5.3.73) !
I I
"")( (RI - y(~ - 13·) - 13· R,.I)'.
(5.~.74)
and the value of the constrained likelihood function is C(y)
NT
T,.
== - - log(2rr) - 2
2
logl~
NT (y)I--.
(5.3.75)
2
Note that the constrained function does depend on y. Fonning the logarithm of the likelihood ratio we have CR(y)
== C(y) - [. :::
-~ [log IE\Y)l-log lEI].
(5.3.76)
The value of y that minimizes the value of the logarithm of the likelihood ratio will be the value which maximizes the constrained log-likelihood fUllction and thus is the maximum likelihood estimator of y. Using the same development as for the Sharpe-Lintner version, the log-likelihood ratio can be simplified to CR(y)
=
T
-2'log
[(
a~
(tlm-y)2+a~
)"o:(y):£, - I o:(y) , +.I]
-~2 log [( (-11m _ a~2 • ~) [jL y +0 .. X
[jL -
y~ -
yL -
13 ([i... - y)]
(3 (il .. -
+ I] .
y)l' E-
1
(5.3.77)
Minimizing CR with respect to y is equivalent to maximizing G where G
= (' _ a~2 + J1...
1 -2) [jL-YL-13(tlm-y)l'E- [jL-YL:"'(3(il ... -y»). yo,.
(5.3.78) Thus the value of y which maximizes G will be the maximum likeIihpod estimator. There are two solutions of aClay = 0, and these are the real rools of the quadratic equation I !I( y)
=
A y2
+ By + C,
(5.3j79)
5. The Cal)i/a/ Anl'l I',irillg Mill""
202
where I
-;-;z (t
-
','-/. m 1.:
-_
(
ii~)
1+ .;
I l
t
-,--I
(i",
(It - {3/1",) - -::-:; (t - {3)
am
(J HI
-
--I
I
-
(t-f3)- .~
(t-f3)':E
/1 ~
+ ii",. a;;' (J1. -
{3"
(t -
m
·-1 • (p.-{3(;.",)':E (jt-{-J(J.",)
alii
Om
(1+ am..;' )
1.:
•
-
(t - j.3)':E
1(",)
-I
, {, -- I . L.-
•
(ji. - {3,i",) 'J -
(,l- ("I",).
If is greater tilan zero, the lIIaximulll likelihood estimator y' is the largest rot, anel if A is less than zero. the II y' is the smallest rool. A will he gl ater than zero if ii", is greater than the mean return on the sample glll!>,,1 minimulll·variance portfolio; thai is, lhe lIIarket portfolio is (lIl the c1~\lcient part of the constrained mean-variance frontier. We can suhstitute Y'I into (5.3.62) and (:1.3.63) to obtain (-J' and -t' without resorting to an itclrativc procedure. I We can COllStruct an approximate test of the lIlack version using relllrns in \bx(ess of y as in ([1.3.GH). Ir y is known then the same methodol0h'Y us(·d to ~onstructlhe Sharpe-l.intnC!' version F-test in (5.3.23) applies to testing the null hypothesis that the zero-heta exn'ss-relllnl nl
=
5.4. Siu (if·/i'.III , This estimator can be evaluated at the maximllm likelihood estimates. and then inferences concerning the vallie of yare possihle giwn the asymptotic normality of y'.
5.4 Size of Tests In some econometric models there arc no analytical re.~uIL~ on the finitesample properties of estimators. III sllch rases. it is UJlllmOn to rely on Iaq!;esample statistics to draw inferences. This reliance opens lip the possibility that the size of the test will be incorrect if the sample sile is not large enough for the asymptotic resulLS to provide a good apPlOximalioll. Because there is no stalldard sample size for which large-sample Iheory can he applied. it is good practice to investigate the appropriateness of the theory. The lIlultivariate F-test we have developed provides all ideal framework fiJI' illustrating the problems that GUI arise if' OIl(' rdies ou as)'mptotic distriL'llion theory for inference. Using the known finite-sample distrihution of tht: F-test statistic }I. we can calclliate the !illite-sample sile for the variolls aSyll1 ptotic tests. Such calculatiolls arc possihk h(,CllISC the ,\syllljltotic test statistics arc monotonic transformations of '/1, We draw on the relations orfl to the 1.lIg('-sanql\c I('sl statistics. COIlIp;'ring equalions (:).:t22) and (:).:1.2:1) 101 ./" WI' II.'v('
./1
('I' - N .- I)
= -
NT
./".
and for }:l from (5.'1.2) and ([1.:1.4 J).
/1
.
=
U-N-Il N
(exp[~] t/-,···~I.
-I).
(5.13)
Under the llltil hypothesis. }o. }~. alld ./1 arc all asymptotically distributed chi-square with N degrees of frecdolll. The exact nuH distrihution of }I is celltral F with N degrees of freedom in til(' numerator and T - N - I degrees of freedom ill the denominator. We caIeulate the exact si/.e ofa test hased on a giv(,11 large-sample statistic and its asymptotic 5% critical value. For example. cOllsider a test usillg }II with 10 portfolios and liO months of dat,\. III this rase, under the lIull hypothesis }II is asymptotically distrihllted as a chi-square randolJl variate with 10 degrees of freedom, Given this distrihutioll. the cdtkal value for a test with all asymptotic sil.C of 5% is I H.31. From (:>.'1.1) this value of 18.31
~1I'1
,A, COITl'SPOII/Is
to a nitical \'ahll' of I A~l:, lill' '/1, Cin'1I that the (');an (Iislrihlliioll 01'./1 is /: wilh 10 dt'grecs offrecdolll illlht' Illlilleralor alld ·I~l degrees olll("nloill ill til(' d("lIollliliator, a lest IIsilig Ihi~ niticd \'ahlt' for ./1 has;1 sil(' of 17.0')!,. Thlls, 1111' aSYIliplotic !,'y" lest has a sil(' of 17.0';' ill a salllpk 01'(;0 11I(llllh~: il reilTls Ihe 111111 hypolhl'sis ilion' Ih;lIlll1r('e lillll'S looofleli. 'Elhle :,.1 PII'S('IIIO; Ihis Cliclll;tliOlilill' .A" J~, alld./I IIsillg Ill, ~Il, ;llId·1O Ii II' valll(,s of ,v all(1 IIsill)!; (iO, 1~(), 1HO, ~40. alld :~(iO f(lr \';II\I('s of r. It is appan'lIl 111;11 Ill(' lillile-s;lIl1ple sil(' of Ihe lesls is largn Ihall Ihe aSYlIlplolic silt, or r,I;;,. Thus Ihe large-s;lIl1plc lesls will n:il'rl Ih(' IHlli h),polhl'sis 100 oft(,lI. This prohll'lll is sn'nl' li,r Ihl' aSYlllplolic lesls hased 011 ./t, alld ,h. Whell N = 10 Ihl' prohl('1Il is 11I0sily il1lpOl'lalil lill' Ihl' low valll(,s of For I'xalllpk, Ihl' lillih'-~;II11pk ~i/l' or a test with
111111
r.
5.5 Power of Test'> \\'111'11 drawing illli-n'lHTs nsillg a gin'lI It'si slalistic II IS 11111)(11 Ian I 10 COIIsidn ils IH,\\,l'r. The pown is lite prohahililY Iltal lite 111111 hypolhesis will Ill' n:jl'cl('d gin'lI Ihal all aitnllaliw hypolhl'sis is 11'111'. Low pown agaillsl all ililen'slillg aiterllalin' slIggl'SIS Ihal Ihe It'sl is 1101 IIsl'fliliO disnil1lillall' 11I'1\\'l'I'1i Ill(' ait('III;lIiv(' alld Ihl' 111111 hypolhesis. Oil Ihl' othn halld, ir Ihl' 1'01\'('1' is high. Ihl'lI till' Il'si elll he very illlill'llIalivl' hili il lIIay also rl'jl'l'I IIII' 111111 hvpollll'sis ;Igaillst all(,)lIali\'('s Ihal an' dose 10 Ihl' 111111 ill ('1'0lIolllie "'I IllS. III Ihis case ;1 rl'j"('lioll llIav hI' dill' to slllall. "COllolllicdhIIl1illlportalll d('li
205
5.5. /'0/(1" of Tests
Table 5.1 .
Finilf'-5f1111/'iP liu oj 1,.. 15 of IIIf Sllflrp,..Ulllwr r.APM wing lnrgt-1am4u. Itsl
.llllli,llir,I,
j,
N
T
j"
}l
\0
60
0.170
0.096
0.051 :
120
0.099
0.070
0.050 \
I
180
0.080
0.062
240
0.072
0.059
0.050 I 0.050 '
360
0.064
0.056
0.050 i
60
0.462
0.211
0.057 ,
120
0.200
O.IO~}
0.051
180
0.136
0.082
0.051
240
0.109
0.073
0.050 :
360
0.086
0.064
0.050.
I
20
I
I
60
0.985
0.805
0.141
120
0.610
0.275
0.059
·180
0.368
0.164
0.053
240
0.257
0.124
0.052
360
0.165
0.092
0.051
40
The exaCt finite-sample size is presented for teslS with a size of 5% asymptotically, The finitesample sire uses Ihe dislribution of J\ and Ihe relation between JI and Ihe large-sample 101 stalislies. jl. fl. and j,. N is Ihe number of dependent portfolio!. and Tis Ihe number of tilllt"-~("rit"s
observations.
should be representative, and it is convenient to document since the exact finite-sample distribution of J) is known under both the null and alternative hypotheses. Conditional on the excess return of the market portfolio, for the distribution of Jt as defined in (5.3.23). we have (5.5.1) where 8 is the noncentrality parameter of the F distribution and (5.5.2)
J) under both tbe null and the alternative hypotheses. w.e need to specify 8. N. and T.
'Ii) specify the distribution of
2qli
5. Thr Ca/Jila/ t\.l.Irl/'ririllg A1(}(h'/
i
I
Under thc null hypothesis a is zero, so in this case 8 is lero and we have Ihe prcvious rcsuh thaI Ihc dislrihution is central F with Nand T - N - I dcigrces offrccdom in the IHllller,ltor and denominator, respectively. Under Ih~ allcrnalivc hypothcsis, 10 specify 8 we need to condition Oil a value of t!~/cr; and spccify thc vaillc of a':E-la. For the value of [1.;''/;';" ~iven a m1>nthly obscrvalion intcrval, wc choosc 0.013 which corresponds to an rx /10.[' annualizcd mcan cxcess return 01'8% and a sample annualized siandard dC[iation of20%. For the quadratic term a':E- 1a, rather than separately spedfyin~ a .lIIel :E, we can usc thc following result of Gihbons, Ross, and Shanken (I !IH!l).1 Rc ailing thaI q is thc tangcncy portfolio and thaI 111 is the market portfolio, wc\havc \ I
:
'I
0-
'I
am~
USing this relation, we need only specify the difference in the squared Sharpe ratio for thc tangency portfolio and the market portfolio. The tangency portfolio is for the universe composcd of the N induded portfolios and thc market portfolio. We consider fOllr sets of valucs lor the tangency portfolio paramcters. For all cases Ihc annualil.ed sland,lnl deviatiol\ of the langcney portfolio is set to lli%. Thc anllllalil.ed expectcd excess return thcn takcs on four valucs, 8.5%, 10.2%. 11.6%, and 13.0%. Using an allnllalized expected excess return of 8% for the market and an allll\lali/.ed standard deviation of 20% for the market's excess return. these four values corrcspond to values of 0.0 1,0.02,0.03, and 0.04 for 8/ T. We consider fIve values for N: I, 5, 10, 20, and 40. For T we consider four values-60, t20, 240, and 360-which arc chosen to correspond to !i, to, 20, and 30 years of monthly data. The power is tabulated for a test with a size of 5%. The results arc presented in Table 5.2. Substantial variation in the power of the test for different experimental designs and alternatives is apparent in T,lble 5.2. For a fixed vallie of N, considerable increases in power arc possible with larger values of r. For .example, under alternative 2 for N equal to 10, the power increases frolll 0.082 to 0.380 as T increases from 60 to 31iO. The power gaill is Slibstalllial when N is reduced for a fixed alternalive. For example, under alternative 3, /01' T equal to 120, the power illtTC'ases from 0.093l0 0.47f) as N decreases from 40 to I. llowcver. such gaillS would not he fea~ible in practice. A~ N is reduced, the Sharpe ratio of the tangency portfolio (and Ihe noncentralily parameler of Ihe F dislriblltion) will declille unlcss Ihe portfolios are comhined in proportion to thcir weightings in th,ll portfolio. The choice of N which maximizes the power will depend Oil the
5.5. /'/)I/I/·/"/I/"'/i·.I/.1
Tab/I! 5.2.
207
/'111/'1''' '1 F·/~.l1 II} SI/lII/It··I.inlll"'/Ii
'J"
0=
r ==
IiI)
'J" == 2·10 "f'= :11;0
Alternative 2:
Il'l
==
10.2%
(1'1
-r = 1;0
O.IH!)
'J"
= 120
O.:t~!1
'f'
=:
2,10
05!17 11.770
-r = :11;0 Alternative :~: "f'
Il'l ""
== (iO
I:.!O "f' = 240 'f'= :1(;0
Alternative 4: 'J" = IiI) "f'= 120 'f' = 2,10 'f'= :\liO
11.li%
(1'1
(1.262 0.47:, O.71l9 O.'lOH
'1'=
Il"
./J.
= I
IU17 0.1 !1I O.:HI OAHO
I~I)
(:~/'t\I /llilll: I/,//il/i,
== 1:1.0%
(1'1
0,:1:1-1 O.:,!I:1 O.H7:1 0.%:,
n.07~,
O.IOli 0.17H 0.2;,!)
1).0Ii', O.OHh 0.1:\,1 0.1!1I1
Il.O:,~)
0.10:\ 0.1:\!)
IU),,?> 0.O1l2 0.OH2 0.10:,
0.OH2 0.1:\0 (1.2·17 IUHO
0.(11;1< II.O'lH 0.17·1 0.21;7
0.0:,7 0.077 11.12,1 IUKI
0,101 I).IHO 0,:17-1 (J,r,70
0,07H 0,12H n.2lil O,'I\(i
0.\)(,1 O.O9?> 0.17:, 0.2HO
D.121 o.:!:17 0.:,02 O,nl;
O.OHI) O.IIi:1 o.:\!,,(j 0.:,(;:1
O.(Hi:, 0.1 III 0.2:\1 O.:IH!l
o.on
= II;')(, 0.111:1 0.17'1 (1.:1,10 n.r,OH
= lIi% 0.1:\,1 O.2~>i
O.!,01 0.711 = lIi%
O.lli7 0.:1:12 o.(iD O.H,I:,
---Th"
;llIcllI.tlivc h)'I)('lh('~is i~ rli;u'aru'li/C'cI
h)' lht'
\'.1111(' of llit' ("xllI'(l('d ('X("('!'oS 1«'1111"11
and
Iht' \'alm' of 11i(' ~I.llld.lni .t.uHLu·({ d{'\i.,.ion oi tht.' l'x(T~S l"C.'t\lrn oi
the 1.tngnH·y ptH ltolio. The lHarkcl portfolio is .\\.'';'UIIH'(} to h.nT .m l'Xpt't In} ("x( t's..11 return of H.O'i; . .111<1 .1 .'it.\Il(LII(} dt'\·i.uioJl or 2U%, UI1(i<'I" th(' 111111 h)'pOlh(,!'ii~ (}at' 11I.\rkt't pOI .folio is 1ht." lallgt'II(\' portlolio, N i!'i the IItluli)('r of portfoli()~ infillti('d ill tile 1('.<.;1 ,Ind r i, lilt, IIIlmh("r 01 IJ}Oll1h., I)f d;\I" illrill
rate at which the Sharpe ratio of the tallgency portfolio dl'riillcs as assets arc ~rourcd tOJ,\e!hcr. While we do JlUl have gellcdl results ,Ibollt the optilllal design of a lIIultivariate test, we c
5.6 Nonnormal and Non-liD Returns III this Sl'l'liOIl \\'t' ~\rt' l"IlIll"l'IIH'd wilh illrcn'llct~s when 1ht,I'(' are dcviations /i'Ofll IIII' asslllllplioll 111011 rellll'llS an'.iointly norfllal and liD throll!!;11 tiflle, We cOllsidn tcst~ whirll al'l'Ollllllolbtc non-norlllality, lll'tl'l'Oskl'dasticity, antitelllporal dependellcc of' rellll"llS. Slich tests are ofilltel't'st fill' two reaSOliS. Firsl, while Ihl' lloJ'lllality asslllll)ltioll is sufficil'llt, it is llotlll'CCSsal'Y to derive the CAI'M as OJ theol't'til'allliodd. Rather, the Ilonllality aSSlIlllptioll is adopled fill' stalislical purposes. Williont this aSSIIIllI>lioll, fillile-Salllpll' )ll'Opt'J'lil's oLlssel pricillJ,!; Illodel IcslS arc dirticult to derive. Sl'colld, departun's of' IlHllllhly scrllrilY ITIlll'llS \'rOIlI Ilormality have heen dOl'llllll:llIed.:1 There is also ahllndalll Cl'idell(,(' of ht'lel'Oskedasticity and telllporal dqJl'ndl'lKC ill stock rl'lllllls.1 Evcn though temporal depl'ndellce Ill"kl's tht' CAI'M ulllikely 10 hold as all cxan IheOJ'etiralllloclcl, il is still of illll'rl'st to exalllille Ihl' ('llIpiril'al )ll'l'fill'lllalll'l' of the lllodei. It is Ihl'l'di)l'e desirahk to ronsidcr tIll' dfl'rts of' rclaxing these statistic,,1 assumplions. Rohllsl IcslS of' IIIl' <:1\»1\-1 rail IIc constructed using a Cl'lleralizeti Method 01' 1\!OIllI'IlIS «;MM) I'rallwwOI·k. We focus 011 tests of Ihe Sharpl'1.'IlllIll'l'\·crsioll: IIm\,('\'('I', rohllsi I('sl s or II II' B1ark vcrsion (';1/ I I)l' rOllsl I'll ned illlhe salllc 111;111111'1'. \\'itllill til<' (;1\11\1 f'rallll'wol'k,llIedislrilllllioll of'relllrllS ('olldilioll;\1 011 IlIl' IlIark('1 rellll'lI CII\ 1)(' bOlh serially sl'l'valions atld N assl'ls. Following the Appendix, we necd 10 setup the vl'ctor of 1l101liellt cOlldil ions with zero expectalion. The reqllired 1II0lllellt l'OluliliollS ",lIo\\' 1'1'0111 1IIl' ('X(·('SS-relllrnlllarkellllOdd. The residual vector provides N III0llll'lII ('ollditions, ant! the product of the ('X(,css I'l'tU/'ll of lite llIarkl'l ;11111 Ihe rl'sidual vector provicles allother N 11IOl1lCnt cOllditiolls. Usillg Ihl' Ilolalioll of' Ihe Appendix, fill' f(O) we have (:,.1;.1 ) where h; = (I /."" (, (", = Z, - n - (J /."''' alld 0' = (0' ,13']. The spl'cilicali,," or IIII' C'x('('ss-retllJ'n market 1l1l)(ld illlplies Ihl' 11101111'111 cOlldilioll 1':1 f,(OIl) I =- 0, \\'111'('(' 011 is the Ii'll<' par;UIlI'I('I' vector. This IlIOllll'nt cOlldition lorllls Ihe hasis lill'l'stilllalion and lesting using a C~'IM approarh. (;Mt-.I chooses till' l'stililalor so that linear cOIlIi>inatiolls or Ihe salllple avc'ragl' oflhis 1lI0llll'IlI (,(llldition arc zero. For thl' salllpl(' avcragl" (I'll;',. 1!17Iil. 111.,"111"1-: ,,,,,ICCI'lC'ell" (1"17·1), l\lfI(·.. ~·t:r.I\I·' a",1 ~kIlClII."eI a,"1 '\:,hl" 1.1 ill (:10,,1"'" I. 'StT (:h"IHC.'I"" '.! .uul I'.!. ,HHI ,IH' I ('klc',u C'!'. 14ln'u in 11\0'\' ("h~'ptt·,~.
"SI'I' F.""., (I"III~I),
).6. Nonnonllai a1ld Non-JlV Helurns
we have
1 T
= T L f (9).
gT(8)
l
1=1
Thc GMM estimator iJ is chosen to minimize the quadratic form (5.6.3) whcrc W is a positive definite (2Nx2N) weighting matrix. Since in this case we have '2N moment condition equations and 2N unknown parameters, the systcm is exactly identified and iJ can be chosen to set the average of the sample moments gT(O) cquallO zero. The GMM estimator will not depend 011 W since QT(O) will attain its minimum o[zero for any weighting matrix. The estimators from this GMM procedure are equivalent to the maximum likelihood estimators in (5.3.13) and (5.3.14). The estimators are
jL -
j3j;.m
(5.6.4)
L;~I(ZI - [J,)(Zml - ft.m) ,\,T
L...I=I (
Z • ~ -ml - 11m)
(5.6.5)
Thc importance of the GMM approach for this application is that a robust covariance matrix of the estimators can be formed. The variances
a.
of and j3 will differ from the variances in the maximum likelihood approach. The covariance matrix of the GMM estimator iJ follows from equation (A.2.R) in the Appendix. It is (5.6~6)
where D and So
=
o
=
E[OgT(8)}
ao
t
+00
L
E[f l (l;l)f l _I(9)'].
1=-00
The asymptotic distrihution of iJ is normal. Thus we have
Tlw application of the distributional result in (5.6.9) requires consistent csrilllarors of Do and So since they are unknown. In this case, for Do we ha~e
~ Dn
= -[ )l", I
(2fJ..m
2)J0IN'
u",+J1 m
(5,6.\0)
210 A consistent estimator OT can easily be constructed using the maximum likt·lihood estimators of Ii", and a;;'. To compute a consistent estimator ors". ;UI assumption is necessary to reduce the summation in (5.6.8) to a Ii nile 1111111' her of terms. Section A.3 in the Appendix discusses possible assumptioll". Defining Sr as a consistent estimator of So. (1/ '1') [O'1'S;-,1 Or J- 1 is a consisten t estimator of the covariance matrix of O. Noting that 0. = RO where R = (I 0) ® IN. a robust estimator of VarIa) is (l/1)R[O'TS;-,'Orl-'R'. Using this we can construct a chi-square test of the Sharpe-Lintner modd as in (5.3.22). The test statistic is (5.1i.11) Under the null hypothesis a
= O. (:>.Ii.I~)
YMacKinlay and Richardson (1991) illustrate Ihe bias in standard CAl'M tes~ statistics that can result from violations of the standard distributional ass~mptions. Specifically. they consider the case of contemporaneous conditiOllal heteroskedasticity. With contemporaneous conditional heteroskedasticiiy. the variance of the market-modd residuals of equation (r).3.:~) depel~ds on the contemporaneous market return. In their example. the as5111 ption that excess returns are 110 aruljointly lIlultivariate Student Ile;\(ls to onditional heteroskedasticity. The lIlultivariate Student 1 assumption for xcess returns can be motivated both empirically and theoretically. One em· irical stylized fact from the distribution of returns literature is that ITtun s have fatter tails and arc more peaked than one would expect from a nor a1 distribution. This is consistent with returns coming from a multivari Ite Stude III I. Further. the multivariate Studellll is a rethrn distribution for vhich mean-variance analysis is consistent with expected utility maximiz tion. making the choice theoretically appealil1g.~· he bias in the size of the standard CAI'M test for the Student I case 'depbnds on the Sharpe ratio of the market portfolio and the degrees of freehom of the Student I. MaeKinlay and Richardson (1991) present sOllie estil~lates of the potential hias for various Sharpe ratios and for Student I degrees of freedom equal to 5 and 10. They find that in general the hias is small. uut if the Sharpe ratio is high and the degrees of freedom slllali. the hias can be suhstantial and lead to incorrect iniCrences. Calculation of the lest statistic 17 based on the CMM fral\lework provides a simple check for the possibility that the rejection of the modd is the result of heteroskedastirity in the data.
5.7.
ImIJ/i'IIII'lIla/ioll o/Tr.ll.l
:lll
5.7 Implementation of TcsL'i thi~ SI"I lion we considcr issues rd.lling III (~Illpiri("al illlplnllcnt
III
5.7.1 SllInmmy oj Eml,iri((l/ ElIitlrnrr An enormolls amolillt of literatllre presenting elllpirical evidence 011 the CAI'M has cvolved since the development of the model in the 1960s. The early cvidence was largely positive. with B1ark •.IellSen. and Scholes (HI72). Fama and MacBeth (1973), and B1l1me and Friend (1973) all reponing evidence cOlISistellt with the lIIean-v"ri"nce efficient·y of the markel portfolio. There was some evidence against the Sharpe-Lintner version of lhe aPM as Ihe esti!nated mean return onlhe zero-heta port[(llio was Iligher than the ri~Hree return. hut this could he accoullted for by the Black version of the l\I(,del. In the late I ~170s less favorable evidence for the CAI'M began to appear inlhe s(}-callcd ,lIIomalies literature. III the contexi orthe tests discllssed in this chaplCf. the ,Illolllalies call he thonght o\" as lirm charallerislics which can be nsed to group asselS together so that the tangency portfulio of the included portfolios has a high I'X 110,11 Sharpe ratio relative to Ihe Sharpe ratio of Ilw market proxy. Alternalively. COlllra!), 10 Ihe preC\inioll of Ihe CAPM. Ihe [inn charaueristics provide explallatory power for lire noss seclion of ~alllple mean returns beyond the heta of lire CAP!'.-!. Early anomalies included Ihe price-earnings-ratio cllect and Ihe si7.e circe\. l\asu (1977) first reported Ihe price-cHrnings-r.ttio elTeu. BaSll's Ii nding is Ihatthe market portfolio appears nol to he l\Iean-variance emdent rel.ttive to portfolios forllled on Ihe hasis of the price-earnings ratios of linns. Firms wilh low price-carnings ralios howe higher sample retllrns, and linlls with high price-earnings ralios have lower lIIean returns Ihall wOllld he Ihe case if the market portli)lio was lIIean-v.u-iallce efficient. The sil.e clfert. which was [irsl doculllented by Bani, '( I ~}H I). is the resllit that low lII,trkct rapilali/,alioll linns have higher sample lIIean returns Ih
J,
Jill'
(,11/1/(111 .. 1,.\t'( I'm'jllg
,1/,,"'"
IkBolld l alld Th;dn (1~1H:1) alld ,'q~adeesh alld Titlllall (I~}~}:,) filld lil;11 a porlli,li o lilllll«'d III bllrill)!; slocks whose \'allll' has dt'c1illl' d ill Iht' 1);1,1 (/11.\/'/'\) ;lIul sellillg slo('k, lI'ilo,,1' \'allll' has risell ill Ihe pasl (lI';lIlIl'n ) h;IS a hi)!;hl'l' an'r;lg(' .1'111111 Ilwl III(' C:/\1'~1 prl'dicls , Failla (1~}~1I) pro\'idl' s ;. good discllssi oll ollhl'sl ' alld 01111'1' ;lIuH.lal il's, Alt hOllgh I Ill" reslllt,\ ill till' a •• 01l.;t1i,'s lill'rat 111'1' Il.a\, si)!;lIal ('rollolll ir;dl\' ill.porla lll d('\i;lIio lls IHIIII till" (:AI'~I, t11('J'(' is lillie IIll'oreli ('al IIIOli\'al ioll li,r Ihl' linll (hara('( '. iSlics "llIdil'd ill Ihis lileralllr e, This opells lip Ih(' possibilil~' Ih"l Ihe ('\'id('IIt' (, a)!;;lil.sl Ihl' CAI'M is O\'l'rslat t'd Iw('allsl ' 01 d;lIasiloopill) !; alld salllpll' sd('('lio ll bia,('s, WI' hridly disCllss Ihl'SI' possibili Li"" I lala-sllo opillg biaSI'S rdi'r 10 Ih(' biasI'S ill slatislic al ill li-rl'II ('(' Ihal r('slIll 11'0.11 IIsill)!; illlillllla Lioll 11'0111 dal;1 10 )!;lIid(' Sll bS('1) 1)('11 I rl'search wiLh Lhl' sallll' or relal('(1 Ibl;1. Thl'''' hi;ISt's an' allllosl illlpossi hll' 10 ,,\,oid dill' 10 Ihl' 1101I('xp ('rillll'lIl al 11,,1111'1' of 1'('OIlOIl.i('s, \Ne do .UII han' Ihl' It.XIIJ'\' of rllllllillg "lIolht'l ' I'xpl'rilll('111 10 (,),I'al(' a III'W dala S('I. 1.0 alld Mad\'ill lal' ( I ~}~}Oh) illllSI raIl' III(' pOlnll i;d 1I.;lglI illule ofdala-sI IOOpill) !; hias('s ill a I('SI of I h(' Sharpe-I .i.II.It'l' \'t'I'sioll 011 hI' ( :AI'M, They cOllsili n Iht' ('ase II'h('l'(' 11ll' d.ara('( nisl ic lI",d 10 )!;I'OII P slocks illlo porI folios (t' ,)!;, sill' or pric(,-I'a rllill)!;s r;lIio) is sd"t'l(,t! 1101 11'0111 Ih('orl' bill 1'1'0111 pre\'ioll s obs('rl'" liolls 01 III ('a II \Itu'k 1'('1 II I'IIs IJsillg ''('/;'.1'<1 <1;11,1. (:"lIIpar isolls "flhl' 111111 dislriblJ lioll 01'1111' Il'sl SI;llislic \\'iLh ;11 It I \\'ilholll d;\I"-Sll oopillg SII)!;g('SIS Ihallhl' IIl"gllill ld(' of Iht' hias('s ('all 1)(' illllllt'lIS t', Ilm\'(,\,('I', ill praclic( ', il is dilli(,1I1t 10 spl'rih' 11t(, adjllslllu 'lIllh;1l shollid Ill' .1I"d(,li, rdala-sIl OOpill)! ;, Thus,lh (' lIIailllll l'ssagt' is a \\';.l'IIillg Ih;11 Ihl' hi;lst's sl,olllt! ;11 leas I b(' cOllsidt' l'l'd ;IS a pol('IILi;t1 t'xplalla lioll for II10dd dl'l'ialio lls, Salllpll' s('Il'('lio ll hi",,'s ('all arisl' II'h('1I daLa a\'ailabi lily leads 10 t't rl;lill Sllhst'ls of slo('k, beillg ('xl'illd( 'd 1'1'0111 Ihl' allalysis , For ('xa.llpl e, KOlh,'ri , Shallkl'l I, alld Sloall ( I ~l~l:,) argllt' I hal dala r('qllin'I IIt'lllS for sliltlit's look i II)!; al hook-lII arkl'l ralios Il'ad Lo bilillg sLo(,ks Iwill)!; ('xcitul( 'd ;lIld a rt'sllitill g slIJ'\'il'o rship hias, Sillt'(, IIII' 1;lilill)!; slorks wOlild he ('xP('('ll 'd 10 han' lo\\' It'IIII'IIS alld high ilook-Ill arkt'l ralios, Ihl' al'erage n'tlll'll ofth(, illcllld(' d high hook-lII arkl'l-ra lio SIOI ks 1I'01iid hal'l' allllpwa rd hias, KOlhari , Shallkl'l I, alld SIO;III ( I ~}~I.',) a rg 11(' I hal !I,i, hi a, is 1;lrgd), J'('spolIs ibl('IIII' lhc previolls ly ci 1('(1 rl'sllit of Falll;1 alld FII'IIl'l, (I~I~}:!, 1~I~n), I I ow('\,e 1', tht' iJllporta llre of II,is pal'li(,lI larslll'\'i n ,rship hia\ is 1101 I II II\' lI'sol\'t'd as Failla ;lIld FrclIl'l1 (I ~}~}(ib) displIl(' Ihl' ('tlllllll\ ioIlS 01 Kolh;lIi, Sh;llIk('I I, alld Sioall, III aliI' I'\'('IIL. il is d .. ;II' 11,;\1 1!',\t';lrt"llt'l'.s sl,ollld 1)(' ;111';11 (' of Ihl' pOI(,lItia l prtlhll'll I\ Ihat ('all arisl' 1'1'011. s;lIl1pll' sl'i('t'lio ll hi;,,(",
'i, 7,2 "III"lIllh -/'
111/1",'"1/'1/1111;01/
WI' pn'St'II1 l('slS 01 II I<' Sh;1I1 11'-1 ,i II L111'1' 1I1t)( 11'1 10 i II11s1 1';11 (' II It' It'sl i IIg II 1<'1 I lOtIolog\', Wt' l'tlllsid n li,IIII('SI slalisLics: ,It frolll (:,.:I,~:I), ,h 1'1'0111 U',,:I,:!:!
)../I
5.7. /1Il/,ll'1l1l'1llalioll o/TrIls
. 213·
from (53.41), and J7 from (5.6.11). The tests are conducted using a thirtyyear sample of monthly returns on ten portfolios. Stocks listed on theiNew York Stock Exchange and on the American Stock Exchange are allocated to the portf(}lios based on the market value of equity and are value-wei~ted within the portfolios. The CRSP value-weighted index is used as a proxy fi}r the market portfolio, and the one-month US Treasury bill return is used for the riskfree return. The sample extends from January 1965 through December 1994. T(:sts are conducted for Ihe overall period, three ten-year subperiods, and six five-year subperiods. The subperiods are also used to form overall aggregale test statistics by assuming that the subperiod statistics are independent. The aggregate statistics for J2, }l, and J7 are the sum of the individual statistics. The distribution of the sum under the null hypothesis will be chisquare with degrees of freedom equal to the number of subperiods times the degrees of freedom for each subperiod. The aggregate statistic for JI is calculated by. scaling and summing the F statistics. The scale factor is calculated by approximating the F distribution with a scaled chi-square distribution. The approximation matches the first two moments. The degrees of freedom of the null distribution of the scaled sum of the subperiod JI'S is Ihe l1umher of snbperiods limes the degrees of freedom of the chi-square approximation. The empirical results are reponed in Table 5.3. The results present evidence against the Sharpe-Lintner CAPM. Using JI' the jrvalue for the overall thirty-year period is 0.020, indicating that the null hypothesis is rejected at the 5% significance level. The five- and ten-year subperiod results suggest that the strongest evidence against the restrictions imposed by the model is in the first ten years of the sample from January 1965 to December 1974.
C.omparisons of the resulL~ across test statistics reveal that in finite samples inferences can difTer. A comparison of the results for JI versus }l illustrates the previously discussed fact that the asymptotic likelihood ratio lest lends to reject too often. The finite-sample adjustment to }l works well as inferences with}l are almost identicalLO those with JI.
5.7.3 UTlObsmlabilily o/the Markel Portfolio
III Ihe preceding analysis, we have not addressed the problem that the rJturn Ihe market portfolio is unobserved and a proxy is used in the tests. Most (eSlS use a value- or equal-weighted basket of NYSE and AMEX stocks '¥' the market proxy, whereas theoretically the market portfolio contains all assets. Roll (1977) emphasizes that tesL~ of the CAPM really only reject the meanvariance efficiency of the proxy and that the model might not be'rejec\ed if (he relurn on the true market portfolio were used. Several approaches rave
Oil
214
5. Tile Callilat AS.ll'll',il·;IIK Modd
Table 5.3. I:m/liriml ".w/I.! for Iroflof of Ihr Shm/lr.l,ill/llrr ,,,,,,,iOIl 11/hr eM'/II.
.J.
Time
!~value
.h
!~valll("
./1
!~vahl('
.h
!>-vahl('
Five-year subperiods )/6~)-12/fi!)
2.0~/l
0.019
20./lli7 0.022
1/l.1~2 O.O4/l
n.IU!', n.u I:.
1/70-12/74
2.1~1i
O.O:~9
I
1.914
O.O(j(i
19.179 OJnH 17.47ti 0.()(i4
2U~/7
1/75-12/79
21.712 O.UI7 1!/.7H4 0.031
1/80-12/84
1.224
0.:\00
13.:\78 0.20:1
1/85-12/89
1.732
O.JOO
IH.164 OJ/52
I1.HI8 0.297 1(i.()15 0.098
I
1/90-12/94
1.153
0.314
Overall
77.224
0.004
I
\,
12.1i1l0 0.212 1(){i58(i
**
11.200 0.:142 94.151 0.003
n.UIH
27.922 0.002 1:I.(Hi!; 0.22U ](UII :. 0.07(; 12.:\79 (1.2(iO 113.78~,
**
ten-year rubperiods
\ I/r"-12174
2.400
0.013
23.883 O.OOH
22.190 O.oJ:1
24.1i19 O.O()!;
1/75-12/84
2.248
0'<)20
22503 O.OI?>
2J.190 0.020
27.192 0.002
1/85-12/94
1.900
0.0,,:1
\9.281 0.037
18.157 0.052
16.373 O.OH'.I
Overall
57.690
O.()(lI
1i[J.{ifi7
fiJ.8:!7 0.001
(iH.215
2.1:,9
0.020
2Uil2 0.017
21.192 0.020
n.17(; O.OJol
**
**
irty-year period 1/65-12/94
\*Le~~ than 0.0005. R~,ult\ are for ten value·weighted portfolios (N = HI) wilh slOe\<..\ assigned 10 III,' 1'''' I(olios ha.\ed 00 market value of equity. The CRSP valuc-w"ighled index is "sed "'" Ill""""',· .. I'll ... n;arkel portfolio and a one-monlh Treasury bill is used as a measure of Ihe risH...·.. fa,,·. Th" lesL. are b.sed on mOlHhly dala frolll.lalluary 196:, 10 D"cemher 1994.
been suggested to considcr if inferences are scnsitive to the lise of a proxy in place of the market portrolio. One approach is advanced in Stambaugh (1982). He examilles the sensitivity of tests to the exclusion of assets by considering a Ilumber of broader proxies ror the market portrolio." Be shows that illkrellces al'e similar whether one uses a stock-based proxy, a stock- and bond-based proxy, or a stock-, bond-, and real-cstate-hased proxy. This suggesL~ that infercnces are not sellSitive to the error in the proxy when viewed as a Illeasure of the market portfolio and thus Roll's concern is not all empirical problem. liRc.'lfflcd work c0I1~idc.·r5 Ih(' pos.."ihility of an:oul1ling for the n'(llrn Oil human Clpit.lI. SCt' May"" (1972), Campllt'll (19!)(;a). ;1I111.1a~;UII"l\han and Wang (1~/!U;).
.5.8.
(:IO.I.I-.\·I'r/;III/I/[ U''J.,T'/I'.i.I;1I1I.\
t\ secolld approach 10 Ihl' prohll'1ll is 11I1'SI'11I<'11 hy KouHld 011111 SlOlIll(19H7) and Shanken (1!IH7a). Their papers ('slim,lIl' an npper hound Oil lite correlalion hetween Ihe lIlarkel PlOxY 1'1'1111'11 and lite lrue lIlarkel I'l'l Ill'll necessary 10 OVl'r!urtltlte rejeclion oflhe CAI'M. Tlte hasic finding is Ihal if the correlation between the proxy and the tnl!' markclexcccds ahoul 0.70, thell Ihe njeClioll of Ihe CAI'M wilh a Illarkel proxy would also imply Ihe rejenioll or Ihe CAI'M wilh lhl' lIue m,lrkel ponfolio, Thus, as long as we belie\'e there is a high correlalioll bCIW(TII lite 11'11(' 1ll,lrk('1 rdurn allel Ihe proxies used, Ihe rc.:jeclions remaill inlact. ball~h
5.8 Cross-Sectional Regressions So far in lhis chapler we have fOCllSc(1 Oil Ihe lllean-varialHT efficiellcy of Ihe markct ponfolio, Another view of Ihe CAPM is Ilral il implies a linear relalion bclwecn expcclcd relUI'llS and markel helas which completely explain tlte cross seClion of expecled reI urns. These implicalions can be les'ed IIsin~ a cross-seClional regrl'ssioll lIIelhodoloh'Y' Fama and Maclklh (1973) lirsl (lcve\oped Ihe cross-sct'lional I{'grcssion "ppm.lrlL The h 0 (positive llIarkel risk prelllium). Because the retllrns arc lIormally dislribuled and Icmporally lID. till' ~allllllas will also he Ilormally dislribuled and lID. llence. ~i\'t'll tillle st:rics or Yu, alld YIt, I = I ..... T, W(' Gill tesl l!tese implic;\liol\s using Ilw IISI"II I-Icst. Ddilling 1"(Y,) ;IS thl' t-St;\liSlir. W(' have Y,
wltnl' Y,
:Wi
allli
The
I, = YII/" -I- YII 0",
+ Y~I " + 1/"
ll,illl-: till' Y~I \ 1'1'0111 (r•. H.:.), \\'" 1'0111 II'SI till' hypolhl'sis Ihal Sill' docs lI,ll han' alii' I'xplall,lIol'l' 11<1\1'1'1' IW\'tllld hCla, Ihal is, Y~ = 0, hy sl'lIill~ j =' '2 ill (:.,H,'2)-(:dU ). The F,III1,I-i\ladklh IIwilllldoloh,)', while IIseflll, do('s hal'" sCl'l'ral prohII'IIIS. Firsl, il CIIIIlOI hc direI'llI' applied hecallse Ihe lIIarkcl helas an,' 1101 knoll'lI. Thlls Ih,' rcgrcssiolls an' COlldllCled IIsin~ helas eSlilllalt',1 froll. the dala, which illlro,liHTS all nrors-ill-variahles complicalioll. Thc l'I"rors-i'lI'ariahlt-s prohlt'lIl elll hc addrl'sscd ill IWO ways, OIlC approach, adopted hy Fama and Macl\elh, is (0 11lillillli/e Ihe errors-in-variahles prohklll hI' ),\rol1ping Ihe slocks illlo portli)lios allel illcr('asin~ Ihe precision or Ihe hCla ('slilllal('s. A secolld approach, devl'iopcd by l.ilzellhcr),\cr alld RaIIlasw;III1Y (1!17!1) alld rdirH'd hy Shallkell (1!l!l2h), is 10 explicilly adjllsl IIII' slalltianl ('1'rors 10 COIT('CI fill' Ihe hiases illlrodll('('d hy Ihe ('1'rors-illvariahles, Shallkl'lI sllgg('SIS lIIultiplyill),\ lr;, ill (!'UI.4) hy all a<\jusIJlleJlI faClor (I + (ji", - f',,)'!/lr;;.). While Ihis approach elilllillales liIe errors-illvariahks hias ill IIII' '-slalistic ill (:•. !t2), il does lIoll'lilllinall' IIIl' possihilily thai olh('\' I'ariahles IIliv,ill ('l\ler spuriollsly ill (:1.115) as a result of lhe \ll!ohservahilily or Ihe lrue helas, The IIl1ohs('l'l'ahilily of Ihe lIIarkel portfolio is also a pOlelllial prohlelll 1i1l'11H' ('I'oss-seclioll,d legressioll approach, Roll and Ross (I !l~H) shill,' Ihal if IhI' I nil' 1II;II'kel (1011 Ii .Iio is I'rticil'lI I, Ihe cross-sel'liOllal rdal iOIl hel\\'(,(,11 l':<'p('(,I('(11 ('111111\ ,lIullwl;IS call hi' "1'1)' sl'lIsilivl' 10 1'1'1'11 small devi,lliolls of IiiI' lIIal'k('1 pOrllolio prox)' frolll till' 11'111' lIIarkel portJ(,lio, Thus el'id('II(,(' of Ihe lack of a I'I'Lliioll IWIW('('II ('xl)('('le(1 relllrn all(1 hela could he Ihe
5.~.
217
CUllriu.lion
result of the fact that empirical work is forced (0 work with proxies for the market portfolio. Kandel and Stambaugh (1995) show that this extreme sensitivity can potentially be mitigated by using a gener.llized-least-squares (Gl.S) estimation approach in place of ordinary least squares. However their result depends on knowing the true covariance matrix ofretums. The gains frolll usin~ GL<'; with an estimated covariance matrix are as yet uncertain.
5.9 Conclusion III this chapter we have concentrated on the classical approach to testing the unconditional CAPM. Other lines of research are also of interesl. One important topic is the extension of the framework to test conditional versions of the CAPM, in which the model holds conditional on state variables that descrihe the state of the economy. This is useful because the CAPM can hold conditionally, period hy period, and yet not hold unconditionally. Chapter 8 discusses the circumstances under which the conditional CAPM !]light hold in a dynamic equilibrium setting, and Chapter 12 discusses econometric methods for testing the conditional CAPM. I Another important subject is Bayesian analysis of mean-variance einciency and the CAPM. Bayesian analysis allows the introduction of prior illformation and addresses some of the shortcomings of the classical approach such as the stark dichotomy between acceptance and rejection of lhe Illodel. Harvey and Zhou (1990), Kandel, McCulloch, and Stambaugh (1995), and Shanken (1987c) arc examples of work with this perspective. We have shown that there is some statistical evidence against the CAPM in the past 30 years of US stock-market data. Despite this evidence, the CAPM remains 'a widely used tool in finance. There is controversy about how the evidence against the model should be interpreted. Some authcls argue that the CAPM should be replaced by multifaClor models with several sources of risk; others argue that the evidence against the CAPM is overstated because of mismeasurement of the market portfolio, improper neglect of conditioning information, data-snooping, or sample-selection bias; and yet others claim that no risk-based model can explain the anomalies of stockmarket behavior. In the next chapter we explore multifactor asset pricing models and then return to this debate in Section 6.6.
Problems-Chapter 5 5.1 Result:l states that for a Illultiple regression of the return on any asset or portfolio Un on the return of any minimum-variance portfolio Rp (except ror the glohal minimum-variance portfolio) and the return of its associated
218
5. 'JIu' Calli/ai A.\Jr/ /'ririllK Mllr/ri
ar~~ro-bCI.a= /32
portfolio
/3np, /31
/~, I?" = /3n+/31/~11+~2/?p+Ei:' /311 ==
=I -
/l"i" and
the regression codlkiellts O. Show tillS.
5.\2 Show that the intercept or the excess-retllrn market model, if;the market portfolio is the tangency portfolio. I
0,
is I.ero
-
5! Using monthly returns from the \D-year period January I!IW) to Dec \IIber 1994 for three individual stocks of your dlOice, a valul~-weighted III rket index, and a Treasury bill with one month to maturity, perform tlw ~ lowing tests of the Sharpe-Lintner Capital Asset Pricing Model. 5.3.1 Using the entire to-year sample, regress excess retlll'llS or each stock on the excess (value-weighted) market return, and perforlll t('sts with a sizc of 5% that the illlercept is zero. Report the point estim.lll's, I-statistics, and whether or not you r<:ject the CAPM. Perrorm regression ragnostics to check yoIII' specification. ~.3.2
. For each stock, perform the saUll: test over each of the tlVO c«uipartitioned SUbsalllples and report the point estimates, [-statistics, and ~whether or not you reject the CAPM in each suhperiud. Also indudc the same diagnostics as above. 5.3.3 Combine all three stocks into a single equal-weighted portfi)lio and re-do the tests for the entire sample and fi)r each of the two snhsaIIIpies, and report the point estimates, I·statistics, and whether or lIot YOli reject the CAPM for the whole sample and in each subsamplc. Include diagnostics. 5.3.4 'Jointly test that the intercepts for all three stocks are zero using the P-test statistic JI in (5.3.23) for the whole sample and for each subsalllple.
5.4
Derive the Gibbons, Ross, and Shanken result in equation (55.3).
6 Multifactor Pricing Models
AT '1'111' END OF CIIAI'TER [} we SUllllllaril.cd cillpirical cvidcllcc illdicatillg that the CAI'M beta does not completely explaill the cross section of cxIHTtecl asset retllrllS. This evidellce suggests that (llle or morc additiollal L("[ors lIIay be required to t:haracteril.c tire hehavior "fexpeoed returns alld nal:lrally leads to consideration of IIIlIlliElctor pricing models. Theoretical arglllllenL~ ;Jlso suggesl lhal more than onl" fa('(or is n·qllired, sin(l~ oilly \llIder strollj.\ ass\llllptiollS willtlle CAI'M ~'pply period l>y pniod. Two main thnm:lical approaches exist. The Arbitrage l'ricillj.\ Theory (APT) developed by Ross (I !J7li) is based 011 arhitrage argunlClits alld the Intenemporal Capilal Assetl'ricillg Model (ICAI'M) developed hy Mellon (I !173a) is bascd on equilihriulIl argumcllLs. III this chapter we will cOllsider Ihe ecollolllclric an.t1ysis of multifaclUr models. Thc chaplcr procceds as follows. Senioll li.1 brid Iy disc\lsses the theoretical background of thc multifanor approachcs. In Scction G.2 we consider estimation and tcsling of the models wilh known faclOrs, whilc ill Scction li.3 we develop estimators for risk premia and expened returns. Since lire factors are 1I0t always provided by theory, we discuss ways to COIIstrllCt lhem ill Sectioll G.4. Scctioll G.!i presenL~ empirical rcsulLs. Becausc of tire Jack of specificity of thc modds, dcviatiolls Gin always be cxplaillcd by additiollal !;rcwrs. This raises all issul" of illlcrprt"lill~ model violations whkh we discuss ill Section (i.G.
6.1 'Theoretical Background The Arbitrage Pricing Thcory (APT) was illtrodllced hy Ross (1976) as all alternative to the Capital A%ct Pricill):( Model. The APT can hc morc general than the CAPM ill lhat il allows for lIIultiple risk (;rclors. Also, unlike the CAPM, the APT does not reCfuire Ihe idelllificalioll of Ihe market ponfolio. Ilowe\'("I", Ihis generality is 1101 wilhollt cosls. III ils mosl gelieral/<>rI1I
6. AI 1/11 i/ill'I!)/" J'ririllg ,\Ji!lll'l.\
lh(' APT provides all 1I/'/lIlIXi/ll(/11' n'lalion for cxpeCled asst'l relurns I\'illl 1I11kllOWll 11111111)('1" or IIl1idt'1I1ifit'd faclors. Al lhis It'I'cI I"ejcclioll oi" lhe lhcory is impossihle (UlIless arhilragt' opportlllliti(~s cxist) ;Illd as ;1 (OIlSl'I(uell(,(' lcslabilil)' or III<' IlIodl'! depellds Oil lhe illlroduClioll of addiliollal assulllptiolls. J Thc AI hilral4t· l'ritilt~ Theory ;ISSltll\l'S lhat tlIarkt·ts an' COlllpt·t itivc ;\IId i"rit'liOlllt·ss alld 111;11 IIII' r('llIrJI I4t'IJ('l"alinJ!; prOfess i"or asst'l n'llIrJlS heill)!; n ltIsidt't't'd is
:III
U,
Elf, I fl
(/, + h; f + E,
«(i.L1 )
0
«(;, I.~)
., (T-
Elf;1
I
<
')
(T-
<
00,
(!i,I.~)
when' H, is Iht· n·II .... 1 fill' assel i, (Ii is the illtercept of Ihe /:tetor model. hi is a (Kx I) \'t'nor of bnor sellsitivities for asset i, f is a (Kx I) V('('\or of COI,1I111011 /;tClor rt'ali/;llions, and f, is the distnrhance tenll. For the sysletll of N aSS(,IS,
R
(IdA)
fl Eiff' I fl
::
o
(G.I.:l)
E.
«;. \.(i)
III the systcllt (''I"ation, R is alt (N x I) Vt'clor with R = [N I R'J ... 1'.\, J', a is an (N x I) ,'('CIOI" I\'ilh a = II/I (/,! '" (l,v)'. B is an (N x K) Illatrix widl n := Ihl h~ ,., h,\' 1', ;\1\<1 f is;\I\ (Nx I) vt'ctor with € = [fl E!? , .. E,'V \', We !'ttl Ihn ass II II II' lhal IIH' i"a("\ol"s atTOIIIII fi'l" liIe COlllillon varialioll ill ;ISSI'I rl'IItI'lIS so that tht· disltllhalt("(' I('rlll lill' largt' wcll-diversilicd portfolios vanishes,'! This n'qllires Ihalllle dislurhaltce I('nlls be slIfficit'lIlly IIlu'olTcialed across asst'ts, C:il'en Ihis slrll("\llIt', Ross (11)7(;) shows that lhl' abst'ut'(' of arhitragc it! tngc CCollolllies iltlpli('s Ih;11 /1. ~" d'l) -I-
n A",
«;,\ ,7)
wll<'n' /t i, Ihl' (,\, xl) "'I",,"ll'd \1'1111"11 \','('tor, Au is lilt, 111011<'1 I.,'\'o-I)('la par;lIlIt'I,'I" alld is ('qlla\ 10 11)(' riskln't' I"elllrn if slIch all assel ('xiSlS, alld Ai: is 0\ (K x I) ,'(Tlor III b.-lor risk preltlia. Ill'n', ;t11d Ihr()lI~h()lIl lilt, ('ilapl('\", 'Thl'''' \0,,, 111'1'11 ,11"","li .• 1 .It-h,''I' Ill' IllI' 1I'.\I"hilil)' III II ..· APT. '-;I>;lIlkt-1l (1~IH:!) "lid Ihh\ig ,11111 H,,,, (I!U"r" 11I1I\ld(' ow' illlt'U"lillg C'xt hangl'. DhrYIIH'.\, Frielld, (;lIl!l·kili. ,1I1l1 (;IIIIt'''-ill (Ptx I) .at . . ., q"t·,IHtli II", c'lIIl'il if ;11 IC"C'\,;lIIn' 01 tlu- lIuulel. ! {\ L""f\' \\dl-di\"I', ... ili",1 p"nlulio i... ;, pu"tulluwith~, lal").~"· n"'l\ht.'ruf~"" k~\\'''h \\'t'igh\i"~s olouk. :y.
fl, J,
'J'IU'lII't'Iim/ JJarkg7Vllntl .,
let ~ represent a cOllfonnin~ vector of ones, The relation in (6.1.7) is proximate as a finite lIumber of assets can be arbitrarily mispriced. Because «(;.1.7) is only an approximation, it does not produce directly testable restric•• ;~,~:': tiollS for ,Isset returns. To obtain restrictions we need to impose additional~,;'~·· 5t l'lIctlIl'e so that the approximation hecomes exact. ~ ,::~~~,; . Connor (1984) presenL~ a competitive equilibrium version of the APT'i,:~!: which h,lscxact factor pricill~ as a feature. In Connor's model the additional·,~~.:.~·.!.• reCJuirelllenL~ are that the market portfolio be well- wealth. The requirement that the factors be pervasive permits investors to diversify away idiosyncratic risk without restricting their choice offactor risk exposure. Dybvig (1985) and Grinblatt and Titman (1985) take a different approach. They investigate the potential magnitudes of the deviations from exact factor pricing given structure on the preferences of a representative agent. Both papers conclude that given a reasonable specification of the parameters of the economy, theoretical deviations from exact factor pricing arc likely to he negligible. As a consequence empirical work based on the exact pricin~ relation isjustified. Exact factor pricing can also be derived in an intertemporal asset pricing framework. The lntertemporal Capital Asset Pricing Model developed in M(~non ( 1973a) combined with assumptions on the conditional distribution of returns delivers a n1Ultifactor model. In this model, the market portfolio serves as one factor and state variables serve as additional factors.! The additional factors arise from investors' demand to hedge uncertainty'about futllre investment opportunities. Breeden (1979), Campbell (1993a, 1996), and Failla (1993) explore this model, and we discuss it in Chapter 8 .. In this chapter, we will generally not differentiate the APT from the ICAPM. We will analyze models where we have exact factor pri!=ing, t~at is,
IL
= tAo + B>'K.
i (p.1.8)
There is sOllie flexibility in the specification of the factors. Most empiri(';\1 illlpl('IlH'nt,ltions choos(' a proxy for the market portfolio as one factor. Ilow(',,('\', different technic]lles are available for handling the additional facrors. We will consider several cases. In one case, the factors of the Apt anef lhe state variables of the I(,APM need not he traded portfolios. In other rases the factors arc returns on portfolios. These factor portfolios arc called lllillli(kin~ po.rtfolios becallsejointly they are maximally correlated with tIlt' factors. Exact factor pricing will hold with stich portfolios. Huberman, K;llldcl, and Stambaugh (19f\7) and Breeden (1979) discuss this issue ill the context of the APT and ICAPM, respectively.
2 2
6. Multi/actor Pdrill/!. MI}{ft./.1
6.2 Estimation and Testing In this section we consider the estimation and testing of various forms of the ex ICt factor pricing relation. The starting point for the econometric analysis of the model is an assumption about the time-series behavior of retul'lls. W will assume that returns conditional on the factor realizations arc II [) th ough time and joilltly multivariate 1I0rmai. This is a strong assumption, bl~t it docs allow for limited dependence in returns through the time-series behavior of the factors. FunhernlOre, this assllmption can be relaxe(1 by casting the estimation and testing problem in a Generalized Method or Moments framework as outlined in the Appendix. The GMM approach for llIultifactor models isjllst a generali7.ation of the GMM approach to testing the CAPM presented ill Chapter 5. As previously mentioned, the multil;lCtor models specify neither the number of factors nor the identification of the factors. Thus to estimate and test the model we need to detCl'minc the bctors-an issue we will address in Section GA. In this section we will proceed by taking the JIlllnber of brtms and their identification as given. We consider fOllr versions of the exact factor pricing model: (I) F,tctors are portfolios of traded asseL~ amI a riskfree asset exisL~: (2) Factors arc portrolios of traded asseL~ and there is not a riskfree asset; (:{) Factors are 1I0t portfolios of traded assets; and (4) Factors arc portfolios of traded assets and the factor portfolios span the mean-variance frontier of risky asseL~. We lise maximum likelihood estimation to handle all fOllr cases. See Shanken (1992h) for a treatmenl of the same fOllr cases lIsing a cross-sectional n:gression approach. Given thejointllormality assumption for the retnrns condition.1! 011 the factors, we can construct a test of any of the fOllr cases using the likelihood ratio. Since derivation of the test statistic parallels the derivatioll of the likelihood ratio test of the CAI'M presented in Chapter 5, we will not re)leat it here. The likelihood ratio test statistic for all cases takes the sal\le gelll"J"al form. DeflllingJ as the test statistic we have
(ti.2.1)
wlwre t and t" arc the lII<1xilllUllI likelihood estimators of the residual cO\'jlriance matrix for the ullconstrained 1I10del and constrained model, resnectivcly. T is the number of time-series observations, N is the number of ihclll
0,2, 1'.:~/i/ll(l/iuli
fllld
'Frs/inK
Ihl' lar).';l' ~;\lIIpll' distriIHltioll.: 1 The hlq.;e sample distrihutiull of} ulldl'\' Ihl' lIull hypothesis will he chi-sqllare wilh Ihe dq.;n·(·s offrecdolll eCjllallo Ihe 1I(11111)('r ofreslrictiolls im(lo!;('d hy Ihe 111111 h),pollH'sis.
10
6.2. I !'orljiJiiu,\
11,1
I·in/o/,I wilh
Il
lIi,I!;}II'I' AI,II'/
We Jirst cOllsider the case where the factors are Iraded (lortJ()lios ;lIld there exists ;\ riskfrce assel. The unconstrained Illodcl will he a /\-bclOr model exprcssed in exccss retllrns. Define Z, as all (N x I) vcnor of excess retllrns for N assets (01' portfolios of asscts). For excess relllrns, the /\-l'al'lor linear III orle I is:
Z, = a
+ HZ!;, + £:,
Elf',!
(li.2.2)
(li.2.3)
()
(li.2A)
EI(Z,,·,
-It!;)
Cov[ZJ\,.
€;J
(Z!;,
-It!;)'j
= 0,
(li.2.!i)
B i~ the (N x 1\) Illatrix ofElCLOr sensilivities, Z!;, is the (/\ x I) vector onaclor (lol'lfolio excess returns, and a ;l\ld (', are (Nx I) I'('ctors of asset return inlercepls alld dislurbances, respectively. 1:: is the variancc-covariance matrix of lile distllrbances, and n/\ is the variance-covarialHT malrix of the factor pOI'l/"olio exccss rellll'llS, while 0 is a (/\ x N) matrix of zeroes. Exact 1,~ll'lor pri( illg illlplics that the demcnts of the v('clor a ill «(;.2,2) will he zero. For lhe IInconstrained model in (li.2.2) lhe maximulll likelihood estimalors ar(' .iIlSt the OLS estimators:
(li.2.7)
(li.2.!/) whcre
.J.L -- -TILr Z' '~l
and
It/-.,
I I -~Z.'
rL '~I
n"
O.
For Ihe (oll,nailled Illodel, wilh a cOllslrailled likelihood ('slilllalllls all'
tIIulli/ilt"/1If I'liring t\llIdl'l.~
10
he zero, Ihl'maximllm
(G. ~. \()) r
+. Z)Z, - n'z"" )(Z, - n·z",,)'.
(Ii.:!. I I )
'~I
The !I II II hypothesis a eljllals I.no CIII he Il'slt'd IIsing the likdihood ratio slatislic j ill (Ii.~. 1). L1l1cler the lIull hYJlothesis Ihe degrees offree
(G.:!.l:?)
Ulld('1' 11\('IIUIl hypolhesis,./I is II II ('01 I! Ii liollally e1islriblltl'd cCIIll'al Fwith N dl'gn'('s of fn'('dolll ill tl\(' 1IIIIIIel'atol' alld ('j' - N - f.:.) dq!;l'cl'S or frecdolll ill IiiI' (kIlOlllillalol'. This tcst call he very IIseflll sillc(' it can elimillate thc prohleJlls thai call an'onlpallY fhe usc of asymptotic distrihutioll tiICory. Johson al\(I Korkil' (I ~1H:» »J'()\'id(' a derivation of jl. 6.2.2 I'(I/'/ji}/im
(II
Fur/on wi/holl/a Ri.lkJrl'l' Anl'/
III tile ahs('llce of a liskfre(' aSSl't, there is a zero-beta JlIocil'l that is a IlIl1ltibnor ('l)lIi\';\\('llt 01'111(' 1~Ia('k \'('l'siOIl of the CAPM,IIl a II\lIltirauor COlllext, the z('I'o·hl'la port/t.lio i~ a port/i)lio with 110 sensitivity to any or the Eiuors, and (,,,))('('\('<1 \'('tllI'l\S ill ('X('('SS of 1)1(' l('('o-beta retllrn ~ll'e linearly rdatl'd to 1)11' ('01 II III liS or Ihe lIIall'i" of faClor SC'lIsitivitil's, Thl' factors an' assullled to hI' I'on/(.)io r('tunls ill ('''('I'SS 0(' III(' 'I.t'I'o-))('la 1'('lul'Il. I)l'Iill(, R, as all (N x I) v('clOI' or I('al returns for N ,ISS('\S (01' »ol't)'olios ofass('ls), For Ih(' 11I1('ollstraill(,d IlIodt'!, W(' h;lv(' a K-/;J('\ol' lilll'ar lIIodd: R, = a
+ n RI\, + 1':,
1':1 f,l
nf",'l
o
«i.~.I·I)
«;.~.)))
(i.:!.I(i)
6.2.
f:~timation
and 'Jesting
• 225 E(RKI - J-LK) (RKI - J-LK)']
COV[RKio €;]
= OK
= O.
(6.2.17)
i
(6.2.18) "
B is the (N x K) matrix of faClor sensitivities, R K , is the (K x 1) vector of factor portfolio real returns, and a and €I are (N xl) vectors of asset return intercepts and disturbances, respectively. 0 is a (KxN) matrix of zeroes. For the untonstrained model in (6.2.14) the maximum likelihood estimators are
a=
{L-BP.K
(6.2.19) 1
Ii =
[t(RI - iL)(R KI - {LK)'] [t(RKI - iLK) (R K1 - iLK),r (6.~.20)
where
1
T I T
= T 2:= R,
P. •
and
J-LK
= T 2:= R K1 .
1=1
1=1
In the constrained model real returns enter in excess of the expected zero-bela portfolio return yo. For the constrained model, we have
R,
= =
Lyo
+ B(RKI -
tyo)
+ €,
(6.2.22)
(t-Bt)yo+BRK1+€t.
The constrained model estimators are:
(6.2.23)
to
=
T T1 l)R /-
.'.
LYo -
•
B (R Kt - LYo)l
/=1
(6.2.24) Yo
= • (6.2.2fi)
2261 .
6. Multifactor Pricing Modl'ls
The:maximum likelihood estimates can hc ohtaincd by iterating over «(i.'L23) to (~.2.2:). B frolll (G.2.20) and :E from (1i.2.21) can bc uscd as starling values for Band :E in (6.2.25). Exact maximulll likelihoud estimators can also be calculatcd without iteration for this case. The IlIcthudulu!,'Y is a generalization of the ,lpproach outlincd for the Black version of the CAPM in Chapter 5; it is presentcd hy Shanken (I985a). The estimator uf Yu is the solution of a fjuadratic equatiun. Givcn y(), the constrained maximum likelihoud estimators or B and :E follow from (6.2.23) and (6.2.24). The restrictions of the cunstrained lIludd in (6.2.22) Oil the unconstrained model in (6.2.14) arc
a
=
(ti.2.2G)
(t - Bt)yu.
These restncltons can be tested using the likelihood ratio statistic .I in (6.2.1). Under the null hypothesis the degree5 of freedom of the null distribution will be N -I. Therc is a reduction uf one degrec of freedom ill comparison to the case with a riskfrec asset. A degree of freedom is used up in estimating the zero-beta expccted return. For use in Section 6.3, we note that the asymptotic variance of Yo evaluated at the maximum likelihood estimators is
Var[y())
+-
(1 + (11" - y()dn~t (jL" -
YUL»)
x [(L-B·L)':E·-t(L-B·LW t .
6.2.3 Macroeconomic Variables
(lJ
«(j.2.27)
Factors
Factors need not be traded portfolios of assets; in some cases proposcd hlctors~:, leludc macroeconomic variahles such as innovations in GNP, changcs in IJ nd yields, or unanticipated ill!lation. Wc now consider estimating and tcstil g cxact facLOr pricing models with such factors. • Again define R/ as all (N xl) vector of real rcturns for N assets (or portfolios ofasscts). For thc ullcollStrained model we have a K-f;IClor linear \Ilo<\fl: (G.2.2H) I I
I !
E[f,]
i
Elf/E,'}
I Elf",) =
E[(f...:,
()
«(i.2.2~)
== :E
-,Ltd (f", -ILf ,,)'}
Cov[f"/. f;1 = O.
(li.2.30)
=
«(i.~.31)
,
(li.2.:·m
6.2. Estimation and TesliTlg
227
f",
B is the
(N x K) matrix of factor sensitivities, is the (K x I) Vl'l'\or of factor realiJ.ations, and a and (;, arc (Nx I) Vl'('\ors of asset return intercepts and distlll'\)'Illl'CS, respectively. 0 is a (/\ x N) lIlat rix of lero(·s. For the unconstrained model in (I;.:!. H) thl' Ill
(ti.:!.33)
«i.2.:H) T
L..,(R , = T1 '\'
• .', a - Bf",)(R, - a - fif"l) ,
(G.:!.:l5)
1=1
wlwre • It =
T
T1 '\' L.., R,
and
'~I
The constraincd model is l1Iostl'ollv('niclltly hll'lllulatcd hy comparing thl' unconditiollal expectation of (11.:!.:!H) wilh (1i.I.H). The IlIlConciitional l'xpectation of ((i.:!.2H) is
/L = a where J.LI" we havl'
+ B/L/".
(ti.2.:~(i)
== Elf",]. Equating the right hand sides or (li.I.H) and (1i.2.36) a
=
LAu
+ 8(>'" -
J.L/,,).
(G.:!.37)
Delining y" as Ihe zero-heta paramctt'l' A/I and dclinillg II as (>'" - J.L/,,) whcre >." is Ihe (Kx I) vcclor of faclor risk premia, for Ihc constrained Illodel, we h,l\'c (G.2.:lS) Thl' constrained model estimators
(C).2.39)
6. MIII/ijf/r/llr Prjr;lIg MIII"'.1
(n. ~.40) (li.~.·11
)
whl'rt' ill (li.:!.·II) X == 11,11'1 and I == I Yo 1'1 J'. The Illaxillllllll likelihood estilllatt's call be obtai lied hr iteratillg o\'er
t
(li.'139) to (li.'1..1 I), B hom «).~.:H) and from (li.2.:);» Gill he \lsnl as stOlrlillg vOIlllcs fill B alld }:; ill (n.~.'II). The rcstriniolls of (1;.2.:IH) Oil (1;.~.2H) are
Thl'sl' n'stl'll"tlllllS em he tested \Ising the likelihood ratio statisti<: .I ill (li.2.1). lhull'!" lIlt" 111111 hypolh('sis the dcgrecs or rrcedolll of the null distrihulion is N - K - l. Th('l'e arc N rcslrictiolls but one degrce orfreectOIll i~ lost t'stilllatiu),( Yo, alld K d('),(I'!'l's of frcl'dom are used eSlimatillg the K ('h-nlt'nls of AA'. The ;1~Yllll'tOlic \';11 i;II",(, 01 :y follows frolll Ihe lIIaxillluln likelihood approach, The 1';1Ii;IIICe ('\';IIl1all'd OIl Ihl' JIIaXilllUIII likelihood I'stilllalOrs is
Applrillg thc partitiolled illVt'lse rule compolIl'nts of:Y \\'c ha\'c cstilllators
to
(1i.2.4:\). fe)r the variances of t)le
+(I + (:y I + ill 1.;>'1i~ (:YI + P-j IJ) (B"t,-llh- 1 I
-1
(B"E,-IU')-IB"E·-\(Y.;'i·{yu))
x
t'}:"
Iihil,,}:·-I B·)-I,
We \Villus(' Ihl'se I'ariallcc n'sulls li)r infercnccs cOllccrnillg Ihc lilnor risk premia ill S('nioll (i.:1. fl, 2,,' 1,(1t lot I''''I/o/im .'i/llllllli IIg Ihl' MfilIl-l'a/'illllC/' 1-I1I1IIit·,.
\,\'hclI 1~\('lOr pOI'I«.lios span Ihe Illean-I'ariall(,(' fronlier, tIlt" inlerccpt 11'1'111 of thl' I'xact pricing rl'ialioll All is /('ro wilhonllhl' !ll'cd lill' a riskfree ass('\.
6.2. i:'sti11l(l/ion and 7fSlill/( Thus this case retains the simplicity of the first case with the riskfree asset. In the context of the APT, spanning occurs when two well-diversified portfolios arc Oil the minimum-variance boundary. Chamberlain (1983a) provides discussion of this case. The unconstrained model will be a K-factor model expressed in real retll\'lls. Define R, as an (Nx I) vector of real returns for N assets (or portfolios of assets). Then for real returns we have a K-factor linear mode!',: R,
=
I
(6.2.4&)
a+BRJ\r+f.,
I
o
(6.2.47 )
E
(6.2.4~)
1
(6.2.49) (6.2.50)
n is the (N x K) matrix offactor sensitivities, RJ\, is the (Kx I) vector offactor portfolio real returns, and a and f., are (N x \) vectors of asset return interr cepts and disturhances, respectively. 0 is a (K x N) matrix of zeroes. The restrictions on (6.2.46) imposed by the included factor portfolios spannin~ the mean-variance frontier are: a
=
0
and
Bt
=
t.
(6.2.5\ )
To understand the intuition behind these restrictions, we can return to :he Black version of the CAPM from Chapter 5 and can construct a spanning example. The theory underlying the model differs but empirically the restrictions are the same as those on a two-factor APT model with spanning. The unconstrained Black model can he written as R,
= a+
f3 0mRol + f3rn Rrn, + f."
(6.2.52)
where ROIl' and Ro, arc the return on the market portfolio and the associated zcro-beta portfolio, respectively. The restrictions on the Black model are a = 0 and f30m +f3rn = L asshown in Chapter 5. These restrictions correspond to those in (6.2.5\). For the unconstrained model in (6.2.46) the maximum likelihood estimators are (6.2.53)
230
6. Multifactor PricillK MrHlrLl
(li.2.:,·1)
where
T
I '" I.L. = TL.....RI
and
1=1
To estimate the constrained lIIodel, we consider the uncolJStrained model in (6.2.46) with the matrix B partitioned into an (N xl) cO\lIlIIn vector b l and an (Nx(K-I» matrix BI illldthe factor portfolio vector partitioned into the first row RI! and the last (K-l) rows R K,!. With this partitioning the constraint B ~ == ~ can be written hi + BI ~ == to For the lI11colIStrailled model we have
Y
Substitllling a jmodel, I 1
\
== 0 and b l ==
L -
BI L into (G.2.56) gives the constrained
R, - ~RII = BI (R h ,! - ~RII)
+ E,.
(li.2.:,7)
Using (6.2.57) the maxillllllll likelihood estimators are
«(i.25H)
b; :to II
==
(G.2.IiO)
~rhe null hypothesis a e«uals I.em c
6.3.
}~,\lill/(/li(}1/ o/UiJ/, /'mllia ami }';x/}('(/t'li UI'IIII7I.~
231
Defining.h as Ihc lest slatistic we haw
h
.
=
(T -- N - h')
N
[1i--;;-:;'1 -I,]
((i.:2.(il)
I~I
U\ld("\" Ih(~ \lull hypothesis, J'l is IIIKoll(litionally disllilHltnl n~ntral F with 2N degrecs o\" frcedom ill the 1I11111crator and :2( '{-N -- K) (\egrees o\" freed011l ill thc dCllolllillator. lIuhcrmall alld KUHlel tion o\" Ih is Icsl.
(1~)H7)
prescnt a deriva-
6.3 Estimation of Risk Premia and Expected Returns AI! the exact/;IClor pricing models allow one to estimate the expected retllrn on a givcII asset. Since the expecte(1 rclurn relation is It = LAu + B>' . , , olle lIe.:ds lIl.:aSllrcs of the faclOr sellsitivily malrix B, Ihe risk\"ree rate or the zcro-bcla expccted retllrJI Au, alld thc LInnI' risk prcmia >'1:. Ohtainillg mcasures of B and the riskfree ratc or the expectcd I,ero-heta return is stra;~lllforwanl. For the given case the constrained maximllm likelihood estimator S' Gill be IIsed lor B. The observcd riskli'cc 1'.11(' is appropriate li)r the riskl'rec asset or, ill the GIS(~S without a riskCnT asset, the maximum likelihood estim,l\or Yo Can be IIscd for the cxpertctl L('w-lJeta r('tllm. Furthcr estimation is lIeccssary to form ('~Iimalcs of the Linor risk premia. Thc appropri,lIe procedure varics across the [(Jill' cascs of cxact htctor pririllg. [II the case where the fa<:lors an~ thc exn~ss retllrllS Oil traded portfolios, til,· risk prl'mia (',111 he estim;l\l'd directly frolll Ihe salllple mealls or the exccss rctllrns Oil the pOrllillios. For this (".I.'C we ha\'c (li.:t \)
An estilllator o\" th(' variance of ).1: is (\).:\.2) [Il tile rasc whlTe j>orLfo[ios are /;\('101" I)\n tlln(' is no riskfree asset, the EIClOr risk prelllia (all be estimatcd IIsing Ihe !lith-rCl\cc bctween thc Slll1lplc IIIcall of the factor portl(Jlios and thc estilllalcd {(To-beta rctllrn:
(i.:D)
III this casc, an estilllator of the v;lriall(T or >.1;
IS
(li.:\.1)
().
ill 11//1/1111111 "ril"illg .\lII/It'll
II'hnl' \-;;;-rl)'ul i\ InJlII «(;.:.!.~7). Till' tic! Ihal alld Yu are indl'pI'llIll'nl lias 1)('1'11 Jllili/l'd 10 WI IIII' covariancl' Il'rlll ill (Ii.:!.'!) 10/('('0. In Ihl' C\SI' I\'hert· Ihl' btlors an' 11111 Iradl'd portfolios, an eSlilllator Ill' till' Vl'nor 01 bClo .. ri~k prl'lIIia AA' is Ihl' S\lIn of Ihl' I'sliJllalor of Ihe lIIeall oflhe 1;1('(01' n';tli/;lliolls alld Ihl' I'slimator of' YI,
ii."
I I
All I'slilnalor of II\(' ";lriaJl(,(' of >-i-: is
I
I
. \'arIAi-:1
=
1.-:rn,,+Varl-rd,
(Ii. :l.Ii)
when' \-;;;'[1'11 i~ lrolll (li.~.'I!i). I\l'callse ilfi-: and 1'1 an' illdl'(lI'llIicllt Ihl' lovarianfl' Il'nll ill «;.:U;) is 'l'fIl. Thl' 1IIIIIIh casl', whert' Ihl' faclor portfolios span Ihl' nll'an-variancl' fr(Jnlin, is IIll' sanll' as 1111: firsl casl' exn'pllhal rcal rl'llIJ'llS are SlIilSlilllll'd
\
\ \
I
lill' exn'ss n'llIrtls. 111'1'1' Ai-: is 1111' I'l'l'Ior of !'aclor pOrlldio sampll' 1Ill',\lIS ;\lHI All is 1.1'1'0. For any assel Ihl' t'X(lI't'lI'c\ reillm call bl' ('slimaled by suilsliluliJlg Ihe ('slilllall's 01 R, Au, alld Ai-: illio (li.I.H). Sinn' «(i.I.H) is nOJllillear ill Ihl' par;III1I'ttTS, calculating;\ stalldard ,.\ 1'01' \'<"Iuirl's \lsill)!; a lilll',lr approximatioll alld ,'slinlaH's or Ihl' '1II',lriaIH'!'s of II\(' paraml'ler ('slilllall's. II is also 01 illll'n'sl 10 ask ir Ihl' tll'lOrs ar(' joilllly prin'd. Cil'l'1I Ihl' ,'('\'lor of risk premia I'slilllall'S ;tllIl ils coval'i;IIICt' lIIalrix, II'SIS of Ihl' lIull hl'polhl'sis Ihal II\(' bClors an' .ioillliv 1101 prin'd rail hI' cOII
./1 ""
cr - /\) .. _.
-r-
'~AA VarIA ...
I'
Ai-:.
(1;,:1,7)
Asymptolical"', IllId\'l the lIulI h\'(lollH'sis Ihal Ai-: == 0,./1 hots ,Ill f dislribulioll wilh 1\ alld '/'-/\ d"glTt,S or frlTtlolll. This dislribuliollal reslllt is all applicalioll of lit!' Iloll'llillg 'r~ stalislic alld will he exaCI ill fillill' salllpil's ror 111\' (';lSI'S whcrt· Iltl' I'slilllalOr or Ai-: is hased ollly 011 IIt I' sa III pll' IIwallS of Ihl' Iit\'lors. \\'(' (;111 also II'SI Ihl' ,igllilicaIH'e or any indi"idllall;\{'lor IIsill)!; A/A
«(i.:I,H)
..[i';, whl'll' ~'i-: is Ill(' lilt 1'1('111('111 or>-i-: :tlld "II is Ihl' (j. jllh dl'lIH'llI 01'\Tt;'1>-i-: I· T('still~ irindi\,idll;tI Lln"rs ;\1'\' prin'd is sl'nsihll' lill' (,
6. -I. Sdrcliull
UJ Faclors
Shanken (I 992b ) shows that factor risk premia can also be estimated a two-pass cross-sectional regression approach. In the firsl pass ,he f;ICtor sensitivities are estimated asset-by-asset using OLS. These estimators represent a measure of the factor loading matrix B which we denote B. This estimator of B will be identical to the uIlconstrained maximum likelihood estimators previously presented for jointly normal and lID residuals. Using this estimator ofB and the (N xl) vector of asset returns for each time period, the ex post factor risk premia can be estimated time- period-bylime-period in the second pass. The second-pass regression is
usin~
(6.3.9)
The re~ression can be consistently estimated using OLS; however, GU) can also he used. The output of the regression is a time series of ex pOSI risk premi,l, )..KIo I = 1, ... , T, and an ex post measure of the zero-beta portfolio return, XOIo / = I, ... , T. Common practice is then to conduct inferences about the risk premia using the means and standard deviations of these ex post series. While this approach is a reasonable approximation, Shanken (1992~) shows that the calculated standard (lrrors of the means will understate the trut standard errors Iwcause they do not account for the estimation error in B. Shanken derives an adjustment which gives consistent standard errors. No adjustment is needed when a maximum likelihood approach is used, because the maximum likelihood estimators already incorporate the adjustment.
6.4 Selection of Factors The estimation and testing results in Section 6.2 assume that the identity of the factors is. known. In this section we address the issue of specifying the factors. The approaches fall into two basic categories, statistical and theoretical. The statistical approaches, largely motivated by the AYf, involve building factors from a comprehensive set of asset returns (usually much laqJ;er than the set of returns llsed to estimate and test the model). Sample data on these returns are used to construct portfolios that represent factors. The theoretical approaches involve specifying factors based on argumen,ts that the factors capture economy-wide systematic risk~. ~
6.4. J S/a/is/iral Approaches Ollr startillg point for the statistical construction of factors is the linear hlclOr model. We present the analysis in terms of real returns. The samf ;In;llysis will apply to excess retnrns in cases with a riskfree asset. Recall th*
2;\4
lilr the linear llIodd we have R, = a
+ B f, + (,
(Id.l)
where R, is the (Nx I) vector of asset rcHlms fill' time period I, f, is Ih(' (Kx I) veclor of faclor realizaliolls fiJI' lillie period t, and lO, is Ihe (Nx I) \'l'ctor ofmodcl dist\lrhan(l~s lor time period t. The numher of assets, N, is 1I0W very hlrge alHlllsllally mill'll \arg,:r thall the IIIlIllber of till\\, pniods, '{'. There arc two primary slalislic~11 approaches, f~IClor analysis and prilll'ipal COlli pone illS.
'1,'(/(/01' AI/alysis
\ ~slil11ali()n using f~IClOr analysis involves a Iwo-slep prOl:c
l \
n ::: 8n" 8'
+ D,
(liA.:\)
n" and r: == D 10 indicate it is diagonal. With the bClors a rotatioual iJl(\clerllliuacy exisls alld B is idclllilinl ouly up to ;\ lIollSingular transformation. This rolalional indeterlllinacy Gill he dillliIrated by restricling' the faclors to be orthogonal III each olher and to have ~ltJit variance. In this case we have 0/\ == I and B is uniquc lip to an 01'thoKonallrans/'Ol'lIIatioll. A1ltralls((JI'IIIS GG are equivalent ((II' an)' (K x K) orlhogollallransforlllalion matrix G, i.e., GG' = I. Wilh th('se r('~lI inions in place we Gill express the rei II I'll covariallc(' lIIalrix as
there
J:'lf, r,l ==
~lIIkuown,
n
= GB'
+ D.
«d.·I)
With Ihe structure ill «(i.4.4) alld the asslllllpliollihal asset I'l'l II I'IIs ;11 (',ioililly IlOrlllal alld t('mporally III>, ('slimalors of B alld D call he fill'llllllated IIsill~ lIIaxillllllll likelihood factor aualysis. Becallse the first-order cOllditiolls ({II' lIIaxilllllllllikelihooc! ar(' highly nOlllincar ill the parameters, solvillg for lhl' estimalors with the IIsual ilerativ(' proccdurl' ('all be slow ancl COnV('IW'IH'l' diflicllIt. Alternative algorithms have hl'en developed by Jiiresko~ (19(i7) and Rubin alld Thayer (I !'H~) which facililate Cluick ('ollvergl'lHT 10 Ihl' l\IaXilll\lII1 likelihood estilllators.
Olle illlnprd
(R, -
Il)
=:
B f, -I- <-,.
Civell ({i.-I 5), a candidale to prox), lor Ih(' 1;lclO(" re;ili/;llions lor lillie pniocl I is Iltt.: cross-s('c'liollal ~elll:rali/,ed leas I squares «:I.S) re~rl'sSi(llleslilllal(lr. Usillg lh(' 1l1'IXillllllll likl"lihllod ('slilllalms or B .\1\11 n we h,\\'(' ror eacll 1 (liA.{i)
I kre
\1'('
are eSlilllatillg f, by ("eg("essillg (R, -
il) 01110
n.
The factor real-
izalioll scries, f" 1 == I, .. , , T, call hI' 1'Il1pl"yed 10 lesl IIIl" Illodl'l IIsillg II\(' app("oach ill Sl'('\ioll {i,~.:t Sillce thl' fano(", are lilll'ar l"ol\lhill;ltioIlS or I'l'tlll'IlS WI' (";111 I"OIlSI\ 1\l"1 !)onf()lios whidl arc \lel'kl"lly t'I)\Tl'Iatl't! wilh lilt' 1;lrlo)"s, Denoling R h", as lhe (I\. x I) Vl'rlor offartor ponfoliolTllIJ"Ils for lillie period I, w(' hav!' ((iA,7)
R h", = AWR" 11'1(('1"('
allel A is ddilll'd as a dia~ollalll\alrix Wilh 1/ II; ;\s till' JIll di;rgollal d<.'llIt'III, wlll'l"<" is till' .ith l'IClllt'lI1 OfWL Thl' !;\\"Io\" portfolio weights oill;lill .. d !'" 1111' /111 1";,,1111 f.-Olll this p.-or .. dlln' .11"(' "'1l1ivaklll 10 Ihe weights Ihal \\'ollid '-"S II I I I"I0lli soh'ill~ lit I' followillg oplilllil:alioll p.-oblem alld lliell II()'-Illali/.ill~ III<" w('ighls to Slllll
1\;
to olle:
«i.,LK)
1\1iIlW'I)w, w,
1
suhjcn to
o
VI! ;" J VI;
I
(d.!) ) ({d./O)
i\l/lIIIJ(/rI()"",il'il/~
(),
,Ilotld{
ThaI is, 1111' bclor pOri folio II'cigllls minimize the residual variant:(' suhject to Ihe COIlSlrailllS Ih;1I each faclor portfolio has a ullit loading Oil its own fal'lor alld it'l'O loadillgs Oil olher bl'lors. The resuiting 1;l('tor porI folio retuflls call he IIsed ill alllhe approaches discussed in Set'lion li,2, If n alld n are kllOWII, Ihl'n the 1;lrtor estimators hased 011 CI.S ",ith the populalioll I'alul's of alld will have the maximum correlatioll I"ilh Ihl' populalioll l;(<'Iors, This IClllows from the miuillllllll-variance unhiased estimalor pl"Opl'l"l)' of gellt'ralized least squares given the assumed lIormality or the dislurhallce 1'1'1'101', But ill pral'lice the l;tcltIJ'S ill (ljA,Ii) alld (1;,4.7) III'('d lIot hal'l' lilt' IIlaxillllllll correlalioll wilh the population common bctors sinn' Ihe}' are hasl'd Oil eSlimales or Band D. I.ehmann and Modest (I QXX) preselll all ;t1lt'1'lIalil't· 10 CI.S, III thl' presence of lIH'asu rt'lIH' II I ermI'. they lilld Ihi, alllTllali\'{' call pmdllct· fa('lor portfolios ",ith a hi)!;lll'r populatioll correlatioll willi Ihe I' 0111111 Oil 1;I('IOfS, They slIggesl for lile jlh Etl'lor 10 IIS(' II'h('l"l' I hI' (N x I) v('ctor is Ihe solulion 10 Ihl' following prohklll:
n
n
w;R,
w,
(GA,I!)
Minw'DwJ' I
W,
slIhjl'('\ 10
o
Vk j
j
(E,<1,12)
I. This approadl lillds Ihl' pllli/olio which has Ihe llIillilllulll residual '!ari;\ll(l' 0(' ;,11 porllcllios orlhogoll;i\ 10 Ihl' olher (K-I) farlms, Ulllike Ill(' CLS pml'l'd\ll"l', Ihis pro('('dlln' igllorl's Ihc illli)rlllalion illihe ElClor loadillgs of Ihl' jill EII'IOI', It is )lossi"'" Ihallhis is bl'llefH'ial became ofllll' Illl'aSllrt'lIlelll error ill 11ll' loadillgs, Illt!I'I'd, I.dllnallll alld Modest lind Ihal Ihis Illelhod offo1'l1lillg tll'lor pllnfolills results ill fa('\ors with less extrel\le weighlings on 1111' ass,' Is allt! a resultillg highl'r ('oITe\;Jlioll wilh Ihl' IInderlying l'0ll1J1l0n faclors,
"'-;11(,;/1(// (:0111/1111/1'11/1 FaCioI' allah'sis n'pn 'SI '1I1s 01111' 0111' sial ist icalllIel hod of fOJ'lll i Ilg 1;11'101' port«,Iins, /\11 ;Jilt· .... alil't· ;ll'pro;lt"h is prillcipal COIlIPOIII'lIls ;lIlalysis. Principal COIIIIHlIlI'IIIS is;1 It·t"hlliqll'· 10 J't'dllIT Ihl' 11111111)('1' ofvariahll's heing sllld· in(wilholll losillg 100 111111'11 illlclllllalioll ill Ihl' ('ovariall!'e lIIalrix, III lite 1'J't'S"1I1 al'plicalioll. lIlt' ohjl'nil'l' is to reduce Ihe dilllensioll frolll N assl'l relllJ'lls 10 1\ Linors, TIlt' pi illcipal (OIll()OIl\'lIls sl'rvI' as IIII' bl'lors, Tltl' lirsl prilltip;tI 'OIIlP"II'·1.I1 is lilt' (lIorlll;t1izec\) lilll'ar I'ollli>illalioll or asscl relllntS lI'illl 11I,I,illllllll 1';lri;IIICt', Tltl' S"('(lIld prill!'ipal ,'OIllPOlH'11I is lilt· (lIonllali"'d) lilll'ar, IIlllhillalioll "I' ;Issel rei II 1'1 IS wilh IIl;1Xilllll1ll \';lri;lIl1,(, "f;l 11'·"111 hi 11;11 i, 'I" IIr! h"~"lIal I, , IIII' Ii rsl pri lI!'i pa I I'OIIlPOII,'1I1. i\ 11<1 SII 1111.
6.4. Sf/fCljolt
~37;'~
oJ Faclors
I
~rl¥'~~'
:
xi
The first sample principal component is xi'R, where the (N x 1) veflor is the solution to the following problem:
I
(6.~.14) !
subjert to (6.~.15)
n
is the sample covariance matrix of returns. The solution xi is the eigenvector associated with the largest eigenvalue oH1. To facilitate the portfolio interpr(~tation of the factors we can define the first factor as R t where WI is xi scaled by the reciprocal of L'xi so that its elements sum to 'ne. The second sample principal component solves the above problem for X2 in the place of XI with the additional restriction xi'x2 O. The solution x 2 is the eigenvector associated with the second largest eigenvalue of xi can be scaled by the reciprocal of L'X 2giving W2. and then the second factor portfolio will be w~Rt. In general the jth factor will be wjR, where Wj is the
w;
=
n.
n.
The rescaled eigenvector associated with the jth largest eigenvalue of factor portfolios derived from the first K principal components analysis can then be employed as factors for all the tests outlined in Section 6.2. Another principal components approach has been developed by Connor and Korajczyk (1986. 1988).4 They propose using the eigenvectors associated with the K largest eigenvalues of the (Tx centered retu-ms crossproduct matrix rather than the standard approach which uses the principal components of the (NxN) sample covariance matrix. They show that as the cross section becomes large the (K x T) matrix with the rows consisting of the K eigenvectors of the cross-product matrix will converge to the matrix of factor realizations (up to a nonsingular linear transfonnation reflecting the rotational indeterminancy offactor models). The potential advantages of this approach are that it allows for time-varying factor risk premia and that it is computationally convenient. Because it is typical to have a cross section of assets much larger than the number of time-series observations. analyzing a (TxT) matrix can be less burdensome than working with an (N x N) sample covariance matrix.
n
File/or AI/(I~YJiJ or Princi/mi Com/lOllmls? We have discllssed two statistical primary approaches for constructing the model factors-factor an,llysis and principal components. Within each approach there are possible variations in the process of estimating the factors. to. qucstion arises as 10 which technique is optimal in the sense of providing the most precise measures of the population factors given a fixed sample of returns. Unfortunatcly the answer in tinite samples is not clear although all proc(,dur(,s can ht' justified in large samples. I
~~4..M.sJ:l.
238
6.
MultiJlI(I()"I,,.ifil/~
Mllllrl.\
Chambcrlain and Rothschild (19H:~) show that consistent estimates of thc factor loading matrix B can bc obtaincd from thc cigcnvectors associated with the largcst cigcnvalues of y-I f!, where Y is any arhitrary positive Odelinite matrix with eigcllvalues bounded away frOllllero anti inlinity. Both istandan.l factor analysis and principal cOlllponents fit into this category, f(lr Ifactor analysis Y D and for principal c01llpollents Y = I. Ilowever, thc finite-sample applicability of the result is unclcar since it is rcrJllired that ,hoth thc numbcr of assets N and the number of time pcriods T go to infinily. i The Connor and Korajczyk principal components approach is also Ulnisistcnt as N increases. It has the further potcntial advilntage that it only requircs T ~ K and does not require T to incrcase to infillity. Ilow("vl'l", whcthcr in tinitc samples it dominates f~\Ct()r analysis or standard prim"ip.1I componcnts is an opcn question.
=
I
6.4.2 Numlier of F([(lors
The underlying theory of the 1IIultif~lc!Or 1IIodels docs not specify the nUIllber of factors that arc requircd, that is, the value of K. While, {(U" the theory \10 hc useful, K should be reasonably s1llall, the researcher still has signilicantlatitude in the choicc. In empirical work this lack of specificalion has \bccn handlcd in scvcral ways. One approach is to repeat the eslimation . ~\IIr1tcsting of thc model for a variety of values of K and ohsern' if the tests ~\re sensitive to increasing the numher of f~IClors. For exalllple I.citlllann and Modcst (1988) present empirical resulL~ for ftve, ten, alltl fincen rac'tors. Their results display minimal sensitivity when the numher of f;IClors increa~es from ftve to ten to ftfteen. Similarly Connor and Kor'~iczyk (I ~lHH) consider five and ten factors with lillie sensitivity to the additioll,,1 five brtors. These rcsults suggest that five factors are adeC"fuate. A second approach is to test explicitly for the adequacy or K factors. An asymptotic likelihood ratio test of the adequacy of K factors rail he COIlstructed using -2 times the difference of the value of the log-likelihood function of the covariance matrix evaluated at the constrained ami lInconstrdined estimators. Morrison (1990, p. 3li2) presents this test. The likelihood ratio test slatistic is
J'> = -
n
(
2)
T-I-t(2N+5)-"3 K
-
---
[loglf!I-logIBB'+DIJ,
«(i.'l.lli)
is the maxim 11m likelihood estimator of f! alld Ii anrl Dare Ihe where maxinllllll likelihood estimators of B ,lIld D, respectively. The lcadill).( terlll is an a(ljustlllent to improve the convcrgence of the finite-sample 1111\1 distrihution to the large-sample distrihution. Under the null hypothesis th"t K factors are adequate.}. will be asymptotically distributed (T -+ (0) as a chi-sC"fuarc vari,ile with ~ I (N - K)~ - N - K I degrees or freerlom. Roll and
6.-1. Sl'il'rlioll O/foill'/IJIJ
Ross (l!lHO) lis(' this approarh alld cOllclllde tll;lt tim'" Ol'/i'llr hlnors arc adequate. A potelltial drawhack ofllsin),( the test rrollllll;\),illlllmlikdihool! fal"lm ev) Ii,,' thl' .llil'qll.ll·y or 1\ tlrtllls under the a~sllmptioll or all approximate raClOr ~lr1Icturl', Their test uses the resull Ihal with an approximale bctor slrllcture Ihe "wrage noss-serliollal variatioll explailled by the 1\ -I- I 'st brtor appro:lrIll'S {('ro as N inrleases,
0,
(()A.17)
where the dependence ofbl(+\ Oil N is implirit. This implies that in a large uo" scnion generated by a I\-far\or lIIodel, Ihl' aWl age residllal variance in :, linl';II' raC\Or llIodel estimaled wilh 1\ + I faclors sllollld converge to the <:\'cragc residllal variance with 1\ (;Ic\ors. This is till' illlplicatioll COllllor and K(,r:~jczyk test. Examining returns hom storks listl'l\ Oil the New York SlOck Exchange and Illl' American Stock Exchallge Ihl'\' conrIllde that Ihere are lip to six pervasive (;Ictors.
(),-I,]
nll'll/"I'/I((/I
A/'/)/I)I/("hl'.\
l'heoretirally based approaLhes 1'01' selecting L\clors /;tli illl<> two m'lin \::Itegories. One approaLh is to specify lIlacroenlllolllic and financial market vari:lbks thaI arc thoughl to C
\\11 ;i1tl'III.llin' illlpkllu'lIlalioli of' tlu' fil!o.l "PI'IO,I('/I i, gi\('11 hr (:,11111'1)('11 ( I !'~Hj.l) ,lIul j, cli.,nl.,~('d
ill Ch;II'I(,1" H.
6. MlIl lljil( /ol'l' r;r;I IR ModI'/.\
Thc scco nd ;Ipp roar h of nl'a ling 1;1("101' port folio s basc d on linll char aClc risli cs h;IS hl"cn IIwd in a 1IIII IIbl'r of stllc\ies. Th(' sc char acle risti cs hav( ' lIIostly sllrf ;u'cd flllli l til(" lill" ralli rc ofC APM viol alion s disc llsse d ill Clla p1('1' Clla r;lIlt "risl ics wllic h hav( ' b(,(,1 1 fOlillel to h(' (,lIIp irica lly illip orta lli illcl ilde IIlark('t \'ahl(" or ('qlliIV, pric c-to -(,ar ning s ratio , ;11111 ralio of hoo k I';lIlIe of ('qllill" 10 111;1/1('1 I'all/(, 01 eqlli ly. The g('II ('ral filld illg is Ihal bno r lIIod els whic h ill.-ll/
r,.
6.5 Empirical Results 1\bll )' I'lllp iricl l silid ics of 1I111 11ifanor nlOd ('\s ('xis!. VV(' will rCl'i('w four of Ihe silld i('s I"lIich IIiI'd )' illus lrall ' IIII' cSlil llali on alltl I(,Slilig 1IIt'l hudoloh'Y wc' hal'( ' disc llsst 't\' 'lim Ctlil lpr .. hcliS il'e siud ies IIsilig stali slica l app roac hcs 10,'l "kct tilt" Lit It'" ;11'" l,t'll IlIal lll ;11111 Mod eSI (I ~IHH) alici (:011 1101' alld Kor;~iCl)'k (19HH). 1,!'IlIlIali ll ;lIId Mod cSI( I.Mj lls(' raCloralla l~'sisalld COli 1101' alld Kor ajol 'k I ( :1\ IlIsc' ( '(' X '/') prill cipa l COIlIPOIlt'IIIS, Two sllId ies usill g t h(' IIH'o r('lic al app roac h 10 faCl or ideli lific atiol l Mt' Faill a alltl Frcl lch (I~'(l:~) alld Che ll, Roll, alld Ross (1~'~ 1;). F;lIlIa and Frt'l ich (FF jlls( ' firlll char acI('ris lics 10 1(,,'111 1;11'101' port lillio s alld ChC lI, Roll , alld Ross [CR R] s!)t'cify IlIanO('COIIOlilic l'ari; lhl('s as racto rs, The first thre e stud ics illcl ude t('s;~ or till' impl icali olls 01 exac i brto r pric illg, wltil<' Che ll, Roll , alld Ross ronl s Oil wh( 'lher or 1101 Ih .. faClors art' pric ed. Tht' evid ('I)(T supp orlil lg t'xac t bno r pricill~ is 1I1ixed, 'J;IhltIi. 1 stull lllar iz('s Ihem aillr esul tsrr olll 1.1\1, CIi., alit! FF. A 11111111)('1' 01 !!;elltTal poil lis (,lII nge froJll Ihis lahlt '. The Slroll~esl nid( ,lIcc ' agai ml exaCI f;lrt or pric illg COIlIt'S rrom tt'sis llsill g dt'(l(,I)(\('lIt port folio s b;Isecl Oil III;U kel I';" lie or ('!(lIiIY alld hook -lo-l IIark el ratio s. EI'(,II Jlllll tibC !or lIIod els h;II'(' difli ( ull)' n:p/ ;Iill illg Ihe "siz(''' ('fre cl alld "hoo k to 1II;lIket" dl('c !. P""f olio s ",hie h arc' lorn it'd has( 'd 011 divi dell d yield alld hast' l! Oil O\VII I'ari allt'" prov ide lilllt- t'\'id ('I)(T agai llsl (,X;Ict fact or pric illg. The CK r(,su lts for ,\;ulual"l' alld 1I01l:I_IIIII;lr)' 1llOIIIhs sligg esl Ihal II,,' ('I'i(\cll ee agai llsi exac l b(lo r pric ing dot's 1101 arise rroJll Iht'J allll ary erfe c!. lisil lg Ille stali slicI ) appr oach ('s, CK alld I.M filld little S(,IIS ilivily to iIlCTt'asill~ til(' IIIII IIhn of bClo rs 1)('\'011(1 fivt'. Oil Iht' olh( 'r hallc l FJ: fill(1 SOliI I' iIIlP""I'('III1'lIt goil l~ from Iwo facto rs 10 fil-(' ';'clo rs. III r('sll lis lIot , illcl uded , FF lilld Ih;II with sioc ks olll\ ' tlm' (' f;Iclors are II('c essa n' alld that wllt'll hOll d pori folio s ;ur illcl uded Ihel l five EH"lors an' II('(' dt'd. The se
Table 6.1.
Summary oj r,su//s Jor {,sis oj eX(lri Jf/r{or priring wing uro-in~t F-test.
N K ___S_'I_u_d~y______'r_i'__ lleperiod_______P_o_rd_'_o_lio __c_h_a_rd_c_t_~_ri_st_ic____________________ ~______ p.vaIue
CK
ti4:0I-ll3:12
markel value of equity
10
10
5 10 5 10 5 10
0.002 0.002 0.236 0.171 0.011 0.019
5 5 5 20 20 20
5 10 15 5 10 15
0.11 0.14 0.42
5 5 5 20 20 20
5 10 15 5 10 15
0.17 0.18 0.17 0.94 0.97 0.98
5 5 20 20 20
10
15
0.57 0.55 0.83 0.97 0.98
2 3 5
0.010 0.039 0.025
CK
10
CK/ CK/ CK N /
10 10 10
CKI'II
63:01-B2:12
I.M
market vdlue of equity
I.M I.M I.M
LM I.M
6:i:O 1-112: 12
I.M
1.M I.M I.M 1.1\1 I.M I.M
dividend yield
--------------------------------------------------------ti:i:()1-1I2:12 uwn va rhll u:e 5 5 0.29
I.M I.M I.M I.M
I.M
FF FF FF
.. ..••
1i:l:07-91:12
stocks and bonds
32 32 32
15 5 10
I
·'I.e" IhJn 0.001. CK rd"rs 10 Connor and Korajczyk (I!lBHl. LM refers to Lehmann and Modest (1988), a~d FF rd"r, 10 Fa,na and French (1993). The CK faClors are derived using ('fxn principal componenl', Ihe LM factors are deriv~d using maximum likelihood factor anal)'5is, and the FF factors are prespecitied factor portfolios. For the FF two-factor case the factors are the return on a portfolio of low market value of equity finn. minus a portfolio of high market value of equity firms and Ihe relurn on a portfolio of high book-to-market value finns minw a portfolio oflqw book·to-market value finns. For the Ihree-faclor case the factors are those in the two-factor c~ pillS II", relllrn on Ihe CRSP value-weigh ,ed slOck index. For Ihe five-faclor case Ihe returns on a IeI'm structure faclor and a default risk faclOr are added. CK include teslS separali'K Ihe in"'rcel" fill' January from Ihe illl~rcepi for other months. CKJ are resullS of leslS of Ihe hypolhesis IhallheJanuary illlercepi is 1.ero and CKNI are re.ults of Ie SIS oflhe hypolhesislhal Ih .. 'lOn:/anllary inlercept is lero. CK and FF work wilh a monthly umpling inlernl. LM use a daily illlerval 10 estimale Ihe faclors and a weekly illlervdl for lesting. The lesl resulu from CK and I.M are hased on leSl' from four five·year periods aggregaled logelher. The porlfolio charact .. ri"i. n·pres,,"I., Ihe finn d,ar~Clerislic "s~d 10 a!iocollt' Slocks inlo Ihe dependent I'Drlh)liu... FF I"e 2:) sUlek ponli)lios alld 7 bOlld portfolios. The stock portfolios are crealed ",ill~ " 11'0 ''''y ,on hast,,1 "" mark"1 value of "'I',ity "lid hook-v~lue·,o-markel-voIlue rollio•. Tht' h,,"d p"nt')lios inrludr five US gOY('rttme"t bond portfi)lios and two corpora Ie bond port'i)li"s. The gowntlllenl hund portfolins art' created hast'tI on mattrrity and Ihe corpordlt' h,,"d p"rt'i,'ios an' ....ealed h«'t'd "" Ih" ll'wl or tI"bult risk. N is Ihe number of depenclenl I'"rt fi,li", alld 1\ i, Ih,' JllIIlIher of!',f1ors. Th,' I~\'"hlt·, an" t'l'tlrtt'tI for the ler<~illlerrel" f~l ..sl.
f42
6.
Multi/(Ir/or I'ririll}; Mor/I'L{
I
~esults are generally consistent with direct tests for the IlIlmber of fartors iSCUSSCd in Section 6.4.2. The LM results display considerable sensitivity to the IItllnher of depenent portfolios included. The Irvalues arc considerably lower with fewer portfolios. This is most likely all issue of the power of the test. For these lcsts with an unspecified alternative hypothesis, reducing the 1I11111ber of >ortfolios without eliminating the deviations fro III the null hypothesis can ead to substalltial increases in power, because fewer restrictions nlll.\t be ested. The eRR paper focuses all the pricill~ of the factors. They usc a crossectional regressionlllethodo\uh'Y which is similar to the approach presellted II Section 6.3. As previously noted they lind evidence of live priced factors. ~'he factors include the yield spread between long and short interest rates for us government bonds (maturity premium). expected inflation, unexpected inflation, industrial production growth. and the yield ~pread between corporate high- and low-grade bonds (default premium).
~
6.6 Interpreting Deviations from Exact Factor Pricing We have just reviewed empirical evidence which suggests that, while lIIultifaclOr models do a reasonable job of describing the cross section of retunts, deviations frolll the models do exist. Civen this. it is important \0 ('onsiller the possible sources of deviatiolls frolll exact factor pricing. This iss lie is importallt because in a given finite sample it is always possible to lilld all additional factor that will make the deviations vanish. However the procedure of adding an extra factor implicitly assumes that the source of the deviatiolls is a missing risk factor and docs not consider other possible explanatiollS. In this section we allalY/.e the deviations frolll exact factor pricing for a given model with the ol~iective of exploring the source of the deviations. For the analysis the potential sources of deviations are categoriled into two groups-risk-based and non risk-based. The objective is to evaluate the· plausibility of the argument that the deviations from the given factor model call be explained by additional risk factors. The analysis relies on all important distinction between the two categories. namely. a difference in the behavior of the maximum squared Sharpe ratio as the cross sectioll of securities is increased. (Recall that the Sharpe ratio is the ratio of the Jlle,1I\ excess return to the standard deviatioll of the excess relllrn.) For the risk-based alternatives the maximuIII sq\lared Sharpe ratio is bounded and for the nOli risk-based alternatives the lIIaxiJlItJIII squared Sharpe ratio is a less usefid construct and can, in principle, Ire unbounded.
243
6.6. Inll'llmlillg /)l'lIia[ioll.l frollll:xf/d l'ill'lor 1'>i";lIg 6.6. J I:'x(lr[ Far[or l'ririllg Modf'll, M"Ill/-\'arim,,"(' Alla/v.liI, fllld tltp O/llilllll/ Orllwgowi/ I'orlfo/;o
For lhe illitial analysis we drop hack to the level of the primary assets ill the economy. Let N be the number of primary assets. ASSIlIll(' that a riskfree asset exists. l.et Z, represent the (N x I) ve(lor of excess ret mIlS for period [. Assullle Z, is stationary alld ergodic Wilh meall IL alld covariance malrix n that is full r;\Ilk. We also take as giVl'1I ;1 set of 1\ fiKtor I'ollf()lios ;llld allalyze the deviatiolls from exact /it("(or pricillg. For til(" (ilctor model,;ls ill (G.:.!.:n, we have (li.Ii.I) Z, = a + BZ At -I- f" Here B is the (N x I\) 111',11 rix of f~l("\or \o,ld'lllgs, Z/:t 'IS the (l\ x I) vector of lime-llilClor portf()lio excess returns, alld a ,lilt! { t are (N x \ ) venors of asset retum illterccpt.\ ;IIHI dislllrballces, respectivcl)'. The variallce-covariance matrix of the disturballccs is E ;IIHI the variallce-covariallce m
=
l~'n IlL) I! I IlL.
lli.li.'2)
In the cOlltex.t of the I\-/";Jctor lIIodel in (li.
/lOrlfolio. A /)(ntjiJlio It will be defllll'd fl.1 IIIf' 0/' (i 11111 1orlhogollill /unl/iJii" Il'ilh [u Ilu'sf 1\ jf/rlor f'O!lfo/im if
re.I/IPr/
(ti.ld) 11.")('(.
Roll (I ~IH{J)
fllr
g<'lH'r.d
1'1 ()p('lli("~ of flllliogolI.1I p!llllo/l( .....
-,, (/1/11
(G.(iA)
jor (/ ( 1\ x I) porljolim. W
II/'t//II W J,
i.1
x 1\) ml//rix 0/ IIHd
lI,h/'/I' WI' i., I hI' (N
Ilu' (N x I)
11/'(/11/ ofal\l'Il{lfi~ht.\ jor Iltf
I{lfi~ht.\ jor Ihf filtlor
o/Jlil/lid orllwgol/II/ Imrtji>iio.
111/(1 w,'
i.\ Iltt'( N x 111'/'(/111' oland II'figh/l./iu·lltf IIII/gnlf) l)/IrljiJli/). I[,JIII' (tjll.\iril'r.l /111/cM wilhlllli 111')'1"(/01' IUJlIIOlio\ (1\ 111m 1"1' nl)li/llll/orllwgol/flllwrlji,lin lI'ill bf lite' lallgnl"'I}()rlj',Ii().
= (\)
1/
TIll' \\,I'ighh of porll •• lio Ii (";1/1 1)(' expressl'd ill lenllS of Ihe parallH'Il'fS of Ihe I\-fa((or llIodel. The vc"("lor of weigh Is is
WJ,
(t:n-1a)-ln-1a
(/.'Eta)-IEta. wh('l"l' 1111'/ SlIpI'lsnipl illdicales Ihe J,!;I'IH'ralil.ed illvt'rst'. TIl(' lIsd"tllll('SS of Ihis ponfi.lio ("OIlIl'S fmlll IIII' (;1("( Ihal whl'1I aclcleclto (lUi. I ) til(' illtlT("Cpt wi II. \'a II isil alld II It' (;uII.r loaelill),!; lIIatrix n willnotlw all('l"('d, The optimality reslriclion ill (1;,li,:I) le;lCls 10 Ihe inl('("("epl v
o
(li,li.7)
E III/u; I (Idi.!)
(:o\"1 ZAr, 11/ I Covl1/,(,
\1/
(li.h.W)
()
I == n,
(1"•. 1"•. 11 )
v.','
("all rciall' Ihe oplilllal orlllC'J,!;ollal portfolio par;lIl1(,\('I"S to Ihl' fal"lor IllOdcl devialions hy "'lJllparillg (/i.li.l) alld ({i,li,li), 'Hlkillg Ih(' 1l11("0lHliliollal ('Xpt'("\;llioIlS 01 hOlh siell's, ((i.(i.I~)
;lIId Ill' 1'C)1I;lIillg IIie
\';11
i;lIltT
~'
'=
_,
01 f
I
wilh IIIl' variallce or f3,,1,,/
+ u/.
'J
4'" I == aa ' -;; C1;; /J,fJJ,r1i, /.,)
+ ....
'i'.
((i.li.I:I)
IIi,
Thl' kl'Y lillk 111'111'('('11 Ihl' Illodd ckl'i;lliolls and IIII' n'sidllal variallfl's ;IIHI ("OI'arialin's l'lIh'lgl'S tnllll ((i,li.I:I). The illlllilioll I'm 11ll' lillk is ~Ir;lighl ((,("wanl. Ik\'iali
•
245.~.~~~·1
6.6. lllll'rprrlilli{ Droialion.! fmm /:'xllcl F(/clol' Prici1lg
,"
cOlllponent in the residual variance to prevent the formation of a portfolio with a positive deviation and a residual variance that decreases to zero as the number of securities in the portfolio grows. that is. an asymptotic arbitrage opportunity.
6.6.2 Squared Sharpe /lalios
The sCJuared Sharpe ratio is a useful construct for interpreting much of the cnsuing analysis. The tangency portfolio q has the maximum squared Sharpe measure of all portfolios. The sCJuared Sharpe ratio of q. is
s;.
(6.6.14) Givell lhat the K factor portfolios and the optimal orthogonal portfolio h can be combined to form the tangency portfolio. the maximum squared Sharpe ratio of these K + I portfolios will be ~. Since h is orthogonal to the portfolios K. MacKinlay (19%) shows that one can express s: as the sum of the squared Sharpe ratio of the orthogonal portfolio and the squared maxilllllill Sharpe ratio of the factor portfolios. (6.6.15)
si
= l1-~nKI 11-1\.7 where s;, = Jli/a~ and Empirical tests of multifaclor models employ subsets of the N assets. The factor portfolios need not be linear combinations of the subset ofasscts. Results similar to those above will hold within a subset of N assets. For subset analysis when considering the tangency portfolio (of the subset), the maximulll squared Sharpe ratio of the assets and factor portfolios, and the optimal orthogonal portfolio for the subset, it is necessary to augment the N assets with the factor portfolios K. Defining Z;, as the (N+Kxl) vector [Z; Z~/l' with mean 11-:' and covariance matrix for the tangency portfolio ofthl'se N+K asseL~ we have
n;,
s;. = 11-;'n:- II1-;.
I
(6.6.\6)
i The subscript s indicates that a subset of the assets is being considered.' If allY of the J;Jctor portfolios is a linear combination of the N assets, it wil\'be 11l'('('ssary to use the generalized inverse in (6.6.16).
~Thi, \'('sul! is r"\OII"d
10
Iht· work of Cihho"s. Ross, OIl1d Shankt'O (I9R9).
i
I
~
.',
I 46
.
6. Multi/artor PlirillK MII/"",~
.
The analysis (with a subset of assets) involves the Cjuaclratie a':E- 1a com)lIted using the parameters for the N assets. Gibbons, Ross, and Shankcn ~ 1989) and Lehmann (19H7, 1992) provide illterpretations of this quadratir terril using Sharpe ratios. A~stlJlling :E is of full rank, the), show
i
(li.li.17)
Consistent with (6.6.15), ror the subset orassets a':E-la is the squared Sharpe ratio orthe subset's optimal onhogonal portrolio h•. TherclillT, fill' a given subset of assel~: ~
',1",
a'.:E~la.
(G.G.IH)
and
.5;,"
2
S",
~
+ J~,
(li.fi.I!J)
Note that the sqmlred Sharpe ratio or the suhset's optimal onhogonal ponfolio is less than or equal to that or the population optimal orthogonal portfolio, that is, «i,G,20) Next we use the optimal orthogonal portfolio and the Sharpe ratios results together with the modd deviation residual variance link to develop implications for distinguishing among asset pricing models. llerealkr the .I subscript is suppressed. No ambiguity will result since, in the suhsequent analysis, we will be working only with subsets or the assets. 6.6.3 InlJllimtions Jor SrJiUmtillg Altematiue Theon'fS
Ira given factor model is rejected a COllllllon interpretation is thatlllore (or dilTerent) risk factors are reCjuired to explain the risk-return relation. This interpretation suggests that one should include additional factors so that the Ilull hypothesis will be accepted. A shortcOllling or this popular approac!1 is that there arc multiple potential interpretations of why the hypothesis is accepted. One view is that genuine progress in terms or identifying the wJight" asset pricing model has been lIIade. nllt it could also be the else that t; Ie apparellt success in identifying a heller model has COllie from linding , good within-sample lit thro\lgh data-snooping. The likelihood of this Ijossibility is increased by the fact th"t the ;ulditional factors lark thet>retical
'~lOtivatioll. This section alll'lllpts to discrilllinate belweell the t\VO interpretatiolls. do this, we compare the distribution of the test statistic under Ihe null h[/'pothesis with the distribution ullcler each or the alternatives. We reconsider the 1.t'ro-intt·IT(·PI F-tl'st of Ihe l1ull hypothesis Ihal tht' i'l tefcept vector a frolll (li,li.l) is O. I.et lin he .the 111111 hypothesis and 11.1 i
'I~)
247 he the allcrualivc:
I III
(,lll
Ilu:
a
11. 1:
a
be tested using the test statistic
()
f-
n.
Ji
hom
«i.~.
I :2): «;.().:21)
where 'j' is the illJlllberoftirlle-scries obsCl'vatiolls, N is the rllllllhlTofassets or portfolios ofassets included. antl/\ is the nlllnbn of 1;lctor portfolios. The hat sliperscripL\ indicate thc maxilllll"r likelihood estimators. U"der the lIuli hypothesis • ./1 is ullcondiliollally distributed Cl'IHr,,1 F with N degrecs of freedom in lhl' III1I11CI'.IIOI· alld ('/' - N - /\) dl'gnTs of fn'cliolll ill thc dell Oil Ii lIat or, To illterpret dcviations frolll the IIull hypothl'sis. wc rClluire a general rej)l'esellt'llion for lhe distributioll of}" COllditiollal Oil rill' buor portfolio rct!p'ns the distribution of Ji is
JI
o=
'/'[1
~ /.:'1.1. N-~('\)'
+iL~il~'iLf..J-Ia'1:
la,
a is the noncl'lItrality paraml'ter of the F distrihlltioll. If 1\ = () thl'll , ... -1 the (crill 11 + iL/o..n,,: f.r.d- t willllot appear ill (t;.Ii.:21) or ill (ti.li.:2:~). alld Ji I,'ill he IIIl(olHlitionally distrihuted 1I0lHTIltral F. We cOllsidl'l' the distribution of./I unell'\' t\\'o different "ltlTllatiVl's. which are separated by their implications for the Illaxillllllll valllc or the squared Sharpc 1·;ltio. With the risk-bascd IIlllltif;ll'tor altl'rnativc thlTl" will he an upper bound on the squared Sharpe ratio, whcl'l';ls with the non riskbased alternarives the maximum squarcd Sharpe ratio i~ IInhollll
Whl'IC
«i.(i.:24) Frolll (i.(i.:!O) .the third tcrlll in (i.(i.:2'1) is hounded ~Iho\'l' hy .17, ~lIld positive. The .,('Co/Hlterlll is hounded hetwlTIi Il"ro and olle. Tlin.s tIlCrc is all upper hound f()r S, ,) < '/'.If, :" rl~. (ti.ti.:2:» The Sl'('Olld inl"Jll
6. Mull/jl/dlll hieillg '\/lIlld,
..:. Ji)
C;il"en a lIIaxinllllll I"ahl<" 1,.1' til(" sqllared Sharpe ralio, Ihe IIpper houlld 011 Ihe lIon("('lllralill" p;II'aIlWI('I" (";111 he illlportanl. Wilh Ihis hOlilld, ind'"!wlld('1I1 01 hOI\" Oil(" ;1I"I;lIlges Ihe assels 10 he included as dq)('ndclII I"ariables ill Ilw pricillg lIlodd regrcssion alld lill' any I"alue or N,lI tll('l"(' i., a lilllit Oil the dislallce I)("tll'('('n the lIull distributioll anel tht' distrihution or the lest st;lliSlic 1111111'1 IllI' Illissillg-Llctnr ;lht'l"IIaliVl'. All Ihe assels call he IlIispric('(1 and I·('t th(' holtlld will slill apply. III cOlltrasl, II'h(,1I Ihe altel"llatil"c Ollt' has in lIIilld is that the sOllrce of del"ialiolls is lIollrisk-has('d, SlIch as dala snooping, lIlark('1 frictiolls, or lIlarkCI irraliollalilies, th(' lIotioll of a lIIaximllm sfJllareci Sharp(, ralio is 1101 liseI'll\. Th(' sqllarl'd Sharp(, ralio (all(1 Ihe lIollcelllralil), paralll('ter) arc ill principlt- IltlhOlll\(l('d h('("alls(' II\(' Ih('ory linking Ih(' (\t-I"ialiolls alld Ih(' r('sidllal varian("('s and ("ovariallc('s does nOI apply. Wh('n comparill!!; alterllalil"es wilh Ih(' illlcrc('pis of ahoul Ihe sallie lIIagllilude, ill gt'nt'ral, 011(' wOllld ('xp('n 10 se(' larg('r lest slalislics ill Ihis lIonrisk-iJaseci case. We exallline Ih(' inforlllalil'l'Ill'ss of the above arJ;llysis hy considering a\ll'l"lI;\liws wilh realislic parallll'll'r vailles. We consider the dislrihulioll of Ih(' I('sl stalislic 1(11 1111('(' casl's: Ihl' 111111 hypOlh('sis, IlIl' lIIissing risk EI("(ors altl'l"nalil"(" alld Ih(' 1I0llrisk-hased alt('malil'c. For Iht' risk-has('d alt('rnative, Ih(' fralllel\"ork is Ilt-sigll('d 10 hI' silllilar 10 Ihal ill Failla alld Fr(,lIch (19!}:1). For Ihe lIolllisk-has('d alt('l"lIalil"(, 11'(' ltS(' a s('IUp Ihal is cOlISisll'lI1 Ivilh liIl' all;llvsis of 1.0 ;11111 tlhd,illl;II' ( 1!'I}Oh) alld the work of I.akollishok, Shkikr, alld Vishll), (I !I~H). COllsid('r a Olw-bclor ass('1 pricillg model ltsill!!; a lilll(, sni('s of the ('X("('SS r('llll"lls I'll" :I~ ponl,.lios as Iht' dqH'lIdelll v;lI·iahlc. The 011(' 1;\\'lor (illdqlt'lldeill I'ariahit') is IIII' ('X("(,SS 1'('1111'11 of Ihe lIIarkel so Ihal Iht' l.l'l 11illt('l"("q>t IIl1l1 hypol\\('sis is Ihl' (:Al'tll. Th(' lellgth of the tilll(' s(,ri(,s is :q~ 11I011lhs. This SI'llIl' ("OIT(,SPOllds 1o Ihal (Jf Failla all(1 Frellch (\ !I~I:{, Tabk ~I, \('gn'ssioll (ii». Th(' lIulI dislribulioll oflh(' I('sl slalislic./I IS ({i.{i.:!{;)
'\i. ddill(' IIIl' disll ihlliioll .. 1'./1 lIlHkr Ihe altl'rll;lIil"l's of illl('l"('si Ol\(' 10 spl'dl'" Ih(' 1':lIall\('\('IS IH'(TSsary 10 t'alclliall' Ihl' lIoIH'('lIlralily I'ar:IIIH'IIT. For Ihl' risk·hased alterllalil"t', giv(,11 a Valll(, for Ihl' squared Sharpl' ralio .. I' Ihe "l'lilll;1\ ." tilOgoll;1\ l'0nl,.lio, Ihe dislrihulioll ("orr('spolldillg 1o Ih(' "1'1'(,1' hOlllld .. I" III(' lIolHTlilralil), parallll'ler frolll ((;.I;.~:.) (";111 h(' Ulllsid('11'11. Th(' Sharp(' ratio of Ih(' oplillial orthogollal portfolio call Ill' ohiailll'd IIsillg (li.I;.1 :.) gin'lI Iht' squarl'd Sharpe ralios of Ih(' lallg('IIl"\' ponfolio ;1111101111(' ill
IH'('d~
~In
pl.II lic
t'
\du'll U"'lll).!.
willi ... "I 11111 1,'lIk.
lilt"
r .!t',{ II \\,1111,(, tH't C:'\.H Y fur S
to
lu.' t.:~, lku\ F- ~:
M' th.lt
~
6.6. 11i/1'/llrftilig Deuiations from f:xac/ Fac/or Pricing MacKinhlY (1995) argues that in a perfect capital markets setting, a' reasonahle value for the Sharpe ratio sfluared of the tangency portfolio for an ohservation interval of one month is 0.031 (or approximately 0.6i for the Sharpe ratio on an annualized hasis). This value, for example,l corresponds to a portfolio with an annual expected excess return of 10%: and a standard deviation of 16%. If the maximum sfluared Sharpe ratio of the included factor portfolios is the ex post sfluared Sharpe ratio of the CRSP value-wciRhted index, the implied maximum squared Sharpe ratio for the optimal orthogonal portfolio is 0.021. This monthly value of 0.021 would I he consistent with a portfolio which has an annualized mean excess return : of R% and annualized standard deviation of Hi%. We work through the analysis lIsing this value. ! UsinR this squared Sharpe ratio for the optimal orthogonal portfolio to l calculate 8, the distribution of 11 from equation (6.2.1) is ~ (6.6.27) This dislribution will be used to characterize the risk-based alternative. One can specify the distrihution for two non risk-based alternatives by specifying values of a, ~,and jJ.~n; 1 jJ.K' and then calculating8 from (6.6.23). To specify the intercepts we assume that the elements of a are normally distrilmted with a mean of zero. We consider two values for the standard deviation, 0.0007 and 0.001. When the standard deviation of the elements ofa is 0.00 I ahollt 95% of deviations will lie between -0.002 and +0.002, an annualized spread of about 4.8%. A standard deviation of 0.0007 for the deviations would correspond to an annual spread of about 3.4%. These spreads are consistent with spreads that could arise from data-snooping. 9 They are plausible and even somewhat conservative given the contrarian strategy returns presented in papers such as Lakonishok, Shleifer, and Vishny (1993). For E we use a sample estimate based on portfolios sorted hy market capitalization • for the Fama and French (1993) sample period 1963 to 1991. The effect of • -I
jJ.~f1.K iLK on 8 will typically be small, so it is set to zero. To get an idea ofa
reasonahle value for the noncentrality parameter given this alternative, the expened value of 8 given the distributional assumption for the elements or a conditional upon E == t is considered. The expected value of the llollCt'lltrality paralileter is 3!l.4 for a standard deviation of 0.0007 and 80.3 for a stalHlard deviation of 0.001. Using these values for the noncentrality par;illll:ler, the distribution of JI is (6.6.28)
LIY
"Willi d.""'''lClC'pinl: II ... dislI'ihlllion .. r}1 i., 1101 <",,,clly,, n,,"c~nlrdl F (se~ I.o and MacKinI ("WI'WI", fur Ihe purl'",es "r Ihis allaly,is, Ihe lIuncelllfdl F will be a good
Il~I(IOhl)·
al)I,nlxill1;(ti t HI.
-;,
~50
6. Multifactor I'ricillK Mot/rls
Null (fi.Ii.:!fi)
AII"rIlatiw I (G.G.'.!7)
..... /
Allernalive '.! (G.li.211)
1
Ii
F 'lali.'lic
Figure 6.1.
/)iJtributiom for Ihp CAI'AI 1.I'lTrllltfrrr/Jt Tr.11 Sta/i.ltic for f'(mr 11)lm/hrJ'.!
whcn a. = 0.0007 and «(;.6.29)
=
whcn a. 0.001. A plot of the four distributions frolll (6.6.26), (6.6.27), (6.6.28). and (~.6.29) is in Figurc 6.1. Thc vertical bar on thc plot rcprcsents thc value 1\91 which Fama and Frcnch calculate [or the tcst statistic. From this figure. Ilt)ticc that thc distributions ullder the null hypothcsis and the risk-based a tcrnativc hypothcsis arc <]uitc closc together. III This reflects thc impact of tl~e uppcr bound on the Ilollcentrality parametcr. In contrast, the lIomiskhhscd alternativcs' distributions arc far to the right of thc other two distrihiltions, consistent with the IInbollndedness urtbe 110nccntrality parameter f(~r thcsc altcrnativcs. Givcn that Fama and French liJld a test statistic of 1.91, these reslllts 5Vggcst that the missing-risk-factors argulllelll is not the whole story. From Flgurc 6.1 one can scc that 1.!11 is slill ill the upper tail whcn the distrihution Of JI in thc prescncc of missing risk factors is tabulatcd. Thc /rvallll' IIsing tl is distrihution is 0.03 ()r the mOllthly d'lla. Hence it secms ulllikely that Il issing bctors completely explaill thl' deviatiolls. I Thc data oncr SOllie support for the Ilonrisk-based alternali\'(' views. test stalistic falls almost in the middle of the Ilonrisk-hascd ,,\tl'ma-
I
TIIC I\
IIIIS('('
MacKilllOlY (1!11I7) for (kl"il("d ,,"al)",is of lil(' I i,k·has('d OIll("rtlOIli,"(".
6.7.
(;ollr ill.lio ll
ti,'c "illl tllc lowe r st'IIH lan\ dcvi aliol l of the deil leill s or a. Seve ral or the lIollrisk-l>asee! views call I{iv!' lhe salll(' lIoll (clIl r.dil y para melC l' alld (('\t- slali slic distr ihwi oll. The resu lts are cOlisisl!'111 wilh lhe e!al.l-sno opinl{ alter naliv e of 1.0 aile! Mac Kinl ay (1\1\101», wilh lhe rt'i
6.7 Con clus ion III Ihis chap ler we have devc lope clth e ('col lolll etric s for ('slillialil ll{ alld testillg- Illlll tibC lor pric ing mod els, The se mod els prol 'ide' lIl allra ctive ahem alin' to the sillg lc-fa clOr CAI 'M. bllt lIsers of slich IIlod ds shol iid he awar e of two serio lls dalll {ers that arise whe ll brlo rs are chos ell 10 lit existinl{ data wi Ihol ll rega rd to ecol lom ic lheo ry. First, the IllOdcls Iliay over fit the data lJeGlllse of data -sllo opin g bias ; in this case Ihey will Ilot hc ahle to pred ict as\': t retu rns ill the futu re. Seco nd, the IIlod eis lIlay capt ure emp irica l regular ities that arc dlle to mar ket illef licic llcy or illve stor irral iolla lity; in this case they iliaI' cont inue to !it the data hili they will i'llpl y Sha rpe ralio s for facto r port folio s that arc 100 high to be cOll siste nt with .1 reas onab le unde rlying 1l1Odd of mar ket equi libri um, Both thes e pwh lem s call he miti gate d if olle deriv es a facto r stru ctur e from an equi lihri lllll mod el, alon l{ Ihe lines disc llsse d in Chapter~. In the elld . how ever , Ihe usef ulne ss of 11111ltifaClor 1110dels will 1101 bc fully kllow li ullti l slIIIicieli1 IICW dat'l I)(,COIIiC avai lable 10 prov ide a true oIlH)I~sal11ple chec k Oil tllci r perf orm ancc .
Pro blem s-C/ lUjJ ter 6 6. I
COl lside r a lIIul tiple regr essio ll of tire rctu nt Oil any assc t or port folio I)n the retu rns of e zero and that the facl or ITl{ressioli (odI ici(' IlI.' for allY asse t will SlIIll 10 llilily, flu
6.2 COl lside r two ccol loll1 ies, ecollolllY 1\ and ccol lom y II. TIr(' mca ll ('XCl'ss-rcturIi \'('cl or and the cova ri'III (T Illat rix is spn ilicd I)('\o w Ii)!' ('ach of the ecol lom ics, Assu mc ther e exist a riskl n't' 'Isset, N risky ass('ls with IIIcali cxcc ss retu r" f.L alld non sing ular ('()\'ariall(,(' Illat rix n, and a risky \'act or port folio with IIIcali ex(e ss ITIIII'Ii II" alld I'aria llcc The faclo r port folio is lIot a lille ar com billa tion of the N 'Isscls, (Thi s nite riol l Gin he IIl{,thy clim inati n),( one oflh e asse ts whic h is illdl ldcd illlh (' 1;lclor ponf c)lio
or
O. Muftil'll·tor/),.iri,,/!. if III'fl·ss"ry.)
Fill"
,Hot!l'/.~
hllill ('CIIIIO\uil's r\ alld II: It =
a + {-JIJI'
((;.7.1)
n = tWa;; + liIi'al~ + /a/. C:i\'l'lI the OII1 .. \,e IIll"all alld c .. \'ariaIIlT lIlatrix and thl' asslllnplioll Ihal Ihl' brtor pllrlhllio /1 is a traded asset. what is the maximulII sCjuared Sharpe ralio 1'01' Ihe gi\'('11 (·C .. II .. llIil's> 6.:1
Relllrllillg 10 11\l" all .. \,l' proillelll. Ihe I'COIHllllics an' rllrther spl'cili('(1. II\(" ("Ie II It'll IS or a 011'(" .... oss-s("((i .. llalJr illdl"()('llIklll alld idl'lllically
ASSlIlI1l'
distli 1111\('''.
i
=
I ..... N.
The sp"(ilicali"l1 or IIIl' disirilllllioll or Ih,' delllents of {j cOllditional differelltiates 1','Olllllllil's :\ alld II. For econolllY 11.:
;\11111'01'
i
=
I. .... N.
i
=
I ..... N.
Oil
a
((i.7.4)
I"l'OlltIlIlY Ii:
1IIH'Olldiliollal\" III<" noss-s,"llioll;t\ dislrihlltion or tl\(" delll("lIts or {j \ViII Ill' lIlt' salllt" lill' 1I.. lh ecollolllies. hilt lill' CCOIIOIII), A conditiollal on a. f5 is lixed. Wh;1I is Ihl' Illaxillllllll sqllarl'd Sharp'- ratio for each ecollolII)'? W:I
7 Present-Value Relations TilE FIRST PART of this book has examined the behavior of stock returns; in some detail. The exclusive focus on returns is traditional in empirical! research on asset pricing; yet it belies the name of the field to study only relurns and not to say anything about asset prices themselves. Many of the i most important applications of financial economics involve valuing assets, ': and for these applications it is essential to be able to calculate the prices I that are implied by models of returns. In this chapter we discuss recent' research that tries to bring attention back to price behavior. We deal with 'I com ilion stock prices throughout, hut of course the concepts developed in this chapter are applicable to other assets as well. The basic framework for our analysis is the discoiinted-cashjlow or presentvalliI' model. This model relates the price of a stock to its expected future cash flows-its dividends-discounted to the present using a cons.tanl or time-varying discount rate. Since dividends in all future periods enter the i present-value formula, the dividend in anyone period is only a small component of the price. Therefore long-lasting or persistent movements in dividends have much larger effects on prices than temporary movements do. ) A similar insight applies to variation in discount rates. The discount rate between anyone period and the next is only a smaIl component of the long-horizon discount rate applied to a distant future cash flow; therefore persistent movements in discount rates have much larger effects on prices than temporary movemenL~ do. For this reason the study of asset prices is intimately related to the study of long-horizon asset returns. Section 7.1 uses the present-value model to discuss these links between movements in prices, dividends, and returns. We men tioned at the end of Chapter 2 that there is some evidence for pr('(linability of stock returns at long horizons. This evidence is statisticaIly weak whell only past returns arc used to forecast future returns, as in Chap. tt'!' 2, hut it hecomes considerably stronger when other variables, such as the dividelld-price ratio or the level of interest rates, arc brought into the
254
7. Prl'.If11I- Va/III' Urialililu
analysis. In Scctioll 7.2, we lise thc fill'lnllias of Section 7.1 10 help interprct thesc findings. Wc show how variolls tcst statistics will hchavc. hoth unclcr thc null hypothesis and undcr thc simple altcrnativc hypoth(,sis that the expected stock return is tilllc-varyill~ and follows a jlnsis\('1\1 lirsl-onkr autorcgressivc (AR( I» proccss. A Ill;!ior Iheme of thc scction i.~ that reccnt empirical findings using longcr-horizon dat;1 arc roughly consistcnt wilh this persistent AR( I) alternativc Illodel. We also dcvelop thc implications or the AR( 1) model for pl'ice behavior. Persistcnt movclllcnts in cxpectcd returns have dramatic effects on stock prices, Illakin~ them much morc volatile than they would he if expectcd returns were constant. Thc source of this persistent variation in expectcd stock rctlll'llS is an important unresolvcd issllc. Onc view is thatthc timc-variation in expcctcd returns and the associated volatility of stock prices arc evidcncc against the Efficicnt Markcts llypothesis (EMil). Bllt as we argucnstructed to fit the data. We cxplore this possibility further in Chapter H. I I
7.1 The Relation between Prices, Dividends, and Returns
Itt this 6ection we discuss thc prescnt-valuc model of stock priccs.
Using tl e identity that relates stock prices, dividends, and returns, Section 7.1.1 P esents the expected-prcselll-valuc forllluia for a stock with constant cxPfcted returns. Section 7.1.1 aSSUJl\es away the possibility that therc arc s(kalkd rational bubbles in stock priccs. but this possibility is considered in Srction 7.1.2. Section 7.1.:~ studies thc gencral casc whcrc cxpccted stock r1111flls vary through time. Thc exact prescnt-value forllluia is Ilonlillear in tltis case, but a loglinear approximatioll yields somc useful insighL~. Scction 7.1.4 develops a simple example in which thc expected stock retllrn is tilne-varying and follows an AR( I) process. i We. first recall the definition of the return Oil a stock given ill Chapter The net simple rCllIfII is
I.,
(7.1.1)
This dclinition is straightli.Hwanl, but it docs me two notational conwntions that deserve emphasis. First, NI + t dellotes the relUrn on the stock held from time I to tillle I + I. The sub~cripl I + I is used because thc return only hccollles knowll at tillle 1 + I. Second, PI denotes the price of' a share of stock measured at the nul of periocl I. or equivalently an cx-dividend price:
Plirchase of Ihe slock al prke f', lod;I)' givn 0111' a cbilll 10 Ill'xl P('l iod's di\'idelld pn share /),+ I bllt nol to Ihis period's dividend /),,1 All ;llternali\,e llleaslireofrCllI1'Il is Ihe log orcoliliIiIlOIlSI),cOlllp(lIll1ded rei 11m, defilled ill Chapter 1 as
==
/i+ I
(7.1.~)
Jog( I -1- 1\". I)·
Here, ;IS Ihrollghollt this chapler. we lise JOW('I'(,
1c1l('I'S 10
7,1.1 'fYl,'f.ill/'lI/' f'w,II'lIt-VlIlul' Iidlliioll wilh L't1l/,'lrlllll::\jil'l1l'd fMum"
III Ihis senioll we explore the cOllse(l'l<'llces of Ihe asslllilplioll Ihat Ihe expecled slock relllrn is eqllal \0 a (,(Hlslalll R:
E,l/(H I I = Ii,
(7.1.:\)
Tak:l1g expectatiolls of the idelltity (7.1.1), illlposi IIg (7.1.:1). and rearranging, we obtain all equation relating th(' (tl1n'nl slock price 10 Ihe Ilexl period's expe('icd stock price alld dividelld:
,
. [/',+1 + /}(II] -1+/I - .
(7.1.1)
1( - .I',
This expectatiollal differcnce cqllation can he solved fi)l'ward hy repcatcdly substituting Ullt flltllrc priccs and using the l.aw of Ileraled l':xpectationsthe rcsultthat E, [E,+t [Xll E,[X], disuissed ill Chapter I-tu eliminate future-dated expectations. Aftcr solving furward 1\ periods we have
=
I', =
E,
[ L,=1~ (-)
I ; ] D,+. I -I- U
+ 1':,
[
I
(-) I + Ii
K
I"+K
]
.
(7.15)
The second term Oil the right-hand side of (7.1.:;) is tlte expeued discoulItcd vallie of the stock price K periuds frolll tll(~ prcscl1t. For 1l0W, we asslImc thai this tcrm shrinks to zero as the horizon 1\ in(Tcases: lim
K~",
E, [(_I )K /',~ /-. ] O. I+ =
('i.I.G)
f{
I These..' limillg a:-'slIlIIpliollS arl' .,I;,IIHl.ud ill tht' lill;tlHT liter.lIllft", 110\\"('\'('" .'onlt.' (Jf,he Ii,· <,,,II lin: 0/1 \'olalilil)' \('>\.', fort'''" II pi .. Shill<-r (I!IIII) .111<1 (:,"111'1,..11 ,,,I
period Iho,('
01"
traded
ill lile
pi i( l' i., IIH';I~III(,d.tI 111«.' IU.'KillninK (II tilt' tile IOIlIlIlI.I~ gi\C'11 ill rlli;.; (h;'pICr and
(1II11-divi
oliKin;d \'obtBilY
p;'pt'rs ;\1('
(hI('
10 tlll .... llilkn·llcc· illlllllillg« OIl\ClllioIiS.
I.
/'11'.1/'/11-\'111,1/' /("illll(l//J
ASSlIlIlplioll (7.1.li) \\'ill II(' salis/ied IIl1less IIII' slock pri(T is expecled 10 grow I(lr\,\,1'I" al 1';\1(' U or b~ln. III S('clioll 7.1.~ Iwlow, WI' discllss IIHHkls of ",Iillllal "IIM/I'I 111;\1 rclax Ihis ;ISSlllliplioll. 1.l'lIillg I\. illl'll'asl' ill (7.1.:.) alltl assul\lillg (7.1.(;), we ohlaill a formula expressillg Ih(' ShIck Illin' as Ihl' expeCled presl'lll value offulure dividcllds 011110 Ihe illfillile 1'111111'1', discollllll'd al a ('ollslalll 1',1\('. For fUIUIT (OIIH'llit'lllT we wril(' Ihis ('xpecled prCSt'll1 \'allll' as 1'0':
/'/
=:
1',"
- F/ [t (_I )/ f)/+/]. 1+ U /~I
<7.1.7)
An 1I111"t';i1islic spt'ci;i1 CISt· Iii'll IU'l'crlhdl'sS provides SOIlU' II SI' I'll I illIuilioll occlirs \\'lIell dil'idcllds art' ('xpt'('Ied 10 grow al a ('ollsl;ml rail' (; (\\'hkh IIlllsl hI' ,slII;i1ln Ihall /( 10 keep Ihe slo('k price flllile): (7.I.H) SUhSlillilillg (7.I.H) illlll (7.1.7), Wt' IIhlailllhe well-kllowlI "Conloll growtll Illodd" (;oHIIIII II!Hi~ I) fill' till' priCI' 01';1 Slock wilh a COllslallt discoullt ratl' U alld di\,idt'lId groll'lh ratl' (;,11'11('1'1' (; < /(:
1'/
E,I Il" II
(I -I- (;)/),
U -- (;
u-(;
(7.1.'1)
The (:01'l101l gnm'th Illodcl shIm'S Ihatlh(' stock price is ('xln'lud), semiti\'(' to a pI'I'lIIall('lli (hallg(' ill thl' clist'otllli ratl' /( wh(,11 /( is dose to C, sill('e Ihe elasticit), or thl' price Idlh n'spl'('\ to thl' discoullt ratt' is «(II'/tlU)(U/ I') .-:: -/(/(/l - c). It is illll'Ol'lali1 til avoid 111'0 ('0111111011 t'frol'S ill illterpretillg Ih('se formulas. First, lIott' Ih;\I \\'t' hal'l' IliadI' lit. assllmptions ahout (,lJuity l'epllrchaM's hv finllS, Equily repurchasl's arrcrt Ihe tillll' pattern of I'xpl'ctcd fUlure di\'idellds 1'1'1' sharI' ill (7,1. 7), hllt 1111'\' do lIot alfeet the validity of t ht' lill'lllllb itst'lL Prohlem 7,1 l'xplnres Ihis pOill1 ill 1II0f(, detail. Se('olld, Ihe hypothesis Ihal lilt' expl'cted stoek n'lurll is COllstallt t h mllg II t ililt' i~ SlIllll't iIIit's kllOWIl a, tI H' 1/11/1/ ill~fllr /llOi/p/ofstock pricC's, ~ Bill a COllstall1 ('xl)(,(,11'1I Slol'k 1t'llIl'Il does not illiply a martingale Ii II" tht, stock pritT itst'lL Rccall that a llIarling-alt, 1<11' thl' priet, rt'(I"in's Ed 1'(+11 = i'/, wllt'n'as (7.1,") illlJllil's Ihal
(I t
m I', - Ed /),., I,
(7,1.10)
:!Sl't' (:I"'lIh'l" ~ lot .1 C\I('lul di'nl ...,j'll\ HI II", IU.n li,,~~,lt· hypn'ht.. :..i~. l.t'ltny ( I~)X~}) 'urn'\', tht' '\\~\I'i,,~~,I\' h,,'I".'\U"· hOUl S;UU\U+.. IlU ('~H{) on. Mort, 14t'II(· .. ,,1 lH.tnillg.ll .. It',nlrs Iflr
Il''-.-lW\llIaH,,',1 pi" (' pilI! t'~'t"
.Ht'
,h."'lll,'c',1 ill (:h"plt"
~).
7. I.
nil' IVialio/l bl'lwl'l'1I Prius,
/)iui(/l'Ilds, alld Helums i
The expected stock price next period docs 110t equal the stock price today ¥ would be required if the stock price were a martingale; rather, the expected lilture stork price equals one plus the constant required return, (l + timcs the current stock price, less an adjustment for dividend payments. 3 To obtain a martingale, we must construct a portfolio for which all dividend payments are reinvested in the stock. At timc I, this portfolio will have N, shares of thc stock, where
m,
N,+ I
=
N, ( I
D,+I) . + -,1,+1
(7.1.11)
The value of this portfolio at time I, discounted to time 0 at rate R, is N,P,
M, -- (\ + R)'
(7.1.12)
It is straightforward to show that M, is a martingale. Even though the stock price P, is not generally a martingale, it will follow a linear process with a unit root if the dividend D, follows a linear process with a ul1it root. 4 In this case the expected present-value formula (7.1.7) relates two unit-root processes for Pr and Dr. It can be transformed to a relation between stationary variables, however, by subtracting a multiple of the dividend from both sides of the equation. We get /),
1',- -
U
: : : (.!.') E, [f (_I ) 6Dr+l+i] . 1+ R
,~O
i
(7.1.13)
R
F.qualion (7.1.13) relates the diITerence between the stock price and l/R tillles the dividend to the expectation of the discounted value of future changes in dividends. which is stationary if changes in dividends are stationary. In this case, even though the dividend process is nonstationary and the price process is nonstationary, there is a stationary linear combination of prices and dividends, so that prices and dividends are coinlegraled. 5 :IIIIIIIe special case where dividellds are expected 10 grow at a constant rate G,lhis5implifies ; = (I + (nl',. The stock price is expected to !(row at the same rdle as the dividend, , h~r;l\lw lilt" dividt"nd'price ralio is conslant ill Ihis case. I ·1~)"'t"ly, 01 Variable follows a Mationary lime-series process if shocks to th~ variabI~ have, It"lliporary hilt 1101 permanent elTeCl5. A vdriahle follows a process with a unit root, also known \ as all ill"lf'tl/ft/ process, if shocks have permanent e!Tects on the level of the variable, but not oll·lht".change in the vdriable. In this case the first di!Terence of the variable is stationary, bUI Iht" ".\"t" I i., nol. A marlingale is a unit·root process where the immediate e/Tect of a shock is lilt" S:1I11t" a.' Ih,' pt"f1nanent e/Tecl. See Chapter 2 nr a textbnok in time.. eries analysis such as . I i;lInillon (I ~)~H) f()r precise definitions of these concepts. r'Two variahles with unit roots are COillle!(rated if some linear combination of the variables is SI"tion"ry. St"e Engle and Gmnger (\!IR7) or Hamilton (1994) for general discussion, or (:"mpl"'11 ,\1111 Shiller (1 !IR7) for this application of the concept. Note thaI here Ihe >lationary lillt'"r fllIllhinOllioll of Ihe variahles involves Ihe COIlSt;lIlt discount rate R, which gener..lIy is 1101 ~n{lwll a /Jrimi.
10 E,I'HI
258
7. PreJellt- Vtllue /lelatiuns
Although this formulation of the expected present-value 1Il0dd has been explored empirically by Call1pbell and Shiller (1987). West (19HHI». and others. stock prices and dividcnds arc like many other macroeconomic lime series in that they appear to grow exponentially over time rather than linearly. This means that a linear Illodel. even one that allows I
o
7.1.2 NIl/iOlwl /Jubbll'.I
In\ the previous section we obtained an expeuational difference cqllation. (7,1.4). and solved it forwal'd to show that the stock price llIust c'luall'/JI. thi:: expected present value of future dividends. The argument relicd on the assumption (7.15) that the cxpected discounted stock price. 1\ periuds in\the future. converges to zero as the hori7.ol1 1\ increases. In Ihis sectioil WI1 discuss models thai relax this assumption. I The convergence assumption (7.1.5) is essenlial for obtaining a unique solution PD, to (7.1.4). Once we drop the assullIptiun. there is all in/inite Ill\mber of solutions to (7.1.4). Any solution can be written in the form
"'~~!
1', = 1'/), + /I"
1J,
=
,
I'.,• [ 11J++ IU ] .
(7.1.14)
(7.1.15)
I
Th1e additional terlll B, in (7.1.14) appears in the price only becausc it is eXI)ected to be prescnt next period. with an expected vallie (I + U) times its Clirrent value. The term Pili is sOllie times called fUlldamental value. and the tcrm B, is often called a rational bubble. The word "bubble" recalls sOllie of the famous episodes in financial histolY in which asset prices rose 1~lr higher than could easily be explained by fllnciamentals. ami in which investors appeared to be betting that other investors would drive prices even higher in the future. 6 The a(ljective "rational" is used because the presence of lJ, in (7:1.14) is elltirely consistent with rational expectations and constant expected returns. fiMCIckay (lH5:l) is a riassif .. clt'n'un' 011 t.·ady t'pi~odt's MICh as tile Dwch tllliplII;lIIia ill Ihe 171h Century ~"t1 Ihe I
7.1. Thr Up/alioll /JrtwrPn Pri(r.l, DillidPlld.I,
tll/I/
Up/llnll
259
It is easiest to illustrate the idea of a ratiollal huhhle wilh an example. Blanchard alld Watsoll (I 9H2) S\l~~('st a huhhle of Ihl' ('!'Itl
with prohahility
IT:
with prob;\bility I --
(7.1. Hi) IT •
This oheys the restnC11011 (7.1.15). provided thaI Ihe shod:. (al satislies E'(H I == O. The Blallchard alld Walsoll huhble has a COllstant prohability, I - 7f, o( hurstin~ ill allY period. If it docs not hursl, it ~rolVs at a rate I ~II _ I, faslcr than H, in ordcr to cOlllpellSate for the prohability ofhursting. Many other bubhle examples can be construCled; Problelll 7.'2 explores an example s\\~~estcd by Froot and Ohstfdd (I ~)~ll), ill which the huhhle is a nonlinear function of the stock's dividend. Altho\\~h rational bubbles have altractl'd cOllsiderahle alt(,lltioll, then~ arc hoth Ihcor(·tical alld nllpirkal argllllH'lllS Ihal CIII b" IIsed to rule out huhhle solutiolls to the dilkrcilce equation (7. J A). Theorelical arglllllcills may be divided illto partial-equilibriulIl argulllellls alld ~clleral-('quilihriulll argulIle .1lS. [11 pMti,,! equi!ihriulII,thc first poilltto lIotl' is tlr,1l tllt're elll IH'wr he;1 Ilcg;lliv(' huhhle Oil all asset wilh limited li;lhility. If a lI('gative huhhle W!'\'(' to ('xist, it would illlply a lIe~ative expecte!l asscI price at sOllie date ill the futllre, ami this wOllld be inconsistenl with lilllite(\ liahility. /\ second illlp0rtant point ()lIows fmlll this: /\ hllhhle Oil \(o ('xisled ill all ass!'t infinite-lived ag('nlS could selllhe assel short, inv(,st \01\1(' or IIIC IlJ()(,(Tds to pay the divide lid strealll, alld have positi\'(' wealth lerl OWl. This "r"ilr;\gc opportunity rilles Ollt bllbbles.
7. /'rl'vl//-\'lI/ul' IMII I illl/.! Tirok (I "H:,) has slllllied 1111" possihility ol"hllhhlt's withill the Dialllolld (I!Hi:,) ovnlappillg-gl'III'Lltions lIIolld. In this IIIOtil'! there is an infinite II til III 11'1' of lillitl'-lived agl'lIts, hilt TirolI' shows that evell hne a hllhhle call1lot ;Irisl' \\'hl'll the illtl'll'st \;Itt' excecds the growth raIl' of tht, t'COIlOlllY, hecallsl' the bllhhle I\'ollid I'vl'lltllally hecolIIl' illlillitcly large rdative to the we;t1th of the 1'1'0110111\'. This wOllld violate SOIlW agl'lIt's hlldget cOllstrailll. Thlls hllhhks CIII (111)' exist ill t1.\'I/{/lIIi({//~\' illl,/Jiril'lll overlapping-gl'n!'!'at it illS t'conOlllil's th,lI h,l\'I' O\'nalTlIllllilatl'd private capital, drivin~ till' intl'l cst 1',111' dowlI Iwln\\' Iht, gn)wlh 1',111' of Ihl' I'COlltllllY, Mall), t't'olltllllislS kel Ih,tI lIyn'"llic ill""licil'n<'\' is IIlllikdy 10 01'1'111' ill practice, allel Ahd, Mallkiw, Slllllllll'rs, and /.l'ckklllScr (I !)H!)) Plt'Sl'lIt I'lllpirical ('vi(II'IKe that it does not dl'scrilw tilt' liS I'COIlOIIl\'. Thnc are ,d~o Stlille I'lllpirical ,IIW"lIt'llIS agaillst th(' existcllt't, of hllhhies, The IIlost illlport,lIlt )Ioillt is that bllhhles illlply explosive hehavior of variolls sl'ries, In tilt' ahSl'nt't' of hllhhll's, if Iht, dividend /)( filllows a lillear PI'III'l'SS with a IInit mot then tilt' stork price 1'( has a IInit rool whik the challge in the price Dol', alld the spread hetween prilT and a IlIl1ltiple of divitlclHls 1'( -- n,l/( an' statiollary, With hllhhks, thest' variahks all have an I'xplosivl' cOllditionall'XIII'ctatio'lI: lilll"_,,,,(1/( I + m")I':" XI-/" J l' 0 ror .\, == I',. /\1'" .. , 1',- /J,j I:. Elllpiricallv, thl'rl' is little Cl'i"I'IIIT of I'xplnsin' Iwhavior ill lhl'sl' sl'ries. :\ can'at is that StOt'\lOlstit' huhhles are nOlllilll'ar, so stallclardlinl'ar IIl1"thods Illa\' Elil to tll'tl'd Ihl' I'xplosil'l' hl'havior or the conditional l'x!,I'I'lalinn ill Ihl'sl' Illl)dds. Fillallr, \\'1' 1I0tl' IhOlt ratioll;11 hllhhll's call1lot I'xplaill thl' ohsl'\'\'t'd pretlinahility 01" stock retllrns. Bllbhll's ITl'atl' volatility ill prices wilhou: creating prl'dinahilit)' ill rl'turns. To Ihl' cxtent that price volalility C'1ll he I'xplaillctl hy I'l'tUI'll prl'dictahilit}', lhl' huhhlt, hypothesis is superfluous, Altho\l~h raliollal huhhles ilia), 1)(' illlplausible. thnl' is nllKh to he It'anll'(1 frolll sludyillg thl'lII. All illlportallt thl'lIIl' of this chapter is thaI slllalllllovCIll('IlIS ill expcrtl'd lI·turns call have Iargl' dICcts 011 prices irthey arc Ilt'rsislt'llt. COllversd)'. Iargl' persistl'nt swings in prin's call hal'e slllall cfJ(-cts on ('xpl't'ted rl'lul'lls ill all)' 0111' period. A ratiollal huhhle CIII Ill' sl'l'n as thl' I'xtn'nll' casl' whl'\'(' pril'l' IllIlVI'llH'lltS an' so persistent-indl'l'd, explosive-that tlln' lIa\'1' 110 "'fl'l'\s Oil I'x(ll'l'\l'd retllrllS at all.
7.1 . ., ,III ,1/'IlItI\'jllllllt'/'/I'\I'III-ljdllt' Udillitlll l(1ilh
Tillll'-\'tll)'ill~ J::\,/lI'rtl'lllll'llllll.l'
So 1;11' WI' hal'\' a~slllllt'd thOlt ('x I'I'CI 1'(1 stock retnrns arc cOllslall!. This as,,"nplitlll is ,lIlalytilalh ctlll\'l'nil·lll. hilt it ctllltraelicts Ihl' 1'\'iell'IIlT ill (:hap"'r ~ alltl ill S('(liOIl 7,'2 th,ll stotk rl'tllrns an' predictahll'. It is n1\\('h Ilion' dirlinllt to wllrk wilh pres('nt-vahl<' rl'latiolls \~hl'n ('Xp('l'Icd stot'k r('tlll'll~ ar(' t i1111'-\';1 r\'i Ilg, lill' Ihl'll the relatioll hl'tWl'l'1I (lrilTs alit! returlls hl'COIIII'S lI11nIiIH';t1'. (hll' ;tpproach is \0 use a loglilll';tr ap-
7.1. Tilr Ul'laliorl bl'lllll'l'II ['rias, /hvidl'rld.l,
111111
Uflums
261
proximation, as suggested by Campbell and Shiller (1988a,b). The loglinear relation between prices, dividends, and returns provides an accounting framcwork: High prices must eventually be followed by high future dividends, low future returns, or some combination of the two, and investors!' expectations must be consistent with this, so high prices must be associated with hiV;h expected future dividends, low expected future returns, or some combination of the two. Similarly, high returns must be associated with upward revisions in expected future dividends. downward revisions in expected future relllrns, or some cornbination of the two (Campbell [1991]). Thus the loglinear framework enables us 10 calculate asset price behav, ior under any model of expected returns. rather than just the model with constant expected returns. The loglinear framework has the additional ad~ v
1
=
P, + 10g(1 + exp(d + 1 -
(7.1.17) PHd}. ' The laS[ term on the right-hand side of (7.1.17) is a nonlinear function of the log dividend-price rj\tio, !(dHI - PHI). Like any nonlinear function !(Xl+l), it can he approximated around the mean of XH[, X, using a first-orderTaylor expansion: (7.1.18) !(XI+t> ~ lex) + !'(X)(XI+I - x). P,+I -
Substitllting this approximation into (7.1.17), we obtain Y,+I ~ k + P PHI
+ (l
- p)d + 1 - Ph '
(7.1.19)
where p and k are parameters of linearization defined by p == I/(l + exp(cI- jJl), where (d - p) is the average log dividend-price ratio, and k == - log(p) - (I - p) logO I p - I). When the dividend-price ratio is constant, [hclI p = I I( I + D/ P), the reciprocal of one plus the dividend-price ratio. Empirically, in US data over the period 1926 to 1994 the average dividendprice ralio lias been about 4% annually. implying that p should be aboutO.96 ill al1l1l1al data, or ahout 0.997 in monthly data. The Taylor approximation (7.1.1 H) replaces tile log of the slIln of the slock price and the dividend in (7.1.17) with a weighted averar;c of the log stock price and the log dividend in (7.1.19). The log stock price gets a weighl p close to one, while the log dividend v;cts a wcight 1- p close to zero because the dividend is on average
262
7. Prt'st'lli- V(/lut' [{rlatiolls
~lUch smaller than the stock price. so a ~iven proport ional chan~c in the divIdend has a much smaller effect Oll the return than the sallie proport ional in the price. I
~:hange I
ANn"Oximation Accurm), the approxi mation (7.1.19) holds exactly when the lo~ dividend -price ratio ~~ constan t, for then dl+l and /'t+! move togethe r onc-for- one and equatio n ~7.1.19) is equival enlto eqllatio n (7.1.17) . Like any other Taylor expansi on. he approxi mation (7.1.19) will bc accuratc provide d that the variatio n in he log dividend -price ratio is not too great. One can get a sellse for the ,ccuracy of the approxi mation by compar ing the exact return (7.1.17) with tl~le approxi mate return (7.1.19) in actual data. Using monthly nominal ( ividends and prices on the CRSP value-w eighted stock index over the peiod 1926: 1 to 1994: 12. for example . the exact and approxi mate relllrns 1 ave means of 0.78% and 0.72% per month. standar d deviatio ns of 5.55% 'lne! !J56% pcr month. and a correlat ion of 0.!)!)99I. The approxi mation <1rror-t he dilTeren ce betwecn the approxi mate and the exact returnhas "mean of -0.06% , a standar d deviatio n of 0.08%. and a correlat ion of 0.08 \~ith the exact return. Usin~ allnllal nominal dividen ds and prices on the CRSP value-w eighted stock index over the period 1926 to 19~11. the exact and approxi mate returns have means of9.20% and 9.03% per year. standard deviatio ns of 19.29% and 19.42% per year. and a correlat ion 01"0.999 93. The approxi mation error has a mean of -0.17% . a standar d deviatio n oI"0.2G%, and a correlat ion of 0.51 with the exact return. Thus the approxi mation misstate s the average stock return but capturc s the dynamic s of stock returlls well. especial ly when it is applied to monthly data. 7
~
Implications for Prices Equatio n (7.1.19) is a linear difTeren ce equatiol l for the log stork price. analogo us to the linear differen ce equatio n for the level of the stock price that we obtaine d in (7.1.4) under the assulllp tion of cOlIStan t expecte d returns. Solving forward and imposin g the conditio n that (7.1.20) we obtain
k
III
co
=- + L.. "fJ'I (I- fJ)dlllf ) I-p
•
li+l+jJ.
(7.1.21)
J~"
7011(' ran also (OlUlldrC l'XiICI ~lIId approxiru alt' re,.d U'UIrI1S. The nnn'fliol l fur iullalic,.. ha. no important clT~
7.1. Till' IMII /io/l
lip/wPI'1l
Prirr-s, lJiliit/nuLI, 111111 UP/lIm.l
2(i3
EfllI ation (7.1 .21) is a dynami(" a(collntin~ idcll Iity; it has heen obta incd mer ely by approximatin~ an iden tity and solv ing forwal d slIhj('ct to a tenn inal cond itiol l. The term illal cond itiol l (7.1 .20) rull' s Ollt ratio llal hllh hles that wOlrld calis e the log stoc k pric e to grow ('xJ>ollclltiall~' «,re ver at rate 1/ p or fa.)tcr. Eqll ation (7.1. 21) show s that if the stoc k pric c is hi~h toda y, thell ther e IlllISt he SOIll C ('om hilla tion of high divid cllds alld I"w stoc k retul 'lls H in the ftllllre. EfllI.lIioll (7.1 .21) hold s ex Im.I/, hut it also hold s PX 1111/1'. Tak ing expe ctatio ns of (7.1 .21) , and noti ng that /'t = E,l/I,I heCIIIS(' fit is kllow n at tillle I, we obta ill
[~.
Ii EI L.., p'[( /'1 = ~+ l-P )d,f .t+, -r,f .lf) ] . p ,=(J
(7.1. 22)
This ,1101I1d be thol lght of as a cOll sistellcy cond ition f(lr expe ctati ons, anal ogol ls to the state mcn t that the expe ctati olls of rand olll varia hles X and Y shol l!d add lip to the expe ctati on of the sum X + Y. If the stoc k pric e is high today, thcn inve stors IIlUSt be expc ctin g sOllle com hin, llion of high futu re divi dend s and low futu re retu rns. Equ atio n (7.1 .22) is a dyna mic gene raliz ation ofth c Gor don form ula for a stor k pric e with cons tant requ ircd retll1'll5 alld divi delld grow th. Cam pbel l and Shil ler (I9HHa,h) call (7.1 .22) -all d (7.1 .24) belo w-th e dynamic Gurd un gmw lil /IIotlrl or the tlillit lrnd -mlio mud d I.ikl' the orig inal Gor don grow th mod el, the dyna mic Cor don grow th mod el says that stoc k pric es arc high whe n divi dend s ,liT expc <·ted to gnlw rapi dly or whe n divi delld s arc disc oun ted at a I(lw rat(~; hilt Ihe elkc t Oil the sioc k pric c of a high divi dend grow th rate (or a low disc oull t rate) IIOW depc nds on how long the divi dend grow th ratc is expe cted to he high (or how IOllg the disc oull t rate is expe cted to be low) , whe reas in thc orig inal mod el thes e rates are assu med to be cons tallt at thei r initial level s forcver. One can lise the ddin ition s of p and " to show that the dyna mic Gor don grow th Illod d redu ces to the orig inal Cor don grow lh 1I10dei whr n divi dcnd grow th rail'S alld disc oull t rate s arc (Oll stan t. For futu re conv cnie nce, we can simp lify tile Ilota tioll in (7.1.~~), rewritillg it as (7.1. 23) whe re /',/1 is the expe cted disc otlll ied v.lllle of (I - p) tinH's !'ruu re log dividcn ds in (7.1. 22) and 11" is thc expe cted dis('oUIIl!'d \'ailic of futu re log stoc k retu rlls. This para llels the nota tion WI' IIsed f(lI' tl}(' cOll stant -{'xp e!'tc dretu rn case in Sect ion 7.1.1 . "C;"" pl>cl i OI'HI Shill "r (I!IHHOI) "I',.I" au' III<' ,n I """ I' III III" 011'1" ",i"1 0'I,"" ill (7.1.'11).
Equatioll (7. I.:!:!) C;III hl' l'('IITilleli ill terllls of thc log dividclld-prit'(, ratio rathn Ihall Ihe log Slock prin':
tI, -
I',
=0
-
Ii
--
1-
fI
1
"\. P 'I - /\ tI'-I It) + /", t-I + ,I . + 1-'., [ '\' L / __ 11
(7. I.:!'!)
Thc log dividelld-price ralio is high whell diviciellds are exp('Cl('d 10 grow ollly slowly, or whclI slock n'llIl'lis art' ('XI}('('lcd 10 h(' high. This cqllalioll is II s('fll I II'hell 111(- dil'idt'llcl folloll's a loglillt'ar IIlIit-root proccss, so Ihat log ciivid('lIds alld log I'rin-s an' IUHlstatio\lary. hI this cas(' change, ill log divi(ic'nds ar(' stational'\', so 1'1'0111 (7.1.~4) till' log- dividend-prict' ratio is stational')' prOl'iclc-d that tht' ('xp('Cl('d stock rellll'll is statiollary. TillIS log- stock prin-s and dividends an' cointegrateci, and thl' stalionarr lillt'ar cOllliJill;Ilil/n or IIII'S(, variahles ilivolVl's IIIl unknown paraml'ters since il is jllst tht' log ratio. This simple strtlt'lllrc makt's the log-linear mociel ('asier to IIS(' in elllpirical work thantht' lint'ar coilllegraled model (7.1.13), So rar W(' hav(' w)"illclI asst't pric('s as linear combinations of t'xpeClt'ci 1'111111'(: divici(,lIds alld )"('IIII'IIS. 'VI' can lise Ihe sallie approach to wrile asst't r('1II I'IIS as lilH'ar cOlllhillaliolls of n'visiolls in t'xpt'('\t'd 1'111 lire divid('ncis and )"elIll'IIS (:alllpl)('llll!)!lI J). SlIhslilillillg (7.1.22) into (7.1.1!), 11'(' OI>I;lill 'iI I --
E,lli III
-. (1-'.'1 I [to fl' rill f/] r' I
E,
[f J~I
p ir,-I
1+/]) . (7.1.~:)
This ('qll;llioll SItIlWS Ikll Illlt'XI'(TI"d Slock ('('lllnlS IllIlSl 1)(' asso('ial('d \"illl challgl's ill l'x),,'n;l1iolls of 1'111111 (' divid(,lIds or ('('al relll)"IIS. All illcrl'asl' ill I'XIll'Cll'd 1111111'(' dil'idl'lIds is associall'd wilh " capilal gaill loday. whill' all illn(';IS(' ill ('XI\tTll'd 1'111 lire \'('\llrIIS is associated wilh a capilal loss \(Hlay, Thl' reasoll is Ihal wilh a gil'l'lI dil'icil'lId slrl'alll, high('r flltllre r('llIl'IIs call ollly h(' W·lu'r;l\(·t! ll\' 111111\1' pritT ;lpl'\'('cialioll frolll a lowl'r ('111'1'('1\1 pl'i('t', For COIII'('lIil'lI(,(', WI' CIII silllplifl' Ihl' lIolalioll of (7.1.~:.) 10 (I.I.:!(;) w!tnl' '/11 1 i, III<' (1IIt'XI,,'cI('(1 .,Iod.. 1'('1111'11,1/01./+1 is Ih(' ('hall1-:(' ill t'xlw(laliollS ofhl\IIIT di\'itl('lItis ill (7. I.:!!'.), ;llIdl/di I is the changl' in I'xpectatiolls 01'1111111'1' I'l'tllntS.
'I'll<' forlllllbs ,It-n'lol'l'd ill III<' )'n'l'i,,"~ s('c'lioll Illay hI' ('asil')" 10 lIIull-rsl:tllci ill Ihl' I'OIII('XI of a ,illll'll' (·x:III1),I(·. LlltT W(' will arglll' that tht, I'xalllple
7. J. Thp UelatioT! be/wpm Prial. /)ivitimtis. (lnd &turns
265
is not ollly simple. but also empirically relevant. Suppose that the expected log stock return is a constant r plus all observable zero-mean variable X,: (7.1.27) We further assume that process
X,
follows the first-order autoregressive (AR( 1)) -I < I/> < I.
(7.1.28)
Whell the AR coemcient I/> is close to one. we will say that the X, process is highly Ilmistent. Equation (7.1.28) implies that the variance of X, and iL~ innovation ~/. which we write as a; and a{ respectively. are related by 2 2 2 a,==(I-¢)a,. Under these assumptions. it is straightforward to show that
[
,
"" p'r,+I+j Ilrl == E, "\"' i...J ,=0
]= - T
1- p
~
+ ---. 1 - pI/>
(7.1.29)
Equation (7.1.29) gives the effect on the stock price of variation through time in the expected stock return. The equation shows that a change in the': expected return has a greater effect on the stock price when the expected: return is persistent: Sinc, p is close to one, a 1% increase in the expectedl return today reduces the stock price by about 2% if ¢ = 0.5, by about 4% ir' rp = 0.75, and hy about 10% if ¢ = 0.9. This example illustrates an important point. The variability of expected stock returns is measured by the standard deviation of X,. If this standard deviation is small, it is tempting to conclude that changing expected returns have lillie influence on stock prices. in other words. that variability in prl ' is sm'lll. Equation (7.1.29) shows that this conclusion is too hasty: The. standard deviation of prl is the standard deviation of X, divided by (l - pI/» , I, so if expected returns vary in a persistent fashion, p,/ can be very variable " even when X, itself is not. This point was stated by Summers (1986), and i particularly forcefully by Shiller (1984): Returns on speculative assets are nearly unforecastable; this fact is the. basis of the most important argument in the oral tradition against a role, for lIlass psychology in speculative markets. One form of this argument claims that because real returns are nearly unforecastable, the real price of stocks is close to the intrinsic value. that is, the present value with cOlIStant discount rate of optimally forecasted future real dividends. This argument ... is one of the most remarkahle errors in the history of ecollomic thought. In ollr example the stock price can he written as the sum of two terms. The first «'nn is the expected discounted value offuture dividends, Pdl; this
··'f:
,
i266
7. Presml- Vttlue UelllliUTlS
\ is not quite a random walk for lhe reasons given in Section 7.1.1 abow. but
Iit is close to a random walk when the dividend stream is 1I0t too hU'ge or I variable. The second term is a stationary AR(I) process. -/1,/. This lwocompollellt description of stock prices is often found in the lilerature (sct: Summers [ 19861. Fama and French [I !l8Hb 1. I'oterba and SUllllllers [ I !lH!{ J. andJegadeesh [19911). The AR( I) example also yields a particularly simple formula lilr the one-pe;iod stock return r/+I. The general stock-return equation (7.1.25) Isimplifies because the innovation in expected future stock returns. 1/,./+1. is ,given by P~,+';(l - p
r,+1
:=
T+ X,
+ '/<1.'+1
P~'+I - ---.
1 - P
(7.1.30)
To understand the implications of this expression. assume for simplicity that news abolll dividends and about future returns. 1/d.I+1 and ';'+1. are uncorrelatedY Then using the Ilotation Var[lld.I+ll = O'j (so oj rqll'csellL~ the variance of news about all future dividends. not the variance of the clITrentdividend). and using the fact that 0(2 := (1-2)0';. we call calculate the variance of T,+I as (7.1.31) where the approximate equality holds when « p and p is close to one. Persistence in the expected return process increases the variability of re realized returns. In the ARMA(l,I) representation the AR coefficient is the positive persistence parameter l/J. hut the MA coellicient is negative. Problem 7.3 cxplores these effects in detail, showing that the latter effect domin
I
"TIIi., lIliglll I", Ihe ('''e. f(>r exalllpk. if expcn,·<1 '-Cll,ms arc d"I,'nnin!'
7.2. I>rl'.mll- Valli I' R,la lioll. l fllld liS S/O(/( 1>/1((' IMlIl lIior
2(;7
he posilively aUIOCOlTeialed ifdivi d('11<1 11('\ " alld ('xl)('( led-r ellln llll'W S have a slIfficiellll), I
7.2 Pres ent- Val ue Rela tion s and US Stoc k Pric e Beh avio r We IlO\\' use the iden titie s disc llsse d ill tlte prev iolls s('('(ion 10 inte rpre t recc nt emp irica l find ings on Ihe limc -sni es I>cll'lvior of US sloc k prke s. Sect ion 7.2.1 disc usse s emp irica l work Ihat pred ins sl()(,k reI urns over long hori zons , lIsin g I()re cast ing varia bles othe r Ihall pasl relll rns Ihem selv es. We pres elll illus trali ve emp irica l resu lts whe n divi delld -pric e ralio s and inte rest rate v;tri;tbles are IIsed to fore casl sloc k reI urns . S('cl ioll 7,'2.'2 relat es long horil.OIl relu rn beha vior 10 pric e beha vior , in parl icul ar sloc k pric e volalililY. Secl ion 7.'2.:~ show s how time -seri es Illod els can he used 10 calc ulale Ihe long hori zon illlp lical ions of shor t-ho ril,o ll asset mar kel I)('havior. 7.2,1 LOllg-IIOlizoll Ul'h7C,I.lio/I,\
Recc lllly thcr e has heen Illllch inle resl ill regr essio ns of retu rns, mea sure d over vario us hori zons , Ollto forec aslill),( v;tri abks . Popl liar filre casl ing variabit' s incl ude ratio s of pric e to divi delld s or earn ings (sec Cam pbel l and Shil ler [1~)HHa,bl. Fam a and Fren ch [1!)HHa], Ilod rick [1!)!J'21. and Shil ler [I !JH·I]) , and vario us illle rest .. ate llleaS\ll('S such as Ih(' yield spre ad beIwecn long - and shor t-ter m raIl'S , the qual ily yield sprc ad h('tw cell low- alld higll-gr;l n, alld IUC;15(11'e S of rccc lll chan gcs in lite /('vel of shor t rates (sec Call1pbt~1I [I !JH71. Faill a alld Fren ch (19K!)!. J-iod rick (I!N 2J. anc! Keill l ;lIId Slal llbau gh (I!JHlij). lIer e we COl lccn lrate on the divi dcnd -prit T ralio . whic h in US dala is Ihe 1Il().~1 sllcc essfu l fore casl ingv aria ble fi>l' long -hor izon relu rns, and (Ill a shor tterlll nOlllillal inln ('sl- ralc varia ble. Wc sl;11"I wilh pri(l 's .lIld divi dcnd s 011
the vallie-weighted (:RSI' illdl'x of stocks Iraded olllhe NYSE, the AMEX, anel Ihe NASDAQ. Till' dividl'lId-price ralio is measured as Ihe Slllll of divide lids paid on Ihl' illdl'x over Ihl' prl'vious Yl'ar, e1ivided hy lilt' clIITelllll'vcl of the indl'x; sUlllluillg dil'idcllds oVl'r a full year rellloves allY seasonal pallerns ill dividt'lld j>aynwllls, hut Ille nll"l"t'lIl siock illcl<'x is lISt'cilo incorporale Ihe ItlOSt n'n'lIt ill(illlll;tliOIl ill sluck prin·s. 11l The illleresi-rall' variahlc is a Irallsformalioll of Ihe Olle-IIIOIlI h nOIll i lIal US Treasury hill rail' lIlolivatl'd hy thl' bCI that unit-rool tests often (;\il to rcjecl Ihe hypothesis thaI Ill<' bill I'ale has a IIllit roo\. We sllhn',l('\ a hackward 0111'-1'1'''1' lllOvillg avcl'agl' of past hill rates fro III the I'lIlTI'nt hill rate 10 get a ,lllIl'illlIlimllv "fir/'lll/rtl illtl'l'est rate that is eqllivalent to a triangularly 1\,l'ightecl Illoving aVI'rage of past changes in bill ratl'S, wherl' the weights dl'clinl' as 0111' IllOves hack in lilliI', Acconlingly Ihe dl,trended interest rate is stationary if changes in hill ratl's arl' stationary. This slochaslic clel rellding 1I11'Ihod has hl'(,11 IIsl'd hy Camphl'll (1!J!lI) andllodrick (1992). 'E\hlc 7.1 shows a I\'pical Sl't of rl'snits whell the dividl'nd-price ratio is lIsl'd 10 (ill'l'casl rl'lllrtls. Thl' lahll' rl'(lorls monlhly regressiolls of log \'l'al stork retllnlS Ollto till' log of Ihe dividend-price ratio at lhe start of lhe holding (lniod. RI'I II rtlS are Illl'asurl'd over a holding pI'riod of K nlOlllhs, which rallgl's frolll 0111' Illollih 10 'IK 1I10111hs (1'0111' YI'ars); Whl'lll'Vl'r l\ > I, Ihl' I'l'grl'ssiolls IlSI' OI'I'Ibppillg llIolllhly clala. Results are J'('porlcd fur the period 1927 to I !}~).( alld also IClI' suhsalllpks 192710 l!l!i I alld 19:1210 1!I~14. For I'ach regressioll Tallie 7.1 rl'ports lill' Il~ stalislic and Ihl' I-slatislic for Ihl' hypolhesis Ihal III<' ('(wHicil'lIl 011 Ihe log dividend-price ralio is z('ro, The I-siatislic is cOITl'ctl'l1 for hl'l('\'oskl'dasliciIY and serial correlali-J1l in Ihe eqllalioll elTor IIsing Ihl' asymplolic Iheory discllssed in Ihe Appell'lix. Tahk 7.1 ((,lIows Failla alltl Frellch (l~}KKa) t'xcl'plthatlhe rl'j.~r('ssor is Ille log dividl'nd-pril'(' ralio ralhn Ihan Ihl' ieI'd of Ihe dividend-prict' r,llio (a challg\' which Jllak('s \'l'I'y lillie dilkrellce 10 thl' results), overlappillg Illolllhly dal;\ an' IIsl'd fc II' allllOrilOns, andlhc sample pniods an' IIpdalcd. Although Ihl' rl'SlIlts ill Ihe lahll' arl' (i,l' I'l'al stork n~tllrus, allllosl idelltical rl'sults ;11(' ohl ;lilll'd fill' I'X('I'SS relllrns 01'1'1' IhI' ont'-Illonth Treasllry hi II 1'01 Ie, AI .1 hori/oll 01'0111' IIIOlllh, Ihl' r~'gression results ill '):Ihlc 7.1 are ralher IlIlilllpn'ssin': TIlt' H~ slalislics 111'\'1'1' I'xn'l'd 2(}{J. alldlhe I-stalislics eXCI'cd 2 clIll)' ill Ih(' l'0sl-Wulld War \I slIhsaJllplc. The slriking LI('[ ;lhoUI Ihe I"hlt' is how Illlich slHlllg,'r IllI' n'slliis hl'colI\\' whl'1I on\' ill('J'e;JSI's IIII' IlDri/oll 1\. AI a Iwo-year !Juri/Oil IIII' N~ slalislic is 14% ((II' Ihl' filII salllpll', 22'X, fClr thl' prewar slIhs;lIllpk, alld :\2'1., ICll Ihe poslwar slI\)salllplI'; OIl a fO\lr-~'l';\l' hori/oll IIII' U~ sialislic- is 2ti% for Ihl' 1'1111 sample alld 42% for earh of Ihe sllhsallll'lcs. III Ihl' 1'1111 sallll'll' and Ihl' pl'l'wal' sllhsalllplc Ihe regressioll 1-
IIITIII' \\',1\' UIIlIt".I'lIIl1lg lilt'
II j ...
al . . .,
101111110111\· ""'.(
ilillu-
di\'iclnul-p' i, C'
I.llill j, ,1;uIII;lId
hu.lllcial iutlu ... llv.
indl(' ;u';uklllic'
lilt'I;IIII1t'.
,11)(1
7.2.
us Siork !',i((' !MullIior
/'fl'.ll'l/l· V(dll~ UI'/(/I;fI//.Illl/r!
269
! Table 7.1.
I AII'K'/lIIriwl/ I"I'K'l'lIio/l.1 of 10K .l/ork 'i+1
+ ... + 'iH
11'1/lI'II\ 01/
= fJ(l\)(d, - /I.)
IIIf loJ!. dil,idnlll-prirl' ratio.
+ 'II+A".I'
FOfl.'cast lIoriwlI (1\)
:{
I:!
0.01:1 0.004 U!21
0.(144 0.0 I:> 1..100
jJ(l\)
0.01:)
II~{I\)
().()(l:~
I(jJ(I\)
0.660
24
36
41\
0.1 !}I
O.:Hl:{
O.lHiH
052!! 0.209
2.07'l
0.144 4.113
,t.t;:~1
O.fi!>4 0.267 3.943
(l.W>!) 0.014 0.1\44
0.27'1 0.074 1.677
0.629 11.207 4.!i'21
O.HHO 1I.:m! '2.967
I.O!iO 0.4'24 3.7H3
n.07!) 0.047 3.0:':>
O.:W!) O.I!JO 3.22f\
0.601 0.344 3.'2'25
0.776 O,12H 3.315
0.H63 0.432 3561
1!)27 10 19!H fo(l\)
H"(/o:) I(~( /0:1)
1\)2710 1!):>1
•
1%2to 1\1\14 fo(l\) !t(K) IIjJ(/o:»
n.n24 0.01:> '2.73:~
or
r is II ... 10); ... ·,,1 r~lurn on a valu,,·weighl~d index NYSE, AMEX, and NASDAQ.locks. (d- PI is lilt' log ralio of divideuds o\'er the last year to lhe current price, Regressions are estimated by 01.';, wilh I 1"list· n alld Hodrirk (I!JHO) siandard errors, calculaled rrom equalion (A.3.3) ill Iii,' APP","lix s<·ttill~ au!Ocovariauccs heyond lall K - I to zem. Newey and Wesl (19K7) ,larul" ... I,·rrms I.ilh 'I K - I or q ';I(K - I) are very ~il\lilar and typically are slightly smaller Ihan Ihm,· r<,ported ill Ihl" labll".
=
=
stalistics also increase dramalically with the forecast horizon, although they are fairly stable within the range 3.0 to 3.5 in the postwar suosample. It is interesting to compare Ihe results in Table 7.1 with those obtained when stock relUrns are regressed onto Ihe stochastically delrended shortterlll inlereSI rate in Table 7.2. The regressions reported in Table ·7.2 are run in just the sallle way as those ill Table 7.1. Once again almost identical results are obtained if real retllrllS are replaced by excess returns over the one·lIlonth Treasury bill rate. Tahk 7.'2 shows that, like the dividend-price ratio, the stochastically detrcndcd short rate has some abilily 10 forecast stock relllrns. However this forl'casling power is very different in IWO respects. Firsl, it is concentrated in Ihe postwar suhsalllple; Ihis is not slIrprising since short-term interest rales were pq~ge(l hy the Federal Reserv(' during I11l1ch of the 1930s and 1940s, and so thl' dl·trl'IHlcd short nile hardly v'lriL's ill these years. Second, the fi>nTasting power of Ihe short rate is at Illllch shorter horizons Ihan Ihe
270
7.
Table 7.2.
l.ol/g.!wtil.ll11
IfJ.."I'.I.lilll/.l IIJ log .1/{)(}{,l'Illrll.l (11/
PreJp.III- VallII' UdaiiOlIS
/Iz, .1/(I(},flS/i((/I~V
IPrIll iI/Inn/ mfr. 'HI
+ ... + 'i,~
= fJ(K)(YI.I - L:~II)'I.I_I/1~)
+ 'Il!u;
Fore .. ;"1 I lori/oil (1\)
-----_._------ ._--------_. :1
I:!
:!4
:lli
-5AG!'! 0.00:. -:2.:29:2
-17.IHI O.Olfi
-·II.fifi~
-4.49~
fU):2:1 -1.:.fi4
0.000 -0.lli4
-:!fi.I·lx -:20.1:2\1 0.004 0.00:2 -OXI!'!' -1.:1-11'
3.144 0.000
7:1.71~
0.:2~~
-fi.! H~ 0.000 -0. Hi:.
0.111 :2 O.f):20
IfIH.9H\) 0.031 I.fi!i:!
-67.f.0:. -f)O.!IOO 0.00:) 0.00:2 -O.li:17' -05XO'
-6547 O.(1l9 -3.263
-IH.ti21 0.047 -3.206
-:)(;.401i
-26.115
O.IO~
O.OI~
-2(i.:)7:\ 0.010
-2.741
-1.354
·IX
1927 to 1994 ft{K) !elK)
I{~(K»
119:2~
\0
1951
Jl(K) le{K)
I{~(K))
:
!
195~ to
-:25H~
1994
~(K) 112 (K)
[(folK))
-~r).X\H
O.OOH -1.092'
I
1
i. the log real return 011 a value.weillhteu index "I' NYSl'.. AMl'.X, and NASDAQ stO! ks. YI.I i. the I·month nominal Treasury bill rate. Rellre"i"tl, are eSlimaleu by OIS, with 11 .. ",etl ",.. d Hodrick (1!l80) Slandanl errors, calculated from e'luation (A.:!.:!) in tht· Appendi" setting llocoV'driances beyond lagK-I lo,ero. Newey and West (19117) Mandanl cr)'Ors, willi '[ = ( • - I), are used when the lIan5cn and 1I0drick (19HO) covariance matrix .. Sli,nalOr is nol 1 "itivc definite. The cases where Ihis OCfurs ar .. marked '.
:f ~'''''"ng
pow« of"" di,idcnd-p"« ,""0. Th, I'""W" 1<' " .. ,i"i" .. "
oomparable to those in Table 7.1 at horizons of one or three JlIonths, but tlley peak at 0.10 at a horizon of one year and then rapidly decline. The regression I·statistics arc likewise insignificant beyond a one-ye,lr horizon. How can we understand the hUlIlp-shaped pattern of R~ statistics and I·statistics in Table 7.2 and the stl'Ongly increasing pallcrn in Table 7.1? At one level, the resulL~ in Table 7.1 Gill be understood by recalling the f<>nuula relating the log dividend-price ratio to expectations of future returns and dividend growth rales, givcn above as (7.1.24):
This expression shows that thc log dividcnd-price ratio will be a good proxy for market expectations of future stock rCllIrns, provided that expectations of future dividend growth rates arc not too variable. Moreovl'l', in gl~ncral
7.2. !'rp.lml-Vtdlll' !lRlalioll.1 allli US SllIrh !'Iia !Mulliiol'
~71
the log dividend-price ratio will be a h('ul'l' prox), 1m cxpectaliolls of IOllghorizoll rellll'lls thall fiJI' expectatiolls of shon-lioril.OlI retlll'llS, becanse Ihe expectatiolls Oil Ihe righi-hand side of (7.1.~'1) are of a discounled vallie of ;\11 retlll'llS illio tile infillill' ('111111"('. This Illay help to explain the improvenH:llt ill forecast power as tilt: hori'l'on incre:ls\'s ill '!;\hte 7. t. [<:\'\'11 ill the ahsellcl' or this dkn, howev('r, i I is I'",,,illl(' 10 "hlaill results like those ill 'I;,hk, 7.1 and 7.'2. 'Ii, sec Ihis we lH>W 1'l'llII'n to our AR( t) example in which the variable X" a pnkc( pmx>' ({,,' Ihe expected sturk retllrn at allY horizon, is observable and (';\11 he Ilsnl as a rq~ressor hy Ihe eCOIIOllletririan. Prohlem 7.4 develops ,I strlll'tllr;Il lIIodl'l of SIlKk priccs a!H1 dividends ill which a mllitiple of the log dividend-price ratio has till' pruperties of the variabk X, ill the AR( I) exampk. W" lise the AR( I) example to show that whcll .\'/ is persistellt, the If 01';\ return rcgression on X/ is very small at a short horil.oll; as the horizon increases, the /{~ (irst increases and thcn eventually d\'tTCases. We also discllss (inite-sample difli('uIties with statistil'al inkl'elHT ill long-hot il()n regrcs.'iions. n~ S,(J/i.,/in First cOllSidn regressing the one-period I'ctlll'll 'i+t on the variahle x"~ For simplicity, we will ignore conslant lel'lllS sincc lilese arc lIot tilt, oi>jccts of interest; constan(s l:(l\Ild be included in Ihe regression, or we could simply work with demeaned data. In population, fi(l) = I, so Ihe filled value is jllst x, itself, with variancc while the vari,lIlce or the letlll'll is givcn by e<]lIation (7.1.:'>1) ahove. It follows thai lhe one-period regl'essioll R~ statistic, which we wrile as /(~(I), is
a;,
(7.'2.1) simplicity we are using Ihe approximate version of (7.1.31) that holds when c/! « p and p is close to one. U~( I) n',Klles an upper hound of (I - c/!)/'2 wlwn the variability of divi(ll:nd IlCWS, is zero. Thus evell when it stock is clTenivc!y a real consolhon
WhCIT fill'
a,i,
'/1,1
+ ... + r'H' ==
/I(I\.).\'( -I- 'I/fAA·
(7.:1.2)
In the AR,( I) example, the ileslliJrccasl of till' otlc-pniot\ n:tllrn j pniocis ahead i, al",;,),s hl,\'( f /_ f I = tjI,-I,\'/. 'I'll<' bcSI fim''';lsl of Ill<' ""l1lllali\'('
I.
• Ir.\t'lI/- V(II/It' /(l'Iall<J//.1
or
/"('111111 OV"I 1\ IIIOlllh~ is IIHllld lIy sllllllnin~ Ihe f(ln'('asls one-period relll""slIplohorilOlIl\,so/l(K) = (J+I/J+",+I/JIi-I) == (1-1/J1i)/(J-I/J). The J(~ sl;lIisli(' IiII' IIII' I\-pniod legressioll is givcn by Val' I/'.~II" II -+- ...
\'all
1'"
I
-I- ...
+ EI [l'lt ...·11 + I'H Ii I
(i.~.:\)
Dividillg hy Ihl' oll"-pniod U~ slalislic alld n'arranging, w,' ohlain U~(f~)
Val
1/':11 I'll I l-+-' .. + \0: 1 [ fIt ",11 )
(
W(\)
x
Val'
( Vad 1'"
i /,.t! li+ I II
) "ar! 'itl I I + ... + r".J\ I .
0.2.·1)
or
Th(' lirsl ralio 011 Ihl' righl-halld side of (i.2.4) is ,illSI the sqllare IiiI' I\-pcriod rl'gn'ssillll cOl'flici"111 divided by thl' sqllare of Ihl' OIlI'-pl'l'iod re)!;rl'ssioll ('(wl'lici"1I1. IlIlh,' AR( I) I'xampk Ihis is (I -I/J"')~ I( I -I/J)~, which is "PPIOXilll;lIdr (''1,,;11 I.. ,,".' lill tll'g(' 1/1\. Pllllill~ Ihe 1wo Il'n11S Oil Ihe right-halld side of (7.2.'1) toget Iwl', w,' find Ihal ifex(>"('(l'd slock I'l'l 111'1 IS all' n'l,\, pnsisl"III, Ihe 1lI11ltipl'riod It stalistic grows at lirst approxilllatdv ill plOpor\ioll to Ihe horizo/l 1\. This bl'havior is w('11 illllstrated hy thl' lI'slilts ill '1:1),1" i.l. Illtuitively, it OCClII'S hel'ausl' Ii >recasts or "x pITh'" Il'tlll"llS sl'vl'ral pniods ahead are ollly slightly less variahk thall th,' !I.n·cast of the 11I'xi period's ,~xpl'l'Ied retlll'll, alld they arl' pl'rfenly (,OlTd;ltl''' with it. SIICCl'ssivl' realized retllrllS, Oil Ihe other halld, are slightly /lc)!;ali\'C/y ('orrl'lall'd wilh 011(' allothl'r. Thlls 011 IIrst IIIl' varianI'(' ofthl' IlIl1ltipl'riod liltnl vallie grows more rapidly thall the vari;IIICl' of thc IIIl1ltipniod rl';dilcd rl'tlll'll, illcrl'asillg Ihl' Illllltip('rio
Dr
273
7.2. PI'f5t'1lt-Value Relations and US Slock Price Behavior simplifying approximation that holds when
U~(2)
U~(1) "'"
«
p and p is close to one
~[2+(1-
(1+
2(l+
.
(7.2.5\
The ratio in (7.2.5) approaches (I + , and aJ/a; ::::: 0, for example, a one-period regression has an R2 . statistic of only 1.5%. but the maximum R2 is 63% for a I 52-period regres-, sion. When the forecasting variable is highly persistent, the R2 statistic can" continlle to rise Ollt to extremely long horizons. '
a; fa;
=\
f)i{fintiliPI wilh lnft>rmcl' in Finill' Saml,uJ : The I-statistics reponed in Tables 7.1 and 7.2 are based on the asymptotic:, theory summarized in the Appendix. There are however a number of pitfalls ~ in applying this theory to regressions of returns onto the information variable XI' A Iirst problem arises from the fact that in the regression of the oneperiod return THI on x" TI+I = P(l)XI + 1]1+1, the regressor XI is correlated with past error terms 1]1-; for i ~ 0, even though it is not correlated with contemporaneous or future error terms 1)1+1+,. These correlations exist because shocks 1O the Slate variable XI are correlated with shocks to returns, and the variable XI is persistent. In the language of econometrics, the regressor X, i5 Imtielt'rmined, but it is not exogenous. This leads to finite-sample bias in the coefficient of a regression of returns on X,. In the AR(1) example. there i5 a simple formula for the bias when the regression horizon is one period:
E[~(l) _ fl(1)] I'
= _ (I + 3et» T
aq,t == p(l
af
(l -
+ 3
(7.2.6)
The term -( I + 3
274 eqllality in (7.2.6). This bias Gill bl~ suhstantial: With p = (1.~1~17, till' cxamplc, it cCJuals :~t)1 T when rp = o.!), 7:\1"}' whcn rp = O.~)!I, and 171/"}' when rp ::: O.!J8," A sccond prohlelll is Ihal Ihe aSyJIIPtolic theory !!;i\'('n ill III(' App('ndix 1 lay hc misleading ill tillitc samples whcn thc horil.OlI 1\ is large rl'iatiVl' t 1 lhe sample size. Hodrick (19~)~) and Nelson allli Killl (I~)~I:\) liS!' MOlllt'
r j
1
i
I
1;+1
=
,.
y(..,,)(x,
+ ... + ·\',+1-.-) + tI,+I.J,,',
(7.2.7)
,
w;hcre Ihe error terlll lit r 1.1\ is now sl'l'ially tmcorrclated. The IIl1nH'rator 01' the regression coefilcicnl y(K) in (7.2.7) is thc samc as Ihc n\\lIlcralor or the regrcssion cocffldellt fi(K) ill (7.~.2), bccausc the co\'ariallcl' or x IlIcasured at OIlC datc and,. mcasured at another date depends ollly on the dilTcrcnce betweellthe two dates. lIencc y(K) = 0 ill (7.2.7) ifan
011(' pC'litld.
Se('
.. 275
7.2. }'mmt-Vaillf HemtionoS anti US Stork Prirf IJrlullIior 7.2.2 Volatility ·li'.l/s
III til{' pr('violls MTlioll we have l~xplon'd regressiolls whoS(' depelllkill variables arc n:llIrliS measllred over IOllg hori/.oIlS. ()I1(, lIlolivalioli for sut:h regressiolls is Ihal assel prices al'l~ influcl1c('d hy n,[)('ctaliol1s of r'l'IUrllS ililO the dislant flHIIIT, so long-horizol1 procedures ar(' lH'cessary if we ;\IT 101111dcrstand price hehavior. We now I\lrn 10 l'llIpiricli work lil;11 looks;1\ price variabilily 1l111l'C (lircClly. LeRoy alld PoneI' (I !Hll) and ShillCl (I ~IH I) slarlnl a he;lled d('hale ill lhe early I ~IH()s hy arguillg Ihal slock pric('s arc 100 volalile 10 hc r;llioll;11 fOlTG\SL~ or rlllllre dividends discollllied al a CllllSl;\1)\ rale. This cOllln>vcrsy has sillce died dOWll, panly becallse il is 1I0W 11101'(' clearly undersloud thaI ;1 r~jeclion of cOllslanl-dis(()UIII-ralc JIlodels is 1101 the sallie as a rejeclion of Ihe Eflicielll Markets Hypothesis. and p,lrlly UC(',lllse regression Ie sis have cOllvinced Illany fmaneial economisls lhal expeC!ed stock relurns arc lillie-varying ralhcr than conslallt. Nonclheless Ih(' volalilily lileralure has illll'odllced sornc importalll ideas Ihal arc closely cOllllected wilh Ihe work un multiperiod rei urn regressiuns discussed in Ihe previous section. Useful surv'?)'s of Ihis lileralllre include Gillcs and I.e Roy (1~191). l.eRoy (1989), Shiller (I ~)H~I, Chapler 4). and Wc~t (I ~)88a). The carly papers in thc volatility lileralnrc IIsl'd levels of slock pritTS and dividends, hUI hcre we reSlale the ideas in logarithmic lill'ln. This is consislenl wilh Ihe more recent literalure and wilh the eXposilion in the rcst of this chapler. We begin by ddining a log 1"'ljl'rljorr.lir;ht .It()(!. 11I1rf, 00
/1; ==
I>l[(l_p)df!lil I-k-
rl.
(7.:!.HI
l="
The perkct-roresighl price g is so namcd becanse frolll Ihl' fX /loJI slock price idcnlily (7.1.:! I) il is Ihe price Ihal would prcvail if ref/liud rCllIrlls were conSlanl at sOllie level T, thal is, if thcre werc 1\0 revisions in expectalions driving unexpected returns. Equivalently. from thc fX alltf slOck price identity (7,1.~':.!) il is Ihc price thai wonld prevail if (,xpcr\cd relnrHs were cunslant and investors had perret:! kllowlcdgc of futllre dividcnds. SubstitUlillg (7.'2.H) inlo (7. I.:! I), we lind Ihal
/I; - /II
'"
= LpJ(li+'+! - rl.
(7.:!.9)
j="
p;
The dilfcrcncc between and III is jllst a dis('olllllcd SIlIlI or Inlml' dl'meaned stuck relllnlS. Irwe now lake expcctalions ;llld usc the definilion givcn in (7. I.':.!:!) ,\II(I (7,1.2:1) oflhe price component/I,,, we lind Ihal (7 .':.!.I 0)
I.
1'11'.11'1I1-\it/1I1'
Udal ilJlIJ
/{I'C alllh al/'" call he illlc rpn' lrcl as tllal CO 1II (>011('11 I oflh (' sloc k priCI' wlli rll is asso cialt 'd with challl!;illl!; ('XP ('cl;u iolis of fulu re sloc k tTlu rns. Thu s Ihe (olld iliol lal ('XI)('CI;lIioli of /': -/" ItIl'aSllreS Ihe t'ffcCI of chal lgill g cxp( 'el('( 1 Sioc k reI urliS Oil lilt' ("1mI'll! SlOck prin '. In I he AR ( I) ('xal llple d('v ('lop ed carl icr, Iht' cond iliol lal I'XI) (Tla lioll of/, : - /I, isjus l x,f(1 -pcp ) froll l (7.1. !!!l). Ifl'x p"(" led sior k n'ltU ns arl' COll Slanl I hrou gh lime , IllI'n Ihl' righ t-ha nd side of (7.~.1O) is /('ro . Tllc rOll slall t-l'x p('C !cd- r('tu nl hypo thl's is imp lies that p; -/" is a fi,,'I' cast ('fro r IlIlrO I"lTl aled with illfo rma tioll kliOWIl attil lll' I. Eqlli val(, lItly , it ililp lil's that thl' stor k prin~ is a ratio llal I'x(l l'cta tioll of the pcrf ",·t-l i,n's il!;h t stoc k pri .. e: (7.~.11
)
Ilow Clli Ihl'S(' iell·a.s hc used 10 !t'SI 111(' hypo lhes is Ihal ('xp ccle d stoc k relm lls are COIISI;iIlI? For silllp lieil) , or I'xp ositi on, w(' heg-ill hy mak ing two llllr( 'alis lic assll mjll iolls : !irsl , Ih:11 IOI!; sloc k pric rs and divi dend s rollo \\' Sl;Ilioll ar), sloc llasl ic pron 'ss,'s , so thai Illey have well -def incd lirsl alld seco lld 1II01lH'lI1S; alld S"COIIII, 111:11 IOI!; divi dl'lI ds are ohse rvah le illio Ihe infil lill' f\llm e, so Ihal IIII' per( i'cl- f(,re sigli l pric c is ohse rvah le 10 IIII' t'con Olll( 'Ilici all. lido \\' \\"1' dis"lI.s~ 110\\ ' II"'SI' assll lllpl iollS arc re\a xed,
g
()rlh"g/lllfllil~ fllld \'flrif lllf'l' -!l"/l ifl/l 'li'lll
E'I" aliol l (7.'2 .11) illip lil's Ihat /,; - I'I is "rlh" K{// w/to infim llali oll vari able s kllow n al lillie I. :\11 mlho ,l\ot lalil ), Il'sl of (7.2.11) rt'l!; ressc s /': - I), 0111 ', infol "lll:t lioll vari ab"' s alld (('SiS for /t'I'( , cod lkie llts. If th,' infil l'lna lio, I v:tri ahlrs ind\ ll'" Ihe Sloc k prilT I'I itsdf , Illis is eq\l ivak llllo a rt'gr ('ssi on of /': onlo /', and olli n \';lri ;lhlc s, II'lIl'rl' III,' hypo llics is 10 be Icsll 'd is 1101,' Ihal /It has a IIl1il cOl'f ficic lII alld lilt, ollll 'r varia bll's have zero cocf ficie nts. The se rt.'l!;r('ssiolls an' "aria llis of Ihe IOIlI!;-II01il.oll n'lu rn regr essio lls Ilisl 'usse d ill Ille pr('v ious sl'fli oll. E'I" alio ll (7.~.!l) show s Ihal g -/1, isjus l a disc oull it'd SIIIII of ftllll l'e 1I1'IIII'alll'd sioc k relUf'IIS, so all orlh ogon alilY I('st of (7.2, II) is a retu rn rl'g-r essio ll wilb all infil lill' Irori llll\, whe re 1\101 '1' lIisla lll relll rns arc I!;co lll('t rical ly dowlI\\'('il!;b ll'd.l~ IlIsl ead ofll'Slill1!; ollll ol!;o llalil Y din' nlr, IIIl1ch oflh e lill'r alllr t· tesls the illlp lical iollS or ol'lh o,l\o llali ty for the ,'ola tilil) , of sloc k pric es. The JIlost f;1I1101lS Slich illlp lic;l lioll . lit-ri ved Ill' I ,I'Ro)' alld POrl ('r (J9H I) alld Shi lln (1!IH I). is Ihl' 1'1111(1/111' illl''i "l/lil \' for IIH' sloc k pric l': (7.'2.I~)
I'.!Thc dOh'II\\'C'igllllllg .dlll\ \\ rill' f('! '1:tli,l ic' ill IIIl' rt'gn. 'ssioll 10 h(' pO!rlilin', WI"'lt ',I" ~hll\\T" ill St'rti oll 7.'.!..11I'.11 \\'(" lilt' N'.! .\',lIi'lil ill ;tlllll l\\'c'i glllt'd lillitt ·-hur i/otl 1('111111 n·gl(· . fOIlH 'lg('\ 10 ,,',0 ,I, lilt, . ~i()11 holil oll illfl(' ;l\('\, 1>1111;1111 alld II.dl (19H~ ••
(:haplt'r I I)
1,,1\(' IlIli It·gn·~,ioll.'
(I!JH~I), SCUll
(If
(IllS .\01 f.
(p.H:,),
.uHI
Shiller
7.2. l'rl'.lt'IIl-\'lliul' Reilliiolls (/lui US Siork l"rire lMulVior
277
The equality in (7.2.12) holds because undcr the null hypothesis (7.2.11) II; - 1'1 must he uncorrelated with PI so no covariance term appears in the variance 0(' I';; the variance inequality follows directly. Equation (7.2.12) can also he understood by noting that an optimal forecast cannot be more variable than the quantity it is forecasting. With constant expected returns the stock price forecasts only the present value of future dividends, so it cannot be more variable than the realized present value of future dividends. TesL~ of this and related propositions are known as variance-bounds lests. A~ Dliriallf and Phillips (1988) point out, variance-bounds tests can be restated as orthogonality tests. To see this, consider a regressiQn of P, on 1'7 - 1'1. This is the reverse of the regression considered above, but it too should have a zero coefficient under the null hypothesis. The reverse regression coefficient is always () == Cov[P; - PI, ptl/Var[p; - pd. It is straightforward to show that Var[p:J - Var[pd Var[p; -
=
PI]
1+28,
(7.2.13)
so the variance inequality (7.2.12) will be satisfied whenever the reverse regression coefficient () > -1/2. This is a weaker restriction than the orthogonality condition () = 0, so the orthogonality test clearly ha~ power in SOIlW situations where the variance-hounds test has none. The justification for using a variance'-hounds test is not increased power; rather it is that a variance-hounds test helps one to descrihe the way in which the null hypothesis fails. lIllil Hool.1
Our analysis so far has assumed that the population variances of log prices and dividcnds exist. This will not bc the case iflog dividends follow a unitroot process; then, as Kleidon (l98!» points out, the sample variances of prices and dividends can be very misleading. Marsh and Merton (1986) provide a particuhlrly neat example. Suppose that expected stock returns an~ constant, so the null hypothesis is true. Suppose also that a firm's managers lise its stock price as an indicator ofMpermanent earnings," selling the firlll's dividcnd equal to a conslant fraction of its stock price last period. In log form, we have (7.2.14) where there is a unique constarlt J that satisfies the null hypothesis (7.2.11). It can Iw shown that both log dividends and log prices follow unit-root processes in this example. Suhstituting (7.2.14) into (7.2.8), we find that the perfert-foresight stock price is related to the actual stock price by 00
p;
== (I -
p)
L )=11
p}Pi+}'
(7.2.15)
~78
7. Presl'1Il- lit/IIII' [{I'llliions
I
"Vhis is just a smoothed version of the actual stock price II" so its variance ([rendS on the variance and aUlOcorrclatioJls of II,. Since autocorrclatiolls c n never be greater than one, g must have a lower variance than flf. The i Ilportance of this result is 1I0tthat it applies to population variances (which a e not well defined in this exalllple because both log prices and log divi ends have unit rooL~), but that it applies to salllple variances in every s;lmple. Thus the variance inequality (7.2.12) will always he violated in the rSh_Merlon example. This unit-root problem is important, but it is also easy to circulllvent. , e variable P; - PI is always stationary provided that stock retuflls arc s tionary, so any test that p; - fll is orthogonal to stationary variables will be wrll-behaved. The problems pointed out by Kleidon (\986) and Marsh and Merton (1986) arise when /'1' - 1'1 is regressed on the stock price 1'1, which has a lunit root. These problems call be avoided by using unit-root I"(~gression tlleory or by choosing a stationary regressor, such as the log divid("nd-pri~e r'llio. SOllie lJlher ways to Ikal with the unit-root problelll arc explored in Problem 7.5. t3
±
Finile-Sample Consideraliuns
So far we have treated the perfect-foresight stock price as if it were .111 observable variable. But as defined in (7.2.8), the perfect-foresight price is unobservable in a finite sample because it is a discounted SUIll of dividends out to the infinite future. The defInition of g implies that T-I-·I
,,; = (1 - p)
L
pj(dH1 +)
+k-
r)
+ pT-I-1 p~.
0.2.16)
)=0
Given data up through tillle T the !irst term 011 the righ t-hand side of (7.2.16) is observable but the second term is not. Following Shiller (1981), olle stalldard response to this dillicu\ty is to replace the unobservable I); hy all ohsCl"vahle proxy 11;:r that IIses ollly illsample information: "/'-1-1 fl;'T
==
(l-fJ)
L
fJ)(dH1'I+h-r)+pr-I-I/,r.
(7.~.17)
1=0
Ilcre the terminal value of the artual stock price, h·, is lIsed ill place of the terminal vallie of tll(· perfect-foresight stork price, fr. Several points arc
'1
"Umlauf and flail (I !JH!/) ;lpply IIl1it·,.oot Il'~,.,·ssio\l Ihl·OI)'. while (;,lIlIpl)("\I ,lIul Shill .. r (I !lRRa,h) replace the 10K Mock price wilh Ihl' lOll \ the ap"road. (li,clIssed
0"
7,2,
}Jrl',II'Il I- Valli I' [(r/a liolls
(I1ul US Stadl I'lif/ ' UI'itt /pilll'
won h noti ng abol lt the vari able II;,T' First, ircxp('ct('(1 retll rns arc COllstanl , (7,~,11) cOll tillll es to holel whe ll /';,T is suhs titut ed for II;, Thu s lests of Ihe cons tant- expe ctcc l-ret lll'll Illod el call use gr' SCCOlld, a ralio llal hllh hle in lhe stoc k pric e will affec t both III aud gT' Thll s tests usin g /I;."/' illcl ude buhbles in the uull rath er thall the aher naliv (' P;,I - II, can he writ len as a disc oun ted hypoth(,sis, Thir d, th(' difk relH 'c SllIll of dl'll H'an cd stoc k retu rns, with tire slim terll lina ting at the end or tire sam ple peri od '[' rath er Ihan at sOllie fixed hori zon from the pres cnt elate I, II; T - /" are just long -hor il.ol l relll ril regr Thu s orlll ogO llali ty tests usin g essio ns, wlln e rutul '(' rclIIrllS arc geol lletr icall y disc oull led and the hori zoll is the clld of the sa III pie perio el. A~ one mig ht expe ct, the asym ptot ic tireo ry for stati stica l illfCrellce in orth ogon ality ami varia nce- houl l, to he a relia ble guid e for stati stica l infc n'lll T. Flavin (E)H:~) gives a part icul arly dea r inlll ition fi)r why this mig ht he a proh lclll in the cont ext of vari ance -hou nds (('sts, SIll' poin ts out that whe ncl'l 'I' a sam ple vari, ulce anJlI lHI a sam ple ml';UI is used to estim ate a popu latio ll vari ance , ther e is SOlll t' dow llwa rd bias c;1I1.~('d hy thl' LId that tht' trill' Illea n of tilt' proc ess is unkn own . WIll 'lltlH ' proc ('ss is wllitc nois e, it is well -kno wn that this bias CUI be corr c('tn l hy divi ding th(' SUIll of squa res by 'f' - I inste ad of 'f'. Unf ortu nate ly, the dow nwa rd bias is Illon~ st'Vl'n' for snia lly corr l'lat ed proc esse s (intllitively, tir('ll' is a sllr; t!kr 11I1I 1ll>n of clfe( live obse rvat ions for thes e proc esse s). so tlris corr et'tio ll he((> lIIt's inad equ;l(l: in the pres ence of seria l corr elati oll. Now g,. is Illor e high ly seria lly corr elate d th;ul III> sinc e g./, chan ges ollly as dil'i l\end s drop out of the pres ent- valu c form ula and disc olln t farlO ls are upda led, while /11 is affe cted by nel\' info rma tion abou t divi dend s. Thu s the ratio of the salll ple vari ance of g, to the salll pk varia llcl' of /'1 is !IowllIv;rrt!-hiasl'd. and this call Cllis e the varia llce illcq ualit y in (7.~,12) to Ill' viol ated too orl!'11 in tillit e s.un pks. Frol ll Ihe cqUiV;rll'lllT orvarianC l'-I>0l111ds alld orth ogon alit) , tests , the salll(' probll'1ll arise s in a regr essio n cont ext.
7.2.]
Vt'r/or'\1I11111'f..,'lI'.llil'l'
tIIdh lldl
The nll'tl rods t\isr mse d in the prev iolls two s('(l ions llan ' tire C0ll1 11101 1 featlln' tllat they try to look dir('('(ly atlo llg-h oriw n prop (,rtil 's of the data . This 1:1 Lel{o )' aile! S(~'ig('r\\';t1c1 (I !t~)!l) naJily ;lIId \'OI .. i'IIICt '·/)otl llCb le.·'IIs.
lIS(' MOIlI (' (:.II"lo IlIelllO ci ... 10
~llId~· I h(' PO\\T I of oJ"lhugu.
7. "r",lt'IIl- 1/1111" {Mal/oIlS call /toad 10 slali~lical dillillllli('s ill lillill' samplt's. All alt('l'lIaliv,' approach is 10 OISSIIIIH' Ihal Iht' dVllalllics or Ih,' dOli a are wdl d,~sniht'd hy a silllpk lilllt'-S('riI'S III CII 11'1; 1,,"~-llOri/oll proPITli('s call th('11 ht' inlplllt'd frolll the short-nllllllodl'l 1'011111'1 Ihall ('slilllall'd dirt'ctly. III lhevariallct'-holillcls lileralm/', this is Iht' approach ofl.l'Ro\' allt!l'ort,'r (1\IHl). Tht'st' allthors 1l01t' Ihat ;t \,;lIiall«·-l!olllld, Il'sl dOl'S 1101 r('qllire ohs('J'vatiolls of the pnkclItll'('si~ht price I'; ilsl'!r; il 1111'1'('1\' n'(llIir('s all estilllale of IIII' \'ariallct' or II;, which call ht' ohlain('d 110111 a IIl1iv;lriall' limc-series I\lodd for tii\'idl'llIis. W('SI (1!lHHh) t\1'\r10PS;1 v;lIialll or Ihis plOet'dllrl'. 'Ii, S('(' how lilis appro;,ch call work, sllppose Ihal 0111' ohserv('s III(' (,(lIl1pl,'\(' \,,', tor of sl;\I(' \';Iri;Ih!,'s X, IIst'd hy mark"l partit'ipants, anti Ihal X, follows a \'Crtor alllOn'~r('ssi\'(' (\fAR) process. Any YAR model call he \\Tillt'll ill lirsl-ordl'l' lill'lll b\' augmt'lllillg IIll' slaic Vl'elol' with slIilahlc lags olthl' original \'ari;Ihll's, so wilholll loss of general it)' we writ!': (7.~.IH)
I Itorl' A is a lIIalrix 01 \'AJ~ codli.-i('lIls, alld f:1I1 is a \'('ctor or shocks 10 the \,AJt WI' han' droppl'd COllstallts 1111' silllpikity; Oil,' call thillk of' the slale ,'('('101' as illd"dil\~ d('II\(';II\("
(7.2.1 ~») This makl's il ('a~\' to Clklll;1l1' Ih(' IOllg-hori/'(1I\ for('clsts that l!t:tnlllinc pricl's in (7.1.~~) alld (7.1.~:\), or IiiI' r(,visions or 10llg-horizoll forecasls lit;1I d('ll'nllilll' rl'lmllS ill (7. I.:!:,) alld (7.1.~(i).
I,'dor '\/1101'1'.1,'11'\\;0111 fllIIll'r;(/' \'o/alilih' As a lirst exalllph-. SIIPPOS(' tital litl' stal(' v('('\or incllldes Ih .. stock pricl' I', as ils lirsl dl'llll'lIt alld litl' di\'idl'lId ii, ;IS its second dl'lI\('lIt, ",hilt: the r('maining ..t('IIIl·lIts an' otlll'r .. ..t ('\'a II I forl'castillg variahl('s. WI' clt'linl' \·l,(·tors d' = II (I (I ... (I I alld e2' = 10 I 0 ... 0 I. Thl'sl' v(,ctors pick olltlh .. lirsl cil'1I1t'1I1 (/") ;lIl(lthl' SI,(,(lIl(l dl'llIt'lIt (iI,) fmlllth(' statl' V('('tor X" llsillg Ih('sl' dl'lilliliolls alld (''I"aliolls (7.1.~~). (7.1.~:\), alld (7.:!.I!l). Ih(' dividl'lId COIIlPOIH'III .,1'1111' Siock p .. i.-(' is '\.
(1
"./oIL/,/('2'A/ " x ,
(1
(7.~.~())
II/
'I h .. slock p .. i, (. ilsdl i, '" -" (., 'XI, ,0 Ih(' ('XpI'CI('t!-n'tllnl nllllplllIl'II1 "I' IllI' ,ltu'k pi i(I' is II,.. ditlnl'lIn' 1)('1\\'('1'11 Ih(' two. If I'xp"('\I'd rl'tllrns aI'\'
7.2. 1',t'.lfllt-Va/III' RI'/atio715 al/l/ US Stork I'rirl' Ill'havior
constant, then III
P,/t. which el'
=
281
imposes the restriction
(1 - p)e2'A(I- pA)-1
(7.2.21)
on the VAR system. This can be tested using a nonlinear Wald test. 1fi So far we have assumed that the vector X, includes all the relevant variables observed by market participants. Fortunately this very strong assumption can be relaxed. Even if X, incluoes only a subset of the relevant information. under the constant-expected-retufll null hypothesis the stock price III should still equal the best VAR forecast of the discounted value of future dividends as given on the right-hand sioe of (7.2.21). Intuitively. when the null hypothesis is true the stock price perfectly reveals investors' information about the discounted value of future dividends. Another way to see the same point is to interpret the restriction (7.2.21) as enforcing the unforecastahility of multi period stock returns. If multi period returns are unforecastable given investors' information, they will also be unforecastable given any smaller set of information variables. and thus the VAR test of (7.2.21) is a valid lest of the null hypothesis. One can also show that (7.2.21) is a nonlinear transformation of the restrictions implied by the unforecastability of single-period stock returns. In the VAR system the single-period stock remrn is unforecastable if and only if (7.2.22) eI'(1 - pAl = (1 - p)e2'A. which is obtained from (7.2.21) by postmultiplying each side by (1- pAl, The economic meaning of this is that multiperiod returns are unforecastable if and only if one-period returns are unforecastable. However Wald test statistics are sensitive to nonlinear transformation~ of hypotheses; thus Wald tests of lhe VAR coefficient restrictions may behave dilTerently when the restrictions are stated in the infinite-horizon form (7.2.21) than when they are stated in the single-period form (7.2.22). An interesting question for future research is how alternative VAR test statistics behave in simple models of time-varying expected returns such as the AR( 1) example developed in this chapter. An important caveat is that when the constant-cxpected-return null hypothesis is false, the VAR estimate of jidl will in general depend on the information included in the VAR. Thus one should be cautious in interpretipg VAR estimates that reject the null hypothesis. As an example, consider an updated version of the VAR system used by Campbell and Shiller (1988a) which includes the log oividend-price ratio and the real log dividend growth rat('. These variahles are lIsed in place of the real log price and log dividend '''Fur
82
7. PreJI'IZI- ValliI' llf/aliollJ
It!
-
('
C"! .
ro
,0
, I '
...
'0 0
...
9 ()l)
9 ~
", oq
11160
1880
Figure 7.1.
HJOO
1920
1940
1960
191:10
',!UOO
Log f/.eat Stork Prite tlwi Dividend Snies, A lin ual US Vatll, 11172 10 1 YY4
in order to ensure that the VAR systcm is stationary. The systcm is estimated with four lags using annual data from 1871 to 1994. Figure 7.1 shows the real log stock price as a solid line and the real log dividend as a dashcd Iinc. Both series have been demeaned so that the sample mcan in the ~-igure is lCro. The two lines tend to Illove together, but the movements of the log stock price are larger than thosc of the log dividend; thus the price-dividend ratio is procyclical and the dividend-pricc ratio is countercyclical. Figure 7.2 again shows the dcmeancd real log stock price as a solid line, but now the dashed Iinc is the demeaned VAR estimate of I'd/> the present v
I \
1"ITh('~(' !\t·rit.·~ hav(' 1101 h('('u dC.'IIH."III('d; 111(' h'vt'l of tht' (\ivi
(.,~." fro", I
tht'
1i~lIrc hy c'pollt'nll"ti,,!: th"
plot(('<1 s"lid lint'. ,
7.2.
/'rl'.lfll/- VII{lIf Uf{lI/ioiIS IIwl US S/mil {'ril/' {Muwior
.,. c c
~!
/'
:: L _____~_.....i.._~ , I Ht;O
I HHO
I !l00
_
__'_~~_'___ _ _ ~___'. _ _
I !J:!O
I!I4O
I!)(ill
I ~lHll
:!(){)()
Figure 7.2. l.oJ.: /Inti SloCR [',ire and blil1lalrd [)hll'dl'lul COIII/wllnll, It 1111 Ii III (IS [)ala, 187() III f'.!'N
or fewer, FII llloves closely with d,. A~ one ilJcrea~es the lag length towards ten lag~, 11,({ becomes much smoother and more like a Irelld lille. The same thillg happens if one adds to the VAR systelll a ratio Dr pritc to a IO-year or 30-year IllOving average of earnings, as suggested hy Call1phell and Shiller ( 1088h). There appears to be sotlle long-run llIean-revCI'sion in dividend growth which is captured by these expanded VAR syslellis. In cOllclusion, the VAR approach strongly suggests that the stock market is too volatile to be consistent with the view that stock J>ri((~s are optimal forecast~ of future dividends discoullted at a cOllstallt rate. Some VAR systems suggest lhat the optimal dividend forecast is close to the ('lIITCllt dividclld, others that the optimal dividend forecast is even sllluuther than the currelit dividelld; lIeither type of system Gill accoullt «II' the tendency of stock prices to 1IIove lIlore than one-ior-()IW with dividends.' ~ Strictly speaking, hol\'('\'('I', onc(' the null hypothesis Ii, == lid' i., r('jected, 'UIY int('l'pretatiolJ 1"1l.,rskr ;IIHllk I.ollg (J!I~U) po,,11 0'11 11t"1 s,urk prir(' h('lI.'l'ior mllJd 1)(' nllioll.,li,,·d if tiH.'re wc:n: ;t ullil-root rOlllpOllclit in dividend growlh. htll Ihey do 1101 pt('~('111 any dil('(t ecol1ollH.'lric (~\'idt'lIn' lor SIKh iI COmpnll('IH.
l)onalthol1 ~lIld K.1l1I'ifl;l (I~J~Hi) argile thAI t" .t.'liL' of long-rull huure
a l1onlilu't\1' dividend lon'casting 1I10dt,) ddi"'('r~ 11101"(' \'oJ.uil('
diviciellds. Thi."i
r('ltIaill'i all .Irlivt' re~\('ar('h
alt',t.
'01
7,
':IH
PI"l'sml-l'allll'llI'lalio/ls
'::'~
<:'.
I
-t:
<:'.
I
~ 0'
1
00
oi I
0
<'i I
::'-!
'"I .,. <'i
1
to
<'i I
7J
or:I
_---'_~
IHliO
f'igrm' 7,3,
IHHO
_ _ ---l _ _ _ _ _ . ........I-_._~_-'-
1!IOO
1'1:.'0
____
1!140
-'-_~
1!)(i0
__
'----_~_-'
1!IHO
:.'000
1.11" lJil'itll'l/lI·I>,ill' /{lIlill 11//11 f.,lill/lIlP11 I>il/itlnul I:OIII/Wllnll, /11/11/111/ (IS
tlltllt, IH7fJIII IIJ<J.J
of Ihe hehavior of i II lilt' VAoR.
/1<11
is (,(lIldilional on Ih(' informalion variahit's inclllded
Ii-dor" 1I100i'gll'\\illll.\ (/1111 [{('Ium \jt/lllilil.V t\ COlllllHln rrilirisnl orvol~llili[)' [('S[s, which applies (~qllally [0 VAR systems incllldin)!; prin's alld dividl'lIds. is [h
~lodi~~li.llli ~ldl(,1 ,11"fllt'lli ... " .... lh.1I Jillll-\,tltU- lII:lxillli/alioli I~' tn'III;I~(·I~ dOl" 1If11 oj dind,'wl poli( \', l.dllll.1I11I (I ~'!J I) "'('.'\ Ihh 10 ;lrRllC" Iha. Ilu' ~llIdl."li(" (h·~t·1 ilJllIg dl\ltlt'llth I.' IIl1hlt"l\" Itl IJC' ,1.lh)('.
1"TIIe'
nUl""lr~tili 1111' 101111 pIUt"(,~~
7.2.
}',-,sl'1Il- Value
IlellltiollS
alld
US Stork Price Behllvior
Recall that the revision in expectations of all future stock returns, '1,.1+10 is defined as
This becomes 00
1), .• +1
(7.2.24)
el'LpiAi€,+1 • j=1
From (7.1.2G) the revision in expectatiolls of future dividends, be treated as a residual:
IJd.l+l'
can
A~
a concrete example, consider an updated version of the system estimated by Camphell (1991) in which the state vector has three elements; the real stock return (r,), lhe log dividend-price ratio (x-u), and the level of the stochastically detrended short-term interest ratc (X31)' Using monthly data over Ihe period 1952:1 to 1994:12, the estimated first-order VAR for these variahles, with asymptotic standard errors in parentheses, is 0.055 (0.053)
0.655 (0.230)
-0.520 (0.166)
-0.038 (0.001)
0.999 (0.003)
-0.000 (0.002)
-0.032 (0.011)
-0.040 (0.046)
0.707 (0.050)
Tit I
( ) X~,I+l
=
X'U+I
+
("," ) f~.l+l
.
() rl
"21
»:II
(7.2.25)
f3.1+1
The matrix in (7.2.25) is a numerical example of the VAR coefficient matrix A. The R~ slatislics for the three regressions summarized in (7.2.2!» are 0.040, 0.99fl, and 0.537, respectively, indicating a modest degree of forecastahility for monthly stock retllrns.~o '!
l<>
,
7.
l'''I'~{'IlI- V(/illI' IMlIliolls
We call suustitutc thc cstimatcd A matrix into thc furllluia (7.:!.:!4) and usc thc estimated variancc-covarian("(' matrix of thc crror V(,(·tor (" I to calculate thc sample varianccs alld nlVari,lI\(c of thc cxpccted-rctllrn ,II\(I dividend componenL~ uf stod, retllrns. The estimatcd expccted-retllrn compunent has a sample variancc cgualto 0.75 times thc variance or rcalized returns, while the estimatcd dividcnd componcnt has a sample variance oniy 0,12 timcs thc variance uf reali/.cd rctlll'lls. Thc rcmainill~ v'lIi.llln: or realized returns (O.l:~ uf thc total) is allrihutcd to covariallcc l)('tIV(,(,1I thc cxpected-return and dividcnd cump()nenL~.~1 The reasun for this rcsult is that thc log dividcnd-price ratio f()\'{:casL~ stuck returns, ,md it is itselra highly persistcnt proccss. Tbus rcvisions in thc log dividend-price ratio arc associated with persistcnt ch'lIlgcs in cxpectcd future returns, and this can justify large changcs in stock priccs. Thc estimatcd VAR process is sumcwhat morc complicated than thc simplc AR( I) example developed earlier in this chaptcr, for it includcs two forccasting variables, each of which is c1use to a univariate AR( I). IlowcvCl' the main effect of the interest-rate variablc is to increasc thc forecastability or oncperiud stock returns; il has a rather mudcst effect 011 the lung-run behaviur of the system, which is duminated by the persistent muvcmcnt.~ of thc log dividend-price ratio, Thus the long-run properties of the VAR system are similar to those of the AR( \) examplc. From this and our prcvious analysis, une wOlild expect that VAR systems like (7,2,2~) could account for thc pattern oflung-horizon regressiun results, and this indeed seems to he the case as shown by Campbell (1991), 1-\mlrick (1992), and Kandel and Stambaugh (1989), Of coursc, VAR systcms impose morc structure on the data; but Hodrick (1992) presents sUllie Montc Carlo cvidence that when thc strltC\re is corrcct, thc flllite-sample behavior ofVAR systcms is corrcspondingly ellcr than t~lat of long-hori LOn rcgrcssiuns with a largc horil.On rclative to t,lC sam pic SIZC,
~
I \
7.3 Conclusion
I I
ihc rcscarch descrihcd in this chaptcr has helpcd to transf(JI'IIl th{' way IiIjandal cconomists view assct llIarkct.~. It lIscd lU bc thoughl that cxpcctcd arsct rcturns werc approximately constant alld that movcments in prices cbuld hc attrihuted to ncws ahout futurc cash )laymcnt.~ to investors. Today tllc importance of tim(,-variation ill exp(,cted rcllll'llS is widd)' re(()gnizcd. 1 11 o\'cr a IOllger period I!Wi 10
1!I/lH, till' Iwo \,a .. i.. "Cl's ;111<1111" covariallre lIa",' rollgilly
('qllal ~hare.s of the overall \'driif,n«~ of Ic.'alil.l'(\ st(Kk H..'tUIIIS. A"'Yluptotic ,lIitandard ern,rs tor the \'~1rianre II~('
decompositiun ( ... " bt., r.,lrutncc.l \lsil\~ tIll' "Idt." 111cthud ex.pla'nt.·d in St:r\ion AA uf Appendix. A", in th,.. pl'in· . .divid('ud VAl{ di~{·'I.~s('d aho\,(', till' dC.'fOllll'osilioli is (ollditiollill
UI~
Ih(' information variahles illdutied in Ihc: VAR ~ysl(·II1.
~H7
"/O/J/t'Il/.1
and this h'ls broad illiplicaliolls fi)r bOlh acadetllirs and in\'('slnH'nl professioll.tis. At th" 'lca(klllic· level. lhere is all {'xpl\l~ioll or res('Mdl Oil lhe d('tenni11.111" or lilll{'-varying experled H'tnrns. Econo\\\ists 'In' exploring a great variel), oride'IS. I'rom macroecollomic IIltHkls OrH"11 husincss cyril'S to lllOlT heterodox IIHHkls of investor psychology. WI' disclIss SOli\(' of thl'se ideas ill Chapter H, AI a more praclical level, dy"amic assel-
")t.
Problems-Chapter 7 7.1 III the laIC I !lHOs corporatiolls begall to repurchasl' sharcs Oil a large scal,'. In this prohklll YOll are asked to allal)'ze the eff(:cl of rqHlrchascs Oil the relation belweell slock prices alld dividends, COllsider a linn wilh fixed cash flow p('r pniod, X. Thl' lolal markel valu(' the firm (including the currenl <'.. of iL~ Gish flo\\' 10 rcpurchasc shares al CUIIHlividcllti pritTs. ant! lhcn IIses a fraclion (I - }.) of ils clsh !low 10 pay dividcnds on Iht' n'lllaining shall'S. The linn has Nt shares ollislanding al Ihe beginning 01 pniod I (In'lol e il repllrchascs shart's).
or
7.1,1 What arc lhe clim-dividelid 11I'iet, pn share anti dividl'nd per share al lillie I? 7.1.2 Derive a relaliun belween lhe dividend-price ralio, lhe growlh rale of dividends per share G. and the discollnl raIl' U. 7.1.3 Show lhal lhe price pCI' share eqllals lhe ('xl'l'('t('d preselll value or dividellds per share. disculIlIled at raIl' II. Explain intuitively why lhis fOl'lnllla is corrn:l, even lhollgh lhe linll is dcvoling ollir a porlioll of its Gl,h flo\\' 10 dividends.
7.2
COIIsitier a slOek WIIOS(' log divide lid tI,
101101\'., ;1
1.llItiolll walk wilh
drifl:
where f,+1 ~ N(O, (]~). 1\'''11111('111.11 tht' n'tl"in'd log 1,IIe oflt'llIl'II onlhe slock is a conslalll r.
I.
/'rt',lf'IIl- \ 'a/lit' IMfllilJlu
7.2.1 WI" lI~I' IIII' lIol;lIiOIl /.; (Ie,r "flilldamenlal vallll"") 10 dellote Ihe I"xpl"ell"d pn'sl"lIl \'allll" or dividl"nds, discollnledllsill~ Ihl" reqllired rale of relllm. Sholl' Ihal I'; is a ("ollslalll mllltiple or Ihe dividend /),. Wrile Ihe ralio 1';/ /), as a fllllnioll oflh" paramelers oflhl" modd. Show th;lt allotlln forlllllla for till' slOck price which ~il'es thl" sallie I"xpenl'd rail' of rellirn is
7.2.2
I', = I';+(I)~, wl!!'rl" A > () is a hlllClioll oflhl' olhl'r parallH'lI'rs of Ihl' lIIodel. SolI'I' Ii,,' A.
7.2.:-\
Ilisnlss lhl' Slrt'II!41hs alld weakllesses of Ihis 1II00Id 01';1 raliollal hllhhle as 1'I1IIIP;III't\ Wilh Ihl' Blallckml-W;lIsol1 hubhll', (7.1. \(i) ill Ihl" II"X\. NOle: This prohlel\l is
7.3
ha~I'd Oil
Flool alld Ohslldd (I ~191).
COllsidl'r a slock whose I'XIlt'CI"d rl'llIrn ohcys
/':,\/'111
= r
+ x,.
(i.I.2i)
ASSlIlII1' Ihal x, /t.liows all AR( I) pron'ss, ·\·t 1 I
70
o
<
cp < I.
(i.I.2!-;)
7.3.1
ASSlIlIl(' Ihal Ell I is III)('olTe/;lIl'd wilh IlI'WS aholll flllllrc dividclld payllll'lIls Oil III!' slo( k. llsillg Ihl' loglillcar approxilllale franll'work Ik\'c\opl'd ill SCl'liOiI 7.1.:\, dl'rin' Ihc aulocovariallcc fUllctiOIl or Ilall/.I'(l stock II'HIlIIS. ,\SSlIlIlI' Ihal Ii> < fl, where p is Ihl' parallll'll'r of lil'earilalioll ill Ihl' loglilll';lr frallH'work. Show Ihal Ihl'
7,:1.2
No", allo", Ell I 10 hI' IOlTciall'd with lIews aholll flllllrc dividl'lId pa)'n II' II Is. Show Ihal III(' alllo('ovariall("I'S ofslork r('llints call hc posilin' if Ell I and dil'idl'lId III'WS ha\'I' a slifficiclllly largt' posili\'t' covariance.
Suppose IIt;1I Ihl' 1014 Ilrtl(,('SS
7.4
I',
= I''''
"1'111111;\1111'111;11
(I ~ ") (tI, -
valllt'" or a slork,
lit-I)
lIto
olw\'s Ihl'
+ I't .. 1 +f,.
wlll'rl' II is a (,Ollst;IIII, I' i~ Ih .. pal';)III1'II'r or lillcari/.alioll ddillCc\ ill Seclioll 7.1.:~, tI, is Ihl'log di\'idl'lltl olllht, ;Isst'l. ;lI1d f t is a whitt'uoist' l'ITor Il'nll.
7.4,1
Sho", liIal ir IiiI' pricl' of IIll' Slock l''Iuals ils flllltlallll'III;t! \'allll', IIII'll Ihl' approxilllall' log Siolk 1I'llIm ddillcd ill S('CliOIl 7.1.:1 is IIlllem'casta!>"'.
289
/'mh{rlll.1
7.4.2 Now suppose that the Illanagers of the company pay di,,:idends acronling to the rule
rt,
= (+1.11/_1+(1-1.)(//_1+1)"
",line ( and A are constants (0 < A < I), and lit is a white noise error Iln('orreialed wilh ft. Managers partially a(!just dividends towards fundaIIlelllal value, where the speed of a(\justmelll is given by A. Marsh and Merton (I !IHli) have argued for the cmpirical relevance of Ihis dividend polie),. Show that if the price of the stock e'luals its fundamental value, the log cli\'i(kncl-price ratio follows an AR ( I) process. What is the persislence of this process as a function of A allel p? 7.4.3 Now suppose Ihal the slock price docs not eqllal fundamental \'alllc,hlltrathersatisfies/lt = 1I1-y(dl-vtl,wherey > O. That is, price exc('eds fllndamental value whenever fllndamental value is high relative to dividends. Show that the approximate log stock return and the log dividend-price ratio satisfy the AR( I) model (7.1.27) and (7.1.28), where the optiJllal forecaster of the log stock relllrn, XI' is a positive multiple of the log dividend-price ratio. 7.4.4 Show that in this example innovations in stock returns are negatively correhtted with il~novati{)ns in XI' 7.5
Recall the deiinition of the perfect-foresight stock price:
V == 'L}::.o (II [< I
- (I)dl + I +/
+ k - r].
(7.2.8)
Thl' hypOllwsis that ('xpected returns arc constant implies that the actual slOck pricc III is a rational expectation of p;, given investors' information. Now consider forecasting dividends using a smaller information set J,. Deline p, == I~[II; I J,l. 7.5.1 Show that Var(jll) ::: Var(j~I}' Givc somc economic intuition for Ihis n'slllt.
Ptl
~ Var(g - M and that Var(p; - PI) > 7.5.2 Show that Var(IJ; Var(/It - I~I)' Give some economic intuition for these results. Discuss cirClllllstallces where these variance inc'l"alities can be more usefullhan ; Ihe illequality in part 7.:>.1.
I
7.5.3 Nowclcfine I-'tl == k+(I ~'+I+( l-p)d'+I-P" 1-'+1 isthereturn that wOllld prcvail Illlder the constant-expected-return model if dividends were' lim'Casl Ilsing Ihe informatioll set j,. Show that Var(r,+l} ~ Var(r,+l)' i (;iw sollie ecollomic illlllitioll for this result alld discuss circumstances' whert' it can he ilion' Ilserlll than the inl'<1"ality ill pan 7.5.1. . Note: This prohlelll is hased Oil Mallkiw, ROlller, alld Shapiro (19H5) and .. West (1!IHHh). '
8 Intertemporal Equilibrium Models
TIllS CHAPTER RElATES assct prices to the consumption and savings deci-
sions of illvestors. The slatic asset pricing models discussed in Chaplers :) alld Ij ignore cOllsllmption decisions. They treal asset prices as being determined by the portfolio choices of inveslors who have preferences defined over wealth one period in the future. Implicitly tht~se models assume that investors cOllsume all lheir wealth alter olle period, or at least that wealth uniquely determines cOlISulIlption so that preferences defined over consumption ,Ire equivalclll to preferences ddilled over weallh. This simplilicltion is ultimately unsalisractory. In the real world investors consider many periods in making their portfolio decisions, and in this intertemporal selling one lJIustmodel consllmption ,lI1d pon/()Iio choices sillluitaneonsly. lnterLel\lporal equilibrinm models of asset pricing haw tilt, potential to answer two qucstions that have heen left llnresolved in earlier chapters. First, what forces determine the riskless int
S. /l/lt'I"II'III/}()m/ fql/ili/llilllll ;\(1I11t·ll snilwd ill (:11011'1<"1 7~ \\". sll.dl Wo,(' illl(TIClllporal cqllililirilllll lIIodds 10 I'xplore Ihest' '1l1nlioll\. SI't'lioli H.I bl'gill\ bl' SI'llilig IIII' proposiliolllhallht'l'1' I'XiSlS a .IIIUIIl1.l/ir tlil(//lIl/l./il(/1I1" slIch lhal IIII' ('xl'l'l'Ied prodllci of allY assl't rl'tllrtl Wilh lhe stochastic dis(olllil ElI'tor I'qll.ils OIH'. This propositioll holds I'tTy gl'lIl'rall\, ill lIIodds thai rille 0111 ;lIhilragt' OPPOrlllllilil's ill fillallcial lIIarkl'ls. EqlliliiJrillllllllodl'ls wilh oplilllilillg illl'l'slOrs imply lighllillks IiI'IWI'('1I the slochaslic dis(,(lIl11l bClm ;\IId IllI' 1l1argillalulililies of inv('siors' COllSlllllPlioll. Thus h)' MlIIlyill).\ Ihl' sloc\laslic disCOlI1I1 bnor 011(' call rciall' assel prices \0 Ihl' Illldl'r\l'ill).\ prdl'rt'lIlTS of illV('stors, III Sl'l'Iioll Itl WI' show how thl' hehal'ior of assl't prices call h(' IIsl'd to ITach COllcillSiollS ahOll1 the hehal'ior of tlie stochastic discoulIl Elctor. III l'arlinlbr WI' desnilw llallSt'1I alld .Jagallll;tth'IIl's (1991) procedllre ror (·aklilatill).\ a 10l\'n hlllllld Oil Ihe volatility of the stochastic discollllt !;\rlor, gil'l'lI allY SI'I 01' assel relllrils. Usillg 11111).\-1'1111 allllllal data Oil US short-term illterl'st rail's alld stock retllrtls 01'1'1' Ihl' pniod IHH!l1O 1!l!l4, WI' I'slimale Ih(' stalldard d(,l'ialioll 01' the stochastic disCOlillt /;lctor to h(, :~()lYt, ptT year or 11101'1'. (:OIlSlIlIlptioll-haSl'd assl't pricillg Illlldds aggn').\ale inv('Slors into a sillgil' n'l'n'sclllalil'e agelll, wh .. is aSSIIIIII'd 10 dnivl' lltilily frolll thl' ag).\ITgatl' COIISllllllltiOIi 01' IIII' ('1'0110111),. III thl'se models the stochastic discOlllI1 Elclor is IIl1' illll'rll'lII/JlJII//IIIIIIPJI/I/! mIl' 1I/.lIIb.llillllioll-the discolllltl'd ratio of Illargillailltililin ill 111'0 SII(TCssil'l' pITiods-fill' th(' rl'(ln'Sl'llialiv(' agellt. '\'hl' 1';/III'I"I'I//1I1/illl/l-11 II' lirs(-I,n In ('( "ulil i( 'liS lill' oplilllal COllSllm pi iOIl allli porlli,lio rhoin's 01' lilt' l\'IIII'sl'lIlalin' agenl-GIIi be \Ised III lillk asset rLIIIrliS alld (OnSlllllplioll. St'clioll H.~ disnlSSl'S a COlllilIOllir IISl'd COIISlllllplioll-hascd model ill which Ih(' rt'pn's('lIlalil'l' agt'1I1 has lillll'-s('parahll' powt'r IHilil)'. III Ihis 11111111'1 a sill).\h' pOI 1'01 II lI'lt'r govt'rtlS hllih I'il" IIlII'T.lioll and IiiI' ,,1/1I/iri/)' III' ;1/Il'I'lr/ll/Jom/lII!J.llillll;IIII-lhl' \\'illill).!;IH'ss or lhe represenlative agellllll al\jllsl planlled (OnSlllllplioli g-rowlh ill r('spolise II) iliveslillelit opportllllities. III b('(, Ihe ('\;tsliciIY orilltl'lll'llll'oral slIhslilillioli is Ihe reciprocal of risk al'erSiOll, so ill Ihis IIl11dd risk-al'erse illl'('siors Illilst also he lIliWillilig 10 adjllsl Iheir COIISlllllPlioli ).\rowlh ral('s 10 (hallg('s ill illll'n'st rat('s. The Illodel \'xpl"ills Ihe risk pl'I'lllia Oil ;tSSt'lS h,· Iheir ("lIvari;\Ilct's wilh .IRRn'g.lle ("011SlIlIIPlioli ).\III\\,lh, IIIl1liiplil'll hI' Ihl' risk-aversion cl)eflici('lll 1'01' lhe rt'pn'selilalil'l' illl'('sior. Usillg 101lg-nlll 01111111,11 I IS dala, \1'(' ('lIIphasize lillll' slyli/ed 1;I('(s. Firsl, Ihe al'l'rag(' exr('ss 1'('1111'11 011 liS slorks 01'\'1' short-II'rIli dl'ht-Ihl' (''1"ily 1'lI'millm-is ;1111111\ (i',V" pCI' year. Sl'colld, aggregalt' COIlSlllllplion is I't'I'Y SlIIoolII, Sll cOl'ari;lIln's wilh conslllnl"ilill growlh a 1'\' small. PlIlling Ihest' l;u'ls logelhn, Ihl' 1'0\\'1'1' Illilill' 1II0dei (all onl>'lil iiII' C'l"ily pn'mililn il'll\\' rocftiril'nl of II'lOIlil'l' risk ;tn'l'sioll is 1'1'1'1' larg(,. This is Ihe I'I/lli/)' /1I1'I1Iilllll
8.1.
'J'Ilf
S/o(/l1l.1/ic J)iJcount
Fac/or
of Meh,.a and I'rescoll (1985). Third, there are some predictable' movemellts ill shon-term real interest rates, but there is little evidence of accompallying predictable movements in consumption growth. This sugg-ests that the elasticity of intenemporal substitution is small, which in the power utility model again implies a large coefficient of relative risk aversioll. Finally, there arc predictable variations in excess returns on stocks over shorl-terlll debt which do not seem to be related to changing covariances of stock returns with consumption growth. These lead formal statistical tests to reject the power-utility model. III Sections 8.3 and 8.4 we explore some ways in which the basic model call he modified to fit these facts. In Section 8.3 we discuss the effects of market frictions such as transactions costs, limits on investors' ability to borrow or sell assets shoTt, exogenous variation in the asset demands o~ some investors, and income risks that investors are unable to insure. We argue that lllany plausible frictions make aggregate consumption an inadequate proxy for the consumption of stock market investors, and we discuss ways to get testable restrictions on asset prices even when consumption is not measured. We also discuss a generalization of power utility that breaks the tight link between risk aversion and the elasticity of intertemporal substitution. III Sectioll 8.4 we explore the possibility that investors have more complica ted prcfcrences than generalizcd powcr utility. For example, the utility fUll('tion of the representative agent lllay be nonseparable between consumption and some other good sllch as lcisure. We emphasize models in which utility is nonscparable over timc bccausc investors derive utility from the level of cOlISumption relative to a time-varying habit or subsistence level. Finally, we consider some unorthodox models that draw inspiration from experimental and psychological research.
/Jlm/!'
8.1 The Stochastic Discount Factor We begin our analysis of the stochastic discount factor in the simplest possible way, by considering the intertemporal choice problem of an investor who can trade freely in asset i and who maximizes the expectation of a timc-separahle utility function:
(R.1.1 )
where l) is the time discount factor, Cl + j is the invcstor's consumption in pcriod I + j, and U( CHj) is the period utility of consumption at I + j. One of til(' lirst-order conditions or l-:u[1'I" equations describing the investor's
,
294
8. intertem/Joml E'luiliiJriwlI Models
optimal consumption and portfolio plan is (H.I.2)
The left-hand side of (8.1.2) is the marginal utility cost of consuming one real dollar less at time I; the right-hand side is the expected marginal utility benefit from investing the dollar in asset i at time t, selling it at tillle t+ 1 for (1 + R;,I+ d dollars, and consuming the proceeds. The investor e(]llates marginal cost and marginal benefit, so (H.l.2) describes the optimum. If we divide both the left- and right-hand sides of (1\.1.2) hy (f'( e,l, we get : I E'[(l + U"I+I)M,+ 1], (8.1.3)
i I
,where Mt+l == lJV'(C,+ 1)/ V'(C, ). The variable M + 1 in (8.1.3) is known as the ' ; stochastic discount/aclor, or /JricillgkemeL In the present model it is equivalent ! to the discounted ratio of marginal utilities .5 V'( C,+d/ V'( Gil, which is called \ the interlem/JOmi marginal rate '1 miJ.llitution. Note that the intcnemporal i marginal rate of substitution, and hence the stochastic discoun t factor, arc always positive since marginal utilities are positive. Expectations in (8.1.3) are taken conditional on information avail,lhle \ at tillle I; however, by taking uncolHlitional expectations of the left- and right-hand sides of (8.1.3) and lagging one period to simplify notation, we obtain an unconditional version: I
== E[(1 + UI/1M,].
(H.I.1)
These relationships can be rearranged so that they explicitly determine Working with the unconditional form for conve~nience, we have E[(1 + /{,,)M/J = E[I + U,,JE[M,] + Cov[U,,, AtlL so
I,expected asset returns.
E[ I + J~tl
I
= - - (I ElM,]
- Cov[/{jto Mtl).
(H. 1.5)
If there is an asset whose unconditional covariance with the stochastic discount factor is zero-an "unconditional zero-beta" asset-then (H. I.!'» implies that this asset's expected gross retum E[ I + Rod = I/E[M,l. This can he substituted into (8.1.5) to ohtain an expression for the excess return Z,I on asset i over the zero-heta return: E[Z,,] == E[Uj/-J~JtJ = -EII+J~JtJCov[UI/,hJ,].
(H.I.t;)
This shows that an asset's expected return is greater, the smaller its covariance with the stochastic discount factor. The intuition behind this result is that an asset whose covariance with MH I is small tends to have low returns when the investor's marginal utility of consumption is high-that is, when collSuinption itself is low. Such an assel is risky in that it bils to deliver
8.1.
niP
Siorlifl.llir /Ji.I((}1l1l1 Farlo,.
295
wealth precisely when wealth is most v,tiuable to the illvestor. Thc invcstor thercfor c deJlJ;\nds a largc risk prcmium to hold it. Althoug h it is easiest to uJlderst and (H.I.:~) by refl-n'IHT to til(' intertell lpor,d choice problem of an invcstor , the equatio n Gill bl~ derived mcrely frolll the ahsence of arbitrag c, without assumin g that investor s m;!ximi7.c well-beh avcd utility function s" We show this in a discrete- state sl~lIing with states .1 == I ... S alld asseL~ i == I ... N. Deline '1, as the price of assct i and q as Ihe (N x I) vector of assct prices, and define X" as the payoff of ,Isset i in state .\ and X as an (Sx N) matrix giving the payolls of each asset ill each state. Provide d that all
=
L,
s
s
1== I>,(I+ !{,,) == L7T,M ,(I-!-!{, ,) = EI(I+!{ ,}M).
(H.1.7)
which is the static discrete -statc equivale nt of (H.1.3). M, is thc ratio of the state price of state .\ to the probahil ity of state .1; hence it is positivc becausc st,lle prices and probabi lities are hoth positive. If M. is small, Ihell stalc 5 is "cheap" ill the s('nse that illvestor s are unwillill g to pay a high price to rcceive wealth in state .1. An asset that tcntis to deliver wealth in cheap states has a rei urn that ('ovaries lIegative lY with M. Such all asset is itself cheap alld has a high n'llllll 011 average. This is the intuilion Ill!" (H. I.!;) within a disnctc- state framewo rk. III tlte dis
(1!1Hi). The lole of fOl1diliollill~ inf(u"m.uiol1 h~l~ ( l!lH7).
ht'('11
('xl'l(II(,(/ by 11.11)\"11 alld Rirh.ud
8. IlIln lt'lIIl w/(/ II:'l/ llililn il/l/l Mod d\
payo n;; ill all ollu 'r SI;II,·S. A 1'1111 111"1 illlp ona ni resu lt is 11I; llthl ' stale prin ' l'l'("(or is IIl1iqIH' ir alld ollly if assl' t Inar kl'ts are COll lpll'l l'. III this case AI is IIl1iqlle, hili with ill("clInplctl' IIw- kets ther l' lIIay exis l llIallY M's satis ryill g eqll atio ll (H.I .:\). This n'slI lt clIll lI' IIl1d ersto od hy cOll side rillg ;III ('con olllY with s"I'('f;,1 IlIilitl'-llIaxilllilill)!; illl'e stors . Till ' lirst -onl er cOll ditio ll (H.I.~) hold s for 1';1( h illl'e stol, so 1';lch illl" 'slor 's llIar gilla l IIlili ties CIII he IIsed 10 COllslrllct a stoc hasl ic disc ollll t bno r Ihal pric es Ihe asst' ls ill thl' l'con 01111'. Wilh COI II)lk ll' III;tr kels, thl' ilin' slol' S' lIIar gilla llllil itics al'l' plTf l'Cll y corr l'lal l'd so Ihn' ;111 I'il'ld Ihl' S;III1 .. , IIl1iqllc stoc hasl ic diSCOlltl i f"Cl or; wilh illl'O lllpk l,' IlIark,'IS i1tl'n ' ilia), 1)(' idios Ylic ralic vari atio n illll larg illal utili lies alld 111'111'1' IIll1hipl .. stoc h;lst ic disc ollll t !;Ict ors Ih;1l satis h' (H.I. :-\).
S.I. I IiI/II/iii/.\' 1I111/1Id.1· All\' lIIod el or exp, ·ct .. d assl' l rl't IIrns Illay he vi(,wl'd as a 1110<11'101 ' thl' stoc hastic disc oull t fal'lC li. IId' .n' WI' disc uss IIll't hods of tl'sli ng pani cllla )' lIIod l'ls, w,' ask ilion ' gnll 'rall y wha t assl' t n·tu .... data llIay hI' ;Ihle to Idll ls ahuu I Ihe IlI'h ;lI'io r or Ihl' sloc hasl ic discO lIlI1 bllo r. IlallSl'1I alld .Iag allll atha ll ( 19!1 I ) haw dl'l'e 1opl 'd ;llo\ \'('r hOli lld Oil thl' \'oIa tility orst ocha stic disc olln tl;lf lors Ihal cOll ld 1)(' cOllsisl"1I1 wilh a givl' li SI'I of asse l relll .... data . The y hl'gi ll wilh Ihe 1IIlI 'olIC lilio liall' qilai ioli (H.I A) alld rewr ile il ill 1','clOr Ic.rlll as (H.l.H) whl 'rl't is all N-I', 'clo" or Olil'S alld R, is Ihl' N-v l'cto r of tillll '-/ assl' t r('tll rllS, with Iypic al l'\I'IIU'lIt N ,. i I lOlliSI'll and .laga lllla thall aSSl lnll' that R, has a nOll sillg lllar varia ll':ecovari;1I1((' llIal rix n, ill oth .... word s, Ih;lt no asse t or com hina tioll ofas scl5 is tilic ollll iliol lally riskl ess. Thl' ll' ilia}, slill I'xis t all IIiK Ollll iliol lall.e ro-i> l'la assl' l wilh g"o~s IlIl'all r('tll rll I'qll al to IIII' reci proc al of thl' UIICOIHlitiollal III ('a II of t III' stoc hast i.. di\co lIllI Lteto r, hili I JallSl'n and .Ia!!;anna than assu lIlc Ihal if tlll'l'(, is SlIl'h all assl' l, its ic\el llily is not kllo wn 1/ 1i/7111 7. I knf l' thl'Y trl'a l Ihl' IIl1l'Ollllitiollal Illea ll or till' sioc hasl ic disc oull t bClO r as all 1111know lI 1';11';11111'11'1' M. For I'ach poss ibll' M. I lOlliSI'll and .lag ann alha ll r01'll 1 a I'alld idal" slO,h;l~ti( (li'c olIlI l 1;1((01' ;\I,'( M) as a lin!' ar cOll lhin atiol l of asSl't rl'llI .... s. Thl' l' shol l' 1";11 1111' " .... i;llln · of 1\1,'(M) plac l's a lowl 'r hOll lld on Iht, l'ari allcl ' of aliI' stoc hasl ic discOlllI1 bCl or that has IIl1'all jl,J alld salis lil's (H.I.H). llall sl'lI a II d.l ;lg;III1I;11 lIa II Ii rSI sllol l' how assl' l pric i n)!; tlll'o ry c\1'1 1'1'1 n ill I'S IIII' ('(Il'ffil'il'lIlS li:l{ ill
,11;( M) If AI,'("/) is 10 1,..;1 ,to(I I;ISl ic di,,, ,,tlli t 1;ll'l or il IIIl1st satisf~' (H.I .H), I
= Flit I H,lM ,'(M )I.
(X,I, 'J)
8.1. TIll' Stodw.rtic Discount Fartor
Expand ing the expecta tion of the product E[(/. + R,)M:(M )]. we have
" = = :=
ME[" + R,] ME[L
+ Cov[Rto M;(M)]
+ R,] + E[(R, -
ME(/. + R,]
+ E[(R, -
E[R,1l(M,·(M) -
M)]
E(R,J)( R, - ElR,J)'f3M] (8.1.10)
where n is the uncond itional variance-("ovariance matrix of asset returns. It follows then that
(8.I.IQ and the variance of the implied stochast ic discoun t factor is . Var[ M;(M)]
= f3~nrJM = (/. - ME[/. + R,»'n- I (/. -
ME[/. + R,]).
(8. 1.I 2). I
The right-ha nd side of (8.1.12) is a lower bound on the volatility of an~ stochast ic discoun t factor with mean M. To see this, note that any othe1 M,(M) satisfying (8.1.8) must have the propert y .
Since M,"(M) is just a linear combin ation of asset returns, it follows that. Cov[Mt (M), M,(M) - M,"(M)1 == O. Thus Var[M,(M)J
=
Var[M; (M)]+V ar[M,(M )-M,·{M )]
+
Cov[M; (M). M,(M) - M;(M)]
=
Var[M; (M)] + Var[M,(M) - M,'(M)]
2:
Var[M, '(M)]'
(8.1.14)
In fact, we can go beyond this inequal ity to show that Var[ M,(M) J
=
Var[M;( M)1 (Corr[M ,(M). M;(M)] )
'I'
(8.1.15)
so a stochast ic discoun t factor can only have a variance close to the lower hound ifit is highly correlat ed with the combin ation of asset returns M;{M).
2!J8
8. IlIil'flt'llljlllmll:'qllililllilllll /IImlt'll'
~
1/1' /Jfl/rlII/WII! Portfolio e can restate these results in a more /;uniliar way by introdurillf( the idea
If a brl/chmnr/t /'orl/olio. We first allgment the vector of risky asscts with all
~trtilicialuncoll(litionally
riskless asset whose return is 1/ M - I. Recall that lve have proceeded lInder the assumption that 110 unconditiollally riskless ~sset exists; but if it were to cxist, its retllrn wOllld have to he 1/ At - I. We lhen (kline the bellchmark portfulio re!llm as I (H. I. I!i)
\ \ 1
1\'1 is straightlill'wanl to check that this retllrn can be obtained by forlllinf( a I ortfolio of the risky assets all() the artificial riskless asset, and that it satisfies t Ie condition (H.I.H) ~ll1owillg propertics:
I
Oil
retllrns. I'roh1c1ll H.l is to prove that Uhf has thc
(PI) lie, is me,lII-variance efficiellt. That is, smaller variance and the salllc mean.
110
other ponfoliu has
(P2) Any stochastic discount /;lctor M,(M) has a gre
lH.1.I7) where /Jib == Cov[ Ri" R ,! / Varll~,,j. When an unconditional zero-heta " substitutl'c! into (H.I.17) to get a (Oll\'cutional asset exists, then it GlIl he beta-pricing equation, Two further propcrties ,liT useful (ill' a geollletric interpretatioll or the IlansclI:!aganllathan bounds. Consider Figure H.1. I'ancl (a) is the f;uniliar mean-standard deviation diagralll f
(~, I,
"fill' SIII("//(I,\I;I' IJi.\UJllIII Ff/I'I"I'
slop.. ' i... Sh;" 1'1' I.lli ..
1/ .II
n(ll) (a)
slope is Sh;\lpt· I."io
(iI)
Figure S,I.
(1/)
l)t'l,lio{;"'I-I\kall
M,'all·SlllIlllrmll h,.if//illll I hl/gli//I/ /111' .1.'.\1'/ lid /III": (") lJiag,atll 10,. Slud/fI.\(;('IJi,\a",,,t Flit 101"\
Im/,Ii,." SIIIIIII",11
disCOllllI fal'lOI'. III pallel (,,). the ((-;!sihle S(,I of risky ;lsscl I ('lIIIIIS is shown, Wc allgllll'II1 Ihis wilh a riskless gross 1'1'1111'11 I/M (III Ihe I'<'llical axis; the lIlillillllllll-\'ariall(,(' sel is Ihcli the lallgclIl lill(, 11'0111 I/;\i 10 Ihe feasihic set or risk)' assels. alld III<' refleclioll of til<' lallgelll ill Ihe verlical axis, PropellY (1'1) 1I11',II1S that the I>cllchlllark porlf()lio 1'1'1111'11 is ill tlie lIIi II illl II 111variallc(' sct, It plots Oil tlie lower I>rallcli 1)(,(,;IIIS(' its posilin' correlatioll with the stochastic discolillt /iH'toi' givcs it a 10\\,('1' 1111',111 gIO." 1'('1111'11 th;1I1 I/M,
Ii. 1111,.,-11'111/'0/111 I:'q II ilibrill III .1/lIdl'l5
VI'I' call 11011' sl;lle 1\\'0 1II0re properties: (1'-1) TIlt" ral in of'siallclarcl devialioll 10 ~ross lII('an for Ihe I)('ndllnark porlll,lio salislies
(H 1.1K)
1\II11he riglll-h;IIHI sid(: oflhis (''1"<1lion isjusilhe slope of lilt" l
all Ell
+ H", I + n",1
a[M,(M)J
<
(R.1.I9)
E(M,(M)]
PropI'rties (1'"1) and (1':» estahlish Ihal Ihe slm:haslic discOUIII !"arlor in palH'I (h) II111S1 lit' ahove Ihe poiul where ;\ ray frolll lhe origin, wilh slope e alld Ihe \'ector 01" ('X(,('SS !"t'lllrns as Z" the hask condilion (K.IA) becomes
or
() = EIZ,M,l. I'\()("t'('dillgas before, w('111I'\1I M;(i\/) liltk is -
lIS('1! --I
to illdicate that
n (- -
=
jj is ddilU'd
(fl,
I.~()
M + (Z,- EIZ,J)'jjM"' where lhl'
wilh excess retlll"lls. W(' lind Ihal
n-
riM = MEIZ,II. \\'1H'!"t· is til!' I'arialll'('-('o"arian('(' malrix ofeXt"('ss )'eIIlI"llS. It ji,llows Ih;11 III<' !OI\'!'I" hOllnd Oil Ih(' variallc!' oi" Ih(' stochastic discollnl LId. II" IS 11011' (fl.l.~ I)
8.1. The Slo,/ta.l/ir DiscouIII Farlor
If we have only a single excess retllrn l,. then this condition -
-2
2
to Var[M,·(M)] == M (E[l,]) /Var[l,]. or a[Mt(M» ::=
M
E[l,] a Ill]'
This is illustrated in Figure 8.2. which has the same structure as Figure 8.1, Now the restriction on the stochastic discount factor in panel (b) is that i~ should lie above a ray from the origin with the same slope as a ray from the origin through the single risky excess return in panel (a).
Impli((llions of Nonnegnlivily So far we have ignored the restriction that M, must be nonnegative. Hansen and Jagannathan (1991) show that this can be handled fairly straightforwardly when an unconditionally riskless asset exists. In this case the mean of M, is known. and the problem can be restated as finding coefficients a that define a random variable (8.1.23) where X+ _ max(X.O) is the nonnegative part of X. subject to the constraint (8.1.24) In the absence of the non negativity constraint. this yields the previous solution for the case where there is an unconditionally riskless asset. With the nonnegativity constraint. it is much harder to find a coefficient vector a that satisfies (8.1.24); Hansen andJagannathan (1991) discuss strategies for transforming the problem to make the solution easier. Once a coefficient vector is found, however. it is easy to show that M,"+ has minimum variance among all nonnegative random variables M, satisfying (8.1.8). To see this. consider any other M, and note that E[M,Mt+]
=
E[M,«L
+ R,)'a)+)
+ R,)M,) = a'E[(L + R,)Mt+) = >
a'E[(L
E[(M,"+)2].
(8.1.25)
fiut if I-:[M,M,·+] ::: E[(Mt+)2]. then E[M,2j ::: E[(M,·+)2] since. the correlation between these variables cannot be greater than one. The above analysis can be generalized to deal with the more realistic case in which there is no unconditionally riskless asset. by augmenting the return vector with a hypothetical riskless asset and varying the return on this asset. This introduces some technical complications which are discussed by I [ans('n and .!agannathan (199 I).
H. JlIll'rlf.lllpOml l~fJllilibl'illlll Mo{lels
1+7.
I.
",(1. ) (a)
a(M)
(h)
Figure 8.2. (a) MNlII·Slmultmi IJ{1liation IJillgr(lmjiJr(l Single F.xrm A.w'l Up/11m; (Ii) /m' /lliNi S/andmri IJroia/ioll·Mrtlll /)itlf..,'TtllIl./ilf S/tlr/UI.,/ir f)i.'OJlIll/ N,r/OIJ
A First I.ook atthr F.quit.1 l'rrllliulII l'u1.ZW The Hansenjaga/l/l,llhan approach call he IIsed to 1Illdersl,lIld Ihe well· ki\IWII equity premium pliZZIe or Mehra alld Prescott (19H:1) ,2 Mchra alld I'lesco\t ,ugue that Ihe average excess return Oil Ihe US slod, markel-the
I !
: ~C:odlral\" .1I1t1 11.111"'1\ (I\I\I~) "1'1"0.1< h 'h,' "'I"il), pn'mill'" plIuk f .... '" Ihis poilll of ,·i,')v. Koehl'oI.I"ol .. (19'11;) sur"'),, Ih,'I'"I\,'li"'I<,,"n' till Ihl' pilule,
"
.. . '
... . 0.91
O.HH
0.%
I.(H
1.00
1.01l
M Figure 8.3. l·i·//.IiM,. /legilill IIII' SIII,./'a.,li,. UNI III I 'JI).J
/)i.l,.I11/ III
I'flllllll III/lilil'" Ii.l' :\ II 111/01 /IS /Jllla,
equity Inrlllinm-is LOO hi~h to be easily explained by slandanl assel pricin~ models. They make this point in the context of a ti~llIly paramctrized cOllSumplion-based model, but it CIII he made more ~l'n(Tally \lsin~ the excess-return restriction (1).1.22). Over the period lHH!) to 1!)94,the annual excess simple return on the Standard and POOl'S stock index over cOlllmercial paper has a standard deviation of 18% and a Illean of 0%.3 Thus the slope of the rays from the origin in Figure 8.2 should he O.OG/O.IS 0.33, meaning that the sLanda~d deviation of the stochastic discoullt factor must be at least 3~% ifit has a mean of onc. A~ we shall see in the next section, the standard consulllption-based lIIodei with a risk-aversion codlicient in the conventional ran~e implies that the stochastic discoullt ('actol' has a lIIean IIcaI' one, hilt an annllal standard deviation IlIlICh less (han :U'Yr,. :\-nh' U'IIII"II (1)
instead of ;,
of this
Tn'
IOllg s;lIllplt·
six·mollih
hill
H'llInl
(,Ol1l1H('lri;d
P;'I"'I,
,oll('d
I)('(';,ms(' TU'''.'HIIY b,1I
p(·riod. Mt'hr.l
0\"('1" ill
d.lla ,11(" 1101
Jill),. i~ IIsed t'ady P;III (OIlIllH·,,·i.II,Mlw(" and
.I.IIII1;UY
.lIId
,I\';,il"hlt·
allCll'n'~c 011 (I!JH:,) .'plil C' loge-du'l
ill Ihe
'J'rl·a.,\lII)' bill r;I«· .... , wllt·lt·a., h{'ll' we lI.\t' (·OIllIlU·I( i.II,"'pcl" I alt'., 1111 ollgliOlll IIH' ~alllpl(' 1'(,1 iod for ('oll!'oi'!('IIC),. Tilt' dlC,ire ol"!
below gi\'e.,
simple
SOIlU' \;ll11pll' 1110111('111.'
r('lllrll~.
Ii,., log
fl.,.'\('1 n'UII"II.', hili
IIIl'
IIloIIU'III.'\
\lal('(1 111'1('
.11('
luI'
Fi~IIIt, ICI, ",lios(' /<mllat "lllo\\,s Il;tlls(,1I alld.lagallllatilan (19~11), ;tlso illllstraies tilt' c'"lIil\' plt'milll}) PIlII.!I'. The lIgllre shows the feasihle region lill lilt' slochaslic dist"OlIlIl /;Ino!" implied hy c'l"alion (RI.I~) and Ihe anIIl1al dala 011 n'al Siock ;11111 ("(Ilnnlt'l"cial pap('\" rt'llIrns OWl' lilt' I)('riod IWII
10 1\1~}.j. Thl' ligllfl' c1oc's 1101 IIS(' 111(' lIoIlIH'g'lIivily rest ric lion c1isctl~sed ill IIII' previolls sl'clioll. The glohal lIIillimlllll stalldard devialiot\ ("or Ihe SIOl"h;ISlic dis("lllllli h\ctor is ahollt 0.:\:\, corresponding 10 a IIll'an stochaslic discOIIIII fador or ahoclI 1l.!lH ;11111 an IIncondilional riskless relllrn or aholll !!'X,. As Ihl' lIlt'all 1II00'I'S awal' 1'1"0111 O.9H, tht' slalldard-dc'viatioll hound rapidly illnc'ast·s. Th .. dilil'r .. tlt"<' hetll"'(,11 Ill,' kasihk \l'gioll in Figure S.:\ ;lIul lilt, n'giotl ahol'(, a r;IY rrom Ibe origin wilb slopt' O.:t\ is callsed hy Iht' facl Ihal Figllre H.:I list'S hOlld alld siock rctllrns sc'paratdr ralher Ihan mcrd), thl' c'xn'ss rt'llIrn 011 siocks 01'('1' honds, The ligllrt, also shows lIIe;\lIs!;\lIII;\I'd del'ialioll poi"ls cOlTc'spolHling to variolls degrc'es of risk avcrsioll alltl a lix(,lllillle discOlIlI1 ral(' ill a COIISlllllptioJl-hasl'd n'pn'sc'lItative ageJlt asst'l pricillg Inocle! or Ihe tvpc' discllssed in Ihe lIext scctioll, The (irst poinl aho\'e th,' !lori/olll;11 axis 1I;ls n'!;lIin' risk aversioll one; sll('(essive points hal'l" risk al'tTsioll of Iwo, Ihree, alld so Oil, The points do 1I0t ellier Ihe ft'asihlc regioJl IInlil relalin' risk aVl'rsioll reaches a v"tlll' of ~!i, III illll'l"pn'lill~ Figllrt, H.:\ alld similar ligurt's, 0111' should "1't'P ill Illilld Ihal hoth IIII' \"(IJatilill' hOlllld for Ihl' slochaslic discollnt Elrlor aud Iltl' poi illS illlplil'd hI" particlliar ass!'1 pricillg lIlodels are t'slinlatc'd willr nr"r. SlalisticalllH'tlHllls arc a\';lil;lhlc 1(1 ICSI \\'hellll'r a paninrlar model salisli,'s Iht' volalililY 11Illllld (st'c' for "";;lInpll' I\llrllsi
or
R.2 Cons\lmption-R.L<;ed Asset Pricing with Power Utility III SI'CliOIl H.I 1\"1' showed hOIl" all (''I"ation rdating aSSl't n'llInlS 10 1/1t' slochaslic di~t 011111 bllor, (H.I.:\), cOlllt1 hl' dni\'('d frolll lite Iirsl-orclcr cOlldilion or a single ill\"('slor's illll'l"lt'IIIJloral COIlSlllllplioll alld portli,lio dlOicl' pmhll'lIl. This (''1"alion is Il'sl;Ill'l1 IInl' for cOII\'enil'lIft': (X.:!. I)
It is COlli ilion ill ("lIIpirical I ('s('arch 10 aSSlIllH' thai illdi\"icillals call he
;Ig-
grq~ated illiO a sillgll' r("pr(,Sl'lIlalivl' illveslor, so Ihal aggrq?;al(' COIlSIIlIlP-
lioll call hI' IIs,'d ill plac(' or Ihl' CIlIISIIIIIIJlioll of allY llOIrlirlll;lr indivicln;d. F.c(lwion (H.:.!.I) ,,'jlh "'III = ,~I"(r:'II)//I'((:,),whl'r(, (;, isaggregalt'wlISIIIIII"ion, is klum'lI ;IS tht' ((JIII/lltl/ilioll t:,II'M, or CCAPM,
X.2. CUllmmjJtiu7I-Bmed AlleE J'/ll/IIK with Pown- Utility In this section we examine the empirical implications of the CCAPM. We begin by assuming that there is a representative agent who maximizes a ' " ,.. time-separable power utility function, so that
U(Ct )
=
c,I-Y -I l-y
,
where y is the coefficient of relative risk aversion. As y approaches one, the utility fUIlClion in (8.2.2) approaches the log utility function U(Ct )
f
10g(Ct).
'
The power utility function has several important properties. First, it is scale-invariant: With constant return distributions, risk premia do not change over time as aggregate wealth and the scale of the economy if/crease. A related property is that if different investors in the economy hav~ the same power utility function and can freely trade all the risks they face, then even if they have different wealth levels they can be aggregated into a single representative investor with the same utility function as the individual investors. 4 This provides some justification for the use of aggregatf consumption, rather than individual consumption, in the CCAPM. ' A property of power utility that may be less desirable is that it rigidly links two important concepts. When utility has the power fonn the elm/icityof i7llrrlfllljlOrai SUblli/utio71 (the derivative of planned log consumption growth with respect to the. log interest rate), which we write as ",. is the reciprocal of tile coefficient of relative risk aversion y. Hall (1988) has argued that thi' linkage is inappropriate because the elasticity of inter temporal substitution concerns the willingness of an investor to move consumption between time periods-it is well-defined even if there is no uncertainty-whereas the coefficient of relative risk aversion concerns the willingness of an investor to move consumption between states of the world-it is weIl-
(8.2.3)
~(;"",,\I;III ;",d Shill"r (I!IH2) show lh.lllhis n·,ull\:CIll'I;lli/cs III a modd wilh nontrJ.d~d ;1,,,,'1, f ..SM·S
(1IIIiIlSlIl.,hle idiosyncratic risks) if consumplion and a.set prices follow diffusion proill ;1 rOlltinllOlls-time mudd.
8. Illlerlempoml Equillbril/JII MIHirLI
06
1
\ hich was Hrst derived hy (;1 osslllan and Shiller ( 19H I ), followill~ the closely I;elated continuous-time lIlodel or Breeden (1979). A typical ()1~eClive or empirical research is to estimate the coefficient orrelative risk aversion y (or its reciprocal Vt) and to test the restrictiollS imposed by (H.2.:~). It is easiest to do this if one assumes that asset returns and aggregate COIISllIlIption are jointly hOllloskedastic and lognormal. Although this itllplies (Ollstallt expected excess log returns. and lhus cannot fil the data. it is useful for building intuition and understanding methods that can be applied to more realistic models. Accordingly we make this assumption in Senion H.2.1 ;lIld relax it in Section 8.2.2. where we discuss the usc of HallScll's (19H2) Generalized Method of Moments (GMM) to test the power utility model without making distributiollal assumptions Oil returns and cOllsllmption. Gur discussion rollows closely the selllin,ll work of Hansell and Sin~lcton (1982. 1983). 8.2.1
J>Ol/J1'r
Ulilily ill fl 1.og71onllfll Model
When a random variable X is conditionally lognormally distributed. it has the convenient property (mentioned in Chapter 1) that. logE,[Xl == E,[log Xl
+ ~ Vartllog Xl,
(H.2.4)
where Var,[log Xl == E, [ (log X - E., [log XlfJ. If in addition X is conditionally homoskedastic. then Var, [101-( Xl = E[(log X - E,[lol-( X])~l = Var[log X - E,[log Xl). Thus with joinl condilional IOl-(llUrmality and !romoskedasticity of asset returns and consumption. we can take 10l-(s of (H.2.3). usc the notational convention that lowercase letters denote logs. and obtain (8.25)
Here th'e notation ax, denotes the IInconditional covariance orinnovations l'~ov[x'+I-E,[xt+tJ.Yt+I-E,[y,+dJ.anda; == a xx • Equation (8.2.5). which was nrst derived by Hanscn and Silll-(leLOn (:1983). has both time-series and cross-sectional implications. In Ihc tillle shies. the riskless real intncst rale obeys
I I
~
I I
~
rJ.t+1 == -lug8 - Y ; '
+ yE,[6'l+lj.
(H.2.G)
I
ljhe riskless real rate is linear in exp(,cted consllmptioll growth. with slope cbcflici~lIt equal to thc roefticicnt of relative risk avcrsion. The equation c~n be reversed to express expected conSulllptioll growth as a linear function ofthe riskless real interest rate, with slopc coefficient Vt = i/y; in hlct this r\'lation hetwecn cxpeclI'd conS\llllptioll ~rowth alld the interest rate is what dl'lilles thc elasticity of inlertemporal suhstitlltioll. I
'
8.2. CClII.lltIII/Jlicm-Ba,\fil A.\.lfll'ri(i,,~ wilh {'(Jwn Ulility The assllmption ofhollloskedasticity m;li... cs tlH'I0f!; risk premiulI\ 011 allY asset over thc riskless real rate constant, so expectcd rcal returns 011 other asscts arc also lillcar ill expected ("ollslImptioll glOwth willi slope codIicicllt y. We have a'.!. EI [l"j.1+ I -
If' .If I
I
-1 -':!
= ya".
(H.:!.7)
The variance term on the left-hand side or (H.:1.7) is a.l~'t1Sen's In~~
Eqllation (H.'2.7) shows that risk prcmia arc determillcd by thc codliciclIluf rclativc risk aversion timcs covariance with conslIInption growth, or course, we have alrcady presentcd evidencc against the implicatioll or this mockl that risk prcmia are constant. Nevertheless wc explore the lIIodei as a lIseful wa)' to develop inluitiun and understand ('('onollletric techniqlles used in mol':: general models. A SI'(olltll.ooh (// Ihr !~I!llily l'rnlliuJIl !'/lUU' Eqllation (8.:1.7) clarifies the argllment or Meltra and l'res(oll (198:» that the equity premiulII is too high to hc consistcnt with ()hserv{~d consumption behavior Ilidess investors arc extremely risk av('I'S('. Mehr.1 and I'rescoll's analy';s is complicated by the Ltd that tltey do lIot IIS(' observed stock. 1'''tums, but instead calcll!;\lc stock retllrns implied hy tIl(' ((Hllllerbnual) aS~lIlllptioll thaI slock dividends c(lllal consllmptioll. Problem H.~ carries out a loglinear version of this calculation. Onc can appreciate the cquity prClllillll1 plll.l.lc tllOl(' dircrtly hy examining- the momcnts of log real stock and cOll1m('l'('ial papcr rClllrJIS and co IISU III ptioll growth shown in Table H.I. The asset rcturns arc measured anllually over the period 188910 1994. Thc mean exccss log rClurn ofSlOCKs over COllllllcl'Cial paper is 4.2% with a standard devinslIlllplion f!;1()wth has :IThi . . illlplicilly asstlllU'S that utility i,1I M'IMI "h'(".H I {)'.'" lili!'o ICIIIII of 'OIl"iIlTIlPlioll alld olht"1' of tllilily. III S('rtiol1 HA we disnl!\.'io W'I),' ill wlli, Ii tllI'I .'\ .... IIIIIIHIOII ("all ht' ,d,iX(.'d.
~()"rn's
.,,,,1
H. /lIlnll'lII/lOml Flfllilibrilllll Mill"".!
'/i,M,' 8, I,
\';11
i;,"I,'
,11,,1//1'1//1
~Ic'all
iiI ((I1i1/l1I//'lil/li /[11111111t (/w/II.I.1t'1 /I'/Un/I. Sialld.lni d"vi.llioll
(:ml'"ialioll wilh n HlSIlillpl i, HI
(:",'"ri,lllrt: wilh fI.IISlllllptioll
~rowth
~ro\l'lh
(:UIIMIIIIIJlioll ~Io\\'lh
0,0 I 7'!.
Il.Il:I'!.~
I,OO(lO
Slo(·k rl'llim
1).(lI;1J1
1I,lli7·1
O..l~)()'!.
O,OO'!.7
<:1' H'llIfII
0.0 lin
O,(I'd·1
-0.11!'.7
-1I.OOIl'!.
Slo('k-(:1' H'lum
IWII H
n,l??1
OA!I7~)
O,OO~!1
(l.OOI I
(:"'\\IIII'l'lillll ~IUIIIII i., II ... , 1•. I1'~'· ill lu~ ,,·,,1, umllllll'lioll of lIollc!IIf.,Ioh-, allel w ...·i(' .. s, TIl(' ,lurk 1I'llIm is II ... III~ "'011 "'lIlm oil lIlt' S&l' :100 ilH\t-x sill(,(' I'/:!Ii, allel III .. n'lum Oil a (0111(>:11'0110'" illelt'\ "Otll (:""'''1\;11\ a'HI Shill"r (l!/HI) h(·for .. l!/:!Ii. Cl' is Ih,' n'al r('lurll '11\ 1~l\Ioulh "Otlllllt',riall'''I''·I. hOllghl ill.l:lllll;1\ r aud IOIl .. eI ol't'r ill.luly. All eI;lla a ...• 'lI\uII"I. IKK'I hI I !I!\.! ,
a slIIall sl.lIIt1anl deviation or only :I,:I'}!,; hence Ihe excess slock rCllIrn hots a .... \'ariall .... with log "OIlS'lIlIptioll g .. mvth or Ollly (),O():~ despile Iii .. !;tn tllal Ihe ('orrl'ialillll or lilt' Iwo serit's is aholll 0.:,. SlIhSlillililig Ihe IIIOJllClIts ill 'Elhlc H.I illlo (H.~.7) shows lhal a risk-awrsion codHcicll1 of 19 is reC]lIircci 10 iiI Iht' ('qllill' \I\,t'l\Iilllll.h This is JllII('h greater thall 10, tht' maximum v;lhlt' I'Clllsi(kred pbllsiblc by Meltr;! alld ['r('SLott. It is worth lIotin).( tklt titl' implied risk-aversioll coefficicnt is sensitive 10 Iht' limillg- (,Olll't'lltioll IIsed I()\' (,OIlSlIlIIplion. While asset prices arc 1II('asllrl'd at lite ellel or ('ach periCHl, (,OIlSIlJllPlioll is IIIt'asul'Cd as a flow durill)!; a period. III ('orrelatillg asset rl'turlls with LOIISlllllptioll gro\\'th 011(' ('all ass II II\(' lhat 1I\('aSllred nlllSlIlIlptio\l represt'll ts hegillllillg-o(~pcriod ('OIlSlIlIIptioll or I'lld-ol~pl'riod (,OIlSIIIIIPlioll, The lilrtlWr assllmplion leads Ol\(' to ('orrdatl' IIII' 1'1'1111'11 OVI'I' a Ill'riotl wilh Ihl' ralio of lilt' Iwxt period's (,OIlSlIlIlptiOillo this pniod's ("OIlSlllllptioll, while tht' lattl'l'asslllllpliolllea
h'(';'hh- K.llqU)1 h 'lie' "lnIlU'III.' 01.1"('1
Ic'llIllI,alld ('OIlSllIlIllIioll
~ro"'lh \\'Iu:n·as'.'qu~"\nn \';,ria,jull in nJlllli·
(K.'.!.7) n't}lliu', Ihc 1I10IlU'III.' fli iIlIlO\'.lIilll1' ill ritt'.\" ~crit.'s.
llo\\'('\'t', I"t·
lion;11 (',(!C'.-h·d IC'IIII''''' ,1I"I"OII'IIIII(llioll J.!.l"owlh '('t'm~ \u IH'
"n;," "Hlmgh
tllal 1111' IIIOlllt'lIl'\
01 iUIIU\";llioB' ~lIt· :-.illIlI.U 10 IIII' IIlolll('UI' ot th\' L\\\' St'11t':'\.
't ~IU,'''iI1l,UI. "'klino .•lIltl Silillt'l (1!11't7) 11.111111(' Ilti~ prohlelll ilion' LII dllll~' h~' ;111
1IIlflc-Ilyilig
(OlllllllllJlh-lilllt' lIIodc"
.11111
;1"i!\IIIIIIIIJ.{
del i\'ill.~ ils illlplic.llicHI"i tOI' liIlH··;t\'('ragc·d lIaLL
8.2. (.'ol/.wlII/Jlion-Basnl Ass!'1 Pricillg with Power Utility These calculations can be related to the work of Hansen and Jagan-!, nalhan (1991) in the following way. In the representative-agent model with power utility, the stochastic discount factor Mt+1 '= 8(Ct+J! C,)-r, and the log stochastic discount factor 17l,+\ == log(8) - y ACt+t. If we are willing to make the approximation 111,+ \ ~ Mt+ \ - I, which will be accurate if M,+ \ has a mean close to one and is not too variable, then we have VarIMt+d ~ Var[ 17l,+il = y2 Varf Ac,+il. Equivalently, the standard deviation oflhe stochaslic discount factor must be approximately the coefficient of relalive risk aversion times the standard deviation of consumption growth. Using the Hansen:Jagannathan methodology we found that the standard deviation of the stochastic discount factor must be at least 0.33 to fit our annual stock market data. Since the standard deviation of consumption growth is 0.033, this by itself implies a coefficient of risk aversion of at least 0.33/0.033 = 10. But a coefficient of risk aversion this low is consistent with the data only if stock returns and consumption are perfectly correlated. If we use the fact that the empirical correlation is about 0.5, we can use the tighter volatility bound in equation (fU .15) to double the required standard deviation of the stochastic discount factor and hence double the risk-aversion coefficient to ahollt 20. The IliskfTf'l~ Rale Puzzle One response to the equity premium puzzle is to consider larger values for the coefficient of relative risk aversion y. Kandel and Stambaugh (1991) have advocated this.1I However this leads to a second puzzle. Equation (8.2.5) implies that the unconditional mean riskless interest rate is
(8.2.8) where Ii is the mean growth rate of consumption. The average riskless inLerest r~\le is determined by three factors. First, the riskless rate is high if the time preference rate - log.5 is high. Second, the riskless rate is high if the average consumption growth rate Ii is high, for then the representative agem has ~\!1 incentive to try to borrow to reduce the discrepancy between consumption today and in the future. The strength of this effect is inversely proportional to the elasticity ofintertemporal suhstitution in consumption; in a power utility model where risk aversion equals the reciprocal of intertemporal substitution, the strength of the effect is directly proportional to y. Finally. the riskless rate is low if the variance of consumption growth is high, for then the representative agent has a precautionary motive for "011,' Il,i~hl Ihillk Ihal illlmsp"Clioll wOllld hl' ",fficil'flIlo rule oul very large vdlue. of y. Iluwt'wr "-alldd alld S""llhal~h (\!19 I) poilll 0111 Ihal illlrmp"n;on call deliver very different ..slilllal<·s "I' risk a'·.. rsion cJ'·p .. ncJinl( on til(" .il." of Ih .. I.:amlll .. con.idered. Thi. sugge'15 Ihal introsp,·rtion C;lII ht- mi.,I("ading or th,lt ~cml(' llIort" J.{t"Ilt"ral IIlnclt-1 of utility i~ n("edect.
310
8. /lllerlem/lOral F:'1uilibrillm MotirLf
saving. The strength of this precautionary saving clfect is proportional to the square of risk aversion, y~. Given the historical average short-term real interest rate of I.WX" the historical average consumption growth rate of 1.8%, and the historical average'standard deviation of consumption growth of 3.3% shown in Table 8.1, a y of 19 implies a discollnt factor li of 1.12; this is greater than Olle, corresponding to a negative rate of tillle preference. Weil (1989) calls this the riskJree rate PUllU. Intuitively, the punic is that if investors are extremely risk-averse (y is large), then with power utility they must also be extremely unwilling to substitute intertemporally (1{1 is small). Given positiw avnage consumption growth, a low riskless interest rate and a positive rate of tillle preference, such investors would have a strong desire to horrow frolll the future. A low riskless interest rate is possible in equilibrium only ifinvestors have a negative rate of time preference that reduces their desire to horrow. Of course, these calculations depend on the exact moments given in Tahie H.I. In some data sets all even larger coefficient of relative risk aVCl"sion is needed to lit the equity premium: Kandel and Stambaugh (I!J!ll), for example, consider a risk-aversion coefficient of 29. With risk aversion this large, the precautionary savings term -y~a;"/2 in equation (H.2.H) n~duces the equilibrium riskfree rate and so Kandel and Stambaugh C\O not need a negative rate of time prderence to lit the riskfree rate. A visual impression 'tlf this elTect is given in Figure 8.3, which shows the lIIean stochastic discount factor first decreasing and then increasing as y increases with a fixed li. Since the riskless interest rate is the reciprocal of the mean stochastic \discount factor, this implies that the riskless interest rate first illl:I"l'ases and \then decreases with y. The behavior of the riskless interest rale is always a :problem for models with high y, however, as the interest rate is extn:mcly rensitive to the parameters g and a 2 and reasonable values of the interest ate are achicved only as a knife-edge case when the elfects of g and a~ Imost exactly offset each other. .
r
s the Equity Premium Puule a HO/lUst Phenomenon? other rcsponse to Ihe equity prcmium puzzle is to arg\le Ih
8.2.
COI/,I/III1/,/illl/-JJIl.lnl
!\.1.11'/l'ririIiK wi/Ii
"1111"'1"
('/ili/v
:111
ma)" appear to he irratiollal ill the sample. While il Ilia), SCCIII illlplilllsihk that this cOlild he all important prohlem ill ]()() ycars of data, Rietz (19HH) arglles that all ('("ollolllic catastrophe thill destro),s '1lmost all stock-lIIarkl'l vallie Gill be extremely lInlikely all(1 yet have it major depressillg clfect Oil stock prices. A related poillt has becnllIade by l\rown, CO('t/manll, alld Ross (199:,). These authors argue that financial econol\lisL~ rOIl({'lItratc 011 th(~ US st()('k market precisdy hccause il has surviv(:d alld growlI to \>c("(l1l1e the world's largest markel. In some other lIIarkt"ls, such as those of Russi,t, illvestors haw had allthcir wealth expropriated during the last 100 years and so there is 110 continuous record of market prices; in others, suclt as the Argentille markct, returns have been so poor that today these Inarkcts 'IIT regarded as cOlllparatively less important elll('rging markets, If" tltis sllr\'ivorship crfect is important, estilllates of average US stock returns an' hi,lsed upwards. Althollgh these points have SOIll(' validity, Ihcy arc IInlikely to he the wholc explanalion for tlte eqllity premilllll plIl.I.k. The diflinilty with the RiLll (I ~)HH) arglllllclIl is that it reqllires Ilot ollly all cconolllic catastrophc, hllt (,Ile which arrccts stock market investors lIlore sniollsly than investors ill Sh('rHerm (!chI instruments. The Rlissian exalnplc suggests that a (alastropll(: causes vcr)' low returns Oil dehl as well as on equity, ill which rase lhe peso prohlem affects estimates of lhe average lewis of retllniS hllt lIot cstilllatcs of tile eqllity premium, Also, there seems 10 he a sllrprisingly I;lrge equit), premiulI\ not oilly ill lhe last 100 years of US data, hilt also in US data from e.lrliei" ill the 19th Century as showil hy Siegel (1l)~14) and ill IIat;1 from other t'o\lntrics as shown hy Call1l'hdi (1!1\l(ih).
Tillll'- Varia/ioll ill J::>:/II'r/ed Asset He/unts IIll1i COII.IIlIII/J/ioll (;,OIo{h Eqllation (8.25) gives a relation betweell rational expectations of asset returns and rational expectations of COIlSUlllptioll growth. II implies that expectcd asset returns arc perfectly correlated with expl'ncd cOllsumption growth, bllt the standard deviatioll of" cXl'eClcd asset retllrns is y timcs as large as the standard devi;lliOIl of expened constlllll'tion growlh. Equivalerlll)" the slandard deviatioll of expected (onsulI\ptioll growth is J/r = I/y limes as large as the standard deviatioll of expected asset retuI"Ils. This suggests all alternative way to estimate y or VI. Ilallsen ,111(1 Sillgleton (I~m:l), followed hy Ilall (1988) alld others. ha\'('l'roposcd illstnllllcntal variables (IV) regression as a way to approach thc prohklli. If we (Idill(' all error tcrlll 1)1.1-1 I == I"l.Itl - E,[ri.,111 - y(6./,·!·1 - 1'.,\6/" II), Ihell wc (all rewrite equatioll (H.25) as a regressioll equatioll,
'i.'+1 =
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(I\') c'!'Ililllollt·, II' ,tH' polI.IlIIt'h'l' l' .Hul 1/1 in l(·gn':-.siuH' (K.'L~') ~HHt (K.1.10} I (,:-'P"fIhel\". Tht' (011111111.' 11('.II,,"d "1i.'.,1 (~.:!.!I)" ,lIlIl"Tt'.,1 (H.!!. 10)" repIJl! Iht' J('!. 'lali!'llic~ arul.1l1illt ,ignilirotllft' 'c', ... I, 1I11}w C'\l'l.llI.II'lr~ \";11 i.lbl('!'1 ill f('gn's.,iou:\ ul IV I(').{n·:\~iull n'.'"dtlu,t!" (X.~ H) alld OC!.IO) UII lilt, It\\lrtUlH'Uh. Th,' ill'" \U\H'llb include' ('jlhe'r uUC' l.lg (ill 10\\'S marked I), or Oil(' awl I\\'o i;lg' (ill lOW, 11I:lrkc·d I .lIul :!) of 'ht, 1'1'<11 COlllllH'rriallMpc:r rate.', tlH.' n..',\\ C"C1l1'illlllplioll gln\\·th 1;lIc', .lIulllu' log di\"idt'lld-prin' r;,lIio.
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lagged ,'al iailltos (orrl'lal('d ",ilh as~l'l rl'lIlI'llS CIII Ill' IIs('d as illSlnlllH'lIts ill ;111 IV 1('gr""iull 10 ('Slilll;II" th!' ,'o!'lIi( i"111 or r<"lati\'(, risk av('rsioll )'. '\;Ihlc H.~ illllstr;II,'s t",O-\t;lg(' Itoast S'lIl;II'l'S ('stililatioll "I' (H.~.~). III Ihis 1;lltk II\(' ;",,'1 \1'1111'1" ;I\'(' th(' 1(';11 ('ollllll('n'ial pap(T 1';11(' alld n'al stock H'tllnl 1'1'0111 'Llhk 1'.1, alld ('U"SIIIII)llillll ~r()\\'lh is lIi(' allllttal growth r;ltc 01' n'al IIIHlllttr;lhlt·s alld s,'n'in's nH"ttlllplioll. Thl' illstnlllll'ttts ar(' dlher OIl(' 1;lg, or 0111' ;11111 tll'O I;lgs, 01' IIII' 1,(,;11 cOlJllllcrl'ial p;tp('r r;lt(', thl' n'al I'ollsttlllilliott glll\\lh r;I!l', ;11111 Ihl' 11114 divid(,tld-price ratio, )0'01 ,',II h ;ISS('t
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8.2. COIl.WIIl/ltion-Based Assrt PririllK with I'owrr Utility
. "sf
"Test (H.2.9) "-the R2 statistic for a regression of the residual on the in~ strulllents together with the associated significance level of a test of the over-identifying restrictions of the model. This test is discussed at the end. of Section A.I of the Appendix. . Table 8.2 shows strong evidence that the real commercial paper rate is; forecastahlt', and weaker evidence that the real stock return is forecastable.: There is very little evidence that consumption growth is forecastable. 9 The i IV estimates of yare negative rather than positive as implied by the underly-, ing theory, but they are not significantly different from zero. The overiden-; tifying restrictions of the model are strongly rejected when the commerciall paper rate is used as the asset. • Olle problem with IV estimation of (8.2.9) is that the instruments are only very weakly correlated with the regressor because consumption growth is hard to forecast in this data set. Nelson and Startz (1990) have shown that in this situation asymptotic theory can be a poor guide to inference in finiw samples; the asymptotic standard error of the coefficient tends to be too small and the overidentifying restrictions of the model may be rejected even when it is true. To circumvent this problem. one can reverse the regression (8.2.9) and estimate (8.2.10) If the orthogonality conditions hold, then as we have already discussed the estimate of 1/1 in (8.2.10) will asymptotically be the reciprocal of the estimate of y in (8.2.9). In a finite sample, however, if y is large and 1/1 is smaIl then IV estimates of (8.2.10) will be better behaved than IV estimates of (8.2.9). In Table 8.2 1/1 is estimated to be negative, like y, but is small and insignificantly different from zero. The overidentifying restrictions of the model {"Test (8.2.10)") are not rejected when only I lag ofthe instruments is used, and they are rejected at the 10% level when 2 lags of the instruments arc used. Table 8.2 also shows that the residual from the IV regression is only marginally less forecastable than consumption growth itself. These results arc not particularly encouraging for the consumption model, but equally they do not provide strong evidence against the view that investors have power utility with a very high y (which would explain the equity premium puzzle) and a correspondingly small 1/1 (which would explain the unpredictability of consumption growth in the face of predictable asset returns).
"tll poslwar 'l""nerly dala Ih~rc is Slroll~cr cvid~lIce of prediclable Ydfialion in cOllsum~ liOl' ~rowlh. Camphdl alld Mallkiw (I !/!/O) ,how Ihallhis varia lion is associaled wilh predictabl<1 ill(,OIlU- ~r{}wlh.
'
8. 11Itertemporal Equilibrium Models
· 314
8.2.2I'ower Utility awl Grllrmliud Method of MOlllrll(.\
So far we have worked with a restrictive loglincar specification and have discllssed cross-sectional and time-series aspects of the data separately. The Generalizcd Method ofMomclIL~ (GMM) of Hansen (1982), applied to the consllmption CAPM by Hansell and Singleton (1982), allows us to estimate and test the power utility modd without makinp; distrihutional assumptiolls and without ignoring either dimension of the data. Section A..2 of the Appendix summarizes the GMM approach, and explains iL~ relation to linear instrumcntal variables. When GMM is used to estimate the consumption CAPM with power utility,using the same asset returns and instruments as in Table 8.2 and assuming white noise errors, the overidentifying restrictions of the model are strongly rejectcd whenever stocks and commercial paper are included together in Y the system. The weak evidence against the model in Table 8.2 becomes much stronger. This occurs because there is predictable variation ill rxrr.I.1 returns I on stocks over commercial paper. III Such predictable variation is ruled out \ by the loglincar homoskedastic model (8.2.5) but could in principle be ex\ plained by a heteroskedastic model in which conditional covariances of as11 I set returns with consumption are correlated with the forecasting variables. I The GMM system allows for this possihility, without lineari/.ing the model or imposing distributional assumptions, so the GMM rejection is powerful evidence against the standard consumption CA.PM with power utililY. Faced with this evidence, economists have explored two main directiollS for research. A first possibility is that market frictions invalidate the standard consumption CAPM. The measured returns used to test the model lIIay nol aClually be available lo investors, who llIay face transactio liS costs and constraints on their ability to borrow or shortsell assets. Market frictiollsllIay also make aggregate consumption an inadequate proxy for the consllmptioll of stock market investors. A second possibility is that investors have more complicated preferences thall the ~implc power specification. We explore each of these possibilities ill the next two sections.
I
!
8.3 Market Frictions We IlOW consider various market frictions that JIIay be relevant for asset pricing. Ifillvestor~ face transactions CoSL~ or limits on their ahililY to horrow I"Reran Ihal i/l Chap"" 7
w('
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('xrcs~~I()(k rt·tllnl~.
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(11.:.1.;.) i/l whirh II ... I'"ria/ln" h"",' lillie ,"h.
8.3.
I\/(lr/(I'I Frjctj(lll.~
01' sell assets short, thell they l\Iay have ollly ,\ limited ,\hility to exploit the empirical pallnlls ill relllnls. III SCl'lioll K.:\. I WI' show how this call aher the basic I Jansen and .J'l!{annatltan (I !)!) I) analysis or Ihe VOI,llility of Ihe stochastic discount faclor. The sallie s\)rL~ offricliolls lIIay lI\ake aggr('g
I/Il/l.\/'lljaglllllllllllllll
lJound.s
The volatility hounds of Hansen and.lagannathall (I !)!) I), discllssed in Section H.I.I, asstlme that investors call freely t raclc in all assets without illcllrring tralls,ll'tions costs alld withollt lill\i\~ltions Oil hOI rowing 1)1' short sales. These assumptions arc ohviously rather eXtrt'IIIC, hut they have hl'clI relaxed by I It: alld Modest (19!)!i) alld l.ullnH"r (I !I!H). '11) understand the approach, note that ifasset j call1lot he sold sllOrt, tlH'lIthc stalldard equality n~strini\}ll E[ (I + H,,) M,] = I must I... rcpLI<"-.! hy ,\II ill<'<1' ,,'(il>' n·,trinio" (K3.1)
If the ineCjllality is strict, lhen an investor would like to sell the asset hill is prel'ellted from doing so by the sllor\saics constrain!. Instead, the investor holds a lero position ill the assel. SllOrtsaies constr.\inLS may apply to some assets but not others; if they apply to ~tll assets, then they Can be interpreted as a .IIJ/lifl/() wIlllmillt, in that they ensure lhat an investor canllolmake investlllents today th,1l deliver negative weallh tomorrow. AsslImillg limited li~,bility for all risky assets, so that the minimum vallie of each assettOIlJOITOW is I.ero,
X. III/"r/t'm/lllm{ 1~IJII;I;b";1I11/
Ml!tifis
dl'JlI'lId Oil IIII' sill' 01 a Iladl', Whl'lI Ih('l"(' are IrallsaCliolls COSls, Ih(' aflcrIrallsanioll-t'oSI Il'tllll1 Oil all ~ISS('t hOIl)!;hl loday and sold IOlllorrow is \lol Ihl' nl'galil'(, of lilt, ;lfln-lralls;lclioll-COSI r('llim Oil Ihe salll(, ass('1 sold loday alld houghl had. tOIIlOITO\\', Thl's(' two r('lllrns can 1)(' 11I(';tSllr('
wller(' 0 is all IIl1kllOWII \'('("101". 'I'll(' 1II0dd i\llpli(~s variolls Il'slliniom on o sllch as the rl'slri("lioll Ihal /I, ~ I lill' all i, Volalilily bOUllds call 1I0W
II(' lillllld lill' (,;I("h ifI hy <"IlOmill)!;, slIhjecl 10 11\1' reslrifliollS, Ihe vaillt' or () thai d(·lil'('I"s thl' loweS( I'ariall("(, for Mt(M). 11(' alld Modl'sl (l~l9!i) lilld Ikll hy cOlllhillill)!; hOlTowillg rOllslraillls, a r('slriclion Oil II\(' short sale or Tn'aslll"}' hills, ;11111 ;ISSI'I-spl'cili(" Irallsaclioll cosls Ih('y call greally 1'1:<111('(' Ihe'l'olalility hOlllld Oil Ihe slochaslic dis('ollrll (;I('(or. This ;llIal),sis is I'xln'lIIl'ly ('ollst'lvalivc illihal 0 is chosl'lI 10 lIIinimize Ihe I'olalilily hOlllld Wilholll askillg whal IIl1dnlyill)!; 1'<Jllilihlilllll wOllld support Ihis dlOit't' fill' (J. If Ihl'fl' al(' SIlIlS\;llIlial lJ'allsaniolls COSIS, for examplc, Ihl'lI (,\'('11 risk-II('lIlral Irad('rs will 1101 sci I 011(' ass('1 10 hlly allolher a\\cl wilh a highn Iclllrll III1I1'SS IIH' n'lIlrtl dilf('ITlltT cxn'('ds Ihe Irallsaniolls cosls. 11111 IIII' OIlI'-pl'liod Ir;1I1sanioll cOSIS will nOI 1)(' relevanl if Iraders call hlly lilt' hi)!;h-n'lllnl assel ;IIHI hold il for lIlall}, periods, or if a Irader has III'I\' wealth 10 ill\'('sl alld IlIliSt p.IY Ihe COS! of pllrchasin)!; Oll(' assl'l or lilt' olhel'. Thus tilt' work 01'11(' alld Modesl (19\)!» alltl I.llItll)('r (1~194) is exploratorv, a war 10 gel a "'lise Ii,,' Ihe ('xtl'lIl 10 which /IIarkel frictiolls loosell Ihe hOlillds illlplit'd hI' a frit-liolll('ss lIIarke\. SOliII' allihors ha\'(' tril'e! 10 solvc I'xplicitly liJr Ihl' assel prices Ihal arc illlpli(,d hy cqllilihlilllll II10dds wilh Irallsacliolls COSls. This is a dinkllit lask ht'C1IIS(, Irallsacliolls ('osls lIIake Ihe investor's decisioll prohlelll 1'0/11paralivd\' illlraclahk I'xccpl ill vcry special cascs (SC(' Davis anti Norman (1'190». Ai\,;lgMi alld (;nllcr (I~)
Th(' )'ej('t'lioll ()1'11lt' slalldard COllSlllllPlioll CAI'M Illay bl' dill' ill parI 10 diflinlhil's ill llle;ISIII illg ag)!;rl')~all' cOllsllmlJlion. Thc cOllsumplioll (J\I'M appli('s loInit' CllllSllllllllilllllllt';ISIII'l'd al a pOill1 ill tillie, hilI lilt' al'aibhle t\;lIa an' lillll'-a)!;gl('gaft'd alld Illt'asllr('d \\'ilh I'rmr. Wilcox (I !l!l~) dl'scrihes the s;llllplill)!; pron'dlllrs IIst'lI 10 cOllslrll('\ ('OIlSlIlIIplioll dala, while Crossmall,
li.J. Ma,.kl'l Frictions
Melino. and Shiller (1987) and Wheatley (1988) have tested the lowing for time-aggregation and measurement error. respectively. N.n,no'hl,,;':':;' speaking. these data problems can calise asset returnsweighted by measured marginal utility of consumption. (1 +ll,.1+ 1)i5 (CHI / C1r- Y• to be forecastable ' ill the short run but not the long run. Thus one can allow for such problems by lagging the instrumenL~ more than one period when testing the mode\.t2 Doing this naturally weakens the evidence against the consumption CAf1,M, but the model is still rejected at conventional significance levels unless very t long lags are used. A Illon~ radical suggestion is that aggregate consumption is not an adequate proxy I<)r the consumption of stock market investors even in t~e long run. One simple explanation is that there are two types of agents ,in the economy: constrained agents who are prevented from trading in asSet markets and simply consume their labor income each period. and unconstrained agents. The consumption of the constrained agents is irrelevant:1O the determinatiOI} of equilibrium asset prices. but it may be a large fraction of;iggregate consumption. Campbel\ and Mankiw (1990) argue that predictable variation in consumption growth. correlated with predictable variation in income growth, suggests an important role for constrained agenr. while Mankiw and Ze1des (1991) lise panel data to show that the consu~ tion of.slOckholders is more volatile and more highly correlated with the stock market than the consumption of nonstockholders. The constrained agents in the above model do not directly influence asset prices, because they are assumed not to hold or trade financial assets. Another strand of the literature argues that there may be some investors who buy and sell stocks for exogenous, perhaps psychological reasons. These noise traders can inOuence stock prices because other investors. who are rational utility-maximizers. mllst he indllc'~d 10 accommodate their shifts in demand. If utility-maximizing investors are risk-averse. then they will only buy stocks from noise traders who wish to sell if stock prices fall and expected stock returns rise; conversely they will only sell stocks to noise traders who wish to buy if stock prices rise and expected stock returns fall. Campbell and Kyle (1993). Cutler. Poterba. and Summers (1991), DeLong et al. (1990a. J990b), and Shiller (1984) develop this model in some detail. The lIIodel implies that rational investors do not hold the market portfolio-instead they shift in and Ollt of the stock market in response to changing demand from noise traders-and do not consume aggregate consumption since sOllie consumption is accounted for by noise traders. This makes the I~c.ullphdl alld Manl<.iw (1!190) discuss this ill the COil text ofa linearized model. Breeden. Gihbolls. alld Lillellber!:er (1!lIl!1) make a rdated poim. arglling Ihal al sharI horizons une shulIl,1 rrplace conslllllplion with Ihe T('lIlrn 011 a portfolio COmlf\lCled 10 be highly Forrelaled with 101l1:"r-nall IIlOvemellls in colslI"'pti,,". IIraillard. Nelson. alld Shapiro (lWI) filld Ihat Ih,' cOllsllmption CAPM works heller at 10111: horizons than at short horizons.
8. J/lIf'l'lempora/ J~ljllilill/illl/l Models
31H
1Il0del hard to test without having detailed information on the investment strategies of diflerent market participants. It is also possible that utility-maximizing stock market investors are heterogeneous in important ways. If investors arc subject \0 large idiosYllcratic risks in their Jabor income and can share these risks only indirectly by trading Va few assets such as stocks and Treasury bills, their individual cOlISumption paths may be llIuch more volatile than aggregate consulllption. Even ifindiIvidual investors have the same power utility functioll, so that any individual's \consumption growth rate raised to the power -v would be a valid stochastic :discount faclOr, the aggregate consumption growth rate raised to the power may not be a valid stochastic discount factorYI Problem H.:~, based on Mankiw (1986), explores this effect in a simple two-period nlOdd. \ Recent research has begun 10 explore the empirical relevance of imIperfect risk-sharing for asset pricing. I-Icaton and l.ucas (l9!H;) calibrate \individual income processes to micro data from the Panel SlIIdy of [ncome nynarn.ics ([>S[D). Because the [,S[D data show that idiosyncratic incollle ariation is largely transitory. Heaton and l.ucas lind that investors can rninrnil.e its c1Tects on their consulllption by borrowing and lending. Thlls they ind only limited effects on asset pricing unless they restrict borrowing or SSlime the presence of large transactions costs. Constantinides ami Duffie (1996) construct a theoreticalmodcl in which idiosyncratic shocks have per~I\anellt eflects on income; they show that this type of income risk can have large effects on asset pricing. 1 Given this evidence, it seems important to develop empirically testable intertemporal asset pricing models that do not rely so heavily Oil aggregate consumption data. One approach is to substitute consulllption out of the consumption CAPM to obtain an asset pricing model that relates mean returns to co variances with the underlying state variables that determine consumption. The strategy is to try to characterize the preferences that an investor would have to have in order to he willing to huy amI hold the aggregate wealth portfolio. without necessarily assuming that this investor also consumes aggregate COlISlllllption. There are several classic asset pricin~ models of this type set in continuous time. most notably Cox, In~erso\l. and Ross (I 9H5a) and Merton ( I973a). But those models arc hard to work with empirically. Campbell (1993a) suggests a simple way to get an empirically tractable discrete-time
I
i-v
t
I:I-n;i, i, an example ol.le",(',,·, 1"'·'I"alil),. Si"n' """l:i"allllilily is "o"li,,('ar. Ih .. average of (OtlMIIHPliou is 1I0t gt'IH'I";'llIy the same as the marKilial utility "r ;I\'('r"gc (omumptio". Thi, prohl"111
(;ro~~man ~lIl(l
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prufe~~r.~
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for individual
fOIl.'iillll1priou MrC'am~
and a. . .O\(·(
prirr~
.Ire.-
c1in1.l~ion~.
8.3. lV/fir/wI Joiidiul/J
319
1II0d<:l IlsillJ.; lhe lIlilily spedlit-'llioll (il-vdopcd hy I·:.l.~lt'ill "nd Zin (!!IH!l, I !l!) I) ;llld Weil (19H!), which wl' 1I0W Sll III III
Hi.l/i
Allt'niull tllUl/lllnlt'm/JOml SUI'.llilulioll
Epsteill, Zill, alld Weil build 011 the approadl of Kreps .1IId l'orll~llS (I !)7H) 10 devclop a IlJOIT lIexible version oflhc basic power IItilil)' llIodel. ThaI model is re.,lrinil'c ill Ihal il makes the elasticily of illtl'rlelllIH)r,d slibslitlilioll, V', the reci procal o/"lhe coerticielll o/" relalivc risk aV('I"sioll, V. Yel it is 1101 (kar thaI Ihese lwo COllcepts should he lillked so liJ.;htly. Risk 'lV('lsion dl'snilws Ihe (OIlSlllllcr's reluClallC<: to sllhslillll(' (,OIISlllllplioll "noss sl,l\l'S of IIIl' world .111(1 is meanill).;ful even III all all~l\\p()l slll~ stilllll' COlISlll11plion over li\l\l' alld is \\\l',lllillghd ,'\"'11 ill
lJ, =
{
(I -
8)('~" -/-.5 ' r
(I':, [(/'\'-1)' I)"
! }
f;
(H.3.3)
=
where Ii == (I - y) / (I - 1/ if!), Whell II I Ihe I<TIIISioll (H.:I.:\) hl'COIIH'S lillear; il COlll thcli hc solved flJrw
wherc j,\~+l is the represel1tative agenl's weallh, and (I + f{m.l> dis Ihe relurn on Ihe "market" portfoliu of all inVl'Sled weallh. This I'orin of Ihe bndgel conslraitll is appropriate fur a complele-markets model in whidl wcallh includes human G\pilal as well as financial asselS. Epslein allli Zin Ilse a complicaled dynamic progralllmil1J.; "'Wlmenl 10 sh()w Ihal (H.3.:\) ,\\HI (H.3A) together imply an ElIler eqllalion of Ihl' fonnl-l
1=1':,
('. - i (j(~) (., [{
}II {·----l 1 I-II (I (I -/-
-/-{{,.I;I)
]
.
{{m./lI)
lfwc aSSIIIIle Ihat assel returns a 11<1 COIlSulIlplion ,Ire h()lJIoskcd
II - I ~ II .> I. = -log8+--n --·_-o--j-I··,II\(,tll. ~
III
'lV''!.
I
VI
(H.:l.li)
HTIH'\"(' ;\n: in Lin typos in t'()u;\\ioi\s (let) lhltHlgh (1~) 1)1 Fp:-'ldu ;Ulll/.iH (I~.~)J) ""hirh f.(iv(' illh'I"III('
cI(', i\'aliotl.
H.
/1I(n(I'III//O,.(/1 /':" II ilil"i II III
All/dl'ls
'I'll<' pn'miulIl 011 rish assels. illclll(\ing II\(' mark('1 portfolio ilst'lf, is
.,
E,I 1;,/111
er-
- ',','ll
o a" V; + (I
+ --'~
(lU,7)
- O)a,.,.
This says Ihal Ihe risk pr('Jllilllll 011 ass!'1 i is a wcigh\('(\ comhinalion of assel i's co\'arialll,(, wilh ('OIlSUlllplioli growth (divided by Ihe (,Iasticity orillIl'I'l(,lIIporal SUhslilulioli 1/1) alld ass('l i's covariance wilh th(' markel re\llrn. Th!' weighls al'l' /I and I - {/ resp!'('li\'('ly. The Epsldn-Zill-Wdllnodd Ihus lI!'sls Ih!' conslllll(>lioli CAPl\l wilh power Illilily (0 = I) alld th!' Iradiliollal SIalic CAPM (1/ == OJ, It is 1t'lIIpting 10 USt' (H.:I.7) logetlrer with ohsC\'wd dala Oil aggrq~al(' l'IllISlIlIIplion and slock lIIark!'1 relnrns 10 estimale 0111(\ test Ihe EpsleinZill-Wcil IIlw\el. Fps\('ill ;\IId Zill (HI!l1) reporl results Ihis type. In a similar spiril. Ciovallnilli and Wei! (!9H9) lise the 1II()(ld 10 rl'inlerprel the results 01 Mankiw and Shapiro (I !IH(;). who found Ihal h!'las wilL the lIIarkel ha\,,' greal!'r I'xplanalon' pow(')' (1I' Ih!' cross-seclion,1I pailI'm or rellirns Ihan do Iwtas wilh consllinplioll; Ihis is ('onsistent wilh a vahll' or (} dose 10 11'1'0. Ilow("'('r Ihis I' 1'0 ('I' d II 1'1' ignorl's Ihl' (;1('\ Ihal Ihe inlt·rt(·IIIJloral hlldgt'l cOllstr;tinl (H.:I.·I) .dso lillks ('onslllll(lIioll alld llIarkl'l lellirns. \Ve now show lhal lhl' hlU(g('1 (,(JIlsIl'ailll ('all II(' usecllo sllhstilllle consulllption (lut (l11\1(' ;ISSI'I pril'illg 111I)(kl.
or
SItb.\lillllil/~ (.'111/11I1II/J/io/l O/lIIl/liJl' M,"11'1 Call1phell (It}~l:tl) POilllS 0\11 thaI one ('an loglillearize Ihe illl(,),It'll'poral hudg('1 cOllsllaillt (H.:I.·I) arolllld Ihe lIIeall log ('OIlSUIll(llioll-weallh !"tlio loohlaill
t\ "'/I
I
""
r",-, t
I
-1 Ii
I· (I - ~) «(, - "',).
(H,:I.H)
wi II' II' fI == I .-, <'xp(;:-:--U;) alld Ii is a ('OIlSI;UII Ihal plays no role ill what ()lIows, (;olllhilling Ihiswilh III(' tri\'ial ('qualiIY t.1II1-11 == t.('I~1 - t.(t',+I11', I_I J. soh'ill)!; tilt' rt'silltillg dilf('l'cII(,(, <'qllatioll forward. alld lakillg ('xP('clatiolls. WI' ('all wril!' Ilw luul)!;t'I (,oll~lrailll ill lhe limll
. [~ , .
I.,
'J
L...-/l (l",,II,- t;I,+,) 1-- 1
(III +~.
(H.:'-!l)
P
This <'qu;llioll ,~a\'s Ihal i1111!' ('OllSlllllplioll-W('alth ralio is high. Ihl'II lire a)!;CIII IIIlIsI I'XP('('( eithn hiV;h r('llIl'Ils Oil w('alth ill Ihe fill III'(' or low ('011SlIlIlptioll growlh ratt's, This Itlllowsjllsl frolll tilt· approxilllale hudgel ('011sl rai III wilholl t i1\1 posi II~ a II\' Iwiravior;t1 aSSll1ll pI iOllS, II is di rl'cll), allalogolls
H.3. Markl'l Frictions
to the linearized formula for the log dividend-price ratio in Chapter 7. Here wealth can be thought of as an asset that pays consumption as its dividend.l~ If we now combine the budget constraint (8.3.9) with the loglinear Euler equations for the Epstein-Zin-Weilmodel, (8.3.6) and (8.3.7), we obtain a closed-form solution for consumption relative to wealth: C( - WI
=
(l - 1/I)E I
L [ ~.
pl r,n.l+i
]
+ p(kI -_ 11m) p .
(8.3.10)
J~I
HereJ~ on
is a constant related to the conditional variances of consumption ~rowth and the market portfolio return. The log consumption-wealth ratio is a constant. plus (l - 1/1) times the discounted value of expected future returns on invested wealLh. If 1/1 is less than one, the consumer is reluctant to substitute intertemporally and the income effect of higher retUrFlS
cl+I-E,[c,+tl :::: r m.,+1 - E,[rm.,+tl
+ (1-1/1)
(8.3.l1)
(EI+I[~pjr"'I+I+j]-EI [~pjrm'l+l+j])'
An unexpected return on invcsted wealth has a onc-foHme effect on con- . surnption. no matter what the parametcrs of the utility function: This fo)lows from the scale independence of the objective function (8.3.3). An increase in expected future returns raises or lowers consumption depend- : ing 011 whether 1/1 is greater or less than one. Equation (8.3.11) also shows' when consumption will be smoother than the return on the market. When ·1 the market return is mean-reverting. there is a negative correlation between . current returns and revisions in expectations offuture returns. This reduces the variability of consumption growth if the elasticity of intertemporal sub- I Stitlltioll 1/1 is less than one bllt amplifies it if 1ft is greater than one. j':C)1I3tion (8.3.) 1) implies that the covariance of any asset return with consumption growth can be rewritten in terms of covariances with the re1"Cunphdl (1\19:~ .. ) and Camphell and Koo (1990) ~xpl()re the accuracy or the loglinear apprnXill\;lIioll ill Ihis (onlext hy comparin)( the approximate analytical ."IUlion for optimal consumplion wilh a numeric;" solulion. In an eX"tnpl" c;"ihr~led 10 US Mod, market dala, Ih,' Iwo SOlllliolls "re close lO)(ether provided Ihal Ihe inveslor's elasticity of intertempor.1I WhslilUlioll is It's, than "hullt :t
II. 11I11'I'll'lllpom/I:'l/ui/iIJll/t1l/ Models
1322 I
\ illlfll on lhe market and revisions ill expectations of fllllln' retllnts on the
Imarket: (H.:~.! ~)
I !where
(8.3.13) ~'I' == Cov, [ri.'+l, E,+l [tPI -E, [fpJ I fIll is dermed to be the cOV
1=1
r,bout future returns
Oil
1'",.1+1+ / ]]'
1=1
the ll\',lrkd, i.e., revisions in expected future re-
~urns.
I
Substituting (8.:1.12) into (tel.7) and using the definition o\'O in terms pr the underlying parameters a all(! Y, we obtain a cross-sectional asset pricing rormu!a that makes no reference to consumption: ~
1-.,' [ ri.,+d -
I'}.I+1
+ 0, 2"' : : :
YO'IIl
+ (y
-
1)0,,,.
(8.:U4)
Equation (8.3.14) has several striking features. First, assets can he priced without direct reference to their covariance with consulIlption growth, using instead their covariances with the return on invested wealth and with news about future returns on invested wealth. This is a discretc-time analogue of Merton 's (1973a) continuous-time model in which assets arc priced using their covariances with certain hedge /J{)rtjo/ios that index changes in the investment opportunity set. Second, the only paramcter of the lIlility function that enters (8.3.14) is the coefficient of relative risk aversion y. The elasticity or intertelllpora! substitution", docs not appear once consumption has been substituted out of the model. This is in striking contrast with the important roll' played by '" in the consumption-based Euler equation (8.3.7), Intuitively, this result comes from the fact that", plays two roles in the theory. A low value of '" rcduccs anticipated /IuCluations in consumption, but it also increases the risk prcmiulIl required to compensate for any contriblllion to these /Il1ctu,\tions. These offsetting efi'cns lead", to cancel out of the ;Isset-hased pricing formula (8.3.14). Third, (8.3.14) expresses the risk premiulII, netoftheJenscn's Inequality a
Ii,},
Mil/III'! i'iirlio/ls
than one, such assets have higher Illean n'tUrllS, The intuitive explanation is that such assets are desirable hecause thcy ('nahl(' the CllilsunHT to profit from illlprovcd invcstmellt opportunities, hut undl'sirahle hl'clllse they n'ducc the cOllsumer's ability to hedge against a deterioration in investment opp()r\lIlli\ie~, Whe,1l Y < 1 the form('l' cfkCl dOlllinates, and consumCl"s arc wiUing to acrept a lower return in order 10 hold aSSI'Is Ihal pay on'when wealth is most productive. When y > I the laller dkct dominates, and consulllers requin' a higher return to hold sudl "ssl'ls. There are several possible cirnlmst,llHTS Ilnder which assets Gill he priceriusing only their covariances with the returll Oil the Illarket portfolio, as in the logarithmic version of the static CAI'M. These cases have heen discmsed illllw lileralure Oil inlerll'mporal asset pr'lcing, but (H.:t I 'I) makes it particularly easy to undersland Ihelll. Firsl, if the cod'licicnt of reialiVl' risk aversion y == I. then the opposing efkcts of covariance wilh inveslment opportunities cancel out so thaI only covariance with the market relurn is rclcvalll for asset pricing. Second, if the in\'eSllll('Il! opportunity set is constant, then 0/" is zero for all asseL~, w again assets CUI be priced using only theil covariances wilh Ihe markel reI 11m. Third, if Ihe relurn on Ihe market fo\lO\;s a univariate stochastic process, Ihen news about future rdurns is perICCII), correlated with the currenl return; thus, covariance with the (,\IITl'lIt retlll'Jl is a slIfficiellt statistic fill' covarialll,!, with lICWS ahout luture returns and l',Ul he uSl'd 10 price all assets. Camphell (I !)!)(;a) aq';lIl's Ihat the lirsl Iwo caSl'S do 1101 Ill-scribe US data evell approxim;lIcl)', hilI Ihat Ihe third case is !'Illpiricall)' relevall!. A 'Iliin/ I.oo/i (lllflr 1:'ljuilJ I'rPllliulIIl'uulr
(H.3.11) 111,
(,;,11
he applied to the risk premiulll
Oil
the ,"ark!'t itself. Whell i =
we gel
When the markel relllrn is unforec;tstahlc, Ih(')'c are 110 revisions of ex pet'lalions ill futllre retllrns, so a/I/I, == O. In Ihis case tl,(' l'<Jllity premium wilh theJensen's InequalilY a7!i/O.031;, = I.H2H. This is the risk-avcrsion coefficient of an investor with power utility whose wealth is entirely illvested ill a portf(>lio wilh all unforecastable rctunt, a risk premium of :,.7;>% per ),ear, ;lIld a variallce of ().():~ l:i (st,tlldard deviatioll of 17.71';;, per ycar). The ('onsulliptioll of wcll 'Ill illVl'stor would .liso haV!' a st;I111\;1I<1 dl'Vi;ltioll of 17.7'1'1" pCI' year, This is Ell' ~realcr thall the \'obtility olllH'asull'd ag~rl'gall' (OIlSIIIllPlioll ill
8.
l"I""/~IIII)(jmll':if"ili/J/'illl/ M()(I~/.\
'nIhIl' H.I, which ('xplaills \\'h)' the risk-aversioll ('slilllal(' is 1I111ch lower than Ihe ('OIlSUllIllIiofl-hased eSlilllal('s discussed earlier. The Frielld ;11\11 BllIllll' ( I ~17!) procedlll'e call hI' sniollsly misleading if Ihe 11I;1I1el relum is sl'I'i;t1ly co ....ebled. If high stock felurns are associaleci with dowllward rl'\'i~iolls ofhtt\l\'(' retllrns, for l'xamplt·, thell a .." is lIegativc ill (H.:I.I !I). With y > I, Illis rl'dlln's 111(' equity risk prellliulII associaled willi all}' 1('\'1'1 01 Y alld illCrl';ISes Ih(' risk-aversioll coefficiellt lIeeded to l'xplail\ a givel\ eqllit)' pn·lIIilllll. 1I\IIIitively, whel\ alii" < II Ihe IOllg-rlIn risk of slock Illarkl'l illvcstllll'lIt is less Ihall Ihe short-rull risk hccause the market t('lIds to 1l1eall-i'l·\·(·rl. IlIv('stors wilh high y Gln~ aholll long-mn risk r"lhn Ihall shorHlI1l risk, so Ihl' Frielld alld Hhlllll' calrlliatiol\ oV('rst.llC5 risk and rOIT('spOlulil\gly tJlIt1t-rst;II('s Ihe risk aVl'rsioll Iwededtojllstify lhl' ('(Illity pn·lIIilllll. Camplll'lI (I!)!ltia) sllows Ihal Ihe estimated coefficielll or relative risk aversioll rises hy a bClOr of 1('11 or 1l10re if one allows fill' the empirically cstilll"ted degree of 1I1(';1l1-rt'\'('rsioll ill postwar monthly US dala. III long-run allllllal US data tile dlc('( is less dralllatic hilt still goes ill the same direrlion. (:alllplll'lI also shows Ikl\ risk-awrsioll eSlilllates illcrcasl' if olle allows for hlllllall capilal as a COII1JlOIIt'lIt of I\'(·allh. In Ihis seIlS(~ onl' can clerive Ihe e'lllil), pn'lllilllll JllIlI.k withollt ;IIIY din'('( n:krellcl' 10 cOllslImption data. till J..'quilibriwl/ II luI/ifill/or ..t.l.l~/I),'i(illg Model Wilh a few IllOIT aSSlIllIptiollS, PCI.I 'I) can he IISl'c\ to c\t'rive all t'f)lIilihrilllll IIllllrif;I('lor ;lss('1 pricillg Illodel of tilt' lyP(' discussed in Chapter Ii. We wrile tht' I'('tlll'll 011 lIlt' markt·t as tIll' lirstl·I(·I\lt'lIl or" K-clcllll'1l1 state vecto;' x,+ I. Tht' othn dt'IIH'nts an' v;lriahks that an' knowll to the market hy the ,lid ofpt'riotl/+ I alld arc rt'h'\'allt "H'II,n'casting I'll till"(' r('turlls on tht' lIlarket. W(' asslll\l(' that tht' \'('("\01' XIII (,,!lows a first-ordt'r v('('\or .\Il1on·grt'ssioll (VAR):
Ax,
+ I' ttl.
UU.I()
Tht' asslullplioll that tIll' "AR is lirst-ordn is nol rt'slriniw, sinn' a highnonk!' VAR 1'0111 ;Ih,'al's ht' stack('d into lirst-onll'r form. N('xt 1\'(' ddilll' :1 ,,'·1'1('111('111 v('('tor cl, whost' first ('lellll'lIt is Ollt' alltl whost' olht'r dl'IlH'llls an' :111 /1'1'11. This Vl'l'lor picks OUI Iht' n'al slock return r""/t I frolH lilt' \'('('101' XIII:
"III,1t
I
=::
el'x/+I,
and
r",.I+1 -
E,
rm,l+1
=:::
Th(' li ..,t-lIllln VAR gCllnat('s simple lI\lIltipt'riod fim'l'asts of fu1111'1' n'llIrns:
d'(III.
F,l/m./1
I I
,I
(i·U.li)
8.3. Markrt Frictions
It follows that the discounted sum of revisions in forecast returns can beI wriuen as 00
== el'LpiAiE/+t J=I
(8.3.18) where t.p' is defined to equal e l' pA(I - pAl -I, a nonlinear function of the VAR coefficients. The clements of the vector t.p measure the importance of each state variahle in forecasting future returns on the market. Ifa particular element IPk is large and positive, then a shock to variable k is an important piece of good news about future investment opportunities. We' \lOW define (8.3.19) where ( •. 1+1 is the ktll element of (1+1. Since the first element of the state vector is the return on the market, ail :::= aim. Then (8.3.14) implies that .
E1[rj.I+!l-
2 Tj.I+1
K
= - a~ + yail + (y - I ) LIPkO'ik,
(8:3.20)
k=l
where 'PM is the kth element of t.p. This is a standard K-factor asset pricing model of the type discussed in Chapter 6. The contribution of the intertemporal optimization problem is a set of restrictions on the risk prices of the factors. The first factor (the innovation in the market return) has a risk price of AI :::= Y + (y - I )ipl. The sign of ipl is the sign of the correlation between market return innovations and revisions in expected future market returns. As we have already discussed, this sign affeclS the risk price of the market factor; with a negative !PI, for example, the market factor risk price is reduCt·d if y is greater than one. • The other factors in this model have risk prices of AA (y - 1)rpk for k > I. Factors here are innovations in variables that help to forecast the return on the market, and their risk prices are proportional to their forecasting illiportance as measured by the elements of the vector t.p. If a particular variable has a positive value of IP., this means that innovations in ; that variable arc associated with good news about future investment oppor- ; tunities. Such a variable will have a negative risk price if the coefficient of . relative risk aversion y is less than one, and a positive risk price ify is greater than OIlC. Campbell (I99(b) estimates this model on long-term annual and postWorld War IIl110llthly US stock market data. He estimates CPI to be negative and lar~(' ill absolute value. so that the price of stock market risk At is much
=
32&
H. JlItl'rtelllIJOral/~qllilill1"i1l1ll Models
smaller than the cocfliciclIl of risk aversion y. The other ('"ctors ill the model have imprecisely estimate d risk prices. Althoug h some of these risk prices are substant ial ill magnitu dc, the other filClOrs have minor effects on the lTIean returns of the assets in the SlIIUY, because thcsc asscts typically have slllall covarian ces with the other factors.
8.4 More Genera l Utility Functio ns One str,ligin forward rcspol1se to the difliclIlties of thc stami
y~.
The Euler equatiol l
1I0W h(,(,0111(, 5
= \"1. [ (I + H',I.e I}/) ((;I+I)-Y T, I (XI+I)X, Yt} ,
Assumi ngjoint lognorm ality and hOl\loskedasticity,
thi~
call he wrillcll as (H.4,3)
Eichenb aum, Hansell, and Singleto ll (I !JHH) have cOllsidercd a Illodel of this form where X, is leisure. A~challer (I!lH!'J) and Stanz (19H9) have dcvelop ed models in which X, is governm ent spendinp; and the stock of dllrilhlc )!;oods,. respectively. Unfortu nalely, nOJ\(~ of these extra variables greatly improve the ability of the consum ption CAI'M to fit the data. Thc difflcuh y is that, at least in data since World War II, these variable s arc not noisy cllough to have much effect on the illtencll lponil llIarginal rale of suhsl itllli()II,I ';
'I
.
lI.4, I If (I{,il JoimlialioH
,
/A llIore promisi ng varialio n of tIll' hasic 1II\;dd is to allow for nOllscpa rahil- ' lity ill utility ovcr time. COI\Sl',II\\illilk~ (I ~l~lO) and S\\lIdal"l~san (I ~lH~l) have i
I
,"'Aho: a~ Call1phdl ;111<1 Mankiw (to~lwar dala,lh"" "i' pn'dicl,\"I~ \larlatiOIl til COn~lIll1ptlOll Krt,wlh th~lt IS unCOITt+.\ lt.'d Wllh pH'(hctahlc V'U"I;,l1um III n';'llll1t("rc~l nll("~("v("n aft«"r Ollt" allow~ for pn'diClahlt' variation ilildstlrt', ROV('rlllllt 'nl ~p(·'HIiIiR. or
g()()(15.
\
8.-/. 1',,1011' (;l'llmtlUtiiify I'itllftillll,~
"rgllcd fil!' the illlportalltiOlI. I len' we discuss SOIlll' sill1ple ways to iJllplellH'lIt this id!'a, S('\'!,!,,11 Illoddillg bsues arise at the outset. Wc write the period utility fUllctioll as U(C" X,). wlwre X, is the lilll('-v'lryillg hahit or sllhsistenn' lI'Vl'l. Till' first isslle is till' fUllctional /(11111 fiJ!' li(,), Ahel (I!I!IO. 199G) has propos!'d thai [/(.) should he a pow('/' fllnction Or the ratio C,;X,. while Call1plwll and Cochralle (I !l9!i). COllstalllillicles (I !I!IO). allcl SlIlHlaresall (19l')~) have IIscd ,I power fllllrtioll of th!' dilf!'!'!'II("(, (;,-- X" The sc(ollcl isslle is lhe cfli.'rl of all a/-:ellt's OWII dl'(isiolls 011 1'111111'(' levcls of hahit. III stanclard illll'/'Ilfl/-/wbillllodcls sllch as thos!' ill COllslalltillidcs (1!)!IO) and SUlldarcsall (I !)H!). hahit depcllds Oil all ag!'llt \ OWIl I'onslllllillioll alld the agl'lIt takes aCcollllt of this whell choosillg ho\\' IllllCh to COIISUIllC. III /'xtl'flllIl-lwbitIllOdcls sllch as Ihose ill Abcl (19!)O. I!)!lti) allel Caillphell 011111 Co(i1raIlC (I!)!)!i). habit depellds Oil aggrq~atc (OIlSlltllptioll whit-h is IIl1al~ I(-t:ted hy allY Oil(' agclIl's decisiolls. AI)('I calls this ""I(hill~ "/, It,itlr flrr jUllf,lrJ. TLe Ihird isslle is Ihe speed wilh whirh hahit r!'a('ls to illdividllal or aggregall' (OIlSlIllIplioll. Ahel (1990, l!l!lti). DIIIlIl allc\ SillgletOIl (I!)H(). alld Fersol1 ,1I1d Constall\inides (I !l!ll) llIake habit dqwlld Oil OIlC lag of rUIlSUIII pliOll, whercas Constantinides (I !)!H), SlInclan'san (I !)H!I), Camphell alld (:ocilr,III!' (I !)!):,), and l\catoll (I !)!):)) IlIak!' kliJil r!'a('( onl), gradually to Ch'llIgCS ill COIlSlllliptioll.
/{alio Motll'!.l FolltJI"illg AI)('I (I!I!IO, I!)!)(;). suppose' Ih,ll as a power functioll of lhe ralio Ci / XI.
li, =
.111
.Igelll', IIlilil)' (,III Ill' wlillell
,~ «(.i~I/X'I-,)I-y ,- I L.-lil
J~,
(HA,'O
1- Y
/=(1
where X, sllllllliarill's the inlluell!T of pasl nlllSUlIllJlioll levels Oil toclay's utilit\', X, rail he specified ,IS all ill\el'llal hahit or as .UI l'xll'lllal hahit. Usilll!; Olll' Llg of COIISlllllptioll for silllplicity, we 111,1)' il,I\'('
x,
= (;;"
the ill Il'I'lI;lI-hahi I specificatioll whn!' ,III agellt's oWIi pasl (,OIISlIlIIllIiolllllat-
x,
=
f;
I'
(HA.(;)
lhl' ('xtl'l'lI'll-h'lilil spnilk'ltioll whn\' aggrcg'I\(, p.IS\ ('omulIlptioll (;1_1 mailers. Sillcl' thnc is a represelltati\'(' agellt, ill l'qllilihrillill the a~ell\'s COIISllllljJlioli IIlIlSt ofrours(' c(l',al ,Iggrq~at(' (OIlSIIIIIPlioll, hilI th!' IWo li)l'JIllllatiolis yield difkn'llI \-:1111'1' (''1l1aliolls. III hOlh l''1l1'lliollS the p'lraml'll'" /( go\'('J'I)s Ihl'
I
8, 1111/'1'11'111/101111 fl/II ililll'i II 11/ '\/Odr/I
\
III Ih(' ililn lI.tI-It'lhil sl'e(itic alioll, lilt' dcriv.lli oll of lilt' EIII('\' "«II,lIio ll is cOlllplic lI('d hI' til(' bCI Ih'lI lillie-I COllSlllllPlioll an,'('[s Ihl' slIlllllla lioll ill (H"I.,I) Ihroll~1t tilt' 1('1'111 dall'd I -I- I as w"'l
I
!
I I
This is r"lIdolll al lilllt' 1 h('c
• J
,
--
(~fJ' .11(,·r '/ .. 1
r .("'I)'-I l(,-y(, /(') '/ 'ttl 'ttl '/'
, , - - uk
(fU,fI)
If Ihis IIHul,,1 is 10 caplllrl' Ihe idl'a of hahil formatio ll, Ihl'lI \\'(' Ill'('d I) ~ 0 to ('l\Slln' Ihal all illcrl'ase ill yeslerda y's COIISlllllplioll inrn'"s,'s IllI' margina l lllilil)' or COllSlllllPlioll Imlay, TIlt' ElIll'f (,(Jllatio ll .. an no", he wrill,'11 as
K(Y -
I
UU,!l)
I
\\'1\1'1'1' Ihe I'XIH'l'I'lliolis operalo r Oil Ihe II'f'I-halid side is IH'('('SS' Il, Iweallse of Ihl' r,lIldolllllt'SS of i/U lilC,. Thl' a/lalysh sillll'lili l's cOllsidt' l'ahly ill the ,'xlt'rna l-hahil specific alioll. III Ihis case (H.-1.H) alld U\.'1.~I) call 1)(' COJllhiIH'" 10 giVl'
,....
'I
(H.4.10)
• I ;'•1
If 1\'" assullle hOlllosk nlaslicil 1' alld joillt lognonn alilY of assel n'tlll'l,S alld rOllsllll lplioll gro\\'lh, Ihis illlplies 11\(' followin g rt'slrinio J\s Oil risk prelltia alld Ihe riskll'ss I'('al illll'reSI I'al,': (HA.II)
....
I I I •I...
,1
I
~
.'
(
'/
"
I I \ I
I
I \
\
(H.4.12) i':qllalio ll (H.·f.II) S;lys Iltal lite riskless real inlt'resl rail' eqllals ils val\le Illldl'r power IIlililY, I('ss ~·(V -- I )1\1'/. I lolding conslIJl lplion today alld expe('\t'd "()\ISIIIIIPlioll 101l101TO\\, conslan t, an ill('f('as, ' in consum ptioll yesIl'nla)' illlTeas( 's til(' IlIai'),:illallitilil), or COIISlllllPlioli loday. This lilah,s Iht' n'pITS('l Ilali\'(' agl'1I1 \\'alll 10 horrow from lhe ruture, driving lip lhe real illlt'l'l'SI rail'. Eqllalio ll (H.·l.l~) dl'scrihi llg Ih(' risk pn~llIi\l1 l\ is I'xanly Ih .. sallie as (H.~.7), lite risk Pl'l'lIlilllll rorlllllia 1'01' Ihe power lIlilily mod"'. The \'xlt'1'Il,,1 hahil Silllpiv adds a 1"1'111 10 Ihe Elller e'lualio ll (HA, 10) whi('h is klloWII al lillie I, :l1l1llhis do('s 1101 alii-I'I Ihl' risk pn'llIilll ll. ,\I\('I (1~1~IO. 1~I(Ui) IH'\'('I'll wkss arglll's Ihal t'OII('hillg lip wilh 1lll'.lollcs('S call "l'Ip 10 n:pl"ill If,,' eqllily p,elllilli ll pIli.!"'. This al).!;III1tt 'lit is hased Oil IWO cOlIsidn alioJls. Fil"I, Ihl' an'rage "'\,1'1 or Ihl' riskless raIl' ill (H.·!. I I ) is
8.4. Afar, Co/fmt Utility FWldiulIs
-log8- y 2 a ; /2+(y-K(y-1 Jig, where gis the average consumption growth rate. When risk aversion y is very large, a positive K reduces the average riskless r,lte. Thus catching up with the Joneses enables one to increase risk aversion to solve the equity premium puzzle without encountering the riskfree rate puzzle. Second, a positive K is likely to make the riskless real interest rate more variable because of the term -K(y-l)ilc, in (8.4.11). If one solves for (he stock returns implied by the assumption that stock dividends equal consumption, a more variable real interest rate increases the covariance of stock returns and consumption O"i, and drives up the equity premium. The second of these points can be regarded as a weakness rather than a strength of the model. The equity premium puzzle shown in Table 8.1 is that the ratio of the measured equity premium to the measured covariance ai, is large; increasing the value ai, implied by a model that equates stpck dividends with consumption does not improve matters. Also the real interest rate does not vary greatly in the short run; the standard deviation of tht ex posl real commercial paper return in Table 8.1 is 5.5%, and Table 8.2 shllwS that ahollt a third of the variance of this return is forecastable, implying a standard deviation for the expected real interest rate of only 3%. Since the standard deviation of consumption growth is also about 3%, large value~ of K and y in equation (~.4.II) tend to produce counter factual voJatilil}i in the expected real interest rate. Similar problems arise in the intemal-h~bit model. This difficulty with the riskless real interest rate is a fundamental problem for habit-formation models. Time-nonseparable preferences make marginal utiliry volatile even when consumption is smooth, because consumers derive utiliry from consumption relative to its recent history rather than from the absolute level of consumption. But unless the consumption and habit processes take particular forms, time-nonseparability also creates large swings in expected marginal utility at successive dates, and this impl~es large movements in the real interest rate. We now present an alternat~e specification ill which it is possible to solve this problem.
Difference Models Consider a model in which the utility function is
(8.4.13) and for simplicity treat the hahit level X, as external. This model difTers from the ratio model in two important ways. First, in the difference model the agent's risk aversion varies with the level of consumption relative to habit, whereas risk aversion is constant in the ratio model. Second, in the
I
i330
l71trrtem/JOmll~qllililJ/illlll
8.
Mot/rLI •
dilference model cOlISulllption must always be above habit for utility to he well-defined, whereas this is not required in the ratio model. To understand the first point, it is convenient to work with the surplus consuJllption ratio St> defined hy Sr
==
Cr - Xr Cr
(8.4.14)
The surplus consumption ratio gives the fr,lCtion of total consumption that is surplus to subsistence or hahit requirements. If habit Xr is held fixed as consumption Cr varies, the normalized curvature of the utility function, which would equal the coefficient of relative risk aversion and would he a constant y in the conventional power utility model, is -CUcr;
Y
(8.4.15)
This measure of risk aversion rises as the the surplus consumption ratio Sr declines, that is, as consumption declines toward habitP The requirement that consumption always be above habit is satisfied automatically in microeconomic models with exogenous asset returns and endogenous consumption, as in Constantinides (1990) and Sundaresan (1989). It presents a more serious problem in models with exogenous consumption processes. To handle this problem Campbell and C.ochrane (1995) specify a nonlinear process hy which habit adjusts to consumption, remaining below consumption at all times. Campbell and Cochrane write down a process for the log surplus consumption ratio Sr == log(.'),). They ,rsumc that log conSllmption follows a random walk with drift g and i.nlloration Vt+l, ~Ct+l "" g + Vr+l' They propose an AR(l) model for 5r:
I I
St+1 = (i-tP)s+tPJr+J..(Sr)Vr+I'
(8.4.16)
~Iere s is the steady-state surplus conslIIllption ratio.
The parameter rp gov' rns the persistence of the log surplus consumption ratio, while the sensiivity function A(sr) controls the sensitivity of .In I and thlls of log habit Xrt I [:o innovations in COlISUllIlJlion growth tlr+l. I Equation (8.4.16) specifics that IOday's habit is a complex nonlinear I'unction of current and past consumption. By taking a linear approxima- . lion around the steady Slate, however. it may he shown' that (H.4.Hi) is ap-
I
1111~a~IIH'd
17Risk a\'("fsioll may .tlso hl' by the Ilormalilcd curvature of tilt, ."allle.' ftlllrlioll maxil1lilen IItility exprc"cd as a fllllClioll of w(';llth). or by the v
j
8.·1. MOH' (;I'III'llIllfliiily hllll'lill//.\ proxilllateiya Iradiliollal hahit-filrlllalioll Jllockl ill II'h ieh slowl)' to 101-{ COIlSIllllptioll,
lo~
hahit respollds
(H.4.17)
where h = III( 1 - S) is the steady stale valu(' of x- f. The prohlem with the traditional model (8.4.17) is that it allows conSulllptiol\ to (;,11 helow hahit, resultillg in inlinite or negative margin'll utility. i\ process for 51 defincd over the reallinc implics that consumption can never (;t!1 helow hahit. Since hahil is cxternal, the marginal utility of consumption is ll'«(~) (C1 - Xtl- Y = SI- r CI- r . The stochastic discounl Elctor is thcn Mt+1
==
8
u'(CI+ I) = 8 u'(el )
('\+1 (;1+1)")' ---'SI
(8.1.18)
(;,
Inl~lc standard powcr utility model SI = I. so Ihe stochaslic discount f;IClor is just consumption growth raised to the power -y. To get a volatile stochastic discount bctor onc necds a larl-{C value of y. In Ih(' hahit-(ill'matioll model one Gill instead get a volatile stochastic di~c()11I11 f;lctor 1'1'0111 a volatile surplus consulllption ratio SI' The riskless real interest rate is rdaled 10 Ih(' slOlhaslic discollnli;ll'tor by (l + H;+ I) = 1/ EI (MI + I). Taking logs. anel using pH. IIi) ,\llIl (1-\.4.11-\). the l()~ riskless real interest rate is
r{1-1
= -log(o)
+ yg - yO -
-~)
~
2
- Y ?-\,\(.I,) I- I]l.
(8.4.19)
The first two lerms on thc right-hand side of (8.4.19) arc familiar from thc power utility model (8.2.6), while the last IWo terills arc new. The third term (Iillear in (SI reflects illlertelllporal suhstitutioll. or mean-revcrsion in marginal utility. If thc surplus conslllllption ralio is low. the marginal utility of COIISIIIIlIllioll is high. However, the surplus COIISUlllptioll ratio is cxpcctcd to revert to its mean, so marginal utilily is expeclcd III f,llI in the future. Therdi)J't', lhc consumer would likt' to borrow and this drives up the cfJuilihrillm risk rree interest ratt'. The lillll'th In", (lint'ar in IA(s,)+1 )~) reflects prccautionary savings. A~ tllH'cnainlY in(Teases, conSlllllcrs hecome more willing 10 savc and this drives down tll(' cqllililllilllll riskless intercst rate. If this mode! is to gCl\crat\~ 51.lhk re.t! intnest r"t(·~ like those observed in the elata. the serial correlation p"r.lIllt·tc(' c/! lIIust he n('"r (JIIC. Also, the scnsitivil), tllnrlion '\(SI) III list declinc wilhl, .'" IIt"t IlIl(Tllaillly is high whell
'5»
I, is low alld 1IlI' prec aUli ollar y savi llg I('rlll offs els the illte r\(,lI Ipor al subs titulioll 1('1"111, 1/1 1;11'1, Cam phel l a/ld Coc hral le para mel rize Ille A(I,) fUllc tioll so thai Ih('s(' 1\\'0 1!'rIllS exac lly oftse t I'ach olhe r ewr ywh ef(', illlp lyill g a ('oll Slal llris kll's s illl(' n'sl rail', EI'I'II wilh a cOIISI,uII riskll'ss illl!' r('si ralc all(lralldolll-\~alk COII SIIlIlPli,," , Ihl' ('xll' l'Ilal -lIah il IIllulcl (';111 prod uct' a larg e equi lY pn'lI li 11111 , vola lile sloc k pric l's, alld prt'd irrah ll' exce ss Sioc k relll fllS, The hasi c mec hall ism is lillH'-Vari,llioll ill risk al'('rSioll. Whe ll COllSlllllPlioll falls n'lal i\'(' 10 habi l, Ihl' resu ltillg illcr casl ' ill risk aver sioll dri\' es lip Ih(~ risk prcm iulII 011 risky ;Isseis such as sloc ks, This also dri\' es dOWII the pric es ofSl ocks , help illg 10 expl ain why sloc k r('lu nls an' so milc h mon ' vola lile Ihal l cons lllllp tioll grow lh or riskll'ss re,ll illll'fl'SI ral(' s. Call 1phl 'li alld (:oc hral le (I~}~}:l) calih rale Ihei r lIIo dd 10 US clala Oil (011 sum plio ll alld divi de/l ds, sol vi II!!; /ill' ('qll ilihr ium sioc k pric es i/l Ihe Irad ilion of Ml'h ,.a .lIl1ll'leSCOI\ (1~)H!'i). Thl' l'l' is also SOIllt' work Oil hahi l form alio n Ihal us('s al"llial sloc k rl'lll l'll dal,l ill Ihe Irad iliol l of I lOlliSI'll allel Sillg le10/1 (I~}H:!. I~}H:I). llea lo/l (Iq~} :l), I,ll' ('xa mpl e, eSli malc s all illlC rtlal -hab il mod i'! allo\ \'illg ((lr lillll '-agg rega lioll of Ihe dala alld for som (' dura bilil Y of Ihos e go(u ls lilrll lally desc rihe d as 1l011l1llrablc in Iht' nali onal inco llle ac(,OIlIlIS, Dllr abili ly (';111 1)(' Ihou ghl O(';IS Ihl' opp osit l' or habi t lilrll lalio ll, ill th"l COIlSUIliptioll (')oq)(,lIl1illll'l' tod" " lowe rs Ihe Inar !!;in allit ility orco llsIl IllPtioll ('XPI'llIlilllr!' IOIlIOITOW. (!ca toll lilld s Ihal dlira hilil Y pred omi nall 's al hi!!;h fn''1u('III'i('s, alld habi l ((lI'I llatioll al lowe r i'rt'q llt'nl 'ics. Ilow cl'cr his hahi l-f(ll 'Illal ioll Illo dd. likl' Ihe silll ple pow ('r ulililY 11111111'1, is l'I:ic('ft'd statislically. BOlh Ihes e app roac hes aSSUllll' Ihat a~grt'R,l\e cons ump lioll is the IIrivill~ pron 'ss ror Illar gina l IIlility, All altcl 'Ilal ive view is Ihal , for reas ons ,lis('uss ed ill S('('lioll H.:\.~, Ihe ('OIlS II IIIpt iOIl or slo(' k mar ket inve slor s lIlay 1101 bl' ad(' '!"a td" pro" il'd hI' Illa( 'l'oc ('oll omic dala on le IIwa n alld siall dard devi alio ll, hUI n('e( \ nOI he hi~ltl)' ('OI..-l'Iall'd wilh ag!!; I'I'gal(' ('OIlSlIllIplion.
S,,,,2 I"wh oillg im/ M(/(II'II IIf !'rrfn 'Pnrr.1 I'SydlOlo!!;iSIS alld ('XPl'l'illl(,llt~tl l'('ol lolll islS haVl' foul ld Ihal ill ('Xpl'I'illlelllal Sl'ltillgs, p('op ll' 111;1\..(' dlOi ('l's that dilfe r ill sew !'al !'I'SP I'ctS frOIll lhe siall dard III(u ld 01 ('X 1)('('\ ('II IIlilil Y. In r('sp olls( ' 10 Ih('s t' lillllill!!;s ullo rtho dox "psydlOlo!!;ical" III1Hlels or prl'f('I'I'III't'S have he(,11 sugg esle d, and SOIll(, 1'1'1'1'111 resl 'ard l l"ls IlI'g ull 10 appl y Ihl's l' tnol lds 10 asse l pricill!!;.I H
1"(1"'1111 ~"I,,'o.,II""'"'II"" ill,
10,,1 .. 11"/:,11""",,1 R.·(", .. (HI1I 7) alld KIt'I " (1\11111),
lJ.4. Mort' Gmaal Utilit., rtmctionl Psychological models may best be understood by comparing them to . standard time-separable specification (8.1.1) in which an investor maximizes (8.4.20)
This specification has three main componen15: the period utility function U( e,), the geometric discounting with discount factor 8, and the mathematical expectations operator E,. Psychological models alter one or 'more of these components. The best-known psychological model of decision-making is probably the prospt'rllheory of Kahn em an and Tversky (1979) and Tversky and Kahneman (l992). Prospect theory was originally formulated in a static context, so it does hot emphasize discounting, but it does alter the other two elements of the standard framework. Instead of defining preferences overconsumption, preferences are defined over gains and losses relative to some benchmark outcome. A key feature of the theory is that losses are given greater weight than gains .. Thus if x is a random variable that is positive for gains and negative for losses, utility might depend on
.
vex) ===
if x ~ 0
{XII-~ly~1 ,_,
\
-A~ 1-)'1
(8.4.21 )
if x < O.
Here YI and Y1 are curvature parameters for gains and losses, which may differ from one another, and A > 1 measures the extent of loss aversion, the greater weight given to losses than gains. Prospect theory also changes the mathematical expectations operator in (8.4.20). The expectations operator weights each possible outcome by itS probability; prospect theory allows outcomes to be weighted by nonlinea~ functions of their probabilities (see Kahneman and Tversky (1979» or by nonlinear functions ofthe probabilities ofa better or worse outcome. Other, more general models of investor psychology also replace the mathematical expectations operator with a model of subjective expectations. See for ext ample Barberis, Shleifer, and Vishny (l996) DeLong, Shleifer, Summers, and Waldmann (l990b), and Froot (1989). I In applying prospect theory to asset pricing, a key question is how the benchmark outcome defining gains and losses evolves over time. Benartzi and Thaler (1995) assume that investors have preferences defined over returns, where a zero return marks the boundary between a gain and a. loss. Returns may be measured over dilTerent horizons; a K-month return' is relevant if investors update their benchmark outcomes every K months. Benartzi and Thaler consider values of K ranging from one to 18. They show that loss aversion comhin~d with a short horizon can rationalize investors'
334
8. IlItrrtfllljJOTilI J~qllili{JI-;1I1II Modell
unwillingness to hold stocks even ill the face of a large eCJuity premium, Bonomo and (~arcia (I !m:{) obtain similar results in a consullllHioll-hascd model with loss aversioll, In related work, Epstein and Zin (1 !)90) have developed a parametric version of the choice theory of Yaari (1987). Their speliliGllion for period Ulility displays jirJt-ordrr 11.111 flVelJioll-the risk premium reCJuired to induce an investor to take a small gamble is proportional to the standard deviatioll of the gamule rather than the v,lriance as in standard theory. This feature increases the risk premia predicted by the model, but in a calihration exercise in tile style of Mellra .\1lU l'rescull (198:», Epstein and Zin find that' they can lit only about one third of the historical equity prellliulII. Another strand of the literature alters the specification of disulIllltinp; in (8.4.20). Ainslie (1992) and l.oewenstein and l'rclee (1992) have arp;ued th,ll experilllen tal evidence suggests not geometric discounting hut hYIIr.rb"lic discounting: The discount factor for horizon K is not /)K uut a functiol\ of the form (1 +/)IK)-~'/~I, where both /)1 and /)2 are positive as in the standard theory. This functional form implies that a lower discount rate is used for periods further in the future. l.aibson (1996) argues that hyperbolic 'jiSCoUntin g is weIl approxi,matedh Y~utility sP,ecification
I I i
U(e,)
+ fiE, [2:>51 U(C,+j)] ,
(8.4.22)
~I
fi
\yhere the additional p.\rallleter < I illlplies greater discoulllinp; over Ie ne\(t period than between allY periods further in the future. Hyperbolic discounting leads to time-inconsistent choices: Because the ( iscount rate between any two dates shifts as thc dates draw nearer, the ptimal plan for those datcs changcs ovcr tillle evcn if no new infi)J'lllation . rrives. The implications for conslImption and portfolio choice depend n the way in which this time-consistency problcm is resolved. I.aibson (1996) derives thc Ellicr equ,llions for conSlllllption choicc assulllinp; that tie individual chooscs each pcriod's conslllllption in lhat period without I eing able to constrain future consumption choiccs. Illlcrestinp;ly, he shows t lat with hyperbolic dis(oullling the elasticity of intertelllporal suhstitution i. less than thc reciprocal of the coefficient of relative risk aversioll evell \~hen the period utility fUlIction has the power form.
~
8.5 Conclusion Financial ecollolllisL~ have lIot yet produced a generally accept(,d model of the stochastic discoullt f;lrlo)"' NOl1l:lheless sul>stalltial prop;n:ss has heen made. We know that the stochastic discoullt f;lr\or must be extrcmely volatile
['/lib/nils
if it is to \:xplaill the cross-sectional pattel"ll or asset retlll"llS, We also know that the cOllclitionalexpectation of the stochastic ciis(,(Hlllt factor IIllist be cOlllp;lrativc1y stable in orcler to explain the stability or the riskless real intert'st rate. These properties pllt st'wrc restricliolls Oil the killcls of asset priring llIoclcls Ihal «Ill he consicinc(\. There is increasing interesl in the idea thai I isk ;tv('Ision lIIay vary over tilll\: with the state of the eCOlIOllly, Tilllc:-v;lryillg risk ;lv\'rSiOIl C',I11 explaill the i
Problems-Chapter 8 Prove that the benchmark portfoli~ has the properties (1'1) throllgh (l'!i) stateel Oil pages ~!IH allel :~OO clf this chaptn.
8.1
8.2 COII,ider all ecollolllY with a rC'I'I'\'S\'lIlalivC' agl'lIl who has power Ulility with coefficicllt of relative risk aversioll V. The agent receives a lIonstorahle endownlellt. The process for the log elldowment, 0)' crl'livaiently the log of cOllsumption (" is
where the coefficient ¢ lI\ay he either positive or ncg
Ihl' 10J.!; ass('1 11'1111"11 (callihis Slllll the "prelllilllll" Oil tltl' assel). is propolliollallo 1111' cOlldiliollal covari;\Ilce o/"Ihe lOR assl't relllrn willt rOllslllllplioll ).\1"O\\'lh, Whal is Ihl' slopl' nll'r1il'il'1I1 ill Ihis rdalilHlship'
" "
!t2.2 assel
To a dosl' ;Ipproxilllalioll. Ihl' IlI11'XPl'l'll'(\ relllnt
011
allY
i call Ill' \\'rilll'll ;IS
1",./11 -
F,I",.,II = I·.'ltl
[tflI6t1,.t+I+i] -EI[tfli6t1,.Itl+l] J=-IJ
/ __ 0
- (Eft
I
['f; pir",tl+i] - I~, [t pir'.lt 1+1])' }.-I
}-I
wh(,11' tI,.1 is IIH' (lividclld paid Oil assel i al tilllt' I, This approxilllatioll was dcvelopcd as (7.1.~:1) ill Chapter 7, i, lise Ihi~ expressio\l 10 caIn date the IItll'xpe('\cd n'llIl'II 011 all (''1l1il\' whil'h pa),s agJ.!;1'('J.!;alt' ('()IlslImption as ils dividend. ii. l I~I' Ihis \'''pre~sio\l 10 Cakllbll' Ihl' IIIH'XPlTll'd t'('tllrtl 011 a r('al cOII~ol bOlld which has a fix('d real dividl'lId each pniod. K.2.:~
i.
(:akll\;tl('llw (''Illill' prl'lIIilllll alld tIll' ('ollsolbolld prl'llIillltl.
ii. Show I hal I h(' boltd pr(,lIIilllll has the opposite sigll to 1> altd is proportiollalto the sqllare o/"y. Cive all econolllic inlerpretalion of Ihis r('slllt. III. Sltow Ihallhe e'l"iIY pn'lIIilllll is always htq-\l'r Ihan the hO\ld prl'lIlilllll. and Ill\' dilfl'1'('\H'C betweell Ihelll is propol'liollallO Y, Ci\,(, all ecolloltlic illl('rpr('lalioll of Ihis r('sllil. i\', Rdall' 1'11111' disClI\~ioll 10 III!' I'llIpirirallill'l'allll'l' 011 Ihe "('QlIil\' pr(,lIIilllll pIlZlk."
R.:l
C:olIsidn a I\\'o-pniod world wilh a cOlltillllllll1 of COIISlIllIlTS. Each COIlSlIlIlI'r has a raltdotll ('llIlowlI\('nl ill Ihe S('coIHI period alld COIISIIIIII'S (111)' ill the sl'nl\HI pl'liod. III till' firsl pl'l'iotl. Sl'l'lIrilil's arc Iradl'd bill 110 IlIotH,\, challJ.!;!'s hallds IInlillhl' Sl'(,(lIld pl'l'iod. All COIlSIIlIll'l'S havl' loJ.!; IItililY OVl'r sl'colld-pl'riod (,Ollstllllplioll,
H.:l.l
SUppOSI' I hal all ('( HISI 1Illl'rs' l'\l
8.3.2 Now suppose that in the second period, with all consumers receive m; with probability 1/2. a fraction (I-b) consumers receive m and a fraction b receive (l-a/b)m. In the" fIrst period, all consumers face the same probability of being in the ". laHer group. but no insurance markets exist through which they, can hedge this risk. Compute the expected return on the claim. defined above. Is it higher or lower than before? Is it bounded b~ a function of a and m? I, 8.3.3 Relate your answer to the recent empirical literature on the, determination of stock returns in representative-agent models. Tol what extent do your results in parts 8.3.1 and 8.3.2 depend on thel details of the model, and to what extent might they hold more! generally? ' Note: This problem is based on Mankiw (1986).
\
•
. '(
9 Derivative Pricing Models
THE PIUCI:,\(; OF OPTIONS, warrants, and other "nll/alilll' securities-financial
secnritics whose payoll's depend on the prices of other securities-is one of the great successes of modern fmaneial economics. Based on the well-known La;v of One Price or no-arbitrage condition, the option pricing models of Blac;;' and Scholes (1973) and Merton (I ~17:)b) gained all almost immediate acceptance alllong academics al\(I investment professionals that is unparalleled in the history of economic science.' The fundamental insight of the B1ack-Schole.~ and Merton models is that under certain conditions an option's payoff (',1lI he exactly n~plicated by a particular dynamic investmcnt stratcgy involving only the underlying s((}ck and riskless debt. This parlicllbr strategy fIIay he constructed to he sl'ljjlwlI/rillg, i.e., reqlliring no cash infllsions except at the start and allowing no cash withdrawals until the option expires; since the strateh')' replicates the optiol\ 's pay ofT at expiration, the initial cost of this sdl~linallcing investment strateh')' mllst be identica\to the option's prict", otherwise ,Ill arbitrage opportllnity will arise. This no-arbitrage condition yields IH)t only the option's price but also the means to replicate the option synthetically-via the dynamic investlllen t stratq,')' of stocks ,lilt! riskless debt-if it does 1I0t exist. This lIIethod of pricing optiolls has since been used" to price literally hundrcrls or othcr types of dcrivative sccurities, sOllie considcrably more COlliplex th,lII a ~illlplc option. In the majority or these cascs, the pricing formula can only be expressed implicitly as the solution of a parabolic partial difrerential equation (PDl~) with boundary conditions and initial values determined by thc contractual tcnlls of each security. 'ji) obtain actual prices, the 1'01': IIIUSt be solved 1Il1llicrica1ly, whirh lIliv;ht have been problematic in 197:~ when mack and Scholes and Merton IiI's! published their papers but is 1I0W cOllllllonp1acc thanks to the hreakthroughs ill rOlllputel' I See 1\(01"11.,,(,;11 (I~'~':i, (:h"I"'" II) lor ,. li"dr "n 'JlIl.' 01 III<' i.llcll,·, 111,.1 hi,'ory 01 I Itt' m"ck·Srhol,·,/t.kI"lOIl Oillioll/ll irill~ /CII"IIIIII,,"
I),
/J,.,.illlllil','/'ririllg '\/11111'11
ll'(hllolo~\' 0\'('1" the p~lsl litree dec~lIks, Ahholl~h a dclailcd discussioll or "('ril'ali\"(' pri( illg IlIo,It'ls is 1)('\'011" Ihe scope or Ihis ll'xl-llH're an' lIIallY ullin ('Xrdlt-III SllllITt'S slll"h ;1., (:ox alld Rlibinsleill (I!IH!,), 111111 (I!I!I:\), alld Mt'ftoll (I ~1'lfI )-W(' do prol'ide IIrid' n'vil'ws Hrowlliall !\lol iOIl ill Se("(ioll ~),I alld f'. Ic-I"I 011 \ deril'alioll or Ihl' Hlack-Scholes Ii 11"111 lila ill Seclioll !l,:! ror rOIll'I'ninl("(', 11"1111 i("a III" ;d IhOllgh I'lil ill).!; d!'li 1';11 il"l' securi t iI'S is oftI' II It igl dy COl lip II tatioll-illtellsin', ill prill("iple it !c;II'es ItTI'lillle room ror Iradiliollal slalistical illfl'n'll.~ WI' sh'll! t'llll-;i(\t'r llollparallH'lrir d('ril'ali\'(' pririll!-\ modl'ls ill Chapll'r I~, and (Ints Oil i,SIl('S SllITOUIHlilll-: p;lrallH'lric Illodds ill S('nioll 9,:t Th(' sl'colld aspl'cI ill\,oll'~'s the pricill~ or path-depelld"111 dl'rivalive's hr MOIlIt· Carlo simulalion, A dnil'alil'l' security is saitilO 1)(' Imlll-rl''IH'IIr1m[ ifils payolfdl'lll'lIlis ill SOil\(' 11';1\' Oil Ihl' ('nlin' /mlll oflht'lllHkrl),illg assel's p,.i(T dllrill).!, Ihe dl'ril'alil'l''s lili-, For (·xample. a pllt optioll which giV<.~s Ihe holdl'r Ihe righl 10 selllhe IIlldlTl~'illg 'IsseI al ils aVt'rag(' prin'-where Ihe avcrag" is calntl"ll'd 01'1'1' lhe lili- or lhl' OpliOIl-is palh-(lt-pelldl'lll I)('('alls(' IIII' awr;l).!,e pri'T is a fllll("lioll of Ihl' IIl1dl'r/yill).!, assl'l's elllirl' price palh, Although;1 ho\\' 'III;III'li('al appl'llXilllalilllls 1'01' pricillg" pal h-tll'j)('1 Hkill tlnil'alil'l's do ....;i,l. hI' Ell' Ihl' IIlosl dkrlil't' melhod Ii II' pridlll-: Ihl'llI is IIv M Illl 11' (:,lI·lo SilllllLilioll, This r .. ises s('\'('ral isslI(,s sllch as ml'asuring Ih" a(Tllral I' or ,illllliall'd I'rin',. Ii<'tl'nllillillg Ih,' 111111111,',. of silllltlatifllls rt"I"irt'l\ lor a dl'sired 1<-1'1'1 01 a("(,lIran, ;IIHI tlesigllilll-: Ihl' simllialiolls 10
or
:'1" lunll.,,' 1n the' lI,ulrlit'II,11 p.II;IIIU'IIU' .II'I"O.lf}) ill h'hic h ,ht' prin' p,.on'" "f ,he" hillg .",,,"t j, IlIlh "I)!'( ili"d "1'111.1 lillilc' IIlIlIIhc', oJ II II I.. 1I4I\\,1I P;II,IIIWII'I'. c.g ... 1 IO!!.llurIII;" dillll,iull \,ilh 1111)..1111\\11 ~hlll .tlut \111.111111\ p.tt.ttHl'klS, ~I (l(H'par~'HIc.'Hic: .'pproac.h dot,!'!.
III It kl
111)1
"p"\"ih"
,t",
1'1"" ))lOt
,q~lIttlil\ t OlitlillOlh,
t"",
"'plu iii\" ;11111 ,IIIt'mp" It) inkr it hUll) Iht·
,tlt.l
\IIHIt"
Iti,Hll~lhlt·
9.1. llmw nian MOlion
mak e the most econ omi cal lise of com puta tion al reso urce s. Ihese issues in Section 9.4.
9.1 Brownian Motion For cert ain financial appl icati ons it is ofte n mor e con ven ient to mod el prices as evolving cOlltinllously thro ugh time rath er than discretely at fixed dates. For ex·,llllpk. Men on's deri vati on of the Black-Scholes opti on-p ricin g foJ. mula relies heavily on the ability to adjust port folio hold ings cont inuo usly in time so as \0 repl icate an opti on's payoff exactly. Suc h a repl icati ng portfol:lio stratCj.,'Y is ofte n impossible to cons truc t in a disc rete -tim e selling; henc~ pricil)g formulas for opti ons and othe r derivative secu ritie s are alm ost alwa ys deri ved in cont inuo lls lime (sec Sect ion 9.2 fora mor e deta iled disc ussi on). l 9.1.1 Constructing Brownian Moli on
The lir~t formal mat hem atic al mod el of 11nanc~a~ asset prices-de~e1oped Bac hell er (1900) for the larg er purp ose of pnc mg war rant s trad mg on the; Paris HOl lrse -wa s lhe cOlltinllO\lS-t iJnc rand om walk, or Brow nian mot ion. ' The refo re, it shou ld not be surp risin g that Brownian mot ion play s such a. cemr,11 role in mod ern derivativ e pric ing models. This cont inuo us-t ime proc ess is closely rela ted to the disc rete-time versions of the rand om walk desc ribe d in Sect ion 2.1 of Cha pter 2, and we shall take the disc rete -tim e r,mdolll walk as a slar ting poin t in our heur islic cons truc tion of the Bro wni an, mot ion. 4 Our goal is to use the discrete-lime rand om walk to defi ne a, sequ ence of cont inuo us-t ime proc esses which will conv erge to a cont inuo ustime anal og of the rand om walk in the limit.
by\
The Dism le-T imf Ran dom Walk Den ote by {IJk I a disc~ete-time rand om walk with incr eme nts that ,tak e on only two values, 6. and - 6.: with probability 1( with probability 1('
EO
1-
1r,
(9.1.1 )
whe re (k is 110 (hen ce p. follows the Random Walk 1 mod el of Sect ion 2.1.1 of Cha pter 2). and pu is fixed. Con side r the following cont inuo us-t ime proc ess JJ,,(t), I E 10, TJ. which can he cons truc ted from the disc rete -tim e process :lTh"r~ an' tlvO n(tlahle excep tions: the equilihriUll1 appr oach uf Ruhinstein (1976). and til .. hillOlni,,1 "ptiOll'pricinl: \node l of Cox. R()ss. and Ruhinstein (1979 ). ~Fo .. ;,11''' ''(' rigoro,,~ cll'riv.uioll . Sf'" nillillgsJry (J~)(;H, Section 37).
342
9. Denlm/illl' [Tiri"g Mtldfis /'(1)
r,6
! I
4~
1
I !
36
26
16
O~----~----r---~r---~-----r-----r----~--~
2"
()
Figure 9.1.
:;"
4"
5"
T= ,,"
Sam/,ie /'al" of II /)i.I(TPle-Time /umtUlm WillI!
I
(p,l. k = 1•...• 11 as follows: each of length h
Partition the
tinl(~
interval [0, T] into
= T/ n. and define the process
E
[0, TJ,
11
pieces. (!1.1.2)
where [x] denotes the greatest integer less than or equal to x. /',,(1) is a rather simple continuolls-time process, constant except at limes / = "h, k I, ... , n (see Figure 9.1). . Although P.(I) is indeed a well-de!lned continuous-time process, it is still essentially the same as the discrete-time process II, since it only varies at discrete points in time. However, if we allow 11 to incr~ase without hound while holding T fixed (hence forcing h to appmach 0), then the lIumber of points in [0, T] .tt which 1,,,(1) varies will also increase without bound. If, at the same time, we a
=
E[j,,,(T) 1
n(rr - rr')6.
Varl/,,,(T) I A.. 1I increases without bound, the lllean ,\Ill! variance ()r P,,(
(9.1.3) (~1.1.4)
on will also in-
9.1. JJmlllllifl/l I\-/olioll
crease without houlld if I:;,rr, and rr' are held fixed. '1.) ohtain a well-defined and 1I01ldegencrate limiting process, WI' must maintain finite mOlllenL~ f()r III/en as 1/ approaches infillilY. III parlicular, sillc(' W!' wish to ohtain a cOlltilluous-lillle version of lhe r,lII(\olll wOlI I<. , we should expclt the llIean and v,lriance or the limiting process II( T) to he lilll'OIr ill T, as in (2.15) and (2.I.G) for the discrcte-time random walk. Therefore, Wl' 111IIst adjllst L\, rr, and rr' so that: T
lI(rr - rr')1:;
-(rr - n')o.
T
41lrr rr' I:; ~
,',
--Inn 1:;-
->
It
ILr
->
It
(T
~
r.
(\1. \'(i)
Ji,.
(9.1.7)
and this is accomplished hy selling: rr
n
,
-I
(
2
~
ILfi,) . 1+-(1
(I _fi,) , 11
2
I:;
(T
(1
The adjustlllt'llts (!1.1.7) imply thai the step sil.!: dClTeases as " inCTeases, but at the slower rate of lifo or fit. 'I'll!' probabilities cOllverge 10 ~,also althe I-ate of fit, alld hence we write: I
n = ~+()(fi,),
rr'
=:
~+()(Jh),
I:;
= 0(,/1;).
(!).I.H)
fi,)
where O( denotes terms that are of the same WYllfllOlic Older as fi,." Therefore, as II incrcascs, thc random wall<. PIICI') v
Ii"'~ I'(/d
1.-.11
<
"'-'.
A lilllnio" f(iI) i, ,aid 10 hl' ol""",,dlt'r a'YIIl!,loli( ordt'l a.' gUIl-dell"led Ill' I (Ill
~
o(/:(II))-if
I (iI)
lilll-- = tI.
1,-." 1'( II) SOIlH' l'X;UIlPIt'·" ora~}'IlIplOli("
ordl'r rc.'i.uiuu,,, al(' ,,- 1'.// -
(Ill,l.
'.J, IJrrilllllil'l'l'ririllg J'Vllltll.fS
,"I
'1111' (,'olllill/lll/I\:rillll' I,illlil
By calndatillg till' IlIoll\l'lIt-gl'neratillg lilllrtion of 11,,( '1') alld taking ils lilllit applOadl(,s illlillil\', il IIla\' he showlI Ihal Ihe dislrihutiou o/" II( 'f) is 1101'wilh IIIt'all/l'/' alld \';11';;111('(' (J~,/,; Ihlls /1,,('1') cOllverges ill dislrihulion to a N(I' 'f'. o~'/') ralltillill \,;lri;\(,II', In bet, a nllllh sll"!lIlg!'r 1I0lioll o/" ('onvergl'nf(' relates 1',,( 'f') 10 I'( which illl'oke.s Ihl' /lIIill'·"il/l/'III;OIlI/I tii,l/l'ihllliol/.\ (FDDs) or Ihe 111'0 slochaslie processl's, ;\" FI)() of a stochastic process /1(1) is the joint distrihlllioll o/" ;111\' lillil(' 1111111\1('1' or ohsl'I'l'aliolls of thl' stochastic pron'ss, i,e" {(/,(lI), /'(/~l."" I'(lm)), 11'11('1'(' () <. II < ,,' < III/ ::: T, It call hI' showlI Ihat 1//11111' F\)lh of IIII' shuh;ISlic (lI'tH'l'SS /,,,(1) (lIoljllsl Ihl' randolll \'ariable 1',,( '1')) nlll\'('1 gl' 10 Ihl' FllI)s or a \\'1'1I·ddin!'d stochaslil' pron'ss I'( I).'; This illlplil's, 1(11' I'xalllplt', Ih;1I Ihl' dislrihlliion of 11,,(1) convl'rgcs 10 Ihe dislrihlliinn of 11(1) Ii II' 01111", 10, '/'I (not just at T), th,lt thejoillt dislribulion of 1/1,,(0),/1,,(:\),/1,,<711 ('olll'ergl's 10 lhcjoillt distrihution o/" 1/,(O)'IICI}./I(711, allds()OIl, IlI'addilioll 10 IIII' lIorlllalil), o/" I'( 'f'), Ihe Jlor/uHlir 1"'IIt'I',1.I /1(1) possl'sses Ihl' lilIlowing IhnT pl'IlpI'nil's:
as
1/
111011
n,
(Ill) For
0111\'
II ;lIul/~
such Ih;1I 0 -:: II
<. I,~
:::
'1': (9,1.9)
(R2) For
0111\' II,
I~, I"
Ihl' ill('\'I'IIU'1I1
I~
::: 1:\ < I, ::: '1',
/'Ud,
11(/1) (RJ) Th!'
alld I" SllI'h Ihal 0 ::: II <
I'( I~) - I'( 'I) is slalislically indepcnc\l'lIt of the intTcllIcnl
S;llIIplc I';tlll~
of 1,(1) al(' ('olltiIIlIOltS.
II is a rl'lIlarkallll' bn liI;1I 1,(1), whit'h is Ihl' n'ldll'al{'c\ arillilllt'lir llm1< l /liflll I/w/ioll or II'imn/mlff'II, is l'ollll'kll'ly cllar"l"tl"ri7.ct\ by these thrt'c properties, If WI' St't /1 == () ;11111 (J == I, WI' ohlaill slal/t/ard IImul1Iiall 11I/11ioll which we shall 111'11011' hI' /lUI. ,V('orllillgly, "'(' lIIay ..c-cxprcss 1,(1) as 1,(1)
=:
111
+
0
I E
/1(1).
[0, 'FI,
(!),1. 10)
'Ii, clt-vc\op !tilthI'I inluition 1
EI I'(/)
I'( I,,) I
1,(111)
\';11"1/,1I)
I'( I,,) I
(T~(J - 10)
1- 11(1 - III)
WI.II) (11.1.1 ~)
.. , lH' (lIH\I'lgl'l)! I' ill i Ilih. touplnl \\ith.\ In hnil "I fotltHtlulMl t .IHI',I II/.:",nn\, 1'1 ,llll'ci ".,."k I on\l'lgl'lif (',.1 jIO\\", 1111 1(111) ill d('II\ill~ ;1""IIIjIIO(if apP"","llIIotiiOlh lur III(' \I;lll,lic ;11 1.1\\" .. IIII,IU\' 1lIIIIlilH',II ,',lilll.lIlIl '. Sn' Billill~''''\' (1~lliH) luI' lunl",1' d"l .. ih.
9.1. IImwTliau Motion
····,.1
I"
Figurrl 9.2.
I II
,'iflll//Jlt I'fllh fI/IIl COJl(lililJlwl !:X/JUlfllioll
Cov[p(ld, PU~))
:::=
'1 ft Jlrorunilln Molion with Drift
COV[p(lI), p(t~) - pUt!
Cov[p
Var[pUI)]
== a
2
(9.I.I3)
PUI)]
Cov[p(lJ),!J(t2) -
+
+ PUI)]
tJ.
(9.1.14) (9.1.15)
As for the discrete-tim~ random walk. the conditional mean and variance of p(t) are linear in t (see Figure 9.2). Properties (82) and (B3) of Brownian motion imply that its sample paths are very erratic and jagged-if they were smooth. the infinitesimal increment B(t+ h)- B(I) would be predictable hy JJ(t)-BU-h). violating independence. In ract. obsen'e Ihat the ratio (JJ(t+h)-ll(l») / h does not converge to a well-defined random variable as h approaches O. since Var [
JJ(t
+ II) II
lJ( t} ]
=
II
(9.1.16)
Tllt,rt·rort" thl' derivative of' Brownian llIotion. Jr (I). does not exist in the ordillary St'llSt·. alld although the sample paths of Brownian motion are t'vl'rywht,rt, cOlllinllolls. Ihey art' lIowhere differentiahle.
46
r
9. Derivative Pricing Mudel.!
9. J. 2 Slocllfl.\lir f)iJJrrl'1ll/(lII~(f1Ult/01l\
Despite this fact, the infinitesimal increJllent of Brownian motion, i.e., the limit of B(t+h)-/J(t) as It approaches an infinitesimal of time (dl), has earned.the notation dB(t) with its own unique interpretation beGlltse it has become a fundamental building block for constructing other continlloustime processes. 7 Heuristically, /J(t+II)-8(t) can be viewed as Gaussian white noise (see Chapter 2) and in the limit as II becomes infinitesimally small, dB(t) is the "continuous-time version" of white noise. It is understood that dlJ(t) is a special kind of differential, a ~turlUlstic ·difft'Tl'1Ilial, not to be confllsed with the differentials dx and dy of the calCII'Ills. Nevertheless, dB(t) does obey some of the mechanical relations that ordinary differentials satisfy. For example, (!l.1.10) is often expressed in dilTerential form as:
I
d/J(t) = Ildt
+ a d/J(t).
(9.1.17)
However, (9.1.17) GunlOt be treated as an ordinary differential equation, and is called a stochastic differl'1ltial equatiun to emphasize this fact. For example, the natural transformation djJ(t)/dl = J1 + dB(I)/dl docs not make sense because d8(t)/ lit is not a well-defined mathematical ol~ject (although dB(1) is, hy definition). Indeed, thus far the symbols in (!l.1.l7) have little formal colltent beyond their relation to (9.1.1 O)--one set of symbols has been defined to be equivalent to another. To give (9.1.17) independent meaning, we mllst develop a more complete ullderstandinl-( of the properties of the stochastic differential dB(t). For example, since dB is a random variahle, what arc its moments? How do (dD)2 and (dB)(dt) behave? To answer these questiollS, consider the definition of dB(t):
d/J(I) == lim IJ(I h-dt
+ h) -
(!l.l.IH)
B(t)
and recall from (3 I) that increments of Brownian motion are normally distributed with zero mean (since J1 == 0) and variance equal to the dillcrencing interval h (since a = I). Therefore, we have E[dlJ] Var[dlJ]
Y
'!
E[(d/J)(dlJ)j
lim E[/J(/
!,-,II
+ II)
- U(t)] = 0
+ II) -
(9.1.19)
/J(t»~]
dt
(!I.I.~O)
liIllE[(lJ(t+h)-/J(t))~1
lit
(!1.1.21)
lim \-:[(/J(I
"-+"1
1,-.. ,11
7/\ rOlllpl"I" ali(I rit;" .... "s "xp",ilio" of II, ow"i"" 1II0tio" all,1 S(,;r1""lir dill""-"li.,1 '-'I"aIlio"" i. heyon" the .r0l'~ of Ihis I<"X I. 11''''1. I'orl. ,,,,,I 510"" (1 \172. Chap(('" 4-li). M<"lIoll ;( I !I!IO. Chapl,'" :1). ;11,,1 S, I""., (I !IHO) 1" ovid,' <"x("dl<"111 ("ov<"""t;<" of Ihi. 111;11,·, i,,1.
9./.
347
1I11111111i(/1i Molio/l .
Val'/ (tI/I)(1I1I) / (!l.I.22)
0(1It)
lim E[ (/J(I
I-:/(IIII)("I)J
),-+11,
lim 1-:/ (/J(I
Var/(dll)("I)J
II_,ll
+ Ii)
- II(I))/d = 0
+ h)
- n(l))~"~ / =
(!).1.2:{) o(IIt). (!l.1.24)
From (B I) alld (9.1.1 !J)-(!1.1.20) and we see Ihal II lIt I) m,IY he viewed as a llormally dislributed random variable with l.e\'O IIIl',11\ ami illlillilcsimal variance dl. Althollgh a variance of til may scem like no variance at all, recall that we arc in a world ofilliinitesimals-after all, accordillg to (9.1.17) the expected vallie of "1)(1) is Ildl-so a variall('e or III is not lIegligihle in a relalive s('nsc. However, a variance of (dt)~ i5nq~ligibk in a rdalive sense-relalive 10 dl-since the sqllare of an infinilesimal is lIluch smaller lhan the infinitesilIlal itself'. Il'we treat terms of order 0(111) as esselltially I.ero, lhen (9.1.21)(:1.1.24) shows lhat (dlJ)t and (diJ)(dl) .... e hoth llOll-slochastic (since the vari'lIlces of the right-hand sides arc of order 0(111) hence the relatiolls (dm'! == til and (dB)(dt) == 0 are satisfied nOljusl in expectatioll bUl exaclly. This yields lhe well·knowll JIIultiplicalion rules lor slochaslic dilkrelliials summ;lrized ill Tllllc !I.I. To sec why II\('s(' rllks ;lle IIs('llIl, OIlSlTI'('lhal we "[able 9.1.
can
1101"
MII/li/dimlio"
""/1'1/'"
II",."(/\Ii, tli/lI'II'"llIdl.
x
till
1111
III
o
III
o
()
ill
calcillale (dl)~: ("I)~
(Wil
+ arilJ)~
Ilt(dl)~ f- fT1(lIm~
atlil.
+ 2Ita(II/i)(rit) (9.1,26)
Thi, siltlPk calcillation shows lhal althollgh 1If! is a randolll variable, (dP)t is llOI. I t also shows Ihal til) docs hehave like a r;Ill
If the ;lrilhllll'lic Browlliall lIlo1ion IJ(I) is I.lkell 10 he Ihe price of sOl1\e assel, I'ro)l('\'ly (III) shows Ihalprin' IhaJlgn O\'l'1 ;111)' iJlIlTval willlJe JlOI~ m;llly
('lit in' rl'al lilll', lIonllall}, di~lrihllll'd pricc ch;lIlgcs imply Ihal pricl's Gill hl' lIl'galiw widl posilivl' prohahility, Ikci\usl' virtually all financial assels enjoy lilllited li;lhililr-111(' Illaxilllllm loss is capped at -I OO/x, or Ihl' total illVl'Sllllt'lIt-IH'g;lIiv(' pric('s ;\1'(' ('IlIJlirically implallSihle, As ill Sl'clioll\ I.'t.:! of Chaptl'r I and 2. I.I of Chapter 2, Wt' lIlay dimilIalt' Ihis prohll'lIl hI' dl'lillillg II( I) 1(1)(' Ihe lIalurallogarilhlll or prin' 1'(1). lIndn Ihis (1I'Iillilioll,/111) call Ill' an arilhnll'lic I\rownian motion wilhollt vio\;lIing limiled lia),ilil\" silln' Ihl' prin' 1'(1) I'/,(I) is always nOIl-ncgalive. TIll' pritT prlln'ss 1'( 1\ == /,/MI is said to ht' a ~I'OI/II'/lj( Umlllll;fll/ 1II111ioll or I()~n/lr II/{/{ diffusion. W(' shall examint' Ihe statislit'al propcl'lics ofholh arithmetic ,Il\(l gt'lllll('lrk Brownian lIIolion ill rOllsidt'rahly lI\on~ detail ill SerliOIl (1.:-\.
=
/11;\ /.1'11111/(/ Although till' (irsl ('olllplelt' nlatht'llIatit'allhelll)' orBrownian lIIotion is dll(, 10 Wil'ner (I\l~:l) ,x il is Ihl' st'JIIinal wnlrihlllion of Itti (19!i I) thaI is largely rcspollsihlt, lill' I he t'IIOrlllOIlS 11111111)('1' orapplir
,il~f
+ ;;-.-" (ti/) • II/r
2
.
(9.1.27)
Thl' lI\odl'st tl'rlll "1('11111101" hardly dOl'sjllstire 10 Ihe widl'-ranging impact (~).1.~7) has had; this I'owl'l'ful tool a\lows liS lO quantify tht' t~voilltion of (olllpll'x stochaslic sysl('nlS ill ;1 singll' stl'p. For example, Il't /) clt'II0le the IO~-l'rin' pmn'ss III' all a\st't alit! SIlPPOS(' it satisfit's (9,1.17); what an' the dynalllics of Ih(' prict' l'roll'ss 1'( /) = I,/'III? Ilt;'s I.t'mma providt,s liS with all illlllll't1i;ltt' alls\\n: ell'
KS,'('
.ll'li'UIi. Sillg"I,
\\,i"IIt,. \. Il':\(',udl ('('(HUllUifs.
j) I' -til) -I-
all
I C
a2 /'
"
-(tI/)·
! a/)~
(\).I.~~)
.lIul SIIIItII"I.. (I~)~th) ItH .tII ('x("('lIclll hi~llJ .. k.tI l(·t,o~p(,l"li\'t' 01
whit h illl
hlllc'~ "'\t'LIl ;Irlil
ks ,lluHII '.Vi('lIcr\ illtlU(,IIf(' OIlIlHHI«'nl
fill;tIU"j.tI
Y.2.
A lJriejR£lJilW of Derivalivt' J'rl(jllJ; MrlJlOds
dP
=
t Pdj)
::=
P(/1dt+adR)+~P(a2dt)
::=
(/1
+ ~ eP(dp)2
+ ~a2)Pdl + a PdB.
(9.1.29)
In contrast to arithmetic Brownian motion (9.1.17), we see from (9.1.29) : tltatthe instantaneous mean and standard deviation of the geometric Brow-' nian 1II0tion are proportional to P. Alternatively (9. i.29) implies that the' instantaneous percenlage price change dP / J> behaves like an arithmetic Brownian motion or random walk, which of course is precisely the case giv~n the exponcntialtransformation. Wc provide a considerably less trivial example of the power of lt~'s Lemma in Scction 9.2: Merton's derivation of the Black-Scholes formula.
9.2 A Brief Review of Derivative Pricing Methods Although we assume that readers are already familiar with the theoretical aspects of pricing options and other derivative securities, we shall provide a very bril"frcview here, primarily to dcvelop tcrminologyand define notation. Ollr derivation is deliberately terse and we urge readers unfamiliar with these models to spend some time with Merton's (1990, Chapter 8) definitive ~ treatmcnt of the subject. Also, for expositional economy we shall confine om attention to plain vanilla options in this chapter, i.c., simple call and put options with no special features, and the underlying asset is assumed to be common stock. 9 Dcnote by G(P(l), I) the price at time I of a European call option with strike price X and expiration date T > I on a stock with price P(I) at time 1. 10 or course, G also depends on other quantities such as the maturity date T, the strike price X, and other parameters but we shall usually suppress these arguments excep~ when we wish to focus on them specifically. I [owever, expressing G as a function of the currenl stock price pel), and not of past prices, is an important restriction that greatly simplifies the task of fincling r; (in Section 9.4 we shall consider options that do not satisfy
" 110""\'<'<, Ih" lechni'lllel reviewed in Ihis sec lion have been applied in simitar fashion II) 1i"·,.,,lIv hlll\,Ir~," or olher Iype. or derivalive secllr;lies, hence Ihey are considerably more ){~II ...."I Ih,," Ihey lIlay appear 10 be. 11I1{l"r,,1I Ih'lI " ((/1/ option gives the hold"r Ihe ri){hl \0 purchase Ihe underlying assel for X alld " /111/ "pliml gives Ihe hulder Ihe righl 10 ,ell Ihe underlying ;USel for X. A EUrr>/Nali oplil)l\ i, \lI1<" 11t"1 call he exercised only on the malurilY dale. An Ammcnn oplion is Ohe thai (';1111)(" t·Xt·n'is~d 011 or I"jo" th~ malUrily ,Iale. For simplicity we 5hall deal only with Ellropt'all oplioll' ill Ihis fh"pl~r. See ('.ox alld Ruhiosleill (I!lW», lIull (1993), alld Merton (l99(» for ill,lillllioll,,1 d"lails and difTerences belweell the priCing of American and Europt'an options.
9. /)rrillillillr I'lirillg Mudrls
this restriction). III ,l
(Al) TllI'rr rollolimilll.l,
1/1"(' (llId
110 mfll/u'l im/II'lji'flioll.l, r.g., 1(1).'1'.1, Imll.mflillll.1 f/).II.I,lhorl.lfIll'.f
lradillli is flllllill /WIlS
(I
lid ji"ifliulllrss.
(A2) Thrrl'iJ IIl1lilllilnl riJhll'.Is blllTIIll'illg IIlId ll'/uiil/g at Ihf {"(/IIlillllllll.l/y rUIII!wlII/til'll mIl' of rPlllm r; 11I'lIff II $1 illllf.I}/111'111 ill sllfh 111/ lI.I.Irl /lilt'/' IIII' lillll illlmlll[ T groW.1 Iu $1 . r or . ALtl'l'l/alillt'ly, if 1)( t) is Ihl' dllll' I /JI'ifl' of II di.\fI)ulIl bUild IIIl1luril/Ii (II rillll' T willi Jllfl' llallll' $1, 011'11 fo)' I E [0, TJ thl' bOlld /JI'ifl' dYlltIIllif.l (1)'1' giIlrII
fry
=
r{)(t) til.
(9.2.1)
9.(AJ) Thf.llork 1"10' 1(1',lI/lllif.1 (Ill' gil'('11 by f/ gl""l1f'1rif JJmwllillll II/lIlillll, 11"..lII/l1lillll
rll Ihr/Ill/moillg lit; .lllIflimlir dij/('Il'11lial ('qllalioll (II' I E [0, '1'1: dl'(1) = Jil'(t) dl
\
+ a 1'(1) tllJ(t),
1'(0) = I'" > O.
(9.2.2)
1 Illhl'),1'
B(I) is
(I
staHriem[ HroWlliflll motioll, alld al [rasl 11111' illlll',\lor Ob.ll'l1lrS
(J
wilhoul rrror. I
\(A4) TIII'!'r is
I/O
(lrbilrage.
l
I
9.2.1 Thp BlUfk-S(hOle.1 (mel MI'!'lull AfJ/nuflfh
he ~oal is to narrow down the possihle expressions for G, with the hope of btaini!lg a specific formula for it. Black and Scholes (l!17:~) and Merton \9731» accomplish this by considel'ing relations among the dynamics of te option price G, the stock price P. ami the riskless bond V. To do lhis, Ie first derive the dynamics of the oplion price by ass\llllill~ thai (; is only a fJunction of the currenl stock price I' and I iL~c1f and applyin~ h(.,.s I.elllma ~see Section 9.1,2 and Merion [I ~m(J, Chapter :~ J) to the function (;( 1'(1). Il. ~hich yields the dynalllirs of Ihe option price:
f
(9.2.3)
where I [ a(; - ItI' (; ill'
(1(;
a~/'~ iI~(;J
+ -ill + -~- -il/'~
(!1.2.4)
I [ ;/(;] - al'-
e;
,JI"
Unforlunatcly. this expression (1m's nol S('eIllIO provide any ohvious reslriclions that might allow us 10 lIarrow dowlI the rhokes for G. One possibilily
I',
is to s('t ~'qu.d to SOIll(' "r~'\l\lirl'd" r.It~' Orll'turll r", 0111' that \'I)III~'S frolll equilibriu\ll eOllsidl'ratiolls of th~' I"
VI C [0, '/'1.
(!I.~Ui)
Portfolios s,lIisfying (\l.~U» ,Irc called mllilmgl' portfolios. 1\11'11011 (1\)(;\) sholl's Ihal Ihe illstantanCOIIS dollar ret Ill'll dllo Ihis arbitrage portfolio is: I" til = - til' I'
+ -I"/) Il/) + -'-I~(; dC.
(9.:l.7)
where Ihe ,to("hastic diflcrcnlials til' alld Ii/) ,liT givl'lI ill rcspectively, alld de; lollows from !tt)'s l.elllllla:
(~).:l.2)
+ 0x(;dll /d'i1G/iJl' + iI(;/iit + "~F-il~(;/iJl'~
I,~r.dt
I', -
(;
a l'iJ G/iJl'
(~).:l.I)
(~).'2.H)
(~). '2.~)
(9.2.10)
(;
SllhSlit\llill~
alld
(~1.:l.7)
,lIId imposing
til = [(I'-·')lt,+r'I,:-")I,:l"t+[rrII,+oK/~ldJi(t).
(!l.'2.11)
til<' dYllamics or 1'(0, /1(1), OIlId (;(1) inlo (!).2.ti) yields
11111 1',11'1;1111,11, I\I,I('~ ,",,1 SriI"I,', (1\17:\) ;)."11111(' tiI;11 Ihl' ( .. \1',\1 h .. ld" ,11,,1 "i1I,lin I;, h\' n·I""11ll whirh link... l'X)U'( a'il 1 t'\11l H~ 10 Ill'l.L f lowen'r.
~\PPt'.\Il1lg 10 tht' ~l'nH Ily~m"\I"kt'I-\itll'
r{'llIrll."! .llId 1I1t'1(" ;II{' .'''"It' ,"hllt, hlll."Iiglliltlu' CAPM .1IIe1 nJlIl111111111,-IIIII(' opliolt.prit iug lIIodd.~ (."( .•.. 101 cxal1lpk, I)Y),\'ig ;lIId Il1gcl!'oooli II !IH:! I). N('\'f.'llIu'k.,:... RllbilJ.,lC"ill (1!17fi) plfl\ id('" .1 drll.llui,' l'Qllilihrillili l1Iudd ill whidl til(' HI.1Ck·SdlOlc·, 101111111.1 III tick
Ihl' C:AP~I i~ 1101.1 dYII.lIllie lIIodd dh'(I'liIiIHitlill
irant
il1("lIlI:..i.,j('JH i('!'\ ht'tWl'('1I
rOlllpl('wly ri,~ldt'ss ill hil rag c' pO l'll illi o to ht' -';1I IlIi: ( li(' n',l 'l lill Now kl liS fili '1'/. Thi.\ /Ilay hI' gu ara lJl eed ' 0/1 is 1I0 IlSI OI' It;I \/i( l1l1 n'lI it,\ l Iha till ' S(,II~(, ,,1111;11 all d II' = III' ('l\IlOSillf.( 1.< "-, W: ?I :!) VI E ((), ',1'/,
to.
I,:
I;
illl pl\ 'ili g /";1 1
,1I' I E
10. TI.
llll hn or sha rI'S of J/~(/}
til!' ('\'( '1'\' ll! ill Ill! ' s('II~!illal lill ll. \'I's IH' I'li\ 'd\' . Ill' lt'. ho th 11;(1) 1111' ~Iock all d IIII' IIl' isl all ! thr oll gh lill nll IIlItc,s {)( ;(I )/a /' is risk les s at all is llio por tfo lio . No ll' Iha l 111<11 thi s po rtli I'-valyillf,( 10 ('!lS lII'( ' il Pis kll ow n )1 (;(1 ,I all d 11;(1) II I lIS 1 Ill' liIH d d~l'd .I\l 10 Ill' /JI'r/i'/'t/y Il/' d .;ai is llio nli po ;1 IIIIl C'S. SlI l'h io as IIII ' III'I~I!" mli". '1'1 yit 'lds a "Yl lf/m if' po rI fol (\I. :!.I :I) fill' al! { E In, lid .. i) '' its :!.l \'('' (\I. S\I ' 1I illg Ihl' jlos l IlII i",' (·st llH 'lll . Bil s all d n'( jlli n's 110 I\('t ens tles s, slr.. tq,·Y Iha l is lis\ ..ks l' Ihe l'!' wo uld ('xi s\ a vis ll'n oll 'O, JI'I he l lls illm urn /iJl '(', nC ret 1. Th 1l0Ils1ot'haslil" leis a POSilivI' \'('(111'1 \11 ';lIl 'g" tha I ri,' lio lO, l'0n ' lllit 'lla «1) l'isklt'ss Iha l , il IIII1SI II(' ill(' caS «' 10 avo id arh iu'; lgc
a
I...i.
, IS' I i
bn \1\; \1 (') to ('m pha si! .c Ihe has Iwt'lI ade ll'd III J{K H'IH ;\\\l aq.? l' lim ';1 Wh l'fl ;t,' 11'1'11. pos sib le il I'ar ies Ihr oll gh lill ll' COJlditioJl n', llI n's tftc sill llll l' Ilo-O\fhilragc s Ihi ', " par abo lic 1';1 1i1l t'r Sur pri sill gll ord Id"(lI a s('( ('x pn' ssi oll wh i .. h is ('h oir es of (; loj llsl Oil"
\ I~
• I...i • I.... I• ...••I...! Ii:
•
j
Pil E II", (;:
i l l ; ; ) ( ; - I'{; =:; II, .. '. if! ( ; - +- - -1'- - - - -I- r/' -ill ill' il/'~ :! =:: (;(/ J( T), 'J') llll dar y cO lld itio lls: ' Iht ' I(.!lowillg 1\\' 0 hO ', IlSI (,(H Ill' is, slIh jl,\, 1 (0 Ihl " IIll i'll ll' sol uti olJ all d qO ,1) :::: 0, Th ~bxu'rn-x, O\' 1\lac);-Scholcs !l1l'1l1ub:
I
-0
I
\
fl
(I'
+ ~n~)('r-I)
j
W~,17)
j
jT ::i
~)('{'-I) log U'( lI/ Xl 1- II' - ~(J
\ l
nJ T- 1
I
I
\ \
j
Hi}
{~l.~.
log U'( !)/ X) -+-
j
(~J.:!.I ~)
j
j
j
9.2. A il,;fJ Rl'view oj Derivative Pricing Methods where <1>(.) is the standard normal cumulative distribution function. and T- t is the time-to-maturity of the option. The importance of assumptions (AI) and (A3) should now be appar- , ent: it is the combination of Browian motion (with its continuous sample paths) and the al~lity to trade continuously that enables us to construct: a perfectly hedged portfolio. If either of these assumptions failed, there,' would be occasions when the return to the arbitrage portfolio is nonzero, and stochastic, i.e., risky. If so, the arbitrage argument no longer applies. Merton's derivation of the Black-Scholes formula also showcases the power o\" Ito's Lemma, which gives us the dynamics (9.2.8) of the option: price {; that led to the PDE (9.2.15). In a discrete-time setting. obtaining i the dynamics of a nonlinear function of even the simplest linear stochastic process is generally hopeless, and yet this is precisely what is required to construct a perfectly hedged portfolio. More importantly, the existence of a self-financing perfectly hedged portfolio of options. stocks, and bonds implies that the option may be synthetically replicated by a self-financing dynamic trading strategy consisting of only stocks and bonds. The initial cost of setting up this replicating portfolio of stocks and bonds must then equal the option's price to rule out arhitrage hCGI\lse the replicating portfolio is self-financing and duplicates the option'S payoff at maturity. The hedge ratio (9.2.13) provides the recipe . for implementing the replicating strategy.
DIllion Sl'1Isilivilies The sensitivities of C to its other arguments also play crucial roles in trading all(1 managing portfolios of options, so much so that the partial derivatives l2 of G with respect (0 its arguments have been assigned specific Greek letters in the parlance of investment professionals and are now known collectively as utlticlll smsitivities or, less formally, as the "Greeks":u Delta
Gamma
Theta
t..
-
r H
-
ac aJ>
a2 c ap~
aG iJt
~ (9.2.19)
(9.2.20)
(9.2.21 )
11'1'1 ... It· fill ""ani;,l d.. riv.lliv~s- in this COll(,'xt rer,·!"s. or collrse. to instantaneous rates or change. This i~ 1Il1fortlll\ate Cuiliricit"I1Ct' or It'nninoloJ.,,ry is usually not a source of confusion, hili f('adtOrs should hewart". nOf nll,r,,·. "\'<'I(a" is not a Gr .... k l~((~r ane! V is simply a script V.
354
9. Dr/iva/ill' Pridng Models
V
VC)!;
For the Black-Scholes formula evaluated explicitly: /),.
-
r
-
(.,)
-
iJG
R -
Rho
(!J.~.I(;),
(~J.~.22)
iJr
aG
-
W~.~:{)
iJa
these option
s(~lIsi'ivi'ics
(!J.~.~4)
4>(dd 4>(d, )
(9.~.~:i)
/'aJT-1
2
1'(1 T-I
J1Ct
call he
Xre-'(T-II(d~)
R
=
(T-t)X,- ,(T-II «(tl
V
-
J>-!T-l 4>(<1 1)
.
)
(9.~.::!G)
(!J.2.27) (9.~.28)
where 1/>(-) is the standarci normal probahility density function. We shan have occasion to consider these l\leaSUres
9.2.2 The Marlingau Approach Once Black and Scholes (1973) and Merton (1973b) presented their optionpricing models, it quickly became apparent that their approach could be used to price a variety of other securities whose payoffs depend 011 the prices of other securities: Find some dynamic. costless self-linancin)!; portfolio strategy that can replicate the payoff of the derivative. and impose the noarbitrage condition. This condition usually reduces to a PDE like (9.~.1:i), subject to boundary conditions that are determined by the specific terms of the derivative security. It is an interesting bct that pricin)!; derivative securities along these lines docs not require any restrictions on a)!;enlS' preferences other than nonsatiation, i.e., agenL~ prclCr more to less, which rules out arbitrage opportunities. Thcrcli>re, the pricing fonnuhl for any derivativc security that call be priced ill this I;lshion mllst he idelltical for all preferellces that do 1I0t admit arbitrage. III particular, the pricing formula Illust he the same regardless of agenlS' risk tolerances, so that an economy composed of riskneutral investors must yield the same option price as an economy composed of risk-averse inV('slOrs. IIlIt undl'r risk-neutrality, all assets Illust earn the
same expected rate of relllrn which, ullcler ;Issumptioll (A:.1) , must e<ju;11 the riskless rate r. This fUlId,lIlwlItal illSi~ht, dm' 10 Cox ;IIHI Ross (I \17(i) , simplifies the computation of optioll-pricill~ f(ll'IIlIlI;IS ('llol"ll1ollsly 1)('('allS(' in a risk-m'lltral ('COIIOIllY the oplioll's pri.-(' is silllply Ih" "xpn led vallie of it:- payoff ,lis«HUllc(1 al lhe riskless raIl': G(l)
=
e-'\I'-')
1':,[MaxIIJ'(/') - X.
011.
(\1.:.1.:.1\1)
Ilowever. the cOllditiollal expeetatioll ill (\1.2.:.1\1) IIl1lst he eV;lluated with respect to the r;1IH\om variable 1"( 'f), not 1'( T), whnl' I" (T) is the termillal stock pricc adjll.lted jor risk-lIeutmlily. Specifically, ullder assllmptioll (A~), tlH' conditiollal distrihutioll of 1'( T) give II 1'(1) is simply a lognol'lnal distrihlltioll with Ellog 1'( T) 11'(1) I = log 1'(1) + (JI - ~)(T-l) and Var[log I'(T) 1/'(1)1 = a~('r-I). Under risknelltrality, the expected rate ofretllrn ((II' all assets IIlIlSt be r, aud hence the cOlldilional distributioll of the ri.l/I-IIPlllmliU'd 1
=
p.-.('f'-I)
1':;[Max[I'(T) -
X.OI].
where the asterisk in E; indicates lhat the expectatioll is to he taken with respect to an adjusted probabililY distribution, adjustcd to be consistenl with risk-nelltrality. In a morc formal selling, Harrison and Kreps (197\) have shown that the a(ljusted probability distribution is preciscly the distribution under which the stock price follows a martingale; thlls they call the a(ljusled distribution lhe equivalent martinKlIle mPll5UTf. Accordingly, the risk-nelllral pricing method is also known as the 1/I{/rliIlKfllt' fJlicillK fer/lIli'll/t'. We shall exploit this procedure eXlensively in Section 9.1 where we propose 10 evaluate expectations like (9.2.30) by MOllte Carlo sillllllalioll.
9.3 Implementing Parametric Option Pricing Modc\s Ikcallse there arc so many diflerellt typcs of options ;lIHI other derivative securities, it is virtually impossihlc to dcscribe a cOlllpictl'iy gCll('J'al method for implementing all derivative-pl'il'illg fill'lllulas. Thl' IMrtinilar features of earh dnivativc security will oftcn pia), a ('('1111';11 roll' ill how ils prit'illg
(),
/)1'/'h'lIlil'l'I'/'irillg
A!1lI11'!.1
IC>IIlIlIla is 10 1)(' appli('d IlloSI dknil'dr Bill Ihl'l'e are s('\'nal ('0111111011 aSI'('CIS 10 e\'('I"\' illlpl"lIH'lIlalioll of' a /HII(/IIIt'/rir o(liioll-pricilll-( /lIodel-a Illodel in ",hich Ihe III ice dYllamics of Ihe ullclerlyill~ s('cllrily. called Ihe IUIlt/I/IIII'1I1111 filII'!, i~ spccifi(,d "I' 10;1 fillite 1ll1l1l1H'r o('parall\ete('s-;u\(1 we shalilill'lis Oil IIH's(' COIIIIIlOII ;\SI\('('IS ill Ihis SC{'(iOIl, 'Ii, ~illiplil\' Il'llllill"I"gy. IIII,,"SS "Ihcrwisl' sl;tll'd W(' shall IIS(' Ihl' I('I III {l1,liollto IIIcall aliI' gell <'I'a 1dni\'alil'(' ~(,cllrily. alld Ihe 11'1'111 ,1/01'1, to lIIeall Ihe dt'lil'alil'e s('nlril\''s IIllderlyillg I'II1I1LIIII('1I1a1 asse\. Althollgh 111('1'1' are c('rlailll), asp('CIS or SOli\(' dl'ril'alil'e se('lilil ies IIial differ dramalically from lho\c of slalldan I (''I" ill' opl iOlls alld (';1111101 1)(' described ill ;lllalogolis lcrllls. Ihey III'('d 1101 1'011('('1'11 liS ;11 Ih(' CIIITI'III 1('\'cI of' gl'lIeralilY, Alkr tll'vclopillg a ('0 h 1'1'1'11 I rrallll'work li,r illlplcllH'lIlillg gell('ral pri('illg Ii,nllllias. wc shall IlIrll 10 IllOcii/iraliolls lailoJ'('d 10 panicllla .. derivalive securilies,
1),1.1 1'1/ll/lIIdl'/' 1',\lillllllioll 0/,;\,\'\1'/1'/11'1' /)yl/lll11in
Till' 1t'l'11I "p;tr;\III\'lril''' ill Ihis sCl'lioll 's lit Ie is lIIealll 10 1'lIIphasil.l' Ihe rc1i;1II ('(' 1)1';1 rlass 01 oplioll-pricillg fOlmulas 011 t\l(' paninilar asslIlllptions ('olll'I'millg the l'll1l1iall\('lIlal assel's plin' dYllamics, Although Ihese lalher SllOlIg assllmpliolls 0111'11 vi('ld c1I'galll and lrartah1l' expressiolls for Ihl' o(llioll's price. IIi('y ar(' lypicallv cOlllradicled hy the dala. which docs 1101 hode wcll for IIIl' plicillJ,!; /ilrmIlLI's SIICC(,SS, III raci. pl'l'haps thl' most important asp('C( of' a SIII'('('ssl'lIl empirical illlplclll('1I1alion of' any oplion-pricinglI\odel is cOIT('nl), id('llIiryinJ,!; Ihe d)'lIamics of the slock prict'. and IIrlCerlainly J'l'J,!;anlillJ,!; Ihcse price dYllamics wilileacllls to consid(,r 1I()11/J(lrt/IIII't' ric alt('l'IIali\'('s in (:lIapl('l' I~, Bllt lill' IlIc IIIOIlIt'III. let liS assert thai the specilic form or Ihl' slock pricc proccss /'( I) is known lip to a V('Clor of unknowlI paralllelers 0 which lie~ in SOIIIl' paralllctl'J' space and Ihat it satisfies Ihe /iIIiOlvinJ!; stochaslic eli/fen'ntial (''Illation:
e.
tll'(I)
1/(/',
I:
n)
til
+ h(l'. I: (3) tll1(l).
I E [0, 'J'). (!U,I)
where 11(1) is a standald Wil'ller proc('ss alld 0 == [0:' (J' l' is a
hI /', I:
tJl
(11'
(!),:U)
357
<).3. /mlJifmmtillK /'IIT1lmftrir 01l/iOIl Priring Models
In this rase. the parameter vector (J consists of only two elements. the con-stants a and fl. H In the more general case lhe functions a(1'. I; 0) and b(P. I; (3) musl be restricted in some fashion so as to ensure the existence of a solution to the stochastic differential e'l\lation (9.3.1) (see Arnold [1974], for example). Also. for tr~\l'tability we assumc th,lI thc codlicient functions only depend on the most recent.price 1'(1); hence the solution to (9.3.1) is a Markov process. This assumption is not as restrictive as it seems. since non-Markov processes can often bc re-expresscd as a vector Markov process by expansion ! of Ihf .1/"lfS. i.t' .• by incrcasing the Iltllnber of stale variables so that the collection or prices and state variables is a vecLOr-Markov process. In practice. however. expanding the states can also create intractabilities that may be more difficult to overcome than the non-Markovian nature of the original price process. For option-pricing purposes, what concerns us is estimating 0, since pricing f'()('(nulas for options on p(t) will invariably be functions of some or all of the parameters in O. In particular, suppose that an explicit expression for the option-pricing function exists and is given by C(P(I), 0) where other , dependencies on obscrvable constants such as strike price, time-to-maturity, I'. and the interest rate have been suppressed for notational convenience. 15 An , estimator of the option price Gmay then be written as G C(P(!), 0), where is some estimator of the parameter vector O. Naturally, the properties of (; are closely related to the properties of 8. so that imprecise estimators of the parameter vector will yield imprecise option prices and vice versa. To quantify this relation between the precision of G and of 8, we must first consider the procedure for obtaining 0.
=
o
MaximulIl Ukelihood t:'slimalion
The most direct method for obtaining 8 is to estimate it from historical data. Suppose we have a se'luence of 11+ I historical observations of pet) sampled at non-stochastic dates' 10 < II < ... < III which are nol necessarily equally Sl)(l(f(/. This is an important feature of financial time series since markets are generally closed on weekends and holidays, yielding irregular sampling intervals. Since P(t) is a continuous-time Markov process by assumption, irregular sampling poses no conceptual problems; the joint density function f of the sample is given by the following product: n
f(I1I.···.I',,;(J)
= ./ilml;9)nf(P..t. .=)
I PH./Jc-I;O),
(9.3.4)
II NOlI' III"t "Ithongll th .. drift awl dilln,io" fnllflioll' dql(,lId Oil distillct parJlllett'r vector.; Ii. th('~(" Iwo \'('('tors m;ty COil rail) sOllie pal.tlllt"ler."i in common. "'h'('11 if (; call1lot be obtained in closed-Conn. (I is a ne,,·s.'ary inpttl Cor numerical solutions or (; ;\llIlllIlIst still he estimated.
(t
;u,,1
358
n.
where p. == 1'(1.). jilU{d is the marginal density lilnction of and j(P•• i. I Ph-I. ik-I; 9) is the conditional density function of I'~ given I'~_I, also called the lramilioll rlfll.lily IUJI(lillll. For notational silllplkity. we will write I(I'h. Ik I Pk-I. Ik-I; 0) simply as ./i. Given (9.:\.4) and the observations 1'", the p.. ramC'tn \"I'ctor llIay be estimated hy the lIIethod of IIlfIxil/l/l//l IiIIP/i}wor! (scc Scnioll A." of the Appcndix and Silvey 11975, Chapter 41). To ddilu' the Illaximlllll
"II .....
Iikclihood estimator O. kt [(0) d('llote thc (0t:-likl'li//IIlld/illldiol/, IIII' lIallirid lugarilhm of Ihe joillt densil)' IiIllClioll of ,~" . , . , "" viclI'ed as a fUllction
of 0: [(0) . ==
L" log ji,. k=1I
Thl' maxinllllll likelihood eslilllalor i.~ Ih('11 given hy
o ==
(he
Under suitable regularilY ('oIHlitions. \Iorlllallimiling distribulion:
J/i(iJ -
9) :.:..
N(o, y-I (0»).
(!I.:I.(i)
arglllax [(0).
iJ
I(O)
is consislcnt and has tht, li,lIoll'ing
I iI~{(fJ)]
=::;
lim -·E - ----; [ " ilOilO ,,~'"
.
(1).:1.7)
where I(O) is called Ihl' il/fiJl'll/lllill1l //llllrix. When 1/ is large, III(' aSYlllplo,ic distributioll in (\I.:t7) allows liS to approxilllal(~ III\' varia lit'(· oj (j 'IS
. I . VarlO] "" - y-l(O).
(\U.H)
1/
andlhc infimllalioll matrix I(O) ma), also he eSlimaled ill III(' lIalliral way.
.
I ;/1 [Ii})
1 = ---. II
ilO i/O'
Morcover. iJ has l)(,l'lI showlI 1(1)(' aSYlllpIOli('all), ertici('111 ill til(' class or all cOlIsiSIC1I1 alld IIlIif(JI'IIII), asymptotically lIonnal (eUAN) I'stimalors: Ihal is, il has Ih(' smalksl asymptolic variallt'1' of all l'slilllalor.~ Ih"l are etlAN. h('lIcc il is Ihl' pr('ll-rr('(llIH'thod oreslilllatioll Wh('IH'\'('r ll-asihl('. ()rcollrsl'. 1Il:lxi III II III likdihood ('slilllatioll is ollly ll-asihle whell Ihl' likdihood fllll(li\)n ran hl' ohtailled ill rlo~('(l fllI'IlI whirl.. ill 0111' I';\S('. Il'qllin's ohuillill).\ Ihc Iransilion (kn~ily fllllctions./. ill dosed forlll. Unfortllllilldy,,, dosl'dform expression for Ji for alhitr"r), clril't alld diffllsion fllllctiolls clIlnol he ohtaillcd in gcneral. IlowevCl'. il is possihle 10 charaClni/.I·[J, illlplicill), a.~ Ihl' sollllioll 10 a PilE. In partinrlar. fix Ihe ('olldiliollillg variahles I'k-I alld 'k I and 11'1/1, he
9.3. /1Il1Iil'1IIt'IIIi11K Parametric Olilioll I'ricillg MOIIt,!., a fUIIClioll oj" 1'" alld I,,; to cmphasize Ihis, we drop the suhscript k from thc argumcJlls alld writc fiJP, t 11',,_1, I.-d. ThclI it follows froll\ lhc FokkcrPlanck orjo/1C1arrl equatioll that .1~ IIIUst satisfy the !i)llowill!!; (scc 1.0 [19HH] fill" a derivatioll):
YA
(!UUO)
ill
with !niti;ti COllditioll (!13.11) where 8(1'-1'" il is the Dirar-ddta (ullnioll lTIIII'ITd OIl I'k-I. Maximum likelihood l'stilllatil)l\ is feasihll' wht'llt'n'!' (\I.:~.IO) (;111 Ill" solwd l'xplirill)' ;\IIclthis c1qJl"llds Oil how rOlllpli .. aled Ihe "'H"ni(iellt hili( liolls (/ alld b ar('. ()II ....
n, ... , I'".
is oblailll'd. iJ ("all be "olllplltcd 11111111'1 i.. ally givell the dat;1 'Ii) illterprct this l'stillialor, II'C IIIIISI .. her\.. Ih.1\ _the regularily
./k
rendiliolls f()I' Ihc cOllsislcncy alld aSYlllplOtit- lIorlllality of 0 are satisfied. III ~Cl\lll' elses of illterest Lhey are IIO\' For exalllpk, a h)!!;lIonll,\1 diffusioll
=
"I' I (I'd I-\- (T I' d Ii violates the st"t iOIlillaliolls of rod!i(it'lll fllll"liolls (/ :11)(1 Ii, (!'.:I.IO) r:lIl1lol he S(llvl'Ci explicitly, hellcc lor th('se caSl's 11I:lximlllll likelihood estimatioll is illkasilJk. All alll'l'llative proposed hI' I bllS(,1I :lIld Schcillkm;tll (19~1:») is to apply I !ailSI'll's ( I!IW2) ).';{'lIcr"li/.l't! IIH'thod or mOIl\{'IIIS «;M M) estilllatOl", whirh Ihl'y I'XI!'IH!IO the rase o('slrirll), st:ltillll,lI), (,llIl1illl\l)\Is-tiIllC ~brkll\' procl'SSt'S (s('('I\t('Appelldix ('or all O,POSilioll o(,(;~li\l). The l'ol'lIs orallY (;MM prOCl'dllre is,oi'COIIISl', thl' 111(11)('111 cOlllliliolls ill \\'hich illl' l'al'alllcit'l' Vl'('(or 0 i" illll'licill)' dl'lilll'd, Till' (;1\11\1 estimator is Ihal [>;II':IIIIl'I('\' \'('clor iJ thatlllillilllil.l's thl' "dislallc(''' \>('1\\'1'1:11 Ihe sampk 1110111('111 concliliolls alld Ihcir popllialion ('Olllllerparis. TIll' pl'o[>cllit's 0(' a CMM eslilllalor dt'lll'lId n ilic:lIl), Oil lilt' choic(' or 11101111'111 COlldilioliS alldthe dislallC'(' 1I\('lril', ;llleilol' sl;1I I< 1:\1'(1 disl'!'('(('-lil\l('
CMt"! ;lppli(;tliOlls, Ihcs(' two isslI(,s
It:!\'('
1)('('11 stlldil'd 1(llit(' tho\\)u).';hly.
i\lollll'lIl COllC!iliollS are I)'pira\ly S\\gg('stl'cl hy thl' illtn
9. /)rril'lIlilw [',-irillg i\J"til'l.l
·II)\)
I{II' exalllple, I Llilliltoll II!I!H, Chapter 14]), and efficiency hounds Gill he obtaillcd (SC(' Ilalls('1l (19W,) alld IlallSen, Heaton, and O~aki (19HH»). Bllt for cOlltillllOlls-tillw applications in linance, especially those il1\'olvill~ derivativc sC('llritics, IlIlICh less is known ahollt the properties of C~IM cstilnators, 111<\('1'(1,011(' of I bllS('1I alld Sc\teinklllan's (I~)~):') lIIain cOlltrihllliolls is to sltow how 10 )!;('I\t'ralt' 1\10111('111 UlllditiollS ({". UlllliIIlIOlIS-(illll' Markov proc('ss('s wilh dis('f('\('lr salllplt'eI data. Although a cOlllplell' l'xposilioll of( :MM eSlilllalioll (til' contillliolls-tilllc procl'sses is he),(lIld Ihe scope of Ihis I('xt, Ih(' n'lIlralthruSI of Ihdr approach can be ilIl1slr;lIed Ihrough a simple exalllplt'. Suppose 11'(, wish 10 cstimatc Ihc parameters Ill(' /(llIo",illg statiollary diffllSioll process:
or
tip =
-Y(/I- JI) tit -1-
(1
till,
/1(0)
= /I"
> 0,
Y > 0,
(!).:~. I~)
This is a cOlltillUOlis-tilllc \'l'l'siol\ of a stalionary AR( I) proccss wilh IlIlconditional Illcall II (SI'(' SC('lioll !).:I.'I for further discussioll), and hCII("<.' il satisfies the h),potl\('sl's of 1hnsl'n anel Scheinkman (199:». To W'IH'ral(' 111(11)('111 COJldiliolls for (jI(l)1. I lallS(, II and Schl'illkillan (1!)9:» IIS(' the ill/illit,.,imtll g"III'mtu,- 11... a\so known as Ihe IrYlll;ill "/11'1'11 tor, which is Ihl' lilll,'-dt'fil'ali\'(' of a condiliollal expectation. Speci(icallv,
, t! . D"I·I == -1""[']' til
(!l.:t 13)
",herc Ihc ('XPCCl;tlioIlS operalor E"I·I is a conditional cxpeclation, ((l'ldit iOllcd Oil 1'( (I) = /'.,. This opcralor has s('vcraJ illlporlani applicatiolls for dcrivillg 1ll01ll1'nt ('()I\(lilioll~ of diffusiol\s. For exalllple, consider Ihl' following hellrislic ralc,,!aliOIl or Ilic I'Xlll'Clalioll of tlr
E"I-y(/, '-11) rltl
+ E,,[a till)
(9.:tH)
-),(E"I/II -
/1}
tit
+ aE,,!dlll
W:I.I:»
tlF .. I/11
--yO':"I/'1 -
JI}
tit
---F"I/I)
-y(E,.I/,) -
It}
(~),:~.17)
'D"I/'I
-y(F,,!I,j -
It}.
(!I.:UR)
E.. I "/11
tI
til
(!).:'Uti)
wherc (!I.:I.I ,I) ~llld (~I.:\, 1:1) folio", fmlll tlte bCltl1t' t'xpcclalioll of a lill,';tr ftlllction is Ih(' lilll'~" (IIJlllions o( III<' ('xpc('\atioll, (!I.:1. It;) 1()lIows frolllille S~IllH' proper! 1'1 01 til(' d i ({(-I(, II IiaJ (Illl'la Ior and from Ihe 1;t<'1 IIta I i IHTl'IllCnls of Brownian nlolioJl lIan' I.('n> ('xp'Ttation, and (!1.;{.17) is ;\Ilotltt'f way of l'xprl'ssing (!U.J Ii).
9.3. Implementing Parametric Option Pricing Models
361
Before considering the importance of (9.3.IS), observe that (9.3.17) is a first-order linear ordinary difTerential equation in E.[p] which can easily be solved to yield a closed-form expression for Earp] (note the initial condition Euf/)(O)] = Po): E.[p] = pu e- yl + )1. By applying silllilar arguments to the stochastic difTerential equations of l, l, and so on-which may be obtained explicitly via Ito's Lemma-all higher
moments of p may be computed in the same way. Now consider the unconditional version of the infinitesimal generator V(·] == dE(·]! dt. Aseries of calculations similar to (9.3.14)-(9.3.1Sl follows, but with one important difference: the time derivative of the unconditional expectation is zero, since p is a strictly stationary process, hence we have the restriction: (9.3.19) VIp] = -y(E(P] - J.t) = 0 which implies
E[pJ
=
)1
(9.3.20)
and this yields a first-moment condition: The unconditional expectation of /' must equal the sleadY-51ate mean )1. : More importantly, we can apply the infinitesimal generator to any well-I behaved transfonhation fO of p and by Ito's Lemma we have: I
(9.3.21) which yields an infinite number of moment conditions--one for each /-: related to the marginal distribution of /(P). From these moment conditions, : and under the regularity conditions specified by Hansen and Scheinkman ~ (1995), GMM estimation may be performed in the usual way. Hansen and Scheinkman (1995 l also generate multipoint moment conditions-i:onditions which exploit information contained in the conditional and joint distributions of f(pl-making creative use of the properties of time-reversed diffusions along the way, and they provide asymptotic approximations for statistical inference. Although it is too early to say how their approach will perform ill practice, it seems quite promising and, for many lti'l processes of practical interest, the GMM estimator is currently the only one that is both computationally feasible and consistent.
9.3.2 f..Stimatingcr in the Black-Scholes Model Til illustrate the techniques described in Section 9.3.1, we consider the illlplelllclIlation of the Black-Scholes model (9.2.16) in which the parameter
cr must he estimated.
62
9. neill/alitit' J'lirill/i Motirll
A common misunderstanding about a is that it is the standard deviation If simple returns U, of the stock. If, for example, the allllllal stalldard leviation of mM's stock return is 3orX" it is often assumed that a = (J.30. 'Iil . ee why this is incorrect, let prices P(t) follow a lognormal diffusion (!I.:!.:!) '. s required by the Black-Scholes model (sec Section 9.2.1) and assume, for <;xposilional simplicity, that prices arc sampled at equally spaced illtervals of length II in the interval [0, T], hcnce Ph == P(lIh), /, = 0, I, ... , II and T = ~III. Then simple retllrns Uk(lt) == (IV Pk- I ) - I arc lognormally distrihuted with mean and varj,lIIce: ~," - I
E\ /lk(h) J
II e~I"I . [0' f
Var[Jlk(It))
(!I.:t2'2) -
I) •
(!l.3.'2:~)
Therefore, the magnitude of IBM's a cannot he gallged solely hy the 30% estimate since this is an estimate of JVar[ Uk ( II)] and not of a. III particlllar, solving (!1.3.22) alld (!I.3.'2:~) fill' a yields the /(Illowing: I
a = [
h
( log
I
+
Var[Hk(lI)] (I
+ E[l4.(h)J)~
)JI/~
(!I.:t24)
Therefore, a mean and standard deviation of 10% and 30%, respectively, for IBM's annual simple returns implies a value of 2li.8% fill' a. While :~O% and 26.8% may seem almost idelltical for 1I10st practical purposes, the fonner value implies a l3Iack-Scholes price of $8.48 for a one-year call option with a$35 strike price on a $40stock, whereas the lallervalue implies a Black-Scholes price 01'$8.1 () on the sallie option, an economically significallt difference. Since most published statistics 1\)1' equity retllrns arc hased on simple returns, (9.3.24) is a IIseful formllia to obtain a quick "ballpark" estimate of a when historical data is not readily available. If historical data al't' available, it is a simple mailer to compute the maximum likelihood estimator of a using continuously compounded ret lints, as discussed in Sections !l.:t I and !1.3,:t In particular, applying !til's l.elJllII
tllogJ' =
(/1-
~a~)"t+ad/i = atll+ad/J
(~l.:~,'2:I)
'her\.' a == Jl. - ~a~. Therefore, contilluously COll1pounde(\ rClUrlls I),(h) ==
~ I
)g(/'k/P~-I)
arc liD lIorlllal randolll variables Wilh mean all and variance ~It hellce the sample variance or ,;.(It)/Jh should he ,j good estimator of riz; ill 1;lct, the sample variallce of Ii,(h)/Jh is the IlI<1XiIllUm likdihood ('stilllator or a, More ro rill
I
.e(a,a)
H '1 1" . = -;-log('27Ta-II} - -.,- "'(th(lt) - all)~ 2 '2a-1I
8
(!I,:~.'2(;)
and ill this case the Inaximllm likelihood ('stilllators fi,r ohl"illed in closed limn:
IX
and (12 can he
(93,27)
(!I.:I,2H)
Moreover, hecause the rdiL)'s are liD nortnal rantiolll variahles under Ihe U)'Il
at
Im',t:;t'/({r1y S({IIIIJi,'r1 nrl/II s('t' why irregularly sampled d"ta poses 110 problems for cOlltillllOus-tillle prO':csscs, ohs('rvt' that the samplillg interval It is arbitrary and can change lIlid-s'lIl1ple withollt affectillg the flllll'tionallill'ln of the likelihood fUllction ·(9,:'\.~(i), Suppose, for cxample, we measure returns ;111 II II all)' for the lirst 1/1 ohs('I'\'ations and thenlllonthly fill' the next 111 ohservatiolls, If II is lIIeasured in 1I11i" of olle y(,ar, so that h = I illdicates a OIH'-),('ar holdillg period. the 111~IXil1llll11 likLlihoo!l estimator of (T'l 1'01' thl' "1 + 112 ohservations is given hy 1~)
when~
Obs(:rvl' that the secolld t(TIll of' (!I,:1.2Q)
111;1\'
1)(' rl'\\'rilt(,11 'IS
which is silllpl)' the varialll'<' est ililator of In( lilt III)' ('( )111 i Illl( >llsly ('olilpollnded retllrns, r('scaled to an annllal f'n'qll('IH")', Thl' cas(' with which irregubrly salilpled (Lit;l Clil hl' ;1( ('ollllllodat('(1 is olle of' thl' greatest adv;ll1tages o\' ('olltiIlIIOIIS-IiIll<' slo('\J;lsli(' processes,
Ilowl'Vl'r, this advalltagl' COUll'S at SOIlIl' cos\: this great flexihility is the reslIl! or strong IMrallletric J'('stri('lions lhal each continllOlls-lime process impost's 011 fill its fillite-dimellsiollal distrihutiolls (see SI'r1ioll 9.1.1 for the definition of a fillite-dillll'llsional dislrihlllion). In p.7), IVPirh lIIay 1)(' ('l"alll;lIed explicit I)" ill tllis case as:
I
"'""
\'arla~1
'""
\';\r I (i
(T'!.
r
(9.:L\O)
2(f'1
(~I.;).:)
I)
II
(>hs('rvl' thai (~).:\.:\ I) docs not dqH'lId Oil th(' sampling illterl"al h. As /I ill\T('asl"S withollt hOlllld while 'f' is fixed (h('lIce It "e{T('ases to 0), a~ h('(OIl\('S 11101"1' precis(·. Tllis sllggests I hat till' "hest" ('slimator 1'01' 11 ~, Ih(' olle with smallest asymptotic v;lriall("(', is the olle hased Oil as l\IallY ohservaliolls as possihle, n'g"rdlcss oflVh;\t tlll"i .. samplillg illlerval is. III t('l"('stillgly, thi,~ reslllt docs lIot hold fill' the ('stilll<\lor of a, whose aSYlllptotic vari;1I11'I' dl'JH'lIds Oil T alld lIot 11. More freq 111'11 I samplillg withill a fixed tilll\' spall-of'tl'n (ailed f"lllIli7l1IOU,\-rf'fmd asymptotics-\\'ill 1101 illcr(';\se IIII' pn'I'isioll or It, ;11 It I till' "llI'sl" I'slima\or 1'01' (1 is 011(' hased Oil as IOllg a lilll(" spall as pilssihk.
Y.3. I111JJ/mutnting Parametric Option J'riri71g ModeLl
Table 9.2a.
365
A~ymt)loli{ SlflllflflTd errors for
" ......
a.
II
"
ii
IT.
I
I ~
r.:i
m
ill
0.4000 0.21128 0.2000 0.1414 0.1000 0.0707 0.0500 0.0354 0.02:'0 0.0177
0.56:)7 0.4000 0.2H28 0.2000 0.1414 0.1000 0.Q7()7 0.0500 0.0354 0.0250
O.Il()O() 0.5657 0.4000 0.2828 0.2000 0.1414 0.1000 0.0707 0.0:'00 0.0354
1.1:,14 0.1l000 0.5657 0.4000 0.2828 0.2000 0.1414 0.1000 0.0707 0.0500
1.6000 1.1314 O.HOOO 0.5657 0.4000 0.2828 0.2000 0.1414 0.1000 0.0707
2.2627 1.6000 1.1314 0.8000 0.5657 0.4000 0.2828 0.2000 0.1414 0.1000
I
2 (1.2000 4 0.1414 H 0.\000 It; 0.0707 :12 0.0500 64 0.0354 12H 0.0250 256 o.m 77 512 0.0125 1.024 O.OOIlS
0.2H211 0.2000 0.1414 O.l(){)() 0.0707 0.0500 0.0354 0.0250 0.0177 0.0125
I
I
I
I
m
I
T:"ffi
S.2000 4.5255 2.2627 3.2~ 1.6000 2.262 1.1~14 1.6000 0.8000 1.1314 0.5657 0.8000 0.4000 0.5657 0.2828 0.4000 0.2()()(J 0.2828 0.\<\14 0.2000
=
A>ymplOtic standard error of [; for various values of nand h, assuming a base imerval of h 1 year alld (1 0.20. Recall thai T '" nh; hellce the values n 64 and h 1/ I 6 imply a sample of 04 oh,("rvalio[Js equally spaced over 4 years.
=
=
=
a
Tables 9.2a and 9.2b illustrate the sharp differences between , first observed by Merton (1980) for the case of geometric Brownian motion. is true for general dilTusion processes and is an artifact of the nOli-differentiability of diffusion sample paths (see Bertsimas, Kogan, and Lo [19!l6) for further discllssion). In fact, if we observe a continuolls', record of 1'(1) over any finite interval. we can recover without error the diffusion coefficient a(·). even in cases where it is time-varying. Of course, in .
!, 3(j(i
9. lJr'rill((lil".I,,.i"i"~ AlIII/eLI
Table 9.2b.
A.\ym/Jlol;r ,\/flllrlflul,'rmn ./to
?
'!.
I, 1/
- - - - - - _ . _ - .. -'--,
_.. _-I
2 0.0400 4 0.O21l:1 II IUJ200 IIi 11.11141 :12 11.01110 li1 !l.OO71 1211 U.IIO:,O 251i, O.OO:!', ,,12 o.om!:, 1,024 IUlOI!!
O.IHOO O.02H:i 0.0200 11.0141 U.OInO 0.0071 0.00,,0 O.(){I3', Il.OO2', Il.OOl1I
0.0400 O.02H:1 (1.020U 0,0141 O,OIUO 11.0071 O.OW,O O.OO:!', O.OO:.!:, 0.00111
.
I
I
iii
1:!
r.:i
O.(HOO O.O:lH:1 (J.0200 11.111·11 O,OIUU I!.OO71 O.OW,O 0,00:1" 0,002" O,OOIH
0.0100 O.O:lH:1 O.O:lOO 0,0141 0,11100 0,0071 0.00,,0 O,OO:!!", 0.002" O.OOIH
0.11100 O.O:lH:1 0.0200 0.0141 0.0100 0,0071 O.OW,O
I
I
l:!x
'!"h
O.(HOO U.O:lH:1 O.O:lUO (1.11141 0.01U0 0.0071 0,00:,0 O,OO~" O.OIJ:i', 0,002" O,OO:.!', o,OOIH O,OOIH
0.11-100 O.O:lH:1 0.0200 11.(11,11 0.0100 0,0071 O.IIW,O 0.00:1:, 0,002" O.OOIH
A'\ymp\otir slc\1\(\anl ('nor of;' '! for \',W11)\\:o\ ""hu':" of n .Hul h, ass\\min~
year and
(1
= 0.20.
of 64 ()hs(~lYJ.\i()ns
Recall Ihal '/' '"
I"'n('" II ... \'alll(,s t'q\lc,\ly :'olMfl'd o\'(.'r .\ Yl'ars. 1/1,;
1/ ' "
I
li4 .111<1 I,
f.\
hasl'
m
I
ilfTi
0.0·1110 11.111011 II.II:!H:I' 1I.0:!01l II.O:!OO 11.0 I,ll 0,01,11 0,0100 0.011111 0,11071 11,111171 11.00:.0 0,011'.0 O,OO:!!". 0,00:1:, 0,0112" 0,002" O,OIlIH O,OOIH
o.o:!ln
''''''1 \',,\ 01 IJ =
1
= 1/ I Ii illll'l\' " "lillI''''
practi<:e we never ubserve continuolls salliple paths-the nOlion or t'oltlinuous time is an approximation, and the magnitude of the approxilllation error varies frOIll onc appliGllion to the next. tis the satllpling interval becollles finer, other problems may arise slich as the effects of the bid-ask spread, nonsynchronous tr'Hling. anll related market microstructure issues (sec Chapter 3). For example. Sllppose we \jdecicle to lise daily data to estitllate a-how sholiid weekends and holidays he treated? SOllie choose to igllUl'c thelll altogether. which is t,\tllalnllllnl to assuming that prices exhibit 110 volatililY when markets arc closed. Wilh I the coulltcrfactual implication that Friday's closing price is always ('qual to I Monday's opening price. Altel'llatively. we may use (~).:t~~l) ill al('o\tlllillg ; fur weekends. but such a procedure implicitly assumes that Iht' prict' proCI'SS Iexhibits the SIIIIIP volatility when markets are closed. implying Il1alll1l' Frillayto-Monday retul'll is three times lIIore volatile thall lhe MOIHla)'-lo-Tttt'sday retttrn. This is also cOllnlcrfactttal. as Frellch alld Roll (l!lH(i) havl' shown. The hetlcliL~ of more frequent salllplitlg must be weighed againsl the coslS, as llIeasttl'ed by Ihe types ofhiases associaled wilh 11101'1' finl'ly sa III pled dala, Unfurtttllalely, Ihere arl' 110 gelll'ral gttidelines as to how 10 lIIakt' such a tradeoff--it musl he iliadI' on an individllal basis with 111l' pal'licular applicatioll alld data at hand. II>
I I I
I
I"S,',· Ik ... ,il\l"s. Ko~,,". ",,,I \... (\\I\lIi). \... "",I M.II·Killl,,)' (1!IH!I), I"'nllll (1\1\11). .11111
Shiller
ill1d
Prrnm (I~}H:) for;1 lHore detailed illl;lIysis o{" III(' inleraniuns lK'tw.... 1I \.lIupling
lilll<'l'\·,,1 "lid ~"t11I'I,' .i/'(·,
1),3,
Imph'IIl/'lllillg /'1//'(;1///'11';(' ()f/lioll I',irillg '\/"dd,
'), J, J (LI/(I/Ili/"lillg Ih",'m'i,iCJI/ CJj (Jjllilll/ ",.i""':,lill/(/lon
()l1rl' IIll' Il1aXilllllllllikdihood ('slimal.,r {} 01'1111' 11IJ(11'I11'i1lf.: assl'l's par.lIl1,'Ins is o\)lail1l'd, Ihl' m'lximllll' li\"~lihood I'slil1lalor 01 IIII' opliol1 pril'\,
(; lila\, hI' r0I1SII'lIl'I('(1 by iIlS .... lil1~ 0 il110 IIII' OI>liol1-I"
il'il1~
lilrll1l1la (or
il1lo Ihe 11 II II H'ril'al alf.:orilhm Ihal W'lH'rall'S Ih" pricc)." Sil1ce 0 cOl1lail1s ,'slilll"liol1 1'1'1'01', (; == (;(1'(1),0) will also cOl1lail1 I'slill1aliol1 I'1T0r, al1d 1'01' Iradil1g al1d hl'df.:il1!!; applicaliol1s il is il1qll'r'lliVl' Ihal IV,' havl' sollie lIIeaSllrl' orils pn'Cisioll. This call 1)(' l'ol1SlrllCil'l1 "y "l'l'lyil1f.:" lirsl-onkr'(;lylor ,'Xp.IIISiol1 10 (;<1) 10 calclllal" ils aSyl1qllolic disil illlllioll (SI'I' Secliol1 A.·I orlh,' Appl'lIdix)
fit(i; -
(n
(!1.:1.:12) (!I.:t:\:I)
1,llerl' (;
==
(;(/'(t), 0) alld I(O) is tilt' illfonll
Th':rd(II'I~, )ill' lar!!;c
1/
the varial1ce or (; may he "Pl'l'Oximalcd hy:
Varl i;J
(9.:~.34)
alld 1'1 Illay he eslimaled ill thc natllral w"y: i!G(/'(I),
0)'
I'
i/(;(/'(I),
in
- - - - I- (0) -------..
i/O
i/O
III 1I111rh Ihl' sa III I' way, the precisioll or IIII' ('stilll"lors of all opliol1's sl'lIsitil'ity 10 its ar~lIll1cllts-thl' oplion's delta, f.:alllllla, Ihl'la, d\(), al1d vega (St'C Sl'ctiol1 D,:!, I )-may also be readily 'ilialltilil'li. 'lJII'li!flrli-SrhlJll',1 (;fl.ll'
As .111 illllslr"liol1 of' Ihl'sl' reslllts, rOl1sider Ihe ,'a,,· or iliad.. alld Schol,'s (I !17:{) ill which I'{/) )(Iilows a ~('om('lri( llrowl1i'lIl 11101iol1, "/,(1) = 11/'(1) "I
+ (] ,'(/) ,/11(1)
alld ill whirh Ihl' ollIy pol 1';\11\1' I.... of illll'l(',1 is (J. Sill\'!' thl' Illaxillllllll likelihood I'slilllalor a~ a~ has all as:'lIll'lolic distrilllltioll f.:in'lI hy
or
17Thi., 1(l1l()\\'~ 110111 Ih<." prillriph' of ill\",11 i;1I1( c; Tht' IIlaxillllllll lildjhClod ('slim.llor of il nonlinear hlllnioll 01 " par.mlt'lel" n'r(nI i~ tlu' IIOlllillt'.11" fllIlflioll of lilt' parallll'It' .. \'('("for's lIIaxillllllll lik(,lihood ('Milllalor. S('t', lor (,X;lIl1pl(', "dlll.1 (I~Jh").
j/i(a~ - ()~) ::.. N(O, ~n I) (~e(' Sectioll 9,3,2), the asymptotic distrihUlioll of the Black·Scholes CllI.oplioll price estimalor {; is '/'-/
')
""
I', == TI'-U)(T-I/I-(dl),
(!).:1.:\7)
whnc 1/1(.) is the slall(i;[rd IIOrlll;t1 prohahility dCllsity fllllction and til is ~i\'t'n ill (!1.~.17). From the asvm)llolic I'ariall(,(' ~ivl'll in (!1.3.:17). SOlll(' simplc comparalive sialic n'sldlS lila\, hL' deril'L'd:
I,
j)
-
I',
r;-;:--.
I', ax
iJ
.•
== -I'n" 'J'-(-i)-(dd d., ~-
iJl'
iI I
I
<1('1'-1)
(9.3.39) TIll' rollowillg- illcqualilies lIlay /111'11 he ('slahlishe
iff
-
:5. {I
(!).'~.'lO)
o
iff
J> -x ~(.) < -
(9.3.41 )
>
(I
if
x"
>
0
if
{:\ < I.
I,
ax ill; i)(T-tJ
I'
()
ill'
X>
rf
iJI'(
ilU'- tJ ",h(' 1'('
" "
.(. cc)II .. /I .
. (.! ~)i'r./I .
IT::j (T'!) - - ( r+. n
~
IUl'qualilY «1.:1.·10) shows liI;11 Ihl' ;lccllracy of i; dt'creases witli Ihe kl'd of Ihe Siock plin' ;IS long ;IS Ill<' r;lIin of' Ihe slock price to thc slrike price is less Ihall 1'1,110\\,('\'('1', ;IS III<' .slock pri«' illcreasl's heyond X(I,lhe acnlracy of
9.3. Implelllnttillg Parametric Option Pricing Models
,.. ,. Table 9.3.
T-t
2 4 H 12 24 48
Cutoff values for co71lparatitJt stalics of \j. (I
C2
1/(,/
C,
1.0015 1.0029 1.0059 \.0118 \.0177 1.0358 1.0729
0.9967 0.9933 O.9H67 0.9736 0.9607 0.9229 0.8518
1.0033 1.0067 1.0135 1.0271 1.0409 1.0835 1.1740
0.0482 0.0682 0.0964 0.1363 0.1670 0.2361 0.3339
Gbegins to increase. Inequality (9.3.41) shows a similar pattern for VI with respect to the strike price. , Interestingly, inequality (9.3.43) does not depend on either the sto~k strike prices, and hence for shorter maturity options the accuracy of G wilT increase with the time-to-maturity T -to But even if (9.3.43) is not satisfied, the accuracy of Gmay still decline with 1'-t if (9.3.42) holds. Table 9.3 reports values of CI through C3 for various values of T -I assuming an annual interest rate of 5% and an annual standard deviation of 50%, corresponding to weekly values of r = log(1.05)/52 and a = 0.50/vfs2. Given the numerical values of C2 and l/c2, (9.3.42) will be satisfied byoptions that are far enough in- or out-of-the-money. For example, if the stock price is $40, then options maturing in 24 weeks with strike prices greater than $12.09 or less than $38.02 will be more precisely estimated as the timeto-maturity declines. This is consistent with the finding of MacBeth and MerviIIe (1979, 1980) that biases of in- and out-of-the-money options decrease as the time-to-maturity declines, and also supports the observation by Gultekin, Rogalski, and Tini~ (1982) that the Black-Scholes formula is more accurate for short-lived options. 18 Through first-orderTaylorexpansions (see Section A.4 of the Appendix), the accuracy of the option's sensitivities (9.2.24)-(9.2.28) can also be readily derived, and thus the accuracy of dynamic hedging strategies can be measured. For convenience, we report the asymptotic variances of these quantities in Table 9.4. .
0\
9.3.4 The EfJl'cts oj Asset Return Predictability The martingale pricing method described in Section 9.2.2 exploits the fact • that the pricing equation (9.2.15) is independent of the drift of P(t). Since 1MTIlt"fC.· alt', of fours~, other po~siblc explanation!'. for such empirical regularities, such thl" pn''''IlC<' or slofhaslic v
as
37()
9.
Table 9.4.
J),.r;l/tlli!'I' I',.il·illg
,bP"I"lIlir 11,lIi,I/l"I'.1 1I//IIlIdl·Sd""I'.I mil/ilia .1I'II.1ilit lily
Estimator
Mllllrll
1'1/111/(1//111.
Asymptotic Variallc!' ~q,t(d,) d~ ~ 1"'(1/, d, - If
~ (.\TI"-I)(' """d'1>(d,»)' ~ (Vd, tI~)~ Th("~(" i\~)1l1ph){ir \'ariaun',1Ii an'
');\S('(\ (HI
11u'
a.'iSIIIIII'liluJ
tllat tilt'
,'at iall( ('
m.lxhnum Hkt'lihontl c:nimator whirh h.\s ;,\~ympt()tif c.li:-.tt ihutiol\ jJi{n ~ -
(':-01illl.'1111"
fT:.')
n'l.
i:'\ thl'
!.!.. lif(H, '20. 1).
'1 h"'l<"e jii(l'(u
t ) - 1'(0"» ::. N(n. 'l0"'(iI/:(ot)/ilot)1) wh.· .. " /o"(01) i, tit.· "I',i,," ,.·",i,i,';ty. Ful)uwinK Mtllutard (nIiVellliolis. thl' l'xpl"{'s,,,ioIlS n~p(}n('d in the tahlt· an' Ih(' ;1... Yltlplolir \'.lri~ ance. oJ" jii(F(a t ) - 1'(0 1 ») and II II 1St ht· divided hy tilt' "'"1plt: ,il.c " to oiliaill tilt' a.'Ylllllllllir
\..uitl.nu:~ of tht' (lIIJ1lonna\i/('d) ~l'Il!'oiti\'il it"s f{o'.!),
I
I
i the drift does not enter into the I'Di': (9.2.15). fi)r purposes of pricing options it may be set to any arbitrary fUllctioll or constant without loss of g'..·lIl'1"ality (suoject to sollle regularity conditions). III particular. under the l'CJuivalent martingale measure in whit:h all asset prices follow Illanillgalcs. tlte optioll's price is simply the present discounted value ofiL~ expected payoff
I
I
til" (I) d log 1"
(I)
:=
rl" (I)
til
"/,'(1)
:=
+ a '"
(I)
tiB
(!I.:H4)
(r - a2~) IiI + a"n.
!Although the risk-neutralized process is not elllpirically ohs....vahk.I~1
It IS
, Ilel'enhcless an extrellll'ly cOllvenient tool for evaluating the prilT of an oplioll Oil the slock wilh a dala-gl'lll'l"aling process givl'1l by 1'(1). Morl'over, lite risk-nl'utral pricing approach yidtls the following illlplication: as long as the llilfllSitlll codlicicllt for the log-price pron'ss is a lixed I'll tuwc"\'c"r, UI\((C.'I ('(.'nat" ("(Iluliliol\:r<. it <:.\11 \1(," ('~tltI1.ttc.·tl: :-'C.'l', fOf ('xalll",,". AII-S,dl;"i .• and 1.,,,( 1!I\l(i) •.Iarkwenh ;lIlIl RlIhi"'t .. ill (1\1\1;.). Ruhi"''''ill (1\1!I<1), Shimko (tll\I:I) .. lIIII S.... tioll I :.1.:1.-1 (II" Chapt"r 1:.1.
I),
J, 11II/,lf'lI/f'lIlillg I'(I/(IIIII'/,.i,. O/Jlillll I'Jirillg ,\111""',
cOllslall1 (J, Ihell Ihe Black-Scholes lill Illliia yidds Ihe to!Ted oplion pricc regardless of' Ihe specificalion alld ,lIgll II 1('11 Is of' lil(' drif'l. This holds 11101(' gCllerally fill' any derivalive assel which C,III he prict'd pllrely hy arhilrage, alld II'hl'le Ihe IIl1derlyilig assel's log-plitT dY"'lIl1ics is desnih('(1 hy all hil dif'rllsiollwilh cOllslall1 diffllsion codficielll: Ihe dl'livalivt"pricillg f(lI'IlIlIla is fliliCliollally illch-pelldelll or Ihe drill ;1I1t1 is dt'It'lIl1illt'<1 pllrely lIy Ihe diffllsio\l coerlicil'lll alld III(' COlllracl spccilicaliolls of Ihe dl'rivaliw assel. This may SI'('1lI p,II',lIloxic,,1 silln' I\I'() sltlcks wil h Iht, sallll' n hilI tlilh-ITIII drins will yidd Ila' SOIiIIt' IIla('k-SdlOl<'s I'rin', yel Ihe slo('k wilh Iht' largn drift has a I:tq!;l'r eXIH'(,\l'd relllnt, implyillg Ihal a call oplioll Oil Ihal stock is 1I10/'(' likdy 10 ht' ill-lhe-lJIolley al IllalllrilY Ih;1I1 ;1 call oplioll Oil Iht' stock wilh lilt' sillall('\'
n/(, 'li('l/(/iIlK (hll.lii'ill-Uh/('lIiJnh 1'1
'I() SI'('
d/I(!)
==
(-Y(/J(t) -- Jill i JI)dl
+n
dll,
(9,3Ali)
Wlll'lT
y
~
0,
I E:
10, (0),
Unlike tilt' ~I'ollll'lric I\rolVnian tllolion dYllamics 01 Ihe original BlackScholl'S llIoclel, which implies that log-prin's f(lllo\\' ,III ;lIilhllll'lir random walk with lID IH.rm,d illerl'lIlt'lIls, lhis log-price I)\(l( I'SS is the slfm ofa I.I'\()IIlI'an stationary ,llllolTgn'ssive (;allssian proccss-;UI ()rtIstl'in-Uhll'lIbcck proc('Ss-allci a dt'llTlIlillislic lillear tlt'IIC!. so WI' cdl this tilt' III'I/{lilig O-l/ proc('ss, /{I'I\Tilillg (!).:t,J(i) as
shows that whclI /IU) cit-I'iatt·s frolll its trelldld , it is pulled hack at a rate proport iollal to its cicl'ialio ll, wh('l'e y is Ihe speed o!(uljUJI 1IIf11 I. This rcvcrsio n to Ihe Ilclld illChllTS prl'dicl;l hilil)' ill Ihl' relurns of this assct. To develop flll'lhn illtuitio n for the propert ies (9.:t4(i) , cOllsid n its explicit SOIUlioll:
or
(9.:l.4H)
frolll which w(' call Oill'lill Ih(' ullcolld itiollal JIIOlllellts alld CO-IIIC IJIIl' II IS of cOlltillllOIlSI), COlllPOlIlHlcd r-period relllnlS rl(r) == P(t)-"(I-r):~o Elldr)1
liT
(9.:\.49)
\'ar! '/( r >I
r > 0
(~),:UO)
.,
( :O\' f Ii, ( r
). li~ ( r ) I
(1'
- - I ' -y(l,-/,- r)
2y
II -/ r
Conl'i( r),
III'
(r) I
[I - I-Y']~ '.
< I',!
(~),::l,:l
I)
I
-- [I - I'-Y'l, 2
Sillcl' (~),:\..J(;) is ;1 (;allssia ll process, Ihe 1Il01l}('nts (~1.:1.·19)-(~),3,:1 I) (Olllpletely charaClt ·rilt· Ihe liJlitc-di lllt'lIsiol lal distribu tiolls of I, (I) (see Sectioll ~),l.l Itlr the defillitio ll of a fillilt'-di lll(,lIsio nal distrihu tion), Ullli,'c the arilhlllc tic Browllia ll IIlOlioll or ralldolll walk which is lIonst,ll iollary 'md ortell said 10 hc tli/li'Il'wl'-I/(/liol/f/}~v or ,I Jlor/uUli c Irnu/, the trclldin g O-U pn)~'('SS is said 10 I~,I' Irl'lltl-,l/l Iliol/(u)' siJlce its dcviatio lls frolll trcnd filII ow a slallona ry proccss, ·1 fOlulitiollC 'cI 011 lIfO) = /1" in dt'lil1in~ Iht, dellc'lule d I()~~prift· pro("('."i.s, ahus(' 01 h'llIIillCilo gy to (all 11It'~t· IIHJlUt'lIts "uJlcullcl irioIMI", Ilo\\'t'\'('r, in this
:.'IISilll (' Wt' 1i;1\'(,
it is a
~Iiglll
fit'" Ih,' di,tillnio ll i.1ri prilllal il)' ~c'JII;lIl1i( ~illn' lilt' c:nllclili()lIin~ v.lriahk is lIlort' of all initial
nmdilioll lhan.1I1 illl"ollll"li oll \,;II·jahlC'-il"\\'(' dt'fillt' Iht' hq~illllillg : oirilllt" as I = () and Ihe flilly oh"'rv"hlt - >1.11 lill~ \',tI"" 01/,(0) '" I'". d"',, ('I,:l,1~1)-(~I,:\,r.~) ;Ir!' 11I"'lIlldil illll.tI IIIOIll('lll' rdativt, 10 ,!I"M' illil;;11 ("OIHlilioll!'oo, \\'e' ~";tli ,uloptlliis definition 01 all1llH'OIH litiol1;IIIII IIIIH'ut tilioligho ul liIc' rc"1I1ailHic-r of llai~, ".IPlt'l. :!I All illlplicllio ll uf 11 "IHI",t;"io ll;U it), is Ihal lilt' variau('(' of r-pt"riod returlls ha.1ri a finite limit as r iIHT(,;IM" \\"illwHI !,Olllld, ill Ilti.1ri ( ... ~" n '.! /y. wht'J"t'as Ihis variallct' increas(,s linearly with r 1IJ1(kr a r;lIIdom \ ....all. \\,I!ik IH'lId-st;lI ioliary prof('sst's art' oht'll simplt·J' 10 (·~tililate. "H'Y Ita\'(' ht'('11 n ili,·i/C'd tI~ 1I111e;lli.lriti( lIuldds of financial ;1~S('t prict's sinn' they do ltol "(ford w('11 with 11i(' (0111111011 illtlliliOlllh~lll()lIg('I-hol i/oll ;,ts:o.t't rt'IIII"J1S ('xhihit llIol"(' .. bk 01 th~lt prirc IOI"l'(";lSt'\ ('xhibillll on' 1IIIr(' I laillt\';ts tlu' 1(11('( ;I.,t hOI i/llU grows. Ilo\\,t'\,(,,,, if lIu' ~ollr('t· of stich i~ ('lIIpiri,.1I 1111,("1 \·~lli'lil. it III.IY wl'll Ill' ({llIsistt'!}1 wilh Irt'lul-M.u ioll;.lrily silln' it is IIOW \\'('II-llloW lIlkll fiu .IIIY fi II Itt' .'1'1 Clf cI.lla, (rc'IItI-M~lIioll~lrily :.uul t1i1f"t'IC·lIft'·!'\I~uiollarity an' \"ittllall\' illtii,lillglli!'ooh;lhl(' (,t"", fUI (,,\.lIllpl,,, St·, lion '2.7 in Ch~lp(('r~. <:ampb,'11
illillilioll
I t'I~III. I t.""iil,,"
II'I~II.
allci 1'(,11011
c :/"'1.11'" 17-IHI, ","1,/,,· """'Y Oil ..... "1I11il 1001" I'",)('r> 11,,·\' cill'),
.
37S'
1).3. illllilfillfllling Paramfl71c 0lilion I'I1cing Moilris
Note that the first-order autocorrelation (9.3.52) of the trending 0U in
-4.
and approachcs - ~ as r increases without bound. These prove to be serious rcstrictions for many empirical applications. and they motivate the alternative processes introduced in l.o and Wang (1995) which have considerably more flexible autocorrelation functions. But as an illustration of thc impact of serial correlation on option prices. the trending O-U process is ideal: despite the differcnccs betwcen the trending O-U process and an arilhmetic Brownian motion. both data-generating processes yield the same risk-neutralized price process (9.3.44), hence the Black-Scholes formula still applics to options. on stocks with log-pricc dynamics given by (9.3.46). I-\owever, although the Black-Scholes formula is the same for (9.3.46). the a in (9.3.46) is not necessarily the same as the (J in the geometric Brownian motion specification (9.2.2). Specifically. the two data-generating processes (9.2.2) and (9.3.46) must fit the same price data-they are, after all, two competing specifications of a single price process. the true" DGP. Therefore. in the presence ofscrial correlation, e.g.• specification (9.3.46). thc Ilumerical value for the Black-Scholes input (J will be different than in the case of geometric Brownian motion (9.2.2). To be concrete. denotc by TI(r), s~[TI(r)J. and PI(r) the unconditional mean, variance, and first-order autocorrelation of Ir/(r)}. respectively. which lIlay be defined without reference to any particular data-generating process.~l The numerical values of these quantities may also be fixed without reference to any particular data-generating process. All competing specifications for the true data-gencrating process must come close to matching thcse moments to be plausihle descriptions of that data (of course. the best specification is one that matches all the moments. in which case the trllc data-gcnerating process will have been discovered). For the arithmetic Brownian motion. this implies that thc parametcrs (p,. a 2 ) must satisfy the following relations: M
rt(r)
p,r
ih(r)] PI (r)
(9.3.51) (9.3.54)
o.
(9.3.55) I
From (93.54). we obtain the wCll-known result that the Black-Scho\es input a~ may be cstimated by the sample variance of continuously compounded
rcturns IT,(r»). Nev<·nil,", .. ,.,. 1.0 alld Wallf: (I ~9:») provide a f:rrwf;tlizalioll of Ihe Irending ()'U process Ihll contains Modla!lotir lrt"lHis. in which case tht" variance of returns will increase wilh the holdirik pt·riod r, :!'!()fClHlrSt·, it IlltlSI
ilar lUll
murt'
il1\'ol\',·d
he as.~llmed thallht" mOIUel)[S t'Xisl. fJ()wever, even if they do not, a sirnmay he ha.\t"d 011 location. seait:'. and association parameters.
argullIt"llt
f
374 In the case of the trending O-U prm:ess, the pai'amt'tt'rs (/1. y, (1~) nllisl satisfy r,(f)
r ::: ()
pdT)
t
I
= -"2[I-r- Y' ] ,
W:t:IH)
tbserve that these relations mllst hold for the 1}()lmlaiioll vallil's of Ihl' parameters if the trending O-U process is to he a plausihle dt:sniplion of Ih(' pGP. Moreover. while (9.3.5li)-(9.3.5H) involve population values of Ihe iJarameters. they also have implications for estimation. In particlilar. under the trellding O-U spccification. thc sample variance or conlillllollsly tom pOllnded rcturns is clearly Iwi an appropriate cstimator fiu' (1 ~. \ I-I()ldi~g thc IIncoIH\itioll.ti variance of ret lints fixed. the particular ~'aluc of (1~ now depends on y. Solving (9.::1.57) and (93':'H) for y and
r
yields:
y
J
-- log( I f 52 (If)
+ ~PI (T J)
y( I -
1'- Y'
r
I
(!l.:\.:I~l)
~
~
.\ (r,) [
T
y T (I -
f - yr
r
I ].
(!I.:UiO)
\ hich shows the dependence of (1~ ()J) yexplicitly. I In the second equation of (9.::I.liO). (12 has been re-explesS(-c\ .IS Ihe llroduct of two terms: the first is Ihe standard Black-Scholes inputuntin the assumption that arithmetic Brownian motion is the data-gencrating process. and the second term is an ,H\jllstlllcnt factor required by the trending O-U specification. Since this a
\)
Holding lixcclthe nncOlH\i lional variance of ret I\I"ns J~ [ Ii (T) J. as the" hsolute value of thc autocorrelation increases frolll 0 10 ~. the vallie of (1 ~ inlTeases without hOUIHI. 2:1 This illlplies that a specificati()~1 error ill the dynalllics of .1(1) call have dramatic cOI1Se(l'wlI(CS for pricil1~ options. ;l"\\'l' lonl~ 011 lh(' "h!'oohlll' valu(' of tli(' aUIOnJlTt"I.lIiOIi lo Hvoid nmlll!'ioll in 1II~,kill~
nJll1pal"i~JI'~ ht-lwf>Cn
'·clllrm. See JAl ;lIul
n'sulls Ic,r Ilc"HOIli\"dy alllnccnrc'I,II('(1 .llld pc)sili\'('ly alllclftJlTt'l.llc.'cl asst't (I!N:.) for fllnller d,-Iai"-
W;III~
9. .>.
jllll'/l'IIlI'nli ng j'fl/w//('/lir 0l'lion I'rici ng Mod!'/.,
.\s Ihe l'ellll'lI intcrval r derreases, il call 1)(' showlI Ihal Ihl' adjllslllwlll faclor 10 .\~lli(r)l/r in (!l.:t(il) appro;H'h('s IIl1ill' (IIS(' 1:llt/pilal's 1'1111'). III Ihe cOlltilluous-lillW limit, thc stalldard devi;llioll of cOlllillllollSly compOll IHlt-d ret u rtIS is a cOllsistellt esl iilia Ior Ii 'I' a ;1I H II he clkcls of predictahilill' OIl a vanish. The illtllitioll (,(III1CS frolll Ihe raI'l th,ll a is a IIwaSllre of /()((J/1/("lIlilil)~lhe volalility of illfillilesilllal pricc challges-alld Ih('l"(' is 110 prcdictahilily OV('I" allY illfillitesimallilllc illll'l'val hy cOllslructioll (see Seclioll 9.1.1). TIH'rd()re, Ihe illfhlcll(,(, of prl'llitlahilily 011 cslimalors tI)r vidc a 11l111l('rical I'x'lIlIpk ill III<' lI('xl Sl'clioll ill which til<' IIlagllilllclt- of Ihcsc cllccls is qllalllili('(lliH'1l1(' Black-Scholes casc. Arlill.l(in~
1111' Ji/(/(l!-Sdw/I',I j'imlllll(/ Jil" j'lI'Ilir(II/li/i(y Expl'l'ssioll (!I.:~.lil) provides thl' 11('('('ssal')' illplll 10 Ihc BIoI('k-Sdloles IilJ'1ll1l10l I'm pricillg opliolls Oil all ass('1 with Ihl' Irending 0-\.1 (IYllalllics. If tlH' IlllCOIHlitiollal variallcl' of daily retllrns is ,1"1,.,( I) I, alld iflhe firsHmier altlo,'ol'l'('i;lIiOIl or r-period retltrllS is 1'1 (r). IhclI III<' pric(' of a call optioll
f
is
gi\'(,\l
by:
1:"" (/'(1). (: , 1\, T.
r
I,
(~U.ti2)
a)
log( I + :!I''' r )) (\ 1 + '2pt\r)}I/. "'::'-1)'
1111r)
Ie
(-~,()I. (!).:l.ti:11
alld rli ,lIId rI~ arc cklilled in (!1.2.17) allcl (!).2.1X), l'('spl'lli\'('I),. Expl'cssion (!I.:{.(i2) issilllply the IIbck-Scholcs forllllllot with all acljusted volatilily input. The a(ljustlllellll~lctor lIIultiplyillgl:!I,.,( I) I/r ill (!I.:Ui:l) is easily labulall'd (sec 1.0 and Wang [I!l\l!">\); h('ll(T ill pranicl' it i~" simple ma \\cr to ,IC!j lISI the Blac k-Scholcs li)l'll1ula Ii II' n('ga Ii\'(' a UloCOtTeial ion of tlH' forlll (!).:t:,H): Multiply thc usual variance ('slilllalor '~I ,.,(I)l/r hy thc appropri,\\(' 100e\or 1'1'0111 Table :~ or 1.0 alld ''\',\llg ( I ~I~I:,), ,IIHI IIS(' this OIS (I" ill [h(' Bbek-Srlloks li,nllllia. NOll' 11101[ 1'01' all valll(,s 01 I'1(r) ill (-1,. Ill, Ih(' lae[(lI IIlllltipl)'ing .I~ [ r, (I) l/r in (!).:l.Ij:~) is greater than or C'lIl,tll:' OIl(' and inlTcas('s in the al>soilltc valliI' or Ihe first-ordcr autocorrelation cocflil'il'1I1. This illlplie~ thaI oplioll pric('s lInder the trcllclillg ()-\ 1 ,'pc('ilil';llioll ,11(' always gr('atn thall or equ;1i to prices under the standard Black-Scholes spccilication, alld titat oplioll pric('s arc an increasing fllll('[ioll or Ihc ahsoilltc 1';lllIC of Ih(' firstor(\n ;lIl\ocolTl'iatioll c()l'niciclIl. TIIl's,' aI'\' pllH'ly !'<'aIIIH'S "fill<' Ir('\\(lillg O-U process alld do lIot gcncrali/,c to othcr sp('C'ilicatiolls or thc clrift (sec 1.0 alld WOIllg 11!)~I!iJ (()I' cxamples orolhn pallerns),
:nli
9. /)n71'rtli!ll' Pricing Modi'l.l
'fhblf 9.5.
Strikl' !'lin'
(J1,lillll
/IIi/h 1/1'~I//il/l't.y f/1//o(()rIl411Inl,1'I1II7IJ.
Tll'lI(lill~ ()-lJ
1I!.11 "·S! h"k,
I'rke. with
l)«iI)' {I,
(I) ==
-
..
!'Iin'
.-.0:.
I -.10 'r '~·;-r--·:~o -r=~~~-T-~·'F'
Tilll"-ltI-/I!atlll ilY '/'-/ == 7 Days
-
:\0
10.0:.'H
:\:)
:).O:~4)
.\1)
O.Hli:1 11.1111 0.000
·1'. :.0
I
10 O:.'H '.0:17 OHW. 11.111:1 0000
10.0:.'1{ :'.0:11{ O.!IIO O.I/lti
11.:l:Ili 7.lilfi .1.11:>1 2.'1:!:! l.tiH7
11.:1!).1 7.7·lti ·1.!17Ii :I.IHH 1.7!17
10.0:1H :>.<14:1 0.!17:1 0.02·1
10.0:.'H :,.0:.1 I.Oli:.' 0.(l·1I
I
11.211'. 7.'.:.11 1.710 :.'.11111
I I
I. r,~)~
.- -.
I
I ! I I
I
I I
I().O:.'~
:,.1 ()~ 1.:11>:-1 0.1 :17
I
I
== IH\! \);I),S ------- -------
·n
115,IH 11.7Hli 1:!.2:~H 1:!.7:!:) 7.!)!)H H.%:. !U1l1 !Ui(iH !,.:!Hli 7.:!-1-1 !1.nH li..l!1I • ') r.t r -r: .\.HI_ -1..>.1.) .,r: ..•11., :\.:\Iil :.'.07;\ 2.·IH:! :1.21 ,I :>'9Ii:~ . __._--- ---- -- -.- - - -- - - - - - - - --'/'-/ = :lli4 Da)'s ...
I
1:.'.7:,:1 '1.\'1:1 li.'(1)11 ·1.'111 :1.111'1
10.0:.'1{ :>.<)7·1 1.2 Iti 0.0112
_~I.~~J _~:~~__I~~~~~. __ ~I~I~ ._(I~~I.~~
Tilll('·I"-~l.lIll1ily
:10 :1'. ·10 .1:. :,0
r~-J···~r~-]~---·
. Tilll<"I0-/vlallllil)' 'J'-/
I
:10 :1:. 10 ·Ir. '.0
Il//iI'\ 1111 1/\.11'/.\
1:.'.11-1',
I:.'.!I:,O
~U)~:2
').7W
7.01i1 ".10:.' :Ui·I:,
7.:.':\·1 :•. :.'H:I :I.H:.'I
-.-
1:1.2IH 111.1 :>:~ 7.liliO :1.7:~j!
·1.:!ti I
..
I :I.ti:!() 1O.lilil H.2W (i.:\7·, ·I.H!Hi
--
-- ---
..
I·U·!!) II.rrH:! !I.:H:. 7.·17H
.
-
1:;.102 12.:)01 ItI.:WI H.:,'ili 7.IOh
(i.()O:~
I
hol(·,( .dl optioll 1'1 i( (', on a hypodwClC..d $·HI ~cod;" UHdt'r an ~u itlHH(.'tic .\ 1"'l\thllg (h l\~H"l\-llhkl\htTk IHun· ....!\. (or log-pl"in.. ~. a~~\l1Billg a ... LUHl.ud ({(·\,',"'UH ui '2 10\ d.ul\' {HHliH\10\l~ly (UI\I})lH\utktl n:lurns, ~Ul(' a daily continuously (umpuu",h'd riskh"('t' ,.Ut' 01 !og( I.O:')j:\t;"L ;\~
(\ItIl)l.1l (..,01\ u( (U.u ~ "\(
(\10\\'''';''' "lo(iOI\ H" ~us 1' {,
:111 /':m/Jirim/I((III/m/i/lJ/
'«. gallgl'
Ih,' "llIpil i.. ;1i .. t'lI'V;III ... · or litis adjllsllIlt'lll {ill' alllocolTl'ialioll, ');lhtt· 'I.:...quu" ;1 (OllIP:lIisoll 01 HIa,'k-SdlOks pric('s Ilndn arithmetic Browllian IllOlioll ;"ul ""dl'r II", In'lldillg Orllsll'ill-Uhl{'nl)('ck proct'ss for \,;11 iOlls holdillg pniods. slrikl' pri",·s. and daily ;tIIlOcoITl'ialions from -!:i [0 --.!:,'Y" {iu' a h"po[h('[il'a! $.J() s[oek. Tht' III1COIldi[ion;1I stalldard dcvialion of t\ailv 1",'!lIIIlS is held flxl'd al :!'J., per day. Tht' Black-Scholes price is eaicll\;I[l'ti a .... ording to (l).~. \til. sl'lIing n t'<jllailo tht' IIllcondilional stalldan!
or
9.3. Implementirlg Parametric Option Pricing Models The first panel of Table 9.5 shows that even extreme autocorre~tion in daily returns does not affect short-maturity in-the-money call option prices very much. For example, a daily autocorrelation of -45% has nQ impact on the $30 7-
9.3.5 Implied Volatility Estimators Suppose the current market price of a one-year European call option on a nondividend-paying stock is $7.382. Suppose further that its strike price is $35, the current stock price is $40, and the annual simple riskfree interest rate is 5%. If the Black-Scholes model holds, then the volatility (1 implied by the values given above can only take on one value--{).200-because' the Black-Scholes formula (9.2.16) yields a one-tCK>ne relation between the option's price and a, holding all other parameters fixed. Therefore. the optioll describen above is sain to have an implied Il()lfllilily of 0.200 or 20%. So COlllmOll is this notion of implied volatility that options traders often quote prices lIot in dollars but in units of volatility. e.g., "The one-year European call with $35.000 strike is trading at 20%.n i . Because implied volatilities are linked directly to current market pri~es (via the Black-Scholes formula). some investment professionals have argtled that they arc betlere~timator!lofv()latility than f'~timlllf(yfll tm!W"tf NNM41ltm·1f1, llala '!leh .... lit . '",...... ,,." 1I(.1·~,.,r,n"( "fI' I~'i'(, '/.011 t"
If( '/,.
"'(fff/(#fi4 "II/hUtll' \
9. /)1'I"illlllitll'
I),.i(il/~
Mlltld,
SiJ1(1they arc based on nllH'lIt prices which presumably have expectations of tl e future impounded in them. -!owevcr, sllch an argullIellt ovcrlooks tlte fact that an illlplied volatility is i11~imately related to a specific /ml'flllll'lrir option·pricing lIIodel-typically thc B1ack:Scholcs model-which, in turn, is intimately related to a particular set ~If dynamics for thc undcrlying stock price (geollletric Brownian motion ill tl e B1ack·Scholes case). llcrein lies tlw prohlem Wilh illlplicd volatilities: If th B1ack-Scholcs forllluia holds, then the parallleter a can he ret'overed with III m-or by inverting the Black-Scholes fOnJllli
{o see this 1II0re clearly, considn Ih(' argument that implied volatilities an'lpeller ()recasts of flltlln~ volatility because changing market t'(IIHlitiolls calls'c volatilitics vary throllgh time stochastically, and historical volatilities canllot acljust to changing market conditions as rapidly. The fillly of this argulllcntlies intire lilCtth,tl stochastic volatility contradicts the assllmptions requircd hy thc Black-Scholes modd-ifvolatilities do change storhasticall), thrungh tillie, the mack-Scholes formllia is no longer the cOlTcrt pricing formula allll all implied volatilit), derived from the Black-Scholes /i>nllllia provides no new information. Of course, in this case the historical volatility estimator is elJuall), IIseless, since it need not enter into the correct stochastic volatility optioll-pricing /(lJ'IlIltla (in ract it does not, as shown by Ilull and White ll9H7 J, Wiggins 119H7J and others-sec Section 9.3.(j). The correct approach is to IIS(' a historical estimator of the unknown parameters clllering into the pricing formula-in the Black-Scholes case, the parallleter a is related to the historical volatility estimator of continuously compounded retllrns, bllt IInder other assllmptions for the stock price dynamics, historical v~)latility need 1I0t phI)' such a celllral role. This raises all interestill!{ iSSllt' n~!{anlill!{ the validit}' of the Black-Scholes /i)flllula. If lhe Black-Scholes formula is indeed t'(lI'rect, thell tht~ implied volatilities orany set of optiolls ont he sallie stock lIIust be 1I11/1l1'rimll)' ilil'lllil'lli. ()f course, in practilT til!'y n('ver ar(,; thlls til(' assllmpliolls of lile IlhH'kScholes lIIodel canllotlitnally be true. This should not cOllie as a cOlllpletc sltrpriS('; aha all the assumptiolls or the Black-Scholes IlIlHld impl)' Ihal options are redundant securities, which e1iminatcs the nccd for organil.cd optiolls markets altog(,ther. The ciifficlllty lies ill d('tnminillg which of the man)' Black-Scholes assmll)ltions arc violatcd. If, for example. tltc Black-Scholes model /;Iib ('111-
(/ J. IlIIjill'llll'lIlillgl'f/rfil/lI'lnr (Jjllillll I',irillg ;\/"r/,./,
piritidl\' I)('callse stock prices do not 1()lIow ,I IOf.:1I01lJl.t1 dilfllsion, we lIIay Ill' ilhk 10 specil)' all alterllate pritT pron'ss Ih.1l lils ill(' data heller, ill which case the "illlplit'll" parallwt('r(s) of options Oil tite SilllH' stod;, may illdel'd he nUlllerically identical. Altenratiq'ly, if Ihe Black-Scitoles model bib l'lllpiricillly hecause ill practic(, it is illlpllssihk to tl;uk cOlltilllHHlsly dUI' 10 transactions costs and other illstitlllioll,t1 cOllstrilillls, thell 111,11 kets arc n('v('r dYllalllically complete, optiolls an' lIeH'I" rnirnHl;llIt serllritit,s, and we sholiid never cxpect "illlplied" parailleters of optiolls on the sallie slock 10 he 1IIIIIH'rically idelltical I()J' any oplion-pricillg I()rlllula. In this case, Ihe degrce to which implied volatilities disagtn' 111,1)" 1)(' iUI indicatioll of h,lI\' "redIIlHLIIII" options really arc. The 1;lcl thill opliolls Iradns quo\(' pri, es in 1"1 IllS olBI,l
(J. 1.6 SI",-!/fI.I/;,·, lii/f/it/;l), ;\I"r/r'/I
ScqTal empirical studies have showlI that the g,'olllctric Browlliall llIotioll (9.~,~) is Ilot all ilppropriate model for (Trtilill st't'mil)' priccs. For cxampk, Ikckers (19K:I). Black (1971i), B1allJH'rg alld (;ollcdes (I ~}74), Christie (19H2}, Failla (I%:>}, 1.0 al\d MacKilliay (i~)HH, I~)~)()c}, ami MalHlelhrot (I ~l(;:~, I ~)71) hilvc dOCulllellted importallt departures flO II I (9.2.2) for US stock returlls: skcwlless, excess kurtosis, snial correlatioll. alld tillle-varyillg voJ;llilitil's. Although each of thesc elllpiried rq,;ularities has implicatiolls for optioll pricillg, it is thc last olle that has rCteivcd tlH" lIloSI attelltioll ill the rcct'llt derivatives litcrature?1 p;trtl), hecau,c volatilit), pla),s such a cClltriti role ill tit(' /\Iack-Scholes/Mcrtoll l(lIIl1Ulatioll alld ill illdllslry practice. II', ill Iltt' gl'olll('tric Browlliall lIlotioll IIl<>dci (~).~.:!) tite (1 is a knowlI det<'lillillistic flillctioll of time aU), thell thl' I\lack-.'icitok, formula still 0(1) ilIon'! till' optioll's ill>plil's btlt with a rl'plared hy the integlal Iift-. I I()\1'l'I'('/', if (T is stochastir, till' situatioll ht'colIH'S 11101 e (olllpkx. For
,I;'
N~ (I~)!U). n.dl.lIld f{tllIl.l (Iqql), B('f"c'" (I~IXII), (AI" (;old('11I ... ,.~ (I!I~II), Ilt-slnll (I!I!I:I), 1101111, ..... , 1'1.,1"11. ,,,,,I ,,1,,\1'11," (I!I~I:l),111I1I "lid l\'loil<' ( t !Illi) ,.101011"111 "lid Sh.IIII'o ( I ~IH7" S. oil II 'IHi, .. , 1101 \\'):gll" ( I'!Hi,. 'lIS"I'. Jol' t'X;UlIph-. Alliin alld
(I~'r.).
i).
"nilllllilll' I'm/11K ,\l/JIM.I
I'\alllpll', supposc' Ih;u Ih(' itliuLtlll«'llIal assl'l's c1)'nalnics are given hy:
til'
1I/'tlI I-
dn
li(n )",
(J
I'dll"
+ /1(n)1I Un,
(!).:I.Ii4 ) (!).:I.Ii:I)
11'11«'1'1' (1 (.) alld 11(·) ;lIt· ;11 hi I1';111' 11111(1 iOIiS or volali lily rr, and /I" and Un arc ,Ialldanl Browlliall 1I10liollS wilh illslalllalH'ous ('olTl'ialioll d/l" dll, fJ ill. III this case, il iliaI' 1101 1)(' possihle- 10 dl'l('rlllillc the pritT or an option hI' arhitral-\(' arl-\llIlu'lIl' alolll', IiII' Ilu' 'illlpic reasoll Ihal Ih('r(' ilia), 1101 exisl ;l dVllalllic sl'll~lillallcing ponl'olio stralc'gy involving SlOcks and riskless bonds Ihat elll p('rl(>('(11' I'l'plictl(' Ihl' opli()II's payolf. I "'"I'isticalll', slodl;lslic I'oblilill' illllOdu('('s a second sourc(' 1111('('1'1;lillll' ililo the rl'plicatillg portli>lio alld if this 11lH·('rtainty (lIn) is lIot perIc'('(11' cOl'relall'd with Ihl' IlIlcertainl)' illhel'ellt ill th(' stock pl'ic(' process (II,.), Ihl' rl'plicalillg portlillio will lIot 1)(' ahle to "spall" the possihll' O\ltCOIIJ(', Ihal all oplioll 111;1\' rl';t1i/(';11 Illallll'il), (WI' Ilarrison ;lnd Kn'ps II D7~) I alld Ililllie ;In
=
or
til'
I'dll"
(9.:U;(;)
+ ~(1~dll".
(cl.:1,()/ )
1lI'tli -I-
(X(1~'"
(J
IIv asslIlllillg Ihal l'obt'''·III' is IIllcorn'LtI('d wilh aggr('gall' ('OIlSlIlIlPlioll, Ihl'y sholl' Ihal el)llililllilllll ()Jllioll price, an' giVl'1l lIy Ih(' ('xp('Clalioll or Ihl' Black-Schoks rontlllLt, 11'1 It ')"1 , Ih(' ('xpt'dalioll is lakl'lI wilh r('sp('rl 10 III(' avera!!;(' volalilily o\'('r Ihl' oplioll\ Iiii-. lJsing Ihe tl\·II;lIl1ic ('(I'lilihrillill llIodels 01' Carlllall (1!)7(ih) alld Cox, III gerso II , alld \{os, (1~IWlh), Wiggills (1~IH7) dl'riws Ihe c<Jllilibrilllll price of vola Ii Iii)' risk ill all ('("0 IlO II II' wh('l"e agt'lIls possess logarilhlllic IIlilily I'lIlIcliollS, \,iddillg;\11 ('(I'lilihrilllll cOlldilioll-ill Ihl' form ofa I'D\<: wilh rCrlaill lIolIlHlarl' cOlldiliolls-ror Ih(' illsl<1I1 1;11 1('0 liS ('xpc('(cd ITIIII'Il or Iht, oplioll pritT. Olh('l" d('l"il';llin'-pri<"illg mod"', wilh siochaslir volalililY lake Sillli\;11' aJlpl'oadICs, ill<' dilli')"t'IJ("(" cOlllillg fro/ll lilt' Iype o\" l'qllilihrilllll 1I10dcl ('1111'101'('<101' Ih(' dlOic(' "I' 1'J'('kn'IHTs Ihal agl'lIls exhihil.
OIJtion Pricing Models Y. J. hlllJ/mlnltillg PtlTametrir j'lIrllllleter Estimation is LS of stochastic-volatility mod els, One of the mos t chal leng ing aspec lo.:m l!" lOn-l OD,t yet e vabl bser ess arc uno that real izati ons of the volatility proc ing a. driv ess proc the of ers met of the para form ulas are inva riab ly func tion s nt orta atte ntio n devo ted to this imp To date . ther e has bee n relatively little use beca arily s like (9.3 .64) -(9.3 .65) prim issue for cOll tinu ous- time proc esse disatin g cont inuo us-t ime mod els with estim in of the dim cult ies inhe rent to ted devo been has n ntio grea t deal of atte crct ely sam pled data . 1Iowever, a edas rosk hete al ition cond ve essi auto regr a rela tcd disc rete -tim e mod el: the 2 J pter 2) and iL~ rnanyvarianLS (see Cha ticity (AR CH) proc ess of EngIe (198 2]). [199 and Bollerslev, Cho u, and Kro ner issues othe r than opti on pric ing. by ed ivat Alth oug h orig inal ly mot g cont inspir it of som e of the corr espo ndin ARC H mod els doe s capt ure the on and Nels ), 994 (J er by Nels on and Fost lIolls-time mod els. Rec ent stud ies orta nt imp e som ide prov ) 1996 (1991, 1992, Ramaswamy (199 0), and Nels on Fos~er and on Nels and 6) (199 on ar, Nels links hetw een the two. In part icul -tim e rete disc ecor d asym ptot ics for several (1994) deri ve the cont inuo us-r esse s proc ime us-t inuo h conv erge to the cont ARC II proc esse s, som e of whic s of ertie prop l irica emp The gins (1987). of !-lull and Whi te (1987) and Wig ect pf subj the be bt dou no will but ored I.hese estim ator s have yet to be expl . arch rese lre iUll
Di.fn·rtr-Timr Models
ilibr ium with a disc rete -tim e dyn ami c equ Ano tha app roac h is to begi n ami cs dyn e pric t's h the fund ame ntal asse llIodel for opti on pric es in whic to ble ossi imp ally typic is el. Alth oug h it arc gov erne d hy an ARC H mod mus t ions vers time ous[inu cOJ1 . lime rete pdc e secl lritie s hy arbi trag e ill disc tilit ; as well in the case of stoc hast ic vola LS men argu ium ilibr equ 10 ,d appe efram time oustinu cOll y in leaving the hel) ce (her e is lillie loss of gene ralit Ng and n Ami by n take h roac is the app wor k alto geth er in this case. This csulas for a variety of pric e dyn ami form g ricin on-p opti ve deri (1 ~)9~). who inic hast stoc , ump tion grow th vari ance stoc hast ic vo[atility, stoc hast ic cons c ami dyn e -tim rete ps- by appl ying the disc tcre st rates , and syst ema tic jum 6). (197 ein inst Rub (1979) and equi libri um mod els of Bre nna n irica lly gene rally easi er to imp lem ent emp also arc els mod time Disc retesactran l arc sam pled disc retel y, fina ncia sinct' virtually all histo rical data and n atio estim rete intervals, para met er tiOl)S
y,
:\82
})nil/{/Iil",
",.;r;lIg Modd.!
of nonlinear functiolls of the data-gclleratilig process an' almost iml)("sihk to ohtain in discrete tillle, hut in rOlltillllOIlS time Itil's dilferl'lItiation nlk gives an explicit expression for such dynamics, Th('~lretical insights into tlw equilibriulII structure of derivatiws prices-I!)I' cxampk, which slalc variables alTcct derivativ(,s prices alld which do not-arc also moJ'(' readily ohtained in a continuous-timc framework such as Cox, Ing('l'soll, ;lIld Ross (19M5b). Thcrclorc, each set of models oilers sOllie vaillahle illsights that arc 1I0t cOlltaincd ill the othcl:
!9.4 Pricing Path-Dependent Derivatives Via Monte Carlo Simulation Consider a contract at date () that gives the holder the right hUI Ilot the ohlig.llioll to sell olle share of stock at date T for a price equal to the maxinllllll of thai stock's price over the period from 0 to T, Such a contract, oftcn called a loohiJ(lcl! oplion, is ('k-ady a pUI oplioll sillce il gi\'('s Ihe holdn Ihe optioll to sell at a particular price at maturity. However, ill this case the strike price is stochastic alld determined ollly at thc maturity date. Because thc strikc pricc dcpends Oil thc /,alh that the stock price takes from 0 to T, and not just on the termillal slock prire I'Cn, such a contra<"l is called a /lIIlh-dr/IPIIIII'III optioll. Path-dependent optiolls have become illneasillgly popular as the hedging nccds of invcstors hecomc cver lIIore complcx. For example, many mllitinational corporations now cxpend greal elTorts to hedge agaillsl exchangc-ratc nllctllations since hu'ge portions or their aCCOlllIlS r('ceivahle and accounts payable are denominated in f()rcign currcllcies. OIlC of' Ihe most popular path-dependenl opliolls arc roreign currency (/tll'mg~ mil' or Alillll options which gives thc holder the right 10 buy foreigll (mrellry OIl a rate cquallo lhe average of the exchange rates over the lire ortl!e ('ol\lr;lcL~:' Palh-dcpendellt options may he priced hy the dynalllic-hnigin h approach orSection 9.2.1, butlhe rcsulting I'DE is often intractabk. The risklIeutral pricing method offers a considerably simpler alternativc in which the pOlVcr of high-speed digital computers mOl)' he exploited. For ('xample, considcr the pricing' of the option to sell al Ihe maximum. If' 1'( t) dellotes Lhe date I stock price alul }1(0) is the inilial value of this put, we havl' 11(0)
,,-.rE'
[1\hXr 0:.../::
1'(1)
-1'(TlJ
(~IA.l
)
'.!:'Tht" I('rm ""-",i.tll" ('OIlH','i fl"ollllh(' Lin Ihal sud I options \"'l'n' tina ,u'ti\'('\ywrillt'll 011 ~llJrk.~
A,ian (·xdlall~f,:!\. BloCIII,,' th,'M.' l'xdl;m~l's an' usually smalh-rll!,1I1 tlwil ElIIC)P(';lIl Hul American rOllntl'rp,trl~. Wilh u'lali\'('ly Ihill trading alld low d'lily \'ohmH', pritT:" 011 slU:h 'x(hangt"~ are ~nll1t"\\'hat (·.,~i(·r 10 II1tlllip1l1aw. To millill1i/.f.' ;111 option's ('SPO!\UU' to tht· risk l.ltlilJ){ 011
t
J'f~'tJ("k-l' ..in' manipulation. ~11Il'\\, option wa.1ri ('I"('ale" witla
111('
flU" optiou's lile' playing till" role olth(' 1t'llIIin.1I Mock prin',
."'(·ra~l' oflill' :o-.1I.d'IH in's 0\'(''-
9,-1, I'ri"illg I'(//It-I ",/It'lIdl'll/1 "',h'lI/iJl'" \ 'ill
" " E' [
" " E' [
,1101111' (.'1/110 S/lIIlIll/lill1l
I
I)']
1\1;"
1'(1)]
1\1;1', I'i
o I
II
I
I
I
"
I, ' 1/ ' 1"1) I
I'i 0).
:1:-1:1
(!I,I.:!)
(!I.-I,:I)
wlwn' F.' i, IIII' "slll"'laliolls "llIT,II"1 willi n"I"TI III II", I i'\..'"l'\III~III' .. oha!Jilily disirillillioll or ('qlli",,1<-1I1 1II""lill),!;;"" 1I1l'~1S11I", OI'SlT\'(' Ih"l ill goillg I'mlll (')"I.:!) III ('),,1.:1) \\',' h.IH' ", ... llhl' 1;lIllh;l1 II,,· n'I"'('I"
1',,,"(, II ,'."
9,-1,1
/)i,\(//'I,'
li'"I1I\
COlllillIlOII,\ '1'/111"
Ih I hl'ir I"'/')' ""1111'1'. digit,,1 ('Ollll"ll''/'' ,II',· illc'I),,,blc or Sillllll;llillg 11'\11)' ,'Ollli"IIOIIS pheIlOIlH'II"; bill as" pI',"'lic;t1 III"II,'/' Ihl'y al'(' 01'1(,11 ('''p"lIk or I'n"'idillj.\ "x(TI1"1I1 "I'I'I'oxill\;\liOl", III p'''li ... d,,,. it w,' dilid" 0"1' lilll(, illl'·''I'"I[O. '1'1 illio II disen'I(' illl(', I'af,; ,.,,' h "rll'lIglh It. ","l ,illllll"ll' pl'icl's ~II ""1'1. dis""l'l!' d"ll' kit. k = 0 ... ,. II. Ihl' rl's\l1t I"ill Ill' "II "pproximalioll 10" cOlllillllOIlS ',\llIple p"lh ",hiI'll (';III II(' 1II;llk slll .... ·ssil'l'l\' lIlorl' pre('iS(' hy ;dlowillg II 10 grow ;llId II 10 ,1I .. illk so .IS 10 I,,('('p lixl'd,
r
For I'x;lmpk, ('oll.sidl'r Ihe CIS" or g,'olll('lri, I\rowlli.1I1 IIIOlioll (!I,:!,:!) 1'01' I,llit'll till' risk-lIl'lItral dYllamics ;11" gilI'll hI' tll"(1) = IP'U) til
I',:
1"(0) ('xl'
+
(J
1"(1l tfll(l),
[t 1;(11)] . .=1
D('spitl' 111(' 1;1,1 tll .. 1 Ihl' ~imIiIOlI('d p;IIII I'" IOIri(" (/1111, OIl 1lIllllipll's 01 It. till' "I'prll:-iimatioll IliaI' 1)(' III;\d,' ,lIhilLI,.ih pn'('i,1' hI' i'H'('easillg '1 ,\lid Il\('rd,)\'(' d('('('('asillg h-as II ill('\('''S('S I,'illlol" hlllllHI. I',; HIIII'('rg('s "'I'lIldy III (~),'I.'I) ('('(' Seetioll ~),1.1 1(,1' 11111111'1' dis(,lIssioll). [Jllfilllllllall'iY.III('I'(' OIrl' 110 g('II('I'OII nd('s fi,l' 1111\\' IOIrg(' II 11111\1 he 10 l'il'ld ;111 ;I(k'l"al(' apl'l'l':-iiIlialioll·-.. iloll.sillg II 11111,1 1)(' dlllll' Oil 01 (',1",·1"'·(',1\(' h.l.si~,
Y. /) ..1"11'1//1111' "til"iIlK i\/oddl
.hl,
1).'/.2//011'
MIIII)'
Silllll/illioll.\
10
I'l'Ijimll
Wc ((III, howcver, plm'ide SOIll(' rlear guidelines for choosillg the lIullll)('r ofn'plicatiolls /I/to sillllllat('. Recall that our MOille Carlo l'st i ilia It, of /1(0) illvolv('s a silllpl(' ;tvcrage aaoss rl'plicatiolls:
I ({(O)
=0
/'
L }",, -' 11/
,J /1/
(9.4.1;)
{'(O),
",I
I
\\'hen'I{'J~ ~=" is Ihe jlt. 1I'I'Iicaiioll or salllple palh oflhl' slock I'ri("('I'I"I)('" IIlIch'r Ihl' risk-lIl'lIlral disiriblliioll which, illlhe case of (9.4.4), implies Ihal =0 r - ~, Bill SillC!' hy l'tIllSlntrtioll Ihe }~" 's ,In~ liD randolll variahles wilh finite po;itive variallc(', the (:clltrall.illJil Thcorem illJplit's thaI ror large 111:
}I
0;(11)
== \'<11["-'1'
}~,,]. (~).4.7)
There/1m', lill' large 11/ an approxilllat(' !I:,% ('ollli
,1.!)()(1,(II) , ( 1/(0) - - - ' - - - < 11(0) :: /1(0)
fo
+
1.%(1,,(11») " r.:. = O,!):).
vIII
(9.4,~)
Tht' elwin' 0[' III thlls depellds dir('nly 011 Ihe desired ac('macy of li (0). If, lill' example, 1\'1' reqllire a /i(O) thai is wilhin $0,001 of /1(0) with !)!i% conflcl('lIn', /1/ nnlst he chosell so that: 1.%(1,( II)
::: 0.001
l\pil'ally \';lll I;" I is 1111\ knowlI, simlliations ill tlJ(' ohviolls \\';1)':
I \'arll'",1 = -
/1/
bIll
>
-
1.% )~ " -(1- II. (0.001 ,()
it (';111 he reaelily ('slimaled
~
L( \', , - _1',,)-,')
froJlI
the
(!I.-1.IO)
1/1 /_1
Sinn' IIII' n'plicltioll~ ;\\(' \II) hy cOIISlntctioll, l'slilll;llol's sllch ;IS (9.4.10) will gClln;lll\' I... "I'I'\' \\'..rI·I ... it;ll'l'd, COlIl'l'lgillg ill prohahilily IIllhdr cxpel'taliolls r;lpidly alld, 11'111'11 JlloJll'I'lv 1I111111;lli/('d, ('/)I\\'('rgillg ill distrihutioll just as rapidly 10 Iheir lilllilillf,!; dislrihuliolls.
I),
-I. 1 (:III11/"IIi.\II/1., luilli
1/
(:/II.I('(/-/-imll SII{lIlioll
III thi' special 1';,,(' 01' Ihl' oJllioll 10 sl'll al Ih(' IIlaxillllllll wilh a gl'ollH'lric Hroll'lIiall lIIotiol\ price plOn'ss,;1 r!os('
9.-1.
j'ririllJ!,j'1I11t-1)I'/JI'IIII1'111 lJnilllllillf.1 ViII MimiI'
Carlo Simulation
385
is given hy (;oldlllan, Sosin, anel Catto (1979): II (0)
=
P(O)t-,T
+P(O)
(_~) oft
[I _
2
0
] -
P(O)
2r
( I+~r2) [ I-
(9.4.11)
where a == r - a:! /2. Thert-fore, in this case we may COlli pare the accuracy of the Monte Carlo estilllator /i((l) with the theOl"etical value fI(O). Table 9.6 provides such a cOlllparison under the following assulllptions (for simple returns): Annual Riskfree Interest Rate
5%
Annual Expected Stock Return
15%
Annual Standard Deviation of Stock Return
20%
Initial Stock Price 1'(0) Time to Maturity 'I'
$40 I Year.
Frolll the entries in Table 9.G, we see that large differences between the con tinuous-time price fI (0) = $4.7937 and the crude Monte Carlo estimator if (0) can arise, even when III and n are relatively large (the antithetic estimator is defined and discussed in the next section). For example. H(O) and H(O) differ by 30 cenlS when n = 250, a nontrivial discrepancy given the typical sizes of options portfolios. ' The difference between /itO) and 11(0) arises from two sources: sampling variatioll ill li(O) and the discreteness of the simulated sample path~ of prices. The former source of discrepancy is controlled by the number of replications m, while the latter source is controlled by the number of ohservations 11 in each simulated sample path. Increasing m will allow us to estimate r:" [Ii (0) 1with arbitrary accuracy, but if n is fixed then E" [H(O)] llced not converge to the continuous-time price H(O). Does this discrepallc), illlply that MOllte Carlo estilllators are inferior to closed-form solutions whell such solutions are available? Not necessarily. This difference hig-hlights the importance of discretization in the pricillg of path-dependent securities. Since ''ie are selecting the maximum ove' /( ('xpollt'lltials of the (discrete) partial slim L~=I where k ranges from 0 to II, as II il1creases the maxinllllll is likely to increase as well. 2 t; Heuristically.
r,·,
~h/\hhllllgh
it is prohahle that the.' maximulII of the partial sum will inrrt"a~ with n. it A. . We illfl('OIS(, II in 'l';,bk ~Ui. Wt" gt'IWfClIt";" llt"W indt")>t"nc\f'lll t-andmu S("'1I11'II(T II; 1;'=1' and ,heft' is ;,dway~sc))nt' challct" IIla1this I)("W sequenre with mure tenns will 1H"\·(·r1h('h-~'\ yic.·}(\ ~ll1al!("J' panial Slims, i...
1I0t gll;U";I1H(·(·(l.
I).
Jah/,' 9.6.
f)rri!lfIti!,,·f'Ii,.;lIg .\I1I11t·11
,\101111' t:llt/" "\/illlll/IOIIII/ /ouiduII!.- o/IIUJII
(:,11<1 ..
" ·I.:IHIH ·1.·1\111 ·1.:,·17!) ·1.:,7·11; .1.I{,!l\) ·l.li·HH l.Ii70li ,1.717:,
t!,o I.OOU :1.1l1l1l :,.flOO
~lunH'
(:,111"
Auli,lo .. , i..
SE(il(O)1
100 :1;,0 :lli:, :,OU
/"itl',
0.011;:, O.Olli·1 O.Olli:, 0.0 II;:, O.Ollili O.Ollili 1l.1l II iI", fl.olli",
SEI i/(II)1
·l.:\li·I·1 ·1.:,1 :lli ·I.:.IiO:1 ·l.liOO7 ·1.li·II·1 ·l.Ii·I\I:1 ·1.701l1 ·1.7:!W
O.Olllili O.OOfrli O.llIlIili 1l.llIlIili O.llIlIili O.llIlIili O.IHlIii fl.Olllili
('~Iilllalol
ollhl' plitt' ul.1 011<"-\"(',11 look-h.It·" pUI opliuil \dlll (Olllillllltll'·IIIIU' Prl(l' II(O)=S·1.7~):\7. E,ldl fu\v "UI"I("polub lu all ilide-pencil'lIl "'I III .. im1lI,lllOII' of 100,0110 u'plicatioll' 01 :o.:lllIpk pilila .. 01 h'l1g1h II. FUI Ih,' .lIl1ilht'lif ·\.111.11'" !'
(~lIhh'h\l\-Sosil\·(;~\lto
1';1111 •. SElli(llll a, HI SElli(l)) I .'1"(' II ..· ";O'HI., ... \ ,."'.,,., "I lillll allel ('1111 .... ·.'1"·. li,d\".
the maximum of the daily dosillg pritTS o/" f' OVIT thc ycar (II = :!:,O Iradillg days) must he lower thall the lIIaxil\lllll\ o\" IIIl' d;lil)' highs o'n Ih;11 S;I/Ill' year (" .... 00). Therc!i)l"e, the cOlltillllOlls-tillle price lJ((l), whi("h is doscr to the maxilllullI of the daily highs, will almost al\\'a)'s cxn'l'd tlH' silll1datioll price {flO) which is disCfl'tized. Whid, price is \Ilore rclt-v;1I1l dqJl'llds of ("ollrsc Oil the 1(T1I1S ollhe particular rOlltr~lft. For exampl(', avcrage r;Il(' optiolls Oil f()("('igll cxchallge uSllall)' spedly ,,,,nit-iliaI' datcs ollwhi("h th('cx("h
'nil'
1).4.'/ (."()II//lIIlrll/ll"a{t'.jjll /1'''1),
two lIIaill ('\1I\("erlls 01 ,IIIV 1\I0llte (:arlo simlliatioll are a("("lIra("y alld 'OIllPllt.,tiOll." cost, alld ill mosl ("a~es thel"(' will he I("adeolls hetll'('I'll the til I• • \s II'C S.IW ill SerliOIl !I.·L:!, til(' ~talldard error of the MOllte (:arlo es1illl'Il.0(" ~i«() is illH'rs!'iy I~ro~)ortioll,d to the sqllare root of Ihc 11 II III her 01 rephcallolls III, hell("(' ,I :,O'lc, rnlllciloll III the stalldard C\Tor rcqlllres "'~I\' till\('s th(' 1II111I1In of n'pliclliolls, ;\lHI sO Oil. This t\'P(' of MOlltc (:+10 prolTlhlll' is olt('11 d('snilll'd as IlIldr' MOlltl' Cado (see I LIIlllIHTsky
C),'/,
I'rit"illg 1'I//h-lh'/lI'lIdl'li/ /)l'Ii1'(//;I'I',1 \';1/ MOil/I" (;ru/o SUIIII/I//ioll
:IH7
alld I \;lI\d~clllllh II~}()'II Ii II' ,'xalllpl('), rllr oll,i()\1\ ""\SOilS, TIll"rdilu', " 11I11I1\)l'r or l/l/ril/IIU'-/nilldio/l tl'dllli'l"('s h"q' h(TIl d",dop('d to illlpro\'l' the l'fficil'lIcy or silllillatioll estimators, Althollgh a thorollgh disClissioll or th,'S(' tCChlli'lIH'S is bl'yolld the scop(' or Ihi, In(\, 1\'(' sh,II\ hridly revi('\,' " 11-11' or 1III'IIl hl'J'l',~7 t\ silllpk I('chlli'llle «II" illlprovillg Ihl' pn«>I"III;II"T "r 1I101l\(' (:arlo l'Slilll;1I0rS is \0 replan' estilllates hy their popilialioll ('('"lllnp"nS Wh('IJeI'tT possihle, fill' Ihis rl'dllces salliplill~ variatioll ill Ihl' ('slilllaIOl, For l'X;1I II pie. whell sillllliatill~ risk-liculralized ass('1 rl'lul"lls, III<' S;lIl1ple Ill<'all or each rcplicatioll will allilost IH'ver he l'qllal 10 ils populalioll III('all (Ihe riskless rate), 11111 W(' elll CO\Tl'('\ this samplillg \';llialioll "'lsi I)' II)' ,,,Idillg I Ill" differ('lilT \)('III'('CII Ihl' risklt-ss raIl' alldlll<' saillplt- IIwall 10 each ohscrvalioll or Ih .. n'plicalioll, Ir this is dOlle fill' each rl'plicalioll, Ihl' n'sult will I)(' a Sl't 01 n'pliLlIiolls wilh 110 samplillg ('\Tor ror the IIIl'a II , The dlicicllcy gaill ("'1'('11(1- Oil Ih(' ('xl(,1I1 10 whit-I! sampling ('ITOrS Ii,," Ih(' 1I,,'au contributes 10 i1,,' ""'Tall sampling varialion or Ill<' SilllUI;lIioll, hUI ill n};lIl), CIS('S III<' iIUill"o\'l'nll'lIl C;1ll he dramalic. ,\ rdalcd Il'chllirple is to exploit olhl'r «))"IllS or populalion inlill"lllalion, For cX;llllplc, suppose we wish tn ('slillla[(' E'I /( X) 1 ;md we lilld a random variable g( l'l sllCh that E"I g( l')j is rlOS(' 10 F.' 1/(.\') 1alld E"1 g( l'll is kllOWIl (this is [he population illfimllatioll to 1)(' n,ploi[nl), 1·:"U(.\'}j mighl 1)(' Ilw pritT or IIcwly rreatl'd palh-dqwnd('II[ dnil'ali\'(' which mllSI be l'Slilll;lIl'd. "lid E· [g( l') J [he markl'l (lrin' or ;111 ('xisling derivative with simil"r \'IJ;lr;IClnistics, hellce a similar eXplTI-pr('ssillg E" [.I (X) I as lite sllill or E'I g( Yl J alld E'U( X)-- g( }') I, Ih .. n'l'('(t;r[ioll 10 he ('sti11I;t[ed is de('ollll'osed illto two t(,rllIS wherc [hc firs[ [nlll is kllown alld thc S(,(,(IlH\ 1
n
rOV(·I.\~t· ollh\s lHiHed.t!. II.\HI"\('I~lt·\· ,\\\<1 llouutsnunh \1\'1 II 11 od;, (1~IH(i) prll\idl,.1 IHol(' dt't.lih-d iuul U}Hi".I\l't\ \'''l)t}~i''tm Ill" :-;.ill1ilar IUeth·riai. Fi:\111l1,Ul \1 ~)~H)) 1:-. fIHl~llh·r ..\h\y lUI)} l' flHllP' t.·h('11~ !-ii,,' and ('0\'('1':\ :-1('\"('1",11 ;ul\'i1l1ct'd topics tlol found ill olh('r ~101l1(, (:.11101('\1' \111"11 ;I.~ M.II "'0' ("h.lill :-';1lllpling. (;ihh, .~;lIl1plil1g. randolll IHIl1'. alld .,iIlIlJi.IIt"d ;III1H',llillg. Fj,IJlII.1II (1~1~lIi) -"'";"S(·\,(·t;tllt'xl:-. plo\'idt., t'Xfdlt'l\1
(I~)(i·l) i.':1 rI~I.'!'
.,bo (ontai1ls mallY applir'lIiolls, t'xplit"it algolitlllll' lor 111;111\' 01 Iht' l('rillliqIU'., lo\"('r"d. ,wd FORTRAN ~oll\\,;lIt' (1111111 ;111 lip silt,) lor ral1dolll I1l1mhc.'1 gc.'IIt'r;llioll, Fill.tll". Fang .lIul \\',lIlg (I ~I~)'I) p"l'~l'llt .1 nlmpan illlroduniollltl" Ill'W ;Ipprn,ldl to ~If)ltlt' ( ·.n It, .. itlllll.Hjoll ha!'
pUldy Ch-tt'llllilli""if' :-..lIl1pling. ,\lIhollgll il i ....,liIlIOf) \0
\lit'
"'nU' tr;Hhlioll,\t \\H'lho
.h,11 10(11.. qllilt' pHlllli,illg,
F.mg
t',l1",,"
10 It'll 110\\' llii, ,'!'IHo.1t It ('OIllIMIC·.,
;\utl \\'."I~ (I~I',H) I)1O\l(k
o.,O\l\t'
IHll\glllug \·~,""plt':-'
Y. /)1'I11Iftlilll' }'ririllK "'oddl
1)(' ....dlectell .. throllgh ils IIwall to produce a mirror-image which has the sallie statistical propcrti('s, This approach yields an added hendit: negative correlation alllOllg p;lirs or replicatiolls. II' the slllllll1<1nds or thl' MOille Carlo estililator ;11'(' 1II011OtOlH' I'lIllctions o('the replicatiolls they will also he lIegati\'dv ('orrd;lted, iUlpl\'ing ;\ sillallt'l' \'ari'lIlCt, ror the estilll;ltol'. This is a simple ('xailiple or a more general Icdllli(lue kllown as 1\11' 1IIIIillll'lir /111111111'.1 1IH'lholl ill which correlatioll is in
:\11 l/I/1.l/mlioll "I li/lll/1/1/' Nl'lillrlillll 'Iil illustrate III(' pOlelllial pow('/' of\'ariall('('-rcdllclion 1('c1l11iqllcs, we ('onstnKt all antithetic·variates estimator of th(' pricc or lhe ())I('-year lookback put optioll ofS('Cliolll).,I.:1. For ('.lch silllulal<'c1 pricc path ~=U' allother CUI he ohlaillcd without furthn Sitllitiation hy reversing Ihe sign o('each or tl\(' LI1HIOllll" g('II('J'ated II\) stalldard lIonllal variates Oil which tl\(' pritT palh is has('d, viddillg a s('('ollcl p.llh T/: which is lIegalively (,OIT«'Ialn\
I}'/. I
Ir IL"
willt th .. lirst. 11111 ';lIlIplt- p;lIh, 01 \I',i.I~=" arc g('II('ralt'd, tIl(' J'('sultill!-(
Cllrlo Simulation
9. -I. }'ririllli }'alh-J)fllrndl'lll J)flil/alivfS Via MUlllf
antithetic-variates estimator f/(O) is simply the average across a1l2m path~ fiCO)
L
LIn) ljJl - 1'(0)
]=1
;=1
I (II< == e-,T 2m ljll +
(9.4.1 ~)
where
The relation between antithetic-variates and crude Monte Carlo can be more easily seen by rew;iting (9.4.12) as 11(0)
(
I r TI~ -2 e- - L..- 011
I TI~_) + -21'-' - L..-lj. -
P(O) (9.4.13)
m j=1
TIl j=1
1 '" y +Y e-,T - ' " ~ - /'(0).
mL..-
') ~
J=t
Equation (9.4.13) shows that ll(O) is based on a simple average of two averages. one hased on the sample paths ~~ ==0 and the other based on Fjit ;~O" The fact that these two averages are negatively correlated leads to a reduction in variance. Equation (9.4.14) combines the two sums of (9.4.13) into one, with the averages of the antithetic pairs as the summands. This sum is particularly easy to analyze because the summands are lID-the correlation is confined within each summand. not across the summands-hence the variance of the sum is simply the sum of the variances. An expression for the variance of ii (Ill then follows readily
I I
! I
Var(ii(O)]
-2,T 1 lj1l+ -Var [ - fj,,] == e m 2
1(12
-2rT ;; == e
'a 2 (n)
-'-(1 2m
Var[ lj"l
+ p)
(9.4.15 )
1
-)
+ 2' Cov[ lj., lj.]
(9.4.16)
where a;(n):=Var(e-rTlj.l==Var[e-rT~II] and p:=Corr[e-rTlj., e-rT~nl. Equation (9.4.15) shows that the variance of II (0) can be estimated by the product of f-rTf m and the sample variance of the IID sequence (OJ.+ ~.)f21. There is 110 need to account for the correlation between antithetic pairs because this is implicitly accounted for in the sample variance of{( lj.+ fj.)f2). Equation (!I.4.IG) provides additional insight into the variance reductioll lhat antithetic variates affords. The reduction in variance comes from
:1!I0 , Iwo SOIlITl'S: a dOllhlillg of Ihl' 1IIIIIIh('l" of r('plicOIlilllls rrlllll III 10 '2111, .lIld Ilw 1;l('tol' 1+p which sItollld h(' less tltall 011(' if tltl' ("OITelOItioll 11<'1\\"("'11 til(' antithetic \'al'iates is lIegali\'e, Notl' that l'WII if the correlatioll i~ positin', Ihl' \'ariallce of li(O) will still he lower thall the lTlIIic MOil\(' (:.1110 ('slilllalor liW) ullkss there is pnfer! correlatioll, ix" p= I, Also, whil(' "'(' ha\'e dOlliJll'd the 11I11Ilh(')" of replicatiolls, 1\'(' ha\'e dOlle so ill a ("(lInp"laliollally Irivial wa)': dlallgillg siglls, Silln' Ihe cOlllp"latiolls illl'oll"!'d ill pselldoI"
t\ (Olll parisoll of Ihe lTude MOIlI!' ( :OIr\O esl imalor i'l (0) 10 II H' ;UII iI hel icvariales eslimator li(O) is prodded ill 'bilk \I.(i, For lIIosl or Ihl' "~illllda liollS, tlte ratio of lite slalld'lI"!l elTor of li(O) 10 Ihe slalldard error of i'1(O) is O,OO(;G/O,()\(;;I=().'lOO, a reduclioll ofahonl (jOW" III comparisoll, a dOIlhlillg of Ihe Illllllhcr of r('plicaliolls 1'1'0111 11110211/ for Ih(' lTlld(' 1\\01111' (:arlo eslimalor wOllld yield a ralio of 1/ ~=(),7()7, ollly a 2!)'J" r('(hlllioll, 1\1or(' 1'""\II01l1y, OhSl~r\'l' 1'1'0111 (!I.,1.7) alld (!).-I, Ili) I\tOlII\t .. rOllio or IIII' slalld'lI"('111'('('11 the alltitltetic pairs of tlte SilllUI;lIiolls ill Tallie !l,(i, a suhslallti.II ,'ahl(' which is rl'spollsihle Ii))" Ihe dr'lll1alic reduclioll ill l'ari;IIH,(, of li((}),
'y
9,-1,5
E>:I!'ll.Iiol/,\ f/I/d !.ill/iff/filil/.l
,
-tIll' MOille Carlo approach 10 pricillg palh-dl'\H'lIdl'lIt .opliolls is qllil!' gl'lI'1ral alld lila), he applied 10 virlll.III), all)' Ellrop('all dl'l"ivatil'l' sl'cllrily, For 'ixalllple, to price an'rage-rall' I(nl'igll nllTl'lH'y opliolls IVl' 1I'01lid sillllliale drire paths as ahOle (pl'lhaJl~ IIsing a dilkn'llI stochastic PIOC(,SS lIlore ap'lropriiHe for exchallge ralt's), l"lllllpllie the 1I1I1'mp/' 1'01' each replicalioll, 1~'Pl'at this man)' limcs, and complltt' the al'nagc 111'101,1 Ihl' replit"'llions, , hilS lhl' power or lhl' Cox-Ross rbk-IH'llIral pricing Illethod is consi(ier: a l\c, However, Ih(',re are St','eral illlportallllilllil'llions 10 this appro'lI"h 111011 S \(lllitl he cllIphaslf.ed, First, th(' MOllte Carlo approach 1\1.\\' (111)' hl' applied to FuroP"'11l oplOllS, opliolls Ihal (01111101 Ill' ,'xl'n"isl'd I'ad\', Th(' !'arly l'x('rci",' I{'ailln' of I 1I1('("":all oplions inlrodllces Ih(' addnl (,(lIl1plicalioll of 1\('ll'rlllillillg ;111 o lIilllal ('x('I'Ciw polin', II'hidl 11111'1 Ill' d01l1' r('cllrsively IIsillg a ',,"II.lIl1icp "ograllllllillg-like allal~'si" III slIch c.I.~('S, 1lIlIlH'I'icd SOllllioll (If II\(' corrl'~11)()ndillg PilE is nllTl'lIll1' Ihl' 01111' ;II'ailahle IIll'lhod for ohlailling pritTS, Second, to apply thl' (:ox-i{oss techlliqll(' 10 a gin'lI c\ni,"aliH' "~(,cllril)', "~' mllsl lirsl prlll'(' Ihal Ihl' SlTllril" 1"11/1 1)(' priced hl' arhilrage (ollsidnali,ms alolle. Recall thai in Ihl' Black-Scho\(', rr;IIIH'\\'ork, 111(' Il
~
!
9, 'i,
Cllllrlil.lioll
opliolls Ihal was riskless, lit effect, this illlplil"s that the optiolt is "spall lied" hr stocks alld hOllds or, lIlore pre('isdy, the optioll's parolf ;It da((' T call he perkctiy replicaled hy a parlinll;lr dY";lIl1ic Iradill)!; slr;II('h'Y illvolvill)!; ollh" SIOCKs and hOllds, The lIo-arhilrage COllclilioll Irallslales illio Ihe reqllirellll'llllhallhe oplioll price mllsl eqllalllw cmt olllw clvlI;lInic Iradill)!; slralq,,)'lhal n'plicall's Ihl' oplioll's parolr. Bllt Ihere are sililatiolls where Ihl' derivative M'cllrit)' (';1111101 he ('('plicaleel hy allY dYllamic strateh')' illvolvillg l'xistillg securities, For example, if we aSSllllle Ihal lite clilfllSioll parameln (7 ill (~l,:!,:!) is stochastic thell illllay he showil Ihal wilhout further restrictiolls Oil r1 there exisls 110 1I0IHIc)!;l'lI' nail' dYIl;lIllic tradillg stratl'h,), involving stocks, hond,. alld optiolls that is riskkss. I kllrislically, hec,l\lse there are 1I0W t\\'o SlIIIIT('S or 1I1ll'l·r\ainty. the oplion is \Ill IO\lgl'l' "spanned" hy a dynalllir pOr\lillio or slocks alld hOllds (see Sectioll ~I,'U; alld Iluallg [ I !)!):ll fill' further disnlssioll), Thcr!'i(.J'(·. hd'.re we (';111 apply IIw I'isk'lu'lItral pricillg IlIl'thod 10 a pal'lintiar derivaliw security, we lIIust fil'st check that it is spalliled hyother Irac:l.~d assets, Sillce Goldman, Sosin, and (:atlo (1!)7!)) delllonstrate thaI the optioll to sell althe maxilllulII is illdeed spalllled, we call apply the Cox· Ross method to thaI case with the assurallce Ihat til(' rcslIltillg pricc is ill ran Ihe Ilo·arbilrage price alld that clevialiolls ('1'0111 Ihis pricc IH'('cssarily illlply riskless prolil opporlllllitics, Bul il llIar 1)(' 11IOIl' cliliicllll 10 \'('1 ily spall IIi II)!; ('or ilIOn' COlllPicx path·clepellelellt e1l'l ivativcs, III Ihose C;IS(,S, \\'l' lIlay have to emhl'll tIll' secllrity ill a mockl of l'colloll1ir eql1ilibril1m, \\'ith,sperilic ass\ll1\ptiolls al>ol1l agellts' prereu~ul'l's auel their il\\'{'stllll'lIl opportullity Sl'" CIS, rill' {'''ClIlll'k, in Ihl' slOeltastic,,,"I"Iilill' Illlldd .. rSl'l'Iioll !l,:\'(i,
9,5 ConcllLo;ion Thl' pricing or dnil',lIil'l' s(,curities is lIue 0(' Iltl' ullqualiliec\ SUllTSSCS 0(' 1110dlTil {,(,OIlOl1lics, It Itas cltangeeltltc \\'a)' e{'t'lloJlli~Is view d)'llamic modcis o('securilics prin's, and itltas had all C'lHlI'IlIOllS ill1P;ICI 011 lite invest11lcllt COllllllllllilY, TIll' ('('eatioll of' ever moJ'(' COlllPlcx lill;111l i;t\ illStl'l111ll'lItS has I>e(,11 all iJ1lport;1I11 stim11l11s Ii.r a(,;1
'c/,
\ \
I I
IJI'IWIlIIlW J'l"Irill~ l\i(Jdd~
lIu'st, cast's, all~llrtiLII expn'ssiolls for hedging positions in these securities do lIot cxist alld mllst .dso be dctt'nllilll~d ~lI\pirically, Tht'l't' arc nl;IIII' IIlIsClllcd isslles ill t he statistical ill/(on:llcc OfCOlltillIlOllStilJlc proct'sses willt dis('I'elelv sampled elata, Clirrently, tile llIost prcssillg iSSlll' is tilt' !lifIin"l\' ill ohtailling tonsistelltcstilllaics of tlte parameters of Itt, processes Ivith nonlincar dril't ;tIl
(nr (· .... \ll\pl,·. Itdl
IHieing 101111111.1
,\\\<1
lUI (Olithill('I~
Totou" (1~1~·n. l~lK:~}.
dillll,itlll,'jllllll'
~1t-nuu (1~17t}h}
PICHt',,,,,,,,,,
Sc'('
393
l'rub/ellu
that volatilities do shift over time ill random fashion, it is clear that issues regarding market incompleteness are central to the pricing of derivative securities. In this chapter we have only touched upon a small set of issues that smrollnd derivatives research, those that have received the least attention in the extant literature, with the hope that a wider group of academics and investment professionals will be encouraged to join in the fray and quicken the progress in this exciting area.
Problems-Chapter 9 9.1 Show that the continuous-time process /JII(t) of Section 9.1.1 converges in distribution to a normally distributed continuous-time process PCt) by calculating tile the moment-generating function of pn(t) and taking limits. Derive (9.3.30) and (9.3.31) explicitly by evaluating and inverting ~e Fisher information matrix in (9.3.7) for the maximum likelihood estimators I'- and ~ of the parameters of a geometric Brownian motion based pn regularly sampled data. !
9.2
a
a
9.3 Derive tile maximum likelhood estimators 1'-, 2 , and y of the parabeters of the trending Ornstein-Uhlenbeck process (9.3.46), and calculate their asymptotic distribution explicitly using (9.3.7). How do these three estimators differ in their asymptotic properties under standard asympLOtiCS and under continuous-record asymptotics? 9.4 You are currently managing a large pension fund and have invested most (lfit in IBM stock. Exactly onc year from now, you will have to liquidate your elllire IHM holdings, and you are concerned that it may be an inauspicious time to sell your position. eLM Financial Products Corporation has come to you with the following proposal: For a fee to be negotiated, th!!y will agree to buy your entire mM holdings exactly one year from now, but at a price per share equal to the maximum of the daily closing prices o~r the olle-year period. What fee should you expect in your negotiations with eLM? Specifically: 9.4.1 Estimate the current (time 0) fair market price H(O) of the option to sell OIl the maximum using Monte Carlo simulation. For simplicity, assume that IBM's stock price P(t) follows a geometric Brownian motion (9.2.2) so that J>(l~)
log - - ~ J>(lI)
N (ll(1~
2 - 'I), a (/2 -
til),
(9.5.1)
and lise daily returns of IBM stock over the most recent five-year period to estimate Ihe parameters 11 and a~ to calihrate your simulations. Assume
394
9. /)nivlltilll' ",icillg Model!.
lhalthere arc 2:)3 tradillg days in a ycar ancllhal market ptin's volatility when markets arc dosed, i.e., weekcnds. holidays.
\1;I\'C 110
9.4.2 Provide a !)!i% cOlllidclln: illlerval for ('(0) and alll'Slilll;Itl' oflhe \lumber ofsilllulatiolls Ileeded 10 yield a price estimate Ihal is wilhin $.0[, of the true price. 9.4.3 How does Ihis pricc co III pare wilh the price givcn hy Ill!' (;oldlllallSosin·Gatto forl\lula? Can you explain the discrepancy? Which price would you lise to decide whether to accept or reject el.M's propos;I!?
\ \ \
\ \
\
1 1
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[:-.: T I (I SC
s tu d ) , b
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Iocool
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c Secu rities
E It
alld t ht o a n ' 1'11 n d s Ih;1l hav\- ~ lII.'X! WI.' \I ll 'l l 0 \1 1 11)' ~I'( ' nu ca ' ,1 \l { ' ll pnJV dlil'd ., ...·III·ili ,I H io i~i"n ill a h a n ~,1 IsseI. H li x c d - il ' d their o \lt t l t t 'n b e ,' p p l i c d ln J ll J ( w n I I lS e v e l o p e d s e to ' a ' f ix e d n s ,' c n lr il i{ p a r 'l l d s c v ( 'r .t li acadL'u 's , F ir i \,('i\SO r Jic s t u w li o ll a l s t r l l s l, Ill( IlS tt l c t u r e a f r o ll i IIU' l'i{ dy of ti lt ' !1Ia ' f ix e d f ix e d - in nd the tlily m rk('\~ ·i ll c o m ,l I 'k e ls ir own Ii)!' TrL come e si/,l.' is , s e c u .- il ';\sury S They tllc; ie s hilS le l' ll li ll o lo g ) ', L 'c u l' il have il K o ll le J~\lr('d h y q u k il s I S .i ,l k iT c x o w n tr a ll li ti e sCCllrit trcllld a d it io n c w is t' t h e sO ics h a v n o c ls y hir){< s , Se( lH lo \Y e a s p e H lS la ll d il lg o r ' rq?;an 'o ll ( l, I ln C l· n c ia l ph lk-ss o f lI lt 'l ll ti s lu d ) ,i a il ll y , \{ w h " ,t il li ll g f i x n '( t' ' s il I l f in a n raded. so Ihd c l' l ·i n c o t:<>\111\ c e ll w o T r p r it T lllc se r a il '' ' w S vary r y h e c a l l i r d , li x e d c u r il ie it h o u t ( 'x ! ,, 'c o s ll u s w ly e th e } ' h'l\'il\ (' <"a ta li o ll
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1 1 1 1 1 1 1 1 1 1 1 1
1(J. 1·IXI ·,,·ll/m lll,· S"(/l l'ilil'' \
Thc litcr atuH ' Oil fixe d-in com e secu ritie s is vas!." Wc brea k it illto two main parts . First, in this chap ter we intro duce basi c conc epts and disc uss elllp irica l work 011 lille ar tillie -seri es mod els of hOlld yields. This work is only loos ely lIIotivated hy theo ry and has the r,rac tical aim of cxpl orin g the fore cast ing pow er of the terll l stru ctur e of inte rest rates . III Cha pter II we tllrn to mor c alllh itiou s, fully spec ified tcrll l-str uctu re lIIod els that call he used to pric e illte rest- rate dniv ativ e secu ritie s.
10.1 Basic Con cep ts In prin cipl e a fixe d·in com e sccu rity can prom ise a strca m orru lure pa)'I1H'llts of any form , hut ther e arc tWO class ic case s. lIra-roll/iIIl/ /JOIli/S, also raile d diJ(V lt1ll bonds, mak e a sing le paym ent at a date ill the flllllre know n as (he IIIl1lurily dalf. The size of this paym ent is the faa llalll f of the bon d. The leng th of tilllc to the matu rity dalc is Ihe mail lrily of the hOlld. US Trea sury hills (Trc asur y obli gatio lls with Illat llrity at issue of up to I ~ IIHlllths) take this li>rlll. ('.oll/}(III /HI//( /' mak e (1111/1/111 /HI,V lllrIl/ J of a give n fraC lioll of facc \,,1111<' at equa lly spac ed dal," up to ;lIId incl udin g the matu rilY datc , whcl I lile I.IC(, valu e is also IMid. US Trea sury no(e s alld hOllds (Tre asur y obli gatio lls "'ith matu rity at issuc abov e 12 lIIon ths) take this fOrtH. Cou pon pa),m cilts on Trea sury Ilote s and bOllds arc mac k ever y six mon ths, but the coup on rales for thes e illStrtlllH'lIts arc norm ally CJllotecl at an anlll lal r;\te; thlls a 7% Trea sury hOlld actllally pays :{.[','J " of face vallie ever y six mon ths up I') alld incl udin g lIIatllril)'.:I Cou pon boncls CIII he thou ght of as pack ages of cliscollnt hond s, olle corr espo ndin g to each COU pOIl paym ellt and olle corr espo ndin g to the fIlial COLI pOll paYlllent log( :ther with the repa yme nt of prin cipa l. This is 1I0t mere ly an acad cllli c conc ept, as the prin cipa l and inte rest com poll cnts of US Trea sury hon ds h;lve beel l tracl ecl sepa r,ltc ly und er the Trea Sllry 's STR IPS (Sep arat e Trad illg or Rq~istered Inte rcst and Prin cipa l Secu ritie s) prog LlIll sinc e 19H!i, alld Ihl' prin :s of such Trea sury slri/l s at allll latu ritie s ha\'( ' heen repo rtecl daily ill thl' \I'(/Il Slrl'l 'l jllltn lfll sinc e HlH9.
2FIlrtlln;lld), il I"" illfl ,'a,,',1 ill '1"ali ly silln' Ed 1\;11 It' '. jlld~t'lIlt'nt: "It is !;"II('rally ,'gree d Ihat, r('lt'ris pari hilS, Ih .. knili lyoi'; .Ii,·ld is roll!;"ly propo rtion al to Ih .. 'I'I;llIlily O!'IIl;III1If'(' lital h;l~ Il!'rn (hllll p".III "lIn il in tilt' /l'''l'lIll'aSI. Ill' Ihi. stand ard, th"te m) ~lfII"lurt' ofin" ,r".,l ra,,'s h;l~ 1,,'wlIl" , .. ;11) "xlr'" l1tiin arily (('nil t'Ii"ld ind"t 'd" (Kall!' 11!l7 ()). S.·,· Mdin o (I~I~H) or Shille r «( !J\)O) Jill' cxrt'Il""1 ...·...·nt.'"r \·t·y~. and SlIIu lar".a n (I !/!/Ii) for a hook·I"II!;lh In"\lI III'IIl . 'S,·t' a l!'Xli>ook s" .. 11 a.' F"holl.; ""ti F"hll lli (I\I\I;l) Ill' F:llltll.zi (t\I\lIi) for !'lInh er ti .. lails un 1h(" mark,'(:o, for lJS TI('~I'lIry St'fw ili('.\.
W.l. nalic Concl'pts IO.l.l lJiS({)Ullt/JOlldl We first define and illustrate basic hond market concepts for discount bonds. The yidd to maturity on a bond is that discoullt rate which equates the present value of the bond's payments to iL~ price. Thus if Pn , is the time t price of a discollnt hond that makes a sin~1c paylllellt 0(" $1 0\\ timc 1+ n, and Y", is the hond's yield to maturity, we have 1'", = (I
+ 1'",)",
(10.1.1)
so the yield can be found from the price as
( I+ V) 1,,(
::::
l'-U) "f •
( 10.1.2)
It is COlllmon in the empirical finance literature to work with log or continuously compounded variables. This has the usual adv:lIltage that it transforms the nonlinear equation (10.1.2) into a linear olle. Using lowercase letters (i)!' logs the relationship betweell log yield OIlHllog price is (10.1.3)
The In7ll slmdure of ill tere.1 I mle5 is the set of yields to maturity, at a given tilllC', Oil honds of different matnrities. The .'lirld JIJread S", == Y", - YI " orin tog terllls .I", = y,,1 - YII, is the difference between the yield on an'n-period bOlld and the yield on a one-period bond, a measure of the shape of the tcrlll structure. The yield curve is a plot of the term structure, that is, a plot of Y"I or y", against 11 on some particular date t. The solid line in Figure 10.1.1 shows the log zero-cou[>on yield curve for US Treasury securities at the end of January 1987. 1 This particular yield curvc rises at first, then falls at longer maturities so that it has a hUlllp shape. This is not unusual, altholl~h the yield curve is most cOllllllonly upw;u!l-sloping over the whole . I rall~(, of maturities. Sometimes the yield curve is illver/ed, sloping down oVfr tlte whole range of malurities.
l1o/tlillg-l'rriod Rrturns The /w/tlillg1Jrriod return on a bond is the return over some holding peri~ less than the bond's maturity. In order to economize on notation,. we specialize at once to the case where the holding period is a single period:~ We ~Tllb rurvc is nOI hased on 'l"ott·d slrip pri(t·~, which M~ r~adily aV;lilable only for rnet yea .. ,. hili is estilllaled frolll Ihe prices of cOIII'0J1-I)('arin~ Treas"ry lx,"ds. FiKure 10.1.\ is clll<' 10 MrCllllorll ane! Kwo" (199:1) :1111111'<"' Mr< :lIl1ocll" (1'171, 1'17,,) estimation melh",\ "-, c1i.,clI"'·cI ill ,t'clion 10.1.:1 helow. \"ihillt'r (19~JO) gi\'t.".~ a milch mon' (olllprelil'Il . . i\'(· rrt',tllllt"IH, \\o'hich requires Inore (Oil)· plil"a[cd JlOI:lliol1.
/II, !-Ix,'d-IIIIII/II/' .'il'fllrilil',1 .c,r-_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ , _ _ ;_ _
.~
c:
~-
r-
-
t..
~(
;;
";; C-
ol' .;:
_.-----------III 1:,
II
---"-----~----
:!u
~',
:\11
\1.lIl1l ily ill Y(';u,
Figltn· 111.1.
11'111
elll,/,I/ll l'IIM 1111/1 1';"11',,111-11111,' (:11' ...,.,\ in/IIIII/(I/)' 1')87
definc U.. ,,+I as Ihc olle-period holdillg-p,'riod rt'llIrn on all 11-l'niod hOlld purchased al lillie 1 and sold;1I lillie 1 -I- I, Since the hond \\'illll!" an (II - 1)pniod hond wht'll il is sold. Ih,· sal" pricc is ",, __ 1.1 f I and Ihc hlliding-pniod relllrn is
(I -\- U", .. d
1'"
(1
I III
+ 1'",)"
(I t- l'"
I',I{
( 10,1.,1)
I,/l I )"
I'
The holdillg-pniod 1,'111111 ill (10,1,1) i, high irlhe hOlld has a high \icld when il is plIn'h;IS\'d al lilll!' I. ,\lId il il 1",,;1 low yidd \\,(,CII il is ,old.II lilllt' t + I (sint''' a lo\\' "it'ld n,I)'\"p"l11b III a high l'rin'). l\1'1\'ing III logs for ,ill'l,li.-il\', ,I,,· Illg h .. ldillg.pt'riod 1\'111111, ' .. I) I ,log( I -I- /{",I) 1 ). i, III \'1/1
I )-"..
1)(-""
I,ll I
1.1)1 --,1'",),
( I 0, I.:,)
Th,' h"l \"plaliIY ill (111,1,:,) ,hll\\" ho\\' lilt, holtlill)!;-pl'l'iotin'lllrn is dl'l('\'lIlill",1 hy Ihe llt'gilll1illg-ol-pl'liod ~'it'I" (posili\'l'ly) all" Ihe !'hang!' ill Ihl' yic-ld llyn Ih,' holdillg 1"'1 ioc\ (1l<",l~;lIi\'t'h-),
III, I 1111,';"
(;""1'1,/111
[:''Illalioll (lll,l.:1) r:lll II\' 1":\lI:III~I'd so 111.11 il 1\'[;11\', I[H' lo~ 1111\111 1'11\'1' l"d:ll'llIlllI' IlIg pli ... · 101110110" ;"ulllll' 1\'1111'1 """ IIII' 11I'~ll'l'Iio,l: /'",
-'I"
/I'
I /'" I" I, 011,' .... 111
"I1I1)'llllllillg (1111 1111111("
IHllill~ 111.11
(,lllel
~:'
-
1.1 /"
,II t I"
til('
log
,"I,,·
Il1i, dilk,,'''''' "'III;lIillll 1011\,:11'\1. i."I 11'~1«'lu'cI
hUlid p .. i((·, Illllii III(' 1I1.1I111il\ d.llc-
I()~ pric," ;11 11I.11i1i il\ ('1)11.11 ... /1'1(1) In ohl~,jl1
tH ill h'l
/Iflf
=
n" or t"t' ",,·hl
,Lit
=
"-! (-;;I) L'"
(I II. J.(i)
11111/'
/:.:0
Tlii" "'I";lIill'l ,1I11"'S 111011 1I11'Iog ~idd 10 1I1:lllllil~'OIl a ~"nl-nllll'''" 110 ... 1 \"IILI!> :iI,' "\'\'Llg" ["!-\ 1<'lllIIlP''\' I'l'I'iod il II ... 11,,1111 is 111'11\ IIIII\:\lIlIil\', h;.';"f/J(II~tlll'.\
1\, "It I" ,I' tiilkl "III III.lllll'ili,·s rail lit', ,""lIi",·d 10 gll;II "III,,\, :111 illl,·n·,1 1':111' "\,,, li,,,d-illl 11111<' i111'CSlIllC1I111l It,· 11I:l<\c ill Ihc !'tlllll":!ill' i"len'sl raIl' 1111 Ihi, i,,\,"IIII\'11i is ",died :1'/IIIWIIHI 111/"." !" ,C:II:II'.IIII',\,:l1 lilll" / 0111 illll'n'sl 1'011,' 011:1 0111'-1"'1 i"d ill\'('sllllt'1I110 h!' IIl.Id,' .,1 lilll<' / -I- II. :III illH'slor rail Pl'o,,(,,'d:ls lilll'"I" 'I'll,' d('sil'('d 1'111111'1' illl<"I""'III\\illl':lI'$! :lllillll'/+ II-!- I "lsh,'Ii"l hlll""III'(II+ I)-period [""Id, :ilis (,,,,Is 1'''11,,;11 lilll" 1 alld 1':I)'s $1,'11 lilll<' (-1- ,,+ I. Th,' ill\'eslor ":1I1h I" 11.111,1,', lhl' ('osl ortllis ill\'('stllll'lli rrllllllilllt' (Itllillle I II; to d"
+
Ih:, ,II" ,,('Ih "" 1 1.,/1'", 11-I'('riO(I hOllds, Thi, pl'o,III(,(,s" posilil'(' cash 1I0\\' or i'", (I' .. , :"jI'",) = 1''''1.1011 lilll\' I, "xal'lll' elltlligh 1001'1:,,'1 the Il('galiv(' lillie / ',I,1t 1111'" 1'1'''"1 Ii ... lil'sl Irallsanioll, Tlte ,;111' or lI-pl'J'iod hOllds illlplies a "":~;Il""
,';"It II till' "I' 1'''IU/I'''I;l\ lill\(' 1 + II. This .... 111
[I(' IlltHlghl .or;ls th,'
('0'' 01 lill' olll'-pnilld illl'("lllll'1l1 10 Ill' lll;ltlt-;l\ lillH' (
+ II,
TIlt' cash IInws
1'\',,"llil.c: 11'11111 lilt'''' I1';1 11 S;t1'1 iOlls al'(' illllsll'al('d ill Figlll'" \()':2, Tl,,' "/1'\\':11 d 1';1\,' is deli 11 ... 110 I... IIII' 1'('1111'11 oil III<' Ii III<' I j- II i\l\'('sllll(,111 I
II i'". ~ ,/1',,/:
(I -1- 1'", I,,)" "I
II -I- 1'",)
(10,1.7)
(I t- 1'",)"
11111", ",'l.lli'>11 1-;" III .. /ir,sl sllhsCl'ipl n'/iTs itl 111,' II II 11 t1){'r orp(,l'io
tI,,· s,'('.oll
I\'k" III tI,,· d.lll· .1' "hi..!IIIH·!ilrwanl 1';11<' is scI. ,\1 lilt' I'osl ora
11'1'
('oilld ;lIso ddill<' I'III'II':IIc1I'.II,·, 1'01' llllIliipl'l'iotl
illl,',II:WIII.\. 1>111 III' <1011111 PIII'SIl(, Illis 1111'111\'1' !IIT",
'II "\,11111'1"
.111,:111111'"
,II I'll \\.11
d
II
1\\1/, d 111.11 kl'l III t ','1 II ",1'0111 \ ,,'I IIfili{'.' .. \I WI" Iwloll' dlt, '(',lilliit" .111' i"llI'd. '\C'c'llIilic"
,u(iug j, Ih,' 1/'/"-,,
111111'\\ 'I"~ III II it', I", ;IIIIIOIIIICTd 1.111
"it"
.IIt' ,',Id,', i HI rlll' 1\ 11'·11·' ...... 111"11111.11 )"('1. hjlll ""111,'1111'111 II I III I III ,,111'11 dl!' ~I'I 1IlIIic'" .In' j .... ucci.
~ .....
J
1"\.1 14 "1/11 0111," • )/"11111(/1'"\
1+
t i 111(' 1
BII)' (1/
II
I
'lbnsa('lic)ns
1+ 11+
I
1
+ 1l-ilC'riod
-1'1111.1
hond
( ~'~) ,,, I'
Sdl/'II+I.I/ I '1I1 II-period honels
/'
J'II~ 1.1
II/
/)", 1'111-1.1
o
Nel
Figure /0.2.
J'ui ('iI.IIt
NIII'" ill
II "'J/WIII" '/i(II/If/l'lillll
Movill)!; to IIl)!;s (ill' simplicitl', tI\(, II-perio
(II -\-
=:
,)'111
-t-
I)YIII'\'/ - IIY",
('1
+ 1)(Y,,+I,1 -
( lO.l.H)
.'Yllt),
Equatioll (IO.I.H) shows Ihat Ihe lill'ward rate is positive whenever discount hond 1)I'in's fall wilh malllrity. Also, Ihe forward raIl' is ahove bOlll the 7/pnioil and tile' (1/ + I )-pc'rind dis(,otHlI hond yields when !ht' (11 + I)-period yield is ahm'(' Ihe' II-pcriod yidel. Ill;!! is. when th(' yielel clirve is upwardsl()pitl~,7 This reb!ioll h('twt'e'n a yield to maturity and the installlalll'OIlS forw,ml rale al 111;11 JII;II uri I)' is ;lIlalo)!;lltJs 10 th(' r('lation h('tweell marginal and averagc' (osl. Thl' yield 10 malurily is the average ('ost orhorrowill~ for II pnio
,I",
\1"'\ . .
7 f\~ tht' lime I\lll)"~ \ d.H1\I' h t liw hculIllH,l1l1rily 11,111(' h""llIul;, ( IIl.I,X) appnl,u"ht,S /'" ":.:. '"", -f tl ;i)'"djl n. Ill1' n pC! ioc\ ridd 1'111, II tillw", tIll" ",lop" 01 yield ("111"\'(' al 111;11111 11\' II.
.It.,
401
JO. J. Basic Concepts
marginal unit, so the average cost rises when the marginal cost is above the average cost. Conversely, the average cost falls when the marginal cost is below the average cost. I
10.1.2 Coupon Bonds
\
A~
we have already emphasized, a coupon bond can be viewed as a pa~kage of discount bonds, one with face value equal to the coupon for each date at which a coupon is paid, and one with the same face value and maturity as the coupon bond itself. Figure 10.3 gives a time line to illustrate the time pattern of payments on a coupon bond. The price of a coupon bond depends not only on its maturity n and the date t, but also on its coupon rate. To keep notation as simple as possible. we define a period as the time interval between coupon payments and Cas the coupon rate per period. In the case of US Treasury bonds a period ,is six months, and C is one half the conventionally quoted annual coupon rate. We write the price of a coupon bond as POll to show its dependence on the coupon rate. . The per-period yield to maturity on a coupon bond, Ycnto is defin~d as that discount rate which equates the present value of the bond's payments to its price, so we have 1""1 ::::
e e l+C (I
+ Y ent )
+
(I
+ Y"'t)
2
+ ... +
(I
+ yent)n
.
(10.1.9)
In the case of US Treasury bonds, where a period is six months, Yenl is the six-month yield and the annual yield is conventionally quoted as twice Yen /. Equation (10.1.9) cannot be inverted to get an analytical solution for Y,"I' Instead it must be solved numerically, but the procedure is straightforward since all future payments are positive so there is a unique positive real solution for Yenl •s Unlike the yield to maturity on a discount bond, the yield to maturity on a coupon bond does not necessarily equal the per-period return if the bond is held to maturity. That return is not even defined until one specifies the reinvestment strategy for coupons received prior to maturity. The yield to maturity equals the per-period return on the coupon bond held to maturity only if coupons are reinvested at a rate equal to'the yield to maturity. The implicit yield formula (10.1.9) simplifies in two important special cases. First, when POll = I, the bond is said to be selling at par. In this case the yield just equals the coupon rate: Yrnl = C. Second, when maturity n "Wilh Ilq:;lliv~ future paymenl'. there can be mllhipl .. positive real solUliol1S tn (10.1.9). or a projcn to il.~ cost is known as tht" i1Jin7lfJ/ mil' oluturn. Wlwil projec~ have some negative cash flow, in rlie flllurt\ tht"rt· CUl hC' multiple ~ollitioll."i for the int~rnal rate of return.
III Ih,' oiliolly';s of ;Ilv..'tllleill projects. th" discolillt rat<· that equate. the pre",nt vdlue
_____ Jime.L_ _ _ _ _ --J.+l- _...--t+-2-W
I
I
(1)
Maturi[)'
(2)
Face value
(3)
Present value discounted at }~'" (1) x (3)
Den'
=
C + Y,nt)
(l
+
(1
+
C (1 + You)
+n-
1
t+ n
I
I
C
(1
t
2
n- I
n
C
C
1+ C
C
C (I + Y,,,,)n-l
(1 + C)
Y,n,)2
2C
(n -l)C
Y,n,)2
(I + Yrn,)n-I
'L((I) x (3») 2:(3)
Figure 10.3.
Calculation of Duration for a Coupon Bond
(1
+ Yrn,)n + C) + Yrn,)n
nO (I
lO.l.
iJll.)i(
COIlrt'/lts
403
(mUIiI \)1' /1n1)f!tuily. In Ihi~ l:OIse the yicldjuSI equab the ratio of the bond price to the coupon raIl': )~."" == C/I'rOiJl'
is infinile, the bond is called a
/Jllmlio/l 1I1/11l/lll/IIWizlltioll
Fordiscounl iJollds, malurity measures the kngth ol'tillle that a hondhulder has invesled money. BUl for cuupon hOllds. malurilY is an illlpafecI mcasure or this Ienglh of lime because lIIuch of' a coupon hond's value comcs frolll paymenls lh
,,('
CL::"I
~ -I-
(Ilel
1T+t;i-l-':'~+"'+I/~
II"+b
( 10.1.10)
POll
The maturilY of the first cOlllponent dis('ounl hond is OIH' lll'riod and this receives a weighl of C/( I -I- l'rlll), the present value of' this hOlld when 1'"" is lhe discounl rale; the maturily of the second discount hond is lwu and lhis rcceives a weighl of C/(I -I- 1'(nl)2; and so on unlil th .. Iasl discounl bund or Il\;llttrily II gt:ts a weight of (I + C)/< I -j 1'",,)". To ('(\lIVer! this illlo ;\11 average. we divide by the sum of the weights C/ (I -j- l'",,) -I- C/ (I -I- r"a)2 -I... -I- (l -I- C)/(I -I- Y'"I)", which from (1O.1.!1) is just the bon(1 price I'"". These calculalions are illustraled graphically in Figure 10.3. When C = 0, the bond is a discounl bond and Macaulay's duralion equals lIIaturity. When C > 0, Macaulay's duralion is kss lhan 1I\;llurity and it declines wilh the coupon rate. For a given cuupon rate, duratiun declines with the bond yield because a higher yield redllCl's the weighl Oil more dislanl paYlllenls in the average (10.1.10). The duration finllluia simplifies wi,en a coupon bond is selling al par or has an ill finite nl.lllirity. A par bond has price 1'",/ = I and yield F,,,, = C, so dlll'alion becumes /),,,, = (I - (I -I- j'",()-")/(i - (I -I- 1',,,,)-1). A consol bond with infinite maturilY has yield Y'CVI = C/I',ool so duration hecomes /),,,,, = (I -I- l"OOI)/l"OOI' NUlllerical examples thal illllstrate these properties are given in 'EIhlc 10.1. The tahle shows Mac,ltllay's dur'ltioll (and Inodiiied dur,lIion, defilled in (10.1.12) helow, in parenlheses) IiII' bonds with yields and coupon ~I~t;lfalll;ty abo sllgg('sl~ that 011 lilt' ('0 11 pOll hOlld
Ollt' could usc.' yit'lds Oil di~("OIIlIt honds r.uil,,!" than th(~ }'idcl to rairul.ttc- the prt"St"JI ,";,dlu' of (,~lfh COUPOII l'aYIl1(,1I1. IloWl'\'('r Ihi.'\
:'lppro;\("h I('qllil('~ IIl;I'
OI}('I11('i\slIre.1 cOlllpl(,l(" Jt'IIH'OtlIHHlIt'JIII.\III1( 1111(',
·HI·.
/0.
"'amlill/.~:' 1/111/ IIItH/illrt!
Table 10.1.
FiXl'll'/lInJlIIl'
SI'I IIritin
dum/i'lll j"r .\f!fftn! !JIII/d,.
Malurily (yt'ars) .J
".-
-. -
otY"
rau-
(:OUI'01l
O(,~,
Yit'ld
rIll
,I",
IO'X.
',.(H)1l
:!.OOO (:!.()I)O)
1.000
(1.000)
(:d)OO)
11l.1l0()
:!.OOO
:>.I)O{)
( 1.%1)
(·I.H7H)
(~).7:,I')
IO.O()O
:!.()OO
".()OO
( 1.~IO")
(·1.762) .. -"-
:IO.()OO
(10.000) C!O.OOO)
1.000
I.OO()
\H.OOO
(~J':':!4 )
:10.000 (:!'I.:!liH) :~().()()()
(:!H571)
-'--~-~'-"-'----.
( :IIl1pllll
OfX.
:.'~J
ll.tlHH
I. ~ 1:1:!
1':',,0
(1l.!IHH)
(I.\J:I:!)
(·I.:.r,O)
HAI7 21.1 ,,0 (HAI7) (:! l.l :,0)
l.tl~H
7.!lH!1 I',.H·II :!o."oo (7.7 t l:,) ( I r •. ·I:,·I) (:.!().()IJlI)
)
(I.HHI)
·I.-IW, (·1.:171\)
(O.tl·to)
1.!I:!·1 (I.!';:!:!)
-1.'1 J.I (-I.:W·\ )
O.~IHH (O.~Hi·1
I o 'X,
O.!IHH
- - -_.._.-
--
..
(:0111'011
ralt'
Yield
-- _.---- --".
r,JI(I I'!
rail'
Vit'll!
--.---
:xc
-----
(O.~)7Ii)
(O.~l":!)
:10
10 --~.-
..
7AH~)
(7.1:~:!)
IO.!I:,7
10.:,00
( 10.·1:\1,) (10.0011)
_-_.-----.--._----------
I f )t}~1 O'y"
O.~)77
I.H7', ( I.H;:,)
(O.~I77)
:,(Ycl
lOry"
,1.2,,0 ('1.2:'0)
7.1,2:,
O.~I77
I.HliH
1.1:,li
7.107
(O.!I',:I)
(I.H:!:I)
('LO,d)
(li.~I:I:1)
O,~J7(i
I,Hli:!
,1.0;'4
6.',4~
«(),~):~O)
(1.77:\)
(:I,HIiI)
(1i.2:-11)
_..
IH.'I~H
0.62:,) ( IH.!l:~H) :!0500 1·1.02" ( l:tliH:~) (:!O.OOO) ~I,!I:~H 10."O() (!)AIi;,) (IO.()OO)
------------------.--
I.. hlt- It'IUIII'' ~l;tLlIIl.ly's dill ;11;011 alld, lit pall'ntlu:s(.·s. l1\odili"tl dur,uinn fur honds \dth st'lt·'"h.'cI yi(·ld . . ,lIul lUallltiti"", Dllration, \'it'1d, ;llItllllalurity an' Mau'd ill ;tlHllIallillics Inll Ihe uudt" lying t·~,h \lbt1tHl'o i\"'~\lIH(' 111;11 hcm
TIlt'
r;ltes ofO'X"
:,'1.., ;III\!
Duraliou is givt' II ill
10'1."
;\11I!
\'(';US 1)(11
\Il;\tlll ities ranging frolll one ycar to inlinity.
is r;tinti;ltt'd using six-lI\onth pniods as wOllld
he approprial(' fi'r I IS Tn';\slIry hOIHk II W(' lake Ihe d('Ji\'ali\'t, of (10. f .II) with rcsp('('( 10 r"", or equivalelltly with respect 10 (I -)- };"tl, WI' filld 11\;11 I-.fa('aulay's duralioll has allolhN \Try
405
10. l. llasic COIlCrflls
illlponant property. It is the negative of the elasticity of a coupon bond's price with respect to its gross yield (I + Y(I//):10
D
_ _
",I -
dP"" d( I
(I
+ Y"II)
+ Y,",) P'"I
(10.1.11)
III illdustry applications, Macaulay's duration is often divided by the gross yield (I + Yrll/ ) to get what is calkd modified dura/ion:
°n.1
dl""1
(lO.l.l2)
Modified duration meaSllres the proportional sensitivity of a bond's Ii rice (0 a sillall absolute change in iL~ yield. Thus if modified duration is 10, an increase in the yield of I basis point (say from 3.00% to 3.01 %) will cause a 10 basis point or 0.10% drop in the bond price. ll I Macaulay's duration and modified duration are sometimes used t~ answer the following question: What single coupon bond best approximates the return on a zero-<:oupon bond with a given maturity? This question is of practical interest because many financial intermediaries have long-term zero-<:oupon liabilities, such as pension obligations, and they may wish to malch or immuniu these Iiabililies with coupon-bearing Treasury bondsP Although today stripped zero-<:oupon Treasury bonds are available, they lIlay be unattractive because of tax clientele and liquidity effects, so the immunization problem remains relevant. If there is a parallel shift in the yield curve so that bond yields of all maturities move by the same amount, then a change in the zero-<:oupon yield is accompanied by an equal change in the coupon bond yield. In this case equation (10.1.11) shows that a coupon bond whose Macaulay duratioll equals the malUrity of the zero-<:oup~l liability (equivalently, a coupon bond whose modified duration equals the modifIed duration of the zero-<:oupon liability) has, to a first-order approximation, the same return as the zero-<:oupon liability. This bond--or any portfolio of bonds with the same duration-solves the immunization problem for small, parallel shifts in the term structure. Although this approach is attractively simple, there are several reasons why it mllst be lIsed with caution. First, it assumes that yields of all maturities move hy the same amount, in a parallel shift of the term structure. We I"Th .... bSlicity of a variable /l with r"'pen [0 a variahle A is defined 10 be the derivative of il with r.. ,p,.(\ to A, lim,.s AI il: (dill dA)(AIIl). ECjllivalemly, il is Ihe derivalive oflo&(B) with rt"p,.n to I()~(A). I' Not,· Ihat if duration is lIleasur,.c1 in six-Illonth time unil-'. then yields should be measurtd Oil a ,iX'1l10Ilth ba,i~. Olle (all ftlllWrl 10 all allllllal basis hy halving dunllion and doubling yield,. Th .. IIl1mb,.rs in T.. blt" 10.1 hav,. h,.en annllali,,.,1 ill Ihis Wdy. l~hllll""lil.ali()l1 was orillinally d .. filled hy R,.c1din~lOn (1952) as "the investment of the ",-,,'Is ill s"ch OJ. WJ.i lhallhe ("XiSlill~ hl"ine" is immllne 10 a ~eneral change in Ihe rdle of i"lt·rn!". FahOl.l.i ami Fahoni (I!l!l:.). Chap"'r 42. ~ives a (olllprehensiYe !lisenssion.
406
10. Fixed-Income Sewri/ifs
show in Section 10.2.1 that historically, movemen15 in shorl-lenn interest rates have tended to be larger than movelllen15 in longer-ternl bond yields. Some modified approaches have been developed to handle the lIlore realistic case where short yields move more than long yields, so that there arc nonparallel shiflS in the term structure (sec I3ierwag, Kaufman, and Toevs [1983], GraniLO [1984], Ingersoll, SkelLOn, and Weil lI9781). Second, (10.1.11) and (10.1.12) give first-<>rder derivatives so they apply only to infinitesimally small changes in yields. Figure lOA illustrates the fact that the relationship between the log price and the yield on a bond is convex rather than linear. The slope of this relationship, modified duration, increases as yields fall (a fact shown also in Table 10.1). This lIlay he taken illlo account by using a second-order derivative. The convexity of a bond is defined as C ",n
,(i+I)
~L-i=l ~
Convexity _
+
II(n+l)
(l+)'F",)·"t
]'",1
qnd convexity can be used in a seeond'
\
dP,n( Y,")
!
P,,,
\
:::::
dP,,, I , - - - d}," Ii Y,,, P,,,
I IF 1'0' I 2 --2- - - (dY,,,) 2 d Y," P,"
+-
(- lnotiiii.ed duration) tl Y,"
I I
+"21 convexity (dY,,,) 2 .
\
(10.1.14)
\
Finally, both Macaulay's duration and Illodified duration aSSlIllle that nows are fixed and do not change when interest rates change. This a. ~umplion is appropriate for Treasury securities but not for callahle seclll'iti s such as corporate bonds or mortgage-backed securilies, or for secmities th default risk if the prohability of default varies with the level of interest r' tes. By'modelling the way in which cash flows vary with interest rates, it is P1Jssible to calculate the sensitivity of prices to interest rates for these lIlore cTnplicated se(:lIritil'S; this sensitivity is known
~
A 'I.oglinear Modd for COli/lOll /JowLI The idea of duration has also bel'1l used in the academic literature to lind approximate linear relationships hetween log cOllpon bOlld yields, holdingperiod returns, and forward rates that arc analogous to the exact rehltionships for zero-colI\lon bonds. To understand this approach, start from the I'See Foibul.li and Fabul.I.i (I\l~):», Chapll'''' '.!!I-:IIl. ami F"lmui (\\J\lti) ('ur a .Ii.n",;,,)) of \'aIiI)U~ methods u5ed
hy
fix("(\·iIlCOll1e al)~I'y~ts II) ralculatt." t"lfl'ctive duration.
407
/0. J. fj(lJir. Conrrl)Lr
/
Slop" = Modilied [)uration
Yield Figure 10.4.
'/1"'!',itf-l'i"''' III'/flli(lI/,hi/,
loglinear approximate return formula (7.l.19) dnivcd in Chapter 7. and apply it to the one-period return r' .... 111 on an II-period conpon bond:
Here the log nominal coupon r. plays the role of the dividt'I\(1 Oil stock. of course it is fixed rather than random. The par,lIneters p and k arc given by p == lie I + exp(r. -I)) and II == - log(p) -- (I - p) log(ll fl - I). Whcn the bond is selling at par, the II its price is SI so its log prilT is J.ero and p =-= III I + C) = (1 + Yrlll ) - ' . It is standard to use this value for p. which gives a good approximation for rctllnts 011 honds selling dose to par. Onc CIII treat (10.1.1;». like Ihe '1Il
bUl
11-1
LP'[h+(I-p)r- r, ... _,.III"I·
(10.1.16)
;=0
This eqll,llion relates the price ofa cOIlj>onlrond 10 its slrcam of co lip on paymCllIs ,lilt! the flllllre returns Oil Ihe IlOlId. 1\ similar 'lpproxim'llion of the
10, FiXfd·/llftllllf S,'('Il,-;I;".1
,101{
IIII-( yidd 10 \I lal lII'i Iv.)', ,,' shows Ih;\l i I sal isli('s all ('<"Illatiol\ of II\(' salll(' lim 11: II""
L p'lhf'
(I -
pic - .1'111/]
1=-0
(t - 1''')
----' I h +(I ", ()
(I - fI)r - ),,,,/ I,
,
(10.1.17)
Eqllatiolls (10,1,1 Ii) alld (I D,I,17) log('11II'r imply Illal IIl(' lI'period COllPOI1
2:;::,:
hond vield salislit,s y,,,, ~ (I _. p)/(I - flll)} pi r'.II_I,Iil+,' Thus alIhough 11I('rt, is 110 ('x:"'1 rt'I;llioll,hip Illne is all approxinlale ('qllalilY hl'IW(,(,II IIII' log vidd III 1II;\llIril~' Oil a COli pOll bond and it weigh led an'r;lg(' or Ihe r('llII'llS 011 lI\e hOlld Wh('ll il is h('ld 10 malurilY, Equalioll (J 0, J,I J) lells liS Illal Macalllay's dural ion fill' a cOIIPon bond is tI\(, (it-rivalivt, orils log prin' wilh n'sp('('llo ils 101-( yidd, EtI'I:llioll (10,1.17) gives I II is dt'li ,,;) Iin' as
n, "
(I - fl") -':;
(1 -
p)
1 - (J
+ 1',",)'"
I - (I
+-
1', ",) -I
(1 0,1. IS) '
wh('It, lilt' Sf 'COIlt! elf";"i IV II\t'S I' == ( 1 -I 1', ",) - I , As 1I0I('d ;\bo\,(', Ih is rda t illn 1)('1\\'('('11 dllralion alld vield holds ('x a 1'1 I)' li)r a bond s(,lIillg al par, SlIbslillllillg (1O.1.17) and (IO,I.IH) illlo (10,1.1:), W(' oblaill a loglin('ar I('blioll IWI\\,(,('II holding-period r('lul'lls and yields [Ill' COUpOIl bonds: ' .. ",'ll
~ IJ,nl,n,-(/),,,-I)Yr,n--I,I-II'
(I n, i 19)
This t"llIaliol1 "'as lirsl tlt'l i\'t'd hy Shillt'r, Camph('lI, anti Schot'nholtl: (19~t\),11 It is allalogolls III (10,1.:,) for zero-coupon bon(\s; maturity in Ihal (''Illalion is rt-pla('('d h)' dllralion IUTC, and o/'('ollrs(' 111(' IWo ('qual ions an' consisll'llI wilh 011(' ,II1011ll'r I'llI' a IITO-COUpOIl bOlld whose duration (,quais ils IlIallllil),. t\ similal an;i1l'sis 1'01' I'ol'\\'anl ral('S sllows Ihal Ih(' II-period-ahead 1Ilt'riot\ forward ralt' implicil in Iht' cOllpon-i>earing lerlll slrllCllln' is
1
(
I
I I
I
I
I
I
/111
\
\
I
:~
1},,1I11
' ',111.,1 - [),'I .,I,", - n,,,
---.-.:..-.-/I, '<\ I
( I (). I.:!()
Tllis rorlllllla, \\'lIich is also dill' 10 Slrilln, Campbell, al1d SrlIOCI1\tO\t1. (I!H'\:\), )'('dllct·s 10 1111' (\i'(01l111 hond 10\'11111'" (IO.I.H) whell dll1'alioll ('<J";lIs lIIalurilY· IISIIilln, ( .. uHphdt ..1Ih\ S, Il\u'l\!\ult, " . . ,. ' ... / ''''It'"tl 01 y, ,," hut thcst· ~\IC.' t'llui\\tlc:nt tu lht' 1i',I-olCkr .IPIIIU-..;illl;Ilioli 11'
~afllC'
, II.
I. Basic Concepts 10.1.3 Estimating the bra-Coupon Term Structure
The classic immunization problem is that of finding a coupon bond or portfolio of coupon bonds whose return has the same sensitivity to s~all interest-rate movements as the return on a given zero-coupon bond. Alternatively, one can try to find a portfolio of coupon bonds whose I:ash flows exactly match those of a given zero-coupon bond. In general, this portfolio will involve shortselling some bonds. This procedure has academic interest as well; one can extract an implied zero-coupon term structure from the coupon term structure. If the complete zero-coupon term structure-that is, the prices of disCOUllt bonds PI ... P" maturing at each coupon elate-is known, then it is easy to find the price of a coupon bond as (lO.1.2l) Time subscripts are omitted here and throughout this section to economize on notation. Similarly, if a complete coupon term structure-the prices of coupon bonds Pel ... P," maturing at each coupon date-is available, then (10.1.21) can be llsed to back out the implied zero-coupon term structure. Starting with a oue-period coupon bond, Pel = PI (l + C) so PI = PcI/(1 + C). We can then proceed iteratively. Given discount bond prices Pi, ... , Pn - t , we can find P" as P _ PC7I-P7I-IC-···-PtC (10.1.22) II 1+ C Sometimes the coupon term structure may be more-than-complete in the sense lhal at least one coupon bond matures on each coupon date anel several coupon bonds mature on some coupon dates. In this case (10.1.21) restricts the prices of some coupon bonds to be exact functions of the prices of other coupon bonds. Such restrictions are unlikely to hold in practice because of tax effects and other market frictions. To handle this Carleton and Cooper (1976) suggest adding a bond-specific error term to ( 10.1.21) and estimating it as a cross-sectional regression with all the bonds outstanding at a particular date. If these bonds are indexed i = 1 '" I, then ,itt' regression is 1"'''' = 1', C;
+ PtC; + ... + p".(1 + C;) + Uj.
j :::::
I . .. I,
(10.1.23)
witer(' (:. is the coupon on the ith bond and 1Ij is the maturity of the ith bond. Thl' regressors are coupon payments at different dates. and the coefficients art' ,ite discount hond prices Ij. j = I ... N. where N == maXj 1Ij is the longest coupon bonrlmaturity. The system can be estimated by au; provided th~l the coupon term structure is complete and that I ::: N. "
410
10. Fixed-lnfOmI'SI'fwitiel
'~'l}line Estimation
In practice the term structure of coupon bonds is usually incomplete, and this means that the coefficients in (10.1.23) are not identified without imposing further restrictions. It seems natural to impose that the prices of discount bonds should vary smoothly with maturity. McCulloch (1!171, 197:,) suggests·that a convenient way to do this is to write POI' regarded as a function of maturity P(n), as a linear combination of certain pres(Jecified functions: J
1'(11)
1
I
+L
(lJ
;;(n).
(10.1.24)
J=I
I
McCulloch calls 1'( n) the di.lfOll1lt fu II ftioll. The jj(lI) in (10.1.24) are known functions of maturity n, and the are coefficiellL~ to be estimated. Since ~)«(}) = I, we must have jj(O) = () for all j. i Substituting (10.1.24) into (10.1.23) and rearranging. we obtain a rc1resSion equation
fl,
\
ni
I
=
t
(lj Xij
+ ui.
i =
I ... 1.
( 10.1.25)
J=I
,.there n; == Pr.... - I - C,II" the difference between the coupon bond price at" the undiscounleu value of iL~ future payments, and X" == jj( II,) + <:\; L;:;'I ;;(1). Like equation (10.1.23), this equation can be estimated by QIS, hut there are now ollly J coefliciellL~ rather than NY' A key question is how to specify the functions ;;(11) ill (10.1.24). One si,mple possibility is to make 1'( II), the discount function, a polynolllial. To do this one sets ;;(11) IIi. Although a sufficiently high-order polynolllial Gill approximate any function. ill practice one may want to use more parameters to fit the discount function at some maturities rather than others. For example one may want a more flexible approximation in maturity ranges where many bonds are traded. To meet this need McCulloch suggesL~ that I'(n) should be a .II/lille fllnction. l6 An rth-order spline. defllled over sOllie fmile illt(~rvai. is a piecewise rth-oruer polynomial with r- I cOl1linuous derivalives; its nh dnivalive is a step function. The poinL~ where the rth derivative changes discontinlIously (including the poinls al lhe hq~inllillg and end of the inlerval over which the spline is defined) arc known as IWIII /Joints. If there arc K knot
i
=
"The hond pricing errors al<' unlikely 10 he hOlnmked"..'lic. McCllllorh .... K'u·.' lhal lhl' Mar"lanl deviation of II, is proportiollal to II,,· hid·ask spread lor bond i. ami 111I1S wl'ights ('arh observdlion by the reciproral of il, sprl'ad. This is not required for ron.,isleney. hili 1Il;'y i'nprnve tl\~ efficiency or the e~timales. I~S\lil'. Mason. amt eh;1\I (1!17H) giV<' ;\n aen'"ihk illlwdllniollio splill(, IIlt'lhoduloKY.
/(). 1. nt/sir COl/rrlll.\
'III
points, there are K - I subintervals ill each of which the spline is a polYlloIlliai. The splint' has K - 2 + r free paramcters, r 1<11' the first suhint('l"val and I (that determines the L1l1festril"lt'd nit derivative) «)r each of the K - 2 followillg subintervals. McCulloch suggest.s that the kllot poillts should he chos(,11 so thaI each subinterval contains an cqualllllllllH"r ofholld 1Jl,I!IHity datc~.
If j"orward rates arc to be contiuuous, the discoullt fUIH"tion must have at least olle continuous derivative. llellce a quadrati! spliIH·. estimated by McCulluch (1~171). is the lowesl-onler spline that c;lIIlit the discount function. lfwe reC]lIire that the forward-rate curve should ,IIsD 1)(" continuously difler(~Illiahlc. thell we lIeed to lise a cubic spline. t'stimall'(lb), McCullodl (197:») and olhers. McCulloch's papers p;ivt' lhe ralhn complicated limllulas for tht' fUllctions .//11) lhalmakc 1'(11) a qll,lIlLltic or l"Ilhic splille. 17 '1fl.\" lj(I'I/I
OI.S eSlilllatioll of (10.1.25) chooses tIl!" parallH'tns ", so t1latthc bond prici:lg errors 11/ arc ullcorrclatcd with the variahles XiI lhat ddille the discount fUI!rlioll. If a sufficiently lIexihle splille is lIsed, thl'lI the pricillg e/Tors will he ullcolTclated with matllrity or any Ilolllinear fUllctioll ofillaturity. Pricing errors may, however. he corrclatnl with the COUpOl1 r;uc which is the other (\elillillg characteristic ofa hone!. IlItlt't'd rvkCIIII"t1l (1~17I) found that his model ll'nded to underprice bone!s tltat IV("I"(' selling below pal' bt'callse of their low coupon rates. McCulloch (I D75) allribules this to ,I lax dkn. US Treasury bOlld COli pOliS arc taxed ,IS ordinary illcollle while priet, apprt'ci~tliol\ 011 a COli POllhe;lring bond purchased al a discollnt is taxed as capital gains. If thc capital gains lax rate T~ is less than lhl' ordinary incolnc tax rate T (as has oftcn been the case historically). then this Gill explain a price prellliulll 011 honds selling below par. For an inveslor who holds a bond 10 maturity the pricing f<>nullia (10.1.21) should be mOllified to
I',,, == [I - TgO -1'".»)1'(11)
+ (I
- riC
L
"(i).
( IO.1.2li)
1=1
The splillc approach call be modified to handle lax dlLTts like lhal in (I ().l.~(;), at the cost of some additiollal complexity ill t'stil1latio\l. Oll(e tax crfects arc included. coupon bond prices lIlust be uscd to construct the variables XI) Olt thc righI-hand side of (I ().I.~:)). This mcalts lhat the bond pricing errors are correlated with lhl' rl'gll'sSOIS so tit!" !"qll;llion Itlllsi hl' 1";:\d;lIllS
and
\';111
De\'elltcr (1~~J4) argile.' 101 tht'
11.'1('
of
.1 101ll11l"0Id('1 "iplillt,.
with tltt'
cuhic 1('1'111 omilH'd, ill order to JIIaximil(' tilt· ...'iIllOolhllt·:-. ...... of III(' for"';lId-l.ttt" nllV(', wh('u' !\l1loolhllt''iS
i;-, ddill('tilo he milltl."i Ih(' ;'I\'('rag" .'iCfllart'd ,\('( oud <1('1 i\';I.i",' 01 IIH' 101 \\,aul-I.Uf'
rur\'(,' with n,':-'IH,'('t to m~\tHr'1y.
J (),
J'/Xt't!-JIlt'()/l/t'St't'llrillt'J
estimated bl' illSlnllllC'lIla\ variables ralher than simple OIX l.ilzenberger alld Rol/il (I ~IHI) 'Ippl)' a lax-adjllsled splille modc'l of this sort to hOlld lIIarket data fro II I sc\'('\'.11 different c(lllIItri('s, Thl' t,lx-adjnslnl splillt' IIlluh-1 aSSllIllt'S that the samt' tax ratt's art' rdevalll fill' all hOllds, Thl' model call1lol halldle "c1ieJltele" efreets, ill which difkrclltl), (;Ixc'd illl'cslClrs spccialill' ill differelll honds, Schaefer (I !lH I, I~IH~) sllggc'sts Ih"t c1iclllcle dkl'ts call Ill" halldled hy first fillding a set of tax-dlicil'llt hOllds for ,111 invl'Slor in a particular tax hracket,thl'lI estimating an implil'dlCT'H'olllHHI vit'ld Cllrvl' from Ihose honels alone, Nlllllillmr /11m/I'll
Despite Ihe f1exihilil)' of Ihc splille "pproach, spline fllnctiolls haw sOllie IIl1appealillg propcrlics, First. sillce splilles are polynomials Iht'y ifllply a discollnt fllllnion whil'h diverges as 1II.llllrity increases r"ther thall goillg to lITO ;IS r('ljllin'll hy Illt'ory, Implied forward ratt's also diverge rathl'l' thall cOllvc'rgillg 10 all)' lixcdlilllit. Second, there is 110 sifllple way to ellsure Ihal the discollllt flilinioll alwa)'s declines with maturity (i,e" that all forw,lrd rates are positive), Thc' fill'ward curve illustrated ill Figllre 10,1 gilt's I\egative al a maturity of '.!.7 YI"lrs. and this hl'havior is not IIIlC0I1111lllll (sc'l' SllI'a 11~IH'11l, Thes,' probkills art' rl'l"tt'" 10 the facl Ihat a lIal Z<'I'O-coupon yield curl'(' illlplic's ,III ,'xpOlll'lllially declining discollnt fllllction. which is not e"sil), approximat('d Ill';I polmOlllial fllnction, Since allY plausihll' yield cllrve lIattcns 0111 at thl' long enolld; th(,), call on I)' he identified hy restricting thc' relation h .. tw('('11 IOllg-horill'lI lill'w
, 141
more difllcultto use than the standard spline and this cost mayoutweig~'the exponential spline's desirable long-horizon properties. In any case, forward rate and yield cUl\fes should be treated with caution if they are extrapolated i beyond the maturity of the longest traded bond. Some other authors have solved the problem of negative forward:rates by restricting the shape of thc zero-coupoll yield CIII\fC. Nelson and Siegel (I !lH7), for example, model the instantancous forward rate at maturity n as thc solution to second-order differcntial equation with cqual ~oots: fen) = flo + fll exp( -a n) + an {J2 exp( -an). This implies that the dis~ount function is doublc-exponential: , !'(n) = exp[-flo n + (fll + {J~)(l - exp(-a1l»/a - nfl2exp(-a1l)].
a
This specification generates forward-rate and yield cul\fes with a desirblc range of shapes, including upward-sloping, inverted, and humIHhaped. Svensson (1994) has developed this specification further. Other recent work has generated bond-price formulas from fully specified general-equilibrium models of the term structurc, which wc discuss in Chapter 11.
10.2 Interpreting the Term Structure of Interest Rates There is a largc empirical literature which tests statements about expectedreturn relati9nships among bonds without deriving these statements from a fully specified eqUilibrium model. For simplicity we discuss this literature assuming that zero-coupon bond prices arc obsel\fcd or can be estimated from coupon bond prices. 10.2.1 The Expectatio1ls HypDthesu
The most popular simple model of the term structure is known as the expectlltiom hY/JOthesis. We distinguish the /Jure expectati01ls hypothesis (PEH) (PEH), which says that expccted excess returns on long-term over short-term bonds are zero, from the expeclati01l5 hypothesu (EH). which says that expected exc('ss returns are constant over time. This terminology is due to Lutz (1940). J)iffmnl J'imns of the Pure EX/Jerlatio1l5 Hypothesis We also distinguish different forms of the PEH, according to the time horizon over which 'expected cxcess returns are zero. A first form of the PEH equates the one-period expected returns on one-period and n-period bonds. The olle-p('riod rcturn on a one-period honel is known in advance to be (I + rill. so this form of the PEH implies
';
414
10. Fixnl-II/((JIIII' SI'(IlIili"J
whle the second equality f()llows from the dC/inition of holdillg-period ret~rn and the fact that (I + YIII) is knowll at time I. A second form of the I'EH equates the II-period expected returlls Oil on I-period and II-period bonds:
j
I
I (I + Y == E,[(I + YII)(I + YI.t+d ... (1 + }'1.1+1I-1)1· (10.\!.\!) Heje (I -+ Yn,)" is the n-period return on an II-period bond, which equals II , ) ·
the expected return frolll rolling over olle-period bonds for 11 periods. It is str,lIghtforward to show that if (I O.\!.\!) holds for all n, it implies 1 + 1';,_1 I .
==
(1
(I
+
+
YII-l.t)"-
l' )"
",
(lO.\!.:~)
I = Ed 1+ l'1 14-11-11. .
Un ler this form of the PEl I, the (II - I )-pcriod-ahead one-period forward ratd equals the expccted (11 - I )-period-ahead spot rate.
lit is also straighlli.)fwanllo show Ihat if (I O.\!.2) holds for all ! .
. (I
+ }'1I1)" =
(I
+ I'll) 1-:, [(I + }',,-I.'ld,,-I].
II,
il illlplies (1O.\!.4)
But (10.2.4) is inconsistent with (10.2.1) whenever interest rates arc random. The problem is that byJcnsen's Inequality, the expectation of the reciprocal of a random variable is llotthe reciprocal of the expectation of that random variablc. Thus thc pure expectations hypothesis cannot hold in both its one-period form and its II-period form. IX One can understand this problellllllore clearly by assuming thaI illterest rates are lognormal and hOlllos\<.edastic and taking logs of the one-period PEH equation (10.2.1) and the II-period PEH equation (1 O.\!.4). Notillg lhat from cquation (10.1.5) the exccss olle-period log relUI'Il Oil all II-period bond is T•. t+t - YII = (YIII - YII) - (/1- I)(Y,,-I.t+1 - YIII),
cquation (10.2.1) implies that E[TII.t+1 - )'11]
-(1/2)\'al'[,.".I+1 - )'11],
(IO.2.(j)
while (10.2.4) implies that E[TII.I+I - )'111 = (1/2)Var["".I+1 - Y111.
The
dif1('ren('(~
I"COX,
(10.2.7)
betw('en the right-hancl sides of (10.2.6) and (10.2.7) is
alld Ross (1!11l1 .. ) ", .. ke Ihis poilll VCI)' dc .. r1y. They also .. r).:IIC Ih .. 1 ill thue, only expected ('quality of iWilantallt'olls n'lurllS (it model rOlTt'~pontiing to (10.:.1.1» i, cn ... i
rontin\1(}\t~
415
10.2. Infl'Ilm'finK till' '/I'rmSlnu'llIIl' 1I/IIIfl'/l'.lf Ullfl'.1
·((dlf,. [().2.
I ('IUI"II
'"./11 - .1'11
(;1"",);<" ill yi<"ld 'il/.Ill -
,.111
Yilt
Ii
Ie
:!·1
II.:\W. (O.liH)
O.,.li·, (I.n'l)
O.IW~
('l.(I,.·I)
0.\)\7 IIi.:! I H)
0.711\1 (11.:1:11
0.010 10.,.'.I'l)
0.010 (0':'71;)
0.010 (0570)
0.0111
11.1)\1
\0.:\ \71
10.·IH:-\1
0.011 \11..1(0)
(\1:\\11)
.1'lt
-lUI!)
IH IUiI·1 11\1.·10)
1'l0 -O.IHH CI7.0H) 11.I11'l
10.liOIl)
10.,.111.)
(0':'7:1)
(O.:):~I:l)
0.011 (0.·111:-\1
0.01 I 111.-110)
0.01'l (0.:110)
0.1\17
0.:1'l" W.:IiI:I)
0.,.70 (004:111)
0.7Ii'. 10.,.\I·\)
O.!F',)-\ (().7~17)
I.t:.:t 11.01'l)
IU:I7)
\,i"I" "1'1'<,,111 )'/11 -
(.III/.i.! bUild lIIal"lfl" (111
:1
<:II.llIgC· 111 yield -0.11111 -,",,-1.1+1 -
(If (1'1111.\/111(/1111' I'1I/;aM,..\.
.,
\',Iri,d,h-
Exn::-. . .
,\In/II.\ ulJI/.\/lIl1dflnl d,.I'lllliOIl\
10.'l1'l)
-0.0:.1;
-0.01·\
1.:lli7
1.llIlg hOlld IlIiilllritit's ale.' lII('il~lII ('d ill mouths. For ("it( Ii \',11 i.lhll' tite t.thlt' 1('pOI h lhl.' ~alllpit' lIU',\\\ autl :-'I\mph' ~1.,,\daH\ dc.'vi"\tion (in p.\lt.·lHh\·:-,\·~) W,\1\~ '''t'Hlhty (LH,\ o\t'r the.' pt.-lim.' I~J:-):':'I~ I q~11
::l. Tilt,
UOiL" .11"('
anllualized pt'l( ('IHagl'
pOilll~.
Tht'
tllldl"llyil1g dala "I(' It'IO·
rOllpo:1 hOlld )'il'ld~ hUIll MrCulloril and K",ol1 (I~J~J:\).
Val'[ 1".,+1 - .vld, whirh lIlea~un:s the quailliialive illlporlallCt' oflheJellscll's Illequalily effect ill a lognormal hOllloskeda~lir model. Table I ().~ reports unconditional sample IllCallS alld standard deviations Ii)!' several terlll-structure variables OVl'\' Ihe period I ~':I~: I to I ~)~II :2.1!' All dala arc 1ll0111llly, bill arc measured ill allllllali/.l'd IH'I('Clllagl' Jloilll~; Ilial is, the r,l\V variables arc multiplied by 12()(). Th(' lilsl 111\\' ~holVs Ihe ml'all and st,\I\danl deviation or excess rdurns Oil II-",ollth I.l'ro-Glupon bonds over olle-Illonth bills. The mean excess retllrn is posili\'l' anti rising with malurity at lirst, bllt it starl!; to rail at a matlllilY of Olle ycar allli b cvell slightly lIegalivc li)r lell-year zero-coupoll bOllds. This pallerll rail be underslOot\ by breakillg excess retul'lls inlo Ihe two l'tJllIJ)onenls on Ihl' right-hand side of l'qllalion (J().2':'): the yidd spread (y", - ),1') bl'lWel'lI II-period and olle-period hOllds, and - (II - 1) times the ch'lnge in yield CV,,--I.'+I - y",) on the II-period bond. Intnesl rates of all lixl'd lIIatllrities rise during the sample pl'l'iod, as shown in the second row ofTablc 10.2 and illllstrated 1()rol1('·monlh alld H'II-Yl'ar ralt's in Figure 105. At the short end of the terlll structurc Ihis dkn is OnSl'l by Ihl' declinc in 1II'lllll'ity frolll II to 11- 1 as the bond is hdd lill' 01\(' 1\\""lh; thlls the r\tallW'
1"·1:II>It· IO.:,! i~ ,11Il'Xp'IIUIt'd "e.'lsioli ofa lahle.' shoWIl III <:;11111'1)('11 (I~)~I:-)), Tilt' IHll1Ih('r~ gin'lI h('Il' .1Il' ~Iighll)" dilh'J"('nt from tht' 11I1I1lj,('I"' ill liIal papt'l 1)1"( .111.... (· Iltt' ~"III"It· pt'liod o.,,,'d ill Ih,lI P''1H"1" \,',is 1~):J1:1 10 19HO::l. _Ihhollgh it W;t~ (,llnllt'oll"'''" Iq)orlC'd 10 ht' I~I:)~: I 10 I\I\II::!.
Ill. h.v
'tlll
xr-----r-----r-----r-----r-----T-----~----~----._----,
l-lIumlhyit'''I "
,
0'1
1~170
1~ II ill
Figllre 10.5.
Shill /- 1/1/11
11)\11)
I\)KO
1./lI/~-·li·'1I/ Il/lrwII Ul/lr,l
1\)%
1952 III I'NI
ill yil'ld (Y,,-l.tt I - )I",), shoWII ill thl' third row of Tahk 10.2, is lIegal:vc for short hOllds, cOlltribllting positively to tht'ir return.~u Al tht' long t'lIrl or th(' t('rlll structure, how('\'('r, the decline ill maturity fmm 11 to 1/ - I is negligible, allel so Ih(' chang(, ill richl (.'1',,-1.11,1 - .v",) is POSiliv(', callsillg capilalloss('s 011 10llg zero-coli pOll bOllds which outweigh Iht' high!'r yields orkn'd hy these hOllds, showlI ill Ihe l(lImh ro\\' ol"I;i1I!e 10.2, The slalldard d('viatioll or ex('('ss retul'lls rises rapiclly with maturity. If ('X("('ss hOllcl r('IIII"IIS an' whitt' IlOis(', Ih('11 Ilw slallclarderroJ" of the sa!llple !IIean is the stallclard (it-Vi;llioll diviclt'cI by the square roOl of the sample si/e (-Hi!llIlOlIlhs). The sianclard ('ITOI' lill" 1/ =: 2 is ollly O,O~(}\" whereas the st;\Ildanl enor (ill' 1/ "'" I:.!O is 1,71 'Xo. Thlls Iht' p;lIlcm or l\I('all retlll'llS is illlpre<"isl'iy t'slilllal('(1 al long 1Il;\Ill1iti('s. Till' slandard (\t-vi;lIitlil of ('x!"t'ss rl'llIrllS also delermines tll(' sizl' of'the w('elge het\\'('('11 Ih(' olH'-p('I'iod allell/-period forms of Ih(' pure exp('natiol1s hypOlll<'sis, 'I'll<' dilf('l'('II("(' l>et\\'('('11 IIl<'ali allllualizC'cll'xcess n'tuflls ullcler (10,2,(,) alld (IO.:.!.7) is (111)' (1.000:\';:, lill" 1/ = 2. It is still olily 0.11 IX, IiII'
'.!\lhl\"('~It,. ~ \dlll
an- ",~,i(\ 'n ht,
"1
,t"''''
10 pfohl" 11111 !IIi, 1t'llth'I)e'\'
illin)!. "u'
"it'l,' ('" \T."
of hlln
to I;IH ,,~ 1Il;lIlIril\'
shrink,
W.2. I7Itnpretillg the 1erlll Structure oj Interest Uates
417
11 = 24. But it rises to 1.15% for 71 = 120. This calculation shows that the differences between different forms of the PEH are small except for very long-maturity zer
E[rn.t+l - YIt]
= o.
(10.2.8)
This model is halfway between equations (10.2.6) and (10.2.7) and can be justified as an approximation to either of them when variance terms are small. Alternatively, it can be derived directly as in McCulloch (1993). IJIIlllimtiom oj the Log Pure Extleetations Hypothesis Once the PEl! is formulated in logs, it is comparatively easy to state its implications for longer-term bonds. The log PEH implies, first, that the one-period log yield (which is the same as the one-period return on a oneperiod bond) should equal the expected log holding return on a longer II-period bond held for one period: . (10.2.9) Second, a long-term n-period log yield should equal the expected sumi of 71 sllccessive log yields on one-period bonds which are rolled over forl n periods: : n-1 Ynt
=
(l/n) LEt[YI.t+;].
{IO.2.1O)
i=O
Finally, the (11 - I )-period-ahead one-period log forward rate should equal the expected one-period spot rate (n - I) periods ahead: (10.2.11) This implies that the log forward rate for a one-period investment to be made at a particular date in the future should follow a martingale: 00.2.12) [I' allY of equations (10.2.9), (10.2.10). and (l 0.2.11) hold for all n and I, thell the other equations also hold for all n and I. Also, if any of these
/0.
FiX I't/- /lIO JII/ I' ,'i/f lllil iI'S
C Illations hold for 1/ == 2 h )Id for n ::::: :1
<Jl sol lie elate t, the n the oth er equ ati on s als o sal lie dal e I. No le how eve r tha t (10 .2. 9)- ( 1O. ill C Ilo t gen era lly ~.II) equ iva len t for p;u ,tk ula r II and I.
.... ...•.
.-...
-•
..-.
...:.a;..
..:.... ....: . ..•... %
A Il'matives to the l'l,re I~XI) /'{t(t!iu/l.I I {ylmtltl· .lis T Ie exp ect ati ons hyp oth esi s (EH ) is mo re gen el'a l tha n the PE it allows the exp ect ed l I III tha t retlll'lIS 011 bo nd s or dif fer ent ma l1u itic s 10 clif C( IIst
/0. 2,2 Yield Spreads allr llnt l'l'l 'st Hate ForcrtL lts We now con sid er empir ic;)1 evi del lce 011 the exp ect ati on s hypothe~b (El l), Sil\ce the El tal low s CO llstillll differences ill the exp ect ed ret urn s 01\ sho an d lon g-t erm bo nd s, it rtcloes /lot rcs tric t COllst< lllt ter ms so for con ven we dro p cOllstaJl\.~ fro ien ce m all eql lal ion s in thi~ sec tio n. So far we hav e sta led the i/llplicati()/I.~ or the exp en< ltio ns hyp oth esi for the levels of lIo mi nal iJltere~t ra\('~. In s pos i-W orl d Wa r II US ina l int ere st rat es see dat a, nO Illm to fol low a hig hly per siS len t pro ces s willi a clo se to IInity. so IIIl1ch roo t very l~l\Ipirkal wo rk liSt 'S yid l! sprCilds 1e1.J'ls.~~ iIlS\(',HI of yie ld
i
11·rhi~
,,,,'/:t' i., Ih,' l rOIll 1I I 11111' ill Ill!' 1i1'· I,II1 1U'. FoWl" amI Bli.~' (19117). lIlos hOl.ever, "'I' "1(''111
t·Xf\·~\ rt·lurn~
lilt
nil louR-tenn hond!\., '~S"r Cha pIN ' 2 alit! 7 f"r a eli,
,
Failla (lm H). FoUll"
(I~)~)I), ;Iud pn'l lIi,, " 10 It''' 'r I" r .... li/l' d. rall lN l\)a ll"' !)I' ot'l l •
e"" jo" 01 (I"il mOl.'. Th,' 1'(''''';,
JO.2. /1I/1'I/1l"l'Ii1l,( Ihl' '1'1'7111 :'i/l1lrlllrr '!f JIII"II'I/I!II1r'.1
11!1
Recall th;ll lhc yield spread hctW('('1I lhl' 1I-pniod yield alld period yidd is .1", == _~,,' - .~I" E
S",
(~)E/[t[(:VI.III-YI/)+(I"'1 1/
(~) E,
i~1
[t
[(11-
i)6)·\.II" -\-
YI.II')]]
1./1,
(1",1
Ih(~ OIlC-
I/H-
)I.IH)]]'
(10.2.1:'.)
The secolld equalily ill cquatioll ( I 0.2.1:~) replan's IIlltllipniod i IIten~st r,IIC challges by Sillns of sillgle-period interest rate challges. Thc e«IIation says Ihal the yield spread equals a weigh led average of ('Xp(~clcd 1IIIIII'e i\llen~SI ratc changes, plus all IIllwcighled aver;lg(' of exp('('tcd IlIlure cxcess relurns 011 IOllg bonds. If changes in inlerest rales arc slaliollary (thaI is, if illlerest rates thelllselves have onc ullit rool hilI 1101 IWO), and il excess relurns are slationary (as wOllld be implied by any Illodel ill which risk aversion and honds' risk characteristics are stationary), lhell the yield spread is also static,nary. This means thaI yields of diffcrellt maturities are willll'gm/l'd. 23 The expectations hypothesis says that the s('('olld term on the righthalld side of (10.2.13) is constanl. This has illq>ortalll implicalions fur the relalioll between the yield spread alld lulure inl('I'esl.r;III'S. It Illeans that lhe yield spread is (lip to a const,lIlt) the ol'tilllall()rn:ast(')' of the challge ill the IOllg-bond yield over the life of the short hUlld, alld the optilllal f(H'CcaSler of changes ill shOrl rates over the lil'c of the long hondo \tecilling that we h,l\'(' dropped all nlllstallt terlllS, the rdaliolls alt'
(n~I)S"1 =
i':lly,,-I,I+t-Yllrl.
(10.2.14)
and S",
:=
E,
L(I - i/Il)6,YI.t+i "-1
[
]
'
(10.2.15)
I~t
Efjllalioll (10.2.14) can be obtained hy sllhstitutill!-: the ddillilion of r".'1- I, (lO.!'!,,», illto (10.2.9) alld rearrangillg. It shows thaI whell the yield spread is hi!-:h, the 10llg rate is expected to risl'. This is hecalls.. a high yield ~jlr('ad gives Ihe IOllg hOlld a yield advalltage whirh IIIlIst he ollset hy an anticipatccl Gtpital loss. Surh a capital loss call only COIllC ahout through all ill('f'ease ill lhe long-bolld yield. Equatioll (10.2.1:1) lilllow~ dir(,ctly 110111 (10.2.1:\) with constant expected excess returns. 1\ shows Ih;ll whell the yicld spread is pro{"('ss
i:-. clisfll.'ist'(\ 11I111u.'r ill <:h''1H('" 11.
:!\~l'c.' (:.IIuph<:ll awl Shitll'r (l!tH7) It)r a di.'i( tI'!'Iioll III (Oilllq!,l.tliclll ill llie h',lII .'itlll( HII(' of illll'n'.\1 ral('$,
1I"gll'\\;1I1I I'II"/litint/l~ ..
'1I,b/.' 10,],
J tI.
J'I,\t'(/'JII(OIlU' ~.'W( UlliU'."
1/11//
Y. ,
1.1I11/!. hOlltlmlltlllity (II)
1)"1"'11<1"111 Llli.,I.1<-
:1
Ii
IHun I II. I ~ II )
·0.11'. (O.:!K:!)
OX\'. lli.·I·I:.!)
d'~tI,g(':-,
o.!"',o:!
( 1Il.:.!,IH)
(O.O!Jli)
1I.·lii7 III.IIK)
II,:I:.!II (O.I,lii)
I:!
:!·I
·IX
1:10
l.""g·Vll'ld rilallg('s ( 10.~.11i)
--I..I:\'.
-I..J.JX
-~.~():!
-·J.~:!(i
10.,.')!/)
(1.00·1)
( I..,.H)
(:!.1I71;)
II.:!?:!
O,:IIi:1 (O.:!:.!:l)
II.·I·I:! III.:IHI)
I,·III:! (O.I·li)
SIUJlI'Llh'
I.ong
10.~OH)
ili"';11 (' 1Il,';I;o..1II (,.I ill 111011111,. Tilt' first row r('port" thl' (':-.limait'd r('gressioll ~" b tlllI (I o.'2.lfi), with .111 ;lWlllplolir stanci,li'd ('nor (ill I);U (")lhl':-'(''''') ctlt'tllatt'd 10
hOllclm;1111I
(udliriC'llI
allow Ilu hCh'ro.,l..,ccl."tit'ily ill III(' malllH'!' dl',n ilwd ill the APpt'lIdi:<, TIle' S('cOJul row fCPOI f." Iht" Iht' (':-.Iilllalt,d n'~n'.,:-,ioll codlit'i"llt VII 110m (IO.~.IH). with all .asymptotic M~lIldaJd ('ITOI' cllrulah'c1 ill lilt" ;o..;lInc II1;UlIU' ... all(lwillj!, also for J ('siciual otlltO(,(HTt"i.uion, The c'xpc'('(;ltioll."i
I"'p"lh"si",",h,' \l'III\ '\1'"·111 ...· illll'li," (hal hOlh~ .. ami Y.. "hOllhl,·II,uloll'·. Th,' 1I1I
high. shOll Lllt·S .IIT ('''I)(,("ll'd to rise so Ihal Ihe average short raIl' over Ihe lifi' of Ihl' long hond I'lJlIals IIII' initial long-hond yield. Nl'ar-ll'rlll increases ill shorl rates an' given grl'aln weighl Ihan furlher-off illcreIS('s. because Ihe), alli'n t hI' 1('\'1'1 of short rates dming a greall'r pan of Ihl' life of Ih-:: IOllg hond, l'idd S/,mllh
1/11//
Fllturf I.ol)g UI/II"
Equation (I ().~.I·I). which sars Ihal high yield sprl'ads should forl'cast in1T1'a.~('.~ ill 10llg ralC.~. f;\rI's poorly in Ihl' dala, Macaulay (I!):~H) firsl nOled Ihl' (;\fl Ihal high yield spreads ;Irtltally lelld 10 precede decreases in long rail'S. Ill' \\'!"OII': "Thl' yidds of bonds of Ihe highesl grade should jflll during a pniod in wllidl shOri-tlTlll rall's arl' highn than Ihe yields of Ihe honds and ril" durillg a pniotl ill which shOIl-Ien\l rales arc lower. Now experience is lllon' Iwarl), tIll' opposill''' (M;lclIII"y jl!):\H. p. :~:~I). 'Elhll' 10.:1 rl'pol Is ('stilllal('s of the cOl'fficil'nt fJ" and ils standard error ill the regrl'ssioll
-""
Thl' malurill' :.' I Fur 111;11111
\.1"
II \,;11
iii""
-
."//.1
== O'lI"'~"
(
.\", ---I /1-
i(" h 11\\1 :1 IlIllIlllts 10 120
,11,11\" II lit· \(',11 tilt,
1.1"1"
Ihe.,
)+
(I(),2.Ui)
€ 11,1-
1110111 hs
( 10 years) .~'I According
til(' approxilllation
.",,,_1.'1 I
=:::: ."'H,H I. Note
LU.~.
11I1t'ltJldlllg lite Ii-nil SllIulllle II} 11I1t'I"fJI Ualt'J
421
10 the expectations hypothesis, we should find f3n = 1. In fact altthe estimates in Table 10.3 are negative; all are significantly less than one, and sOllle arc significantly less than zero. When the long-short yield spread is high the long yield tends to fall, amplifying the yield differential between long and short bonds, rather than rising to offset the yield differential as required by the expectations hypothesis. The regression equation (I 0.2.1 G) contains the same information as a regression of the excess one-period return on an n-period bond onto the yield spread s",. Equation (10.2.5) relating excess relUrns to yields implies Ihat Ihe excess-return regression would have a coefficient of (I - f3n). Thus the IIcgatiw estimates of f3n in Table 10.3 correspond to a strong positive relationship between yield spreads and excess returns on long bonds. This is similar to the positive relationship between dividend yields and stock returns disclIssed in Chapter 7.~~) One difficulty with the regression (l 0.2.16) is that it is particularly sensitive to measurement error in the long-term interest rate (see Stambaugh [ I ~lHH]). Since the long rate appears both in the regressor with a positive sign and in the dependent variable with a negative sign, measurement error wOllld tend to produce the negative signs found in Table 10.3. Campbell and Shiller (1991) point Ollt that this can be handled by using instrument;\1 variahles regression where the instruments are correlated with the yield spread bllt not with the bond yield measurement error. They try a variety of instruments and find that the negative regression coefficients are quite rob\lst.
YiPld Spreads and Future Shorl Ralps There is mllch more truth in proposition (10.2.15), that high yield spreads should forecast long-term increases in short rates. This can be tested either directly or indirectly. The direct approach is to form the ex post value of the short-rate changes that appear 011 the right-hand side of (10.2.15) and to regress this on the yield spread. We define n-I
s;', -
L(I - i/n)6.Yl.t+i.
(10.2.17)
j~l
i )
I that this is lIot the same as approximatinf( p._I.I+1 by p•. 1+ I. The numbers given differ slighlly from th"", ill Campbell (I!J!J!'J) because that I"per IIses the sample period 19!i1:1 to 19')0:2. t'rrollt"ollsly r<,()()ned as 19:'2: I to I !J!JI :2. iI ~ -"Calllpbell and Ammer (199:1). Fama and French (I9R9). and Keirn and Stambaugh (I9~6) show that yit"ld spreads help to forecast eXces., returns on bonds as well as on other long-ttlnn ass,·ts. (:;lInpb"1l and Shiller (I \191) alld Shiller. Campbell. and Schoenholtz (19R3"> show tllal yield spH'ads tend lO forecast declines ill long·bond yields.
i
1"
~J1d rim
10.
FiXI'tl-/II('(IIIII' .vm,.itir,~
the regression
J:', =
J1."
+ y" .';", + fill'
(IO,\!,IH) tH
The expectations hypothesis implies that y" == I for all /I. Table 10.3 reporlS estimated y" coemcienL~ with standard errors, correcting for heteroskedasticity and overlap in the equation errors ill the manner discussed in the Appendix. The estimated coef[jcienL~ have a U shape: For small n they arc smaller th;11I one bllt significantly positivt'; lip to a year or so they decline with 11, becoming insignificantly dilferent fl'Om zero; beyond one year the coellicienlS increase and at ten years the coefficient is even significantly greater than olle. Thus Table 10.3 shows that yield spreads have forecasting power ror short-rate 1ll0vemenlS over a horiwn of two or three months, anel again over horizolls of several ye'II's. Around one yt';lr, however, yield-spread variation sn~l1Is almost lin related to suhseqllent JIIovemenlS in short rates. The regression equation (I O.:!.I H) cOlllaills Ihe same infill'lnalioll as a regression of (1/11) times the excess II-period return Oil :III II-period hond onto the yield spread s",. The relation hetween excess [etunls and yields implies that the excess-relllrn regl'essiolJ would h;lve a coefficient of (I - y,,), Table 10.3 implies that yield spreads forecast excess returns olltlO horizons of several years, but the forecasting power diminishes towards ten years, There arc several econometric difficulties with the direct approachjust descrihed. First, one los('s II p('riods or dala at the (~IJ(I of the salllpk period, This can be quite serious: For example, the len-year regression in 'Iitble 10.3 ends in 1981, whereas the three-mollth rq;ression ends in 1991. This makes a substantial difference to the reslIllS, as discussed hy Camphell alld Shiller (1991). Second, the error term {IO' is a movillg average of order (n - 1), so standard errors must he conected in lhe mallller descrih(~d in the App\:ndix, This can lead to IInite-sample problems when (II - I) is not small relative to the sample size. Thinl, the regressor is serially correlated ai'll correlated with lags of the dependellt variahle, and lhis too call ('allse fil/ite-sample prohlems (see M,lIlkiw and Shapiro [1~8fiJ. Rkhanlsoll alld St6ck [1990], ;lI\d Stambaugh 11!)H{ij). Although these econollletric prohklllS are imj)onallt, they do lIot seem to laccount lor the U-shaped lJ<\tlern of coefliciellls. Campbell and Shiller (1 ~191) find similar results \lsillg a vector alltoregressive (VAR) met hoe!ology likf that described in Section 7,2,:l of Chapter 7, They rind Ihat the IOllg11'1\111 yicid spI'I'ad is higltiy ("lInci;Ited with all IIl1l'estrirt('d VAR (U'('Clsl of 1111'111"(' ,~hon-ral(' IIIOV(,III('lIts, while the illtenllediate-lenll yide! spread is IIIltch IIlOre weakly correlated with the VAR f()I,(,Gls!.
i
t';Fallla (19H4) and 5l1il1<-r, Calli 1'1,..11 , and ScllO('lIh"hl. (l\IH:\) "'" Ihis approadl alllo",lo"n ("(1<, of Ihe (("rm ~lnl('Clln·. while Fall1;l anti Bliss (1 ~lH7) ('xtt'lld it lO Ih(' long ('lIti. (:;IIBphdl ;lll(, ShiUt'r (J!ml) pnwiclc .a ("(Jlll()rt'ht'IlSi\'(' U'\'it'w.
10.1.
4\13
(;Olldll.\iO/l
To illlerprel ·r.lblc rewrilc il as
IO.:~,
it is helpful
\0
relurn to \·(lll.lIioll (I O.:!.I :~) and
( to.:!. I!))
"YH' and .11",
..... (-;;I) E, [.'!~(1/
E,[<"I
==
(D [t E,
(/,,11_1.11, -
-
;)6'\'1./1, ] .
)'114 ,)].
III ~cllcrallhe yield spread is the SlIlIl oftlVo (,OlllpOIlCIIL~, olle Ihatlilrecasts interesl r~lll' changes (.~y",) and Olll' thaI rorl'c~lsls cxcess 1'l'lllrllS Oll long honds (.,r",). This means that the regrcssion ('()('flicicnl y" in ('cluation (I O.:!.I H) is
VH
COV('\·~f' .f",]
Var[s"tJ
Var[.ry",] + COV[.I.~"I' .\r"ll Var!"Y"ll + Var!.lr,,11 + :! CoV!-'.'~""II.. ,!·
(IO.\1.:!O)
For any givl'n variance of excess-rcturn filrecasls .\1,,,, as lhc variance of in\erest rale !ill'CGISlS .\y", goes to ~.('ro the cocfJid('lll y .. gO\~S 10 ~,cro, hUI as Ihe variallce of .IY,,1 increases the coeflicienl y" goes to onc. The Ushaped pallel'll of regression cod'licicnL~ in -Lillie I o,:~ lIlay be explained hy redu('cd forccastabililY of interesl ral(' movemcllts at \loril.ons around onc ycar. There lIlay he somc short-run forecaslahilily 'lrising from Federal Reservc operaling procedures, <1l1cl some long-run forecastahililY arising from husinl'ss-cycle effects on interest rates, hUI al a one-yeai' hori~.on Ihe Federal Res('\ve lIlay slIlooth interest raIl'S so Ihatlhe variniods whell illlCl'esl ratc Illovcments have been highly rorecaslahlc, slIch as Ihe perioc\ illllll<'C\ialdy Iwfi,re Ihe roullC\illg or Ihe Fednal I{cs('\'ve System.
10.3 Conclusion Thc results ill Table I ().:~ illlply thatnaivc inveslOrs, whojlldge honc\s by their yield., 10 lllatlll'ilY alld buy IOllg honds when Iheir yields al'l' rcl;llivdy high. havl' Il'lldec\lo ('arn superior relurns ill III<' poslwar period ill Ih(' Iinill'd
III.
hxnl-IIIWI/II' .'''"1"11 ullr.\
Slal(·s. This findillg is n'llIillisc('nt of III<' filldillg disCIIss('d ill Chapt
Problems-Chapter 10 10.1 YOII an' lold Iilal all H-year lIolllillalll'rO-<"OllPOII hOlld has a lo~ yield to maturilY of~),1 'Y... alld a ~)-\,<,ar nOlllillall('r<>-<"OIl)lOIl hOlld has a 10;; yield of H,W!." 10,1.1
Call Ihl' plll"l' ('xpcctaliolls Iheory o/"th!' t(,!'I1l slructure d,'\crihl' IllI's(' tlala?
10.1.2 ,\ )"(',Ir gO('S hI'. alld IiiI' hOllds ill !l;1rI (a) slill hal'(' IIi(' S;III1(' yil'lds to llIalllril\', Callthl' plll"l' expectaliolls theory o/"Ih(' It'rlll strllllllrc dl'snii>e 1111's(' 11('", .Ltla~ IO.I.:l I lllw wlluld VilliI' allsw('rs dlang(' ifY(lIl \\1('1'1' told thai the h()uds ha\'(' all H';;, COli pOll I;lle pn I'('ar. raliler thall zero COUpOIlS?
10,2 SUP)lOS(' th,ll lit" 1IIIlIH'(;UY aUlhorilY «lIIlro\s sl\orHcI"I\I illtl'J"('sl ral('s hy sclling \"11
_00
.\'1 I I -\- A\.\'.!I -
)'11)
+ f"
with A > O. 1IIIIIili\('II'. IIII' 111011<'1,11'), alilhority Iries III S\llIlIllh interest ral('s hili 1;lises 111<'111 \111<'11 Ihe ,i,'ld CIIII'(' is Sl('('p, SIIPpO~(, also Ihal Ih(' tl""-pl'riod hout! "it'ld ';l1i,li('\ I'll
=
1.1'11
I F,I.I'IIIII)/~+xl'
425:"
I'roblrms
'
~
-:.-t\
where Xl is a term premium that represents the deviation of the two-period ;,'...:: i yield from the log pure expectations hypothesis, equation (10.2.10). T h e ' '.' variahle Xl follows an AR( I) process Xl
The error terms each other.
fl
=
r/lxI-l
+ TI/·
and '1/ are serially uncorrelated and uncorrelated with
10.2.1 Show that this model can be solved. for an in terest-rate proc~ of the form ;
= YI.I-I + yXI +
YI/
fl'
Express the coefficient y as a function of the other parameters in l the lIIo(k!.
i
10.2.2 The expectations hypothesis of the term structure is often te~ted in the manner of equation (10.2.17) by regressing the scaled chanie in the short rate onto the yield spread,
I
and tc~tillg thc hypothesis that the coeflicient f3 = 1. If the m6del described above holds, what is the population value of the regression coefficient fJ?
10.2.3 Now consider a version of the problem involving n-period bOllds. The monetary authority scts short-term interest rates as .
•
and lhe n·period bond yield is determincd by
y.. 1 - YII
=
(n - l)EIlY ... ttl
-
y,l/)
+ x"
where XI now measures the deviation of the n-period yield from the log pure expectations hypothesis (10.2.14). (This formulation ignores the distinction between y.. 1 and y.. _I,I.) As before, XI follows an AR( 1) process. What is the coefficient y in this case? What is the regression coefficient fJ ill a regression of the form (l 0.2.16), (y"./+I - Y.. ,)
= ex + t3(y", -
YI/)/(n - I)
+ U,+I
?
10.2.4 Do you find the model you have studied in this problem to be a plausible explanation of empirical findings on the term structure? Why or why not? Note: This problem is based
011
McCallum (1994}.
11 Term-Structure Models
Tills ClIAI'TER EXPLORES the large lIIo(\cm lilcrallllT Oil hIlly spn'ilie(\ gelleral-eCJuilihrium models of the term structure or illtercst rates. Much of this litl'r,rture is set in continuous tillie, which simplilies sUllie of the theoretical analysis but complicates empirical implementation. Since we fucus 011 the econollletric testing of tire models ,\lui their I'lIIpirical implications, W(' adopt a discrete-time approach; however we tak(~ care to relate all our results to tlreir continuuus-time equiv'llents. We follow the literalllre by lirst developing models lur real bonds, but we discllss in SOIlIC detail how these lllodels Clll he IlsC'd to price nOlllinal honds. All the lIlodels in this chapter start from the gt'neral assct pricing COIIdilioll introduced as (8.1.3) in Chapter H: I = E,[ (l + ({,.II dM1+ I J. whcn~ R,.I+ I is the rcal relllrll on sOllle asset i and M,t I is the .I/(}cil(l.llir tiiJ({) 11II I }f,rlUI. A, we explaincd in Scction H.I or Chapter H, this cOlldition implics that thc cxp~ctcd return on any asset is ncgatively related to its covariance with thc s(()chastic discount factor. III models with utility-maximizing investors, the stochastic discount factor measures tire margin,lllltility or investors. Asscts whose returns covary positively with tht: slochastic disco\lnll~lctor lend to payoff when marginal utilily is high-they deliver wealth at linles when wealth is most valuable lO investors. Investors are willin).( to pay high prit-es aIHI accept low ret lirns on such assets. Fixcd-illcollle securities arc particularly easy to pritT lisill).( this rralllework. When (
II.
li'IIII-Slrudllll' ,\(/ltldl
way: (I + 11".1 t I) =:: I'" __ 1.11 1/ I '"I, Suhstitutillg- this illto (H.I.:~), we lind that the r('al pri!"t' 01' all II-pniod r('al hOlld, {',i/, satisfies
( I I. 0.1 ) This "'1,,;11 iOIl kIlt Is itsr! rto;\ I t'cursivt, ;Ipproach. WI' model }'''I as a funCl ion 01' Ihose slale variahl,'s Ihat an' rd('valll fOl" fill"eCaslillg- Ihe A'i'+1 prtln:ss. Civl'n that prtlct'Ss ~\nd till' fllllClitln rt'latin~ [',,-1.1 to state variables, \\'l' can C\!culat<, Ih(' I'Ullctioll relalillg ['''I Itl stale variahles. We slart Ihe calndalioll hy nllting Ihal 1\" = I. Equalioll (II.D.I) (';111 also ht, solved forwartl 10 express Ihe II-period hond price as Ihl' ('xl't'l'led prodllrt 01' II stochastic dis(,Olllll bclors:
( 11.0.2) Althollgh w,' l'lIlphasilt' tilt' It'cursive approach, in some lIlodels it is more ronvenit'llt to work dircctly witlt (Il.(),~). Section 11.1 explores a class of simple models in which all releyant variables are cOlldilioll.dly log-normal and log hUlld yields arc lillear in stal<' variables, These II/Ii Itf-yiPlrllllodd.1 include all the Illost cOllllllonly used terlllsl 1'1I1'1ure IllOdds. Se('1 iOIl I I.!! shows how Ihesc 1I10dds ('all he fit to 110m i nal interest r;\\('
11,1 Affine-Yield Models To kl'('p lIIall('rs simpie-, 1,'(' aSSllUll' Ihrougholll this st,rtion Ihat the dislrihution of the slocha,tic discoulll EKlor Mit I is contlilion;t1ly lognormal. We sJll'dl)' IIlodel, ill which bond prices an' jointly lognorlllal \Vilh M I + J. We call Ihcn take !o~s of (11.n.l) to ohlain
1',,1
= FIIIIIIIII/'"
1.1//1
i-(I/!!)Varllllllfl +I',,-I.ltll,
(11.1.1)
",htTt· as lIsn.t! IOI,'t'lelse kll('rs tlt'llott' II\{' logs of t!\{' COlTl'spollding IIPJlercase kilns so 101' t'xalllpk 1111\ I =' log-(Mlt I). This is Ihl' hasic eqllalioll Wt' shall liSt'. WI' hegin wilh I\\'o Inodds in which a singlt' statl' vari;lh!e fOI'I'C
11.1. AfJilll'-Yil'ld Modl'ls
discrete-time versions of the well-known models of Vasicek (1977) and Cox, Ingersoll, and Ross (1985a), respectively. Section 11.l.3 then considers a more general model with two state variables, a discrete-time version of the model of Longstaff and Schwartz (1992). All of these models have the property that log bond prices, and hence log bond yields, are linear or afftllf in the state variables. This ensures the desired joint lognormality of bond prices with the stochastic discount factor. Section ILIA describes the general properties of these affine-yield modfiJ, and discusses some alternative modelling approaches. I 11.1.1 A llomoskrdaslic Singll'-Faclor Model
It is convenient to work with the negative of the log stochastic discollnt factor, -11l1+1. Without loss of generality, this can be expressed as the sum of its one-period-ahead conditional expectation x/ and an innovation tl+l: -m,+I==Xt+tHI.
(11.1.2)
We assume that f/+1 is normally distributed wilh constanl variance. Next we assume that Xt+1 follows the simplest interesting time-series process, a univariate AR( 1) process wilh mean J1. and persistence cpo The shock to Xt+1 is written ~/+l: (11'.1.3)
The innovations to write Et+1 as
11l1+1
and
XI+I
1'1+1
may be correlated. To capture this, we
== fJ~1+1
+ 1]1+1,
(11j1.4)
where ~t+1 and 1]/+1 are normally distributed with constant variances and are uncorrdated with each other. The presence of the uncorrclated shock 1]/+1 only affects the average level ofthe term structure and not its average slope or its time-series behavior. To simplify notation, we accordingly drop it and assume that fl+l = j3~'+I' Eqllation (11.1.2) can then be rewritten as
•
(11.1.5)
Tlw innovation ~I+I is now the only shock in the system; accordingly we can write its variance simply as (}2 without causing confusion. Eqllations (11.1.5) and (I!. 1.:\) imply that -m,+1 can be written as an ARMA( 1,1) process since it is the slim of an AR( 1) process and white noise. 10111' di"frt·l<··tilll~ I'r~"t,"tation {(,llow" Singleton (I!J!IO), Sun (1992), and especially 1I.u·kll' (I \1\1:\). SlIn (1\1\12) t'xplor~s the r.. lation h .. tw.. en discrete·time and (onlinuom·tim .. II1lHh·ls in more' detail.
430
J J. 'Jinl/·S/mr/1II1'Mlld"LI'
In fact, -m'+1 has the same strllctllre as asset retllrns did ill the (·x.uIIJllc of Chapter 7, Section 7.1.4. As in tl1
y 1
JII'
=
Ellllll+il+(1/2)Var,llll,+il
=
-X, +/32a 2/'2.
(1I.I.li)
I
IThe one-period bond yield )'11 = -/'1" so i
),1'
=
X, -
/3 2 a 2 /'2.
(11.1.7)
IThe short rate eqllals the state variahle less a constant tcrlll, so it inherils thc AR( I) dynamics of the Slate variable. Indeed, wc can think of the short \ rale as mcasllring the state of the ecollomy ill this modcl. Note that thnt: 'liS nothing in cquation (11.1.7) lhat rules out a negative short rate. is Wc now guess thai the f01'1II of the price funclion for an It-period bond
-/1"1 = A"+B"x,,
I
(II.I.H)
=
Since thc n-period bond yield y", -11"ti II, we arc guessing thal the yield on a bond of any maturity is linear or affine in the state variahle XI (Brown and Schacfcr [1991). We alrcad), know that bond prices for 11 = 0 and 11 = I !satisfy cCJuation (11.1.8), with Au = Bo = 0, A, = -/32a~/'2, and III = I. We Iprocccd to verify our gucss by showing Ihal il is consistent with Ihe pricing 'relation (1 I. 1.1). At the salll(, time we can derive recursive fornlllias (t)r Ihe coellicients A" alld Il". Our guess for Ihe price function (II.I.H) implies Ihal the two It'II\IS Oil Ihe right·hand sidc of (11.1.1) arc
!
( 11.1.9) Substituting (II.I.H) alld (11.1.9) illto (11.1.1), we
A.
gc~
+ U"x,-x,-A"._,-II,,_dl-eJil/L-/J,,_,eJix, + (fl + /I/I-Jl~(T~ /'2 = O.
(11.1.10)
11.1. AJIIIII~ l'id" Morldf
Thi, IIlIISI hold for allY x" so Ihe (,O('lfu'icllls Oil x, 11111,1 SIIIII relll;lillillg coefficiellts IIIlISI also Sllill to /.(TO. This illiplics II"
1+<1>11"-1 == (I
10
I.(~nl alld the
We ha\'(' now verilied Ihe g"ess (11. U\), sillce willi the codf'lcienL~ ill (11.1.11) Ihe price functioll (II.I.H) satislies the asset pricing equation (11.1.1) alld ils aSSllmptiol1 that hOlld rclul"lls arc cOllditiol\;tlly logllorm;tI. IIIIIJlimtioll.1 oj thf I [omoshetill.llir Mudd The hOllloskedastic bond pricing lIIodel has several interesting illlplicatiolls. First, the cod'licient iJ" measures the bll ill the log price of an II-period iJolid when there is an illCl'Case in the state variahle x, or e'luivillelltly ill Ihe ()lIe-period illterest r")/( 1 - <1». As II increases, /J" approaches a limit /J I/( 1- cp). Thus hond prices bl! whell shOlt rates rise, and till' semitivity ofhond returns to short rates increases with maturity. NOll.' that iJ tI is different from duration, ddined ill Sectioll 10.1.2 of Chapter 10. Duration measures the sellSitivity of the II-period bond retulII to the ll-period bond yield. and for zero-coupon honds duration nJuals 1Jl'ltlll'ity. iJ" lJleasures the scmitivity or the lI-peri()(1 bond retllJ'n to the olJe-period interest rate; it is alwap less than matlirilY hera lise the II-!,('['iod bond yield moves less than one-/i)r-olle with the one-period ilJterest rate. A second implication of the model is tllat the expe('\('d log excess return 011 all II-period bond over a one-period bond. E, l r".ll-1 J -),1' = E/ [IJ"-I./~ll I'", + jll,. is given by
=
E,l /'".I-j-11 -
}II
-COV/l'·"./+I,III,j-IJ -V;II-,!r"./fI1/2 13,,_1 Co\', (x"
I. II/Il I
I-
Ii;'
1\'.11, (x, H 1/2 (11.1.12)
The first equality in (11.1.12) is a general result, disl'lIssed ill Chapter H. th,lt holds for the excess log return OIJ any asset over the riskfree interest rate. It ran he obtained by taking' logs of tile fnlJdallll'ntal rdatioll I == E,l(1 + U,.,+IlM,+d for the 11-period hond and the short interest rate. and then taking the difl'erence between Ihe two eqnations. It says that the expected excess log return is the sum of a risk premillm 1('1'111 and a./ensen's Inequality t(,),111 in the own variance which appcars I)('callse we arc working in I()~s.
II,
(n II/-Slnll 1111<'
,1,,,,11'1.\
TIll" "'(1111(1 I'qll;dill' ill (II,I,I:!) IIS(,S Ille /;11'1 Illal III(' 1lIlt'Xpe('(eci (0111POIII"III 011111' log 1('111111 Oil .III II-pnioci hOlld isjllsi -11,,_1 lilllt's Ih(' illIII "';11 i011 ill IIII' ,1;111' \,;lIi.lJ,I .. , Th .. Ihird "l(lIalily ill (I I.I.I~) IIS(,S IIII' LI(I I h;11 IIII' (olld i I iOIl;i\ 1;1 I i;IIII'" "I ;11 ('
,I, I
I a lid i Is ('olldi liolla I cOI'alia 11('(' lI'i III
(Ollsl;lIlts 10 ,hOI,' 111.11 11ll' ("p(Tled IOf!; ('xcess n'llIl'II
III, \
I
bOlld is
Oil ;\111'
cOlIsl;tll1 ol'n liltl(', so Ih.IIIIll' log ('\p,'dalions h\'pollll'sis-blllllOllh(' IOf{ p"l'e e\l)('cl;\liO\lS l"pOll!l'sis-holds, - II" I i, lht, t'lH'IIil'i"1l1 1'1'0111 .I j'('gr('ssioll of 1I-1)(,liod 10f{ hOlld ITIIIIIIS IlII ,1;111' \';lli;lhll' illllo";llillIlS, 'I' I\'(' "all illl('rplt'l -1/" I as Iltt' hOlld's 111;1(1illg Oil IIII' sillgle '0111('1' oil is!.. alld lin'! as IliC r(,ward Ii,r i>e;lrillg a IIlIil of ris!... ,\111'1'1 lal i\'('h', ronll"'illg \';Isin').. (1!177) alld Ollll'lS,
11'('
Illiglll (';IIclllal<'
Ill(' pri .. e orli,k as Ill<' r;lIio 01'1111' ('\p('('«('d ('xcess log 1'('1 Ill'll Oil a hlllld, pillS OIl!' h;df its IIII'll 1·;lriall .... III adjtlsi /ilr ./(,IIS(,II 's 11I(,«lIalil\', 10 Ihe sialldanl
",s log 1'<'1111'11 1111
d(,l'ialioll of 1111" "XI
ill(' hOlld, IlI-/ill('d Ihis lI'a)" Iht' prin'
0/ risk isjltsl ,fin i II I h is IIwcI .. 1. The hottlmkedaslit' hond pricing IIICHIc! ;1!sO has illlplir;lliolis for lilt, pall<'t'li of /ill'\\';11'11 I';II"S, ;Ind 111'11\ ... lor IIII' shapl' of Ih(' l'ic'lt!
1'111'1'(,.
'Ii)
,It-ril',, lllt'sl' illiplit'aliolls, ,,',' 110'" IIt;11 ill all)' 1(,I'II1-slJ'tlClIll'e IIlodl'l Ih,'
11-
IH'riod';IIll';ltI 101 11,11 d I al(' /'" s;lIisli,'s /'"
/'",
\'11
/'" , I.'
+ (1-:'/,." I I., I I I -
\'1') -
(I,:, [/1".1 f I [
-
( I I. I. 1:1)
/1",).
III Ihis Illodel F,I /1".'1 I 1..- /1", = - 1/" 1-:, [ t,..I', ~I I. alld 1-:, r""'11.'1 I 1- .1'1' is gin'lI (II.I.I~), SlIhslilitlittg illio (11.1.1:1) alld IIsillg II" = (I - CP")/( I - CP),
hI'
\l'C
gCI
/'"
/1 -,
[/il
(.!.. - (f>")]~ n~ + CP"l.\', I - l'
I /1" --[ ( If" 1 - (i'
/1)
:!
)~n~] -;- + [(I', .!
/1)
+
('+fin-4») ., (' - CP)-
"J cP
n-
/I
(11.1.11) Tllt'lilSl ,'cl'l;tlill ill (11,1,11) sltol\" ilI;1I lilt' challg" illil ... II-p('riod ((,rwanl 1';11,' i~ (/." lillit's IIii' 1'1t;lIlgi' ill ,\, Till" 11I00'CII1<'lIls ill lite I"r\\'anl 1';11,'
Tllis ,;lIl 1)(' lIlldcrsloot\
I." Ilolillg Ih,ll lit"
log "\llI'(,(;llillll' Illpolltl'sis Itolds itlillis IlIodd,.so lil .. wanl-ral\' 1ll0\','11II'\lls
n'Ih-, I 11III\"'IIIt'III.' ill lilt' ("1)(,(,(,'<1 IUIII('(' short ral,' w\tidl lilllt',~
IIIOI·(,IIt<'1I1.s ill lit .. 1'1I1T"lll s\torl LII",
;1('('
givt'tI Il\' "
11.1. AJjilLe-rield Mudels
As malllrity n incre,l~es, the forward rate approaches
a constant that does not depend on the current value or the state variable X,. Equation (1 Ll.7-) implies that the average short rate is J1 - fj 2a 2 /2. Thus the difTerellce between the limiting fonvard rate and the average short ~ate is I -(1/(1 - if;))2a 2/2 - (,8/(1 - if;))a 2. This is the same as the limiting expected log excess return on a long-t~nn bond. Because of the Jensen's Inequality effect, the log forward-rate curve tends to slope downwards towards its limit unless fj is sufficienuy negative, fJ < -1/2(1 -I/J). • A~ XI varies, the forward-rate curve may take on different shapes. The seconct equality in (11.1.14) shows that the forward-rate curve can be written as the sum of a component that does not vary with n, a component that dies out with n at rate if;, and a component that dies out with n at rate if;2. The third component has a constant coefficient with a negative sign; thus there is always a steeply rising component of the forward-rate curve. The second component has a coefficient that varies with x" so this component may be slowly rising, slowly falling, or flat. Hence the forward-rate curve may be rising throughout, falling throughout (inverted), or may be rising at first and then falling (hump-shaped) if the third component initially dominates and then is dominated by the second component further out along the curve. These are the most common shapes for nominal fonvard-rate curves. Thus, if one is willing to apply the model to nominal interest rates, disregarding the fact that it allows interest rates to go negative, one can fit most observed nominal term structures. However the model cannot generate a forwardrate curve which is falling at first and then rising (inverted hump-shaped), as occasionally seen in the data. It is worth noting that when if; :::: I, the one-period interest rate follows a random walk. In this case the coefficients An and Bn never converge as n increases. We have Bn :::: n and An - An-I :::: -(fj + n - 1)2a 2 /2. The forward rate becomes Inl = XI - (fJ + n)2a 2/2, which may increase with matllrity at first if fJ is negative but eventually decreases with maturity forever. Thlls the homoskedastic bond pricing model does not aIlow the limiting forward rate to be both finite and time-varying; either if; < I, in which case the liJlliting forward rate is constant over time, or I/J = I, in which case there is liD finite limiting forward ratc. This restriction may seem rather countcriIltllitivc; in fact it follows from the very general result-derived by Dybvig, IngersoIl, and Ross (1996)-that the limiting fonvard rate, if it exists, ran lIever fall. In the hO!l1oskcdastic model with if; < I the limiting forward rate never falls because it is COllstallt; in the homoskedastic model with tP :::: I the limiting forward rate does not cxist.
434
11. 1er1ll-S/ruflur" Mot/I'l{
Thc discrctc-timc model developed in this scction is closely relaled 10 the continilolls-tillle model of Vasicek (1977). Vasicek specifies a continllolls-timc AR( I) or Ornstcin-Uhlcnhcck proccss for thc short inteJ"('sl rate r. givcn by the following stochastic differential eqnation:
y
Ilr = K(IJ-r)d/+adlJ.
I
~here
K, 0, and a arc constants,~ Also, Vasicck assumes that the /)/1(1' of In/errst m/e rish--the ratio of thc cxpectcd excess return on a bond to the standard deviatioll of the excess return on the honcl-is a constant that docs \lot depend on the Icvel of the short interest rate, The model of this section ~Ierives an AR( I) proccss for the short ratc and a constant price of risk from ,)rimitive assumptions on the stochastic discount factor.
~::qui[jbrjum Interpretation oj the Mot/el G)ur analysis has shown that the sign of the coefficient f3 determines the . gn of a\l bond risk premia. To understand this, cOllsider the effects of . positive shock ~'+I which increases the state variable X,+I and lowers all I ond prices. When f3 is positive the shock also drives down 1111-11. so I>ond turns are positively correlatcd with the stochastic discount factor. This orrclation has hedge vallie, so risk premia on bonds arc negative. When fJ i~ negative, on the other hand, bond retllrns arc negatively correlated with the stochastic discount factor, and risk premia arc positive. We can gct Illore intuition by considering the case where the stochastic discount factor reflects the power utility fllllctioll of a represelltative agent. asinChapter8.lllthiscaseMt+1 = li(CI+J/C,)-Y.whereliisthediscollllt factor and y is the risk-aversioll coefficient or the representative agenJ. Taking logs, we have (11.1.\(i)
~
It follows that X, :E E,[-m,+il = -log(c5) + yEI[tH,+tl, and (HI == -mt+1 - E,[-ml+tl y(tH,+1 - E, [L':>.(I+ I J). X, is a linear fUllclioll of expected consumption growth, aud E/+I is proportional to the innovation in consumption growth. The terlll-structure model of this senioJl then implies that expected consumption growth is an AR( 1) process. so that realized consumption growth is an ARMA( 1,1 i process. The ('odlirielll ~ governs the covariance hetween consumption innov
=
'lA... in Chapter 9. dB in (11. L' :) dt'HOh";-; till' hlt"rctlWllt to;,l nrnwnlan 11lt)tlo11~ it ~hn\lh' 11111 h" (Ullr",,,
11.1. AJJi1/e-YiddMode~
435
Ii
is negative. a positive shock to COIISUllIptioll lowns illtncst rates so honds have positive risk premia. Campbell (19HG) explores the relatioll betweell boml risk premia and the time-series properties of COIISlllllptioll ill a related model. Camphell's model is similar to the one here ill that cOllslImptioll .lIld
(11.1.17) whae K, i~ capital at the start of the pniod, (1\, -- (;,) is invested capital, and X, V,+I is the return on capital. Thi~ budget C
The hOllloskedas(ic model uf tlte previous sectioll is appealing becallse of' iL~ simplicity. but it has sevcral IIlIattrarti\'e features. First, it assullles that interest rate changes have cOllstant variance. Secolld, Ihe model allows interest rates to go negative. This lIIakes it applicahle to real interest rates. hut less appropriate for nominal interest ratt's. Third, it implies that risk premia
II. '/i'r/I/-Sll"llrlwl' Alorlfll
are COllstalll Ol'("f lillI!", cOlllrary 10 Iht' evidellce presented in Section 10.2.1 of ChapIn 10. ()Ile CUI alter Ihe llIodel (0 handle these prohlems, while rel;linin~ 11111<"11 of Ill<' silliplicily of lht' hasic Slrllf\lIre, h)' allowill~ the slale l'ariaIJh·.\', 10 1()IIol\' a (ondilionally IO~I\Orlllallllll hel('\'Os\:.cdaslir .H{lIIlH'-mo/ process. Tl1is (11;lng(' is (,lIlird), cOllsislenl Wilh III(' t'quilihriullI foundalions I(Hlll(' modd gi\'('n ill Ill!" prcviolls scclion. Tht' sqllan'-root IIIO(h-I, whid! is a disITetl,-tinll' version or tlil' blllOllS Cox, IIIf,!;crsoll, alld Hoss (I ~)H!'Ja) contilluouS-lilllc lIIodel, replan's (II. ! .:-») alld (11.1.:\) willi
(11.1.1 ~J)
-. 111,,1
(II.I.~O)
Thc new clelllellt herc is lhal lhc shock ~'+I is lIlultiplied hy x,t/~. To IIll
/'", == E,IIII"I'
·'·'lll""I+(I/~)Var.l/lllll+···+IIII1·"J.
(II.I.:!I)
Calculations h:ls('(1 11I1 (11.1.21) ;\H' IHOIT t'llIlIhcrsOllll' thall lhl' I. ',his IIIt';IIIS that 011(' can 0111, allalyl.t' lit(' s'lllar('-roollllodelusill)!; III(' rt'('llrsi\~ t'cl'lalion (11.1.1); IIII' lI-pl'l'ioel Illglilll'ar relatioll (11.1.21) docs not holel ill the s<)llart'-root lIIodel. Procl'cding wilh Ihe rccursiw analysis as he/iJre, we call delermine the prict' of a olle-pt'l'iod bond hy suhstiluling ( 11.1.1 !) illlO (11.1.1) 10 g('l (11.1.~2)
The olH'-pniole x,. ()lIn' ,Igaill Ihe shOll rail' nlt',lsUrt·S Ihl' SIal!' or Ihc CCOIiOlln' ill the IIl1ldd. Si 11(,1' Ilit' sh or! 1<1 lI' is prllport iOIl:tl \(l Ihe stall' vari;lhle, i I in heri IS the property Ihal its cOlldilional v;lrianc!' is proportional 10 its lew!. Many ,\lIt hoI'S have lillII'd Ihat illtl'l'l'st I
11.1. Ajjine-YieldMoaels
show that negative interest rates are ruled out ill the cOlltinuous-time version: of this model, where the instantaneous interest rate follows the process elr == K((J - r)dt+crr l / 2 dlfl Tilllc-variation in voIatiIityalso produces timevariation in term premia, so that the log expectations hypothesis no longer holds ill this model. We now guess that the price fUJlClion for all II-period bond has the same linear form as before, - POll = A" + fIll X" equation (11.1.8). In this model Au == l~, ;::: 0, AI == 0, and BI == 1 - fJ 2 a 2 /2. It is straightforwarclto verify the guess and to show that All and BII obey
, l
Comparing (11.1.23) with (11.1.11), we see that the term in (J2 has been mowd from the equation describing An La tbe equation describing Bn. this is because the v;riance is now proportional to the Slate variable, so it affects the slope coefficient rather than the illtercept coefficient for the bond price. TIl(' lilllitin~ value of B", which we write as B, is now the solution to a CJuadratic equation, bllt for realistic parameter values this solution is c1.se to the limit 1/(1 -
Etlr".,+il- YII
== -Covl[r".I+I. B,,_I
mt+d -Var,[Tn .,+d/2
COV,[X1+I.
mt+d -
R!_I Var.[x/+I1/2 (11.1.24 )
Thc first two equalities here are the same as in the previous model. The third equality is the formula from the previolls model, (11.1.12), multiplied hy the state variable XI' ThllS the expected log excess return is proportional to tlw state variable X, or, equivalently, to the short interest rate YI/. This is the expected result since the conditional variance of interest rates is proportional to XI' Once again the sign of f3 determines the sign of the risk premium term in (J 1.1.24). Since the standard deviation of excess bond returns is proportional to the square root of XI> the price of interest rate risk-the ratio of the expected excess log return on a bond, plus one half its own variance to acljust for Jensen's IneC]uality, to the standard de\iation of :Itkpending on the par-.unett'rvailles. it lIlay be possible forthe interest rate \0 be zero in the "omin\\\""-tin\t' ""l
,.n
}'
438
11. 'limn-Structure Mot/ell
,the excess log return
011
the bond-is also proportional to the square root
lof x,.
I
The forward rate in the square-root lIlodel is given by
/." =
YI,
+ B,,(E,(t.X'+I)
- COV/(X/+I. 111/+1 J) - n~Var/[x'l-ll/2
== (l-/Fa 2 /2)x,-IJ,,(I-1>)(x,-tl)+x,{ja 2 ) ( 11.1.2!i) "he first equality in (11.1.25) is the sallie as in the hOllloskedastic lIIodel. vhile the second equality lIIultiplies variance terms by X, where appropri\le. It can be shown that the square-root model permits the sallie range of hares for the yield curve-upward-sloping, inverted, alld humped-as thl' IOmoskedastic model. . Pearson and SUIl (1994) have shown that the square-root model can I}(: generalized to allow the variance of the state variable to be linear in the level of the state variable, rather than proportional to it. One simply replaces the x,1/2 terms, multiplying the shocks in (11.1.19) and (11.1.20) with terms of the form (ao + al x,) 1/2. The resulting model is tractable because it relllaillS in the affine-yield class, and it nests both the homoskedastic model (the case ao == I, al == 0) and the basic square-root model (the case a" = 0, al :::: I).
l
11.1.3A 11u()-Fac/orMotiel
So far we have only considered single-factor models. Such models illlply lhat all bond returns are perfectly correlated. While bond returns do tend to be highly correlated, their correlations are certainly not one and so it is natural 10 ask how this implication can be avoided. We now present a simple model in which there are two factors rather than one, so lhal bond returns are no longer perfectly correlated." The model is a discrete-lime version of the model of Longstall and Schwartz (1992). It replaces (11.1.19) with (11.1.2G)
and replaces (11.1.20) wilh a pair of equations for the st,\te varia hies: I/~
Xu+ 1
(1-1>1)/11 +1>l x l,+X"
xv+,
(I - 1>~)IL~
~I.t+"
+ 1/1'2 X~I + '>:'21/2 1 ~2.H-I.
( 11.1(27) (11.1.'2H)
~AhhulIKh hond relllfllS are "," I'<'r I<-r II)' rorrd;ol"d ill Ihis IIlt)(I"I, III<" fovari,,"n' ",ald. (,rhuncl rt'l1lfn~ h.u rtf.llk tW() alull11'11f(' i~ sillglllar WIIC."IIl'V('r W(' (,hsl"lv(' m(,n'lllilil Iwe, IUJllcls. We disc,,'" Ihi. poilll rllrlll<"l' ill Senioll 11.1.·1.
,.
/ /. 1. Afji Ilr- Yirlel M Oelfts
439
Finally, Ille rdalion IWlw~~el\ the shod.s is ftll == fl~I.III'
(11.1.29)
alld Ihe shocks ~I.I+I al1(l ~V+I an' uncoITt'iale
The ollc-period bOlld yield YII == -Jill is 110 IOllg('\' proportional 10 lltc slalC variable XII, bccause it depellds also OIJ X~/' The shorl ilileresl rale is 110 1()Il~cr sufficienl to IIIcasure lhe slale or Ihe ('COIlOIll), ill Ihis model. l.ongsla{f alld Schwartz (l992) poillt olll, however, Ihal lhe condiliollal variancl' or Ihl' ,hort raIl' is a di ffl'l'e II \ linear function of Ihe Iwo SI /JIll> alld H2u ohey
Thus Ihe s!lort
(11. U~?)
J I,
Irllll-,\Innlllll' .II{)(/,'I,I'
Th(' dilf('f('lIc(' ('/(II.lIioll lill" III" is Ih(' saJlle as in Ihe singlc-hlclol" squarerool Jllod('l, (II,I,~:\), hUI 11t(' dilf('I'('II('(' equation I(lr H!" includes only a \('rlll ill Ihe own v;lri;II\(,(, of x~ I)('caus(' X~ is IIIlCOIT('\atecI wilh 111 anel clc)('s 1101 alf('cl 11)(' v;II'i;IIH,(, of
III,
Thc dilkrcllcc ('qllatioll ((II'
A" isjllslthe Stlill
of IWo lenllS, each of which has Ih(' bmiliar I(ll'lll from lite sillglc-bclor sqllare-fOOI II10de!, The exp,'('I('d (')\('"SS log hOlld r('llim ill Ihe Iwo-I;telor IIwdcl is gil'ell
Ill'
Ed ".. ", II
-
-- (:0",11'"", I,
,\'1'
1II1t"
I - V'lI',1 1'",1+1 I/~
II\." I <:OI',hUII, 1I/1t11 - /t~,"_1 Vard,"1.1+11/2
- n~"1 V;lr'!~·!.Ild/'!.
! -HI." Ifln~
-
U~,"_'la~ /'!.JXI/
I n.;
( I 1.1.:1:1)
_.11
This is III<' S'III1" ;1\ ill III<' s'l";lr(,-lOol Ili0dd, wilh Ihe addilioll of .11' exlla 1('\'111, arisillg I'rolll.lI'IISI'II\ i\lI'')II;IIil)', illihe varian\'(' of X~,ltl' '1'11(' r"I"'OInl r;II(' ill III<' 111'0-1'01<'101 III"dd is givell hy /;"
.\'1, ~, /II" (E,! 1\ 1"1.11 I I - <:01'1\"'1.1 I I,
- Ii;" \'OIr,1 1"1,/1 II/~ -
111/\
I Il
+ H!" Ed LI. X~,lf-I I
Iii" \'arl\x~,It_ll/2
(1-/1~(J~/'2)I"I'+.\'~'- 1I1,,(l-c/>I)(XIt-III)
-- /i~" (I -, tjl~) ( ,\-~ I
--
II ~) -
/II"
,\'1/
fJ
(1
~ (11.1.:\,1 )
This is III<' oill'iOll' g(,lIer;IIil.;lIioll of Ihe square-rool Jllodel. IlIlponalltl~', il ran gelll'LII(' 1\\01'1' (,(lIlIplicl\('d shapcs for 11\1' yield rllr\,l', illcllldillg illl'l'I'lt'd hllilip sllapl", ;IS III<' illdqH'llIlclll 1II01'1'II\elliS of hOlh XI, aile! -"2, .IIkCI Ihe 11'1'111 slrllclllrl', Tire all;t1ysis of Ilris III"dd iIlIlSII;II('s all illlportanl prillciplt-, I\s Cox, !lIgI'lsoll, alld Ross (I~IW>;,) alld Dybvig (19K~)) haw ('lllphasi'l.I't\, un(\1'1 ,'('naill .. i),"'"I1SIOI""('S 0111' (';Ill (,,,"Slfll.-t IIulltif;lClor 1I'J'lII-sIJ'lICllliT lIIodels silllpl)' Ill' "addillg lip" sillglt'-bnor llrodds, Whl'lIc\'('I' Ihe siorhastic dis('OUIII fartol /II" 1)(' ",rilll'n as Ih .. Slllll of Iwo indepl'ndl'lIl procl'sseS, Ihl'lIlll1' rl'sulting 11'1'111 stru('IUlt' is IIII' SUIII of IIII' 11'11Il sl1'\l('\un's Ihal wO\lld ('xisl ulldn ('"d, III' tll<'SI' III "(,('SSI'S, III Ihl' I,ollgstalf alld Srhwartl (I !)!)~)
I ".11'
1lI0del lilt' slo('h"Sli(' dis("11I11 farlo)' is lite SIIIIl of -XII - x~:2fi~I,1t I and '-x~" ;md Ihese (,""IPOIII'III>; ;tIl' illt\I'(lCllt\I'llI of each ollteL IlIspe('(ioll or (II.I.:H) shows Ihatlh" resultillg tl'l'IIl SlfllCllll'(' i,sjusllhc Slllll ofa gl'lH'ral
J I. I. Affine-Yield Modds
square-root term structure driven by the XII process and a special term structure with parameter restriction fJ = 0 driven by the X2, process, ' .
I I. 1.4 Beyond Affine-Yield Modell We have considered a sequence or models, each or which turns out to have the property that log bond yields are linear or affine in the underlying state variables. Brown and Schaefer (1991) and Duffle and Ran 0.993) have clarified the primitive assumptions necessary to get an affine-yield model. In the discrete-time rramework used here, these conditions are most easily stated by defining a vector XI which contains the log stochastic discount factor ml and the time t values or the state variables relevant ror forecasting ruture ml+i, i = 1 ... n. Ir the conditional rorecast of x one period ahead, EI [XI+ I], is affine in the state variables, and ir the conditional distribution or x one period ahead is normal with a variance-covariance matrix Varl[xl+d which is afflOe in the state variables, then the resulting term-structure model is an arfine-yield model. To see this, consider the steps we lIsed to derive the implications of each sllccessive term-structure model. We first calculated the'log shortterlll imerest rate; this is affine in the underlying state variables if mt+l is conditionally normal and E/ [m/+ tl and Varl [ml+tl are affine in the state vari'lbles. We next guessed that log bond yields were affine and proceeded to verify the guess. If yields are affine, and if X is conditionally normal with affIne variance-covariance matrix, then the risk premium on any bond is affIne. Finally we derived log forward rates; these are affine if the short rate, risk premium, and the expected change in the state variable are all affine, Affll1e forward rates imply affine yields, verifying that the model is in the arrme-yield class. Brown and Schaerer (1991) and Durfie and Kan (1993) state conditions on the short rate which deliver an affine-yield model in a continuous-time setting. They show that the risk-adjusted drift in the short rate-the expected change in the short rate less the covariance of the short rate with the stochastic discount ractor-and the variance of the short rate must boch be affine to get an affine-yield model. The models of Vasicek (1977), Cox, I ngersoll, and Ross (1985a), and Pcarson and Sun (1994) satisfy these requiremcnts, butsornc other continuous-time models such as that orBrenna,n and Schwartz (1979) do not. . Arline-yield models have a number or desirable properties which help to cxplain their appeal. First, log bond yields inherit the conditional normality asslimed ror the underlying state variables. Second, because log bond yields are linear runctions or the state variables we can renormaIize the model so that the yields themselves arc the state variables. This is obvious in a onc-factor model whcrc the short ratc is thc statc variable, but it is equaIly
442
11. 1mll-Slruclltrf Modrls
possible in a model with any Illllllher of factors. Longstaff and Schwanz (1992) present their two-factor model as one in which the volatility of the short rate and the level of the short rate are the factors; the moe!e1 coule! be written equally well in terms of any two bond yields of fixed maturities. Third, affine-yield models with K state variables imply that the terlll structme , of interest rates can bc summarized by the levels of K bond yields at each : point in timc and the constant coefficients relating other bond yields to the K basis yields. In this sense affme-yield lIlodels arc lin car; their nonlinearity is confined to the process governing the intertemporal evolution of the K basis yields and the relation betwecn the cross-sectional coefficients and the underlying parameters of the model. Affine-yield models also have sOllle disadvantages. The lincar relations among bond yields llIean that the covariance matrix of hond returns has rank K--cquivalently, we can perfectly Ilt the return on any bOlld using a regression on K other contemporaneous bond returns. This implication will always he rejected hy a data set containing lIIore than K homb, unless we add extra error terms to the model. Affine-yield models also limit the wa), in which interest rate volatility can change with the level of interest rates; for example a model in which volatility is proportional to the square of the interest rate is not afline. Finally, as COlIStantinides (1992) emplwsizes, single-factor affine-yield models imply that risk premia on long-term honds always have the salTle sign. If we move outside the affine-yield class of models, we can no longer work with equation (11.1.1) but must return to the underlying nonlinear difference equation (11.0.1) or its n-period representation (11.0.2). In general these equations must be solved numerically. One common method is to set up a binomial tree for the short-term interest rate. lllack, Derman, and Toy (1990) and Black and K;lrasinski (1991), for example, assllme that the simple one-period yield YII is conditionally lognormal (as opposed to the assumption ofafITne-yicld models that (I + YII ) is conditionally lognormal). They use a binomial tree to solve their models for the implied term structure of interest rates. Constantinides (1992), however, presents a model that can be solved in closed form. His model makes the log stochastic discount factor a SIIIlI of nOllcentral c.hi-sqllared randolll variahles rather than a normal randolll variable, and Constantinides is thell ahle to ('akulaIC 11)(' 'txpeCl~tiollS in ( 11.0.2) analytically,
I
I I
I
11.2 Fitting Term-Structure Models to the Data 11,2.1 ]{eal/Jolll[.I, NOl1lill(lllJolld.{, (lnrl1njl{(lioll
the term-structure models descrihnl so f~lr apply to bonds whose payol1s arc 'liskless in real terms. Almost all aCllIal honds insteacl have payofh that an~
11.2. filli/l~ '1n-III-Slruclul"f' M()drLI I() lilt" /)flla
riskless in llolllinallCnlJs.~> We 1I0W discuss how llic models call he adapted deal with this fact. To study nominal bonds we need 10 inlroduce slime lIew nolatioll. We write the nomill·.\1 price index at til\lt~ I ",IS ClJ. ;\Ill! tht" gros~ ralc of inll,lIioll frolll I 10 1+ 1 'IS n,; I == {2,; 1/{2,. We have ;Ilre;uly ddined 1'", to he the rcal pritT of all n-period n'al bond whidl P;IYS Ollt' goods unil OIL lillie I + II; we now ddille I>!I to he Ihe nominal price of ;111 n-pcriod nominal bond whirh pays $1 at Ii III(' I + II. Frolllthcsl' IkfinitiollS il follows Ihallhe IlOlllinal price o/" an II-pcriod real hOlld is I'", (~, and the real price of all to
II-period nominal bond is I~'; (li. We do nol adopl allY spccialllotatioll /CU" Ihese last two (on(cpts. If we now apply the gelleral asset pricing condilioll.
10
lile real relul'll on all II-p('riod nominal bond. WI' filld Ilwl
:0
1,$1
= E,
[I'S'~I~'I\-I
Mit I
]
.
(11.2.1)
Multiplying through by Q,. we have
( 11.'.!.2) where M:+I == M'+I/ n l + 1 can he thought of as a nOll/illal s\ochaslic dis(Ollllt t;lctor that pri(es nominal returns. The clllpiricalli\eralUre onllolllillal honds uses Ihis result in aile of two ways. Thc firsl approa(h is to take lile primiliV!" assllllll'liollS Ihal W(' made abollt M,+ 1 ill Sec lion ILl ami 10 apply Ihem ill'le'IlI \0 M~+ I' The real lerlll-SlrllrLlIre JIlodels of the lasl scclion aJ"(~ l!Jen n'inll"rpreled as numinal tnll\-stnIC[IlrC JIlodels. Brown and Dyhvig ( I ~IK()). tor exalllple. do this when ~'SOIIH' gOVCTIIIIH"It,"i, lIot;lhly 1110.\(' "I (:.III.tcl.l. b"wl, .11111111" \ lh.. h,l\"(' 1.'~IIt'd hOlld., \\1111\(' lIominall'ayofls arc linked to a nomillal prill' index. III I!J9ti IIi(' US TIC'astlry is considerillg-
isstling- ,'\imiiar securities. These judex-Ii liked hOllds approximate I('al hOTlch hilt an' nU('ly e{\llivalelll t() rca I bonds. Un,wI\ ~lIld Sc:il,l(.'lc:r (l~I~H) gi\t';' 1\l("l(( (ti~nlssi(Jll (,I Ilu,' impl·rt(·niot\s in the UK il1dl'xin~ ~ysh·lU. anet ~'pply tht' (:I)X, l1\g('I~ml1, anti R()~, (I'JH:M) 1I\{)dl'\ III UK illdt"X-link"t\ hllml •. S'T "Iso II"". .IIHI \:;11111'1 ... 11 (I~I\I:» ",HI \:""'l'bell "lid Slrilln (I~I%)"
('x~lClly
I I, 'J'nlll-S/rUr/url! ,'vlodels 11i('}, appl}' IiiI' (:0:\, Ingnsoll, alld Ross (I !IH!'ia) sCjllare-root lIlodel directly liS III Hllillal hOlld prices, Thl' sqllare-root model restricts interest rales 10 he POSilil'l', and in Ihis n'slH'n il is JIlore approprial(' for nominal inlen'sl r.II('s Ihall f(H' 1'('.11 illll'rl'si rail'S, The secolld ''IlplOarh is 10 ;ISSllnlt' Ihat Ihe two COIIIJlOI1('nts of Ihe nOlninal slodl;I~lil' disl'ounl bl'lor, 1\11/ I and II n H I, art' independent of each otht'r. 'IiI SC(' hllw Ihis .ISSlllllptioll f;u'ilitates empirical work, take lo~s ()f tht' llol1linal slodl'lstic discollnt fade))' t() get 10 dara 011
(11.2,3) \Vh(,11 tilt' COIIIIHlJlt'lliS ,It,) I ,llld IT/II ;tr(' illdq)(,lldent, we ran price nominal honds hy IIsing Ihl' illsights of Cox, lngt'rsoll, alld Ross (I!lH!'ia) and Dybvig (1 !IH!I), R('call fronl St'nion 1I,I,:~ Ihdr r('snlt that the log hond price in a l1l()del with two illd('p('Il(\cllt ('OmpOll('llts of the stochastic discount factor is the Slim ()f Ihl' log hOlld prices illlplicd hy each compont'nt. We can, for example, ''1lpl}' I he l.ollgslaff alld Schwartz (1 !)!)2) JIIodd to nOJllin.l1 bonds hy assllmillg Ihal lilt) I is desnilll'd hy a sqllare-root single-factor model, -1/11/
I = .l'Jr
+ XIII/:! 11~1.t) I, alld
Ihat
IT,) 1
,
IS known at I and ('qllal to a state
\'al'iahk,\'~I.v\'I'lhl'lIgl,t
111;)1 = "I11I1I+rr"11 .'(Jr+x:;:!fl~l.lt,+x:!1> and Ihe l.ollgslan~S('hl\';11'I1 IIllldl'l (ksl"rihl's lIoJl1illal bonds. Mort' )!;(,IIt'J'al/)" 11)(' asslIllIplioll I hal M, f I alld 1/ n 11,1 art' ill(kpcnde~tl iJl1pli\'s thaI prin's Ill' 1I011lill
=
=
I'~, = Elli\I~)11
= FIIMIl11E/[--I-]
nil I
= 1'11f.!.JF'/[_I_],
(11,2.4)
f.!.J+l
sillc\' I'll = E,l M'II' alld I/n l l l ::;;: f.!.JIGt-I' Tltlls Ihc 1I0minai price of a hotld whirh pal's $1 IOl\lorrow is Iht' lIominal price of a hond which pays 0111' ullit 01' good~ IOlllorrow, liJl1es Iltt' ('xp('elalioll of Ih(' rcal vallie of $1 IOllloITOW,
WI' W('
1I0W
glll'SS Ih'l1 a sillliLlr l('bliollShip holds for a\l JIIalurilies n, and
pro\'1' I his Ill' i 1)(11\('1 iolt, I I' 1111'
(1/-
I )-I'nioc/ relalionship holds,
I'" u(hF"II/
Jo: I
[/'~ 1/
1.,) 1 /\1'11
_(h (6 \ l ]
1':1["" 1,')I(!;"EI)I[_I_]!ltlll-~] (h (h I "
I I
r:-I.t
==
11.2. Filling 'lmn-Struclure Models to tlU'Data
=
Q, EI [1',,-1.1+1 M I+ 1 E I+ 1
=
J>"I Qs
[(LJ]
E, [_I_J . (b+"
where the last equality uses both the independence of real variables from the price level (which enables us to replace the expectation of a p r o d u f t ' i by the product of expectations). and the fact that POI == E/[P._ 1•1+ 1 MI+I~' Equation (11.2.5) is the desired result that the nominal price of a bond which pays $1 at time t + n is the nominal price of a bond which pays one unit of goods at time 1+ n, times the expected real value ofS1 at time 1+ n. Dividing (11.2.5) by 0. we can see that the same relationship holds between the real prices of nominal bonds and the real prices of real bonds. Further, (11.2.5) implies that the expected real return on a nominal bond equals the expected real return on a real bond:
EI [1~_~I+l ~] J "I
Qs+1
(11.2.6) Gibbons and Ramaswamy (1993) usc these results to test the implications of real term-structure models for econometric forecasts of real returns on nominal bonds. Although it is extremely convenient to assume that inflation is independent of the real stochastic discount factor, this assumption may be unrealistic. Ibrr and Campbell (1995), Campbell and Ammer (1993), and Pennacchi (1991), using respectively UK data on indexed and nominal bonds, rational-expectations methodology applied to US data, and survey data, all find that innovations to expected inflation are negatively correlated in the short nm with innovations to expected future real interest rates. More directly, Campbell and Shiller (1996) fwd that inflation innovations are correlated with stock returns and real consumption growth, proxies for the stochastic discount factor suggested by the traditional CAPM of Chapter 5 and the consumption CAPM of Chapter 8. 11.2.2 Em/)ineal Evidmre on Affine-Yield Models
All the models we have discllssed so far need additional error terms if they are to fit the data. To see why. consider a model in which the real stochastic' discount factor is driven by a single state variable. In such a model, returns on all real bonds are perfectly correlated because the model has only a single shock. Similarly, ret urns on all nominal bonds are perfectly correlated in any'
I
44()
I I. 'Irnll-SlruriIll1' !IIm/t'll
model where a sin!!;lc slat(' variahle drives the nOlllinal stochastic discount IiI(' \I 1/'. In reality there are 110 d(,terlllinistic Iincar relationships alllOn!!; returns Oil dillcrent honds, so these implications are bound to he n:i('Ctcd hy the data, Adding extra state variables increases the rank of the variallcccovariance matrix of bon
M/)dfl~
Stambaugh (1988) and Heston (1992) show that undcr fairly weak assulllptiOl~ about the additional bone! price errors, an affine-yield model illlplies a I tmt-variable structure for bond relllrns. Variables that forecast I>olld relt rns can do so only as proxies 1(11' the underlying statc variables of the lllo~c1; if there are fewer state variables than forecasting variables. this pUL~ testable restrictions on I()recasting equations for bond returns. 'A general anine-yield model with K state variables takes the f(mll
I
I
-/)",
=
A"
+ Ill" XII + ... + Ill\" XI\"
(11.'2.7)
whe 'e Xk" k == I ... K, arc the state l"ariahJes, and A" alld lJ.". /1 = I ... K, arc :onstants. The model also illlpl.ies that expected exn'ss returlls Oil long hOIl Is over thc short interest ratc (an he wrillclI as
1
E,l r".H I -
,vI,)
A;, + n;" XII
+ ... + Il~" XI\"
(ll.'2.H)
whe e A~ and B;". k = I ... K, are constants. The model puts n \lSSsect ollal restricliolls 011 these constants which arc related to the tillie-series prm\css driving the state variahles. hut we ignore this aspect of the lIIodd
hen"
11.2. Filling '/i'flll-S/rurlll;l' M(I(It-/.1 /(/111/'
/)1//1/
Now ~IIPpO~C that we do not ohscrn' thc 11'111' I'XI'I'SS rcturns on lOll!!; hOllds, hili illsl!';ul ohsl'rvl' a lIoi~y IIl1'aSllrc 1'".1+1
=
(II.:!.!/)
1'".1.1 - .\'11 !·II".I.I,
\Vh I' rc 'I ",1+ I is a II ClTor tcrill. Wc aSSlllliC Ikll // ",II I i~ h, conlainillg.l inslnlllll'lliS "/" j = I ... .1:
01'1
hogolla I loa VI'I'lo.'
(I U.IO)
Till: \'cclor h, lIli!!;ht contain la~~l'(1 variahlc~, for I'xal1lpk, if the rcturtl error '/.../+1 is serially 1Il1corrclated. We further aSSIIIIle Ihat iiII' ea('h statc vari;lhlc XI .. " = I .. , K, the cxpl'ctation or the statl' \',11 iahk conditiollal 011 Ihl' illSll'IllIlClliS is linear in the illSlnlllll'IIIS: I
E
IXk' I
h,]
(\ 1.2.11)
LOk/hjl /=1
for
~ollle
cOllstallt coe/Iiciellls (l.)' These ;tsSlllllptions imply that the expectatioll of I'",tt I conditional on the illstruiliellts, which from (I I.:!. 10) is thc sallic as Ihl' expcctatioll orthe true ex('c~s rCIUI'Il 1'",'1 1 - YI, condiliollal 011 Illc illSll'llllll'lItS, is lillcar in the illstrlllllellts:
EI"",'+I I
h,l
=
E[I'".'~I-.\'II I
h,l =
,...
II:. -I- L JJ;"E lx.,
h,l
:=
"
I
.=1
,=1
,1:.+ LJJZ . LOI,hjt.
k=1
defille C'II to he the vector [/'1.111 .. , ",V.,. 11101' asselS thcli (II,:!, I:!) call he rcwritten in veftor ()l'J1I as
11'11'1'
C,+I = A' -I- Ch, -I-
1/ 11,1'
11
(11.:.1.1:.1)
I .. ,N.
(\ 1.2.1:\)
where A' is a vt'('(or whos(' IIlh dl'lIll' II I is II;, alld C is a 11I'lIrix of('ocnicicllls whose (II, jl dClllellt is (.',,/ =
L'" 1I;"lI
k/.
( I I.:!. 14 )
k=1
Equatiolls (II.:!,I:{) alld (I I.:!. H) dl'lillc a lalclll-varialllt- IIlode! ((,.. n,p(Ttnl cX('css hOlld returns with K lalclIl variahles. E
II.
ir,Il/·Slrllt"{Wi'M(}(idl
("o!'fli("iellls) ;lIltllhc 10k ofllw sial!' variahles ill d!'II'nllining l'X("!'ss hond n'lUrlls (llH'aslIll'd hI' IIJ(' II;" coefficil'llIS). Thc systCtll is particlllariv easy 10 1Il11lnslalld in Ihl' singll'-faClor cas!', IItTt' K == I, WI' CIII drop tht' It sllhsnipls. aud (II.:.!.I·I) 1ll'("Ollll'S
(.',,/
1-:,\\1;\1 ion ( 1 I. ~. I:») SOl\"' ,11;\, <';Ie II row or 11\1' III 011 rix C is pl"Opor\ iOlla 1 '() cae h 01 hn row, and" w ("oef"ficieilis of pJ"OpOri ionality are ratios oflj~ coeflirien ts. Note that the r;\llk of Ihe 1ll
II;"
449
11.2. Fitting Term-SITUcture Models to the Data
Evidence on the Short-Rate Process If one is willing to assume that there is negligible measurement error in the short-term nominal interest rate, then time-series analysis of short-rate behavior may be a useful first step in building a nominal term-slructure model. Chan, Karolyi, LongstalT, and Sanders (1992) estimate a discrete· time model for the short rate of the form YI.I+I - Ylt
ex
+ /3YII + f,+I.
01.2.16)
where
(11.2.17) This specification nests the single-factor models we discllssed in Section 11.1; thc hOlTloskeoastic model has y = 0, while the square-root model has y = 0.5. It also approximates a continuous-time difrusion process for the ins tan taneous short rate r(t) of the form liT = (flo + /31 r)dt + a rY dB. Such a diffusion process nests the major single-factor continuous-time models for lhe short rale. The Vasicek (1977) model, for example, has y :::; 0; the Cox, Ingersoll, and Ross (19B5a) model has y = 0.5, and the Brennan and Schwartz (1979) model has y == 1. 6 Chan et al. (1992) estimate (11.2.16) and (11.2.17) by Generalized Method of Moments. Theydefine an error vector with two elements, the first being YI.I+I - (l + ,B)Ytl-a and the second being (YI.t+1 - (1 + ,B)YII - ex)2 7"' a~y;;. These errors are orthogonal to any instruments known at time t; it. constant and the level of the short rate YII arc used as the instruments. If! monthly data on a one-month Treasury bill rate over the period 1964:6 t~ 19R9: 12, Chan et a!. find that ex and fl are small and often statistically iri· significant. The short-term interest rate is highly persistent so it is hard t6 reject the hypothesis that it follows a random walk. They also find that y is large and precisely estimated. They can reject all models which make y < I, and their unrestricted estimate of y is about 1.5 and almost two standard errors above I. To understand these results, consider the case where ex = fJ = 0, so the short rate is a random walk without drift. Then the error term €'+I ill (11.2.16) isjust the change in the short rate YI.I+I -YII, and (11.2.17) says that the (~xpectation of the squared change in the short rate, E,[(YI.I+I - YII)21 .) 2y F . I 1 iT"YII' .qlllva ent y,
=
(11.2.18) whclI the change in the short rate is scaled by the appropriate power of the short rate, it hecomes hOlfloskeclastic. Figllf(~s 11.1a through d illustrate
SO
"NOI(' how("'('" Ihat (11.2.1 Ii) and (11.2.17) do nol nest th., Pearson-Sun m(~lel.
.".,.
J
I i
r
-
I
r
t
\
{
{ =f
4-
--=:s:::-
?'
~~~--~~--~~--~~--~-
~JI
rol
Mil
til
1111
I 0-
~tt.....
(rI-
'I I-
I I
·... 0
':: II ;:. -.
::'2.
\."
,.-
I ,. ~ ;
j
,- -
Ir:
-II
:>.. -0
11,2, FilljllK '/i,,./II-Slntrlitrf MfI/(rL\ In II,,· (Jt/la
Ihe resulls of Chan et al. by plOl\illg challgl's ill shOl'l raIl'S sl'alcd by various powers or shorl rales, The figures show (YLIt I - .1'I,l/(yi,l for y := 0,05, I, and l.!i, usillg the dala of McCulloch alld Kwoll (I ~'~I:\) OVl'!' IIIl' period I ~1:):2: I to I ~I~ll :~, Over the period sinc!' I ~l(i'l slll ill (1I,~,17), The l'slilllaled value of l.!i lakcs one outside Ihl' Iran;lhll' rLlss or alline-yidd lI\o
';Althollgh llli ... i~ 1!1!lIh,
1101
~ItO\\,11
ill
the: liglllt'~. IIIf' Y
1.:1 lIIodd .11 ... 0 III {".II...:\ dowlI ill lilt"
II. '[;'1"111-.\"1,.,,(/'111' Mlldd\'
a ..e allowt'd. (; .. al' (I ~)~Hi) ('"plon's Ihis possihility hilt ('stilllal("s only slightly lower val lit'S "I y Ihall (:h;lI\ ('1 aI., whil,' Naik and 1.('(' (I~)~\
a/)';;
desnilwd hy (11.~,17), They repin/_ I • a standard (;ARCII(I,f) l1Iodel. Tht'y lind thai a 1II0dd with y = 05 lits the short rate s(· .. ies '1llite well once GARCI-I t'frccts '\IT inclllded in Ihe 1Il001d; h()\~eve .. they do nol explore Ihe illlplications or this Ii, .. hond hond-oplion pricing.
a/
0"
Cmu-Snlilllud UI',\lri,.lilll/.\
(11/
Ihl' '/lnl/ Slmrl/lll'
So br Wl' ha\'(' (,lIIpha~ill'd tilt' tilllt'-snil's implications of aflilll'-yidd lIIodds and hav(' ignored Iheir cross-secliollal implicalions. HrowlI alld Dyhvig (19Hli) and Brown and Schadi'r (I !19'1) lake tltt' opposite approach, ignoring tI\(' IIwdcls' tillle-sl'\'ics implicatiolls and eSlimating all tht' pal'alllt'\('rs frollllhe Icrlll Sll'llCllllt' ofin\(T('sl raIl'S observcd al a poilll in lilll(", If Ihis procedll .. c is repealed 0\'('1' I!lany lilllC periods, il g('n('ralcs a scqllCIIC(" or parameter estilllates whi.h shollid ill theory hl' idl'ntit',,1 for all tillll' \ll'l'iods hilI which ill practi( (. varies 0\'('1' lilliI', Thl' procedllre is analogolls 10 rhe COlIIlIHlIl practice or calculating illlplicd volalility hy ill\'('rting the BlackScho\('s t<JrI\\l\la \lsillg tradcd optioll pric('s; tllne 100 thl' model rcq\\ires Ihal volalilil), 1)(' con~lallr ov('/' Ii 111('. hilI illlplied volatililY tellds to ,\lOVl' OVl'I' ti IIIC, Of ('()\\rSl', hOlld pricillg e!Tors I\\ight calise ('still\al<'d parall\Ctn~ to shift O\'lT Iilll(, ('\'l'1I if rnl<' IIl1dcrl)'i ng paraillerers are COI\S!a1l I. Bllt ill silnl'ie 1I'rIIl-stnlctllre IIIIHids lill'r(' also appear to he SOllll' syslelllalic diffe!'l'I\('('s hI'IW('ell thl' p"rallll·tcr \';!Il1('s Ill'l'lled to lit cross-sectional t''\'III-strllnllre data ali
or
.)
a-
J J.2. FillillK Tmll-S/rllr/lIl'f' Mudd, /0 II" 1)11/11
: : _(_I )~ a~/2 _ (_fI )(J~ l-rp
I-I/l
(11.2. 19)
The first-order autocorrelation of the short rate identifies the autoregressive parametcr I/l. Given I/l. the variance of the short rate then identifies the ill!lo\"ation variance a~. Givell I/l and a~. the average excess return 011 a vcr)' long-term bond. or equivalently the ;lVerage difference between a very long-tcrmfc)rwarcl rate and the short rate. identify the parameter fl. Finally. givell CPo a~. and fl. the mean short rate identifi(~s J1.. III the zero·coupon yield data of McCulloch and Kwon (1993) over the period I ~)52 to 1991. the monthly fir.~t-order autocorrelation of the short ratc is O.9H. implying I/l :::: 0.9H. The stand~\rd deviation of the short rate is :~.(Hi4% at all anllual rate or 0.00255 in natural \lniL~. implying a == 0.00051 ill n
454
II. 1imll-Slmrllll1' M{I(/c·ll
,~' , ...... "',- TIH'OITlkal ;l\'l'r;.tg:t' f(HWarci·...H(· (,lIIV(" "
Sample i;n'cr.lg:e fon"ard-rall'
2
(Urn'
I:!
f.
4
MalurilY in years
I
Figure 11.2.
Smnpll' (1/1(/ Ti'fol1'Ii<(I/ Avrmgr [o'OIwcur/·U"lr
CWW\
de iatioll of the forward rate declines at rale <1>. In the 1952 to I ~l~ll lwrio
=
2
(l -
2
2
2
a It
fl a 1'2) I _ (P
(11.2.20)
11.3. l'rjl'il'K Fix/'c!-I"('(Jllu' I krill/llitlt' Sl'fur;I;f'.\
wh('re IJ is th(' limitinl-; vallie 01' iJ" frolll ('qllatioll (11.1.:.t\). As helill'e, we Gill identilyc,/> = O.!lH from the estimated lirst-ordn alltocorrelatioll ofthc shorl rate, but now lhe olher pOIl'allldel's of the llIudd arc simllltalleously detCl'mined. One call of course eSli1llOlIe Ihe1ll by Ccnerali/.cd Melhod or MOJllenls. The s(luare-ruol model, like lhe homosknlaslic 1II0del, produces an av('ra!-:c forward-rale curvc IhOlI approaches ils aSYlllplO!(' vCI),slowlywhell Ih(' shoJ'l rale is highly persislelll; Ihlls Ihl' lIIodel hOI, lIlany 01' Ihe sallie elllpiriral lilllilations as lhe homoskedaslic IIlOdl'!. III sllllllllary, lhe Sill~lc-racl()r aflille-yidd llIodds WI' haw dl'sl'rihc,1 ill Ihis ch;ljllcr arc 100 restrictive 10 IiI Ih(' hl'hOlvior of nOlllinal interest rates. Th(' Ialelll-variahk struClllre of Ill<' data, Ihe lIalllre of Ihe shOl'l ralc process, aile! the shape of Ihc avera~e lerlll slrllClllrc ar(' all hanl 10 Iii wilh Ih('se models. III responsc to this rcsearchers are "xplorill~ IIlore I-;ell('ral III(Hlels, illrludilll-; aHille-yield models ill whi('h Ihe sill~le slale variahle lilllows a hil-;her-order ARMA process (Backus alld Zill [I !194j), alline-yield models Wilh several state variables (l.olll-;st.d'f a III I Schwan/. [ I!1!l2)), regimes\\:t,hing lIIodels (Gray [19961, Naik and l.ee [1994)), and GARClllllodcls of il!terest rate volatilily (Brcnner, Haljes, and Kroner [199G)), No one 1II0dl'l has yCI emerged as lhe consensus choile liJi' 1Il0delin~ Ihe nominal IeI'm slructure. We nole however lhal Brown and Schadi'r (I !l94) ,lIld Gil>hOlls alld Ramaswamy (199:1) haY<' ;lchiewd SOllIe SUl'U'SS in (ittill~ simple II10dcls 10 prices of UK index-linkcd bonds alld l'conollll'lril' I(,recasts of real retlll'llS on US nOlllinal bonds, Thus sillgle-Ctnor alline-yield lIIodels ilia\, be 1II0re appropriale for lIIodelilll-; r('al illterest ral,'s thanl(II'lIIodelilll-; Illlillinal interest rail'S.
11.3 Pricing Fixed-Income Derivative Securities One of lhe main reasons for the explosioll of interesl in terlll-siruclure models is the practical need to price alld hedge lixed-illcollle derivative securities, In Ihis section we show how terlll-strlll'lllrl' lIIodl'ls call bc used in this context. Section 11.3,1 begins by dis(lissinl-; ways to .IlJl-;lIlelll standard terlll-structure models so that thcy lit the current yield curve exactly. Derivalives traders llsually wanl to take this yield curve as ~ivell, alld so lhey wallt to llSl' a pricing model lhal is fully cOllsis\{'nt wilh all (,lII'renl bond prices. We explain lhe popular approaches of 110 alld Let: (I !IHG), Black, Derlllan, and Toy (1990). and lIeath,.Jarrow, and Molton (1!I!l2). Sectioll 113.2 shows how lerlll-slrlll'lure 1II0dels (';UI he 1IS1'd til pricl' li"ward and IlItllr('s COli tracts 011 fixed-incollle securitics, while Sl'l'lion 11.:1.:1 e)'plon's option pricing ill the context of a lel'ln-SIl'lulIlll' 1110<\1'1.
1.'"
II, J. I Fillillg 1111' (,'II/Trlll '/i'rlll
Slru(/l/If
/':,\'(/(11\,
III g('II('Ld a IlIod,') girl's all ('xact (it to as IIlany data POilllS as it has parallwt('lS, TI\(' hOIlI",I,,('d,lstic sillgk-(;I('(or mod('1 pn'st'llIed ill S('ctioll ILl, (Ill' ('x;lJlJpl(', has 1'0111 P;ILIJII('I('rS, cp, fl. a~, alld II. IlIcvitahly this model does 1I0t fil tI\(' whol(' 1('1111 ,tlll('(I1/(' ('xactly, To allow 1'01' this th(' I'Il1piricalwork of Ih(' pn'\'iou\ s('('(ioJl ad,lt-d ('rror I('1'1 liS, refleclillg Illode! sp('('ili(,;lIioll I'ITor alld 1I\(';\Sun'1I1I'111 I'ITO), ill hOlld pric('s, III pricillg (ixl'd-iIlCOIIIl' dnivative securities it ilia), bl' dl'sir;lhlt- 10 h,l\'(' a IIlodeilital dol'S Iii Ill(' t'\1IT1'llt lel'lll structurl' l'xanly, 'IiI achil'vl' Ihi", I,'l' (,;UI USt' Iltt' /('Slilt or Cox, I 11 gl' rsoll , alld Ross (l9Wla) alld Dybl'i!!; (I~)~~)) that Ollt' ('all add illdq)('IHll'lit tl'rllI-structure lIIodels logelher, A silllplt, approach, dul' ori!!;illally to 110 alld I.ee (IDHIi), is to break oi>s('J'\'cd /'orwald r;llt's '/;" illto Iwo (,OIlIPOII('IIIS:
( 11.:\,1 )
I,;;
wilt'll' is Ihl' 1'<11 ward ralt' illlpli('d hy a slalldard Ira('lahle IIlOdd alld /,~', is Ihl' rl'sidll;11. Thl' r('sidllall'IIIll»III11'1I1 is Ihell attributed to a dl'terminislic 1t'I'III-strtICIU\'(' Inodl'l. Sillc\' ;\ dl'ltTllIillistic proc('ss is illdl'Jlt'lldetll II!' allY slocliaslic IIJ'Oc('s" IiiI' dl'nlllll'"silioll (11.:1.1) is always kgililllat(', '('itnt' is a ('orrespllllding d(,(,Olllposilioll of Ihl' siochastic discolillt b(')or, (I t,:~,:n
III a del('l'IlIillislic 1l10llt-1. Ih(' ahselll't' or arhi I rage r(''1"in's 1h;'1
==
h
YI.lt
II
=
h WIllI'
Tllus IV<' aI'\' l'o'lllbliJlg Ihal I'tlllll'l' slodlastic disCOllll! (~I('\O\'S cOlltaill a tit-Iel'lllillislic COIIlP(I/H'1l1 lital is rell('(')et! ill future SilOrl-lt'l'lli illle\'esl )'alcs alld CilITt'II1 (iI/ward r;tles, Allholl!!;h Illis proc('d II II' \VOl ks w('11 ill allY Ollt' period, IIi ere is nOlhillg 10 (,IlSIl/'(' tltat il will 1)(' (,OIiSiSII'IlI frolll pcriod to p('riod, A typical applicatioll Iitt' a)lpro;)( h S('IS = 0, so Iltal IIt(' CIllTt'll1 short ratt' is Ils('d as .111 illplll illio lilt, st<,,'h'lSlic \('rtll-slrllrllll't' \llodel withoul allY adjllstlllcill (ill' a tll'll'lllIillbli(' (,OIIiPOlH'1l1. Ikll'l'lllillislic ('olll»ollcnis or flltllrl' ~hort rail'S l;,II" ;11'1' 11)('11 'I'I 10 11<1/11<'1'0 I'alll(,s to iii till' tillle I t('rlll SI),IlC\lI)'l',
()r
vi,
\\'h('1l lilll(, I -I- 1/ ;Ini,'('s, h(l\\'l'I'('\', this pro('('dllrl' is r('pealcd; 1l0'" )";,1+,, is ~('Ilo 1('1'0 alld d(,lt'!'lIlillisti(' ('olllpOnl'lllS oflllore dislallt {'utlln' shon r;lI(,s a\'(' llIad(' 1I01l1<'!'O 10 Iii III(' lilll(' I + 1/ /('1'111 stnH'llIrl', As [)yhl'i~ (1~IH~I) ('llIpltasi/('s, Ihis lillI(' illltlllSisl('lln' is lI'OIIhil'sOIlH' althollgh lite procedure IIlal' work \VI'II ror SOl Ill' pllrpOSI'S, It is ;Ilso illlpoll;11I1 10 1II1c1e)'sl'lIullh;1l lilling onl' sci ofassel prices ('x
J J. 3. l'riring Fixl'd-Inromr DPTilinJilJf Sr(unlies Backus, Faresi, and Zin (1996) illustrat e this problem as follows. They SlIllle that the homosk edastic single-f actor model of subsect ion 11.1.1 with a mean-re verting short rate so if! <: 1. They show that one can exactly ,,' lit the current term structur e with a homosk edastic random walk model, a " lognorm al version ofHo and Lee ( 19R6). The model uses equatio n (11.1.5), hili replaces equatio n (11.1.:~) with \ (11.3.4) , where 1';'-1,j is a determi nistic drift term that is specifie d at time ~, for all
fUllIre dates 1+ i in order to fIt the time I term structur e of imerest rates. and as before ~I+j is a normall y distribu ted shock with constan t variance 2 0 • Hackus, Foresi, and Zin (1996) show that this model does not capture the conditio nal means or variance s of future interest rates, and so it mi~prices options on bonds. Problem 11.1 works this out in detail. A somewh at more sophisti cated procedu re for fitting the term struc· ture of interest rates specifies future determi nistic volatilities of short rate tnovem ents, as well as future determi nistic drifts. Black, Derman , and Toy (1990) do this in a lognorm al model for the short rate. In the present morlel one can replace the constan t variance of ~t+j, 0 2 , wilh a det~rmin· i~lic
-
YII1.
(11.3.5)
l I. -limn-Structure MculeLf
458
1
1e expected log excess return on a bond of any maturity over the onep riod interest rate is minus one-half the variance of the log excess return. Now recall the relation between an n-period-ahead I-period lo~ (lI"\vard r~te f., and log bond prices, ~iven as (10.1.8) in Chapter 10: fll' = 1)",fJ:.+l.t. This implics that thc chan~c from timc t to time t + I ill ,I forward rate for an investment to be made attimc t + 11 is
II
(1),,-1.1+1 -1)1I.,+Il- (I)", -1)"+l.tl
f.-1.1+1 - fn,
r".,+1 - rll+I.lfI
( 11.3.6)
T1ki~g expectations of (ll.:~.(j) and using (ll.:t!i), wc find that Et[J.-I.I+1 - /..,] =
(2)
(Var/[r,,+I.t+1 - )1,1 - Var/[r"./+I - )I,l).
(I I.:l.7) The conditional varianccs of futurc cxcess bond rcturns determine the expJcLcd changes in fonvard rates, and these cxpectcd changcs together with tHe current forward-rate curvc dctcrminc thc forward-ratc curvcs and yield curvcs that are cxpected to prevail at every date in the future. Similar properties hold for the risk-adjusted forw
11.3.2 Fonvllrds alld Futures A particularly simplc kind of derivative securilY is a forward COlllracl. Au n-period forward contract, ncgotiatcd at timc t on an undcrlyin~ secmilY with price SI+" at tilllc I + 11, specilies a pricc at which the secmity will hc purchased at time t + n. Thus the forward price, which wc writl' C"" is determilled allillle t but 110 money chan~es hands ulltiltillle 1+ II.N Cox, Ingersoll, and Ross (I ~H 1b) show that thc forward price (;"1 is the tillle t price ofa claim to a payoffof S,+,,/ 1'111 at time t+1!. Equivalently, (;"1 1'''1 "The II-period forward ral,' ddillcd ill S"rlioll 10.1. t o("Chaplcr 10 is Ihe yield 011 a f," w.. ,,1 «IIlIrao 10 huy a 1.ern",0Ilpon hond wilh lIlalllrily dal,· I + /I + I ;1\ lin", I + II.
45!I
II,}, I'ririllK Fixf'lI-IIIWilll' I)nillalilll' Snllrilil'"
is the price 01' a claim to a payorI' or -"/+/1' Intuitively, the 1'111 terms appear because no llIoney need be paid until time 1 + II; thus th(' purchaser of a lonvard contr;lCt has the lise of mOIH'Y (,ctwC\'n I and I + II, (:OX, Inl-:('rsoll, and Ross establish this proposition using a simple arhitragl' argument. They ronsider the fi)lIowing investment strategy: At tilll!' I, take a long positioll in 1/1'", fi)l'ward contracts and put (;", into II-period hOllds. lIy doing this olle can purchase G"t! 1'", bonds. The payoff from this stratch')' at tillle 1+ 1/ IS
( II.:tH)
\Vhere the first terlll is the profit or loss on the f()I'wanl contracts alld the se(olld terlll is the payofl'on the honds. Sill!'e this illvestlll('lll strateh')' costs (;", at time I alld pays ,,"1+11/1'111 at time 1 + II, the proposition is estahlished, It ('all also he stated using stochastic-dis!'oullt-factor 11Otatioll as
G",
=
Et!M""~"S,~,,/I',,,I,
(11.3,9)
wl.cre the n-pl'l'iod stochastic discount f;lctor M",'+II is the product 01' 11 successive olle-pcriod stochastic discoullt ractors: MII,I\II 0= M'+I'.' M'+1I' A futlln's contra(( differs frolll a ((lrward (,Olltr;lLt in one illlportant res]leo: It is I1wrimilo marh/'Ieach period during the Jill' orlhe flllltraCl, so th,lI the (lurchaser ofa futurcs contract receives the rUllIITS pi ire illCl'ease or pays the futures price deCl'C
This call he established using a similar argumelll tu thai of Cox, Ingersoll, and Ross. Consider the following investmcnt stratchT At time I, take a long positioll in 1/1'11 fultlres contracts and put /I", into one-period honds, By doing [his one can purchase 11",/1'11 honds, At time I + I. liquidate the flllllrcs COil tracts. The payofffroJII Ihis strateh,)' at tilllC 1 -1- I is I -I' f /1,,-1,1+1 - /I tl
II",
tl + --
" Plt
11,,_1.1 I,' ---
(11.:\.1 I)
1'1,
!'Ttw Tn·;,:-.ury·holltl.\lul Tn'asury-nuh,' 1,,\\11"\':-' fDIHI,\rh lLH\t',\ nil tht' {:hirilgn l\u.1I"1I01 Tradl' al:-.o han' a tlluuher of spedal option fe.tHlI(':-' tkH ,\((('rt tlwir prlc.:c.'s. A uader with a shOll position can ('hoo~(' 10
c1divcr Oil
allY
day ,\·ithill
thc M'It!(,IIU'1I1 lIloluh
and
Gill dUH)s('
to delin'!" a Humhl',. of "ltl'rll.Hive hOlUb. Tht' shOll tradt· .. ..tI~o h...'\ a "wild card option" 10
delivery al ;1 paniflilar day's sclIl('lIIt'lIl I" in' ,IllY lilllt' ill th ...\jx 1111111." alfel" fh~tt pi ict' is d('('rl1lill('d. Tht, c1i.Ii(II~si()n IU'l"c ah.'HI";l{"l,'i frolll liu'st, oplioll k'HIlIt's; sec' Ilull ( I!'9~t. Chaplt·1' '\) for an illtroduftioJl 10 111('111,
.1I111011lH"t'
11.
'l;"III /-.)/, ."rl"r " /lJlld"['~
whe n' till" lirst 11'1111 is till" IIlar k-to- III;lr ket payl llell t 011 the flit II res cOll tract s (lurd lase cl at till\(" I ;111<1 the SITOlld tcrll l is the payo ff 011 the hOll ds. Becaus e the hllll n's cOll lract s arc IIlar kl'd to Illar ket, the 1~lIlire posi tioll can he liqu idat ed;\ I lillI<' I + I wilh oul g-en erati ng- any flllt her cash 1I00,'S at tillle I + II. Sillc e Ihis illl'I'Stllll'lIt stral< '!W cost s H", at time I alld pays 11,,-1.1,1/1'11 allil lie I I, WI' hal'l ' show n Ihal (11. :\.10 ) hold s. Furt herm ore, we call solv e (113 .10) (())'w ard to tillll ' I + II, IIsillg- thl' fac,t that H II . I +" .'\,+", to obta in
+
=
( 11.:1,12)
COlllparill)!; 1''111atiolls (11.: I.!l) alld (11. :\.12 ), we call S(T Ihat ther l' are sOllie Cirl"lllllsl'IIll'!'S whe rl' forw ard conl racl s alld rUIUrl's COll lract s wrilt l'lI 011 Ihc' salll l' untll 'll}'i llg- ;lsS!'1 witll Ihe sam e llIat urilY hal'! ' l"lua l pric es. First , ifho nclp rilT s are 1101 ralld ollli hell ahse ncl' ofar hilra g-e rl'ql lires thai 1'", == n:'~-III/)I,'I" so (;", = If"" This lIlea ns that forw ard and flllll res pril' es ;Irl' eqll al in allY Illod cl with a cOll stan t intCJ'est rate . SCl' olld, if ther e is only Ollt' pni od 10 lIIal lll'ily Ihl'n I'", == I'll anel ag-ai ll (;", :=: II",. Silln ' flliu res COlllral'lS arl' lIlar kl'ti 10 llIar ket daily Ihl' peri od hl'rl ' 1I1IIst hI' 0111'
sillce
1'111
= E, ( ,,1
show Ihat II", =
11 ,1 f II
I. l j nc\cr Ihe saille fouciilioJls
I' if 11,,_ 1,'/1
III r ::::: V, alld w(' ,.. an
\' bl'Cl IlSe (11.:1.10) hl'l'Olll('S
(11.:1,14) Thll s /I", == \' lor all II, so !(II'w ard ;lIld flllll r('s pric ('s ar(' cqlla \. Mor e gC'I)('1 allv, It/M cITr , forw ard alld fllltl iTS pric cs may dille r. III the case wllC'rl' Ihe lInde rlyin g- assc t is all 1/ + T-pe riod zero -cot lpon hon d at Ii Ill«' I, whic h will hc a r-l'( 'Jiod hon d al lilll! ' 1+ 11, we can wril( ' Ih(' ()rw are! pric (' as (;, "I alld lh(' fllill res pric t' as 1/"" , The forw ard pric (' is easy 10 calc ulal e ill Ihis case :
Whe ll r = I IIII' I'iell! OIl Ihis for"';1J'(1 COlllraCI is Ihc I()\,w anl rate defi n('d in Scct ion 10,1. I of Cha pin 10: [':,, == II (;1 ",, The fllltl res pri, I' IIIIISI hc calc llial eci r(,l"llrsil'c1y froll l t''1I1;lIioll (I I .:1.10). III a l'arli cliLI I I('I'IIISI 1'11('\ 11)(' lIlOc ld OliC (';111 do Ihe calc lllal ioll t'xpl icill y alld sol\' (' 1'01' I\)t' n·\;t lioll ht'lw l'('n !(,rw ard alld fillll res pric es.
II. J. I'ri(illg
Fixnl-I"«l/II~ IJn;vlllivt' .'il"l"/lr;tit'.~
Problem 11.2 is to do this filr the hOllloskedastic single-factor model opec! in Section 11.1.1. The problem is to show that the ratio of ~ to futures prices is constant in that Illodel. and that it exceeds one so lhal forward prices are always greater than futures prices. 11.3.3 Oll/ioll 1'n"riIlK ill
1/
Tmll-S/niC/IlTI' Modf'l
Suppme one wants to price a European call option written on an underlying se("\lrity with pdce S,.IH Irthe option has II periods to expiration and exercise price X. then its terlllinal payoff is Max(S,+ II-X. 0). It can be priced like any other II-period ,lsset !Ising the T/-perioo stochastic discount factor Mil.,+," A'I'TI ... M'+II' Writing the option price as C'",(X). we have (;,,,(.'<)
==
E,[MII.'+II Max('\;+11 - X.O»)
i I
E,[M".,+"S,+"
S'h > X)
\
-X E,[MII .,+"
.11,+" > X).
(I 1.,.16)
III general equation (11.3.16) lIIust be evaluated using numerical rriethoels. bllt it simplifies dramatically in one special case. Suppose that M~.I+II aile! S,+" art' jointly lognormal conditional on time I information, with ~on elitionalexpectations of their logs /lm and /l" and conditional variances~nd covariance of their logs am"" a". and am" All these moments may depFnd on I and II, hut we suppress this for notational simplicity. Then we have E,[M".,tll S'tll
I
exp ( 11", x
(
S'+II > X)
+ /1. +
allllll
+
a"2 + 2a,.,)
/l.+a ,,-x +a, ) . ll
a.
I I
•
(11.3.17)
alld E,IMnl+ n I S,+"
~
XJ
==
exp(ll",+a~II/)(Jl'+:~"-X).
(11.3.18)
when' <1>(.) is the cUlllulative elistriblltioll function of a standard normal ralldoJll variahk. anel x == log(X). I I EqlJations (11.3.17) and (113.1 Il) hold for any lognormal random variahles M and S and do nol derencI 011 any otllt'r rroperties ofth('se varia hIes.
",,'<1
1"'1'1. .. "ol;o,i"" lin,' di/l'·I.< 1'1'0"1 'til' l"I';llioll ill (:hal'l .. r ~J. Tlrrn' /', i, list·" for II", hili IIl'r" we 1"( .... (.1"\"(. /' rill' 1('&"fH:ollpon hond prin's ~lIHllI~(·.\ fur a J..:,C'IIC" if' '''("(,lIdly pdn" t tTl ....,,· ... ·'''Ih w.. r",,,"';,,,,,, It\' R"himi<"i" ( I!Iili): "',' also 11,,;0"1( allli l.il/ .. "I>'·I'I(,·r (J 'JXlI). IIlld(,II~'illg !'Ic'nll ily prirt',
I
'4'i2
II. ·1i'I'II/·SlnHIIIIl'M"dd,
nut W(' kllow frolll asset jlril-in):: 1IIl'IIry 111011 11ll' ulHkrlying s'Tllrily pri, (" S, I"IIISI salisfy
I Sf
~
==
.
,
(
== ('xp /1",+/1,+ rr""" + a"2 + 2a",,) '
1,·tlt\llf,f-I"·\f~,,1
(11.:~,I'I)
We also kilo\\, Ihal Ille pi ice of an II-pel iod lero-coupon hond, I'"f. III1ISI ';lIisIY
I
I'''f
=
EtlM".f+,,1
== eXP(ll",+
a~",).
(11.:~.20)
I
Usillg (11.3.1 !)) alld (11.:t20) losilllplify (11.:\.17) alld (11.:U H), ;1\111 slIhSliIlIlillg inlo (11.:~.!(i), we gel all expression for Ihe price ofa call ol'lioll whell Ihe IInderlyillg securily isjoilll\Y \ogllol'lna\ wilh Ille Illllhipniod stochaslic dis(,Ollllt I;IClor:
, (/I,+a",,-.\'+a,,)
.'i,
,fiT:.
"
- XI",
(Il,+a",.-x) ~ " a \\
. (s, - .\'-/1", +a,,/'2)
'\f
- XI
r;:yyaH
,
Ifl
("f - X-/1", r;:y-
0,,/2)
(11.:1.21 )
.
" (1"
To get Ihe sialldard oplioll pricillg IClI'IlIu\a of Hlack and Sci" lieS ( I !17:1),
we lIe(~d two fllnhel' asslIlI'IlIions. Firsl, assl\lII"lhallhe cOlldilion;" \,;lri;I\\('(' of Ihe ullderlying securilY price /I periwis ahead, a,,, is prop"rtioll,,1 III II: au = lIa~ for sOllie COllslall1 a~. Secolld, assullle Ihal Ihe lerll1 slruclure
=
is lIal so Ihat I'lff 1'-111 Ie)!' sOllie conslalll inleresl rail' I'. Wilh tllese ,l
.,
.
<'",( X) == .\,
""I"A
.,,\('
,,>
(\,-x+(/'+a~/2)1I) r::
ylla
(,,-x+(/,-rr 1/ 2)1I) .J/i rr
.
( /1.:1.22)
For fixed-illcollle deri\'alives, however, Ihl' extra asslIillpliollS 1I('('deli 10 I{el the UhlCk-SdlO\cS IC)l'Inula (11.:1.22) arc nol reasonahle. SIIPPOs(' lltal Ihe asset 011 which Ihe call oplioll is wrillcil is a I.ero-coupoll I>olld II'hicl, cUlTelltly has 1/+ r p('riods 10 malurily. IfillI' oplioll has ext'l'cis(' prin' X alld l~jl)('riOtiS 10 expir.1I iOll, III,' Opl ion's 11OI),o1LIl "X piral ion willII(' ~101 x( I'r.' I 1 / I:!Of f(}I1I'~(". for allY gi\('11 " we rail ~11\\';,y., cldillt' n:! ::::::; n,./ II alld I ::::: - {,,,,/,, '" Ih.1I II (' l\1~IC·k·SdlOlt's lonl1111;aOlpplit·s lor Iha, II. Tht, ;l~.'lIl11lJ(i()lI~ gi\'('u ;11(' IU'nll'fllor,lit' HI.,d.. ~ S; I!O!c',1Iii 101'1II111a tu apply 10 all II wilh 111(' 'i..III)(, , ;1IIc1 n:!. I
II.J. 1'l'il'illJ; Fbml-III((I/IU' Dmlllllil". Sl'l'lIl'ilil',l
41i~
.\,0). TII{' r{'\evant hOlld prin' at expiLII inll is Ihl' T -p,'1 iot! bont! prin' sillt'(, Ihl' Ill
Var,[-A, -Ii, x,\,,1
CT" .
/J~ Var, [x, f,,1 =
(1-t/>')~CT~(I-tP~")
.,
l'
(I-¢)-(I-,p')
(11.3.23)
This expression for a" docs not grow linearly with II. I knte if one uses the Black-Scholes j(mllula (11.3,22) and cairlllatl's illlj)licd volatility. the illlplied vol'ltility will depend 011 the lIl;lllll'ity uflhe option; therl' willlw a trIm ,Iimrlllrp oj im/)iil'li VOilllilii.~ Ihal will depend 011 Ihl' parameters of Ihe IIndl'rIving Il'rlll- structllre llIodl'l. JallIshidian (1 !lH!l) prl'scllts a cOlltinuous-limc versioll of Ihis result, and Turnhllll alld Milnc (I !1!H) <JeriV<' it in disCTctl' tillle alollf,?; wilh nUlllcrous rcsults I<JI' <JtitCI' typl'S of' l!cri"ativl' securilies, Oplioll pricill~ is cOllsiderahly 1II0Il' dillintll ill ;\ ~;q\larl'-ro()1 model. hlll Cux, Ingersoll. and Ross (1985a) present sOllie lISdll1 rl'sllIL~. Invl'stml'nl professionals orten wallt to pricr optiolls in a way that is exactly consistent with the CllrrCllttenll strllrlllre of interest rates. Tt) do this. we can hreak Ihe II-pcriod stochastic disCOlltll {;Irtor illtu tWI) cOlllpollenL~: (I
U.24)
where. as ill Sectiol\ 11.3,1. the ll-nll\\pOI\I'11I i, sll)('h
(;",(:\) == E,[M;:.I+ft M::. II " Max(/';~" " 1';',1+
M::.I+>I p;.,+)·:/l 1'.1;:,/+11 1'"n+r.1 ('" (XII'"r./t-n ). '",
>I
M,IX (/';', 1+ "
-
-
X, 0) 1
XI I':',ff ",0) J
(I1.:t2!i)
where {.''' is the cdl optioll pricc that would prevail ifthe stochastic discount brlor weI"!' M". III othn words optiolls c;ln he priced using the stochaslic lertll-slnll'tlll"!' IIIOd..!, using Ihe delermillistic model only to a(Ullst Ihe exercise pritT alld 1111" filial solulion ((II' the oplion price, This approach was firsl IIsed hy 110 ;llId Lee (I (IH(i); however as nyhvig (19WI) POilllS OUI, 110 ;uull.(T (hoOSI' ;1' t1wif II-IIIOdd Ilrl' Sillgk-f;\l"lOf 1r00\Ioskedaslic 1II0del wilh t/> = I, \"lrich Ira, 1I1111}('IOIIS unappealillg properties, Black, lklmall, alld Toy (1~1!IO), Ilealh, .larrow, and Morton (1992), and 111111 and \\'hile (I!l!/Oa) IIS(, similar ;lpplOaciws wilh differenl choice.~ fill' Ihe a-lIIodel.
I 1.4 Conclusion In Ihis chapler liT ha\'(' thorollghly I'xplorcd a Iraclahle class of illttTestrale models, Ihl' so-called affine-yield models. In Ihes(' models log hond yidds arc lincar in slate variables, which simplifies Ihe .nlalysis of Ihe term slructure ofilllcresl raIl'S and offix(,d-illcoll\e derivalive securilies. We have als"o S('('II Ihat affilw-yil'ld IlIo(ld, have SOUl(' limilaliollS, particlliarly in desnihing thl' dynamics or the short-term nominal interesl rate. Th('l"(' is 'Iccordillgll' I-(reat int('l'esl ill del'dopilll-( more f1exihle models Ihal allo\\' for slich pheIlOI\\('II;1 as 1I11111ipic 1'('l-(illll'S, nonlinear nwan-n'l'ersioll, and serially corrd;tled illl('l'esl-rale l'olatililY, and thaI fully exploil IIII' informatioll ill Ihe yidd 1'1Ir\'('. A~ Ihe lel'lll-slrll(llire lill'rallllT \IIoves forward, il will he imporlanl 10 intel-(r.\I(' il with Ihe I'esl or the asset pricilll-( lileralllrc. WI' have secn thaI lenll-sll'll('\IIJ'(' lIIodds can 1)(' dewed as lilllc-series models filr the sl')chastic discollllt bctol', The research on stock retllrns discllsscd ill Chaplel H also sl'('k.~ (0 chara('\('ri/e Ihe hehavior of Ihe slochastic disco\lnl f;lclor. Ry COlllhining lilt' inlimll;!tioll in til(' prices of stocks and lixed-illcolI\e securities it sholllcll)(' possihle 10 I-(aill a hel\er IInderstanding of Ihe economic forces that d('tennil\(' Ihe prices of financial assets,
Problems-Chapter J 1 11.1 AsslllIll' Ihallhe hOllloskl'(bstic IOl-(nol'lnal I>ond pricing llIodel giV('1I h)' eqllatioJ\s (I I. I.:\) alld (11.1.:») holds wilh rp < I. 11.1.1 SIiPPOSI' VOlllillhc C\ll'rl'lIllerlll structure of ill (t'res( rates Il~illl-( a randolll w;rlk lIIodd ;\lIgllll'lIted by I\t-terrllillis(ir drift ('\'IllS, clju;lIioll ( 11.:~.4). Ileri\'(' all ex pn'ssioll rdat illl-( the drift terllls 10 Ihe stale \'ariahle .\', and the paranlt'll'Is pf lilt' trill' I>olld pricing Iliodd.
11.1.2 (:OIIlP;III' Ill<' 1"'1)('CII'<1 1'111111'1' 101-( short raIl'S illlplil'd hy tl\(' Il'It(' hOlld pricillg Iliodd ,,"e1111t' ""ldwnwalk luodel wilh detel'lllillislic drifts.
465 11.1.3 Compare the time I conditional variances of log bond pryces al limc I + I implied by the true bOlld priciIlI{ model and the random walk l1lodel with deterministic drifts. ' I
1l.1.4 Compare the prices of bond optiollS illlplierl hy the true! bond I pricillg Illodel and the random walk lIlodel with deterministic drifis. Note: This questioll is based
011
fiackus and Zin (1994).
,
11.2 Define (;'"1 to be the price at tilllc I of all II-period forward contract on a zero-coupon bond which matures at time 1+ n + r. Define H'"1 to he the price at time I of all n-period futures contract on the same zero-coupon hond. AsslIlIle that the hOIll()skedastic single-factor term-structure model of Sec t iOIl I 1.1.1 holds. 11.2.1 Show that both the log forward price g'"1 and the log futures price It, ", are affine in the state variable x,. Solve for the coefficients determining these prices as functions of the term-structllre coefficients All and Jill' 11.2.2 Show that the ratio of forward to futures prices is constant and gn'atcr thall one. Give some economic intuition for this result. 11.2.3 For the parameter values in Section 11.2.2, plot the ratio of forward prices 10 futures prices as a function of maturity n. Note: This question is based on fiackus, Foresi, and Zin (\996).
12 Nonlinearities in Financial Data
TilE ECONOMETRIC METHODS we discuss ill this tt~xt an~ allllost all designed to detcct linrar structure in finam:ial data. In Chapter 2, for example, we develop lime-series tests for predictability of asset relllrllS that use weighted ccmbin,lliuns of return autocorrclations-linear predictability is the fucus. The event study uf Chapter 4, and the (,APM and APT uf Chapters 5 and 6, arc based on linear models of expected returns. And even when we broaden our foclls inlaler chapters to include other economic variables slIch as consumption, dividends, and interest rates, the lIlodcls remain linear. This emphasis on linearity should not be too surprising since many of the economic models lhal drive financial econometrics are linear models. However, many aspects of economic hehavior llIay not he linear. Experimental evidence and casual introspection suggest that investor's' alii tudes towards risk and expected return are nonlinear. The lenns of many lill
12, NOI/ /iI/t'l /rilit' J il/ Fil/I /I/ril /II ),1/1/
ical systc llls Ihco ry. lIoll lille ar lillH"-s('ries allal ysis, stoc hast ic-\'o latili lY mod els, lIon para lllel ric slali slics , aile! artil ieial ncu ral netw orks have fu('l ed lhl' 1'('(,(' 111 illl(' r('sl ill lIoll lill(' arili ('s ill fina ncia l dala , alld wc shal l expl oce each of Ih('s (' topi cs ill th(' follo wing sedi ollS , Sect ioll I ~,I r!'l'i ,its SOIll!' of lhc isslI('s raisc d in Cha pler ~ rcga rdin g pr('diclOlhilil\,. hill hOlIl ;1 lill!'; II-\'t"rSIiS-lIolllill('ar pnsp ('cti \'(', \Vl' pr('s cill a laxOIIOIII)' of IllO dds Ihal dislill)!;l Iisli('s h('lw eell mod els llial arc Ilon lill(' ar ill III ('a II alld 11<'111'(' d('pa l'l froll llli( ' lIlal 'ling ai!' hypO lh!'s is. alld mod els Ihal arc lIoli lilic ar in "OIriOlIl(,(' alld II<'IICC depa l'l froll l indq H'lI dl'IH T hili nol froll l lh!' llIal'lill)!;ait- h\'POllI('s is, S('('( ioll I~,~ ('xp lm(' s ill gr(' aln d(,ta il IIlOd('ls thaI arc Iloll lilic ar ill \'ari ancl '. incl ildil lg IIl1i\'arial(' and Illui tivar iat(' C(,l lcra lilcd AIIlOJ'(')!;rc\sil'(' COl lditi ollal ly IlclcJ'()skedOlstie ( ;ARCII) and stoc hast ic-v olati lity lIlod els, In S('ct iolls I :!,:I alld I~..I wc nIOl '(' beyo nd para lJl(, tric tillle -s('ri cs llIod e1s to ('xp lor(' nOli para lll!'l ric lIl('t hods for fittin g non lille ar r('la tioll ship s 1)('IW('('1I \'aria lll!'s , illdu dillg Sillo othi llg tech lliqu es and artif icial IlI'lI lal lI!'tw orks , Alth ou)! ;h Ih('s(' t('ch lliqu !'s arc able [0 llllCOI'('r a \'ari ety of nOIllin!' arili !'s, thn' ;rr<' IIl'OI"ill' dOll a-d!'(l!,lId!'nt alld com puta tioll ally illll'IISil'!', 'Ii) illlls lralc th!' POIl'('I' oftll ('S!' l!'ch lliql l!'S, we pres !'lIt all appl icati oll 10 lhe (llicill)!; .111(1 h(,d ging of dnil'OIlil '!' sl'cl lrilie s OIlId 10 eSlil llali llg Slal!'-pi ic(' dell sili! ", W!' .!lso disc llss SOIlI(' of III<' lilllilOiliolls of Ihes e lech lliqu l's ill S!'clioll I ':2,:1. The IlIosl illl!, ollal il lilllilOiliolls al(, the lwill proh lem s of ol'er filtillg alld data -sllo opil lg. whic h plagll<' line ar mod els too but 1I0t lIear ly 10 the salll !' c\eg n'e, Ullfill'lIIl1atc ly, 11'(' have I'cry lillie to say abo ut holl' te deal lI'ith thcs c issue s exce pt ill vel')' sp!'c ial case s, hCll ce this is all area with 1l1:11lY ope n rese arch qU(, Slio m to 1)(' allswl'!'I'C\.
12.1 Non linc ar Stru ctur c in Uni vari ate Tim e Seri es A lypi ct\li lllt'- snie s lIlod d relat es all obse rvcd tilll(, serie s x,IO all ulld erly ing S('II'II'II('(' of shoc ks f" III lilll'f lr tillll '-ser ies allal ysis the shoc ks an' assu lIled 1(1) (' IIIH'OITdall'ci hili arl' 1101 1Il'I,(,~"II'ily aSSllllll'd 10 Iw lID, By lh(' \Vol d i{epn'sl'IlIOIlioll TIH'Of('1I1 all\, tilll!' snil 's call he writ tcn as all infil lile- orde r linl' ar IIlOl'ill)!; 'lI'I'!'a)!;1' of slIch shoc ks, alld Ihis lilll' al' mov illgav('r age r('pres! 'nlal ion SllllllllOlri/c's tilt' llllc olHl ilion al vari ance and auto cova riall C<'s of th(' snic s, III IIOlll il/l'I1 I tilll! '-sni ('s all;i1vsis tIl!' IlIll krly ing ,~hocks ar(' typic ;i1ly asSllllH'd to hc 111>,11111 W(' s('ck a pms illly non line ar fUllc tioll r(,la ting tile scri(',~ XI to tlie histo ry of lire shoc ks, ,\ gl'lH'rOlI r(,pr esen tatio n is (I ~,I.I)
I
I'
J2. J. Nonlinf(lr Structure in Ullillflrillif Time SflifS
where the shocks are assumed to have mean zero and unit variance, an~ f(·) is some unknown function. The generality of this representation makes it very hard to work with-most models used in practice fall into a somewhat lllore restricted c\a~s that can be written as
i
(12.1.2) The function g(.) represents the mean of XI conditional on past information, since E'_I [x, 1 = g(E '-I. f,_~ •... ). The innovation in XI is proportionallo the shock fl' where the coefficient of proportionality is the functiQn h(·). The sCJlIare of this function is the variance of X, conditional on past information, since r.1_1 (xI-Et-I [X,])2) = h(fl_l. f l-2 •. ' .)2. Models with nonlinear gO are said to be nonlinear in mf(ln, whereas models with nonlinear h(.)2 arc said to be nonlinear in variance. To t.lIlder~tand t~e restrictions ~mposed by (12.1.2) on.( 12.1.1), coryider expandlllg (I 2. l.l ) III a Taylor senes around fl 0 for given (I-I. (1-'2' ••• :
=
XI
=
/(O,ft-l .... )+flji(O.f/_I .... )
(12.1.3) where Ji is lh~ derivative of J with respect to fl' its first argument; iii is the second derivatiw of / with respect to f,; and so forth. To obtain (12.1.2), we drop the higher-order terms in the Taylor expansion and set g(fl_I •... ) = /(0. EI_I .... ) and h(f/_I .... ) = ji(O, (I-I>' .. ). By dropping higher-order tenllS we link the time-variation in the higher conditional moments of XI inflexibly with the time-variation in the second conditional moment of x" since for all powers p~2, E/_I[(XI - E/_I(x,])PJ :::: h(.)PE[€fJ. Those who arc interested primarily in the first two conditional moments of XI regard this restriction as a price worth paying for the greater tractability of (12.1.2). ECJuation (12.1.2) leads to a natural division in the nonlinear timeseries literature between models of the conditional mean gO and models of the conditional variance h(-)2. Most time-series models concentrate on one form of nonlinearity or the other. A simple nonlinear moving-average model, for example, takes the form (12.1.4) Here f.{(') = cui_I and hO = 1. This model is nonlinear in mean but not in variance. The first-order Autoregressive Conditionally Heteroskedastic (ARCH) model of Engle (19R2), on the other hand, takes the form
(12.1.5) Here g(.) = 0 and h(·) not in Illean.
= j(U;_I'
This model is nonlinear in variance but
.;~;:~~.
J2. NOlllinearitieJ ill Fil/flllrifl/IJtl/1l
470
Onc way to understand thc distinction bctwecn nonlincarity in lIlean and nonlinearity in variancc is to considcr thc moments of thc x, proccss. A~ we havc cmphasizcd, nonlinear Jllodels can be constructed so that second moments (autocovarianccs) E[x, x,_;] arc all zero for i>O. III the two cxamplcs abovc it is easy to conlirm that this is thc casc provided that f., is symmetrically distributcd, i.c., its third momcnt is zen>. For thc nonlincar moving averagc (12.1.4), for examplc. we have Elx, X,_I] == E[ Cf.,+af.;_1 )(15,-1 +af.;_2) J == aE[f!.:-ll = 0 whcn EI f.~_1 J=O. Now considcr thc bchavior of highcr momcnts of thc form
Models that arc nonlincar in the mcan allow these higher 1ll0mCIllS to he nonlcro whcn i, j. k, ... >0. Models that arc nonlincar in variance bllt obey thc martingalc propcrty havc E(x, I X,-I •.. . J=O, so their highcr momcills arc lCro whcn i. j. k • ... >0. These models Gill only havc nonzero higher momcnts if at Icast onc timc lag indcx i. j. h• ... is 1.Cro. In thc nonlinearmoving-avcragc cxamplc, (\ 2.1.4). the third momcnt with i= j= I, E[ (f,+af ;_I)(f,-I +af;_t)2 J 2 aEk/_ 11+2a E[f~2J Elf;_11
i' O.
In} tc nrst-o~dcr ~CH cxample. (12.1.5); the same third momellt [[x, x;_11
== ·(f.,Jafi_l)fi_Iaf.~21 =
o.
fillt for this model the fOllrth IIlOlllent with
i= ,j=k=l, E(x; x;_.1 = Elf; at f.:_ 1f;_21 i' O. Wc discuss ARCH and other models of changing variance in Sectioll 12.2; for ;the remainder of this scction we concentratc on nonlinear models or thc Iconditional mean. In Section 12.1.1 we explore scveral alternative ways to ~arametrize nonlinear models, and in Section 12.1.2 wc usc thcse parame ric models to motivate and explain somc COllllTlonly IIsed tests for nonlin arity in univariatc time series, including thc test of Brock, Dechert, and Sch inkman (1987).
I
J2. J. I S011lr I'llram~tri(: /I1otidl
impossible to provide an exhaustive aCCOllill of all nonlinear specilicati ns, cven when we restrict ollr alleillioll to the slIbset of parametric 1II01.els. Priestlcy (1988), Ter;isvirta, 'lj0stheilll, and Granger (\ 9!14). and Tong (1990) provide excellcnt coveragc of mailY of thc lIIost poplllar 11011Iinellr time-series lIIodels, including IllOre-speciali/.ed lIIodels with sOllie very intrigUing names, c .g., .v/frxritillg Ihrr.~/lOltllllltort'gTr.\.\i(m (SETAl{) , flm/J/ill/tirdrJJmcll'1li I'x/Jonential a utoTrwe.uio 11 (EXPAR), and Jttltr-dr/Jenlil'1lt "'f}(lrll (SDM). To provide a sense of the hreadth of this area, we discllss fOllr examples in
.17\
12. I. NtmJil/i'l/r S/rurllHl' in (/lIil>lIIill/1' Timl' Sl'ril'.1
this sectioll: polYIIOIlliallllodels, pic(Twisc-lillear lII(xlds, Illodels, alld deterministic rhaotir lllOllels.
Mark()v-llwitchill~
['o/)'Ill/millt Mor/pLf Om' way to reprcselll the fUllction g(.) is cxp.\l\d it ill .1 'bylor scrics arollnd (/_I=(/_~='" =0, which yields a
"-'
'"X.;
LII/(-1 1+ LL",,(I"('I', ,;::::1
I
"-'
'"X..:
I
r'l
.......,
(1~.l.Ii)
+LLL("kf/.,fl ,(I k+· ... 1=1 F' k=,
SUIllIll.ltioll in (12.I.G) is a slalldanl lillcar Illovill~ average, the sUllllllatioll captllres thc cITects oflag~ed cross-prodllcts ortwo innovatiolls, the triple slllllmatioll capturcs the eCkels of la~~ed cross-products of (hree innovations, and so 011. The sUllllllatiolls indexe(1 by j st.art at i, the SlIllllllatiollS indexed by k start at j, and so on to avoid cOllntill~ a ~iven cross-product of innovatiolls /IIore thall OIHT. The idea is (0 represcllt the tnit' nonlinear function of past innovations as a weighted SUIII of polynomial runnions of (he innovations. Eqllatioll (I ~.I.1) is a Silllple example of a 1I'lOdei of this fonll. Robinson (1979) and Priestley (I~)HH) m,lke o:tensive II,,' of this specification. Polynomial models lIIay also hc writ tell in autoregressive «»"III. The fUllction g-(fl_l,
TIl('
sill~le
dO~lble
0V
"-
'>.!
LIl~XI-I+
LLI';,xl-,xl._,
1;;:;1
1=1 j:.-:/ no
+L ;=1
ro
ro
,=,
k~)
LL
(~k X,_,
XI_,
X'-k + ....
( 1~.1.7)
It is also possible to ohtailllllixed autore~ressive/ll1()villfi-avna~e representatiolls, the nonlinear equivalent of ARM/\ iIlodds. Thl' hili\lear lIloclel, for example. IIses la~g('d vailles of XI. I.. g~ed v;t1ll(,S of f " and (Toss·products or illl' 111'0:
""" La,E /_;+ Lfi,xl "'-
1=1
1=1
-t-
"'v
"-
,=I
,= I
L Lv" XI_,f /·
l•
(1~.l.H)
} 2, NOI/!iI/t'l/rilil'v il/ Fil/tll/ritll
})1I111
This lIlodel call capt 111'1' lIolllilll'arities parsillloniously (with a fillite, short lal-( 1{'lIl-(th) II'hl'lI PUI'(' lIolllille;II' lIlovilll-(-;\Vcral-(' or Ilonlinl'ar alltor('~rl's siVl' lIlodds f;lillo 110 so, (;rangl'l' ant! Andersen (197H) anti Suhha K;'o ;mel Cabr (1!IH'I) ex pIon' bilillear lIIodels ill delail. l'il't'l7l1i.wl.i 111'11,. II/oddl Allothn popllLlr \\';\\' to lit tlotllitlt'
if XI-l < Ir
(l ~,1.9)
if X,_I > Ie. lien' the illtenqll
lr,~
fir
it' .\,
+ fl~ x, + f~,
if v,
-I /ll_\, _ -t-
=0
(1~,I.I()}
12.1. Nonlinear Structure in Univariate Time Series
473
where 5, is an unobservable two-state Markov chain with some transition probability matrix P. Note the slightly different timing convention in (12.1.10): 5, determines the regime at time t, not 5,_1. In both regimes, x, is all AR( I), but the parameters (including the variance of the error term) differ across regimes, and the change in regime is stochastic and possibly serially correlated. This model has obvious appeal from an economic perspective. Changes ill regime are caused by factors other than the series we are currently modeling (5, determines the regime, not x,), rarely do we know which regime we are in (5, is unobservable), but after the fact we can often identify which regime we were in with some degree of confidence (5, can be estimated, via lIamilton's [1989] filtering process). Moreover, the Markov-switching model does not suffer from some of the statistical biases that models of structural breaks do; the regime shifL~ are "identified" by the interaction between the data and the Markov chain, not by a priori inspection of the data. Hamilton's (1989) application to business cycles is an excellent illustration of the power and scope of this technique.
iMrnllilli-ltir Nonlinear Dynamical.'>~Y5terllJ Thnt' have been many exciting recenl advances in modeling deterministic noniiTll'{lrtiynamicaisysterns, and these have motivated a number of techniques for estimating nonlinear relationships. Relatively simple systems of ordinary (Iifferential and difference equations have been shown to exhibit extremely complex dynamics. The popular term for such complexity is the Butterfly FJFrt, the notion that "a flap ofa butterfly's wings in Brazil sets offa tornado in Texas".1 This refers, only halfjokingly, to the following simple system of deterministic ordinary diflerential equations proposed by Lorenz (1963) for modeling weather patterns:
X
10(y - x)
(12.1.11)
y =
xz + 28x - y 8 xy + '3z.
(l2.Q2)
i:
=
i
(12.1.13)
I.orenz (1963) observed that even the slightest change in the starting values of this system-in the fourth decimal place, for example-produces dramatically different sample paths, even after only a short time. This sensitivity to initial conditions is a hallmark of the emerging field of chaos theory.
•
IThis b adaplrd from Iht' lilk of Edward l.orellz's addr..", to the American Association for Ihe Advancement of Science in Washilll{ton, D.C., December 1979. See Gleick (191l7) for a livl'ly and t'ntertaining layman's account of the emerging science of nonlinear dynamical systellls. or chaos theory.
474
12. Nonlinearities ill Hlumrial Data C!
co
o· <0
0
><
...
0
""0
~I
0.2
0
U.4
0,(;
1.11
O.H
!
1
I
Figure 12.1.
'O,e Ten/ Mal'
An even simpler example ofa chaotic system is the well-known telllmap: X,
=
if X,_I < ~ if X,_I > ~
-,\, E
(0, I).
(1~.1.11)
Th' tent map can be viewed as a first-order thrcshold autoregression with no shock f, and with parameters (11=0, f31=2, (12=2, and f32= -~. If X,_I Iie~ between 0 and I, X, also lics in this intcrval; thus thc telll map maps thc unIt interval back into itself as illustrated in Figure 12.1. Data generated by k12.1.14) appear random in that they are uniformly distributed on the unit interval and arc serially uncorrclated. Moreover, the data also exhibit selisitive dependcncc to initial conditions, which will bc vcrificd in Prohlcm 12.1. Hsieh (1991) prcsents several other leading examples, while Brock (1986), Holdcn (1986), and Thompson and Stewart (1986) provide lI10re formal di.~cussions of the mathematics of chaotic systems. Although thc JIIany important breakthroughs in nonlinear dynamical systems do have immediate implications for physics, biology, aIHI other "hard" scienccs, the impact 011 econoJllics and finance has been less dramatic. While a number of economic applicatiollS have bcen cOIISiliercd,2 nonc arc especially compelling, particularly fmm an empirical pnspenivc. 2&e, for example, Boldrill alld WoodlOld (1990), lI,ork alld Sayers (19HH), (:raig. I\ohl.lw. and Papdl (1991). Day (19H:I). (;',111(11110111 alld Mal!(,allt(· (19M!;). Ihidl (199:1). 1\('11"""
12.1. NOlllilll'llr Sirtlrillll' iii Ultillllli(//t' Tilllt, Snit'.1
There are two serious problems ill IlH)(ldillg I'COIlOlllic plu'lloml'lla as detennillistic Ilolllillear dynamical systems. First, unlike the theory that is available in mallY natural sciences, ecolloillic theory is gt'lllTally Ilot specific about fUllctional forms. Thus ecollomists rarely have theoreticil reasons for expectillg to find one form of nOll linearity r;llhlT thall allother. Second, economists arc rarely ahle to t:onduct controlled experiments, and this Inakes it almost impossihle to deduce the par;lIlH'tlTS or a deterministic dynamical system governing econolllic phenomena. eve II ifsuch a system exists and is low-dimellsional. When controllnl expniml'nts are kasible, e.g., in p;micle physics, it is possible to recover the dynaillics with great precision hy taking many "snapshots" of the system ;It r1osl'iy spaced tillle intervals. This technique, knowll as ,I .Iim/}(}.IIO/Iit lilli/lora I'"illl'lllt; I/'rlioll, has given elllpirical content to even the llIost ahstral'lnotions of nonlinear dynamit:al systems. hut unlilrtunatdy cannot he applied to non-expnimental data. The possihility that a rdatively simpit' set of nonlinear deterministic equations can generate the kind of complexities we see in finallciallllarkelS is tantalizing. hut it is of lillie interest if we C
l/lIill(lnllit'
·li·.I!.1 Jor NOlllilll'r/r '\/1'1/('/111/'
Despite the clveat.~ of thc previous section, the mathelnalit:s of chaos theory has motivated several ncw statistical tests liJr indcpcndence and nonlincar structure which arc valuablc in their own right, and WI' now discuss thcse tests. '/I'.Ii.1
/Ifl.\rd
Oil
Higher Momenis
Our earlier discllssion of higher mOlllenl.~ of nonlinear lIlodeis can scrve as the basis lill' a statistical test of nonlinearity. Iisieh (I !IH!I), Ii)!' cxample, ddines a selinl titirumolllent:
E I X,
.\'/-/ X,
E[
J
I
x/ 1:1/ 2
(1~.I.Ei)
ane! observes that O firr liD data, or data generated hy a m;lrlin)!;;11c modclthat is nonlillear only ill variall!'l'. I Ie suggests estimatin)!;
and O'Brien (I\I~I:I). I'c""',,,, ,lIul 1'011('" Schelllkmall alld Woodlord (1\.194).
(l(r~I:!).
SdH'inkmall .11111 l.ell.lIlJll
(I~IH~I).
alld
/2. NllfI/illl'tltilil'.1 il/ l-'illfl1ll'illl /)11111 (p(i, j) ill IIII' olll'io\ls way:
.p(i, jl -
+- LI X, [ .L"
r
X,_; X'_j 'l]:If'!
'--I X,
•
(I ~,l. Iti)
l 'lI(kr Ihe IIIIIlII\')\OII\('sis Ihal
\' --
(12.1.17)
Ilsi('h \ lesl IIS('S 011(' parrindar third 1I\0ll1CIII or Ihl' data, but it is also possible 10 look at Sel'crallilollll'lIts silllullallt'ously. The alltoregressiVl' polylIollliallllocll'l ( I:!.1.7), lill' I'Xalllplt·, suggl'sts that a simpll' lI'st of nOli Ii Ill' arit)' ill the mea II is to rq-\n'ss x, Ollto its OlVlllagS alld crOSS-PI oducts of its 0\\,11 lags, ;1111110 \('SI for Ihe joilll sigllilicllI(,(' ofthl' 1l0nlillCar I.-nilS. Tsay (l!lH()) proposl's a \t'SI of Illb son using S('eOllll-oHler lerlllS alld ,\1 lags for a lolal of M (M + I )/'2 lIonlilH'ar regressors. ()nc call ca!rulale hdnoskt't!;ISlicilYcOllsiSlt'll1 slandarcll'ITOI'S so Ihal Ihl' lesl becolllcs robusl to the prcsence or Ilolllillcarity in variancl'. '1111' Cllndlllillll IlIltgml II lid /111' Cllndl//illll J)illlfn.lillll To distinguish a I\t-tel'lllinistic, chaotic process rrom a lruly ralldom process, it is esselltial to view the data in a sllllirielldy high-dimellsiollallimll. In lhe case of Ihe t('111 Illap, for l'xalllpl(', lhl' data appear randolll if one plots x, Oil lhl' IInil inlerval sillc(' x, has" IlIIifonll dislriblliioll. If Olll' ploLs X, all(l -"'_1011 the ullit squarl', howl'vl'r, lhl' data will all fall 011 tit I' tellt-shapccllilll' showlI ill Figure I ~.I. This straiglillill'wanl approach call yil'id surprising insights, as \\'1' saw in allal)'l.illg slork pric(' dis(TI'I('IIt'ss ill (:hapll'r~. Howevl'r it becollles diHiclIlt 10 illlplcllH'lIllVlu'1I high('r dilliellsiolls or more cOlJlplicate
Th(' parallll'l('r II is kllo\\,11 as Ihe "IIt/mldilll; dillll'lHioll. Thl' lIext ~t('1' is 10 Lrinlbll' tl\(' i'raclioll of pairs of II-histories that arc "do,,'" to OIl(' ;11 11>1 hIT. 'I'll III1'
,<'
..~
...
k: /lIaxi=o ..... II~ I IX.-i-Xt-il < k. We define a closeness indicator K" that one if the two n-histories are close to one another and zero otherwise: , K,t
=
{I
o
if maxi=O ..... n-1 \X'-i - xt-il < k otherwise.
(12.1.19)
We define CII,'r(k) to be the fraction of pairs that arc close in this sense, in a sample of n-histories of size T: (12.1.20) The coTTe/alion integral Cn(k) is the limit of this fraction as the sample size increases: 02.1.21) Equivalently, it is the probability that a randomly selected pair of n-histories is close. Obviously the correlation integral will depend on both the embedding dimension n and the parameter k. To see how k can matter, set the embeddin!!; dimension n= 1 so that n-histories consist of single data points, and consider the case where the data are lID and uniformly distributed on the unit inten-al (0, 1). In this case the fraction of data points that are within a distance k ofa benchmark data point is 2k when the benchmark data point is in the middle of the unit inten-al (between k and I-k), but it is smaller when the benchmark data point lies ncar the edge of the unit interval. In the extreme case where the benchmark data point is zero or one, only a fraction k of the other data points arc within k of the benchmark. The general formula for the fraction of data points that are close to a benchmark point b is min(k+b, 2k, k+l-b). As k shrinks, however, the complications caused by this "edge problem" become negligible and the correlation integral approaches 2k. Grassberger and Procaccia (1983) investigate the behavior of the correlation integral as the distance measure k shrinks. They calculate the ratio of log CII(k) to log k for small k: ' v"
=
. log CliCk) hm ----"'---k~() log k
(12.1.22)
which measures the proportional decrease in the fraction of points that a~e close to one another as we decrease the parameter that defines closeness. In the lID uniform case with n= I, the ratio log CI (k)/ log k approaches log 2k/ log k=(log 2+ log k)/ log k= I as k shrinks. Thus VI =1 for lID lIllilimll data; for small k, the fraction of points that arc close to one anothpr shrinks at the same rate as k. I
478
12. NonlillPflTilip.l il/ Fil/(/I/ri(/l /)(/((/
Now consider the behavior of the correlatioll integral with highn ("111bedding dimensions II. When 1/==2, we are plotting 2-histories of till' ll~\ta 011 a 2-dimensional diagram such as Figure 12.1 and asking what fraclion of the 2-histories lie within a square whose center is a benchmark 2-history and whose sides are of length '211. With unifi>nnly distributed IlL> data, a fra(r.ion 4k2 of the data points lie within such .1 square when the hendllllark 2-hi tory is sulliciently far away rrom the edges or the ullit sqllilre. Again we landle the edge problem by letting II shrillk, and we lind tl ... t the ratio log Pl(k)/ log k approaches log 4kt / log k = (log 4+21og Ii)/ log Ii == '2 as k sh!rinks. Thus 11'1==2 for IlD unifOl"Ill data; for slIIall k, the fraction of p~'irs of I~oillts 'that are close to one another shrinks twice as fast as k. In general v"9n for lID uniform data; for small k, the fraction of Tl-histories that are clo* to one another shrinks 71 times as f~lst as k. \The correlation integral behaves very differently when the data an' generaied by a nonlinear deterministic prucess. Tu see this, consider data geJl~rated hy the tent map. III olle dimcusioll, such data rail Ulli(()\"fllly Oil the IlIIit lille so we again get VI:::: I. But in two dimensions, all the data points faIltn the tent-shaped line shown in Figure 1'2.4. For slIIall k, the fraction of I irs ofpoinL~ that are close to one .mother shrinks at the same I"ilte as k so 1 I. In higher dimensions a similar argulllent applies, and V,,= I lilr all 1/ wF,len data are generated hy the tent map. he correlation dimension is ddined to he the limit of v" ,IS /I illlleasl's, whe 1 this limit exists:
=
r
v::::
linl
11".
(12.1.~~')
11-00
Nonlinear deterministic proccsses arc char'lCterized by linite v. The contrast between nun linear deterministic data and lID uniform data generalizes to lID d data regardless of the distrihlltioll. The dfed or the distribution averages out because we take each lI-history in tlll"n as a benchmark n-history whell c random data by calculating V" ror different 11 and seeing whether it grows wilh II or converges to SUIIIC lixed limit. This approach requires l
The Ilro(h-Decherl-Schrinhlllllll '/"51 Brock, Dechert, and Scheillklllall (19H7) have developed all altemative approach that is better suited to thc Iilllited allloullts uf data typically availahle ill economics and finance. They show that evell when k is fillite, ir the data arc Ill) then for any n ( 1'2.1.24)
/2.2. 1\1111"".1 n/CllIlIIgil/g i't,/alilily
47!1
'Ii. IInderstand this result, note that III<' ralio (:"I.(/I)(C,,(k) call he illlnpreted as a conditional probability: C"II(l1) C,,(h)
ItI;l)~
Ix. ,- x, ,I
1-::.1 •..• 11
lIIax Ix.-, -- x, . ,I 1=1..,.,11
That i~, C,,+I (h)/ C,,(h) is the prohahility Ih'lI Iwo data poillts are dose, givell that the previolls II data poillts are close. Ir till' laill (12.1.2'1). Brock, Dechert, and Scheinkmall (1!IH7) plOpose Ihe /WS lest slalislic,
.I", r(ll)
= "
~J' C"./(II.) - CII(h!~, I (1 ". /
(I '2.1.'21;)
(/1)
whITe C", /,(11) ;'IHI (;1. rUt) arc the s'lIl1ple correlatioll illtegrals ddilled ill ( 12. ; .20), alld n", r(k) is an estimator of the aSYllIptotic standard deviation of C".r(ll)- (:1./ (h)". The nDS statist'l(, is aSYIlIJ>lot'lcdly standard normal under the liD Ilull hypothesis; it is applied ;lIld n;plail\t'd by I !sid I (1~IH9) alld Scheinklll'1I1 alld LeBaroll (1!IH!l), who provide t'xplicitexpressiolls lor a".r(/i). Iisieh (I !lH!I) and Iisieh (I !l!ll) report Monte Calio lesults on the si/,e an(1 power of the BDS statistic ill linite sampks. 'A'hill' thele are sOllie pathologicaillolllillear Itwdds fOI which C,,(k)= (;1 (h)" as in liD dala, the BDS statistic apJ>('ars to ha\'(' good power againsl Ihl' IIIOSI cOllllnonly \lsed nonlinear IllOdels. It is importallt to understand th.1l it has power against models that are IHllllinear ill \',\rian('(~ hut not in mean, as well "s models that arc nonlinear in meall, Titus" liDS Icjeuion docs not necessarily imply that a time-series has a tillle-v.Il'ring conditional mean; it could simply he evidellu' 1(.1' a time-var),illg conditiollal variance. IIsieh (1901), 1'01' example, strongly n:jnb the hypot hl'sis that (0111111011 stock returns arc liD using the liDS test. I Ie tltcn estimates models or the tiIlH:-v;lr)'illg conditional variallcc of returlls and g(,ts 11 11K It weaker e\'idencc ag;lillsl the hypothesis that tlte r('siduals 1'10111 such IIlOdcls ;1)(' III>.
12.2 Models of Changing Volatility III this sn:tiol\ W(' cOl\sider alterllati\,t' ways to IlI
j.<..
"'J1JiJlJt'IJfJ/J/'.1
JIj
"Jlj{Jlfj
wi
/)((/1/
is lillkt'd (0 tilllt'-\';Iriatioll illtht' cOllditiollal variallce; tht'se models art' 11011lillear in holll Ine;lll alld varialll'l'. III ordn to (,(lIIlTlltrall' Oil volatility, we asslIme Ihal lilt I is all illn(),,;\lioll, lhal is, il has lIIt'an/el'O condilional 011 time / information. III a finance application, /11/ I lIIight 1)(' the illnovatioll ill all assct rt'lul'll. Wt' defille to hI' tht' time' I conditioll;11 \,;II'i;ItH'C of 1111 I or I'CJuivalcntly thc conditional expcctatioll of //;'1 I' W(' assltllll' that cOllditiollaloll lillie I illforlllatioll, lhe illllOV;lIioll is lIorlllall\' distrillllt('d:
0/
'iltl
~
N(o, 0/).
The I\IICOIlIliliultal variance oflh(' iltllovatiolt, eXllI'ctatioll of 0/::1
(I:!.~.I
0
2
•
)
isjllst the IInconditiollal
a/
Thlls variahility of arolltHI its lIIt'all does 1I0t changt' the IIIICOllditiol1al . " vanalHT a- . . The variahility or al~ do('s, how('v('r, affect higher l1J(llllCl1ts or tht' 1111(,(lIl1litiollal distrihltlioll or 'ilt I, III particlllar, with time-varyillg the 1111COllditiollal distJihlltioll 01'111 I I has (;\lter t
a/
(I~.~.~)
I is an liD randolll vari;lhle \.itll /.l'I'O \\lean aJld IInit vari:ln('(' (as ill Ihe pre\'iolls s('('tioll) that is lIorlllally distrihllled (all aSSlllllptiol 1 we did lIot lIIake ill till' pre\'iolls senioll), As WI' Ilisnlsse(\ ill (:hapIl'f I, a IIsefltllllt'asllrt' of tail thicklll'sS for lht' distrihlltioll or a ralldolll I'ariahk V is tht' Ilortllalized fOllrth 1lI01l1t'IlI, or kurtosis, defillet! hy I\(y) '" E[/liEI/F, It is well kllown that the kurtosis ora Illlllllal ralldolll variahle is :1; hell(,c K(~/+t) Bllt for innovations
wltl'n' fit
= :\.
'I/~I, WI'
ha\,(' 1\ (III I
Ela;t I E[ f:+1 1 I)
Wla/])2
:11-:10/1 (Elo,~ l)~
:\<E[an)~
(Elo/IJ~ :tThb I ('!\lIh Iwleh 11111\ }we .111 .... '· \\'(' .1It" h"tlll..illg with all (II'IH) (ullclilion;d 1I1t',tll. hJl ,I \;\li:IIH(' j, IItJlllu' ";11111'
.1'
IIIC"
"',ic·,wilh a
:~.
inltor.llioll
~('I i,':"o Ih~u hil~;1 ("oll'.. l;lIll 1I11ffilulillilflal
lillH'-".lryilig-foll
IIIIt"CHlclilinll.d
,·.xpc·n.lIioll 01111" rOlldiliuUOII
\'ari;lIln',
12.2. Mod.els oj Changing Volatility
where the first equality follows from the independence of a, and €t+I. and the ineCJuality is implied by Jensen's Inequality. InlUitively. the unconditional distribution is a mixture of normal distributions. some with small variances that concentrate mass around the mean and some with large variances that put mass in the tails of the distribution. Thus the mixed distribution has fatter tails than the normal. We now consider alternative ways of modeling and estimating the process. The literalUre on this subject is enormous. and so our review is inevitably selective. Bollerslev. Chou, and Kroner (1992), Bollerslev, Engle, and Nelson (1994), Hamilton (1994) provide much more comprehensive slIrveys.
al
12.2.1 Univariate Models
Early research on time-varying volatility extracted volatility estimates from asset return data before specitying a parametric time-series model forv9latility. Officer (1973), for example, used a rolling standard deviation-the~stan darel deviation of returns measured over a subsample which moves forward through time-to estimate volatility ateach point in time. Other researchers have IIscd the difference between the high and low prices on a given day to estimate volatility for that day (Garman and Klass [1980), Parkinson [1980)). Such methods implicitly assume that volatility is constant over some inierval of tillle. These methods are often CJuite accurate if the objective is simply to measure volatility at a point in time; as Merton (1980) observed, if an ,asset price follows a diffusion with constant volatility, e.g., a geometric Brownian Illotion, volatility can be estimated arbitrarily accurately with an arbitrarily short sample period if one measures prices sufficiently frequently."- Nilson (19!12) has shown that a similar argument can be made even when volatility changes through time, provided that the conditional distribution of returns is not too fat-tailed and that volatility changes are sufficiently gradual. It is, however. both logically inconsistent and statistically inefficient to lise volatility measures that are based on the assumption of constant volatility over some period when the resulting series moves through time. To handle this. more recent work specifies a parametric model for volatility first, and then IIses the model to extract volatility estimates from the data?n returns. AHCf{ Models
A basic observation about asset return data is that large returns (of either sign) tend to be followed by more large returns (of either sign). In other ~ S"t' Sectioll 9.3.2 of Chapt~r 9. Note however that high-frequency price data are often '~v~r~ly
Iill\it~d
(I\IHO).
affected by microstructure problems of the sort discussed in Chapter 3. This has the ll,~fllllles., of Ihe hil\h.luw IlI"lhlld of Carman and Klass (\980) and Parkinson
12. NOlllillmrilips ill Fillflllfi,,{ /)(11"
f
J?, Ii I I
bII'": ,)
o L-____~______~____~______~____~__~__~____~______~
,. 1920
1930
1940
Figure 12.2.
I !ISO
Monthly
I9fiO Year
EXWJ l,0t:
1970
I!lMO
1!190
~WOO
US S/(}rk lle/llmJ. /926 /() 1 <)<)4
words. the volatility of asset returns appears to be serially correlated. This can be seen visually in Figure 12.2, which plots monthly excess relllrns on the CRSP value-weighted stock index over the period from Inti to I !I!I1. The individual monthly returns vary wildly. but they do so within a range which itself changes slowly over time. The range for returns is very wide in the 1930s, for example, and lIluch narrower in the 1950s and I<JGOs. An alternative way to understand this is to calculate serial correlation coefficients for squared excess returns or absolute excess returns. At 0.2:1 and 0.21. respectively, the first-urder serial correlation coclliciell IS for these series are about twice as large as the first-order serial correlation coefIiciellt for returns themselves. 0.11, and arc highly statistically significant since the standard error under the Ilull of no serial corrt"iation is 1/ fl = O.O:\(). The dilTerence is evell more dramatic in the average uf the first 12 autocorrelation coefTicienL~: 0.20 fill' squared excess returns. 0.21 fur ahsolute excess returns. and 0.02 for excess returns themselves. This rcilecL~ the l;tCl that the autocurrelations of squared and absolute returns die out only vcr}' slowly. To capture the serial correlation or volatility, Engle (I !lH2) proposed Ihe class or Autoregn'ssiv(' COlulitionally Ilctcl'Oskedastic, or ARCll, Inod-
9
12.2. Mc)(JeL\ ()rClUIll~illi V(){lIlilil.~ cis: TileS!' writ(' ('ollditiollal variall(,(, as a dislrihlll('(liag or pasl s'llian'd inllov'llions: (1\1.\1.1) where a(l.) is a polynomial in the 100g Op(T.llor. To k('('p the conditional variallce posilive, wand the coeflil'i(,llts in aU.) mllsl hI' nOllllegative. As a way to model persistelll movements in volatility without eslimating a very large IIIlmber of coeflkients il\ a high-unler polYI\\)lllial a(/.). Bollerslcv (I ~IH(i) sllggested the (;cnerali1.ed AIIIOI('gl('ssiw (:onditionally I-lcteroskedastic, or GARCH, lI\odd: o,~ =
lJ.)
-+- fJ(I.)a/
1
f- a(/.)IJ7,
(I~.~.!i)
wilerI' fJ(l.) is also a polynomial in Ihe lag operator. By allalogy with ARMA Illodels, Ihis is called a GARCH(/), '/) llIodd whel\ the onit-r of the polynomial fJ(l.) is /' and the order oflhe polY\lol\li.1I a(!.) is 'I. TIlt' I\lOst ('o\ll\llonly IIsed model ill Ihe CARCI1 class is the silllple CARCII (t ,I) which can be w,illcn as
( 12.2.G)
In the sccond equality in (12.2.6), Ihe terlll (11; -0,1. 1) has IIIC
(\ ~.~.7) This representatioll makes it clear that the CARel I (1,1) model is an AR:vtA(I,I) model for squared innovatiolls; but a standanl ARMJ\(I,I) model has homosked'l.~tic .~hocks, while hel'l' thl' shocks (II;!. 1 -0/) arc themselves hcteroskedastic. 1"'1 li.ltl'llt/' lint! Stlltionarity
In the CARCH( 1,1) model it is easy 10 construct lIlultipniod I<JI'('casts or volatility, When a +fJ < I, the unconditional variall<'l' or '/,+1, or equiv
J L.
NOIl/illl'llri/iI'J ill Fill{/I/ri(/l /)(//(/
illg ill (I ~.~.ti), alld IIsing Ihe law of ileraled expectations, the fondi lional expectalioll 0(' ml;lIilil)' j pl'riods ahead is (I~.~.H)
Thl' IIIl1lripcriod \'o!alility lim'Clst r('verts 10 its III1COIHliliollallll(';ln ,II ral(" (ll' + Ill, This rl'Lilioll 1)1'(\\"('(,11 sill~k-Jleriod and IIIl1hipcriod forecasls is (hI' S,III1,' as ill a lill('ar ARMA( 1,1) model wilh alllor(,gr(,ssive cod!ici('nl (a + Ill. l\Itlitipl'fiod (im'c"sls (";\11 Ill' ,'ollslnICl('d ill a similar bshioll «Ir hi~l\l'r-onlcr (;J\IU :11 IIHHI,.}s. When a -I- Il= I, I hI' condil iOllal I'xIll'Clal ion or volatili I)' j periods alll'ad is insll'ad
(12.2.9) Thc GARCI I (1,1) Illodd wil h a + Il = I has a IIl1il allllllTgrcssive root so Ihat IOdav's VoblililY affl'Cls lim'casls ofvoialililY inlo Ihe itHh'finilc fllt\ll"l'. II is t\lI'I"I'('orl' kllo\"11 a~ an inll'grall'd CARel I, or ICARCI I (1,1), lIJodl'\. . Th(' :AR(:I I (1,1) process ror looks very 1I1\ll:h lik" a lilH'ar randolll walk wilh drift w. Ilo\\'('\'('r Ndsoll (19!H») shows Ihal Ihis allalo!-,'Y lIlust he trl'all'e1 wilh caulioll. :\ lillcar ralldolll walk is lIollstalionar), in IWO S(,II'1'S. Firsl, it has Ill) sialiollary dislrihulioll, h(,lIcl' Ih,' pro(('ss is IlI)I ,,/ridly ,1/f/liOI/(/1"\', Sl'colld, il has 110 IIIll"olHliliollal first or s('cond IIJOlllelllS, h(,lIce it is 111'11 rOlI(/rial/rn/f/liol/(/fY. flllh(' I(:ARCll( 1,1) lIloelel, olllh(' olher halld, is strictly staliollary ('\'('11 Ihollgh ils sialiollary
«
(J;
(J/
=0;,
a/
(J/,
~lN('I."on show, Ih.11 Iii",,' III
,,+ Ii .,
I hilI \\'illo
FII"~lfi 'j
(lV'"
Iii"
hllid ilion- g('lIeraJly for (;:\JH :11 ( 1.1) Illo
1t1;11 .- 0,
• 485
12.2. Modfis ofCha7lgi1lg Volatility
AltmlfltiTlf FU7lctio7lal Fonns In the standard GARCH model, forecasts of future variance are linear in current and past variances and squared returns drive revisions in the forecasts. An alternative model, sometimes known as the absolute value GARCH Jllodel, makes forecasts of future standard deviation linear in current and past standard deviations and has absolute values of returns driving revisions in the forecasts. An absolute value GARCH (1,1) model, for example, would be (12.2.10)
Schwert (19R9) and Taylor (1986) estimate absolute value ARCf"I models, while Nelson and Foster (1994) discuss the absolute value GARCH(I ,l). The models we have considered so far are symmetric in that negative and positive shocks ft+t have the same effect on volatility. However Black (1976) and many others have pointed out that there appears to be an asymmetry in stock market data: Negative innovations to stock returns tend to increase volatility more than positive innovations of the same magnitude. Possible explanations for this asymmetry are discussed in Section 12.2.3. To handle this, one can generalize the absolute value GARCH model to (\2.2.11)
where f(f / )
= If, -
bl - r(fl - b).
(12.2.12)
Here the shift parameter b and the tilt parameter c measure two different types of asymmetry. b is unrestricted but we need lei ~ I to ensure that !(f I)"~O. When c=O but b=/=O, the effect of a shock on volatility depends on its distance from b, so that volatility increases more when there is no shock than when there is a shock of size b. When b=O but c=/=O, a zero shock. has the smallest impact on volatility but there is a distinction between positive ane! negative shocks; a shock of given size may have a larger effect when it is negative than when it is positive, orvice versa. Following Hentschel (1995), a nice way to understand (12.2.12) is to plot !(f/) against flo as in Figure 12.3.6 Panel (a) of the figure shows the absolute-value function (b=O, c=0); this is plotted again as a dashed line in each of the other panels. Panel (b) sliows the shiftee! absolute-value fUlIction (b=0.5, c=O), panel (c) shows the ti'lted absolute-value function (b=O, c=0.25), ami panel (d) shows a shifted ~nd tiltee! absolute-value function (b=0.5, c= - 0.25). Hentschel (1995) further generalizes (12.2.11) to allow a power of f(f,), rarhn than j(f / ) itself, to affect volatility, and to allow a power of a" rather IThis is silllilar 10 Ihe "Ilews illlparl fll,,'t·" or Pagan and Schwert (1990) and Engle:and Ng (I !1!1:1). ,,·hid. pIOL' againsl ry,. holdillg any olher relcvaIll state vdriables at their lI~con diliollallllt"alls.
a/
•
4 Ii
12. NOlllinearitif.l;1I
....
...,r----:---...,...--~--..,
"i
eN
,i
,,
,,
S.
I I
I
,
1"""
'.
-2
/ /
"; I I
0
2
Fil/al/rial /)ata
-I
/ /
I
,./
2
'. (b)
(a)
'"
I
~CN,
.1
~
'"C'l
,
I
I
+ "1-
:
;/[
/ / / /
/
/
/
/
~
2
/ /
0
-2
0
-I
2
£.
(d)
(e)
Figure 12.3.
Shifl,,1 allIl nUnl Ab.lllh,lr-Valllf fo/lII(lilJl/
than a, itself, to be the variable that follows a linear difference equation. The resulting equation is (!~.~.13)
Equation (12.2.13) defines a !;lIllily of models that incllldes most of the popular GARCII-type models ill thc lilcraturc. 7 Thc standard GARCII model seL'I A=v=2, and b=c=O. GlostClI, .Jagannalhan, and Runkle (19!13) have g('lerali1.ecllhe slanclarcl GARCII lIIodei to allow non1.ero (. Engh: and Ng (I! 93) have instcad allowcd nOll7.cro b. Thc absolute valuc GARel1 model set A::::: \l = I with frec /1 and f. Anolhl'r partirularly illlpoftanlllll'lnl>n or Ihl' rall~ily (12.2.12) is the expollclltial (;ARCII or ECARCH model of Nelson (I!~~>O), which is uhlained hy selling 1.=0, 1'= I, al1(1 b=() to gel I I
rs. .
log(a,) =
lti
+ Ii !og(a,_1 ) + a
"I", l>i"l\. (;'''''1\'''' .• ,,<1
E,,~I.- (I~I~':\)
[If II -
(f
I) .
I ...... ..t .• lnl !;olllil)' .. flllo,"-'"
(l~.:!.l
,I)
J2.2. M(lIlrLl
ol Cllllllgillg V()latility
4H7
This modcl i, appealing hec;ltlse it docs not leqllilc any paralileter re~tric tions 10 cnSlire Ihat the conditional valiallce ofllw letllill is always positive. Also it becolllcs hoth strictly nonstatiollary ;md covariallce l\(lIl~tationary whell a + fJ= I, so il docs Illll share lhe 11l1l1Sllai Sl
We have introduced an almost bewildninl-( varicty of mbtility llIodels. Til discover which katllfes of these models arc importallt ill fillinl-( financial data, OIlC must he able to estimate the Illodeb' parallll'tcrs. Fortllllately this is bidy strail-(Illforward f(lf (;ARCII lIIodels alld olher lI10deis in the class defined hy ( 12.2.1:~). Conditiollal Oil the paraflleler~ of lile llIodel alld all illitial vafiallce e~tilllate. the data are norlllally distribllted and we call C()(1stl'lIct a likelihood functioll recursively. We write the vecto!' of model pal,ulleters as (). define r(' log(g(J/,+I/a,(O)) -log(a,~(O»/2 - log(J2rr) - II; t I /'2a,~ (0) -log(a/
(12.2.1 G)
where the laM term is aJacobian tCfm thal appears hecause we ooserve 1/1+1 and not 1/,+1 /a,(8). The log likelihood or Ihe whole data sell/I •... , tiT is r
L(I/I.·· .• 1/T)
L
f,(IJr.I; 0).
(12.'2.17)
'~I
The maximulIl likelihood imizes (I '2.'2.17).~
c~tilllator
is the choice or paramell'rs () th,ltmax-
lilll pr;.Klln: OUl' lH.'('(h. an initial a,; to hq~ill c.lkul.ltillg lhe (ollciitioll.d likt·lihood ... ill (1!!.:!.IIl). Tlu.' infhu'lHT of lhe: initial condilion clilllilli~h(" .. on', filiI(' .111(1 h('fOIlH'~ 1Iq.~ligi. hit· .\,"\'IIII)(Olic ally; Ihlls Ill(" choice of jllifi~" (fllldirioll do('.\ 11111 all("(, the t ollshlt'llf), 01 III(" (·~tilll.lloi.
J1. Nlllllilll'(/I';t;I'.\;1I "'lIlwollI {Jllt(/
Allhollgh il is (,'IS), 10 sholl' Ihal Ihe maximum likdihoodeslilllalor is l" Ihe po\\,tT thr('(' halvl's. The volatilily proCI'SS is highly pcrSiSll'll1 ill ;rllll ... Illockis eSlilll;II('d, ;rllhollgh ,he degn'(' ol"pI'rsis'('IIIT is st'llsilin' Itl SI)('( ili,',lIioll ill IIII' p,,,I-Wodd War \I period, Additillllllll'.Ox/df/III1/III\' \ 'f/rillh"', lip 10 Ihis pOill1 \1'(' kll'(' 1110111'1("(1 volalilily IIsillg ollly Illl' pasl hislorl' of n'lIlrns tlll'lllScll'cs, II is slr;Jighll()rward 10 add olher cxplanalory \"Iriahles: For l'xalllpk, (llll' LIB II'riw all allgllH'llled CARel I (1,1) Illodd as ( I:.!:.!,IKI whe\"(' X, is ;111\" ,'ari"hl(' kl\m"l\ al tillle /. Provided thaI X,::,,:O alld )' :::0, Ihis Illodd still ("ol"lr"ills ,'o!alilill' 10 1)(' positiVI'. Altel"llalivdy, OIlC rail add ex pl'lI 1;1 101"\' vari;rhl('.~ 10 Ill(' H:i\R( :llllIodel Wilholll allY sigll reslriniolls, (;loSII'Il,J;lg'III1lalhall, alld RUllkle (I !)!):I) aclcl a shorl-Ierlllllolllinal intl'l"esl rail' 10 various (;.-\\{( :'1 I\lodds alld show Ihal il has a sigllilical\1 posiliVl' clll'!"1 1>11 slmk Ill~nk(', "oLililitl', (:/IIl/litilllllll.'VlIlIl/ll/lllldity
The CARel I llltl(I..t~ 1\'1' hOI\"(' (ollsid(')cd imply ,hal Ihc dislrii>ution or rclurus, ("(llldilioll;rl Oil til(' 1';'" hislOIY or rellll'llS, is lIonlla!. Eqllivalelltly, Ihl' sl'lIuLrnli,,'d r('siduals 01 IheS(' 1I10dds, f ' l .c0I=I/H ,In,(O), should Ill' IlOnlla\. l '1I",rlll\l;lleh', ill I'r;\clin' Iherl' is ('XCI'SS kllrtosis ill Ih(' slall(brdi/nl \"('sidllals 01 (::\\{( :11 Illllllds, ;1I1)('il kss Ih.uI ill thl' raw relllrllS (SCI', lor ('x;\lIlph-,1\011("J,I('\'II~)H71 .11111 Nelsoll 11\'\)( I),
!489
12.2. ModeLl oj Changing Volatility
Onc way to handlc this problcm is to continue to work with the conditiollal normal likelihood function defincd by (12.2.16) and (12.2.17); but to iuterpret the estimator as a quasi-maximum likelihood estimator (White [ 19H2]). Standard errors for parametcr estimates can then be calculated using a robust covariance matrix estimator as discllssed by Bollerslelanrl Wooldridge (1992). Alternatively, one can explicitly 1lI0dclthe fat-tailed distribution of the shocks driving a GAReH process. Bollerslev (1987), for example, suggests a Stll(\t-nH distribution with k dcgrees of frcedom: K(E/+I(O»=f
(
k;
1)
f
(k)-I 2
(k-2)-1/2
( l+f~+~2 (O»)-(A+I)/2 '
. (12.2.19) where fO is the gamma function. The t distribUlion converges to the normal distribution as k increases, but has excess kurtosis; indeed its fourth moment is infinite when k :s 4. In a similar spirit Nelson (1991) uses a Gcneralized Error Distribution, while Engle and Gonzalez-Rivera (1991) estimate the error density non parametrically. GARCH models can also be estimated by Generalized Method of Momellts (GMM). This is appealing when the conditional volatility can be wrillen as a fairly simple function of obsel\led past variables (past squared retllrtlS and additional variables sllch as interest rates). Then the model implies that squarcd returns, less thc appropriate function of th~ observed variables, are orthogonal to the obsel\led variables. GMM estimation has the usual attraction that one need not specify a density for shocks to returns.
a?
Slor/wslie-Volatility Models Another respollse to the lIunllorlllality of relUrns conditional upon past returns is to assume that therc is a random variable conditional upon which returns are normal, but that this variable-which we may call stochastic volati/ity-is not directly observed. This kind of assumption is often made in continuous-time theorcticalmodels, where asset prices follow diffusions with volatility parameters that also follow diffusions. Melino and Turnbull (1990) and Wiggins () 987) argue that discrete-time stochastic-volatility models are natural approximations to such processes. If we parametrize the discretetime process for stochastic volatililY, we then have a filtering problem: to process the ohsel\led data to estimate the parameters driving stochastic volatility and to estimate the level of volatility at each point in time. A simple example of a stochastic-volatility model is the following:
where €/~N((). a,2), ~/~N(O. a/), and we assume thal f, and ~, are seri~lly uncorre\ated and independent of each other. Here a, measures the dif-
I
4~0
/2. NOlllinearities in Final/rial /)(/t(l
I
ference between the conditional lo~ standard deviation of returns and its mean; it follows a zcro-Illean AR( I) process. We can rewrite this system hy squaring the return equation and taking logs to get 10g(IJ~)
::=
a, + log(E;),
a, = cpa,-l + ~,.
(12.2.21 )
This is in linear state-space form except that the first equation of ( 12.2.21 ) has an error with a log X2 distribution instead of a normal distrihution. To appreciate the importance of the nonnormality. one need only consider thl' fact that when f., is very dose to zero (an "inlier")' log(E~) is a very large negative outlier. The system can be estimated in a variety of ways. Melino and Turnhull (1990) and Wiggins (1987) use GMM estimators. While this is straigillforward. it is not emdent. Harvey. Ruiz. and Shephard (1994) suggest a (Iuasimaximum-likelihood estimator which ignores the non normality of 10g(E;> and proceeds as if both equations in (12.2.21) had normal error tenlls. More recentlY,Jacquier, Polson, and Rossi (1994) have suggested a Bayesian approach and Shephard and Kim (1994) have proposed a simulation-hased exact maximum-likelihood estimator. 12.2.2 Multivariate ModeLl
So far we have considered only the volatility of a single asset return. More generally, we may have a vector of asset retuJ'llS whose conditional covariance matrix evolves through time. Suppose we have N asseL~ with return innovations l)i.,+I, i= I . " N. We stack these innovations into a' ctor '71+1 =[ 1)1.1+1 ... I)N.l+tl' and dc/Inc aii,,= Var,(T/i.,+d and a'j.'= V,(l)i.I+I.l)j,I+I); hence l:,=[a,j.,] is the conditional covariance matrix of a I the returns. It is often convenient to stack the nonreliun<1ant clements o E,-those on alld below the main diagonal-into a vector. The operator which performs this stacking is kllown as the verI! operator: vech(E,) is a vector with N (N + I) /2 clcmen ts.
~ I
A-fultivariate GAllCI I Models Many of the ideas we have considered in a univariate context translate n and to v<jch(E,_tl:
J
vech(E,)
=w+
IJ! vcch(E,_d + A vech('7,'7;).
(12.2.22)
H re w is a vector with N(N+i)/2c1emenls, and IJ! and A are N(N+I)/2 x (N + 1)/2 matrices; hence the total number of parameters in this model
J2.2. Milt/I'll IIf (.'IulI/gillg Vola/ili/.r
4!1I
is N~(N+I)~/'2 + N(N+I)/'2 which /-:I'OWS Wilh lhe /illlillt pow('/' of N. II is dear lltal lhis llIodei becollIcs IInlll.lIlag(·OIhl(· wry qllickly; III1ICIl 01'111(' Ii ler.ilur(' Oil 1I11111iv
(12,'2.23) Til i.' lIIodd is ohlained frolll (12.2.2'2) by llIakin).;' lhe lIIalrices A and W diagonal. The impliecl cOllditiollal covarianc(, nl
( 1'2.2.24) where C is a [ower lriangular malrix wilh N(N + 1)/'2 paramcters, anti Band A arc squarc IIlatrices with N 2 parameters each, Ii II' a IOlal parameler COUllt of (:'N 2 +N)/'2. Weak reslrictiolls Oil n and A ).;'11 a r
1:, =
etc + .xX[flw'l::,.(W + u(W'll,)~ \.
Here Cis reslricted as ill lhe previous equal ion. We call illlpos(' one normalizing reslrictiOIi Oil this model; il is ('ollv(,lIielll (0 scI t'w= I, where t is
/2. NOII/iIlNIIl/i/'{ ill FiIlI/II(ill/ /)010
a v('clor of Olles. Th(' v('Clor w call IhclI he Ih()lI~hl of as a v('cwr of port\Vci~hls. W(' til-fill(' 11",==w'1J, alld (T/'J,.,==w'"E,w. The modd call IIUI\" he reslaled as
")fio
(I,/.,
...
w'l /' liJl'/'
t-
A, A/ (TN',' /1(1"".,_1
+ (OIf,.I'
Th(' covari.III ... ·S of .11 I\' 1\\'1l as,,·t retllrns II IUI'{' throlll-\h tilll(' only \\'ilil the varianc(' of the portf(llio retllrll. which follows a univariate (;AR(:I I ( I .1) llIudd. The sill~le-t;lrtor CAR( :11 ( 1.1) 1Il()<1r1 is a special else of tht' \I EKK model when' Ih(' lllairin's A ,lIld l\ haw rank OIlC: A == fowX alld B == JiiwX. It has (N~+:IN+'!.)/'!. rret· paramelers. The lIlodel call h(' ext('lIded strai~htrorwanll\' to allow ror llIultiple filC!ors or a hi~htT-order C;AI~ClI strlll'lure. Filially, BIIII(,lsl('\' (I~)')()) has propost'd a constaut-corr(,lation IIlOtl<-: ill which each ,!Sst"! r('tlll'll varialHT f()llows ,I univariate CARCII( I ,I) modd aud the covarian ... · h.. tw .... 11 ally two ,IsseiS is ~iven hy a constani-corrcblillll cot'rticielll lIIultiplyill).; Ih(' conditional siandard dcviatiolls or Ihe 1('1\11'11<
(1/1.1
(0Il
{l'1
+ fi,; (1;l.I-1 +
(ill
lJ~/
J(lII,' (IIi.' .
This model lias IV (IV +:1 )/!! par,lllletl'l'S, I t ~i"es a positive defill ite c()"a;'iallce malrix provided tilat IiiI' corr.. latiolls (l"lllake up a w(,ll-defined cOlrda'ion matrix alld Ih(' P"I,Ull<'tl'l" (V". «II' aud fill an: all posiliv{·. '1'(1 ulldnstalld the din"'II'IHTS hetwecll these Illodds, it is inslJ'lluivc 10 consider whal happells to 111(' cOlldilional covariance helween Iwo
493
12.2. Moddl O/,CllIlIIgillg \'11111 I iJit)'
MII/til/I/ria//' S/or//{/.\tir-Vol(//ili/y M(}{M.I' 'I'll(' uni\'ariate stochastic-volatility model V;ivclI n;\l'lIded III a IIIllhivariale sCllinv;. Wc ha\'('
III
(I2.2~) i&
also easily
02.2.2R) II'h('l'(, 11t, ('t, Qt, and ~t arc now (N x I) vectors and is all (N x N) matrix. This lIlo(kl has N~ paralllt'tl'l's ill th(' matrix , N(N+I)/2 parameters in lil(' co\'arianc(' lIlatrix or ('" and N(N+ I )/'2 parameters in the covariance 1lIatrix of'7J" so the totalnl1lllber ofparallleters is N(2N+l). There is no IIc('d to rcstrin 0., to lw positive and it is straiv;htrorwarcl to estimate the (', and 7]t covariancc parameters ill sql1are-root forlllio ensure Ihat Ihe implied ('()v;lriallcc matrix is positive definite. Ilarvey. Ruiz, and Shephard (1994) sU!2;!2;('st restrit'l('d versiolls of th is model in which is diagonal (reducing the Iltl1IJilt'r of paramelers to N(N+'2» or is evell the identity matrix (further reducing the !\lImber or parameters to N(N+I». Ev('n without such extra restrictions, it is important to understand that the sp(,cification (12.2.2H) imposes constant conditional correlations or a!iset re!lII'lls. Itl this respect it is as restrictive as Bollerskv's (1990) const~nt corrdalion CARel-! Illodel, and it has lllore parameters than that model whelll'vl'r N>?.
A CO/ldi/iolllll A/arli/'t MOIlPl E\'en the lIlost restrictive of the models we have discussed so rar are hard to apply to a large cross-sectional data set because the lIumber or their parameters grows with the square orthe number of assets N. The problem is that thest' lIlodl'b take the whole conditional covariance matrix or returns as the ohject 10 he studied. An alternative approach, paralleling the much earlier development of static mean-variance analysis, is to work with a conditional lI\arkl~1 lIlo(kl. Continuing to ignore llonzero mean returns, we write ! '1i.I+1
== fill l1m.ttl + (i.t+I,
•
(12.2.29)
whnc {J,t == ai",.t/a"",,.1 is the conditional heta of asset iwith the market, and (,.1+ I is an idiosyncratic shock which is assumed to be uneorrelated across assets. Within this framework we Illightlllodel 0 11"0.1, the conditional variance of thc lIIarket rctul'll as a univariate GARCH (I, I) process; we might model {Jim .• or equiv;t1cntly ai,".t as depending on amlll.1> fJim.l-h and the retllrns 1/i/ alld '111"; and we might nlodd till' cOllditional variance or the idiosyncratic shock III \'l'tlll'll as ;\lIothl'l' IInivariate CARCI-I( 1,1) process. The covariance matrix implied by a \1Iodl'l of this sort is guaranteed 10 he positive definite, alld the 1I111lli>('r of parameters ill thl' lIlodel grows at rate N rather than N'l , whit'h lIIakes Ihe lIIodd applicable to Illuch larger 1\lImbers orasscts. BraulI,
494
12. NUllii1lfaritil'.1 ill Fi,ul/u'i(// /)(/1(/
Nelson, and Sunicr (1995) take this approach, using EGARCII fUllctional forms for the individual componenL~ of the model.
12.2.] Links between First
01/(1
Second MUlllenls
We have reviewed some extremely sophisticated models of tillie-varying second moments in time series whose first moments are assumed to be cunstant and zero. But the essellce of finance theory is that it relatcs the first alld second moments of asset returns. Accordingly we now discuss models in which conditional mean returns may change with the conditional variances and covariances. The GARClI-M Modd Engle, Lilien, and Robins (I 9S7) suggest adding a time-varying intercept to the basic univariate model (12.2.2). Writing r,+1 for a continuously nllllpounded asset return which is the tillle series of interest (since we no longer work with a mean-zero innovation), we have
( 12.2,:~()
'rI
I where 1+ t is an lID random variable as befure, and a; can follow any (;AR( :11 I process. This GARCi/-in-mean or GARCH-M model makes the conditional €
; mcan of thc return linear in the conditional variance. It can be straiglllforI wardly estimated by maximum likelihood, although it is not known whethn thc model satisfies the regularity conditions for asymptotic normality of the maximum likelihood estimator. ' The GARCH-M model can also bc specified so that the condition'llmean is linear in the conditional standard deviation rather than the conditional variance. It has been generali7xd to a multivariate selling hy Bollerslev, Englc, and Wooldridge (1988) and others, but the number of paralllet('\'s increases rapidly with the numher of relums and the model is typically applied to only a few assets.
I
The Instrummtai V(/ri(/bles ANm)(lch As an alternative to the GARCII-M model, Campbell (1987) ali(I Ilarvcy (1989,1991) have suggested that one call estimate the parameters linking ; first and second moments by GMM. These authors stan with a lIIodel fi)I' 'the "market" return that makes the expected markct retunt linear in its OWII variance, conditional Oil sOllie vector HI cOlltaining I. inslnlllH'lIts or forccasting variablcs:
E[ r,.,I+ dH, J
(I~.:DI)
12.2. Motif/I t!/ OWlIgi IIIi Voill/iiil.v CaJllphell alld l/arvey assullle that cOIl
(~XP(Th'd
n'llIl"IIs are lillt'ar
r"'.1f I - lI,b",.
( 12.2.32)
1"111,1+1
Here bIll is a vector of regressioll codlicicl\L~ of Ihe market relurn on the illslrlllllenL~. The error lim.ltl is the dilfnellce betwe('11 the market retllrll and a linear comhination of the instrIlJllenL~, while Ihe error em.ltl is the dilference betweell the market retllrn alld a lillear fllllriiOIl of ,tl' The JIlodel (12.2.:{ I) implies that the errors IL",.'II alld P'".H I ~IIT both orlhogon;lI to the instrulllenL~ H" With I. instrumellts, Ihne air 2/. orthogonality condilions aV;lilable to estimate I.·t-2 paralncters (Yo, YI, ali
1I;•.
E[r,.II_1 1 H,] = Yu
+ YI COV['i,I,I' I'''',lfl I H,l.
(12.2.:tl)
IIthcre arc N such asseL~, we ddilli' ;, wctor T'II eo' l'l,ll I ..... IN,If I J'. The cOllditiollal expectatioll OfTH I is given by Elr", 11,1 = II,B, where B is ;l matrix with NI. coefficiellts. We dcfille ('!Tors
u," ,
r,t' - H,B, ([2.2.31)
and we geL 2NI. extra orthogonality conditions to identily NI. + I eXlra paraJlleters. The tOlalnlllHber of orthogon;lIity cOllditions in (12.2.33) and ( 12.2.34) is 2(N + 1)1. and the total number of parameters is N (I. + I) + I. + 2. Thus the model is identified whenever two or lIIore instruments are available. Harvey (I !IH9) further gcncralizes lhe 1II0del to allow for a time-varying price of risk. lie replaces (12.2.:\:1) hy
where Y" varies through tillle but is holds f(JI' lhe market portfolio itself,
YI,
=
COIlIIlIOIl
ID all assets. Sillce (12.2.35)
I 11,1- y" Var\ 1'",,1 t' I 11,1 .
E[r,,,
1-11
(I2.2.3G)
alld Harwy uses lhis tu eSlimate the IIwdcl. I Ie substitutes (/2.2.311) into (l:!.:!.:I:',). lHultiplies through hy Var\ I'm, I f I I H,I. ;lIl1lllsl's E[ 1'",,11 till,} =
12. NOII/iIlNllili,.., ill FiI/({lIrial lJi111l
II,h",
;11)(1
FI", I ! 11,1 "" V,II
II,B 10 COllslruct a new eITor vector
(11I1.'t I
(r"
I
.
II,U)(f"'./l1 - H{h,,.)(ll,b,,, .•
)'11).
(I ~.~.:\7)
I Ltl"\"('\" I .. pLIi·(·' ,'" I ill (I ~.~.:l-I) \\'illl
V'I I ill (I ~.2.:\7), alld drops 111(' error ,'",./11 ill (I~.~.:\~). Tlli-;giv"-;;I s)'SIt'1Il willi I. fcwt'l'orlilogollalilycolldiliolls alld Ol\(' I,'" p;lr,III1('11T 10 (·"illl;II" (sill(,(' YI drops (HII of lite lIlodel). Tlte 1I11111hl'l 01 o\('rid""lil\'ill~~ 1I',"i
nil' (:lIlldiliOIllt! (.',1/',\/ ({lIrllh" {'l/mll1li/illll'" (;AI'M Fqll;lliOIl' (I ~.~.:VJ) alld (I ~.~.:\(i) (';111 Ill' I'e\\'rillell as ( 12~.1H)
11" . Co\I'",I.r,,,.I<1 11,1/\'.111,.",.141 II,!, liIe cOlldilional h"L1O!',""·li"itl'I!t""I.lk,·I"·III"I,;,"dA, '''' Fir",."I 1II,I"'Yo,lhl' "'1'('('\ .. <1 "S(l'SS ,,'111,11 oil Ill<' 111;11 }.. .. I OV('I .1 risklt-ss relllrll. J;lg;IIlIl;IIh.1I1 ;111<1 W,lllg (I ()()(;) eillpila,i/{' 111<11 Ihis cOlldiliollal v('I'sioll ollil(' (:,\I'r-.ll1el'd 1101 ill'lll\' liIe IIllcolldili"llal CAPM Ihal was disCllSs('d ill Chapin :1. If I,',· 1;lk,· III)(Olidiliollalexpenaliolls of (12.2.:\H), we get
\,)\('1('
( 12.2,19) I "'IT FlAIl i., lilt' IIIICOlidilioll;1I t'xp(,(,I,'d ,'x('('ss relllrll Oil lile mark("t. F.\fJ"1 is II\(' III1COIUliliollal expectalioll or Ihe condilional bela, which need llill 1)(' llie ';1111(';" liI(' IlIlCOlldiliollal hela. althollgh Ihe dilli'renee is likely 10 he slIlall, Mo,1 ill'l,ollalll, tI\(' covariance helwecn the conditional beta alld Ill<' e"pl'lll'(1 (',,!,S' IlIark .. 1 rellint AI appears ill (I ~,~.;l~l), Ass('ls whose !>"Ias alt' itigit wlll'lI till' IIlark!'1 risk I'lelllillll1 is itigh will have hir.;her 1111· ('olldilioll;t1 IIl1'all n'llints Ihall wOlild he predicled hy lite ullcondiliollal C\PM,Jlg;IIII1:1Ili;1I1 ;111<1 \V:lllg (I!)')(i) argile thai Ihe high av(,rage r('llirns Oil siliall sllltk, Illighl Ilt'l'''plaill('d hI' Ihis ('ffeet ifsllIall,slock bctas 1("lId 10 rise al lillie, wll<'11 1111' ('''IIt'( 1('<1 ('x(,(',s r('llIrll Oil Ihl' slock IIlark('1 is high, Thcy l' I'CS(, II I Slllll(' illdin'('\ "\'i1 litis rd:lliollship (SC(' \lollel'sl,'\', Ellgle, and
497
J2.2. Mudels uf Changing Volatility
Wooldridge (1988), French, Schwert, and Stambaugh [1987], and Harvey [ 19R9]), hut other papers which use the shon-term nominal interest rate as an instrument find a negative relationship between the mean and volatility of rellIrns (see Campbell [19R7] ami Glosten, Jagannathan, and Runkle [ 1993}). A~ French. Schwert. and Stamhaugh (19R7) emphasize, there is much stronger eviden'ce that positive innovations to volatility are correlated with negative innovations to returns. We have already discussed how asymmetric (;ARClllllodeis can fit this correlation. At a deeper level, it can be explained in one of two ways. One possibility is that negative shocks to returns drive up volatility. The lroerage hypothesis, due originally to Black (1976), says that when the total value of a levered firm falls, the value of its equity becomes a smaller share of the total. Since equity bears the full risk of the firm. the percentage volatility of equity should rise. Even if a firm is not financially levered with debt, this may occur if the firm has fixed commitments to workers or suppliers. Although there is surely some truth to this story, it is hard to account for the magnitude of the return-volatility correlation using realistic leverage estimates (see Christie [19R2] and Schwerl [1989]). An alternative explanation is that causality runs the other way: Positive shocks to volatility drive down returns. Campbell and Hentschel (1992) call this the lIultitilityjeedback hypothesis. If expected stock returns increase when volatility increases. and if expected dividends are unchanged, then stock prices should fall when volatility increases. Campbell and Hentschel build this into a formal model by using the loglinear approximation for returns (7.2.2tl): (12.2.40) where
I)d.I+1 == E/+1 [fpitJ.dl+l+j] - EI }=o
[fpj.tJ.dHl+i] }=o
is the change in expectations of future dividends in (7.2.25), and
I)r.I+1 == E/+1
[f }=I
piT1+1+i] - EI
[f
piTHl+i]
}=l
is lhe change in expectations of future returns. Campbell and Hentschel model the dividend news variable I)d,,+1 as a GARCII(l.I) process with a zero mean: TJd.HI-N(O, (//> , where 0,2 (tJ + 1 + aTJ~/Y They model the expected return as linear in the variance
/ll1;:
=
"111 !;,n Ihry """ a morr I:ell"rdl asymmelric Illodel. Ihe quadrdlic GARCH or QGARCH IIln,It·, nf Srnlan:, (1991). Thi, i. 10 allow Ihe moclrl 10 fil asymmelry in retllrns even in
Ihr "hsrllcr (If volalililY ferclback. However Ih" h",ic iclea is more simply illlIslQled using a SI'll"\:mll:ARClllllod,,l.
}2. NOlllilleanlin ill Fillflllrilli })lIlf/
498
of dividcnd ncws: E, [ rltl J = Yo + YIO? Thcsc assumptiolls illlply that the revision in cxpectatiolls ofa11 futllrc returns is a multiple oftoday's volatility 2 2 • , shock (1/".1+1 - a l ). 11 •. HI
E'+I
[t, pJr,t i] - E, [tPJrHI+J] It
J~I
J=I
(I :1.:1.41)
where O=Ylpa/(I-p(ex -\- Il)). The codTlcieut 0 is large when YI is large (for then expected returns move strongly with volatility), whcn ex is large (for then shocks feed strongly into future volatility), and whell ex + Ii is large (for then volatility shocks have persistcnt c!fecLs on expccted returns). PSubstilliting into (12.2.-10), tlH" implied process for returns is
l I
(I ~.~.1~)
,This is .not a CARel I process, but a quadratic function ofa CARel I process.
'I'It implies that returns arc negatively skewed because a large negative realiza,tion of I/d,l+l will be amplified by the quadratic term whereas a large positive ,I realization will he damped by the Cjlladratic term. The intuition is that any
\large shock of either sign raises expected future volatility and required reIturns, driving down the stock rellirn today. Conversely, "no news is good i news"; if IJd,I+I=O this lowers expectcd future volatility and raises the stock I\return today. Campbell and Hentschel lind lllllCh stronger evidence for a positive price of risk YI when they estilll;IlC the model (I ~.2.12) than whcn ithey simply estilllate a standard GARCII-M model. Their results suggest that !hoth the volatility feellback effect and the leverage clfect contribute to the iasvlllllletric behavior or stock market volatility. , '
12.:{ Nonparametric Estimation III somc financial applicatiolls we Illay be led to a functiollal relation between two variahles Y and X witll<,ut the bCllcfit ora structural model to restrict the parametric form of the relatioll. III these situations, we call use llVll/mrtllllelric estimation techniqucs to capture a wide varicty ofnonlillcaritics without recoursc 10 anyone particular specification of the nonlinear relation. In contrast to the relatively highly strtlctllred or /wTameln'c approach to estimating nonlinearities described in Sections 12. I and 12.2, lIonparaml'tric (·stim.llion requires few assulllptions ahoutthe nature of the nonlinearities. However, this is not without cost-non parametric estimation is highly
J2. J. NOII/){Irt/lIll'/lir b/illlOliol/
4!1!I
I'crllap~ tll(' lIlost COllllllOllly tiSI'd nonparalll!'tric !'stilllators ilre SlnIHJ'''il/K estimators, in whkh ohservational CJ"rors are n'dlln~d hy aver,lging the data in sophisticated ways. Kernel regression, orthogonal series expansion, pr<~jl'Clion ptlrsuit, nearest-neigh hoI' estimators. average derivative estimators, splines, and artificial neural networks arc al\ examples of smoothing. To understand the motivation for stich ilvcraging, sllppose that we wish to estimate the relation he tween two variahles 1', and X, which s;llisfy l'1
=
m(X/)
+ fl'
1=
I ..... T.
(12.3.1)
whert' m(·) is an arhitrary fixed btll tlnkllown nonlinear fllJlction and If,) is a zero-mean 1/ n process. COllsidn eSlil\laling 111(.) al a parlit-Illar dal(' In (ill' which X4,=')'(/, and suppose thilt li,r this one observation XI", WI' can ohtain 11'/II'flifli independt'nt ohservations of the variahle YI " say yl l,=YI .. , .. l',::=.v". Then a natural eSlimator of 11\1' fllndion 111(') atthl' point X(, is " I" -I I> = -L n ;=1
[III(X(,)
+ E; J
( 12.3.2)
II ;=1
=
( 12.3.3)
and by the Law or Large NUJllbers, the second tcrlll in (12.3.3) becomes Jl(,gligible for large II. Of ('ourS!', if I I',l is a til\le series, WI' do not haY(' II\(' Iuxllry of n'pl~al('d observations for a given XI' However, if we assnllle Ihatthe Iilllction m(·) is slIfflciently smooth, then for tillie-series ohservations X, Ilear thc vallie Xth Ih!' corresponding values of YI shollid be close to m(.),(/). In other words, if III ( .) is Sllf/iciclltly smooth, then in a smallneighhorhood around .),(" m(~/) will be nearly COllstant and may be estimated by takillg an average of the Y,'s that correspond 10 those XI'S ncar .),(,. The closer the X,'s are to the value .),(,. the doser all averagc of corresponding Y,'s will he to m(xo). This argues for " lIJriKfltrd "wrage of Ihe Y/s, where Ihe weigh Is dedint' as Ihe X/s get farther away from .),(/. This weigh led averagt' procedllre of estimating m(x) is lhe essell!'e ofsllloolhing. More formally, (()r allY arhitrary x, a slIIoothing estimalor of lIl(x) llIolY be expressed as T
. lIl(x)
I L... ""' (UI. r!x) r,. == T
( 12.3.4)
I~I
where the weights Iwt.J·(x») arc large for Ihose )'I'S paired with X,'s lIt'ar x, alld small (Ill' lhose Y,'s wilh X,'s f;lr frolll x. To impkllll'lll sllch a procedllrl', Wl'\JJllst dl'fill(' what WI' ml'an hY"llcar" and "br". If we choose too large a nl'ighhorhood arol\nd x to (011111111/' thl'
J2.
Ntmlinf(I/-jlil'.~ ill Fi/liI/l/"iaflJa/a
aVl'I';lgI' will bl' 100 slIlooth and will not exhibil Ihe IlOlllilll'arilil's or III ( .). II' WI' choose 100 sm;,l1 ;t nei~hb()rh()od aroulld x, thl' wl'ighll'd avna).:t' will hI' 100 variahll', rt'llntillg Iloisl' a, well as tIll' varialions in 11/(.). Thnd(lI"t·, the weights (lv,.-r(x)llllust hI' chosen can'IIIII~' to halall(,(' th('s(' two cOllsideratiolls. W(' shall ;,ddn'ss lhi\ and other rl'latl'd issll(,s I'xpliritlv in S('('\iolls I ~3.1 to 123.:1 ;11111 St'ction 12.r). ~en\lille
An illl)lorlant silioothill!-:' tl'dlJliqlll' 1'01' I'stimalill~ 11/(') is !(I'm/'fogll·.llioll. III thl' ktTlld n'!-:,ressioll 1I10tl('\, lilt' wl'i~lIl runnioll wu·(x) is cOllstructl'd ('mill a prohahilily dl'nsity rllllt'liOIl K(x), also c
!
K(x) :::. 0,
K( u)du
= 1.
Dl'spite thl' 1;I('t th;1I KCd is;1 proh;lbility dellsilY runction, it Flays IlO pro!>· O, WI' ran ch,"1~e its spread hy varying It ir WI' (It-rlnt': K,,(II)
==
I
(I~.:ttl)
-K(II/It),
It
Now WI' cm ddilll' Ihe \\'l'i!-:,hl function lO be Ilsed in tilt' wcighlcd ( 12.:1.'\) as
,\\'l'J';I~C
( 1:!.:-I.7)
I
g,,(.\)
r
1- L
K,,(x - X,),
( 12.:~.H)
/~I
If II is V('IS ,lIIall, th(' r 1;1" (x) ()f /tI( x):
J 2.3. Non/Jllwmetllc Estimation
. t .. lI.
lI.
...
A
~
£I.
l}.lI.
lIfIrl.
. ..
£I.
~ lI.
C.
01 lloll.
t:. lloA
A lI. A
~
"
'"~I
1(L)--~~--~~"~~--~~~-w~~--~--~~~~.--~------~"w 2 " T ~"
x
• Figure 12.4. Simulation of Y,
Sin(X,) + O.5f,
Unclcr certain regularity conditions on the shape of the kernel K and the magnitudes and behavior of the weights as the sample size grows, it may be shown that mh(x) converges to m(x) asymptotically in several ways (see Hardie [1990] for further details). This convergence property holds for a wide class of kernels, but for the remainder of this chapter and in our empirical examples we shall usc the most popular choin' of kernel, Ihe Gaussian kernel: I .' K,,(x) = --e-:;:E. (12.3.10)
h.,ffii
Au Illustration oj Kernel Regression To illustrate the power of kernel regression in capturing nonlinear relations, we apply this smoothing technique to an artificial dataset constructed by MOille Carlo simulatioll. Denote by {X,} a sequence of 500 observations which lake on values between 0 and 27T at evenly spaced increments, and leI {r, }Ill' rdalcclto {X,} through the following nonlinear relation:
( 12.3.11) where {f,} is a sequellce of liD pseudorandom standard normal variates. Using the simulated data IX" Y,} (see Figure 12.4), we shall allemptto estim,lIe the conditional expectation E[ Y, I X,) = Sin(X,), using kernel
502
12. Nonlinl!(lrilies ill HI/'II/rilll I )ata
regression. To do this. we apply the Nadaraya-Watson estimator (12.:t!l) 'th a Gaussian kernel to the data. and vary the bandwidth parametn It h tween 0.10, and 0.50, where is the sample standard deviation of (X,!. y varying II in units of standard deviation. we arc implicitly nOrlnalil.illf.( the e~planatory variable XI by its own standard deviation. as (12.3.10) SIlf.(f.(cstS. I For each value of II. we plot the kernel estimator as a function of X" and t~ese plots are given in Figures 12.5a to 12.5c. Observe that for a bandwidth 0.10•• the kernel estimator is too choppy-the bandwidth is too small t provide sufficient local averaging to recover Sin(XI). While the kel'1lel e timator does pick up the cyclical nature of the data. it is also picking up dndom variations due to noise. which may be eliminated by increasing the bjllldwidth and consequently widening the range of local averaf.(inf.(. Figure 12.5b shows the kernel estimator for a larger bandwidth 01'0.30 A. W lich is much smoothcr and a closer fit to the true conditional expectation. As the bandwidth is increased. the local averaging is per/imned over M ccessivcly wider ranges. and the variability of the kernel estimator (as a \ function of x) is reduced. Figure 12.5c ploL~ the kernel estimator with a b\mdwidth ofO.50'x. which is too smooth since some of the genuine variation or the sine function has been c1imiuated along with the noise. In thc limit. tIle kernel estimator approaches the sample average of I YI ). and all the variability of YI as a function of XI is lost.
~
a.
0t
12.3.2 Optimal Bandwidth Seleclioll
It is apparent from the example in Section 12.3.1 that choosing the proper bandwidth is critical in any application of kernel regression. There arc several methods for selecting an optimal bandwidth; the most cOlllmon of these is the method of crvu-validalioll. popular because of its robustness and asymptotic optimality (see Hardie [1990. Chapter 5] for further details). III this approach. the bandwidth is chosen to minimize a weighted-average squared error of the kernel estimator. In particular. for a sample of T observations IX" Y,l:~r. let mh.j(X,l =
TI "" L cvu( Xj) Y
I
(12.:tI2l
It- ]
which is simply the kernel estilllator based 011 the datasct with observatiolJ j deleted. roaluated at the jth observation Xj' Theil the cross-validatiolJ function CV(h) is defined as (''V(h) =
.,. I"" T
.
L[ YI - mh.I(XI )] 2 o(XI ).
02.:1.1:\ )
1=1
wllC:re a(X,) is a nonnef.('llivl' weif.(bt function that is required to redlll'l' bOundary clTcCIs (sec IHrdll' I 1\I~lO. p. Hi2] for furth('r (\isntssion). Thl'
,
o.
~'.::,~I 1
I·
'''''\'1
I
~. L----~----~-·---~,.i"""'"-
'l.'f
(a)" = 0.1,;,
L-_ _ _
~
___
~
_____
. \. ::::,\. i ~.;---
_
.. '
:, OS:£1iii~'~<~'~G:;
.r r-
,"
~.'" II
._._---------"
Figure 12.5.
I\,.II,I'I,.;,I;lIIlIlm
._..
!
1_.
""'1/1111,'(1//1/1',1 III "1111111'
wi 'h,ll/
flillnioll C\'(/I) is (;lIkd IIII' /Toss-validalioll fUllctioll I)('callse it \'alid,ltes Ihe SIHH'SS 011111' k('nll'l eslillJalor ill fittillg ! Y,j across Ih(' T sllhsailiples
! XI'
i'dl l " (,;ll'h \villl 011(' oilservation olllitted. Th(' optilll;d halldwidth is IIII' olle Ihal IIlillillli/('S Ihis fllllctiOll,
12,
>. ., ,\pl'/age /)l'/illu(illl' I',\(ill/u((lr\
For mall)' fillancial applicaliolls, W(' wish 10 relale l'1 10 ,\I'llI'm! variahks Xli, ' . , , XAt 1I0llparallll'lri('ally, For ('"alii pie, we lIlay wish to lIIodel th(' ('''1)('('1('(\ H'tlln!'; of stocks alld honds as a nonlinear fllnnion or Sl'HTal bClors: Ih(' 11Iark('1 r('IIIJ'II, illl ....('si rale spreads, dividelld yield, ete. (se(' i.o and MacKilll;l), 11~I~Hil), SlH'h a task is cOllsiderably llIore alllbitiollS tltall Ihe tlllivariate ('X.I 0; h('IICC WI' would lI('ed at !cast 100,000 OhS('\'V;ltiolls to ('nSIII'(' an avcrag(' ofjllst o\le dat;1 poillt ]leI' lleighhorhood! This ('w\t'"/tlillll'/lIiol/ulil.Y call 0111)' he soh-ed hy placillg \('stri( tinllS till tl((' killds .. r 1I(1l1lill('arili('s that are allowilhk, For ('"alllple, SIlPP"'" ,I lilll'f1r ('oillhillatioll of the .'l,t's is relaled I" )'1 IHHlparalll('tril'all\', This has tl\(' adv,lIIlage of captllrill~ illlportallt 11011lilll'arili('s ",hilt- prm'idillg SlIliicil'lll slrllclure to ]ll'I'IIIil eSlilllalioll wilh n'asoll
wh('\'e XI = I XI,.,. Xlt!' is IH>\\' IIO\\ill~ Iwo cOlldiliollS is I nit': (I) tht' X/s an' IlIltltivariatl' lIorlllal \'('('tors; or, ilIon' g('lll'rall\', ('2) 1",1 X" 1 X;/11 is lil\(';II' ill X;/-J li>r i I" .. , h,ll II' Il('illt('\' oflh('s(' cOlltiiliolls holds, Sto[..t'I' (I(JH(i) propos('s all ill~('lIiolls estilllalor, Ihl' 1/11,'mgt' 1II'I'i1'II1i1'I' ('slilll;llor, whidl rail ('stillla\(' {3 rOllsistl'llllv (SIT ,"so St"k('r 11~1!1~ll,
=
II nih .. ,'e IIlulllllllllllllll I" ... I1I'1lit'd
h\'
IlIld,i\.1I i.llt' 1I01"l1l.1i
X, \
1.111
i... tl .. "
"Ili .. li('d
lor
110(1-
ntH u\.\\ dhlHit ~\n\" :-.\"unu"" it tll .. u ilm'iHu~. St't' (:hamh"rbiu (I~U,\:\h). (:hul\).!. .\lul (~,)ldht'l ~t'r 11~IH I), IlI-,II"" ,11111 It i,II II"H I), ,.. III RUIIII II~)H:\),
12.3. NOn?llrametric Estimation
Average derivative estimators are based on the fact that the eX1pe<:taCloI11 of the derivative of m(·) with respect to the X, 's is proportional to f3:
02.3.15) Therefore, an estimator of the average derivative is equivalent to an estiIllator of (3 up to a scale factor, and this scale factor is irrelevant for our purposes since it may be subsumed hy 1ll(.) and consistently estimated by kernel regression. There are several average derivative estimators available: the dire~t, indirer/, and slope estimators. StoKer (1991, Theorem I) shows that they are all asymptotically equivalent; however, Stoker (1992, Chapter 3) favors the indirat J/O/Je estimator (lSE) for two reasons. First, if the relation between Y, and X, is truly linear, the indirect slope estimator is still unbiased whereas the others are not. Second, the indirect slope estimator requires less.precision from its nonparametric component estimators because of the ~SE's ratio form (see below). 1 Heuristically, the indirect slope estimator {J'SE exploilS the fact thaf the \tI\kllOWn parameter vector (3 is proportional to the covariance between the dependent variable Y and the negative of the derivative of the logarithm of the marginal density of independent variables X" denoted by 1(-). Therefore, by estimating Cov[ Y, 1(·)], we obtain a consistent <;stimator of (3 up to scale. This covariance may be estimated by computing the sample • I covariance between Y and the sample counterpart to IC·). . More formally, {JtSF. may be viewed as an instrumental variables ~IV) estimator (see Section A.I of the Appendix) of the regression of Y, on X, with the instrument matrix H: (12.3.16) whereY -
[YI ... YT ]',
X'I
H-
X=
X'I
(12.3.17)
X'T i(.) is an estimator of the negative of the derivative of the log of the marginal dellsity of X" and IbCx) is an indicator function that trims a portion of the sample with estimated marginal densities lower than a fixed constant b: (12.3.18)
506
J2. N0111inearilif.s in Fill(lI/(ia/ [)lIla
In most empirical applications, the cOllstant II is set so that betweell I % and 5% of the sample is trimllled. To ohtain i(.), ohserve that if !(x) denotes the marginal dcnsity of XI, then the Gaussian kernel estimator or !(x) is given byl2 .
X
!()
""
I
L K (X-XI) --T
I
--
T '" I 1=1
(I :l.:1.l9)
I ' I
where ( 1:l.3.20)
(:lJT
)-k/~ ,:xp [-~(X :lit'
-X,)'(X -
XI)]'
(I :l.:t:ll)
Therefore, we have
IlL K ,(X-XI) -T
-T hlr+1
/(x)
\
) -I- "J' 11r+1 I
I I I
( 1:l.:t2:l)
h
1=1
LT K 1=1
(X-XI) -II
·(x-X,)/II
'
•
and we can define /(x) to be /(X) _
j'(X)
j(x)
( 12.3.24)
Despite the lIIultivariatc nature of !(-), obscn-e that there is still unly a s ngle bandwidth to adjust in the kernel estimator (12.3.19). As ill the ulivariate case, the bandwidth controls the degree of local averaging, but no over multidimensional neighborhoods. ~a practical maller, the num rical properties of this local averaging procedure may be improved by I~O!JllaliZillg all the Xi/'S hy their own standard deviation~efore ('Oll\llIllillg !(. ,and then llIultiplying each of the Iii'S by the standard deviation or the ("ol,rcsponding XII to undo the normalization. I
I tl Nole Ihallh" ball
j(X)
i,. ill g('I1,.,.al, (lilf.. relll 1I01ll1h .. b,,"<1,,"i
Ih(' lIullparillHclrir ('Milll;t.lor of m(·) in (12.:\.1 ,1). <:ros~-\'aliclalion terhlliqlU'~ may he' ",{'ct to ,,'I<-rr hOlh; how",'rr, Ihis lIIay he fOIllj>III.lIiollally lOll
may suflirr.
12. J. NOII/Jllmmrtrir /<:stilllllliml 12. J." iI/J/Jliffll iOJl:
507 1·.~\li 11/(// IJlg S/a/r·I', I, I'
1)1'1/ Iii il'.\
( ) IlC or the most ilIIportalll t 1I~'on'l ical ;t< ku Ices ill tI H" enll I< II lIics or ill\'cstIIICllt IIl1dcr IIIHTrtaillty is thc till\('-st;\I(' pn'kn'IH (. lIIodd ol"Anuw (I!Hi4) alld IkiJrclI (I !);I!) ill which tllcy illtrodllcc prilllitiv(' St,("lIIitics, each payillg $1 ill olle specilit: state ur lIatlllc allc\ nothillg othcrwise. Now kllOWII as Anow-Dr!""ll securities, they ;u·(· Ihe fllll(blllctltal htlildillg hlocks rrolll which we have derived 1I111t:h or otlr ClllTCllt tllHic-rslatldillg or c'collolllic equilibrilllll ill all uncertain cllvirolllllcllt. III practice, since true Arrow-Ikhrt'll sc:curitics arc not ),cttradeti Oil any org;mi/.ec! exchange, Arrow-Dehn'\1 prict's arc nol ohsnvahk. I:1 Ilowewr, usillg nonparallletric tet:hniqnes-spt'("ilically, IIlultivariatc kernel reg ressioll-Ait-Sahalia and 1,0 (I !)!)(;) develop estimators It)!' sllch prices, known as al/ail"jJ1'irf(lrllJily (SPD) in the cOlltinllous-state casc. The SI'D contains a wc;,lth of illr()flllation confeflling the pricing (;\llCl hedging) or risky assets ill all (,("OIlOIllY. In principle, it can he IIsed to price olhcr assets. evell assets thaI are flllTently not traded (sec Ait-Sahalia alld 1.0 II !)!);, J lin exalllples). H More importantly, SPDs containllIllch formation ahollt preferent:es and as.;('1 prke dynalllics. For exalllple, if parametric restrictions arc illlposed Oli Ihe data-gcnerating process of asset priccs, the SI'D estimator may he used to illrer the preferences or the represcnlatiV(' agellt ill all \~«(llilihrillm model or asset prices (sec, ((II' example, Bick [I ~)~)() J and lie alld Lelalld [I ~)!)3 J). Alternatively, if specific preferellces arc imposec\, the SPD estimator may he used to infer the data-generating process of asset (lIkes (sec, f"r example, Derman and Kani II !)!)4 J. Dupin' II !l!l4 J. .Iackwcnh alld Rubillstein [I!)!);,], LUllgstalf [I!!!!:!, 1!l!)4J, Rady 11!)~)4J. Rui>illstein II!!H5J, ,111(1 Shimko [1991, 1993]). Indeed, Rubinstein (1~)Hr,) 1i;ls observed that allY two or the following implies the third: (I) the represelltative agent's preferences; (2) asset price dynamics; alld el) the SPD. /)rjlJlilicJ/l
of Ihe Siale-Price Dmsily
To ddine the SPD formally, consider a standard dynamic exchange economy (see Chapter 8) in which the equilibrium price /1, of a security at date I with a single li(l'lidating payofr Y( Cd al date T that is a rUllctioll or aggregate cOllSlImptioll Cr is givell hy: 1', =
E,l Y(Cr)Mu·J.
(, r"lJ'( C r ) (I'( (;,)
Ii",
(12.3.25)
UThi~ lII.lY !oo(ioll rhallgt' with tht' 'Hi\"t'llt 01 W/wu}ulln, IHolll' . . cII hy (~"rlll.\II (197H) alld Il.lk;.II.\sOIl (1~l7Ij, t !J77) and fllI .... ·lllly IIl1clel dndoj>IIH'1I1 hy 1.(,1,111<1 ()"!Irirll Kllhinsleill :\.':-;oCictH.· . . , 11Il'. St'l' M.tSOIl, Merton, Pl'IOld. ,wei TIlt.1II0 (I!t~):)) 101 fllllli('" dl·I.lil~. I~Of((lursl·. lU ..ukt·t~ lH\\s.l he dyt\~u"i<:<.llly «Hllpi(.'t<.· tot Mt<. h ~"in'!' HI 11<" 111<.·O\nin~I\lI~(·<.·. l()rt'~;lIl1l'l('. c:OIl~f""tinid(,'5 (19H~), This ;l\\llIlIptioli i~ .t1I1ICI"1 .!IW;(VS ;l(loplC'
12. NVlIlillt'milil's ill J-Illuwia/ lJn(a
where Mu i~ Ihe m;Irgillal rale o/' suhSlilutioll hetween dates I alld T, alld is the rale or liule prefcrcllu·. This well-known equilihriulll assel-pricing rd;lIion ('quail's CIIITl'llI price o/, IIII' secllrity 10 its cxp('cted disc()llnlul /'lItllre payorf, discouilled "sing Ih(' stochastic discoullt I;Inor. I.ucas (1~}7Hl (,hwrI'('s Ihal (I~.:t~rl) Il('cdnol implya martillgale process fill (I', I. supporting I.eroy's (I ~}7:\) ("()lItelltiou Ihat Ihe martillgale plOp('ny is IH'ilh('J';I Il('cessary 1I0r sllfliril'nl conditioll for rationally delerlllined asset prices. Ilowl,\,("-. assumillg that the conditional dislrihlllion of future consulllillioll has a d('mitl' representalion f,C'l. Ihe condiliollal expectalion ill C12.:t2!il call he Ie-expressed ill Ihl' filliowing way:
I)
J I"
0=
1)1'-1(1.'.«(;..)
r({:,)
",I r "
I' ",II
I
U; ((;, 1
I
f,( Cd dC r
J
nCr) f,*(Crl dC,
/)1':;1 l'U:rlJ,
(I ~.:\'~Hl
.wht·I'(·
alld Ii. J is the cOlllillllously ('OlllpOlllHlcd 111'1 rale of rl'lurll bl'twl'en / and T o/'an assel pnllnisillg 0111' ullil o/' COIISUIIlPlioll al T; i.e .• it is Ihl' relllrn Oil the riskll'ss ass(·1. This version of IIll' Ellll'r eqll;lIioll shows Ihal an assel's (lnTl'nl price r111'1' I,' is ohlaillcd, it Cllllw IIsed to pricl' allY asst' I at datI' / with a single liqllidalill)!; pa\'olf ~\I d;\I(' 'J' liI;1I is all ;Irhilrary fllllrtiOIl fOIlSlllllptiOIl Cr· II; Sl'lh also prol'ide Ihe lillk h('I\\'('('1I pl'l'l'cn'IH'('-has('<1 eqllilihrillill Illodds ollhl' 11'1)(' discussed ill Chapin Hand al'hilrag('-has('d dl'rivalivl' pricillg 11l0dl'Is or lilt, 1\'1)(' disn"s('d in Chapter ~}. Indl'ed. implicil ill IiiI'
r
or
"'S"t' Ilu.lIIg .IIHll.it/cllh'·lgl'l (I~IHK. (:11;11'1('1 !",) for" moft' dl'IOIilt·d di~nl., . . i()11 o,-SPD .....
'''S('nllili('~ widllllul1iplc' p.I\,Cllb ,11111 infillilc'itori/olls call also h(' priced hy tIle" SPIl. hilt ill 1111"(' fa",,',... lIu' SI'D lilli" Iw :lpl'ltlpri;lId\' It'e1din('d to (""pilin' 1IIl' til1H'-\';u';;nioli ill !lit, margillal r.II('\ 01 ,"b ... lilillioll-.... CT Brec·dc'" .HHI LIIIt'lIht'rgcf (I~J7H) and R.uitwr (I!JH:!) lor
furth",.
di.~(·IIS ... iol1.
12.3. NOlltJarumetric Estimation
prices of a// financial securities-derivatives or not-are the prices of ArrowDebreu securities, and these prices may be used to value all other securities; '. " 1\0 llIatl('r how complex. . I'lirill~ Derivalives
wilh SI'J)s Under some regularity conditions, we may express as an explicit function of I and T so that a single SPD (C T ; I, T) may be used to price an asset at any date t with a single liquidating payofT Y(C T ) at any future date T > I (s("t' footnote 16):
r
r
(12.3.30) and we shall adopt this convention for notational simplicity. For example, a European call option on date- T aggregate consumption Cr with strike price X has a payofTfunction Y( Dr) = max [Dr - X, OJ and hence its date-I price G( is si III ply (12.3.31 ) EVt'1l the most complex path-independent derivative security can be priced and hedl!;ed accordinl!; to (12.3.30). For example, consider a security with the highly nonlinear payoff function:
Y(C)
a-b
------:--~
1 + exp[ -(l(e - a)1
a > 0, Ct
+ b,
b < 0
(12.3.32)
I c + -log( -a/ b).
(12.3.33)
fJ
This payofT function is a smoothed version of the payoff to an option portfolio commonly known as bullish verli((l/ spT1'ad, in which a call option with a I,ow strike is purchased and a call option with a high strike price is written (see Figure 12.5 and Cox and Rubinstein [1985, Chapter 1] for further details).
ExtmrtinK SPDJ from Derivativl'J Prias There is an even closer relation between option prices and SPDs than ( 12.~.30) suggests. which Ross (1976), BarlZ and Miller (1978), and Breeden alld Litl.enberger (I978) first discovered. In particular, they show that the second derivative of the call-pricing function C( with respect to the strike
12. NOlllineanlies in Financilll /)lIla
510
tI
----.--------.--.-r-----
()
~--------~----------
b 1-----'
Figure 12.6.
fl
--.----.----------------------
() ~--------~----------
b
------------------ - - --- -- - - ---
liul/ish Vrrlim/ S/m·(/(ll'ayoJ! I'iml"lioll tlwl SlIwolh,rI Ih,j'JII
price X must equal the SPD:
()~
'\ I I
G,
iJX~
(l:.!.:I.:H)
r.
-therefore. impounded in eve I]' option pricill!:: formula is the SI'D i To estimate the SPD using (12.3.34). we require a call option priciAg formula. Although many parametric pricing formulas exist (sec Ilull l}993. Chapter 17] for some poplllar examples). Ait-Sah'lli" and 1.0 (19%) clmstruct a nonparamellic pricing formula that places fewer restrictiollspritnarily smoothness and weak dependence-Gn the data-generatin!:: procbs of the underlying asset's price. While parametric formulas such as t~ose of Black and Scholes (1973) and Merton (1973) olTer !::reat advant· ges when the parametric assumptions (e.g" geometric Brownian motion) a e satisfied. nonparametric methods arc robust to violations of these ass Imptions. Since there is some empirical evidence that casts doubt Oil such a sumptions, at least for stock indexes. 17 the Ilonparametric approach may 1 I~ave some important advantages}X Giv!=n observed call option prices {Gj • Xj. r;} (wtTt:re rj == '1; - I,), the p,rices of the underlying asset {I';}. and the riskless rate of interest (r" /. we may construct the smooth nonpara,uetric call-pricing functft>u as
i
G(l',X,r.r,) = E[G I P.X.r.r,]
(12,3.35)
lIsing a multivariate kernel K. formed as a product of d=4 univariate kernels:
( 12.:t3ti)
17Secl.o and MarKinlay (1!)tlH).lor exam!'1<-. I"Sec IIl1tchimon. 1.0, ~nd POf\~i() (I !)!)4) ;lIul Ail-Sah"li" ( 19!){;a) rur IIIher nonl'ar;III1('lrir "plion pririnl! ahe .... aliv",_
/2. J. NOIl/mmlllelric fJli1flaiioll
511
alld hellcc L;~I II,,~(I'- /',)II",(.\: - X,)iI",(T - r,)iI",(r, - r,,lC; L:~I I/,,~ (I'
-
1',)11", (X - .\')/('" (r _. r,)I1", (r, - r f ,)
•
(1:!.:l.37) TIll' optioll \ deita alld Sl'\) l'~lilll'llor dlCll follow h)' difkn'IIliating I;;
6(1'. x. r.
I' (l'r
I I', r.
iJ(;(I'. X. r.
r, )
If)
(I:!.:UH)
ill'
( I :!.:l,3!1)
(,'It
Ii)
Under ~tandanl regularity assulliptiolls Oil the (bt.t-generating process as well as slI\oothness assumpliolls 011 the Irlle call-pricing hlllnioll, AilS.thalia alld 1.0 (J99G) show Ih,lI the e~ti1l1ator~ of the option price. Ihe optioll's della, alld the SPD an.: all cOllsistelll
n(.
1
00
G(I'. C r. r, ) =
e-·,r
Y«(.I'1'lj'U'r
I I'.
T, If)
dl'r.
( 12.3.40)
u
If tht: payoff fUllction YU is twice"difkrl'lIti"bk ill 1'1'. th(,11 (;(1'. (.
T. r,)
e-','l""
"" 1 1
l'(.l'rl/'U'r
iJ~i; Y«(, 1'/ ) - " 111'1' iJ I'i
U
00
o
I 1'.r,I',llll'r
(1:!.3.41)
u
(j2
Y«(. I'rl A.
,
------''--,2,--- (,dl 1'.
(I :!.3.42)
(1 :!.3,43)
;)/'1'
Integr,\tillg against Ginstead of it:> second derivative speeds up the convergence rate of the cstilllator-G converges at speed 111/2 ht/~ and its integral ;tj4ainst a slIIoo[h fUlIctioll I'.,. ('onvcq.;l'S a[ spcl'd 1/ 1/2 1t:1/ 2 , whereas the
or
secOlHI derivative of t; converges at 111/2 hK/~ and its intl'gr,,1 against a smooth fUlIction of VI' at /1 1/ 2 h6 / 2 • A fanor of h: I/ 2 is gainl'd in thl' speed of convergence by integrating the second derivative of thl' payoff ftull'tiol1-when it Cxisls-agai1l51 i; instead ()fintegrating tht' payoff ftlll!'liOIl itwlf against the second derivative of i;,
J L.
N()lIitllt'lm//l'.1
ill
hlltlllnllJ / )I/{II
Ail-Sah;di;1 alltll.o (1~}~1I;) ;Ipplv Ihis ,'slimalor 10 Ih(' pricillg alld dell;lhedgillg of SK'I' r,oo call alld pili opliollS IIsin!!; daily dala oblailled frolll liI(' {:hicago Hoard ()pliolls Ext'hallgl' for Iht' salllpll' pnill(l frolll.laIlILIl'\· ·1, I ~1!l:Ilo 1I"',(,lIlhl'l :\ I, I ~I!I:\, liddillg a lol;d sampk silt' of 1·1,,1:\ I olN'r";lliolls. The ('slilll;ll('s of Ihe Sl'lh exhihit lI('!!;;lti\'(" SkeWllt'SS ;llId exct'ss kllrlosis, a conmlon 1"';111111' or Itislorical slock 1'<'llIrllS (se,' Chapl('r I 1'01' !'xalllpl!'). AI,o, IInli",· nl;\!I)' paraml'lric oplion pril'ing models, IIie SI'I>· g"lInall'
12.4 Artificial Ncural Networks All ahnllalil'" 10 lHIlIP;ILlllll'tric rl'!!;rl'ssioll Ihal has refeil'l'dlllllCh IT(,'1I1 alll'lItioll illlhl' ('ngilll'lTing and hllSill!'ss commlmitit's is Ih(' IIrli/irilll 1II'IIml lIt'lwllIk Artilici;lIl1('III;tllll'tworks llIay 1)(' I'i('w('d as a \lo\lpar;\lI11'tric t('('h· niqllt', h"II(,(' IhI'S(' llIodels wOllldl;1 qllile naillrally ill Sertion 12.:1. 110\\'· (,I'('r, h,'('alls(' inilialll' Ihl'\' dre\l' IIi!'ir llIolivalioll ('rolll biological plll·1I01ll· ('\la-ill IMl'linllar, frOl1l II\(' pl\\'siolo).\)' of 1I1'!,I'" n'lls-II\('y ha\'(' Iwco\l\(' pari of a sl'l',II'al(', dislilll'l, alld blllV;l'onillg literalllr!' (s!'e I I!'I'l I., Krogh, and l'alllH'r I I'I!III. 1IIIId,illsoll,I.o, alld Poggio 1l!I!I·1l.Poggio alld (;iI'IIsi II!I!IOI, and Whitl'11!11)21 ((,I' o\'('I'"i!'\Vs of this lilerature). To 11lI1h-rs,'o!'l' till' COllllllOll lHlllparanll'lrir origins of artilicial Ill'llr;t\ lIelworks, \1'(' d('snilll' Ilm'l' killd, of lI('tworks in Ihis section, colll'Oilel1' known as Il'IImillg 1I1'lll'O,.b (Sl'" HalTolI ;\IId l\;ll'l'OIl [I ~IHH J). III ~:t'rti()1l 12...1.1 \1'1' illl!'(ultlcl' Iltl' Illtlltibl'l'!' pl'!'ccpll'On, pl'l'haps (\11' Illost pO:)Illar Iypl' ofarlilicialllCllrallll'llI'ork in IIII' rl'n'llllilerallll'l'-t his is whal IIt(' (( rill "lH'llrallH'lwork" is IIslIalll' lakl'lI 10 llH'all. III Sectiolls I ~A.~ alld I ~A.:~ Wl' presl'1l1 111'0 olher lecltlliqll('s Iltat also Itave lH'lwork illll'l'pn'I;lliolls: raelial basis ftlnctiolls, alld proj!'clion pmstlit regression.
/2.'/.1 ;\II/Ililaw',. 1'I'I'fp/Jlmm
I'nltaps IllI' silllph-st ('Xaillph- of' all arlilicial 1ll'Ilrai Ill'lwo!'k is IIII'
bil/tII)'
Ihll'\h"ft/I/I/II!t-i Ill' 1\1« :l1l1"dl alld Pills (1!1·1:1), ill wltit-It all o/ll/JIII variahk )'
lakillg Oil 011 Iv IIi(' "a!tu's 01'.1 iII/JIll variahlt-s "" i
;11)(1 (IIII' is lHlltlillearly relaled 10 a ('(lII"I'lioll 1•..• ,.1 ill Ih" rllllowill!!; way:
liTO
""
r
(ttl,,,, -II) iI
g
(12.'1.\ )
I-I
gill)
if' if'
/I
> ()
/I
<
n.
( 12.·1.2)
J2.4. Artijicial Neural Netwurks OUIPUI
It'pm
Layer I
\
X5 Figure 12.7.
•
fli1lary Thmho'" MotUl
According to (12.4.1). each input ;So is weighted by a coefficient Pj. called the ronnection strrngth. and then summed across all inpulS. If this weighted SIIIII exceeds the threshold J.I.. then the artificial neuron is switched on or ar/iva/f(l via the activation junction gO; otherwise it remains dormant. This simple network is often represented graphically as in Figure 12.7. in which the in/ilIl [{/yl7 is said to be connected to the output Layer. Generalizations of the binary threshold model form the basis of most current applications of artificial neural network models. In particular. to allow for continuous-valucd outputs, thc Heaviside activation function (12.4.2) is replaced by the logistic function (see Figure 12.8): g(u)
1
=-. 1 + e-
(12.4.3)
U
- - - - - - -I
-J()
x Figure 12.8.
-5
o x
COlll/J(lriJOII oJ /hm,iJitl, and l.ogi.lti, Arlillalion /oIl1I,liofU
10
514
J2. Noniinearilies i" FiIUl/lrillilJlllll Output
---------"
I
Hidden Llyer
I I
I
I
Input Layer
Figure 12.9.
Multilfl)'I'''I'em~/Jt/l)1l
with
f/
Sillgw llidtlml.flyr,.
Also, without loss of generality, wc setlL to zcro since it is always possihle = I ill which case thc negative of that input'S cOIll\l:ction strength - fll hl~(,ollles \ thc activation lcvel. \ But pcrhaps thc most important extension of thc hinary threshold imodel is thc introduction of a hidden lflyer bctwccn the inl'ut layer and ; thc outputlaycr. Spccifically, let to model a nonzcro activation level by defining the first input XI
y
h(t.a~g(.B~X))
{34
[.B~I PAt ... PAl
(1~.4A)
r.
X = [XI X~ ... XJ
r.
whcrc h(·) is anothcr arbitrary nonlincar activation function. J.M...this case, thc inputs arc connectcd to multiple hidden units, and at each hiddcn Hnit they are weightcd (dilTcrcntly) and transformcd by thc (same) activation function gO. Thc output of cach hiddcn unit is thell weighted ye~again this time by the a;s-and summed and transformcd bY'a second activation function h(·). Such a network conliguration is an example of a lIlullilfl)'t'r percef!tron (Ml.P)-a single (hidden) layer in this case-and is perhaps thc l1Iost com 10011 type of artilicial nellral network among recent appliclliolls. In contrast to Figure 12.7. the multilayer perrcptron has a more complex ,utwork topology (scc Figure 12.9). This can be gcneralized in the ohvious way by adding more hiddcn layers. hence the term lIlultilflyrr percept roll. For a givcn sct of inpuL~ ,md outputs (X" Y,), Ml.I' approximation amollnts to cstimating the par,ullcters of the Ml.I' nctwork-the vectors .B. and scalars 0'4, k= I •... , K-typically hy minimizing the sum of squared dcviations between the output and the network, i.e., L,( YI - L~ 0'4R«(:J~X) I~.
12.4. Arlijirilll Neuml NelworiLi
!)I!)
In the terminoloh'Y of thi~ literature, the proces~ of parameter estimation is called ImininK the nctwork. This is Icss pretcntious than it may appear to bc-an carly mcthod of parameter estimation was IlIu'k/m>/mgnlion, and this does mimic a k.ind oflcarning hchavior (alheit a wry simplistic ol1c).19 llowever, White (I !)~2) cites a numher of practical disadvantages with hackpropag-ation (numcriCII instahilitics, on:asion'll IH>lH.:onvt'rgence, etc.), hcnce thc preferred method for cstimating the parallll'lcrs of (12.4.4) is nonlincar least-sCjuarcs, Even the singlc hidden-layer MU' (12.4.4) possesses thc 1I11i1>mnl n/~ jnoxilllntionjlrojll'Tly: It can approximate atly nonlinear function to an arhitrary degree of accuracy with a suitable nluBher of hiddell ullits (see White [1992]). Howevcr, thc univcrsal approximatioll property is shared by lIIany nonpar.llnetric estimation tcchniques, illcludinl!; the nOllparametric regression cstimator of Section 12.3, alld the techniques ill Sertiolls 12.4.2 alld I ~.'1.:~. or course, this tclls liS lIothing ahotll the pcrfonnalllT of such techttiqltl:~ ill practice, and for a given set of data it is possihk for one tedltli(llIe to dOlllinate anothcr ill accuracy and in other ways. Perhaps the most importallt advantage ofMI .f's is their ahility to approxilll.tle complex nonlillear relations through the compositioll ofa network of relatively simple functions, This specilication lends itself naturally to /mml{P1 !})of/'.lSing, allli althongh there are ollTl'ntly no financial applications that exploit this feature of MLl's, this lIIay soon change as parallel-processing softw'lre and hardware become lIIore widely available. To illustrate the MLP model, we apply it to the artificial dataset generated hy (12.:i.II). For a network wilh olle hiddell la}'"r alld live hidden unil.~, dl'tlotn\ by MLI'( I ,:J), with H(·) set to the idetltity futlction, we obtain the fol!owing l\Iodel:
- :J.<111g(-'2.6'28 + O.lHL\/) - :-I.07Ig( 13.288 - 2347 X/)
+ 7.8()2gl-3.3I{) + 2.484.\/)
-+
(i.:-I20g-( -2,009
+ 4.00!lX/) (I '2.45)
where g( II) = I/( I + p- "). This modd is plottnl ill Figure 12.10 ami compares well with the kerllcl estimator dcsrril)('d ill Sectiol1 12.:1.1. Dcspit(" the fan that (12.4.5) looks nothing like Ihe sil1e IUII("Iiol1, I1l'Vel thcless the MI.I' per/i)J"Jlls quite wcllllllll1erically and i~ relaliVl'ly e
PIl\arkpropag.llioll i:'\ ('s"('lItial1y 1ht" lII('tho(\ ot .\/o(/UI\III" flNmJXWIIII'''1I III"it l)lopo~'fI hy Rohbi", .11111 Momo (1!I:lla). Set' Whil" (I'I'I:!) I,,, I"rlher del.lIl.,.
J _.
~ ""1111111'(111111'.\ III
"/lltlile /(11' Jella
7'
•,
f
....
.\
.Y.:
""
"-
t,'"
"
7'
I
MI.I' n,ll.1
1
~11l(.\)
1 II~------------~--------------------------~------------~ IT ~;r .1,
~
:!
X
"'gun'12./O.
"'1,1'(/.51,1/",/1'/ ofT,
12, ·1, 2 J:wlill{ I!I/\i,\ FIl//dill//\ The class or mdilll h(/\;\jll/J('li()l/\ (RBFs) wcre firsl IIscd 10 solvc iltlf"jJlJ/tlli()1t prohlcllls--lillilll-\ a ('lIn'(' cx:tcll), I!trollgh a sCI of POilllS (scc Powcl: ( I ~)S71 Ii)ra rc\'icw). ~Iorc I'('('CIIII)" RBFs hav(' hecncxlcnded hy scv(,ral rcsc;tl'ciH"rs 10 pcr((>I'tII Ihe tIIon' gelleral task or approximalion (sec Broollll\cad anti Lowe II ~IHH I. to. lood), and DarkclI II (IWI La lid I'ol-\gio and (;i rosi 119~10 1), III parlic"lar, I'ol-\I-\io alld (:irosi (I ~1~1O) show how RBFs (,<1II1lt'
)', = III(X,1 If"
The reglllari"lli,," (III' lIollpar;lIl1l'lrir ('slilll'lIioll) prohlelll ilia), !l1l'11 hc vie\\'ed as IIIl' IIlillillli/;lIiOIl or 1111' Ii lllowi "1-\ ohjcl'liv(' I'lIl1rtiollal: I
L (II i', ,
I
lII(x/)f
+ AII'flIl/(X,)II~),
il. .... Artijlcilti Nwml Nl'tlllurh
where II . II' is some vector norm and 'D is a diITerential operator. The' terlll of the slim in (12.4.7) is simply the distance between m(Xt) and observation Ylo the second term is a penally function that is a decreasing', function of the smoothness of mO, and A. controls the tradeolT between' sIlloothness anrl fll. In its ruost general form, and under certain conriitions (see, for example, Poggio and Girosi [1990]), the solution to the minimization of (12.4.7) is given by the following expression: K
lil(Xt) :::: L.B.rll.(IlXt - V.II) .=1
+ P(Xt),
02.4.8)
where IV.} are d-dimcnsional vector cl'1!lers (similar 10 the knots of spline functions), I.B.} are scalar coefficients, 1rn.} are scalar functions, PO is a polynomial, and K is typically much less than the numher of observations T in the sample. Such approximanlS have been termed hyperbasis Junctions by Poggio and Cirosi (1990) and are closely related to splines. smoothers such as kernel estimators, and other non parametric estimators.2° For our current purposes, we shaH take the vector norm to be a weighted Eurlidean norm defmed by a (d x d) weighting-matrix W, and we shall take the polynomial term to be just the linear and constant terms, yielding the following specification for rll('):
ih(X /)
=
au
•
+ 0'1 XI + Lfl.lilk (Xt - Vkl'W/W(Xt - V.»'
(12A.9)
k=1
w!tnt" au and 01 are till' coefficients of the polynomial P(·). Miccheli (1986) shows that a large class of basis functions "lkO are appropriate, bill the most ('olllll\on choices for basis fUllctions are Ca\\ssi~ms e-,/(J' and mUltiquaqrics r-;--;; , yx+a-. ; Networks of this type can generate any real-valued output, but in applications where we have some a priori knowledge of the range of the des/red O\ltp"ts, it is computationally more efficient to apply some nonlinear transfi:r fUlictioll to the olitpuL~ to reflect that knowledge. This will be the case ill our application to derivative pricing models, in which some of the RBF lIetworks will he augmented with an Olltput sigmoid, which maps the r~nge (-00, (0) into the fixed range (0, I). In particular, the augmented network will he of the form g(ln(X») where g(u) :::: 1/(1 + ('-U). • A~ Wilh MLPs, RBF approximation for a given set of inputs and out(llllS (X" }'/), involves estimating the parameters of the RBF network-the ~IITo ,·"oIlOll1il.(· on terllIinolo).,'Y. hert' we m" RBFs to ('nmlll"a,s hoth the interpolation 1t'chlliqlle" 11-,<,<1 hy I'OWl'1l (19117) a'Hillll'ir "'\""'1'1('111 gt·neralil.alions.
"~,j:."
.
118
J2. NOlllinearities ill Fi1lfI1lr;a[ /)111(1
I
tf(d+ I )/2 uni<jue entries of the matrix W' W, the dh clemcnts of thc (ellters ~Ull, and the d+k+ I coeflicicnts au, 010 and (till-typically by nonlincar I 'ast-squarcs, i.e., by minimizing LI [ Y/-m(X/»)2 numerically. 12.4.3 I'rojrrlioll Pur.wit UPt,,,,·ps.lioll
'rojection pursuit is a lIlethod that emcrged from thc statistics l"Ollllllllllity or analyzing high-dimcnsional datasets hy looking at their low-dimcnsional rojections. Friedman and Stueu.le (HIHI) developed a version for thc I\tonlinear regression problell\ called projcctiOl1 pursuit regrcssioll (PPR). Limilar to MLPs, PPR models are composcd of projections of the data, i.e., I roducLS of the data with estimatcd cocfficicnts, but unlike MU's thcy also stimate the nonlinear combining functions from the data. Following thc I)otation of Section 12.4.1, the formulation for PPR with a univariate output ~an be written as
~
1\
III(X,)
=
au
+ Laiml(.i3~X,)
( 12.4.10)
k:1
where the functions mk(-) are estimated from the data (typically with a smoother), the (all and (.i3 k l are coemcien\..~, K is the number of l"l1jrr/jOllJ, and ao is commonly taken to be the s'lIl1ple mean of the outputs m(X,). The similarities between PPR, RllF, and MI.» nctworks should be apparent frolll (12.4.10). J2.4.4 /.imilaliom of LeaminR Nplworks
Despite the many advantages that le'lrning networks possess for "pproximating nonlinear functions, they have several important limitations. III particular, there are currently no widely accepted procedures fo~tcnllill ing the network arehitecture in a given application, e.g., the number of hidden layers, the number of hiddcn units, the specification of thc activation function(s), etc. Although some rules of thumb have eIllCl"!;l'd from , casual empirical obscrvation, they arc heuristic at best. Dilliculties also arise in training thc network. Typically, network parameters are obtained by minimizing the SUIll of squared errors, bUI bel',IlISl' of the nonlincarities inhcrent in these specifications, the ohjective function may not be globally convcx and Gill have many local minima. Finally, traditional tcchniqucs of statistical in/erencc such as si~nifi('ancc tcsting cannot always he
J2. -I. Arli/irilll Nt'uml Npllllurh
12.-1.5 AN,limlioll:
519 1.l'lImil/~
IIII' IIIwi<-.'i,-/wl", 1';1111111111
(;il'('n 11j(' POW('I' anrIll. III particular, I lutdlinsoll, 1.0, alld Poggio (I ~I~H) pos(' Ihe fi.!Iowillg chaliellge: If optioll prices were truly detennillcd hy Ihl' Black-Scholes It)rl1Iula exactly, can lIellral networks "leal'll" thl' Black-Scholes limllula? 111 11I0re slandard statistical jargoll: Call the Black-Scholes formula he estim,ltcdllonp,lr,ullctrically via learning nctworks wilh a sllrJiril'nt dq!;l'cC of an:llracy to be of practical usc? Hutchinson, 1.0, and Poggio (1994) bee this challenge by performing MOlltc Carlo simulation experimellts ill which various nelll'a\networks arc trained Oil artificially gencratee! Black-Scholes option prices and then compared to the Black-Scholes formula hoth allalYlirally and ill ollHlf-sample hcdg-illg- cxperilJlenL~ to sec how close they come. Evell with training seL~ of ollly six months of daily data, Ieamillg- network pricillg formulas can al,proximate the B1ack-Scholcs formula with reasonable accuracy. Specifically, thcy bcgin by simulatillg a two-year samplc of daily stock pric~s, and crcating a cross-section of options each day according to the rules uscd by the Chicago Board OptiollS Exchange with prices given hy the Black-Scholes formula. They refer to this two-year sample of stock and (Illultiple) option priccs as a single Imillillg /ialh, since thc nctwork is trained on this sample. ~I Given a simulatcd training path (1'(1)} ofdaily stock prices, they construct a corresponding path of option prices according to the rules of the Chicago Hoare! Options Exchangc (CBOE) for introducing optiolls on stoc ks. A typical training path is shown in Figure 12.11. IkGlliSC the options gellerated for a particular salllple path arc a function ofthc (random) stock price palh, the size of this data matrix (in terms of lIumber of options and tot;t! numher of data points) varies across samplc paths. For thcir training Sci, the number of options pcr sample path rang-e from 71 to 91. with an a\crag-e of HI. The total llll/nhcl' of d.lla points rangl' from :),~~7 to G,H47, wilh all average ofG,001. The nonlincar Illodeb obtaincd frolll Ill'lIral Iletworks yield l'stilllates :!l They as.'mlla'
th.H the lUHlerlyillK
CI..("st't tor
the.' simulation
{'xpet illu.'tlls i!lri a
lypical NYSE
stork, ,,-.. ilil an initial price /'(0) ofS!">O.OU. all aTlnual fOlllillll()II.'"ly rOll1pollllfit'd «''''preted f.\lr of 1('1111"11 J{ of I WYt" and all alJlHlal \'olatililY n of '2IJfYr,. lllldc'l lilt' gl.lc J,..·S( hole . . OI~."lInptioll of .\ ~t·\\HH·\rir
l\ro\\,ul.Ul
llH)\\tm,
til'
=
/I/'tli
•1Ild (,Iking 111(' IItllllher of day."i pCI' Y('ar to
hom the.' dblrihlltioll N(Jl/:l:):l. retllnt"i, which an' ("{Jove'fled to 1>11.
(1
'1 /:l!"):l)
1)('
+ o/'tlU .
L:):~.
'he}'
If,)
10 ohtaill two yc.'.us oftl.lily rOlltilHlOlI.,lyrolllpolllld('d
prin'~ wilh
til(' lI.'iIlOl' r('tllioll I'(t)
=
/J(O)
«'xplL::=-, f,J
for
J":.
,\01///111'(/1///1'.1/1/ "'IIf/lIl)I,, 1),{/II
-or 'i.
"..
"', "<- ::;:
'C C-
O':
~
~ c.: ,...
"''l.
"' Mar!IU FUIIII"(,S (:lIlIlr~H·1
figure
,2""
·li,/.im/Sill/lI/ol,·d '/iolllillg /'IIlh (Vf II" Irxl j""/lfIn""I'1"n) I)(/.\},,,I lillr 1'1"'-""11 .11". I, 1"';I'r, ",hi/I' Iii, I/i'/{Ill" Ifl",.iI'lIl Ilu 0ll/ioll.1 {III Iii, .1/01'11.
FiI, .I'·aHI/dilluh- ott/II'li/, o/Ihl' (/011/1' iut/ili/h'., lit., I/,-ilu'/"ia (ill'/IIWI (/I, .lluulnllolllldu- tli//"'I'I/I ;"IIIH/lldi"" (/1/(/1'.\,/.;"11;",, tloll'I ";.,ib/I').
or optiOIl plices "lid ddl;" Illal ;11(' diHindt to dislill!!;lIisll vislI"lI), rmlll IIII' tllll' Hbrk-Srhoh-s "allies. All (,xample or the estimates alld e!Tors 1'01 ;\Il RIIF lll'tIVOIk is showlI ill Fi!!;lIrc I:!.I:!, The estimated (,flllatioll fClI' tllis particlllar RIW IH'twork is
-O,O:IJ ["/'\ - I.:I:'J I
11.0.'I
~
--------
n.·F,
-I-
:!.",;-,
----
[/·IX-I.IK]'[:,l).7~1 r . II.:! I
T -
-0.0:1
-0.0:1] ["/ X.- I.IHJ. .',-, I (I.:!-\
.
T -
II.:!.'
f- I
,
.521
/2.4. II rtiJiritlI Neural Netwul1i,
(a) NI'lwork call price (;j X
(c) Call price error
Figure 12.12.
ejk -
(d) Delta error
\W
¥s - ¥s
Typical BehavioroJFour-Nunlintar-Term RBFMOlkI
o 10 [PI X -
+ .
Gj X
(h) Nelwork delta
1.05]' [59.79 -0.03
r +0.10
+ 0.14PI X -
0.24r - 0.01,
-0.03] [PI X - 1.05] + 1.62 10.24
r
+ 0.10
.
(12.4.H)
where r ==:: '1'-1. Observe from (12.4.11) that the centers in the RBF IllOde! arc nOI constrained to lie within the range of the inputs. and in fact do Itot ill the third and fourth centers in this example. The largest errors ill Ihese lIetworks tend to occllr at the kink-point for options at the money at expiration. and also along the boundary of the sample points.' . While the accuracy of the learning network prius is obviously of great illterest, this alolle is not sufficient to ensure the practical relevance of the 1\()l\parametric approach. In particular. the ability to hedge an option position is as illlportant. since the very existence of an arbitrage-based pricing [(H"llIula is predicated on the ability to replicate the option through a dyItalllic heclging strate).,'!' (see the cliscussion in Chapter 9). This additioo;al
f)22
12. Nonlinearilies ill Fillal/cial /)a/II
j
. onstraint provides additional motivation for regularization techni
I
~~In fact, it i, well known Ihal Ihe prohklll "f 1lI11l1erir,,1 clilkrcllli;l\i,," i., ill-po.,,"\. '1'1", ,'.. "I.,iral approach I Reinsrh (l!lfi7) I i., 10 rq:lllariw il hy lindin!: a wflicie,"ly sn"",lh IlIlIClioll I al ",Iv(', IIII' varialional prohl"111 in (t 2.4.7). A, we disf\"sed earlier, RBF nelwo. ks as wi'll 'plilles alld ,ever;,1 1111'1'" of MI.I' '1l'lwurks follow di\{'l'!ly fru\\l Ihe rc~\\lari"\lioll app .... Mh a III arl' Ihereforl' exp"C!"d (0 approximall' 1101 onty Ih" pririn!: lemnll);' hIll also i" dl'ri,· •• lin·, (provided the ba.~is function corresponding 10 a smoo(hnes..~ prior is ora ~llf1iri~llt (h-gr('(', ."i('(' PlJK~i() and Gir()~i (1990): in particular. the Gaus",ian is certainly sufficiclllly ~t1Iootll lor 01lr prublem). A'pecial cas" "flhi. !:cllcral ;1If:uII"'nl is Ihe result .,fGallan! and While' (t !J!l2) alld linrnik. Slinchcomhe, and Whit<, (1~)~ln) wlu) show that singh·-hidden-Iayt·:, Ml.1' 'It'lwflll~ f
l
;'t
or I
J 2.5.
{)pl'1jllling and I )ala-SI/()o/Iing
slrlleted, thaI can provit\c real-tim(, l'slilll;l\l's of al'l)loximatioll ('/Tors ill milch the S;lIl1C way that stalldard elTors lIIay hl' oht;lincd lill' typicil statistical estilllators. Alld filially, the lIeed for he II ('I' pnf(>nllann~ lIH'a~lIreS is dear. While typical 11IeaSlires of /{oodness-ol~lit such as J{l. do olfer some /{uidance for lIlodd s('\(octioll, they an' only in('tllllpkt(, llH'"SIIITS of pl'rforlllancc, Moreovl'r, the Ilotioll of de/{rees of freedom is 110 IOIl!!;('r wt'll-defincd for nonlinear lIlodels, and this has illlplicitiolls for ;111 statisticil III('aSIIITS of fil.
12.5 Overfilling and Data-Snooping While each of lhe nonlinear lIIethods discllssed ill this dlaptet' has iL~ own coslS and bellefits, the prohlems of Ollr>jllii/lg and dtl/fI-l//(/o/,i7lg alfed all of thelll to Ihe sallie d('WTe, Overfillill),; on:llrs wh(,11 a IlIodt'llits "too wdl," in the sCllse thaI the lIlodel has captllrcd both ralltiollllloise as well as genuille !lOtllinearilies, Heurislically, lhe primary suurce of overfilling is having 100 few "degrees of frcedom" or too mallY parameters relative to the l1umher of datapoints, and a typical symptom is an excellent in-sample iiI hUI poor ollt-of-sample perforlllance,2~ Data-snooping is a related prohlelll that can lead to excellent bUI spurious oUI-()I~saJllpk pnforlllancc. Data-snooping biases arise wlten we ignore lite hlCI thal many sp{~cillcatioll searches have becn conducted to obtain lhe final specification of a model we arc filling to the data?1 Even if a model is in fact incorrect, by searchill),; IOllg enough over var'lUlis tialase\s and/or parameter valu('s, we arc likely to filld sOllie combination thaI will fillhe data, llowevn, this {it is spuriolls and is merdy a s)'mptoJll of our eXlensivc search procedure, Unfortunately, lhere arc no simple reJlledics to these two problcms ,ince lhe procedures lhal give rise to them are the sallie procedures lhal produce genuine empirical discoveries, Thc SOllrce of both problems is lhe inability to perform cOlllrolled experiments and, (Ol\Sc<j\\{~ully, lhe heavy n:liance on statistical infercnce for our understanding of thc data, As with all fi>rlns of statistical inference, there is always a margin of error, .lIId this llIar),;ill is often sensitive to small chatlgcs in the IVa), we process lhe data and revise our models,
:!:tTIIl' d('grt'l'~ of In,'l'dolll of a 1I01l1ilH';1I lIlockl ,II (' ollt'll clilli( 1111 10 c!t'U'llIIim' 1H.·C:'III~(· ,IH' lIotioll of a Mp.U.JlIlC'tcr" ll1.ty b(~ hluru'd. For example, lilt' )"'c'lll('l fcgrl'ssion may ~('elll to have only OHe fnOt· p~'ramctC'r-tlu." h.uul ..... i,hh "-hu, ,h~s i:-. flt',\1"Iy ml ... h-<1thng :\;lH t' t' ..... h
d.H.lptliul ~l'n'('S as a n'IHt'r for 10lal aH'I.tging. S(T 11.11I1J1{'1 (I~IXI)) .lIul \\'.thh,t (l~mO) 101 1111
tht'r
(lisnl~.,ioll.
:!.IS{'(' 1.(,;1111('1' (I
!17H)
;111(1 1.0 al1<1 ~I;H "'illl,l), (I
!1!IOhJ
101 1o!
Black (l!)!):~) lor ,11('("('111 ('x;lIliple ill Iht' fill,1I1( ('I1I('/",HIII('.
111;11 ;1I1,iI\,c' Cli
'11(
It hi.\,('!Io .. Hul
/2,
iliul/ lil/nl l/lll', l 11/ Fil/a l/ri(l l J)al( l
N('v crlh clt's s, Ihc/ (' an' s('v('r;ll wa~'s 10 miti~ale the l'I'i(oCls of OVt'rfillill~ aile! e!ata-slloopill~, For ,'xal llplc , the imp act of syst ema tic spec ifica tioll se;lr t'ilt's lIIay ofll' n he clln rlat( 'd ('xpl icitly , as ill to alld Mac Killl ay (1990b), III slIch illst ance s, IIsin~ a ('o!T ect(' d stati stica l distr ibut ioll for illfcJ'(.'llcc will safe~lIare! a~ainst findin~ si~lli(i('allt resu lts whe re 110111' exis !. Care ful Olltol~sallll'lc peri'OJIII;IIII'(' el'al llali on (';111 IIIlCOVCr ovcrfillil1~ proh lem s, ;1I1c\ if rdat ivt'l y few oill-of~salllpl(' lesls are ('oll lillc led, or il' lirey are COl ldll!'lcd over dilf en'n t (;11111 ",('akl), corr elal ed) dala s('ts , Ihis will lllill imiz e the eUi' us of data -sllo opil lg, Bill pnh aps till' II lOSt cl'kC li\'c IIH'allS of recltl('ill~ the imp act 01' 0\'('1 'lillill~ alld dala -sllo opil lg is to imp ose SOIlI(' disc iplil le Oil the sp('c ific' llion sear ch Il)' 1/ I"i",-; Iheo r('lic ;t\ COII Si(i<'raliolls, The se cOll sidc ratio lls may he ilJ Ihe f011ll or wi'll-art i('JILrlec\ IlJalll('lJIatical mod els o[ ('COIlOll lic beha vior, or hcha \'ior allJl Odd s lJIot ivale d hy (>sydJOl()~ical ph(' nom ena, or silJlply hellr istic rlllc sofl hlllJ lh hasc d ouju dgll J('nt , illill ition , and past c'xp cricn ('(', Whi le all of thcs c SOIlIH'S an' also afli'c tee! hy clata-snoopin~ and over li 1I i Il~ to SOIlH' c'xtC 'lIt-l Io form of illli' n'lIc e can esca pe thes e pro hlem s-th ey are kss slIsc q>ti hlc alld offt- r a !t'ss (Llta -dep (,lId ('nt mea ns 01' mod ('1 vali dali on, AI! this SllggcSIS the l1el'd fill' all (/ Illio n [ralJ l('wo rk or spec ifica tion f(lI' Ih(' 1IJ0del hefo n' cOll frolllin!-\ IIlI' (Llta, By prop osil lg slIch a spcc ifica lioll , aloll!-\ wilh lire kind s of plll'n OIIl ('lIa ont' is seek illg 10 capi llre alld Ihe r(')('valll varia hll's to hc used ill lite sear ch, Ihe chaJl(,(, or cOlI Jing IIpOIl a spur ious ly slIcc(,ssful lIIod el is r('du ccd,
12,6 Con clus ion NOl llill( 'arili ('s ;tn' cI('a ll\' play illg a lIIor e prol llilW lll role in fina ncia l appl icalio ns, Ihal lks 10 illl'\'('aWS ill COIl lPJllill!-\ pow er and the avai lahil ilY ofla rg(' dala s('ls , Unli k(' tltc Illal nial pn's ('llle d ill ('arl iel' chap lers , SClIlle of till' id('a s ill this chap ll'l' arc less well -('sia hlisl lt'd and 11101'(' 1(,lllaliv( ', With ill a shor t tillJe Illall), 01 Ihe I('ril l Iiqlll 's we hal'( ' cove red will h(' r('li lied , and sOllJe lIJay h('co llll' ohso lete, Nl'v('J'tl ll'lcss, il is imp orta nt to dl'v clop a sells e Ihl' dire t'lio ll of rese ard I alld Ihe opel l ques tioll s 10 h(' addr esse d, ('spe ciall \' at till' carl y St;I~('S of thes e n:pl or;lt ions , Des pite Ihe /lexihililV oflh e lIoll lille ar lIlod e)s we hav( ' cOll si(lc r('d, 111('\' do have SOIllC s('rio lls lilllil;tlio Jls, TlI('y are typic ally mor e dif'li( 'J11t to eslilIJate prec isely , ilIOn ' sCIISili\'e to ollll iers, IlIlIlH~rically less stah lc, alld mOl (' pron (' 10 oV('l'liII illg alld dala -sllo opil lg hias(,s Ihal l rom para hl(' lille ar mod (,Is, COl ltl'ar y 10 pop lllar 1)('lief'. lJoll lilll' ar I1wc\els reql lir(, /lion ' ('col lolll ic Slnl ctllr c alld f/ I"'iori cOll side ralio lls, 1101 I('ss, Aile! Ihei r illie rpre lalio ll 01'1(,11 J'('()Ilin's ilion ' ('Hi"'1 alld carl' , ll()\\'('\'er, 1I0nlillL'arili('s ar(' OftC II a faCI or ('('Oll(llllil' lik, ,llId for lIIallY lilla llcia l appl icali olls Ihe sOUJ'('('S alld ll.tlllI'!'
or
525
l'r"hlelll,1
of nonlinearity can be readily idelltified Of, at the very least, characterized in sOllie fashion. In such situations, the techniques described in this chapter are powerful additions to the arlllory of the financial econometrician.
Problems-Chapter 12 12.1 Most pseudorandolllnumbergenerators implemented on digital computers are multiplicative linear (ongrumti£ll gmerators (MLCG), in which Xn '= (aX,,_1 + r) mod m, where a is some "well-chosen~ multiplier, cis an optional constant, and 1/1 is equal to or slightly smaller than the largest integer that can he represente.d ill one computer word. (For example, let a = 1664525, r = 0, and 1/1 = 232.) In contrast to MLCG numbers, consider the following two nonlinear recursions: the tent map (see Section 12.l.1) and the logistic map, respectively: \
)\" XII
=
{2X,,_1 2(1 - X,,_tl
if X,,_I < .,! if XII-I
1Xn - l (1 - X,,-I).
::: Xo
I
I
X.
E
(0, 1) (12.6.(1 )
'l
E (0, 1).
(12.6.2) "
These recursions are examples of chaotic systems, which exhibit extre~e sensitive dependence to initial conditions and unusually complex dynamic behavior. • 12.1.1 What are good properties for pseudorandom number generawrs to have, and how should you make comparisons between distinct generators in practice (not in theory)? 12.1.2 Perform various Monte Carlo simulations comparing MLCG to the tent and logistic maps to determine which is the better pseudorandom lIumber generator. Which is better and why? In deciding which criteria to USl', think about the kinds of applications for which you will be using the pseudorandom number generators. Hint: Use 1.99999999 instead of 2 in your implementation of (12.6.1), and 3.99999999 instead of 4 in your impkmentation of (12.6.2)-for extra credit: Explain why. 12.2 Estilllate a multilayer perceptron model for monthly returns on the S&P !'iOO index from 1926: 1 to 1994: 12 using five lagged returns as inputs and one hidden layer with ten units. Calculate the in-sample root-mean-squarednror (RMSE) of the one-step-ahead forecast of your model and compare it to the cOITesponding ollt-<>f-sample results for the test period 1986:1 to I !)!)·1: 12. Can you explain the differences in performance (if any)?
I
526
12. Ntmlinearities ill Fi1lf1ll(ial /)ata
12.3 Usc kernel regression to estimatc the relation between the 1I\0\IIhly returns of IBM and the S&P :)00 rrolll 1%:>:\ to 1994:12. How wllulcl a convclIliona! bCLa be calculated rrom the results of the kernel estil\lator? Construct at!easttwo measures that capture the incremental v
\
I I I
\
Appendix
TillS ApPENDIX PROVIDES a briefilltro
tilll'llion techniques in financial econollletrics. Many olher ).\oud reference texts cover Ihis material in more delail, hellce we foclis only on those aspects that are must relevant for our imlllediate purposes. Readers lookin).\ for a Ill( .re systematic and comprehensive treatment shuuld Cl»lSl1lt Hal\ (199~), lla:niltoll (1~I~H), O~aki (199~). and White (19H4). We he~in hy [l)\lowin~ Hall's (l99~) expositiun of linear instrulllent,lI v,lriahles (IV) estimation in St~rlion 1\.\ as an intuitive introduction 10 I btlSt'n 's (I ~IH~) GCllerali/.ed Mcthod of MOIIWllts (CMM) {~Mim'\lor. We dcvelop the GMM method itself in Senioll I\.~, atld discuss lIIethods f()r handlin).\ scri
A.I Linear Instrumental Variables
=
COlISider a linear relationship h{~twl~en a scalar Y, and a vector "I: Y, X;Oll + E,(Ou). I := I .. , T. Stackill).\ Ilw T oh,ervatiolls, this can he written as
y
:=
XO"
+ dOll).
(1\.1.1 )
where y is a (Tx I) vector containiJl~ T ohsl'I"valioJls of )'1. X is a (Tx N.,) matrix cOJltaiJliJlg T ubservations of the N.\ indepeJldent variables in "" 011 is an (Nxx I) parameter vector, alld dOu) is a (Tx I) V('ClOr containing T OhSlTV;l(iollS of (he error (crill f" The l'nor (1'1111 is wri((('11
'1'11 .... 1' all' also ,lI',li lahl, ' N" illSlrllIll !'lllS ill all (N" x I) col 1111111 v('C lorh " Tht' T ohs, 'n'al iOlls of I hi,s V('(' lor 1'01'111 a (Tx N,,) ilia tri x II, TII(' illStJ'l11l11' IllS have III!' prop erlv tllOll F{h; f ,(Oil )) is all (N" x I) vecl or of zero es; Ihat is, IIII' illSlrtllllt'II1S an' ('OIl l<'Ill POL llll'o lisly III1C olTd aled wilh Ille erro r ("I The slalt 'lIll' lIl Ihat a 1"lIl iclli ar illSl nJlIU 'llt is IInCOITl'ialCd with the e<jJ ldlio n erro r is kilOI\'II as an ,"lho gol/( tlil)' ('IJIIf/ilioll, and IV rq,:- ressi oll Jlses til<' .'II" avai lahle onho ).\"o llalil ), cOll dilio ns 10 estil llale the lIlod el. (;i\'('11 .111 arhi ll;II \' ('()('I'ticiellt \'('('t or 0, W(' call rorll l Ihl' corrl 'spolirliJl).\" resid llal (,(0 ) =0 v, -- x;£J. Slac killg Ihis r('sid llal illlO a V('('IO I', W(' gel (0) =0 Y - XO. WI' call also tid i III' all (N" x I) colll JllJl ve!'l or con lain ing IIII' t'l'os s· prod ll('l of Ihl' inSll 'lllll l'nl vecl or wilh tl)(' I'esicillal,
Tht, expe ('lal ion of Ihis cros s-pr odli ci is all (N" x I) ve(,l Or of ,,'ro es al Ill(' 11'111' P;ll"1II 11'11' 1 \'I'('i or:
FI 1',(011 ) 1
== o.
(A,l.:~)
The hasi c id,'a of 1\' ,'slil ll'lli oll is 10 cho ose c()(' fliri enls 10 salis l)' this cOll di· tion as dose h' as poss ihle. ()i'C ()lIrS ", W(' do nol ohs( '/,vl' 111(' I nit' I'xpl 'l'I,\ lioll offa lld so W(' IIIl1sl wo/' k illsl ead lI'ill llhe salll ple aver age off. W(' wril e this as gr(O), IIsili g Ihl' slIhs .... ipl T 10 indi cate clc(>elldC'II(,(' 011 Ihe salll ple: 1
r
gdO ) -
I
L 1',(0) TI Lr h,( ,(0) ,~
I
=
T- I H'dO ).
'~I
(A.I .'1)
Alil/i ll/lIl1 / Oi,'/lI l1f'1' C;,.ill'l'i(J1I
III g(,II ('/'al lh('l' 1' nla), he nlon ' e1"IIII'IIIS of grlO ) than (her e arc coef lici( 'IIIS , and so in gell nal il is IIO( poss ihle 10 sci all titl' ('It'l Ilt'll Is of gr(O ) to I.t:ro. I IIs(e ad, \v" III illill lile a qJla dral ic {(mi l, a weig htec i SIIIll of sCJllan's alld (TOss· prod Jlcls 01'111<' ci<'lll<'IIIS ofg l (0). WI' cldi lll' thl' CJlla drati c fonl l Qr(O ) as
\\'11<'1'1' WI is all (N" x ,VII) S\'III 1II1'I/,i,', !losili\'1' c1di llilt, weig lilill g lIIal ri);. IV rl'gn 'ssio ll dloo,,'~ () I OlS IiiI' \,~IIIII' or 0 Ihal llIill illli/ ('S Q,(O ). Sill>· slillilill~ Ilwt ldill iliol l off'( {J) illto ('\.1.:1),1111' lirst ·orc \l'r(' ollti iliol i/(II 'lile 'IIIIII ,IU\ ;ll'l'\ic .lIillll .... , I ;UIV.l llft', tUllI j, ••
al lillie' I·
1'"
i. .
,,'1111 '( ,1"1 ('11111 Ih.11 j, 1Il1l' ond,l It'" willi ;111\' \.11
Ih(' i",1111I11("1I1 \1" Inl
l 01 ,',II lie-I. NUlw llwlc' ''''
\\C'
i;lhl('." kllllh 'lI
ill
h, \\.11111( 'wit' ollly tagge!l \';11 i.III"· , liI;11 ;11"{' kill t\\'Il WI ill' ., .• \ h, luI' II01;l Iiolla l ~illlpli('i'\' ;11)(1 ~(·ll("rftlil\'.
A. J. l.iruar/rlSl,.umenlal Variables
minimization problem is
X'HWTH'y
= X'HWTH'XO T.
(A. 1.6)
When the n umber of instruments, Nil, equals the number of parameters to he estilllated, Nx, and the matrix H'X is nonsingular, then X'HWT cancels out of the left- and right-hand sides of (A.1.6) and the minimization gives (A. 1.7)
This estimate is independent of the weighting matrix W T , since with Nil = Nx all the orthogonality conditions can be satisfied exactly and there is no Ileed to trade off one against another, I t is easy to see that (A.I.7) gives the uSlIal formula for an OU; regression coefficient when the instruments H are the same as the explanatory variables X. More generaIly, NJlmay exceed Nx so that there are more orthogonality conditions to be satisfied than parameters to be estimated. In this case the I model is overidentified and thc solution for 0·[ is I
(A.
~.8) I
AS'yIll/Jlolir Disln'bulion Theory The next step is to calculate the asymptotic distribution of the parame~er estimate OT. Substituting in for y from (A.l.1) and rearranging, we find t~at I
fi(OT - Do)
=
(7'-'X'HW T r'H'X)-1 r'X'HW T T- J/ 2H'£(00).
!
(A.I:9) Now suppose that as T increases, y-1H'H converges to MIIII. a nonsingular momen.t matrix, and T-'X'H converges to MXlI. a moment matrix of rank Nx . Suppose also that as T increases the weighting matrix WT converges to some symmetric, positive definite limit W. Because we have assllllled that the error £(Oll) is seriaIly uncorre!ated and homoskedastic, when the orthogonality conditions hold T-1/tH' f(OO) converges in distribution a normal vector with mean zero and variance-covariance matrixo 2 MIIIl. ~e lise the notation S for this asymptotic variance-covariance matrix:
io
S
== lim
T~oo
Var[T-'/~H't(Oo)l
= a 2MIIII'
(A.l.IO)
Using (A.I.4), S can be interprcted more generally as the asymptotic variallc(' oj' '1'1/2 limes the ~alllple average off, that is, TI/2 times gT:
(A.Ul)
530
With these convergence assumplions. (A.1.Y) implies Ihat
Jhor - ( 0 )
~ N(O. V).
(A.l.l 2)
where
v =
= and MI/x
(M.wWMl/x)-lMsI/WSWMI/.dMxI/WMl/x),-l a2(MsJ/WMI/.d-lMxI/WM/lI/WM/lx(M.\:J/WM/I.d-l. (A.!. 1::1)
== M'x//'
O/)Iimal Weighting Mahix We have now shown that the estimator O-r is consistent and asymplOtiGIl\y 1I0rmal. The IInal step of the analysis is to pick a weighting matrix W that minimil.es the asymptotic variance matrix V and hence delivers an aSylllptotically e!licielll eSlimalOr. It turns out that V is minillli/.ed hy picking W equal to any positive scalar times S-I. Recall that S is the asymptotic variance-covariance matrix of the sample average orthogonality conditions ~T(O), Intuitively, one wants to downweight lIoisy orthogonality conditions filI(I place 1II0re weight on orthogonality conditions that arc precisely mcal;lIred. Since here S-I = a-2MI/~/' it is convcnient to set W cqualto I
1
(1\.1.14)
~he formula for V thcn simplifies to \
V"
=::
a 2 (M.\:J/
M,,~/ M/lx)-t.
\
\11 practice one call choose a weighting lIlatrix
~
S
(
W;. = (T-1H'H)-I. T increases, Wj, will converge to W', With this weighting matrix the formula for
(A. I. Hi)
Or hecollles
I 0;. = [X'H(H'H)-III'X]-IX'H(H'I-WIH'y == d('xrIX'y. I
.
(A.1.17)
,yhere X == H(H'H) -I H'X is the predicted V;tllle of X in a rC!~ressi()1I of X 011 H. This is the well-known two-stage least squares (2SU;) estimator. II call be thought of as a two-stagc procedure in which olle first regresses X on H, Ihell regresses y on the fitted value from the first stage to estimate the parameter vector 0 0 • Alt'emativc\y. one can think of ~SI.S as regrcssing hOlh X .11\(\ yon H in the fJrststage, and thell regressing the lilll'd value of yon the filled value of X;
11.1. UI/I'{l/' IlI.linwwi/a[ Varillblr.f
531
exactly thc same codlicient estimale (A.I.17) is iIJlpli('d. Note that IInd('r this alternative interprctatiulI, lhe second-stage regression asymptotically has ,lIl Il~ slalistic Ofllllity becaus(' lhe error I('rm ill (A. I. I ) is orlhogollallo lhe illslnllllellL~ alld llwrefore has a filled v"llle of I.ero whell projecled on lhe illstfllmcllls. This implies lhal asymptoticall)" if (A.I.I) and thc orthogonality cOllditions hold, thcn the co\'l'liciellt cSlilllal('s shouleJldeJlI variable in (A.I.I) allli which are chusen to be regressors. A~ymptotically, tlte saltlt: codliriel\l estimates will be obtained (lip tu a nOl'lnali/.ation) whichl'l'('t' way the regressioll is writtcn. Thc variaIlCl:-covariance matrix or ~SI.S c()('i'lici('nl cstimates, V', can be cSlimatcd by substilllting consistenl cstiJllales of the various momellt matric(,s into (A. I. I !i) to obtain (A. 1.1 H)
wherc a~ is a consistent estimate or the v,lriann: of the (,£Illation errur. This formula is valid for just-icientilit:d IV and OI.S coefficicnt estimates as wdl. In place of the weighting matrix W' defined ahove, it is always possihle to use hW' where h is any positive scalar. Similarly, ill pl,\('(' of th(' weighting 1ll,llrix one ran use hrW;" wherc iiI' i~ all)' positive sCllar that cOllverges to Ii. This rescaling does not afleclthe formula lill' the inslrumental variablcs estim;ltor (A.I.17). Onc possihle choice lill' the s(alar II is (T .. 2, tlte reciprocal of the variallce of the equatioll error f/; tltis makes tlt(' IVeil-\ltlillg matrix equal to S -I. The correspolltiilll-\ dloice I()r the S('abr k r is sOllie nmsistt'lil ('stim" te -~ of (T -2. Hallsell ( 1!IK2) h"s shown t h" t wi III 1his sc"ling. T Iilll(,s the Illinimilt'd value of the ohjective fUllctioll is asymptotically distributed X ~ with (.'11,,- Nx) degrees offreedolll under til!' 1I1111ltypothesis that (A. I. I ) holds and the illstrumellts arc onho!-(ollal to the equation error. Iiallsell's test of the null hypothesis is related to thc intuitioll thatumier the l1ull. the residual from the IV rcgrt~ssion t~qllati{)lI shuuld he unconc\,t\cd with the imirumellls and a regressioll of the r('si(lual on the illstruIn(,lIts should ha\'(' a "small" U~ statistic. To IlIldcrst;Uld tltis. note that whell WI' == (a~T-IH'H)-I, tlte minimil.cd objt'oi"e fllllclion is
W;.
a
(A. 1.1 !l)
No\\' consider regressing the residll;11 f(i}~) on the illslllllllellis H. The filled v.tllle is H(H'H)-IH'€(O:.). and lltc U2 statistic of the regn.:ssioll cOllwrg('s to the salllc lilllit as (A.1.19). Tltlls I bIlSCII'S resliit illlpli('s that T timcs tlt(' U~ ill a regressioll of the residual on Ihe illslllllll('llls i., asymptotically X~ \Ii 1h (Nil - N,) dewecs of fret:riom. Tltis is a st;ItIlLm It ('St of OI'Clidcn tifying i'esl rirl iOlls ill tIVo-stal-\e IC;lst squares (Engk ( 1!IH'll),
A.~
Gl'Ill'ralizl'd Method of Momen ts
'IIII' (;('Il(, .. &li/('d I\klhod "r.\IOIIH 'lIls (I lOlliSI'll [1!IH~I) (';11l h(' Illlcl('rsl o"cI as ;IIlI'XI(' llsioll of ill!' lillc;lr 1\' rq~r('ssioll w(' hal'(' disclIsse d, SIlPPOS(, 1101,' 111;11 11'(' ha\'(' ;1 IIlod..! IIhidl "dilles a 1'1'('101' £1 £(x l , fJ). II'I)('r(' x, IlOII' illdJl(ks ;&11 III!' ,Lila r .. lel';1111 lor III(' IIlodd (i1lal is, WI' ha\'(' droppec lll'" dislillCl ioll 1)('111'1"'11 .I, ;111<1 x,), ;111<1 () is a 1'('('lor or Nfl I'IH'lIicil'IIIS, This li,rlllllL ilioll gCIII'r;a1i/('S ill!' lillcar 1\' r('gl'l'ssi oll ill Ihn'(' Il'al's. Firsl. fix"~ 0) call 1)(' a 1'011111111 1'('Clor lI'iill .V, 1'I('III1'lIls railler Ihall a scalar. SI'('ollcl. f(X I , (}l (';1111)(' a Ilolllilll 'ar 1;1I1,,'r Ih;1I1 a lill(,ar fUlll'liol l oflhl' d;II;1 ;llId Ih .. par;IIII1' I('rs. Third, f(x" 0) ('all 1)(' 1)('I('I'os kl'dasli(' alld s('('iall\' corn'lal .. d ralh('1' Ih;11l hOlllosk l'lLlSlic II'hill' Iloisl'. 0111' lIIodel 1('lIs liS Oil II' Ihal Ihnl' is SOIl\(' IIIH' sl'I or pal'allll'l l'J's Oil 1'01' which f(X , Ou) is orlhogo llal 10 ;1 '1'1 I or iIlS(I'IIIl\('IIIS; as '1I'f(,1 (' Ihl'sl' arl' II'rilll'lI ill all (Nil x I) 1'011111111 1'('('(01' h" HI' ;lIlalogl' wilh (,\.I.~) Ill' ddil\('
=
f,((})
,.= hI ® £(x{o 0).
'I'll(' Ilolaliol l W d('llOll'S Ih .. l\'II/{l'lhl 'l' /I/'odlll'/ or Ih(' IWO \'('(,loI'S. Thai is, f is a 1'1'1'101' 1'IIIIIail lillg Ihl' (Toss-pr odIH'1 or I'adl illSlnlllH 'll1 ill h wilh (';1"'1 1'1('111('111 Orf, fis 11"'11'1.,1'(' a 1'011111111 \'('('Ior wilh Nt = N, Nil 1'I(,11I('IlIS. alld Ih(' mod('1 implil's II)' allalo)..,'\' wilh (,\.1.:\) Ihal
E I J',({J,,) I == O.
(,\,~,:!)
.I"SI as ill (,\.1"1). \\,1' "dill(' a 1'('1'101' g,(O) ('olliaill illg (he salllpl(' ;:\'('1'ag('s COITI'sp olHlillg (0 Ih(' 1'1('1111'11 Is orr ill (A,I.I!» : I
J.: I
((})
r
I
Lf,(OI .
(.\.:!,:\)
'~I
HI' all;llogy Idlll (,\.1.:,). (;1\1.\1 IlIillilllil l'S Ihl' qlladral ic filllll (.\.:! ,.1) Silln' Ih .. !,lohl('11 I is 11011' 1""a1ill( "II. Ihis lIIillillli /;llioll IlIIISI he "l'Ilill'll l('d 1Il1ll1cri Cllh. 'I'll(' lirsl·"nl( '1 "Clll
(.\.:!.I.)
,\. 2. (;l'IImdiwl MI'I/wd of M(I/III'IIII tI.I."IIII,Ioli. /)isirilllliioll Till'Ory
The aSYJllptotic distrihutioll or the coefllciellt estimate
no, - 0
vr.: -
0 )
"
.......
61' is
N(O. V).
(A.2.7) I (A.2.H)
These ('xpressions arc directly analogous to (A.I.12) and (A.I.13) for the,linear illstrlllllclltal variahl(·s case. Du is a generalization ofM"x in those e(ll1ations and is defined by Du == E[Clf(x/o Oo)/aO u]. Dr(O) converges asymptot, ieall)' to Do. S is defined as in (A. I. I I ) hy S ==
}~~I~ Var [TI/~
t
f/(OU)]
=
}~l.!., Var! rl/~gr(OIl)J.
(A.2.9)
1=1
OIl/iII/til Wfighlillf,; Mllirix
.I"st as in lhe line.lr IV case, the optimal weighting matrix that minimizes V is any positive scalar times 5- 1 • With an optimal weighting matrix the aSYlllplolic variance of '1'1/2 tillles lhe (oef(icielll estimate iJ~ is (A.2.1O) Also, whcn the weighting matrix S-l is used, T times the minimized objective functioll is distribull'd X2 with (~- Ntll degrees offreedom. where Nj is the Iltllllher of orthogonality conditions and Nfl is the number of parameters to he estimated. III practice, of course, 5 and the other <Juantities in (A.2.8) must he eSlilllal('cl. To do this, one starL~ with an arhitrary weighting matrix WT; this could he Ihe identity lIIalrix or could be chosen using some prior informalion ahoul lhe rclalive variances of the difTerent orthogonality conditions. Usillg W" on(' lIIinimizes (A.2.4) to get
rh.
DII can hI' r(,placed hy D'de'd, W can be replaced hy W T , and S can be
replan'd hy a consistent estilllate s,dJr). Given these estimates, one can (OIlSlrul'l a n('w weighting lIIatrix Wj. = S,CO'r>-1 and minimize (A.2.4) agaill 10 gel a second-stage estilllate iJ~ .. The ;lsymplOtic variance of '1'1/2 lillles th(' second-stage estilllate can he estimated as (A.2.11) alld Ih(' sec()lld-~tage minilllized ohjective fUllnioll is distrihuted X2 Wilh (N, - Nfl) degre('s 0(' ('reedolll. AlillOlI~h a two-stage procedurt' is aSYIllploliedl" ('ffici('llI, il is also possihle 10 iterat(, Ihe procedure further un.til
534 the parameter estimates and minimized ol~ective function converge. This eliminates any dependence of the estimator 011 the initial weighting matrix, and it appears 10 improve the finite-sample performance ofGMM when the lIumber of parameters is large (Fcrson and Foerster [ I!l!l4J).
A.3 Serially Correlated and Hcteroskedastic Errors One of the most important step5 in implementing GMM estimation is eMimat~ng the matrix S. From (A.2.9), \
S
=
f~I~E[rl (tf,(OO») (tf,(OO),)]
=
ro(6 u) +
I I
00
L (r,(6
0)
+ rj(6 o») ,
(A.:~.I)
j=1
1
whde
II
(A.3.2)
is th jth autocovariance matrix of [,(Bo). The matrix S is the variancecova 'iance matrix of the time-average of[,(Ou); equivalently, it is the spectral dens ty matrix of [,(0 0 ) at frequency zero. It can be written as an infinite slim f autocovariance matrices of [,(0 0 ), fthe autocovariances off,(Oo) are zero beyond some lag, then olle can simp ify the formula (A.3.1) for S by dropping the l.cro autocovariallccs. The auto ovariances of f,(Ou) are zero if the corresponding autocovariances of E(X" !Bo) are zero. In the linear IV case with serially uncurrclated errors discyssed earlier, for example, t:(x,.Ou) is white noise and so £,(0 0 ) is white for j > 0 and S noise; in this case one can drop all the autocovariances is just o, the variance of f,(Ou). The same result holds in the conslimption CAPM with one-period returns studied in Chapter 8. However in regressions with K-period returns, like those studied in Chapter 7, K - 1 autocovariances of [,(Ou) arc nonlero and the expression for S is correspondingly Il\ore complicated.
r,
r
The Newey-West Estimator
To estimate S it would seem natural to replace the true autucovariallccs of [,(0,,), r j (90 ), with sample autocovariances '1'
rj.7·(o·,)
== r
I
L
f,(Or) [,_ ,{Or>'.
,=,+1 and substitute into (A.3.1). Ilowever there arc two difficulties that 1II11st be faced. First, in a finite sample one can estimate only :t finite number
A. J. Sl'lil//~v (;ond(/II'II tlllt! J JI'I"l1I.l/it'dtl.l/ir 1':mll.1
of alltocovariallces; and to )!;et " consis\('nt estimator of S Ollt' (";11I11I)t "lIow the 11I1II1ber of estimated autocov.lrialln's 10 ill(T('aSI' 100 rapidly wilh Ihe sample size. Sccond, there is 110 )!;lIaralltec that all ('stim'ltor of'S limlled hy substituting (A.:13) illto (A3.1) will bl' posiliV(' c1dinilc. 'Ii) halldle thcs(~ two problems Newey ;llId West (1!IH7) sugg('sl('c1lhl' following ('stilll;ltor: • Sr('1,fh) ==
•
'/- I
'I ( ro.r(lI'[')+L -,-'.) }=I
.
.
(rj.rw/) + r;rWrl) ,
(A.:U)
I
",herc '{ iIHTC'1S('S with the salllple sil.(' but 1101 100 r;lpidly. ,,-I is the II1;1Xinlllll1 lag length that receives a lIonl.('ro weight ill Ihl' Newcy and West (19H7) estimator. The estimator );lIar;lntccs positive definitcness hy downweighting higher-order autocovarianccs, and it is consistcllt hccallse the downweighting dis;lppears asymptotically. In models where aUlOcovariances arc known to be I.em beyond lag K - I, il is lempling to \lse the Newey and West (19K7) eSlilllator with II == K,
Thl.' is icgitirn;lte when K= I, so that oilly the variance 1'(1.'1·(0],) appears in the e,lim;llOr; but when K> 1 this appro;lch can severely dowllweight sOllie 1I0llzero autocovariances; depending 011 the sample size, it m;ly he better to use 'I> K in this situatioll. Although the Newey and West (I !IH7) weighting schclllc is the most ('OIllIllOllly lls('d, there arc several altcrnative estimators in the litcratllr(" indlldillg those of Andrews (1991), Alldrews alld MOllah,1II (I!19'l) , and C.lllallt (19K7). Hallliiton (1994) pl'Ovi(\('s a llscflll overvicw. Thr j,illt'flr jl/.\lrlllllfllial Variablf.5 emf
The gl'llcr;d formulas givcn here apply in hOlllnonlinear and lincar 1J1l)(lels, hut they can hc understood lIIore silllply in liJlcar IV regressioll lIIodels. Retllm to the linl'ar model of Scction A.I, hill allow th(' enor tCI'I1l f,(Ou) to he serially corrc:lated and hetcroskcdastil'. EquatioJl (A.I.IO) I)('collles S
= lim Var[ r
I/~H' 1'(00) 1
[-':Xl
""
lilll T Ill'!:1(O,,)Il,
T-':X.J
wllerc H(lJo) is IIII' variance-nlVariallu' matrix of llIal cd h)'
,,(011)'
(1\.3.5)
This (';\11 bc ('sti(A.3.tJ)
will'l'c H,
(0 r)
is
.111
cstimator of HWo). Eqllation (A.:!.II)
IIOW
hccomcs
(A.:t7)
III the homoskedastic white lIoise case cOllsidcred carli("r, 12 == (1 "I r so wc used all estimate n"'{o'rl == where = ,[,-I 'E.::I (;cilr). Suhstituti;lg this illto (A.:U) gives (A.I.IH).
a"l/
a"
W!at'll III(' I'ITOI 11'1'111 is scriall" IIIH'olTclalcd hill hcll'roskl'
H is a dia){ollalll\allix wilh
For I'aell dl'lIlt'llI II'hill' ('a("11
SlIlIill!!; lIIatlix U I (0 I) illto (:\.:l.Ii)
(1I1l'
!!;I'ts a 1'0llsisll'Illl'stilllalor ofS I (ill.
alld SlIilslillllilig il ililO (A.:I,7) (~II(' gelS a ('onsiSI('1l1 l'slilllator
V'r.
This is
n(o
n
11111' ('VI'Il IhOIl)!;hllll' 11\;lIrix I) is 1101 ilSl'lra ('ollsis«'1l1 I'slilllalor of ",'('ails,' Ill<" Illlllllll'r .. I' dl'lIwnh "I' n Ihal IIlllsl he eslilllaled elj\l;t!s lil,' sa III pll' sill'. WIll'1I II", cnor 1t'lllI i.s snially ('oITclalt't! alltl h(1I110sknlaslil, IIIl'II (>Ill' ("all ("ollsll'llli I'alli cI"llI('111 of thl' 11l:ltlix
H, "UI,) =
"I,'1
I
L...,~f.11
o.r(O rl
as tClllolI's:
(L.!.) (") 0" 'I f,( r)f,-,( r)
to
III == I1-.11 < ({,
"
olhl'lwiSt,
(A.:I.Hl wlll'rc Ihl' N('w<'\'
"
0.",(0,) = . .
n I (() I)
as:
{ ('L~) (',Ii) ,) 0
'I
f
(0,)
if I
== II - ,II
<
1/,
olll('I'II'isl'
whl'll' IIII' Nl'\\'I')' alld \<\'<,st (1!)H7) Wl'i){htillg sclll'lIlt' is IIsl'd, Again 0111' call Il'plan' till' lriallglliar wcigll1s willi IIllit lVt'i!!;II1S 10 g<'l ~hl' ('slilllalor of I LlIISI'II alld Ilodriek (I!)HO), III I'aeh CISI' SUhSlilll1ill!!; 0.(01') illlo (A.:~.(j) !-:iws a (,()IlSiSIl'1I1 ('slillla1<' ofS, alld SUhslilutillg il illto t'<jIl,llioll (A,:I,7) givl's an mSiS\l'1I1 1'~1 illlalor V~, 1'\'1'11 I hough t hI' 1ll'l1rix o.(il'rl is lIot ilsdfa ('ollsis1l'lIll'Slilll;lIor (11'0 h"CIIIS" IIII' 11I11lI1)('r OrllOlll.I'rO dellll'lIts illcreases too r;lpidll' with 11)(' sa III 1'''' Sill', Whill' (1!IH,l) !!;ives a rOlllprcll('llsi\'1' Irl';ltlll('111 of Ill(' lill,'a)' lllt"lI'I willi s''l'i;,lIv ('(11'1',,1;,1<'11 ;lI\d IW\l')'(lskt'
A.4 GMM and Maximum Likelihood Followillg 1I;II11ill(lil (1!)!)I),wl' \lOW .show how SOIll<' WI'II·kIlOWII projll'lIil's 01 Maxillllllll l.i"dih",,,1 .-slilll;IIO)'S ([\11.1-:) rail 1)(' IIIHkrslo(ld ill relalioll to (:[\1M. WI' lil,t I,I\' 'lilt SOIIl(' Illll;ltioll, We liSt' I., to d('lIotl' 1I11' dl'lIsi!\' of
:\.·1. (;i\L\l X/+I
fllld
537
Maximum l.ikelihood
conditional on the history of Xt and a parameter vector 9: (AA.l )
We lise the llotation i, for the log- of Lt. the conditional log likelihood: i,(X'+I.l/J
=
log- L,(x'+I> 0).
The log likelihood [ of the whole data set XI •...• conditional log likelihoods:
XT
(AA.2) is just the sum
of~he \ I
(AA.3) Since I., is a conditional density. it must integrate to I:
f
I.t(Xt+l. £J)dxt+1
= I.
(AA.4)
Given certain regularity conditions. it follows that the partial derivative of L, with respect 10 0 must integrate to zero. A series of simple manipulations then shows that ()
= (A.4.5)
The partial derivative of the conditional log likelihood with respect to Ihe parameter vector. ait(xt+l. O)/ar}. is a vector with NfJ elements. It is known as the score vector. From (AA.5), it has conditional expectation zero when the data are generated by (AA.I). It also has unconditional expectation zero and th us plays a role analogous to the vector of orthogonality conditiolls f, in CMM analysis. The sample average (11 T) 2:.;"'1 3l,(XH;t. 9)/39 plays a role analogous to gT(O) in CMM analysis. The maximum likelihood estimate of the parameters isjustthe solution to Max [(fJ) == L:~I i,. The ftrst-order condition for the maximization can he written as
o
T
gT(lJ)
= rl L ili,(x,+l. O)/(){J = D.
(AA.6)
,=1
whirh also characterizes the GMM parameter estimate for a just-identified model. Thlls the ML1~ is the same as GMM based on the orthogonality conditions in (AA.5).
538 Ih.~ml)totic
Distribution
71U'1Jr)
'1~lle asymptotic distribution
or ML parameter estimates is given by the I()I-
" wing result:
I
fl(O -
00 )
~
N(o.r'(Oo)).
(1\.4.7)
wi,ere I given by:
I
I
[I ()~£.(O)J
lim -E - -.- .-, • 11 ,JOiJO
1(0) =:
a'~d is known as the
(A.'Ul)
,,~oo
jlljOn/Ulti()11 matrix. I
Gill
be estimated by tht' sample
Clbullterpart: _
I
I"
iJ~e,(iJ)
T
L
-7
=
iJOiJO' .
(A.'I.!I)
1=1
The information matrix gives \\S;I measure of the sensitivity of the vallie of tIle likelihood is to the v,llues of the paramcters in the neighborhood of tl e maximum. If slllal\ changes in the parameters produce large changes ill likelihood ncar the maximum. then the parametcrs can bc precisely esti/natcd: Since the likelihood function is lIat at the maximum. the local s~nsitivity of thc likelihood to the parametcrs is mcasured by the local curvaturc (the sccond derivative) of likelihood with respcct to thc parameters. cvaluatcd atthc maximulII.
Infimnrltioll-Matrix Equality \ An alternative estimator of thc information matrix. Tb • uses thc average ollter product or sample variance of the score vectors:
i
=:
rl ~
al,(O) iJl,d})'.
ao
L,
b
,=1
(A.'I.I
ao
t.
To see why ib convergcs to the salllc limit as differcntiate the third equality of equation (A.45) with rcspcct to 0' to gct a~e(XI+I. 0)
o =
J
iJOaO'
f
ilOilO'
a2e,(X'+I. 0)
. ()~l,(xHI.O)
I'.(
=
1" dX'+1
iJOiJO'
.
1"
+
J
iJf,(X,+I. 0) iJI-,(XI+I. 0)'
f
ao
ao
dX'+1
aC,(X,+I. 0) afl(x,+I. 0)'
ao
ilO
.
1"
rI.\( \ I
. af,(xl+l'O) ilf,(x".I.O)'
+1'.,
E iJ~f,(x'+I.O)
ilOao'
rI'\111
+
+
iJO
iJO
E ;)f,(XHI.O) iJf,(X,+I.O)' ;)0 iJO
(A.4.II)
I\. ·1. (;Mt\I find II/flxi/llllllll.illtlill/l/ld
This is known as the illjimlullil//H/IIl/rix "'II/fllily, ,11111 ililpli('s lh"l lhe exp('claliolls 10 which lh(' s,ulIpk aV(,)"lg('s and II. ('oll\','rg" ;II'l' ('l(lIal. TIl(' illj'oi'lllalioll IIlalrix (''lu,dily holds 1111111-1' Ih(' 'ISSlIIlIl'lioll liJ,lI III(' d'II" .11" gl'IH'I'alcd hy (tvl.l), CMM allalysis gives ,Ill allel'llaliw lorlllllla 1(,,' illl' disll'illlllioll of MI. paralll~~lcr ('still\al~'s, RccllI frolll (A.':L II) Ih"III,,' (;MM {'stim"lor is a5YIIII>IOlicdly llormal wilh aSYlllpll>li<' V
i"
III lhis (
a,l.tr
S
ao
'
cJ-t'/(O) ilOilO'
1111
(A.4,12)
I
(A.4.1:H
",hill51
(i/
I
)
-
-
rl ~
S
ilC,d)
ao
il£,dJ)' -
;)(}
-
"
silll''' lhl' scorc vcclor is serially uncolTdal('d so S rail h(' ('slilllalt'" 1'1'0111 ils s;llIIpk v'll'i'III(,('. Thereforc, llw dislriblllioll "j'11I1' CMM ('slilll,llll\' 1'<111 hI' exp\'('ssl'd as: (A.4.14) where In alld If, are lhe limits of i" and i" as '{' increases wilholll bound, ('vaillated al the true parameter vector 0 0 • \,\'hen the lIIodel is correctly specified, in and II, bOlh cOllwq~e 10 the informalion matrix I, hence (AA,14) simplifies in Ihe limit 10 (11- 11)-1 := 1-1 which \'educes to Ihe cOllvenlional t'xpressiotl for the asymptotic variallce in (A.4. 7). Therefore, dlher ia or i" (or bOIIl) ,
l/Y/Jnllil'li,1 Tr.llill/!. The aSYIll»tolic variallces ill (A.4.7) ,11111 (A..I.I·I) call 1)(' IIwd ill a straightf
.1 "1\1
Tht' idea of sud I It'SIS is 10 SI,(, \\'1It'lIl1'r Ihe ullrestricted paraml'ter estimal('S an' siglliliralltly dirh-n'lIt from IIl1'ir H'Slrit:led values, where the variance of Iht' IIlln'slriC\l'd t'Slilll;lIt'S is caIndall'd wilholll imposing lilt' n'slriCiiolls, Ahefllaliwly, 0111' ilia), W"1I1 10 It'sl restrictions IIsing estimates 0111)' of III<' restrictt'd modcl. OIlCC n'striCliolls "rc imposed, thl' minimizl'd (;:-'1\1 ol~jccti\'c fUllction is 110 IOllgn idclltically Zl'ro, IlIstl'ad, Ihl' I IaIlSI'll (1~IH~) rl'sllil is that 'f' tillles Ihl' IlIillillli/.l'd ohjt,t"livl' funClioll has a X~ distributioll wilh dl'grcl's of fn'cdolll eqllalw tht' IIl1mht'J' of restrictiolls, III Ihis casl' T tilllcs Iht' IIlillillli/('d oi>jcni\'(' fUllClioll isjllst
whirh is tht' l,agr;llIgt'llIllltiplin tl'St slatistic rora resiricletllllodd estimatl'd by lIlaXillllllll likdihood,
nil' /)"//11 /l1"th,," MOl"<: cOlllpliralt'd illh-n'llt"t's I'or ;nhilrary lion linear funniolls of Iht' l'Slilllalor 0 lIIay \)(' p .... lill·llIt'd via Taylor's Thcorem or tht, (Mtll 111ft/lilt!, If
fldJ -
0 0 ) .~ lllO, I;d, tht'll ;\ 11t)lIlilll'ar functioll
I
aSYlIlptotic dislI iilttliotl:
Vf
ilf
ilf
== -\",ao' aO
(1\.4, Hi)
which follows frolll a tirst-ordn 'Elyl"r series approximalion tilr f{O)
• 541 \
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IlI/m'IIt·I'.1
I\.I('\,IIS. ll .. '" (;I"~O")', alld S. I.ill. I\IH\I. "Ri,1... 1'1"llIiIlIIlS ill Ih,' '1<-1111 SIIII!'IlIn': h'id"lIn'lrolll Anilidal 1-:('1'"01 II i,·s."./III/II/(// 0/ M/",f'lI",\'/':",I/OIII;, I. 2·1. :171:1\1\1. B.lllrill.lllt. S .. "alt-. J. ;IIHI T. Nm·. 1\1\),•. "()I SIt"I'It,·"". SI\('q'. ;IIHI lite (:""';\111111'1)1'1 !'i;lIinns in Eqllit)' R"I\II liS." H/1';I1/' "/ fil/(/IH ;(//.\/I,,/;n. H. ·11l1-·I:\Il. B.I~l'IIt)I.
W.. (;l.k.a ..Ial'k Tn'plo,,), 1!171. "Tit,· (lilly (;'""" ill '1'011'11." /'//11/1" ;11/;\1/(//\'111 ./,,1/1'1,,11. '.!'2. 1'2-1·1.
Ittl...,lti. (;" alld I.. (:1"'11. 1\1\11 •. "Tlte Spi .. il 01 (:.ll'il.III,," .lIl1l Slo, I... :\/1/1'1';1'111/ /':WI/II/II;I' 11,1';"/'. Hli. 1:1:1-1:,7.
~1.1I1...'·1
1'1 il'n:
Ibldlll.li. 1'.. t;. 1\1'I'10la. alld S. Fmcsi. 1\1\1:1. ",\ ~11I1"'1 III '\;lIg"1 Cltallg'·s alllllh,' '1(01'111 SI"I/('IIII'(' 01 1IIIt' ..esl Rail"." \\'l1ll...illg 1"'IlI'l'·1:\ 17. NIIER. (:alllh .. idg". ~I'\.
Il.tll. (:" I\)HH. "hlilll;lIion Bias Indlll'('d hy lJi"'\"'I(' S"I IIlil' I'lin·"'·'/II//ll/lli 1/1/11(('• •1:1. H·II-HI""
II/ h-
Il.tll. ( "" .I\\(\ A. RI)IlI.I. \\1\).\. "Swclt",ti,' VIII;\lili" ( )1'1'1<>11 1'1 i, illg." ./lI//lI/(// II/ /"i 1/11 1/1';11/ 1/1/1/ (LI/(/I/I;I(/I;",' tll/lI/pi.l. '2\1, [,H\)-Ii07. lI.tll. 1:.. alld W. TO,..,IIs. I !II{I. "A Silllplifi .. d .I I\IIl I' 1'\Il«'" 10' COllllllOn Siock R.. IIII'IIs,"'/III/('I/lli II! /'/1/(/1/(';11/ (lilt! (LIIIIIII;llIliI'I' :\1/(//\'1/1. IH. :,:I-li:,. - - - . I 'IH!i. "( )nJ II III Jls in (:olllllllln Slol'k I' .. in·, and TIt .. i, 111111;11'1011 (:;tli Oplioll I'l'il'ill),;:·./"'"1II1i
4 hll II III 1', '10,
1:."._17·1.
- - - , I \)HH. "11I\'('sligaling SeclI .. ily-P";I'(· I',· .. fi" 111.1111',' ill lit .. 1'1'1'''''"''- 01 E\'('III1l.llc l" 1('(· .. lai III),," '/111/1'1,,11 "I/'/I/(/II/ ;(// "'1'111///111;1'.1. 22. 1'2:1-"..... lI.tll. R..• II,d 1'.11"011'11. 1\lIiH, "All Elllpi,,;cll b'alllalioll 01 A(,(,l1l1l1lilll\ 111('('1111' NIIIIIh(' ..,." ./"/(I'/ull /// ;\u'II/il/I;I/K 1II'.I/·I/I,}I. 1,,\I--17H. I\all. C" W. T'>lI)\Is. ;tlld 1\. Tsrhocgl, \\IW" "Th .. I><-g"'" 01 I',ir .. R",ollllillll: Thl' ( .. I'" oiIIH' (;111<1 Markel," ./"1/11/(//"/ F//I//II'.I M(/'/uof.,. ~" '2(I~I:I. Balli. R" I \IHI. "Th .. I{l'ialioll helweenl{('11I1'II alld Ma .. kl'1 \'alt,,· of'( :0111111011 Slocks." }!/I/'I/ld "/ i'l1/01/(';0/ /:'((J/IIJ/Il;CI, !/. :I-IH.
H,,'II. R. •. IIId M. Milll'l', 1\I7H. "Prices f(,r Slate-Conlingent LI.lilll.': SOIll(, Eslimal('s "nd A\lplie;lli()ns," ./0111'1/(// 'if' !l1ll;'If'\!, :11. (;:,:1-(;7'2. 1\;" hni.,. 1\: .• A. Shlciler. ;IIHI R. Vishny. I\/!}(;. "A Modd "I I1\ \'l's (or Senliment with holh ['ndl'IT(,"l'lioll and OV'·IT,·anioll." IlIlp"hli,I"'d 1'''1'' .... ['lIi\'l'I'sil), 01' (:IIi(';'go a,"lllarl'ard llniv('I'sil),. lI;II'r\;t\',
~I
.. ,,1111 R.I.iIl.('nh('J'gt'r. I!/HH, ""nn,lI\1\('('II\('1I1 Eik ... , "I'N('w 1-:'1"il)' I,,"('s .llId du' lIse (If Illlraciay PritT I>.lI~t." .I0IiJlud (~f FUUlll(ill1 r:fUlwmio, 'll, 711110.
I\"rl...('\'. (:" I \/:,Ii. "1·JfI'rti\'(, Stork Splils," //(/II'/IIt! /1'";""11 1111'1/'/1'. :',1 ( I) . .l.lIl1lal),Fehl'll"r),.IOI-lOti. - - - . 1!):.7. "Slorl... Splils ill a Bull r-Lllk'·l." 1/(/1I'1IIt! 11'";111'11 1i11"''II'. :1:,(:\). r-Lt)'.1"1"" 7'.!-7\). - - - . 1\l'iH, "E\'"I"alioll "I'Slork Di\'idends." I/(/I/"/lt! 11",/1/1'.11 IIn'I/·II'. :IIi(-I),.I"I\,. 1\II.l:U'I. \)!)-114.
1i1'/1'1i'liffl
.1'1
BaIT, D., and/. (:;""1'11('/1. 1'1'1;1, "Inflalion, R"allnlt'f('sl R'llt'S, anelllll" Bonel ~f:tr kl'l: ,\ SI"d\' 01 (IK NOlllinal alld hult'x-Lillk"d COVt'l'IIlIIt'lI! lIund p .. in·,." I )j"'II,,iulI 1',1/)1'1 17:\'!., I b.-\,.lnl Inslilllll' o\" Ecol1olllic Rt''''arch, 11.,,·\'anl (llli\' ....,il\'. lIartoll. A .. I~I'I:\. "(I"i\,l'Isal I\Pl'lOxilll'llioll IIOIll1(\S (il\' SlIpt'rposiliollS o\" a SiglIloi,"'1 FIIII('lion." IHJ·: 'Jilllll. 111/0. nlnll)" :{!I, !I:\O-!H:I. I~I~H.
"'\PllIoximalilln and Eslimalioll BOIIII,ls wllIks," ;\/111'11;111' /.l'IInJ;lIg. 1·1, 11:,-I:l:t
Ibll 1111. A., awl R.
B;"T"".
((II'
Arlilici;II Nt'mal
""1-
I ~IHH. "SI,lIisli .. all.t'anrill~ N"lwork'jf,A ( IlIir)'ing Vit'\\'." in
20lh S\'/11IItIl;II/IIIIIII//I'/ull'll"rr: f.'1I1II1'lIlillgSrimrf IIntlSIIII;llin."". I!I'!.-'!.O:I,
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1I1O(·k. W.. W.lkl 1"'11. ;111(1,1. Schl'i"k,"all. 1!IH7. "AT"Sllill'IIIII"(I"lIdl'lI("(' Based Oil til(' (:o,,!'blioll lliIlH·II,ioll." llllp"hlislll'ti papl·r. lJlliwrsilY "I' Wis("OIl,ill ;11 to.l.,di" )1\. I' lIinTsil), "I Ilollslol\. aUll lIuivl'I"sil}, of Chir;l~o. IImck. \\'.• ,1. I.a~olli,ho~. alit! B. I.dlarOl'. 19!12. "Simple Tl't"llllic;1l Tradillg Rilles alit! Ihl' SII)(it;I,lic 1', o(l('lli,'s u("Slock ({(·IUnts.".!"",.",d ,,(foIllllllff. ·17. 1,:11171;1. IIrodsk),. II .. I'l'n . .\'1I1I/'IlII/IIII·/lil ,\11'//'1111, I'lIbli.,III·".II""11I1.
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Ih hl'ig, 1'.. and.J. Ingersoll. IVH'2, "Mean-Vari'\lllT Theory in Complete Market.': .Iolll"llfli of /Jll.liIlP.IJ. 55. '2.33-'2!,>'2. Ddn·ig. 1'., and S. Rms, I \JH!i. "Yes. Thl' APT Is ·I'·slabl,· ... ./""1"1111/ II/I'/Ilill/ir. ,10. I 17:'>-11 HH, D)'hl'ig. p,,} Ingnsoll . .Ir.• alld S, Ross. l\l!)ti, "Long Forward .1I1111.1·lo·Conpoll Rail'S LUI Ncver ~"IIl." j,mmal '1 /I/(Iilln.<. li9, 1_'2r,. Ea.,lcy. D .. and M. O·Hara. 19117, "l'rke. Trade Sit.e. and IlIf,tnll"tioll ill Securities Markels," journal o/Fillflll(itlll':nlllOlIlin. I ~). ()~)-~)O.
- - - , I \)9'2.. "Time and the Process "fSecurit)' PritT Adju'llllcnt ..../IIIlI7WI 'ifl'II/(lIla.
47. !i77-liO!'>, Eberlcin. E., and M. Taqqu. 19!!!i. Vr/wI/(IPIICP ill ['/lIvllbili(y wul S(II(i.l(in: A SlIn'l')' of Iv(l'II( IImlil, I'ro!-:rcss in Prohahility and Statistics. Vol. II. Birkhallscr. Boston, Erl..l>o. II., 1\IH:~. "IImiJ.lmlal MerKel'S. Collusion. and Storkhohler Wealth," ./olln",/ of 1'111 II I/(illl 1:'(0/10111 irs. II, '241-'27Ii. Erkho, E.. V. M .. ksimovic, and J Williams, 1\I\W, "( :on.,islcnt Estimation of CrossSl'l"Iion,,1 Models in Evcnt-Stndies: IIn,i,.w of /'II/(IIIn'''' S(/Ulif.!, 3(3), 343%!'J. Edwards. R., andJ Magee. I!Jti6. '/;'(/III;mlllllllh.lil John Magee. Bostoll.
11 S{illh '/im",
(n'vi"'d !',th (',1.).
Eidll'llilallm, M .. 1.. IIans('n. and K. Singh-Ion. 19HH. ",\ Tillie S('lil's Analysis 01 Representalive A!-:enl Models of (:onsumption and I,cisure (:hoice under Uncertainty." Qllarleriyjolll7llll 'j/:·mnfllllin. 10;1. rJl-7H. EikebooOl. A., I \)93. "The Dynamics of the Bid-A.sk Spread." wOlkin!-: papn, Slo,1Il School of Mana!-:emcnt. Mass"chusl'lIS Institutt' o)"fi.orhnoloh')'. Call1hrid!-:('. MA.
EiIlSh·ill. A.. I~IWI ... \ 1"h"1 di,' \'1111 ,ll'r Illllkklllar-killt'lisch"1I TIlt'oric <1<"1" W;inlll' ~dlll d"III' 11"I\"'~III1~ 11111 illl"llll"lIdcll FlilSSil-:kcill'll SIlSPl'lIdi"1 1"11 T,·ildwlI." .11/111/11'11 ",') l'II)'1il... 17. r,·I~)-:,liO. F.II~"'.
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557 - - - . 19H4. "The Infonnation in the Term Structure," journal o/FinanciJJlRconomics, 13,509-521. - - - , 1990, "Term-Structure F()recasL~ of Interest Rates, Inflation, and Real: Returns," journal of Mon,t(lry 1:'cOllUmics, 25, 59-76. \ - - - , 1991. "Ellicient Capital MarkeL~: II," Journal of Finan a, 46, 157:r-1618. • - - - . 1993. "Multifactor Portfolio Efficiency and Multifactor Asset Pricing Models," working paper, CR..W, University of Chicago, Chicago, IL. \ I
Failla. E.. and R. Bliss, 19H7, "The Informatioll in \.ong-Maturity Forward Rares," Alltrrican &D1wmic l&uiew. 77, 6RO-692. I Failla, F.., and M. Blume, 1966, "Filter Rules and Stock Market Trading profits," journal of Rusint.!.!, 39, 226-241. . Failla. E., and K. French, 1988a, "Dividend Yields and Expected Stock
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jOllnwl ofFinmlCial J:conomics, 22, 3-27.
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- - - . 1!)96a, "Multifactor Explanations of Asset Pricing Anomalies," Journal of h'IUI/ICf. 51. 55-H4. - - - , 1!l%b. "The (,.APM Is Wanted. Dead or Alive," journal of Finanu, forthcoming. Failla. E., andJ. MacBeth, 1973. "Risk. Return, and E<Juilibrium: Empirical Tests," jOIl1711l1 oj Political Hconomy. 71, 607-6:~6. amI R. Roll, 1971, "Parameter Estimates for Symmetric Stable Distributions: journlll of tit, Amniran Stati.!tiwl As.!ociation, 66,331-338:
1",\11\01. E.•
F"lIla, E., 1.. Fisher, M.Jensen, and R. Roll, 1969, "The Adjustment of Stock Prices to New Information," Intt'T1!alional r.col/omic RnJinD, 10, 1-21. Fall!(, K, and Y. Wang, 1994, NU7llbl7'-Tizeortlir Mrthods in Statutirs, Chapman and Hall, 1.(llIoon. F'llIst •.J., El92, "When Are Variance Ratio Econom,lnw, fiO, 1215-1226.
Test~
For Serial Dependence Optimal?,"
F,·II<-,., W., 19tiH. All 'ntnJlillcticm to I'mbabilit) nlfOry anti It.! APPliCIIlions, John Wiley alld Sons, New York. FI'ISOII.
W., and C. Constantinides, 199\, "Habit Persistence anel Durability in Ag-
!(fl'gate Consumption: Empirical I ~l~I-210. Fns()lI,
Tesl~,"
journal of Financial t:Conomin, 29,
W., and S. Foerster, 19!1'1, "Fillite S,\1l1ple Properties of the Generolli/.ed
Method or MOlllellts in Tests ore :onditional
"i "'ill/lllfir!l EnJllumics, :~(). 2!1-:,:,.
A~set
Pricing Models," jounlfll
558 Fielit1., B.• 1976, "Further ResulL~ on A~ylllllletric Stahle Distributions ofStod, I'rin's Changes,' jo/mUlI of Fill(/I/('i(/I (/1/(1 Q/lfllllilfllitlt' AII(IIY.li" II, :I!J-;,:,. Ficlit1., B., aud.J. R01.ell, 1983, "Stahle Distriblllions and Mixtures of Dislrilnllions Hypotheses for Common SlOck Returns," .I0w.",/I o/Iht' AII/I'/;mll SII/lillillli Auocill/ion, 7M, 2!\-3G. Fisher,!'., I !Inn, "Sollie New Stock Mark"l Indexes," jO/l/'IItlI11Iill,lil/f',I.I, :I!), I!) 1-2:!:,. FisQman, G., 1996, MOl/If' Gil rio: COl/a/,I", Verlag, New York.
I
A/~IJI;lhIl/J,
alld ANJiimlioJ/.I, Sprillger-
Flayin, M., 1!J113, "Excess Volatility in til<: Financial Markels: A Re.lsseSSIIll'1I1 ~~npirical Evidence ," jUl/rllal 'if I'ulitim/ fWllumy, !l I, !l2~1-%li.
I
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Frailkel, J.. G. Galli, and A. Giovannini, elis., 19!){i, '1111' Mif'lllstrrr(lwf' of l-i1/"i~1/ I';xI (hange Mark,t.! (NnER Conkrence Repoll), University 0(' Chicago I're,s, Chicago, IL. FreLch, K., and R. Roll, I\JHli, "Slock Return Variallces: The Arriv'11 0(' InfOllllalioli and the Reaction of Traders." jounUlI '1 Fillflllciall:"ollomin, 17, !i-2Ii,
I
FretlCh, K., G. Schwer!, and R. Siamhallgh, 1!)H7, "Expected Sw,'k R"l11l'11s alld Volatility: jOlmlll/ of Nnfllll'illl/:rf)/lIIl1lie" l!l, 3-:\0.
I
Fri+lrnan,.J., and W. Stuetzle, I!JH I, "Pl'!~jection Pursuit Regression," .1011111111 of lIlt' Amnican S/alistiw/ As,'Dcia/iDII, 76(376), December.
I
Fritlnd, I., and M. Blume, 1975, "The Demand f()r Risky linlil'lll, 65, 900-!J22.
I
A~seL~: AlllniulIl
I';mlwlllie
FI'O,!t, K., 19H9, "New Hope for the Expeclalions Hypolhesis of the Term SUIIt'Ulre I or Interest Rates: jounltl/ 0/ Fi IlIIII a, 44, 2H3--:10:,. Fro,!" K., and M. Obslfcld, 1U91, "Intrinsic nubbles: The Case of Siork Prices: Amniran l:i'OllOmir Rroirw, 81, II H!J-I ~ 17. Fullt'r, W., 1976, iniroducliollio .'illlli"liml Tilllt' Srrir." Wiley, New York. Flirbush, D., andJ. Smith, I !J96, "Quoting Behavior 011 NASDAQ: The Determinants of Clustering and Relative Spreads," EconomislS Inc., Washinglon, D.C. (;alai, D., 1977, "TeSL~or Market Erflcicncyorthe Chicago noard Options \'.xc-hange,"
journal oj Bus;n,.,s, 50, I li7-1 !)7. ---,197M, "Empirical Tests of UOlindalY Conditions oJFinancial J-:collomin, Ii, IH7-~11.
"n' enol': Options," ./11111'11111
Gallant, A., 19H7, NOlllinrar Slali.,lira/ Modrll,John Wiley and Sons, New YOI k. t;"lIant, A., and H. White, 1!J!J2, "On Learning Ihe Derivatives of an Unknown Mapping with Muhilayer Feedf(lnvard Nelworks: Nrum/ Nrlwlllk,l, ", 12H13H. (;"II"nt, R., 1'. Rossi, and G. '/:Iuchcn, I ~)!Il, "Stock I'rit:es and Volullle," Iil1lil1t1 11/ Fillallria/ Sludies, 5, 1!)!J-:.!4~. (;arher, 1'., 19H9, "Tulipm'lIli'l,"./lIl1l'1lil{ vJ {'o/ilim/ fn)//IIIIl)" 97, :,:~:)-:)li\). Garman, M., 1!J76a, "Market MicrostntCllln': j"'(l1II11 of fillllncial Em '10 ",in, :1, '157275. 1!J76h, "A General Thcory of Assel V"III"lion IInder Diffllsion 51,,1(' Processes." Working Paper :)0, Uni\'t:lsity of California, Ikrkell~y.
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(;,·,k,·. It. "lid K. Sh'lstri. I !/H!'" "Vah""ioll hy "l'l'l'Oxillo;"ioll: i\ COIllI'''1 isoll or ( )I't i, III Valll"tio" TI'('h lIi'lIl<·S. "'/011'1111111/1'111(/111 IIrllllltf (l"ll"lllllli.'!';\ II II Ip II, ~Il,
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~I., "lid K. Ramaswamy, 19!/:\, "ATest ortbe Cox, Ill~elsol\, and Ross Mudd of Ihe Tel'''' Structure: H"";,,II' tif H,lIlIIc;1l1 SIUt/i,.I, Ii, Ii 19-ti:,H.
Clhh,,"S, M., S. Ross, alld J. Shallken, I!lH!/, "A Test of thl' Elliciellcy of a Givt'll l'ort!(,lio," 1,'COllolII,trim, ;,7, 1121-11:,2. (;i1ks. (:., and S. l,eRoy, I!l!ll, "Econollletric /\spcets 01' thc V,II ian('(' 1I0unds Tests: A SllrVl'Y." IVlIil'w oj Financial Studips, 4,7:,:\-7\11. Ci()\'''"llilli, A., ali(I 1'. Wcil, I!lH!l, "Risk i\version and Inll'riCllll'OIal Substitution in the Capital A~sl'tl'ricillg Modl'l: Workin~ l'ap!'J' ~H~'I, NIIER, Cambridge, ~Ii\.
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CloStCll, 1... 1!IH7, "Componenl~ of the !\id-A.sk Spr('ad "ll(ltlie Statistiral Propertil's "fTransaclion Prices: jounJllllIlh,/Ill!cp, '12, I 29:{....I:107. (;I""I,·n. 1.., and I.. Harris, I!lIlH, "Esti\11atin~ thl' Components of the lIid/ Ask SpIl'ad," jll/l/'l/al of /-llUl/lrjall,;'()/wlllirs, 21. 12:\-142. (;lost("lI. 1.., and 1'. Mil~rom, I !lH:" "lIid, A.sk and TI allsaetioll I''';('('s in a SpCli,llist Market witli !letero~enl·o\l.sly Infol'l)\l'(\ Tr;,,\l'I's: .Il11lnll,i IIf Fi,lIIl1rilif 1·.'IIlIlIlIIin, 14.71-100. (;10.'1"11, 1... R. .l,,~annathan, and D. Rllnkle. I \/\/:1. "On Ihl' Rl'lation 1I"twl'en thl' Expe('(l'd Vallie and the Volatility ol'tll,· :--JOl1lill,tI Exress Rellllll on Stocks: .I"II11UIl lifFi'lIl11rf, 41l, 177!l-IHIlI. (;Ild,·k. 1'..
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Crundy. B.. and M. McNichols, 1989, "Tr
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Il:lgnman, It, 1978, "More Evidence on the Distribution of Security Returns," Jour· Hal of Finance. 33, 1213-1220. I bkansson. N., 1976. "Purchasing Power Funds: A New Kind of Financial Intem1ediary," Filla1!cialAnnlys/sJollrnal, 32, 49-59.
i
1977, "The Superfund: Emcient Paths towards Emcient Capital Markeu ill l.arKe and Small Countries," in II. Levy and M. Sarnat (eds.>, Financipl • Or,ision Making Under Uncertainty. Academic Press. New York. 11"leI. A .• 1990, A History of Probability ami Statistics and Their Applications ~fcm 1750. John Wiley and Sons. New York. Hall. A.. I!}92, "Some Aspecl~ of Generalized Method of Momenu Estimation: in C. Maddala, C. Rao, and H. Vinod (eds.), Handbook of Statistics, Volunu II: l:i"OllOllletriCS. North·Holland. Amsterdam. Hall, R., I!l88, "Intertemporal Substitution in Consumption," Journal ofPolitical Economy. 96, 221-273. ""Iperll. 1'.. and S. Turnhull. 1985, "Empirical Tests of Boundary Conditions for Toronto Stock Exchange Options: JO 1< 17lal of Finance, 40, 481-500. Ilalllilton. J.. 19119, "A New Approach to the Economic Analysis of Nonstalionary Time Series and the Business Cycle," l:ro1!ometrica, 57, 357-384. I!}~}O. "Analysis or Time Series Subject !-."({J/llllllftrin. 4:>, 39-70.
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Changes in Regime: Journal of
I q~I:\. "Estimation, Inference, and ForecastinK of Time Series Subject to (:h"ng("~ ill ReKillle." ill C. Maddala. C. Rao. and H. Vinod. eds .• Hant/book (I/Stllti.,lin, Volumt' II. North-lloUand. New York.
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I ht;llq,~. (:,. and R. Litzl'nberg,'r, 1\IHH, l-il/llllll/lilJlIl (r," Fillnllrial f:((I/lomin, NOI'thHolland, New York. . • 111I"lIg, R" alHI H, Stoll, 199:>a, -Th .. COlllpOlll'nts of the Bid-Ask Spread: A Cen('rOIl Approach: Financial Mark("1S Research Celller Working Paper 94-~~, OW("1l (;r'lduatt· School of Manag('J1ll'nt, Vanderbilt University, - - - , I \)\):>h, "Dealer Verslts Auction Markets: A Paired Comparison of Execntiull Costs on NASDAQ anrl the NYSE: Financial Market~ Research Center W"rking Paper \):>-1/), Owen Cra(\uatl' School of Manag~m~nt, Vandl'rhill llni\'('rsity, llulll'r, 1'.. I !)W>. "Projection Pursuit." AllllfliJ "ISlalislin, 13(2) • .j35--~2~, IltlhcrJnall, (;" 19H2, "A Sill1pk Approach til 1'.i()lIl1mir 771f(1I),. 2H, I H:{-I!I I.
Arhitrag~
Pricing
Th~ory:
j()unlalllf
.
II Uhl'lJ1l;ln. C., S. "-mdel. ,1Ild R. StaJllhaugh, I!lH7. "Mimicking Portfolios and Exact Arhitrage Pricing," jOlll7wl iif Fi/l(llllf. 42. 1-9. 111IllI'rJnan. (; .. and S. Kandel, 19H7. "Mean-Variance Spanning: jounUlI oj hnflllU, '1~(4). H7~HHH.
J IIIII . ./.. I '1~}3, 0l,liolls. l·illllrl',I, 1I11t! 011",1' /)1'17I'lIliw Sfrun'lifl (2d ed,), Prentice-Hall. Englewood Clitls. New.lersey. 111111, ,I., and A. Whitt', 19H7, "The Pricing of Options OJl V"latilities ... ./llurlllll t1'Fil/fll/ff, 42, 2KI-~()0.
A~set~
with
SlOcha~tic
- - - , I '}\IOa. "I~ricing (Jlt,'resi-Ratt'-Deriv
l'I~I()h.
"Valuing Derivative S"nlrilic~ Using th" Explicit Finile Difference Method: JOl/m1l1 of Fillnllrilll flllIl QWII/lillllillf Analysis, 2~. H7-IOO.
Illls q, II.. I ~I:>I. "l.ong Term Storage Capacity of Resen'oirs," 'li-nnsllrliolls Ilf tI" Alllflil'fll/ Swirl), '1Th,il Hligil/mI. 1 Iii. 770-799, IIIllChillsoll . .l .. A. 1.0. alJ(l T. Poggio. 19~)4 ... A Nonparametric Approach to the Pricillg and lIedgill~ of Dl'rivativc Secllrities Via l.~arning Networks," JOlin/fit "I Fil/Illllf. 4!1. H:>I-KH!I. Illgl'lsoll.J, I!IH7, 'f1/fIll)' of I-';'/III/(il/[ /)1'(iJiOIl fIIl/killg, Rowman & LiuldicJd. Totowa, ~I· 111),\"1',011,
J.I r.. J. Sk,'holl. ,Illd R. \\'(·il.
I!17H, "[)ur,ttion ForlY Years Lu,'r: .llIunUlI (If "
I-/l1ll1/rilllal/" QIIIII/lilll/it'r /\l/lIl)'.\i.I, 1:,.1;27-1;:>0.
It .... K.. 1'1:>1. "Oil StorhaSlir Dillt'n'lliial Equations," MOl/oin IIf IIIl/lim/.'iorirly, '1,1-:>1.
a,r AIII"i,."" MIIIII,...
JII\-II'('Iill.J. ,lIld M. Rubin.stein, III!):" "Rt'co\'t'rin~ I'rohahility Di,'lrihutioll,S ffl/,111 (:"'lIt'lllporary S('curity Priccs," working p.lpcr, Jlaas School of l\mill(~ss, lllliYt'rsilY llfCalii'ortlia al Ikrl
I\I~)'\.
"11,1\'('siall Analysis ofStodl
~1"dcls.'·./lI/lmlll 0llllllilll'\\ 1/1111 fllillolflit .'ill/lil/in, l~, :171-:IH~..
I
!JGG .Iagal nathan, R., and Z. Wang, 1!I!)(i, "Tht' Conditional CAPM and the Cross·S.... tioll of Expected Returns," jOUnlrtl '1 Fillllllff, [, I, 3-!>3. Jain, P., 19HG, "Analyses of thc Distribution of Sel'llrity Market Modd Predictioll Errors fur Daily Returns Data,"jollntfll oj Armuntil/g l/Pmll"h. 21. 71;-!)\; . .Iamsl idian: F., )91l9, "An Exact Bond Option Forlllula ... .Ioltl7llll lif FiIUllltr• .1-1, 20;)I 209. I
Jan'cO, G.• and A, Puulscn, I !J!l9. "The Returns to Acquirin~ Finns ill Tender 011"1 s: Evidence from Three Decades." FiIUlI//itll !lltll/tlli'lIIm/. HI. 12-1\). Jarrell, G.,j. Bricklcy, and]. Neller, 1!)HH, "The Market (('1' Corporate Control: Th .. Empirical Eviclence Since I!)HO.".Ioltnllli if frtlllfllllir J>rn/,rr/irw.I, 2, li:I!)-h:)H . .Iarrow. R., and A. RUlhl, I!l1l2, "Approximate Option Valuation fi.r Arbitrar), Still hastic Pron·s.~es: jour/1II1 '11'llIlIlIfilll /':mllo/llir.I, 10. :117-:16\1 . .Il'gadeesh, N.. 1990, "Evidence ofPredictahlc Bchavior of Security Rt·tlllllS ... .It1l1nud oj f1llanu. 45, !:I!:I)-!:I9H. - - - , 19\11, "Seasonality ill Sto.. k Price Ml'all Revl'I'sioll: Evi(\eIlCl' frolll tlt(· \ I.S. and thc U.K.: jOllmll1 lif /oI'lIlIIr" 4G, 1427-14·14 . .Iegadecsh, N., and S. Titman, 1993, "Returns to lluyin~ Winners and Sellill~ I.osns: Implications for Stock Markct Efficicncy: jounUlI lif Filltmr,. 4H. li:l-!ll. - - - , 1995, "Ovcrrc;lction, Ddayed Reaction. and Contrarian Pru(its." /(11,il1<' Fillallrial S/udifJ. Il, 973-993 .
til
.It·nsen, M., and R. Ruh;lck, 19H3. "The Market Ii,,' Corporatc Control: Thl'S";"IIIi1ic Evidcncc." Joumal oj Fillt/llrilll hOllolllirJ, II, :/-:/0 . .It·rison, D., I. Singcr. and D. Stroock (cds.), I !)!)(j, 'I'll' I '}C)4 Wimrr SY7ll//(}liulII 1'/lIIt','dillgs, American Mathematical Socicty. Providcnce. RI. .loh~(m,
D., amI R. Kur\<'ie, I!)H'2, "Potentiall'erfonll
- - - , 19!:1!>, 'Somc
Tesl~
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A~set I'rkin~
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CanadianJoumnl of Atlmillil/m/i"f Srimm, 2.111-138.
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(:han~ill~,"
.lones, 1.., 1!IH7, "On a Conjecturl' of IllIh"r COIll'ernin~ the COllver~elHT or Proj .. ction Pursllit Re~ression." AUl/tlll flfSttl/i.lti/,I. I[)(2). HIlO-HH'2. ,F)r"skoll, K., l!)li7, "Some COlltrihutiolls to M'I>;;lIIulII Likl'lihood F'l(lo .. "1I"lysis,"
1~'.vrh~I/."t:l/(fl. 3.1, IH:~202~ .. ., .. Jndge. ( ' .. W. (.nfhths, (,. lilli, II. (.Ikql< .hl, and I. (.('t'. I !IH;•. IIJ,' Illt'tlly (//ull'/(/I/'/I' l I oIJo:nlllom,lrir.I.Johll Wiley and SOliS, Nt'w York.
\t '
Kagd. ~I., and A. Roth. cds., 1!)9[). /lIIIItI/J(lOH University l'ress. l'rincelon, N.J.
til J:'x/It'lill/l'/I/1l1
fo.'((IIIIIII/;I'I, l'riIHTt()n
K.;lhne~lan, D., and A. Twrsky, I!17!), "Prospect Thl'ory: An Allal}"is or Dl'rision
'i
LInder Risk," frl/lllJlllrtllffl, '17, '21;3-'29\.
Ntft'/'{'/1 {I',I I\.;d"" M .. and 1', Whitlock, 1~)Hli, /11111/1" (;111/11 Aldllllt!.,. lidulllt, /: 1I,l.\;n • .John Wiky and SIIIlS. Nl'W York. I\.;lIl1kl. E .. 'lIlcll.. Marx. 1~)~l(i. "NASDAQ M,Irkl,t Stll.ClIIII· alld Spread Patterns: IIIlIJl.hlislll·d working pap ..... SilllOIl (;radllal<' School 01 I\lIsi,"'SS. lIIIivl'r,it)' or Rochester, K.I"dl'l. S,. I !IIH. "The I.ikelihood R.llio 'I"sl 01 ~k.III·V.lli.IIIII· I'Jlil il'lIl)' without a Riskll'!\s Assl"t," .Jourllul of Fillilllf;all':nJlwm;t \, I :~. ;, 7:,-:,~)~, 1",.,111\'1. S,. R, McCulloch. alld It Stalllhallgh. 1\1\);,. "1\.1),<'si'"1 IIIIlTI'nc<' ,"111 1'011lolio Elfiriency." n,~/;t'l/I o! I-/lIl/l/f;l// SIIIt/It". H. 1-;,:1, Kandel. S" and R. Stalllhaugh. I!lH7, "On (:ol'll'I"lions '"l1llllnl'll( es aholll Ml';I1IV"l'i'"I('(' I'Jliciency," .Imm"t! 0lhl/l/llf;1/1 fUJlllllltln. I H. Ii I-\)(). I')H\), "M\)(lc\\in~ Expl'cted Stock Rl'turns lor I.on~ ,lilt! ShOll \loriwns: WOl'kill~ Papn 42-HH, Rodnl'Y I .. Whitt, (:l'lIlel'. Wh"rlon School. Uni\'t'r~itl' or I'enllsylvania. - ' - - . 1\)\)0. "1\ Mean-Val'i;III1:C I'r;lllll'wllI k forT,·,ts 01 Asse' 1'1 iring Mo"ds." Un/;ntl 11/ /'/1/1/ I/f;1/1 SIll/lit'S, 2, 12;'-1 ;,Ii. I \)!I I. "Asset Retllrns and /ntert"lIIpor;d I'rd'·f(·nl'l's." 1,'uJlIfII/I;n,
./11/11 11ft!
or "'nl/t'l",y
27. 3\)-71.
I !)')!i. "Portllilio InetlicienC)' and the CIIISS-SITli"II 01 Expl'cted R,'wl'lls," ./0111'111/1 'fl'/lIIlI/a, ;,0, 1!",7-IH·1. I".,,,,, L, 1\170, "The TI'I'1ll StI'IICI,"'1' of 11111'11'," R'IlI''' '\11 Atto'lll!,1 10 R''foIKill' Teal'hillg with Practice," 10111'1111111/ i'lI/III1IP, 2:,. ~Lt)'. 3Iil-:17·1. - - - , I \)W>, "Nl's,ed Tesls of Altel'llative '('1'1111 Slrul'III"" Thl'mi,'s." 1I"1;'~" lltJ1uin and .')Ial;.,lir.\, (i;), 11 [)-I ~:t
or Fro-
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lI"min. :17. :171-:>!'H. I !)!):,h. "EX(Tlilioll (:osls al1d 111\'('sll1l1'nl PnIOl'ln.lllce: A" bnpiril'al Anal)'sis "I' IllSlilliliollal I-:qllil)' Tlad,'s." 1'01 killf( pal"", S, h",,1 "I III1,illl''' Administral ion, UniVl'l'sity of SOllt h"111 (:"IiIOl'l1i,1. 1!1\J(i. "The Upslairs Markel li,r LIlf(,'-lIlo, k TI."I.,,,cli,,"s: Al1al)"i, .1I1l1 ;\ll'a""'1'I1I1'11 0,I'l'l irl' Elkl'ls." 1I,~,1t1l' 11/ /-iu,f/It illlSIWllI'l, 'I, I-:Iti, hl'JIlI. ()" and R, Slalllb,,"~h, I !lHIi, "('I'I'dining R('IIII'IlS in Siock "llllllolld M.II kl'ls../11I1171ft! o(hllllllr;al Frollolll;n. 17. Y,7-:I\H),
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569 I.l'l', (:., ami M. Ready, 1991, "Inferrin~ Tt,l(\t, Din'nioll from Illtr.\day Data," jau";lal 4Fil/Illla, 46, 73:\-746. l.et', S.,
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Ollie!
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l~-n
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R., and.J. Rolh), 19H1, "An International Study of Tax EtrecL\ .... lIl1lelll Bonds: .Im,"1fI1 of Fillflllrr, :~~), 1-'2'2.
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\l.leNnl"y, i\. C .. al\(i M. Richardson, J!I!II, "U,illg (;l'lIn,tli/l'll M('lhod~ 01 MIlIIH·III.' 10 Tl'st ~lean-Variance Elficiellry," .f/ll/ll/(// 01 h,ll/l;",. ·lIi, !'II-:I:.o. \bdd"la, C., 19H3, I.imilfl/-/)r/}{',ulelllllllt/ Qualifalit". li"i,,/JI," ill 1,·wllolllrl>1r.1, (;01111hridl{e University Press, C,II11hridl{e, UK. \I,ldh""'\ll, A., and S. Smidt, 1991, "A lIayesian Mod,,1 01 Inlrad"y SI'l''';"list l'ricill!{: .f0Il,.",,1 'if foi'llIlIIriall:(OIl/IlIIin, 30, 9!1- 1:14. \Lt~nlls,
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\1;11111.'/"1'01, B., '"I1I. J. \';111 ,'\lc·". IClfil<. "Flac 'lioll al Brow lliall ~IOlioll, Fl'an iolia l Nois .. s '"111 .\ppl i, ,"iol l'," S.I.,\ .i\I. /(nlin c', 10, '1:!:! -I:I7. \\.I1I 1"'lh l'ol, II., :11,,1 J. \\'all i" I~II;X. "No. dl . .lo"· ph allel Opn ;"iol l.rI 1(~·dl'olo.l\Y." . \1'1111'1 !It'11I1 /1l ['I /(nl'l l/I It. ·1. ~/(I~I~II 1<. -'-- -, 1~lli~I.I, "(:OI lI('"l c'r (':,(,n iIIlI' IIIS wilh Flafl iolla l (;alls .siall I\:oisc·s.l'al'l.s I.:!. :~." Ittllf' " U,'\o llnn
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MYlT.s,.J.. ami A.Bakay. I!J4H. "llIlh,ell (" ofSlock Splil-llps on Markl·ll' rict'." II"n,(/,,1 /l'U;III'.!.! Un,in". ~:11-~6:I.
'nl Ikci,ion s S .. and :-.I. Maillir. 1!IX·I. "Corpora t!" Fill;lIll illf.: alld III\","-,tllll -,,",,",,1 II/ h· · .. II"n· NOI no Inn-sto,., that tion 11iIorllla a\'(" I I When Finlls '/i/I/I·illll· .illlllllllin. 1:1. IH7-2:!1. ()plion I'li,," "ilh \Ii.':\aik. v.. ,",,1 :--1. I.ee. I!I!).\. "Thl" Yield CIII\'I' alld 1I0lid II IIi,",""iIY 01111 ilish p.ll',"l. hl"d IIllpllhlis " R'"gillll"s, c EnlllOllli in Shins n,'I,' (:ollllllili a aIHllllli\"l"rsil)" orSaskat l
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577 Pl·tt'rs~n. M.. and S. Umlauf, 1990, "An Empirical Examination of the Intract..y Behavior of the NYSE Specialist: working paper, Massachusetts Institute ofTechnolob'Y' C..ambridge, MA. Phillips, S., and C. Smith, 19RO, "Trading Costs for Listed Options: The Implications lelr Market Efliciency," journal of Filloncial HcollomicJ, H, 179-20 I. . Pog~io,
T., and F. Girosi, 1990, "Networks for Approximation and l.earning: Pro-
crPtlill{!;S of Ihe IF.rl~·, special issue: Neural Nelworks I: Theory and Modeling, 78, 14HI-1497. l'()t~rba .
.J., and 1.. Summers,
19Hf>, "The Persistence of Volatility and Stock Market Fluctuations: A1IIl!riwnl:conomic Review, 76,1142-1151.
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I'ri ..,tl<·y. M., I !IHH. Non-f.inrny and Non-Stationmy Time Snit.1 Analysis, Academic Press, San Diego. Ibchll'r, R., 1!IH2, "Equilihrium Under Uncertainty." in K. Arrow and M. Imriligawr (eds.), }fandbook oj MathfmatirnU;conomics, Volume II, North-Holland, New York, 923-1 00f>. Raely, S., 1994, ~State Prices Implicit in Valuation Fonnulae for Derivative Securities: A Martingale Approach: Discussion Paper 181, L5E Financial Markets Group, London, UK. Ranellt's. R.. and D: Wolfe, 1979, Introductio1l to Ihe Theory of Nonparametric Statislics, .lohn Wiley and Sons, New York. Itc-ddington, F., 1952, "Review of the Principle of Life-Office Valuations." Journal of the 11I.ltittete of ActteflrifS. 7H, 2Hf}-:l4U. Rl'insrh, c., 1967, "Smoothing by Spline Functions," Numer Math, 10, 177-183. Itc,ln),. F., and P. Weil, 1993, "Approximale Equilibrium Assel Pricing." unpublished
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"A Unificd Appmarh It) Tt"Slillg for Serial Correlation in Stock Re-i tllrns.".Immlfll oj IIwiwH, f>7, :~71-399.
578
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RichanlsIln, M" andJ. Stock. Im\!l, "Drawing Inferences frolll Statistics 1\;lSedoll Multi-Yeilr A...~et Retlll'lls," jllllmfll (lIFil/flll(;flll~i'lll/(}III;rs. 2:,. :12:1-:I'IH, Riel/., T,. 19HH. "The Equity Risk Prcmium: A $ollllion?," ./0111'11/11 of /l/lIlIl'IlIry /:'mIIOIII;C5. 21. 117-1 :~2, Ritchken,
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"On Option Pricillg UOllnds ... ./llunwl oj Fillflllrp,1(), 121!1-12:I:I,
Riller, J.. 1990. "Long-Run i'erf
Robbins. B,. and S, Monro. 19:, I a. "A Stochastic Approximation Metho(I," AIII"tI, ol Mnlhnnfllirnl Slflli.l/;n, 2:" 7:17-744, - - - . 195 I h. "A Stochastic Approximation Modl'l," A II II II II IljMlllhpllllllimlSlllli,l/in,
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Robens, II., I !l:,!l. "SIO<,I<.-l\Iar"('1 'l'al:('llIs' .,"d Fillall' ial AII,lly"is: l'>ft-Ihlldlll,,>\i, al Su!;);estiolls ... .1I11I1'1"t/ oj l'i'l/ul/l'/', 1,1. 1-10, IY67. "Statistical l'erSllS C1illkal Prediclioll of the Stock 1\1;11 kel," III1I",i>, lishcd m;llIlIscripl, Celller 1,,1' Researd, in Se~uril}' Prin's, l'lIin'rsin' 01 Chicago. May, I{o'timon. M" I~IH!-I, "lIIol'kTradl'sol\ Iltl' t-.bjorCanadi'lll amlll,S, Slock EXI h,I1'>\('S: A Slud), or Pricing Behal'ior alld l\larkcl Ellicienc)'." 11Iljll,hlish .. d "h,ll, disserlalion. S~hool of IIIIsi IIt'SS Adll\inisl rOll illll. U lIiversi I)' of W('sll"" ()Il' lario, Onlario, Callada, RolJ,illS()n. p" 1979. "The Estimalion or a Non-Linear Moving 1\"'1'''>\l' Siorhaslic PlIJrr,III'.\ 111111 Thfir "N,limlionl, 5, HI-!Hl,
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Roll,. R,. 1!177. "A Criliqul' of Iltl' ASSl'l Pricing Th<,ol'),'-' 'Ie:;(,,: Pal'l I." ./11111111" II/ ! Fillnl/c;1I1 I:'rmllill/ir", 4, 129-17(j, --;-, I!tHO. "Onhogollal Ponrolio","./IIIII'IIII/ '1 1'II/111/rilllllllli C!."l1l1liIIlI1J /P .. llIlI/pi", 1:"IO():,-IO:.n, - - - . 19H1. "1\ Simpll' Iml'li<:il Ml'aSIII(' of Ihe Elkl'live Hid-A," Spl(';ul ill ;tn • Ellil'it'llt Marl:.el:.I0lll'llll/o/FiI/IlI/If, :I!I, I 1:!7-1l'\fl, Roll. R,. and S, Ross. 1!IHO. "An Empilkallll\'esti>\'lliotl of the A,hill'''gl' ",iling Thl'or),," .11111/'111/11111'/11111/0', :I;l, 107:1-1 10:\' - - - , 19H,I. "A Crilkal Rl'l'xaminalion of lit" Empirical El'i
J.,
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Working l'apt'r, Ullivt'rsilY of Tl'xas, AIISlill.
Sicgd,.J., 1!J!J4, SII/rkJ )ilr Illp LOIIK HIIII, Norton, Nt'w YOlk. Sih'!-r, S" I !175, .'i11l/i.l/im/llI)nr'III', (:hapIllan and Iiall, l.ondon. !
Simkowil/., M" and W, 1\c"llIcs, I !IHO, "A'rIllII"'lric Slahlt' Disl/illlll('d SI'<1I1 ill [('" : turns." jOll.nu,[ oj lilt, 1\1I1('17'(flll Slali.'i/ioti ;\.\.wrifllioll, 7[>, :~Oti--:\ I~. ' Silllr'
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I I
Sillglcloll, K., I !J!JO, "Specification alld EsliIllalioll 01 IIIIUlt'II'p .... al ASWI 1', ici"J.: I Modds," ill B. Friedma" alld F. lIahll (cds.), /Il1l1d/lI/ok of ,\/"",'1111.1' FI'" 1I01II;r., North-Ilolland, Alll.~lerdam, SllIi!h, A., 1!l/iH, Sllli~h,
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SI;II\lhallJ.:h, R., 19W1, "On thc Exclusion of Ass~ls from Tcsts of thl~ Two P
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Whitt', II., and I. DonlOwitz, 19H4, "Nonlincar tions," /:ronomflrim, 52, 14:\-it>2.
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Rothschild, 92, 238. 4!11 Roy, :15 Rozell,I7 Ruhack,I79 Rubin, 2::\4 Rubinstein, :l10, ::\41. ::\49. 351. 370, :l81. 461, 507,509 ~ Rudehusch, 423 Ruiz, 190. 493 Runkle, 423, 486, 488. 497 RUlld.501
Ib
Ramaswamy, 216, ::\!B. 445. 446, 455 Rt'ady, I:~t, Rt·,hlingl(}n. 405 Rt'dt'l". :~:~'2 Rt'insch. 522 Rich,ml, '2!I:1 Rkil;ml\on, ·17~1!l, "H, 7!l. 1M. 210. 27·1.4'2'2 Ri"I/, :110, :111 Rillt'l, 1:,6 Robhin-, :1 I!l Rohn
Roll, 17,7'2,101-103.106,128,134, 1:~5. 143. 145, 150. 184, '213, 216, '2:~!I, 240. 243, :~66 ROIlla, 379 Rom.lIlo, 17 ROIlier. 27H. 2H!l Rosenfeld. I'll, 123-125, 143 Ross, D., 110 Ross. S., 31.156, l!l::\, 196,206,216. 219.220,2::\9.240,245, 246, ~II, ::\IH,::\41.355.3HO,::\82,414,429. 433,435,436,440.441,443.444. 44!l, 451l, 458, 463, 508. 509 Rossi,490 Roth. 84
i
I !
Salandro, 102 <;'9. :>'50. :V,i, :~54, ::\56. ::\67, 462. 510 Schull/.I07 Schuss. 346 Schwam. E .• 429, 438, 439. 441, 442. 144.449,455 Schwam., R., H4. R5, RR, 104. 107 Schwei7.er, 379 Schwer!. G .• 149. 4R5. 497 Sclovt'.472
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Scoll, :176, 37!1 SEC, H4, IIIH Scnlana, oJ!!7 Shankt'Il, H5, 1!13, 1!lIi, 19!I, 200, :101;, 212,21[,-217, :12(), :122, 221;, :1:n. 2·15, 241i, :1[, I Shallllo,37!1 Shapim,,J.,107 Shapi..." M., :17H, :1H!I, :117, :120, 12:! Sharpc, 14, 15[" 1[,Ii, IHI Sh~\.~tri, 10:1 Shea,412 Shephard, 4!10, 4!1:I Shiller, :1[)[" :157, 2[,H, :1(il-:1(j:l. 2(i[), :1(;7, :17[), :17C" :17H, 2H I, :1H:I, :1(0), :lOli, 30H, 317, 31 H, :Ililj, :1%-:llI7, ·IOH, ·119, 121,122, 11:1, -14:) Shimko, :170, :,77 Shkifcr. 2·IH, 21!1. 317. :1:1:1 Sias. 1:14 Siegel, A.• 413 Sit·gt'i,.J., :11 I SiC)!;III111HI.4n Sil\'l')'. 7, 12:1. 35H Simkowill., 17 Simonds. H:, Sims. !I!) Sillgl"',:HH Sillglcllill, :lOti. :III, :11-1, 3:11;, :127, 332. :41 H. 4:1!! Ske1toll,;406 Sloan, 2~:1, 2[, I Smidl,l;07 Smilh, 2:1 Smilh, (r" 1110 Smilh,J,. 110 Smith. ~.. '17.7\1 Soriano" 107 Sosill. :I/" <1,,~ lIIultivariate, 490 p('rsistl'lIft', 4H:\ Q( ;ARCII 1II0tlel, 4~17 sillglc-lilftOl' (;ARCII (1,1) lIIodd, 491 slationar), distributioll, ·IH·I liS stock r<'tllrns, 4HH VE(:II lIIodel, '1\11 (;,\R(:I l-ill-lIl<'all Illotlel, '1\).1 (;ARCIl-~1 lIIodd, ·I~)'\ (;'III"iall knlH'I, "II I (;(,IH'raliled AUlon'gressiw (:ollditiollall), 1iel<'l'oskedaslic modd" ·IH:I. S"t' (//.III (;AIH :11 models (;('II('''lli/('1I En 01 Di"l ihutioll, ·IH\I
g('lIerali/.cd illwrS(~ of a 1II.ltriX, :l+I, \!·I:, (;cll('r.tlil.l'd Ml'thod of MOllI<'lIls «;MM), 174. :lOH, \!:l\!, :IH, :I:,~I, 'HH, <1-1\1, 4:,:), 4H!.l, 4~H, :,:12 asymptotic disirilllllioll, ,,:1:1 as}'rnptotic \'ariann... :-l:\:\ Nl·Wt..·)'~\\'t'SI (· ... tirnator, !",:\;, SIDcilasli .. dilI,-,r(,lllial <''1l1alioll, :I:,~)
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ilia I tingall' pric"ing t(·chniqul'. :,:,.! In~lIurity, 3!1(, ltIaXillllllll ('orn:latioll portf,.lio, 29H. .'iff aLI" IlanscnJav;ann
:\:.H, :,:~6 ;ls)'IIl)llOtic distrihutioll, :.:,H ('"ltinuous-record asymptotics, :-\Ii.!
factor analysis, 2:H (;AR( :11 lIlodels, 487 illfOfltl:tlioll matrix, 53H illf •• nllatioll-matfix cquality, ,.:-\!I inq~ularly sampled data. 3(,:, oplion prict', 3li7 'I'""i-maximllm likdihoo
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portfolio, I W" '217 Ilansell:Jagannathan volatility houlld,2'JH minilllum-variancc frontier, I Wi Sharpe ralio, IHH tangcncy portfolio, I HH, I !JG, 21 H {('ro-Ilt'ta portfolio, 18'2, IW., 21 H m-historil's, 112 mixe(1 distribution, .JHI mixture of normal distrihutions, 'IHI MI. estimation. Sf' maximum likelihood estimation M1.1'. SI'I' multilayer perceptron modified duration, 40!l_ Srr al.l{) duration nIOIII('IIt conditions, 3:,9
Monte Carlo simulation methods, :HO, 382, 386 antithetic variates method, 388 comparisons with closed-form solutions, 384 computational cost, 386 control variate method, 3R7 nude Monte Carlo, 3R6 discretiJation, 383, 3R5 efficiency, 386 importance sampling, 3H8 limitations, 390 IlIl1llher-theorelic method, 38H path-dependent option pricing, 3H2 stratifird samplinv;, 3HH variallce-reduction techniques, 387 mortv;age-backed securities, 406 lIlultifactor models, 219, 324. !Su (Ill" I Arhitrage Pricing Theory, exact factor pricing, Intertempor.tl Capital Asset Pricing Model, selection of factors Hlack version of the CAPM"1224, 229 I cross-sectional regression \ approach, 222, 233 empirical studies, 240 Epstein-Zin-Weil recursive utility model,324 eSlimation of expected returns, 231 estimation of risk premia, 2~1 factor portfolios spanning tile mean-variance frontier, 22R Generalized Method of Mor'enL~, 222 Ilotelling 'rl statistic, 232 macroeconomic variables as factors, 226 portfolios as factors, 223 term-strttrture models, 440 multilayer perceptfOn (Ml.P), 512 multiplicative linear congruential generators (MLCG), ~'2:) multipoint moment conditions, 361
,'iubj,.,.t 11II1t'.'(
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Nadara}'a-Wal,oll kernl'l estilliatur. 51)!)
IIl'\\\,\,rk lUpolo),'Y, ~, H Il<'t"'I'l'ks, f, Il!. S"" tllw learnin); Ilf(\vorks nCllqllnclworks. ;,Il! . .'iPl'IIL", Idarnill)( networks Nl'wqy-Wcstestilllator, [,:{!i 1It'\\,s:i"'pact CIIrvc, 4X!i II-hi'lmit's, 47(i nO-oll!hitraKl' collditioll, :\:19 'lOisl' traders. 317 lIominal bonds, 442 lIominal stochastic discolIlII factor, '1-13. Spr tlllO sto(\lastic disl'ount lac lOr nonlille"r ARMA nlodc1s, 471 nonlincar ,autorq~rcssi\'c models, 471 nonlilll'ar dynamical S)'S'l'IJlS, -17:t S" tiL", dctertllillislic lIonli'l<'ar d)'lIamical s}'stellls nonlinear-in-mean time-series Illodels, 41i9 nonlilll~alcin-variance timc-snies lIIodels, 4(i!1 nonlinear least sqllares estilllation. :,1:" ;,IH nonlinear llloving-m'l'raKI' Illolkls. ,lti!! l1onlinl'ar tirnc·st'ric"s analysis. '1(iH lIolllinearity h"till)(. S,'p testill): h,r
nonlincar strllelllrl' nonparametric cstimation, 4!'H. ~,I" IInivcrsal appn,ximatioll propl'rty. ;,1;, nOllparanll'trir option f'ricin~ mcthods, :HO. :192. [II 0 nOllparamcllic lesL<, 172 nonpcriodic cycles, li3 nonseparabilit), in utilit}" :{2Ii nOlls},lIchronolls tr,((ling, H,I. 177
l'llIpiril'allimli.i).:,. 12H nontrading prOl'l'''. I,I!', nonlraclin~, S"" nons),'lI'hronolls tradin~ 11 ....111,,1
distribution, 1;, 'llnnher-tilcorctic nll,thod. :IHH
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