Jaroslaw Morawski Investment Decisions on Illiquid Assets
GABLER EDITION WISSENSCHAFT
Jaroslaw Morawski
Investment Decisions on Illiquid Assets A Search Theoretical Approach to Real Estate Liquidity
With a foreword by Prof. Dr. Heinz Rehkugler
GABLER EDITION WISSENSCHAFT
Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.
Dissertation Universität Freiburg, 2008
1st Edition 2008 All rights reserved © Gabler | GWV Fachverlage GmbH, Wiesbaden 2008 Editorial Office: Frauke Schindler / Anita Wilke Gabler is part of the specialist publishing group Springer Science+Business Media. www.gabler.de No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the copyright holder. Registered and/or industrial names, trade names, trade descriptions etc. cited in this publication are part of the law for trade-mark protection and may not be used free in any form or by any means even if this is not specifically marked. Cover design: Regine Zimmer, Dipl.-Designerin, Frankfurt/Main Printed on acid-free paper Printed in Germany ISBN 978-3-8349-1004-2
To Helena
Foreword
The Portfolio Selection Model developed by Markowitz in the 1950s offers a theoretically founded approach to combining securities into a utility-optimizing investment portfolio, that is, a portfolio leading to the best possible trade-off between the expected return and investment risk. However, when applied in practice, the model often reaches its limits as soon as certain central assumptions are not fulfilled. This holds especially for the assumption of perfect liquidity of all assets in the portfolio, that is, the ability of selling or buying any of these assets at any time immediately and without influencing its market price. A serious additional source of investment risk may arise when no organized and centralized market for a specific asset exists that would disclose the currently prevailing price level, and when the valuations of the asset differ strongly among market participants. Examples of such assets characterized by limited liquidity and valuations’ heterogeneity are direct real estate investments, private equity, and many other privately traded goods. A number of rather simple extensions to the Markowitz model allowing for listed securities with limited liquidity have been developed to date. They are usually based on a price discount due to the lacking ability of an immediate sale. The work of Morawski, however, goes further and offers a complex model of market participants’ behavior on illiquid markets with heterogeneous expectations. Morawski uses his model to analyze the consequences of imperfect liquidity for the price building on the affected markets. Furthermore, he formulates the optimal trading strategy for buyers and sellers and derives several approaches to measuring and managing liquidity risk on the level of individual investments and on the portfolio level. The Theory of Search, which has received only little attention in the theoretical finance so far, constitutes the foundation of this innovative approach. The search model is formulated very generally, and as such it is not restricted to any specific asset class. The assumed way of reasoning and the exemplary practical application focus, however, on private real estate. Illiquidity of property markets is one of the main reasons why portfolio selection based on formal models is still not widespread there. The scope and the depth of the analysis are unique both from the national and the international perspective. The work comes fully up to the claim of providing „the first
VIII
Foreword
coherent framework for analyzing, modeling, and managing problems associated with investing in relatively rarely traded private assets like real estate“ (p. 346). Especially the methodically and analytically challenging central parts, which address the application of the search theory to the liquidity problem and the connection to the classical portfolio theory, are highly innovative and seminal. The propositions regarding the analysis of asset liquidity and the inclusion of a liquidity criterion in portfolio optimization models will doubtlessly stimulate further research in this field. Also, despite the high level of sophistication, they are likely to be applied by professional investors for improving strategic decisions. Therefore, I am certain that this book will be given due attention among researchers and practitioners alike.
Prof. Dr. Heinz Rehkugler
Acknowledgments
This book originated during my time as a scientific assistant and a Ph.D. student at the University of Freiburg, and has been accepted as a doctoral thesis by the Faculty of Economics and Behavioral Sciences. Accomplishing it would not have been possible without the aid and support of many people. At this place, I would like to express my gratitude to all those who contributed to it directly or indirectly. My most profound thanks go to Prof. Dr. Heinz Rehkugler, my academic teacher and doctoral advisor. His uncountable comments and suggestions inspired many of the ideas presented in this book and helped me to avoid numerous pitfalls. I am grateful for his deep insights, his openness, and the freedom he granted me in the final phase of the work. I am also indebted to Prof. Dr. Dr. h. c. Hans-Hermann Francke, who was the second referee of the dissertation. Furthermore, I would like to thank Prof. Dr. Rehkugler and Prof. Dr. Dr. h. c. Francke for the possibility of absolving the Degree Program in Real Estate Appraisal at the Deutsche Immobilien Akademie (DIA) in Freiburg. An important contribution to this work has been made by my colleagues and fellow Ph.D. students from the Department of Banking and Finance: Dr Isabelle Jandura, Dr André Schenek, Dr Ulrike Jedem, Simone Glunz, Pascal Schnelle, Felix Schindler, and Tobias Rombach. They accompanied me during all the years at the University helping with critical comments and discussions. To them I owe the great atmosphere at the Department – inspiring and relaxing at the same time – that made the work so much easier. I will always associate Freiburg with their company. A special word of thanks goes to Karin Leppert, our department secretary, for her friendship and help in formal matters. This book would not have been possible without the support from my parents, Hanna and Franciszek. They guided me through the first years of my education and always had faith in me after I had taken my own way. I would also like to thank my brother Slawek, my best friend and advisor. I could always count on him, whenever I needed help of any kind.
X
Acknowledgments
My deepest gratitude and love, however, is reserved for my wife, Helena. Without her patience, her enthusiasm about my work, and her encouragements in the countless moments of doubt and frustration, I would never have been able to reach this point. She gave me the strength and endurance to finish what I have started. Therefore, I dedicate this book to her.
Jaroslaw Morawski
Contents
Figures ....................................................................................................................... XIX Tables ..................................................................................................................... XXIII Abbreviations .......................................................................................................... XXV Symbols .................................................................................................................. XXIX Introduction ................................................................................................................... 1 Why liquidity? ............................................................................................................ 1 Goals of the Analysis.................................................................................................. 2 Structure of the Analysis ............................................................................................ 4 Chapter 1: The Concept of Liquidity .......................................................................... 9 1.1. Definition of Liquidity ..................................................................................... 10 1.1.1. Review of Liquidity Definitions............................................................... 10 1.1.1.1. Asset Liquidity .................................................................................. 13 1.1.1.2. Market Liquidity ............................................................................... 18 1.1.1.3. Corporate Liquidity ........................................................................... 25 1.1.1.4. Relations between the Notions of Liquidity ..................................... 27 1.1.2. Liquidity Risk ........................................................................................... 29 1.1.2.1. Liquidity Risk of Privately Traded Assets........................................ 29 1.1.2.2. Liquidity Risk in Public Markets ...................................................... 32 1.1.3. A Two-Dimensional Definition of Liquidity ........................................... 36 1.2. Sources of Liquidity ......................................................................................... 40
XII
Contents 1.2.1. Preliminary Considerations ...................................................................... 41 1.2.2. Transaction and Opportunity Costs .......................................................... 44 1.2.2.1. Commissions and Taxes ................................................................... 44 1.2.2.2. Indirect Transaction Costs ................................................................ 48 1.2.2.3. Opportunity Costs ............................................................................. 51 1.2.3. Market Organization and the Search for a Trading Partner ..................... 55 1.2.3.1. Forms of Market Organization ......................................................... 55 1.2.3.2. Market Organization, Search, and Liquidity .................................... 59 1.2.4. Diversity of Valuations............................................................................. 64
1.3. Review of Illiquid Assets ................................................................................. 73 1.3.1. Characteristics of Illiquid Assets .............................................................. 74 1.3.2. Real Estate ................................................................................................ 76 1.3.3. Private Equity ........................................................................................... 81 1.3.4. Alternative Investments ............................................................................ 84 1.4. Economic Relevance of Liquidity ................................................................... 88 1.4.1. Money and Liquidity Preference .............................................................. 88 1.4.2. Liquidity in Investment Decisions............................................................ 92 1.4.2.1. Investment Goals .............................................................................. 92 1.4.2.2. Expected and Unexpected Liquidation ............................................. 98 1.4.2.3. Liquidity in the Sale Case and in the Purchase Case ...................... 100 1.4.2.4. Individual Liquidity and Portfolio Liquidity .................................. 102 Chapter 2: Search in Illiquid Markets .................................................................... 105 2.1. The Theory of Search..................................................................................... 106
Contents
XIII
2.2. Introduction to Search Models ....................................................................... 110 2.2.1. Search Framework .................................................................................. 110 2.2.2. Search Strategy ....................................................................................... 114 2.2.3. Basic Search Model ................................................................................ 116 2.2.4. Karlin’s Model........................................................................................ 121 2.3. The Real Estate Search Model ....................................................................... 123 2.3.1. Framework Modifications ...................................................................... 124 2.3.1.1. Distribution of Offers...................................................................... 125 2.3.1.2. Continuous Time ............................................................................. 126 2.3.1.3. Opportunity Cost and Discounting ................................................. 128 2.3.1.4. Rental Revenues.............................................................................. 129 2.3.1.5. Market Uncertainty ......................................................................... 131 2.3.1.6. The Relative Approach ................................................................... 135 2.3.2. Model Design ......................................................................................... 137 2.3.3. Limitations and Possible Extensions ...................................................... 142 2.3.3.1. Bounded Search Horizon ................................................................ 143 2.3.3.2. Dynamic Market ............................................................................. 146 2.3.3.3. Unknown Offer Distribution and Learning .................................... 149 2.3.3.4. Offer Recalls ................................................................................... 152 2.3.3.5. Intensity of Search .......................................................................... 153 2.3.3.6. Listing Price .................................................................................... 154 2.3.3.7. Search for the Best Seller................................................................ 156 2.3.4. MCS Solution ......................................................................................... 158 2.3.5. Liquidity within the Model ..................................................................... 163 2.4. Search and the Functioning of Illiquid Markets ............................................ 166
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Contents
Chapter 3: Liquidity Measurement ........................................................................ 171 3.1. Traditional Measures of Market Liquidity..................................................... 172 3.1.1. Liquidity of Public Markets.................................................................... 173 3.1.1.1. Bid-Ask Spread ............................................................................... 173 3.1.1.2. Market Depth .................................................................................. 177 3.1.1.3. Price Reversal ................................................................................. 179 3.1.2. Application to Real Estate Markets ........................................................ 182 3.1.2.1. Implicit Bid-Ask Spread ................................................................. 182 3.1.2.2. Quick Sale Discount ....................................................................... 185 3.1.2.3. Market Depth .................................................................................. 186 3.1.2.4. Market Resiliency ........................................................................... 189 3.2. Time- and Probability-Based Measures ......................................................... 191 3.2.1. Probability of Sale and Time on the Market .......................................... 191 3.2.2. Proportional Hazard Ratio ...................................................................... 195 3.3. Measures of Liquidity Risk ............................................................................ 197 3.3.1. Principles of Risk Measurement ............................................................. 198 3.3.2. Volatility ................................................................................................. 202 3.3.3. Asymmetric Measures ............................................................................ 209 3.3.3.1. Default Probability .......................................................................... 210 3.3.3.2. Semivolatility and Lower Partial Moments .................................... 214 3.3.3.3. Value at Risk ................................................................................... 216 3.4. Alternative Measurement Approaches........................................................... 220 3.4.1. Liquidity Performance Measures ........................................................... 221
Contents
XV
3.4.2. Utility-Based Measurement .................................................................... 223 3.5. Relations between the Measures .................................................................... 226 Chapter 4: Liquidity as a Decision Criterion ......................................................... 233 4.1. Investment Decisions in the Mean-Variance Framework.............................. 234 4.1.1. The Efficiency Criterion ......................................................................... 234 4.1.2. Diversification and Mean-Variance Portfolio Selection ........................ 238 4.1.3. Portfolio Selection with Risk-Free Assets.............................................. 241 4.1.4. Limitations of the MV-Criterion ............................................................ 242 4.2. Strategic Liquidation ...................................................................................... 244 4.2.1. Literature Review ................................................................................... 245 4.2.2. Efficiency of Liquidation Strategies ....................................................... 247 4.2.3. Liquidation Strategies for Portfolios of Assets ...................................... 254 4.2.3.1. The Liquidity-Diversification Effect .............................................. 254 4.2.3.2. Covariance of Gains........................................................................ 255 4.2.3.3. Portfolio Effect of Liquidation ....................................................... 258 4.2.4. Simultaneous Liquidation of Liquid and Illiquid Assets........................ 266 4.3. Optimal Liquidation and the Notion of Liquidity .......................................... 270 4.3.1. Liquidity in Terms of Efficient Liquidation Strategies .......................... 271 4.3.2. Liquidity with Liquid and Illiquid Assets .............................................. 272 4.3.3. Liquidity of Assets in Portfolios............................................................. 274 4.4. Portfolio Selection with Illiquid Assets ......................................................... 276 4.4.1. Literature Review ................................................................................... 277
XVI
Contents 4.4.1.1. Asset Allocation with Non-Tradable Assets................................... 278 4.4.1.2. Models with Trade Restrictions ...................................................... 280 4.4.1.3. Liquidity as an Independent Decision Criterion ............................. 283 4.4.2. MPT with Illiquid Assets........................................................................ 286 4.4.2.1. Planned Portfolio Liquidation ......................................................... 287 4.4.2.2. Unexpected Portfolio Liquidation .................................................. 292 4.4.3. Optimization Algorithms ........................................................................ 299 4.4.4. Sources of Biases .................................................................................... 302 4.4.4.1. Effects of Search Model Imperfections .......................................... 302 4.4.4.2. Risk Measurement Issues................................................................ 304
Chapter 5: Liquidity of German Condominium Markets .................................... 307 5.1. Determination of Model Parameters .............................................................. 308 5.1.1. Volatility of Offers ................................................................................. 308 5.1.2. Offer Arrival Frequency ......................................................................... 311 5.1.3. Other Parameters .................................................................................... 313 5.2. Condominium Liquidity Analysis ................................................................. 314 5.2.1. Data Material .......................................................................................... 315 5.2.1.1. RDM/IVD Data ............................................................................... 315 5.2.1.2. GAA Data ....................................................................................... 316 5.2.1.3. Determination of Model Parameters ............................................... 319 5.2.2. Liquidity Measurement .......................................................................... 323 5.2.2.1. Market Depth .................................................................................. 323 5.2.2.2. Market Breadth ............................................................................... 324
Contents
XVII
5.2.2.3. Time on Market............................................................................... 327 5.2.2.4. Liquidity Risk ................................................................................. 331 5.2.2.5. Liquidity Risk Reward .................................................................... 336 5.2.2.6. Two-Dimensional Liquidity Assessment........................................ 337 5.2.2.7. Comparison of the Measures .......................................................... 341 5.2.3. Condominiums in Portfolio Decisions ................................................... 347 5.2.3.1. Planned Liquidation, Return Characteristics, and the Efficient Frontier ............................................................................................ 348 5.2.3.2. Portfolio Selection in a Multidimensional Decision Framework with Liquidity Criterion .................................................................. 352 5.3. Discussion of the Results ............................................................................... 357 Concluding remarks ................................................................................................. 363 Appendix .................................................................................................................... 367 References .................................................................................................................. 397
Figures
Figure 1-1: Liquidity as a price-time locus ................................................................ 16 Figure 1-2: Ambiguousness of relative liquidity in terms of the price-time locus .... 18 Figure 1-3: Liquidity as a trade-off between execution costs, execution time, and order size .......................................................................................... 19 Figure 1-4: Temporary and permanent price effects of a seller-initiated large sale transaction ........................................................................................ 23 Figure 1-5: Market breadth and market depth ........................................................... 24 Figure 1-6: Optimal cash balance .............................................................................. 27 Figure 1-7: Optimal liquidation time without liquidity risk ...................................... 31 Figure 1-8: Liquidity risk as the uncertainty of the liquidation value ....................... 32 Figure 1-9: Variations of market liquidity on NYSE and AMEX ............................. 34 Figure 1-10: Exogenous and endogenous liquidity risk .............................................. 36 Figure 1-11: Structure of the two-dimensional liquidity definition............................. 37 Figure 1-12: The effect of a fixed commission on the price-time locus of liquidation ............................................................................................... 46 Figure 1-13: Price-time loci for organized and non-organized markets ...................... 63 Figure 1-14: Diversity of valuations and the price building room .............................. 71 Figure 1-15: Decreasing marginal utility and risk aversion ........................................ 96 Figure 1-16: Interrelations between the main investment goals .................................. 98 Figure 2-1: Scheme of the search process ............................................................... 113
XX
Figures
Figure 2-2: Determination of the optimal reservation price in the real estate search model ......................................................................................... 141 Figure 2-3: Scheme of a Monte Carlo Simulation for the real estate sale process .. 161 Figure 2-4: Monte Carlo estimation of the distribution of net receipts from sale ... 162 Figure 2-5: Locus of expected net sale receipts and search duration in the real estate search model ............................................................................... 164 Figure 2-6: Reservation prices of buyers and sellers and the price building in illiquid markets ..................................................................................... 169 Figure 3-1: Alternative notions of risk referred to the probability distribution of the goal variable .................................................................................... 201 Figure 3-2: Volatility of net sale receipts as a function of the reservation price ..... 208 Figure 3-3: Total variability and downside risk under a right-skewed (a) and a left-skewed (b) distribution of the goal variable. .................................. 210 Figure 3-4: Default probability as a function of the reservation price .................... 212 Figure 4-1: First degree stochastic dominance ........................................................ 236 Figure 4-2: Mean-variance efficiency and investment choice ................................. 237 Figure 4-3: Diversification effect in a two-asset portfolio ...................................... 239 Figure 4-4: Efficient frontier and portfolio selection............................................... 240 Figure 4-5: Portfolio selection with a risk-free interest rate .................................... 241 Figure 4-6: Stochastic dominance of liquidation strategies ..................................... 248 Figure 4-7: Expected net sale receipts and receipts’ volatility in the search framework ............................................................................................. 251 Figure 4-8: Locus of expected net receipts and receipts’ volatility ......................... 252 Figure 4-9: Liquidity efficiency frontier .................................................................. 253
Figures
XXI
Figure 4-10: Volatility of net receipts from liquidation of a portfolio of identical assets as a function of the reservation price and the number of properties ............................................................................................... 259 Figure 4-11: Efficiency of liquidation strategies for portfolios consisting of multiple identical assets ........................................................................ 260 Figure 4-12: Liquidity risk reduction and the optimality of reservation prices for two identical assets liquidated using the same strategy ........................ 261 Figure 4-13: Expected net receipts (a) and receipts’ volatility (b) for two identical assets liquidated using different reservation prices................ 262 Figure 4-14: Liquidity-efficient frontier for two identical assets liquidated using different reservation prices.................................................................... 263 Figure 4-15: Liquidity risk reduction for two different assets liquidated in different proportions ............................................................................ 264 Figure 4-16: Simultaneous liquidation of liquid and illiquid assets .......................... 268 Figure 4-17: Asset liquidity in terms of liquidity efficient frontiers ......................... 271 Figure 4-18: Asset liquidity in terms of liquidity efficient lines ............................... 273 Figure 4-19: Portfolio decision room with a non-marketable asset according to Brito (1978) ........................................................................................... 279 Figure 4-20: Optimal portfolio weight of a liquid stock as a function of the illiquid fraction for various levels of the illiquid stock’s beta in the model of Kahl et al. (2003) ................................................................... 281 Figure 4-21: Examples of liquidity-filtered (a) and liquidity-constrained (b) efficient portfolios according to Lo et al. (2003) .................................. 285 Figure 4-22: Stochastic dominance with return and liquidity goals .......................... 295 Figure 4-23: Efficient plane and portfolio selection with a one-dimensional liquidity goal ......................................................................................... 297
XXII
Figures
Figure 5-1: Expected ToM of selected German condominium markets (days)....... 331 Figure 5-2: Two-dimensional liquidity measurement of selected German condominium markets ........................................................................... 340 Figure 5-3: Ranks of selected German condominium markets with respect to their liquidity according to different measurement approaches............ 344 Figure 5-4: Efficient frontier with returns corrected for liquidation effects ............ 351 Figure 5-5: Efficient frontiers with separate liquidity criteria: Implicit Spread (a), LRR (b), and expected net receipts and receipts’ volatility (c) ...... 356
Tables
Table 1-1: Definitions of liquidity on the internet ..................................................... 11 Table 1-2: Definitions of euro area monetary aggregates .......................................... 15 Table 1-3: Elements of real estate transaction costs in selected European countries.................................................................................................... 78 Table 1-4: Stages of private equity investments ........................................................ 82 Table 1-5: Rankings of selected investors’ goals....................................................... 93 Table 3-1: Cross-sectional means of time series correlations between liquidity measures for individual stocks on the NYSE. ........................................ 228 Table 5-1: Return statistics for selected German condominium markets ................ 320 Table 5-2: Parameters of the search model for selected German condominium markets.................................................................................................... 322 Table 5-3: Estimated number of transactions per inhabitant in selected German condominium markets ............................................................................ 324 Table 5-4: Market breadth measures for selected German condominium markets.................................................................................................... 326 Table 5-5: Expected ToM of condominiums in selected German condominium markets (days)......................................................................................... 328 Table 5-6: Volatility of net receipts on selected German condominium markets ... 332 Table 5-7: Liquidity risk measures for selected German condominium markets .... 334 Table 5-8: Liquidity Risk Reward of selected German condominium markets ...... 337 Table 5-9: Rank correlations between liquidity measures of selected German condominium markets ............................................................................ 345
XXIV
Tables
Table 5-10: Characteristics of investment returns on selected German condominium markets with optimal liquidation at the investment horizon .................................................................................................... 350
Abbreviations
$
– U.S. dollar
ADIG
– Allgemeine Deutsche Investmentgesellschaft (mbH)
AMEX
– America Stock Exchange
APT
– Arbitrage Pricing Theory
ATM
– Automating Telling Machine
BauGB
– Baugesetzbuch (German Construction Code)
c.d.f.
– cumulative distribution function
CAPM
– Capital Asset Pricing Model
CDAX
– Composite DAX (Deutscher Aktienindex)
CLA
– Critical Lines Algorithm
DEP
– Quoted Depth
DP
– Default Probability
e.g.
– for example (lat. exempli gratia)
ECB
– European Central bank
ESPR
– Effective Spread
et al.
– and others (lat. et alia)
EVCA
– European Venture Capital Association
f., ff.
– following page, following pages
FN
– footnote
FSD
– first degree stochastic dominance
FTSE
– Financial Times Stock Exchange (Index)
FV
– fair value
XXVI
Abbreviations
G
– good (quality)
GAA
– Gutachterausschuss für Grundstückswerte (Appraisers’ Committee)
GDP
– Gross Domestic Product
HARA
– Hyperbolic Absolute Risk Aversion
HNWI
– High Net Wealth Individual
i.e.
– that is (lat. id est)
IPD
– Investment Property Databank
IPO
– Initial Private Offering
IS
– Implicit Spread
IVD
– Immobilien Verband Deutschland
L-Beta, Lß
– Liquidity Beta
LIAC
– liquid-illiquid asset correlation
LPM
– Lower Partial Moment
LR
– Liquidity Ratio
LRR
– Liquidity Risk Reward
LV
– Liquidation Value
LVaR
– Liquidity Value at Risk
M
– medium (quality)
MBO
– Management Buy Out
MEC
– Market Efficiency Coefficient
MERP
– maximal expected return portfolio
MPT
– Modern Portfolio Theory
MSC
– Monte Carlo Simulation
MV
– mean-variance, mean-volatility (criterion)
MVP
– minimal-variance portfolio
Abbreviations
XXVII
NASDAQ
– National Association of Securities Dealers Automated Quotations
NCREIF
– National Council of Real Estate Fiduciaries
NVCA
– National Venture Capital Association
NYSE
– New York Stock Exchange
p., pp.
– page, pages
p.d.f.
– probability density function
PE
– Private Equity
PESPR
– Proportional Effective Spread
PHR
– proportional hazard ratio
PoS
– Probability of Sale
PQSPR
– Proportional Quoted Spread
QSD
– Quick Sale Discount
QSPR
– Quoted Spread
RDM
– Ring Deutscher Makler
RICS
– Royal Institution of Chartered Surveyors
RV
– reward-to-volatility
S&P
– Standard and Poors
SSD
– second degree stochastic dominance
TEGoVA
– The European Group of Valuers Associations
ToM
– Time on the Market
TSD
– third degree stochastic dominance
U.S., USA
– United States, United States of America
UK
– United Kingdom
VaR
– Value at Risk
VAR
– Vector Auto-Regressive (Model)
XXVIII
Abbreviations
VAT
– Value Added Tax
VBA
– Visual Basic for Applications
VC
– Venture Capital
Symbols
Due to the large amount of mathematical notations, it was not always possible to assign each variable a different symbol. Hence, in some cases, the meaning of a symbol depends on the context in which it is used. Roman alphabet arg max (.) – argument maximum operator A
– random deviation from the trend (market uncertainty parameter)
Ã
– continuous random deviation from the trend
Â
– random deviation from the trend in a representative period
A, Α
– random deviation from the trend referring to the earlier or the later period
Add1, Add2 – addends of sums (auxiliary variables) b
– parameter of the hazard function
c
– unit observation cost
CFt
– cash flow (net operating income) between t-1 and t
cov(.,.)
– covariance operator
corrs
– Spearman correlation coefficient
D
– expected duration of the search
Dist
– probability distribution from the set of possible distributions
e
– Euler’s constant
E
– absolute purchasing expense
E(.)
– expectation operator
E(.|.)
– conditional expectation operator
f(.)
– probability density function operator or p.d.f of the offer distribution
XXX
Symbols
F(.)
– probability function operator or c.d.f of the offer distribution
fN(.)
– normal probability density function operator
FN(.)
– cumulative normal distribution function operator or c.d.f. of the offer distribution in period N (depending on the context)
fR(.)
– p.d.f of the relative offer distribution
FR(.)
– c.d.f. of the relative offer distribution
FT(.)
– c.d.f. of time intervals between offers
Gt
– effective sale receipts in period t
Gi
– absolute net receipts from the decision on offer i
GM
– minimum absolute net receipts required to avoid default
h
– rental revenues
h(.)
– hazard function
H
– capitalized rental revenues (auxiliary variable)
H(.)
– auxiliary function
i
– indexing or enumeration
I
– number of offers
k
– indexing or enumerator
ki
– rank of the ith element
ℓ
– price impact parameter
ℓ
– liquidity metric
~ l
– liquidity measure underlying the liquidity metric ℓ
l0
– threshold level of the liquidity metric ℓ
L
– point in the MV-room corresponding with perfect liquidity
Liq
– liquidity measure (in particular, Mok’s liquidity measure)
ln(.)
– logarithm operator
Symbols LVaRM
XXXI – market level based LVaR
LVaRrelative – market level based LVaR n
– realization of the random number of offers, number of elements, period, or grade of the LPM (depending on the context)
N
– random number of offers or number of shares traded at a given bid/ask (depending on the context)
N(.)
– normal distribution operator
New
– new offer
O(p*)
– point in the MV-room corresponding with the reservation price p*
Old
– set of past offers
P, p
– random price, realization of the random price
p ∗E ,i
– reservation price applied in the planned liquidation of asset i
p ∗UE ,i
– reservation price applied in the unexpected liquidation of asset i
pL
– listing price
p*
– absolute reservation price
p*i
– absolute reservation price for offer i
p*T
– tangential reservation price
p*T
– tangential reservation array
PM
– market portfolio
PM,i
– minimum absolute sale price required to avoid default in period i
PM, PNM
– price of a marketable and non-marketable asset
POpt
– optimal portfolio
Pr(.)
– probability operator
Pr(.|.)
– conditional probability operator
R, r
– (random) return, realization of a random return
rc
– continuous return
XXXII
Symbols
rd
– discrete return
rt
– return in period t
rte
– excess return over the market (market index) in period t
RF
– risk-free interest rate
RM, RNM
– returns of a marketable and non-marketable asset
RP
– (random) portfolio return
~ Rt
– corrected total return of a property investment in period i
S(.)
– standard deviation operator
Si
– market state i
SV(.)
– semivolatility operator
t
– point in time, time interval, the realization of a random time interval, or target of a risk function (depending on the context)
T
– random time interval
~ ~ T, t
– total random time or search duration, realization of the total random time or search duration
~∗ t
– optimal duration of search
U(.)
– utility function operator
Vi
– value of further search if offer i is rejected
V(.)
– variance operator
V*
– future minimal value of an asset
V0
– current value of an asset
Vopt
– value of an optimally conducted search
Vt
– value of a search terminated within the first t periods
wi
– weight of an asset i in a portfolio
x%
– confidence level
Symbols
XXXIII
X, Y, Z
– symbols denoting different assets, alternatives, scenarios etc., or auxiliary variables
x 1, x 2
– parameters (especially in a utility function)
x(.), y(.)
– auxiliary functions
Greek alphabet ß
– Beta coefficient or parameter of the PHR (depending on context)
γ
– relative rent
Γ
– relative net receipts from sale
ΓM
– minimum relative sale receipts required to avoid default
δ
– discounting factor
ε
– capitalization factor (auxiliary variable)
ζ
– risk aversion parameter
η
– relative inspection /appraisal cost
λ
– frequency of offer arrivals
µ
– expected value; in particular expected value of an offer,
µ0
– expected net receipts from an immediate liquidation
µr
– expected value of continuous returns
ν
– volatility coefficient of offers
Ξ
– relative purchasing expense
ξt
– error term in period t
π*
– relative reservation price
πB*
– relative reservation price in the purchase case
π S*
– relative reservation price in the sale case
Π, π
– random relative price offer, realization of the random relative price offer
ΠM,i
– minimum relative sale price required to avoid default in period i
XXXIV
Symbols
~ Π
– pi-constant
ρ
– discounting rate
ς
– relative known (non-random) purchase price
σ
– standard deviation; in particular standard deviation of offers,
σA
– standard deviation of the market uncertainty parameter
σr
– standard deviation of continuous returns
σXY
– covariance between asset X and asset Y
τ
– deterministic linear market trend
ϑt
– trade volume in period t
ϕ (.)
– p.d.f. of the standard normal distribution,
Φ (.)
– c.d.f. of the standard normal distribution.
Ψ, ψ
– Markov transition matrix, element of the Markov transition matrix
Introduction
Why liquidity? “Of the maxims of orthodox finance none, surely, is more anti-social than the fetish of liquidity, the doctrine that it is a positive virtue on the part of investment institutions to concentrate their resources upon the holding of ‘liquid’ securities. It forgets that there is no such thing as liquidity of investment for the community as a whole” wrote Keynes in “The General Theory of Employment, Interest and Money” (1936, p.155). And still, there is little doubt as to the importance of liquidity for financial investments, neither among researchers nor among investors. The “ease of liquidation”, or more generally the “ease of trading”, which is the simplest (though imprecise) connotation of this term, has been discussed by a number of leading economists, among them Marschak, Tobin, and Hicks,1 and it is an always present subject in the daily news from financial markets. High interest of the theoretical economy in this phenomenon is driven by the desire to understand the differences in the economic role of different assets and, consequently, in the differences in their valuations, which are not always apparent and rational on the first sight. Why should a house, which has a very direct practical use to any individual, be more difficult to sell than a piece of paper containing only a very vague promise of future gains? And does this fact affect the value of the former or the latter? In contrast, the interest of investors is more practical and focuses on the consequences of liquidity for investment activity. For this purpose, the fact of an asset being more or less liquid than another asset can be taken as given; the key question is how to allow for this fact in the investment strategy in order to ensure optimal results? Yet, despite different perspectives both researchers and investors are concerned with one and the same issue and their interests are, in fact, complementary. Hence, although the main goal of the analysis lies in the provision of a practicable method for coping with problems associated with investing in illiquid asset, it can only be achieved by exploring the nature of this phenomenon in an adequate theoretical framework. In effect, the work remains on the edge between theoretical economic considerations and practical investment analysis.
1
See Marschak (1949, 1950), Tobin (1958), and Hicks (1962, 1974).
2
Introduction
Goals of the Analysis Two main problems encountered by practitioners willing to invest in illiquid assets are the determination of the level of liquidity and its incorporation in the decision processes. These issues have been already discussed in the literature with respect to publicly traded assets, mainly stocks, but they still remain unsolved for most of the privately traded illiquid investments, especially for real estate. For example, it is quite clear that the stocks traded on the New York Stock Exchange are much more liquid than the building at 11 Wall Street in New York, just as the stocks traded on the Deutsche Börse are more liquid that the building at Neue Börsenplatz 4 in Frankfurt. It is more difficult, though probably still feasible, to determine which of these stocks are more liquid, but it is nearly impossible to state with the available methods which of the two mentioned buildings in New York and Frankfurt would be easier to sell. Hence, a general method for comparing different assets with respect to their liquidity is necessary. However, even if the relative liquidity of two investments can be assessed, it is still seldom possible to state how high the difference between them in this respect is. The latter problem requires the implementation a well defined quantitative measure that would be general enough to be applied to a class of different investments. The development of a liquidity measure is the first step to allow for liquidity in investment decisions. They refer, on the one hand, to the way individual investments are conducted, and, on the other hand, to the way different investments are combined in portfolios. The majority of existing models used for optimization of investment decisions, including the most popular mean-variance portfolio selection framework proposed by Markowitz (1952 and 1959), require that all analyzed assets are perfectly liquid. This means that either the range of application of these models is highly limited, or that the results achieved by their application to illiquid assets yield unreliable results. The latter case can lead to a false allocation of capital and have disastrous consequences for the affected investors. Thus, an adequate method of assessing the consequences of assets’ imperfect liquidity for the outcome of an investment as well as for the properties of an investment portfolio is inevitable. In order to obtain satisfactory results, the liquidity measure on which such a method is based should posses certain properties. On the one hand, it should take into account the effects of investor’s actions, which can potentially affect her liquidity position; on the other hand, it should be applicable not only on the level of individual assets but also on the portfolio level. Fur-
Goals of the Analysis
3
thermore, the measurement method needs to be compatible with other decision variables and with measures of other investment goals. All in all, a coherent decision system, of which the liquidity criterion is an integral part, becomes necessary. The approach to the measurement and management of liquidity followed in this work is based on the Search Theory, also known as the Theory of Optimal Stopping. This branch of mathematical stochastics deals with a family of problems that can be interpreted in terms of a search process. The choice of this method arose from the theoretical analysis of the nature of liquidity and its determinants. It its course, it turned out that the character of the search for a trading partner during the liquidation of an investment is crucial for its liquidity. The environment of the search and the manner of its execution are the prime determinants of the ability to sell quickly and at a low cost. A formal description of the corresponding process can, therefore, be used as the reference point for analyzing liquidity. In particular, it can be utilized for the derivation of statistical parameters of the liquidation value and liquidation time as well as for the estimation of the uncertainty associated with the liquidation, i.e., liquidity risk. Due to its good extendibility, the search theoretical framework allows for the inclusion of a number of different factors relevant in this context and can be used for modeling a number of different situations. It is also compatible with many of the existing decision models, in particular, with the standard portfolio selection model. Finally, due to the possibility of deriving easily computable closed-form solutions, the approach seems to be an excellent tool for the application in practical investment decisions. The focus of the analysis on the practical aspects of investment activity requires that both the measurement of liquidity and its inclusion in portfolio decisions can be achieved on the basis of information available to investors. The latter point turns out to be especially problematic. In particular, smaller investors may find it difficult to access the necessary market data, which is often scarce or proprietary in intransparent private markets. For this reason, the book is addressed mainly to institutional investors. Furthermore, the methods proposed here are designed mainly for the application to direct real estate investments. Also this choice has a practical background. On the one hand, a separate consideration of every possible private illiquid asset would be far beyond the scope of this work. On the other hand, real estate is by far the most relevant private illiquid asset and the lack of an adequate method to cope with its illiquidity is most pressing. Nevertheless, the proposed methods are very general in their nature and can
4
Introduction
be easily redefined for the application to any asset provided the required parameters can be assessed with sufficient precision.
Structure of the Analysis The book is structured in five Chapters. The first Chapter has an introductory character and addresses the issue of the precise meaning of the term ‘liquidity’. What was originally intended as a brief presentation of a widely accepted definition turned out to be a complex discussion of the numerous approaches present in the literature, which not only differ from each other but are also partially contradictory. In effect, a single notion of liquidity that would be valid throughout the work needed to be chosen from among the different existing ones. However, in the course of the analysis, it became apparent that most of the existing approaches focus on certain aspects of the problem only. In particular, the uncertainty associated with buying or selling assets on private markets remains mostly disregarded. Therefore, a two-dimensional definition referring to assets rather than to markets and encompassing both the expected outcome of the liquidation and the uncertainty about it has been formulated. In the next step, the Chapter addresses the sources of liquidity. Also at this point researchers do not speak with one voice. A large number of possible direct and indirect factors that may influence liquidity of assets or markets have been thoroughly discussed and categorized into three groups. One of the main conclusions from this analysis is the recognition of the central role of search. This finding was formative for the approach followed later in the analysis. The next section of the Chapter offers a review of assets affected by the liquidity problem. The purpose of such a review is to refer the abstractly discussed sources of liquidity to real assets and to demonstrate how they affect investors active on the respective markets. The main result is the reinforcement of the intuitive thesis that real estate is by far the most important class of illiquid investments. The Chapter closes with considerations regarding the role of liquidity in the economy. Starting with a short discussion of the relevant economic theories, the section concentrates on the importance of liquidity for investment decisions. In particular, its place among investment goals is highlighted and several special aspects of this subject are discussed. After the extensive discussion of the actual meaning of liquidity, its sources, and its consequences, the second Chapter is devoted to the search theoretical model, which is central for most of the later considerations. The presentation of the theoretical background of the “Theory of Search” is offered at the beginning. It summarizes the main
Structure of the Analysis
5
features of the theory and its fields of application. In the following section, the principles of the construction of search models are outlined including two most popular basic model forms. The subsequent section, in which a specific model of real estate transaction process is developed and discussed, constitutes the central part of the Chapter. After proposing several extensions to the basic search models, which allow more accurate capturing of the specific features of property investments, the design of the actual real estate search model is presented. It is then thoroughly discussed with respect to its limitations as well as the possibilities of its improvement. While most of the solutions within the model are offered in a closed form, the analytical approach cannot always be applied. The utilization of Monte Carlo methods, which receive special attention in a separate subsection, can provide an approximate solution in such cases. The Chapter is concluded with an analysis of the relations between the notion of liquidity formulated earlier and the features of the (real estate) search theoretical model. It demonstrates that all of the relevant aspects of the problem are captured within the model parameters. The next, third Chapter deals with the problem of liquidity measurement. As already stated, this is the first and probably most important of the problems that need to be solved in order to allow for liquidity in investment decisions. Despite the fact that a number of researchers have already approached this challenge, still no well structured and coherent framework is available that could be applied to a number of different asset classes. Most of the existing ones focus only on publicly traded assets, and, thus, remain on relatively high levels of liquidity. The presentation of these traditional measures is offered at the beginning of the Chapter. Simultaneously, however, a method of their application to real estate investments is proposed by reinterpreting the respective measures in terms of the real estate search model. The second category of measures, also present in the literature though much more seldom, is based on the assessments of liquidation probabilities and times. It is presented in the subsequent section. A separate section is also devoted to measures of liquidity risk understood mainly as the uncertainty of liquidation, which has been identified as an independent dimension of liquidity in Chapter 1. These measures are largely a novel development and result from the combination of popular methods of investment risk measurement with the real estate search model. Finally, the last group of measures is discussed under the caption “alternative approaches”. Their common feature is the inclusion of both di-
6
Introduction
mensions of liquidity –“marketability” and “liquidity risk” – within one figure. The Chapter closes with the analysis of the relations between the discussed measures. The fourth Chapter concentrates on the actual goal of the work – the specification of methods for including liquidity as a criterion in investment decisions. Like in the earlier parts of the thesis, also here institutional real estate investors are addressed in the first line. The central idea is to combine the real estate search model with the meanvariance decision model. This way, the search theoretical approach developed in Chapter 2 can be implemented in the widespread and well researched investment decision framework. For the sake of a structured presentation, the principles of decision making (portfolio selection) in the mean-variance room are presented first. The brief introduction of the main features of the Modern Portfolio Theory (MPT) should, on the one hand, provide the theoretical background for applying analogous methodology to the liquidation problem and, on the other hand, ensure a coherent terminology in the following parts of the Chapter. The next section deals with strategic liquidation of assets. In particular, a method for determining the optimal liquidation strategy for portfolios containing illiquid asses (real estate) is proposed. Pure real estate portfolios and portfolios containing both liquid and illiquid investments are considered separately. The conclusions from this section have consequences for the understanding of the term ‘liquidity’. It turns out that the level of liquidity may differ depending on whether an investment is viewed on a stand-alone basis or whether it is viewed in the portfolio context. These issues are discussed in the subsequent section. In the final section of the Chapter, an extension to the standard portfolio selection model is proposed in which liquidity is allowed for as a separate decision criterion. After the presentation of solutions for two different liquidation cases and the discussion of optimization algorithms in the extended model, possible sources of biases and their effects are discussed. The main goal of the final Chapter of the book is to demonstrate how the techniques discussed in the preceding chapters can be applied in practice. Condominium investments in selected German urban areas have been chosen for this purpose. The analysis of their liquidity gives also the opportunity to discuss problems arising in the implementation of the search theoretical approach. Determination of the values of model parameters constitutes the main difficulty. Although their interpretations within the search model are relatively straightforward, some of them cannot be observed directly. The necessity to fall back on measurable proxies results in further sources of biases.
Structure of the Analysis
7
Having discussed the related problems theoretically, the Chapter turns to the empirical analysis on the basis of a unique data set for condominiums provided by Appraisers’ Committees (Gutachterausschüsse für Grundstückswerte) in five German cities. After the demonstration of various liquidity measurement techniques as well as optimization methods for portfolios containing investments in condominium markets, the achieved results are compares and discussed. This step closes the Chapter. The concluding remarks offered at the end of the book summarize the main findings, point out their contributions to research on liquidity, and offer an outlook for further studies in this field.
Chapter 1 The Concept of Liquidity
The main goal of the initial chapter of this work is to formulate the concept of liquidity and, thus, to prepare the foundation for the entire following analysis. Unambiguous definitions of the key terms are not only necessary for delimiting the discussed problems from other related issues in finance but also essential for ensuring the consistency of the measurement and management approaches proposed later. The review of the literature reveals a substantial disagreement among researchers concerning the nature of liquidity, its sources, and its consequences for investors. The use of liquidity related terms is chaotic and definitions are partially self-contradictory. This Chapter is an attempt to provide a structured concept, though a number of issues are still left open. In the first place, however, it should be viewed as the basis for the analysis in the following parts of the book. The term “liquidity” is widely used in the literature, though its understanding varies significantly. The goal of the first section of the Chapter is, thus, to provide an unambiguous definition, which could be referred to throughout the analysis. Firstly, existing notions of liquidity are reviewed and classified into two main approaches: one focusing on the characteristics of assets (asset liquidity) and the other one regarding the subject from the market perspective (market liquidity). In the second step, the concept of liquidity risk is introduced and integrated into these approaches; a two-dimensional notion of liquidity results. The second section of the Chapter is devoted to the analysis of sources of liquidity and liquidity risk. In particular, the main characteristics of investments and investment environments, which may have a significant impact on different aspects of liquidity, are discussed. The following, third section adds a practical dimension to the discussion. A review of assets with respect to which liquidity problems are especially severe is offered and their characteristics are analyzed. The identification of illiquid assets is also necessary to determine the scope of application of the methods developed later. Finally, the fourth section deals with the economic relevance of liquidity. Its role in the economy is discussed concentrating especially on individual investment decisions.
10
Chapter 1: The Concept of Liquidity
1.1.
Definition of Liquidity
Paraphrasing the statement of the US Supreme Court on pornography some researchers state that “liquidity, like pornography, is easily recognized but not so easily defined”2. Indeed, both the investment theory and the practice seem to have substantial difficulties in formulating a clear definition. The term is mostly used without further explanations under the assumption any reader would know what is meant. In many cases, especially in the day-to-day trading, it is not really necessary to theorize extensively on the meaning of liquidity. Most investors can intuitively qualify the main markets or asset types as liquid or illiquid and recognize the related effects. However, the lack of a formal definition causes serious problems when developing formal approaches considering liquidity as an investment criterion. Therefore, this task has to be dealt with at the very beginning of the first Chapter. 1.1.1.
Review of Liquidity Definitions
Doubtlessly the most frequently cited definition of liquidity, dating back to 1930, is the one provided by Keynes in his “Treatise on money”. Referring to bills and call loans of a bank he describes them as “…more ‘liquid’ than investments, i.e., more certainly realizable at short notice without loss…”3. A lot of research referred to this definition since then,4 but also numerous new approaches have arisen. Today, searching the internet for the term “liquidity” yields a huge number of hits in various online lexicons on finance. A selection is presented in the Table 1-1.
2 3 4
See, e.g., O’Hara (1997), p. 215, David Porter in Caginalp et al. (2002), p. 40, or Roll (2005), p. 8. Keynes (1930), p. 67. E.g., Hicks (1962), Miller (1965), or Lippman/McCall (1986) offer extensive discussions on the precise meaning of the Keynes’ liquidity definition.
1.1 Definition of Liquidity
11
Table 1-1: Definitions of liquidity on the internet5 • •
•
•
• • •
• • • • •
•
5
The ease with which an asset can be turned into cash. http://www.morganstanley.com/im/glossary/ …ability to buy or sell an asset quickly and in large volume without substantially affecting the asset's price. Shares in large blue-chip stocks like General Motors or General Electric are liquid, because they are actively traded and therefore the stock price will not be dramatically moved by a few buy or sell orders. ... http://www.pbucc.org/pension/tools/glossary.php The ability of the market in a particular security to absorb a reasonable amount of buying or selling at reasonable price changes. Liquidity is one of the most important characteristics of a good market. http://www.4insurance.com/annuity/glossary.asp The proportion of cash or cash equivalents in a company's assets. Sometimes used as a measure of the near term financial health of a company. Also a measure of the volume of shares being traded, which may affect the ability of buyers or sellers to build/unwind large holdings without a substantial impact on the price. http://www.scottish-newcastle.com/sn/investor/services/glossary/ The ease and speed with which an investment can be converted into cash. http://www.tiaa-cref.org/pubs/html/financial_terms/kl.html The ability to buy or sell an asset quickly or the ability to convert to cash quickly http://www.reliancemutual.com/mportal/VirtualPageView.jsp The ability to have ready access to invested money. Mutual funds are liquid because their shares can be redeemed for current value (which may be more or less than the original cost) on any business day. http://www.mtbfunds.com/education/glossary.php Refers to the ease with which an investment may be converted to cash at a reasonable price. http://www.globefund.com/centre/Glossary_IFIC.html The ability of a market to accept large transactions. http://www.fx-forex-trading.com/glossary.htm A measure of the ease with which an asset can be converted to cash without the loss of principal. http://www.reidepot.com/Glossary/l.html The ability of an asset to be converted into cash quickly and without discount. http://www.smartfunds.ie/glossary.html The ease with which something can be bought or sold (converted to cash) in the marketplace. A large number of buyers and sellers and a high volume of trading activity are important components of liquidity. Depth, or the ability of the market to absorb either a large buy or a large sell order without a significant price change in a security, is also crucial to the liquidity of the market. ... http://www.sia.com/capitol_hill/html/glossary.html How quickly and easily an asset can be converted into cash. http://print.smallbusiness.findlaw.com/starting-business/starting-business-moretopics/starting-business-buying-glossary.html
Selected results generated by „Google“ (http://www.google.com) on 20th January 2006 after entering the search term „define: liquidity”. Links may be subject to change.
12
Chapter 1: The Concept of Liquidity •
• •
•
•
•
•
• •
•
• •
• •
Refers to an investor's ability to sell an investment as a means of payment or easily convert it to cash without risk of loss of nominal value. http://www.fhwa.dot.gov/innovativefinance/appf_04.htm The ability of an individual or business to quickly convert assets into cash without incurring a considerable loss. http://www.partnersmortgage.com/glossL.htm Measure of a firm's ability to quickly convert assets into cash. A firm is said to be liquid if its liquid assets exceed short-term liabilities. http://www.mastercardbusiness.com/mcbizdocs/smallbiz/finguide/glossary.html This refers to how easily securities can be bought or sold in the market. A security is liquid when there are enough units outstanding to allow large transactions without a substantial change in price. Liquidity is one of the most important characteristics of a good market. Liquidity also refers to how easily investors can convert their securities into cash and refers to a corporation's cash position, i.e. how much the value of current assets exceeds current liabilities. http://www.geoshares.com/glossary.htm The ability of the market for a security to absorb a reasonable amount of buying or selling without major price changes. http://www.msc.gov.mb.ca/education/glossary.html The ability to convert assets into cash (or cash equivalent) without significant loss. If a business has good liquidity they will be able to meet their maturing obligations promptly, earn trade discounts, benefit from a good credit rating, etc. http://www.americancashflow.com/bcnucapital/Glossary.html A measure of the ability of an individual, business, or institution to convert assets to cash without significant loss at a particular point in time. http://www.ncbuy.com/credit/glossary.html The quality of being readily convertible to cash. http://www.amark.com/faq/glossary.asp The ability of a business to generate cash to meet its financial obligations as they become due. http://www.extension.iastate.edu/agdm/wholefarm/html/c3-05.html Depth of market to absorb buy and sell interest of even large orders at prices appropriate to supply and demand. The market must also adapt quickly to new information and incorporate that information into the stock's price. Liquidity is one of the most important characteristics of a good market. http://www.candlestrength.com/stock-trading-glossary.htm The ability of an insurer to convert its assets into cash to pay claims if necessary. http://www.insweb.com/learningcenter/glossary/general-l.htm Refers to the ability to buy and sell with little or no impact on price stability. The number of players in a market/security has a direct impact on this ability. The FX market is the most liquid market in the world. http://fxtrade.oanda.com/help/glossary/glossaryL_R.html being in cash or easily convertible to cash; debt paying ability http://wordnet.princeton.edu/perl/webwn Market liquidity is a business or economics term that refers to the ability to quickly buy or sell a particular item without causing a significant movement in the price. The term is usually shortened to liquidity. http://en.wikipedia.org/wiki/Liquidity
1.1 Definition of Liquidity
13
A similarly diverse picture arises from the review of liquidity definitions in textbooks on finance and financial management. E.g., Reilly/Brown (1997, p. 106-107) describe it as the “ability to buy and sell an asset quickly and at a known price”; according to Bodie/Merton (1998, p. 35), it is “the relative ease and speed with which an asset can be converted into cash”, and according to Arnold (2005. p. G18), liquidity is “the degree to which an asset can be sold quickly and easily without loss in value”; Schwartz/Francioni (2004, p. 60) call an asset liquid if it is “readily convertible into cash” and Brealey et al. (2006, p. 1000) if it is “easily and cheaply turned into cash”, and for Begg et al. (2003, p. 315) liquidity is the “cheapness, speed, and certainty, with which asset values can be converted back into money”. Slightly different formulation is used by Sharpe/Alexander (1990, p. 804), who see it as “the ability of investors to convert securities into cash at a price similar to the price of the previous trade in the security, assuming no significant new information has arrived since the previous trade”, or by Harris (2003, p. 394), who defines liquidity as “the ability to trade large sizes quickly, at low cost, when you want to trade”. On the other hand Brealey et al. (2006, p. 789) describe a company that can “easily lay its hands on cash” as liquid; and for Moyer et al. (1998, p. 609) it is “the ability of a firm to meet its cash obligations as they come”. This short review of various approaches reinforces the earlier formulated statement that there is no general agreement on the precise meaning of the term “liquidity” in the financial community. Nevertheless, there is some regularity in its understanding. It seems possible to classify the most frequent definitions depending on the object to which they refer: assets, markets, and companies.6 These three approaches are discussed more closely in the following subsections.
1.1.1.1. Asset Liquidity Liquidity understood as a characteristic of assets reflects probably the most popular approach to this subject dating back to Keynes.7 In this case, it is understood as the ease of converting an asset into cash. In other words, the easier an asset is sold the more liquid it is. Consequently, cash itself is assumed to be most liquid and other assets can be classified in reference to it as more or less “cash-like”. 6 7
See also http://www.riskglossary.com/articles/liquidity.htm (viewed on 15.01.2006). See FN 3.
14
Chapter 1: The Concept of Liquidity
Though intuitively simple, this definition contains several disputable elements. First, the definition of “cash” needs to be provided.8 The simplest one refers only to coins and banknotes issued by the central bank. However, it seems obvious that this approach is far too narrow – a number of alternative means of payment are in use in most countries. Electronic money on banking accounts, checks, or bills of exchange are practically just as liquid as cash, since they can either be converted into banknotes immediately at no cost, or are directly accepted as cash substitutes. Thus, it seems reasonable to extend the meaning of cash to other media of exchange circulating in the economy. Also further positions, like saving deposits or treasury bonds, can be liquidated very quickly at (almost) their face value, so that they are practical equivalents to cash in most cases.9 The broad category of all that can be considered as money is usually denoted as liquidity supply. For the purpose of its measurement, which is inevitable for practical monetary policy, different money aggregates are defined by central bankers.10 E.g., according to the European Central Bank (ECB), the narrowest M1aggregate of money supply contains apart from currency in circulation also overnight deposits; M2 encompasses additionally deposits with maturity of up to 2 years as well as those accessible at 3 months notice; finally, repurchase agreements, money market fund units, and debt securities with the maturity of up to 2 years are also included in M3 (see Table 1-2). Aggregates used in other regions are generally similar.11
8 9
10 11
See also the discussion of the functions of money in section 1.4.1. On various forms of money see Mishkin (2006), pp. 48 ff., Howells/Bain (2005), pp. 228 ff., or practically any text-book on banking and finance. On the measurement of money supply see, e.g., Mishkin (2006), pp. 51 ff. For an international overview of money aggregates’ definitions see Howells/Bain (2005), pp. 231 ff.
1.1 Definition of Liquidity
15
Table 1-2: Definitions of euro area monetary aggregates12 Liabilities
M1
M2
M3
Currency in circulation
X
X
X
Overnight deposits
X
X
X
Deposits with an agreed maturity of up to 2 years
X
X
Deposits redeemable at notice of up to 3 months
X
X
Repurchase agreements
X
Money market fund shares/units
X
Debt securities issued with a maturity of up to 2 years
X
Which of these categories is to be considered as cash in the sense of the liquidity definition is unclear. Currency issued by the central bank definitely does account as such. In fact, the M1 aggregate (or respective equivalents) should be appropriate in most cases. However, specific contexts need to be considered, as some types of money may lose their “cash-like” character under certain circumstances. E.g., deposits with commercial banks may prove highly illiquid in times of banking crises. Also the personal situation of the investor may play a role. While an overnight money market deposit might be as good as an on-sight deposit for some individuals, one day delay in the access to the funds is already a potential liquidity bottleneck for others. This subjectivity in the perception of cash leads to the respective subjectivity in the relative assessment of liquidity. Assets may be viewed as more or less liquid just because they can be converted into different sorts of money. E.g., selling an asset against a banknote and selling it against a cash check may make no difference to some investors but a substantial difference to others. Obviously, investors’ individual positions, especially the level of time pressure, play a great role at this point. The term “easily convertible” causes even more problems than the term “cash”. In the most intuitive meaning, the conversion should require no or only little intellectual and organizational effort. In this sense, treasury bonds are liquid because selling them only requires a telephone call to the stock broker, and a Picasso painting is illiquid because selling it requires the organization of an auction or a tedious search for an interested buyer. However, many of the definitions in the literature additionally remark that the conversion into cash should be quick and without loss. In fact, if the selling effort, independent of its actual nature, is understood as a time and money consuming process, 12
See ECB (2004), p. 37.
16
Chapter 1: The Concept of Liquidity
the ease of sale can be viewed in terms of a combination of the selling duration and the selling cost.
Price
The above considerations lead to the notion of liquidity as a function of time and the realized sale value. This duality has been recognized already by Hicks (1962) and Miller (1965) as an interpretation of Keynes’ definition cited earlier.13 According to it, an illiquid asset is one that can be sold promptly only at a discount; on the other hand, a longer liquidation period has to be accepted if the fair value is to be preserved. This approach can be presented graphically as a locus of time and liquidation value (see Figure 1-1). A perfectly liquid asset (X) can be sold at its current fair value immediately. Extending the duration of the liquidation process has no effect on the realized price; the price-time locus is therefore a straight line in this case. On the other hand, less liquid assets (Y and Z) can be sold immediately only at a discount – converging closer to the fair value is possible if longer liquidation periods are accepted. Thus, the liquidation price is an increasing function of the liquidation time.
Fair Value X Y Z
Time Figure 1-1: Liquidity as a price-time locus
The “price-time approach” reveals the multi-faceted character of liquidity. It cannot be described with reference to only one variable without a loss of certain crucial properties. Ignoring the price aspect would indicate the impossibility of a quick liquidation. This is, however, not true with respect to most assets. Almost everything can be sold quickly if the price is reduced far enough to convince an individual who was actually 13
See also Hasbrouck (1991a), p. 8, or Dubil (2003b), p. 2.
1.1 Definition of Liquidity
17
not intending to buy at the given time to change her mind. Note that in some extreme cases this may result in negative prices. They may occur, when the asset in question is very special and, in fact, worthless to most market participants; they would only be ready to own it if they are adequately compensated. One could paraphrase this aspect of liquidity with the words “there are no unsalable assets, there are only improper prices”. On the other hand, ignoring the time aspect would make liquidity undistinguishable from pure discounts on the nominal value. There would be no practical difference between liquidity and price reducing effects such as commissions or turnover taxes. Although the latter can affect liquidity (see section 1.2.2.1), they only constitute one aspect of the problem. In particular, they are definite and do not depend on the liquidation time, while the nature of liquidity lies in the possibility of avoiding discounts by prolonging the liquidation process. The price-time definition has also important consequences for the comparability of assets with respect to liquidity. In the sense of Figure 1-1, the asset Y can be unambiguously identified as more liquid than Z because the respective time-price locus of the former one lies completely above the latter one. In other words, for any liquidation time a (relatively) higher price can achieved for Y than for Z or the same price can always be achieved quicker for Y than for Z. However, it is theoretically possible that the loci of two assets intercross each other so that none lies always above the other – this case is depicted in Figure 1-2 for assets Y’ and Z’. While a quick increase in the price of the former one is possible up to a certain level, it takes very long to achieve further improvements above this level; on the other hand the asset Z’ allows a more steady improvement of the liquidation price with longer liquidation times. In effect, there is no possibility to state which of the two assets is objectively more liquid. The relative liquidity can only be assessed subjectively on the basis of investors’ preferences. Those attaching more importance to time would probably consider Y’ to be more liquid and those to whom the price is more important would judge Z’ as more liquid. By expressing investor’s time preference with a discount rate, it is possible to redefine the loci in terms of present rather than nominal values. It may then turn out that such modified liquidation present value-time locus is higher for Y’ or Z’ in its full length. But even then a group of investors may exist to whom the assets’ relative liquidity remains ambiguous. Thus, by assuming the price-time approach to asset liquidity, one has to accept the eventuality that a clear cut-off between more and less liquid investments is not possible.
Chapter 1: The Concept of Liquidity
Price
18
Y’
Z’
Time Figure 1-2: Ambiguousness of relative liquidity in terms of the price-time locus
Note that the definition of liquidity formulated in this section can and should be understood very broadly. The discount to the fair value is defined as encompassing all sorts of “efforts” including, e.g., the cost of employing professional brokers, appraisal cost etc. In the extreme case, even the (abstract) cost of learning how to operate an ATM can be interpreted as a liquidity discount associated with converting money on a banking account into cash. On the other hand, the time aspect of liquidity encompasses the entire duration of the selling process from the moment of the decision to sell to the moment of cash receipt. This includes the time necessary to find a trading partner, to deal with all formalities, and to realize the payment check; also the time required for walking to the nearest ATM falls under this category. Due to its generality and practical convenience, this definition of liquidity is the starting point for most of the following considerations.
1.1.1.2. Market Liquidity The second possible meaning of liquidity refers to markets. According to it, a liquid market is one on which trading is possible at any time, and, independent of the transaction size, it does not induce fundamentally unjustified price changes. In this sense, the market for U.S. Treasury Bonds is liquid since practically any amount of the bonds can be absorbed at the current price, and it is most unlikely that the market would react to such a large transaction. On the other hand, selling a large number of flats in one resi-
1.1 Definition of Liquidity
19
dential area at one time is usually extremely difficult, and it would certainly affect the level of house prices on the local market. Despite its brevity, the above definition encompasses a number of dimensions. The main of them are time (how long does it take to liquidate a position?), quantity (how large positions can be liquidated?), and price or liquidation cost (how high discount on the fair value has to be accepted at liquidation).14 Depicting liquidity as a trade-off between the execution time, order size, and execution cost yields a three-dimensional plane presented in Figure 1-3.
Figure 1-3: Liquidity as a trade-off between execution costs, execution time, and order size15
The time, size, and cost dimensions refer to single transactions. In order to speak of market liquidity, one needs to translate them into properties of markets. Literature on liquidity of organized financial markets usually identifies its three main components:
14 15
See Moorthy (2003), p. 29, Lo et al. (2003), p. 56, or Hodrick/Moulton (2007), p. 1. See Moorthy (2003), p. 30.
20
Chapter 1: The Concept of Liquidity
breadth (also referred to as width or tightness), depth, and resiliency.16 They correspond with the characteristics of markets which determine the possibilities of closing any transaction at any time without moving the prices, and also reflect three different possible sources of illiquidity. Although widely used, these terms are often understood in slightly different ways. Garbade (1982, p. 420) defines market breadth as the situation in which “there exist orders, either actual or easily uncovered, both above and below the price at which a security is actually trading”17; Kyle (1985, p. 1316) defines tightness as “the cost of turning around a position over a short period of time”; and to Fernandez (1999, p. 10) breadth is “the distance from the mid-market prices to the transactions that actually occur”. Mostly, however, breadth is associated with the bid-ask spread. A more general definition is assumed in this work; according to it, breadth is understood as the (abstract) distance between the sellers and the buyers with respect to the valuation of the asset traded on the market. Larger distances result in larger discounts to the (objective or subjective) fair value that sellers need to accept in order to sell the asset quickly or the in larger premiums that buyers need to pay in order to purchase the asset quickly. Thus, larger distances result in less liquid markers since it is more difficult for traders to find acceptable prices without deferring substantially from their own valuations. The form of manifestation of market breadth depends on the form of market organization.18 In dealer type markets, in which dealers or market makers are present, the bidask spread is the natural measure of breath.19 This is possible because market makers are obliged to trade any asset with any investor at any time provided that she accepts the prices quoted by them. The price at which the market maker is ready to buy (bid) and the price at which he is ready to sell (ask) determine the price levels for both sides of the market. In the absence of market makers or dealers, breadth is more difficult to observe. However, it is straightforward that finding a trading partner becomes more difficult when valuations of sellers and buyers go apart. Low breadth in auction mar-
16
17
18 19
The identification of the three liquidity components was probably introduced by Garbade (1982), p. 419-422, for the first time, but it is also used by a number of other authors including Kyle (1985), p. 1316, Bernstein (1987), p. 55-56, or Schwartz/Francioni (2004), p. 60-61. Note that the terms “breadth” and “depth” are switched in Garbade (1982); thus Garbade’s “depth” corresponds widely with what in this section is referred to as “breadth” and vice versa. See section 1.2.3.1 for an overview of the forms of public market organization. The bid-ask spread is discussed in more detail in section 3.1.1.1.
1.1 Definition of Liquidity
21
kets with market and limit orders would mean that buy orders are placed at low limits while sell orders are placed at high limits, leaving little room for price setting. In the extreme case, no single market price may exist at which any trade would be possible. Although the concept of market breadth is usually associated with public markets, the same principle should also apply to private markets without organized trade structures. Wherever the prices demanded by sellers are far above the prices offered by buyers the market can be described as “broad”. Under such circumstances only relatively few trades can be executed, and an investor determined to sell or to buy will be able to do so only if she accepts less favorable conditions. Hence, larger gaps between buyers and sellers (i.e., larger market breadth) indicate lower liquidity.20 The concept of market depth is based on the number of market participants ready to trade rather than on their readiness to pay. Yet, also here the definitions found in the literature go slightly apart. Most researchers understand depth either as the availability of counteroffers, or as the quantities quoted by market makers, or as the maximum trading volume not affecting the prices.21 In either case, the size of the market (however defined) plays the central role. The reference to market liquidity is relatively straightforward – finding a trading partner offering an acceptable price should be relatively easier in markets on which numerous traders are active simultaneously than on markets with only few active traders. E.g., it is definitely easier to sell shares on the NYSE with ca. 6 million transactions a day, than on the Warsaw Stock Exchange with only ca. 40 thousand transactions a day;22 but selling shares on either of these stock markets is still incomparably easier than selling apartments on a real estate market in a small town with only few transactions a year. Hence, large markets are considered to be deeper and as such more liquid than small ones. The size of the market, though undoubtedly highly important for liquidity, is, as noted by Persaud (2002), not per se sufficient for market liquidity. It has to be accompanied by sufficient diversity of traders to unfold its effect. Persaud illustrates it by comparing two idealized markets: one with only two and the other one with thousands of traders. 20
21 22
An analogical approach is followed by Garbade (1982, p. 420-422). He defines a broad market as one, in which traders differ with respect to their valuations and place (limit) orders at different values. However, Garbade’s conclusion is contrary to the one formulated here – he considers a broad market to be generally more liquid. See Kyle (1985), p. 1316, Fernandez (1999), p. 9, or Harris (2003), p. 398. See http://www.nyse.com and http://www.gpw.com.pl (estimated figures as of January 2006).
22
Chapter 1: The Concept of Liquidity
He assumes in his example that whenever one of the traders on the first market wants to sell, the other one wants to buy and vice versa. In contrast, whenever one of traders on the second market wants to sell (buy), all the other ones want to sell (buy) as well. In effect, despite the much smaller size, the first market is perfectly liquid and the second one, though much bigger, is perfectly illiquid.23 Hence, only if both sides of the market are balanced, more depth leads to higher liquidity. In normal times, this balance is ensured by the activity of noise traders. While insider or information traders act upon their superior knowledge about the true value of the asset, noise traders trade for other reasons, which arise from their personal situations or specific expectations about the future.24. While the former ones become active only if the current market price drifts away from the fair valuation of the asset, the latter ones buy and sell (to some extent) independently of the market price level. As noted by Black (1986), a market with full information and without noise, no matter how big, would be extremely illiquid. If prices fell below the fair value, everybody would like to buy; if prices rose above the fair value, everybody would like to sell; and with prices at the fair level, nobody would have an incentive to trade. The existence of investors who trade for exogenous reasons provides the necessary diversity and consequently the depth of the market. Thus, noise trading is essential for market liquidity. The practical relevance of market depth lies not only in the possibility of finding a buyer or a seller for a single unit of an asset but also in the ease of trading larger quantities of assets. An investor willing to accomplish an unusually large transaction, e.g., a sale of a large stock holding, is forced to fall back on buyers offering less favorable prices. On an auction market like a stock exchange it means that not only orders at the current market price but also orders below it need to be utilized. In consequence, the realized price per share deteriorates and the after-trade market price decreases. This effect is denoted as the market impact of trading.25 For any market a certain transac23
24
25
Note that the diversity in the sense of Persaud (2002) is similar but not identical with market breadth. The latter is about the divergence of valuations among investors while the former one is about their willingness to buy or to sell at these valuations. The differentiation of the information and noise traders was probably made by Black (1986) for the first time. It constitutes an important part of many theories of market microstructure and is present in numerous models; see Copeland/Galai (1983), Kyle (1985), or Glosten/Milgrom (1985). See also O’Hara (1997), Chapter 3, for a review. The impact of large transactions has been researched mainly in the context of security block transactions, i.e., transactions of larger blocks of shares. See Dann et al. (1977), Holthausen et al. (1987), or Keim/Madhavan (1996). See also O’Hara (1997), pp. 233 ff., for a review.
1.1 Definition of Liquidity
23
Price
tion size exists above which an inferior price has to be accepted and the impact of trading becomes relevant. However, it is reached quicker on smaller markets than on larger ones. With sufficient depth it should be possible to meet on enough demand (supply) to sell (buy) even a substantial amount of an asset at the usual market price; without it, the trade will cause a change in the market price level, which does not necessarily need to be associated with a change in the asset’s fundamentals. In this context, it is useful to differentiate between the temporary and the permanent impact (see Figure 1-4). The first one disappears after a certain period of time when new traders arrive on the market, while the second one does not – it is the result of a change in the general opinion of market participants about the value of an asset, which has been induced by the trade. The delimitation of both effects depends, of course, on the applied time horizon.
P0
Permanent effect
P2
Temporary effect
P1
t0
t1
t2
Time
Figure 1-4: Temporary and permanent price effects of a seller-initiated large sale transaction26
Note that the consequences of market breadth and market depth for an individual investor are similar. Both aspects of liquidity refer to the discount on the fair value that must be beard in order to accomplish a transaction within a reasonable period of time. However, while breadth stands for the cost of trading in typical situations, depth determines how these costs depend on the size of the deal (see Figure 1-5). In effect, the former one is relevant for investors trading frequently and the latter one for investors trading large quantities. Basing on this distinction, Bangia et al. (1999, pp. 3-5) intro26
See Holthausen et al. (1987), p. 241, or Dubil (2002), p. 68.
24
Chapter 1: The Concept of Liquidity
Volume
duced the terms of exogenous and endogenous liquidity. The former one refers to the market normal state and is determined by technical, political, or economic factors. The latter one refers to the effects of trading on the market price level. Thus, up to a certain order size only exogenous liquidity is experienced; the endogenous liquidity occurs above the volume at which the spread between the selling and the buying price starts to widen – it corresponds roughly with the crook of the price-volume curves depicted in Figure 1-5.
Breadth
Depth
Selling price
Purchasing price
Figure 1-5: Market breadth and market depth27
With the market impact of trading a dynamic dimension of liquidity has been introduced. However, from the market perspective even more important than the sole price change is the speed at which the price level recovers after a large transaction or some other event causing a temporary order imbalance. It is denoted as market resiliency.28 Since fundamentally unjustified price changes are not accompanied by any changes of factors which determine assets’ values, they should be outbalanced by additional orders from traders willing to take advantage of the temporary over- or underpricing. How quickly such counter reactions occur is central for the resiliency of markets. One would suspect that the flow of new orders should be sufficient to restore the market equilibrium within a short period of time in liquid markets, but it may take much longer in small and illiquid markets. In Figure 1-4, resiliency can be associated with the length of the time gap between t1 and t2, i.e., with the duration of the temporary price 27 28
See Buhl (2004), p.12, or Schmidt-von Rhein (1996), p. 148. See Garbade (1982), p. 422, Kyle (1985), p. 1316, or Schwartz (2004), p. 61.
1.1 Definition of Liquidity
25
impact of trading. Note, however, that resiliency refers to the speed of reactions to any unjustified price changes and not only to those induces by large transactions. At first sight, resiliency seems to be closely related to market depth. Deeper markets should not only be less prone to market impacts but also react more quickly to trading activity. However, this mechanism works only if investors can learn about the unjustified price change and its backgrounds (e.g., a large transaction) quickly enough.29 It requires that channels exist through which such information can be forwarded to interested market participants. While it is usually unproblematic in organized markets in which prices and trading volumes are quoted publicly, it might be a serious problem in non-organized markets. Without a central source of information some investors who would be interested in trading at the (temporarily) favorable prices will not be able to do so simply because they will not learn about the order imbalance on time. In effect, the traders on the “underrepresented” side of the market are forced to either postpone their transactions or to accept unfair prices. Thus, what distinguishes resiliency from other aspects of liquidity is the role of the flow of information among market participants.
1.1.1.3. Corporate Liquidity The third group of liquidity definitions refers to the ability of companies (or other institutions) to meet their financial obligations. A liquid company has no difficulties with settling their accounts on time, while an illiquid one may have problems with timely payment running the risk of insolvency. Thus, the lack of liquidity is a possible bankruptcy reason and is therefore one of the central issues in the corporate financial management. Liquidity of financial institutions is of particular importance in this context – insolvency in the banking system could have serious economic consequences. Liquidity of companies depends on two aspects: the availability of means of payment and the term structure of liabilities. The first one can be understood twofold: as “cash on hand”, i.e., the balance on the corporate bank account available for making payments, or as the ability to provide sufficient funds within a short period of time. While the ability to meet obligations is given immediately in the first case, it may require some time consuming and possibly costly actions in the second case. These may encompass the sale of some of the company’s assets, obtaining quick financing through 29
See Garbade (1982), p. 422.
26
Chapter 1: The Concept of Liquidity
standing lines of credit or other financing facilities, or obtaining repayments from outstanding accounts with customers. On the other hand, longer terms of corporate liabilities leave more time for providing sufficient funding. The complex character of corporate liquidity is also reflected in various liquidity ratios used in financial analysis; among the most popular are:30
Cash Ratio =
Cash Current Liabilities
Quick Ratio =
Current Assets - Inventories Current Liabilities
Current Ratio =
Current Assets Current Liabilities
(1.1) (1.2) (1.3)
Ensuring sufficient liquidity is the key task of short-term financial planning and cash management.31 They encompass a wide field of problems ranging from the formulation of exact financial plans on the basis of operating cycles to cash budgeting. One of the central issues is the optimal size of cash holdings, or more generally, the optimal financial slack. With an increased holding of cash or highly liquid assets the risk of becoming illiquid and the “shortage costs” associated with it – the costs of emergency financing, loss of reputation, production interruptions etc. – are reduced. On the other hand, however, the opportunity costs increase as no or only small interest is earned on these assets. The trade-off between these types of costs is considered in numerous models to determine the optimal balance of liquid assets (see Figure 1-6).32 However, considerations about the optimal level of company’s liquidity should also include such issues as the availability of credit lines33 or the payment behavior of customers.
30 31
32
33
See Samuels et al. (1995), pp- 54-56, or Brealey et al. (2006), p. 792. See Brealey et al. (2006), Chapter 31, Ross et al. (2005), Chapters 26 and 27, Moyer et al. (1998), Chapter 16, Drukarczyk (2003), pp. 91-110, Rehkugler/Schindel (1994), pp. 210-225, as well as virtually any textbook on corporate finance. The inventory model developed by Baumol (1952) is probably the most distinctive of the optimal cash balance models. Better access to short-term financing facilities is one of the reasons why larger firms with sound financial standing tend to hold lower liquid asset balances while retaining high levels of solvency. See Moyer et al. (1998), p. 617.
27
Cost
1.1 Definition of Liquidity
Total costs
Holding costs
Shortage costs Optimal cash balance
Cash
Figure 1-6: Optimal cash balance34
Only selected aspects of the wide subject of corporate liquidity could be discussed in this section – a full presentation is far beyond the scope of this work. Nevertheless, it should be sufficient to give the impression of the complexity of this issue. It depends on many factors. Some of them are exogenous, like the payment behavior of customers or the availability of credit lines, but there are also factors that can be influenced by the management, among them mainly the holdings of cash and liquid assets. Thus, corporate liquidity is subject to corporate policy.
1.1.1.4. Relations between the otions of Liquidity Although the three different definitions of liquidity concern different areas of economics, they are not unrelated. A link between them can be recognized even intuitively. A closer look confirms that, in fact, they refer to the same basic phenomenon. However, in each approach liquidity is viewed from a slightly different perspective, and different aspects are put in the foreground, so that no full identity is given. Consider the asset and market liquidity first. Obviously, liquid assets are usually traded on liquid markets. For example, it is easier to sell an asset quickly at a fair price on a market on which a large number of traders are active. On the other hand, high liquidity of an asset induces more investors to trade more frequently establishing a liquid market. Yet, despite this similarity, the two approaches are not synonymous. In the 34
See Moyer (1998), p. 617, or Ross et al. (2005), p. 739.
28
Chapter 1: The Concept of Liquidity
first place, the focus of asset liquidity is on the situation of an individual investor while market liquidity refers to the average situation of all investors. This means that investors with different access to a certain market will also perceive liquidity of assets traded on this market differently. Different time preferences will also lead to different assessments of asset liquidity. In contrast, market liquidity is independent of any individual investor characteristics. Furthermore, while only the conversion into cash (i.e., a sale), is important for asset liquidity, both sides of the market, the sellers’ and the buyers’ side, are relevant for market liquidity. In consequence, the existence of a liquid market should guarantee high liquidity of assets traded on this market, but the opposite does not necessarily need to hold. Under certain circumstances an asset can be liquid even if no liquid market for it exists.35 Moreover, in the extreme case, it is even conceivable that asset and market liquidity contradict each other: market A may be (objectively) more liquid than market B, but some investors may consider the asset traded on the market A as (subjectively) less liquid that the asset traded on the market B. Summing up, liquidity of assets seems to be a more general concept than liquidity of markets. A link can be also established between asset or market liquidity and corporate liquidity despite the different focus of these approaches. Since it is only possible to meet financial obligations payable in cash if sufficient cash is available, company’s ability to sell its assets quickly is crucial for maintaining liquidity. Hence, “one firm would be considered more liquid than another firm if it has a greater portion of its total assets in the form of current assets”.36 In other words, a company that holds a sufficient stock of liquid assets can be as liquid. Also in this case, however, there is no perfect identity between the liquidity of the company’s assets and the liquidity of the company itself. As noted in the former section, the availability of liquid assets is an important but not the only factor determining the ability to meet financial obligations. The term structure of liabilities and the access to credit lines are largely independent from the marketability of assets, but they do influence corporate liquidity. Nevertheless, the availability of cash is central to both approaches.
35
36
Units in open end funds can be named as examples for liquid assets without liquid markets. Even if they are not publicly traded, they can be easily converted into cash by reselling them to the fund. Besley/Brigham (1999), p. 539. An illustrative example of the link between the grade of liquidity of the assets held by a company and the probability of its insolvency is provided by Duffie/Ziegler (2003).
1.1 Definition of Liquidity
29
To sum up, although various definitions of liquidity are not quite equivalent, they are closely related. In the core of the problem is the need for flexibility, i.e., the ability to decide about investments and disinvestments freely at any time without bearing additional cost. Since the main mean of financial flexibility is cash, all liquidity definitions are related to transactions against cash. Thus, investors wish to be able to buy and sell according to their individual needs independent of the circumstances. On the one hand, assets or markets that allow them to achieve this goal have the quality of being liquid. On the other hand, a company that is able to meet its obligations at any time at a low cost, i.e., is able to act flexibly, is also considered liquid. 1.1.2.
Liquidity Risk
In the former section, liquidity was treated as a quality of assets or markets, which is more or less stable over time. However, a number of researchers noted that it is not an unchanging constant. Most assets (apart from cash) and markets go through phases of higher or lower ease of trading. Thus, there is definitely some grade of uncertainty about liquidity, which should not be neglected by investors. In this section, the definition of liquidity is extended to include this aspect, which is referred to as liquidity risk. It can be identified with respect to both asset and market liquidity. Although these two views generally refer to the same issue, their character is slightly different. This difference is especially distinct when considering privately traded assets on the one hand and public markets with organized and centralized trading mechanisms on the other hand. The discussion in this section is therefore structured according to this scheme.
1.1.2.1. Liquidity Risk of Privately Traded Assets An indication of liquidity risk is already present in the Keynes’ liquidity definition cited in section 1.1.1.1, but it was highlighted by Hicks (1962) for the first time. Hicks argues (p. 788) that the mere ability to sell an asset quickly without discount, which he denotes as “marketability”, is still not entirely identical with liquidity itself.37 While marketability refers only to the expected outcome of the sale, liquidity encompasses also the level of certainty about the outcome. Thus, referring to the Keynes’ original definition saying that one asset is more liquid than another asset when it is “more certainly realizable at short notice without loss”38, Hicks stresses the term “more certain37 38
See also Hicks (1974), Chapter 2, for an in-deep discussion of liquidity risk. See FN 3.
30
Chapter 1: The Concept of Liquidity
ly”. This certainty, or rather its lack, is here referred to as “liquidity risk” in contrast to “marketability”, which refers only to the expectation about the result of the liquidation. In order to work out the precise meaning of liquidity risk, consider first a hypothetical asset, which liquidity is known and constant over time. As discussed in section 1.1.1.1 its grade of liquidity can be described in terms of a price-time locus, which is depicted as an increasing curve in Figure 1-1. According to it, a quick sale of an illiquid asset is only possible at a discount to the fair value, but a higher price can be achieved if a time consuming liquidation process is conduced. The price, which is referred to in this context, is nominal. However, investors make decisions concerning future payments on the basis of discounted present values. Given time preference expressed as a discount rate, nominal prices can be discounted yielding a present value-time locus specific for each seller. Since the nominal selling price is asymptotically approaching some upper bound for long liquidation periods, its present value has to possess a maximum for a finite duration of the selling process. Thus, there is some optimal time topt that the investor should take to sell the asset in order to maximize its present value (see Figure 1-7). The time topt is not only asset-specific but also dependent on the investor’s current situation – in particular, it is shorter in periods of higher time pressure when a higher discounting rate is applied. Nevertheless, knowing the asset’s typical nominal price-time locus, the optimal liquidation time and the corresponding present value can be easily determined in each situation. The liquidity discount corresponds then with the difference between the maximal present value and the fair value. It depends only on the investor’s time preference.
1.1 Definition of Liquidity
31
Price
ominal value
Present value
topt
Time
Figure 1-7: Optimal liquidation time without liquidity risk
Note that liquidity is deterministic in this approach. Since there is no uncertainty about the shape of the locus, the liquidation problem is trivial. The time horizon should be set to topt in order to maximize the value of the asset and minimize the discount. However, this case is obviously not very realistic. The dependence between the realized selling price and the time necessary to achieve it is by no means functional or deterministic. More likely, it is of stochastic nature with the price-time locus being (at best) the set of expected liquidation values. This means that longer marketing periods tend to lead to higher liquidation prices but do not guarantee them. Suppose, for example, that a house is to be sold and assume that the optimal liquidation time (e.g., 1 month) and the corresponding price (e.g., $ 100,000) could be computed. Obviously, it is highly improbable that the house will sell exactly at the targeted price after exactly one month of selling efforts. Although it might be typical for such properties to transact at about $ 100,000 within approximately one month, the seller can also be lucky to find a buyer willing to pay this (or even a higher) price already during the first week or be unlucky not to find one within a year. Thus, although some dependence between the price and the liquidation time does exist, it is uncertain. The grade of this uncertainty might be different for different assets, but it never disappears entirely. The consequence of the stochastic character of the price-time locus, illustrated in Figure 1-8 with the zigzag line, is the non-existence of an unambiguous optimal liquidation time. A
32
Chapter 1: The Concept of Liquidity
typical or expected duration of the selling process might exist, but it will always be burdened with uncertainty.39
Price
ominal value
Present value
Time Figure 1-8: Liquidity risk as the uncertainty of the liquidation value
Above considerations make clear that it is essential to allow for uncertainty in order to capture the full consequences of assets’ illiquidity. Note, however, that liquidity risk is not necessarily related to Hick’s marketability represented by the deterministic pricetime locus. While the latter one corresponds with the (expected) dependence between the liquidation value and time, liquidity risk is about the possible deviation from this expectation. In fact, it is the two-dimensional nature what makes liquidity a practical problem. Ignoring the risk, as it was done in the example at the beginning of this section, would result in a deterministic optimal present value, which could be treated as equivalent to the immediately receivable price of a perfectly liquid asset. There would be only little economic difference between liquid and illiquid assets in such case. It is the uncertainty about the liquidation outcome that makes liquidity so difficult to include in investment decisions.
1.1.2.2. Liquidity Risk in Public Markets Since asset liquidity and market liquidity are closely related, it can be expected that liquidity risk also occurs on the market level. A closer look reveals, however, that it 39
Similar considerations concerning the uncertainty about the liquidation outcome can be found in several models of optimal liquidation; see, e.g., Almgren/Chriss (1998, 1999, 2000/2001) or Dubil (2002). See also section 4.2.1.
1.1 Definition of Liquidity
33
has a slightly different character from this perspective. Not the individual uncertainty about a single transaction but the average uncertainty about all transactions is important in this case. Moreover, liquid risk on the market level can be unstable over time; thus, uncertainty itself is uncertain. It is the latter aspect that is given special attencion in this section. The fact that liquidity is not constant in public markets is obvious to most market practitioners. Even most liquid assets, like popular securities or currencies, experience times when they are more or less difficult to trade or are tradable at a higher cost than usual.40 This is illustrated on the basis of different liquidity measures in Figure 1-9 with respect to large U.S. stock markets, which are considered to be among the most liquid worldwide. Even without going into the precise meaning of these measures – this is done in Chapter 3 – one can easily observe that the level of market liquidity indicated by them fluctuates strongly both in the long and in the short term. Furthermore, also the intensity of these fluctuations is not constant and extreme outliers occur from time to time. One can expect the instability of the liquidity level to be even stronger in less liquid markets. It may result from a number of different sources. For one, technical matters such as opening hours of major stock exchanges or fluctuations of prices around popular stop-loss levels often cause short-term (intra-day) variations in liquidity. E.g., it is widely known that trading becomes significantly more difficult (and costly) outside the official trading times.41 Also changes in the number of liquidity providers may affect market liquidity – e.g., mergers between banks, who act as market makers in many important markets, reduce the number of potential trading partners.42 Another group of factors leading to varying liquidity refers to the general state of economy. Chordia et al. (2001a) find that liquidity of US equity markets (measured as trading activity) is influenced by variables such as interest rates or macroeconomic announcements. Interestingly, they also report significant decreases of liquidity in down markets but only slight increases in up markets. Furthermore, several researchers state high levels of commonality in changes of liquidity attributes, which may indicate common underlying (economic) determinants.43.
40 41 42 43
See Fernandez (1999) for a review of liquidity changes in various markets. See Barclay/Hendershott (2004). See Richmond/Crawford (2003), p. 7 ff. See Chordia et al. (2000 and 2005), Huberman/Halka (2001), or Hasbrouck/Seppi (2001).
34
Chapter 1: The Concept of Liquidity a)
b)
c)
Figure 1-9: Variations of market liquidity on YSE and AMEX44
Summing up, varying liquidity is not unusual and has to be treated as a natural state of economy. However, in some cases the variations may assume levels at which, despite 44
Following measures have been applied: (a) quoted and effective bid-ask spread [source: Chordia et al. (2001), p. 508], (b) illiquidity innovations (unexpected changes) according to Acharya/Pedersen (2005) and (c) aggregate liquidity according to Pastor/Stambaugh (2003).
1.1 Definition of Liquidity
35
the seemingly sufficient number of active traders, the ability of trading disappears. Such situations are denoted by Persaud (2002) as “Liquidity Black Holes” and may occur even in big markets due to the herd behavior or traders attempting to sell or buy all at the same time.45 They result mainly from the lack of diversity among traders, who similarly anticipate future price changes. Such “Black Holes” could be observed, e.g., during the Asian financial crisis of 1997-98 or in a number of smaller turmoils after the Russian default in 1998. Such events illustrate that uncertainty about the level of liquidity affects all public markets. Moreover, the deviations from the normal state can assume levels at which trading is impossible over longer periods of time. Thus, the consequences of sudden, unexpected changes in liquidity are by no means marginal. Obviously, uncertainty about market liquidity refers to all aspects of this feature. In particular, it means that market breadth and market depth are also uncertain. Thus, on the one hand, an investor willing to trade cannot be sure about the exact selling price (bid-ask spread) at the moment of the trade; on the other hand, she cannot be sure, whether she will be able to sell the whole amount at the quoted price. In consequence, there is uncertainty about the impact of trading on the price level and, accordingly, on the shape of the price-volume curve in the moment of the transaction. These points are illustrated in Figure 1-10 with a zigzag instead of a smooth line symbolizing the stochastic nature of the dependence between the transaction volume and the realized price. Furthermore, the decomposition of market liquidity into the exogenous and the endogenous component implies that also liquidity risk can be split into exogenous and endogenous one.46 The former arises from the uncertainty about the normal level of liquidity (normal market breadth), and the latter results from the uncertainty about the reaction of the market price to the placement of a large order.
45
46
For an extensive discussion of Liquidity Black Holes see Persaud (2003) and a number of papers therein. See, e.g., Jarrow/Subramanian (1997), Almgren/Chriss (1998, 1999, and 2000/2001), and Bertsimas/Lo (1998), who focus on endogenous liquidity risk, or Bangia et al. (1999), who address the exogenous liquidity risk.
Chapter 1: The Concept of Liquidity
Volume
36
Endogenous liquidity risk
Exogenous liquidity risk
Selling price
Endogenous liquidity risk
Purchasing price
Figure 1-10: Exogenous and endogenous liquidity risk47
Note that there is not necessarily a direct relation between the level of market liquidity and the uncertainty about it. On the one hand, the fact that markets are liquid does not mean that the high level of liquidity cannot fluctuate substantially. On the other hand, highly illiquid markets may retain the same (low) level of liquidity unchanged over longer periods of time. In effect, two largely independent dimensions of market liquidity can be identified – one referring to the expected level of liquidity, and the other referring to the uncertainty about it. They are analogical to marketability and liquidity risk identified for privately traded assets in the former section. 1.1.3.
A Two-Dimensional Definition of Liquidity
The considerations in the former section introduced a new dimension to the liquidity concept, which has only rarely been given attention in the literature on this subject. Liquidity risk, viewed either on the level of assets or on the level of markets, constitutes its integral part. Ignoring it, could lead to situations in which an asset or a market would be falsely qualified as liquid although the level of its liquidity would be highly uncertain. Incorporating this aspect in the liquidity definition formulated in section 1.1.1 leads to an extended, two-dimensional notion of liquidity, which is used throughout the analysis. Its main features are summarized in this section. A scheme of the relations between the respective terms is offered in Figure 1-11.
47
Based on Bangia et al. (1999), p. 5.
1.1 Definition of Liquidity
37
Asset Liquidity
Time
Marketability (expectation about the liquidation outcome)
Price
Market-wide factors affecting the outcome of liquidation
Expected Liquidity (expectation about the ease of trading an arbitrary amount of an assets)
Characteristics of the investor
Breadth Depth Resiliency
Liquidity Risk (uncertainty about the liquidation outcome)
Randomness of the search process
Liquidity Risk (uncertainty about the ease of trading an arbitrary amount of an assets)
Market Liquidity
Figure 1-11: Structure of the two-dimensional liquidity definition
Since the main focus of this work is on illiquid privately traded assets, the approach focusing on asset liquidity has been chosen as the starting point for the further analysis. As already noted, its main feature is the explicit distinction of marketability, defined as the expected outcome of a specific transaction, and liquidity risk, defined as the uncertainty about this outcome. Both can be applied to sales as well as purchases. Thus, if only a sale (liquidation) is addressed in certain contexts, it is done only for convenience. In most cases, analogical considerations can be applied to problems arising when buying an asset. The two-dimensional liquidity definition can be further refined by analyzing the components of the dimensions. Since the liquidation outcome refers to both the liquidation price and the liquidation time, also marketability and liquidity risk can be addressed this way. Hence, marketability can be understood as the combination of the expected
38
Chapter 1: The Concept of Liquidity
effective liquidation value and the expected time required to achieve this value.48 Analogically, liquidity risk is the uncertainty about these two aspects. Furthermore, the dimensions of asset liquidity can be traced to their sources. Marketwide factors affecting the ease of trading by all investors in a similar manner can be pooled into one category. On the one hand, the expectation about the state of the market at the moment of the decision to sell constitutes the marketability component. On the other hand, the uncertainty about the market state is a source of liquidity risk. However, when liquidity is considered on the asset level and viewed from the perspective of an individual investor, also her individual characteristics influence the level of marketability and liquidity risk. In particular, the ease (and the cost) of accessing the market or contacting other market participants as well as the attitude towards time (time preference) and risk (risk aversion) are of relevance in this context. Finaly, in the case of privately traded assets, liquidity risk arises also from the fact that the search for a trading partner is a random process. The structure of the definition of market liquidity is roughly similar. Two dimensions can be identified here as well – the “expected liquidity” referrs to the expectation about the possibilities of a quick liquidation of an arbitrary amount of an asset, and the “liquidity risk” referrs to the uncertainty about these possibilities. Market liquidity can be further split into three relevant aspects: breadth, depth, and resiliency of the considered market. These are the characteristics of markets which determine the ease of trading and, in particular, the relation between the prices that can be achieved in transactions involving certain quantities of assets and the duration of the trading process. Breadth, depth, and resiliency usually have some typical levels on specific markets, but they are also subject to fluctuations. Considering the sources of market liquidity leads to the conclusion that it is determined only by general, market-wide factors. Neither individual characteristics of investors nor the randomness of the search process affect is relevant in this case. This is intuitively conceivable. Firstly, since market liquidity refers to the average situation of all traders, it describes an objective feature of the considered market – any kind of individual information must therefore remain irre48
Note, however, that the distinction of the two components (time and price) is not necessary as soon as a concrete investor with a well defined time preference is considered. It is than possible to “translate” the time aspect of the transaction into value by applying discounted prices. Nevertheless, for theoretical considerations conducted on the abstract level, it may still be convenient to view these elements separately.
1.1 Definition of Liquidity
39
levant. Secondly, the notion of market liquidity is usually applied to public, centralized markets on which no individual search takes place. But even if the search for a trading partner was possible and purposeful, its effects would be cancelled out by the aggregation of transactions. Hence, while there are many analogies between the dimensions of market and asset liquidity, one should not consider them as fully interchangeable terms. Going deeper into the comparison of the two approaches, reveals even more fundamental differences.49 Consider marketability and expected market liquidity first. As discussed above, the latter refers to the average “ease of trading” on the market. However, this does not mean that expected market liquidity is equivalent to the average marketability of the asset traded on the considered market. In fact, it also comprises the “expected uncertainty” about the outcomes of transactions on this market. This is best visible when analyzing the components of the bid-ask spread, the most widespread measure of market liquidity – uncertainty about the possibility of closing an open position is priced by the dealer when setting the spread.50 Thus, expected market liquidity contains not only the (average) marketability but, to some extent, also the (average) level of asset liquidity risk. In consequence, market liquidity risk, which is defined as the uncertainty about the level of market liquidity, contains the uncertainty about the average marketability but also the uncertainty about the average level of asset liquidity risk. On the other hand, as noted earlier, due to the aggregation of individual transactions in the notion of market liquidity, all kinds of individual investor characteristics or search effects, which are present in the asset approach to liquidity, remain disregarded. The above discussion shows that although the core of the liquidity problem remains the same, the precise meaning of this term differs within different approaches. Thus, the choice of an unambiguous definition is of crucial importance for the consistency of the following analysis. The notion of liquidity assumed in this work builds mainly on asset liquidity presented in section 1.1.1.1, which corresponds to a large extent with the liquidity concept of Keynes. The definition is, however, extended to include the purchase case as well. Thus, a liquid asset is one that can be bought and sold quickly, at a good price, and with little uncertainty. References to market liquidity, which appear in several contexts throughout the book, are based on the definition provided 49 50
See also the discussion in section 1.1.1.4. See also section 3.5. See section 3.1.1.1 for the discussion of the components of the bid-ask spread.
40
Chapter 1: The Concept of Liquidity
above. Although it is not always possible to discuss the differences resulting from following this approach, the reader should bear in mind that it is not fully identical with asset liquidity. In contrast, the issue of corporate liquidity is not further followed explicitly. However, since it is a function of the liquidity of company’s assets, it does not remain entirely disregarded. The complex concept of liquidity presented in this section has important consequences for its measurement as well as for including it as a decision criterion in asset management models. In particular, it implies that the two dimensions, marketability and liquidity risk, need to be considered separately. In consequence, the application of multidimensional methods is necessary, what substantially increases the complexity of the analysis. This issue is addressed in the following chapters, especially in Chapter 3 and Chapter 4. Yet, before the development of concrete methodical approaches can be undertaken, a closer look at the notion of liquidity presented here is necessary. In particular, its determinants, which have been mentioned only briefly in this subsection, require a more thorough examination
1.2.
Sources of Liquidity
The general definition of liquidity has been formulated and discussed in the preceding section. According to it, liquid assets can be expected to sell more quickly without substantial discounts opposed to illiquid ones, which liquidation requires either time or the acceptance of less favorable prices. Moreover, not only the expected liquidation outcome but also the uncertainty about it is relevant for investment decisions. Hitherto, these issues have been considered only on the abstract level, and assets have been assumed to have certain degrees of liquidity by their nature. However, for an in-depth analysis, it is important to identify factors which determine liquidity. Thus, the question “what makes an asset illiquid?” is addressed in this section. Sources of liquidity are classified into three groups, which should cover most of the important factors. These are: transaction costs, organization of trading, and diversity of valuations. However, before they are discussed in more detail, some general issues need to be clarified in order to ensure a structured analysis.
1.2 Sources of Liquidity 1.2.1.
41
Preliminary Considerations
Before the sources of liquidity are discussed in detail, it is necessary to identify the channels through which the characteristics of assets, markets, and market participants affect the ease of trading. The starting point is the definition of asset liquidity provided earlier in this Chapter in section 1.1.1.1. In order to be able to sell quickly and without discount, two conditions need to be fulfilled: 1. The investor has to be able to meet a buyer willing to purchase the asset; quick liquidation is technically impossible otherwise. 2. The duration of the liquidation process should have no or only little impact on the (nominal) transaction price. The above conditions refer mainly to marketability. With respect to liquidity risk, the third condition can be formulated: 3. The outcome of the liquidation process should be predictable to some extent. When selling or purchasing an asset is impossible, its liquidity is per definition zero. Such situations are often feared by investors; however, true “unsalability” is relatively seldom. By saying that a mansion, a painting, or a luxurious car is impossible to sell, the owner usually means that it is impossible to sell at the desired price. In most cases, such assets could be liquidated if the price requirements were (substantially) lowered. According to this logic, everything can be sold; the question is only at what price. Same applies to the purchase problem – nearly everything can be bought if a sufficiently high price is offered. The rare cases of true “unsalability” (or “unpurchasability”) encompass mainly situations in which trading is legally restricted. This refers, on the one hand, to “assets” which are definitely excluded from being sold or bought or even owned – human beings belong to this category. On the other hand, trading restrictions may be laid upon some otherwise tradable goods – national currencies are sometimes subject to such regulations. However, since these cases are rather exceptions in the day-to-day investment activity, they are disregarded in further considerations; the first condition can therefore be considered as fulfilled. The sources of asset liquidity are, thus, to be seeked among factors affecting the duration of the marketing process and its outcome. The main one is the way buyers and
42
Chapter 1: The Concept of Liquidity
sellers search, find each other, and agree on prices.51 When the search can be accomplished quickly, i.e., it is possible to identify a potential trading partner with an acceptable price limit within a short period of time, liquidity is (ceteris paribus) higher than when longer search is required. Also the impact of the search process on the final transaction value is relevant. In this context, two main factors affecting the character of search can be identified: the number of potential trading partners on the market (market depth) and the flow of information about trading possibilities between market participants. They should be given a closer look. The more investors are interested in buying or selling, the better are the chances of finding an acceptable offer quickly. The number of potential buyers or sellers is, however, not exogenous and depends on a number of factors. To some extent, it depends on the characteristics of the asset itself. Some goods, like rare collectibles, have a positive value only to a limited number of individuals; other goods, like cars, are widely used and demanded by larger groups. However, this criterion is of little relevance with respect to financial assets, i.e., assets held only for the purpose of generating returns. Since only this kind of assets is regarded in this work, their intrinsic utility is irrelevant for further considerations. It can be realistically assumed that every investor would be ready to buy (or to sell) any asset if it offered her sufficient profitability. Hence, in the ideal case, the number of potential market participants should be a function of expected (risk adjusted) returns. Among the numerous factors which affect assets’ return characteristics, transaction costs are especially relevant for liquidity considerations.52 Due to their one-time character, they affect short term investments stronger than long term investments and discriminate frequent trading. In consequence, trading activity should be weaker in markets with higher transaction costs. Similar effects can be caused by organizational obstacles, especially by formal requirements affecting the speed of executing transactions, and by imperfect divisibility of assets, which prevents optimal allocation of capital.
51
52
The role of search in the organization of trade and liquidity of markets is discusses in Harris (2003), pp. 394 ff. To be more precise, market imperfections, of which transaction costs are an important part, are the main source of differences in risk adjusted returns. Theoretically, if there were no obstacles in the access to markets or information, arbitrage should lead to identical returns on all investments. Frictions preventing such “perfect markets” result in differences in returns. See e.g. Fama/Miller (1972), p. 21, or Fabozzi/Modigliani (2003), p. 109.
1.2 Sources of Liquidity
43
A different reason for difficulties in finding a trading partner is the lack of knowledge about his existence, or more precisely, the high cost of gaining such knowledge. This problem is best illustrated by the comparison of financial markets hundred years ago and today. The informational revolution enabled investors to conduct transactions with individuals on the other side of the globe of whose existence they would otherwise never learn; this led to the globalization of markets. Aside from the technical issues, also the form of market organization plays a big role in providing means of contact between buyers and sellers. It includes, in particular, the existence of institutions assisting market participants in their search efforts. Due to professional organization, economies of scale, and similar factors such institutions may be able to substantially reduce the cost of searching for trading partners. Finally, differences in opinions about the fair value of an asset will also influence the ease of finding a trading partner. Suppose that an investor wishes to sell a share of a company and is certain that its value (arising from expected future earnings) is no less than $ 120, while other investors on the market value it between $ 90 and $ 110. Since the investor would rather not sell at all than sell below the believed true value, she will not be able to find an individual with whom she could agree on the sale price. On the other hand, if the range of valuations by market participants was between $ 75 and $ 125, a chance of finding a suitable buyer would exist. To sum up, it seems that it should be easier for an average investor to find a trading partner offering an acceptable price when beliefs about the true value vary stronger. However, with a larger spread of valuations also the uncertainty about the outcome of the liquidation is higher, i.e., liquidity risk increases. The above discussed factors affect the average (expected) marketing time and the realized transaction price, i.e., the liquidity price-time locus of an asset as well as the uncertainty about its shape. Transaction costs lower the liquidation value and lead to less frequent trading; trading mechanisms determine the duration of the transaction process and are also crucial for the evolution of prices; finally, the divergence of opinions about the value of an asset is important both for the chances of finding a trading partner and for the predictability of realized prices. In view of the importance of these three groups of factors, they are discussed more thoroughly in the following subsections.
44
Chapter 1: The Concept of Liquidity
1.2.2.
Transaction and Opportunity Costs
In most cases, investors are able to trade assets only at a certain cost. The existence of such costs is often associated with liquidity. In fact, many researchers go as far as to set an equality sign between these terms.53 Schwartz/Francioni (2004, p. 63) state even that “illiquidity and trading costs are two sides of the same coin”. As discussed further in this section, this opinion seems to be justified only with respect to certain types of transaction costs and even then not unconditionally. Furthermore, it seems that the literature on the postulated link between liquidity and transaction costs often fails to distinguish the costs that affect liquidity from the costs that are the effect of lacking liquidity. This section attempts to clarify some of these issues. Transaction costs may have a number of different forms. They are usually classified into direct commissions and a broad group of indirect costs.54 The latter ones encompass, in particular, the bid-ask spread and the market impact. Also opportunity costs of trading are often considered to be a form of transaction costs.55 From the trader’s point of view, all these costs have a similar effect – they result in the payment made at purchase being effectively higher and the payment received at sale being effectively lower than the nominal price of an asset. Consequently, they can be referred to as a premium or a discount on the asset‘s current (fair) market price. However, depending on the origin of the costs, their impact on liquidity is different.
1.2.2.1. Commissions and Taxes Various commissions, which need to be paid in order to trade, constitute the most distinctive part of direct transaction costs. This diverse category encompasses a number of various payments resulting from different sources. Most often are different types of legal fees charged by notaries or other administrative bodies for legal procedures as well as brokerage and intermediation fees necessary to obtain access to markets. They 53
54 55
This is indicated by the presentation of the subject of transaction costs in chapters on financial market liquidity (or the subject of liquidity in chapters on transaction costs) in many finance text books, as, e.g., Francis/Ibbotson (2002), pp. 151 ff., or Schwartz/Francioni (2004), pp. 63 ff. Also a lot of “journal” research uses these terms as exchangeable; see Marschak (1949, 1950), Constantinides (1986), Grossman/Miller (1988), or Grossman/Laroque (1990) as well as the literature in FN 294, 295, and 296. See Fabozzi/Modigliani (2003), pp. 259 ff., or Wagner/Edwards (1993), p. 67. See Schwartz/Francioni (2004), p. 66, Fabozzi/Modigliani (2003), p. 262, Francis/Ibbotson (2002), p. 151, or Maginn/Tuttle (1990), p 12-35.
1.2 Sources of Liquidity
45
vary strongly depending on the type of asset or market and range from negligible amounts to substantial portions of invested capital.56 Another important category of direct trading costs consists of various taxes. On the one hand, turnover-based taxes are imposed on certain assets – e.g., the land transfer tax belongs to this category, but also the definitive value added tax (if applies) must be treated as a transaction cost. On the other hand, various income taxes have a similar effect. Since they are imposed on (positive) capital gains, they affect the effective sale prices of assets, which yield positive holding period capital returns. The above mentioned costs arise from clearly defined regulations and are, at least to some extent, time invariant. The fact that direct transaction costs can be determined with certainty before a transaction is carried out is one of their most important characteristics – they constitute the fixed component of the total trading costs.57 The impact of direct transaction costs on liquidity is less straightforward than it may seem at first sight. The analysis in this section is therefore conducted in several steps. Consider first the introduction of a transaction fee (commission) from the perspective of a single investor holding an asset as a capital investment. The effect of such a fee on the asset’s liquidity depends on whether it is levied as a fixed amount or proportionally to the size of the transaction. As it seems, a direct liquidity effect occurs only in the first case. It is best illustrated on the basis of the price-time locus of liquidation. Liquidity, or more precisely marketability, has been defined in this context as the slope of the respective curve (see also Figure 1-1). While liquid assets can be liquidated quickly at the fair value, so that their liquidation prices increase quickly with the duration of the liquidation process, achieving the maximum possible price for an illiquid asset requires more time resulting in a slow increase of the price. The introduction of the commission results in an immediate downward shift of the whole locus as illustrated with the grey line in Figure 1-12. Note, however, that not the position of the locus but only its shape (slope) is relevant for the asset’s marketability. Thus, in order to correctly assess the effect of the commission, it is necessary to relate liquidation prices resulting at different liquidation periods to the maximum achievable price after the commission and not to the one that could have been gained before the commission was introduced. In other words, it is the price-time relation that matters in marketability considerations and not the level of prices. Since a fixed fee accounts for a larger 56 57
See sections 1.3.2, 1.3.3, and 1.3.4 for empirical examples. See Collins/Fabozzi (1991), p. 27.
46
Chapter 1: The Concept of Liquidity
fraction of the lower prices at the beginning of the liquidation than of the higher prices resulting at longer liquidation periods, the “after-commission” relative price-time locus lies below the original one. The dotted line in Figure 1-12 shows the shape of the locus scaled to the new (lower) maximum achievable price. It clearly implies lower marketability.58
fixed commission
Price
max
── ── ----
original locus (unscaled) after-commission locus (unscaled) after-commission locus (scaled) Time
Figure 1-12: The effect of a fixed commission on the price-time locus of liquidation
In contrast, a proportional commission has no effect on the relative price-time locus. For example, if extending the duration of the liquidation process results on average in an x% higher sale price, the proportional gain will be the same with a y% commission on the realized price. Thus, marketability is not affected by a proportional commission. Same holds also for liquidity risk. While a fixed fee leads to higher uncertainty about the realized price, especially for shorter liquidation periods, a proportional commission has no effect on the fluctuations of relative prices – liquidity risk remains unchanged. The above conclusions refer to liquidity of assets and hold for commissions or taxes imposed on the level of single investors. However, there is another, less direct effect on liquidity, which can only be recognized when a market is considered as a whole, i.e., as an interaction of numerous investors. Commissions applying to all traders affect 58
Note, however, that there would be no liquidity effect in the described sense if the asset was perfectly marketable before the introduction of the commission, i.e. the same price could be achieved independent of the liquidation time. Introducing a fixed commission in this case reduces the achievable price, but it still can be achieved immediately, so that formally marketability remains perfect.
1.2 Sources of Liquidity
47
the price-time loci of their individually held assets, but they also determine the profitability of frequent trading. The cost of buying and selling an asset directly influences its holding period return. Thus, quick liquidation may not be favorable even if the liquidation price is independent of the liquidation time, i.e., the locus is flat, but lies below the purchasing price. Higher direct transaction costs result in effectively lower returns of short-term investments enforcing longer investment horizons and leading, in consequence, to less frequent trading. Hence, they lead to lower trading volumes and reduce market depth.59 For an individual seller, this means a lower chance of meeting a trading partner willing to trade at a certain price and, thus, the necessity of accepting either a longer duration of sale or a lower price. Note, however, that this only holds when all market participants are affected by the direct transaction cost in the same manner – discriminating commissions or taxes applying only to individuals or to small groups of investors have no such effect. The above discussion makes clear that the liquidity effect of commissions comes less through the direct reduction of the realizable sale price – the most popular proportional commissions have no impact on liquidity on the level of individual assets – but mostly indirectly, through the reduced intensity of trading on the affected market. This conclusion concerns in the first place marketability (expected liquidity); the effect on liquidity risk is less obvious. There seem to be no straightforward argument why less frequent trading as such should lead to higher uncertainty about the realized sale or purchase price (marked to market). An increase in the overall investment risk is, however, probable due to higher exposition to market risk – with longer intervals between trades it is more difficult to react to new information.60 This additional market risk arising from direct transaction costs can, and in fact should, be viewed as a part of liquidity risk. However, since it is essentially identical with the risk arising from opportunity costs, it is discussed more thoroughly in section 1.2.2.3.
59
60
The formal model by Lo et al. (2004) leads to an analogical result. In the presence of fixed transaction costs heterogeneous agents, who would trade continuously in a frictionless market in order to hedge their non-tradable risks, choose to trade only infrequently. The inability to trade at will results in a reduced demand for the asset and gives rise to significant liquidity discounts. The analysis of various measures of liquidity risk introduced in section 3.3 generally confirms this link. In particular a higher variance of the net receipts from liquidation arises when lower trading intensity (arrival of offers) is assumed.
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Chapter 1: The Concept of Liquidity
1.2.2.2. Indirect Transaction Costs Indirect costs of trading, which consist mainly of the spread between the buying and the selling price (bid-ask spread) and the impact of trading on the market price, constitute a separate group of transaction costs discussed in the literature. Although these costs are less explicitly visible than direct commissions, they often outweigh them61 – Wagner/Edwards (1993, p. 67) illustratively refer to direct transaction costs as to the “top of the iceberg” of total transaction costs consisting mainly of the indiect ones. They are sometimes presented in form of the cost of a “round-trip”, i.e., the total cost of a simultaneous purchase and sale of an asset.62 This approach allows assessing the portion of the asset’s value that an investor forfeits in the investment process due to factors other than pure changes in supply and demand. Demsetz (1968) was probably the first to refer to the bid-ask spread in terms of a transaction cost. In the strict sense, the spread defined as the difference between the selling and the buying price exists only in dealer type markets, i.e., in markets in which a specialized dealer or market maker publicly quotes prices at which he is ready to trade. On the other hand, also many directly traded assets with no organized trading systems may demonstrate some kind of a spread. For real estate, collectibles, or second-hand cars average prices are usually higher for buyers willing to buy quickly than for sellers willing to sell quickly. This arises from the fact that the active party, i.e., the one actively searching for a trading partner, is usually under a higher pressure to transact and has less bargaining power than the passive one. Another component of the indirect cost is the impact of trading on the market price. Especially in shallow but wide markers selling or buying larger quantities is only possible by falling back on individuals asking or bidding less favorable prices. Thus, an attempt to execute a large transaction leads to an adjustment of the market price forcing the trader to sell at a lower price or to buy at a higher price than usual. This applies especially when the transaction is to be executed quickly – with sufficient time, an investor can split the order into several smaller ones avoiding the reduction of the market price. Thus, the market impact can be treated as the (additional) cost of trading large amounts quickly. 61
62
See, e.g., the empirical data in Loeb (1983), Sharpe/Alexander (1990), pp. 52-53, Jones (2002), or Schwartz/Francioni (2004), p. 66-67. Note, however, that virtually all empirical studies refer to publicly traded equities. The assessment of indirect transaction costs for other assets, in particular for real estate, is hardly possible. Ibidiem.
1.2 Sources of Liquidity
49
For an individual investor, the bid-ask spread and the market impact have similar effects to commissions or taxes – the effectively realized purchasing price is higher and the selling price is lower than the mid-price, which would result otherwise. This analogy allows analyzing the effect of indirect transaction costs on the price-time locus in a similar manner. From the point of view of an individual investor, the spread, which is proportional by its nature, affects the absolute price-time locus of an asset but not the relative one; i.e., marketability on the individual level remains unchanged. However, higher “round-trip” costs reduce returns from short term investments and prevent frequent trading leading to less market activity and lower market depth. In contrast, the existence of the price impact forces investors willing to maximize assets’ sale prices to liquidate larger quantities over longer time periods. Hence, the liquidation value increases more slowly over time indicating lower marketability. These conclusions demonstrate that liquidity effects of direct and indirect transaction costs are to a large extent analogical when referring to marketability (expected liquidity). The difference is more substantial with respect to liquidity risk. While there seems to be no clear reason, other than opportunity costs, why commissions or taxes should increase the uncertainty about the outcome of liquidation, it is different with respect to indirect costs. Unlike commissions, spreads and price impacts are not predetermined and their precise magnitudes depend on the decision of a dealer or on the reaction of a market. Both these determinants are associated with uncertainty – even shortly before the transaction one cannot be absolutely sure about the effectively realized price. The final effect of the spread and the market impact is always burdened with additional uncertainty. Thus, the existence of indirect transaction costs increases assets’ and markets’ liquidity risk. A closer look at the origin of indirect transaction costs reveals another, even more substantial difference to the direct costs. While the latter are exogenously enforced by authorities, it is much more difficult to identify one unique source of the indirect costs. In fact, it seems that they are determined to a large extent by the same factors as liquidity itself. For example, Keim/Madhavan (1998) classify the determinants of (equity) trading costs into stock-specific factors, like relative (not precisely defined) liquidity, stock volatility, market design, and traders skills and reputation, and transactionspecific factors, like order size, order type, and investment style. Note, that most of these factors are discussed in subsequent sections as the actual sources of liquidity
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Chapter 1: The Concept of Liquidity
(marketability and liquidity risk). Also, a number of market microstructure models address the sources of the bid-ask spread and the market impact coming to similar conclusions. For example, the spread in the models of Stoll (1978) and Ho/Stoll (1981) is determined by the costs of order processing and the costs of holding an inventory; these costs correspond with the direct trading costs discussed earlier and the opportunity costs discussed later, respectively. Garman (1976) concentrates on the role of the stochastic nature of supply and demand in setting the bid and ask prices, which is related to the problem of (random) search for the trading partner discussed in section 1.2.3.2. Finally, models of Copeland/Galai (1983), Glosten/Milgrom (1985), or Kyle (1985) focus on the impact of information asymmetry on the bid-ask spread, which seems to show some analogy to the impact of heterogeneous valuations among market participants discussed in section 1.2.4. The similarity of the sources of indirect trading costs and liquidity justifies their treatment by numerous researchers as two (nearly) exchangeable terms. Indeed, if one assumed that there were no indirect costs of trading in a perfectly liquid market, all such costs incurred in reality could be attributed to the lack of perfect liquidity making these two concepts equivalent. This approach was postulated by Miller (1965) who noticed that both the price and the time dimension of the liquidation process can be expressed in money units and interpreted as costs as soon as time is considered valuable.63 However, a more rigorous analysis raises some doubts about the nature of this link. Firstly, a reference point is necessary in order to translate asset’s present value into cost – a cost means a reduction of the value, but from what basis? At least several possibilities seem plausible here: the presumable fair value in the state of perfect liquidity, the maximal achievable value, and the value realized in the case of an immediate liquidation. Each of these alternatives leads to a different notion of transaction costs and liquidity, and it is unclear which of them is most accurate. Secondly, Miller’s postulate of equivalency between indirect transaction costs and liquidity seems to refer mainly to the expected liquidation outcome; liquidity risk is not explicitly mentioned. Theoretically, uncertainty about the outcomes of individual trades should be reflected in indirect trading costs to some extent; in particular, it should be prices in the bid-ask spread. However, the exact mechanism is still not clear and empirically little researched. The cited market microstructure models contribute a lot to the explanation of
63
This approach is referred to as the present value-time locus in sections 1.1.1.1 and 1.1.2.1.
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51
this problem, but doubts about full equivalency between indirect transaction costs and liquidity still remain. To sum up, despite the seemingly similar effects for individual investors, the relation between indirect transaction costs and liquidity is significantly different from the relation between direct costs and liquidity. While commissions and taxes constitute only one of the factors determining the ease of liquidation, indirect costs seem to refer to the same phenomenon as liquidity itself. Thus, one can either consider indirect costs to be the consequence of lacking liquidity or illiquidity to be the result of high indirect costs of trading. In any case, both terms are closely related, although more research would be necessary to state whether full equivalency is given. Avoiding further extensive discussion of this subject, the analysis concentrates on liquidity understood primarily as the “ease of liquidation” rather than the “cost of liquidation”. Consequently, indirect transaction costs are treated mainly as the result and not the source of liquidity.
1.2.2.3. Opportunity Costs The importance of opportunity costs has been already mentioned in the context of introducing liquidity risk in section 1.1.2.1. Although they are often classified as indirect transaction costs, it is reasonable to discuss them separately. Opportunity costs can be most simply described as the “costs of waiting”. They denote all losses incurred and gains missed that would have been avoided or achieved if the transaction had been accomplished earlier than it actually was. Despite the common caption, this group of costs is not homogenous and (at least) two types of opportunity costs can be identified.64 The first one refers to the effects of postponing a transaction on the effectively realized price; the second one encompasses the consequences of missing alternative investment opportunities. While searching for better liquidation (or purchase) possibilities, investors run the risk that the market situation may change. This change may be both to investor’s advantage as well as to her disadvantage. However, assuming that the moment of liquidation is not chosen accidentally and that the investor follows a timing strategy, delaying the 64
A number of various classifications of opportunity cost can be found in the literature; in particular, they are often classified as indirect transaction costs (see Keim/Madhavan, 1998, p. 54 ff., Francis/Ibbotson, 2002, p. 151, or Schwartz/Francioni, 2004, p. 66). The approach followed in this section is especially purposeful for the analysis in the following chapters.
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trade will possibly corrupt this strategy. This type of opportunity costs is usually referred to as timing costs and this term is also used in this book.65 However, researchers usually fail to differentiate between its two separate components: the expected market development during the liquidation process and the additional uncertainty arising from unexpected market fluctuations, i.e., additional market risk. The fist one means that postponing the sale can be advantageous in bull markets with expected price increases but disadvantageous in bear markets, where prices tend to fall (the opposite holds for the purchase case). Thus, the expected cost of a delayed liquidation can be either negative or positive.66 In terms of the (expected) time-price locus of an asset, marketability is higher in the former case and lower in the latter case in comparison to the situation of absolute price stability. Additional market risk is, however, incurred in either case. It results in a higher uncertainty about the terminal value of an investment. Although its nature is the same as that of the “regular” market risk, i.e., the uncertainty about the change in the value of an asset within the investment horizon, it seems to make more sense to attribute it to liquidity risk rather than to market risk. In consequence, the market timing effect leads to an increase in liquidity risk. Its level depends on the level of asset’s price volatility on the one hand and on the frequency of trading possibilities, i.e., the frequency of sale or purchase offers, on the other hand. The chance of a substantial change in the market price level is higher in more volatile markets than in more stable markets, but with more opportunities to trade the liquidation process can be accomplished more quickly reducing the exposure to these changes.67 A different type of opportunity costs are the benefits that could have been gained if the transaction had been completed more quickly – it is referred to as the opportunity cost in the narrow sense. Such costs arise both in the sale and in the purchase case. The rea65 66
67
See Collins/Fabozzi (1991), p. 28, or Wagner/Edwards (1993), p. 68. A practical approach to capture market timing costs is the “implementation shortfall” proposed by Perold (1988), which is computed as the difference between the notional return that would be earned if an order was executed immediately and the actual return. Perold states that this type of costs constitutes a substantial portion of the total transaction costs, what is also confirmed by Keim/Madhavan (1997). Schwartz/Francioni (2004, p. 69-70) note that low liquidity alone can affect the volatility of prices in certain markets. Prices in such markets may tend to “bounce” between higher values paid by buyers and lower values received by sellers; lower trading activity also results in larger errors in price discovery (i.e., finding an equilibrium price through trading). Since, as argued above, higher volatility means higher liquidity risk and, thus, lower total liquidity, the whole process might be to some extent self-accelerating. This issue is not further followed in this book leaving an interesting field for further research.
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son for selling an asset is always the ability to use the capital in a more efficient way – this could be an attractive investment opportunity, repayment of depth, or consumption. Independent of its type, the alternative use of capital is viewed by the investor as more valuable than the income from the original investment. Thus, by continuing the ownership of the latter one unwillingly, for example, due to the inability to trade, she misses the benefits of the alternative.68 On the other hand, assets are purchased because the gains from owning them are higher than the gains from the current allocation of capital; delaying the purchase results in the loss of these gains. Note, however, that in both cases it would be incorrect to define all resulting losses as opportunity costs – only the difference between the desired and the actual state is an effective cost. In the sale case, it is the difference between the alternative use of capital (e.g., the rate of return of an alternative investment) and the gains provided by the asset scheduled for liquidation. Analogically, the difference between the current use of capital and the gains provided by the asset constitute the opportunity cost in the purchase case. With respect to financial assets, opportunity costs in the above sense are usually allowed for by discounting future payments. While the level of market timing costs depends solely on the characteristics of assets and markets, opportunity costs in the narrow sense are a function of investor’s individual situation. Obviously, the availability and the type of alternative uses of capital differ not only among investors but also change with time. There is therefore no universal, always valid discount rate that can be used to allow for these costs; it needs to be set by each investor individually according to her current situation. In this context, it is useful to differentiate between two situations, which with high probability lead to significantly different opportunity cost levels. The first one is a liquidation (or a purchase) that has been anticipated and is a part of the investment strategy. This is the case when the investment goal has been achieved or the planned holding duration of an asset has been reached. The sale of stocks from a portfolio, which has been optimized for one year holding time, or the sale of an accomplished real estate project by a developer are examples of planned liquidations. In this case, there is usually no urgent need for a quick sale. The difference between the alternative uses of capital, like a new 68
Baldwin/Meyer (1979) provide an excellent demonstration of how opportunity costs in the narrow sense affect liquidity. They model the situation of an investor who commits to an illiquid investment not knowing whether a more lucrative opportunity arrives later (opportunities arrive in a random sequence). The central conclusion is that a larger liquidity premium should be demanded for longer-lasting investments in order to compensate for lost opportunities.
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equity portfolio or a new development project, is not excessive, so that the investor can take more time to optimize the liquidation process. In consequence, a lower discounting rate can be applied. A substantially different situation arises when the liquidation is unexpected, i.e., forced by external circumstances. This is usually the case in times of liquidity bottlenecks, when events such as a sudden default of a large debtor or cancellation of a credit line rapidly increase investor’s demand for cash. Failure to gather the required funds on time results usually in a default on own liabilities and, in consequence, in insolvency. It is therefore highly probable that the opportunity cost of postponing the sale is significantly higher in such cases than it would be if the sale was expected and planned. In particular, the investor might be forced to fall back on expensive emergency financing or even accept bankruptcy costs. Note, however, that the latter, although high, are not infinite;69 thus, it might be preferable to accept bankruptcy rather than to liquidate an asset far below the normally achievable price. Analogical considerations can also be applied to the purchase case. It may sometimes come to situations when the necessity of buying arises unexpectedly, e.g., when an option for delivery of securities or commodities is executed. The cost of the inability to deliver is then most probably far above the one faced in the case of a scheduled purchase. The differentiation between a planned and a forced liquidation is of high relevance for liquidity management and is discussed in several contexts in this work. Probably the most important conclusion arising from the discussion of opportunity costs is the subjectivity of this parameter. The fact that each investor sees a different set of opportunities results in differences in personal valuations of the receipts achievable at sale or expenditures necessary for purchase. Obviously, those with better alternative investment possibilities should put more weight on the time aspect – the optimal liquidation time will be shorter for them than for investors with poor investment opportunities. Analogically, individuals facing sudden liquidity bottlenecks will be more prone to sell quickly, even accepting a substantial discount, than those who planned the sale earlier and are less in hurry. In consequence, the grade of assets’ liquidity will differ depending on the individual situation of the investor and the opportunity costs related with it. This is best illustrated with a comparison between a bank savings account and a piece of fine jewelry. The savings account can be liquidated at any time at the nominal value; it may take, however, up to several working days before the amount 69
The assessment of bankruptcy costs meets on numerous practical problems; see Kalaba et al. (1984) or Bris et al. (2006) for approaches to this problem.
1.2 Sources of Liquidity
55
is available in cash. On the other hand, jewelry can be sold (pawned) immediately against cash, but the seller usually has to sacrifice a substantial portion of the value achievable in a regular sale. To most people in common everyday situations, the savings account is far more liquid than jewelry. However, if an individual faces a situation in which she urgently needs cash within one day facing severe consequences otherwise, jewelry may prove to be more liquid than the savings account; the latter may, in fact, seem perfectly illiquid in this situation. Thus, the level of opportunity costs is highly relevant for asset liquidity and its subjectivity makes also liquidity a partially subjective characteristic of assets. 1.2.3.
Market Organization and the Search for a Trading Partner
A subject that is often discussed in the context of market and asset liquidity is organization of trading. Central issues include, on the one hand, the way sellers and buyers are brought together to trade and, on the other hand, the way price agreements are reached. Their relevance is straightforward – they affect the time necessary to accomplish a transaction as well as the price effectively realized; consequently, they affect the price-time locus of liquidation. Before conducting a closer analysis of the role of market organization for liquidity, it is purposeful to review its main existing forms. They differ not only with respect to the technical procedures involved but also with respect to the character of the search for a trading partner. The latter aspect turns out to be of key relevance for liquidity and receives therefore special attention.
1.2.3.1. Forms of Market Organization The numerous various approches to the organization of trade can be classified in four main groups: direct search markets, brokered markets, dealer markets, and auction markets.70 The first two can be associated with less structured trading and are referred to as non-organized markets, and the latter two are characterized by the existence of specialized institutions assisting investors in their trading activity and are referred to as organized markets.
70
This classification is based mainly on Garbade (1982), Part 7, but similar ones can be found in most handbooks on security markets; see, e.g., Reilly/Brown (1997), Chapter 4, Santomero/Babbel (1997), Chapter 19, or Fabozzi/Modigliani (2003), Chapter 7. Although it originally refers to stock trading, same principles are also valid for other assets.
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Traders in direct search markets (also referred to as direct markets) act on their own in their efforts to find a trading partner. This form of market organization is most common on markets for rarely traded goods but also on markets for goods that are difficult to standardize in a large scale. Among them are many commodities, arts, wine etc. Also real estate as well as shares of smaller companies are widely traded in direct markets. The lack of organized trading structures induces the necessity of contacting each potential buyer (when selling) or seller (when buying) individually and negotiating the price separately in each case. Due to the relatively high cost of reviewing trading opportunities, it is seldom economical to conduct the search among all potential partners. The result is therefore more or less dependent on the luck of meeting the right one quickly. High search costs and the lack of reliable information about trading possibilities are doubtlessly the main weaknesses of direct markets. However, this type of market organization does not only have disadvantages. Firstly, it enables individuals with strong bargaining positions to negotiate relatively better prices. Secondly, it does not preclude finding a trading partner offering above-average conditions. Due to the latter property, it is possible that transactions are executed which would never come to execution under a different market regime. The lack of complete knowledge about the current market situation by all market participants may, thus, be advantageous for some of them, particularly for those with better market overview. A natural further step in the development of direct search markets are brokered markets in which brokers offer search services to individual investors. This allows a reduction of search costs. However, there will only be demand for such services if economies of scale occur, i.e., if brokers are able to search at a lower cost than individuals. There are a number of possible reasons why this can be the case in certain markets. Firstly, specialized information channels are often available against a fixed fee. By utilizing them to conduct a larger number of searches, the unit cost of obtaining information about a potential buyer or seller can be substantially reduced. Another reason for lower search costs is the possibility to fall back on information about past offers not executed in earlier trades or to combine searches for several similar clients. In either case, the marginal cost of finding another trading partner is lower than the cost of a separate search. Furthermore, since brokers usually have a good overview of the market, they are able to judge the quality of offers better and quicker than individual traders. In effect, they are in the position to offer search services against fees that are smaller than the cost of an individual search. Nevertheless, one can expect brokers to
1.2 Sources of Liquidity
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occur in direct markets only if the scale of trading is large enough to ensure sufficient profits. A special form of market organization, which can be classified as a subtype of brokered markets, is an auction, i.e., a system provided by an auction house in which an item is sold to the individual placing the highest bid within a certain time horizon.71 The classical “English” auction is conducted as an open bidding in which subsequent bidders know the previous bids and can react to them. As soon as no higher bid arrives, the item is “hammered down” at the last bid provided it exceeds the sellers “reserve price”. A different possibility is a closed auction in which bids are supplied within a given time period and remain secret until opening. There are also further variations of auctions, with different rules, which mainly aim at the maximal utilization of buyers’ rent, i.e., at achieving possibly highest prices. Auction systems are mainly applied for commodities, arts, and other real goods, but they are also used on financial markets – e.g., the allocation of treasure bills is done through an auction in some countries. The function of dealers in dealer markets is generally similar to that of brokers – they facilitate the search of individual investors for trading partners. However, instead of only intermediating between sellers and buyers, they trade on their own account. They are ready to purchase assets from willing sellers and to sell them to willing buyers at any time at predetermined prices. The costs and risks associated with this readiness to trade are covered by the difference (spread) between the higher selling price (bid) and the lower buying price (ask). They include, in particular, the costs and risks of finding counterparties to close open positions as well as potential losses resulting from trading with better informed investors.72 The existence of dealers gives investors the ability to trade instantly at any time. The duration of search can be reduces to zero provided the quoted bid or ask price is accepted. Thus, dealers, and even more so market makers, who oblige themselves to quote prices on demand and trade at any time with any investor, “bridge the time gaps between asynchronously arriving public purchases and sales”73. They provide markets with liquidity services, the price for which is paid through the spread. Of course, an individual investor may always seek and possibly 71
72 73
See McAfee/McMillan (1987) or Klemperer (1999) for excellent reviews of auction forms and theory as well as for literature references. Determinants of the bid-ask spread are discussed more thoroughly in section 3.1.1.1. Garbade (1982), p. 429. See also Garman (1976), p. 258.
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find a trading partner offering better conditions, but they will never accept prices worse than those quoted by dealers. In this sense, the “official” bid-ask quotations constitute the natural boundaries for price building in dealer markets. Although there is no compulsion for all dealers to quote precisely the same prices, they usually do not differ substantially. Competition and the ease of obtaining quotes from several parties should effectively direct investors to those offering lowest possible spreads. In practice, differences between quotes are marginal, especially in more lively markets. A standard auction, as described earlier in this section, despite the similar name, has only little in common with an auction trading system, which is the main form of organization of security markets. The term “auction market” refers to a number of different forms of trading,74 all of which have one common feature – buyers and sellers are paired directly, without an intermediary, through an organized trading system. Interested parties place their orders with the exchange specifying desired quantities. Two types of orders are usually allowed: limit orders, in which a price limit is set, and market orders, which are executed at any price.75 After collecting all orders, a single price is set, which applies to all transactions.76 The resulting market price is the one at which demand and supply are in equilibrium; simultaneously, this is the price that leads to the maximal trading volume.77 This means that only sell limit orders below the market price and buy limit orders above the market price as well as market orders are executed. Depending on the frequency of order execution, two types of auction markets can be differentiated. On discrete (periodic) markets, transactions are executed in fixed time intervals between which orders are collected. In continuous markets, orders are executed as they arrive provided sufficient counterorders are available. Execution times of transactions should be typically shorter in the latter ones. 74 75
76
77
See Benesch/Prüggler (2005) for an international review of stock markets’ organizations. See Francis/Ibbotson (2002), p. 131 ff., Fabozzi/Modigliani (2003), p. 254 ff., or Bodie et al. (2005), p. 78 ff. Two further often distinguished types of orders are stop orders and stop-limit orders, which are executed as market or limit orders, respectively, only if the market price reaches a predetermined stop price. The mechanisms of “order collecting” vary from one market to another and are sometimes highly complex. Individuals are usually allowed to place their orders through a brokerage house only. Furthermore, specialists exist in most stock exchanges overseeing the order flow process and, to some extent, steering it by buying and selling on their own account. The details of stock exchange trading systems are, however, not crucial for the analysis in this book, so the reader is referred to the extensive literature on the subject. See, e.g., Garbade (1982), Chapter 22, or Fabozzi/Modigliani (2003), Chapter 13. See Schmidt (1988), p. 11-12, for an example on how stock exchange market prices are set.
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1.2.3.2. Market Organization, Search, and Liquidity The form of market organization plays a big role not only for the time necessary to buy or to sell an asset but also for the price level. Obviously, it affects the price-time locus of liquidation and the certainty about its shape, the constructs on which the concept of liquidity used in this work is based. The dependence between the way an asset is traded and its liquidity is discussed in the following. There is substantial literature on the role of market organization (trading mechanism) for different aspects of market functioning, mainly in the area of market microstructure. It concentrates predominantly on the characteristics of returns under different regimes;78 some of it, however, also refers to liquidity. Among the latter are such significant papers as Garman (1976), Garbade/Silber (1979), Grossman/Miller (1988), or Hasbrouck (1991a). However, the literature on this subject seems to cover only dealer and auction markets; refereces to non-organized (i.e., direct or brokered) markets are practically non-existent. As it seems, most authors assume that the former ones always provide superior liquidity and see no need to discuss this issue. Although this thesis surely holds in the majority of cases, the analysis in this section shows that its validity is not as straightforward as it may seem. Two central issues related to market organization are the search for a trading partner and the negotiation of prices. Search means in this context the process of obtaining information about available trading possibilities. An investor willing to sell an asset has to identify at least one potential buyer; identifying more than one opens the possibility of choosing the one that offers best conditions. In this sense, search has a positive value as it allows achieving potentially higher sale proceeds. This interpretation of search as a process of obtaining valuable information has been introduced by Stigler (1961) initiating research in information economies.79 Search, however, is also costly. The magnitude of the search costs depends on a variety of factors. Firstly, the characteristics of assets play a big role. In particular, complexity and weak comparability of assets result in the necessity of individual assessment of the true value of each investment, which might prove very expensive. For example, in a real estate transaction an extensive check of the true state of the property (building) is necessary, buying private 78
79
See Cohen et al. (1978, 1980), Amihud/Mendelson (1987), Madhavan (1992), or Affleck-Graves et al. (1994). See also the discussion and the literature review in section 2.1.
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equity requires an in-depth analysis of the company, and selling a Picasso painting is hardly possible without a genuineness expertise. In all these cases, the verification of the true characteristics of the traded asset is a time and money consuming process, so that reviewing a large number of alternatives may prove uneconomic. Secondly, search costs depend on the intensity of trading. The more buyers and sellers are active on a market, the easier is the identification of possible trading partners. Finally, the cost of communication between traders plays a big role. This is clearly visible in the age of widely available electronic telecommunication systems. The ability of cheap communication with geographically remote parts of the world significantly increases the range of potential trading opportunities. The form of market organization affects the “ease of trading” by determining the mechanism of the search process and the level of search costs. On direct markets, investors need to search individually caring the full costs of finding potential buyers or sellers, communicating with them, verifying the asset etc.. Brokers can help to simplify the search process and reduce these costs by taking advantage of economies of scale and utilizing their expertise knowledge. However, although they take over the task of identifying potentially interesting sale or purchase offers, the investor still needs to choose the preferred one herself. This problem disappears to a large extent in dealer markets – by offering immediate sales and purchases, dealers assume the full risk of finding appropriate trading partners to close their open positions. Investors’ (residual) necessity of search is, thus, reduced to finding the dealer who offers the best quote; yet, due to the proximity of quotations, it is usually hardly remunerative. Finally, there is neither need nor possibility of search in auction markets – orders from interested buyers and sellers are gathered in one place and paired automatically without any additional effort. Summing up, the need to search for a trading partner as well as the cost of conducting it diminish as one moves to “higher” forms of market organization. Another channel through which market organization affects liquidity is the intensity of trade. In general, better and cheaper possibilities of marketing an asset induce a larger number of investors to trade more frequently. So, forms of market organization providing better possibilities of finding a trading partner at a lower cost should also lead to higher trading activity. On the other hand, more intensive trading increases the chance of finding a trading partner. This, in turn, leads to a further reduction of the search costs and attracts further traders. Due to this self accelerating process, the increase in
1.2 Sources of Liquidity
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liquidity resulting from the implementation of a more efficient trading system is larger than the mere reduction of the search costs. However, certain levels of trading activity need to be reached in order to enable higher forms of market organization.80 The existence of brokers depends on the possibility of sufficient earnings from offering this kind of services, which is only given if the number of transactions exceeds some minimal level. Similarly, dealing activity makes only sense when open positions can be closed relatively promptly. Otherwise dealers would unwillingly run too high risks on suboptimal asset portfolios that could not be covered by reasonably high spreads. Also auction markets cannot function efficiently without a continuous flow of supply and demand; the cost of their maintenance would not be justified with only few trades per period. Another hurdle is the possibility of standardizating the traded assets.81 Full comparability of all trading units is one of the main conditions for dealer and auction markets. Without it, no unified system of price quotations or a single market price valid for all traders is possible. The form of market organization affects not only the duration and the cost of searching, but it can also affect the level of prices achievable for certain assets. To this end, it is necessary to note that the term “market value” has a different meaning under different trading regimes. In fact, a single “market price” exists only in auction markets; there is no such concept under other market forms. Dealer markets know at least two price levels that are potential candidates for the market value. The mid-price is usually regarded as the presumed single equilibrium price. However, this only holds if the demand and the supply side of the market are perfectly symmetric; in particular, the costs of inspection, trade execution, and delivery need to be identical for both sides. In direct and brokered markets, no distinguished prices exits that could be interpreted in the sense of a market value. Each transaction is negotiated individually and results in a different price. The average transaction value may be treated as an indicator for the unobservable “fair value”. For the lack of a better solution, this is done throughout this book, though it seems to be a highly imperfect proxy. In particular, it might be dis-
80
81
Note, however, that trading activity should be associated with the number of transactions rather than with the trading volume. Garbade (1982), pp. 426-427, points out that only the combination of the volume and the average transaction size is relevant for the feasibility of “higher” forms of market organization. Garbade/Silber (1979) refer to the number of market participants in this context. Garbade (1982), pp. 499, stresses the importance of the existence of close substitutes between securities for market liquidity, which is analogical to the role of standardization discussed here.
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torted by only few unusual transactions (outliers), especially, if the overall trading volume is low. In view of the discrepancies in the meaning of market value and in the formation of transaction prices, it is by no means obvious that an asset should have the same price among the same group of traders under different forms of market organization. In particular, the maximum achievable price, which, as noted in section 1.2.2.1, is especially relevant for liquidity analysis, can and most probably will be different under different regimes. From the point of view of an individual investor prices on a dealer or an auction market are set exogenously and are not negotiable. Thus, the market price or the bid price is simultaneously the highest possible one. In contrast, traders in nonorganized markets can negotiate freely, so the only limit to the transaction price is the ability to find a trading partner willing to pay it. In consequence, the achievable price level should be regularly higher on direct or brokered markets than on dealer or auction markets.82 In terms of price-time loci, it means that while organized markets allow achieving the market or the bid price more quickly or even immediately, nonorganized markets may allow reaching higher liquidation values at the cost of longer liquidation periods, as depicted in Figure 1-13. In this situation, none of the markets can be denoted as clearly more or less liquid (in terms of marketability) and ambiguousness analogical to the one discussed in section 1.1.1.1 and depicted in Figure 1-2 occurs.83 Note, however, that this holds only when referring to two parallel markets for the same asset. When two different assets with different market organizations are considered, their price levels are not directly comparable – it makes little sense to deliberate about the hypothetical price a stock exchange traded share would achieve if it was traded directly. In this case, it seems reasonable to scale the prices to the respective maximally achievable levels. Doing so would result in the price-time loci of assets traded on non-organized markets lying below the loci of assets traded on organized markets as indicated with the dotted line in Figure 1-13. From the investor’s point of view, the latter would be perceived as more marketable.
82
83
Note, however, that this thesis refers to the prices achievable by an individual investor and not necessarily to the level of prices on the market. In this respect, simulations by Morawski/Schnelle (2006) indicate that the average price level of an asset should be higher on an auction market than on a direct market if traders stick to their valuations, i.e., do not sell (buy) under (over) the individually assessed values. See also the analysis in section 2.3.2.
Price
1.2 Sources of Liquidity
63
non-organized markets (unscaled)
organized markets non-organized markets (scaled to the maximum achievable price)
Time Figure 1-13: Price-time loci for organized and non-organized markets
Finally, the role of market organization and search for the level of liquidity risk incurred by market participants needs to be highlighted. Liquidity risk has been defined as the uncertainty about the effectively achieved transaction price or its deviation from the market value in the moment of the decision to transact. Obviously, the amount of liquidity risk associated with a certain investment depends on the ability to transact quickly, which, as discussed above, is influenced by the organization of trading. Moreover, since the transaction value is usually the result of a search, also liquidity risk depends on the characteristics of the search process. Because search is obsolete in organized markets, there is only marginal uncertainly about the deviation of the transaction price from the market value of an asset. On the other hand, search is necessary and reasonable in non-organized markets. Its outcome depends to a large extent on the luck of finding a “good” or a “poor” trading partner. This property makes direct and brokered markets clearly subject to higher liquidity risk. Furthermore, if one assumes that transactions on most of the large dealer or auction markets are executed nearly immediately, also the chance that the level of market prices would change during the time between the placement and the execution of an order is minimal.84 Thus, one could consider such markets as lacking any liquidity risk. This assumption is fairly realistic when comparing, e.g., stock exchanges with real estate markets, and it is used 84
This holds especially for institutional investors under normal market conditions; for smaller investors and in times of turmoils, the delay in the execution of an order and the risk that the market price would change in the meantime may be significantly higher.
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Chapter 1: The Concept of Liquidity
in the formal analysis further in the book to provide a reference point for measuring liquidity risk. Further aspects of market organization relevant for liquidity can be named. They concern mostly the details of trading regulations. For example, the “tick size”, i.e., the minimal possible change of the market price may play a significant role in some situations. Narrowing of the bid-ask spread has been observed as the result of a tick reduction in several cases.85 Also the “clearing frequency”, i.e., the frequency of order execution, is potentially relevant for liquidity.86 The latter one can be also understood more generally as the typical time gap between the agreements to trade and the execution of a transaction. It results, in particular, from legal regulations, which encompass the average duration of formalities that need to be fulfilled before the ownership change is effective and the payment is made. Thus, although the sale of shares on a stock exchange can be accomplished within minutes, it can take up to several working days until cash is paid. Similarly, formalities associated with the notary act and the entry in the land register (cadastre) when selling a property may require time. Although these and other organizational details may play an enormous role in certain special situations, they should be less important in the “standard case”. They are therefore largely omitted in the further analysis. 1.2.4.
Diversity of Valuations
As already indicated in the preceding section, the existence of an objectively correct price or value is a purely theoretical concept with respect to nearly any economic good. In fact, different investors asked about the value of any asset will give varying answers. Some of them will provide higher estimates and will be ready to pay higher prices, while others will be more conservative in this respect. Although varying individual valuations are in many cases results of methodical errors, many of them are still subjectively correct. In effect, there is no unique and objective value but rather a distribution of values among market participants. Even though a single price exists in certain markets, it is only a derivate of the respective value distribution; its existence is by
85
86
See MacKinnon/Nemiroff (1999) for the American Stock Exchange or Ahn et al. (2007) for the Tokyo Stock Exchange. Garbade/Silber (1979) analyze the role of the clearing frequency on liquidity and risk.
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65
no means a sign of a perfect agreement among investors. This fact has significant consequences for liquidity, which are the subject of this section.87 There are a number of reasons, why different individuals attach different values to one and the same asset. The three most important ones seem to be: divergence of information, divergence of expectations, and divergence of use possibilities. The first one is also referred to as information asymmetry and arises from different access to non public information. Differences in the quality of possessed information result in differences in the rationales underlying individual valuations. However, not only the access but also the interpretation of information is relevant. Hence, different forecasts concerning the future development of an investment can be derived by different investors even if they are based on the same set of data. The resulting divergence of expectations can refer to future cash flows but also to risks associated with an investment. Note that this source of valuation diversity is based on more or less random errors. While some tend to overestimate the future prospects, others underestimate them coming to different values. These differences are of subjective nature; each individual may be convinced of his or her accuracy, but eventually only some of them will prove to have been right. Yet, there can also be objective reasons why investors attach different values to one and the same investment. If they have different possibilities of using the asset, it may yield different cash flows and cause different risks to each of them. A good example of such divergence of use possibilities is the case of an industrial property, e.g., an automobile assembly. While some investors, especially car producers, may be able to use it efficiently in order to optimize their revenues, many others would hardly be able to take advantage of the special features of that property. Also differences in tax regulations applying to different groups of investors fall under this category. The latter point is often the reason for different statuses of individual and institutional or domestic and foreign investors. Another objective reason for differences in valuations is the divergence of tastes defined as individual utility gained from possessing an asset. A standard example is the valuation of a Picasso painting by different individuals – it varies from “worthless” to “priceless”. Different possibilities of use as well as different tastes refer in the first place to real assets, which can be used for pro-
87
The role of investors’ heterogeneity for the functioning of financial markets has been recognized by early economists but “rediscovered” relatively recently by authors such as Miller (1977), Williams (1977), or Mayshar (1983). Newer studies on this subject include Miller (2001), Fama/French (2005), or Sadka/Scherbina (2006).
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duction or consumption. They seem to play a smaller role with respect to financial assets, which are treated only as sources of cash flows. However, as Fama/French (2005) note, there is some empirical evidence that also financial investors follow certain seemingly irrational tastes. For example, they tend to invest more in domestic stocks (home bias) or in own employer’s stocks than it would be justified by the risk-return characteristics.88 Also the tendency to “socially responsible investing”, i.e., the refusal of investing in companies, which activities are considered immoral or socially unwelcome, 89 can be defined as “taste”. A number of market microstructure researchers have studied the effects of asymmetric information on certain aspects of market liquidity, mainly market breadth and maket depth. The main focus of these studies is on the consequences of insider activity, i.e., the activity of individuals possesing superior information and attempting to take advantage of it by selling fundamentally overpriced and buying fundamentally underpriced securities. Such frameworks have been modeled in the seminal papers by Glosten/Milgrom (1985) and Kyle (1985) as well as in numerous successive studies.90 These models formalize the intuition of Bagehot (1971) and demonstrate how market makers need to compensate for the possibility of insider trading by widening the quoted bid-ask spread to include the expected losses from such trades. Furthermore, market makers also react to the order flow assuming the possibility that it is information motivated. These reactions result in the impact of trading on the spread, which should be stronger when more informed individuals are expected to be active in the market and weaker when more uninformed, “noisy” investors are active. The possibility of insider trading and the reactions of market makers make strategic behavior of market participants worthwhile. On the one hand, investors possessing valuable private information choose trading patterns allowing them to maximize their profits; on the other hand, uninformed investors try to trade in a way that exposes them as little as possible to price reactions evoked by insiders.91 Such strategies are based mainly on the timing of orders, what may lead to fluctuations of the trading volume over time. Summing up, larger information asymmetry among market participants leads to larger 88 89
90 91
See French/Poterba (1991) or Cohen (2006). See http://www.socialinvest.org or http://www.domini.com for reviews of “socially responsible” investment alternatives. See O’Hara (1997), Chapters 4 and 5, for an excellent review of research in this direction. See Kyle (1985) or Back (1992) for models of strategic behavior of insiders and Admati/Pfleiderer (1988) or Foster/Viswanathan (1990) for model of strategic behavior of uninformed traders.
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67
market breadth as both buyers and seller must take into account the possibility that the trading partner is better informed. It also leads to larger market impacts of trading as it is more probable that larger trades are based on insider information. Finally, it may lead to a reduced trading volume in some periods as uninformed traders postpone their orders in order to avoid being mistaken for informed ones. The latter point makes it clear how investor’s ability to signal that she does not have superior information may influence the liquidity of her assets. As soon as the dealer is sure that the trade is not information motivated, he might be willing to tighten the spread and offer better conditions.92 The above considerations stemming from the market microstructure analysis of organized public markets are more difficult to apply when private markets for heterogeneous assets are regarded. Also the nature of information asymmetry is slightly different there – while it refers to non-public information about a public company in the former case, it occurs mostly in the relation of the current owner and a potential buyer of the private asset in the latter case. The position of the buyer is regularly weaker in this situation as she buys more or less the proverbial “pig in the poke”. Nevertheless, the asymmetry can be overcome by a professional valuation. Such appraisal services are offered (and demanded) in most private markets (real estate, arts, automobiles etc.); also the due diligence process in private equity transactions falls under this category. Although expensive, they allow reducing or even eliminating the differences in the information levels between buyers and sellers in a way that is practically unavailable in public markets. Thus, information asymmetry in private markets can be quantified as appraisal costs making it, in consequence, similar to direct transaction costs. The liquidity effect of information asymmetry is based on the fact that some investors know the correct value of the asset more precisely than the others, what makes the others fear that the informational edge could be used to their disadvantage. However, in the case of diversity of expectations, use possibilities, or tastes no one really knows the objective true value. The differences in valuations arise either from errors – though it is not possible to tell ex ante who is wrong and who is right – or from differences in individual utilities. These sources of investors’ heterogeneity are exogenous and can92
The role of investor’s reputation for the level of indirect trading costs has been highlighted by Keim/Madhavan (1998). Due to the close relation between the indirect transaction costs and liquidity – see section 1.2.2.2 for a discussion – the importance of reputation for the individually experienced level of liquidity can be derived per analogy.
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not be easily removed by providing better access to information. They result in a distribution of valuations among investors, which can be supposed to concentrate around the (however defined) fair value; however, a unique and objective “true value" does not exist. In this sense, heterogeneity of valuations exists in both organized and nonorganized markets, although it is obscured by the existence of central price setting systems in the former ones. In order to clarify how the heterogeneity of investors’ valuations affects liquidity, it is useful to consider its effects on the buyers’ and sellers’ side of the market separately. It seems rational to assume that the valuations’ distribution of potential buyers lies more to the left than the respective distribution of the sellers’ valuations.93 This assumption is intuitive – since potential seller are the actual holders of the assets, they should on average assign them higher values or they wouldn’t hold them otherwise. Such presentation of the market situation allows several convenient interpretations. On the one hand, the distance between the distributions, measured, for example, as the difference between the means or medians, corresponds with the “distance” between the supply and the demand side of the market and can be interpreted as market breadth. On the other hand, the intersection of the two distributions determines the room in which trade agreements are possible. It is denoted as the price building room and corresponds with the shaded areas in Figure 1-14. A larger area indicates a larger number of possible buyer-seller pairs and, thus, a larger number of possible transactions. In this sense, it corresponds with market depth. Hence, larger price building room in the intersection of valuations’ distributions indicates higher expected liquidity of the market and better marketability of the respective asset.94 The price building room indicates only the potentially possible transactions. The number of transactions that indeed occur depends, however, largely on the way trade is organized. On non-organized (direct or brokered) markets in which buyers and sellers
93
94
See Geltner (1997), p. 424, or Fisher et al. (2003), p. 273. Note, however, that with the possibility of short sales every individual can be considered as a potential buyer and seller at the same time, so that both sides of the market stem from one and the same distribution of valuations among all market participants in this case. These and the following considerations are based on Fisher et al. (2003), pp. 275 f., especially FN 8. Note, however, that Fisher et al. refer to the distribution of reservation prices rather than valuations. This implies strategic behavior of market participants searching for trading partners and seems, in fact, more realistic. For better tractability this issue is omitted here and returned to in more detail in the next Chapter in section 2.4.
1.2 Sources of Liquidity
69
seek each other and agree on prices individually, it is theoretically possible that all buyers and sellers in the price building room transact if they match themselves perfectly; however, it is also possible that with “bad luck” only few of them find a suitable partner. The actual result depends largely on the applied means of search, such as a specialized communication system or the assistance of a broker, and will most probably vary from one period to another. Similarly, the scope of realized transaction prices is varying in non-organized markets, so that individual traders cannot be certain about the eventually paid or received prices. Both of these sources of uncertainty – about the ability to transact and about the final execution price – are related to liquidity risk (see section 1.1.2). On the other hand, organized trading systems determine a single execution price, which is either different for buyers and sellers (dealer markets) or identical for both sides of the market (auction markets). This means that only a part of the potentially possible transactions is actually executed – only the buyers “left” to the single price and the sellers “right” to the single price are comforted. There is, however, less uncertainty about the number of transactions and about the prices. Comparing the two main types of market organization leads to the conclusion that organized markets trade the manifold possibilities of transacting arising from investors’ heterogeneity against the certainty of trading. In other words, auction markets offer lower liquidity risk than direct markets, but they also reduce the potential for trading agreements.95 Association of liquidity with the area of the price building room allows a relatively straightforward analysis of the effects of changes in the traders’ valuation heterogeneity on the liquidity of privately traded assets. Consider the case in which the diversity of opinions or tastes suddenly decreases, but all other market parameters remain constant. In the sense of valuations’ distributions this would mean a narrowing of the respective sellers and buyers distribution curves – this situation is depicted in Figure 114 in charts a) and b). One of the effects is the reduction of the intersection area indicating a smaller number of potentially possible transactions. With less dispersion in investors’ valuations, there are fewer buyers ready to pay higher prices and fewer sel-
95
This conclusion has been confirmed in a simulation study by Morawski/Schnelle (2006). The authors set up a finite number of investors with heterogeneous beliefs about the true value of an asset, who are making random decisions to trade. They examine two different trading systems: a direct market, in which traders are allowed to review one random trading partner per period, and an auction market, in which a single price is set on the basis of (limit) buy and sell orders. The results of multiple simulations indicated higher trading volumes as well as higher variability of transaction prices on the direct market.
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lers ready to accept lower prices, so that it becomes more difficult for two individuals who could theoretically agree on a common transaction price to actually meet each other. In effect, the expected ease of liquidation and the marketability of the assets traded on this market decrease. However, also the dispersion of transaction prices is lower. It is therefore possible that those investors who would be able to trade in the new situation would be facing lower uncertainty about the outcome of the liquidation, i.e., lower liquidity risk. Thus, the overall liquidity effect is unclear.
1.2 Sources of Liquidity
71
Frequency
a)
Buyers
Sellers
Price building room
Valuation
Frequency
b) Buyers
Sellers
Price building room
Valuation
Frequency
c) Buyers
Sellers
Price building room
Valuation
Figure 1-14: Diversity of valuations and the price building room
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Chapter 1: The Concept of Liquidity
The above considerations were based on the assumption that market parameters other than valuation diversity remain unchanged. Yet, in the real world, sellers and buyers cannot be seen as fully separate groups. In fact, it is probable that lower dispersion of valuations would not only affect the shapes of the buyers’ and sellers’ distributions but also bring them nearer to each other as demonstrated in the charts b) and c) in Figure 1-14. This, in turn, would result in lower market breadth, larger price building room and, consequently, in a larger number of possible transactions. In total, it is possible that despite the decrease in the heterogeneity of market participants’ valuations the size of the price building area remains unchanged as in the charts a) and c). Still, the levels of liquidity in these two cases (i.e., high diversity of valuations combined with large market breadth versus low diversity of valuations combined with small market breadth) are not identical. Although the number of possible transactions is the same, the dispersion of possible prices is smaller in the second case. This means that an individual trader is less uncertain about the final transaction price achieved in the sale of her asset, i.e., she is exposed to lower liquidity risk. Thus, despite the same level of asset’s perceived marketability, its liquidity is higher in the case depicted in the chart c) than in the case depicted in the chart a).96 The final conclusion from the above considerations is the non-trivial character of the relation between the diversity of valuations and liquidity. It can depend on a number of factors, such as the form of market organization or market breadth. Moreover, the ambiguousness of the liquidity effect can be even larger when possible changes in the shapes of the distributions, which may be the result of, e.g., some sellers changing their minds and becoming buyers or vice versa, are considered. Still, it is one of the crucial issues in modeling liquidity and will be addressed on several occasions in the later chapters.
96
An interesting side effect of presenting different market states in form of distributions of buyers’ and sellers’ valuations is the demonstration of the relation between market breadth and market depth discussed in section 1.1.1.2. It follows from the analysis of Figure 1-14 that lower breadth (i.e., a smaller distance between the buyers’ and the sellers’ distribution) results in larger depth denoted as the intersection area of the distributions. However, the same does not necessarily hold in the opposite direction, i.e., larger depth is not necessarily a result of lower breadth but may also be caused by larger diversity of valuations among market participants.
1.3 Review of Illiquid Assets
1.3.
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Review of Illiquid Assets
Having formulated the definition of liquidity and analyzed its main sources, it is now necessary to clarify which assets or asset classes are affected by this problem. While there seems to be common agreement about which investments are to be considered as illiquid, it is based mainly on intuition or practical experience. Concrete features of these markets that lead to their low grade of liquidity are relatively rarely a subject of comprehensive discussions. Thus, one goal of this section is to provide a theoretical rationale for the existing intuitive classification. An even more important reason for reviewing illiquid assets is the preparation of the grounds for the application of the search theoretical approach, which constitutes the core of this work. Although the idea behind this model, presented in Chapter 2, is very general, its specification, which is necessary for a practical application, must be based on properties of concrete assets. Therefore, it is necessary to explicitly identify their key features. While it is difficult to draw a line delimiting liquid assets form illiquid ones, it is relatively easy to name those that are clearly highly illiquid. This section focuses on the latter ones only. This means that security markets are omitted at this point. Of course, to investors who trade mainly on stock exchanges some stocks (e.g., “small caps”) are clearly less liquid than other stocks (e.g., “blue chips”). Also certain exchanges can be considered less liquid than others. Nevertheless, in each of these cases, low liquidity is still incomparably higher than the liquidity of most direct investments. Since the focus of the analysis in this book is on higher levels of illiquidity, the exclusion of security markets seems justified. Before reviewing concrete illiquid assets, general characteristics of markets, which at least potentially can be suspected of high illiquidity, are discussed. They are derived on the basis of considerations from the former section. The presentation of the most prominent examples of illiquid assets follows. In the first place, real estate investments are considered. This doubtlessly most significant illiquid asset remains in the main focus of the analysis. The second broad category is composed of privately traded company shares, i.e., private equity. Finally, a number of alternative real investments are considered, among them arts, collectibles, and wine; they receive increased attention as financial investments in the recent years.
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1.3.1.
Characteristics of Illiquid Assets
The previous section provided a catalogue of sources that can lead to the lack or at least a reduction of liquidity. Among the most important ones were direct transaction costs, timing costs, market organization, and valuations’ diversity.97 Additionally, all other effects should be allowed for as soon as they significantly affect times and prices at which transactions are accomplished. On the basis of these sources of liquidity, it should be possible to draw an outline of the characteristics of an illiquid asset. The first straightforward characteristic refers to the level of transaction costs. Clearly, this element is negatively related to liquidity mainly because it negatively affects the benefits from frequent trading and discourages investors from providing market depth. Thus, investments that require the payment of high commissions or are subject to overaverage taxation can be expected to be less liquid. Note that this criterion applies to assets (asset classes) as well as to markets on which different assets are traded. In particular, it can refer to national markets meaning that countries with higher tax rates should, ceteris paribus, be less “liquid” than low-tax countries. Timing costs refer mainly to the possibility of a change in the market situation during the time between the decision to trade and the execution of the transaction. Hence, assets with high price volatility should also tend to be less liquid. However, not the absolute but much more the relative volatility compared to the frequency of trading possibilities is to be considered at this point. Clearly, if an asset X can be traded only once a day and an asset Y can be bought or sold every minute, given the same level of daily volatility, the first one is associated with much higher timing costs. Thus, only the volatility between subsequent trading occasions is relevant; in the above example it is higher for the asset X. This explains why some highly volatile assets can still be considered highly liquid – the volatility measured between subsequent transactions, which can occur even many times a second, is relatively low. To sum up, illiquid assets are to be sought among those having high price volatility on the one hand and being traded only infrequently on the other hand.
97
Note that the indirect transaction costs, such as the bid-ask spread or the market impact, are not regarded here, although, as discussed in section 1.2.2.2, there will be a strong correlation between these costs and liquidity. Nevertheless, they have been identified as results rather than causes of illiquidity and as such do not make an asset or a market illiquid.
1.3 Review of Illiquid Assets
75
A further characteristic of potentially illiquid assets or markets can be derived from the form of market organization. Generally, liquidity should be higher when it is easier for market participants to trade. This implies the superiority of those forms of trading under which market entry barriers are lowest. The review of different forms of market organization in section 1.2.3.1 led to the conclusion that organized public markets should be generally more efficient in bringing buyers and sellers together and, thus, more liquid. On the other pole are direct markets with each investor searching for a trading partner on her own – the overall difficulty in finding a trading partner should be higher there indicating lower liquidity. Hence, one would expect privately traded assets to be less liquid. However, one should not overlook the fact that a direct market can offer the possibility of achieving better prices than an organized exchange with a single market price. The superiority of organized markets with respect to their liquidity is therefore only conditional – they allow achieving the maximum achievable price quickly and with low uncertainty (liquidity risk), but they do it at the cost of giving up the chance of achieving above-average prices. Nevertheless, in practice, the majority of markets regarded by investors as illiquid are private. When considering the form of market organization as a criterion for the identification of illiquid markets, one should bear in mind that it is not a purely endogenous feature and that it arises from the characteristics of the asset in question. Most important seem the comparability of items within the asset class and the ease of ownership transfer. On the one hand, highly homogenous assets are easy to standardize, what makes the organization of wide and anonymous trading systems possible. On the other hand, heterogeneous and hardly comparable assets enforce non-organized trading systems as there is no easy way of a quick and effective assessment of their true values. Same logic, though with lower severity, applies with respect to legal regulations – the more difficult an ownership change is, the less probable is the development of an organized trading system. Thus, heterogeneous assets with highly regulated transaction procedures are natural candidates for illiquid investments. Finally, the divergence of asset’s valuations among investors has proven to be relevant for liquidity, but the direction of the effect was not clear. Given constant market breadth, more heterogeneity among buyers and sellers increases trading possibilities, but given a constant price building room, i.e., a constant number of potentially possible trades, lower divergence of valuations improves liquidity. As already noted in sec-
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tion 1.2.4, the second scenario seems to be more probable in reality. Since buyers and sellers are not really two distinctly separate groups, it seems more likely that a change in the overall investor heterogeneity would affect both sides of the market simultaneously moving the valuations’ distributions closer together (see Figure 1-14). In this case, markets with less heterogeneous investors should also tend to be more liquid.98 The indicators formulated above should provide the first hints of possible illiquidity of assets or markets. However, they are by no means definite evidence of this quality. Therefore, only those asset types are discussed in the following subsections with respect to which illiquidity indications are especially strong. These are: direct property investments, private equity, and a broad class of alternative investments. They all seem to fit perfectly in the scheme presented here. 1.3.2.
Real Estate
Although it seems intuitively clear what real estate is, there are at least several approaches to defining it. Probably the most famous definition is the economic one provided by Graaskamp (1972, p. 513) and characterizing real estate as “a manufactured product of artificially differentiated cubage with an institutional time dimension designed to interface society with the natural recourse land”. This short sentence recapitulates the main dimensions of this investment: room and time. The use of room over time generates utility, which in the investment perspective can be measured in money units.99 Thus, a real estate market can be characterized as a market for the usage of delimited room, while usage can also include rental. On the other hand, the legal definition treats real estate as a bundle of rights encompassing the right to trespass, to use, to rent, or to change (develop) it.100 These rights may be limited (e.g., by land use restrictions) and also partially divided and distributed among individuals. These two approaches are mainly relevant from the investment perspective. Note that the “technical” side (e.g., architecture or technical facilities) is only of secondary relevance in this case. 98
99
100
Note at this point that although asset heterogeneity and investor heterogeneity are two different terms, some relation between them is probable. Weaker comparability increases the chance that investors disagree on the true characteristics of the asset giving another reason for different valuations. Pyhrr et al. (1989), p. 4, describes real estate in short as “money flow over time”, which is, in fact, identical with the definition of any cash flow generating financial investment. See Ling/Archer (2005), p. 5.
1.3 Review of Illiquid Assets
77
The economic importance of real estate as a capital investment is manifested by its high share in the total assets of national economies. Even though such estimations are extremely difficult, and the results obtained from different sources vary strongly, their message is clear. For example, Federal Reserve’s assessments of the share of real estate in the household wealth in the USA are between 20% and 45%;101 the analogical assessment for Germany is about 50%102. On the other hand, the preliminary result of the Luxembourg Wealth Study – a comparative analysis of household wealth – provides estimates of real estate in household portfolios (net of house secured debt) in several countries at between ca. 40% and 70%.103 Ibbotson/Siegel (1983) assess the portion of real estate in a “world wealth portfolio” at over 50%; also in the “world investable wealth”, real estate has the largest portion. Data on real estate holdings by institutional investors is even less reliable. Estimates of the optimal share of this asset in a mixed-asset portfolio by various researchers range between ca. 10% and 20%.104 These figures, despite the lack of precision, demonstrate that real estate is doubtlessly the most important illiquid asset, if not the most important asset at all. There is common agreement that even the most liquid (direct) property investments are still far less liquid than any other traditional financial asset. Exceptions are few and occur practically only in extraordinary market situations. The analysis of the features of various property markets confirms these observations. To start with, a look at the transaction costs associated with real estate trading reveals that they are, in fact, extraordinarily high. They encompass a long list of positions including appraisal fees, brokerage commissions, legal fees, mortgage-related fees, or transfer taxes. Precise estimates are difficult and vary strongly not only internationally but also among different property types. Several researchers assess the transaction costs associated with homeownership (house sale) in the USA at between 6% and 13%.105 CMS (2005) provides an overview over real estate transaction costs in Europe listing 13 different categories – a selection is presented in Table 1-3. Although precise figures for the total costs are difficult to quote due to the complexity of the regulations and partial negotiability of the fees, they amount to several percent of the value in most cases. In compar101 102 103 104
105
See Ling/Archer, 2005, p. 10, and Ibbotson/Siegel (1983). See various monthly reports of Deutsche Bundesbank. See Brandolini et al. (2006). See Fogler (1984), Firstenberg et al. (1988), Giliberto (1992), or Kallberg et al. (1996). Webb et al. (1988) consider an even higher real estate share in investment portfolios (up to 60%) as eligible. See references cited in Haurin/Gill (2002), pp. 564 f.
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ison, direct trading costs of publicly traded equities are nearly negligible; Domowitz et al. (2001) assessed the average explicit trading cost (commissions and fees) for institutional investors in the USA at about 0.2% and worldwide at about 0.4%.106 Table 1-3: Elements of real estate transaction costs in selected European countries107 Survey/Valuation Fees
Transfer Duty
Land Registry Registration Fees
otary Fees
France
Approx. €3,800€14,000
5.09% if no VAT applies (VAT applies on first sale within five years of building completion).
Where VAT is due, 0.615% real estate property tax applies
0.825% of value
Germany
Survey: hourly rates starting at €38.5. Valuation fee depends on value/difficulty, e.g., value €1m: €1,800 to €3,000
3.5% of purchase price
Scale fees, e.g., value €1m: €1,557
On contract, e.g. value €1m: €3,500 to €3,900. On conveyance, e.g. value €1m: €800. Signature affirmation: €130
Italy
Approx. 0.2%0.4% of purchase price, depending on nature of property
Depends on the status of the seller and the nature of the property: €168 if no VAT applies, 7% of the value otherwise
Depends on the status of the seller and the nature of the property: €336 if no VAT applies, 3% of the value otherwise
Scale fees (approx. 0.15%-2% of the transaction value) according to value and title enquiries required
The etherlands
1%-2% of purchase price
6% of purchase price
Maximum €455
Contract sale, e.g. value €1m: from €1,131 to €3,678.
United Kingdom
1%-2% of purchase price
Up to 4% of total purchase consideration
Maximum £1,120 (Scotland max. £12,000)
Not applicable
Another liquidity related issue concerns the opportunity costs associated with real estate investments, in particular the timing costs. As discussed earlier, they increase with the instability of prices, which is usually measured with price or return volatility. In the common opinion, property prices remain relatively stable over time compared to other investments. The annual volatility of real estate returns reported in the literature ranges from about 2% to about 9%, whereas the volatility of returns of major stock 106
107
However, the costs for individual investors are higher as they include brokerage fees. See Bodie et al. (2005), p. 86. Source: CMS (2005).
1.3 Review of Illiquid Assets
79
indices exceeds 20%.108 This would indicate that the timing component of the transaction costs is relatively small in the case of real estate. However, it has been noted in the prior discussion that not the absolute volatility but rather the volatility between subsequent trading opportunities is relevant in this context. Since the frequency of real estate transactions is very low in comparison to other assets, the effective price variability between time points at which a transaction can be accomplished may still be relatively high. To illustrate this consider a real estate market with 5% annual volatility and a stock market with 20% annual volatility of total returns. On the first glace, the property market seems to be much more stable. However, assuming that transactions occur there only once a week while stocks are traded every minute leads to a different picture. Recalculating the volatilities in terms of the variability between subsequent transactions yields approximately 0.7% for the real estate market (given 52 transactions a year) and only 0.03% for the stock market (given 500,000 transactions a year). This means that the timing cost associated with real estate is much higher than the cost associated with listed stocks. Although the figures assumed in this example are fictional, they are not unrealistic. So, the risk of a substantial unfavorable price change between the moment an investor decides to sell a property and the moment the trade is accomplished (the price agreement is met) is, in fact, far higher than suggested by the overall real estate price variability. Also the form of market organization predestines private real estate as a highly illiquid inestment. It is traded on non-organized markets practically without exception. By far most popular are brokered markets. They are typical for residential, office, and retail properties, although both the number of brokers and the scope of services offered differ strongly. Direct markets often function parallel, since not all individuals are ready to bear high broker provisions; their share in the total turnover is, however, usually smaller. On the other hand, for some less active markets, like special properties, where the trading volume is too small for a profitable brokerage activity, direct search is the only possibility of finding a buyers or a seller. An alternative system, which
108
E.g.: Maurer et al. (2004) report annual volatilities of the German, American, and British real estate at 2.34%, 5.99%, and 8.96% respectively; the annualized volatilities of the American S&P 500, British FTSE 100, and German CDAX for the last 10 years (1996-2006) were 21.5%, 21.5%, and 26.5%, respectively (computed on the basis of data from Thomson Financial Datastream).
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gains increasing popularity, is a property auction.109 Such auctions are organized either as sealed bids systems – the vendor sets a minimum price, invites non-public bids and selects the buyer after all bids have been placed – or as “English” auctions with open ascending bids. This form of sale organization enables the seller to exhaust the buyers’ willingness to pay to a higher extent than it would be possible in the traditional “search and negotiate” system.110 However, it is also possible that the cost of organizing an auction exceed these additional gains. Furthermore, since auctions take place only infrequently, there is no regular trading. In effect, finding a trading partner in a real estate transaction is practically always connected with a costly and time-consuming search. Finally, also the heterogeneity of real estate contributes to its illiquidity resulting in higher information costs (appraisal fees), weaker market transparency, and consequently, higher disagreement about properties’ values. The latter can arise from information asymmetry – sellers can be supposed to possess better information than buyers. However, it may also originate from different possibilities of using a property or from different tastes of investors. While the grade of the “fit” with entrepreneurial goals plays a big role in the case of commercial real estate, the notion of a “beautiful house” is highly relevant on residential markets.111 The resulting diversity of valuations by market participants leads to the impossibility of a precise estimation of a properties’ sale values. This is reflected in relatively high deviations of appraisals from subsequently realized prices – they often exceed 10%.112 As discussed in section 1.2.4, higher valuation diversity among investors does not necessarily lead to lower marketability – larger disagreement may even allow achieving better prices on the individual
109
110
111
112
Real estate auctions are widespread in Scotland or Australia (see Lusht, 1996, p. 518, or Pryce/Gibb, 2006, pp. 380-382), but also gain popularity in the USA, especially as online auctions (see Dymi, 2006). Interestingly, there is only ambiguous empirical evidence that properties sell at higher prices in auctions than in individual negotiations. On the one hand, Lush (1996) and Ashenfelter/Genesove (1992) confirm this theses; on the other hand, Mayer (1998) and Ong (2006) came to the contrary conclusion. The role of different use possibilities for the value of real estate is reflected in the discussion about the appraisal principles, in particular, in the differentiation of the “highest and best use” and the “existing use” value. See, e.g., TEGoVA (2003), Standard 4, RICS (1995), PS 3, or Appraisal Institute (2001), pp. 24 ff. and Chapter 12. Royal Institution of Chartered Surveyors (RICS), a British association of real estate appraisers, reports average absolute differences between valuations and sale prices of commercial properties in the UK ranging from 6.9% to 12.4%. These figures were even higher in the past. See RICS (2005).
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level – but increases the uncertainty about the liquidation value, i.e., liquidity risk. It seems, therefore, that more heterogeneous real estate markets should tend to be less liquid. The above review of the liquidity relevant characteristics of real estate reveals that literally all aspects of this problem, which have been identified on a purely theoretical basis in section 1.2, apply to this asset. This, combined with the enormous role of real estate in the global economy, supports the initial statement that it is by far the most important illiquid asset worldwide. The lack of satisfactory methods to deal with illiquidity is surely one of the key reasons, why real estate still remains at the margin of modern financial theories. Therefore, the main focus of the analysis in this book is placed on this asset class, although, as already mentioned, analogical reasoning can be applied to other similarly illiquid investments. 1.3.3.
Private Equity
Private equity (PE) represents a very wide category of investments. Generally, it can be denoted as “investing in securities through a negotiated process”113 and encompasses shares of non-public companies. Delimitation from a “normal” ownership of a company is difficult. Among the key features of private equity are: focus on companies in critical phases of development, high investment risk with no sufficient securities, and provision of managerial assistance by the investor.114 A number a various types of transactions fall under this heterogeneous category. Depending on the stage in the corporate development, one differentiates between venture capital (VC), i.e., highrisk investments in young companies with high growth potential,115 buy-outs, i.e., acquisitions of large portions of companies entailing control takeover,116 and special case financing, encompassing a broad spectrum of situations ranging from the repayment of distressed debt to one-time opportunities.117 Most frequent is, however, the classification of different types of PE investments based on financing phases, as presented in Table 1-4. This broad definition is followed here without going into its detailed com113 114 115
116 117
Bance (2004), p. 2. See Bader (1996), pp. 10 ff., Rudolph/Fischer (2000), pp. 49-50, or Jesch (2004), pp. 21-23. Note that the term “Venture Capital” is sometimes used as synonymous to “Private Equity”, especially in Europe. See Röper (2004), p. 23. The frequent case of management buyouts (MBOs) also belongs to this category. See Bance (2004), pp. 2-3, Bader (1996), pp. 7-9, or Jesch (2004), Chapter 6.
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ponents. Also purchases of shares in PE or VC funds are often considered to be PE investments. From the perspective of this book they are classified as such only as far as they are not publicly traded. Table 1-4: Stages of private equity investments118 Private Equity (= Venture Capital in the “European” sense) Early Stage Seed
Expansion Stage Start-up
Expansion
Late Stage Bridge
Buy-out
The role of PE as both a financing source and an investment opportunity is increasing not only on the national level but also internationally. However, while it experienced its main growth phase in the late 1990 together with the IT industry boom, the sums invested in this form decreased after 2000 as the crises of this branch came.119 Nonetheless, despite the significant drop, the scale of private equity is still much larger in the USA and UK than in other countries. Even there, however, it is far behind the share of wealth invested in real estate or in public stocks, staying below 1% of the national GDPs.120 Still, the effective capital invested in non-public companies is probably significantly higher since official figures on private equity provided by various associations (e.g., NVCA or EVCA121) are usually based on institutional investments, mainly those by private equity funds. Though non-public equity investments are definitely considered as highly illiquid,122 it is more difficult to specify the precise sources of their illiquidity than in the case of other illiquid assets. The direct transaction costs are, in fact, not significantly different than in other cases of corporate ownership. They vary from one country to another but are seldom high. In fact, a relatively inexpensive registration of the ownership change with the appropriate authority is often all that is formally necessary. The main component of the transaction costs is less apparent and is accrued during the investment selection phase. The lack of sufficient information about the target company makes a 118
119 120 121
122
Based on Schefczyk (2000), p. 24. See Bader (1996), pp. 103 ff., Jesch (2004), pp. 79 ff., or Grunert (2006), pp. 6 ff., for detailed discussions of the stages. See Haemming (2003), p. 69. See Baygan/Freudenberg (2000), p. 19. NVCA: National Venture Capital Association (http://www.nvca.org); EVCA: European Venture Capital Association (http://www.evca.com). See Bader (1996), pp. 75-77, 92-95, and 203, as well as the literature references cited there.
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costly and time consuming “due diligence” process necessary.123 During it, internal enterprise data is evaluated in order to obtain sufficient information for the assessment of the company’s value. In this sense, it is similar to the role of appraisals in real estate transactions and can be considered as a mean of overcoming information asymmetry. This type of costs can be immense and encompasses not only the expenditures entailed directly during the due diligence but also timing costs, i.e., the costs associated with the risk that the market or the corporate situation may change during the process. Due to the heterogeneity of private equity seen as an asset class as well as the confidential character of transactions, it is hardly possible to quantify the effective transaction costs, but one can expect them to amount even up to several percents of the total investment value. Also other characteristics of private equity reinforce its reputation as an illiquid asset. Trading takes place practically only in form of direct search markets. The players are mainly specialized PE or VC funds, but also wealthy individuals (“business angels”), industrial corporations, and public-sector institutions are active in this market.124 Due to the complex nature of this form of corporate financing, there are no clear market structures as in the case of other assets. The initial investment (purchase of shares) is usually made in a very early stage, partially even before the actual foundation of the company. The duration of the commitment is typically middle- to long-term; however, it varies strongly depending on the goals of the investor and the characteristics of the corporation.125 Returns are achieved mainly through a successful exit (sale), which is conducted after the growth potential has been skimmed. There are several possible exit alternatives.126 An initial public offering (IPO) is considered most attractive, but is not as dominating as sometimes considered.127 Other options encompass: sale to another PE investor (fund), sale to a corporate investor, sale to the management, resale to the original owned (initiator), or, if the company proves unsuccessful, discontinuation and bankruptcy. The heterogeneity of players and assets results in the lack of market structures as they are known for other assets. Transactions often occur within a relatively limited group of participants and are structured very individually. Thus, an investor 123 124 125 126 127
On due diligence see Berens (2005) or Bing (1996). See Weitnauer (2001), pp. 8-10. See Bader (1996), p. 14, and Weitnauer (2001), p. 7. See Schefczyk (2000), pp. 29-31, Jesh (2004), pp. 97-109, or Vance (2005), pp. 152-154. According to Vance (2005, p. 154), only 6.3% of the VC exists in the U.S. in 2002 were through an IPO. Higher figures (up to over 40%) were only observed during the “dot com” bubble in 2000.
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willing to include private equity of a very specific type in her portfolio may have extreme difficulties to find an adequate opportunity. Similarly, it may be even more difficult to sell shares of a private company, especially one that offers no extraordinary value growth perspectives. The heterogeneity of companies, which shares constitute the asset class “private equity”, results in the heterogeneity of opinions about the true values of these companies. It may arise from very different sources. For example, in the case of a management buyout (MBO), the management can be expected to have better information about the actual state of the business, so there is information asymmetry between the board and other potential buyers. There may also be differences with respect to use possibilities – a corporate buyer can possibly achieve synergy effects that are not available for other investors. Finally, there is also uncertainty about the economic variables determining the value of the enterprise. Different assessments of the future profitability of the business lead to different valuations. Hence, also this aspect increases the illiquidity of PE investments. To sum up, private equity is definitely one of the most illiquid asset categories, but it is also a very heterogeneous one, so that high differences in the grades of liquidity can be expected. A particularly disturbing consequence of this fact is the extreme difficulty to assess the typical values of liquidity related parameters, which are necessary for the specification of the model formulated in the next Chapter. Application of the liquidity measurement and management techniques developed in this book seems therefore hardly realizable. For this reason, private equity is not regarded in further discussions. 1.3.4.
Alternative Investments
This section encompasses a number of goods, which, although originally regarded as consumption goods only, have evolved as financial investments as well.128 “Assets” in this category encompass a variety of different items, among them various works of art, wine, jewelry, antiques, collectibles (e.g., stamps or coins), and even personal belonging of famous individuals. While many of these items play only a marginal role as financial assets, some of them have gained high popularity in the recent years. The fine arts have been considered an interesting investment for a relatively long time already. 128
Note, that a wider meaning of the term „alternative investments“ is often used, which includes also private equity, hedge funds, commodities, and other non-standard financial assets.
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Cases of paintings reaching astronomical prices in auctions are reported regularly in the press,129 but more thorough studies based on appropriate indices do not confirm that art would offer particularly attractive risk-adjusted returns.130 Also wine is often discussed in terms of financial investments.131 Several researchers stated that adding this “liquid” asset to the traditional security portfolio may allow reaching higher efficiency frontiers, i.e., achieving higher returns at lower risks.132 The sizes of the markets for various types of alternative investments are extremely difficult to assess. Publicly available data is limited practically only to figures published by the leading agents, so it surely underestimates the true volumes. There is also a practical difficulty of delimiting investments from consumption motivated purchases. Nevertheless, the share of global wealth invested this way is most probably diminishingly small; it might be somewhat higher only with respect to wealthy individuals, the so called HNWIs (High Net Wealth Individuals), who are the main players on these markets.133 The level of liquidity of alternative assets is determined mainly by the form of trading. Probably the most prominent one is an auction. Famous houses, such as Sotheby’s or Christies, organize auctions on a variety of collectibles including all kinds of arts, jewelry, motor cars, or even teddy bears.134 This form of trading allows a relatively effortless liquidation of such assets, usually at a good price and with little uncertainty – values predicted by the auctioneers deviate rarely strongly from the actually achieved ones.135 However, auctions have also a number of drawbacks. Firstly, they are available practically only to sellers. Buyers have no guarantee that they will be able to purchase the chosen item at a reasonable price and may suffer from the “winner’s curse” if they attempt to precipitate the purchase – a winner willing to buy quickly may often 129 130
131 132 133
134 135
See http://www.artcult.com for examples of prices reached by various collectibles. Although investments in arts (paintings) brought in the last century significantly higher returns compared to other financial assets, also their return volatility was much higher; see Anderson (1974), Stein (1977), Bryan (1985), Baumol (1986), or Goetzmann (1993). Interestingly, most researchers state high correlations of paintings’ returns with returns on other financial assets, especially stocks. See Wilkinson (2004), Laschinger (2004), or May (2006). See Kumar (2004) and Engelskirchen (2006). According to CapGemini/Merrill Lynch (2006) about 20% of the HNWIs’ wealth was invested in alternative assets in 2005, and the portion is still rising. This figure includes, however, structured products, hedge funds, managed funds, foreign currency, commodities, and private equity. See http://www.christies.com. See Ashenfelter (1989) for the study of auctioneers’ price prediction quality (pp. 33-35) as well as certain aspects of wine auctions’ functioning.
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find himself having paid too much.136 Another problem with auctions lies in the extremely high transaction costs. At Christies, for example, buyers’ premium (i.e., the amount that is due in addition to the “hammer price”) is 20%; charges on the seller’s side depend on the client’s activity during the year, but usually amount to several percents and include insurance and shipping.137 Hence, the total cost may be as high as one third of the actual value. Smaller houses may offer better conditions than the big ones, but the overall cost is still high. Finally, an auction is not necessarily a quick way of selling. At Christies, it takes about two to three months to schedule an auction and approximately 35 days until the payment is made thereafter. Furthermore, auctions for certain items are conducted only infrequently. Even wine, which is comparatively popular, is offered at only about 300 auctions per year worldwide.138 Thus, depending on the type of the alternative asset, it may take several months or longer to liquidate it. An alternative way to buy or sell an alternative investment is by addressing a specialized dealer or by performing ones’ own search for a trading partner. The former solution may be the quickest one. In fact, there are numerous individuals and institutions specialized in trading various items – the number of art dealers is assessed at about 9,500 compared to only about 750 auction houses.139 One can expect that a sale or a purchase can be completed nearly immediately this way; however, since dealers often buy and sell at auctions themselves, their selling prices will usually be higher and purchasing prices lower than the auction prices.140 On the other hand, searching for a buyer or seller on one’s own account may yield more favorable prices but induce high search costs. Apart from the necessity to advertise in specialized press, the mere contact with potential trading partners scattered throughout the world may prove costly. The thinness of the markets for particular assets may eventually result in extremely long marketing durations.141 The fact that a lot of trading is within relatively small 136
137 138 139 140
141
The issue of the „winner’s curse” is, in fact, a much more general phenomenon referring to different fields of economics. Its consequences and the conditions under which it may occur have been intensively discussed in the literature. For reviews of this subject and for further references see Thaler (1992), Chapter 5, Varian (2003), Chapter 17, or Kagel/Levin (2002). See http://www.christies.com and Burton/Jacobsen (2001), pp. 348-349. See Burton/Jacobson (2001), p. 339. See Prickett (2004), p. 25. Anderson (1974), p. 13, estimates the mark-up of the dealers’ asking or bidding prices at as much as about 20% to 50% above or below the auction prices. For example, an average wine auction is attended by about 100 to 150 buyers bidding on bottles offered by about 30 to 40 sellers. Given the small number of auctions (about 300 per year world-
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groups of specialized collectors may also make it very difficult to enter the market as an outsider. As discussed above, none of the trading forms available for alternative assets contributes to improving their liquidity. On the one hand, they are associated either with long marketing durations or with significant discounts (premiums) on the sale (purchasing) prices. On the other hand, high transaction costs need to be accrued forcing potential investors to long holding periods and diminishing the trading intensity on the affected markets. The catalogue of liquidity reducing factors is even longer. In addition to the already discussed direct trading costs, expertise expenses need to be borne; they include, e.g., examining the genuineness of a painting or opening some of the wine bottles for tasting. Long liquidation periods may induce high opportunity costs; timing costs, however, seem to be rather insignificant in this case. Finally, the extremely high diversity of valuations also contributes to low liquidity, mainly in the sense of higher liquidity risk. The uncertainty about the outcome of an auction is immanent to this form of trading and is also significant in the individual search. It can be reduced by resorting to a dealer but only at the cost of an inferior price. Moreover, the diversity of valuations has also a very specific character. Unlike in the case of real estate or private equity, it is mainly the divergence of tastes that leads to different valuations of collectibles among market participants. As such, it cannot be effectively reduced by better information of more transparent market organization. Furthermore, as noted by Baumol (1986), the prices of works of art and similar items are “unnatural” in the sense that they cannot be derived from any objective notion of value based on the means of use or future cash flows. In effect, they may have no natural equilibrium levels. The “aimless floating” of such prices, resulting from randomly changing tastes and trends, leads to the instability of the characteristics of these markets. Similarly as in the case of private equity, the heterogeneity of alternative investments makes it difficult to estimate market parameters necessary for the practical application of formal models. The problems are amplified by the extreme thinness and the fundamental lack of transparency of these markets as well as distortions in the price building caused by varying tastes (fashions) or consumption oriented trading. For these reasons, wide), these figures give the impression how thin the global wine market actually is (see Burton/Jacobson, 2001, p. 339). The markets for other, less popular items are even thinner.
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the application of quantitative methods, like those based on the search theoretical approach discussed in the subsequent chapters, though theoretically possible, is extremely difficult. Hence, also this group of illiquid assets is not further followed here.
1.4.
Economic Relevance of Liquidity
The question addressed in this final section of the first, introductory Chapter could very well have been asked at the very beginning: What role does liquidity play in the economy and why should investors give it special attention? So far, the importance of the issue was treated as given. The hitherto considerations were based on the assumption that liquidity is relevant for investment decisions and that more liquidity is better than less. These statements are, however, not self-evident and require a more thorough examination. Although the scope of this section is not sufficient to provide an extensive answer, it attempts to outline the key aspects. The discussion starts at the very origin of liquidity, at the role of money. This question has been addressed by countless researchers for more than a century. The most widespread approach can, however, be traced back to Keynes and his notion of liquidity preference. Basing on it, the significance of liquidity for a single individual in her everyday life as well as for an investor in her capital allocation decisions can be derived. Since the focus of this work is mainly on investment analysis, liquidity in the context of investment goals receives special attention. 1.4.1.
Money and Liquidity Preference
Since money is the only perfectly liquid asset, any discussion about the role of liquidity must start with the question about the role of money in the economy. Its definition has already been provided in section 1.1.1.1 showing that the term it not as unambiguous as it may seem. Based on the broad understanding of money, its three functions are usually identified in the literature depending on whether it is treated as a mean of exchange, store of value, or a unit of account.142 In the first case, it is thought of as a vehicle through which exchange of goods and services becomes possible without the necessity of perfect equivalence of their values in each transaction. In comparison with a pure barter economy, using money increases trading efficiency by reducing the effort and time that would otherwise be necessary in order to accomplish the exchange of 142
See Borchert (1998), pp. 21 ff., Burda/Wyplosz (2005), pp. 174 ff., Howells/Bain (2005), pp.228 ff., Mishkin (2006), pp. 45 ff., or Mankiw (2007), pp. 77 ff.
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goods.143 Treating money as an asset implies that it can be used to store value. Since its nominal value remains constant over time, it is unaffected by changes of economic variables such as risk premiums or interest rates. However, in how far money can be used to transfer purchasing power into future periods depends on the stability of prices. Inflation reduces its value and deflation increases it. Finally, money is used as a unit of account to measure values of commodities and services. Comparability of different goods in the economy can be improved this way leading to lower information costs and easier organization of trade. The first of these functions, i.e., the mean of exchange, is usually seen as the primary one distinguishing money from other financial instruments; the other two are considered to be secondary.144 Indeed, application of alternative units for value measurement is conceivable, and the use of money for storing wealth, especially over longer periods of time, plays only a marginal role in the economy. Thus, the main reason why people wish to hold money is for the sake of using it for payments. Going deeper into the matter, one can ask about the motives that drive individuals to hold certain amounts of money. This issue is discussed thoroughly by Keynes in his probably most famous work “The General Theory of Employment, Interest and Money”.145 He states that the only reason for the willingness to hold money instead of other return generating assets is the fear of the uncertain future. In their struggle for the accumulation of wealth, individuals are aware that the consequences of their acts are uncertain. Keeping a part of the personal wealth in form of money allows them to react to unexpected developments more flexibly and, thus, to face the future with more confidence. In effect, “our desire to hold money as a store of wealth is a barometer of the degree of our distrust of our own calculations and conventions concerning the future.”146 Preference towards liquidity is, thus, the result of uncertainty.147 While trying 143
144
145
146 147
See Brunner/Meltzer (1971), Alchian (1977), and Clower (1977) for in-deep discussions of advantages of using money as a mean of exchange. This conclusion, expressed in many textbooks (see the references in FN 142), was backed with a formal analysis by Marschak (1950). However, Sawyer (2003) points out that a different function may prove to be central depending on the assumed definition of money. See Keynes (1936), in particular Chapters 13 and 15. For extensive comments see Patinkin (1976) or Maclachlan (1993). Keynes (1937), p. 216. This conclusion has been stated and restated by numerous later authors; see Jones/Ostroy (1984), Miller (1986), or Rochon (2003) and the literature cited there. Brunner/Meltzer (1971) discuss additionally the advantages of using money under incomplete information.
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to cope with it, individuals follow various motives, which can be classified into three groups: the transaction motive, the precautionary motive, and the speculative motive.148 The first one corresponds with the general need for money to conduct day-today purchases of goods and services. Depending on whether it refers to a private individual needing to bridge the interval between the receipt of income and its expenditure or to a company needing to bridge the interval between the moment costs are incurred and the moment products are sold, it can be further classified as the income motive or the business motive, correspondingly. The precautionary motive arises from the desire to be prepared for unexpected future expenditures caused by unforeseen events. Finally, the speculative motive is based on the fear to miss profitable investment opportunities that may arise in the process of time. While the transaction motive is of purely practical nature, both latter ones arise from the need to counteract the effects of uncertainty. Which of them prevails with respect to a certain market is partially dependent on the ease of entering and exiting it; in particular, “…in the absence of an organized market, liquidity-preference due to the precautionary-motive would be greatly increased; whereas the existence of an organized market gives an opportunity for wide fluctuations in liquidity-preference due to the speculative-motive.”149 The three motives for holding money translate directly into a non-negative liquidity preference and, consequently, into the preference for assets that are “closer” to money. Thus, storing wealth in goods that can be sold quickly and without discount should be preferred to storing wealth in forms that are less easily convertible into money. On the other hand, surrendering the state of perfect liquidity must be rewarded. Keynes sees this reward in the rate of interest, which is also “a measure of the unwillingness of those who possess money to part with their liquid control over it.”150 The higher the reward, the more prone are individuals to give up liquidity; thus, liquidity preference is a function of the interest rate. However, it must be noted that only the speculative motive is sensitive to the interest rate. Since the rationale behind the precautionary motive does not refer to financial markets but to the general future personal situation of the individual, there is no reason why the amounts held for this reason should fluctuate with the rates of return on other assets. In contrast, depending on the current situation, individuals may decide to take advantage of the current opportunities or to wait for the 148 149 150
See Keynes (1936), pp. 170 ff. Keynes (1936), pp. 170-171. Keynes (1936), p. 67.
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market situation to improve. Hence, changes in the economic prospects may lead to fluctuations in liquidity preference due to the speculative motive. The main conclusion from the above considerations, which give only a very brief outline of the Keynesian theory, is the importance that individuals attribute to the possibility of flexible disposal of money understood mainly as a mean of exchange. The preference of the state of liquidity against the state of illiquidity results from the uncertainty about the future and the wish to secure oneself against unforeseen unfavorable developments, which can lead either to unplanned (or only uncertain) expenditures or to new profitable investment opportunities. The identification of the precise “psychological and business incentives to liquidity”151 helps to understand the mechanisms and the consequences of this phenomenon. Note that, although not explicitly stated in the “General Theory”, the considerations regarding liquidity preference are based on the assumption of “certainty preference”, i.e., the assumption that individuals prefer certain states to uncertain ones. A similar preference for certainty arises also from the expected utility model and is the reason for risk aversion.152 According to it, an individual prefers (i.e., derives higher utility of) a moderate but certain pay-off to an uncertain one having the same expected value, even though a chance of a higher gain exists in the latter case. Thus, it seems that the very basic reason why individuals should prefer, ceteris paribus, liquid to illiquid assets is essentially similar to the reason why they should prefer, also ceteris paribus, low risk to risky assets.153 The lack of precise knowledge about the future motivates them to take measures against it by either choosing ways of storing wealth that are less affected by uncertainty or by preparing to react to unexpected events (opportunities) by holding sufficient funds in a liquid form.154 Giving up either of these ways of cop151 152 153
154
This is the title of Chapter 15 in Keynes (1936). See section 1.4.2.1 for a brief description of the expected utility model. The relation between liquidity preference and the aversion to investment risk has been stated already by Tobin (1958); he does it, however, for the purpose of explaining why cash holdings should be negatively related to interest rates rather than for the investigation of the nature of liquidity. Note, however, that the uncertainty underlying investment risk and the uncertainty underlying liquidity are not quite of the same nature. Hicks (1974, pp. 38 ff.) writes: “For liquidity is not a property of a single choice; it is a matter of a sequence of choices. It is concerned with the passage from the known to the unknown – with the knowledge that if we wait we can have more knowledge” (pp. 38-39). Thus, while investment risk refers to the uncertainty about the consequences of a single investment decision, liquidity refers to the uncertainty about a series of potential future de-
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ing with uncertainty must be rewarded – a risk premium is demanded in the former case and a liquidity premium in the latter case. 1.4.2.
Liquidity in Investment Decisions
While the former subsection dealt with the role of liquidity in the economy in general, the focus of this section is on its role from the investor’s point of view. Although the main conclusions from the former considerations still hold in this case, the focus is more on returns from investments rather than on wealth accumulation and consumption. In particular, the position of liquidity among investment (and investors’) goals needs to be discussed. Furthermore, the precise delimitation of situations in which liquidity may become problematic needs to be clarified. In the latter case, it is useful to distinguish between the cases in which such problems can be anticipated and the cases in which they arise surprisingly. Finally, the positioning of the “liquidity goal” in portfolio decisions needs to be highlighted. All three aspects, i.e., liquidity as an investment goal, “expected” and “unexpected” liquidity, and liquidity in portfolio decisions are discussed in the following sections.
1.4.2.1. Investment Goals Although highly relevant from the practical and theoretical point of view, the discussion of investors’ goals rarely goes beyond the “terminal value” or “return and risk” level. The expected profitability and the uncertainty about it are doubtlessly the key criteria for the majority of investment decisions, but they are surely not the only ones, especially when non-public assets are considered. Even intuitively, one can easily name further goals that can be highly relevant for at least some groups of investors. Interesting references in this case are the studies of investors’ motives conducted in Germany. ADIG (1974) conducted a survey on the most important features of a “perfect” investment; Ruda (1988) performed both a theoretical, literature based, and an empirical analysis of goals of private investors;155 Oehler (1990) questioned financial advisors on the characteristics of investments that are considered important by their clients. A larger number of various, not always clearly defined criteria were recorded
155
cisions. By holding liquid assets an investor assumes the possibility of learning about new, yet unknown or uncertain opportunities, each of which may be subject to investment risk. This idea is also expressed in the model by Jones/Ostroy (1984). See Schmidt-von Rhein (1996) for a discussion of Ruda’s (1988) results and their comparison with other studies.
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93
and evaluated in each of these studies. Rankings of investments goals, which looked somewhat different in each case, resulted. A selection of the most relevant ones is presented in Table 1-5. Table 1-5: Rankings of selected investors’ goals156 Goal
ADIG (1974)
Ruda (1988)
Ruda (1988) Lit.*
Oehler (1990)
Dividend-Return
4
6
2
1
Divisibility
10
19
11
9
Ease of management
9
-
6
5
Retirement provision
-
5
-
-
Influence on corporate policy
-
17
5
-
Information and labor input
6
13
11
-
Liquidity
3
3
3
4
Long-term capital growth
2
4
4
2
Prestige
-
16
6
-
Real value preservation
-
11
-
3
Security
1
1
1
-
Short-term profitability
7
8
-
8
Tax-benefits
-
7
8
10
Total-Return
-
2
-
-
* based on the literature survey by Ruda (1988)
The first look at the presented goals reveals that they are highly heterogeneous, partially inconsistent, and often not sufficiently precisely defined. Nevertheless, the summary of the results allows the specification of only few issues that seem to be of central relevance to investors:157
156
157
A compilation based on ADIG (1974), p. 94, Ruda (1988), pp. 20 and 219, and Oehler (1990), p. 496. The surveys encompassed also further categories, which not always could be classified as true investment goals (e.g., “investments recommended by my bank”). They have been omitted in the Table, so that gaps in the rankings are possible. It is also possible that different criteria have the same rank; they are then to be viewed as equally important. Furthermore, not all categories were identically named in each survey. Closest equivalents have been chosen where it was possible in such cases. See also Schmidt-von Rhein (1996), pp. 109-110, for a similar review. For similar notions see Schmidt-von Rhein (1996), pp. 111, Müller (1995), p. 138, or Thiele (1977), pp. 31-40.
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Chapter 1: The Concept of Liquidity • return (encompasses such goals as dividend- and total-returns, short-term profitability, long-term capital growth, and tax benefits), • risk (encompasses security, real value preservation, and suitability as a retirement provision), • liquidity, • manageability (encompasses the necessary information and labor input, ease of management, as well as divisibility).
The return and the risk goal are probably the most widely recognized in the literature.158 The first one refers usually to the increase in wealth expressed as a percentage of the invested capital. For most investments, total return can be split into income return arising from the current payments (e.g., interest, dividends, rental revenues, or cash flows) and capital growth arising from the increase of the asset’s value. Furthermore, depending on the logic behind the concrete approach, several types of returns can be defined. They include mainly holding period or discrete returns, which assume periodic payment and capitalization of interest (income), and continuous returns, which are based on the assumptions that interest is capitalized infinitely frequently. The latter approach, apart from avoiding the necessity of specifying a uniform interest payment interval, is also convenient for mathematical operations. The return goal always refers to the future – past returns can only be considered as achieving of missing past goals. Therefore, it is always associated with uncertainty. The grade of this uncertainty is different for different investments and is referred to as risk. One must differentiate between uncertainty understood as the lack of any reference point concerning the future and uncertainty in the sense of a probability distribution for different future states – only the second one is denoted as risk and can be reasonably analyzed as an operational investment goal.159 A clear delimitation of these two types of uncertainty is, however, difficult; yet, at least a subjective assessment of 158
159
See Reilly/Brown (1997), Chapter 1, Bodie et al. (2005), Chapter 5, or virtually any investment text-book. The classification of the general uncertainty into risk (with known state probabilities) and uncertainty in the narrow sense (with unknown state probabilities) goes back to Knight (1964), chapter VIII (first published 1921). See also Levy/Sarnat (1984), pp. 104-106, Kupsch (1973), p. 26, Rehkugler/Schindel (1990), p. 105, or Bamberg/Coenenberg (1991), p. 17.
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probabilities should usually be possible. Furthermore, investment risk can be classified according to its source. Thus, one speaks of credit risk, market risk, political risk, currency risk etc. Both, the return goal and the risk goal can be derived from the expected utility theory.160 According to it, as already noted in the previous subsection, individual’s utility is typically an increasing function of wealth or consumption161, but marginal utility is its decreasing function, i.e., it is concave. This shape of the function implies that individuals can be saturated – an increase in the holdings of each good separately and all goods jointly contributes less and less to the overall utility. Furthermore, the utility of uncertain wealth is equal to the expected utility over all possible scenarios. These assumptions are sufficient to derive the return and the risk goal. On the one hand, the fact that the utility function is increasing results in investor’s preference for investments leading to higher terminal wealth, i.e., yielding higher rates of return. On the other hand, the fact that the function is concave leads to the preference for more certain outcomes. To illustrate this, three investment alternatives are depicted in Figure 1-15: the first one yields a certain wealth level of X, the second one yields an uncertain wealth of Y or Y’, and the third one yields an even more uncertain wealth of Z or Z’. The expected wealth from investing in any of these alternatives is equal X. Since the expected utility (U) of an uncertain outcome is simply the utility of the possible outcomes weighted with their probabilities, it lies on the straight line connecting U(Y) with U(Y’) or U(Z) with U(Z’). In effect, the utilities of investments yielding less certain terminal wealths are lower. Hence, given identical expectations in all cases an investor prefers the alternative with the lowest uncertainty, i.e., lowest risk.
160
161
The (expected) utility theory and the issue of risk aversion is discussed in virtually every test-book on microeconomics; see Mas-Colell et al. (1995), Chapter 6, or Varian (2003), Chapter 12. For a more formal presentation see Ingersoll (1987), Chapter 1. Since wealth can be considered as delayed consumption, the choice of the variable for the utility function does not affect the conclusions.
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Utility
96
U(X) U(Y,Y’ U(Z,Z’)
Z Y
X
Y’
Z’
Wealth
Figure 1-15: Decreasing marginal utility and risk aversion162
Liquidity as an investment goal is, of course, especially relevant for this work. Its extensive definition has been provided earlier in this Chapter. Although it is most frequently described in the literature as the “ease of conversion into money”163, it follows from the hitherto discussion that this is only a common heading for a bundle of goals encompassing the time and the value aspect of liquidation. They can be summarized in two sub-goals: the expected present value of sale receipts, either in absolute terms or in relation to the (however defined) “fair value”, and the uncertainty about it. This means that investors should strive to maximize the present value they expect to receive at sale (i.e., marketability) and simultaneously to minimize the uncertainty about it (i.e., liquidity risk). The analogy to the return and risk goals is apparent; however, while the latter refer to increases of wealth, marketability and liquidity risk refer to liquidation outcomes. The two basic liquidity sub-goals can be further categorized depending on the investor’s situation and also extended on the purchase of assets; this is done in the following subsections. Manageability is the last of the four and a far less frequently recognized investment goal.164 It refers to the effort necessary to conduct an investment and can have a num162 163
164
For similar presentations see Mas-Colell et al. (1995), p. 186, or Varian (2003), p. 225. See Thiele (1977), p. 33, Fank (1992), pp. 206, Schmidt-von Rhein (1996), pp. 104-105, as well as the literature references in section 1.1.1. In fact, many authors omit the manageability goal reducing the list of investor’s goals only to return, risk, and liquidity; see Fank (1982), pp. 206 ff., or Lerbinger (1984), pp. 142 ff.
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ber of different forms. In the first line, it may be the effort of gaining sufficient information to evaluate and to monitor an investment but also information that the specific investment alternative is at all available. Furthermore, some assets require more engagement of the investor than other assets. For instance, regulations may require that certain legal steps are conducted personally, or the nature of the investment may require frequent revisions of the initial decisions. Obviously, investors should prefer (ceteris paribus) investments that can be accomplished with as little effort as possible. This can be done by choosing an asset that has the corresponding characteristic or by transferring the efforts on a hired manager. In the last case, the management fee would diminish the payout; thus, manageability can be considered as a return reducing factor. The four main investment goals identified here are not entirely independent. The directions of the relations between them are outlined in Figure 1-16 in form of an “investment goals’ square”. The negative link between expected return and risk is a commonly recognized rule – for accepting risk investors must be rewarded with higher returns.165 Similarly, lower liquidity and poor manageability need to be compensated with additional return premiums to cover the additional costs. The risk increasing character of illiquidity has been discusses in section 1.1.2; also high complexity of an investment may result in increased risk due to a higher possibility of wrong decisions. Finally, as discussed in section 1.2.3.2, poor liquidity of an asset may make a more thorough search for a trading partner necessary (or worthwhile) increasing the effort necessary to conduct and to conclude the investment.
165
See literature references in FN 158. The relation between expected returns and risk is usually justified with the CAPM (see section 4.1.3).
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Return
+
Risk
– –
– – Liquidity
+
– positive influence
–
– negative influence
+
Manageability
Figure 1-16: Interrelations between the main investment goals
After presenting the general position of liquidity among investment goals, it is necessary to state more precisely what the concrete objectives are. It has already been highlighted that liquidity is, in fact, a dual goal encompassing the expected outcome of liquidation (marketability) and the uncertainty associated with it (liquidity risk). Still, there are at least three different situations in which an investor is confronted with this problem, each of them resulting in a slightly different specification of the original goal. They are discussed in the following sections and play a significant role in the development of the search model in later chapters.
1.4.2.2. Expected and Unexpected Liquidation The first important differentiation of the liquidity goal is with respect to the moment of liquidation. Investors are usually facing liquidation problems associated with an illiquid asset at the end of the investment horizon. When the planned holding period expires and the asset is to be sold, the issue of its liquidation value and liquidation time arises. The second type of situations in which investors are concerned with liquidity is an unexpected sale caused by exogenous circumstances. Due to unplanned and possibly unforeseeable events, quick access to larger sums of money may be required in order to meet payment deadlines. This is often achievable only by selling some of the investor’s asset holdings. These two situations are basically similar in their nature – investors are interested in a quick and cheap liquidation in either case. However, differences
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in the backgrounds of the sale decisions can lead to emphases on different aspects of the liquidity goal. Consider the expected liquidation case first. Since most investments are conducted with a certain intended time horizon, it is apparent that their liquidation is scheduled more or less precisely already at the moment of the investment decision. Its outcome is relevant with respect to the price achieved and to the duration of the process. On the one hand, the sale price significantly affects the total return from the investment; on the other hand, timely sale may be necessary to meet the scheduled times of subsequent investments. Thus, the first aspect is important for maximizing the profitability of the investment, and the second aspect determines the incurred opportunity costs defined as the profits lost due to delays in the investment schedule. The fact that the sale time is known (at least approximately) should lead to relatively higher emphasis on the sale price. Since the liquidation deadline and the consequences of missing it can be assessed in advance, they can also be adequately mitigated. In particular, the liquidation process can be initiated accordingly earlier in order to minimize the chance of a delay. Furthermore, emergency financing necessary to bridge the time gap between the sale of the illiquid asset and the new investment in case of a delay can be arranged in due time at an acceptable cost. The planning certainty can be additionally enhanced by a forecast of the market situation at the moment of the sale. All in all, since the time aspect of liquidation can be relatively well managed in this scenario, investors’ main concern should be obtaining an adequate sale price. In contrast, investors’ focus in the case of an unexpected liquidation should be more on the time aspect. The liquidation is then the result of an unforeseen event, a liquidity shock forcing the investor to sell the asset earlier than actually intended. This has two profound consequences. Firstly, due to the discontinuation of the investment, its original goal cannot be achieved; and secondly, failure to sell the asset in due time may result in severe consequences. These consequences may encompass a variety of costs, including lost customers and, in consequence, lost future revenues, financial penalties, like the penalty interest for delayed installment payments, or missed opportunities of particularly profitable investments. In the worst case, also bankruptcy due to the inability of timely debt repayment may follow. Thus, the investor needs to compare the return lost due to the premature termination of the investment with the cost resulting from forgoing the liquidation; obviously, liquidation is only rational if the former one
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does not exceed the latter one. Although the nature of this problem is similar to the planned sale, there are two significant differences: the opportunity costs of not selling are much higher, and the timing of the liquidity shock is unknown. Hence, time is more important in this case than in the case of an expected and planned liquidation inducing a more time-oriented attitude with respect to the liquidity goal. The distinction of the two types of “liquidity situations” has profound practical consequences. While relative certainty (or at least expectation) concerning the timing, the opportunity costs, and the state of the market can be assumed for the planned liquidation, all these variables are either unknown or only imprecisely assessable for the unexpected liquidation. In fact, the only statement that can be made in the latter case is that opportunity costs are likely to be high. This makes the unexpected liquidation far more problematic than the excepted one. For this reason, the liquidity goal formulated by investors will usually refer mainly to this case.166 It does not mean, however, that liquidity in the planned sale case can be ignored. Since each scenario requires the application of a different analysis framework, they are often discussed separately in the following chapters.
1.4.2.3. Liquidity in the Sale Case and in the Purchase Case Another two types of situations that may lead to different notions of the liquidity goal are sale and purchase of an (illiquid) asset. Hitherto, only the former case has been considered. Yet, it has already been stated several times that liquidity considerations may also become relevant when buying an asset – the definition of liquidity formulated in section 1.1.3 includes also this eventuality. At this point, it is necessary to clarify whether and in how far investor’s goals differ in these two cases. The general objectives when buying or selling an illiquid asset are symmetric. In the former case, the investor aims at maximizing the effective sale price, and in the latter case, she aims at minimizing the effective expense. Minimization of uncertainty is present in a similar manner in both cases. However, differences become apparent when the decision variables are considered more closely.
166
This conclusion is in line with the Keynes’ notion of liquidity preference due to the precautionary and the speculative motive (see section 1.4.1). The possibility of a timely liquidation in an emergency case seems also to be in the focus of the literature on the subject.
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In the first place, as already discussed in section 1.2.2, the transaction costs are not necessarily symmetrical in both considered cases. Direct fees and taxes are usually charged on one side of the transaction only; also inspection and evaluation (appraisal) costs are usually higher on the buyer’s side. Yet, while this problem can be solved by allowing for the respective expenses in price negotiations, the issue of opportunity costs is much more problematic. As mentioned in section 1.2.2.3, “lost opportunities” in the sale case encompass revenues that could have been earned or losses that could have been avoided if the liquidation was accomplished promptly. In contrast, income flow and value appreciation of the prospecttive investment are lost or postponed if the transaction is delayed in the purchase case. This means that while opportunity costs are solely a function of the personal situation of the concrete investor in the former case, they depend mainly on the characteristics of the asset in the latter case. In other words, they are predominantly investor-specific when selling and predominantly assetspecific when buying.167 Due to this property, liquidity shocks resulting in the necessity of an unexpected sale are in practice far more relevant than liquidity shocks resulting in the necessity of an unexpected purchase. The latter may occur in certain less typical situations, such as delivery of the underlying in a forward agreement, but are less frequent and usually of less severe consequence than unplanned, forced sales conducted to bridge liquidity bottlenecks. Also, certain types of market organization may lead to different positions of sellers and buyers. For example, real estate brokers usually represent sellers and actively seek for buyers. Although a buyer may also turn to a broker, she will usually be directed to the already known sellers – an active search for a property to purchase is rarely offered. Similarly, auctions, such as the typical English one, are only organized in order to sell an item but not in order to buy one. Hence, some forms of trading seem to favor sellers. Finally, characteristics of market participants do not necessarily need to be the same on the supply and on the demand side of the market. This refers both to the availability of sale offers or buy bids and to the diversity of valuations among potential buyers or sellers. Especially in unusually hot or cold markets, it may be easier to sell or to buy, respectively. 167
This statement is only valid for typical transactions. Counter-examples of special situations, in which transaction costs are asset-specific at sale or investor-specific at purchase, can be easily found. This is, e.g., the case when the sale is a short-sale, i.e., a speculation or a hedging strategy based on a decrease in the asset’s value, or when the value of the purchased asset results from its special function for the concrete investor.
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To sum up, there are a number of reasons why liquidity considered from the seller’s point of view may be different than liquidity considered from the buyer’s point of view. For practical investment analysis and decision making, this means that considerations referring to liquidity as an investment goal need to be structured differently in these two cases.
1.4.2.4. Individual Liquidity and Portfolio Liquidity The definition of asset liquidity refers to only one asset; the definition of market liquidity includes considerations about the amount of liquidated assets, but it still refers to only one asset type. Recognizing that the concept of liquidity can also be applied to the entire wealth (or a part of it) of an individual or an institution introduces a new aspect to the problem. The dual nature of liquidity as an investment goal has been emphasized in the discussion in section 1.4.1. According to it, investors should prefer (ceteris paribus) assets yielding higher expected liquidation values as well as assets offering more certain liquidation values. This set of objectives can be applied to investment portfolios in the same manner as to individual assets. It would then mean that investors prefer portfolios that can be liquidated at higher expected values as well as portfolios which liquidation values are more certain. Although these two notions of the liquidity goal appear to be nearly identical, there is one important difference, which becomes especially apparent when an unexpected sale is considered. Coping with a liquidity shock requires that a certain amount of money is put to investor’s disposal at short notice; this is achieved by selling some of her asset holdings. The concept of individual liquidity assumes that one concrete asset is sold in such case. However, as soon as the investor holds more than only one asset (asset type), the possibility of selling any one of them occurs. Liquidation in this sense refers to a part of the total wealth without determining precisely which part is to be sold. The only condition is that the total proceeds are sufficient to cover the unexpected expenses. Also liquidity risk can be redefined in this manner. It refers then to the certainty about the effective outcome of the liquidation of any of the held assets. The liquidity goal in the portfolio context implies that the investor attempts to sell more than one asset simultaneously. In this case, expected sale proceeds are equal to the (conditional) sum of expected proceeds from one or, if necessary, several assets
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that sell first. Thus, the expected liquidation value does not essentially differ from the single asset liquidation case. However, the same does not hold for liquidity risk. The chance of successfully selling any asset is always higher, or at least not lower, than the chance of selling some particular asset. Thus, the uncertainty about obtaining the required sum quickly is lower in this case, or alternatively, a higher sum is obtainable within the given time horizon at the given level of uncertainty. This means that liquidity risk is generally lower for a portfolio than for any single asset in this portfolio. Thus, holding a mixture of less liquid assets may result in a similar liquidity position as holding one more liquid asset.168 Note that the above notion of portfolio liquidity is similar to the corporate liquidity discussed in section 1.1.1.3. Companies that have the ability to cover all their expenses on time are denoted as liquid. This ability is usually judged on the basis of assets and liabilities disclosed in the balance sheet. In fact, assets of a company can be regarded as a portfolio that can be liquidated in an emergency situation. However, while corporate liquidity focuses mainly on holding sufficient stock of liquid assets, portfolio liquidity is also about the variety of assets. In this sense, the liquidity position of a company owing only one large piece of equipment is worse than the respective position of a company owing a larger number of smaller devices. The latter one is facing lower liquidity risk, because it can choose to liquidate only a part of its assets in an emergency case. Intuitively, portfolio liquidity seems to be a more appropriate investment goal than liquidity of individual assets. Like in the case of other goals, in particular the return and the risk goal, an investor should be concerned about liquidity of her investments viewed as a whole and not separately.169 Thus, this aspect of the problem receives special attention in the following chapters, especially in Chapter 4. *** The main objective of this Chapter was to clarify the meaning of the central term of the book. Despite its widespread use, no unambiguous and generally accepted liquidity definition seems to exist. The literature review demonstrated that the term is used in at least three different contexts – referring to assets, markets, and companies – but even 168 169
“Liquidity risk diversification” is analyzed more closely in sections 4.3.3 and 4.2.3. See section 4.1 for portfolio considerations referring to the return and risk goal.
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within these contexts there is no consensus about its precise meaning. Thus, the task was, on the one hand, to categorize the existing definitions, and on the other hand, to work out a precise description of the phenomenon denoted as liquidity. On this basis, the foundation for the development of measurement approaches and investment decision tools, which would take liquidity into account, was to be prepared. The extensive analysis led to several conclusions of which three are of particular importance for the following chapters. In the first line, the definition of liquidity was addressed. The central result from the analysis of the existing approaches, extended by own considerations, was the recognition that it has two distinct and largely independent dimensions. On the one hand, the expected outcome from liquidation is relevant; on the other hand, the uncertainty about it is also important. These dimensions have been denoted as marketability (referring to assets) or expected liquidity (referring to markets) and liquidity risk, respectively. In the second step, the sources of the problem have been examined. A number of variables, which in a more or less direct way affect the possibilities of liquidating an asset, have been classified into three main groups: transaction and opportunity costs, organization of trading, and heterogeneity of opinions about the value of the asset. An important “side effect” of this analysis was the recognition of the importance of the search for a trading partner. As it seems, liquidity understood as the ability to sell quickly, without discount, and with little uncertainty can be interpreted as the ability to find a trading partner offering a “good” price within a short period of time and with high confidence. In this sense, modeling the “seek and sell” process is equivalent to modeling liquidity itself. Finally, the review of the main types of illiquid investments led to the recognition of the role of real estate, which proved to be by far the most important one. Volumes invested in properties are of magnitudes comparable with other popular assets like stocks or bonds. In contrast, other illiquid assets, among them private equity and various collectibles, are only of marginal practical relevance. For this reason, further analysis in the book is focused primarily on real estate.
Chapter 2 Search in Illiquid Markets
As stated in the previous Chapter, the necessity to search for a trading partner seems to be the key factor in understanding liquidity. It directly affects the possibility of quick liquidation at a reasonable price but is difficult to specify in a way that would allow its inclusion in the investment decision process. This Chapter concentrates on the formal description of the search process. A sequential search model has been chosen for this purpose. The function of the model is twofold: on the one hand, it should allow the quantification of the influence of various market parameters on the final result of liquidation; on the other hand, it should provide methods for analyzing various search strategies and identifying the optimal one. However, these two issues cannot be entirely separated – the effect of the market situation on the outcome of search depends on the chosen search strategy, and the optimal strategy is to a large extent determined by the characteristics of assets and markets. Thus, the central issue of the analysis is the dependence between the liquidity of an asset and the strategic choice of an investor. The Chapter starts with a brief presentation of the “Theory of Search”, a branch of mathematics and operations research dealing specifically with search processes. An introduction to search models follows. Two seminal models that constitute the foundation for most of the search theoretical analysis are in its core: the basic model with observation costs and Karlin’s model with discounting. The prime objective is to provide the methodical foundation for the formulation of a real estate search model in the next section, but also to give the reader a better impression of the nature of the problem. A model of search conducted by an investor liquidating a real estate investment constitutes the central point of the Chapter. It builds on a simple model with observation and opportunity costs but is refined by adding several more realistic assumptions about the key parameters. Main modifications include continuous time and market uncertainty as well as the redefinition of the variables in a relative manner. A closed form formula for the expected net receipts is derived and conditions for the existence of a gain maximizing search strategy are considered. Recognizing the limitations of the model, a number of possible extensions are discussed in the following section. Finally, a simulation approach to the search problem is presented. It allows for more freedom with regard to
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restricting assumptions but yields only approximate results. The Chapter is closed with some more general considerations about the implications of strategic search for the functioning of illiquid markets.
2.1.
The Theory of Search
Sequential search models have been developed and applied in economic sciences for the last several decades already. They are also known as optimal stopping models since they focus on the optimal stopping of a sequence of random variables, which can be interpreted as a search process. The main interest is usually in finding a stopping rule that optimizes the expected outcome of the search. Numerous variations of this problem have been analyzed in the literature. Despite differences in constraints and objectives, their core structures remain similar. However, like in many mathematically complex problems, even seemingly slight modifications can lead to significant complications or even to the insolvability of the model. Research on the sequential search theory follows therefore a twofold aim: application of the models to certain real world phenomena and development of further variants of the problem. The pioneering work in the field of search theory was done by Wald (1947a) and Arrow et al. (1949). They analyzed sequential decision problems and formulated several basic theorems useful in the analysis of series of random variables. The originally intended application was in statistical estimation and testing based on sequential sampling, but more general application possibilities were also indicated.170 The rapid development of the actual “Theory of Search” started about 1960. Mathematical elaborations constitute the largest part of the search theoretical research.171 One of the first papers formulating the standard search problem was MacQueen/Miller (1960). The authors use the house buyer’s dilemma as an example of a search problem. They derive the optimal reservation price, i.e., the minimal price that should be accepted, and formulate conditions for its existence and uniqueness. Among later significant mathematical contributions are: Chow/Robbins (1963), Karlin (1962), Elfving (1967), or Siegmund (1967). The paper by Karlin is especially interesting for the analysis in this book. The author studies the optimal behavior of an individual sell170 171
See Wald (1947), p. 280. A lot of the early development in this field is summarized in DeGroot (1970), pp. 265 ff., Chow et al. (1971), Lippman/McCall (1976a), and Ferguson (2000).
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ing an asset and willing to maximize the expected receipts. He analyzes the optimal search policy in several frameworks including discrete and continuous time as well as finite and infinite horizon and also introduces discounting of future payments. Further papers on the search theory consider special cases of the problem, e.g., when the searcher is allowed to “recall” past offers,172 when the distribution of offers is unknown, and the searcher “learns” from subsequent observations (adaptive search),173 or when the search environment is dynamic174. The possibilities of generating different search frameworks are almost infinite,175 but their solvability is often problematic. Therefore, a separate branch of the search theory addresses the formulation of practicable solutions to the existing problems. Two frequent approaches include analytical derivation of explicit solutions under simplifying assumptions176 and derivation of approximate solutions.177 While the former allow more precise answers, the latter are more general. Most papers on the “mathematical” search theory consider the sequential search problem as a class of abstract mathematical problems. Its different variations are referred to as the “secretary problem”, the “parking problem”, the “one-handed-bandit problem”, or the “house selling problem”. Although these names suggest relations to real live situations, the papers remain mostly on an abstract theoretical level. A series of random variables on which a target function is applied and optimized is always in the centre of the analysis. The relevance of these works for the subject of this book is therefore mainly due to the methods of analysis of various search problems developed therein. The domain of economics most influenced by the theory of search is information economics. This course of research was introduced by Stigler (1961). In this groundbreaking paper the author concentrates on the role of asymmetric and heterogeneous infor172
173
174
175
176 177
See Yahav (1966), DeGroot (1968), Rosenfield/Shapiro (1981), and Rosenfield et al. (1983), as well as section 2.3.3.3. See Telser (1973), Kohn/Shavell (1974), Rothschild (1974a), Albright (1977), Rosenfield/Shapiro (1981), Rosenfield et al. (1983), or more recently Einav (2005). See Karlin (1962), McCall (1965), Lippman/McCall (1976c), and Ondrich (1987a), as well as section 2.3.3.2. Examples of other special cases are the “search and evaluation” model of MacQueen (1964) or the “Pandora’s Problem” of Weitzman (1979). See Telser (1973), Gastwirth (1976), Albright (1977), and Feinberg/Johnson (1977). See DeGroot (1968), Kennedy/Kertz (1991), and Kühne/Rüschendorf (2000 a, b).
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mation in economic processes, especially in the market price ascertainment. Stigler points out that in the absence of an organized market no single price exists. An individual wanting to buy or sell an asset will therefore not accept the first quote, but search for the best one she can obtain. “Searching” means in this case observing different offers and gathering information about others’ opinions on the asset’s value. Thus, it can be seen as a method to cope with insufficient information. In this sense, search is valuable and its value is equal to the additional gain achieved by searching. This idea inspired numerous authors to study the role of search as a source of information. One of the most significant following papers is McCall (1965) where valuation of information is recapitulated and generalized. The far reaching consequence of Stigler’s idea was the development of market models with imperfectly informed participants searching for best alternatives. Numerous works deal with market organization and equilibrium in this context building directly on Stigler.178 Others, following Nelson (1970), analyze consumers’ behavior in markets with non-homogenous products.179 Several papers address the problem of price setting by companies acting under incomplete information.180 Another major field of application of the search theory is labor economics. This course of research was introduced by Stigler himself (1962) who was followed by McCall (1970) and other researchers in the 70s181. The search for a job is analyzed in these works. The key issue is maximizing the expected future earnings of a potential employee. Most of these studies concentrate on the optimal search behavior and on the explanation of the unemployment phenomenon as well as other characteristics of labor markets. The research on this subject is still lively. McCall (1994), who examine the effects of job heterogeneity, or Pissarides (1994), who analyze an on-job-search, are examples for the more recent developments in this field.
178
179 180 181
See Rothschild (1973), Burdett/Judd (1983), and several articles published in the Swedish Journal of Economics, Vol. 76, 1974. See Gastwirth (1976), Wilde (1981), or Cressy (1983). See Rothschild (1974b) or Axell (1977). See Alchian (1970), Mortensen (1970), Pissarides (1976), and Lippman/McCall (1976 a, b, c). Several significant search theoretical contributions in labor economics can be found in Lippman/McCall (1979). A good review of the research on this subject is brought in MacKenna (1985).
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Reference to real estate economics has been present in the search theory from its very beginning and is best reflected in the “house selling problem”182, which deals with the dilemma of a house owner wanting to sell a property and searching for the best buyer. However, despite the relation to a property transaction, the analysis of this scenario concentrates mostly on the general search problem using the house only as an example and ignoring the unique characteristics of real estate markets. Specific applications of the search theory in real estate economics are relatively recent. One course of research is the analysis of the search behavior of market participants.183 Several papers focus on the role of broker in this process.184 Another often discussed problem is the marketing time of properties. The relevance of search in this context was mentioned already by Miller (1978, pp. 165-167), but it was directly applied by Haurin (1988) for the first time. The latter author utilizes the search theory to study the effects of residential houses’ atypicality on their marketing times. His followers study relations between listing price, marketing time, and the eventual selling price.185 Studies on liquidity (illiquidity) belong to the most recent applications of the search theory in real estate economics. Research on this subject started with Lippman/McCall (1986) and has been followed by relatively few researchers so far.186 There have also been numerous applications of the search theory in many other branches of economics.187. The relevance of various applications of the search theory in fields of research not necessarily directly connected with liquidity is given by the fact that the nature of the search process and the encountered mathematical problems are very similar in all markets with imperfect information. Their formulations vary, but the core structures remain analogous – sequential search is conducted and should be optimized. This quality allows the application of the same methodology in most cases.
182
183
184 185 186 187
The “house selling problem” is defined and analyzed among others by MacQueen/Miller (1960) and Albright (1977). See also Ferguson (2000), pp. 1.4-1.5. See Courant (1978), Chinloy (1999), Cronin (1982), Quan/Quigley (1991), Kim (1992), or Yavaş/Colwell (1995). Yavaş (1992) and Arnold (1999) incorporate bargaining between the buyer and the seller in their search models. See Yinger (1981), Wu/Colwell (1986), Geltner et al. (1991), or Yavaş (1996). See Yavaş/Yang (1995), Glower et al. (1998), and Green/Vandell (1998). See Chinloy (1999), Krainer (1999, 2001), and Mok (2002 a, b). Further applications of the search theory include i.e. modeling research and development activity (see Nelson, 1961, Marschak/Yahav, 1966, or Reinganum, 1982) or valuation of financial instruments (see Karatzas, 1988, or Beibel/Lerche, 1997).
110
2.2.
Chapter 2: Search in Illiquid Markets
Introduction to Search Models
Two models of sequential search for a trading partner constitute the core of the following section. They are based on MacQueen/Miller (1960) and Karlin (1962), respectively, and have been widely discussed in the literature so far. Due to the structural simplicity, they allow for only few variables and include a number of simplifying assumptions. However, the simple structures of these models should not mislead about the full potential of search-theoretical approaches. By implementing additional variables and modifying the search framework, they can be used to study highly complex problems. Therefore, the models in this section should be treated as an introduction to the actual subject of the Chapter. 2.2.1.
Search Framework
Before the mathematics of the search models are introduced, it is necessary to provide precise definitions of all relevant variables and parameters in order to avoid confusion when the analysis becomes more complex. The explicit presentation of the model framework is also useful to give the reader a better impression of what kind of problem is dealt with. For this reason, the setting of the model is held as simple as possible at this stage. The underlying search process is summarized in Figure 2-1.188 A sale of an illiquid asset is considered. It is not necessary to specify the type of the asset at this point, but several assumptions concerning its key features as well as the characteristics of the seller are necessary.189 Firstly, it is assumed that no objective valuation of the asset in question is possible. As a result, potential buyers differ in their assessments of the asset’s true value. Several different sources of this heterogeneity of opinions are possible: it may result from different possibilities of use, which generate different income streams, or simply from different subjective tastes. The owner does not use the asset for her own purposes and regards it purely as a financial investment. In particular, it may be a part of her investment portfolio. This assumption implies that
188
189
The trade framework presented in this section is based on standard search models of asset sale as analyzed by MacQueen/Miller (1960), Stigler (1961), Karlin (1962), McCall (1965), or Rosenfield et al. (1983). Its core structure is the same as in the job search models presented by McCall (1970), Lippman/McCall (1976a), Lippman/McCall (1979), pp. 2-3, or MacKenna (1985), pp. 4-5. Analogical frameworks of real estate trade are used by Miller (1978), Haurin (1988), or Tryfos (1981). For better tractability, the seller (investor) is assumed to be female while potential buyers (market participants) are assumed to be male.
2.2 Introduction to Search Models
111
the seller does not need to combine the sale with any other activity or transaction – it is viewed on a standalone basis. For the analysis conducted in this Chapter, it is also convenient to assume that the seller (investor) is risk-indifferent, i.e., she judges investment opportunities only by expected gains ignoring other characteristics. This results in the notion of an optimal search strategy as the one that maximizes the expected receipts. This assumption is, however, only temporary and will be dropped in the subsequent chapters. Trying to sell the asset, the seller searches for the best buyer (or bidder), i.e., one that offers the highest price. It is assumed that no other feature is of importance. Especially, the seller is not concerned about what happens with the asset after the sale – the time after the transaction is beyond her consideration. The seller has also no preferences concerning the characteristics of the buyer, including his creditworthiness. All potential buyers are assumed to have the same perfect financial standing and will be able to pay the price they offer without delay. Reassuming, the only thing that matters is the price. Furthermore, the seller is concerned about the duration of the search only as far as it influences the gain from the sale. This means that no deadline is set for completing the sale and the search can continue as long as needed to maximize the effective sale price. The seller continues the search until the asset is sold. During this time, prospective buyers bid on the property placing their purchase offers sequentially.190 It is assumed that only one offer can be viewed at a time and that the search starts with the arrival of the first one.191 Offers are enumerated with an index i according to their arrival order starting with i=1. Intervals between subsequent bids are assumed to be non-random, identical, and equal to t. Each potential buyer offers a price of Pi, which is based on his subjective opinion about the value of the asset. The seller can either accept it or reject it; however, she has to make her decision immediately, and if she chooses to reject an offer, it cannot be recalled afterwards. Since potential buyers do not know and do not 190
Note that the term “offer” is used throughout the analysis in the sense of a bid placed by a prospective buyer, as is typically done in the search theory. In particular, it should not be confused with the “offer-price” demanded by a seller; the term “ask-price” is used in this context when necessary. 191 In later sections, intervals between subsequent bids are assumed to be random rather than constant. The beginning of the search is then associated with the decision to sell the property. In this case, the starting point of the search process does not necessarily need to (and indeed usually doesn’t) coincide with the first offer. The reader should bear in mind some minor differences in model’s equations arising from this fact when comparing the results in this book with those in other works.
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Chapter 2: Search in Illiquid Markets
influence each other, their opinions are independent. Under the assumption that they arrive in a random order and that their valuations are unknown before the arrival, a price offer Pi can be viewed as random variable and a series of price offers (P1, P2, …) as a sequence of independent random variables.192 Probability distributions of the bids arise from the distribution of opinions on the asset’s value among market participants. If valuations of potential buyers are constant over time (static market), offers are independent and identically distributed (i.i.d.) and can be treated as realizations of one random variable P. The search is assumed to be costly. Two types of search costs are possible: observation costs and opportunity costs. The two standard models presented in this section use either the first (basic model) or the second alternative (Karlin’s model), although both cost types can also be combined, as is done in the real estate search model later on in this Chapter. Observation costs are connected with obtaining further offers. They may encompass, for example, advertising and representation expenditures, costs of visitations, travel expenses etc. Not included are, however, typical transaction costs, like fees, taxes etc., which are assumed to be already comprised in prices. The unit observation cost of c is constant and is accrued each time an offer arrives, i.e., regularly at intervals of t. So, if the search is terminated after the nth offer, the cost of c is borne n times. Opportunity costs arise from the fact that the seller cannot dispose of the capital bounded in the asset for other purposes as long as it remains unsold. This cost of delaying the sale is reflected by the discounting factor δ used to calculate the present value of future payments. It is assumed to be positive and smaller then 1. The search for the best buyer runs according to a certain strategy, which may remain unchanged or vary depending on the offer flow. The seller defines a set of criteria according to which she decides whether to accept or to reject an offer. The possibilities of different search strategies are almost infinite. The strategy considered in most sequential search models and also followed in this work is to set a reservation price pi*, a critical price above which any offer is automatically accepted and below which every offer is automatically rejected. While p*i applies only to the ith offer, a constant reser-
192
Note that random variables are labeled with capital letters and their realizations with small letters. Thus, Pi represents a yet unknown price offer (ex ante) and pi a price offer that has already been made (ex post).
2.2 Introduction to Search Models
113
vation price valid for the whole duration of the search is assumed in many models; it is denoted as p*.
Beginning of the search
Offer P1
Cost of c
Offer P2
Cost of c
Offer P3
…
…
P1>p*1
P1
Cost of c
P2>p*2
P2
t2 TIME
t3
P3>p*3
P3
Pool of potential buyers
t1
t4
Total search cost
…
Sale price of Pi Discounting
et receipts from sale
Figure 2-1: Scheme of the search process
It must be stressed that the described selling problem is only an example for a family of similar problems. The same framework can be used, e.g., to describe consumer’s search for the best product, employee’s search for the job, or other mentioned in section 2.1. What underlies all these problems is in fact a much more general case of acting in a heterogeneous environment.
114 2.2.2.
Chapter 2: Search in Illiquid Markets Search Strategy
Before the actual models are presented, it is necessary to discuss several general issues encountered in the analysis of each search problem. The central question is: which search strategy is the best? Two basic possibilities are a fixed sample strategy and a sequential search strategy.193 In the first case, the searcher reviews a predefined number of offers before she decides to choose the highest one. The second possibility is to decide immediately on each arriving offer whether to accept it and complete the sale, or whether to reject it and continue searching. It seems intuitive that the latter method should generally produce better results. While collecting a fixed number of offers, each of which generates a unit observation cost, it is possible that the search would continue even after receiving an unexpectedly high offer. The sequential procedure allows avoiding such unnecessary search costs by terminating the search as soon as no prospects for a better offer exist. It is economical to continue the search only as long as the expected marginal revenue from viewing further offers exceeds the marginal cost.194 It fact, it can be proved that the sequential approach generally dominates the fixed sample strategy in terms of a higher expected revenue.195 Therefore, only the sequential search strategy is considered in the following analysis. It must be stressed, however, that its dominance only holds under certain conditions, which are similar to those assumed in section 2.2.1. Under different model assumptions other search strategies may be advantageous.196 Especially, if a mixed strategy is allowed, i.e., one in which whole samples rather than single offers are reviewed and decided on sequentially (sequential sample picking), it may be the optimal one for a wide class of search problems.197
193
194 195 196
197
The fixed sample strategy is considered by Stigler (1961) and Stigler (1962); the sequential search strategy is used by MacQueen/Miller (1960), McCall (1965) as well as in the majority of the search theoretical literature. See Rothschild (1974a), p. 691 or MacKenna (1985), p. 10. For the poof see DeGroot (1970), pp. 267-272, and MacKenna (1985), pp. 7-13. Feinberg/Johnson (1977) demonstrate for several specified distributions of offers that the superiority of the sequential search depends on the level of search costs. Wilde (1977), p. 375, argues that economies of scale leading to lower search costs may cause a fixed sample strategy to dominate the sequential one. See also FN 195. The optimality of the mixed strategy is indicated by Gastwirth (1976) and analyzed by Gal et al. (1981), Morgan (1983), Morgan/Manning (1985), or Harrison/Morgan (1990). Mok (2002b) applies a mix strategy search model on real estate.
2.2 Introduction to Search Models
115
Another problem of sequential search models is the existence of an optimal procedure. Optimality is understood in terms of the maximum expected sale proceeds in virtually all search models. This holds for a risk neutral investor assumed in this Chapter; however, one should bear in mind that also other definitions of optimality, including risk minimizing, are conceivable. This subject is discussed more closely in the following chapters. It can be proved that under certain conditions a receipts maximizing search procedure (stopping rule) always exists and is fully described by a stopping variable.198 This variable is a function of past observations and determines whether to continue or to terminate the search. The stopping variable in the asset selling problem takes the form of a reservation price, i.e., a price above which each offer is accepted and under which each offer is rejected. Search frameworks with so defined optimal search procedure are said to posses the “reservation price property”.199 Only this type of search procedures is considered in the following analysis. The third question to be cleared is whether the search always ends with a sale transaction for a finite reservation price. The sale situation presented in section 2.2.1 sets an infinite time horizon for the search process, which means that the seller is not bound to terminate the search at some fixed deadline in the future. Does it, however, mean that under circumstances the process may run infinitely? Under a finite reservation price receiving an acceptable offer is not impossible as long as the at least one offer value at or above the reservation price has a positive probability of occurrence. Thus, as long as for each x in the domain of the assumed probability density function of offers f(.) a y exists such that y>x and f(y)>0, there is a positive probability of an acceptable offer for any (finite) reservation price. In this case, the probability that the search will be successful (i.e., ends with a sale) approaches one when the number of offers approaches infinity. The search will end almost certainly after a finite number of offers. However, for some distribution functions not fulfilling the above condition, the search process may never end if the reservation price is set too high. The latter case is excluded from the following considerations. It seems realistic to assume that there is al198
199
The proof has been brought by Chow/Robbins (1963). See also DeGroot (1970), pp. 278-293, Chow et al. (1971), pp. 41-61, and Ferguson (2000), pp. 3.1-3.9. The proof of the optimality of reservation price based stopping rules utilizes the martingale theory and is presented in DeGroot (1970), pp.278-289, or McCall (1965), pp. 310-312. The reservation price property holds, when the distribution of offers is known and has a finite second moment, as shown in DeGroot (1968), p. 108, Lippman/McCall (1976a), pp. 158-159, Rosenfield/Shapiro (1981), pp. 2-3, or Rosenfield et al. (1983), p. 1052. For the discussion of the reservation price property when the offer distribution is unknown see Karni/Schwartz (1977), pp. 45-48.
116
Chapter 2: Search in Illiquid Markets
ways at least a slight chance of meeting some (irrational) individual ready to pay an extremely high price. In this case, accomplishing the sale is only a question of time independent of the reservation price.200 2.2.3.
Basic Search Model
The simplest model of the market for illiquid assets is one with observation costs only.201 The consequence of the lack of opportunity costs is that the value of money remains unchanged in time and no discounting is necessary. This assumption is rather rarely fulfilled in reality. One case when it may approximate a true sale situation is when the search process can be performed relatively quickly. After receiving an offer the seller has to choose whether to accept or to reject it. Acceptance means receiving the offered price, rejection means continuing the search. In order to choose the economically superior of these two options, their values need to be compared. The value of the first one is simply the price offered. The value of the further search is more difficult to determine; it is denoted as V. Since the seller always chooses the more valuable alternative, her net receipts (gain) from the decision on an offer Pi are defined as:202
G i = max[Pi , Vi ] with: Gi Vi
(2.1)
- net receipts from the decision on offer i - value of further search if offer i is rejected
It is straightforward that in order to maximize the receipts from the sale, an offer Pi should be accepted only if it exceeds Vi. Hence, the optimal reservation price is equal to the value of further search. Finding the optimal reservation price is, thus, reduced to the calculation of the corresponding search value.
200
201
202
The problem of the finiteness of search is more complex if one assumes that the searcher does not know the distribution of prices. The proof that also in this case the search ends after a finite number of offers is presented in Rothschild (1974a), pp. 699-701. The model presented in this section is based mainly on MacQueen/Miller (1960) and Lippman/McCall (1979), pp. 2-6. This type of search framework is, however, the most often analyzed one and is referred to in most search theoretical publications. See Stigler (1961), DeGroot (1968), Kohn/Shavell (1974), or Gastwirth (1976). See MacQueen/Miller (1960), p. 364. For the derivation of the basic model see also McCall (1970), pp. 115-117, Lippman/McCall (1979), pp. 2-6, and Ferguson (2000), pp. 4.2-4.4.
2.2 Introduction to Search Models
p *i = Vi with: p*i
117 (2.2)
- reservation price for offer i
Since the time horizon of the seller is infinite, it is not possible to analyze reservation prices for each period separately. Only a general rule for determining their values can be derived in this case.203 The assumptions that the world is static and that all offers come from the same probability distribution are helpful at this point. They result in the seller being confronted with exactly the same problem at each stage of the search. It doesn’t matter whether she decides on the 1st or the 100th offer; the future outlook remains constant. Since the past offers are lost in the considered framework, they are not relevant for the decision. This means that the problem is invariant in time (myopic) and that the optimal search policy can be determined by analyzing a single representative period. The optimal decision rule for some offer i is then optimal for any other offer as well, and the same reservation price should be applied during the whole search process.204 p *i = p *
(2.3)
for all i
The value of further search in period i for a risk-indifferent investor is equal to the expected net receipts achieved when she continues according to the assumed strategy (Ei(G)). Due to the time-indifference of the problem, also the expected receipts from future search are constant during the whole search. For simplicity they are denoted as E(G): Vi = E i (G) = E(G) with: E(.)
for all i
(2.4)
- expectation operator
E(G) is computed from the definition of the expected value as an integral over all possible outcomes weighted with their probabilities.205
203
204 205
The analytical derivation of optimal reservation prices in the infinite horizon case is possible only for certain cases including the myopic one. No general solution exists. Attempts have been made to derive approximate solution (see, e.g., Kühne/Rüschendorf, 2000 a, b). This problem is less severe if the search horizon is limited; see section 2.3.3.1. See DeGroot (1970), pp. 376-377, or Ferguson (2000), pp. 4.2. The value of search is typically derived in the literature using the iterative approach presented later in this section. Mok (2002a), pp. 8-9, and Mok (2002b), pp. 4-5, use a method similar to the one applied here.
118 E (G ) =
Chapter 2: Search in Illiquid Markets ∞
∞
−∞
−∞
∫ g ⋅ f (g) ⋅ dg = ∫ g ⋅ dF(g)
with: f(.)
(2.5)
- probability density function
Net receipts from sale in the period i depend on the value of the received acceptable price offer and on the total search cost incurred up to this point: G i = Pi − i ⋅ c
(2.6)
For the considered case, the probability of selling at a certain price pi composes of three terms: the probability of receiving an offer pi provided it is higher than p*, the probability of receiving an offer higher than p*, and the probability that all former offers were lower than p*. The second term ensures that the ith offer is acceptable and the third term ensures that the process comes as far as to the ith offer. Applying this to the equation (2.5) yields: ∞ ∞
⎡
−∞
⎣
E(G) = ∑
⎤
i −1
Pr(Pj < p*)⎥ ⋅ dp i ∫ ⎢(pi − i ⋅ c) ⋅ Pr(Pi = p i Pi > p *) ⋅ Pr(Pi > p*) ⋅ ∏ i =1 j=1
with: Pr(.) Pr(.|.)
⎦
(2.7)
- probability - conditional probability
Because all offers have the same probability distribution, the probability of any offer being under or above the reservation price is constant end equals to F(p*) and (1F(p*)), respectively, where F(.) is the cumulative offer distribution function (c.d.f.). Also the conditional expected value of an offer, provided that it exceeds p*, is constant and defined as follows: ∞ ∞
E(P P > p*) = E(Pi Pi > p*) = ∫ p ⋅ Pr(Pi = p P > p *) ⋅ dp = −∞
Hence, the equation (2.7) can be rewritten as follows:
∫ p ⋅ dF(p)
p*
1 − F(p*)
(2.8)
2.2 Introduction to Search Models ∞
[
119
]
E (G ) = ∑ (E(P P > p*) − i ⋅ c ) ⋅ (1 − F(p*)) ⋅ F(p*)i−1 = i =1
∞
∞
i =1
i =1
(2.9)
= E (P P > p *) ⋅ (1 − F(p*)) ⋅ ∑ F(p*)i−1 − c ⋅ (1 − F(p*)) ⋅ ∑ i ⋅ F(p*)i−1 Noticing that the sums in this formula are sums of infinite geometric series allows further simplification: E(G) = E(P P > p*) −
c 1 − F(p*)
(2.10)
The value of search is simply the expected value of an offer under the condition that it exceeds the reservation price less the expected costs incurred during the search. The result in (2.10) can also be achieved by following an iterative approach, which is most common in the literature and may be helpful in understanding the logic of the search model. Instead of considering all future stages of the search, one may look only one period ahead and “pretend” that the searcher has only an option to terminate the search immediately or to move to the next period.206 In this case, the value of search can be regarded as the expected result of two events: the value of the next offer if it is accepted and the expected gain in the next period if the offer is rejected. Because of the time invariability of the problem, the value of search is the same in all periods, and so are the expected net receipts. The observation cost of c for the current period is borne in any case. In effect, the following relation results:207
E(G ) = E(P P > p*) ⋅ (1 − F(p*)) + E(G ) ⋅ F(p*) − c
(2.11)
Transformation and simplification leads to the equation (2.10). Having received the simplified formula for E(G), optimization techniques can be applied to find the value of p* for which the value of search is maximal. Since no specific probability function of P has been defined, no closed form solution is possible. However, also with a defined p.d.f., an explicit solution for p* is an exception rather 206
207
A search problem that can be solved by looking only one stage ahead (i.e. is myopic) is referred to as “the monotone case”. This class of search problems was defined by Chow/Robbins (1961). See also Chow et al. (1971), pp. 54-55, or Ferguson (2000), pp.5.7-5.19. Equations of this type are often called “Bellman equations”.
120
Chapter 2: Search in Illiquid Markets
than a rule. Yet, this is not a serious practical problem as the precision achieved with numerical optimization techniques is very high even when applied on a common personal computer. The application of these methods allows finding a local maximum; however, it does not assure that this maximum is a global one at the same time. The certainty that the result is correct is given only when a single maximum (local and global) exists. Fortunately, this is always the case for the basic model independent on the p.d.f. of offers. A brief proof is presented below.208 It follows from the equations (2.2), (2.3), and (2.4) that the optimal reservation price is equal to the search value, i.e., p* = V. Substituting this in the equation (2.10) and additionally substituting for E(P | P>p*) using (2.8) yields: ∞
p* =
∫ p ⋅ dF(p) − c
p*
1 − F(p*)
(2.12)
Further rearrangement yields:209 ∞
∫ (p − p*) ⋅ dF(p) = c
(2.13)
p*
The left-hand-side term in this equation approaches infinity for small values of p*, approaches zero for large values of p*, and is strictly decreasing210. Since the right hand-side is constant, a unique root exists if c is positive; otherwise the equation has no solution.211 This means that E(G) as defined in equation (2.10) has a single maximum value with respect to p*. Finding any value of p* maximizing the expected gain locally is equivalent to finding the absolute maximum of E(G). This fact ensures that
208
209
210
211
See McCall (1970), pp.117-119, Lippman/McCall (1976a), pp. 159-161, or Lippman/McCall (1979), pp. 3-4. The same result can be achieved by deriving (2.10) with respect to p* and setting it equal to zero as the necessary condition for the existence of extreme value. The derivate of the expression on the left-hand-side of the equation is (F(p*)-1), which is negative (or at least non-positive) for all p*. It follows that the whole right hand-side term is strictly decreasing. If the seller is able to search “unpunished” as long as she wishes, she would choose an infinitely high reservation price expecting an infinitely high return. The search would last infinitely long in this case.
2.2 Introduction to Search Models
121
the application of numerical computer algorithms to calculate the optimal reservation price will provide correct results. The equation (2.13), apart from its convenience in proving the existence of a unique optimal reservation price, has also an appealing economic interpretation. Its right hand side can be interpreted as the value by which the next offer is expected to exceed the reservation price. In other words, this is the marginal revenue expected when the search is continued for one more period. In the optimal case, this marginal revenue should be equal to the marginal cost of continuing the search for one more period, that is, the cost of observing one more offer. This is in accordance with the general economic theory – the expected surplus receipts from one more search should be equal to the additional cost accrued when it is conducted. The reservation price satisfying this equation is the optimal one.212 2.2.4.
Karlin’s Model
The next model, like the previous one, utilizes the search theory to analyze the sale of an illiquid asset. It ignores, however, the observation costs allowing for discounting instead. The stress is mainly on the time aspect and on the opportunity costs of search. Fixed search costs are treated as negligible and are omitted. The model was presented by Karlin (1962) for the first time and generalized by McCall (1965). Considerations regarding the optimal reservation price from the former section are also valid for this model. When deciding on the acceptance or rejection of an offer, a rational, wealth maximizing investor will always choose the higher of two values: the offered price and the value of further search. It follows that the optimal reservation price is equal to the value of further search. However, in this model, the value of money is not constant over time. An investor is therefore concerned about the present value of future cash-flows rather than their nominal values. This means that possible future payments need to be discounted with the factor δ in order to compare them with the current offer. Analogue to (2.2) and considering (2.4), the reservation price equals the discounted expected gain from further search: pi * = δ ⋅ Vi = δ ⋅ E i (G)
212
See McCall (1970), p. 117, or Lippman/McCall (1979), p. 4.
(2.14)
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Chapter 2: Search in Illiquid Markets
If the investor rejects an offer and moves on to the next period, the base period of the discounting procedure changes as well. The choice of the optimal reservation price is then made on the basis of the discounted value of further search from the new point of view. So, the situation of the seller remains the same in all nodes of the search. Like in the former model, the search problem is time invariant. From this, it follows that the reservation price and the expected net receipts remain constant. The equation (2.14) can be rewritten as follows: p* = δ ⋅ E(G)
(2.15)
Receipts from a successful sale in a period i are defined as: G i = δ i ⋅ Pi
(2.16)
Considering that all offers are identically distributed and (2.8) is also valid for this model, the value of search can be computed as follows: ⎡ ⎤ i −1 ⎢ p ⋅ δ i−1 ⋅ Pr(P = p P > p *) ⋅ Pr(P > p*) ⋅ Pr(P < p*)⎥ ⋅ dp = ∏ j i i i i i ∫⎢ i ⎥ i =1 −∞ j=1 i >1 ⎣⎢ ⎦⎥ ∞ ∞
E (G ) = ∑ ∞
(
)
[
= ∑ δ i−1 ⋅ E(P P > p *) ⋅ (1 − F(p*)) ⋅ F(p*)i−1 i =1
(2.17)
]
Since the final expression in (2.17) is an infinite geometric series, the equation can be simplified to: E (G ) =
E ( P P > p*) ⋅ (1 − F( p*) ) 1 − δ ⋅ F( p*)
=
∞
1 ⋅ p ⋅ dF( p) 1 − δ ⋅ F( p*) p∫*
(2.18)
Another way to derive this equation is by using an iterative approach analogue to (2.11). The value of search in each period is composed of the expected receipts resulting from the acceptance of the next offer and of the present value of further search if the next offer is rejected:213
E (G ) = E(P P > p*) ⋅ (1 − F(p*)) + δ ⋅ E(G ) ⋅ F(p*) 213
The iterative derivation was used by Karlin (1962).
(2.19)
2.3 The Real Estate Search Model
123
Solving (2.19) with respect to E(G) yields (2.18). The optimal reservation price for a risk-indifferent seller can be calculated from (2.18) by maximizing the search value. However, also in this model it is generally only possible by applying numerical techniques, what makes it necessary to prove that there is a unique solution for the optimal p*. Substituting for E(G) in (2.18) according to equations (2.15) yields after rearrangement: ∞
p* − p * ⋅F( p*) − ∫ p ⋅ dF(p) = 0 δ p*
(2.20)
It can be proved that, provided δ<1, the derivative of the expression on the left handside of the above equation is strictly positive (the expression is increasing). The expression is also negative for small values of p* and positive for large values of p*. Thus, it is straightforward that a unique root to (2.18) exists and that E(G) possesses a single maximum.214 Karlin’s model demonstrates how sequential search can be optimized under opportunity costs. Like the former model with observation costs, it provides a method to specify the optimal reservation price and the proof that a unique solution to the problem exists. A possibly serious and so far unsolved problem with Karlin’s model is that the sequential search strategy does not necessarily need to be optimal in this case. If the observation costs are low compared to the discounting effects, a fixed sample strategy, which allows viewing multiple observations in one period, may be more cost-effective than the time-consuming sequential search.215
2.3.
The Real Estate Search Model
Search models presented in the former sections served as an introduction of this analysis tool. Basing on these concepts, a search theoretical approach is developed in this section to model problems arising during the liquidation process of direct real estate investments. The use of search models in this field is not new and has been discussed a number of times in the literature.216 Most papers, however, refer only indirectly to the 214 215 216
For a formal proof see Appendix A.1. See Morgan (1983). See Courant (1978), Cronin (1982), Glower et al. (1998), Balvers (1990), or Arnold (1999).
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problem of liquidity concentrating rather on various aspects of market operation. In the following sections, the standard search models are adjusted to include certain aspects of property markets, which are especially relevant in the context of their limited marketability. The resulting real estate search model constitutes the basis for the analysis in the following chapters. One of the main dilemmas encountered when constructing a search model, and actually any specific model, is the choice between realism and practical applicability. The first goal requires that subtle interdependencies between model variables and processes occurring in real markets are accounted for. The second goal requires directly observable input variables and robust methods of analysis, which are insensitive to possible misspecifications and observation errors. Unfortunately, these two goals are to a large extent mutually exclusive. Enhancing the model with more realistic features leads inevitably to an increase in complexity that makes the specification of parameter values on the basis of easily available data very difficult. Because practical relevance of the results is one of the main aspirations of this work, simplicity has been chosen as the general principle of model construction in this Chapter. The objective was to include only as many variables as were absolutely necessary to allow for the key aspects of liquidity in real estate markets. It should also be possible to estimate model parameters only on the basis of data available to professional investors. As a result, some less important aspects of the search process, which may play some role in reality, have been omitted. This is the main source of the model’s major limitations, which are discussed at the end of the section. Also several possible extensions are outlined, but the issue of their full implementation is left open for further research. 2.3.1.
Framework Modifications
The real estate sale framework is based on the standard search framework from section 2.2.1. The abstract “asset” is now specified as a real estate property, but its key characteristics remain the same. It is assumed that the seller considers it as a financial investment, i.e., does not inhabit it and is only concerned about the selling price ignoring other characteristics of prospective buyers. Offers arrive sequentially and are accepted or rejected on the basis of the reservation price. The seller operates in an infinite horizon framework in which recalls of past offers are not allowed. Unlike in the discussed basic models, search costs including observation and opportunity costs are allowed for simultaneously. This framework is further extended with a number of additional as-
2.3 The Real Estate Search Model
125
sumptions concerning, in particular, the distribution of offers, rental revenues, market fluctuations, and continuous time. Since they require a more detailed discussion, separate sections are devoted to each of them.
2.3.1.1. Distribution of Offers In the ideal case, the distribution of offers in the model should be based on the real one adequate for the studied market. However, since in most cases the real distribution either cannot be precisely assessed or it has a very complex mathematical form, an acceptable approximation is the only practicable solution. Like in many economic issues, the Gaussian normal distribution is the first and most natural choice. The probability of receiving an offer p and an offer lower than p when offers are normally distributed is determined by the following probability density function (p.d.f.) and cumulative distribution function (c.d.f.), respectively:
1 f (p) = e ~ 2⋅Π ⋅σ
−( p −μ )2
p
2⋅σ2
1 F(p) = ∫e ~ 2 ⋅ Π ⋅ σ −∞ with: µ σ ~ Π
(2.21)
−( x −μ )2 2⋅σ 2
dx
- expected value of P, - standard deviation of P, - pi-constant217.
These functions can by alternatively denoted using the standard normal distribution: 1 ⎛p−μ⎞ 1 e f (p) = ⋅ ϕ⎜ ⎟= ~ σ ⎝ σ ⎠ 2⋅Π ⋅σ 1 ⎛p−μ⎞ F(p) = Φ⎜ ⎟= ∫e ~ ⎝ σ ⎠ 2 ⋅ Π ⋅ σ −∞ p
−( p −μ )2 2⋅σ2
−( x −μ )2 2⋅σ2
dx =
1 ~ 2⋅Π
(2.22)
p −μ σ
∫
e
−x 2 2
dx
−∞
with: ϕ (.)
- p.d.f. of the standard normal distribution,
Φ (.)
- c.d.f. of the standard normal distribution.
217
~ Π is used to denote the pi-constant (3.1415…) in order to avoid confusion with the relative price variable used later on in this chapter.
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A useful feature of the normal distribution is the irrelevance of its higher moments. The distribution is fully determined by only two parameters: the mean and the standard deviation; other moments, including the skewness and kurtosis, are constant. This characteristic simplifies mathematical applications in many cases; only two parameters need to be analyzed. They also have intuitive economic interpretations. With respect to the value of a property, µ can be viewed as the average valuation by market participants or the price level on the market, and σ can be interpreted as the divergence of opinions about the value of the house. Since property prices are never negative, one could argue that normal distribution (at least in its left tail) cannot comply with the true distribution of offers, and a rightskewed distribution with non-negative values would provide a better fit. Yet, there are at least two arguments to still support the use of the normal distribution. Firstly, in the vast majority of cases, real estate prices are far away from the zero mark, and for higher values many skewed distributions, like the lognormal, t-Student, or χ2, can be approximated by the normal one. Secondly, what actually stands behind the distribution of offers are not prices but opinions about the value of a property, which by all means can be negative. One can easily imagine that someone would accept a house only if she’s paid for it. This would be the case, for example, when the personal situation of the individual wouldn’t allow her to use (or rent) the house in any way, but she would be forces to carry the maintenance costs and taxes associated with it. Of course, negative prices are never observed in reality; it doesn’t mean, however, that there are no negative appraisals by some market participants. In this sense, a positive probability of receiving a negative offer from the left tail of the distribution means meeting a potential buyer who would accept the burden of owing the property only against an appropriate compensation.
2.3.1.2. Continuous Time One of the key aspects of trading illiquid assets is time. Since the search procedure is time consuming this aspect is also present in most search models. The duration of search in the basic models is equivalent to the number of reviewed offers and deter-
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127
mined by the reservation price – the expected duration of search (D) is simply the reciprocal of the probability that an acceptable offer arrives:218 E ( D) =
1 1 − F(p*)
(2.23)
However, the equivalence between the number of offers and the duration of search is given only for fixed and identical time intervals between offers (t) as they were assumed in the original search framework. The expected time until sale is then t·E(D). This approach is easy to handle but not very realistic. Random offer arrival times are much closer to the true sale situation.219 They also comply better with the nature of the time aspect of liquidity. A standard approach to modeling stochastic arrival processes is the assumption of a Poisson process. The numbers of events (offer arrivals) in any two non-overlapping intervals are independent and no simultaneous events are possible in this process. Its characteristic feature is the lack of “memory” – the probability of the next event occurring within a given time span is independent of when the last event occurred. In other words, the probability of receiving an offer does not depend on the time elapsed since the rejections of the last offer. This approach is adequate when describing the arrival of offers in markets with independent buyers. However, if buyers’ independency is not given, the assumption of the Poisson process becomes problematic. This can be the case in very narrow markets with only few potential buyers. It is then possible that all offers are placed simultaneously, so that no real “offer flow” takes place. Also extensive marketing actions undertaken before or during the liquidation may distort the independency of offer arrivals. The realism of the model may be limited in such cases. Within a Poisson process, the Poisson probability function describes the number of events occurring during a fixed period, and the exponential probability function describes the time between two subsequent events. Thus, the latter one can be used to 218
The expected duration of search is, in fact, equal to the expected value of a geometric distribution and can be computing directly from its definition: ∞
E ( D) = ∑ i ⋅ (1 − F( p*) ) ⋅ F(p*) i −1 i =1
219
Search models with random intervals between offers are often referred to as “continues time models” as opposed to “discrete time models”. They are used by Karlin (1962), pp. 151-153, or McCall (1965), pp. 308 ff.
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describe intervals between subsequent offers (T1, T2, T3 etc.). Since they are independent and identically distributed, they can also be treated as realizations of a random variable T. Its p.d.f. is defined as follows: f ( t ) = λ ⋅ e − λ⋅t
(2.24)
The parameter λ in this formula represents the “intensity” of offer flow. It can be interpreted as the number of offers expected in a unit of time (e.g., one year). On the other hand, the reciprocal of λ is the expected time between two subsequent offers. The assumption of random offer arrival times affects all time-dependent variables in the model, in particular, the discounting factors, rental revenues, and market changes. Note that the unit of λ should be identical with the base time unit of these variables.
2.3.1.3. Opportunity Cost and Discounting A discounting factor has been already introduced in the Karlin’s model. Its main function remains the same in the real estate search model – it is used to discount future payments in order to determine their present value. The discounting effect depends on the discount rate ρ and on the discounting period t. It is calculated in the continuous time framework as:
δ = e −ρ⋅t
(2.25)
As a consequence of random time intervals introduced in the former section also the discounting factor is random. Its expected value is:220 ∞
(
)
E (δ) = ∫ e −ρ⋅t ⋅ λ ⋅ e −λ⋅t ⋅ dt = 0
λ ρ+λ
(2.26)
From the economic point of view, the (non-random) discounting rate imbedded in the discounting factor represents opportunity costs, i.e., the costs of delaying the sale for one period. In finance, ρ is usually equivalent to the profitability of the second best use of the capital bounded in the analyzed investment project. This can be the profitability of an alternative investment or the financing cost. This is also the sense of the discounting rate in the search model. One the one hand, as long as the property remains 220
See Lippman/McCall (1976a), pp.164-165.
2.3 The Real Estate Search Model
129
unsold, the investor cannot use the bounded capital for other purposes and looses the potential interest. On the other hand, if the sale proceeds are to be used for debt repayment, penalty interest has to be paid or additional costs of financial bridging arise when the sale is delayed. It is hereby assumed that financing facilities are always available at the same conditions, i.e., failing to sell within some time horizon does not affect the overall financial standing of the investor. Note that the above traditional notion of opportunity cost is valid only if a planned liquidation is considered. It can be assumed then that the investor is prepared for the eventuality of a long search and sufficient financing is secured. As discussed in the Chapter 1 in section 1.4.2.2, this is the largely unproblematic case. The lack of perfect liquidity becomes much more disturbing when a liquidity bottleneck arises suddenly. When the sale is forced by unforeseen circumstances, it is rather improbable that sufficient funds are available at normal conditions. Beside the extremely high emergency crediting costs, which are very likely in such case, not completing the sale quickly may also have other severe consequences such as the deterioration of credit rating, loss of independency, or even bankruptcy. Since these are the true costs of delaying the sale, they should be used for discounting. However, determination of appropriate values may prove very difficult in this case and require an individual analysis of investor’s situation in an “emergency” event.
2.3.1.4. Rental Revenues One of the main features of property investments is the twofold source of returns: they came from value appreciations and from rental revenues. This allows treating real estate as a combination of equity and debt221 and should be accounted for when modeling liquidation of property investments. On the one hand, an investor attempts to realize the best achievable price – sale means the flow of funds corresponding with the accepted offer. On the other hand, as long as the property remains unsold, she receives the monthly rent – sale means in this context the cease of the income cash flow. The analogue situation arises at the purchase of real estate: sale means paying the price, but also starting to receive the rental income. The easiest way to implement rental revenues (denoted as h) is by combining them with observation costs. It requires the assumption that rent payment periods coincide 221
See Booth et al. (1989) or Jandura (2003), 28 ff.
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with offer arrival times. But even if no perfect coincidence is given, it is always possible to split the payments and attach them to the respective offers without incurring major errors. It is also theoretically conceivable to consider these factors separately, but it would lead, at least for certain subperiods, to the loss of model’s time invariance and unnecessary complicate the analysis. Since a practicable approach should be kept as simple as possible, it is assumed that each offer arrival is associated with a net payment of (h-c). In most cases of real estate investments, pure observation costs are very small compared to other amounts involved. The cost of making a house available for visitation or placing an advertisement in a newspaper is practically near to zero. The costs of employing a broker who organizes visitations and reviews the offers may be slightly higher. However, since the brokerage commission is usually calculated as a percentage of the sale price, it is easier and more precise to consider them together with other transaction costs and make respective corrections of offer values. Altogether, observation costs during the search for a buyer may be treated as negligible in the real estate case allowing for further simplification.222 Recapitulating, for the sake of simplicity the observation cost c is ignored in the following considerations. Only the rental revenue h is allowed for as a “negative observation cost”. A slight problem with the coincidence between rent payment periods and offer arrival times arises when the latter are assumed random. In order to retain the assumption that the rent is payable each time a new offer is received, variable, time dependent rental revenues need to be introduced. This is done by redefining h as a periodical (e.g., monthly or annual) rental rate, which is multiplied with the length of the time interval to determine the rent due at the moment of an offer arrival. Thus, h·Ti stands for the rent received between the (i-1)th and ith offer. This approach still leads to a slight bias since the capitalization of an irregularly incurred cash flow does not yield exactly the same result as the capitalization of payments made in fixed time intervals. However, errors generated by this effect should be relatively small in the typical case.
222
However, one needs to bear in mind that this may not hold for the search for a buyer since the assessment of offers requires a separate appraisal of each property, which may proof costly.
2.3 The Real Estate Search Model
131
2.3.1.5. Market Uncertainty Up to this point the market has been assumed stable in the sense that the distribution of arriving offers remained unchanged over time. Yet, the possibility of market changes during the search constitutes an important factor in the liquidity analysis; it is one of the main sources of liquidity risk formulated in section 1.2.2.3. There are numerous possibilities how market uncertainty can be introduced in the search problem. A relatively simple approach based on a linear trend with random distortions is presented here. Some more sophisticated alternatives are discussed in section 2.3.3.2. A deterministic linear market trend of τ is assumed.223 On the one hand, it affects the valuation of the property by each market participant in the same manner – a shift in the whole offer distribution results. On the other hand, it also applies to the rental income. The fact that both prices and rents are subject to changes at the same rate allows an easy implementation of the trend in the model by adjusting the discounting factor δ. Since the discount rate ρ and the trend rate τ affect the (present) value of future payments, their effects, despite different directions, are similar. The trend-corrected discounting factor is thus: ~ δ = e τ⋅t ⋅ e -ρ⋅t = e -(ρ−τ)⋅t
(2.27)
Along with the deterministic trend random, deviations are allowed for with respect to property prices, or more specifically, to the distribution of offers. They are interpreted as unexpected fluctuations of the real estate market between offer arrivals, which drive the price level away from its expected path. Although these deviations can be modeled as a series of random, independent variables, it is more convenient to capture the total effect within one variable referring to the moment of sale; it is denoted as A and referred to as “market uncertainty”. The effect of A on the offer distribution is assumed to depend linearly on the time until sale and is captured with the factor (1+A·T). The compound rather than the (otherwise common) continuous time calculation has been chosen at this point for easier model calculations. Given the sale at the ith offer, the combination of the deterministic trend with the random deviation results in the following total change of the price level:
223
Search models with trend are often applied in the valuation of financial options. See Jacka (1991) or Beibel/Lerche (1997).
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Chapter 2: Search in Illiquid Markets i
Pi = P ⋅ (1 + A ⋅ ∑ Tj ) ⋅ e
i
− τ⋅ ∑ Tj j=1
(2.28)
j=1
The deviations from the trend occurring between offers and, consequently, the market uncertainty factor A are assumed to have the expected value of 0. This means that the seller expects the price level to follow the general trend. However, she is not certain whether this expectation will come true. The level of this uncertainty can be expressed as the standard deviation of A (σA). The deviations from the trend are assumed to be known immediately as they occur. Thus, the seller always knows the current distribution of offers when setting the reservation price in each period including all expected and unexpected changes up to that time. Note that random fluctuations affect only the price but not the rental income, which strictly follows the deterministic trend. This approach seems to comply with reality. Properties are usually appraised on the basis of discounted future cash flows;224 it is therefore plausible that prices and rents change approximately at the same rate in the long run.225 In the short term, however, rental revenues should be much less volatile than property prices. The latter can be subject to sudden changes resulting from economic shocks (e.g., stock market turmoils may lead to the flight of funds into real estate) or legal changes (e.g., new tax regulations). Furthermore, the price level can also be influenced by more or less random fluctuations in the supply and demand or by changes in the attitude of users to certain types of properties. On the other hand, a stable increase of rents in place is often secured by appropriate indexing clauses in lease contracts as well as long durations of the leases.226 To sum up, it is rather improbable that the income from the running leases would vary in response to random short-term market fluctuations, like property prices do, but it is quite probable that it would be steadily adjusted according to an indexing clause, which reflects the expected trend. The variability of rents can therefore be neglected.
224
225
226
On the income approach to real estate appraisal see Appraisal Institute (2001), Chapter 20, or TEGoVA (2003), Appendix 1. A practically important advantage of the assumption that prices and rents follow the same trend is the preservation of the myopic property of the model. Note that the stable increase of rents due to indexing refers only to the existing rental contracts; new lettings are made on the basis of the current market situation. However, only the former case is relevant in the considered liquidation framework.
2.3 The Real Estate Search Model
133
Nevertheless, the use of the uncertainty parameter A to model market uncertainty is not entirely unproblematic. Assuming that only unexpected events cause departures from the trend (information efficiency thesis227) allows treating price changes in terms of a stochastic process. Modern capital market theories demand that such price processes have certain properties if a long-term equilibrium is to be preserved; in particular, they should follow a random walk. However, the assumption about the expected value of the deviations from the trend being equal to 0 turns out to be inconsistent with the random walk hypothesis. The information efficiency thesis implies that assets’ returns in subsequent periods are independent, that is, price movements are not influenced by the past. If returns are also assumed to be identically distributed than, according to the central limit theorem, compound returns over longer periods follow a random walk and are asymptotically normally distributed. Since the convergence to the Gaussian curve is very quick, normal distribution can be assumed for continuous returns (rc). This results in a logarithmic normal (lognormal) distribution of discrete returns (rd).228 rc = ln(e rc ) = ln(1 + rd ) ~ N(μ r , σ r ) with:
(2.29)
µr - expected value of continuous returns σr - standard deviation of continuous returns N(.) - normal distribution operator
In consequence, the expected price change is given as: E(1 + rd ) = e μ r +σ r
2
2
(2.30)
which is larger than 1 for any non-negative µr. So, even if the market was stable in terms of a zero expected (continuous) return, market prices would follow an increasing trend.229
227
228
229
In general, a market is efficient if all information that influences the value of an asset is reflected in its market price immediately as it occurs. Depending on the type of information the literature differentiates between weak, semistrong and strong form of efficiency. See Fama (1970), Levy/Sarnat (1984), Chapter 19, or Elton et al. (2003), Chapter 17, as well as the literature cited there. For a review of the stochastic properties of market prices see Osborne (1959) or Fama (1965), as well as a number of papers reprinted in Cootner (1964). The source of this effect is the asymmetry of upward and downward market changes in the discrete approach. An increase of x% and a subsequent decrease of x% do not lead to the initial value. E.g.,
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Applying the above logic to the search model leads to the conclusion that the expected deviation from the trend E(A) = 0 implies in fact an expected continuous trend of less than τ. To demonstrate it, denote the continuous deviation from the trend as Ã. If price fluctuations around the trend where independent and identically distributed than A and à should be lognormally and normally distributed, respectively. In analogy to (2.30), the expected effect on the price level in the time period T would be then given as:
(
)
~
E (1 + A ⋅ T) ⋅ e τ⋅T = E(1 + A ⋅ T) ⋅ e τ⋅T = e T⋅E(A )+T
2
~ ⋅V ( A ) 2
⋅ e τ⋅T
(2.31)
Note that E(1+A·T) = 1, and consequently E(A) = 0, only if E(Ã) = - T2·V(Ã)/2 < 0. Thus, the expected discrete deviation from the trend is zero only if the expected continuous deviation is negative. The latter leads in the long run to a negative drift away from the assumed trend – the actual trend rate is then less than τ. An adjustment of the market trend is a possible solution to the above problem. One could compensate for the above distortion by increasing τ by T2·V(Ã)/2. A drawback of this approach is, however, that the adjusted trend is also applied to rents, so that the source of the error is shifted from the price to the rental component. Another solution is to assume A to be normally (instead of lognormally) distributed accepting the resulting inaccuracy. This approach seems to be more appropriate for modeling real estate liquidity. The reason is twofold: Firstly, the lognormal distribution can be often well approximated by the normal one; simulations have shown that in most cases differences between price paths resulting from sequences of discrete market changes generated from a normal and a lognormal distribution become practically relevant only after several hundreds or even thousands of periods. It is rather improbable that any search in a real estate market would take so long. Secondly, the thesis of normally distributed continuous returns is controversial even for publicly traded, highly liquid stocks;230 it is even more so for real estate returns. Empirical studies in the USA, UK, Germany, and Australia found evidence of leptokurtosis and asymmetry (skewness) in the distributions of property returns in these countries. The results were significant enough to re-
230
starting with the price level of 1 a sequence of market changes -10% and +10% result in 1·0,9·1,1 = 0,99 < 1. This asymmetry does not occur in the continuous approach due to the logarithm; subsequent returns offset each other (e10%-10% = 1). See Mandelbrot (1963, 1967), Fama (1965), Clark (1973), and more recently Chunhachinda et al. (1997), Rehkugler et al. (1999), pp. 157-160, Aparicio/Estrada (2001), or Harris/Küçüközmen (2001).
2.3 The Real Estate Search Model
135
ject the normality hypothesis.231 Thus, most probably, errors could not have been avoided even with a random walk conform approach. Therefore, discrete price changes in the search model are assumed to be normally distributed. Due to the good approximation of the lognormal distribution by the normal one for realistic durations of the search process, the resulting bias should be acceptably low.
2.3.1.6. The Relative Approach A serious practical problem with the traditional search models is that they operate on absolute variable values. Specification of model parameters necessary for its practical application may then turn out to be a difficult forecasting problem. Especially, if liquidity is to be allowed for in investment decisions, the respective analysis needs to be accomplished long before the sale process begins, and possibly even before the asset is purchased. Thus, the analysis of strategic liquidation at the end of investment horizon, which is usually very long term for real estate, requires the prediction of the market situation many years ahead; the quality of such forecasts is usually very low. The problem becomes even more severe in the case of an unexpected liquidation. Without the knowledge of the presumed timing of the liquidation, a forecast becomes practically impossible. In order to mitigate the above problem, it is advantageous to redefine the model in a way in which it relies as little as possible on precise predictions of the future. The most straightforward way is to use relative rather than absolute variable values. The state of the market at the beginning of the search, measured as the average valuation of the asset or the expected value of the offer distribution (µ), is a reasonable reference point for this purpose. Thus, model variables in the relative approach are computed by relating the original ones to µ: Π, π
- relative price offer,
π*= p*/µ
- relative reservation price,
Γ = G/µ
- relative net receipts from sale,
231
See Miles/McCue (1984), Myer/Webb (1993), or Young/Graff (1995) for studies of the real estate returns distribution on the U.S. market, Maurer et al. (2004) for the English and German, Morawski/Rehkugler (2006) for the German market, and Graff et al. (1997) for the Australian market. See Morawski/Rehkugler (2006), p. 21-22, for a review of research on the normality of real estate returns.
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ν = σ/µ
- volatility coefficient of offers,
γ = h/µ
- relative rent.
The new variables in the model receive respectively new interpretations. Offers are now quoted not as absolute values but as fractions of the average valuation by potential buyers – an offer of 1.1 means 10% above the average and an offer of 0.9 means 10% below the average. In consequence, the distribution of offers needs to be redefined. The mean of the new “relative” distribution, with the c.d.f. denoted as FR, is equal to 1 and the standard deviation is ν = σ/µ. The values of the relative distribution function are, however, the same as the respective values of the absolute distribution function, i.e., FR(π) = F(p). The assumption about the normal distribution of offers in section 2.3.1.1 becomes especially advantageous at this point. It is fully defined by the mean and the standard deviation. Since the mean always equals 1 in the relative approach, only the standard deviation of the relative offers matters. Thus, the relative heterogeneity of opinions among potential buyers remains the only relevant parameter. With the new notion of an offer, reservation prices and sale receipts also receive new interpretations – they are now expressed as fractions of the average valuation of the property. One of the main practical advantages of the relative approach is higher stability of the main model parameters. In the first place, relating all variables to the average market valuation “neutralizes” the role of the price level for the results of the analysis. This parameter, which was probably most difficult to forecast in the absolute version of the search model, is now absent. Furthermore, it seems plausible that the relative dispersion of opinions about the value of a property is more stable than the absolute one. An argument in favor of this thesis is the stability of the observed appraisal errors understood as percentage deviations of appraisals from the subsequent transaction prices.232 Assuming that most real estate investors obtain and follow such appraisals when buying or selling properties, the dispersion of offers should also display a similar stability. Same applies to the “relative rent”, which can be interpreted as the ratio of the rental revenue to the value of a property; it corresponds with the property’s income return.233 This parameter should to be also less volatile than the absolute rent. Although it may 232 233
See, e.g., RICS (2005). For other illiquid assets, for which observation costs are not negligible, this parameter would take the form of the “relative observation cost” (h-c)/µ.
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137
fluctuate in the short term, one would expect that real estate rents and prices should not go too far apart in the long run. This is due to the role of rents as key value drivers of real estate; their long term increases or decreases must result in respective (though possibly lagged) adjustments in property prices. The only reason for a sustainable change of γ is a change in the fundamental characteristics, especially the risk, of the considered property market. This, however, is much less frequent than common market fluctuations. To sum up, the relative approach seems to allow for more stability in the main model variables and, thus, be less prone to estimation and forecasting errors. It also seems to be more suitable for analyzing liquidity. By relating all variables in the model to the average price level at the beginning of the liquidation process, the effects of changes in the market situation, which may occure before the sale is initiated, are disregarded. Thus, it does not matter where the market stands at the moment of the liquidation; the only issue that matters is the relation of the net sale receipts to the current average valuation of the property. 2.3.2.
Model Design
The real estate sale model results as a combination of both basic models from sections 2.2.3 and 2.2.4 modified according to the propositions in the previous sections.234 The model analysis follows the same scheme as in the former sections. Since many of the arising problems are similar to those already dealt with, the presentation is abbreviated at these points. The present value of the net sale receipts, provided that the sale was accopmlished at the ith offer, is composed of the discounted accepted price after expected and unexpected market changes and the sum of discounted rental revenues incurred so far: i ~ ~ ~ −( ρ− τ )⋅Tj Γi = Π i ⋅ (1 + A ⋅ Ti ) ⋅ e −(ρ−τ)⋅Ti + ∑ γ ⋅ Tj ⋅ e
(2.32)
j=1
i ~ with: Ti = ∑ Tj
- total search duration provided the ith offer is accepted
j=1
234
A similar, though not identical problem is dealt with by Lippman/McCall (1976a), pp. 163-166, and Lippman/McCall (1986).
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Since the expected value of the market uncertainty factor is 0, i.e., the market is ex ante not expected to depart from the long term trend, the seller is faced with the same situation at each offer arrival. Also opportunity costs and rental revenues do not allow distinguishing one period from another. The search problem remains therefore time indifferent and the model retains its myopic property. However, due to the possibility of a market change, the nature of the reservation price is slightly different. At this point, the assumption that the seller knows the current market situation when setting the reservation price in each period is relevant. The past market development is then already known and the past market uncertainty factors as well as time intervals are not random any more. Thus, the reservation price can be defined as referring only to the “base offer” Π.235 The acceptance condition can be denoted equivalently as Π > π* and is identical with the one in the former models. However, its meaning is now different ~
~
and should be read as: “accept any offer at or above π*·(1+a· t )”, with a and t being ~ the known realizations of A and T , respectively. Furthermore, the randomness of offer
arrival times requires that also the moment of receiving the first offer is uncertain. Thus, the search is assumed to begin with the decision to sell – the first offer arrives after a random period of time T1. Given the c.d.f. of the market uncertainty factor denoted as FA(.) and the c.d.f. of the time intervals denoted as FT(.), which is exponential (i.e., FT(t) = λ·e-λt), the value of search is analogue to (2.7) and (2.17) and given as follows:236 i ~ ~ ⎛ ~ −( ρ− τ )⋅Tj ⎞ ⎟= E (Γ) = E⎜⎜ Π ⋅ (1 + A ⋅ Ti ) ⋅ e −( ρ−τ)⋅Ti + ∑ γ ⋅ Tj ⋅ e ⎟ j=1 ⎝ ⎠ i i ∞ ∞ ∞ ∞ ⎛ i ∞ ⎡∞ i ( τ −ρ ) ∑ t j ( τ−ρ )⋅ ∑ t kj ⎞ j=1 j=1 ⎜ ⎟⋅ ⎢ + ∑γ ⋅ tj ⋅e ∑ ⎢ ∫ ... ∫ ∫ ...∫ ∫ ⎜ πi ⋅ (1 + a ⋅ ∑ t j ) ⋅ e ⎟ j=1 i =1 −∞ −∞ 0 0 −∞ j=1 ⎝ ⎠ ⎣
(2.33)
i ⎤ Pr(Π i = πi Π i > π *) ⋅ (1 − FR (π*)) ⋅ FRi−1 (π*) ⋅ dπi ⋅ ∏ dFT ( t j ) ⋅ dFA (a )⎥ j=1 ⎦
235
236
Since the base offers have the same distribution in all periods, i.e. Πi are independent and identically distributed for all i, they can be denoted as one random variable Π. Note that the terms “search value” and “expected (relative) net receipts” are used as synonyms throughout the analysis. They are also abbreviated as “expected receipts” in some context. In all these cases, the expected value of all discounted payments experienced by the investor during the liquidation (i.e., E(Γ)) is meant.
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139
The simplification of the integrals and sums in this formula is presented in Appendix A.2. The same result can be achieved by utilizing model’s myopic property and following the iterative way of reasoning, as shown below. Nevertheless, the “long” approach is still useful for the derivation of the conditions for the existence of a finite solution as well as for the derivation of other statistical parameters discussed in following chapters. The value of search (i.e., the expected net receipts from sale) in period t results from the expected outcome of the decision on the next offer and on the value of the further search. Thus: E t (Γ ) ˆ ) ⋅ E(Π Π > π *) ⋅ (1 − F(π*)) + E (Γ) ⋅ F(π*) + T ⋅ γ ⎞⎟ ⎞⎟ = E⎛⎜ e −T⋅(ρ−τ) ⋅ ⎛⎜ (1 + T ⋅ A t +1 ⎝ ⎠⎠ ⎝
(2.34)
with T and  referring to the time interval and the market uncertainty effect in one representative period, respectively. Since E(Â) = 0 and T is exponentially distributed, the expected value of Γ is:
E t (Γ ) =
λ λ λ⋅γ ⋅ E (Π Π > π *) ⋅ (1 − F( π*)) + ⋅ E t +1 (Γ) ⋅ F( π*) + λ+ρ−τ λ+ρ−τ (λ + ρ − τ)2
(2.35)
Time invariance of the search problem requires that Et(Γ) = Et+1(Γ) = E(Γ). Considering it yields after rearrangement:
E (Γ ) =
⎞ ⎛ λ γ ⎟ ⎜ E(Π Π > π *) ⋅ (1 − FR (π*)) + ρ − τ + λ ⋅ (1 − FR (π*)) ⎜⎝ λ + ρ − τ ⎟⎠
(2.36)
The final step in the construction of the model is the implementation of the assumption about the normal distribution of offers as defined in section 2.3.1.1. It is accomplished by substituting the general c.d.f. FR(.) with the cumulative normal distribution function FN(.) having the mean of 1 and the standard deviation of ν:237
237
The analogue formula for the search model with observation cost can be found in Gastwirth (1976), p. 76 or Feinberg/Johnson (1977), p. 1595. Also the solution for the uniform offer distribution is derived there.
140
E (Γ ) =
Chapter 2: Search in Illiquid Markets
⎛ ⎞ λ γ ⎜ (1 − FN (π*)) + σ 2 ⋅ f N (π*) + ⎟ ρ − τ + λ ⋅ (1 − FN (π*)) ⎜⎝ λ + ρ − τ ⎟⎠
(2.37)
The formula can be analogically rewritten in terms of the standard normal distribution:
E (Γ ) =
⎛⎛ ⎞ γ π * −1 ⎞ ⎞ ⎛ π * −1 ⎞ ⎜⎜ ⎜1 − Φ⎛⎜ ⎟ (2.38) ⎟+ ⎟ ⎟ + ν ⋅ ϕ⎜ ⎛ ⎛ π * −1 ⎞ ⎞ ⎝ ⎝ ⎝ ν ⎠⎠ ⎝ ν ⎠ λ + ρ − τ ⎟⎠ ρ − τ + λ ⋅ ⎜1 − Φ⎜ ⎟⎟ ⎝ ν ⎠⎠ ⎝ λ
The reservation price maximizing the expected net receipts can be computed numerically provided a unique solution exists. This was the case in both earlier discussed models but only under certain conditions: c>0 (standard model with observation costs) and δ<1 (Karlin’s model). The second condition becomes now: τ < ρ. It is fulfilled only as long as the trend does not exceed the opportunity cost. Otherwise future offers become more and more valuable with time, and infinite net receipts are possible.238 The first condition takes the form of γ < 0. Since rental revenues are positive, it is clearly never fulfilled in the real estate case. However, it turns out that it is not necessary for the existence of a unique solution in this framework. Since γ is subject to discounting, one can expect that its influence on the value of search weakens as far future is considered. This can be formally proved by analyzing the function (2.36). It reveals that E(Γ) is increasing for small values of π* and is decreasing for large π*. Since E(Γ) is continuous, it has always at least one maximum. Further analysis reveals that independent of the distribution of offers only one maximum exists.239 The typical plot of the expected net receipts as a function of the reservation price is depicted in Figure 22: it takes the value of λ/(λ+ρ-τ)·(E(Π) + λ·γ/(λ+ρ-τ)) for very low reservation prices and the value of λ·γ/(λ+ρ-τ)·(ρ-τ) for very high reservation prices. The former one is, in fact, the expected outcome of accepting the first incoming offer and the latter one is the (discounted) value of infinite rental payments. Hence, a single absolute maximum value exists.
238
239
This condition arises from the finiteness condition of integrals as well as infinite sums occurring in the computation of the search value. See Appendix A.2. An extensive proof of the existence of a single maximum of E(Γ) is brought in Appendix A.3.
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1,4
Expected Net Receipts
1,2
1
0,8
0,6
0,4
0,2
π*opt
0 0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
1,6
1,8
2,0
Reservation Price
Figure 2-2: Determination of the optimal reservation price in the real estate search model240
From the economic point of view, it is important that no economically pointless behavior results from following the optimal strategy. Especially, it should be ensured that the accepted selling price is not below the searcher’s valuation of the property. The individual valuation arises from the capitalized cash flows generated by the investment; in the model setting, it corresponds with the sum of discounted rental revenues. Recall from equations (2.2), (2.3), and (2.4) that the reservation price maximizing the value of the search always equals the expected net receipts from continuing the search. In the considered case, it means that π* = E(Γ|π*). Since the maximal value of E(Γ) cannot be smaller than the discounted infinite annuity of rental payments, the same must be valid for π*. Thus, the expected receipts from holding the property infinitely long determine the lowest possible reservation price. In other words, the reservation price always lies above seller’s valuation of the property. The formula (2.38) allows also analyzing the influence of other model parameters on the expected receipts from search. In some cases the effects are clear: an increase of the rent (income return) or the trend factor leads to an increase of the expected re240
Following parameters were used: offer volatility (ν) = 15%; trend (τ) = 5%; rental income (γ) = 5%; discounting rate (ρ) = 15%; offer frequency (λ) = 52.
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ceipts, but an increase of the discount rate decreases E(Γ). The effects of changes in the dispersion of offers or in the offer frequency are, however, ambiguous and depend on the values of other parameters. A higher ν increases the probability of receiving extreme offers. Thus, for reservation prices above the average valuation (π*>1), the probability of receiving an acceptable offer increases with ν. One would intuitively expect a higher expected sale price and a higher value of search in this case. However, this effect may reverse in some situations. Since a (constant) reservation price is easier to achieve with a higher ν, the search is completed more quickly. This means smaller opportunity costs but also lower rental revenues. If the rental income is high compared to the effective discount rate (i.e., discount rate less trend), shortening the duration of the search may negatively affect the total expected receipts. For reservation prices below the average valuation (π*<1), the probability of receiving an acceptable offer decreases with ν, so the effects of changes in the dispersion of offers are unclear even with low rental income rates.241 The impact of changes in the offer arrival rate is similarly ambiguous. Without rental income, higher offer frequency leads to shorter search durations reducing the discounting effects and, thus, increasing the expected value of the receipts. However, if γ is high compared to (ρ-τ), shortening the search may negatively affect the expected gain.242 Summing up, the consequences of changes in the model parameters are often results of complex interactions between variables, and their final effects can only be evaluated in the context of a concrete situation. 2.3.3.
Limitations and Possible Extensions
As already noted before, the real estate search model presented here is the result of a compromise between realism and practicability. The central idea was to design a model that would capture the main features characterizing direct real estate investments but remain simple enough to allow easy practical application. This led inevitably to simplifications, which in certain circumstances may be considered as going too far. They can refer to the characteristics of the analyzed markets (e.g., the regime of the price process), to the person of the investor or to the nature of the search strategy. Still, 241
242
The effects of changes in the dispersion of the offer distribution are analyzed by Venezia (1980) or Balvers (1990). They consider, however, the effect with the simultaneous adjustment of the reservation price. At first glance, it may seem startling that the level of market uncertainty does not affect E(Γ). Since the expectation about the value of this parameter is always zero, it does not occur in the computation of the expected net receipts. In fact, only the volatility of A (σA), will play a role when the risk aspect of liquidation is considered in the forthcoming chapters.
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143
many of these problems can be solved on the basis of the search framework by applying respective adjustments to the model. In most cases, however, such adjustments considerably increase the grade of model’s complexity, require additional assumptions about the relations between variables, or prevent a closed-form analytical solution. Therefore, this section provides only an outline of propositions how certain special features can be incorporated into the model; their full implementation is beyond the scope of this work. For better tractability and easier comparability with the literature, the basic search model with observation costs is used as the reference point in most cases, but analogue results are valid for any other type of search model including the real estate one.
2.3.3.1. Bounded Search Horizon No limitation to the duration of the search has been set in the basic model. The investor can search for the buyer as long as needed in order to optimize the receipts from sale. This is unproblematic when the liquidation is planned and the only incentive to sell quickly is the loss of income from an alternative investment. Often, however, investors face some kind of a more or less explicit deadline up to which the sale must be completed. The introduction of such a deadline in the model results in certain complications; in particular, time invariance is lost, and no universal search strategy valid in all periods exists any more. The simplest way to allow for a liquidation deadline is to set a maximum number of offers that can be viewed.243 A deadline means in this case that after rejecting all allowed offers, the investor is forced to accept the last one, no matter how good or bad it is. The solution to this problem is, in fact, more straightforward and more general than in the case of the infinite horizon. The complete strategy, including reservation prices and expected receipts, can be derived using “backward induction”.244 Optimal reservation prices are determined for each stage of the search separately starting in the future, at the deadline, and moving back to the beginning of the search. Since the value of the search is equal to the expected net receipts from accepting the next offer and from continuing to search, reservation prices in all periods can be obtained recursively. Deter243
244
The finite horizon version of the search problem with a maximal number of offers – the so called Cayley-Mosel-problem – is, in fact, the oldest one. See Cayley (1896), p. 587; for a discussion see also Ferguson (2000), pp. 2.7-2.10. See DeGroot (1970), pp. 277-278, Chow et al. (1971), pp. 49-50, or Ferguson (2000), p. 2.1.
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mining the optimal reservation price and the value of search for the basic model with observations costs proceeds in this case as follows:
there is no strategy if the seller gets as far as to the deadline – any offer has to be accepted then, so the expected net receipts are simply: E(GN) = E(PN) – c
one period before the deadline the seller has only the choice between the next and the final offer; she should accept the next one only if it exceeds the expected net receipts from the last offer; the value of search is then: E(G N-1) = E(PN-1| PN-1> pN-1*)·(1-F(pN-1*)) + E(GN)·F(pN-1*) – c
in earlier periods, offers should be accepted if they exceed the expected value of further search; expected receipts can be determined recursively using search values of the later periods: E(Gn) = E(Pn| Pn> pn*)·(1-F(pn*)) + E(Gn+1)·F(pn*) – c
Thus, the reservation price in this framework is not constant. It comes closer to E(P) as the deadline draws nearer. This effect is less distinctive when more offers are left until the deadline. Nevertheless, one cannot assume that the same strategy can be applied in all periods and that the value of search is constant. With an increasing number of periods the calculation effort increases as well. The existence of a deadline becomes more problematic when continuous time with random arrival times is considered.245 The deadline is then defined by the maximum duration of the search in time units rather than by the number of offers, which becomes random in this case. With the offer flow being a Poisson process, as assumed in the real estate search model, there is always a positive probability that any offer is or is not the last one. Thus, the search strategy can be based only on the expectation about the number of offers remaining until the deadline. The value of search at the moment t can be then defined as the expected net receipts over scenarios with 0 to infinitely many remaining offers: ∞
E(G t ) = ∑ E(G N = i) ⋅ Pr( N = i t )
(2.39)
i =0
245
To my best knowledge, the case of search in a finite search horizon with continuous time has not been analyzed in the literature yet.
2.3 The Real Estate Search Model
145
Note that providing an analytical solution to the above formula is very difficult, if at all possible. In particular, the determination of the expected net receipts conditioned on the number of remaining offers requires considering a separate finite horizon search problem for each value of i. Since it is not possible to review all different scenarios explicitly, it seems also impossible to determine the expected result of further search and thus, the optimal search strategy analytically. Hence, no clearly defined optimal search strategy seems to exist in this case, and merely an estimation converging to the optimal solution may be possible.
Both finite and infinite horizon models have their advantages and disadvantages. The choice of the preferred model type is therefore not straightforward. It may seem at first sight that investors indeed set deadlines when selling assets like real estate. They may be enforced by legal regulations or result from investors’ internal rules. A deadline may also have an informal character – be a kind of investor’s obligation to herself to have the matter dealt with up to some date. Despite the fact that some kind of a deadline is often present when selling an illiquid asset, its nature does not fully correspond with the limited horizon in a search model. The latter one cannot be exceeded in any case – the search process ends no later than with the last offer. Deadlines faced by investors in reality cannot be broken without incurring (substantial) costs. This means that an investor could continue to search after the deadline if she accepts the consequences. These could be legal fines, additional holding costs, or even bankruptcy if not-selling would lead to the loss of solvency. These additional costs may be very high, but they are not infinite, so that it might even be optimal to accept bankruptcy rather than a very low sale price. In this sense, a deadline is a point at which the opportunity cost of continuing the search increases rapidly. The correct way to handle it would be therefore to split the search in two phases: a finite horizon phase ending with the deadline and an infinite horizon phase applied after the deadline. The usual opportunity costs (i.e., the profitability of an alternative investment or financing costs) apply in the first phase; in the second phase, however, deadline-breach costs (e.g., bankruptcy costs) are to be used. The value of search after the deadline can be applied as the receipts from rejecting the last offer in the finite horizon phase. The above considerations can be further generalized by assuming that investor’s time preference changes not only at the moment of the deadline but is generally timevarying. This probably complies with the attitude of sellers in many situations. As time
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passes, the necessity of accomplishing the sale becomes more and more urgent. This corresponds with an increasing discount rate in the model’s nomenclature. In consequence, the value of further search and the reservation price decrease; the seller is more prone to accept lower offers.246 The discounting rate in this setting is a function of time (duration of search), and the existence of a deadline is its special case. Although the idea of time varying opportunity costs is very appealing from the theoretical point of view, its practical implementation is extremely difficult, if at all possible. With an infinite search horizon on the one hand, and the lack of time invariance due to the varying discount rate on the other hand, no definable search strategy is in sight.
2.3.3.2. Dynamic Market It has been assumed so far that the distribution of offers is constant or changes according to a predefined linear trend with random deviations. However, in some markets non-linear changes are more typical taking most often the form of a cycle. Cyclical patterns in prices, rents, and other variables are also observed on many real estate markets.247 A linear trend may be still used to approximate their development if the search period is short compared with the length of the cycle, but the error will increase with an increasing duration of search. A cycle and other regular patterns can be incorporated in the search model by introducing a limited number of feasible offer distribution functions. Changes in the market are reflected by switching between these functions. Thus a cycle means a recurrent order of market regimes. Using an iterative notation analogue to (2.11) one may describe the dynamic version of the model with a set of equations, one for each state of the market. For a strictly cyclical economy, the market state denoted as S1 is always followed by the state S2, which again is always followed by S3 etc. The last state SN is followed by S1 starting a new round of the cycle. The market states are described by the corresponding probability distribution functions of offers: F1, F2, …, FN. A search
246
247
The effect of a decreasing reservation price when the search takes longer than expected may, however, also result from the revision of seller’s idea about the distribution of offers. The latter issue is discussed in the section 2.3.3.3. For studies on real estate cycles see Pyhrr et al. (1999) and the literature cited there.
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147
model with observation costs and a dynamic environment can then be defined by the following set of equations:248
(
)
E(G1 ) = E(P1 P1 > p1* ) ⋅ 1 − F1 (p1* ) + E(G 2 ) ⋅ F1 (p1* ) − c
(
)
E(G 2 ) = E(P2 P2 > p ) ⋅ 1 − F2 (p ) + E(G 3 ) ⋅ F2 (p*2 ) − c * 2
...
* 2
(
(2.40)
)
E(G N ) = E(PN PN > p*N ) ⋅ 1 − FN (p*N ) + E(G1 ) ⋅ FN (p*N ) − c Note that this approach is similar to the inverse deduction in the finite horizon model – the expected gain from search in one period is defined by the conditional expected receipts form the next offer, provided it is accepted, and the expected value of further search if the next offer is rejected. The difference is only in the last term of the last equation – instead of the expected value of the last offer, when the search deadline is reached, the expected value of search from the beginning of the cycle is used. By solving the equations with respect to pi*, the optimal strategy can be determined. The optimal reservation prices in different points of the cycle are not equal, but if the search continues for more than one cycle, the order of subsequent reservation prices remains the same. For further generalization, it can be assumed that the order of market states is not deterministic and that random jumps are possible. This type of market dynamics can be described in terms of a Markov chain with a specified transition matrix Ψ:249 ⎡ ψ11 ψ12 ⎢ψ ψ 22 21 Ψ=⎢ ⎢ M M ⎢ ψ ψ N2 ⎣ N1
L ψ1N ⎤ L ψ 2N ⎥ ⎥ O M ⎥ ⎥ L ψ NN ⎦
(2.41)
The elements of the matrix are the probabilities of switches from one state to another in the following period. E.g., ψ12 is the probability of a change from the state S1 to S2; values on the diagonal of the matrix are the probabilities that the market remains unchanged in the next period. The transition matrix can also be used to assess the stabili248
249
Analogue formulas can be derived for other versions of the search model, including the model with opportunity costs and the real estate model. See Karlin (1962), p. 154 ff., and Lippman/McCall (1976c) for Markov-type search models. For general properties of Markov processes see, e.g., Meyn/Tweedie (1993).
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ty of the market – the more probability mass is concentrated on the diagonal, the more probable it is that the market will not change. Using the transition matrix, the equation set (2.40) can be rewritten as follows: N
(
)
(
)
(
)
N
N
E(G1 ) = ∑ E(Pi Pi > p*i ) ⋅ 1 − Fi (p*i ) ⋅ ψ Ni + ∑∑ E(G j ) ⋅ ψ ij ⋅ Fi (p*i ) ⋅ ψ Ni − c i =1 N
i =1 j=1
N
N
E(G 2 ) = ∑ E(Pi Pi > p*i ) ⋅ 1 − Fi (p*i ) ⋅ ψ1i + ∑∑ E(G j ) ⋅ ψ ij ⋅ Fi (p*i ) ⋅ ψ1i − c i =1
i =1 j=1
(2.42)
... N
N
N
E(G N ) = ∑ E(Pi Pi > p*i ) ⋅ 1 − Fi (p*i ) ⋅ ψ ( N−1)i + ∑∑ E(G j ) ⋅ ψ ij ⋅ Fi (p*i ) ⋅ ψ ( N−1)i − c i =1
i =1 j=1
Solving the set of equations with respect to pi* yields the optimal search strategy as a vector of reservation prices. The value of search changes in the cyclical frameworks and depends on the cycle phase. Thus, the expected net receipts from liquidation at the beginning of the search depend on the state of the market at this moment. This may be a serious practical problem, because liquidity analysis often needs to be accomplished long before the search for a buyer begins – the starting state is, thus, unknown. One possible solution is the application of the most probable state; another way is the estimation of the value of search as the expected value over different possible initial market states. However, in either case, the probabilities with which the search will begin in each of the states need to be known. In the Markov approach, these probabilities can be obtained from the steady state vector, which results from the transition matrix Ψ when the process is running very long. Otherwise, an empirical estimation of the state probabilities or a forecast of the market state (cycle phase) at the beginning of the liquidation process is necessary. The application of dynamically changing markets in search models surely adds more realism. However, it has a noticeable effect on the results only when the average duration of the search process is comparable with the length of the cycle. This is seldom the case in reality – sales are usually accomplished much too quickly for any economic cycle to take a noteworthy effect even on extremely illiquid markets. Nevertheless, the
2.3 The Real Estate Search Model
149
Markov approach with multiple randomly changing regimes can be used to model search in markets in which abrupt structural breaks are possible.
2.3.3.3. Unknown Offer Distribution and Learning The assumption that the distribution of offers is known to the seller is surely one of the most unrealistic ones. It is practically impossible to determine the precise probabilities of offers since it would require the knowledge of individual valuations of all market participants. In reality, sellers’ knowledge ranges between a good assessment and total uncertainty about the relevant distribution. The most straightforward way to improve it is to use the offers received during the search to revise the opinion about the distribution of offers. Thus, a learning process takes place. A search problem with an entirely unknown offer distribution is unsolvable in the general case.250 It is straightforward that no optimization is possible if absolutely no information about the state of the market is available. In reality, however, investors should be able to determine at least a set of possible offer distributions as well as assign them (subjective) probabilities with which they are expected. Learning means in this case accumulating information indicating which of the possible distributions the seller is actually facing. The search strategy remains unchanged in its core – offers are accepted or rejected on the basis of a reservation price. However, the way the reservation price is determined changes substantially. It varies from one period to another as the presumed offer distribution is updated with new information. Updating may refer to its functional form as well as to any of the parameters. It seems, however, that most frequently only the mean is corrected. In this case, an effect of a falling reservation price is likely to occur. If the seller initially overestimated the level of the average valuation of the property, she will probably gradually reduce her assessment as lower offers arrive; but if she underestimated the value, the property is likely to be sold before any correction of the reservation price is possible.251
250 251
See Tesler (1973), p. 44. Note that a similar effect may also result when updating refers to the dispersion of offers. Since, as discussed in section 2.3.2, the value of search and the reservation price might increase with ν in the real estate search model under certain realistic conditions, π* would be set too high if ν was overstated. This would lead to lowering the reservation price as the true value of offer volatility is revealed in the course of search. In the opposite case, when ν was underestimated, the sale would occur too quickly to allow substantial corrections. However, since the effect of ν on the result of the search and on the optimal strategy is partially ambiguous and depends on the values of other model
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The seller’s main problem in the situation of an unknown offer distribution is to choose the optimal learning procedure. There are several alternative strategies discussed in the literature. Tesler (1973) suggests dividing the process in the “learning phase” and the “decision phase”. The first n observations are used only to estimate the distribution of offers and are always rejected. Starting with the (n+1)st offer the search proceeds as in models with known distribution. The determination of the optimal number of observations to pass, i.e., the optimal length of the learning phase, is central in this approach. The goal is to increase the efficiency of the search in terms of expected receipts compared to a “naïve search” without learning. An economically reasonable way to accomplish it is to set n so that the (expected) additional search cost resulting from passing the first n offers equals the expected additional gain resulting from the improved estimation of the true offer distribution. The result is dependent on the set of possible distributions and their probabilities, but also on the unit cost of offer observation.252
In the above learning scheme, some observations remained unused in the sense that they cannot be accepted. This may be uneconomical in some cases. It is generally possible to receive a very high offer, which would be high under any possible distribution, already during the learning phase. Rejecting it would mean losing a unique opportunity. A more general learning rule is, thus, to use all offers both for learning and for searching. The concept of Bayesian updating can be applied for this purpose. According to it, the distribution having the highest posterior probability after observing a new offer should be chosen from the set of possible offer distributions. Posterior probabilities result from corrections of prior probabilities assigned to different distributions according to the Bayes’ decision rule:253 Pr(Dist | Old ∩ New) = Pr(Dist | Old) · Pr(New | Dist ∩ Old) / Pr(New | Old) with: Dist - probability distribution from the set of possible distributions Old - set of past offers New - new offer
252
253
(2.43)
variables, decreasing reservation prices due to learning about the diversity of valuations on the market are do not necessarily need to be observed. Tesler (1973) demonstrates in his example (search for the lowest price) that learning only pays when the unit search cost is not too high. For further reading on Bayesian analysis see Press (1989), Bernardo/Smith (1997), or Congdon (2003).
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151
Several authors have analyzed the properties of searching strategies implementing Bayesian learning.254 In the general case, this setting leads to severe problems. Firstly, the reservation price property does not necessarily need to hold.255 This is apparent in the following example: if only two probability distributions of offers are possible, the first one assigning 100% probability to the value of 100 and the second assigning 1% probability to 200 and 99% probability to 300, than the seller should accept an offer of 100 or 300 but reject an offer of 200.256 There is no ex ante defined reservation price in this case. The second difficulty is that the problem may not be time invariant (myopic), what means that no general search strategy can be defined for the infinite case.
However, the above problems do not occur under certain restrictive conditions.257 One of such cases is when offers are normally distributed with a known variance but unknown mean, which is updated. It can be shown that both the reservation price and the myopic property (for appropriate parameters) still hold then, so that the solution can be derived using standard procedures.258 This is probably the most relevant case for real estate sale modeling. Normal distribution of offers has been discussed in section 2.3.1.1, and it seems plausible to assume that its location (mean) is the main parameter adjusted by searching sellers. Although the assumption that the distribution of offers is not fully known doubtlessly complies with reality, its application in the real estate search model is still problematic. The first difficulty is associated with the intended application of the model. Determination of the optimal selling strategy (reservation price) is only one of the relevant aspects; much more important is the application in investment decisions on illiquid assets. In the latter case, the goal is to assess the grade of liquidity and to optimize the choice of investment alternatives with respect to this feature. Considering liquidation strategies that remain undefined until the sale procedure is actually conducted makes it difficult (if at all possible) to assess the liquidity of an asset before it has even been purchased. Another reason why learning has not been implemented in the real estate 254
255 256
257
258
See DeGroot (1968), DeGroot (1970), pp. 336-345, Kohn/Shavell (1974), Rothschild (1974a), Rosenfield/Shapiro (1981), or Rosenfield et al. (1983). See Rothschild (1974a), p. 701. See Rothschild (1974a), p. 701, Kohl/Shavell (1974), p. 102, or Rosenfield/Shapiro (1981), pp. 4-5, for similar examples. Rosenfield et al. (1983) review the conditions under which the properties that hold for the search with a known distribution of offers also hold when the distribution is unknown. See DeGroot (1970), pp. 345-353, Rosenfield/Shapiro (1981), and Rosenfield et al. (1983).
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model is its partial ambiguity in the relative approach. As mentioned above, it seems realistic that not the whole form of the offer distribution but only its parameters, in particular the mean, are unknown. Thus, learning from subsequent observations is equivalent to sequentially correcting the mean. Variables in the real estate search model are, however, defined in a relative manner and are independent from the average level of offers. Adjustments of the mean during an adaptive search would have no effect on the relative reservation price π*. In this sense, learning with respect to the level of market prices is already incorporated in the real estate model.
2.3.3.4. Offer Recalls No recalls are allowed in the real estate search model; yet, the possibility of recalling past offers is given, at least to some extent, in most cases of selling or buying illiquid assets. Allowing for this feature could therefore make the analysis more realistic.259 However, it can be demonstrated that the possibility of recalls does not affect the optimal search policy in simple models with stable offer distributions and positive search costs like those presented in sections 2.2.3 and 2.2.4. Moreover, it never pays to recall any offer in these frameworks even if it is possible.260 If an offer pi was not optimal in the period i, it is also not optimal in the period i+1. In fact, waiting only induces additional search and opportunity costs, so if an offer is acceptable, it should be accepted as soon as possible. Otherwise, its value decreases with time due to discounting, and an acceptable offer may become unacceptable. In other words, if pi
259
260
Search or sampling with recall is analyzed by Kohl/Shavell (1974), Rosenfield/Shapiro (1981), Morgan (1983), or Rosenfield et al. (1983). See Rosenfield/Shapiro (1981), pp. 2-3, or Ferguson (2000), p. 4.3.
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Another framework in which recalls can be optimal is a cyclical market, as discussed in the section 2.3.3.2. The optimal reservation price is not constant in this case. It is therefore possible that an offer, which was unacceptable in the “old” market state, becomes acceptable in the “new” state; however, recalls make only sense within one market cycle in this case. The recall problem is also nontrivial when the searcher does not know the distribution of offers and is learning during the search (see section 2.3.3.3). Each incoming offer influences then her opinion about the buyers’ valuations. These changes affect the optimal reservation price, so that a formerly unacceptable offer can turn out to be acceptable after the revision of the market assessment.
An even more realistic possibility of allowing for recalls in a search model is an uncertain recall. In this case, the seller may recall past offers, but they are not always available after being rejected for the first time. In other words, the seller can fall back on a former bidder, but she cannot be sure if his offer is still valid. A past offer that is still available at a later time can be treated as a random variable with an appropriate (binominal) probability distribution. Alternatively the duration of an offer can be viewed as a Poisson or a Weibull process.261 Although widely discussed in the mathematical literature, search problems with full recalls are not easily applied in practicable models. They usually contain a maximizing function to choose the best of the past offers, which is relatively difficult to model analytically. A simulation approach is handier in this case.
2.3.3.5. Intensity of Search In most search models, including the real estate one, the seller has no influence on the behavior of potential buyers. In particular, she cannot influence their interest in the property on sale. However, in many cases, it is realistic to assume that the number of received offers is not exogenous to the liquidation process. The seller is usually able to conduct marketing actions or actively search for buyers. These measures intensify the flow of offers. The offer arrival intensity in the continuous time model with Poisson offer flow can be used to incorporate the possibility of influencing buyers’ interest. It is achieved by expressing λ as a function of seller’s efforts. The more active she is in the search for a 261
See Karni/Schwartz (1977) and Landsberger/Peled (1977) for search models with uncertain recall.
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buyer, the more offers can be expected to arrive per unit of time. This brings another strategic variable into the model – the intensity of search. It has two opposite effects: on the one hand, a more intensive search may allow accomplishing the sale more quickly and reducing opportunity costs, but on the other hand, it also generates additional costs (e.g., marketing expenses). Thus, also the search cost c has to depend on the search intensity. The optimal “effort level” can be determined by maximizing the value of search with respect to the intensity variable.262
The intensity of offer arrivals can be also utilized to allow for the economic state of the analyzed market. More interested potential buyers can be expected during hot market periods resulting in a higher λ. The frequency of offers can also depend on the characteristics of the sale object; e.g., a flat in a popular location will probably raise more interest than one in the outskirts. However, introducing these additional features may cause problems with the myopic property of the model. With a time varying λparameter no time invariance of the decision situation can be ensured.
2.3.3.6. Listing Price It has been assumed so far that only interested buyers make price offers. The seller remains passive by only accepting or rejecting the incoming bids without making any suggestions about the price that she would be willing to accept immediately. This puts the seller in a better bargaining position because she does not preclude any offer from being placed – neither a very low, nor a very high one. However, in many illiquid markets, e.g., residential real estate, it is common that the seller quotes her preferred price, the so called listing price, which is the starting point for further negotiations. The listing price is usually published together with the sale offer and is the price that the seller would accept on the spot. This does not, however, preclude bargaining – a potential buyer with a personal valuation of the property below the listing price may try to convince the seller to lower the price. An important consequence of quoting a listing price is that it constitutes the upper limit to the achievable selling price. No rational buyer would offer more than the listing
262
The strategic choice of search intensity is treated by Benhabib/Bull (1983). Search efforts have been discussed in the real estate literature mainly in the context of seller’s or broker’s behavior and marker equilibrium; see Yinger (1981), Wu/Colwell (1986), Yavaş (1992, 1996), or Anglin (1997).
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price even if his valuation was higher.263 For the model’s mathematics, this means cutting off the right tail of the offer distribution and allocating its whole probability mass to one point – the listing price. Another consequence is that the listing price is treated by a potential buyer as a hint at the seller’s reservation price. He can use the gap between his valuation of the property and the listing price to assess the chances of successful bargaining. If the chances are low, he would probably not engage at all, i.e., not place any offer. This behavior results in the dependence between the listing price and the frequency of offers (λ), which influences the opportunity costs of the search (i.e., the discount rate δ). The level of the listing price may also influence the distribution of offers. Higher offers are relatively more probable with higher listing prices since potential buyers with lower valuations are then discouraged. The listing price becomes therefore another strategic variable in the model and can be optimized.264
In the model with opportunity costs (Karlin’s model) allowing for the listing price results in the following expected net receipts from sale analogue to (2.18): E (G ) =
⎛ pL ⎞ δ( p L ) ⎜ p ⋅ dF(p p ) + p ⋅ F(p p ) ⎟ L L L L ∫ ⎟ 1 − δ(p L ) ⋅ F(p * p L ) ⎜ p* ⎝ ⎠
with: pL F(·|pL) δ(pL)
(2.44)
- listing price - distribution of offers conditional on the listing price - discounting factor as a function of the listing price
If there was no impact of the listing price on the frequency or the distribution of offers, an infinitely high listing price would maximize the expected receipts.265 However, as soon as such dependence exists, an infinite listing price would also mean infinitely long intervals between offers – sale would not occur because of the lack of interested parties. The result would have been the same if the search had not been undertaken at
263
264
265
Empirical studies show that sales above listing prices are very rare. Such cases amounted to 3.8% of sold houses in the sample of Horowitz (1992, p. 118) and to 9.3% in the sample of Anglin et al. (2003, p. 100). In most cases this was probably the effect of two or more potential buyers with high valuations bidding simultaneously on the same property. The strategic role of the listing price in real estate markets has been analyzed by Horowitz (1992), Yavaş/Yang (1995), Miller/Sklarz (1997), and recently by Anglin et al. (2003). This can be easily shown by assuming δ and F(.) in the formula (2.44) independent of pL and deriving the equation with respect to pL. The result is positive for all values of pL meaning that E(G) is an increasing function of pL.
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all. An infinite listing price would also be suboptimal if it influenced only the offers’ distribution and not the frequency, as shown by Horowitz (1992). The implications of the listing price are not further followed in this book. It is, however, clear that an optimal listing price exists. It has to be at least as high as the seller’s optimal reservation price, and it has to be finite; otherwise a contradiction would arise or the search would become worthless. As soon as all necessary functions are specified, it should be possible to find at least approximate values of both the listing and the reservation price maximizing the value of search.
2.3.3.7. Search for the Best Seller The final question relevant for the liquidity problem is the search for the best seller. As already noted in section 1.4.2.3 in Chapter 1, investors are not only facing liquidity problems when selling an illiquid asset, but also when buying one. Finding an investment with required characteristics may be a time-consuming undertaking, leaving the investor with the typical liquidity dilemma between buying quickly and buying at a reasonable price. The search problem resulting in this case is analogue in its nature to the search for the best buyer.266 Several adjustments to the original model are, however, necessary. Taking the buyer’s point of view requires redefining the “offer distribution”. Since the buyer is now reviewing potential sellers, each offer refers to a different object. However, an offer distribution can be defined only with respect to identical or at least similar alternatives; otherwise, comparability of prices cannot to be fully maintained. This is difficult for many heterogenic assets, including real estate, where each property is more or less unique. Hedonic methods using econometric modeling to attribute prices paid for real estate to the characteristics of the sold properties (number of bedrooms, location quality, construction year etc.) may be of help at this point.267 On this basis, a hypothetical price distribution for a concrete investment can be simulated. An alterna-
266
267
See Anglin (1997) and Chun (2000), pp. 327-329. The search for the best buyer is discussed in the real estate literature mainly in the context of market models. Sellers and buyers search simultaneously generating market equilibrium in Wheaton (1990), Yavaş (1992), Williams (1995), Krainer (1999), or Krainer/LeRoy (1997, 2002). Hedonic methods are often used for the construction of real estate indexes. See Ferri (1977), Wallace (1996), or Thomas (1997, p. 162).
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tive approach could be based on some kind of an atypicality measure, like the one developed by Haurin (1988). The issue of inspection or appraisal costs is related to the above problem. Information asymmetry between buyers and sellers is immanent to heterogenic assets. In most cases, the seller is substantially better informed about the true characteristics of the traded asset than the (potential) buyer. This makes it difficult for the latter one to properly assess the quality of the received offer. The typical way to cope with this problem is an independent inspection or an appraisal of the transaction object. The resulting costs are accrued whenever a new offer is considered increasing the overall search costs. Summing up the above considerations, the expected expense (E) resulting from the search for an illiquid asset in the basic search framework with observation costs can be computed as follows:
E(E) = E(P P < p ∗B ) + with: πB* η
c+η 1 − F(p∗B )
(2.45)
- reservation price in the purchase case - appraisal cost
For the relative version of the search model and normally distributed offers, the following formula results:268
E (Ξ ) =
⎛ ⎛ π * −1 ⎞ ⎛ π * −1 ⎞ γ−η ⎞ ⎜ Φ⎜ ⎟ ⎟⎟ + ⎟ − ν X ⋅ ϕ⎜⎜ λ + ρ − τ ⎟⎠ ν ⎛ π * −1 ⎞ ⎜⎝ ⎜⎝ ν X ⎟⎠ X ⎠ ⎝ ⎟⎟ ρ − τ + λ ⋅ Φ⎜⎜ ⎝ νX ⎠ λ
(2.46)
The final adjustment concerns the substitution of the maximization of the value of search with its minimization. Obviously, a buyer is interested in the lowest possible expense; thus, the way of reasoning in the whole model changes respectively. The reservation price in this case is the maximal price, which the buyer should accept. This adjustment is relatively simple and encompasses only a slight change in the optimization algorithms.
268
See Appendix A.2 for the derivation.
158 2.3.4.
Chapter 2: Search in Illiquid Markets MCS Solution
The considerations on the search problem presented so far were based on analytical models, which can be specified in a more or less general form by using analytical formulas. This approach has serious limitations considering its solvability in specific cases. The search framework behind the model, like the basic sale situation presented in section 2.2.1 or the real estate liquidation framework presented in section 2.3.1, contain numerous simplifications and assumptions. Many of them can be overcome (see section 2.3.3) or result in only negligible misspecifications. Nevertheless, in some points, the realism of the model may be insufficient to allow acceptable approximation of real markets. A simulation approach, often referred to as the Monte Carlo Simulation (MSC), is an interesting alternative in such cases.269 Its great advantage is the possibility of easy implementation of practically any thinkable additional condition; the only limit is the computation power. Monte Carlo methods have been used in optimal stopping problems mainly for the valuation of American options, where an optimal realization of an option is crucial for its value.270 Other applications in the search theory are rather rare, which is probably due to the predominantly theoretical analysis in this field. Nevertheless, despite several problems, the use of MCS in the search for the best buyer may allow an insight in the effects of certain additional conditions, which otherwise would be difficult to analyze. The sense of a simulation approach lies in repeating a predefined scenario under varying environmental variables and observing the results. In the case of the search for a buyer, it means rerunning the search and sale process. Each replication represents some hypothetical scenario and leads to a hypothetical realized selling price. By generating a large number of such replications, it is possible to assess the distribution of the net sale receipts. Of course, a simulation can never perfectly reflect the reality, but the fit increases rapidly with an increasing number of replications. With today’s data processing technology, a sufficient level of accuracy for most practical applications can be achieved even with a personal computer. Each simulation uses two types of variables: the certain ones, remaining constant in all runs, and the uncertain ones, varying from one run to another and depending on the 269
270
An excellent review of Monte Carlo methods offers Gentle (1998). See also Robert/Casella (1999), Jäckel (2003), or Glasserman (2004). For a comparative study of MCS algorithms for pricing American options see Fu et al. (2001).
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assumed nature of their uncertainty. The former ones represent either facts, like fixed notary fees or tax rates, or assumptions, like the personal discount rate. Since they remain constant, they constitute the general simulation framework. In contrast, the varying variables are the ones that are to be studied. By assigning them different values, their influence on the target is analyzed. In particular, price offers fall under this category in search models. The decision which variables in the process should be simulated and which should remain constant depends on the goal of the analysis. The more uncertainty is allowed, the more precise are usually the results. There is, however, a tradeoff between the range of simulated variables and the computation effort necessary to preserve the desired accuracy. The number of runs necessary to retain a constant level of precision increases exponentially with every additional simulated variable. Characteristic for Monte Carlo approaches are computer generated random variables. This issue is simultaneously a serious technical problem and a great advantage of these methods. Depending on the knowledge about the probability distributions of the simulated variables, two different approaches can be followed. If the stochastic process behind the variable is known, and its distribution function can be assumed with sufficient certainty, random number generators can be applied to obtain series of values with required characteristics. Such generators exist for most of the popular distributions.271 When no distributional assumption is possible, the only source of information is an observation sample. The first straightforward approach is to use the sample data directly in the simulation by scrambling the association of the data points, i.e. combining different values of the variables.272 This approach has two major drawbacks: firstly, only a limit number of scenarios can be generated this way, and secondly, some of the available observations may be older and not necessarily comply with the current situation. Two alternative techniques available in this case are jackknife and bootstrap.273 The first one is used mainly to obtain better estimations of distributional parameters of variables. Single (or multiple) observations are removed from the sample, and parame271
272
273
A simple method of generating variables from any continuous probability distribution includes the generation of a uniformly distributed series and the application of the distribution’s inverse c.d.f.. Numerous other approaches are discussed in the literature. See Gentle (1998), pp. 1-38 and 85-118, or Glasserman (2004), pp. 39-77. This approach corresponds with the randomization techniques in randomness tests (see Edgington, 1995) or with the historical simulation used for computing Value at Risk (see Jorion, 2001, pp. 221 ff). For more extensive presentations of the jackknife and bootstrap techniques see Shao/Tu (1995) or Efron/Tibshirani (1998).
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ters are recalculated on the basis of a reduced sample. Using these results a jackknife estimator is computed having a reduced bias compared to the original parameter. The central idea of the bootstrap is to treat the sample as if it represented the whole population.274 Observations are drawn randomly from the original sample with replacements building bootstrap samples. The number of “new” samples that can be generated this way depends on the size of the original sample. An especially interesting application of the bootstrap is the possibility of using the averages from bootstrap samples as additional (artificial) observations. Adding them to the original ones increases the “density” of the sample allowing better assessment of the variable’s probability distribution. An advantage of both jackknife and bootstrap is their easy implementation, which can be accomplished even in a common calculation sheet.275 A possible MSC algorithm of the real estate liquidation problem analyzed in this chapter is presented in Figure 2-3. The simulation process itself is contained within the shaded area, and the assumptions are depicted on the left side of the Figure. The solid arrow lines indicate the direction of the algorithm, and the dotted ones mean that a variable or a partial result is to be applied. Before the simulation can start, its parameters need to be set. They include the reservation price and other deterministic variables (discount rate, trend, rental revenues) as well as the distributions of the random variables. The latter can be based on assumptions or derived from observation samples. The process begins with the generation of a random offer from the distribution of buyers’ valuations. The offer is compared with the reservation price and dependent on the result either a new offer is generated, or the sale is concluded. In the former case, the number of offers (periods) is increased by one. In the latter case, a market change and time intervals for all periods are generated from the respective distributions. They are used to calculate capitalized rental revenues and the discounted sale price. The value of the total net receipts from sale results as the sum of the price and the income component. After updating the receipts’ distribution, a new replication starts from the beginning.
274 275
The bootstrap was proposed as an extension of the jackknife by Efron (1979). See Woodroof (2000) for the application of bootstrap in the Lotus 1-2-3® environment.
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Simulation parameters
Simulation process Start
Basic offer distribution
Generate an offer π Update the number of offers i
Reservation price (π*)
No
Is π > π*?
Yes Distribution of market changes
Distribution of time intervals
Interest rate (ρ) Market trend (τ)
Rental revenues (γ)
Generate a market change a
Generate time intervals t1…ti
Calculate discounting factors
Calculate discounted rental rev.
Calculate discounted sale price
Calculate net receipts
Process flow
Update the distribution of receipts
Variable application
Figure 2-3: Scheme of a Monte Carlo Simulation for the real estate sale process
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The final result of the simulation is an estimation of the distribution of net receipts for a given set of parameters. An example of such distribution obtained with a MCS is presented in Figure 2-4. Its accuracy depends strongly on the number of replications, but it should be sufficient after a few thousand runs in most cases. Various statistics, including probabilities of certain outcomes, moments of the distribution, as well as complex ratios and other measures can be computed on the basis of this distribution. Another important advantage of the MCS is the possibility of easy implementation of further features. New conditions and additional environmental variables can be added, or the character (e.g., randomness) of the existing parameters can be changed by simply adding new nodes in the simulation algorithm. Also constraints, which are present in the search model in order to ensure its solvability, do not apply here. Especially, no restrictive assumptions about the probability distribution of offers or other random variables are necessary. Due to the possibility of using empirical distributions, this problem is practically nonexistent.
9,0% 8,0%
Frequency
7,0% 6,0% 5,0% 4,0% 3,0% 2,0% 1,0% 0,0% 0,36
0,49
0,62
0,75
0,88
1,01
1,14
1,27
1,40
1,53
1,66
Net Sale Receipts
Figure 2-4: Monte Carlo estimation of the distribution of net receipts from sale276
276
Monte Carlo Simulation with 10.000 runs and following parameters: reservation price = 1.2, offer volatility = 15%; rent = 5%; trend factor = 5%; discount rate = 15%; offer frequency = 52 p.a.; normally distributed market changes (A) with the volatility of 5%.
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Despite the numerous advantages, a possibly serious problem with the simulation approach is the determination of the optimal reservation price. The distribution of net receipts determined in the simulation is only valid for one concrete value of π*. In order to “try out” other reservation prices, one needs to rerun the whole simulation. This may lead to an enormous computational effort: with an increasing number of different values of π*, the number of replications increases geometrically. Since it is not possible to analyze receipts’ distributions for all π*, the only reasonable alternative is to concentrate on ranges in which the optimal reservation price is suspected. This approach will probably yield good results in most cases, especially, if the existence of a unique optimal π* can be proved analytically. However, when a modified search model is considered, or when the optimality criterion is other than maximizing expected receipts, this problem becomes a serious hindrance.277 It is even more severe when a varying reservation price is assumed. 2.3.5.
Liquidity within the Model
In order to complete the presentation of the real estate search model on which most of the analysis in this book is based, it is necessary to state explicitly how different aspects of liquidity are captured within it. The two dimensions of asset liquidity and the sources of liquidity identified in Chapter 1 constitute the reference points for the discussion. To begin with, recall the liquidity definition of Keynes: an asset is more liquid if it is “more certainly realizable at short notice without loss”.278, 279 The two main aspects of liquidity can be derived from it: the duration of the liquidation process and the price realized at liquidation. The presence of the latter aspect in the search model is obvious – the (relative) net liquidation receipts Γ can be regarded as equivalent to the liquidation price. Interpreted in the sense of Keynes, E(Γ) should be not lower than 1 or a “loss”, i.e., a discount to the fair value, would result. The reference to the liquidation time can be considered in terms of the expected duration of search (D) – in the model, search can end only with a sale, so that the two notions are, in fact, identical. Further277
278 279
Especially, when a risk-minimizing reservation price is of interest, the identification of its optimal value using MCS may prove to be difficult. See section 3.3, especially 0. Keynes (1930), p. 67. See also Lippman/McCall (1986) for the discussion on the consistency of the search theoretical approach to liquidity with the Keynesian liquidity definition.
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more, the formal definition of the expected value of D formulated in (2.23) implies also the identity of the expected duration of search and the reservation price. Since E(D) is a function of the reservation price p*, each reservation price always corresponds with a specific expected duration of search; and since the probability function F(.) is by definition continuous and monotonous, each value of the expected duration of search corresponds with a specific reservation price (or prices). A non-ambiguous relation between the duration of the sale process and the expected net receipts from sale is the consequence of this identity. Since the reservation price determines both the expected receipts and the search duration, it can be represented by a locus of E(Γ) and E(D). For the real estate search model, the locus takes a form similar to the one presented in Figure 2-5. Note that it highly resembles the relation between the present value of the selling price and the selling time depicted in Figure 1-7 and Figure 1-8.
1,4
Expected Net Receipts
1,2
1
0,8
0,6
0,4
0,2
0 1
100
10000
1000000
100000000
1E+10
1E+12
Time on the Market
Figure 2-5: Locus of expected net sale receipts and search duration in the real estate search model
Having recognized the main features of the liquidity (marketability) definition in the search model, it is purposeful to check whether it also adequately captures the sources of liquidity. If it wasn’t so, the model would provide only an incomplete picture of the problem. The presence of transaction costs (direct and opportunity costs) is relatively
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straightforward.280 On the one hand, direct costs are allowed for in the levels of offers. Since the prices offered by potential buyers are assumed to be net of all costs, commissions and taxes are deducted already at this point. On the other hand, the costs that are born during the search and the evaluation of offers are regarded as observation costs and netted with rental revenues (see section 2.3.1.4). Opportunity costs are captured in the model through a number of factors. Since the discount rate expresses the decision maker’s time preference, it also reflects the returns from an alternative investment that could have been earned if the property was sold more quickly. Market timing costs, i.e. the costs resulting from a negative change of the market situation during the liquidation, are expressed by the (expected) market trend and the (unexpected) market uncertainty factor. The expected change of the level of arriving offers during the search is equal to the market trend parameter. On the other hand, the risk of an unexpected development is considered by the inclusion of the market uncertainty parameter A. Another source of liquidity identified in Chapter 1 is the diversity of valuations of the asset in question by market participants. Also this aspect is very apparent in the search framework. Since the level of the arriving purchase offers is random, one can treat them as the effect of random drawing from the distribution of property’s valuations among investors. Hence, the mere fact that different offers are placed by different sellers during the search reflects the valuation diversity. The allowance for market organization is less clearly visible in the model. In fact, the core idea of the sequential search is identical with that of a direct market on which sellers and buyers search, find each other, and negotiate transaction prices individually. Hence, the search theoretical approach proposed here seems to be most appropriate for this type of markets. However, brokered markets can also be easily described in the search framework. Also here, the seller reviews a number of sequentially arriving offers and decides to accept one that complies with her requirements. The main difference to a direct market lies in the values of model parameters. Since a broker is usually in a position to establish contact to a larger number of potential buyers, the frequency of offers should be higher. A broker should also be able to filter out unserious traders, 280
Indirect transaction costs have been recognized in section 1.2.2.2 in Chapter 1 as a phenomenon caused by illiquidity, or even equivalent with it, and not as a source of liquidity. Therefore, this type of transaction costs should be explained by the model rather than enter it as an explanatory variable. In fact, the real estate search model can explain the existence of certain indirect transaction costs, as demonstrated, e.g., by the implicit bid-ask spread in section 3.1.2.1 in Chapter 3.
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so that their offers would not be reviewed at all. In effect, the seller would be facing a slightly different distribution of offers compared to the direct market. Interpretation of the search model in the context of a dealer market is more difficult. The search is then conducted not among potential individual buyers or sellers but among different dealers. Application of the model would require several more fundamental adjustments. Firstly, due to the relatively small number of dealers active on the market, it might be more appropriate to assume a finite search horizon (see section 2.3.3.1). Secondly, the impact of market fluctuations on the realized liquidation values would be relatively high, so that the issue of appropriate modeling of this aspect should be given special attention (see discussion in section 2.3.1.5). Nevertheless, in view of the presumably very low dispersion of offers, the usefulness of the search model in this case is rather doubtful. The application of the model to assets traded on auction markets is even less purposeful. Since no search takes place there, and traders are matched automatically, there is little point in analyzing search processes. However, if one interprets the flow of orders arriving on the exchange as a flow of offers and regards the short-term (intraday) variations of these offers as the result of investors’ heterogeneous valuations, the trading activity on an auction market can indeed be interpreted as an intense search process. Limit orders have the function of reservation prices and market orders are equivalent to selling to the first incoming bidder. Nevertheless, also in this case, there seems to be little practical use from the application of the search theoretical approach. Hence, the main intended fields of application of the search model are non-organized markets with less trading activity and lower levels of liquidity.
2.4.
Search and the Functioning of Illiquid Markets
The analysis in the former sections concentrated on the optimal behavior of an investor selling an illiquid investment, in particular real estate. The main issue was the possibility of influencing the outcome of her actions by optimizing the search strategy, which in this case took the form of a reservation price. It followed that a rational investor should not treat the search for a buyer as fully exogenous but act strategically. The search problem has been approached from the perspective of an individual investor. This was in line with the goal of the analysis formulated in the Introduction, that is, the provision of a practical tool to include liquidity as a decision criterion. However, the
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167
results of the search theoretical analysis have implications not only for individual investment decisions but also for the functioning of illiquid markets. The main point of interest when considering the effects of search in illiquid markets is the analysis of the situation of market participants compared with their situation in liquid markets. The starting point of such considerations is the relationship between the maximum expected receipts from sale and the hypothetical price that could be realized in a liquid market. Due to the lack of organized trading systems, determination of hypothetical market prices is a major problem for most illiquid assets. It seems plausible to assume that the average valuation of the asset by market participants would constitute the single market price if perfect liquidity was given. Thus, the mean of the distribution of buyers’ opinions about the value of the asset can be treated as a “quasimarket price”. On the other hand, expected net receipts from sale correspond with the effective price achievable on average by a seller following the assumed liquidation strategy. If she set her reservation price at minus infinity (i.e., she accepted the first offer to come), she could expect to receive the asset’s average valuation adjusted by one-time cost (or income) arising from the necessity to wait for the first offer.281 Thus, when search costs are negligible, setting an infinitely low reservation price results in expected receipts equal to the quasi-market price. However, by optimizing the search strategy, higher expected net receipts are usually achievable. In fact, simulations with plausible parameters for the real estate search model have shown that “beating the average” is a rule rather than an exception. This effect diminishes with increasing costs of search or decreasing heterogeneity of buyers, and it disappears if the former is too high or the latter is too low. Nevertheless, the maximum expected receipts from sale should still lie above the quasi-market prices in many markets. The straightforward conclusion from the above considerations is that, in certain circumstances, illiquidity can allow the seller to achieve higher gains than it would be possible when the asset was perfectly liquid. In organized markets, like stock exchanges, only one price exists, and it is equal for all investors. There is no direct possibility to take advantage of the fact that some buyers would also be ready to buy at higher prices. This “consumer surplus” can, however, be utilized if no organized market exists, and potential buyers have heterogeneous opinions about the asset’s value. It 281
The fact that an infinite negative reservation price is economically unreasonable and that the seller’s personal valuation is the actual lower limit is ignored at this point.
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turns out that illiquidity does not necessarily need to be disadvantageous. In some cases, it may even be more favorable for certain market participants to act in an illiquid market with no organized trading systems and no transparency regulations. However, this conclusion must be interpreted with caution. Even if the model indicates that the maximum receipts from sale are above the average level of offers, it does not necessarily mean that they are above the hypothetical liquid market price. Firstly, it must be noted that “beating the market” in the above sense has a solely subjective character. It has been already argued in the previous sections that, in the ideal case, the distribution of offers applied in the search model should arise from the valuations of the asset among all potential buyers. Yet, determination of the true distribution is a practical impossibility. Only an omniscient investor could know it, and hence only a prophet could truly “beat the market”. The ordinary seller can only fall back on her assessment of the distribution, which is always subject to individual information. It may be possible for her to outperform the market from her point of view, but there is no guarantee that her view complies with objective reality. There is always some risk that the assessment of the offer distribution was wrong, and what subjectively seemed to be a good value was, in fact, far below the average achievable price. The matter becomes even more complex when not only sellers but also buyers are allowed to search for the best price.282 Buyer’s search for the best seller when there are a number of comparable houses on the market was discussed briefly in section 2.3.3.7; apart from several slight adjustments in the formulas, the same principles as in the sale case apply. This means that also a buyer can (subjectively) “beat the market” by optimizing the search strategy. If search costs are not too high and potential sellers are heterogeneous enough, she can expect to obtain the desired asset at an effective expense below the average valuation on the market. Thus, same as for sellers, illiquidity can be advantageous for buyers. This, however, leads to a logical problem: how can buyers expect to buy below the average market price and simultaneously sellers expect to sell above the average? This contradiction can be explained within the model by the already mentioned subjectivity of the parameters used for determining the optimal policy. Market participants do not all face the same distribution of offers. Since sellers 282
Search models with both sides of the market analyzed simultaneously are considered by Wheaton (1990), Quan/Quigley (1991), Yavaş (1992), Krainer (1999), Krainer/LeRoy (2002), or Fisher et al. (2003); however, they do not quite correspond with the two-side strategic search problem mentioned here.
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Frequency
search for bids above their reservation prices and buyers search for offers below their reservation prices, the distribution of buyers’ bids should tend to lie further to the left than the sellers’ offer distribution. It is than by all means possible that at least some of the buyers’ reservation prices lie above the sellers’ reservation prices making transactions possible in which both parties subjectively outperform the market (see Figure 26).283 The final distribution of offers on both sides is, however, the result of a complex market game and may be unstable.
Investors’ valuations of the asset Sellers’ reservation prices
Buyers’ reservation prices
Direction of the search
Price building room
Reservation Price / Valuation
Figure 2-6: Reservation prices of buyers and sellers and the price building in illiquid markets284
The above considerations only touch the issue of price building in illiquid markets. Strategic search by all participants generates a market wide game, which is too complex to be fully analyzed in this book. Yet, even these introductory notes allow the conclusion that illiquidity may be advantageous to some investors. This applies especially to those, who are able to assess the distribution of offers on the opposite side of 283
284
Note that these considerations refer to the reservation prices, which are always at least as high (low) as sellers’ (buyers’) valuations of the asset. The fact that a price building room between the reservation prices exists ensures the possibility of a trade profitable for both sides of the market also in terms of individual valuations. See Geltner (1997) and Fisher et al. (2003).
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the market more precisely. They should be able to take advantage of their superior knowledge at the cost of the “naïve” market participants. *** Models presented in this section, though doubtlessly extremely interesting as a source of knowledge about strategic behavior of investors in direct markets, were intended mainly to provide the methodical framework for further considerations. Since search turns out to be crucial for the phenomena discussed in this book, search models seem to be perfectly suited to cope with the problems arising from the lack of perfect liquidity. As shown in the course of the analysis in this Chapter, they allow examining the effects of the main sources of liquidity, such as the heterogeneity of market participants, the lack of market organization, or the existence of search costs. Furthermore, asset and investor specific characteristics, such as modalities of the search process, time horizon, or market dynamics, can also be taken into account. Application of the search theory in liquidity measurement and management techniques developed in the subsequent chapters confirm the enormous flexibility of this approach. It turns out that nearly any liquidity related problem can be presented in terms of a search model. In this context, further research on methods dedicated not only to real estate but also to other illiquid assets gains particular importance.
Chapter 3 Liquidity Measurement
The definition of liquidity provided in Chapter 1 is crucial for the delimitation of this feature from other relevant characteristics of investment alternatives. However, it is still insufficient for the application in formal decision models – a more precise notion is needed for this purpose that would allow an unambiguous comparison of investment alternatives with respect to their grades of liquidity. Having to choose between two otherwise identical assets, a rational investor will prefer the more liquid one, but she has to be able to identify it. A quantitative measure of liquidity is, thus, inevitable if it is to be considered as a decision criterion. However, due to its multifaceted character, it is very difficult, if at all possible, to define a single ratio that would adequately encompass all aspects of this phenomenon. This part of the book summarizes and discusses different possible approaches. While the main goal of the Chapter is the presentation of various liquidity measures, it is done against the background of search theoretical considerations. As argued earlier, the necessity of a more or less intensive search for a trading partner when selling or buying certain assets is the main source of their illiquidity. It should be therefore possible to interpret various liquidity measures in terms of a search problem. Indeed, it turns out that a link to search processes in illiquid markets can be established for many of the existing measurement approaches, even for those which seem to have nothing in common with search at first glance. Since liquidity of real estate markets is the main focus of the analysis, the real estate model is applied throughout the Chapter, so that the search-based versions of the discussed liquidity measures refer mainly to the case of direct property investments. However, like in other parts of the book, the restriction to real estate should be treated in terms of an example – analogical considerations could also be applied to other illiquid assets. The Chapter begins with the presentation of traditional liquidity measures referring to the liquidity of organized public markets. This subject has been researched for several decades already, mainly in the context of market microstructure. A number of different methods have been developed of which the bid-ask spread is undoubtedly the most popular one. Although these measures have originally been developed for publicly
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traded assets, they can also be used with private investments to some extent. This is demonstrated in the discussion of the possibilities of their application to direct real estate. The next group of measures refers to the probability of successful liquidation; hence, they are labeled as “probability-based” approaches. Unlike the traditional ones, they measure liquidity on the asset rather than the market level and have been applied mainly to assets traded in direct markets. Both traditional and probability-based approaches refer predominantly to marketability – they focus on the expected outcome of liquidation either with respect to the price dimension or with respect to the time required to accomplish it. They fail, however, to account for uncertainty about the liquidation value, which is immanent to illiquid assets. A separate section is therefore devoted to measures of liquidity risk. Only few specific approaches have been developed in the literature for this purpose. The section consists therefore mainly of own propositions made on the basis of general risk measurement principles and the real estate search model. Finally, a family of measures combining various aspects of asset liquidity in one figure is presented. Several approaches classify to this section. Among them are, on the one hand, liquidity performance measures, designed in analogy to return performance measures, and, on the other hand, utility-based measures based the concept of investor’s personal utility. Especially the latter group constitutes a bridge to the concept of liquidity management presented in Chapter 4.
3.1.
Traditional Measures of Market Liquidity
The problem of liquidity measurement was traditionally discussed in reference to public markets, especially stock exchanges, and the development of specific methods was mainly for the purpose of studying the effects of liquidity on stock prices, market returns, and related issues. Most of the existing approaches are therefore empirical proxies for unobservable liquidity rather than autonomous measures. They can be classified according to the three dimensions of market liquidity discussed in section 1.1.1.2 as measures of breadth, depth, and resiliency, referring to immediacy costs, large-trade effects, and the speed of market reactions, respectively. After reviewing the most popular approaches in their original forms, possibilities of their application to direct real estate investments are considered. The analysis on the basis of the search model from section 2.3 yields several possible measures, which allow capturing liquidity of nonpublicly traded assets in a similar manner as it is done for more liquid public markets.
3.1 Traditional Measures of Market Liquidity 3.1.1.
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Liquidity of Public Markets
3.1.1.1. Bid-Ask Spread As already discussed in section 1.1.1.2, market breadth is generally understood as the “distance” between the sellers’ and the buyers’ side of the market. On markets on which dealer services are offered, its natural measure is the difference between the bid (price for immediate buying) and the ask (price for immediate selling). Demsetz (1968) was probably one of the first to interpret the bid-ask spread in terms of a liquidity measure.285 From the investor’s point of view, it constitutes the cost of an immediate execution and, thus, the cost of turning an illiquid asset into a liquid one. In order to buy quickly, one has to pay a premium on the presumed fair price, and in order to sell quickly, one has to pay a commission reducing the effective sale value. Analyzing the spread from the dealers’ or market makers’ point of view allows a deeper insight into its connection with liquidity. For an individual (or institution) who commits oneself to trade at any time with any investor who accepts the quoted prices, the difference between the bid and the ask constitutes the profit margin. In a competitive market, it should be set just to cover the costs caused by providing liquidity. The determinants of these costs have been analyzed by many researchers; the general agreement is that they consist of three main components:286 • order processing costs, including the costs of being a market maker (e.g., the price of a seat on the exchange), • inventory costs, i.e., the costs of holding a sufficient inventory to bridge the time gap between buy and sell orders,287
285
286
287
Strictly speaking, Demsetz made the connection between the bid-ask spread and transaction costs; the reference to liquidity is indirect, expressed in the definition of the spread as the cost of making transactions without delay. See Demsetz (1968), p. 39. See Tinic (1972), Stoll (1978), p. 1133, Glosten/Harris (1988), Stoll (1989), p. 115, Glosten (1987), p. 1293, or Iversen (1994), pp. 28-51. For an early analysis of the spread components see also Demsetz (1968), pp. 40-45. For inventory models explaining the bid-ask spread see Garman (1976), Stoll (1978), Amihud/Mendelson (1980), or Ho/Stoll (1981).
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• adverse selection or information costs, i.e., the costs of exposition to the activity of informed traders who may take advantage of their superior knowledge at the cost of the market maker.288 While the first component is directly observable und widely unproblematic, both latter ones require a closer consideration. Lower liquidity of a public market means, in particular, less frequent trades. Thus, the market maker needs more time to match both sides of the market, and the size of the inventory must increase if the market making function is to be conducted without interruptions. This forces the market maker to hold a portfolio deviating from the optimal one and justifies a higher premium for his activity.289 The problem of informed traders additionally amplifies this effect. With less trade a single transaction gains more weight, and the impact of informed trading is stronger. The reward for the risk of losses incurred in such transactions widens the spread. The absolute distance between the bid (PB) and the ask (PA), i.e., the quoted spread, though doubtlessly closely correlated with liquidity, is still an imperfect measure. Its main weakness is the lack of direct comparability between different assets (stocks). In order to correctly assess assets’ relative liquidity, price levels need to be accounted for. Hence, the relative spread is more informative at this point. Another problematic issue is the fact that market participants are often able to trade within the official spread. The bids and asks quoted by market makers denote only the upper and the lower bound for individual transaction prices (Pt).290 Thus, more precise liquidity measurement can be achieved with realized effective spreads. Considering the above drawbacks several alternatives have been proposed in the literature:291
288
289
290
291
On the role of asymmetric information in the determination of bid-ask spread see Copeland/Galai (1983), Glosten/Milgrom (1985), or Kyle (1985). See Stoll (1978) for the analysis of dealer’s inventory holding costs in terms of deviations from the optimal portfolio composition. See Roll (1984), p. 1127, Grossman/Miller (1988), p. 629, Stoll (1989), pp. 115-116, or Huang/Stoll (1996), pp. 326-333. Bertin et al. (2005) find that ca. 17%-18% of REIT (Real Estate Investment Trust) transactions and ca. 22%-23% of non-REIT transaction transactions on the New York Stock Exchange are made within the quoted spreads. Empirical data on the relation between the quoted and the effective spread is also presented in Chordia et al. (2001a), pp. 508-509. See Chordia et al. (2000), p. 8, Hasbrouck/Seppi (2001), p. 401, or Christie/Huang (1994), p. 307 for definitions of different variants of the bid-ask spread.
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175
Proportional (percentage, relative) quoted spread = 2 (PA – PB) / (PA + PB)
(3.1)
Log spread = ln(PA/PB)
(3.2)
Effective Spread = |2Pt - (PA + PB)|
(3.3)
Proportional effective spread = |2Pt - (PA + PB)|/ Pt
(3.4)
Another spread-related measure is the amortized spread defined by Chalmers/Kadlec (1998) as the half of the total daily effective spread, i.e., the sum of all effective spreads in all transactions, scaled by the market value of the company at the end of the day. It can be approximated as the proportional effective spread times share turnover. Hasbrouck/Seppi (2001) proposed a further alternative, the quote slope, defined as:292
Quote slope = (PA - PB) / (ln(NA) + ln(NB))
(3.5)
where NA and NB denote quantities (number of shares) traded at the given bid and ask prices. In the graphical interpretation it is “the slope of the […] line connecting the bid and ask price/quantity pairs. If more quantity is added at either the bid or ask, or if either quote moves closer to the other, the line flattens and the market is more liquid.”293 Empirical studies reveal that the bid-ask spread is one of the determinants of returns in dealer markets. Amihud/Mendelson (1986a, b) and many others294 stated a significant correlation between spreads and stock returns interpreting this result as a liquidity premium, i.e. a reward to the investor for holding an illiquid asset. The effect remained after controlling for other possible sources of increased returns, including risk, company size etc.,295 and it could also be observed for other publicly traded assets.296 This led to the general opinion that the spread is indeed a good measure of liquidity in organized markets. Nevertheless, it has its limitations. Grossman/Miller (1988, pp. 628292
They also suggest a “log”-version of the slope defined as: ln(PA/PB) / (ln(NA) + ln(NB)). Hasbrouck/Seppi (2001), p. 402. 294 See Amihud/Mendelson (1989), Eleswarapu/Reinganum (1993), Kadlec/McConnell (1994), Eleswarapu (1997), Chalmers/Kadlec (1998), or more recently Porter (2003). 295 The role of the bid-ask spread was tested together with other factors (market and firm size, book-tomarket ration, beta coefficient, investor recognition) by Amihud/Mendelson (1986b, 1989), Kadlec/McConnell (1994), or Brennan/Subrahmanyam (1996). 296 See Amihud/Mendelson (1991) for treasury securities or Dimson/Hanke (2004) for equity indexlinked bonds. 293
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630) point out that the difference between the bid and the ask is not quite equivalent to the cost of providing immediacy services even if the effective spread is used. This would only be the case if the dealer was able to close positions immediately. However, positions opened with today’s transactions are usually closed in the nearer or further future. Thus, the “true” spread should be defined as the difference between the present purchase/sale price and the future repurchase/resale price, the latter one being uncertain. By ignoring this issue the time aspect of liquidity is neglected. Furthermore, the empirically measured spread contains several components that have more to do with the organization of market making than with liquidity. These include, e.g., “seat” costs or limitations of feasible price changes (minimal “ticks”). A more operational drawback of the bid-ask spread as a liquidity measure is its limited applicability – it can be computed only for dealer-type markets. Although many important markets, like currency, bond, or money markets are organized this way, many other relevant ones have other forms of organization. In particular, no bids or asks exist in security auction markets. Attempts have been therefore made to derive market breadth measures for these cases. Among the most widespread ones is the implicit spread proposed by Roll (1984).297 It is computed as:
Roll’s implicit spread = 2 − cov( R i , R i +1 )
(3.6)
with cov(Ri,Ri+1) being the (negative) covariance between successive price changes or returns (serial covariance). The proposition is based on the fact that there is no possibility of a price decrease after a trade at a bid or a price increase after a trade at an ask if the bid-ask spread is stable. E.g., if the last transaction was a sale, than the next one can either also be a sale accomplished at the same price or a purchase accomplished at a higher ask price. This asymmetry should lead to a negative serial correlation of price changes if the observed interval is long enough. Roll shows that the covariance of subsequent changes depends on the width of the spread and equals -spread2/4. Reversing this logic, he proposes an implicit measure of spread based on the return covariance, which can be computed in the absence of dealers. However, there are at least two serious problems with Roll’s proposition resulting from the assumptions behind his model. The first one is the efficiency of the analyzed 297
Choi et al. (1988) extent Roll’s concept by allowing for serial correlations in transaction types.
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177
market. Only in efficient markets all information is always incorporated in prices, and the width of the spread can remain constant. This also means that no adverse information component is present in the implicit spread.298 Another requirement is that price changes should be stationary during the observation period in order to provide an unbiased estimation. Both these constraints can be regarded as fulfilled for highly liquid assets, but they could be problematic for many less liquid ones. Thus, despite numerous successful applications of Roll’s implicit spread to stocks,299 the possibility of its application to less efficient and less liquid markets is highly disputable.
3.1.1.2. Market Depth As discussed in section 1.1.1.2, market depth directly influences the consequences of trading. Deep markets are those with many active traders, opposed to thin (or shallow) markets, where only few traders act at a time. In consequence, an individual investor may not be able to meet on sufficient supply or demand when attempting to buy or sell larger quantities, and execution of a large order may make a change in the market price (or in the bid-ask spread) necessary in order to absorb the trade. Thus, a single transaction has a greater impact on the market if liquidity is imperfect, and the price effect increases with the size of the transaction. Thus, an adequate measure of this aspect of liquidity should consider the gap between supply and demand, i.e., it should encompass not only the accomplished transactions but also the transactions that could not have been carried out due to insufficient market depth. However, extensive information about the intentions of all market participants necessary for the construction of such a measure is a serious hurdle.300 The most direct measure of depth in dealer type markets is the quantity available for trading at the quotes; it can be expressed either in currency units or as a number of shares.301 Such quantities are announced by market makers together with bids and
298 299
300
301
See Glosten (1987). See Roll (1984), Choi et al. (1988), Haller/Stoll (1989), Pagano/Roell (1990), pp. 79-83, or Iversen (1994), pp. 115-150, as well as references in Iversen (1994), pp. 108-110. See Fernandez (1999), p. 10. A unique study of market depth based on outstanding supply (i.e., the trading volume that could not have been realized due to insufficient demand) is provided by Silber (1975). The author utilizes the ability of the trading system on the Tel Aviv Stock Exchange to provide this kind of data. See Chordia et al. (2000), p. 8, Hasbrouck/Seppi (2001), p. 401, Huberman/Halka (2001), p. 165, Degryse et al. (2004), or Bertin et al. (2005), p. 160.
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asks, and trading within these limits does not affect prices. Other often used proxies available also for auction markets are the average trading volume and the turnover.302 One can expect that balancing supply and demand is more difficult when there is less trading (in absolute terms or in relation to market capitalization). Some of the sellers or buyers will therefore be able to accomplish transactions only by accepting less favorable conditions. In effect, the impact of a single transaction on the market price should be higher. On the other hand, higher volumes indicate better chances of finding a trading partner without affecting the price level. However, both the quoted maximal transaction size and the level of trading activity have serious limitations as depth measures. By using the first one, the possibility of splitting larger transactions among multiple dealers is not taken into account, and by using the latter one, the average transaction size is ignored. In both cases, the resulting biases may distort the conclusions about markets’ relative liquidity. Nevertheless, several studies stated higher average returns in times of lower trading activity interpreting them in terms of liquidity premia.303 Another popular measure based on return fluctuations due to trading is the “liquidity ratio” – the ratio of the (daily) trading volume to the (daily) price change.304 Even a relatively low market activity leads to major price fluctuations in illiquid markets; on the other hand, deep markets can absorb high trading volumes without fundamentally unjustified price effects. Thus, the liquidity ratio should be high in the former ones and low the in the latter ones. Though simple in computation, this measure has several major drawbacks. One of them arises from the fact that in many markets a lot of trading is between dealers. E.g., inter-dealer trading constitutes up to 40%-45% of the total volume on the London Stock Exchange and on NASDAQ, and it can even reach 85% in foreign exchange markets.305 This means that in some cases, despite the seemingly high market activity, an “outside” investor may find it difficult to finding a trading partner. Another possible problem is the disproportional reaction of the price level to
302
303 304
305
See Silber (1975), Brennan et al. (1998), Datar et al. (1998), Jones (2002), Chordia et al. (2001b), or Baker/Stein (2002) for applications of trading volume and turnover as liquidity proxies. On the link between trading volume and liquidity see Pagano (1989). See Datar et al. (1998), Brennan et al. (1998), or Jones (2002). See Khan/Baker (1993), pp. 225-226, Cooper et al. (1985), p. 25, or Bernstein (1987), pp.57-59. A reciprocal of the liquidity ratio is used by Porter (2003), pp. 7-9, and Brunetti/Caldarera (2004), p. 12. Amihud (2002, p. 34) additionally adjusts the ratio by the time period to which it refers. See Viswanathan/Wang (2004), p. 987-989, and the references therein.
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179
the size of the trade. Although a correlation between the volume and the price change is often observed,306 it is not proportional – markets seem to react relatively stronger to smaller transactions.307 Hence, when comparing two equally liquid markets with different average transaction sizes, lower liquidity ratio of the one with smaller transactions would result. To ease this problem, Marsh/Rock (1986) suggest using the number of transactions instead of the trading volume. Finally, the most severe problem with the liquidity ratio, as indicated by Grossman/Miller (1988, p. 630), is the fact that the volatility of prices can and does have many sources other than illiquidity. Frequent fundamental information shared by all market participants moves prices without necessarily increasing the trading volume. Fundamentally more volatile markets will therefore have higher liquidity ratios. Market depth can also be measured as the order flow necessary to induce a unit change in the market price or spread. This notion is used, e.g., in the market microstructure model by Kyle (1985), where it is denoted as “λ”. It is also utilized in a number of theoretical and empirical studies in both dealer and auction markets.308 The practical computation is usually based on an econometric estimation – trading volume is regressed on price changes. Brennan/Subrahmanyam (1996) found a positive relation between Kyle’s λ and average returns interpreting it in terms of a liquidity premium. Proper interpretation of this measure may, however, be biased by other factors affecting the relation between trading activity and returns as discussed earlier.
3.1.1.3. Price Reversal Resiliency refers to the dynamic aspect of trading (see section 1.1.1.2). It is “the speed with which price fluctuations resulting from trades dissipate or how quickly market clears order imbalances”309. A direct measure of resiliency, indicated by Grossman/Miller (1988, pp. 627-628), is the correlation (or covariance) between successive price changes. In illiquid markets, trading activity induces changes of the price level (see former section). Since they are not fundamentally justified, a counter-reaction should follow bringing the price level back to equilibrium. In consequence, negative 306
307 308 309
See Karpoff (1987). The link between volumes and price changes is intuitive – new information induces more investors to trade resulting in stronger market reactions. See Marsh/Rock (1986), as well as Bernstein (1987), pp. 58-60, for a comment. See Glosten/Harris (1988), Hasbrouck (1991b), Foster/Viswanathan (1993). Fernandez (1999), p. 10.
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serial correlations should be observed. Due to quicker reactions, the measured correlation level should approach zero on liquid markets with large numbers of traders. Grossman/Miller (1988) demonstrated this effect using a three period market maker model and also stated the dependence of resiliency from the variability of the market, hedging demand, and the number of market makers. These theoretical considerations have been confirmed to some extent by empirical studies. While positive short term (low order) autocorrelations between daily stock returns have been often reported, correlations for longer terms (higher orders) were mostly negative.310 Furthermore, Campbell et al. (1993) found a negative relation between serial return correlation and trading volume, which seems to confirm the postulated link to liquidity. In this context, the implicit spread proposed by Roll (1984) and discussed in the former subsection can also be viewed as a resiliency measure as it is directly based on serial covariance of returns. However, it has to be noted that this measure is based on an entirely different idea. While resiliency is about the speed with which markets reacts to changes caused by single transactions, Roll’s spread reflects the consequences of the mere existence of buyers’ and sellers’ on the market. Nevertheless, since only series of effective prices and their correlations can be observed in practice, differentiation between “normal” trading within the (implicit) spread and “resiliency” trading resulting as a reaction to liquidity-induced price movements will usually be impossible. Therefore, Roll’s implicit spread will inevitably have a resiliency component. Also the realized spread (RS) defined by Huang/Stoll (1996) can be interpreted in terms of resiliency. This empirical measure is defined as the difference between prices observed within a time interval Δt provided the first price was an ask (RSA) or a bid (RSB): RSA = - [(pt+Δt - pt)|pt = askt] RSB = - [(pt+Δt - pt)|pt = bidt]
(3.7) (3.8)
In liquid markets with quick reversals, realized spreads should be lower than in illiquid markets where the price impact of a single transaction lasts longer.
310
See Fama (1965) or Conrad/Kaul (1988).
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181
One of the more recent approaches to capture market resiliency is the empirical price reversal suggested by Pastor/Stambaugh (2003).311 It is computed as the least square estimate of the parameter l in the following regression: rte+1 = θ + φ ⋅ rt + l ⋅ sign(rte ) ⋅ ϑ t + ξ t
with: rt e t
(3.9)
- return in period t
r
- excess return over the market (market index) in period t
ℓ
- price impact parameter
ϑt
- trade volume in period t
ξt
- error term
Since excess returns resulting solely from the flow of orders are practically absent (or are balanced almost immediately) in liquid markets, no influence of trading volume should be observable there, and the estimate of l should be close to zero.312 On the other hand, on an illiquid market, order imbalances leading to contemporaneous excess returns are expected to be reversed in the future – slower reversals indicate lower liquidity. Hence, high values of l indicating strong reversals occurring in subsequent periods are symptomatic for illiquid markets. A problem common to all above measures is the choice of the time horizon necessary to observe a reversal. Huang/Stoll (1996, pp. 327-32) argue that the considered periods should be long enough to give an offsetting transaction an opportunity to occur, but short enough to preclude noise resulting from the general return volatility. This problem leads to a serious limitation in the practical application of the discussed measures with respect to the comparability of different assets. While one day seems to be appropriate for stocks, longer periods may be necessary for other investments. The Market Efficiency Coefficient (MEC) proposed by Hasbrouck/Schwartz (1988) may perform better in such cases.313 It is computed as the ratio of the long term variance of logarithmic returns to their short term variance. Liquid assets’ returns should fluctuate 311
312
313
See also Porter (2003), pp. 5 ff. A related approach to analyze resiliency on the basis of a simulation with a Vector Auto-Regressive (VAR) model is applied by Degryse et al. (2004), pp. 9-11 and 17-19. The inclusion of the trading volume in the regression is motivated by the results of Campbell et al. (1993) who state and model the dependence between trading volume and serial correlations of stock returns. See Pastor/Stambaugh (2003), p. 647. For comments see Bernstein (1987), p. 60.
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within relatively short periods remaining more stable over longer time horizons. On the other hand, illiquid assets, due to the slow return reversion, should vary more strongly over longer periods. MEC should therefore be low for the former ones and high for the latter ones. Since the measure is based on two different time horizons, it should be less sensitive to the above mentioned problem and allow more general conclusions. 3.1.2.
Application to Real Estate Markets
Liquidity measures reviewed in the previous section are (more or less) standard approaches designed for various types of organized public markets or assets traded in these markets. This environment is distinctly different from the one valid for direct real estate investments as well as for most direct markets. A straightforward application of depth, breadth, or resiliency based measures is therefore in most cases not possible. Nevertheless, they have some desirable advantages, which would also be welcome in the measurement of private markets’ liquidity: they are widely known and well researched, and they fit well in many existing liquidity management systems. It might be therefore worthwhile to attempt a translation of the “public” measures for the use in direct markets. This is done in the following sections on the basis of the search model developed in Chapter 2. Apart from the pure theoretical appeal, this step may be also interesting from the practical point of view, providing analysts with measures they are familiar with.
3.1.2.1. Implicit Bid-Ask Spread No bid-ask spread exists in the absence of market makers or dealers. This applies obviously to direct real estate and most private markets – there are no institutions from which properties could be bought or to which they could be sold immediately provided a premium or a concession was paid. Same applies to private equity, wine, arts, and other assets traded on direct markets. Thus, there is no direct possibility of observing or computing the spread. However, using the search model presented in the previous chapter, an implicit measure of a hypothetical spread is feasible. Its theoretical derivation is, in fact, simple and follows the same logic as various inventory models with search, like the one by Stoll (1978). If some hypothetical risk-neutral institution had the technical and financial possibility of dealing on a real estate market, it would set bid and ask prices in the same manner as it is done on security or currency markets. It
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would be ready to buy at a price no higher than the expected expense that could be achieved with the optimal search strategy – only then the hypothetical dealer would be indifferent between buying immediately from an investor and searching for a seller by his own. Similarly, the immediate selling price should be no less than the expected net receipts from a liquidation conducted according to the optimal strategy. Hence, the implicit spread can be assessed as the difference between the expected price paid and the expected receipts received at the respective optimal reservation prices. The former one should typically lie below the average market valuation, and the latter one should lie above it. Assuming that buy and sell offers come from the same probability distribution, following implicit relative bid-ask spread (Implicit Spread, IS) based on the real estate search model from section 2.3 results:314 IS = max E(Γ π ∗S ) − max E(Ξ π ∗B ) ∗ ∗ πS
with: E(Γ πS∗ ) = E(Ξ π∗B ) =
(3.10)
πB
⎛ λ γ−η ⎞ ⎟ ⎜⎜ E(Π Π > πS∗ ) ⋅ 1 − FR (π∗S ) + ∗ λ + ρ − τ ⎟⎠ ρ − τ + λ ⋅ (1 − FR (πS )) ⎝
(
)
⎛ γ+η ⎞ λ ⎟ ⎜ E(Π Π < π∗B ) ⋅ FR (π∗B ) − λ + ρ − τ ⎟⎠ ρ − τ + λ ⋅ FR (π∗B ) ⎜⎝
πS*, πB* - optimal selling (S) and buying (B) reservation prices γ - relative rental income η - relative inspection /appraisal cost (referred to the mean offer value) Reservation prices in this formula are set to maximize expected receipts (πS*) or minimizing expected expenditure (πB*), which is the consequence of the assumed risk neutrality of the hypothetical dealer. Although the idea behind this approach is relatively straightforward, its practical realization meets with serious difficulties. If an objective measure valid for any investor is to be derived, optimal buying and selling prices also need to be objective. While the structure of offers can be assumed to be the same for all investors, the choice of an appropriate discounting factor constitutes a serious problem. It was argued in section 2.3.1.3 that investor’s personal opportunity costs should be applied. On the other hand, 314
The expected net receipts are defined in section 2.3.2, formula (2.36); the expected expense corresponds with the one derived in the section 2.3.3.7, formula (2.46).
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when considering a professional (though hypothetical) risk neutral dealer, a risk free or money market interest rate would be probably more in place. Also observation costs should be set at a level appropriate for a dealer. The issue of appraisal or inspection costs arises in this context. They have been ignored in the initially analyzed real estate sale problem since the seller can be expected to know the characteristics of her property. This is obviously inappropriate when the purchase case is considered, as discussed in section 2.3.3.7. However, the necessity of inspecting and valuing the property does not disappear in the sale case either; it is only transferred on the other party of the transaction. Thus, the question arises if the respective costs should be accounted for in an objective hypothetical bid price. Their inclusion seems justified if the same offer distribution is to be used for both sides of the market since prospective buyers can be expected to correct their offers for appraisal expenses. An alternative approach to computing the Implicit Spread may be borrowed directly from Roll (1984). Since Roll’s spread requires only the estimation of the covariance between subsequent changes of transaction prices, it is significantly simpler to compute than the search theory-based spread. The amount of transaction data necessary for a precise estimation is, however, its main drawback. While easy to obtain in organized public markets, reliable transaction prices are rather rare in the real estate branch. Moreover, properties in the existing databases are usually only partially comparable, and prices are reported in irregular intervals. Attempts to estimate serial correlations would be therefore subject to substantial biases. A problem immanent to any indirect estimation of the bid-ask spread is the issue of information risk. As noted in the section 3.1.1.1, information asymmetry is one of the components of the premium required by dealers who have to be rewarded for expected losses arising from trades with insiders. With respect to real estate markets, this aspect will most probably play a substantial role on the buyers’ (bid) side, but it will be relatively unproblematic on the sellers’ (ask) side. Unfortunately, this issue remains disregarded in Roll’s implicit spread giving another reason against the use of this measure with real estate investments. On the other hand, the cost of inferior information is, at least partially, included in the search-based approach by allowing for appraisal expenditures. Assuming that appraisals reveal any hidden characteristics of properties allows their interpretation as monitoring costs necessary to overcome information asymmetry.
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3.1.2.2. Quick Sale Discount An alternative concept related to the spread but based on a slightly different idea is the “discount attending premature sale” proposed by Lippman/McCall (1986).315 For simplicity, it concentrates only on the seller’s side of a transaction, though incorporating the buyer’s side is trivial. The measure reflects the cost of a quick but suboptimal sale. As indicated in earlier chapters, illiquid assets are characterized by the necessity of a more or less intensive search. By conducting it according to a certain strategy, the outcome can be optimized; in particular, the expected receipts can be maximized. An immediate sale to the first potential buyer is therefore in most cases not optimal. The discount, which the investor is expected to incur when she ignores the optimal search policy and insists on a quick liquidation, may be used to measure liquidity. It is here denoted as the Quick Sale Discount (QSD) and defined as follows: QSD = 1 - Vt/Vopt, with: Vt - value of the search terminated within the first t periods Vopt - value of the optimally conducted search
(3.11)
A more specific definition of the QSD can be provided by applying the real estate search model. For simplicity, quick sale can be defined as sale within one period, i.e., sale to the first interested buyer. It is equivalent to setting an infinite negative reservation price. The expected net receipts can be then derived by computing the limit of the expression (2.37) for π* approaching minus infinity. The following QSD-formula results: ⎞ ⎛ λ λ⋅γ ⎟ ⎜ ⎜ ρ − τ + λ + (ρ − τ + λ )2 ⎟ ⎠ ⎝ QSD = 1 − ⎡ ⎛ ⎞⎤ λ γ max ⎢ ⎜⎜ E(Π Π > π *) ⋅ (1 − FR (π*)) + ⎟⎟⎥ π* ( 1 F ( *)) ρ − τ + λ ⋅ − π λ + ρ − τ ⎝ ⎠⎦ R ⎣ = 1−
315
⎡ ⎤ ρ − τ + λ ⋅ (1 − FR (π*)) γ ⋅ min ⎢ ⎥ ρ − τ + λ π* ⎣ E(Π Π > π *) ⋅ (1 − FR (π*)) ⋅ (λ + ρ − τ) + γ ⎦
(3.12)
See also Mok (2002a), p. 12, and section 3.4.2 for the “utility version” of the Quick Sale Discount.
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Perfectly liquid assets are marketable without a discount. Search is impossible or pointless in this case, so the first available offer should be accepted. Highly illiquid assets are characterized by high divergence of subjective valuations making search worthwhile. Discounts for these assets should therefore be high. A different version of the discount is the spread between the retail and the wholesale price proposed by Krainer/LeRoy (1997 and 2002). The retail price is defined as the effective realized sale price, and the wholesale price is the value of the asset (house) to its owner just prior to the arrival of the successful buyer. The term „wholesale “ is used to indicate that the seller would be ready to give up the house immediately without further search to a wholesaler (if one existed) at this price. In other words, “this spread measures the capital gain the seller experiences when a house sells”316, or equivalently “it equals the price sacrifice a seller would have to accept in order to achieve immediate sale”317. Krainer and LeRoy show that their spread is zero for perfectly liquid assets, and that it increases with a decreasing frequency of offers. This measure is rather difficult to interpret directly in terms of a search process, as it arises from an equilibrium model designed by the authors. It seems, however, that the “retail price” corresponds with the value of the accepted offer and the “wholesale price” with the value of search. In terms of the search model, the expected value of the former is E(P|P>p*), and the latter is equal to the maximum net receipts from search (max E(G)) for a risk neutral investor. In the traditional search setting, the difference between these terms is, however, equivalent with the periodical search cost (see section 2.2.3, p. 120). An investor would be ready to give up the next offer only if the expected additional gain would exceed the additional cost. Thus, the measure of Krainer and LeRoy seems to be, in fact, a measure of search costs.
3.1.2.3. Market Depth Obviously, no direct measure of market depth referring to the maximal volume of trading at a given bid-ask spread is applicable to direct investments. One could try to proxy it as the average transaction size for which the Implicit Spread defined above is still valid. A slightly more practicable alternative includes the use of empirical turnover. In real estate markets, the latter can be understood as the ratio of the overall trans316 317
Krainer/LeRoy (1997), p. 5. Ibidiem, p. 2.
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187
action volume in a (sub)market to the total value of properties in this market. High turnovers implicate higher liquidity; however, a correction for the number of transactions or the size of a typical transaction is necessary. Given two real estate markets with equal turnovers, the one with more frequent transactions, and hence with lower average transaction sums, can be expected to offer better liquidation chances. Ignoring this could lead to misleading results. For instance, it is possible that in a certain area one of two shopping malls has been sold generating a turnover of 50%, which would probably exceed the turnover in the local residential housing market; but this would not mean that the market for malls was more liquid than the market for houses. Thus, inflating the turnover ratio by the size or by the number of transactions or even substituting the transaction volume with the number of transactions seems reasonable. Even then, however, the ratio would be only a rough proxy of market depth. Equally difficult are attempts to construct an empirical measure of impact of large transactions on the price level. The most straightforward difficulty is connected with the lack of sufficient data. There is rarely a possibility of observing a sufficient number of transactions within a short period of time to assess the price effect of a large trade. But also the stickiness of real estate prices may complicate impact measurement; it is by all means possible that a large transaction leads to a lower trading activity rather than to an adjustment of subsequent prices, so that no price reaction would be observable at all. Further difficulties arise from the relatively underdeveloped market information systems in real estate markets. It is conceivable that a substantial fraction of market participants learns about the unusually large transaction with a delay, or even does not learn at all, so that market reactions may be stretched over long periods of time and blurred by other events. Another factor is the role of the construction industry, which also reacts to changes in the market situation and may influence it by providing new properties. All in all, direct empirical measurement of reactions of prices to trading activity, which is difficult even for public markets, seems nearly impossible for real estate. Bearing in mind the complications discussed above, it is still tempting to attempt a definition of some kind of an implicit depth measure applicable to real estate, preferably by using the search framework. The implicit liquidity ratio seems to be the most promising reference point. Similarly to the original one, it should measure the reaction of prices (spreads) to high trading volumes, and it should be small in liquid markets
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capable of absorbing large trades without price impacts and large in illiquid markets in which even a relatively small transaction affects the overall price level. In the search framework, such reactions may be viewed in terms of changes in the offer flow, especially in the distribution of offers. Consider the situation of a seller willing to put up for sale an apartment in a neighborhood in which a large sale transaction took place very recently – e.g., a residential real estate trust liquidated a portfolio of identical apartments. The pool of potential buyers faced by the seller is smaller than usually, because many of those who considered buying an apartment already bought one from the trust. Furthermore, assuming that the trust sold its properties optimally, fully exhausting buyers’ willingness to pay, the apartments were bought predominantly by investors with high valuations. In consequence the seller cannot expect as many unusually high offers for her apartment as in the “normal” market state. This situation can be simplified by assuming that the new offer distribution is the original one with a part of the right tail cut off.318 Knowing the typical trading volumes and the transaction frequencies on the market, it should be possible to assess the dimension of this effect, and by applying the adjusted probability distribution of offers in the search model, the corresponding reduction of the expected receipts from sale can be estimated. The implicit liquidity ratio LRi could be then computed as the ratio of the (proportional) change in the expected receipts due to the large transaction to the volume of this transaction: LR i =
E (Γ FR ,post −sale (π) ) − E (Γ FR ,pre−sale ( π) )
E (Γ FR ,pre−sale (π) ) ⋅ Transaction Volume
(3.13)
Note that this measure does not need to be based on any empirical observations of large sales – a hypothetical trading volume and its estimated effect on the offer distribution can be used. This is advantageous if ratios for various markets are to be compared as the same base volume can be applied.
318
Note that the effect of a recent large transaction on the distribution of offers is analogous to the effect of setting a listing price (see section 2.3.3.5) – in both cases the right tail of the p.d.f. is cut off at a certain point. The only difference is that the whole probability mass of the tail is allotted to the cut-off point in the “listing price” case, while it is distributed over all remaining offers in the “large transaction” case.
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189
3.1.2.4. Market Resiliency The resiliency measures presented in section 3.1.1.3 are mostly based on the relation between subsequent returns. High serial covariance of daily stock returns is interpreted as an indicator of low resiliency. This approach is obviously not applicable to privately or even less frequently traded assets, and, thus, to higher levels of illiquidity. The main problem is the already mentioned choice of the proper time window for observing return reversals. While liquid stock or currency markets react within minutes, so that one day can be already viewed as a “long period”, it is clearly too short to register any response in a real estate market. As a result, short term return covariance at a zero level can be expected for most property markets, which is definitely not due to their extremely high liquidity. However, even with adequately long measurement periods, it is rather doubtful that serial covariance can be a useful resiliency measure for privately traded assets. Its unambiguous interpretation when comparing different markets seems hardly possible. Imagine, for instance, that a very low covariance was measured for monthly returns in the (real estate) market A but a very high negative one in the market B. Without a deeper analysis it can only be stated that the reversal period in A is other than one month; lower covariance may result from the fact that this time interval was too short to allow any reaction or that the reaction was much quicker and was already overlaid with other sources of volatility. This problem is slightly less severe for Hasbrouck and Schwartz’s (1988) Market Efficiency Coefficient since two time windows are used there, but it does not disappear entirely. In view of the problems connected with the use of serial correlations in private markets, time needed to balance a large transaction seems to be a more reasonable alternative. If a typical reaction time exists, it should be possible to identify it by analyzing autocorrelations of various orders and, thus, various time windows. The window length for which the highest negative correlation is observed can be interpreted as the time required to reverse liquidity induced price fluctuations. Markets with short reaction periods can be considered as more resilient than those with longer ones. However, though methodically very appealing, this concept is rather difficult to apply for most real estate markets. The reason is again the lack of sufficient data. With annual or at best monthly indexes, no reasonable market reaction periods can be determined – weekly data seem to be the minimum for this purpose. Another problem, already mentioned in the previous section, is the possibility that not the price level but the trading
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volume reacts to liquidity events and is reversed. Thus, an additional analysis of volume autocorrelations might be necessary. A further bias may arise from the existence of real estate cycles, which also induce (positive) autocorrelations of market variables. In consequence, even if the application of reaction times to proxy real estate markets’ resiliency was possible, it would be a very imprecise measure. An alternative approach to estimate the resiliency of real estate markets is the use of offer arrival rates.319 Since resiliency stands for the dynamic aspect of liquidity, markets with more intensive offer flows should also react faster to short term, unexpectedly high trading volumes. However, this approach does not entirely capture the idea of resiliency. Not the arrival rate itself but rather the reaction of the arrival rate to the possibilities of profitable trading arising from fundamentally unfounded price movements is of interest. As discussed in the previous section, high sale volumes take buyers with higher reservation prices out of the market and result in (contemporaneous) changes of the offer distributions. In effect, the achievable gains from searching fall, and sellers should correct their reservation prices in response. This again opens for buyers the possibility of acquiring properties below their usual prices. The offer arrival rate should increase in such situations, as even those potential buyers who would not attempt a purchase under “normal” conditions are attracted. Back on the sellers’ side, an increased offer arrival rate may result in lower opportunity costs making up for the changes in the offer distribution and driving the expected receipts from sale, reservation prices, and finally also the transaction prices to higher levels. Of course, this effect is only temporary, and the initial situation is restored as the offer distribution comes back to normal. It follows from the above reasoning that the price reversal is initiated by the adjustment of the offer arrival rate. A possible measure based on this concept could include, e.g., the sensitivity of expected sale receipts to changes in the offer arrival rate. Unfortunately, there is no easy way of implementing this idea in the search model, as it assumes strategic trading on both sides of the market; the model utilized in this work allows for strategic behavior only on one of the sides. Changes in the arrival rate expected within a certain period of time after the initial transaction would therefore need
319
The role of the offer arrival rate (in terms of a Poisson process) for various aspects of public markets’ microstructure, including liquidity, is modeled by Garman (1976) and Amihud/Mendelson (1980).
3.2 Time- and Probability-Based Measures
191
to be estimated empirically. Considering the amount of information about market activity necessary for such estimation, it seems rather impossible to conduct in practice. However, even if this problem could be overcome, changes in the offer distribution resulting from the arrival of new buyers would still remain disregarded. Thus, also this approach gives little hope for a practicable resiliency measure based on techniques developed in this book.
3.2.
Time- and Probability-Based Measures
Illiquidity measures presented in the former section were related to the traditional approaches concentrating typically on various aspects of liquidity in public markets. Among liquidity measures used in the literature for non (or not necessarily) public investments, those based on the chance of successful liquidation or on the time required for liquidation are most popular. Contrary to the traditional ones, they focus on the liquidity of assets rather than the liquidity of whole markets, and they tend to concentrate only on the possibilities of sale disregarding the purchase case. 3.2.1.
Probability of Sale and Time on the Market
The probability of sale (PoS) is a very intuitive measure of liquidity.320 It refers to the chances of a successful liquidation of an asset within a given span of time. Clearly, it should be higher for more liquid assets. Time-on-the-market (ToM), also referred to as time-till-sale or marketing duration, is directly related to the probability of sale.321 It is defined as the time it takes on average to sell an asset, or the time the asset remains on the market. Liquid assets can be marketed quickly, illiquid ones require more time; hence, ToM decreases with liquidity. The ease of interpretation is probably one of the reasons for the popularity of this measure among researchers. It has been estimated from empirical data using econometric models322 as well as derived from formal market models323. The link between PoS and ToM is straightforward – higher probability of sale should on average lead to shorter liquidation periods. However, while ToM can be defined either as the expected future marketing time (ex ante approach) or as the 320
321
322 323
See Haurin (1988), pp. 406-407, Krainer/LeRoy (1997), pp. 5-7, Krainer (1999), pp. 17-19, Krainer (2001), pp. 42-45, Krainer/LeRoy (2002), pp. 232-233. Some researches go even further and set an equivalence mark between liquidity and ToM. See Fisher et al. (2004), p. 241. See Miller (1978), Asabere et al. (1993), Forgey et al. (1996), or Anglin et al. (2003). See Lippman/McCall (1986) or Krainer (1999).
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average marketing duration observed in the past (ex post approach), PoS makes sense only as an ex ante measure. Note that even if sale probabilities are derived from past transaction frequencies, they are, in fact, estimations of (implicit) past ex ante sale probabilities. An important aspect in the research on the marketing time on real estate markets is its relation to the listing price and to the realized sale price. It has been empirically verified by numerous researchers and interpreted in terms of a price-time locus specific for each market.324 The observed slope of the locus is usually positive – the listing price and the sale price increase with ToM.325 This result is in line with the traditional view of liquidity as the positive dependence between the duration of the liquidation process and the realized value (see section 1.1.1.1). Which point of the locus is chosen by an investor depends on her preferences – more time concerned investors prefer shorter ToMs, and more price concerned ones choose longer ToMs.326 Expressing time preference in form of a discounting rate leads straight to the definition of liquidity as the selling time corresponding with the highest achievable present value of liquidation discussed in section 1.1.2. According to it, investors with higher time preference choose quicker selling procedures than less time concerned ones, but they do it at the cost of a lower nominal sale price. Further evidence on the link between ToM and liquidity arises from the analysis of the determinants of marketing duration. One of the central statements is that the duration of search and ToM increase with property’s atypicality.327 The more unusual the property is, the more difficult it is to find a buyer. This again complies with the notion of illiquidity as the result of heterogeneity of asset’s valuations among potential buyers (see section 1.2.4), which is higher for less typical properties.
324
325
326
327
See Belkin et al. (1976), Trippi (1977), Miller (1978), Asabere et al. (1993), Forgey et al. (1996), Glower et al. (1998), or Anglin et al. (2003). Cubbin (1974) found an opposite direction of the relationship between the selling price and the speed of sale on the analyzed housing market; he admitted, however, that this should not occur in a normal market situation. Anglin (2003) studies the consequences of market changes on investor’s choice of the preferred ToM decomposing the total effect in a liquidity and a value component. The first one results from the time preference and the second one from the price preference. Cubbin (1974), pp. 183 ff., reverses this logic theorizing on the consequences of investor’s choice for the market. See Haurin (1988), Forgey et al. (1996), or Glower et al. (1998).
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Both PoS and ToM can be easily interpreted in terms of a search model. The former one is nothing else than the probability of receiving an acceptable offer within some time horizon, i.e., the probability that an offer above the reservation price arrives during this time period. The simplest approach is to define the time horizon as the number of reviewed offers (i). Using the basic search framework, the following PoS formula results:328 i
PoSi = ∑ (1 − F(p*)) ⋅ F j−1 (p*) =1 − Fi (p*)
(3.14)
j=1
Defining the time horizon in time units and assuming random arrival times complicates the computations since the number of offers (I) within the considered time period ~
( T ) is random in this case. The respective PoS formula is then as follows:329 ∞ ~ PoST~ = ∑ 1 − Fi (p*) ⋅ Pr(I = i T)
i =0
(
)
(3.15)
In this context, ToM corresponds with the duration of search and is related to the time horizon in the definition of PoS.330 The expected ToM is, thus, equivalent to the expected duration of search. In the basic models with discrete time, like those discussed in sections 2.2.3 and 2.2.4, the duration of search corresponds with the number of offers and is determined by the reservation price – the expected duration of search as well as the “discrete” time-on-the-market (ToMD) is simply the reciprocal of the probability that an acceptable offer arrives:331, 332
328
329
330
331
332
The definition of the “probability of purchase” is analogue; it is the probability of finding a property with the demanded price lying below the reservation price. Also other considerations in this section can be applied to the buyer’s search for the best seller. In the real estate search model with a Poisson offer arrival process, the probability of receiving i ~ offers within a time span of T is determined by the Poisson probability function, which is defined ~ ~ −λ⋅T i ~i as P(I = i T )=e ⋅ λ ⋅ T ⋅ i! with λ being the offer arrival rate. Note that ToM refers to the exact duration of the liquidation process, while the time horizon in the definition of PoS is the maximal liquidation time. ToM defined as the number of offers is, in fact, identical with the expected duration of search E(D) defined in section 2.3.1.2. See also FN 218. See Cubbin (1974), p. 178, Haurin (1988), p. 399-400, or Glower et al. (1998), p. 723. Krainer/LeRoy (1997), pp. 5-6, Krainer (1999), p. 18, and Krainer/LeRoy (2002), p. 232-233, define ToM as the expected remaining marketing duration, which corresponds with the reciprocal of the hazard ratio defined in the next section.
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E(ToM D ) =
1 1 − F(p*)
(3.16)
The unit of E(ToMD) in the above formula is the number of offers reviewed until an acceptable offer arrives. The corresponding duration of search expressed in time units depends on the lengths of the time periods between offers, which can be viewed either as fixed (discrete approach) or as random (continuous time framework). With λ being the frequency of offers (fixed or expected in terms of the mean of a Poisson process), the “continuous” ToMC is defined as follows: E (ToM C ) = E (ToM D ) / λ
(3.17)
The possibility to apply the search-based ToM for liquidity measurement was indicated already by Miller (1978) and extensively formulated by Lippman/McCall (1986).333 The latter authors propose the optimal expected selling time as a liquidity ~ measure. The optimal duration of search ( t ∗ ) for a risk neutral seller is defined as:334
(
~ ~ ~ ~ E (G t *) = max E (G t ) : t ∈ T
)
(3.18)
~ The variable t * is random, and its expected value is used as the indicator of asset’s ~ liquidity.335 It is easily noticed that E ( t *) is equivalent to the expected duration of
search (or ToMD) resulting at a reservation price which maximizes the expected net receipts from sale.336
~ E ( t *) =
1 1 − F(p ∗opt )
(3.19)
Lippman and McCall demonstrate the compliance of their measure with a number of various notions of liquidity. In the first place, they re-interpret Keynes’ concept of a 333
334
335
336
Further applications of the search framework for the analysis of ToM can be found in Haurin (1988), Forgey et al. (1996), Glower et al. (1998), Krainer/LeRoy (1997, 2002), or Krainer (1999, 2001). Note that the optimal duration of the search arises from the optimal stopping rule applied during the search and is therefore a random variable. See Lippman/McCall (1986), p. 46-47. The authors choose the mean of ~t * but indicate that also other distributional parameters can be used for this purpose. See Lippman/McCall (1986), FN 10. See also the discussion of the equivalence between the expected duration of the search and the reservation price in section 2.3.5.
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liquid asset being “more certainly realizable at short notice without loss”337 as: “has a higher probability p of being sold in one period in accord with the optimal policy”338. It follows from the equation (3.19) that a shorter expected search duration implies a ~ higher sale probability (1-F(popt*)), so lower E ( t *) means higher liquidity. The measure is also positively correlated with investor’s time preference rate. This is in line with the role of investor’s impatience for the perceived level of liquidity: the more pressing the need for money is, the more disturbing the inability of quick liquidation becomes.339, 340 Furthermore, the predictability of the realized price, understood as the divergence of opinions about asset’s value, and the thickness of the market, understood as the frequency of offers, are positively related to the optimal expected marketing time. However, the latter statement holds only if either the interest rate is near zero or the frequency of offers is very high.341 Finally, the authors provide the proof of a link between their measure and the discount due to prompt liquidation (as derived in section 3.1.2). Thus, a measurement concept as simple as the optimal ToM can encompass a variety of different approaches to liquidity. 3.2.2.
Proportional Hazard Ratio
Kluger/Miller (1990) propose a measure of liquidity based on Cox’s proportional hazard model.342 This methodology is generally a useful tool for describing durations in various processes.343 The concept of the proportional hazard ratio (PHR), also denoted as the odds ratio, is closely related to the probability of sale; the measure provides, however, a relative rather than an absolute notion of liquidity. 337 338 339
340
341
342 343
Lippman/McCall (1986), p. 46 after Keynes (1930), p. 67. Lippman/McCall (1986), p. 46. An analogous result can be derived for the real estate search model: a higher discounting rate leads to a lower expected net receipts, a lower reservation price, and finally, according to the equation (3.16), to a higher expected duration of search. This way of reasoning is not strictly correct as it ignores the adjustment of the optimal reservation price resulting from changes in the discounting factor. However, as Lippman/McCall (1986), pp. 46-47, show, the conclusion also holds when the search strategy is varied. See also the discussion of the ambiguous effect of changes in the frequency of offers on the value of the search in the real estate search model in section 2.3.2. See also Kluger/Stephan (1997) who apply the proportional hazard measure of liquidity on stocks. Hazard and proportional hazard models have been used for the analysis of ToM by Genesove/Mayer (1994), pp. 9-15, Anglin (1997), pp. 579-581, Glower et al. (1998), pp. 728-731, Krainer (1999), pp. 19-20, or Pryce/Gibb (2006), as well as for the analysis of other duration problems within real estate, e.g., mortgage defaults (Vandell et al., 1993, and Simons, 1994) or investment property holding periods (Collett et al., 2003).
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The starting point is the hazard function, defined as the conditional probability of sale ~ ~ in period t provided that the property has not been sold yet. If f ( t ) and F( t ) are the ~ p.d.f. and the c.d.f. of the selling time (search duration) t , respectively, than the hazard ~ function h ( t ) is defined as:344 ~ h( t ) =
~ f(t) ~ 1 − F( t )
(3.20)
The idea of the proportional hazard ratio is based on the constant relation between hazard functions of various property types. This means that the hazard function for each ~ specific submarket can be derived from the original function h ( t ) common for the global real estate market or, in simple words, that the relation between the probabilities of selling properties in markets A and B, provided they have not been sold yet, remains constant over time. If the house A is twice as easy to sell as the house B in the first week, than it is also twice as easy to sell in the second week. The relation between hazard ratios depends only on the characteristics of markets. Kluger and Miller assume a functional dependence of the market-specific hazard function from some market characteristic X (e.g., number of bedrooms, lot size, neighborhood quality etc.) with the following baseline hazard function:
~ ~ h ( t , X) = e b⋅X ⋅ h ( t )
(3.21)
The parameter b defines the effect of X on the hazard function. The resulting relation between hazard functions (proportional hazard ratio, PHR) of markets delimited by the characteristics X1 and X2 (e.g., one- and two-bedroom houses) is as follows:
PHR (X1 , X 2 ) =
344
~ ~ h ( t , X1 ) eβ⋅X1 ⋅ h ( t ) = = eβ⋅( X1 −X2 ) ~ h ( t , X 2 ) eβ⋅X2 ⋅ h (~t )
(3.22)
The hazard function can also be defined in terms of the standard search model with respect to the number of observed offer (i):
h (i) =
1 − F( p*) 1 − FR ( π*) = F i (p*) FRi (π*)
Since a geometric process is considered here, it an increasing exponential function.
3.3 Measures of Liquidity Risk
197
An interesting and practically relevant feature of this relation is its independence from the form of the baseline hazard function. The only required parameter is b, which can be estimated from empirical data using the maximum likelihood approach.345 It always refers to a certain characteristic of properties meaning that a separate estimation is necessary in each case. As mentioned, PHR is closely related to ToM. In fact, the relative sale probability corresponds directly with the relative expected remaining marketing time. This means that an odds ratio of, e.g., 2 indicates that the conditional probability of sale of the first property is twice as high as that of the second property; at the same time, it says that the second property is expected to remain on the market twice as long as the first one. Estimation of ToM from a known PHR is, however, not possible unless the baseline hazard function is either known or assumed. ToM can be then computed using the formulas (3.20) and (3.21). One of the main advantages of the proportional hazard ratio is the simplicity of its application. After estimating b-parameters for all relevant characteristics, relative liquidity of submarkets delimited according to any arbitrary set of criteria can be computed on the spot. Apart from quantitative characteristics, like size, number of rooms, or age, also qualitative criteria, like location quality, architectonic style, or social structure, can be incorporated as dummy variables. Furthermore, it is also possible to use biased data for the estimation of b-parameters, provided the same bias is present in every sample (i.e., with respect to every characteristic). The fact that only relative measurement of liquidity is possible with PHR may be viewed as a drawback; however, in many cases the comparison of liquidity levels for different markets is all an investor requires.
3.3.
Measures of Liquidity Risk
As discussed in Chapter 1 in section 1.1.2, one of the most important and often underestimated aspects of liquidity is the uncertainty of the liquidation value. This issue is particularly important with respect to privately traded assets for which no organized market structures exist. Due to the heterogeneity of the asset’s valuations among market participants, an investor cannot be sure about the effective receipts from sale even if she allows for sufficiently long liquidation time. In fact, it is this uncertainty that 345
For the description of the estimation procedure see Kluger/Miller (1990), p. 150.
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constitutes the main problem when investing in illiquid assets. Therefore, it is necessary to extend the measurement of liquidity by the risk aspect. There is only limited literature on liquidity risk measurement. It operates mostly on the level of market liquidity and concentrates on organized public markets. As already noted in section 1.1.2, most of the popular approaches address the risk arising from changes in the market-wide liquidity level (exogenous risk) and from price reactions to large transactions (endogenous risk). Apart from several variance based methods, incorporation of liquidity in the Value at Risk (VaR) concept is most frequent. Finally, several more recent papers consider liquidity as a factor that indirectly increases investment risk by affecting volatilities and correlations of returns346. All of these approaches refer, however, to public markets – liquidity risk of real estate and other privately traded illiquid assets remains an open issue. In view of the importance of the risk aspect for the rational management of liquidity, the state of research on this subject is clearly insufficient. Therefore, the focus of the section is on the derivation of possible approaches to capturing liquidity risk associated with investments in real estate and other privately traded assets. In particular, the necessity to search for a buyer when selling these assets should be adequately taken into account. The classical principles of risk measurement, as they are known from the finance literature on market risk, have been chosen as the reference point. They are presented at the beginning of the section. The discussion of the most popular measures including volatility, lower partial moment, default probability, and Value at Risk follows. The search theoretical approach is applied in each case to derive practical measures of liquidity risk. 3.3.1.
Principles of Risk Measurement
Measurement of risk has been discussed by various researchers in uncounted papers. In general, its purpose is to determine the grade of uncertainty about reaching the goal intended with an investment. Thus, one can only speak of risk when there is more than one possible outcome of an investment and at least one possible scenario can be interpreted as missing the goal. Measurement is only possible when dealing with risk in the narrow sense, i.e. when objective or at least subjective probabilities can be assigned to the alternative scenarios. Otherwise, in the state of total incertitude, no useful state346
See Buhl et al. (2002), Brunetti/Caldarera (2004), or Budhraja/de Figueiredo (2005).
3.3 Measures of Liquidity Risk
199
ment can be made about the level of risk associated with an investment making rational management of this aspect literally impossible. The latter case is therefore excluded from further considerations.347 The numerous existing approaches to risk measurement can be divided into qualitative and quantitative ones. The former are usually based on a verbal description of the situation, including identification of different sources of risk and their consequences. The active role of an analyst, whose opinion is the main judgment criterion, is especially characteristic for these measures. One of their advantages is the possibility of allowing for soft criteria, which are difficult to measure or even to express in simple words or symbols. Furthermore, the result can be adjusted according to the analyst’s personal experience or even unpronounced feelings. The possibility of profiting from the cumulated knowledge of highly qualified specialists is one of the main strengths of qualitative measures. However, the “human factor” is at the same time the source of their weaknesses. The quality of the results depends on the abilities of the analysts and may vary from one measurement to another. For the same reason, different qualitative measures may be difficult to compare, especially when they originate from different institutions applying different standards. Finally, qualitative measures are difficult to implement in formalized investment decision systems, which require that risk is expressed with a figure calculable for single investment as well as for portfolios. Mainly for the latter reason qualitative liquidity risk measurement is not further followed in this book. In contrast, quantitative approaches to risk measurement usually utilize statistical parameters of a random variable corresponding with the goal of the investment (goal variable). This implies that the goal can be quantified (i.e., expressed as a figure) and that probabilities can be assigned to its values. However, even with a known probability distribution of the goal variable, it is still unclear which of the numerous theoretically computable statistics is most appropriate. The choice depends greatly on the precise definition of risk. Since investors not always mean the same when they include “risk” in the catalogue of their decision criteria, specification of the actual meaning of this word is of key importance in determining the proper measure. It is straightforward that a different approach will be apt depending on the individual altitude of the deci-
347
See FN 159.
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Chapter 3: Liquidity Measurement
sion maker. It follows that the existence of a universal risk measure is practically impossible.348 Figure 3-1 summarizes different possible understandings of risk and relates them to the probability distribution of the underlying variable.349 The arrows indicate the directions of the deviations from the reference value relevant for the investor. A point target (a) means that any variation is interpreted as risk. The desired result is precisely specified and the investor fears both under- and overshooting it; the intensity of the perceived risk increases with the distance from the target. An interval target (b) is similar in its nature; however, the objective is to reach a span instead of a single value of the goal variable. With a minimum requirement set as the target (c) the investor expresses her pursuit of reaching certain minimal result; only underperformance is viewed as risk. Finally, an investor considering any result lower than the maximum possible one as negative follows the “maximization target” (d). Further possible ways to understand risk can be enumerated. Generally they can be classified into two categories: symmetric approaches, in which risk is defined as any deviation from the selected target, and asymmetric approaches, in which deviations in only one direction are regarded as risk. Analogue classification applies to risk measures; one distinguishes between symmetric, variation based risk measures and asymmetric, one-side risk measures. The latter ones are further decomposed into downside and upside risk measures, depending on the direction of deviations they refer to.
348
349
On the discussion of various notions of risk see Kupsch (1973), 26-33, Levy/Sarnat (1984), pp. 235-239, or Schmidt-von Rhein (1996), p. 159-165, as well as the literature cited there. The presentation in this paragraph is based on Schmidt-von Rhein (1996), p. 165-168.
3.3 Measures of Liquidity Risk
201
A. Point Target Risk = the possibility of missing a target t Probability
t
Goal Variable
B. Interval Target Risk = the possibility of missing an interval (tmin, tmax) Probability
tmin
tmax
Goal Variable
C. Minimum Requirement Target Risk = the possibility of underperforming a minimal target tmin Probability
tmin
Goal Variable
D. Maximization Target Risk = the possibility of underperforming a maximal achievable tmax Probability
Goal Variable
Figure 3-1: Alternative notions of risk referred to the probability distribution of the goal variable350
350
Based on Schmidt-von Rhein (1996), p. 167.
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The different notions of risk and the respective types of risk measures defined above are very general, without reference to any concrete goal variable. In the investment practice they are usually applied to returns or asset values; however, they can also be applied to liquidation receipts or to purchase expenses in the same manner. In this sense, a symmetric notion of liquidity risk encompasses any deviations from some targeted net receipts or expenses and an asymmetric notion is equivalent with receiving less or spending more than intended. Thus, depending on the precise definition of risk, a number of different measures are possible. Several alternatives are proposed in the following sections. Of course, they do not exhaust even a fraction of all possibilities; however, they correspond with the most popular measures of market risk used in the investment theory and praxis. Volatility is an example of a variation based measure; default probability, semivolatility, and Value at Risk are downside risk measures. Each of them measures liquidity risk from a slightly different perspective and is appropriate for a different type of investor. 3.3.2.
Volatility
Volatility is doubtlessly the most popular statistical measure of risk. Although most frequently applied to returns, it can also be computed for any other investmentrelevant random variable. Low volatility means that the variable’s values are not expected to deviate substantially from its expected value; high volatility indicates that they will probably vary very strongly. Formally, volatility corresponds with standard deviation, i.e., the square root of variance: S( X ) = V (X ) = E ( X − E (X )) 2 =
∞
∫ (X − E(X))
2
⋅ f ( X ) ⋅ dX
(3.23)
−∞
Variance, which is a risk measure equivalent to volatility, can be alternatively computed as variable’s square expected value minus the expected value of the squared variable: V(X) = E(X 2 ) − E 2 (X)
(3.24)
Volatility is a symmetric measure; it measures the level of uncertainty referring to both the risk of performing weaker than expected (downside risk) and the chance of outper-
3.3 Measures of Liquidity Risk
203
forming the expectations (upside risk). As long as the distribution of the goal variable is symmetric, like the normal distribution, the deviations from the mean in both directions are equally probable and volatility is proportional to both the upside and the downside risk. Multiple volatility corresponds then with a certain confidence interval determining the range of variable’s fluctuations around the mean that is not exceeded with a given probability. However, this property does not hold as soon as an asymmetric distribution of the goal variable is considered. As soon as the probabilities of performing better or worse than expected are not equal, it is possible that the “wrong side” of uncertainty is measured. This drawback of volatility is discussed more thoroughly in the next section. Volatility is used in the literature mainly for measuring exogenous and endogenous liquidity risk of public markets. With respect to the former one, it is usually based on market breadth or depth measures. This approach is applied to the bid-ask spread,351 to trading volume or turnover,352 or to the liquidity ratio353. In fact, as soon as a concrete measure of market liquidity is agreed on, its volatility can be used to assess (exogenous) liquidity risk.354 In the measurement of endogenous liquidity risk, volatility (or variance) is usually applied to the realized liquidation value or to the liquidity cost. In either case, the source of uncertainty is the change of the market price resulting from executing a (large) order.355 However, practically all of the above approaches use volatility to correct the overall Value at Risk rather than as a stand-alone measure; they are therefore discussed more thoroughly in section 3.3.3.3. A related but more general application of volatility as a liquidity risk measure has been introduced by Garbade/Silber (1979). The prime goal of the paper is to demonstrate the influence of the clearing frequency on liquidity risk. For the purpose of this study, the authors define liquidity risk as the variability of the difference between the equilibrium value of an asset at the moment of the decision to trade and the equilibrium value at the moment the transaction price is paid; variance is used to quantify this varia351
352 353 354
355
See Bangia et al. (1999), Duffie/Ziegler (2003), Jorion (2001), pp. 343-351, Le Saout (2002), François-Haude/Van Wynendaele (2001), or Angelidis/Benos (2005). See Chordia et al. (2001a). See Acharya/Pedersen (2005). E.g., Pastor/Stambaugh (2003) analyze the variability of their reversal measure (see section 3.1.1.3) and interpret it as liquidity risk, though they do not quantify it explicitly with variance or volatility. See Almgren/Chriss (1998, 2000/2001), Dubil (2002, 2003a), or Moorthy (2003), pp. 32 ff.
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bility. It is derived on the basis of a formal market model developed by the authors and is a function of the variance of traders’ reservation prices, the variance of changes in the equilibrium value of the asset, the rate of transactions’ exposure to the market, and the clearing frequency. By referring directly to realized prices, this approach is not limited to organized markets and can be applied for any type of market organization. Hence, the idea formulated by Garbade/Silber (1979) can be used to develop a volatility-based liquidity risk measure for privately traded assets. The search theoretical approach provides a method for computing the volatility of realized transaction prices or sale proceeds even without the existence of an organized market. In particular, the real estate search model can be used to derive the volatility of relative net receipts from selling property investments.356 For the derivation of the variance formula, it is convenient to recall the equation (3.24). The variance of relative sale receipts can be expressed using the expected receipts and the expected square receipts:
V ( Γ ) = E (Γ 2 ) − E 2 ( Γ )
(3.25)
Since the expected net receipts formula has been already derived for the real estate search model (see formulas (2.36) and (2.38)), derivation of the expected square receipts remains. j j ⎛ ⎞ −( ρ− τ ) ∑ t k −( ρ − τ ) ∑ t k k =1 k =1 E(Γ 2 ) = E⎜ Π ⋅ (1 + A ⋅ ∑ Tj ) ⋅ e + ∑ γ ⋅ Tj ⋅ e Π < π *⎟ ⎜ ⎟ j j ⎝ ⎠
2
(3.26)
As assumed in section 2.3.1.5, the expected value of the market uncertainty factor equals 1, and its standard deviation equals σA. Following the same logic as in the calculation of the expected receipts, i.e., applying the formula for the sum of an infinite geometric series, yields the following result:357
356
357
An analogous approach is used by Mok (2002a, b). This is, to my best knowledge, the only attempt to use the volatility of sale receipts with respect to non-publicly traded assets. See section 3.4.2 for a closer discussion of this approach. See Appendix A.5 for the full derivation. Notice that the formula is valid only under certain conditions, especially the nominators must not be negative.
3.3 Measures of Liquidity Risk
E (Γ 2 ) = +
205
E (Π 2 Π > π *) ⋅ X 2 ⋅ (1 − FR ( π*)) 1 − X 2 ⋅ FR (π*)
+
σ 2A ⋅ E(Π 2 Π > π *) ⋅ Z 2 ⋅ (1 − FR (π*))
2 ⋅ σ 2A ⋅ E (Π 2 Π > π *) ⋅ Y22 ⋅ (1 − FR (π*)) ⋅ FR ( π*)
(1 − X 2 ⋅ FR (π*))2
(1 − X 2 ⋅ FR (π*))3 2 ⋅ E(Π Π > π *) ⋅ (1 − FR (π*)) ⋅ γ ⋅ Y2 + (1 − X1 ⋅ FR (π*)) ⋅ (1 − X 2 ⋅ FR (π*)) γ 2 ⋅ (Z 2 ⋅ (1 − X 1 ⋅ FR (π*)) + 2 ⋅ Y1 ⋅ Y2 ⋅ FR (π*)) + (1 − X1 ⋅ FR (π*)) ⋅ (1 − X 2 ⋅ FR (π*)) ∞
with: X1 = ∫ λe − t (ρ−τ ) e −λt dt = 0 ∞
λ (ρ − τ) + λ
λ 2(ρ − τ) + λ
X 2 = ∫ λe −2 t ( ρ−τ ) e −λt dt = 0
∞
Y1 = ∫ λte − t (ρ−τ) e −λt dt = 0
∞
0
λ
((ρ − τ) + λ )2
Y2 = ∫ λte −2 t (ρ−τ) e −λt dt = ∞
λ (2(ρ − τ) + λ )2
Z 2 = ∫ λt 2 e −2 t (ρ−τ) e −λt dt = 0
(3.27)
2λ
(2(ρ − τ) + λ )3
Hence, the variance of the relative net receipts from search equals: V (Γ ) = +
E (Π 2 Π > π *) ⋅ X 2 ⋅ (1 − FR ( π*) ) σ 2A ⋅ E (Π 2 Π > π *) ⋅ Z 2 ⋅ (1 − FR ( π*) ) + 1 − X 2 ⋅ FR (π*) (1 − X 2 ⋅ FR (π*))2
2 ⋅ σ 2A ⋅ E (Π 2 Π > π *) ⋅ Y22 ⋅ (1 − FR ( π*) ) ⋅ FR ( π*)
(1 − X 2 ⋅ FR (π*))3 2 ⋅ E (Π Π > π *) ⋅ (1 − FR ( π*) ) ⋅ γ ⋅ Y2 + (1 − X1 ⋅ FR (π*)) ⋅ (1 − X 2 ⋅ FR (π*)) γ 2 ⋅ (Z 2 ⋅ (1 − X1 ⋅ FR (π*) ) + 2 ⋅ Y1 ⋅ Y2 ⋅ FR (π*) ) + (1 − X1 ⋅ FR (π*)) ⋅ (1 − X 2 ⋅ FR (π*)) 2 ⎛ E (Π Π > π *) ⋅ X1 ⋅ (1 − FR ( π*) ) + Y1 ⋅ γ ⎞ ⎟ ⎜ −⎜ ⎝
1 − X1 ⋅ FR (π*)
⎟ ⎠
(3.28)
206
Chapter 3: Liquidity Measurement
Further substitutions are possible for normally distributed offers:358 ⎛ ⎛ π * −1 ⎞ ⎞ ⎛ π * −1 ⎞ E (Π Π > π*) ⋅ (1 − FR (π*) ) = ⎜1 − Φ⎜ ⎟ ⎟ + ν ⋅ ϕ⎜ ⎟ ⎝ ν ⎠ ⎝ ν ⎠⎠ ⎝
(3.29)
⎛ ⎛ π * −1 ⎞ ⎛ π * −1 ⎞ ⎞ 2 E(Π 2 Π > π*) ⋅ (1 − FR (π*)) = ν ⋅ (π * +1) ⋅ ϕ⎜ ⎟ + ν + 1 ⋅ ⎜1 − Φ ⎜ ⎟⎟ ⎝ ν ⎠ ⎝ ν ⎠⎠ ⎝
(
)
(3.30)
The new term σA appearing in all above equations requires additional explanations. The volatility of the market uncertainty factor A refers to the uncertainty about the market change until the property is sold. As assumed, the searcher expects the market to follow a deterministic trend until the next potential buyer arrives, but she is not sure whether some unexpected events will not cause unforeseen changes in the offer distribution. Hence, higher σA means that a significant deviation from the trend is more probable. In the practical application, finding an adequate proxy for this variable may be difficult. Using the variance of prices observed at intervals corresponding with the base interval of the model (i.e., the interval to which the trend, the discounting factor, the offer arrival rate etc. refer to) is a possible solution. It seems also plausible to differentiate between the “typical” variance and the “worst case” variance. The latter approach would be especially worth considering in the case of an unexpected liquidation, when potential liquidity problems occur in times of unexpected market shocks. For liquidity analysis, it is especially interesting how the variance of sale receipts depends on the search strategy and, in particular, on the reservation price. Consider setting a very low reservation price first. The seller is then ready to accept any offer independent of its value. Intuitively one can expect the volatility of receipts to be approximately equal to the volatility of offers. Setting an infinitely low π* in the formula (3.28) results in the receipts’ variance of: V (Γ π* → −∞ )
(
)
= ν 2 ⋅ ( X 2 + σ 2A ⋅ Z 2 ) + 2 ⋅ γ ⋅ (Y2 − X1 ⋅ Y1 ) + γ 2 ⋅ Z 2 − Y12 + σ 2A ⋅ Z 2 + X 2 − X12
(3.31)
Note that this is in fact the original relative variance of offers ν² corrected for the uncertainty about the arrival time of the first offer, for the possible market change up to
358
See Appendix A.4.1and A.4.2 for the full derivation.
3.3 Measures of Liquidity Risk
207
this point, and for rental revenues achieved. For relatively high offer frequencies, low discounting factors, and relatively stable markets V(Γ) approaches ν². On the other hand, the receipts’ variance should approach zero for very high reservation prices. In this case, the sale becomes practically impossible and the investor can only expect an endless flow of rental revenues. The outcome of the search is then practically predefined and barely varying.359 The extreme values of the reservation price describe investors willing to sell an asset at any price or not willing to sell it at all. Yet, from the practical point of view, intermediate values of π* are most interesting. Especially relevant is the question whether a unique reservation price exists for which the variance is minimal. Unfortunately, an analytic proof of this property is not possible. However, results of multiple simulations for parameter values, which seemed to be realistic for real estate markets, allow some conclusions concerning this issue. Firstly, in most cases a clear local minimum existed; secondly, a clear local maximum followed each time. The typical pattern is depicted in Figure 3-2. It is assumed to be valid in “normal” market situations, though the lack of a formal proof is disturbing at this point. However, since this property is not really crucial for further reasoning, the issue is not further followed.
359
Formally, the limit of V(Γ) for π* → ∞ is positive; this is, however, due to the assumption that the rent is payable at the arrivals of offers. The positive variance results only from the uncertainty about the timing of the rental income.
208
Chapter 3: Liquidity Measurement 0,4 0,35
Receipts' Volatility
0,3 0,25 0,2 0,15 0,1 0,05 0 0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
1,6
1,8
2,0
Reservation Price
Figure 3-2: Volatility of net sale receipts as a function of the reservation price360
Note that the volatility of relative net sale receipts corresponds with the approach proposed by Garbade/Silber (1979). The “variance of the difference between the equilibrium value of an asset at the time a market participant decides to trade and the transaction price ultimately realized” (p. 577) formulated in terms of the search model is the variance of the difference between the receipts from sale and the average valuation of the asset at the beginning of the search. Since the latter one always equals one in the relative approach, Garbade’s and Silber’s liquidity measure defined in this context is simply: V(Γ–1) = V(Γ). There are several reasons why the variance of receipts is an appealing measure of liquidity risk. The first one is the popularity of this statistical mass of dispersion in many fields of finance. Researchers and practitioners are used to using it, what surely helps to avoid misinterpretations in many cases. This advantage should not be underestimated, especially if practical application is considered. The second reason for using volatility is the possibility of its analytical computation, which is not easy, if at all 360
Net receipts’ volatility has been computed for search model in the relative approach with normally distributed offers. Following parameter values have been assumed: offer dispersion = 15%; market volatility = 5% p.a.; interest rate = 15% p.a.; income return = 5%; offer frequency = 52 per year (or 1 per week).
3.3 Measures of Liquidity Risk
209
possible, for many other measures (see following sections). The formula provided in this section is quicker and handier than numerical or simulation-based approximations. Finally, problems concerning the distribution of random market changes (deviations from the trend) arising from the assumption of the expected value of the market uncertainty parameter being equal to 1 (see the discussion in section 2.3.1.5) are avoided since no assumption about the functional form of the distribution of A is necessary. The standard deviation or variance of this parameter is all that is required. 3.3.3.
Asymmetric Measures
Volatility, and symmetric risk measures in general, are based on the notion of risk regarded as the possibility of any deviation from the assumed target, in particular, from the expected value of the goal variable. However, investors are typically concerned only about the possibility of underperforming the target.361 They usually prefer to avoid achieving low returns or suffering high losses, but the possibility of performing better than expected, i.e., realizing extraordinarily high returns or revenues, can be hardly viewed as disadvantageous. As long as the goal variable is symmetrically distributed, the risk of performing worse and the chance of performing better than expected are equal. Volatility is than proportional to the downside risk. However, it under- or overestimates the true risk under a skewed distribution (see Figure 3-3). Application of downside risk measures is advised in such cases.
361
See Sortino/van der Meer (1991).
210 a)
Chapter 3: Liquidity Measurement b)
Figure 3-3: Total variability and downside risk under a right-skewed (a) and a left-skewed (b) distribution of the goal variable.362
The estimations of the distribution of net sale receipts obtained using a Monte Carlo Simulation based on the real estate search model give rise to the apprehension that its asymmetry is a rule rather than an exception.363 The grade of skewness depends on the relation between the income return, the trend rate, and the discount rate. A roughly symmetrical distribution of receipts results when the discount rate is equal to the sum of the income return and the trend rate; higher discount rates lead to right-skewed distributions, and lower discount rates lead to left-skewed distributions. Thus, under certain conditions, volatility may be misleading when the investor is downside risk oriented. This justifies the consideration of downside risk measures for the purpose of liquidity risk measurement. Apart from the integration of liquidity risks in the Value at Risk, there have been, to my best knowledge, no attempts to measure liquidity risk in the downside-risk framework so far.
3.3.3.1. Default Probability Default probability (DP) in credit business is defined as the probability that a debtor will not be able (or willing) to repay outstanding debt with interest as scheduled. In investment risk management, it describes the probability that the return from a certain investment falls below some critical level.364 Using the same logic, one can define de362 363 364
See Morawski/Rehkugler (2006), p. 13. See section 2.3.4. See Steiner/Bruns (2002), pp.64-65. Most distinctive application of the default probability in investment risk and portfolio analysis is in safety-first models. See Rudolf (1994), Breuer et al. (1999), pp. 336-372, or Elton et al. (2003), pp. 235-241, for a review.
3.3 Measures of Liquidity Risk
211
fault probability with respect to liquidity risk. It corresponds with the probability that the achieved net sale receipts are below some minimum value. Within the search framework, a default in the above sense occurs when a sale price is accepted that after adding up all search costs and revenues and after discounting leads to a sale value lower than the minimum required one. Thus, for each period of search (i), a minimum price exists that needs to be achieved in order to avoid a default; it is denoted as PM,i. A default is not possible as long as the minimum price is smaller than the reservation price – any acceptable price guarantees total net receipts higher than the minimum level in this case. Also, a default can occur in a period i only if the asset has not been sold until then. Thus, the probability of default in terms of liquidity risk can be expressed as follows: i −1
DP = Pr(G < G M ) = ∑ Pr(Pi < PM,i Pi > p *) ⋅ Pr(Pi > p*) ⋅ ∏ Pr(Pj < p*)
(3.32)
j=1
i
For the real estate search model from section 2.3, the default probability is defined analogously. However, since the model is formulated in terms of relative values, also the minimum (relative) net receipts and the minimum price are related to the average market valuation. i −1
DP = Pr(Γ < ΓM ) = ∑ Pr(Π i < Π M,i Π i > π *) ⋅ Pr(Π i > π*) ⋅ ∏ Pr(Π j < π*)
(3.33)
j=1
i
ΠM,i in this formula is defined as follows: i ~ ~ ~ −( ρ− τ )⋅Tj ΓM = Π M ,i ⋅ (1 + A ⋅ Ti ) ⋅ e −(ρ−τ )⋅Ti + ∑ γ ⋅ Tj ⋅ e j=1
i
⇒
Π M ,i =
ΓM − ∑ γ ⋅ Tj ⋅ e
~ −( ρ− τ )⋅Tj
(3.34)
j=1
~ ~ (1 + A ⋅ Ti ) ⋅ e −( ρ−τ)⋅Ti
Unfortunately, no further simplification of the DP-formula is possible, at least not without highly simplifying distributional assumptions. This is a serious difficulty in the practical application of DP common to all downside risk measures. A numerical approximation should be possible in certain cases. A more general alternative includes
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a MCS conducted according to the scheme presented in section 2.3.4. Applying it allows assessing the distribution of sale receipts; the fraction of scenarios that led to values below the minimum receipts gives the approximated DP value. However, a potentially serious problem with computations based on numerical or Monte Carlo methods is the required knowledge of the p.d.f. of the uncertainty factor A. On the one hand, normal distribution is inconsistent with the hypothesis of a random walk in returns and would lead to a negative drift in prices in the long run. On the other hand, log-normal distribution is compatible with the classical capital market theories but causes problems with the myopic property of the search model. Following the argumentation in section 2.3.1.5, normal distribution of A is assumed wherever necessary. For relatively low volatilities of A and/or relatively short durations of search, the aberration caused by this assumption should be negligible. This is also apparent from Figure 3-4.
1,2
Default probability
1
0,8
0,6
0,4
0,2
Logrnomally distributed A Nomally distributed A
1,98
1,91
1,83
1,76
1,69
1,61
1,54
1,47
1,39
1,32
1,25
1,17
1,10
1,03
0,95
0,88
0,81
0,73
0,66
0,59
0,51
0,44
0,37
0,29
0,22
0,15
0,07
0,00
0
Reservation price
Figure 3-4: Default probability as a function of the reservation price365
365
Default probabilities were computed for the relative search model with normally distributed offers. Following parameter values were assumed: minimum receipts = 1; offer dispersion = 15%; market volatility = 5% p.a.; interest rate = 15% p.a.; income return = 5%; offer frequency = 52 per year (or 1 per week). The simulation was conducted with 1000 runs for each reservation price.
3.3 Measures of Liquidity Risk
213
Another critical issue in the practical application of the default probability is the choice of the minimum search value or the minimum required (relative) net sale receipts. Since it is a subjective parameter determined by what the investor considers to be a sufficient revenue, the concrete choice depends on her personal attitude; however, one can attempt to define economically plausible values. The most straightforward approach is to derive it from the minimum return required for the analysed asset during the planned holding period. If the asset was acquired at a price P0, and the minimum return is RM, the minimum absolute net receipts would be GM = P0·(1+RM). Correspondingly, the minimum relative net receipts would be ΓM = P0·(1+RM)/P1, where P1 is the asset’s average price at the end of the time horizon. Reasonable values of RM are: zero (nominal wealth preservation), inflation rate (real wealth preservation), risk free, or risk adjusted interest rate.366 Another possibility is to use the average market valuation. It corresponds with setting ΓM = 1 when DP is computed on the basis of the real estate search model. Investor’s goal implied by this approach is to perform at least as well as an average investor would. If the average valuation is interpreted as the “would be” liquid price (quasi-market price), one can also interpret this approach as the endeavor to perform at least as well as it would be possible if the asset was perfectly liquid. The complex form of the default probability formula makes it difficult to analyze it analytically. However, some of its properties can be guessed intuitively. Setting the reservation price very low (infinitely low) has the effect that the first incoming offer is accepted. The probability of a default is then simply the probability that the minimum price is not reached with the first offer. If the observation and opportunity costs are low, and the market is relatively stable, it should not diverge far from the c.d.f. value of the offer distribution for the minimum receipts’ level (i.e., F(GM) or FR(ΓM)). Setting an infinitely high reservation price results in an infinite search, and the receipts are equal to the present value of infinitely generated observation costs and revenues. If the latter value is lower than GM or ΓM, a default is sure and PD=1; otherwise, it will never occur and PD=0.367 However, if the reservation price is finite, no straightforward general conclusions are possible. The outcome depends largely on the level of market 366
367
On the choice of a return benchmark for downside-risk measures see Schmidt-von Rhein (1996), p. 427-431. Note that the postulated default probability of 1 for an infinitely high reservation price and low rental revenues is in conflict with the formal definition of default in equation (3.33). A default in this equation can occur only when a sale is accomplished. Since this never happens when the reservation price is infinite, the equation provides a misleading result in this case.
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uncertainty. With a relatively stable distribution of offers, one could expect a decreasing DP for low reservation prices as very low sale prices are excluded, and an increasing DP for high reservation prices as the probability of a successful sale diminishes. This would suggest the existence of a local minimum. However, higher reservation prices also lead to longer search durations increasing the potential magnitude of market changes. It is not clear which of these effects, the falling probability of sale or the potentially large negative deviations from the trend, outweighs, especially if highly volatile markets are considered. Above considerations are based on intuition rather than on formal analysis. To support them, several simulations assessing the DP in representative real estate scenarios have been conducted. An example of the typical plot of the default probability as a function of the (relative) reservation price, with the average market valuation used as the minimum receipts’ level, is presented in Figure 3-4. The results are in line with expectations: a DP of ca. 50% results for very low values of π*, and it approaches one for large π*. A unique minimum has been observed in most cases. However, these results can only serve as an indicator and are no proofs of general properties of the PDfunction. Additionally, due to technical limitations of simulation techniques in probability tails, they should be interpreted with caution for very high and very low reservation prices.
3.3.3.2. Semivolatility and Lower Partial Moments Default probability provides information about the chances of missing the liquidation target. However, used as a liquidity risk measure, it only differentiates between sale receipts above and under the minimum level – the extent to which the target is missed remains disregarded. In effect, two investments would be judged as having the same liquidity risk as soon as the DP values were equal, even if one of them would lead to substantially higher shortfalls below ΓM than the other one. Such conclusion would be clearly inconsistent with investors’ perception of risk. Not only the mere fact of not reaching the minimum required target matters, but also by how much it has not been reached. Hence, it seems more appropriate to use measurement approaches that include the distance between the realized and the minimum receipts. Semivolatility and lower partial moments, of which semivolatility is a special case, meet this requirement.
3.3 Measures of Liquidity Risk
215
Semivolatility is defined in analogy to volatility; the only difference is the fact that only negative deviations from the mean are regarded. Appling it to the relative net sale receipts yields the following formula:368
SV(Γ) = E (max(0; E (Γ) − Γ) 2 )
(3.35)
For symmetrically distributed receipts, semivolatility equals volatility divided by the square of 2, i.e., SV(Γ) = V(Γ)/√2. Since the relation between the two measures is then proportional, they yield identical assessments of the relative liquidity risk for different assets. However, this does not hold for an asymmetric (skewed) distribution of receipts. As already noted, semivolatility can be considered as a special case of the lower partial moment (LPM) developed by Bawa und Fishburn.369 LPM constitutes a generalization of the downside risk concept and can be defined for relative sale receipts as follows:370 LPM(ΓT , n) =
ΓT
∫ (ΓT − Γ )
n
f(Γ) dΓ
(3.36)
−∞
ΓT denotes the target or the minimum net receipts from sale. It can be equal to the expected net receipts, as it is in the case of semivolatility, but any other fixed value can also be used. The choice of this parameter follows the same logic as the choice of ΓM in the case of default probability. The parameter n defines investor’s risk aversion. For n = 0, one receives the default probability discussed in the former section. With n = 1 and the target equal to the expected receipts, LPM becomes the mean downside deviation or the expected loss; and with n = 2, it is the semivariance or, when a root is extracted, semivolatility.371 The values of n do not, however, need to be integers. This allows a very precise adjustment of LPM to the individual attitude of a specific investor. For high n values relatively more weight is assigned to higher deviations from the target – this characterizes a strongly risk averse investor who tolerates none or only slight shortfalls. On the other hand, low values of n characterize an aggressive inves368
369 370 371
See Harlow (1991), Sortino/van der Meer (1991), Schmidt-von Rhein (1996), pp. 445 ff., Füss (2004), pp. 410 ff., or Morawski/Rehkugler (2006). See Bawa (1975) and Fishburn (1977). For a review of the concept see also Nawrocki (1999) See Harlow (1991), pp. 30 and 40. See Nawrocki (1999), p. 14.
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tor. Her prime objective is not to miss the target; however, if the target is missed, she may even prefer higher losses to lower ones (for n < 0). Such attitude may be described as “conditional risk seeking” opposed to “normal” risk aversion.372 In addition to the compliance with the more natural, downside-oriented attitude to risk, an important advantage of LPM measures is their compatibility with the utility theory, especially with many of Neumann-Morgenstern utility functions.373 As Fishburn (1977) shows, they are also equivalent with the concept of stochastic dominance.374 The already mentioned ease of adjustment to individual requirements of investors is a further reason to use LPM for liquidity risk measurement.375 However, the relative complexity of computations needs to be outweighed against these advantages. The problems are especially severe if LPM (or semivolatility) is to be computed not for historical liquidation data but on the basis of a search model. No analytical formula can be derived then, so that the use of simulations is inevitable.
3.3.3.3. Value at Risk A practically relevant risk measure, closely related to Default Probability, is Value at Risk (VaR). It stems from credit risk management, but has become widespread also in other business areas.376 The logic behind this measure is to disclose “the worst expected loss over a given time interval under normal market conditions at a given confidence level”377. In other words, it answers the question: how much can I loose with a given probability over a given time horizon? Thus, instead of the probability of missing a target, the minimum target with a given probability of missing it is determined. VaR can be defined either as the difference between today’s (V0) and minimum future value of an asset (absolute VaR) or as the difference between its future expected and future minimum value (relative VaR):378
372 373 374 375
376 377 378
See Nawrocki (1999), pp. 14-15, and Fishburn (1977), p. 119. See Nawrocki (1999), p. 16 and Fishburn (1977), pp. 120 ff. See Nawrocki (1999), p. 15, and Fishburn (1977), pp. 122 ff. A method of adjustment of the LPM to investors’ preferences is presented by Nawrocki/Staples (1989). See Jorion (2001) for an extensive review of VaR measurement. Jorion (2001), p. xxii. See Jorion (2001), p. 109.
3.3 Measures of Liquidity Risk
VaR absolute = V0 − V * VaR relative = E(V) − V *
217 (3.37)
There have been several attempts in the literature to allow for illiquidity in the VaR framework. Most of them consider the problem in terms of liquidation costs in organized stock markets and concentrate on the exogenous liquidity risk only. This course of research was initiated by Jarrow/Subramanian (1997) who address the optimal liquidation of stocks in a market on which trading large quantities affects prices.379 The investor (seller) can chose a liquidation strategy defined by quantities sold at multiple transactions each of which affects the market price. Her choice determines the total discount resulting from the sum of discounts at single trades as well as execution lags. Basing on a continuous time model with prices following geometric Brownian motion, the authors propose a modification to the traditional market risk VaR. Analogous approaches to the adjustment of VaR for liquidity risk are applied by Almgren/Chriss (1998, 2000/2001) and Dubil (2002, 2003a, 2003b) who additionally divide the market impact of liquidation in the temporary and the permanent component.380 While the above researchers consider only endogenous liquidity risk induced by the execution of a large transaction, Bangia et al. (1999) concentrate on exogenous liquidity risk arising from the fact that the bid-ask spread is subject to fluctuations. They use the cost of liquidity defined as the maximal spread at a certain confidence interval to adjust the standard market risk VaR. Several authors extend the model of Bangia et al. to include both exogenous and endogenous liquidity risk in VaR, among them Le Saout (2002), François-Heude/Van Wynendaele (2001), and Angelidis/Benos (2005). VaR corrections in the above mentioned papers are based on the fact that investors are concerned not only about value deterioration due to a general market decline but also due to unusually high liquidity costs. Since the illiquidity discount is assumed to be normally distributed (it is the consequence of the random walk in returns), Liquidity VaR (LVaR) can be computed for a given liquidation strategy as the upper bound of the discount’s confidence interval:381 LVaR=E(discount) + ζ·S(discount) 379 380 381
(3.38)
See also Subramanian/Jarrow (2001). See also Moorthy (2003), pp. 32 ff. The definition of LVaR can also refer to the liquidation value instead of the liquidation cost as it does in Dubil (2002, 2003a, 2003b); the results are analogous.
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where the risk aversion parameter ζ corresponds with a certain confidence level x% satisfying the following condition:382 Pr(discount < LVaR(ζ))=x%
(3.39)
An interesting feature of this approach is the possibility of interpreting the LVaR in terms of a utility function. This aspect is discussed more closely in section 3.4.2. Several other researchers deal with liquidity risk in the VaR context.383 Literarily none, however, (to my best knowledge) go beyond public markets. Consequently, none of the existing models can be practically applied when no unambiguous and publicly known market price exists. This drawback can be solved by using a search model to derive Liquidity VaR for private markets; in particular, an operational LVaR measure for real estate investments can be computed using the real estate search model. It is then the difference between the minimum liquidation value realized with a given probability and the reference value. The first component corresponds with the minimum net sale receipts at a given confidence level and can be determined by reversing the default probability formula (3.32) or (3.33). Similarly as for the default probability, no analytical computation will be possible in most cases, but a numerical approximation or estimation on the basis of a MCS should be feasible. The choice of the reference value is less obvious. Asset’s current market value (absolute VaR) or its expected future value (relative VaR) is used in the original VaR concept (see equation (3.37)); however, these two approaches lead to entirely different results when applied in the search framework. Using assets’ current market values in LVaR computations is obviously not directly possible in private markets; the expected (current) offer or the expected net receipts from an immediate liquidation (µ0) could be applied instead. This leads to the following definition of the absolute LVaR: LVaRabsolute = µ0 – Gmin, x%
382
383
(3.40)
For banks, the implicit risk aversion in the context of VaR arises from the required confidence intervals set by supervision authorities usually at ca. 99-99.5%. See Berkowitz (2000), Cherubini/Della Lunga (2001), or Giot/Gramming (2002); on the Liquidity VaR see also Jorion (2001), pp. 339-357.
3.3 Measures of Liquidity Risk
219
The minimum receipts from sale at the end of the investment horizon achieved with the confidence of x% (i.e., Gmin, x%) correspond with the receipts defined in the expression (3.32). Since µ0 is an absolute value, this approach makes only sense when the absolute version of the search model is used. The minimum liquidation value is then defined as an absolute value and depends not only on the confidence level but also on the market state at the beginning of the search process, in particular, on the level (mean) of offers at this time (µt).384 As soon as future liquidation is considered, µt refers to the future average valuation of the asset and is uncertain. In consequence, also Gmin is uncertain and subject to possible market changes during the holding period. This means that the absolute LVaR encompasses liquidity risk arising from the necessity to search for a buyer as well as market risk arising from possible changes of asset’s valuation among market participants. Ignoring the latter aspect, i.e. assuming some deterministic state of the market at the end of the investment horizon (e.g., the expected value of µt), could yield a higher LVaR value and, thus, indicate lower liquidity risk for assets that are expected to yield higher returns over the investment horizon. On the other hand, assuming an uncertain µt leads to a VaR measure encompassing total investment risk and not only liquidity risk. Asset’s future expected value seems to be more appropriate reference for the search model based LVaR. Since both values in the formula of the relative VaR (3.37) refer to the same moment in the future (i.e., the beginning of the liquidation process) in this case, there is no need to regard the market risk component. A practical benefit of this approach is the possibility of applying the relative version of the search model. The minimum sale receipts at a given confidence level x% (Γmin,x%) are then expressed as a fraction of the (future) average valuation by potential buyers (µt), and the asset’s expected future value can be interpreted as the expected net receipts under the optimal liquidation strategy. The latter one results in the real estate search model from optimizing the equation (2.38) with respect to π*. Thus, the relative LVaR is defined as:385 LVaR relative = max[ E(Γ π*)] − Γ min ,x% π*
384
385
(3.41)
Of course, also the reservation price and other model parameters determine the minimum liquidation value; they are omitted for better tractability. Maximization of expected net receipts is assumed in the formula (3.41). Note, however, that also other notions of optimality are possible here, e.g., risk minimizing.
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This version of the measure answers the question: “how far, relative to the average market valuation and at a given confidence level, can the result of the liquidation lie below the expected one when the optimal search strategy is applied”. Note that due to the relative definition of prices, the relative LVaR is independent of the average price level and is valid as long as other parameters of the search model (especially the divergence of opinions, market trend, and offer arrival rate) remain unchanged. There is therefore no need to precisely determine the time horizon, and the relative LVaR can be applied to both the expected and the unexpected liquidation. Another interesting alternative for the reference value in the LVaR formula is the asset’s future average valuation. In the real estate search model, it corresponds with the expected relative value of offers, which is by definition equal to one. Thus, the market level based LVaR (LVaRM) equals: LVaRM = 1 – Γmin, x%
(3.42)
Assuming that the absolute average valuation among potential buyers corresponds with the hypothetical market value of the property if it was perfectly liquid (see discussion in section 2.4) allows defining LVaRM as the maximal possible loss arising from the fact that the asset is illiquid.
3.4.
Alternative Measurement Approaches
Each of the hitherto discussed liquidity measurement concepts concentrated on one facet of the problem only. In the case of the traditional measures, it was the (expected) market liquidity and the cost of its lack. The time- and probability-related measures focused on the duration and the probability of successful liquidation of an asset, and risk measures regarded the uncertainty about the outcome of the liquidation. Considering the substantial differences between these approaches, it seems rather impossible to capture the full scope of liquidity within one figure. Nevertheless, in many cases a measure combining several various aspects would be desirable. This refers especially to the two main dimensions of asset liquidity: marketability and liquidity risk. Two possible approaches to this task are discussed in this section. The first one borrows from the concepts of performance measures, which are very popular in the analysis of stocks. Although to my best knowledge no attempt in this direction has been made so far, it seems possible to derive several “liquidity performance measures” on the basis
3.4 Alternative Measurement Approaches
221
of the search framework. Another possibility builds on the concept of utility. The few existing approaches following this idea are presented and discussed in the light of the search theory. Since all measures addressed in this section assume individual search for a trading partner, they refer mainly to asset liquidity. 3.4.1.
Liquidity Performance Measures
The most popular approach to combine various characteristics of investment alternatives is the computation of performance measures.386 The goal is usually to present the return position of an investor after correcting for investment risk. In general, the construction of performance ratios follows the following principle:387 Performance = (Expected return – Benchmark return)/Risk
(3.43)
An often used interpretation is the unit price of risk, i.e., the additional return achieved for each unit of risk that is accepted with the investment. Using this general principle, a number of different performance measures can be defined depending on the notions of return and risk and on the type of the benchmark. An analogous approach can be applied to liquidity. The analogy between the expected liquidation value and the expected return as well as between liquidity risk and investment or market risk, which is discussed in more detail in Chapter 4, is very useful at this point. Thus, one could ask about investor‘s reward for accepting liquidity risk associated with a certain asset. The natural benchmark for measuring the reward is the case of a perfectly liquid asset for which liquidity risk is zero. A liquidity performance measure in this sense could be interpreted as the additional expected liquidation value arising from liquidating an illiquid asset instead of a liquid one computed per unit of liquidity risk involved in such liquidation or, simply, as the unit reward for assuming liquidity risk. The most straightforward way to implement the above idea is by defining the expected liquidation value and liquidity risk as the expected net sale receipts and the volatility of sale receipts, respectively. Both expressions can be computed on the basis of the real estate search model. The benchmark – i.e., the liquidation value of a perfectly liq-
386
387
See Levy/Sarnat (1984), pp. 515-559, or Wittrock (1995), part C, for an extensive review of performance measures. See Steiner/Bruns (2002), p. 597.
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uid asset – is always equal to 1 in the relative search model. The resulting Liquidity Risk Reward (LRR) is then defined as follows:388
LRR(π*) =
E(Γ π *) − 1 S(Γ π *)
(3.44)
Since both expected net receipts and receipts’ volatility are functions of the reservation price, also LRR depends on the reservation price. Thus, the liquidity-performance of an illiquid asset can vary depending on the liquidation strategy of the seller. In order to use LRR for comparing various illiquid assets or markets, it seems reasonable to consider only those reservations prices which lead to the maximal performance. This reasoning leads to the following asset-specific and liquidation strategy-independent LRR*:
LRR* = max[LRR(π*)] π*
(3.45)
An important property of this measure is its consistence with the optimal search strategy in the case of simultaneous liquidation of liquid and illiquid assets. As discussed in section 4.3.2, the optimal reservation price is then independent of investor’s preferences, and consequently, LRR* becomes (at least to some extent) an objective measure of liquidity. Basing on the above idea, further liquidity performance measures can be developed by applying different measures of liquidation value (marketability) and uncertainty. In particular, different liquidity risk measured discussed in section 3.3 can be applied, among them especially the asymmetric measures. Furthermore, Liquidity Beta (LBeta) defined in section 4.3.3 can be used for this purpose. However, one must note that its interpretation does not quite correspond with that of the market Beta. L-Beta refers always to a specific portfolio containing illiquid assets and is different for each investor. As such, it is not suitable for comparisons of assets or markets and allows only determining the relative liquidity performance of an asset within a given portfo388
Note that there seems to be some analogy between Liquidity Risk Reward and Sharpe Ratio. The latter measure is defined as the relation of the expected excess return of an asset over the risk-free rate to the asset’s return volatility (see Sharpe, 1966, as well as Sharpe/Alexander, 1990, pp. 750752, Sharpe (1994), or Elton et al., 2003, pp. 626-628). By substituting liquidation receipts for returns as the central variable and considering a perfectly liquid asset to be liquidity risk-free, the LRR ratio results.
3.4 Alternative Measurement Approaches
223
lio. Hence, also liquidity performance measurement based on L-Beta would be valid only within a certain portfolio. Before concluding the presentation of the liquidity performance measures, the issue of their correct interpretation needs to be highlighted. It is important to note that these measures refer explicitly to assets and not to markets. Thus, they are based on individual optimization of reservation prices and take higher values for assets which allow individual investors to achieve higher sale receipts. In other words, given a constant level of liquidity risk, an asset that sells at a higher premium to its average valuation when marketed under the optimal selling strategy is considered to be more liquid. In contrast, traditional liquidity measures refer to the (expected) market liquidity, i.e., to the average ability to buy or sell assets. In consequence, both groups of measures may under certain circumstances yield inconsistent results. This is best visible in the comparison of the Liquidity Risk Reward with the bid-ask spread. While higher expected receipts increase (ceteris paribus) the LRR ratio, they lead to higher bids, higher trading costs, and lower market liquidity with respect to the spread. This difference is due to the implicit assumption about the investor’s ability to act strategically, which lies behind the concepts of market and asset liquidity. While the Implicit Spread, QSD, and other related measures assume that the seller is passive, i.e., she accepts the sale price offered by the dealer (bid) or by the market (average valuation), the performance measures presented in this section allow for an active search for a buyer. In effect, the seller is never passive – even under a very high time pressure, she attempts to maximize the present value of sale receipts. The increase in the expected outcome is, however, evaluated in relation to the involved uncertainty. Thus, in contrast to the traditional measures, performance measures address an “active” investor willing to measure the subjective liquidity of her assets rather than the objective liquidity of markets to which these assets belong. As such, they seem especially appropriate for assessing liquidity of privately traded investments. 3.4.2.
Utility-Based Measurement
Since the characteristics of sale receipts depend on the reservation price and as such can be controlled to some extent by the seller, subjective liquidity of an asset is also affected by investor’s preferences. It is therefore straightforward to use the personal utility function to quantify the grade of liquidity. Few researchers followed this approach; among them most distinguishable are papers by Mok (2002a, b). Also the
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course of research concerning the interpretation of Liquidity-VaR in terms of utility can be associated with utility based liquidity measurement. In both cases, the basic idea is to measure liquidity with the highest utility level achievable by optimizing the liquidation strategy. Mok’s starting point is a simple search model with discounting similar to the one by Karlin (see section 2.2.4). Additionally, Mok (2002b) adds the possibility of multiple offers per period interpreting this parameter as market thickness. She considers two sale settings, a finite horizon with 2 or i periods and an infinite horizon; for each of them the expected value and the variance of net liquidation receipts is derived. Both statistics are functions of the reservation price, which defines the search strategy. Mok analyzes the effects of changes in model parameters on the outcome of the sale process demonstrating, e.g., that searching can be more attractive in thicker markets. Up to this point, the analysis does not significantly differ from the standard search analysis presented in Chapter 2. Conceptually new is, however, the introduction of utility from liquidation, which depends on both the expected sale receipts and the standard deviation of sale receipts.
U(p*) = E(G p *) − ζ ⋅ S(G p *) with: ζ
(3.46)
- risk aversion parameter
The optimal search strategy in this setting is the one that maximizes the utility of the seller. Obviously, it is different form the strategy maximizing the expected receipts considered usually in the literature and depends on the grade of risk aversion.389 Conceptually related to the Mok’s approach is the utility interpretation of Liquidity Value at Risk (LVaR).390 Given normal distribution of liquidity discounts (or in fact any symmetric distribution), the upper bound of the confidence interval for the maximum liquidation cost can be presented as the mean less the standard deviation scaled with a parameter ζ, which corresponds with the confidence level x%.391 Defining the
389
390 391
In fact, Mok’s concept is closely related to the main idea of the Modern Portfolio Theory discussed more closely in the following Chapter – it requires the assumption that either investor’s preferences are quadratic with respect to sale receipts or that the distribution of receipts is normal. See especially section 4.1.4. See Almgren/Chriss (1998, 2000/2001), Dubil (2002, 2003a, 2003b), Moorthy (2003), p. 33. See also expression (3.38).
3.4 Alternative Measurement Approaches
225
liquidation discount as asset’s non-random fair value (FV) less the random realized liquidation value yields: LVaR = E(FV − LV) + ζ ⋅ S(FV − LV) = FV − E(LV) − ζ ⋅ S (LV)
(3.47)
It results in the minimal value at an x% confidence interval of: FV − LVaR = E(LV) + ζ ⋅ S (LV)
(3.48)
Note that this notion of LVaR is equivalent to the quadratic utility function similar to the one utilized by Mok. Since both the discount and the liquidation value are conditional on the liquidation strategy, they can be influenced by the investor. Minimizing LVaR and, thus, maximizing asset’s minimal liquidation value corresponds in this case with utility maximization. Basing on the concept of liquidation utility, Mok defines the liquidity cost as the decrease in utility due to a shorter sale horizon. It is the difference between the utility resulting from selling quickly within i periods (Ui) and the utility resulting from selling ~ optimally within the time horizon T ( U ∗T~ ):
Liq t ,T~ = U i − U ∗T~
(3.49)
~ Typical choices of i and T would be 1 and ∞, respectively; U1 is then the utility of an immediate sale, and U∞ is the maximal achievable utility with an unlimited search horizon.
Mok’s liquidity measure is, in fact, analogical to the Quick Sale Discount from section 3.1.2.2 applied on the level of utility. An investor forced to sell quickly has to give up some of the utility she could gain when she searched optimally. This loss comes, on the one hand, from the possibly reduced expected net sale receipts and, on the other hand, from the possibly increased receipts’ volatility. In illiquid markets, a quick sale would result in larger discounts in overall utility; in perfectly liquid markets, search is not optimal anyway so Mok’s measure would be zero. As indicated by Dubil (2002, p. 72), introduction of utility to liquidity measurement opens nearly unlimited possibilities for designing new measures. The form of the utility function does not need to be limited to the quadratic one; in particular, one could
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pick HARA (Hyperbolic Absolute Risk Aversion) or any other widely acknowledged type of preferences.392 Also the set of utility-relevant variables can be extended; e.g., the skewness of sale receipts’ distribution could be included. Further possibilities open when traditional liquidity measures are redefined in terms of utility, e.g., as a “utility spread” – the sum of utility losses from simultaneously purchasing and selling an illiquid asset. Properties of such measures depend on the applied utility measurement approach, which should be chosen according to the preferences of the decision maker. They are therefore always subjective measures, tailor-made for a specific investor. This may be a serious limitation for the practical application when the characteristics of the investor are unknown. Introduction of utility is one of the crucial issues in the construction of rational approaches to liquidity management. In particular, it plays a central role in most asset and portfolio selection models including the Modern Portfolio Theory. The standard parametric approach to portfolio optimization, the so called mean-variance approach, is based on the maximization of investor’s utility defined as a quadratic function of returns. When applied to liquidation decisions or to portfolio selection with illiquid and privately traded assets, it turns out to be closely related to Mok’s approach presented above. This issue is discussed in more detail in Chapter 4.
3.5.
Relations between the Measures
After a large variety of partially very different approaches to liquidity measurement have been presented, the question arises to what extent they lead to identical results and, thus, are exchangeable. Since they generally refer to the same phenomenon, a certain linkage can be expected. Nevertheless, a closer look reveals that it is by no means perfect, and that especially the measures of liquidity risk clearly distinguish themselves from the other approaches. This result is in line with the concept of two dimensions of liquidity discussed in Chapter 1 in section 1.1.2. Analyzing relations between the measures also helps to get a better idea of what is actually measured with different ratios. Probably best researched are relations between the traditional liquidity measures referring to public markets. They are also relatively easy to establish on the theoretical ba392
For a review of the utility measurement techniques see, e.g., Ingersoll (1987), pp. 39-40, as well as most handbooks on decision making.
3.5 Relations between the Measures
227
sis. Especially relevant in this context is the link between the bid-ask spread and the quoted market depth (transaction size for which the spread is valid) – the two most popular measures. They are set by a market maker simultaneously and constitute a trade-off: larger volumes are acceptable only at larger spreads.393 But there are also easily explicable links to other measures. The spread between the bid and the ask price depends, among others, on the cost of holding an open position until a countertransaction occurs. Low trading activity, indicated by low volumes or turnovers, gives fewer possibilities to close a position and leads to larger spreads; dealers are less willing to accept large transactions unless they are adequately compensated. On the other hand, higher spreads favor longer holding periods and less frequent trading. Investors are then not eager to react to relatively slight fluctuations of market prices even if they consider them to be fundamentally unfounded. As a result, price reversals occur only slowly. Summing up, spreads should be negatively correlated with market depth and positively correlated with market resiliency; the correlation between depth and resiliency measures should be positive. The above theoretical considerations have been often tested empirically. The observed links were, however, often weaker than expected. Lee et al. (1993) stated a highly significant negative dependence between quoted spreads and quoted depths on the New York Stock Exchange (NYSE) using a nonparametric test. Furthermore, a regression analysis yielded a strong positive relation between normalized trading volumes and spreads and a strong negative relation between volumes and quoted depths. Chordia et al. (2000) looked at the correlations between quoted and effective spreads (absolute and proportional) and quoted depths on the NYSE receiving values ranging from ca. 0.16 to 0.87 (see Table 3-1). The analysis of correlations between changes of different market breadth and market depth measures computed by Chordia et al. (2001a and 2005) yielded similar results. Using a ranking procedure Chalmers/Kadlec (1998) stated that “stocks with high amortized spreads have both high effective spreads and high share turnover, while stocks with low amortized spreads have both low effective spreads and low share turnover.”394 The cross-sectional correlation between the amortized and the quoted spread remained, however, on the level of 0.54.395 Huang/Stoll (1996) compared quoted and effective spreads with realized spreads receiving correla393 394 395
See Lee et al. (1993), pp. 49-51 Chalmers/Kadlec (1998), p. 167. See Chalmers/Kadlec (1998), p. 167.
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tions between 0.7 and 0.8,396 and Brennan/Subrahmanyam (1996) stated literally no correlation between their empirical price impact measure and the proportional spread.397 Finally, Choi et al. (1998) tested the dependence between the quoted spread and an extended version of Roll’s implicit spread on an option market and were able to explain over 80% of the variation of the latter one by the variation of the former one.398 Table 3-1: Cross-sectional means of time series correlations between liquidity measures for individual stocks on the YSE.399 QSPR
PQSPR
ESPR
PQSPR
0.844
ESPR
0.665
0.549
PESPR
0.555
0.699
0.871
DEP
-0.396
-0.030
-0.228
PESPR
-0.156
QSPR – Quoted Spread; PQSPR – Proportional Quoted Spread; ESPR – Effective Spread; PESPR – Proportional Effective Spread; DEP – Quoted Depth
Time on the market, probability of sale, and the proportional hazard ratio seem to concentrate on a different aspect of liquidity than the traditional measures. Liquidity is viewed from the perspective of a single transaction rather than from the perspective of a whole market. The stress is also more on private rather than public trading. Nevertheless, a relation to the traditional measures should exist. It is best recognized in the link between the probability of sale and the trading volume. The more probable the sale of an asset is, the more frequently transactions should occur. In the ideal case, if trading activity is viewed in terms of a Poisson or Bernoulli process, the average number of transactions per unit of time should be equal to the reciprocal of the sale probability within this time interval. Thus, the probability of sale multiplied with the market size (i.e., the number of assets available for sale/purchase) should provide an estimate of the trading volume. A link to the bid-ask spread can be established using formal market models of dealers’ and market makers’ behavior, as demonstrated by Garman
396 397
398 399
See Huang/Stoll (1996), p. 345. See Brennan/Subrahmanyam (1996), p. 451. A correlation of 0.03 between the price impact measure and the proportional spread was found. See Choi et al. (1998), p. 227. Chordia et al. (2000), p. 8.
3.5 Relations between the Measures
229
(1976) or Amihud/Mendelson (1980).400 But also without a formal proof, it is straightforward that sale probabilities and expected marketing times influence the cost of market makers’ operation and must be taken into account when setting spreads. Furthermore, with more frequent trades and shorter marketing periods, also reactions of markets to large transactions should be quicker; thus, resiliency should increase. These considerations lead to the conclusion that a relation between the probability-based and the traditional measures can be expected, although, to my best knowledge, it has not been analyzed empirically so far. Although these measures differ in their constructions, they seem to refer mainly to the aspect of liquidity that corresponds with the expected outcome of liquidation, i.e. to marketability. In the context of asset liquidity viewed as the relation between the sale price and the sale time (see section 1.1.1.1), the traditional measures seems to concentrate more on the price dimension, while the probability-based measures, and in particular ToM, focus on the time dimension. As expected, the connection between the traditional or probability-based measures and the measures of liquidity risk is much less obvious. While the former concentrate on expectations, the latter focus on the uncertainty about these expectations. At first sight, there seems to be no apparent reason why they should influence each other. For example, the level of market activity in highly risky markets (e.g., derivative markets) is not systematically different than in low-risk markets (e.g., money markets); it is not clear, why trading volumes should react differently to liquidity risk. A closer look reveals, however, that some link may exist. It is best visible in the analysis of the components of the bid-ask spread. In the ideal case, competitive market makers should act risk neutrally. This means that, ignoring the issue of asymmetric information, bid and ask prices should be based only on expectations; fluctuations of the effectively realized prices (after opportunity and holding costs) would offset each other due to the large scale of market maker’s operation reducing bankruptcy risk practically to zero. However, in reality, full risk neutrality is usually not given. The reasons are the lack of perfect competition between market makers on the one hand – there is only a limited number of market maker seats on most exchanges – and the inability to extend the scale of activity to a level at which fluctuations of the order flow can be fully outbalanced. Thus, 400
In terms of the real estate search model, the link between the probability of sale and the bid-ask spread can be identified by considering the dependence between the implicit spread defined in section 3.1.2.1 and the frequency of offers, which is one of the determinants of the probability of sale within a certain time interval. For low levels of rental revenues, both the expected net receipts E(Γ) and the expected net expenses E(Ξ) should increase.
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the observed spreads may be affected by liquidity risk, especially when the market is thin and there are fewer active dealers. In such cases, also other liquidity measures could be affected in the same manner – investors would need to account for the additional source of uncertainty in their decisions to trade. In consequence, there may be less turnover, and price reversals may occur more slowly. The extent of this effect is, however, unclear and, to my best knowledge, no empirical research on this issue exists. While the above discussed measures concentrate on certain aspect of liquidity only, the alternative approaches are attempts to combine various aspects. Therefore, by their nature, they should show connections to all other measures. The extents and the directions of these links may, however, vary very strongly. As already noted in section 3.4.1, the liquidity performance measures may even lead to an opposite assessment of liquidity than the traditional measures, which is due to the underlying assumption about the active, strategic behavior of the investor. While bid and ask prices are valid for passive investors who forgo their own search and choose to trade with a dealer, the LRR is more appropriate for investors who are willing and able to conduct a strategic search for a trading partner. In the former case, low expected purchasing expenses and high expected sale receipts are to investor’s disadvantage – they constitute trading costs and reduce returns. In the case of LRR, however, the investor is assumed to take advantage of the high heterogeneity of market participants and low market transparency. This way, she should be able to obtain a higher expected sale price; at the same time, however, liquidity risk is also higher. Since both the nominator and the denominator of the LRR ratio increase, the final effect is unclear. It depends not only on the precise characteristics of the asset and the market but also on the characteristics of the investor, in particular, her time preference. The analysis of the relations betweens utility based approaches and other liquidity measures is even more difficult. It depends almost entirely on the utility function used in the concrete measure. The results achieved with the approach proposed by Mok should generally lead to similar results as the application of LRR when applied on the level of utility levels or QSD when applied on the level of utility decreases due shorter sale horizons. However, the conclusions may be entirely different when the assumed preferences are not quadratic. ***
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A number of approaches to measuring liquidity and liquidity risk have been proposed in this Chapter. The main conclusion from the analysis is the classification of the measures into two groups: those concentrating on the expected outcome of liquidation and those concentrating on uncertainty. This corresponds with the decomposition of liquidity presented in section 1.1.2 of Chapter 1 into marketability and liquidity risk. Since no complete picture of the problem is possible without considering both dimensions, neither the traditional, nor the probability-based, nor the liquidity risk measures can be viewed as sufficient for all types of liquidity related decision problems. For the same reason, the alternative measures should be viewed as most universal. Their quality depends, however, on the compliance between the method according to which various aspects of liquidity have been combined and the significance of these aspects for the concrete decision problem. In particular, the relative importance of marketability and liquidity risk as well as the freedom of action available to the investor (ability to conduct a search) need to be taken into account. Summing up, the choice of the best liquidity measure is at least as difficult as the choice of any other measure of a complex phenomenon. The accuracy achieved with a certain approach depends not only on its construction but also on investor’s understanding of the terms that are to be measured. In practice, it might turn out that a simpler ratio will give the decision maker a better rationale for her investment decision than a theoretically elaborate but intuitively intractable concept. Mainly for this reason, the analysis in the next Chapter falls back on relatively simple, though possibly theoretically imperfect measures of marketability and liquidity risk: expected net receipts from liquidation and receipts’ volatility, which are derived on the basis of the real estate search model. Still, the issue of liquidity measurement remains a wide field for further research.
Chapter 4 Liquidity as a Decision Criterion
Considerations in the former chapters concentrated mainly on the definition of liquidity, methods of its appropriate description, and its measurement. Yet, the main reason why investors are concerned with this issue is its relevance for investment decisions. Since liquidity is one of the standard investment targets, ignoring it will most probably result in a suboptimal allocation of capital. Despite this fact, the vast majority of the existing investment analysis and decision tools skip on this issue narrowing the scope of admissible assets to those considered to be perfectly liquid. Obviously, this is inappropriate for a wide range of assets including real estate. This drawback is dealt with in this Chapter. In the course of the analysis in Chapter 1 two independent dimensions of liquidity have been identified: marketability (or expected liquidity) and liquidity risk. As demonstrated in Chapter 3, various popular measures of liquidity can be classified according to this scheme. It seems therefore plausible to base the liquidity management concept on these two dimensions as well. In terms of the search model formulated in Chapter 2, they correspond with the expectation and the uncertainty about liquidation receipts, which are random for illiquid assets. This approach resembles the profitability-risk approach to investment analysis in the traditional finance; in fact, the only difference is that the liquidation value instead of the rate of return is used as the underlying variable. A straightforward advantage of this analogy is the possibility of applying the same analysis tools, in particular, the most popular and well researched meanvariance analysis. Combining the latter approach with conclusions from the search theoretical analysis allows determining the optimal liquidation strategy of a liquidity risk-averse investor as well as enhancing the traditional portfolio optimization techniques with the liquidity criterion. The Chapter is organized as follows: The general mean-variance decision framework is presented at the beginning followed by the application of this framework to the problem of strategic liquidation. By defining a “liquidity efficient frontier”, it is possible to identify the set of rational liquidation strategies. The implications differ substantially when liquidation of whole portfolios rather than of single assets is considered.
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The conclusions have also consequences for the notion of liquidity. In particular, the subjectivity of this characteristic vanishes as soon as a simultaneous sale of liquid and illiquid assets is allowed. This is the subject of section 4.3. Finally, liquidity is implemented in the portfolio selection framework. Planned portfolio liquidation at the end of the investment horizon and unexpected, premature liquidation are considered separately. In the first case, liquidity is allowed for by correcting expected returns and market risk; in the second case, measures of liquidity are introduced as separate optimization criteria. Like in former parts of the book, also in this Chapter the analysis is limited to the liquidation case only and focuses on real estate. This is mainly for better tractability of the results as the same logic applies to the purchase case and the conclusions are also valid for other highly illiquid assets.
4.1.
Investment Decisions in the Mean-Variance Framework
The mean-variance decision framework, which is the most distinctive one within the Modern Portfolio Theory (MPT), dates back to Markowitz (1952). It was enhanced by Tobin (1958) and further developed by numerous researchers becoming probably the most popular tool of portfolio analysis and selection. The model combines the idea of risk reduction by diversification, which is crucial for the most of today’s finance thought, with a relatively simple methodical approach allowing its implementation even on personal computers. It is therefore appealing for both scientists using it as a starting point for further research and practitioners seeking optimal (or at least reasonable) combinations of investments. The presentation of the concept is conducted in three steps: definition of the notion of efficiency with respect to multiple decision criteria, presentation of the mean-variance-based portfolio selection problem, and its extension by allowing for risk-free investments. It is based mainly on Levy/Sarnat (1984) and Elton et al. (2003), but the foundations of the portfolio theory can be found in practically any investment text-book. The purpose of this section is to establish the foundation for an analogous liquidity management concept presented further in this Chapter. 4.1.1.
The Efficiency Criterion
If investment alternatives could be evaluated on the basis of only one criterion, choosing the optimal one would be trivial. Imagine an investor concerned only about investment’s expected profitability – obviously, she should invest her whole capital in
4.1 Investment Decisions in the Mean-Variance Framework
235
one asset that offers the highest expected rate of return. Individuals with such simple attitudes are, however, only seldom in reality. In most cases, further criteria need to be considered constituting a non trivial trade-off. The decision problem refers then to the optimal combination of the relevant criteria achieved with the concrete choice of assets. In consequence, the notion of optimality depends on investor’s subjective preferences with respect to the considered criteria. Even then, however, some alternatives are likely to exist that would not be preferred by any rational individual; they are said to be inefficient. In contrast, the group of investments that might be desirable for at least some investors is said to be efficient. The efficiency principle is a decision rule for dividing all potential investments into two exclusive sets: the efficient and the inefficient. Thus, the choice of investments or portfolios under multiple decision criteria can be accomplished in two steps: first, the set of efficient alternatives is identified, and then, the final choice is made from among this set according to investor’s preferences, which are usually described by a utility function.401 The key decision variables in the original portfolio theory are returns. They are considered to be random and accordingly described by a probability distribution. This assumption allows defining a very general efficiency principle: stochastic dominance. It does not require the knowledge of investor’s precise preferences but implies some of their properties. According to the first degree stochastic dominance (FSD) one alternative dominates (is more efficient than) another alternative if for each return level the cumulative probability of achieving it is higher for the first one than for the second one. Following this logic the alternative X in Figure 4-1 dominates Y and Z since FX(R)>FY(R) and FX(R)>FZ(R) for all R, but neither does Y dominate Z, nor does Z dominate Y. Hence, X is efficient and Y and Z are inefficient. The only assumption behind this criterion is that investors prefer higher returns to lower returns.
401
See Levy/Sarnat (1984), pp. 178-181.
Cumulative probability
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X
Y
Z
Return Figure 4-1: First degree stochastic dominance
Two further degrees of stochastic dominance can be identified. The second one (SSD) requires additionally that investors are risk averse, and the third one (TSD) adds the requirement of a decreasing absolute risk aversion. However, as long as an investment alternative is efficient according to the first degree stochastic dominance principle, it is also efficient according to the second and the third degree.402 Though very appealing as a theoretical concept, stochastic dominance requires the knowledge of the entire probability distribution of returns. Such knowledge is rarely available in reality; an assessment of returns’ main characteristics is usually all one can get. Another group of efficiency principles concentrates therefore on selected distributional parameters (parameter preference models).403 According to these principles, only those alternatives that are inferior with respect to all parameters simultaneously are denoted as inefficient. Although any number of decision relevant parameters can be used, most common is their reduction to measures of profitability and investment risk. The mean and the volatility of returns are predominantly used for this purpose constituting the Markowitz’s mean-variance (MV) criterion.404 According to it, an al402
403 404
For the derivations of the stochastic dominance criteria see Hadar/Russell (1969) and Whitmore (1970). For an extensive research review see Bawa (1982). See Schmidt-von Rhein (1996), p. 224. Although the MV-criterion in its most popular form was proposed by Markowitz (1952 and 1959), it must be noted that the relevance of the mean and the variance of returns for investment decisions has been proposed much earlier, e.g., by Keynes (1937), Marschak (1938), or Hicks (1946). See also: Levy/Sarnat (1984), p. 236.
4.1 Investment Decisions in the Mean-Variance Framework
237
ternative is efficient when no other alternative exists having simultaneously a higher expected return and a lower return volatility (variance). Hence, the alternatives X and Y in Figure 4-2 are efficient, and the alternative Z is inefficient since it has both a lower expected return and a higher risk. In terms of preferences, the MV-criterion requires that either the investor’s utility function is quadratic (i.e., it depends only on R and R2), or that the probability distribution of returns is normal and, thus, fully defined by the mean and the variance.405
Expected Returns
U’3
U’2
U’1
U’’3
Y U’’2 U’’1 X Z
Standard Deviation
Figure 4-2: Mean-variance efficiency and investment choice
As discussed in the section 3.3.1, the choice of the appropriate risk measure is always connected with the understanding of this characteristic. Volatility is only one of the possibilities, which owes it popularity to the easy handling. However, other parametric approaches use alternative risk measures instead (e.g., semi-volatility). Also further distributional parameters (e.g., skewness) can be added as decision criteria. By doing so, a three- rather than two-dimensional efficiency principle is considered.406 Although the approaches differ with respect to the assumptions about the nature of investor’s preferences, the general way of reasoning about the efficiency of investment alterna405
406
The fact that returns of publicly traded assets are often considered to be normally or nearly normally distributed is mostly named as the reason for the choice of mean and volatility as decision parameters for portfolio selection. The argument basing on the quadratic utility function, brought up originally by Markowitz (1959), seems rather weak and is mostly rejected. See Markowitz (1987), pp. 52-56, and the literature cited there. See Breuer et al. (1999), part III, for a review of alternative parametric approaches to portfolio selection.
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tives remains unchanged – an efficient alternative is the one for which no other alternative exists that is superior with respect to all parameters simultaneously.407 As soon as the set of efficient investment alternatives has been identified, the optimal choice can be made on the basis of investor’s personal preferences. This procedure can be depicted in the MV-diagram by drawing the respective indifference curves. They represent combinations of expected returns and return volatilities that lead to equal levels of utility. Higher curves (lying further in the direction of the upper left corner of the diagram) indicate higher utility levels and “steeper” curves indicate more riskaverse investors. In Figure 4-2, the alternative X would be preferred by more riskaverse individuals (U’), and the alternative Y would be preferred by less risk-averse individuals (U’’); neither group would, however, prefer the alternative Z. 4.1.2.
Diversification and Mean-Variance Portfolio Selection
Determination of the efficient set and identification of the optimal investment complicates when the investor is allowed to invest in more than one alternative. The central idea of the MPT is the fact that the risk of an investment portfolio is not equal to the sum of risks of its components. As long as the assets in the portfolio do not follow exactly the same return path, some of the random negative (positive) fluctuations of one asset may be offset by positive (negative) fluctuations of another asset. Thus, fluctuations of the total portfolio value are usually lower than the sum of fluctuations of its components – some of the risk vanishes due to diversification. If risk is measured with volatility, this effect arises from the fact that the standard deviation (variance) of a sum or a weighted average of random variables depends not only on the standard deviations (variances) of these variables but also on the covariances between them. On the other hand, there is no such effect with respect to expected returns. Thus, the parameters of a two-asset portfolio with wX and wY percent of the total capital invested in X and Y, respectively, are: E( w X ⋅ R X + w Y ⋅ R Y ) = w X ⋅ E(R X ) + w Y ⋅ E(R Y )
(4.1)
S( w X ⋅ R X + w Y ⋅ R Y ) = w 2X ⋅ V(R X ) + w 2Y ⋅ V(R Y ) + 2 ⋅ w X ⋅ w Y ⋅ Cov(R X , R Y ) (4.2)
407
See Rehkugler/Schindel (1990), p. 94.
4.1 Investment Decisions in the Mean-Variance Framework
239
The respective general formulas for N assets are: N
E(R P ) = ∑ w i ⋅ E(R i )
(4.3)
i =1
S(R P ) =
N
N
N
i =1
i =1 j=1 j≠i
∑ w i2 ⋅ V(R i ) + ∑∑ w i ⋅ w j ⋅ Cov(R i , R j )
(4.4)
The portion of risk eliminated by adding an additional asset to an existing portfolio depends on its correlations with other assets.408 It changes proportionally for the correlation of 1, so that no diversification effect occurs in this case, and can be reduced to zero for the correlation of -1. But even for the correlation of 0 an over-proportional risk reduction is possible (see Figure 4-3). Since neither the correlation of 1, nor the correlation of -1 occurs in reality, it can be expected that the total portfolio risk decreases with every additional asset, but it is not possible to fully eliminate risk this way.
Expected Returns
Correlation = 0 Y
Correlation = +1
Correlation = –1
X
Standard Deviation Figure 4-3: Diversification effect in a two-asset portfolio409
The possibility of portfolio building has a direct effect on the determination of efficient investment alternatives. It has to be viewed not in terms of single assets but in 408
409
Since the correlation coefficient equals the covariance divided by the standard deviations of both variables, it can be treated as a standardized covariance; it takes values between -1 and 1 corresponding with precisely parallel variable changes and precisely reverse changes, respectively. Based on Levy/Sarnat (1984), p. 291.
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Expected Returns
terms of their portfolios. This results in a practically infinite universe of different alternatives – additional points in the MV-room become accessible by varying assets’ proportions in the portfolio. In effect, the graphical representation of available investment alternatives as combinations of means and variances of returns resembles a dense cloud rather than scattered points. Only portfolios lying on the upper left boundary of this cloud are efficient – they constitute the so called “efficient frontier”. Among the efficient portfolios, there is one having the minimal achievable risk and denoted as the minimum-variance portfolio (MVP), and one having the maximal expected return (MERP). While the MVP is a combination of assets that yields the highest diversification effect, the MERP consists always of only one asset – the one having the highest expected return.
MERP
U3
POPT
U2 MVP U1
Standard Deviation Figure 4-4: Efficient frontier and portfolio selection
The choice of the optimal portfolio follows the same procedure as in the case of single assets. Basing on investor’s preferences, the combination of assets is selected that offers the highest utility. It can be represented graphically by drawing the respective indifference curves. Obviously, the investor prefers the highest possible curve, which in the considered case is the one having only one common point with the efficient frontier. The optimal portfolio is the portfolio that corresponds with the MV-locus at the tangential point between the indifference curve and the efficient frontier.
4.1 Investment Decisions in the Mean-Variance Framework 4.1.3.
241
Portfolio Selection with Risk-Free Assets
In the standard MV portfolio selection, all investments are assumed to be more or less risky. Since no perfect negative correlations between assets are possible in reality, it is also not possible to reduce risk to zero by building portfolios. However, apart from risky investments, investors usually also have the possibility to purchase (almost) riskfree assets. In particular, the interest earned on treasury bonds, especially those emitted by the US government, is usually considered to be free of any risk. Also the risk connected with most of the money market investments is practically negligible. The effects of including a risk-free alternative in an investment portfolio have been considered by Tobin (1958).410 Since the risk-free interest rate (RF) does not vary within the investment horizon, its variance and volatility are zero; hence, its risk-return locus lies on the y-axis. Furthermore, also the covariance of RF with any other investment equals zero. In consequence, the risk of a combination of a risky asset with the risk-free interest rate is proportional to the amount of the risky asset. Due to this property, the risk-return locus of a portfolio containing a risk-free investment can be presented graphically as a straight line connecting the respective risky portfolio with the point on the y-axis corresponding with the risk-free interest rate.
Expected Returns
U
POPT
PM
RF Standard Deviation Figure 4-5: Portfolio selection with a risk-free interest rate
410
See also Sharpe/Alexander (1990), pp. 166 ff., or Elton et al. (2003), pp. 84 ff.
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Chapter 4: Liquidity as a Decision Criterion
Considering the modified problem, the question arises: which risky portfolio should be combined with the risk-free investment? According to the efficiency principle, an investor always prefers the highest expected return at a given volatility level. Hence, it is optimal to choose a portfolio P on the efficiency frontier that yields the highest RF-Pline when combined with the risk-free rate. This is the portfolio for which the mentioned line is tangential to the efficiency frontier (see Figure 4-5); it is denoted as the “market portfolio” (PM). Assuming that not only risk-free investing but also borrowing at RF is possible allows extending the line beyond the PM point. It turns out that (unless PM was already investor’s choice) it is always preferable to purchase the market portfolio and combine it with the risk-free investment rather than to invest in any other portfolio on the efficient frontier. Thus, investors should select PM and “adjust” it to their own needs by adding an appropriate portion of the risk-free asset. Note that the choice of the optimal risky portfolio is independent of investor’s individual preferences in this case. This effect is denoted as Tobin’s separation theorem as it separates the portfolio selection problem form investor’s attitude to risk.411 4.1.4.
Limitations of the MV-Criterion
The MV-efficiency principle and the portfolio selection methods arising from it have become standard investment analysis tools. Aside from their theoretical appeal, the ease of practical implementation was surely among the crucial determinants of their great popularity. Statistical measures of returns, such as expected values, variances, and covariances are usually assessed from historical data, which are easily available for most public markets. Also the optimization algorithms allowing the computation of the minimum-variance and the market portfolio are relatively simple and operate even on personal computers. Nevertheless, the model is subject to rigorous restrictions,
411
Basing on the Tobin’s separation theorem, Sharp, Lintner, and Mossin developed the Capital Asset Pricing Model (CAPM); see Sharpe (1964), Lintner (1965), and Mossin (1966), as well as Elton et al. (2003), p. 293 ff. Note, however, that while the CAPM is a descriptive market model, the portfolio selection framework discussed in this section is a normative model of investor behavior. Hence, the separation theorem refers only to investors facing the same set of investment alternatives.
4.1 Investment Decisions in the Mean-Variance Framework
243
which often seem to be overlooked in practice. The following four assumptions are usually considered to be most important:412 (A1) investors are concerned only about two moments of the return distribution: the expected return and the variance; (A2) no market access restrictions exist; in particular, there are no transaction costs or legal restrictions; (A3) all assets are perfectly divisible, i.e., it is possible to purchase them in any desired quantities; (A4) all assets are perfectly liquid, i.e., it is possible to sell and purchase them at will without discounts or premiums and without affecting prices. The first assumption implies, as already mentioned, either a quadratic utility function or normally distributed returns. Assuming the first eventuality would enormously limit the scope of addressable individuals; it seems rather unrealistic that a utility function in the form U(R) = x1·R + x2·R2 (with x1 and x2 being respective parameters) is a very common one. The second eventuality is, at least theoretically, easier to accept; it, requires, however an efficient market.413 In this case, subsequent very short term returns are independent and yield an asymptotically normally distributed random variable when compounded to longer periods.414 However, since efficiency is disputable even for highly liquid stock markets, also normal distribution of returns is problematic.415 It is even more problematic for private markets that are widely known to be inefficient, in particular, for real estate markets.416 Application of alternative risk measures, especially those referring to downside risk, may be advantageous is some cases, although it does not entirely solve the problem.
412
413 414 415 416
For an extensive review of the MPT constrains see Truxius (1980), pp. 44-50, or Schmidt-von Rhein (1996), pp. 230-232. Note that market models based on the MPT, such as the CAPM, require further assumptions, in particular, market participants need to be homogonous. See FN 227. See also the discussion in section 2.3.1.5. See the literature in FN 228 for empirical studies. For empirical studies on the efficiency of real estate markets see Guntermann/Smith (1987), Gau (1987), Case/Shiller (1989), or Clayton (1998).
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The assumption about the lack of market access restrictions is obviously unrealistic. There are both regulations preventing investments in certain assets (e.g., limiting the access of foreign individuals to domestic markets, or preventing pension funds from high-risk investments) and substantial transaction costs encompassing broker or administrative fees. The latter can be especially severe in the case of direct real estate investments. Fortunately, this limitation can be solved relatively easily by correcting returns for additional costs and introducing side constraints in the optimization algorithms.417 The latter method can also be applied to ease the assumption of perfect divisibility of investments. By allowing only investments of certain fixed amounts in each asset, a more realistic optimal portfolio can be determined.418 Probably most severe and especially important for this work is the assumption of perfect liquidity. This issue has already been discussed in Chapter 1. In its original version, the MPT is a one-period model and refers to a certain fixed investment horizon, which corresponds with the assumed return period. However, if limited liquidity prevents the investor from selling the investment as scheduled, the statistical parameters of returns may alter resulting in a suboptimal asset allocation. A possible solution is the correction of the original return series for the discounts accrued in an immediate sale. Yet, as discussed in former chapters, immediate liquidation of an illiquid asset may be, and in most cases is, not optimal. Furthermore, the uncertainty component of liquidation still remains disregarded. Thus, there is no easy way to cope with imperfect liquidity in the MV framework, and the MPT is therefore applied practically only to highly liquid publicly traded assets. Finding a method to adequately allow for liquidity in the standard portfolio selection model is one of the central issues in this Chapter.
4.2.
Strategic Liquidation
Although the mean-variance decision framework presented in the former section has been developed for the purpose of rational asset selection, it can be modified for application to other decision problems, in particular those arising from the lack of perfect liquidity. The first and probably most straightforward issue that requires a strategic approach in the situation of imperfect liquidity is liquidation of assets. Considerations regarding liquidation strategies for perfectly liquid assets are pointless; since there is 417
418
For portfolio selection algorithms with transaction costs see Pogue (1970), Patel/Subrahmanyam (1982), Davis/Norman (1990), Gennotte/Jung (1994), Li et al. (2000, 2001), or Liu et al. (2003). See Jacob (1974) or Levy (1978).
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only one market price that can be realized immediately, there is no need for a strategic behavior. However, with heterogeneous opinions about the asset’s value, the lack of an organized trading system, and the necessity to search for a trading partner, the choice of the best buying or selling strategy becomes nontrivial. The following section offers an approach to optimal liquidation of illiquid assets based on a combination of the mean-variance decision framework with the search theoretical model. Despite the different character of portfolio selection and liquidation decisions, the main principles remain the same; the only change is in the target variable – optimization is not applied to assets’ rates of returns but to liquidation values. A short review of the existing approaches to strategic liquidation is presented first. The issue of efficiency of liquidation strategies follows. By analyzing the loci of expected receipts and receipts’ volatilities associated with different reservation prices, it is possible to isolate inefficient strategies, which should not be preferred by any investor and as such can be excluded from further analysis. The remaining ones form a liquidity efficient frontier. These considerations are then extended on liquidations of whole portfolios containing either only illiquid or both liquid and illiquid assets. It can be shown that the notions of efficiency and optimality of sale strategies change as soon as more than one asset is liquidated at a time. 4.2.1.
Literature Review
Optimal liquidation is one of the central issues of the search theory; it has been extensively discussed in Chapter 2. Since a liquidation strategy is usually defined by the reservation price applied in the search process, the problem is reduced to finding the optimal reservation price. This issue has been addressed in the context of the basic search model in section 2.2.3, in the context of the Karlin’s model in section 2.2.4, and in the context of the real estate search model in section 2.3.2. In all these approaches, the optimal liquidation strategy is interpreted as the one that maximizes the expected net receipts from sale: this is the consequence of the risk neutrality of the decision maker assumed in Chapter 2. A different way of addressing the issue of strategic liquidation of financial investments in organized markets was proposed by Bertsimas/Lo (1998). The focus of this approach is not on the search for a trading partner, which is not necessary in an organized market, but rather on the impact of trading on the market price. The authors con-
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sider a strategy minimizing the expected cost of buying a fixed number of shares within a given time horizon when prices are affected by random exogenous shocks as well as executed trades. They define a trading trajectory as a sequence of amounts of shares purchased in discrete time intervals. They show that the optimal strategy (trajectory) can be presented as a combination of naïve strategies of breaking the total amount into equal portions and correcting for new information. Almgren/Chriss (1998, 1999, and 2000/2001) use a similar framework to define efficient liquidation strategies in public markets. However, in addition to the minimization of trading costs, they assume that the trader is also concerned about the risk of liquidation. Furthermore, they assume that trades have both a temporary and a permanent effect on prices. The latter one is the only source of the drift in the equilibrium price level, which would otherwise follow a random walk. Almgren and Chriss use the expected liquidation cost and the variance of the liquidation cost to define the efficient frontier of optimal liquidations. The optimal trajectory results from investor’s preferences with respect to these parameters; a linear utility function is assumed.419 A related approach is applied by Dubil (2002). However, he operates in a continuous time framework and also solves for the optimal liquidation horizon. Within this horizon, the investment is liquidated at a constant rate per unit of time, so the trajectory is a straight line. Dubil also allows for different forms of impact functions. Furthermore, he also considers the consequences of a correlation between the stochastic market impact and assets’ returns. The presented approaches, though granting valuable insights in the problems encountered when liquidating illiquid assets, have major weaknesses. The search theoretical models based on the maximization of expected receipts ignore the risks arising from the lack of perfect liquidity. They can yield acceptable results when the investor is risk-neutral but are clearly inappropriate for risk-averse individuals. The methods focusing on liquidation trajectories mostly include risk considerations, but they refer only to public markets and only to certain aspects of liquidity, i.e., to depth and partially to resiliency. These shortcomings can be overcome by merging the two approaches. On the one hand, by applying the search theory to model liquidation receipts, it is possible to obtain very general results that are valid also for direct markets. On the 419
The linear utility function used by Almgren and Chriss allows the interpretation of the result in term of Value at Risk; see also section 3.3.3.3.
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other hand, a two dimensional mean-variance approach based on both the expectation and the uncertainty about outcome of liquidation ensures that investors’ attitude to risk is adequately regarded. 4.2.2.
Efficiency of Liquidation Strategies
An optimal liquidation strategy is the one that results in the best possible outcome. However, in order to identify it, a target variable, i.e., a clear criterion for the evaluation of outcomes is required. In the case of asset selection, rates of return are used for this purpose – assets yielding higher returns were preferred to those yielding lower returns. In the case of strategic liquidation, net (discounted) receipts from liquidation are a natural candidate for a target variable. Alternatively, its variations, such as the liquidity discount or the implied spread, could be used. As soon as investors are assumed to be risk-averse, the one-dimensional notion of optimality based on expeted sale recipts is not sufficient, and the uncertainty about the liquidation outcome needs to be considered as an additional decision criterion. However, with more than one criterion, evaluation of liquidation strategies becomes substantially more complex. Although possibly superior with respect to the expected outcome, some strategies may proof inferior with respect to risk. Thus, the combination of both criteria needs to be considered. In consequemce, similarly as in the case of asset selection, the notion of optimality depends on investor’s preferences. However, it is also possible that certain strategies lead to lower expected receipts and higher uncertainty than other strategies and are therefore not preferred by any investor. Thus, similarly as assets or portfolios of assets, liquidation strategies can be classified as efficient and inefficient. The latter should be excluded from further analysis. Regarding the liquidation outcome as random, the analysis of liquidation strategies’ efficiency should be based on probability distributions of net receipts resulting from these strategies. In the most general approach, whole distributions should be considered since they contain all possibly relevant information. The concept of stochastic dominance can be applied at this point. According to the FSD, one strategy dominates another strategy only if the probability of achieving any level of net sale receipts is higher for the first one than for the second one; further degrees of stochastic dominance can be defined analogically. The main practical problem is the determination of the respective probability distributions. It can be solved by referring to a search model
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of sale. This approach allows computing, or at least estimating, the probabilities of sale receipts for certain strategies defined as reservation prices. Thus, with sufficient computational effort, it should be possible to plot cumulative distribution functions of net receipts for different reservation prices and to determine whether some of them are dominated by other. An example of such analysis conducted using a simulation technique is presented in Figure 4-6. The conclusion is that the strategy Z is dominated by the strategy X for lower receipts’ levels and by the strategy Y for higher receipts’ levels.
100% 90%
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Figure 4-6: Stochastic dominance of liquidation strategies420
Since search theoretical models allow estimating the probabilities for different levels of sale receipts without falling back on large amounts of empirical data, the application of the stochastic dominance concept seems to be easier than for asset returns. Nevertheless, analytical proves of efficiency will usually not be possible leaving a simulation as the only available solution. In this case, however, only a finite number of 420
Computation on the basis of a Monte Carlo Simulation with 10.000 runs and following parameters: offer volatility = 15%; rent = 5%; trend factor = 5%; discount rate = 15%; offer frequency = 52 p.a.; normally distributed market changes (A) with volatility of 5%. X corresponds with the reservation price of 1.4, Y with the reservation price of 1, and Z with the reservation price of 1.2.
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search strategies and receipts’ levels can be analyzed. Even with an enormous computational effort one can never be sure whether some other, not reviewed strategy exists that would dominate any other. Hence, it seems more practical to concentrate only on certain parameters of receipts’ distributions. In analogy to portfolio selection models, it seems reasonable to limit their number to only two: one referring to the expected outcome of liquidation (marketability) and one referring to liquidity risk. Like in the traditional mean-variance approach, the expected value and the variance (or the standard deviation, i.e., volatility) of net sale receipts can be used as respective decision citeria. In this sense, only those liquidation strategies are efficient which do not simultaneously yield a lower expected value of receipts and a higher receipts’ volatility than any other strategy. While using expected receipts as a measure of marketability is largely unproblematic, the use of volatility as a liquidity risk measure is subject to several difficulties. The main problem arises in the context of investor’s preferences. As discussed earlier, the MV criterion holds only if either investor’s utility function is quadratic with respect to the considered (random) target variable, which in this case is the outcome of sale, or the probability distribution of this variable is normal. While the first condition leads to a very restrictive assumption about investor’s preferences and is therefore mostly rejected, the second one is fulfilled only in certain cases. As discussed in section 3.3.3, simulations show that sale receipts resulting from the real estate search model are roughly symmetrically distributed only if the discount rate is level with total expected returns from the property (i.e., expected appreciation plus rental income). Otherwise, the distribution is skewed and clearly not normal. Hence, in some situations, volatility may not be compliant with investor’s notion of risk. This will especially be the case when the sale is forced by unexpected liquidity problems or by unusually profitable alternative investment opportunities since discount rates are high then. Alternative approaches, especially those based on the idea of downside risk (see section 3.3) should provide better results in such cases. Nevertheless, the MV approach is followed in this Chapter mainly due to the possibility of obtaining analytical solutions. Furthermore, what should not be underestimated, it is by far the most popular risk measure in quantitative finance, well established in the financial community, what makes it more easily acceptable by practitioners. Using volatility also makes it easier to borrow from existing approaches, like the MPT. Despite the mentioned drawbacks, it should still yield satisfactory results in a number of typical liquidation situations. In other cases, one
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should consider substituting volatility (variance) with an alternative risk measure. The general way of reasoning would not change and most of the conclusions valid for volatility should also hold for other measures, but model mathematics is likely to complicate substantially preventing an analytical solution. Still, it might be an interesting subject for further research. Further considerations build on the real estate search model presented in Chapter 2, although other versions of search models can be applied as well. Conclusions can therefore be considered as representative for a whole family of analogical problems, not necessarily referring to real estate. Receipts from sale are the key variable; in the considered case they are defined relative to the average valuation of the property on the market. The goal of the real estate seller is to achieve possibly high expected receipts while keeping uncertainty low. In terms of the assumed measures, it means maximizing the expected relative net receipts E(Γ) at a given level of receipts’ volatility S(Γ), or minimizing S(Γ) at a given level of E(Γ). The steering (strategic) variable is the reservation price π*, which is set by the investor arbitrarily and determines the liquidation strategy. Figure 4-7 plots E(Γ) and S(Γ) dependent on the reservation price.
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Figure 4-7: Expected net sale receipts and receipts’ volatility in the search framework421
Since there are no infinities in the expected receipts and the receipts’ volatility in the typical case, and there are regions in which these statistics change in opposite directions, investor’s choice of the reservation price is not trivial. Some combinations of E(Γ) and S(Γ) clearly dominate others, but at least for some of them, expected receipts maximization and risk minimization constitute contradictory goals. A locus of expected receipts and liquidity risk similar to the traditional return-risk locus results. It can be depicted in the liquidity-MV room and takes usually the form of a (slightly slanting) letter J as in Figure 4-8; each point of the curve corresponds with a certain reservation price.
421
Combination of Figure 2-2 and Figure 3-2.
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X
Y
Expected Net Receipts
1,2
1
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Figure 4-8: Locus of expected net receipts and receipts’ volatility
Obviously, not all points in this chart are efficient. Upper left points are always better than lower right ones as they simultaneously indicate higher expected and more certain sale receipts. Hence, the point X in Figure 4-8 corresponding to the (relative) reservation price of 1.32 dominates the point Y corresponding to the reservation price of 1.39. Thus, only the points at the upper left peak of the J-curve as well as in the lower left arm are efficient. However, the latter correspond with very high reservation prices. Though formally efficient, they do not seem to be a reasonable choice for a typical investor. Very low levels of liquidity risk are in this case achieved at the cost of sacrificing a large part of expected receipts. Only an extremely risk fearing individual could consider taking this path. Moreover, the high reservation prices practically preclude a sale, so that net sale receipts arise mainly from an (almost) infinite stream of rental revenues in this case. Thus, an investor choosing a strategy from the lower left arm of the J-curve would express her preference of not selling at all. Since the focus of the analysis is on “typical situations”, and the considered seller is assumed to be truly interested in selling, this type of efficient sale strategies is omitted in further considerations. Concentrating on “serious sellers” leads to the definition of the liquidity efficient frontier encompassing only combinations of E(Γ) and S(Γ) located in the upper left peak of the J-curve (see Figure 4-9). Since the points on the curve refer to certain res-
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ervation prices, this is equivalent to defining the set of efficient liquidation strategies (reservation prices). It contains the strategy maximizing the expected receipts (MaxE) and the strategy minimizing the receipts’ volatility (MinV).
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Figure 4-9: Liquidity efficiency frontier
The concrete choice of the reservation price depends on investor’s utility function. Knowing it would allow determining the solution that is optimal in the analyzed case. More risk-averse investors with steeper utility functions would prefer reservation prices locating them in the left part of the E(Γ)-S(Γ) chart and, thus, accept lower but more certain receipts; less risk concerned sellers would choose reservation prices leading to higher expected receipts at the cost of higher risk. This choice can be presented graphically by drawing the respective indifference curves in the liquidity MV-room, as presented in Figure 4-9. The reservation price corresponding with the single tangential point of the highest possible indifference curve (here U1) and the liquidity efficient frontier is the optimal one (π*opt). The mean-variance approach of identifying optimal liquidation strategies is closely related to the utility based liquidity measurement presented in section 3.4.2, especially to the approach used by Mok (2002a, b). These measures are based on a specific utility
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function and the choice of the reservation price aims at maximizing it. Thus, liquidity is measured by the maximal achievable utility. If the utility function is quadratic with respect to sale receipts, as assumed by Mok, it is fully determined by the expected value and the volatility (variance). Maximizing utility is then equivalent to finding the combination of E(Γ) and S(Γ) (or E(G) and S(G)) lying on the highest possible indifference curve – it corresponds with the tangential point in Figure 4-9. The strategic choice of the reservation price derived in this section on the basis of the mean-variance decision framework is therefore implicitly incorporated in the utility based measurement. An advantage of the two step approach, in which the efficient set of reservation prices is identified before the individual decision on the basis of investor’s preferences is made, is the possibility of avoiding the choice of a concrete utility function. It is sufficient to state that sellers should choose only the reservation prices on the efficient frontier. This enables more general applications. 4.2.3.
Liquidation Strategies for Portfolios of Assets
Liquidation of only one illiquid asset was considered so far. As shown by using the search model and applying the MV-efficiency criterion, the range of rational liquidation strategies (reservation prices) can be limited to the efficient ones only. However, the question arises whether the liquidity-efficient frontier remains unchanged when not one but several illiquid assets are liquidated simultaneously. The traditional MPT demonstrates that the portfolio perspective can change the notion of efficiency utterly. Is it then, per analogy, possible that certain reservation prices, which were efficient when a single sale was considered, will not be efficient when a portfolio is liquidated? Or can an inefficient reservation price become efficient in such situations?
4.2.3.1. The Liquidity-Diversification Effect Consider for simplicity a simultaneous liquidation of two properties: X and Y. The investor is obviously interested in the total receipts from selling both houses and not in the separate receipts from each transaction. Since the results of the sales are random, the total outcome can be viewed as a sum of two random variables. Hence, its expected value is simply the sum of expected values, but the variance depends also on the covariance between the sale receipts. Analogically to (4.1) and (4.2) one can write: E (ΓXY ) = w X ⋅ E (ΓX ) + w Y ⋅ E (ΓY )
(4.5)
4.2 Strategic Liquidation
S(ΓXY ) = w 2X ⋅ V(ΓX ) + w 2Y ⋅ V(ΓY ) + 2 ⋅ w X ⋅ w Y ⋅ Cov(ΓX , ΓY )
255 (4.6)
with: wX, wY - relative values of X and Y respectively Obviously, as long as ΓX and ΓY are not perfectly correlated, the volatility (standard deviation) of ΓXY is less than the sum of weighted volatilities of ΓX and ΓY. Thus, expected receipts from portfolio liquidation change proportionally to the proportions of liquidated assets, but the volatility can be reduces more than proportionally – a liquidity-diversification effect occurs. This formal result is very intuitive: with two independent liquidations, it is possible that while one of the properties sells below the expectations, the other one exceeds them, so that the total receipts fluctuate less than the receipts from selling each portfolio component separately. This liquidity risk reduction effect should be even stronger when multiple assets are liquidated, but it will probably approach some undiversifiable limit.
4.2.3.2. Covariance of Gains Like diversification of market risk, diversification of liquidity risk depends to a great extent on the correlations (covariances) between portfolio components. In the analyzed problem, however, not the co-movements of returns but the dependences between realized sale receipts are relevant. In consequence, the interpretation of the correlation coefficient is slightly different – it can be equal to one practically only if all assets in the portfolio are sold as a bundle. The buyer has then the choice between taking all of them or none, and the same conditions apply to the whole stake. However, this eventuality means that, in fact, only one asset, consisting possibly of several parts, is liquidated. As soon as each liquidation is conducted separately, i.e., there is an independent flow of bids on each asset, a perfect correlation of receipts, either positive or negative, is virtually impossible. If the liquidation processes are entirely detached, the correlations are likely to be low, but there are at least two reasons why they should still differ from zero. Firstly, sale receipts are not entirely decoupled from the development of the respective markets. As the search for a buyer proceeds, subsequent bids are subject to changes in the market situation, so that also the eventually realized sale price is affected by this factor. Hence, correlations between price levels on the markets to which the liquidated assets belong are relevant for the correlations between the liquidation receipts. In particular, when similar properties are liquidated, the link between the respective (sub)markets is
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likely to increase the receipts’ correlation. Note, however, that even with all assets belonging to the same market (e.g., flats in one block) the correlation will be less than one. The second reason for the mutual dependence of net sale receipts is the fact that liquidation processes run simultaneously by the same investor are seldom fully separated. The seller will probably utilize the available information channels to notify potential buyers about all assets that have been put on sale. This holds especially for the liquidation of real estate portfolios – if similar properties are to be sold, prospective buyers will most probably show interest in more than one property. In effect, bids on different assets (properties) may not be independent with respect to the arrival times and the valuation tendencies of the bidders leading to a positive correlation of the sale receipts. Summing up, a non-zero correlation between receipts from parallel sales of multiple assets may arise from three sources: common market development, simultaneous offer arrivals, and similar levels of parallel offers. The first one can be expressed as the correlation between price changes (i.e., discrete returns) in analyzed markets. In terms of the real estate search model, it corresponds with the correlation or covariance between the respective uncertainty factors A. The two latter sources are, however, very difficult to capture since they depend on individual marketing methods of the sellers. They should play a bigger role when similar objects are liquidated (e.g., two similar residences) but should be negligible for very different objects (e.g., a residential and an industrial property). As it was not possible to define reasonable proxies for these sources of receipts’ correlation, they are omitted in the further analysis. Still, one has to bear in mind that it may result in an underestimation of the actual correlation values. The full derivation of the covariance between net sale receipts in the real estate search framework is presented in Appendix A.6.1. The covariance between market coefficients AX and AY of cov(AX,AY) = σXY is assumed. The following formula results for the real estate search model:
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cov(Γx , ΓY ) ⎛⎛ ⎛ π* − 1 ⎞ ⎞ ⎛ π* − 1 ⎞ ⎞ ⎛ π* − 1 ⎞ ⎞⎟ ⎛ π* − 1 ⎞ ⎞⎟ ⎛⎜ ⎛ ⎟⎟ ⎟ + ν Y ⋅ ϕ⎜⎜ Y ⎟⎟ ⎟ + ν X ⋅ ϕ⎜⎜ X ⎟⎟ ⎟⎟ ⋅ ⎜1 − Φ⎜⎜ Y = σ XY ⋅ ⎜ ⎜⎜1 − Φ⎜⎜ X ⎜ ⎟ ⎟ ⎜ ⎝ νY ⎠⎠ ⎝ νX ⎠⎠ ⎝ ν Y ⎠ ⎟⎠ ⎝ ν X ⎠ ⎟⎠ ⎜⎝ ⎝ ⎝⎝ Y1,X Y1,Y ⋅ ⋅ 2 2 ⎛ ⎞⎞ ⎛ ⎛ * ⎞⎞ ⎛ * ⎜1 − X1,X ⋅ Φ⎜ π X − 1 ⎟ ⎟ ⎜1 − X1,Y ⋅ Φ⎜ π Y − 1 ⎟ ⎟ ⎜ ν ⎟⎟ ⎜ ⎜ ν ⎟⎟ ⎜ ⎝ X ⎠⎠ ⎝ ⎝ Y ⎠⎠ ⎝ with: X1,X = Y1,X =
(4.7)
λX (ρ − τ X ) + λ X λX
((ρ − τ X ) + λ X )2
and X1,Y and Y1,Y defined respectively It is apparent from this presentation that the covariance is zero when σXY is zero, i.e., when considered markets are independent. Furthermore, by analyzing the respective correlation coefficient, i.e., cov(ΓX, ΓY)/(S(ΓX)·S(ΓY)), it is easily stated that it is always strictly smaller than 1 and larger than -1. For practical applications, a proxy of σXY is required. The easiest way to obtain it is by computing covariances between discrete returns of price indexes for the considered markets. However, due to the way the receipts’ covariance has been defined in the formula (4.7), this approach is problematic. Although the A’s are annualized and as such comparable, they refer to market changes occurring during the search, which can have different durations for different assets. Thus, it is possible that AX refers to a change within a year and AY to a change within a month. On the other hand, the empirically measured covariance (correlation) between index returns always refers to exactly the same period of time. This means that σXY is not strictly compliant with the covariance of assets’ (discrete) returns. To ensure such compliancy only temporarily coincidental changes in A’s should be considered; hence, only the time horizon until the sale of the first property should be regarded. In the computation of the receipts’ covariance, this can be allowed for by splitting the market change into two components: one referring to the time while both assets remain unsold and the other one referring to the time after the first sale; the empirical covariance of returns refers then only to the first component. An attempt to follow this approach is presented in Appendix A.6.2. It was possible to derive conditional receipts’ covariances provided that one of the prop-
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erties is sold first. In order to compute the unconditional covariance, however, it is necessary to know for each property the probability that it will be sold as first. Unfortunately, it was not possible to find analytical solutions for these probabilities and, thus, to provide the unconditional covariance formula. Using a numerical approximation should allow accomplishing this task; however, due to the computational complexity of this approach, the simpler formula (4.7) is used in the following analysis. One has to bear in mind that it probably overestimates the receipts covariance if the empirical return covariance is used for σXY. However, numerous simulations conducted for realistic cases of real estate investments resulted in only negligible deviations of the simplified approach from the numerically estimated correct receipts’ covariance even when relatively large differences between assets’ search parameters were assumed.422 Hence, the simplified approach should generally yield satisfactory results, especially when only pure real estate portfolios are considered.
4.2.3.3. Portfolio Effect of Liquidation This section addresses the consequences of liquidity risk reduction due to the portfolio effect. For better tractability, several cases are considered separately. Liquidation of identical assets using identical selling strategies is viewed first; different strategies are allowed for in the next step; finally, liquidation of two different assets is analyzed. When liquidating several identical properties (e.g., identical flats in the same building), it seems intuitive to apply the same strategy to all them. Since the sale receipts are not perfectly correlated, a reduction of risk due to the liquidity-diversification effect can be expected. Assuming that all N properties have equal shares of 1/N in the portfolio, the following volatility formula results:423
S(ΓP ) =
1 N −1 ⋅ V (Γ ) + Cov(Γi , Γj ) N N
(4.8)
with: i ≠ j 422
423
The simulation framework was analogical to the one discussed in section 2.3.4 in Chapter 2. On the one hand, a series of approximately 1000 net sale receipts was generated with a predefined contemporaneous covariance between market uncertainty parameters. On the other hand, the simplified covariance formula with respective parameters was applied. The simulation was repeated for different combinations of search parameters and reservation prices. The differences between the correct and the estimated covariances rarely exceeded 10%. The formula is easily derived from (4.4). See also Poddig et al. (2003), pp. 157-159.
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Figure 4-10: Volatility of net receipts from liquidation of a portfolio of identical assets as a function of the reservation price and the number of properties424
The volatility reduction depends on the liquidation strategy and on the number of simultaneously liquidated assets. S(Γ) decreases with N for all reservation prices, as presented in Figure 4-10; however, not only the general volatility level but also the form of the volatility as a function of the reservation price changes with N. Simulations have shown that the local minimum of S(Γ), which could be observed for single sales, 424
This and the following examples are based on following parameters: offer volatility = 15%; rent = 5%; trend factor = 5%; discount rate = 15%; offer frequency = 52 p.a.; normally distributed market changes (A) with volatility of 5%. Properties account equal shares of the total portfolio value and the correlation coefficient between their uncertainty parameters is always one.
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disappears when the number of assets is large.425 This fact has consequences for the efficiency and the optimality of reservation prices. The results of simulations for portfolios containing 1 to 100 assets are presented in Figure 4-11. Increasing the size of the liquidated portfolio leads to a shift of the E(Γ)-S(Γ)-locus to the left. Moreover, also the form of the locus and the set of efficient strategies change slightly. Thus, an individual selling one flat should possibly proceed differently than an institutional investor selling a portfolio of several hundreds or thousands of identical properties. A strategy efficient in the first case can prove to be inefficient in the second case.
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Figure 4-11: Efficiency of liquidation strategies for portfolios consisting of multiple identical assets
Although the effects of portfolio liquidation are most distinct for large portfolios, the preferred selling strategy may also change when only few assets are sold simultaneously. In this case, however, despite the shift of the efficient frontier to the left, the set of efficient reservation prices remains in most cases nearly unchanged. As long as investor’s indifference curves are parallel, there should be no or only a slight effect on the optimal liquidation strategy. However, if investor’s risk aversion changes for higher utility levels, it is possible that a different reservation price corresponds with the 425
See section 0.
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tangential point of the higher indifference curve with the “new” liquidity efficient frontier (see Figure 4-12). In effect, although the set of efficient reservation prices remains practically the same, a different strategy is optimal.
1,34
One Asset
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π*opt, 1
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Receipts' Volatility
Figure 4-12: Liquidity risk reduction and the optimality of reservation prices for two identical assets liquidated using the same strategy
Though it seems intuitive, it is not obvious that setting the same strategy for all liquidated (identical) assets is optimal. It is by all means possible to use different reservations prices. This should allow an investor to access new points in the E(Γ)-S(Γ) room. The question is, however, whether such points would be efficient. The analysis of this case is more complex since the choice of the reservation prices affects not only the expected receipts and receipts’ volatilities but also the covariances (correlations) between receipts. No analytical solution has been derived for this case, and the analysis was restricted to a simulation of simultaneous liquidation of two properties.
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Figure 4-13: Expected net receipts (a) and receipts’ volatility (b) for two identical assets liquidated using different reservation prices
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Since the liquidation strategy in this case is defined by a pair of reservation prices, the E(Γ)-S(Γ) charts analogical to Figure 2-2 and Figure 3-2 are now three-dimensional as depicted in Figure 4-13. However, each strategy still corresponds with certain values of expected receipts and receipts’ volatility. Considering all possible pairs of reservation prices and depicting them in the expected receipts-volatility room yields the set of feasible liquidation positions; this time, however, it resembles a “cloud” rather than a curve. Multiple simulations of this problem led to very similar results – the E(Γ)-S(Γ) combinations accessible by applying a different reservation price to each property seem all to lie within the borders set by the same-reservation-price strategy, as depicted in Figure 4-14. This means that the “upper” part of the efficient frontier consists of pairs of identical reservation prices – they remain efficient and probably optimal for average investors in most cases. However, new efficient points can arise for lower levels of expected receipts and volatilities (e.g., the point π*opt in Figure 4-14). Especially more risk averse investors can achieve higher utility levels by choosing these points.
1,4
U
Expected Net Receipts
1,2
1
π*opt
0,8
0,6
0,4
0,2
Different Reservation Prices Equal Reservation Prices
0 0
0,05
0,1
0,15
0,2
0,25
0,3
Receipts' Volatility
Figure 4-14: Liquidity-efficient frontier for two identical assets liquidated using different reservation prices
Another relevant case is the liquidation of multiple different assets. The first difference to the liquidation of identical assets is that the parameters of the receipts’ distributions
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are no longer the same. Obviously, this leads to different patterns of expected receipts and receipts’ volatilities and also to different optimal reservation prices. The shapes of the E(Γ)- and S(Γ)-planes regarded as functions of the reservation prices (analogue to Figure 4-13) are therefore asymmetric in this case. Another new issue is the possibly different weighting of the assets in the portfolio. Under certain circumstances, this may introduce a new strategic variable to the liquidation problem. If the investor is in a position to decide on the value or on the number of liquidated assets, she can use it to reach additional points in the E(Γ)-S(Γ) room. The extent of the liquidity diversification effect depends then not only on the applied liquidation strategies but also on the proportions of liquidated asses (see Figure 4-15). In practice, however, the choice of the weights is likely to be very limited, if at all possible. In the case of real estate, large investment sums and lacking divisibility will prevent the arbitrary choice of proportions in which properties are to be liquidated diminishing the importance of this variable.
1,4
Expected Net Receipts
1,2
1
0,8
0,6
0,4 Property X 0,2
Property Y Portfolio XY
0 0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
Receipts' Volatility
Figure 4-15: Liquidity risk reduction for two different assets liquidated in different proportions
In effect, determination of the liquidity efficient frontier when multiple different illiquid assets are liquidated becomes very complex. It requires the analysis of all possible reservation prices as well as all feasible weights for each of the assets. Simulations led
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to the conclusion that although the efficient frontier consists in this case mainly of the reservation prices that are efficient in the one-asset-liquidation case, they are now efficient only in certain combinations. So, it is possible that, for example, relative reservation prices of 1.15 and 1.25 are efficient when separate sales of the properties are considered, but a combination of these prices becomes inefficient when properties are to be liquidated simultaneously. Furthermore, combinations of reservation prices which were formerly inefficient may turn out to be efficient and even preferable for some investors. Further, more complex liquidation scenarios can be considered by introducing additional decision variables. For example, one can allow splitting an asset into several parts and liquidating each part separately. This way, the initial problem of single asset liquidation becomes a portfolio liquidation problem. An additional strategic issue arises then – how should the asset be split in order to reach the highest efficiency frontier and the highest utility from liquidation? Another possibility is to allow arbitrary timing of sales. Due to the random nature of the search process, it is not possible to determine the sale time, but one can decide on the moment when the sale process starts. In terms of the search model, this corresponds with rejecting some of the first offers. Starting the search later than initially intended means that potential sale chances, which would occur during this time, are missed. In the traditional search framework, this kind of delay is pointless – by foregoing incoming offers, a chance of receiving an unusually high one is missed as well. Even if one expects the market situation to improve in the near future, it is still better to inspect any incoming offer and perhaps reject it than to ignore it. However, there is one possibility when timing may be advantageous. If selling an asset or even placing it on sale results in a temporary change of the offer distribution, it might be preferable not to liquidate the whole portfolio at once but to do it in several steps. Such situations may occur when the number of buyers is limited and new ones arrive only slowly. Selling an asset of a certain type, e.g., a residence in a certain area, takes then some of the potential demand for further properties of this type out of the market. The remaining potential buyers may have different valuations of the asset and, thus, offer lower prices. This effect is closely related to the depth and partially to the resiliency dimensions of market liquidity as discussed in section 3.1.2.4 – if depth and resiliency are low, i.e., the distribution of valuations by market participant is prone to rapid changes due to trading activity and recovers only slowly to the initial
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state, it might be advantageous to increase time intervals between subsequent sales.426 Finally, one could extent the analysis to include the problem of optimality of the remaining portfolio. If the investor is allowed to choose freely which of the assets held in the portfolio are to be liquidated and in what proportions, she should also consider her situation after the liquidation. While optimizing the selling strategy may lead to the decision about selling certain assets, it may turn out that the remaining holdings are suboptimal with respect to other investment goals, e.g., they result in a very high market risk exposure. Implementing such considerations would require a criterion of comparing the improvement of investor’s liquidity position with the possible deterioration of her position in other aspects – the utility theory seems to offer the most promising approach to this problem. Since these and other possible enhancements substantially increase the complexity of the model without modifying the central conclusions of this Chapter, they are ignored in further considerations. Nevertheless, for certain practical applications, they might increase the precision of the analysis. The analysis of the effects of simultaneous liquidation of multiple illiquid assets led to the conclusion that efficiency as well as optimality of liquidation strategies need to be reviewed in such situations. Depending on the types and weights of the liquidated assets, different strategies may prove to be efficient and, in consequence, different strategies may be optimal. The implication for investors holding portfolios of illiquid assets, such as real estate companies or funds, is that the portfolio aspect should be taken into account when considering liquidation strategies. By selling each property “on its own”, possible efficiency benefits can be lost. Another, more general consequence is that the notion of liquidity needs to be redefined – in certain cases it should refer to portfolios of assets instead of single investments. This issue is discussed more thoroughly in the following sections. 4.2.4.
Simultaneous Liquidation of Liquid and Illiquid Assets
Only liquidation of illiquid assets has been considered so far. However, in the reality investors seldom hold only illiquid assets. Their portfolios usually contain some por426
Note that this aspect of liquidity is in the hub of the models by Bertsimas/Lo (1998), Almgren/Chriss (1998, 2000/2001), and Dubil (2002). Since they consider the problem of liquidating an asset portfolio on a market on which trading results in changes of the price level, market depth and resiliency play the central role. However, as already noted in section 4.2.1, these authors limit the scope of the analysis to organized markets only. Thus, search for a buyer and strategies defined by reservation prices are irrelevant in their models.
4.2 Strategic Liquidation
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tion of perfectly or nearly perfectly liquid investments that are necessary to maintain solvency. Among them are bank deposits, publicly traded stocks, or other securities. Many institutional investors are even legally obliged to such holdings. Strategic liquidation should be therefore considered also with respect to portfolios containing both liquid and illiquid assets. According to the definition provided in Chapter 1, perfectly liquid assets can be always sold (almost) immediately at the current market price. Search for a buyer is pointless since no market participant would ever offer any other price; hence, the relative sale receipts are always equal to the asset’s market value, i.e., E(Γ) = 1. Furthermore, the lack of deviations from the market value means that there is no liquidity risk involved in the liquidation, i.e., S(Γ) = 0. In the E(Γ)-S(Γ)-chart any perfectly liquid asset is therefore represented by the point (0,1); it is labeled as L in Figure 4-16. Another consequence of the lack of liquidity risk is that the correlation with the receipts from selling any other not perfectly liquid asset equals zero. This fact is of relevance when a combined sale of a liquid and an illiquid asset is considered. Since the covariance and the correlation between them is zero, and the volatility (variance) of the liquid asset is zero as well, both the expected receipts and the receipts’ volatility of such a portfolio change linearly with the proportions of the assets. It can be presented graphically as a straight line connecting the (0,1) point with the point of the J-curve of the illiquid asset corresponding with the chosen reservation price (straight line in Figure 4-16). Points on this line correspond with liquidation positions of an investor who sells the illiquid asset using a certain strategy and simultaneously sells the liquid asset at the market price. Furthermore, the line can be extended beyond the point O by liquidating more of the illiquid asset than actually desired and reinvesting the surplus receipts in the liquid asset. This corresponds with simultaneous restructuring of the investment portfolio and increasing liquid holdings.
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Chapter 4: Liquidity as a Decision Criterion 1,7 U
Expected Net Receipts
1,5
1,3 O(π*T) 1,1 L 0,9
0,7
0,5
0,3 0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
Receipts' Volatility
Figure 4-16: Simultaneous liquidation of liquid and illiquid assets
Note that the described effect is analogue to the one occurring in the combination of risky assets (portfolios) with a risk-free investment in the traditional MPT (see section 4.1.3). Also here, the higher lines dominate the lower ones as they allow higher expected receipts at the same level of uncertainty. The highest possible, and thus the only efficient, is the line tangent to the J-curve (liquidity efficient frontier). The tangential point O corresponds with a certain reservation price denoted as p*T and referred to as the tangential reservation price. Note that (in most cases) it is neither the one maximizing expected receipts nor the one minimizing receipts’ volatility. Instead of choosing a reservation price corresponding with any other point on the liquidity efficient frontier, the investor is able to reach higher expected receipts or lower receipts’ volatility by applying p*T and liquidating a part of the liquid assets’ holdings at the same time. The proportion of the liquid and illiquid investment depends on investor’s preferences – it corresponds with the tangential point of the highest possible indifference curve with the OL-line. The same logic can be applied when not a single illiquid asset but a portfolio of illiquid assets is considered. The only difference is that the tangential point does not correspond with a single liquidation strategy but with a set of strategies for all assets. In
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terms of the search theoretical approach, it is an array of reservation prices. In analogy to the tangential reservation price, it is referred to as the tangential array and denoted as p*T (bold). Note that the reservation prices in p*T do not necessarily comply with the tangential reservation prices valid when the assets are sold successively rather than simultaneously. Thus, the main conclusion from the former section saying that a different liquidation strategy may be optimal when selling a portfolio of assets holds also in this case. Note that an objectively optimal reservation price exists in the above case – any investor, independent of her individual attitude, should choose p*T. Preferences are relevant only for the choice of the proportion in which liquid and illiquid assets are to be sold. If this reasoning was valid for all investors on the market, it would mean that anyone selling the asset should demand at least p*T. In effect, this should be the only price paid on the market and the divergence of valuations should disappear. Obviously, this idealized scenario does not occur in reality, and the objectivity of the tangential reservation price is disturbed by other factors. Indeed, there are several reasons why investors’ reservation prices should deviate from p*T. Firstly, the parameters of the search model are to a great extent subjective. Constants such as the discounting rate or the arrival rate of potential buyers depend on seller’s attitude and her efforts. In effect, also the form of the J-curve and the position of the liquidity efficient frontier are subjective, and so are also the tangential reservation price. Secondly, the above conclusion does not necessarily hold if not a single liquid and a single illiquid asset but a portfolio of (multiple) illiquid and liquid assets is considered. As noted above, the tangential reservation price becomes then an array of reservation prices that are not necessarily identical with those which would be optimal in the single-asset-liquidation case. Since each investor holds and liquidates a different combination of illiquid assets, a different tangential reservation price (array) applies in each case. The idea of a universal reservation price common for all investors has to be therefore rejected. Still, objective optimal reservation prices may be true in many realistic cases. Especially in the real estate branch, only one property is often liquidated at a time making portfolio considerations unnecessary. Also the characteristics of institutional investors are often similar, so that similar parameters of the search model can be applied. Some kind of a common p*T value may exist in such cases.
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There are two practical difficulties connected with the implementation of a strategy comprising of the liquidation of a liquid and an illiquid asset. Firstly, realizing the required proportion between these two assets (asset classes) may prove problematic due to the limited divisibility of illiquid assets. It is rather impossible to adjust the amount of the real property that should be sold – it can either be sold as a whole or not at all. Therefore, only the portion of the liquid asset can be adjusted. The strategic decision in this case is not about the reservation price that should be applied but about the amount of the liquid asset that should be sold together with the property. The second issue concerns finding the actual value of the tangential reservation price. Techniques known from the traditional portfolio theory may be applied here.427 The simplest approach seems, however, to look for the reservation price maximizing the slope of the OL-line: ⎡ E (Γ p *) − 1⎤ p *T = arg max ⎢ ⎥ p* ⎣ S(Γ p *) ⎦
(4.9)
This problem can be solved numerically. As long as the expected net receipts and the receipts’ volatility are finite, a unique solution exists. It is also worth noting that p*T computed this way is closely related to the reservation price maximizing the Liquidity Risk Reward defined in section 3.4.1 (see also section 4.3.2).
4.3.
Optimal Liquidation and the otion of Liquidity
Considerations in the former sections involved the issue of a liquidation strategy that is optimal not only with respect to the expected receipts but also with respect to the uncertainty about the receipts. The strategic liquidation problem resulting from this twodimensional approach is more complex than the sole maximization of the expected search value discussed in Chapter 2. In particular, more than one reservation price can be efficient at a time. The problem becomes even more complex when not a single asset but a portfolio of assets is liquidated. However, implications regarding the strategic behavior of investors selling illiquid assets are not the only conclusion from the search theoretical approach. It also adds a new perspective to the liquidity problem itself. This issue is discussed in the following subsections.
427
See Rudolf (1994) for a review of portfolio optimization techniques.
4.3 Optimal Liquidation and the Notion of Liquidity 4.3.1.
271
Liquidity in Terms of Efficient Liquidation Strategies
The starting point for the search theoretical definition of liquidity is the concept of efficient liquidation strategies. As stated in the former sections, a set of efficient liquidation strategies can be determined for each illiquid asset. Each of these strategies leads to a different combination of E(Γ) and S(Γ), and none of them is clearly superior or inferior to any other efficient strategy; the choice of the best one depends solely on the preferences of the investor. Hence, as long as the analysis is not addressed to a concrete individual, it cannot be limited to a single strategy but needs to encompass all potentially interesting, i.e., efficient ones. Following this logic, one can attempt to judge assets’ liquidity by the form and position of their liquidity efficient frontiers.
U’’
Expected receipts
U’3 U’2
U’1 U’’ U’’
X
Z Y Receipts’ volatility Figure 4-17: Asset liquidity in terms of liquidity efficient frontiers
Efficient liquidation frontiers for three exemplary assets are depicted in Figure 4-17. It is clear that asset X dominates asset Z – it allows achieving higher expected receipts at a lower risk for any liquidation strategy. However, no such statement is possible for other pairs of assets (X and Z, and Y and Z). X sells on average at a better price than Y for higher receipts’ volatilities, but a lower level of receipts’ volatility is possible with Y. Analogically, Z is burdened with higher liquidity risk than Y for lower levels of expected receipts, but allows achieving high values of E(Γ), which are not accessible with Y. Basing on the above considerations, one asset can be considered more liquid than another asset if it yields at least as high net receipts from sale and at least as low re-
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ceipts’ volatility as the other asset independent of the reservation price. As soon as one of these conditions is not fulfilled, no definite statement about assets’ relative liquidity is possible without assuming investor’s preferences. In this sense, X is more liquid than Z, but the liquidity relation between X and Y as well as between Y and Z is indefinite and depends on investor’s preferred reservation prices. For example, a more liquidity risk-averse individual (with utility levels U’) would consider the alternative Y more liquid, and a less liquidity risk-averse investor (with utility levels: U’’) would regard X as more liquid. At this point, it is again necessary to stress the difference between the two-dimensional approach based on expected sale receipts and liquidity risk and the traditional notion of liquidity based on market breadth, depth, and resiliency. Note that the assumption that the investor is able to influence the receipts from selling an asset by adjusting the reservation price implies the ability to conduct an active search. In contrast, the traditional liquidity concepts assume that the investor accepts the market situation as it is, including the price level. Hence the notion of liquidity presented in this section is addressed predominantly to individuals who are in a position to behave strategically. This may not apply to many smaller investors, as they have neither the means nor sufficient information to apply a strategic search. The traditional approach may be still more appropriate for such investors. 4.3.2.
Liquidity with Liquid and Illiquid Assets
The above notion of liquidity needs to be modified when simultaneous sale of a liquid asset is allowed. As discussed in section 4.2.4, reservation prices associated with the liquidation efficient frontier are then not efficient any more. Due to the possibility of combining the sale of an illiquid asset with the sale of a liquid one, new, more efficient combinations of expected net receipts and receipts’ volatilities become accessible. The only efficient liquidation strategy, independent of investor’s preferences, is then the tangential one. A liquidity efficient line representing portfolios with different proportions of the liquid and the illiquid asset results. Such efficient lines can be drawn for all illiquid assets and correspond with the respective tangential reservation prices. Since their efficient liquidation frontiers usually differ, also the lines are asset specific.
Expected receipts
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p*T, A X p*T, C p*T, B Z
L
Y Receipts’ volatility
Figure 4-18: Asset liquidity in terms of liquidity efficient lines
As stated in the former section, one asset is considered to be more liquid than another asset if it leads to higher expected net receipts from sale and lower receipts’ volatility for any liquidation strategy. With the simultaneous liquidation of liquid assets, this definition requires that the liquidity efficient line of the first asset lies higher than the efficient line of the second asset. Since all efficient lines start at the point L and cannot intersect, it is always possible to unambiguously identify more and less liquid assets in this case. According to this modified notion of liquidity, asset X in Figure 4-18 is the most liquid one, and asset Z is the least liquid one. In general, it holds that the steeper the slope of the tangential line, the higher asset’s liquidity. In this sense, the slope can be treated as a liquidity measure. It is easily computed as: Slope =
E(Γ π *T ) − 1 S(Γ π *T )
(4.10)
Note that the slope of the efficient line corresponds with the Liquidity Risk Reward defined in section 3.4.1. This property makes LRR an especially appealing liquidity measure. Not only it can be interpreted in terms of a liquidity performance measure, but it also corresponds with the optimal liquidation strategy in the case of a simultaneous sale of a liquid and an illiquid asset. An important advantage of this approach is the independence from investor’s preferences – it allows comparing assets with respect to liquidity without the necessity of assuming the shape of seller’s utility function.
274 4.3.3.
Chapter 4: Liquidity as a Decision Criterion Liquidity of Assets in Portfolios
As discussed in section 4.2.3, a different liquidation strategy may be optimal when an asset is liquidated as a part of a portfolio than when it is liquidated on a standalone basis. It is therefore straightforward that also its liquidity may be different if other assets are sold at the same time. The general conclusions from the two previous sections retain their validity in this case. One portfolio can be viewed as more liquid than another portfolio if it offers higher expected receipts and lower receipts’ volatility for any combination of liquidation strategies (arrays of reservation prices). Furthermore, if a simultaneous sale of a liquid asset is allowed, a portfolio is more liquid if the tangential line corresponding with it is higher than the line of the other portfolio. Note that as soon as a portfolio of assets is considered, only the liquidity of the whole portfolio matters. It is pointless to compare expected receipts and receipts’ volatilities of its components because the reservation prices assigned to them have been optimized in the context of the portfolio and are not necessarily identical with those which would have been chosen if the assets were sold separately. Hence, due to the liquidity diversification effect, it is possible that a portfolio of highly illiquid assets is more liquid than some other asset, which, in turn, is more liquid than any single component of the portfolio. Nevertheless, it might be worthwhile to ask about the contribution of each of the assets to the liquidity of the whole portfolio, or, in other words, to ask how much of the portfolio’s total liquidity risk can be associated with each single asset. The answer would provide a different notion of liquidity than those regarded so far – it would refer to the relative liquidity within a group of assets. This approach can be interesting for investors willing to identify the sources of their portfolio’s (il)liquidity. The contribution of an asset to the liquidity risk of a portfolio is twofold: through the inherent liquidity risk associated with the asset when it is sold separately, and through its contribution to the liquidity diversification effect. The latter component is determined by the level of correlations between the sale receipts of the considered asset and the sale receipts of other assets in the portfolio. In this sense, relative liquidity risk in the portfolio context can be defined as the “stand-alone” risk reduced by the additional liquidity diversification caused by the assets’ inclusion in the portfolio. Hence, even if an asset has a high level of own liquidity risk, it still might prove to be relatively less risky due to the low correlations of its receipts with the receipts of other assets.
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An interesting possibility to translate the concept of relative liquidity within a portfolio into an operational measure is based on the Beta-coefficient from the Single-Index Model.428 Beta is defined as the coefficient in the regression of asset’s returns on the returns of some underlying index. It is interpreted as an indicator of the “systematic market risk”, i.e., it measures the undiversifiable market risk associated with the asset. In a more general sense, Beta measures the contribution of an asset to the risk of a perfectly diversified portfolio (market portfolio) containing also the analyzed asset. It is estimated as the covariance of asset’s returns with the returns of the market portfolio divided by the variance of the market portfolio’s returns.429 The reasoning behind Beta can be relatively easily applied to liquidity risk. In only requires substituting returns with sale receipts and market portfolio with the portfolio that is to be liquidated. The measure obtained this way expresses liquidity risk of a single asset in a portfolio in which all reservation prices have been set optimally. In this sense, Liquidity Beta (LBeta, Lβ) is a measure of the undiversifiable liquidity risk. However, contrary to the original market risk Beta, L-Beta refers only to a concrete portfolio of a concrete investor and as such is portfolio-specific. Applying the definition of Beta to sale receipts yields the following formula for the Liquidity-Beta of an asset i within a portfolio P: Lβi ,P =
cov(Γi , ΓP ) S2 (ΓP )
(4.11)
Since the asset i is a part of the portfolio P consisting of N assets, the L-Beta formula can be alternatively presented as:430 N
w i ⋅ S2 (ΓP ) + ∑ cov(Γi , Γj ) Lβi ,P =
428
429 430
j=1 j≠i
S (ΓP ) 2
(4.12)
See, e.g., Levy/Sarnat (1984), Chapter 10, Sharpe/Alexander (1990), section 9.1, or Elton et al. (2003), Chapter 7. See Levy/Sarnat (1984), p.429, Sharpe/Alexander (1990), p. 204, or any investment handbook. The transformation of (4.11) into (4.12), although not standard, is easily accomplished by presenting the portfolio return in the nominator as a weighted sum of components’ returns.
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The covariances of the asset i with other components of the portfolio can be computed within the real estate search model with the formula (4.7), and the volatilities of the asset i and of the portfolio P are given with the formulas (3.28) and (4.6), respectively. Note that L-Beta can be computed for any set of reservation prices – it does not necessarily need to correspond with an efficient E(Γ)-S(Γ)-combination. However, for practical application, it seems reasonable to limit the scope of considered liquidation strategies to the efficient ones. If a simultaneous liquidation of a liquid asset is allowed, there is only one efficient set of reservation prices – the tangential one. Thus, in the most general case of liquidation of a portfolio containing liquid and illiquid assets, LBeta should be computed on the basis of p*T. Only then its interpretation as a measure of assets’ relative liquidity risk in an optimally liquidated portfolio holds without additional assumptions about investor’s preferences.
4.4.
Portfolio Selection with Illiquid Assets
Former sections of this Chapter dealt with the problem of strategic liquidation, i.e., with the choice of a strategy (reservation price) that leads to the optimal outcome of the sale. It has been demonstrated that such a strategy depends not only on the characteristics of the liquidated asset but also on the circumstances of the sale. However, this issue is only relevant when the illiquid asset has already been purchased. The question whether such an asset should at all be included in a portfolio and in what proportion has not been addressed so far. Yet, this issue is highly relevant for most investors to which liquidity is one of the central investment goals (see section 1.4.2). Therefore, the problem of optimal portfolio selection with illiquid assets is addressed in the following subsections. The mean-variance framework presented in section 4.1 is a standard tool used for portfolio selection. Still, as discussed in section 4.1.4, the scope of its applications is limited due to strict assumptions. Many of them have been loosened in various extensions to the basic model proposed by numerous researchers. For example, transaction costs can be allowed for by correcting historical returns, and the problem of imperfect divisibility of assets can be coped with by using integer optimization.431 However, imperfect liquidity of assets still remains to a great extent an open issue. Some of the existing approaches are presented in the next section. However, as is seems, they do not 431
See literature references in FN 417 and 418.
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fully allow for the effects caused by the inclusion of illiquid assets in investment portfolios. A two-dimensional search theoretical approach offers a solution to this problem. In the course of the analysis, it turned out that different approaches are necessary when a “planned” and an “unexpected” liquidation is considered. While a correction of expected returns and return volatilities is sufficient in the first case, an extension of the decision framework with further variables is necessary in the second case. 4.4.1.
Literature Review
While literature on liquidity is generally extensive, there is surprisingly little research on the impact of asset illiquidity on portfolio selection. The existing approaches can be divided into three groups. The first and probably the largest one addresses illiquidity understood as “impossibility to sell”. However, since unsalability can be seen as an immanent characteristic of very few assets, practical relevance of these methods is limited. Approaches in the second group allow for trading illiquid assets but impose certain restrictions, usually with respect to the transaction size or the liquidation time. Although they use a more appropriate definition of illiquidity, they are mostly based on strongly simplified market models intended to demonstrate certain effects associated with the lack of liquidity. Thus, they are not applicable to concrete decision problems. Finally, the third group consists of approaches treating liquidity as a separate decision criterion; to some extent they also provide tools for practical implementation. Unfortunately, there are only few papers addressing portfolio selection with illiquid assets in this manner. Before reviewing relevant literature, it must be noted that a number of papers deal with related problems. In particular, some of the literature on portfolio selection with transaction costs refers to liquidity; the liquidity discount is then treated as a component of the total transaction costs.432 Since this approach is not quite in line with the notion of liquidity as a random result of a search process assumed in this work, this path of research is not addressed here. Also approaches referring to portfolio structuring with random cash demand are related to the problem of portfolio selection with illiquid assets.433 Liquidity is defined there as the “need for cash”, which is analogical to the definition of “corporate liquidity” in Chapter 1 (see section 1.1.1.3). Since the latter issue 432 433
See Pogue (1970), Constantinides (1986), or Grossman/Laroque (1990). See Chen et al. (1972 and 1975) or Thakkar (1976).
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has explicitly been excluded from the analysis, also this group of approaches remains disregarded.
4.4.1.1. Asset Allocation with on-Tradable Assets One of the first to discuss non-tradable assets in terms of capital assets pricing was Mayers (1972).434 He modified the standard Capital Asset Pricing Model (CAPM) to allow for assets that cannot be sold.435 Such assets, like human capital, constitute a fixed part of an investment portfolio. Since they yield returns which are not uncorrelated with the returns of marketable assets, a correction of the capital pricing formula is necessary. Not only the systematic risk arising from the (perfectly diversified) market portfolio has to be rewarded but also the risk of the non-marketable asset, which cannot be diversified in the usual manner. The following modified CAMP formula results:436 E(Ri) = RF + βi*·(E(RM) – RF) with: β∗i =
(4.13)
cov(R i , R M ) + ( PNM PM ) ⋅ cov(R i , R NM ) V( R M ) + ( PNM PM ) ⋅ cov(R i , R NM )
RNM - returns of the non-marketable asset PNM/PM - relation of the value of all non-marketable assets to the value of all marketable assets Brito (1977) further develops this concept and formulates the “three fund separation theorem”. According to it, each market participant, apart from holding the (corrected) market portfolio and the risk-free asset, which are identical for all investors, also holds a corrective portfolio of marketable assets outbalancing (“diversifying”) the effects of the non-marketable one. The main conclusion from this approach is that investors’ liquid portfolios differ in equilibrium even when they are facing the same universe of accessible liquid assets. Since each of them holds different non-marketable assets, their corrective portfolios need to be designed individually.
434 435 436
See also Mayers (1973 and 1976) See also Elton et al. (2003), pp. 321-323. See Mayers (1973), p. 266, and Elton et al. (2003), p. 322. Note, however, that the correction of the Beta factor in the presence of non-marketable assets is highly dependent on model assumptions. Stapleton/Subrahmanyam (1979) show for a range of utility functions that the degree of marketability has no effect on the price of risk or on the level of prices.
4.4 Portfolio Selection with Illiquid Assets
279
While the above discussed approaches consider the equilibrium state of a world with non-marketable markets, Brito (1978) concentrates on portfolio decisions of individual investors in this framework. He shows that rational capital allocation decisions need to be made in a 3-dimensional room on the basis of the expected returns and return volatilities of liquid (marketable) assets, and additionally on the basis of correlations of these assets with the illiquid (non-marketable) one (see Figure 4-19). However, in the presence of a risk-free interest rate (RR), the decision problem can be reduced to two dimensions. For a given level of the liquid-illiquid asset correlation (LIAC) it is optimal to choose the portfolio that maximizes the reward-to-volatility (RV), i.e., the ratio of the expected return to the return volatility; it is the portfolio X in Figure 4-19. Thus, the efficiency criterion can be formulated on the basis of the RV-ratios and LIAC only. For an individual investor, the choice of the optimal portfolio of liquid assets depends on the RV-LIAC-efficient frontier and on the RV-LIAC-preferences.
Figure 4-19: Portfolio decision room with a non-marketable asset according to Brito (1978)437
437
Modified after Brito (1978), p. 593. Symbols used in the original paper have been changed to avoid confusion with other variables in the book.
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Chapter 4: Liquidity as a Decision Criterion
Several other authors use analogical methods to analyze portfolio decision with nontradable assets; among them are Svensson/Werner (1993), Henderson/Hobson (2002), and Schwartz/Tebaldi (2006).
4.4.1.2. Models with Trade Restrictions Approaches discussed in the former section, although a step in the right direction, do not reflect the liquidity problem usually encountered in reality. Generally, illiquid assets can be traded, but the trade is possible either with a delay or at a discount. It is more realistic to assume that an illiquid asset purchased today is restricted from selling only for a certain period of time. Selected works following this idea are presented in this section. Longstaff (2001) analyzes a continuous time model in which an investor divides her wealth between a risk-free asset and a risky one. While the value of the first one is constant, the value of the second one is random with stochastic volatility. The investor maximizes the utility of her wealth. Since the volatility is not constant in this setting, the optimal portfolio weight of the risky asset changes over time, so that it is optimal to trade as frequently as possible. Liquidity is introduced by restricting the number of shares of the risky asset that can be traded. In this situation, the initial portfolio choice is of high significance as later adjustment possibilities are limited. The effects of liquidity are modelled numerically. The analysis shows that the optimal share of the risky asset is smaller when liquidity constrains are introduced. The deviation from the unconstrained case (perfect liquidity) is larger when the trading limit is smaller, when market volatility is less stable, and when investor’s time horizon is shorter. Longstaff also documents that trading restrictions may lead to substantial value discounts of up to 80%.438 Kahl et al. (2003) consider portfolio decisions in a three-asset universe.439 An investor can allocate wealth in a risk-free bond or a risky stock market; both investments are perfectly liquid. Apart from it, she is given a certain amount of an illiquid stock at a time zero (“today”), which she cannot sell until some future point in time. During this period, however, the illiquid stock can be traded by others, so that its price, like the 438 439
See also Longstaff (1995). The model by Kahl et al. (2003) can be viewed as a more general version of the model with nonmarketable assets by Henderson/Hobson (2002).
4.4 Portfolio Selection with Illiquid Assets
281
prices on the liquid stock market, follows Brownian motion. Furthermore, returns of the liquid market and the illiquid stock are correlated, and the investor is allowed to take unlimited short positions in all liquid investments. Her goal it to choose the share of wealth invested in the liquid stock market and the level of consumption that maximize her total (power law) utility. The optimal solution depends on the assumed values of model parameters, in particular, on the level of risk aversion, the volatilities of the risky assets, and the time horizon. The correlation between the liquid stock market and the illiquid stock is of particular importance in this setting; it can be expressed as the beta-coefficient of the illiquid stock. High beta-values (either negative or positive) allow the investor to hedge the risk of the illiquid stock by taking opposite, if necessary short, positions in the stock index (see Figure 4-20). This result is in line with the conclusions formulated in the literature on portfolio optimization with non-marketable goods – the inability to liquidate an asset, either permanent or temporary, influences the selection of liquid assets, as this is the only way to hedge risks that cannot be avoided by terminating the illiquid investment. Furthermore, Kahl et al. illustrate on the basis of their model, how trade restrictions can lead to substantial reductions of stock’s values.440
Figure 4-20: Optimal portfolio weight of a liquid stock as a function of the illiquid fraction for various levels of the illiquid stock’s beta in the model of Kahl et al. (2003)441
440
441
See also Silber (1991) for a study on discounts on restricted stocks and the impact of illiquidity in this case. Kahl et al. (2003), p. 401.
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Chapter 4: Liquidity as a Decision Criterion
The model of Longstaff (2004 and 2005) is a revised version of Kahl’s et al. (2003) model. Two players are assumed to be on the market: a patient one with a lower discount rate, and an impatient one with a higher discount rate. These investors can allocate capital between two assets: a liquid one and an illiquid one. Like Kahl et al., Longstaff assumes that the illiquid asset cannot be traded for a certain period of time; it experiences a trading “blackout”. Both assets yield dividends following geometric Brownian motions. Investors choose the shares of their total wealth invested in each asset by maximizing their initial utility from consumption. The main conclusion of the model is that portfolio structures change substantially as soon as one of the assets is assumed to be illiquid. With an increasing duration of the “blackout”, investors give up diversification and change to highly polarized portfolios; in particular, the impatient individual tends to allocate nearly all of her capital in the liquid asset. On the basis of this model, Longstaff illustrates how a temporary lack of marketability can lead to substantial discounts on illiquid assets, even if they are otherwise identical with the liquid ones. Further two models dealing with trade restrictions have been offered by Koren and Szeild. Koren/Szeidl (2001), like Longstaff, assume an economy with two assets: a liquid and an illiquid one. The liquid asset is instantly tradable at any point in time but yields a lower rate of return; the illiquid asset can be bought at anytime but selling it might be temporarily impossible due to the lack of buyers on the market (buyers are assumed to arrive according to a Poisson process). Furthermore, the investor experiences times of increased need for cash due to randomly occurring alternative investment possibilities – the inability to sell the illiquid asset in this situation may result in the loss of a profitable investment opportunity.442 The authors calibrated the model with reasonable parameter values and analyzed the optimal (utility maximizing) portfolio choice of the investor. They show that even a small increase in illiquidity, modelled through a lower arrival rate of buyers and longer average marketing periods, can result in substantial changes in the optimal portfolio allocation. For example, an increase of the average waiting period from 3 to 5 days led to an increase of the share of capital invested in the liquid asset by 4 to 6% points. Koren/Szeidl (2003) consider a slightly altered framework in which the illiquid asset can be liquidated only after a certain fixed lag from the moment of placing an order. The analysis of the model leads 442
In this respect, the model by Koren and Szeidl resembles portfolio selection models with stochastic cash demand; see Chen et al. (1972 and 1975) or Thakkar (1976).
4.4 Portfolio Selection with Illiquid Assets
283
to similar conclusions – portions of capital invested in the liquid and illiquid asset respond strongly even to slight changes in the grade of liquidity. Additionally, a positive dependence between investor’s risk aversion and the share of wealth invested in the illiquid asset as well as between investor’s impatience (discounting factor) and the share of wealth invested in the liquid asset could be stated; long horizon investors should hold larger shares of the illiquid asset.
4.4.1.3. Liquidity as an Independent Decision Criterion Approaches presented above give interesting insights in the nature and the effects of liquidity, but they are only of subordinate relevance for practitioners willing to optimize their portfolios with respect to this factor. In fact, only very few researchers address the subject of practical portfolio selection with imperfect liquidity. Sharpe/Alexander (1990, p. 233-236) propose an extension of the CAPM to allow for liquidity as an additional characteristic determining expected returns of assets in efficient financial markets. They state that, like in the case of market risk, only the marginal contribution of a security to the liquidity of an efficient portfolio should affect expected returns. A security market plane rather than a line would result from this modified version of CAPM. However, liquidity (defined only briefly as the cost of selling or buying “in a hurry”) is in this approach only an example of an additional return-determining characteristic that modifies the original CAPM in the direction of a multi-factor model like the one used in the Arbitrage Pricing Theory (APT). Among the first to propose the inclusion of liquidity as a separate decision criterion in the individual portfolio optimization framework was Schmidt-von Rhein (1996, pp. 333-336). He regards liquidity as the (relative) liquidation cost and suggests three methods how is can be implemented in the MPT. The first one is to treat it as a separate random variable. This approach is especially appropriate when the liquidation cost is uncertain and as such a source of risk. Schmidt-von Rhein proposes to use its expected value and variance as two additional decision parameters, but he notes that also other risk measures can be applied. The main difficulty in this approach is the necessity to define a multi-dimensional utility function that determines investor’s preferences about the trade-offs between the return-related and liquidity-related parameters. In view of the expected practical problems, the author advocates for the use of simplified methods. By assuming a constant liquidation cost, it is possible either to apply it as an
284
Chapter 4: Liquidity as a Decision Criterion
independent decision variable in a three-parameter portfolio optimization, or to subtract it from assets’ returns and apply the standard two-parameter optimization using net returns. Finally, the third possibility proposed by Schmidt-von Rhein is to define a side constraint to the standard optimization algorithm requiring that efficient portfolios have some minimum acceptable level of liquidity (maximum liquidation cost). Lo et al. (2003) propose a method to introduce simple measures of (public) market liquidity to the standard mean-variance portfolio optimization approach; the idea is in its core similar to the approach proposed by Schmidt-von Rhein (1996). A number of different measures can be used in this model, including trading volume, turnover, or bid-ask spread. The authors define a very general liquidity metric ℓ, which allows ~
normalizing the (arbitrary) underlying liquidity measure l . The metric for security i in month t is defined as follows: ~ ~ lit − min lkT k ,T l it = ~ ~ max lkT − min lkT k ,T
k ,T
~ ~ with: max lkT , min lkT k ,T
(4.14)
k ,T
- the maximal and the minimal value of the liquidity meas-
~ ure l computed over k securities in T months. For portfolios, the respective liquidity metric is defined as the weighted average metric of individual securities. Two further methods for measuring portfolio liquidity are also proposed: one appropriate for the case when short sales are allowed (absolute weights are used then), and one for the case when interactions or cross-effects among securities with regard to their liquidity occur. Having defined a portfolio liquidity metric, Lo et al. propose three ways to include it in the portfolio decision process. Liquidity filters are used in the first approach. Optimization techniques are only applied to portfolios with liquidity is greater than some arbitrarily specified threshold level ( l 0 ). Efficient portfolios have than a guaranteed minimum liquidity level, but they are not further differentiated above this level. An alternative approach is to impose an additional side constrain in the optimization problem. In this case, portfolios on the efficient frontier have exactly the required level of liquidity. Finally, in the third approach, liquidity is incorporated directly in the utility function. In this case, an additional parameter needs to be employed to define the
4.4 Portfolio Selection with Illiquid Assets
285
weight placed on liquidity by the decision maker. Lo et al. assume a linear preference with respect to the liquidity metric; different functional forms seem, however, equally feasible. The three approaches are demonstrated in an empirical example on the basis of selected US stocks. As illustrated in Figure 4-21, the results vary depending on the method used. However, even the simplest liquidity-based portfolio optimization procedure yields distinctly more liquid portfolio that the standard MV-technique. The method proposed by Lo et al. (2003), though undoubtedly improvable as noted by the authors themselves, is, to my best knowledge, the only one of this kind to be found in the literature.
a)
b)
Figure 4-21: Examples of liquidity-filtered (a) and liquidity-constrained (b) efficient portfolios according to Lo et al. (2003)443
Also the resent approach of Acharya/Pedersen (2005) is worth noting in the discussed context although the issue of optimal portfolio selection is not directly addressed there. The authors consider the influence of uncertain liquidity costs on the pricing of capital investments. They assume that the liquidity discount is random but correlated with the market-wide liquidity and with the returns of the asset and the market portfolio. On this basis, they derive a liquidity-adjusted version of the CAPM in which net returns after liquidity costs are considered. According to this model, assets’ expected excess returns are functions of the systematic market risk and the systematic liquidity risk. Acharya and Pedersen identify three types of liquidity risk associated with: (i) the commonality between asset liquidity and market liquidity, (ii) sensitivity of asset re443
Lo et al. (2003), pp. 25 and 35.
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Chapter 4: Liquidity as a Decision Criterion
turns to market liquidity, and (iii) sensitivity of asset liquidity to market returns. The first type of liquidity risk results from the empirically stated fact that there is a common factor affecting liquidity of securities on financial markets;444 the second type of liquidity risk is the effect of investors’ preference for securities with higher returns in times when market liquid is low; the source of the third type of liquidity risk is investors’ readiness to pay larger premiums for liquidity in times of low returns (i.e., in bull markets). In the adjusted version of the CAPM, each of these risks is associated with a specific Beta factor and priced separately. Though not explicitly stated by Acharya and Pedersen, their approach implies that individual investors diversify their portfolios not only with respect to market risk (on the basis of return correlations) but also with respect to the three types of liquidity risk. The latter can be diversified due to the commonality in liquidity, the relation between securities’ returns and market liquidity, and the relation between securities’ liquidity and marker returns. Thus, efficiency of assets needs to be considered with respect to net returns, and the efficient frontier results from the optimization with respect to all sources of risk. With a risk-free asset, which is also free of any liquidity risk, the optimal portfolio corresponds with the tangential line connecting the risk-free rate of return with the liquidity-adjusted efficient frontier – this line is described by the Acharya’s and Pedersen’s liquidity-adjusted CAPM. 4.4.2.
MPT with Illiquid Assets
The approaches presented in the preceding sections can be considered as steps in the direction of the actual goal of this Chapter, i.e., the development of a practicable method allowing rational portfolio decisions when investing in illiquid assets. However, in view of the considerations about the nature of liquidity conducted in earlier chapters, they do not seem to capture all aspects of the problem. For instance, liquidity risk is to a large extent omitted in the model by Lo et al. (2003) – the measures employed by these authors focus mainly on the expected liquidity. In contrast, Acharya/Pedersen (2005) disregard the possibility that the investor could be forced into a premature liquidation of her portfolio. Furthermore, practically all approaches concentrate only on liquidity of organized public markets. The following two sections demonstrate how these problems can be dealt with on the basis of the search theoretical approach. The 444
See Chordia et al. (2000 and 2005), Hasbrouck/Seppi (2001), or Huberman/Halka (2001).
4.4 Portfolio Selection with Illiquid Assets
287
situation of a planned liquidation, which is in fact similar to the one considered by Acharya/Pedersen (2005), is discussed first. The analysis of rational decisions when an unexpected liquidation becomes necessary, which is somewhat related to the approach of Lo et al. (2003), follows. In general, the considerations presented in the following sections can be viewed as further development and application of the ideas formulated by Schmidt-von Rhein (1996, pp. 331-336).
4.4.2.1. Planned Portfolio Liquidation In the planned liquidation case the investor does not expect to sell any of the assets in the portfolio before the assumed time horizon elapses, or she is simply not concerned about such possibility. In this case, the liquidation problem arises at the same moment as the realization of returns. Hence, returns are directly affected by the outcome of the sale process. Due to this temporal coincidence, there is no need to consider liquidation separately – since the investor is only interested on the total effective return, it is sufficient to concentrate on market returns corrected for liquidation effects. The usual MV optimization techniques can be applied for portfolio selection. The main problem in the considered case is the determination of expected returns, return volatilities, and correlations that are necessary to apply the portfolio optimization algorithm. Unlike in the case of perfectly liquid investments, using historical market data will yield only biased results for illiquid assets. This is because the historical returns, especially when computed as average returns over multiple transactions, do not contain the effects that result from the search for a trading partner. For illiquid assets these effects may occur both at purchase and at sale. Pooled historical data or market indexes provide information about average prices achieved by market participants in the past; however, as demonstrated in Chapter 2, by searching strategically, an individual may achieve (expected) purchase expenditures or sale receipts differing from the average market valuation. Furthermore, each historical transaction price or index value represents only one possible outcome of the random search process. It is treated as known and certain from the ex post perspective, but it is uncertain when viewed ex ante, i.e., before the search is accomplished. Hence, the total uncertainty about the rate of return comprises of both the uncertainty about the future market situation and the uncertainty about the effective expenditures and receipts realized at the beginning and at the end of the investment. Therefore, historical volatility underestimates the true volatility experienced by the investor. In consequence, also correlations with other as-
288
Chapter 4: Liquidity as a Decision Criterion
sets are biased when based on historical data. The variability of the search outcome superposes the variability of market values adding an additional source of noise. Correlations between effectively realized returns are therefore lower than correlations between market returns. In order to demonstrate how return statistics can be corrected for search effects typical ~
for illiquid assets, corrected (effective) returns are denoted as R while index returns (or market returns) are denoted as R. Furthermore, total index returns are split in the appreciation component RApp and the income component RInc. For convenience, all considerations refer to real estate. This allows the application of the real estate search model and the formulation of an explicit solution; generalization on other classes of illiquid investments is straightforward. Moreover, it is assumed that only purchase and sale prices are affected by search effects; renting income is identical for all market participants at all times. The starting point for the derivation of corrected measures of expected returns, risks, and correlations for illiquid property investments is the definition of the total return in period t:445
G − Ε t −1 + CFt G CF ~ Rt = t = t + t −1 Ε t −1 Ε t −1 Ε t −1 with: Et-1 Gt CFt
(4.15)
- effective expense at purchase in period t-1. - effective sale receipts in period t. - cash flow (net operating income) between t-1 and t.
Thus, the return from an investment is a compound effect of the random expense at purchase (Et) and the random receipts from sale (Gt). In terms of the relative search model, Et can be expressed as the average market valuation or the expected offer E(P) multiplied with the relative expense Ξt, and Gt is equal to E(P) multiplied with the relative receipts Γt. Since both Ξt and Γt are functions of the respective reservation prices, so is the expected effective return in t:
445
The use of discrete instead of, otherwise standard, logarithmic returns is due to computational problems arising in the latter case. The logarithmic return cannot be presented as a sum of the appreciation return and the income return: this makes the application of the search model to the appreciation component difficult.
4.4 Portfolio Selection with Illiquid Assets
⎛G ⎞ ⎛ E (P t ) ⋅Γ t ⎞ CF CF t ~ E (R t ) = E⎜⎜ t + t − 1⎟⎟ = E⎜⎜ + − 1 π ∗S, t , π ∗B, t −1 ⎟⎟ Ε Ε E ( P ) E ( P ) ⋅ Ξ ⋅ Ξ t −1 t −1 t −1 t −1 t −1 ⎝ t −1 ⎠ ⎝ ⎠ ⎛ 1 ⎞ E(P t ) ⋅ ⎛ Γ t ∗ ∗ ⎞ CF t = E⎜⎜ πS, t , π B, t −1 ⎟⎟ + ⋅ E⎜⎜ π∗S,t , π∗B,t −1 ⎟⎟ − 1 E (P t −1 ) ⎝ Ξ t −1 ⎠ E (P t −1 ) ⎝ ⋅ Ξ t −1 ⎠
289
(4.16)
Note that the ratio of the average price level in t to the average price level in t-1 corresponds with the index-based appreciation return, and the ratio of the operating income to the average price level in t-1 corresponds with the index-based income return. Substituting for these variables and omitting the reservation prices’ conditions for better tractability yields: ⎛ Γ ⎞ ⎛ 1 ⎞ ~ ⎟⎟ − 1 E(R t ) = (1 + R App,t ) ⋅ E⎜⎜ t ⎟⎟ + R Inc,t ⋅ E⎜⎜ Ξ ⎝ t −1 ⎠ ⎝ ⋅ Ξ t −1 ⎠
(4.17)
It is apparent from this presentation that not only the appreciation return but also the income return needs to be adjusted. Assuming that in the typical case the search for the best seller leads to a purchasing expense below the average market level (and thus below E(P)) and the search for the best buyer leads to sale receipts above the average market level, both components of the effective return are higher than those assessed on the basis of average market prices. The expected effective return over the whole time horizon is estimated as the average return from all periods: ⎛ Γ ⎞ ⎛ 1 ⎞ ⎞ 1 N ⎛ ~ ⎟⎟ − 1⎟ E (R ) = ∑ ⎜⎜ (1 + R App, t ) ⋅ E⎜⎜ t ⎟⎟ + R Inc, t ⋅ E⎜⎜ ⎟ N t =1 ⎝ ⎝ Ξ t −1 ⎠ ⎝ Ξ t −1 ⎠ ⎠
(4.18)
The adjustments of further statistics are done in an analogical manner. The variance of the effective return is composed of the fluctuations of the index return and of the fluctuations of the expenditures/receipts realized at purchase/sale:
290 ~ ~ ~ V(R ) = E(R 2 ) − E 2 (R ) =
Chapter 4: Liquidity as a Decision Criterion N 1 N 1 ~ ~ ⋅ ∑ E ( R 2t ) − ⋅ E 2 (R ) = N −1 N −1 N − 1 t =1
N ⎡ ⎛ Γt ⎞ ⎛ 1 ⎞ ⎛ Γ2 ⎞ 2 ⎟ ⎜ ⎟ ⎜ ⋅ ∑ ⎢(1 + R App,t ) 2 ⋅ E⎜⎜ 2t ⎟⎟ + R Inc , t ⋅ E⎜ 2 ⎟ + 2 ⋅ (1 + R App , t ) ⋅ R Inc , t ⋅ E⎜ 2 ⎟ Ξ Ξ ⎢ t =1 ⎣ ⎝ Ξ t −1 ⎠ ⎝ t −1 ⎠ ⎝ t −1 ⎠
(4.19)
⎛ Γ ⎞ ⎛ 1 ⎞ ⎤ N ~ ⎟⎟ + 1⎥ − − 2 ⋅ (1 + R App,t ) ⋅ E⎜⎜ t ⎟⎟ − 2 ⋅ R Inc,t ⋅ E⎜⎜ ⋅ E 2 (R ) Ξ Ξ N − 1 ⎝ t −1 ⎠ ⎝ t −1 ⎠ ⎦ Finally, following the same reasoning, the correlation coefficient between the effective returns of two illiquid assets can be defined as follows: ~ ~ ~ ~ ~ ~ cov(R A , R B ) = E(R A ⋅ R B ) − E(R A ) ⋅ E(R B ) =
N 1 N ~ ~ ~ ~ ⋅ ∑ E ( R A , t ⋅ R B,t ) − ⋅ E (R A ) ⋅ E(R B ) N − 1 t =1 N −1
=
N ⎡ ⎞ ⎛ Γ ⋅Γ 1 ⋅ ∑ ⎢(1 + R App,A ,t ) ⋅ (1 + R App,B,t ) ⋅ E⎜⎜ A ,t B,t ⎟⎟ Ξ ⋅ Ξ N − 1 t =1 ⎣⎢ ⎝ A ,t −1 B,t −1 ⎠
⎞ ⎛ ΓA ,t ⎟ + (1 + R App,A ,t ) ⋅ R Inc,B,t ⋅ E⎜⎜ ⎟ Ξ ⋅ Ξ ⎝ A ,t −1 B,t −1 ⎠ ⎛ ⎞ ⎛ Γ ⎞ ΓB,t ⎟ − (1 + R App,A ,t ) ⋅ E⎜ A ,t ⎟ + (1 + R App,B,t ) ⋅ R Inc,A ,t ⋅ E⎜⎜ ⎟ ⎜Ξ ⎟ Ξ ⋅ Ξ ⎝ A ,t −1 B,t −1 ⎠ ⎝ A ,t −1 ⎠ ⎛ 1 ⎞ ⎛ Γ ⎞ ⎟ − (1 + R App,B,t ) ⋅ E⎜⎜ B,t ⎟⎟ − R Inc,A ,t ⋅ E⎜⎜ ⎟ ⎝ Ξ A ,t −1 ⎠ ⎝ Ξ B,t −1 ⎠ ⎞ ⎤ ⎛ ⎛ 1 ⎞ 1 ⎟ + 1⎥ ⎟ + R Inc,A ,t ⋅ R Inc,B,t ⋅ E⎜ − R Inc,B,t ⋅ E⎜⎜ ⎟ ⎜ ⎟ ⎝ Ξ B,t −1 ⋅ Ξ A ,t −1 ⎠ ⎦⎥ ⎝ Ξ B,t −1 ⎠ N ~ ~ − ⋅ E(R A ) ⋅ E(R B ) N −1
(4.20)
All of the above formulas contain the ratio of the relative sale receipts to the relative purchase expense under the expectation operator, what makes them analytically unsolvable. However, an approximation is possible using a Taylor series representation of the expected value and the variance of a quotient of two random variables provided by Mood et al. (1974, p. 181). Limiting the series to the first three terms, what should be sufficient for most practical applications, yields the following formulas:
4.4 Portfolio Selection with Illiquid Assets
⎛ Γ ⎞ E (Γ ) cov(Γ, Ξ ) E (Γ) ⋅ V (Ξ ) − + E⎜ ⎟ ≈ E 2 (Ξ ) E 3 (Ξ ) ⎝ Ξ ⎠ E (Ξ )
291 (4.21)
2
V (Ξ ) cov( Γ, Ξ ) ⎞ ⎛ Γ ⎞ ⎛ E (Γ ) ⎞ ⎛ V (Γ ) ⎟ ⎟⎟ ⋅ ⎜⎜ 2 + 2 +2 V ⎜ ⎟ ≈ ⎜⎜ E (Γ ) ⋅ E (Ξ ) ⎟⎠ ⎝ Ξ ⎠ ⎝ E (Ξ ) ⎠ ⎝ E (Γ ) E (Ξ )
(4.22)
These formulas can be easily rearranged to provide estimations for the problematic terms in the equations (4.18), (4.19), and (4.20). The complexity of the above problem can be substantially reduced by slightly redefining the decision framework. The assumption that both the purchase price and the sale price are random corresponds with decision making in terms of abstract asset classes. In other words, the investor makes the decision of allocating capital in a certain asset class (a real estate submarket) before starting to search for concrete opportunities within this class. However, in practice, decisions are often made on the basis of concrete investment opportunities. After receiving a proposition of buying an investment at a certain price, the investor considers the effect of such a purchase for her portfolio and makes the decision on the basis of these considerations. In this framework, the purchasing expense is not random at the moment of decision making – it is known and can be expressed either in absolute units or as a fraction of the current average market valuation. The latter one can either be objective (i.e., based on perfect knowledge of the market) or subjective (i.e., resulting from investor’s personal market assessment). Only the exit side of the investment (i.e., the future sale receipts) is affected by uncertainty in this situation. If the relation of the initial purchase price to the average market valuation is denoted as ς, the expected return, volatility, and the covariance of the effective return can be redefined as follows: R ⎞ 1 N ⎛ (1 + R App,t ) ~ ⋅ E(Γ t ) + Inc,t − 1⎟⎟ E(R ) = ∑ ⎜⎜ ς ς N t =1 ⎝ ⎠
(4.23)
~ V(R ) =
2 N ⎡ (1 + R (1 + R App,t ) ⋅ R Inc,t R2 1 App ,t ) ⋅ E(Γt ) (4.24) ⋅ V(Γt ) + E 2 (Γt ) + Inc2 ,t + 2 ⋅ ⋅∑⎢ 2 N − 1 t =1 ⎣⎢ ς2 ς ς
− 2⋅
(
(1 + R App,t ) ς
⋅ E(Γt ) − 2 ⋅
)
R Inc,t ⎤ N ~ + 1⎥ − ⋅ E 2 (R ) ς N − 1 ⎦
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Chapter 4: Liquidity as a Decision Criterion
~ ~ cov(R A , R B ) = + − −
N (1 + R ⎡ 1 App ,A , t ) ⋅ (1 + R App,B, t ) ⋅ (cov(ΓA ,t ⋅ ΓB,t ) + E(ΓA ,t ) ⋅ E(ΓB,t ) ) ⋅ ∑⎢ N − 1 t =1 ⎣ ςA ⋅ ςB
(1 + R App,A ,t ) ςA ⋅ ςB (1 + R App,A ,t ) ςA R Inc,B,t ςB
+
⋅ R Inc,B,t ⋅ E(ΓA ,t ) + ⋅ E(ΓA ,t ) −
(1 + R App,B,t )
R Inc,A ,t ⋅ R Inc,B,t ςA ⋅ ςB
(1 + R App,B,t ) ⋅ R Inc,A ,t
ςB
ςA ⋅ ςB ⋅ E(ΓB,t ) −
⋅ E(ΓB,t )
(4.25)
R Inc,A ,t ςA
⎤ N ~ ~ + 1⎥ − ⋅ E(R A ) ⋅ E(R B ) ⎦ N −1
Since analytical formulas for all expressions in the above equations are available (see sections 2.3.2, 3.3.2, and 4.2.3.2), a quick and efficient computation is possible. In the considered case of a planned liquidation, return statistics are adjusted on the basis of concrete search strategies, which the investor plans to apply when purchasing and liquidating the asset. This means that the corrected expected returns, return volatilities, and correlations are valid only for the assumed set of strategies. Since the choice of the strategy depends on investor’s preferences, the resulting return statistics are only valid for the concrete investor. Thus, no objectively optimal portfolio can be designed.
4.4.2.2. Unexpected Portfolio Liquidation As discussed in section 1.4.2.2 in Chapter 1, illiquidity is problematic not only because of possible difficulties when purchasing and selling an asset according to the assumed investment strategy, but also, or even mainly, because of the impossibility of a quick liquidation in an emergency case. An investor experiencing unexpected liquidity problems and forced to sell a part of her investment portfolio in order to settle due payments is therefore usually better off using liquid rather than illiquid assets for this purpose. However, there are still strong arguments for investing in real estate and other illiquid assets – these include mainly advantageous risk-return profiles as well as contributions to the overall portfolio diversification. Thus, a rational portfolio selection strategy should consider the trade-off between the possibility of a quick liquidation without incurring major discounts and the advantages resulting from including an illiquid asset in the portfolio.
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The conclusions from the previous section still hold in this case. Expected returns, volatilities, and correlations of illiquid assets computed on the basis of average prices or indexes always contain biases resulting from the omission of search effects at the end of the time horizon. Thus, return parameters need to be adjusted in any case. However, even corrected returns are only valid when assets are held until the end of the planned investment horizon. Should an unexpected, premature liquidation be necessary, none of the computed statistics will hold. In such cases, the possibility of a favorable sale of a single asset or the whole portfolio at any time becomes relevant. Although the nature of the search problem is generally the same independent of the moment of the liquidation, the receipts from a premature sale cannot be combined with market returns due to the lack of time coincidence. This makes it necessary to consider asset liquidity in the case of an unexpected liquidation as a separate decision criterion in addition to the expected return and market volatility.446 The main practical problem when modeling an unexpected sale using a search model is the unknown liquidation time and, consequently, the unpredictable state of the market at this moment. While a forecast can be used in the analysis of a planned liquidation, this approach is rather inapplicable when no date can be defined to which the forecast would refer. Thus, the best available solution is to fall back on the relative search model discussed in section 2.3.1.6. By defining offers in reference to the current market price level, the influence of the market state (i.e., the average level of offers) is neutralized. Provided that other model parameters including the dispersion of offers, offer arrival frequency, and market trend remain stable, the results should be valid throughout the whole investment period. Furthermore, one should consider that investor’s time preference is different when the sale is due to unexpected liquidity problems than in the case of a planned and expected liquidation. Since the main priority is then to sell quickly, the investor calculates with a higher discounting rate. In its principles, the portfolio selection framework allowing for liquidity in terms of an unexpected sale is not very different from the “common” MPT selection framework. Like in the traditional method, the selection process can be split into two steps: identification of objectively efficient alternatives and choice of the alternative subjectively preferred by the investor. The only difference to the MPT is the larger number of decision variables. It leads, however, to difficulties with the application of existing optimi446
See Schmidt-von Rhein (1996), p. 333.
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zation tools. With respect to the efficiency principle, both the stochastic dominance and the parametric approach can be followed. According to the first degree stochastic dominance (FSD) criterion applied to market returns, one investment alternative dominates another alternative if for any return level the cumulative probability of achieving it is higher for the first alternative than for the second one. In this sense, efficient are only those alternatives which are not dominated by any other. Referring to asset liquidation, the FSD criterion defines an efficient strategy as the one for which no other strategy exists that would allow achieving any level of net sale receipts with a higher cumulative probability than the considered strategy. By combining these two definitions, a two-dimensional stochastic dominance criterion can be formulated: one investment alternative dominates another alternative if any combination of market returns and (relative) net sale receipts can be achieved with a higher cumulative probability for the first alternative than for the second one. The respective cumulative probability functions can be represented graphically as plains assigning each combination of returns and liquidation receipts the probability of its occurrence. Thus, one asset dominates another asset if the plain characteristic for the first one lies entirely above the plain characteristic for the second one. In this sense, asset X in Figure 4-22 dominates Y, but asset Z is not dominated by any other. In consequence, X and Z are efficient, and Y is inefficient.
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Cumulative probability
4.4 Portfolio Selection with Illiquid Assets
X Y Z
Ne t sa le r ec e ipts
t rke Ma
rn retu
Figure 4-22: Stochastic dominance with return and liquidity goals
As already discussed, the main problem with stochastic dominance is the required knowledge of the whole probability function, which is usually very difficult if at all possible to determine. Parametric approaches concentrating only on selected parameters of the probability distribution are therefore easier in application though less general.447 Expected returns and return volatilities are used in the Markowitz’s portfolio selection model. The simplest way to allow for liquidity in this framework is to extent the set of decision variables by an additional variable referring to assets’ and portfolios’ liquidity, as was done by Lo et al. (2003). The key issue in this approach is the choice of the adequate liquidity measure. As already discussed in Chapter 3, no single measure can fully capture all aspects of this problem. Therefore, any single measure can be only a “second best” solution. It seems that measures combining several aspects of the problem are best suitable for this purpose. Among them are the utility based measures as well as the performance measures. Since the first group requires the as447
For analogical considerations see Schmidt-von Rhein (1996), S. 333-334.
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sumption of a specific utility function, it is rather difficult to apply in a general model. Liquidity performance measures have the advantage of being objective, independent of investors’ individual preferences. Among them, the Liquidity Risk Reward is an especially appealing candidate for a decision variable. Due to its interpretation as the slope of the efficient line in the strategic liquidation of an illiquid asset together with a liquid one, it is a good measure of objective liquidity understood as the possibility of efficient liquidation.448 LRR is therefore used in the following considerations. The course of reasoning would, however, be the same if other measures were applied instead. The general principle of portfolio selection based on three decision variables is the same as in the traditional model.449 Efficient alternatives are identified first, and the optimal one is chosen from among them on the basis of investor’s individual preferences. The definition of an efficient portfolio is slightly modified and encompasses its LRR – a portfolio is efficient when no other portfolio exists having simultaneously a higher expected return, lower return volatility, and higher liquidity. Portfolios can be presented graphically as points corresponding with respective combinations of E(R), S(R), and LRR – the efficient ones form the efficient plain as presented in Figure 4-23. Since investor’s utility also needs to be defined with respect to all three variables, the levels of constant utility can be presented as indifference planes. The optimal portfolio results as the tangential point of the efficient plane and the highest possible indifference plane – it is the portfolio that maximizes investor’s utility with respect to all three decision variables.
448 449
See sections 3.4.1 and 4.3.2. Introduction of additional dimensions in the portfolio optimization problem is not new. Probably most discussed was the enhancement of the decision rule with further moments of the return distribution, especially the skewness; see, e.g., Samuelson (1970), Konno/Yamamoto (1993, 2005), Liu et al. (2003), or Füss (2004), pp. 374-410, and the literature cited there.. Also other goal variables are considered, e.g., the “ethicalness” of investments (see Beal et al., 2005).
4.4 Portfolio Selection with Illiquid Assets
297 Retu rn
Vola tility
Expected Return
Utility Indifference Planes Efficient Plane Optimal Portfolio
0 id Liqu
RR) ity (L
Figure 4-23: Efficient plane and portfolio selection with a one-dimensional liquidity goal
In the above three dimensional asset selection framework only illiquid portfolios having positive LRR values can be considered. Note that no perfectly liquid portfolios can be analyzed explicitly in this framework because the liquidity reward measure is not defined in such cases – both the numerator and the denominator of the LRR ratio are then zero. However, this does not mean that liquid assets are generally excluded from the analysis. As demonstrated in section 4.3.2, LRR is a good measure of liquidity especially in the case of a simultaneous sale of illiquid and perfectly liquid assets. It can be then interpreted as the marginal expected reward for accepting liquidity risk. Thus, the application of LRR implies that the investor considers selling both liquid and illiquid assets and is concerned about the optimality of the liquidation strategy in such case. Nevertheless, the impossibility of analyzing purely liquid portfolios and comparing them with illiquid ones is a serious limitation of this approach. Therefore, it seems to be useful only to those investors who definitely have illiquid assets in their portfolios and decide only about their optimal combination. Among others, most real estate funds and companies belong to this group.
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Using other liquidity measures, which assume finite values for liquid assets, may provide a solution to the above problem. However, new difficulties are likely to arise due to the general limitations of one-dimensional liquidity measurement as discussed in Chapter 3. Thus, three-dimensional optimization can only yield proper results in certain cases, especially when no perfectly liquid portfolios are allowed, or when the investor is not concerned about certain aspects of liquidity, e.g., liquidity risk. To allow for a more general approach, the concept of strategic asset liquidation presented earlier in this Chapter can be fully integrated in the traditional MPT framework. Combining these two approaches requires that the set of decision variables includes measures of expected liquidation outcome and liquidity risk in addition to the expected return and return volatility. Expected net sale receipts and receipts’ volatility can be used here, so that portfolio decisions are made on the basis of four variables: E(R), S(R), E(Γ), and S(Γ).450 Assuming that all rational investors prefer higher expected returns and higher expected net sale receipts to lower ones and prefer lower risks to higher ones allows defining the set of efficient alternatives. An efficient portfolio is in this context the one for which no other portfolio exists having simultaneously a higher expected return, higher expected sale receipts, lower return volatility, and lower receipts’ volatility. Due to the large number of variables, it is not possible to represent the so defined efficiency principle graphically. One can, however, imagine a four dimensional space in which each portfolio can be represented as a point corresponding with the respective combination of the four decision parameters. Efficient portfolios form a fourdimensional hyperplane, which is analogous to the efficient frontier in the standard model. Rational investors should choose only portfolios lying on the efficient hyperplane. Selection of a concrete portfolio requires the definition of a utility function, which also needs to be extended by two additional variables: E(Γ) and S(Γ). The indifference hyperplane consisting of combinations of the decision parameters leading to the same utility level is therefore also four-dimensional. The optimal portfolio corresponds with the single tangential point of the highest possible indifference hyperplane and the efficient hyperplane.
450
This approach corresponds with the proposition of Schmidt-von Rhein (1996), p. 333. Of course, other measures of liquidity risk can be used. In particular, it is possible to apply asymmetric measures defined in section 3.3.3.
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An advantage of this approach is the possibility of analyzing any combination of any assets. For portfolios consisting only of perfectly liquid ones, relative expected sale receipts are equal to 1 and receipts’ volatility is zero; thus, they are all located on a two-dimensional E(R)-S(R)-plane and correspond with the “liquid” efficient frontier. On the other hand, illiquid assets have S(Γ) values higher than zero; they should also have E(Γ) values higher than one as strategic choice of the reservation price should generally lead to higher receipts than using no strategy at all. Furthermore, if any arbitrary combination of liquid and illiquid assets can be liquidated, the corresponding graphical representation is a line connecting the “liquid” and “illiquid” points. A “flat” efficient hyperplane, analogical to the efficient line in the traditional MPT, results. The four-dimensional approach to portfolio selection is in line with the liquidity definition formulated in Chapter 1. It is very general and allows identifying efficient portfolios for all assets for which parameters of the search model can be assessed with sufficient accuracy. In particular, it allows an adequate analysis of portfolios containing direct real estate investments when the real estate search model is applied. The elimination of the limitations of the traditional MPT arising from the assumption of perfect liquidity of all assets is, however, achieved at the cost of increased model complexity. The lack of analytical solutions and the necessity of extensive numerical computations, high grade of mathematical complexity, and the impossibility of a graphical presentation of the outputs may make the intuitive tractability of the results difficult and, in effect, negatively affect the acceptance of the model. It seems that in many cases it is sufficient to use only one liquidity measure (e.g., LRR). Hence, the decision to use the full, four-dimensional model is, as in all cases of practical application of theoretical concepts, a trade-off between easy tractability, low-cost implementation, and adequate modeling of reality. 4.4.3.
Optimization Algorithms
From practitioners’ point of view, the possibility of a quick and effective determination of efficient portfolios is of key importance. A number of optimization algorithms allowing automated computation on the basis of easily accessible market data are available for the original MPT model. The Critical Lines Algorithm (CLA) is probably the most popular portfolio selection tool, but also alternative techniques applicable to
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different variations of the decision problem have been developed.451 They are usually based on quadratic programming and allow determining the composition of assets leading to the lowest possible volatility of portfolio returns at a given level of expected returns. Unfortunately, they are not directly applicable to the modified version of the model presented in the former section. The decision problem behind the efficient hyperplane can be formally formulated as follows:452, 453 N
N
N
i =1
i =1 j=1 j≠i
Minimize: V(R P ) = ∑ w i2 ⋅ V(R i p∗E,i ) + ∑∑ w i ⋅ w j ⋅ Cov(R i , R j p∗E,i , p∗E, j )
(4.26)
subject to constraints: N
E(R P ) = ∑ w i ⋅ E(R i p ∗E ,i ) i =1
N
E(ΓP ) = ∑ w i ⋅ E(Γi p ∗UE ,i ) i =1 N
N
N
i =1
i =1 j=1 j≠i
V(ΓP ) = ∑ w i2 ⋅ V(Γi p ∗UE,i ) + ∑∑ w i ⋅ w j ⋅ Cov(Γi , Γj p ∗UE ,i , p ∗UE , j ) N
∑ wi = 1 i =1
with: p ∗E ,i p ∗UE ,i
- reservation price applied in the planned liquidation of asset i - reservation price applied in the unexpected liquidation of asset i
Thus, the investor seeks a portfolio that allows her to achieve a minimal variance of returns at a given levels of expected returns, expected net sale receipts, and receipts’ variance. Alternatively, other parameters can be optimized; e.g., the variance of liquidation receipts can be minimized for given levels of the remaining three parameters.
451 452
453
For a review of portfolio optimization techniques see Rudolf (1994). This formulation is analogical to the standard one used in the traditional MPT. See, e.g., Levy/Sarnat (1984), pp. 341 f. The constraint requiring nonnegative share of each asset is often added; however, it is not necessary if short sales are allowed. Strategic purchase of assets is omitted for better tractability.
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Since the result is the same in each case, the choice of the optimization scheme is only the question of computational convenience. Independent of the optimization approach, the goal is to find a combination of strategic variables that leads to an efficient portfolio. However, while assets’ weights are the only variables in the original MPT, two additional variables for each asset need to be determined in the extended model: relative reservation prices applied in the case of a planned liquidation, and reservation prices applied in the case of an unexpected liquidation. They can but don’t necessarily need to be identical. In fact, since the time preference is usually different in these cases, it seems more likely that different liquidation strategies, and thus different reservation prices, are optimal. In effect, with N available investments the number of strategic parameters increases from (N-1) in the original portfolio selection framework to (3N-1) in the modified framework with strategic liquidation. This makes the optimization formula incomparably more complex. In particular, the analytical insolvability of the equations describing statistical parameters of sale receipts with respect to reservation prices (formulas 2.38, 3.28, and 4.7) results in the unavailability of a closed form solution for the optimal portfolio. In consequence, neither quadratic optimization used in the original portfolio selection model nor any other known technique can be applied to find a portfolio corresponding with a concrete combination of the decision criteria. Numerical approximations or Monte Carlo Simulations seem to be the only available methods. Summing up, there seems to be no straightforward method allowing a quick, effective, and precise identification of the efficient frontier when illiquid assets are included in the portfolio without sacrificing at least some of the key elements of the model developed in Chapter 2. However, in many cases the number of possible portfolios is limited. Since many illiquid assets have high unit values and are indivisible or divisible only with substantial difficulties, they are very often available only in large lumps. This applies especially to real estate but also to arts and many private equity investments. Hence, with limited funds at investors’ disposal, the share of illiquid assets of different types in the portfolio cannot be set at any arbitrary level. In consequence, only a limited number of asset combinations, and thus a limited number of feasible portfolios, are possible. Limiting the analysis only to realistic scenarios substantially reduces the range of numerical computations necessary for the estimation of the efficient hyperplane. The computational effort can be further reduced by introducing side
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constraints. This especially refers to the reservation prices. As demonstrated earlier in this Chapter, only a relatively narrow range of π*-values leads to efficient combinations of expected receipts and receipts’ volatilities. Reviewing reasonable ranges for each asset before running the optimization should help to keep the size of the simulation within reasonable limits. Thus, it should be possible to accomplish the analysis at a satisfactory precision level using a common personal computer in many practically relevant cases. 4.4.4.
Sources of Biases
Like all models that simplify reality, also the approach presented in this Chapter has its weak points. Some of the problems resulting from the assumptions made in the search theoretical model have been already discussed in section 2.3.3. On the other hand, the limitations of the MPT have been the subject of section 4.1.4. Obviously, they also apply to any investment decision framework building on these models. Although many of them can be solved or at least mitigated, some of them are still unavoidable. Identifying them is especially important for practitioners – the decision maker should be aware of the direction of possible biases that may result from the application of the model. Two possible sources of biases, which seem to play the largest role, are discussed in this section: those resulting from the assumptions met during the construction of the search model, and those resulting from the choice of volatility as the risk measure.
4.4.4.1. Effects of Search Model Imperfections The way in which correlations between sale receipts are computed in the search theoretical model is one of the main sources of biases in the composition of portfolios optimized on the basis of this model. This issue has already been discussed in section 4.2.3.2. The fact that market uncertainty factors (A) for different assets refer to possibly different durations of search makes their assessment on the basis of correlations between empirically observed prices or returns disputable. Unless average search durations are similar for both assets, this approach leads to a possibly substantial overestimation of the true receipts’ correlations, what may distort the optimal portfolio composition in several ways. Firstly, the volatility of portfolio liquidation receipts might be falsely estimated since the actual liquidity diversification effect is either larger (when the affected correlation is positive) or smaller (when the correlations is negative) than
4.4 Portfolio Selection with Illiquid Assets
303
assumed. Secondly, the volatility of portfolio returns may also be biased due to an incorrect correction of market returns for search effects in the case of a planned liquidation at the end of the time horizon. In effect, the method may tend to assess the quality of portfolios with large stakes of illiquid assets too pessimistically or too optimistically depending on the direction of the correlation between assets. Another problem related to the way market changes are incorporated in the model is its incompatibility with the random walk thesis (see section 2.3.1.5). This is not quite in line with the traditional portfolio theory, which requires that asset returns follow random walk. Since the search model is unsolvable if randomness in continuous returns is assumed, some inconsistency of the modified portfolio selection approach is unavoidable. The direction of the resulting bias is difficult to assess in this case. Simulations indicate that both the expected receipts and the volatility of receipts might be slightly lower if normal and not lognormal distribution of changes in the market price level is applied. In consequence, the model may suggest a slightly lower weight of illiquid assets in the optimal portfolio than actually justified. However, the difference should not be significant and of marginal practical relevance. The assumed equivalence between the average valuation of potential buyers, the average value of offers, and the presumable market price of the asset if it was liquid may also prove problematic. Obviously, as discussed in section 2.4, no perfect equivalence of this type is to be expected. When not all market traders are assumed to be naïve and at least some of them behave strategically, the observed transaction prices (and the hypothetical market price) should tend to be higher than the average of bids received by sellers and lower than the average of offers received by buyers. This misspecification can lead to an overestimation of the receipts from liquidation and possibly to a too high share of illiquid assets in the optimal portfolio. The deviation of the real bidding system from the one assumed in the model may have a similar effect. In particular, the existence of listing prices in most real estate markets should lead to lower sale receipts and higher purchase expenditures compared to the situation when investors do not reveal their preferences and decide only on incoming offers. The model might suggest a too high share of real estate in the optimal portfolio in this situation. Constant grade of time preference during the search is a further feature of the search model that may lead to biased results. As already discussed in section 2.3.1.3, the discounting rate will most probably increase as the asset remains unsold over a longer
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period of time. By ignoring the changes in the time preference, the (real estate) search model overestimates the gains from searching. Especially in the case of a forced liquidation, which is particularly relevant in the portfolio selection approach, a constant and certain discount rate is not quite realistic. Optimally, it should reflect the level of delay costs expected in such case. However, since the reason for the unexpected sale is unknown ex ante, also the appropriate discounting factor is uncertain. Ignoring this source of uncertainty in the model leads to an underestimation of the receipts’ volatility. The consequence might be a too high portion of illiquid assets in the portfolio.
4.4.4.2. Risk Measurement Issues Sources of biases may also lie in the statistics used as decision variables. This refers especially to the measures of market and liquidity risk. This issue has been already referred to in section 3.3.3. However, due to its relevance for the portfolio selection problem, it is necessary to stress the effects of using volatility (variance) in this context. As discussed in section 3.3.3.1, volatility is an adequate risk measure if either the investors understand risk as any deviation from the desired or expected outcome or if the risk variable is normally or at least symmetrically distributed. The first case seems to be an exception rather than a rule – most investors fear that the realized market returns and sale receipts could turn out to be lower than expected, but they would have nothing against unexpectedly high returns or receipts. Only seldom both too high and too low outcomes are regarded as negative. Also the shapes of the return and receipts distributions are problematic. While normal distribution of market returns is disputable for publicly traded stocks,454 it has to be clearly rejected for many other assets including real estate.455 Also the distribution of net sale receipts or purchase expenses is rarely normal (see sections 2.3.4 and 3.3.3.1). The deviation from normality is higher when the difference between the discounting rate, market’s growth rate, and offer arrival frequency is larger. Hence, especially in the case of an unexpected liquidation, in which the investor is under a high time pressure and calculates with a higher discount rate, non-normally distributed receipts can be expected. Moreover, simulations show that they tend to be right-skewed. 454 455
See references in FN 230. See references in FN 231.
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305
The above mentioned problems lead to the lack of conformity between the measured risk and the risk actually experienced by the investor.456 In consequence, optimization techniques lead to portfolios that are formally efficient but with respect to wrong goals. If the true probability distribution of market returns is left-skewed, variance underestimates risk, and with a right-skewed return distribution, risk is overestimated (see Figure 3-3). In effect, seemingly efficient portfolios contain too much or too little of the affected asset, respectively. Same holds for liquidation receipts. However, since in the most relevant case of a forced liquidation, the receipts’ distribution tends to be right-skewed due to the high discounting rate, variance can be expected to underestimate liquidity risk leading, in consequence, to a too high share of illiquid assets. The best solution to the above problem would be to replace variance with an asymmetric measure. This issue is well researched with respect to portfolio selection with liquid assets. Return volatility is usually substituted with semi-volatility or other lower partial moment. An alternative approach is the introduction of skewness as an additional (third) decision parameter.457 Analogically, asymmetric measures of liquidity risk discussed in section 3.3.3 can be used instead of volatility, or receipts’ skewness can be introduced; however, respective formulas or computation algorithms would need to be derived in both cases, what may prove difficult. Furthermore, including skewness as an additional parameter would substantially increase the number of decision variables and additionally complicate the portfolio selection process – with three return-related and three liquidity-related parameters, the decision room would become six-dimensional. Summing up, there are a number of reasons why a portfolio selection model with illiquid assets based on the search theory, and especially on the real estate search model, may yield biased and, in effect, suboptimal (at least partially) results. Most of the above discussed problems lead to a too high portion of illiquid assets. The extent of the distortions depends mainly on the discrepancy between the features of the market on which the investor is active and the assumptions of the model. While the application of the model to portfolios containing real estate should be possible with no or only slight errors, the inclusion of other illiquid assets can be highly problematic. In any case, a 456
457
There is a large amount of literature on the consequences of the incompatibility of variance as a risk measure for portfolio selection; see Ortobelli et al. (2005) for a review. See literature references in FN 449. There also exist approaches introducing higher moments (e.g., kurtosis) to the decision framework; see Breuer et al. (1999), pp. 177-217, for a review.
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detailed analysis of the grade to which model assumptions are fulfilled is necessary. Even then, however, some inaccuracies will be unavoidable; they are immanent to any theoretical approach that attempts to simplify reality. The question is therefore, how large errors are still acceptable. For more accurate results, one can attempt to develop a more realistic version of the search model, e.g., by implementing the propositions made in section 2.3.3. This, however, would lead to a substantial increase in the complexity of the approach. Thus, there is no easy solution to the trade-off between practicability and realism. *** Methods presented in this Chapter were intended to offer practicable tools allowing rational decisions on illiquid assets. The main innovation is the merger of the traditional mean-variance optimization framework with the search theoretical approach. This way, the advantages of search models in the analysis of illiquid assets can be implemented in the probably most popular decision technique allowing the optimization of the liquidation processes on the one hand, and the optimization of portfolios containing illiquid investments on the other hand. The price for the broader scope of application is, however, the high level of complexity. With the larger number of decision criteria and additional strategic variables, no simple optimization method can be applied. An analytical solution to the problem is also not in sight leaving numerical approximations and simulation techniques as the only possibility. Nevertheless, it seems that in many cases, especially in the analysis of direct real estate investments, these limitations can be overcome. The results, though not of point precision, should be acceptable for practitioners and in any case preferable to the results achieved with methods that ignore liquidity considerations. The next chapter offers a practical demonstration how these methods can be applied to real estate investment decisions.
Chapter 5 Liquidity of German Condominium Markets
A number of techniques for dealing with low levels of liquidity of privately traded assets, in particular real estate, have been developed in the previous chapters. They are based on the model of search for a trading partner, which allows a formal analysis of the selling or purchasing process and the resulting net receipts or expenses. Large parts of the discussion focused on the derivation of closed form solutions that could be applied in practice using data available to investors. However, most of the hitherto considerations were only theoretical. It is possible, and even probable, that in the course of their practical application to concrete problems obstacles occur that have not yet been addressed. This Chapter is intended to uncover and discuss some of such obstacles and to demonstrate possible ways of dealing with them. This is done by applying the search theoretical methodology to analyze liquidity of residential condominiums in selected German urban areas. This exemplary application should be viewed as a step toward the practical implementation of the concepts developed in this book. The main problem with the application of the search theoretical approach to real estate liquidity is the determination of the model parameters. Although it is relatively easy to define variables such as the “divergence of valuations” or the “frequency of offer arrivals” on the abstract level, it may be extremely difficult to assign them concrete values when addressing a specific real estate market. This issue is discussed in the first section of the Chapter. Further sections demonstrate the implementation of the model on German residential real estate markets. The data is presented in the second section – it consists mainly of large samples of transaction prices for residential condominiums in selected large German cities. The third section is devoted to the comparison of liquidity levels of the analyzed markets using different approaches from Chapter 3. The fourth section utilized the methods developed in Chapter 4 to derive optimal liquidation strategies as well as optimal portfolio decisions with respect to investments in the analyzed markets. Finally, the last section summarizes the main findings and offers a critical discussion of the methods.
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5.1.
Chapter 5: Liquidity of German Condominium Markets
Determination of Model Parameters
Since all liquidity measurement and liquidity management methods presented in this book are based on the search model, their application depends on the ability to determine the parameters of the underlying model. While some of them are relatively straightforward, other are not directly observable and need to be derived implicitly from the available data. The latter, by their nature, are always more or less approximate, what results in errors affecting the quality of the decisions made on the basis of the model. This problem is, however, immanent to any methods attempting to capture complex processes within simplified model structures. The question is therefore not how to entirely eliminate errors arising from the imperfect estimation of the parameters but rather how to minimize these errors. Yet, due to the novelty of the search theoretical approach presented here, no special error-minimization techniques are available, and developing such techniques is beyond the scope of this work. The section deals therefore mainly with the mere ability to determine, at least roughly, the values of the required parameters. There are six parameters in the relative real estate search model that are required for the computation of marketability and liquidity risk measures. These are: relative offer volatility (ν), frequency of offer arrivals (λ), market trend (τ), market trend volatility (σ), relative rental revenues (γ), and discounting rate (ρ). Furthermore, as soon as the purchase case is considered, values of the respective parameters from the buyer’s point of view are required. In the following, the possibilities of obtaining each of these parameters are discussed. Since the most severe problems are encountered with respect to the volatility of offers and the offer arrival frequency, separate subsections are devoted to these parameters. The remaining ones are far less problematic and are addressed in the last subsection. 5.1.1.
Volatility of Offers
The nature of the parameter ν has already been discussed in section 2.4 in Chapter 2. In general, it should correspond with the dispersion of valuations of the asset in question among potential buyers. Assuming that bids are placed in a random order makes the selling process similar to drawing randomly from the distribution of valuations. Hence, instead of determining the volatility of offers, it would be sufficient to determine the standard deviation of the values assigned to the property by potential buyers.
5.1 Determination of Model Parameters
309
Yet, obtaining such information may still be highly problematic. In fact, there is only little literature on the distribution of valuations for real estate.458 If the number of market participants is small, it is theoretically possible to estimate the valuation of the property by every single potential buyer. This may be the case, e.g., for specialized industrial properties, which only few investors can be interested in. In such cases, market participants may know each other well enough to be able to assess the value of the property for each of the parties on the basis of their individual situations, their corporate structures, or strategic orientations. On the other hand, however, it is questionable, whether it is at all worthwhile to apply search theory with such a limited number of bidders. In particular, the real estate search model from Chapter 2 is not suitable in such cases as it does not limit the maximal possible number of offers. A modification of the model allowing for a bounded search horizon (see section 2.3.3.1) would be necessary; this, however, would lead to the non-existence of a single reservation price and would significantly complicate the computation of statistical parameters for liquidation receipts. A more common situation, in which the application of the real estate search model is more suitable, is the sale (or purchase) of investments traded in markets with large groups of participants. Popular types of real estate, like residential or office properties, as well as many other illiquid assets (e.g., popular collectibles) belong to this category. In this case, the population from which the offers are “drown” is too large to review all its members and can be analyzed only by sampling. One way to estimate the distribution of buyers’ valuations is by questioning. A traditional survey is, however, not likely to provide satisfactory results in this case; one can hardly expect potentially interested investors to reveal truthfully their estimations of the property’s value. A possibly more accurate estimation of offer dispersion can be achieved on the basis of professional opinions. Experienced market players, like big real estate agents, are likely to be able to provide fairly precise statements on the distribution of valuations for certain types of properties in certain locations. This approach would be probably preferable for large real estate companies whose employees have expert knowledge about the markets on which they are active. However, expert opinions are far more difficult to obtain by new or less experienced investors. They can also be biased due to specific 458
See Merlo/Ortalo-Magné (2004) or Leung et al. (2006) for studies addressing the dispersion of real estate prices and valuations.
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Chapter 5: Liquidity of German Condominium Markets
fields of competence of the individuals or institutions who issue them. Thus, estimation based on objective empirical data might be preferable in some cases. The most straightforward way is to fall back on prices realized for comparable properties. Yet, the question arises in this case, to what extent realized prices reflect buyers’ valuations and their readiness to pay. According to the economic logic, a transaction can be closed only when the buyer’s valuation is not lower than the seller’s valuation; the transaction price lies between these two values and depends on the bargaining powers of the parties. In effect, the distribution of prices should tend to lie further to the right than the sellers’ valuations and further to the left than the buyers’ valuations.459 Thus, the estimation of the dispersions of valuations based on transaction prices may be biased. The extent of the bias depends mainly on the distance between the sellers’ and the buyers’ side of the market (market breadth). However, one could intuitively expect that the bias should not be large under normal market conditions and that in would increase in times of temporary market inequilibriums. Another issue that arises when the dispersion of valuations is estimated on the basis of transaction prices is the comparability of the latter. Theoretically, all prices should refer to properties identical with the analyzed one. However, heterogeneity is one of the main characteristics of real estate, and it is also distinct with respect to other privately traded assets. Full identity of properties is therefore practically impossible alone because of the uniqueness of the locations – no two properties can exist in one and the same place. Thus, when considering a specific property, only transaction prices achieved for similar objects can be regarded. Identifying a sufficient number of such comparable properties is not difficult when large amounts of data are available. However, this is possible with respect to only very few real estate markets in which transactions occur frequently and their outcomes are publicly known. Most property markets suffer from very low trading frequencies and poor data availability. In effect, the number of data points may be too small to allow a meaningful estimation of the valuation diversity when only comparable properties are selected from among the available ones. A possible way to cope with this problem is the application of hedonic techniques analogues to those used in the construction of real estate indices.460 By regressing the features of heterogeneous properties on the transaction prices achieved for them, not only the average contributions of these features to the total value but also their standard 459 460
See also the discussion in section 2.4 and Figure 2-6. See FN 267.
5.1 Determination of Model Parameters
311
deviations can be computed. The variability of prices for a hypothetical property with predefined characteristics can be estimated on the basis of these deviations. This approach allows the utilization of information contained in the prices of objects not directly comparable with the analyzed one, which would be discarded otherwise. The final possibility to estimate the volatility of offers is to build on the observed volatility of real estate appraisals or the appraisal errors. This approach is founded on the assumption that the majority of buyers and sellers acting in real estate markets judge the fair values of properties on the basis of opinions issued by appointed appraisers. In this sense, the distribution of valuations can be viewed as derivative to the distributions of appraisals obtained by market participants. It follows that the divergence of appraisals should be related to the divergence of sale or purchase offers received during the marketing process. This approach may, however, contain a substantial bias. Firstly, in many cases the valuation of a property depends on the possibilities of using it by the concrete individual or institution or on its “fit” in her existing portfolio. In contrast, appraisers are concerned about the average value achievable on the market. Secondly, many of the real estate transactions, especially the transactions for residential properties, are made (at least partially) on the basis of individual tastes, which are also not covered by appraisals. Hence, this approach can be useful only in real estate markets dominated by professional investors; in other markets it may lead to an underestimation of the true offer volatility.461 5.1.2.
Offer Arrival Frequency
The determination of the offer arrival frequency (λ) is insofar different from the determination of the dispersion of offers as it cannot be easily derived from a characteristic of buyers’ (or sellers’) population such as the distribution of valuations. The decision to bid on a property is a function of investor’s current situation as well as the effect of more or less random incidents (e.g., finding a sale advertisement). In effect, the probability of placing an offer can be hardly analyzed on the individual level, and the population of potential buyers needs to be considered as a whole. Furthermore, there is little sense in assessing the offer arrival frequency in markets with only few players. In such markets, like the already mentioned special industrial property market, offers 461
Empirical results on the divergence of appraisals or the appraisal errors (i.e., the deviations of appraisals from the subsequent transaction prices) lead to very different results ranging from 2% to 20%. See, e.g., Kain/Quigley (1972), Diaz (1997), or Graff/Young (1999).
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Chapter 5: Liquidity of German Condominium Markets
would most probably be placed all at the same time meaning that the search does not have a sequential character. Hence, the application of the real estate search model is pointless unless it is redefined in terms of a fixed sample search strategy (see section 2.2.2). This step would, however, lead to significant complications regarding the computation of liquidation receipts and other relevant variables. For this reason, the field of model application is limited to markets with large (practically infinite) numbers of participants. The most straightforward approach to the estimation of the offer arrival frequency is the analysis of this variable in past transactions on similar properties. Given a sufficient number of sales with full histories of bids that have been placed during the marketing periods, the average number of bids per unit of time or the average time between bids can be used as a proxy for the required model parameter. Of course, the quality of such a proxy depends on the comparability of properties and on the time period in which the sales were observed. The latter point requires particular caution; not only sales in the far past but also sales that occurred during, before or after important market events may not reflect the typical offer arrival frequency for the analyzed property. If these problems are given adequate attention, estimation on the basis of transaction histories should provide best results. However, the availability of such data may prove highly problematic. One would expect large players, like real estate companies, funds, or big brokerage houses, to have sufficient own transaction histories at their disposal to be able to use them for that purpose. In order to further improve the estimation of λ, a hedonic approach could be used - the number of offers per period could be regressed on the characteristics of properties in the databank. This way, all available information could be utilized without the necessity to discard noncomparable properties. However, transaction histories are usually proprietary, so that smaller players with less experience and no sufficiently large databanks on real estate transactions may have significant difficulties in accessing them. Only a very rough assessment of the offer arrival frequency on the basis of public information is possible in this case. There seems to be no straightforward solution to this problem; merely a number of rough hints for the approximate value of λ can be given. The simplest approach is to assess the relations between the respective parameters for different markets on the basis of trading intensities on these markets. It seems plausible that one property type that is
5.1 Determination of Model Parameters
313
traded twice as frequently as some other property type should also have approximately twice the number of offers per period. However, one should be aware that such a statement can be burdened with very high inaccuracy. On the one hand, the relation does not need to be linear – a complex analysis would be necessary to state its functional form. On the other hand, there are also a number of other factors affecting λ, like the marketing channels of the seller. Nevertheless, with no better alternative, this is probably the only approach allowing any approximate determination of the offer arrival frequency when lacking more precise transaction data. 5.1.3.
Other Parameters
The remaining model parameters are easier to estimate than the offer volatility or the arrival frequency. In particular, they can be estimated on the basis of publicly available data such as real estate indices. The main ones, such as the U.S. American NCREIF index or the European indexes provided by the Investment Property Databank (IPD), are subdivided into types of real estate as well as into appreciation and rental components. This allows an almost direct determination of both the expected price change (market trend, τ) and the volatility of price changes (volatility of the trend, σ) as well as the relative rental revenues (γ) for the required property type. A drawback of this approach is the relatively gross delimitation of real estate markets; indices usually differentiate only between the main property types (e.g., residential, office, retail, industrial) and between very broad geographic regions (e.g., countries in Europe or states in the USA). More accurate results can be possibly achieved by using regional indexes published for more narrowly defined markets; yet, it is also possible that these are based on smaller data sets, what would negatively affect their accuracy. The final parameter, the discounting rate ρ, is by its nature subjective and needs to be determined for each investor individually. When specifying its value, one needs to bear in mind that it depends on the character of the liquidation. In particular, one needs to differentiate between a planned and a forced liquidation, as discussed on several occasions in earlier chapters.462 In former case, an illiquid investment (property) is held until the end of its holding period; in the latter case, the liquidation is forced prematurely due to an unexpected liquidity event. The discount rate should be regularly higher in the second case. 462
See especially sections 1.4.2.2 and 4.4.2.
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Chapter 5: Liquidity of German Condominium Markets
To sum up, the determination of model parameters can cause serious practical problems, especially with respect to offer volatility and arrival frequency. These problems are especially severe for investors with poor access to detailed market information. In contrast, large market participants, who have been active on the analyzed market for a long time and have detailed histories of numerous transactions at their disposal, should have no problems with the practical application of the search model. It follows that the techniques of measuring and managing liquidity of real estate investments proposed in this book are most suitable for large institutional investors, like large real estate companies, real estate funds, or agents. Smaller market participants and investors who act on real estate markets only infrequently can be expected to have more difficulties achieving satisfactory results. But also in their case, an imperfect method of coping with illiquidity is still better than no method at all.
5.2.
Condominium Liquidity Analysis
In this section, German residential condominium markets have been chosen to demonstrate the application of search theoretical methods in practice. Basing on this example, two central questions are addressed: Do different measurement approaches lead to the same or at least similar classification of real estate submarkets with respect to their liquidity? And does the inclusion of a liquidity criterion result in a different optimal portfolio than the one that would be selected without liquidity considerations? A positive answer to these questions would mean that special attention needs to be given to the choice of the appropriate method when the real estate search model is to be applied for dealing with liquidity. In contrast, negative answers would indicate (but not prove) higher independence of the result from the chosen method of analysis. In view of different structures of the approaches presented in earlier chapter as well as different notions of liquidity underlying them the first possibility seems more probable. This expectation is also confirmed by the empirical analysis in this section. In the first subsection data material is presented. It consists mainly of information available publicly, though not free of charge. The second subsection deals with liquidity measurement. Selected methods based on the real estate search model and discussed extensively in Chapter 3 are applied to German condominium markets. Finally, portfolio optimization is conducted. The traditional mean-variance approach is extended to include the possibility of a strategic liquidation.
5.2 Condominium Liquidity Analysis 5.2.1.
315
Data Material
The empirical study is based on data referring to condominium markets in five large German urban areas: Cologne, Duisburg, Frankfurt, Hanover, and Stuttgart. It comes from two sources. One of them encompasses average residential condominium transaction prices assessed by brokers who are members of Immobilien Verband Deutschland (IVD) or Ring Deutscher Makler (RDM, before 2005).463 The other source consists of transaction prices for condominiums provided by German Appraisers’ Committees – Gutachterausschüsse für Grundstückswerte (GAA) – official bodies collecting, processing, and providing to the public information on local real estate markets.
5.2.1.1. RDM/IVD Data IVD and previously RDM is (and was) the biggest German association of real estate brokers with over 6000 members.464 Since 1971 it publishes a review of real estate prices (Immobilienpreisspiegel), which contains information on average rents and sale prices for residential and commercial properties in over 250 selected cities. The data is provided by member brokers as their assessments of average rents and price levels made on the basis of transactions by their clients.465 Since the study in this Chapter refers only to residential real estate, only this type of IVD/RDM data is used. All figures refer to a (hypothetical) standard condominium with 3 rooms and about 70 m² space. From about 350 towns included in the reports only Cologne, Duisburg, Frankfurt, Hanover, and Stuttgart have been selected. These are all large urban areas with populations exceeding 500,000, lying in different geographical regions of Germany, and having different economic backgrounds. For this reason, they can be regarded as separate markets. Furthermore, IVD/RDM differentiates rents and prices according to the quality of condominiums distinguishing good, medium, and poor quality. These categories include both a location component and a facility component; they can be used to delimit further submarkets. However, only the good and the medium category are considered in the following study; the poor quality category is ignored due to insufficient data. In effect, ten condominium markets are 463 464 465
IVD has emerged in 2005 as a merger of RDM with Verband Deutscher Makler (VDM). See http://www.ivd.de for further information. See comments to the IVD/RDM Preisspiegel (1972-2006).
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Chapter 5: Liquidity of German Condominium Markets
defined referring to two quality categories in each of the five cities. For simplicity, the categories are denoted as G and M for good and medium quality, respectively. The differentiation between age classes, which is also provided by IVD/RDM, is disregarded here – only condominiums constructed after 1949 but prior to 2006 (no new buildings) are considered. Rents and prices for the ten submarkets are available annually for the time period 1972-2006.
5.2.1.2. GAA Data “Gutachterausschüsse für Grundstückswerte” are appraisers’ committees founded on the basis of the German Construction Code (Baugesetzbuch, BauGB) and committed to ensuring transparency of real estate markets.466 Official regulations require that every real estate transaction is reported by the notary to the regional GAA. The report includes the price, the location, as well as the main characteristics of the property. On the basis of this information, GAAs fulfill their statutory obligations including especially the provision of summary information on land prices and other data necessary for the appraisal of properties in the region. Certain types of data are usually provided free of charge, while others, especially prices for comparable properties, are provided against a fee. The fact that GAAs collect all information about real estate transactions in their range of authority makes them the potentially most powerful source of information in the branch with probably no similar counterparty worldwide. The use of the vast databanks for commercial or scientific purposes is, however, limited for two reasons. Firstly, the data is scattered – theoretically, a separate GAA should exist in each of over 14,000 German towns and communities (Gemeinde). Not only there is no unified central databank system, but also the form of records differs from one GAA to another. Secondly, inquiries about transaction prices are usually charged on “per price” basis. Obtaining hundreds or thousands of prices, which would be necessary for a statistically significant parameter estimation of any model, even if it was granted by the GAA, would result in an enormous cost that would be probably only seldom justified. Hence, these enormous amounts of data are still, to a great extent, unavailable for market research. However, thanks to the kind assistance of the GAA Cologne, GAA Duisburg, GAA Frankfurt, GAA Hanover, and GAA Stuttgart summary statistics or lists of 466
See BauGB §§192 ff.
5.2 Condominium Liquidity Analysis
317
transaction prices for condominiums in these cities have been made available for this study.467 This unique set of data can be used for the estimation of search model parameters, in particular, for the estimation of the divergence of valuations among market participants. The data for Cologne encompasses summary statistics of condominium prices for the time period 1996-2006. Separate reports are available for each city district including the number of evaluated transactions, average prices, and standard deviations of prices. Estimation of the location quality was possible on the basis of general opinions about the qualities of the districts. The data for Duisburg consists of a selection of nearly 3300 condominium transactions between 1996 and 2006. Apart from transaction prices, construction years, location qualities, and condominium sizes are reported in each case. The data for Frankfurt encompasses the number of transactions in each year between 1996 and 2005 as well as annual means and standard deviations of transaction prices in years 2004 and 2005. The statistics are reported separately for poor, middle, and good location quality and refer to condominiums in building constructed between 1949 and 2005. The Hanover data is comprised of a selection of nearly 3000 transaction prices in the time period 1997-2006 together with the associated information on the construction year (being after 1949), condominium size, average land price in the respective location, and the quality of facilities (being poor, middle, or good). Finally, the Stuttgart data includes a list of nearly 14,000 transactions for condominiums constructed between 1949 and 2005 and sold in the time period 1995-2005. Apart from transaction prices the data includes assessments of the flats’ locations and facilities’ quality, as well as sizes; the quality information is reported in form of points that sum up a number of various features. In order to ensure the compatibility of GAA data with IVD/RDM data, two submarkets have been defined for each of the five urban areas: a submarket for medium quality condominiums and a submarket for good quality condominiums. The classification was made on the basis of all available quality information. For Cologne, Duisburg, and Frankfurt, it was limited to the location quality only; for the Hanover and Stuttgart also the quality of facilities has been included.468 Furthermore, for the computation of the 467
468
The author would like to express his gratitude to the GAA Frankfurt, GAA Hanover, and GAA Stuttgart for providing data for this study free of charge. In Hanover, location quality has been qualified on the basis of land prices associated with the locations of properties – the upper 1/3 quantile has been qualified as good, the middle 1/3 quantile as
318
Chapter 5: Liquidity of German Condominium Markets
offer volatility in Duisburg, Hanover, and Stuttgart, the data was filtered to include only condominiums with sizes between 50 and 90 m², which is approximately -/+ 30% of the standard size of 70 m² to which the IDV/RDM data refers. In consequence, the amount of data for these cities was reduced to ca. 2300, 1900, and 4300 records, respectively. Due to the lack of sufficient information, no such adjustment could be made for Cologne and Frankfurt. In addition to the information on condominium transactions, broadly defined turnover data for local condominium markets in 2005 or 2006 (depending on the availability) was used.469 It originated from the publicly available property market reports published by the GAAs in the analyzed cities.470 On this basis and on the basis of the information about the populations of the cities in 2005/2006, the average number of transactions per inhabitant was calculated. The values for different quality categories were determined according to the proportions in which different qualities were present in the samples of the transaction data received from the GAAs. Estimated numbers of sales of different condominium types per inhabitant were applied to estimate offer arrival frequencies. It is apparent from the above description of the data that no perfect comparability of information on condominium markets in the five analyzed cities can be ensured. In the first place, the delimitation of the sub-markets was arbitrary and partially conducted on the basis of different quality criteria, i.e., a “good condominium” in Frankfurt is most likely not the same as a “good condominium” in Stuttgart. Secondly, the sample sizes differ strongly; while only 2 years are included in Frankfurt, 11 years are considered in Stuttgart. Finally, the types of condominiums included in the samples are slightly different; e.g., the adjustment for the condominium size could be made for Duisburg, Ha-
469
470
middle, and the lower 1/3 quantile as poor. The combination of location quality and facilities’ quality to obtain the overall condominium quality followed the following scheme: good + good = good, good + middle = middle, good + poor = middle, middle + middle = middle, middle + poor = poor, poor + poor = poor. For Stuttgart, quality information was available in “quality points”; the overall condominium quality was computed as the average of the location quality and the facilities’ quality and translated into good, middle, and poor according to the upper, middle, and lower 1/3 quantile of the resulting average points. Although the detailed transaction data provided by the GAAs also included numbers of transactions, the figures referred apparently to sample sizes rather than to total turnovers. In order to ensure comparability of the data, it was necessary to fall back on more general figures published in the GAAs’ annual property market reports. See: GAA Cologne (2006), GAA Duisburg (2006), GAA Hanover (2006), GAA Frankfurt (2006), and GAA Stuttgart (2006).
5.2 Condominium Liquidity Analysis
319
nover, and Stuttgart, but it was not possible for Cologne and Frankfurt. In effect, one has to bear in mind that the criteria according to which markets are delimited in this study are not fully consistent. Such problems are, however, typical to real estate markets and are encountered both by researchers and practitioners alike. Property markets, and especially residential property markets, are local by their nature and the criteria of their classification are always defined with respect to the local conditions or customs. Therefore, the data is used “as it is” without an attempt to unify it. Slightly more disturbing is the lack of full consistency between GAA and RDM/IVD data. In particular, the former source includes real transactions on real properties while the latter one refers to an abstractly defined standard condominium. Unfortunately, this problem can be hardly coped with without significant reductions of sample sizes (especially the lengths of returns time series), so that the resulting biases have to be accepted. However, since both sources of data refer in the core to the same condominium markets, the biases should not be higher than in typical studies of real estate investments.
5.2.1.3. Determination of Model Parameters The data presented in the former subsection was used to determine the parameters of the real estate search model and the portfolio selection model for the ten property submarkets delimited on the basis of location and quality criteria. On the one hand, RMD/IVG data was applied to determine return statistics (expected returns and return volatilities) as well as long term price trends (τ), uncertainty about the trends (σA), and relative rental revenues (γ). On the other hand, GAA data was used to estimate offers volatilities (ν) and offer arrival frequencies (λ). Returns have been defined as total returns encompassing the rental and the appreciation component. The following return formula was used:471
471
The application of discrete rather than continuous (logarithmic) returns, which are otherwise more common in finance, is dictated by the methods applied in the analysis of a planned strategic liquidation in section 5.2.3.1. A correction for search effects would be impossible with log-returns.
320 R=
Chapter 5: Liquidity of German Condominium Markets Pt − Pt −1 + 12 × H t Pt −1
with: Pt Ht
(5.1)
- price in the year t - monthly rent in the year t
On the basis of total returns, expected values, standard deviations, and correlations were calculated; they are necessary for the application of the standard portfolio selection model. The results are summarized in Table 5-1.
Cologne M
Duisburg G
Duisburg M
Frankfurt G
Frankfurt M
Hanover G
Hanover M
Stuttgart G
Stuttgart M
E(R) [in %]
7.25
7.78
6.08
6.31
7.66
7.42
6.62
7.10
6.69
8.27
S(R) [in %]
17.17
15.94
9.76
11.64
17.30
9.60
17.41
11.11
13.02
21.05
0.46
0.34
0.10
0.29
0.21
Parameter
Cologne G
Table 5-1: Return statistics for selected German condominium markets
Correlations Cologne G Cologne M Duisburg G Duisburg M Frankfurt G Frankfurt M Hanover G Hanover M Stuttgart G Stuttgart M
1.00
0.62 1.00
0.42
0.51
0.35
0.30
0.28
0.17
0.59
0.14
0.27
0.38
0.36
1.00
0.60
0.13
0.34
0.34
0.08
0.49
0.33
1.00
0.24
0.44
0.12
0.11
0.62
0.45
1.00
0.20
0.76
0.10
0.26
0.09
1.00
-0.03
0.46
0.74
0.67
0.08
0.05
-0.07
1.00
0.41
0.28
1.00
0.75
1.00
1.00
The time series of average condominium prices per square meter were also used to determine the expected trends of liquidation prices (τ) and the trend uncertainty parameters (σA) required in the search model. In each case, the former parameter was calculated as the average continuous appreciation return ln(Pt)-ln(Pt-1), and the latter was defined as the standard deviation of the appreciation returns. Furthermore, relative rents γ were estimated as average ratios of rental revenues to condominium prices (12×H/Pt-1).
5.2 Condominium Liquidity Analysis
321
The remaining search model parameters were estimated on the basis of GAA data. The volatility of offers was approximated by the average annual dispersion of transaction prices. In the first step, standard deviations of transactions in each sub-market were calculated for each available year; they were averaged in the second step yielding an estimation of ν. Estimation of offer arrival frequencies proved to be most difficult. Since neither average times between offers, nor average numbers of offers during a liquidation process were available for the analyzed markets, the choice of this variable’s values had to be arbitrary. One offer per day, or 365 per year, was assumed as the starting point.472 This value was associated with the condominium sub-market exhibiting the highest activity, which was the Hanover M (medium quality) market. For other markets, the arrival frequency was assumed to be proportional to the market activity measured as the approximate annual number of transactions per inhabitant. The result is, of course, only a very rough estimation, highly dependent on the accuracy of the initial “guess” and the grade of linearity in the relationship between λ and the number of transactions per capita. Such a rough approximation is sufficient for the exemplary computations intended in this Chapter; for serious practical applications, however, a more precise method would be necessary. As discussed in section 5.1.2, it should be preferably based on long term experiences of professional investors. Discounting factors (ρ) were set arbitrarily at the level of 5% for the planned liquidation and 25% or 50% for the forced (unplanned) liquidation. Table 5-2 summarizes the estimations of the main search parameters for the analyzed condominium sub-markets.
472
The base arrival frequency of one offer per day is roughly based on the experiences of several interviewed real estate brokers in large German cities and refers to good quality condominiums in preferred locations.
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Chapter 5: Liquidity of German Condominium Markets
Stuttgart M
Stuttgart G
Hanover M
Hanover G
Frankfurt M
Frankfurt G
Duisburg M
Duisburg G
Cologne M
Cologne G
Parameter
Table 5-2: Parameters of the search model for selected German condominium markets
Τ
1.6%
1.9%
1.2%
0.8%
2.2%
2.4%
0.2%
0.9%
1.7%
1.9%
σA
17.6%
16.0%
9.6%
10.6%
14.8%
8.8%
16.0%
10.0%
12.5%
19.5%
γ
4.2%
4.7%
4.4%
4.9%
4.3%
4.6%
5.1%
5.7%
4.2%
4.4%
ν
32.7%
26.1%
23.4%
32.4%
31.1%
36.5%
22.6%
24.7%
18.3%
19.6%
λ
121.67
244.33
19.31
99.11
24.07
197.82
92.87
365.00
148.87
296.44
As already mentioned, the precision of estimations could be improved if more precise data was available. The main problem is the imprecise estimation of the offer arrival frequencies. Since the following study is conducted mainly for the purpose of demonstrating the developed methods rather than for analyzing concrete markets, these issues are not of crucial importance here. However, in a serious practical application, a more thorough analysis would be necessary at this point. Only then the result could be expected to provide good rationale for investment decisions. Due to the novelty of the approach, no specialized software was available for the search theoretical analysis. Therefore, Microsoft Excel® was used in most cases. Routines for the necessary calculations, in particular algorithms for calculating expected values, variances (standard deviations), and covariances (correlations) of net sale receipts, were programmed in Visual Basic for Applications (VBA) attached to the Excel package. Wherever numerical estimation or optimization was necessary, the included Solver routine was used. This was necessary, in particular, for the determination of optimal reservation prices and for portfolio optimization in the last section. Thus, the precision of the results depends on the effectiveness of the Solver. In this context, it must be noted that this software conducts only local optimization, which is highly dependent on the starting variable values. Unfortunately, it was not possible to verify whether all solutions were also global optima, so it is theoretically possible that a recalculation with different initial values could lead to different results. The general conclusions from the analysis should, however, not be affected by these minor inaccuracies.
5.2 Condominium Liquidity Analysis 5.2.2.
323
Liquidity Measurement
The following section deals with the measurement of liquidity for the residential condominium markets delimited and described earlier. Although it is highly interesting whether and to what extent these markets differ in this respect, the main purpose of the section lies in the comparative analysis of different measurement approaches developed and presented in earlier chapters. In total, eight different measures are tested, representative for each of the groups discussed in Chapter 3. Their comparison is based, on the one hand, on the implications regarding the nature of trading in respective markets and, on the other hand, on the relative classification of the markets with respect to their liquidity.
5.2.2.1. Market Depth Probably the most popular measure of market depth and simultaneously the simplest indicator of market liquidity is the average trading volume observed within a certain time window. However, as already discussed in section 3.1.2.3, trading volume or turnover may yield misleading results when applied to real estate markets. Large transaction sizes, which are the effect of poor divisibility of properties, may lead to immense trading volumes caused by only few trades. In fact, however, there may be only few players active on such markets making them very shallow and their liquidity comparably poor. This effect may also occur in residential property markets, although it is probably less severe there than in the case of more specialized properties like industrial real estate. Thus, it seems that the number of transactions is a much better liquidity indicator in this case. More frequent transacting implies a larger number of interested buyers and sellers and seems to be more suitable to estimate the chances of a successful liquidation. Yet, it must be noted that the numbers of transactions in different real estate markets are only comparable if the considered markets are of similar sizes. If it was not the case, the figure would misleadingly tend to indicate higher liquidity of more broadly delimited markets – e.g., there are obviously more condominium transactions in London than in Frankfurt, but it does not necessarily mean that the London condominium market is more liquid. Hence, it is the transaction intensity rather than the absolute number of transactions that is relevant for liquidity. It can be defined as the ratio of the realized property transactions in a certain market to the total number of buyers and
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Chapter 5: Liquidity of German Condominium Markets
sellers willing to transact. Unfortunately, the number of properties available for trading during the considered time interval is not directly observable in most cases. The second best solution would be the utilization of the total value or the total number of properties of a certain type in the considered geographical submarket, but even this kind of data is rather difficult to obtain and possibly instable over time. The third possibility, which is conceivable for residential real estate, is to fall back on the population of the considered region. A close relation between the population and the local demand for living space seems plausible. Thus, the number of transactions per inhabitant can be considered as a proxy of the transaction intensity. One should note, however, that biases may result from differences in the local social structures. With the same total population, the demand for high quality condominiums, and consequently their liquidity, can be expected to be higher in wealthier regions. However, this effect should be of only marginal importance for the cities analyzed in this Chapter. Since they are all large urban areas in western Germany, the social structures should not differ significantly. Hence, estimated numbers of transactions per inhabitant are used as the first liquidity indicator. They are summarized in Table 5-3.
Cologne M
Duisburg G
Duisburg M
Frankfurt G
Frankfurt M
Hanover G
Hanover M
Stuttgart G
Stuttgart M
Number of transactions
Cologne G
Table 5-3: Estimated number of transactions per inhabitant in selected German condominium markets
231
464
37
188
46
376
176
693
283
563
Most striking in the above results are high differences between the transaction intensities in the analyzed markets. In particular, it is possible to classify the cities in two separate groups with distinctly different levels of transactions per inhabitant: the first one includes Duisburg and Frankfurt and the second one includes Cologne, Hanover, and Stuttgart. This may be the effect of the differences in the social and/or economics structures, but it may also be due to the specific delimitation of the sub-markets.
5.2.2.2. Market Breadth The Implicit Spread and the Quick Sale Discount, both based on the real estate search model, have been proposed as measures of breadth for real estate markets in sections
5.2 Condominium Liquidity Analysis
325
3.1.2.1 and 3.1.2.2, respectively. The first measure is computed as the difference between the maximum selling price and the minimum purchasing price valid for a hypothetical risk-neutral dealer. The purchase price offered by such a dealer (bid) would not be higher than the minimum expected expenditure achievable under the optimal search strategy; analogically, the sale price (ask) would not be higher than the maximum expected liquidation value. This logic implies that the hypothetical dealer is not under time pressure when purchasing or liquidating according to the optimal strategy. Hence, also his opportunity cost should not be high, and the discounting rate at a lowrisk investment level seems appropriate; the rate of 5% has been chosen for this purpose. While the Implicit Spread takes both sides of the market into account, the Quick Sale Discount refers to the sellers’ side only. It attempts to answer the question: how much of the asset’s value the investor is expected to lose by selling quicker than optimal? A “quick sale” is defined in terms of the search model as a sale accomplished at an infinite negative reservation price, or in other words, a sale to the fist interested buyer encountered in the liquidation process. The so defined discount is highly sensitive to the situation of the investor, in particular, to the pressure under which she is selling. Allowing for alternative scenarios, three different (annual) discounting rates are applied: 5% for a fully flexible investor with no pressing need to sell, 25% for an investor under moderate pressure, and 50% for an investor who is likely to face severe consequences if the asset is not sold promptly. The computations of the measures are conducted according to the formulas (3.10) and (3.12), respectively. The results, including reservations prices for the sale and for the purchase case, are summarized in Table 5-4.
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Chapter 5: Liquidity of German Condominium Markets
1.05
1.74
1.52
2.23
1.19
1.51
1.06
1.22
45.7
35.4
47.0
45.4
54.3
37.6
44.1
35.7
39.4
QSD (ρ = 25%) in %
39.5
36.5
22.5
38.4
30.6
44.5
29.1
36.3
26.3
30.2
QSD (ρ = 50%) in %
Stuttgart M
1.59
48.4
Stuttgart G
1.83
QSD (ρ = 5%) in %
Hanover M
Implicit Spread
Hanover G
Duisburg G
Frankfurt M
Cologne M
Frankfurt G
Cologne G
Duisburg M
Table 5-4: Market breadth measures for selected German condominium markets
36.0
33.5
17.8
34.8
25.6
41.3
25.7
33.6
23.2
27.4
π*: Sale
1.94
1.84
1.55
1.89
1.83
2.19
1.60
1.79
1.55
1.65
π*: Purchase
0.11
0.25
0.50
0.15
0.31
-0.04
0.41
0.28
0.50
0.43
The immense dimensions of the spread are striking at first sight – they lie between one and two times the average valuation of the condominium. This is obviously the result of very high sale reservation prices and very low purchase reservation prices that allow the hypothetical dealer achieving respectively high expected liquidation receipts and low expected purchase expenses. In the extreme case of the medium quality condominiums in Frankfurt, it is optimal for the dealer to buy at a negative price and to sell at over the double of the average valuation. This result is caused, on the one hand, by the high dispersion of valuations (historical transaction prices) and, on the other hand, by the low time preference (discount rate = 5%). In effect, the reduction of the liquidation value incurred when postponing the sale is practically compensated by the rental revenues earned in this time. Hence, as long as the dealer is risk neutral and concerned only about expected values, he would act rationally by waiting patiently for an extremely good offer rather than selling or buying at an average price. He would have no incentive to quote narrower spreads than those presented in Table 5-4. This result may seem absurd at first sight, especially when compared with the dimensions of spreads in financial markets. Nevertheless, even if lacking realistic interpretations, these figures give a good impression of how illiquid direct real estate investments in fact are. Furthermore, it becomes increasingly clear that liquidity risk cannot be ignored even by a very large investor (dealer), so that using only the Implied Spread, in which risk neutrality is assumed, leads to a biased picture of the true liquidity of real estate investments.
5.2 Condominium Liquidity Analysis
327
The results for the Quick Sale Discount confirm the conclusions from the analysis of the Implied Spread. At the discount rate of 5%, quick selling leads to losses of up to 50% compared with the value that could be realized with the strategy maximizing the expected net receipts. The discount decreases, however, when a higher pressure to sell is assumed. This is understandable – higher opportunity costs faced by the investor in such cases decrease the value of search, so that a quick liquidation is comparably cheaper. Yet, even with the discounting rate as high as 50%, the discounts are still very high reaching up to 40%. This means that even if waiting leads to a rapid loss of value, as it could be expected, e.g., in an insolvency case, it is still suboptimal to sell without a thorough search for a buyer. However, also this measure does not take liquidity risk into account. The “optimal sale strategy” is here the one that maximizes the expected net receipts; yet, it is possible that the “quick sale” and the “optimal sale” involve utterly different levels of uncertainty.
5.2.2.3. Time on Market Representative for the group of time- and probability-based measures, the expected duration of the liquidation process is used. It corresponds with the Time-on-theMarket; however, while ToM is usually estimated empirically, the measure used in this section is implicit and computed on the basis of the search model. It is defined as the reciprocal of the probability of receiving an acceptable offer (i.e., one that exceeds the reservation price) times the expected time between offers, which equals the reciprocal of the offer arrival frequency (see formula 3.17). The main problem with the computation of ToM is the choice of the adequate reservation price. According to the proposition of Lippman/McCall (1986), the price leading to the maximum expected net receipts should be used. However, considerations regarding liquidity risk have shown that also other reservation prices may be rational, like e.g., the one leading to the minimum liquidity risk or the one for which a subjectively optimal combination of expected receipts and risk is achieved. To avoid these problems, two versions of the measure were computed: the reservation price maximizing the expected net receipts was used in ToMmax, and the reservation price minimiz-
328
Chapter 5: Liquidity of German Condominium Markets
ing the receipts volatility was used in ToMmin. The results are summarized in Table 55.473
Cologne G
Cologne M
Duisburg G
Duisburg M
Frankfurt G
Frankfurt M
Hanover G
Hanover M
Stuttgart G
Stuttgart M
Table 5-5: Expected ToM of condominiums in selected German condominium markets (days)
ToMmax (ρ = 5%)
1454
2372
1978
1204
4063
3327
1008
1439
2016
2697
ToMmin (ρ = 5%)
43
35
100
67
78
68
42
45
40
25
ToMmax (ρ = 25%)
129
105
161
132
182
130
110
95
93
88
ToMmin (ρ = 25%)
38
30
74
49
70
45
36
31
31
22
ToMmax (ρ = 50%)
68
54
90
71
100
67
59
49
48
44
ToMmin (ρ = 50%)
28
21
51
32
52
28
26
19
20
16
In the above results, the large discrepancies between the expected ToMs computed at minimum and maximum reservation prices are particularly surprising. According to them, an investor acting without time pressure (under 5% discount rate) and following the strategy that maximizes expected net sale receipts should expect to be searching for a buyer for several years - the extreme value is reached for medium quality condominiums in Frankfurt with over 10 years expected search duration. On the other hand, following the risk minimizing strategy leads to expected durations of the liquidation process of only few months. Both ToMmax and ToMmin decrease rapidly when opportunity costs increase. In most cases they are below three months and one month, respectively, when a discount rate of 50% is assumed. Even under such extreme time pressure, however, it is still not purposeful to shorten the liquidation below two weeks independent of the chosen strategy. High dependence of the expected ToM from the assumed reservation price makes it very difficult to compare the liquidity of property markets on the basis of this measure. To illustrate this problem, ranges of liquidation times resulting for rational sale strategies have been depicted in Figure 5-1. Not only the measured ToM values but also the assessments of markets’ relative liquidity are subject to the assumed reservation prices.
473
Reservation prices applied in the computation of TOMmax and TOMmin are identical with those used for the computation of the minimum and maximum receipts’ volatilities summarized in Table 5-6.
5.2 Condominium Liquidity Analysis
329
Only in extreme cases and only for high discounting factors some of the condominiums are clearly more liquid than the others. For example, good-quality condominiums in Stuttgart always have a shorter expected liquidation period than respective condominiums in Frankfurt at ρ=50%. In most cases, however, no such clear cut relation can be observed. Hence, also with respect to this measure, no simple, onedimensional delimitation of more and less liquid markets is possible. Contrary to the opinion of Lippman/McCall (1986) who proposed the expected duration of search as an economically well founded operational liquidity measure, it proves to yields only vogue results when merely the maximization of expected receipts but not the uncertainty about the outcome of the liquidation is taken into account.
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Chapter 5: Liquidity of German Condominium Markets
4000 3500 Cologne G Cologne M
3000
Expected ToM
Duisburg G 2500
Duisburg M Frankfurt G
2000
Frankfurt M Hanover G
1500
Hanover M Stuttgart G
1000
Stuttgart M 500 0 1,21
1,33
1,45
1,57
1,69
1,81
1,93
2,05
2,17
ρ=5%
Reservation Price
200 180 Cologne G
160
Cologne M
Expected ToM
140
Duisburg G Duisburg M
120
Frankfurt G 100
Frankfurt M
80
Hanover G Hanover M
60
Stuttgart G Stuttgart M
40 20 0 1,16
1,23
1,30
1,37
1,44
1,51
1,58
Reservation Price
1,65
1,72
1,79
ρ=25%
5.2 Condominium Liquidity Analysis
331
120
Cologne G
100
Expected ToM
Cologne M Duisburg G
80
Duisburg M Frankfurt G 60
Frankfurt M Hanover G Hanover M
40
Stuttgart G Stuttgart M 20
0 1,08
1,15
1,22
1,29
1,36
1,43
1,50
1,57
1,64
Reservation Price
ρ=50%
Figure 5-1: Expected ToM of selected German condominium markets (days)
5.2.2.4. Liquidity Risk One of the main messages of this book is the necessity of including risk considerations in the liquidity analysis, especially when highly illiquid assets like real estate are considered. Hence, this section is devoted to the application of several types of liquidity risk measures to the German condominium markets. In the first place, the volatility of net sale receipts is computed for each submarket. This measure is also central for the analysis in the following sections. Additionally, two asymmetric risk measures are considered: default probability, i.e., the probability of achieving a liquidation value below some target value, and semivolatility of net liquidation receipts, i.e., the lower partial moment of grade 2. Formula (3.28) was used for computing volatility. Like in the case of the Quick Sale Discount and the expected ToM, the results highly depend on the reservation price assumed in the search model. Depending on investors’ preferences, different alternatives are feasible. While some investors may strive for high expected sale outcomes, more risk-averse individuals may prefer to keep the risk low. Each group would choose a different search strategy and obtain different values of the receipts’ volatility.
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Chapter 5: Liquidity of German Condominium Markets
In order to allow for this fact, two extreme cases were analyzed – on the one hand, reservation prices minimizing the volatility of net sale receipts were used, on the other hand, reservation prices maximizing the expected receipts were applied. These two values can be considered as the upper and the lower limit for liquidity risk involved under rational liquidation strategies. Furthermore, different grades of time pressure were allowed for with discount rates ranging from 5% to 50%. The results are summarizes in Table 5-6.
Vmin (ρ = 5%) Reservation Prices Vmax (ρ = 5%) Reservation Prices Vmin (ρ = 25%) Reservation Prices Vmax (ρ = 25%) Reservation Prices Vmin (ρ = 50%) Reservation Prices Vmax (ρ = 50%) Reservation Prices
Stuttgart M
Stuttgart G
Hanover M
Hanover G
Frankfurt M
Frankfurt G
Duisburg M
Duisburg G
Cologne M
all figures in %
Cologne G
Table 5-6: Volatility of net receipts on selected German condominium markets
0.14
0.10
0.12
0.13
0.16
0.13
0.10
0.09
0.07
0.08
1.48
1.45
1.21
1.52
1.27
1.70
1.30
1.50
1.28
1.32
1.41
1.70
0.72
0.69
2.16
1.47
0.74
0.68
0.99
1.97
1.94
1.84
1.55
1.89
1.83
2.19
1.60
1.79
1.55
1.65
0.14
0.10
0.12
0.13
0.16
0.14
0.10
0.09
0.08
0.08 1.31
1.46
1.43
1.15
1.46
1.24
1.64
1.28
1.46
1.26
0.20
0.15
0.15
0.17
0.21
0.18
0.14
0.12
0.10
0.12
1.65
1.57
1.28
1.62
1.43
1.80
1.41
1.57
1.35
1.43
0.14
0.11
0.14
0.14
0.17
0.15
0.11
0.10
0.08
0.08
1.41
1.38
1.08
1.39
1.17
1.55
1.23
1.40
1.21
1.28
0.18
0.13
0.16
0.17
0.20
0.18
0.13
0.12
0.10
0.10
1.56
1.50
1.19
1.53
1.32
1.70
1.34
1.50
1.30
1.38
As it seems, the minimal volatility remains to a large extent unchanged for different discounting factors, but the maximal volatility reacts much stronger to the time pressure. Especially for low opportunity costs, the uncertainty associated with the liquidation according to the reservation price maximizing expected net receipts is far above the respective minimal value. The discrepancy is smaller when higher discount rates are assumed. The conclusion can be drawn from this result that the minimal volatility is a more robust measure of liquidity risk; it is less sensitive to the investor’s personal situation. Yet, in certain situations, e.g., for less risk-averse individuals acting under small time pressure, it may be only a biased indicator of the true uncertainty. Attempt-
5.2 Condominium Liquidity Analysis
333
ing to increase the expected receipts from sale, such investors are likely to run risks that are far above the minimum risk level. A methodical problem related to the use of volatility or variance is the symmetric definition of risk implied by these statistics. This may lead to a false interpretation of the chance of achieving unexpectedly high receipts as undesired by the investor. In order to control for this possible problem, two alternative asymmetric measures were computed. Unfortunately, as discussed in section 3.3.3, there is no easy way of determining these measures analytically; hence, a simulation-based estimation was necessary. A liquidation scenario with parameters as they were applied in the search model was set up and re-run 10,000 times. The structure of the simulation corresponded with the one proposed in section 2.3.4. The distribution of randomly generated net liquidation receipts was obtained as a result. Due to the large scope of computations, no optimizing with respect to the reservation price could be conducted; i.e., no minimum value for the default probability or semivolatility was available. Instead, the volatility minimizing reservation price was used to estimate the minimum values of the measures. Parallel, the reservation price maximizing the expected net receipts was applied to obtain the maximum values. While only negative deviations from the expected receipts were included in the computation of semivolatility, the probability of default was additionally computed with regard a fixed target value of 1, which corresponds with the average valuation of condominiums on the market. The results are summarized in Table 5-7.
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Chapter 5: Liquidity of German Condominium Markets
Stuttgart M
Stuttgart G
Hanover M
Hanover G
Frankfurt M
Frankfurt G
Duisburg M
Duisburg G
all figures in %
Cologne G
Cologne M
Table 5-7: Liquidity risk measures for selected German condominium markets
Def. Prob. (t=1)
1.1
1.9
3.8
0.2
11.2
0.8
0.7
0.1
2.2
4.7
Def. Prob. (t=E(Γ))
55.3
57.3
51.1
54.0
55.2
53.7
54.2
54.9
54.6
57.3
Semivolatility
24.9
27.2
24.3
14.8
67.8
28.4
14.6
11.2
18.2
33.3
Def. Prob. (t=1)
0.0
0.0
0.1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Def. Prob. (t=E(Γ))
59.5
60.7
58.1
60.3
57.7
61.3
60.1
60.9
60.0
59.5
Semivolatility
6.7
5.0
6.5
6.7
8.4
6.8
4.9
4.4
3.7
3.8
Def. Prob. (t=1)
0.0
0.0
0.3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Def. Prob. (t=E(Γ))
60.6
60.8
57.6
60.3
58.2
60.8
59.5
60.7
60.1
61.0
Semivolatility
7.0
5.2
7.2
7.0
8.6
7.2
5.0
4.6
3.9
3.9
ρ = 50%
ρ = 25%
ρ = 5%
E(Γ) maximizing reservation price
ρ = 50%
ρ = 25%
ρ = 5%
V(Γ) minimizing reservation price Def. Prob. (t=1)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Def. Prob. (t=E(Γ))
60.4
60.6
59.7
60.7
59.3
60.9
60.5
60.5
61.0
61.3
Semivolatility
7.1
5.2
6.3
6.8
8.4
7.0
5.3
4.6
3.9
4.0
Def. Prob. (t=1)
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Def. Prob. (t=E(Γ))
60.3
60.3
59.4
60.2
59.6
60.3
60.0
60.7
60.6
61.0
Semivolatility
7.4
5.3
6.7
7.1
8.5
7.2
5.3
4.8
4.1
4.0
Def. Prob. (t=1)
0.0
0.0
0.4
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Def. Prob. (t=E(Γ))
60.3
61.1
58.5
60.7
58.7
60.9
59.9
60.1
60.1
60.3
Semivolatility
7.5
5.7
7.7
7.7
9.3
7.9
5.7
5.0
4.4
4.4
What strikes at first sight with respect to the default probability is that it only rarely exceeds zero when a fixed target of 1 is set. In fact, only for E(Γ) maximizing reservation prices and for the discount rate of 5% somewhat higher values have been recorded. In other cases, it was almost certain that condominiums would sell above their average valuations. However, the probability that net receipts were below their expected values was significantly higher and lied at about 60% in most cases. This result indicates clearly that the distribution of net sale receipts is not only non-normal but also asymmetric – the probability of lying below the mean should be at about 50% otherwise. The fact that the relative assessment of liquidity risk in the analyzed markets is
5.2 Condominium Liquidity Analysis
335
highly sensitive to the chosen target level of receipts and to the assumed reservation prices and discounting factors is another important result. E.g., while Frankfurt G could be classified as a highly (liquidity) risky market according to the default probability with a fixed target at ρ=5%, it was among the least risky ones when E(Γ) was set as the target. The results of liquidity risk measurement with semivolatility reinforce these conclusion – ranks of the analyzed markets differ from those arising from default probability, even when the latter is referred to the expected sale receipts. This means that the liquidity risk of the condominium markets is assessed differently by an investor who is only concerned about not selling below the expectations than by an investor who also cares about how much below the expectations she could sell. In view of the differences in the result achieved with different liquidity measures, the question arises, which of them allows the most precise measurement. The answer depends, of course, on the attitude and the personal situation of the particular decision maker; however, some objectively valid remarks are also possible. As argued in section 3.3.3, it is more appropriate to assume that investors are averse only to underperforming the target and also perceive risk this way. The superiority of the asymmetric measures follows directly from this assumption. Furthermore, it seems also plausible that investors are concerned about the scale of the underperformance and do not only differentiate between achieving and missing the goal. Hence, higher grade Lower Partial Moments should usually provide better results. It consequence, semivolatility should be preferred to volatility and default probability. An additional argument against volatility is the asymmetry of the receipts’ distributions stated during the simulation. However, a closer look reveals that the relative assessments of liquidity risks with volatility and semivolatility is nearly identical. This can be explained by the fact that the direction as well as the intensity of asymmetry is very similar for all condominium markets. On the one hand, this is indicated by the low variation of default probabilities computed with respect to expected receipts. On the other hand, also the skewnesses of the respective receipts’ distributions obtained in the simulation did not deviate much from each other.474 This result seems to restore the validity of receipts’ volatility as a good liquidity risk measure. Since it leads to a similar relative assessment 474
The skewnesses of the distributions of liquidation receipts for different markets lied between 1.1 and 1.5 for receipts volatility minimizing reservation prices. The dispersion was higher for expected receipts maximizing reservation prices and discounting rates of 25% and 50% with skewnesses lying between 0.9 and 1.5; for the discounting factor of 5%, the respective values varied more strongly between 0.1 and 1.2. For more detailed results see Appendix B.2.
336
Chapter 5: Liquidity of German Condominium Markets
of liquidity risk as the theoretically superior semivolatility and is simultaneously much easier to handle in decision models, it is used in the following sections. It should, however, be stressed that this conclusion refers only to the analyzed condominium markets and does not necessarily need to hold in the general case.
5.2.2.5. Liquidity Risk Reward The Liquidity Risk Reward defined in section 3.4.1 is a very appealing measure, which also seems to have a firm theoretical foundation. On the one hand, its construction is very similar to that of the widely used and accepted Sharpe ratio. In this sense, it can be interpreted as the unit price of liquidity risk. On the other hand, the liquidation strategy (reservation price) that maximizes asset’s LRR can be considered as optimal for a wide group of investors. In particular, if not only illiquid but also liquid assets can be sold in order to deal with a liquidity shock, investors should always apply the LRR maximizing reservation price independent of their attitude to liquidity risk. Individual preferences can be allowed for by optimizing the proportions in which the liquid and the illiquid asset are to be liquidated. Hence, LRR measures liquidity risk as the ability of an asset to provide adequate receipts in relation to the liquidity risk involved in the liquidation process in a broader context. The results of the application of LRR to the German condominium markets are summarized in Table 5-8. The measure was computed separately for three different discounting rates; in each case, only the maximum values (with respect to the reservation price) are reported. Large differences between the condominium markets in the measured levels of liquidity are striking. While the ratio of the liquidation premium (i.e., the excess of the expected net receipts over the average valuation of the condominium) to the standard deviation of the receipts measured at a 5% discount rate is nearly 7 in Hanover G, it lies only at about 3 in Frankfurt G. Since higher time pressure reduces the reward for accepting liquidity risk, LRR values range between 1.3 and 5 for the discount rate of 50%. The relative assessment of the markets’ liquidity remains, however, unchanged.
5.2 Condominium Liquidity Analysis
337
Reservation Prices
3.09
5.28
3.07
6.59
4.33
6.96
5.09
5.49
1.27
1.58
1.36
1.76
1.35
1.54
1.32
1.36
4.36
5.18
2.06
4.34
2.46
5.50
3.71
5.92
4.27
5.02
1.52
1.47
1.21
1.51
1.32
1.68
1.32
1.49
1.29
1.34
3.67
4.40
1.27
3.47
1.76
4.52
2.98
4.95
3.40
4.32
1.46
1.42
1.14
1.44
1.24
1.60
1.27
1.43
1.24
1.31
Frankfurt G
Duisburg M
5.75 1.50
Duisburg G
4.86 1.55
Cologne M
Stuttgart M
Reservation Prices LRR (ρ = 50%)
Stuttgart G
LRR (ρ = 25%)
Hanover M
Reservation Prices
Hanover G
LRR (ρ = 5%)
Cologne G
all figures in %
Frankfurt M
Table 5-8: Liquidity Risk Reward of selected German condominium markets
As already mentioned, the good theoretical foundation of the Liquidity Risk Reward makes it an interesting liquidity measure for a variety of different investor types in different situations. Still, one must bear in mind that it can be biased under certain conditions. In the first place, as the underlying liquidity risk measure is volatility, LRR may not perform well when the distributions of liquidation receipts have significantly different skewnesses – using semivolatility instead of volatility could yield better results in this case. However, as demonstrated in the previous section, this problem is not severe with respect to the analyzed condominium markets. Another important point is the correct interpretation of the measure. In particular, investor’s preferences can be ignored only if a parallel sale of a liquid asset is possible. Otherwise, if no liquid assets are available, investor’s risk attitude does not necessarily need to correspond with the inverse relation assumed in the measure. For the same reason, LRR seems to be less suitable for the planned liquidation, when a stand-alone sale is in focus. Thus, the values computed for the discounting rates of 25% and 50% seem to be more meaningful.
5.2.2.6. Two-Dimensional Liquidity Assessment The final approach to liquidity measurement, proposed in section 5.2.2.6, goes away from expressing this quality with a single figure and regards marketability and liquidity risk as separate components in a two-dimensional room. This procedure does not ensure that unambiguous identification of more and less liquid asset is always possible, but it allows for more robust conclusions. In particular, it is possible to state which investments are definitely illiquid to any investor independent of her preferences, and
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which can be differently assessed by individuals with different levels of risk aversion. The logic behind this approach complies with the main conclusion form the considerations about the nature of liquidity in Chapter 1. According to it, liquidity is not an absolute quality, and its level depends strongly on individual preferences. Another advantage of this approach is its flexibility. It can be easily adjusted to different measurement methods of marketability and liquidity risk. Following the discussions in earlier Chapters, expected net receipts and receipts’ volatility are used in this section. However, it is also possible, though more computation intensive, to apply other (e.g., asymmetric) liquidity risk measures. The two-dimensional approach in this section is based on the graphical analysis of E(Γ)-V(Γ)-loci resulting for efficient liquidation strategies. After determining the range of reservation prices that lead to minimum liquidity risk at the given level of marketability (or to maximum marketability at the given level of liquidity risk) the respective expected net receipts and receipts’ volatilities have been plotted. They produce liquidity efficient frontiers specific for each condominium market. The analysis has been repeated for discounting rates of 5%, 25%, and 50%. The results are depicted in Figure 5-2.
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2,2
Expected Net Receipts
Cologne G Cologne M
2
Duisburg G Duisburg M 1,8
Frankfurt G Frankfurt M Hanover G
1,6
Hanover M Stuttgart G Stuttgart M
1,4
1,2 0
0,5
1
1,5
2
ρ=5%
Volatility of Net Receipts
1,9
1,8
Expected Net Receipts
Cologne G Cologne M
1,7
Duisburg G Duisburg M
1,6
Frankfurt G Frankfurt M
1,5
Hanover G Hanover M
1,4
Stuttgart G Stuttgart M
1,3
1,2 0,05
0,1
0,15
Volatility of Net Receipts
0,2
0,25
ρ=25%
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Chapter 5: Liquidity of German Condominium Markets
1,8 1,7 Cologne G
Expected Net Receipts
1,6
Cologne M Duisburg G
1,5
Duisburg M Frankfurt G
1,4
Frankfurt M Hanover G
1,3
Hanover M Stuttgart G
1,2
Stuttgart M 1,1 1 0,06
0,08
0,1
0,12
0,14
0,16
Volatility of Net Receipts
0,18
0,2
0,22
ρ=50%
Figure 5-2: Two-dimensional liquidity measurement of selected German condominium markets
The “lengths” of the efficient frontiers depend strongly on the discount rate and become “shorter” with an increasing time pressure. Nevertheless, in each case, some markets turn out to be “liquidity inefficient”, i.e., they yield lower expected net receipts at a higher liquidity risk than other markets. The “inefficient set” consists of Cologne G and M, Duisburg G, Frankfurt G, and Hanover G in the scenario with 5% discount rate, of Duisburg G, Frankfurt G, and Hanover G in the scenario with 25% discount rate, and of Duisburg G and M, Frankfurt G, and Hanover G in the scenario with 50% discount rate. Hence, certain markets, among them Frankfurt M, Hanover M, and Stuttgart G and M are so superior with respect to their liquidity characteristics that they are efficient in all analyzed scenarios. Apart from determining the superior and inferior sets of condominium markets, the two-dimensional approach also allows comparing pairs of investments. So, e.g., under the assumption of a very high time pressure (ρ=50%) a good quality condominium in Duisburg can be considered as less liquid than a good condominium in Hanover; however, their relation to good quality condominiums in Frankfurt is ambiguous and can only be determined on the basis of preferences of a concrete investor.
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The method applied in this section is unorthodox as it does not always allow ranking the analyzed markets with respect to their liquidity. Although it was possible to name objectively more liquid markets in some cases, ambiguousness still remained in many other cases. However, as soon as investor’s preferences between marketability (expected liquidation receipts) and liquidity risk (receipts’ volatility) are formulated and quantified as indifference curves, the level of utility associated with the liquidation of each property can be determined. It can be then used as a subjective liquidity measure, as proposed in section 3.4.2, and enables a strict identification of the most and the least liquid condominium markets. Since such a measure would reflect not only the features of the specific property market but also the individual attitude of the investor, it would only be valid for this single investor. Because the analysis in this Chapter is not addressed to any specific individual, this step has not been taken here. Nevertheless, the preference-based two-dimensional method seems to be the most appropriate approach to liquidity measurement.
5.2.2.7. Comparison of the Measures The results of the liquidity analysis of German condominium markets presented in the former sections were highly dependent on the applied measurement method. Hence, the question arises in how far the choice of the method affects the conclusions about the relative levels of liquidity on different markets. The discussion of the relations between different measures in section 3.5 revealed that some connection can be expected between most of the measures; it should, however, vary depending on the aspect of the problem on which the focus it placed. In particular, measures of liquidity risk may display only a weak relation to the more traditional measures of marketability. Already a superficial analysis of the results in this Chapter reveals that the matter is more complex – the relations between the measures depend also on the assumed environment of the decision maker, in particular, on the time pressure. Thus, it is very difficult to identify the ultimate liquidity measure that could be applied in any situation. This result is not surprising – the subjective character of liquidity has been already stated earlier in this book. In practice, however, identification of the appropriate approach and the parameters on which it should be based may prove very difficult. In particular, liquidity analysis might need to be conducted without knowing the investor to whom it is addressed – this is the problem often faced by market researchers – or the parameters of the model might be uncertain. The latter difficulty is probable when the liquidation is
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expected to occur in the far future. In this context, it is highly relevant whether and to what extent different measures lead to the same ranking of assets or markets with respect to their liquidity. The relative liquidity of German condominium markets resulting from the application of the seven different measures is presented in Figure 5-3. Wherever several variants of a measure were computed, only one of them has been chosen for better tractability. Thus, only the minimum liquidation duration (ToM) is regarded, only liquidity risk measures resulting for volatility minimizing reservation prices are considered, and the target for the default probability is limited to expected net receipts. Furthermore, for better comparability of the results, only the ranks of the markets resulting for different measures are depicted. In this sense, 1 means the highest and 10 the lowest liquidity rank; equal ranks are possible. Due to the impossibility of producing an unambiguous ranking, the two-dimensional approach is omitted.
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Figure 5-3: Ranks of selected German condominium markets with respect to their liquidity according to different measurement approaches
The first impression from the comparison of rankings obtained on the basis of different measures confirms the high dependence of the relative liquidity from the applied approach. While some of the measures lead to contrary conclusions (e.g., the number of transactions and the Implicit Spread), others match fairly well (e.g., the Implicit Spread and the Quick Sale Discount). In order to quantify these differences, rank correlations according to Spearman (corrs) have been used. This statistic can be interpreted similarly to the standard Bravais-Pearson correlation coefficient (it ranges from -1 to 1) and is computed for observation vectors X and Y as:475 n
corrS (X, Y ) = 1 − with: ki
475
6 ⋅ ∑ (k X ,i − k Y ,i ) 2 i =1
n ⋅ (n ² − 1)
- rank of the ith element
See Kendall (1962), p. 20 f., Hartung et al. (2002), pp. 553 ff., or any statistics handbook.
(5.2)
5.2 Condominium Liquidity Analysis n
345
- number of elements
The results are summarized in Table 5-9. The upper value in each cell refers to the discounting rate of 5%, the middle value to 25%, and the lower value to 50%.
Liquidity Risk Reward
Liquidity Risk Reward
Receipts' Semivolatility
Receipts' Semivolatility
Def. Probability
Def. Probability
Receipts' Volatility
Receipts' Volatility
Time on Market
Time on Market
Quick Sale Discount
Quick Sale Discount
Implicit Spread
Implicit Spread
Number of Transactions
Number of Transactions
Table 5-9: Rank correlations between liquidity measures of selected German condominium markets
1.00 1.00 1.00
-0.25 -0.35 -0.43
-0.25 -0.35 -0.47
0.61 0.79 0.85
0.50 0.48 0.53
-0.66 -0.91 -0.55
0.56 0.54 0.64
0.90 0.92 0.92
1.00 1.00 1.00
1.00 1.00 0.98
0.10 0.04 -0.01
0.64 0.56 0.42
0.07 0.25 0.81
0.59 0.53 0.27
-0.39 -0.65 -0.71
1.00 1.00 1.00
0.10 0.04 -0.09
0.64 0.56 0.32
0.07 0.25 0.75
0.59 0.53 0.20
-0.39 -0.65 -0.75
1.00 1.00 1.00
0.66 0.75 0.87
-0.61 -0.76 -0.27
0.68 0.76 0.90
0.36 0.55 0.62
1.00 1.00 1.00
-0.61 -0.53 0.09
0.99 0.98 0.98
0.36 0.15 0.20
1.00 1.00 1.00
-0.64 -0.56 -0.07
-0.65 -0.78 -0.68
1.00 1.00 1.00
0.42 0.22 0.32 1.00 1.00 1.00
The above comparison reveals that although different measures lead to different rankings of the markets, they can be associated in several groups. One of such groups encompasses the Implicit Spread and the Quick Sale Discount. This similarity is not sur-
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prising as the construction of these measures is based on the same principle – maximum expected net receipts from sale are central to both of them. The only relevant difference is the additional inclusion of expected net expenditures at purchase in the Spread. However, since both expected values are roughly symmetric around the mean offer value of 1, the resulting differences in the rankings are only marginal. In effect, both measures describe the liquidity of the respective property markets, or equivalently, the situation of a passive investor, who does not attempt a strategic liquidation and either trades with a (hypothetical) dealer, or sells to the first interested buyer. The second group consists of volatility and semivolatility of liquidation receipts. Also this result is in line with expectations. As discussed earlier, both measures should yield consistent relative assessments of liquidity if the underlying distributions of net receipts are symmetric or the skewness is similar for the analyzed investments (markets). Although asymmetry was present in the distributions generated in the Monte Carlo Simulation in section 5.2.2.4, its level measured with the skewness coefficient was similar for most markets. Hence, volatility and semivolatility seem to be relatively good substitutes in the analyzed case. However, this does not hold for the default probability which leads to partially opposite rankings of the markets compared to the volatility measures. Thus, a decision maker concerned only about not reaching a certain liquidation value may act against her own preferences if she chooses volatility or semivolatility to measure liquidity risk. Analogically, an investor concerned about not deviating much from the expectations about the outcome of the liquidation will perform poorly by applying the default probability. Somewhat surprising is the similarity stated between the rankings of markets resulting from the application of the Liquidity Risk Reward and the number of transactions. As noted earlier, the former measure is, in my opinion, most adequate when subjective liquidity of an asset to an active investor following individually optimized selling strategies is considered, and liquidity measurement needs to be performed on the basis of a single ratio. Not only does it combine the expectation and uncertainty dimensions of liquidity, but it also corresponds with the objective optimal liquidation strategy in the most general case of an investor holding both liquid and illiquid assets. In this light, it is astonishing that a measure as simple as the average number of transactions observed on the market in the past leads to nearly the same result as the much more elaborate Liquidity Risk Reward. Yet, it is not clear whether this effect is due to the
5.2 Condominium Liquidity Analysis
347
actual similarity of the approaches or only the result of a more or less coincidental combination of parameters for the analyzed markets. A more thorough analysis would be necessary at this point. In contrast, the negative relation between LRR and the Implicit Spread or QSD could be expected. As discussed in sections 3.4.1and 3.5, high expected sale receipts (and low expenses) are a positive feature when the investor is able and willing to perform a strategic search, but they are a negative feature when she passively accepts the terms dictated by the market. Thus, the difference between these measures is more fundamental and arises from a different notion of liquidity. Summing up, it can be stated that the choice of the method for liquidity measurement is crucial for the quality of the decisions made on its basis. Although LRR and the two-dimensional approach seem to be most adequate and have the best theoretical foundations, investors who are concerned only about certain aspects of liquidity may achieve better results using other approaches. In either case, however, the first step should be a precise analysis of the actual goals of the concrete investor. The quality of the measurement depends wholly on the correspondence between individual preferences and the rationale behind the measurement approach. 5.2.3.
Condominiums in Portfolio Decisions
When a decision on a single investment is to be made, available alternatives need to be compared with respect to their relevant characteristics, in particular, their expected profitability, risk, and liquidity. In this context, the issue of liquidity measurement becomes particularly important. The analysis of condominium markets in selected German urban areas demonstrated how various measures derived in Chapter 3 can be utilized for this purpose. However, investors often conduct more than one investment simultaneously and are confronted with the problem of optimal portfolio selection. Hence, the liquidity criterion needs to be considered not with respect to a single investment but with respect to the whole portfolio. Combining it with other criteria can (and in most cases will) lead to different results than the traditional optimization techniques based solely on portfolio’s expected returns and return volatilities. The following two sections deal with changes in the composition of optimal portfolios resulting from the introduction of a liquidity goal based on the search theoretical considerations. German condominium markets are used as the reference point and portfolios consisting of real estate investments in these markets are considered. The main
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purpose of the analysis is, however, not the formulation of a concrete investment strategy but a demonstration how the notion of efficiency changes when the portfolio selection model is extended to include the liquidity criterion. The effects of a planned liquidation and possible liquidity problems in an emergency case are considered separately. On the one hand, it turns out that the expected profitability of a real estate portfolio increases rapidly when the possibility of a strategic behavior during the planned liquidation is allowed for, but the level of investment risk remains relatively unaffected. In effect, the efficient frontier shifts upwards. On the other hand, the scope of feasible portfolios that can be regarded as efficient and may be optimal for certain groups of investors widens rapidly when the possibility of favorable liquidation in an emergency case is included in the analysis.
5.2.3.1. Planned Liquidation, Return Characteristics, and the Efficient Frontier As discussed in section 4.4.2.1, the fact that a liquidation strategy can be applied at the end of the time horizon requires a correction of the return statistics applied in the portfolio optimization. In particular, historical returns of heterogeneous assets, which are usually based on average changes of market prices, can be expected to underestimate both the profitability and the risk of such investments. This effect is even more distinct when the purchase is also subject to a random search process. In consequence, an investor following optimal buying and selling strategies should be able to achieve higher returns than the returns of an index based on average market valuations. On the other hand, also the uncertainty about the realized returns experienced by the investor is higher than the variability of the index returns – it is increased by the uncertainty about the outcome of the search process. Finally, due to the independency of different searches, also the correlations between the returns of different assets should be lower than the correlations between the respective index returns. These effects lead not only to different assessments of the performances of different investment alternatives but also to a shift in the efficient frontier. The extent of the shift for investments in German condominiums is analyzed in this section. Since considerations about changes in return characteristics of illiquid assets are only purposeful in the case of a planned liquidation, only the discounting rate of 5% is applied in this section. Furthermore, search effects during the purchasing process are disregarded in order to keep the analysis tractable. Thus, return statistics are corrected according to the formulas (4.23), (4.24), and (4.25), and the purchasing price is as-
5.2 Condominium Liquidity Analysis
349
sumed to be on the level of the average market valuation (i.e., ς=1). Additionally, a correction for the length of the holding period is necessary. The returns of the condominium investments computed in section 5.2.1.3 are based on annual data. In reality, however, engagements in such investments are much longer than one year. Disregarding this fact would lead to a drastic overestimation of the true return levels experiences by investors. Thus, the strategic liquidation effect needs to be adjusted for the supposed holding period. A time horizon of 10 year is assumed in the following analysis. According to this assumption, relative liquidation receipts are adjusted as follows:
~ Γ =1+
Γ −1 Time Horizon
(5.3)
The key statistics of the liquidation receipts (expected values, standard deviations, and covariances) are adjusted correspondingly.476 Another problematic issue in the adjustment of market (index) returns for liquidation effects is the choice of the adequate selling strategies. Since it depends on the preferences of the decision maker, no definite answer can be given here. The following analysis is therefore conducted for maximum and for minimum reservation prices to give a better impression about the range of achievable results. Note, however, that while the maximum expected return is achieved for the same set of reservation prices which maximizes expected net receipts, reservation prices minimizing the return volatility do not necessarily need to be the same as the prices minimizing receipts’ volatilities. Hence, the ranges of reservation prices that are rational with respect to total returns from condominium investments differ from the ranges resulting from sole liquidity considerations.477 The key statistics of the corrected returns are summarized in Table 5-10.
476
477
Computation of the expected value of the corrected relative liquidation receipts is trivial. It is also easily shown that the volatility (standard deviation) of the receipts equals the original volatility divided by the time horizon, and the correlation between receipts of two investments equals the original correlation divided by the square product of the respective time horizons. As noted in section 4.4.2.1, the minimal return volatility results per definition for an infinitely high reservation price, as the variability of net liquidation receipts is then zero. This case, however, is not regarded here as it implies that the asset (condominium) is not sold at all. Hence, only local minima of the receipts’ variances are considered.
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Stuttgart M
Stuttgart G
Hanover M
Hanover G
Frankfurt M
Frankfurt G
Duisburg M
Duisburg G
all figures in %
Cologne M
Cologne G
Table 5-10: Characteristics of investment returns on selected German condominium markets with optimal liquidation at the investment horizon
Return Characteristics Without Liquidity Considerations Expected Return (%)
7.3
7.8
6.1
6.3
7.7
7.4
6.6
7.1
6.7
8.3
Return Volatility (%)
17.2
15.9
9.8
11.6
17.3
9.6
17.4
11.1
13.0
21.1
Liquidation Strategies Maximizing Expected Returns Reservation Price
1.94
1.84
1.55
1.89
1.83
2.19
1.60
1.79
1.55
1.65
Expected Return (%)
16.9
16.5
11.7
15.3
16.3
19.7
12.7
15.1
12.4
15.0
23.5
24.5
12.8
14.9
32.2
19.4
20.1
14.1
17.4
31.7
0.24
Return Volatility (%)
Liquidation Strategies Minimizing Return Volatilities Reservation Price
0.36
0.46
0.78
0.69
0.36
0.88
0.38
0.72
0.63
Expected Return (%)
7.5
7.9
6.8
7.3
7.8
9.7
6.6
7.7
6.8
8.3
Return Volatility (%)
17.5
16.2
10.0
12.0
17.6
10.1
17.6
11.4
13.2
21.2
The conclusions from the correction of market returns are twofold. On the one hand, applying a sale strategy that maximizes the expected return (and simultaneously the expected net sale receipts) led in all cases to a substantial increase of both the expected profitability and the risk. On average, the expected return was higher by about 10%points, and the return volatility nearly doubled in some cases (Frankfurt G). On the other hand, the application of a reservation price that minimizes the return volatility induced only marginal changes in the return characteristics of the condominium markets. Hence, as it seems, average returns and risks experienced by investors investing in German condominiums are higher than indicated by real estate indices based on average market values; the extent of this effect depends, however, on individual preferences. A similar effect occurs also with respect to the covariances and correlations between the condominium markets’ returns. While they remain at a level similar to the original one when the reservation price is set to minimize return volatility, they are much lower for reservation prices maximizing expected returns. This fact is highly relevant for the optimal structuring of portfolios consisting of condominiums in the selected markets.
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Lower correlations indicate that a higher risk reduction due to diversification should be possible. However, the position and the shape of the efficient frontier depend eventually on all relevant return characteristics: expected returns, return volatilities, and correlations. Taking this into account, two alternative efficient frontiers are presented in Figure 5-4: one resulting from high reservation prices maximizing the expected return and the other resulting from risk minimizing reservation prices. For comparison, also the original efficient frontier without strategic liquidation is depicted.
22% 20%
Expected Return
18% 16%
Strategic liquidation maximising expected returns Strategic liquidation minimising return volatility
14%
(Frankfurt M excluded) Original efficient frontier
12% 10% 8% 6% 0%
1%
2%
3%
4%
5%
Return Volatility
Figure 5-4: Efficient frontier with returns corrected for liquidation effects
The strong upward shift of the efficiency frontier in the scenario that assumes a sale strategy which maximizes expected returns is striking, though not unexpected in the light of changes in the return characteristics discussed above. While the volatility range remained roughly similar, the expected returns have more than doubled. In contrast, applying a search strategy that minimizes the return volatility led to a smaller increase in expected returns but also narrowed the volatility range. The latter effect was, however, mainly due to the high expected return of the Frankfurt M market; in effect, it “crowded out” Stuttgart M from the maximum return portfolio. Portfolio optimization without this market yielded an efficient frontier that was nearly identical with the one obtained without liquidity considerations.
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Summing up, the main conclusion from the portfolio optimization based on returns corrected for strategic liquidation at the end of the time horizon is the improvement of the resulting efficient frontier. This means that higher returns can be achieved at a lower risk. However, it also means that a different combination of investments may be optimal for a specific investor. E.g., while an investment in Stuttgart M is necessary to achieve high levels of expected returns in the original portfolio selection framework, this market does not enter any efficient portfolio in the extended scenarios unless Frankfurt M is excluded. In contrast, the composition of the minimum variance portfolio for the original scenario and for the scenario assuming risk minimizing liquidation is nearly identical. Hence, it seems that the relevance of liquidity considerations for the allocation of capital on the analyzed condominium markets decreases with the riskaversion of the investor. Since risk minimizing reservation prices are low, the expected outcome of the liquidation does not deviate much from the average market price level. However, for more speculative investors, ready to accept higher risks, investments in German condominiums seem to be much more attractive than it would follow from the analysis of the respective market indices.
5.2.3.2. Portfolio Selection in a Multidimensional Decision Framework with Liquidity Criterion The final part of the liquidity analysis addresses possible changes in the composition of the optimal portfolio compared to the traditional mean-variance framework when liquidity is allowed for as an additional decision criterion. Three alternatives are considered: in the first one, liquidity is captured with the Implicit Spread, Liquidity Risk Reward is applied in the second one, and in the third one, a four dimensional approach is followed with marketability (defined as expected net receipts from liquidation) and liquidity risk (defined as volatility of net receipts) included as separate criteria. Each of these approaches represents a slightly different understanding of liquidity. Using the Spread implies that the investor does not sell strategically, but accepts without bargaining the prices that would be offered by a hypothetical risk neutral dealer. This is equivalent to assuming that the investor is passive and takes market conditions as given. In contrast, LRR and the two-dimensional approach are based on the assumption of a strategically acting investor. The latter of the two approaches additionally allows for any arbitrary preferences towards liquidity risk.
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Since regarding liquidity as a dimension separate from market returns is only purposeful in the case of an unexpected liquidation, the discounting rate is set at 25%; it corresponds with an investor selling under a moderate time pressure. Furthermore, the correction of return statistics, as it was conducted in the former section, is disregarded for better tractability of the results. Hence, the returns of the condominium investments are assumed to be at the level suggested by the IVD/RDM data. The inclusion of marketability and liquidity risk in portfolio selection, either combined within one figure like the Implicit Spread or LRR or considered separately, leads to a multidimensional optimization problem which is not easy to solve analytically. It was therefore necessary to apply numerical optimization. However, the amount of calculations in this approach increases rapidly with the number of analyzed markets. The case of 10 condominium markets in 5 selected German urban areas proved to be too vast for the available software; computations would still be possible, but they would take up to several months. Since such effort would not be justified by the purpose of this Chapter intending merely to demonstrate the practical application of the search theoretical approach, the number of considered markets has been reduced to 5 – only the markets for good quality condominiums have been selected in each of the cities. The number of explicitly identified efficient portfolios has been limited to about 500 for the optimization with the Implicit Spread, to about 750 for the optimization with the Liquidiy Risk Reward, and to about 3000 for the four-dimensional optimization with expected net receipts and receipts’ volatility. Another problem was the presentation of the results. They encompass not only the expected returns and return volatilities of the alternative (efficient) portfolios, as it is in the case of the traditional MPT framework, but also one or two variables describing liquidity. Hence, at least a three-dimensional chart would be necessary to depict the efficient portfolios graphically. Yet, for better tractability and better comparability of the results, a classical two-dimensional presentation has been chosen. In this case, efficient portfolios form an area rather than a line; it results from the projection of the efficient plane (or hyper-plane) on the two-dimensional coordinate system.
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The results of the optimization in the above described framework are presented in Figure 5-5. In addition to the return- and liquidity-efficient area also the original returnefficient frontier is depicted in each diagram.478
478
Note that the liquidity efficient portfolios are only selected points from a continuous efficient hyperplane. Since the optimization was based on a limited MSC, this selection is not necessarily fully representative. In particular, certain regions (e.g., portfolios near the maximum expected return portfolio) are underrepresented.
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7,8%
a) Liquidity = Implicit Spread 7,6%
Expected Return
7,4% 7,2% 7,0% 6,8% 6,6% 6,4% 6,2%
Liquidity Efficient Portfolios Original Efficient Frontier
6,0% 0,0%
0,5%
1,0%
1,5%
2,0%
2,5%
3,0%
3,5%
Return Volatility
7,8%
b) Liquidity = Liquidity Risk Reward 7,6%
Expected Return
7,4% 7,2% 7,0% 6,8% 6,6% 6,4% 6,2% 6,0% 0,0%
Liquidity Efficient Portfolios Original Efficient Frontier 0,5%
1,0%
1,5%
2,0%
Return Volatility
2,5%
3,0%
3,5%
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Chapter 5: Liquidity of German Condominium Markets 7,8%
c) Liquidity = Exp. Receipts and Receipts’ Volaility 7,6%
Expected Return
7,4% 7,2% 7,0% 6,8% 6,6% 6,4% 6,2%
Liquidity Efficient Portfolios Original Efficient Frontier
6,0% 0,0%
0,5%
1,0%
1,5%
2,0%
2,5%
3,0%
3,5%
Return Volatility
Figure 5-5: Efficient frontiers with separate liquidity criteria: Implicit Spread (a), LRR (b), and expected net receipts and receipts’ volatility (c)
It is immediately apparent that the scope of efficient portfolios is much larger when a liquidity criterion is allowed for. This means that portfolios which would be considered highly inefficient if the optimization was conducted only on the basis of index returns may turn out to be efficient when liquidity is taken into account. Such portfolios offer superior chances of advantageous liquidation in case of a liquidity bottleneck, and this compensates for their possibly inferior return characteristics. E.g., a portfolio consisting of condominiums in Cologne, Duisburg, Frankfurt, Hanover, and Stuttgart in proportions: 10.8%, 33.9%, 9.9%, 14.4%, and 31%, respectively (weights are rounded) offers the same expected return of about 5.6% as a portfolio with 20.3%, 10.7%, 9.6%, 23.5%, and 35.8% invested in the respective cities but at a lower return volatility of only 0.7% instead of 0.8%;. However, the latter portfolio offers a higher LRR of 7.8 instead of 6.3 in case of a forced liquidation. Thus, both portfolios are efficient, and it depends solely on investor’s preferences which of them should be preferred. Clearly, each of the three presented approaches leads to a different efficient area and a different set of efficient portfolios. Moreover, since “portfolios” are here defined not
5.3 Discussion of the Results
357
only in terms of portions of capital invested in different markets but also encompass reservation prices applicable in an unexpected liquidation, the differences refer also to this aspect. This fact makes is difficult to state which investments are actually to be considered efficient. The answer depends on the appropriateness of the applied liquidity measurement concept in the concrete case. The interpretations and the fields of application of the measures have been discussed extensively in Chapter 3 as well as in section 5.2.2 earlier in this Chapter. The Implicit Spread can be of interest only to absolutely risk-neutral investors that are rather rare in the real estate branch. In contrast, optimization with the LRR criterion is eligible to a wide group of liquidity risk-averse investors holding both liquid and illiquid assets. Most universal, but also most difficult to implement, is the optimization in the four-dimensional framework with marketability and liquidity risk considered separately.
5.3.
Discussion of the Results
The practical application of the search theory to the measurement and management of liquidity undertook in this Chapter is novel and requires a critical discussion. In particular, it has to be cleared to which extent these results can be considered as realistic and practically relevant. Can an investor deciding on the allocation of her limited funds in one or more of the analyzed condominium markets draw valuable conclusions from the estimation of various liquidity measures and the analysis of portfolio efficiency performed on the basis of the search theoretical approach, or is it only a purely theoretical concept? To answer this question, it is useful to confront the results with the observed behavior of market participants and consider if the implications are in line with common sense. In the first place, the validity of the results is highly dependent on the quality of parameter estimation. As already discussed in section 5.2.1, the determination of model parameters from publicly available data is highly problematic, particularly with respect to the dispersion of valuations and the arrival frequency of offers. The former parameter has been estimated in this Chapter on the basis of the observed dispersion of condominium transaction prices. As already noted, this is only an imperfect proxy. On the one hand, condominiums in the samples are similar but not identical. This means that the variability of transaction prices is not only due to different valuations by market participants but also due to differences in the characteristics of the condominiums. Thus, the standard deviations of offers used in this model are probably overestimated.
358
Chapter 5: Liquidity of German Condominium Markets
On the other hand, each transaction price arises from (at least) two valuations: that by the seller and that by the buyer. Hence, the prices should vary less than the valuations themselves. This may to some extent mitigate the supposed overestimation of the offer volatility. Another issue is the accuracy of the estimated offer arrival frequencies. For the lack of a better method, they have been derived from the trading intensity observed on the markets. This was, however, a rather intuitive “guess” and not a strict estimation, so that inaccuracy can be very high here. Nevertheless, the levels of the figures seem not entirely unrealistic, as confirmed by several real estate dealers. The first remarkable result has been obtained in the computation of the Implicit Spreads. The reservation prices maximizing this measure proved to be extremely high and so were the values of the measure itself. On the one hand, the optimal purchasing strategy implied that condominiums should be bought far below the average valuation – in the extreme case of Frankfurt M even at a negative price. On the other hand, optimal sale was to be conducted at prices as high as the double of the average. No such behavior, not even anything in its proximity, can be observed on residential real estate markets. A conclusion that it would be possible to realize an expected spread of up to 200% of the fair value is therefore clearly misleading. In fact, these results may be too high due to the overestimation of the offer volatility and the offer arrival frequency mentioned earlier. Reducing the former parameter by a half with respect to all submarkets would result in Implicit Spreads between 0.5 and 1.3 (about half of the original values). Yet, even these values are unrealistically high. The actual source of the problem seems to lie in the risk neutrality implied in the Spread. Such attitude is seldom, if ever, met among real estate investors. The sole maximization of expected expenditures or receipts is only a hypothetical concept, which leads to levels of liquidity risk that are evidently unacceptable to any investor. Hence, there is only a limited possibility of a meaningful interpretation of this measure with respect to real situations. Same applies to other measures computed under the assumption that reservation prices are set to maximize expected receipts. Also the respective values of the Quick Sale Discount or the Time on the Market were unrealistically high in the analogous cases; the latter reached selling durations of several years. In contrast, the results obtained for reservation prices minimizing the volatility of sale receipts were much more realistic – e.g., the ToM could be measured in weeks instead of years. A similar effect can be expected with respect to liquidity risk measures, though it is not easily verified on the basis of empirical data.
5.3 Discussion of the Results
359
Summing up, the unrealistic results achieved for different reservation prices indicate that the risk attitude of condominium investors is far from neutrality and much closer to high levels of liquidity risk aversion. This may be due to the fact that German residential real estate is generally considered to be a low risk investment, but also due to the fact that much of the trade is consumption and not investment driven. The consequences for liquidity measurement in these markets are twofold. On the one hand, measurement based on maximizing the value of search does not seem to be appropriate; measures based on risk minimizing reservation prices can be expected to perform better. On the other hand, liquidity risk is of high importance, so that omitting it can lead to a false picture of the markets. Measures combining marketability and liquidity risk are therefore preferable. This conclusion favors the Liquidity Risk Reward and the two-dimensional approach, though the former seems easier to implement due to its brevity. According to it, Hanover M, Frankfurt M, Cologne M should be considered most liquid, while Duisburg G and Frankfurt G least liquid.479 However, it must be once again stressed that this conclusion is subject to individual preferences of the decision maker. In this sense, the results achieved in this Chapter cannot be generalized in terms of suggesting an objectively optimal liquidation strategy. It is also not possible to state on this basis which of the markets are objectively more or less liquid. The differences resulting from the application of different reservation prices and different discount rates make it clear that liquidity is a highly subjective quality and can be experienced differently by different investors. However, as soon as the preferences of the individual investor are adequately allowed for, search theory based measurement can provide very detailed analysis of investors’ situation in the case of planned or forced liquidation of an asset. Another remarkable result that requires a more detailed discussion is the effect of introducing search theoretical liquidity considerations into portfolio optimization. In the first place, allowing for strategic liquidation at the end of the investment horizon resulted in increased expected returns without a significant increase of investment risk. The effect depended strongly on the assumed strategy and, similarly as in the above discussed case of liquidity measurement, was unrealistically strong when reservation
479
Note that the most liquid markets have simultaneously low estimated dispersion of offers, high frequency of offers, and low market uncertainty; in contrast, the least liquid markets are those with the (by far) lowest offer arrival frequencies. This result complies with the theoretical considerations about sources of liquidity in Chapter 1.
360
Chapter 5: Liquidity of German Condominium Markets
prices maximizing expected net receipts were assumed. Hence, also in this case, it seems that high (liquidity) risk aversion and, thus, the application of liquidity risk minimizing strategies is more appropriate. Even in this case, however, there was a clear upward shift of the efficient frontier for condominium portfolios. In consequence, the analyzed condominium investments have proven to be more attractive than it would follow from the pure index analysis. The robustness of this effect was not explicitly tested here. Should it, however, be of more general nature, it could provide a plausible explanation to the puzzle stated, e.g., by Giliberto (1992) who came to the conclusion that returns of real estate indices are not sufficient to justify the inclusion of this asset class in investment portfolios.480 It seems conceivable that, at least for certain groups of investors, the possibility of strategic liquidation leads to more favorable characteristics of real estate portfolios than indicated by aggregated data.481 Finally, a comment is necessary on the widening of the scope of efficient portfolios resulting from the optimization with a liquidity criterion compared to the traditional Markowitz mean-variance approach. Also here, the results differ depending on the assumed approach; yet, in each case numerous portfolios, which would be inefficient in the standard framework, become efficient due to their favorable liquidity characteristics. This is especially apparent in the most general, four-dimensional approach with marketability and liquidity risk treated as separate criteria – portfolios with expected returns below the level of the original Minimum Variance Portfolio and risks above the original Maximum Return Portfolio were still among the efficient ones. This striking result may be an explanation why certain real estate markets, despite their poor performance, still enter investment portfolios in higher proportions than suggested by the analysis of their past returns. It might be advantageous for certain investors to hold a part of their capital in less profitable and/or riskier assets, which, however, can be liquidated at a good value with little uncertainty in case of an unexpected liquidity bot-
480
481
Also the results of other researchers confirm this puzzle. Fogler (1984), Firstenberg et al. (1988), or Kallberg et al. (1996) assess the optimal allocation of capital to real estate at about 10%-20%. This is less than the total share of this asset in the investable wealth estimated by Ibbotson/Siegel (1983) at over 25% and far less than the overall share of real estate in world’s wealth (see section 1.3.2 in Chapter 1). Validation of this hypothesis should be possible by comparing returns achieved by institutional investors on concrete properties with index returns. An indication that the former might be higher is given by the relatively high returns of real estate companies (REITs) compared to the returns of direct real estate market indices. See, e.g., Geltner/Kluger (1998), Geltner/Rodriquez (1998), or Pagliari et al. (2005).
5.3 Discussion of the Results
361
tleneck. Sacrificing a part of potential returns can be viewed as an “insurance” against solvency problems. In this context, higher expected performance of portfolios lying on the original mean-variance efficient frontier compared to that of the “liquidityefficient” portfolios can be interpreted as a premium paid for their inferior liquidity characteristics. *** As already stated at the beginning of this Chapter, its main goal was not the analysis of concrete condominium markets but rather a demonstration of the search theoretical concepts developed in earlier Chapters and the analysis of the practical problems arising in the course of their application. With respect to the former point, methods of model parameter estimation have been discussed and computation routines set up in a popular MS Excel framework. The application to German condominium markets was accomplished as far as the available data and software allowed it. Yet, it is clear that much more precise results could be achieved with means available to professional, institutional investors. While it was not possible to estimate all model parameters with maximal precision (e.g., the offer arrival rate could be assessed only very roughly), this problem can be largely overcome by using large real estate databanks and relying on longer investment experience in the relevant markets. Also the limitations arising from the computation power are not a serious obstacle for a large investment company. Hence, the methodology proposed in this book is not only a pure theory and can be implemented in investment decisions on real estate and other illiquid assets.
Concluding remarks
The initial impulse for the choice of the subject for this work was the discussion about the applicability of the existing risk measurement and portfolio management methods to real estate.482 In turns out that the main difficulty with this asset class is the lack of perfect liquidity, which is required by most capital market theories. While researchers generally agree that this issue is of high importance, there is surprisingly little related literature; operational approaches that could help investors to improve their decision quality are practically absent. The ambitious goal of this book was, thus, to propose a way of coping with the illiquidity problem. Moreover, it should not only remain a theoretical concept based on non-measurable features but also have a potential of practical implementation in the future. Preferably, it should also be compatible with the existing decision frameworks, like the mean-variance approach to portfolio optimization. In the light of the enormous complexity of the problem, a methodical approach was needed that would allow capturing the full scope of liquidity within one consistent and easily handled model. The solution was provided by the mathematical apparatus of the Theory of Search. This approach, suggested already by Lippman/McCall (1986), proved to be most promising for coping with problems encountered when liquidating privately traded assets. The main goal of the analysis was, thus, to translate and, if needed, to amend the search theoretical methodology for modeling liquidation processes. I believe that the resulting approach is unique and novel in a number of different aspects. The first Chapter of the book demonstrates how complex the issue of liquidity actually is. The simple definition referring to it as the “ease to sell” turns out to be insufficient for any operational implementation of the liquidity criterion in investment decisions. The analysis led to the identification of its two distinctly different dimensions: the first one refers to the expected outcome of liquidation and has been denoted after Hicks (1962) as marketability; the second one refers to the uncertainty about the liquidation outcome and has been denoted as liquidity risk. Moreover, it seems that the notion of liquidity can be also extended to the purchase of assets as a similar problem arises then. In the course of the analysis, a number of sources of the so defined liquidity have 482
See, e.g., Draper/Findlay (1982) or Jandura (2003), pp. 59 ff.
364
Concluding remarks
been identified, a review of assets most severely affected by this problem was provided, and its major economic consequences were discussed. In total, the Chapter offers an extensive presentation of the subject, which (to my best knowledge) has been only rarely, if at all, presented in the literature in this scope. The development of a search theoretical model of the liquidation process is in the hub of Chapter 2. Although a lot of literature is devoted to the solutions of analogue search problems, only a handful of papers contain references to liquidity. On the one hand, the Chapter provides a fully specified model of the selling process for property investments; on the other hand, it proposes a number of possible improvements and enhancements to the model. The analytical derivations of the variance (volatility) of net sale receipts and the covariance (correlation) between receipts from selling different assets are of particular novelty. Although they are presented in later chapters, they can be considered as amendments to the model from Chapter 2. The result is a unique, coherent framework allowing the analysis of the liquidation process under different environments and different selling strategies. Moreover, its parameters can be interpreted as observable characteristics of assets and markets, what makes it eligible for practical applications. Chapter 3 is devoted to the development of concrete propositions for quantifying liquidity. A large spectrum of alternatives based on different understandings of the problem and different attitudes of investors have been reviewed. On the one hand, they are based on already existing approaches, which were modified to fit in the search theoretical framework. On the other hand, several new approaches have also been proposed. Among them are measures of liquidity risk as well as measures encompassing both marketability and liquidity risk. Especially the latter offer, in my opinion, the most adequate description of the problem. In contrast to the majority of existing measures, which concentrate on the marketability aspect only, they allow for the fact that the outcome of the sale (or purchase) process on illiquid private markets cannot be assumed as certain. However, regarding liquidity as a two-dimensional phenomenon leads to the dependence of the measurement result on the preferences of the investor. In effect, no absolute assessment of this feature is possible. This result is in line with the conclusions from Chapter 1 stating that liquidity is a highly subjective characteristic of assets and markets.
Concluding remarks
365
While only quantification of liquidity has been addressed in Chapter 3, the possibility of influencing it is considered in Chapter 4. One of the conclusions from the search theoretical analysis of the sale process is its dependence on the chosen search strategy, which is defined in terms of a reservation price. By altering the reservation price, both the expected outcome of the process and its uncertainty can be influenced to some extent. This means that liquidity is not an intrinsic feature of assets but can be subject to individual decisions. This conclusion has important consequences for investors willing to optimize the execution of investments and the allocation of capital. Firstly, liquidation strategies for assets held by an investor can be adjusted so that the liquidity of her total holdings rather than the liquidity of single assets is optimized. This is especially relevant when unexpected solvency problems are to be minimized. On the other hand, liquidity can be considered in asset allocation decisions not only as an additional decision criterion but also as a strategically adjustable feature. Hence, portfolios containing illiquid assets can be optimized not only with respect to assets’ weights but also with respect to liquidation strategies assigned to them. This extension of the classical portfolio selection model introduces a new dimension to investment decisions on illiquid assets. According to it, these investments offer more “degrees of freedom” in optimizing investment portfolios than stocks and other highly liquid assets. Summing up, this work offers, as I believe, the first coherent framework for analyzing, modeling, and managing problems associated with investing in relatively rarely traded private assets like real estate. The source of its novelty is twofold: firstly, it is the recognition of the complex, multifaceted nature of liquidity and the role of search in this respect; secondly, it is the application of the Theory of Search for liquidity analysis. The developed model contributes to better understanding of the processes occurring during the liquidation of an illiquid asset. Moreover, it can be implemented in practice to improve the quality of investment decisions. The latter point is of particular importance in view of the initial goal of the analysis, which, as stated in the introduction, was to provide an operational approach to liquidity. The analysis of German condominium markets in Chapter 5 demonstrated that the methods proposed in this work can be applied even on the basis of publicly available information. However, due to the high requirements regarding the quality of data and the computation power, the approach is addressed mainly to institutional investors.
366
Concluding remarks
Concluding the results, an outlook for further research in this field can be given. In the first place, the proposed methods can be refined. The search model can be extended following the propositions in Chapter 2, and analytical solutions for further statistics can be derived. Another issue that deserves a more detailed investigation is the impact of the characteristics of assets and markets as well as the attitude of the decision maker on the results of the liquidity analysis. In particular, the sensitivity of the results to various errors and misspecifications may be of high practical relevance. A different field for further research is the empirical validation of certain statements made on the basis of the search model. This refers, e.g., to the behavior of market participants in illiquid markets, to the possibilities of “beating the average” on the basis of optimized liquidation strategies, or to the role of liquidity risk and liquidity risk aversion. Finally, also other applications of the search model are conceivable including pricing of illiquid assets and equilibrium analysis in illiquid private markets. Hence, although the approach presented in this book is relatively mature and offers solutions to numerous problems associated with imperfect liquidity, it can also serve as a starting point for further research in several directions.
Appendix A
A.1.
Unique Solution of the Standard Search Problem
It is to prove under what conditions the following equation has a unique root in p*: ∞
p* − p * ⋅F(p*) − ∫ p ⋅ dF(p) = 0 δ p*
(A.1)
Define the function H(p*) as the left hand-side of the above equation and calculate its derivate: dH(p*) 1 1 = − p * ⋅dF(p*) − F(p*) + p * ⋅dF(p*) = − F(p*) dp * δ δ
(A.2)
Since F(p*)∈(0;1), the derivate of H(p*) is always positive for δ < 1. Further, it holds that: ∞
p* − lim p * ⋅F(p*) − lim ∫ p ⋅ dF(p) p*→−∞ δ p*→−∞ p*→−∞ p*
lim H (p*) = lim
p*→−∞
(A.3)
1 = lim p * −0 − E (P) = −∞ δ p*→−∞ and
p*→∞
∞
p* 1 − lim p * ⋅F(p*) − lim ∫ p ⋅ dF(p) = lim p * − lim p * −0 p*→∞ δ p*→∞ p*→∞ p*→∞ δ p*→∞ p*
lim H (p*) = lim
(A.4)
Thus, the limit of H(p*) for p* approaching +∝ is positive when δ < 1. Recapitulating, function H(p*) is strictly increasing for δ < 1 the, it is negative for small values of p*, and it is positive for large p*. The equation (A.1) has a unique solution in this case.
368
Appendix A
For δ ≥ 1 H(p*) has a single global maximum at which F(p*) = 1/δ. By substituting for F(p*) one receives the function’s maximal value, which is: H (p*) =
∞
∞
∞
p* p* p* − p * ⋅F(p*) − ∫ p ⋅ dF(p) = − − p ⋅ dF(p) = − ∫ p ⋅ dF(p) < 0 δ δ δ p∫* p* p*
(A.5)
Since the maximum and both limits of H(p*) are negative for δ ≥ 1, no solution to (A.1) exists in this case. This accomplishes the proof.
A.2.
Closed Form Solution for Expected et Sale Receipts and Expected et Purchase Expense
A closed form formula for expected net receipts is to derive: j ( τ−ρ ) ∑ Tj ( τ−ρ ) ∑ Tj ⎞ ⎛ j j E(Γ) = E⎜ Π ⋅ (1 + A ⋅ ∑ Tj )e + ∑ γ ⋅ Tj ⋅ ∏ e Π > π *⎟ ⎟ ⎜ j j k =1 ⎠ ⎝ i i ∞ ⎡ ∞∞ ∞ ∞ ⎛ i i ( τ−ρ ) ∑ t j ( τ−ρ )⋅ ∑ t kj ⎞ j=1 j=1 ⎟ = ∑ ⎢ ∫ ∫ ...∫ ∫ ⎜ πi ⋅ (1 + a ⋅ ∑ t j ) ⋅ e + ∑γ ⋅tj ⋅e ⎜ ⎟ i =1 ⎢ −∞ 0 j=1 j=1 0 −∞ ⎝ ⎠ ⎣
(A.6)
i ⎤ ⋅ Pr(Π i = πi Π i > π *) ⋅ (1 − FR (π*)) ⋅ FRi−1 ( π*) ⋅ dπi ⋅ ∏ dFT ( t j ) ⋅ dFA (a )⎥ j=1 ⎦
Integrating over πi yields: ⎡∞ ∞ ∞ E(Γ) = ∑ ⎢ ∫ ...∫ ∫ i =1 ⎢ 0 ⎣ 0 −∞ ∞
i i i i ⎛ ( τ−ρ ) ∑ t j ( τ−ρ )⋅ ∑ t kj ⎞ j=1 j=1 ⎜ E(Π Π > π *) ⋅ (1 + a ⋅ t ) ⋅ e ⎟ + ∑γ ⋅tj ⋅e ∑j ⎜ ⎟ j=1 j=1 ⎝ ⎠
(A.7)
⎤ ⋅ (1 − FR (π*)) ⋅ F (π*) ⋅ ∏ dFT ( t j ) ⋅ dFA (a )⎥ j=1 ⎦ i
i −1 R
Integrating over a and applying the assumption that E(A) = 0 yields: ∞ ⎡∞ ∞ E(Γ) = ∑ ⎢ ∫ ...∫ i =1 ⎢ 0 ⎣ 0
i i i ⎛ ( τ −ρ ) ∑ t j ( τ −ρ )⋅ ∑ t kj ⎞ j=1 j=1 ⎜ E(Π Π > π *) ⋅ e ⎟ + ∑γ ⋅t j ⋅e ⎜ ⎟ j=1 ⎝ ⎠
⎤ ⋅ (1 − FR (π*)) ⋅ F (π*) ⋅ ∏ dFT ( t j )⎥ j=1 ⎦ i −1 R
i
(A.8)
Appendix A
369
Finally, assuming exponential distribution of t1 to ti and integrating yields: ∞ ⎡∞ ∞ E(Γ) = ∑ ⎢ ∫ ...∫ i =1 ⎣ ⎢0 0
j i i ⎛ ⎜ E(Π Π > π *) ⋅ ∏ e t ( τ−ρ) + ∑ γ ⋅ t j ⋅ ∏ e t ⎜ j=1 j=1 k =1 ⎝ j
k
( τ −ρ )
⎞ ⎟ ⎟ ⎠
i ⎤ tλ ⋅ (1 − FR (π*)) ⋅ FRi −1 (π*) ⋅ ∏ λ ⋅ e dt j ⎥ j=1 ⎦ j
(A.9)
i j−1 ⎡⎛ i ⎛ ⎞ ⎛ ⎛ ⎞ ⎞ ⎞⎟ λ λ⋅γ λ ⎟⎟ ⋅ ⎜⎜ = ∑ ⎢⎜ E(Π Π > π *) ⋅ ⎜⎜ ⎟ + ∑ ⎜⎜ ⎟ ⎟ ⎟ 2 ⎜ i =1 ⎢ ⎝ λ + ρ − τ ⎠ j=1 ⎝ (λ + ρ − τ) ⎠ ⎝ λ + ρ − τ ⎠ ⎟⎠ ⎣⎝ ∞
⋅ (1 − FR (π*)) ⋅ FRi −1 (π*)
]
Consider both addends separately. The sum in the first addend is clearly a sum of an infinite geometric series, so the expression can be rewritten as follows: i −1
∞ ⎛ λ ⋅ FR ( π*) ⎞ ⎞ ⎛ λ ⎟ ⋅ (1 − FR ( π*) ) ⋅ ∑ ⎜⎜ Add1 = E(Π Π > π *) ⋅ ⎜⎜ ⎟⎟ ⎟ λ ρ τ + − i =1 ⎝ λ + ρ − τ ⎠ ⎠ ⎝ ⎞ ⎛ ⎛ ⎞ (λ + ρ − τ) λ ⎟⎟ ⎟⎟ ⋅ (1 − FR ( π*) ) ⋅ ⎜⎜ = E(Π Π > π *) ⋅ ⎜⎜ ⎝λ +ρ− τ⎠ ⎝ ρ − τ + λ ⋅ (1 − FR ( π*) ) ⎠
=
(A.10)
λ ⋅ E(Π Π > π *) ⋅ (1 − FR ( π*) ) ρ − τ + λ ⋅ (1 − FR ( π*))
In the second addend the interior sum is a sum of a finite geometric series, hence: j−1 ∞ ⎡ i ⎛ ⎞ ⎤ λ⋅γ λ i −1 ⎜ ⎟ ⋅ − ⋅ ⋅ ( ) Add 2 = ∑ ⎢ 1 F ( π* ) F ( π*) ∑⎜ λ + ρ − τ ⎟ ⎥ R R 2 i =1 ⎢ (λ + ρ − τ) j=1 ⎝ ⎠ ⎥⎦ ⎣ i ⎡ ⎛ ⎞ ⎤ λ ⎜ ⎟ ⎥ − 1 ⎢ ⎜ ⎟ ∞ λ+ρ−τ⎠ ⎥ λ⋅γ = ∑ ⎢⎢ ⋅ (1 − FR ( π*) ) ⋅ FRi −1 ( π*) ⋅ ⎝ 2 ⎥ λ i =1 (λ + ρ − τ) 1− ⎥ ⎢ λ+ρ−τ ⎥ ⎦ ⎣⎢ i ∞ ⎡ ⎛ ⎛ ⎞ ⎞⎟⎤ λ ⋅ γ ⋅ (1 − FR ( π*) ) i −1 λ ⎟⎟ ⎥ = ∑⎢ ⋅ FR ( π*) ⋅ ⎜1 − ⎜⎜ ⎜ ⎝ λ + ρ − τ ⎠ ⎟⎥ i =1 ⎢ (λ + ρ − τ) ⋅ (ρ − τ) ⎝ ⎠⎦ ⎣
=
i −1 ∞ ⎛ F ( π*) ⋅ λ ⎞ ⎞⎟ λ ⋅ γ ⋅ (1 − FR ( π*) ) ⎛⎜ ∞ i −1 λ ⎟⎟ ⋅ ∑ FR ( π*) − ⋅ ∑ ⎜⎜ R (λ + ρ − τ) ⋅ (ρ − τ) ⎜ i =1 λ + ρ − τ i =1 ⎝ λ + ρ − τ ⎠ ⎟ ⎝ ⎠
(A.11)
370
Appendix A
The expressions under both sums in Add2 are again infinite geometric series.
⎛ ⎞ ⎜ ⎟ λ ⋅ γ ⋅ (1 − FR ( π*) ) ⎜ 1 λ 1 ⎟ ⋅ − ⋅ Add 2 = (λ + ρ − τ) ⋅ (ρ − τ) ⎜ 1 − FR ( π*) λ + ρ − τ − FR ( π*) ⋅ λ ⎟ 1 ⎜ λ + ρ − τ ⎟⎠ ⎝ =
⎞ λ ⋅ γ ⋅ (1 − FR ( π*) ) ⎛ 1 λ ⎟ ⋅⎜ − (λ + ρ − τ) ⋅ (ρ − τ) ⎜⎝ 1 − FR ( π*) ρ − τ + λ ⋅ (1 − FR ( π*)) ⎟⎠
=
λ⋅γ (λ + ρ − τ) ⋅ (ρ − τ + λ ⋅ (1 − FR ( π*) )
(A.12)
The sum of both addends yields the simplified formula for E(Γ): E (Γ ) = =
λ ⋅ E(Π Π > π *) ⋅ (1 − FR (π*)) ρ − τ + λ ⋅ (1 − FR (π*))
+
λ⋅γ (λ + ρ − τ) ⋅ (ρ − τ + λ ⋅ (1 − FR (π*)))
⎛ ⎞ λ γ ⎜ E(Π Π > π *) ⋅ (1 − FR (π*)) + ⎟ ρ − τ + λ ⋅ (1 − FR ( π*)) ⎜⎝ λ + ρ − τ ⎟⎠
(A.13)
Alternatively, it can be denoted as:
E (Γ) =
E(Π Π > π *) ⋅ X1 ⋅ (1 − FR (π*)) + Y1 ⋅ γ 1 − X1 ⋅ FR (π*)
with: X1 =
λ λ+ρ−τ
and
Y1 =
(A.14)
λ
(λ + ρ − τ)2
Finally, assuming that offers (Π) are normally distributed yields:
E (Γ ) =
⎞ ⎛⎛ π * −1 ⎞ ⎞ γ ⎛ π * −1 ⎞ ⎟ (A.15) ⎜⎜ ⎜1 − Φ⎛⎜ ⎟ ⎟ + ν ⋅ ϕ⎜ ⎟+ ⎛ ⎛ π * −1 ⎞ ⎞ ⎝ ⎝ ⎝ ν ⎠⎠ ⎝ ν ⎠ λ + ρ − τ ⎟⎠ ρ − τ + λ ⋅ ⎜ 1 − Φ⎜ ⎟⎟ ⎝ ν ⎠⎠ ⎝ λ
The respective formula for the expected net expense can be derived analogically. Noting that an offer is accepted if it is below (and not above) the reservation price and conducting analogical derivations yields the following result (appraisal costs are netted with rental revenues):
Appendix A
E (Ξ ) =
A.3.
371
⎛ ⎛ π * −1 ⎞ ⎞ ⎛ π * −1 ⎞ γ ⎟ ⎟⎟ − ν X ⋅ ϕ⎜⎜ ⎟⎟ + ⋅ ⎜⎜ Φ⎜⎜ ⎟ ⎛ π * −1 ⎞ ⎝ ⎝ ν X ⎠ ⎝ νX ⎠ λ + ρ − τ ⎠ ⎟⎟ ρ − τ + λ ⋅ Φ⎜⎜ ⎝ νX ⎠ λ
(A.16)
Unique Maximum of Expected et Sale Receipts
It is to prove that E(Γ) defined as:
E (Γ) =
⎞ ⎛ λ γ ⋅ ⎜ E(Π Π > π *) ⋅ (1 − FR ( π*)) + ⎟ ρ − τ + λ ⋅ (1 − FR (π*)) ⎜⎝ λ + ρ − τ ⎟⎠
⎛∞ ⎞ λ γ ⎟ = ⋅ ⎜ ∫ π ⋅ dFR (π) + ρ − τ + λ ⋅ (1 − FR (π*)) ⎜⎝ π* λ + ρ − τ ⎟⎠
(A.17)
possesses a unique global maximum with respect π* when τ < ρ. Computing the derivative of E(Γ) with respect to π* yields: λ ⋅ f R (π*) dE(Γ) = dπ * (ρ − τ + λ ⋅ (1 − FR (π*))2 ⎛ ∞ ⎞ λ⋅γ ⋅ ⎜⎜ λ ⋅ ∫ π ⋅ dFR (π) + − (ρ − τ + λ ⋅ (1 − FR (π*)) ⋅ π *⎟⎟ λ+ρ−τ ⎝ π* ⎠ with: fR(x) = dFR(x)/dx
(A.18)
Since τ < ρ by assumption, the sign of dE(Γ)/dπ* is the same as the sign of the expression in brackets. It is therefore sufficient to consider the latter one only, which is denoted as H(π*): ∞
H (π*) = λ ⋅ ∫ π ⋅ dF(π) + π*
λ⋅γ − (ρ − τ + λ ⋅ (1 − FR (π*))) ⋅ π * λ+ρ−τ
(A.19)
Since all expressions apart from π*, including the integral, are limited and always positive, the derivative of H(π*) approaches minus infinity for high (positive) values of π* and plus infinity for low (negative) values of π*. Furthermore, the derivative of H(π*) with respect to π* equals:
372
Appendix A
dH(π*) = −(ρ − τ + λ ⋅ (1 − FR (π*))) dπ *
(A.20)
and is strictly negative. Recapitulating, H(π*) is positive for low values of π*, negative for high values of π*, and strictly decreasing. This indicates that H(π*) has a unique root (zero) in which it changes from positive to negative values. Obviously, same applies to dE(Γ) /dπ*. It follows that E(Γ) increases for low π*, decreases for high π*, and possesses a unique maximum. This accomplishes the proof.
A.4.
Conditional Expected Values
A.4.1. Conditional Expected Offer The conditional expected offer under normal distribution of offers is to be computed: ∞
E(P P > p*) =
∫ p ⋅ dF(p)
p*
(A.21)
1 − F(p*)
Consider the integral expression substituting normal distribution density function for F: ∞
− 1 ∫ p ⋅ 2Π~ σ e p* ∞
( p−μ )2 2 σ2
− 1 ~ e 2Π σ
= ∫p⋅ p* ∞
dp
( p−μ )2 2 σ2
− 1 = ∫ ( p − μ) ⋅ ~ e 2Π σ p* ∞
∞
p*
( p−μ )2 2σ
2
∞
( p−μ )2 2 σ2
( p −μ )2 2 σ2
∞
∞
dp + ∫ μ ⋅
− 1 dp + ∫ μ ⋅ ~ e 2Π σ p*
− ( p − μ) 1 e ⋅ = −σ ∫ − 2 ~ σ 2Π σ p*
2
− 1 ~ e 2Π σ
dp − ∫ μ ⋅
p*
− 1 ~ e 2Π σ
( p−μ )2 2 σ2
dp
( p−μ )2 2σ
2
dp
− 1 dp + ∫ μ ⋅ ~ e 2Π σ p*
( p −μ )2 2 σ2
dp
Substituting with y = - (p-µ)²/2σ² and noticing that dy = - (p-µ)/σ²dp yields:
(A.22)
Appendix A ∞
1 ~ e 2Π σ
∫p ⋅
p*
= σ2 ⋅
373
−
−
( p −μ ) 2
(p*− μ )2 σ2
−∞
[ ]
1 y ~ e 2Π σ
− 1 =σ ⋅ ~ e 2Π σ
−
( p*− μ )2
−∞
σ2
∞
+μ⋅ ∫
p*
(p*− μ )2 σ2
2
∞
+μ⋅ ∫
p*
∞
− 1 1 y ~ e dy + μ ⋅ ∫ ~ e 2Π σ p* 2Π σ
∫
dp = σ 2
2 σ2
− 1 ~ e 2Π σ
− 1 ~ e 2Π σ
( p −μ ) 2 2σ2
dp
( p −μ ) 2 2 σ2
( p −μ ) 2 2 σ2
dp
(A.23)
dp = μ ⋅ (1 − F(p*)) + σ 2 ⋅ f ( p*)
Defining φ(p) and Φ(p) as standard normal density and standard normal distribution functions, respectively, yields finally: E(P P > p*) =
μ ⋅ (1 − F(p*)) + σ 2 ⋅ f (p*) f (p*) = μ + σ2 ⋅ (1 − F(p*)) (1 − F(p*))
⎛ p * −μ ⎞ ⎛ ⎛ p * −μ ⎞ ⎞ = μ + σ ⋅ ϕ⎜ ⎟ ⎜1 − Φ ⎜ ⎟⎟ ⎝ σ ⎠ ⎝ ⎝ σ ⎠⎠
(A.24)
For the relative version of the model the conditional relative offer is computed by the division of E(P|P>p*) by µ: E(Π Π > π*) = 1 + ν 2 ⋅
f R (π*)
(1 − FR (π*))
⎛ π * −1 ⎞ ⎛ ⎛ π * −1 ⎞ ⎞ = 1 + ν ⋅ ϕ⎜ ⎟ ⎜1 − Φ ⎜ ⎟⎟ ν ⎝ ⎠ ⎝ ⎝ ν ⎠⎠
(A.25)
with FR(π) and fR(π) being the relative offers’ normal density and normal distribution functions. A.4.2. Conditional Expected Square Offer The conditional expected square offer under normal distribution of offers is to be computed: ∞
E(P P > p*) = 2
∫p
2
⋅ dF(p)
p*
1 − F(p*)
(A.26)
Consider the integral expression substituting normal distribution density function for F. Define functions x(p) and y(p) as follows:
374
Appendix A
x ( p) =
− 1 ~ e 2Π σ
( p −μ ) 2 2 σ2
y( p) = p
⇒
x ′(p) = −
⇒
y′(p) = 1
( p − μ) − ~ e 2Π σ 3
( p −μ ) 2 2 σ2
(A.27)
According to the principle of integration by parts, the following equation holds for the above functions: −
∞
1 p2 σ 2 p∫*
− 1 ~ e 2Π σ
∞
( p − μ) − = −∫ p ⋅ ~ e 2Π σ3 p*
( p −μ ) 2
dp +
2 σ2
∞
− μ 1 p⋅ e 2 ∫ ~ 3 σ p* 2Π σ
( p−μ ) 2 2 σ2
dp
( p−μ ) 2 2 σ2
dp
∞
∞
p*
p*
= ∫ x ′(p) ⋅ y(p) ⋅ dp = x (p) ⋅ y(p) − ∫ x ( p) ⋅ y′( p) ⋅ dp
(A.28)
∞
( p −μ ) 2 ⎤ ( p −μ ) 2 ∞ ⎡ − − 1 1 2 2σ2 ⎥ − e e = ⎢p ⋅ ∫ 2Π~ σ 2σ dp ~ ⎥ ⎢ 2 Π σ ⎦ π* p* ⎣
= −p * ⋅
− 1 ~ e 2Π σ
( p*− μ ) 2
∞
−∫
2σ2
p*
− 1 ~ e 2Π σ
( p −μ ) 2 2 σ2
dp
Rearranging yields: ∞
∫p
2
p*
− 1 ~ e 2Π σ
( p −μ ) 2 2 σ2
dp
− 1 = σ ⋅ p *⋅ ~ e 2Π σ
2
( p*− μ ) 2 2 σ2
+σ
∞
2
∫
p*
− 1 ~ e 2Π σ
( p −μ ) 2 2σ2
∞
− 1 dp + μ ⋅ ∫ p ⋅ ~ 3e 2Π σ p*
Applying (A.23) allows the computation of the last integral:
(A.29)
( p −μ ) 2 2 σ2
dp
Appendix A ∞
2 ∫p
p*
− 1 ~ e 2Π σ
375 ( p −μ ) 2 2 σ2
dp
= σ 2 ⋅ p * ⋅f (p*) + σ 2 ⋅ (1 − F(p*)) + μ 2 ⋅ (1 − F(p*)) + μ ⋅ σ 2 ⋅ f (p*)
(
)
= σ 2 ⋅ (p * +μ ) ⋅ f (p*) + σ 2 + μ 2 ⋅ (1 − F(p*)) ⎛ p * −μ ⎞ 2 2 = σ ⋅ (p * +μ ) ⋅ ϕ⎜ ⎟+ σ +μ σ ⎝ ⎠
(
(A.30)
)⋅ ⎛⎜1 − Φ⎛⎜ p *σ−μ ⎞⎟ ⎞⎟ ⎝
⎝
⎠⎠
Defining fN(p) and FN(p) as normal density and normal distribution functions, respectively, and φ(p) and Φ(p) as standard normal density and standard normal distribution functions, respectively, yields finally: E (P 2 P > p*) = =
(
)
σ 2 ⋅ (p * +μ ) ⋅ f N ( p*) + σ 2 + μ 2 ⋅ (1 − FN (p*)) (1 − FN (p*))
= σ 2 ⋅ (p * +μ ) ⋅
(
f N (p*) + σ2 + μ2 (1 − FN (p*))
)
(A.31)
(
⎛ p * −μ ⎞ ⎛ ⎛ p * −μ ⎞ ⎞ 2 2 = σ ⋅ (p * +μ ) ⋅ ϕ⎜ ⎟ ⎜1 − Φ ⎜ ⎟⎟ + σ + μ ⎝ σ ⎠ ⎝ ⎝ σ ⎠⎠
)
For the relative version of the model the conditional relative offer is computed by the division of E(P²|P>p*) by µ:
(
) ⎛ ⎛ π * −1 ⎞ ⎞ ⎜1 − Φ ⎜ ⎟ ⎟ + (ν ν
E (Π 2 Π > π*) = ν 2 ⋅ (π * +1) ⋅ f R (π*) + ν 2 + 1 ⋅ (1 − FR ( π*)) ⎛ π * −1 ⎞ = ν ⋅ (π * +1) ⋅ ϕ⎜ ⎟ ⎝ ν ⎠ ⎝
⎝
⎠⎠
2
)
+1
(A.32)
A.4.3. Conditional Expected Offer and Conditional Expected Square Offer Below the Reservation Price The conditional expected offer and the conditional expected square offer below the reservation price (purchase case) can be computed analogically yielding: E (Π Π < π*) = 1 − ν 2 ⋅
f R (π*) ⎛ π * −1 ⎞ ⎛ π * −1 ⎞ = 1 − ν ⋅ ϕ⎜ ⎟ Φ⎜ ⎟ FR ( π*) ⎝ ν ⎠ ⎝ ν ⎠
(A.33)
376
Appendix A
(
)
E (Π 2 Π < π*) = ν 2 ⋅ (1 − π *) ⋅ f R (π*) / FR (π*) + ν 2 + 1 ⎛ π * −1 ⎞ ⎛ π * −1 ⎞ 2 = ν ⋅ (1 − π *) ⋅ ϕ⎜ ⎟ Φ⎜ ⎟ + ν +1 ⎝ ν ⎠ ⎝ ν ⎠
(
A.5.
)
(A.34)
Closed Form Solutions for the Variance of et Sale Receipts and the Variance of et Purchase Expense
The closed form formula for V(Γ) is to be derived. From the general property of variance, it holds that: V(Γ) = E(Γ²) – E²(Γ)
(A.35)
The formula for E(Γ) has been derived in Appendix A.2, so that only the formula for E(Γ²) needs to be derived: −( ρ− τ )⋅ ∑ Tj −( ρ− τ )⋅∑ Ti ⎞ ⎛ j< i i + ∑ γ ⋅ Ti ⋅ e Π < π *⎟⎟ E(Γ 2 ) = E⎜⎜ Π ⋅ (1 + A ⋅ ∑ Ti ) ⋅ e ⎠ ⎝ i i ∞ ⎡∞∞ ∞ ∞ = ∑ ⎢ ∫ ∫ ...∫ ∫ ⎢ i =1 −∞ 0 0 −∞ ⎣⎢
i j i i ⎛ −( ρ− τ )⋅ ∑ t j −( ρ− τ )⋅ ∑ t k ⎞ j=1 ⎜ π ⋅ (1 + a ⋅ t ) ⋅ e k =1 ⎟ + γ ⋅ ⋅ t e ∑ j ∑ j ⎜ i ⎟ j=1 j=1 ⎝ ⎠
2
2
i ⎤ ⋅ Pr(Π i = πi Π i > π *) ⋅ (1 − FR (π*)) ⋅ FRi−1 (π*) ⋅ dπi ⋅ ∏ dFT ( t j ) ⋅ dFA (a )⎥ j=1 ⎦
⎡∞∞ ∞ ∞ = ∑ ⎢ ∫ ∫ ...∫ ∫ i =1 ⎢ −∞ 0 0 −∞ ⎣ ∞
i i ⎛ −2 ( ρ− τ )⋅ ∑ t j j=1 ⎜ π 2 ⋅ (1 + a ⋅ t ) 2 ⋅ e ∑ j ⎜ i j=1 ⎝
i
+ 2 ⋅ πi ⋅ (1 + a ⋅ ∑ t j ) ⋅ e j=1
i
−( ρ− τ )⋅ ∑ t j j=1
i
⋅∑γ⋅tj ⋅e j=1
j
−( ρ − τ ) ∑ t k k =1
j ⎛ i −( ρ− τ )⋅ ∑ t k ⎞ k =1 ⎟ + ⎜∑ γ ⋅ t j ⋅e ⎜ j=1 ⎟ ⎝ ⎠
i ⎤ ⋅ Pr(Π i = πi Π i > π *) ⋅ (1 − FR (π*)) ⋅ FRi−1 (π*) ⋅ dπi ⋅ ∏ dFT ( t j ) ⋅ dFA (a )⎥ j=1 ⎦
2
⎞ ⎟ ⎟ ⎟ ⎠
(A.36)
Extending the above expression yields three addends that are considered separately:
Appendix A
377
⎡ ∞∞ ∞ Add1 = ∑ ⎢ ∫ ∫ ...∫ i =1 ⎢ − ∞ 0 0 ⎣ ∞
i i ⎛ − 2 ( ρ − τ )⋅ ∑ t j ⎞ j =1 ⎟ ⎜ E (Π 2 Π > π *) ⋅ (1 + a ⋅ t ) 2 ⋅ e ∑ j ⎜ ⎟ j=1 ⎝ ⎠
(A.37)
⎤ ⋅ (1 − FR (π*)) ⋅ FRi −1 (π*) ⋅ ∏ dFT ( t j ) ⋅ dFA (a )⎥ j=1 ⎦ i
∞ ⎡∞ ∞ ∞ Add 2 = ∑ ⎢ ∫ ∫ ...∫ i =1 ⎢ − ∞ 0 0 ⎣
i j i i ⎛ − (ρ − τ)⋅ ∑ t j − (ρ − τ)⋅ ∑ t k ⎞ j=1 ⎜ 2 ⋅ E(Π Π > π *) ⋅ (1 + a ⋅ t ) ⋅ e ⎟ k =1 ⋅∑γ ⋅ tj ⋅e ∑ j ⎜ ⎟ j =1 j =1 ⎝ ⎠ (A.38)
i ⎤ ⋅ (1 − FR (π*)) ⋅ FRi −1 (π*) ⋅ ∏ dFT ( t j ) ⋅ dFA (a )⎥ j =1 ⎦
∞ ⎡∞ ∞ Add 3 = ∑ ⎢ ∫ ...∫ i =1 ⎢ 0 ⎣ 0
2
j i ⎛ i −( ρ− τ )⋅ ∑ t k ⎞ ⎤ k =1 ⎟ ⎜ γ ⋅t ⋅e ⋅ (1 − FR ( π*) ) ⋅ FRi−1 ( π*) ⋅ ∏ dFT ( t j ) ⎥ ∑ j ⎟ ⎜ j=1 j=1 ⎦ ⎠ ⎝
(A.39)
For further derivations, it is convenient to define the following variables: ∞
X1 = ∫ λe − t (ρ − τ ) e − λt dt = 0
λ (ρ − τ) + λ
∞
X 2 = ∫ λe − 2 t (ρ − τ ) e − λt dt = 0
λ 2(ρ − τ) + λ
∞
Y1 = ∫ λte − t (ρ − τ ) e − λt dt = 0
λ
((ρ − τ) + λ )2
∞
Y2 = ∫ λte − 2 t ( ρ − τ ) e − λt dt = 0
∞
λ
(2(ρ − τ) + λ )2
Z 2 = ∫ λt 2e − 2 t ( ρ − τ ) e − λt dt = 0
(A.40)
2λ (2(ρ − τ) + λ )3
X1, X2, Y1, Y2, and Z is equal for all time intervals ti and can be employed to simplify the notation of the addends Add1, Add2, and Add3 when computing integrals. Furthermore, the fact that the expected value of the market uncertainty factors is zero means ∞
that
∫ a ⋅ dFA (a ) = 0 .
−∞
378
Appendix A
The first addend can be simplified as follows: ∞ ⎡∞∞ ∞ Add1 = ∑ ⎢ ∫ ∫ ...∫ i =1 ⎢ −∞ 0 0 ⎣
i i ⎛ − 2 ( ρ−τ )⋅ ∑ t j ⎞ j=1 ⎟ ⎜ E(Π 2 Π > π *) ⋅ (1 + a ⋅ t ) 2 ⋅ e ∑ j ⎟ ⎜ j=1 ⎝ ⎠
i i ⎤ ⋅ (1 − FR (π*)) ⋅ FRi−1 (π*) ⋅ ∏ dFT ( t j ) ⋅ ∏ dFA (a )⎥ j=1 j=1 ⎦
⎡∞∞ ∞ = ∑ ⎢ ∫ ∫ ...∫ i =1 ⎢ −∞ 0 0 ⎣
(A.41)
i 2 ⎛ ⎞ ⎛ ⎞ i i ⎜ E(Π 2 Π > π *) ⋅ ⎜1 + 2 ⋅ a ⋅ t + a 2 ⋅ ⎛⎜ t ⎞⎟ ⎟ ⋅ e −2(ρ−τ )⋅ j∑=1t j ⎟ ∑ ∑ j j ⎜⎜ ⎜ ⎟ ⎟⎟ ⎜ ⎟ j=1 ⎝ j=1 ⎠ ⎠ ⎝ ⎝ ⎠
∞
i ⎤ ⋅ (1 − FR (π*)) ⋅ FRi−1 (π*) ⋅ ∏ dFT ( t j ) ⋅ dFA (a )⎥ j=1 ⎦
Computing integrals over a as well as t1 to ti and substituting according to (A.40) yields: ∞
[
Add1 = ∑ E (Π 2 Π > π *) ⋅ X i2 ⋅ (1 − FR (π*)) ⋅ FRi−1 ( π*) i =1
∞
[
]
+ E (A 2 ) ⋅ ∑ E (Π 2 Π > π *) ⋅ i ⋅ Z 2 ⋅ X i2−1 ⋅ (1 − FR (π*)) ⋅ FRi−1 ( π*) i =1 ∞
[
]
+ E (A 2 ) ⋅ ∑ E (Π 2 Π > π *) ⋅ E (A 2 ) ⋅ i ⋅ (i − 1) ⋅ Y22 ⋅ X i2−2 ⋅ (1 − FR (π*)) ⋅ FRi−1 (π*) i =2
(A.42)
]
Since ρ > τ, X2 is always positive and smaller than one, and the infinite sums are always finite. As long as the variance of A, i.e. E(A²) = σ²A is finite, Add1 is also finite and equals: Add1 = +
E (Π 2 Π > π *) ⋅ X 2 ⋅ (1 − FR (π*)) σ 2A ⋅ E (Π 2 Π > π *) ⋅ Z 2 ⋅ (1 − FR (π*)) + (1 − X 2 ⋅ FR (π*)) (1 − X 2 ⋅ FR (π*))2
2 ⋅ σ 2A ⋅ E (Π 2 Π > π *) ⋅ Y22 ⋅ (1 − FR (π*)) ⋅ FR (π*)
(1 − X 2 ⋅ FR (π*))3
The second addend can be simplified as follows:
(A.43)
Appendix A
379
∞ ⎡ ∞∞ ∞ Add 2 = ∑ ⎢ ∫ ∫ ...∫ i =1 ⎢ −∞ 0 0 ⎣
i j i i ⎛ −( ρ−τ )⋅ ∑ t j −( ρ− τ )⋅ ∑ t k ⎞ j=1 ⎜ 2 ⋅ E (Π Π > π *) ⋅ (1 + a ⋅ t ) ⋅ e k =1 ⎟ t e ⋅ γ ⋅ ⋅ ∑j ∑ j ⎟ ⎜ j=1 j=1 ⎠ ⎝
i ⎤ ⋅ (1 − FR (π*) ) ⋅ FRi−1 ( π*) ⋅ ∏ dFT ( t j ) ⋅ dFA (a ) ⎥ j=1 ⎦ = 2 ⋅ E(Π Π > π *) ⋅ (1 − FR (π*)) ⋅ γ
⎡∞ ∞ ∞ ⋅ ⎢∑ ∫ ...∫ ⎢ i=1 0 0 ⎣
(A.44)
j ⎤ i ⎛ −(ρ−τ)⋅ ∑i t j i −( ρ−τ )⋅ ∑ t k ⎞ j=1 ⎜e k =1 ⎟ ⋅ FRi−1 (π*) ⋅ ∏ dFT ( t j )⎥ ⋅∑tj ⋅e ⎟ ⎜ ⎥ j=1 j=1 ⎠ ⎝ ⎦
Integrating and substituting according to (A.40) yields: i ∞ ⎡ ⎤ Add 2 = 2 ⋅ E(Π Π > π *) ⋅ (1 − FR (π*)) ⋅ γ ⋅ ∑ ⎢FRi−1 (π*)∑ Y2 ⋅ X 2j−1 ⋅ X1i− j ⎥ i =1 ⎣ j=1 ⎦ i i ∞ ⎡ ⎤ X X − 1 = 2 ⋅ E(Π Π > π *) ⋅ (1 − FR ( π*)) ⋅ γ ⋅ Y2 ⋅ ∑ ⎢FRi−1 (π*) 2 ⎥ X 2 − X1 ⎦ i =1 ⎣
(A.45)
Since due to the assumption that ρ > τ both X1 < 1 and X2 < 1, the infinite sum is always finite and can be simplified as follows: Add2 =
2 ⋅ E(Π Π > π *) ⋅ (1 − FR (π*)) ⋅ γ ⋅ Y2 (1 − X1 ⋅ FR (π*)) ⋅ (1 − X 2 ⋅ FR (π*))
(A.46)
The third addend can be simplified as follows: ⎡∞ ∞ Add3 = ∑ ⎢ ∫ ...∫ i =1 ⎢ 0 ⎣ 0 ∞
⎡∞ ∞ = ∑ ⎢ ∫ ...∫ ⎢ 0 i =1 0 ⎣⎢ ∞
2
j i ⎛ i −( ρ−τ )⋅ ∑ t k ⎞ ⎤ k =1 ⎟ ⎜ γ ⋅t ⋅e ⋅ (1 − FR (π*)) ⋅ FRi−1 (π*) ⋅ ∏ dFT ( t j )⎥ ∑ j ⎜ j=1 ⎟ j=1 ⎦ ⎠ ⎝
2 j ⎛ i ⎛ −( ρ−τ )⋅ ∑ t k ⎞ ⎜ ⎜ k =1 ⎟ ⎜∑⎜γ ⋅ t j ⋅e ⎟ ⎜ j=1 ⎝ ⎠ ⎝
(A.47)
⎞ i i-1 ⎛ −( ρ−τ )⋅ ∑ t k ⎞ ⎛ −( ρ−τ )⋅ ∑ t k ⎞ ⎟ k =1 ⎟ ⎜ k =1 ⎟ + 2 ⋅ ∑ ∑ ⎜ γ ⋅ tm ⋅ e ⋅ γ ⋅ tn ⋅ e ⎜ ⎟ ⎜ ⎟ ⎟⎟ m =1 n =m +1 ⎝ ⎠ ⎝ ⎠⎠ i >1 i ⎤ ⋅ (1 − FR (π*)) ⋅ FRi−1 (π*) ⋅ ∏ dFT ( t j )⎥ j=1 ⎦ m
n
380
Appendix A
Rearranging the sums and substituting for integrals over t according (A.40) yields: Add 3 = γ 2 ⋅ (1 − FR ( π*) ) i ∞ ⎛ i −1 i ⎛ ∞ ⎛ ⎞ j−1 ⎞ k − j−1 j−1 ⎞ ⋅ ⎜ ∑ ⎜⎜ FRi−1 ( π*) ⋅ ∑ Z 2 ⋅ X 2 ⎟⎟ + 2 ⋅ ∑ ⎜⎜ FRi−1 (π*) ⋅ ∑ ∑ Y1 ⋅ Y2 ⋅ X1 ⋅ X 2 ⎟⎟ ⎟ ⎜ i=1 ⎟ j=1 i=2 ⎝ j=1 k = j+1 ⎠ ⎠⎠ ⎝ ⎝ i ∞ ⎛ ⎛ X −1⎞ ⎟ = γ 2 ⋅ (1 − FR ( π*) ) ⋅ ⎜ Z 2 ⋅ ∑ ⎜⎜ FRi−1 ( π*) ⋅ 2 ⎜ X 2 − 1 ⎟⎠ i =1 ⎝ ⎝
(A.48)
i i ∞ ⎛ X ( X − 1) + X 2 ( X1 − 1) − X1 + X 2 ⎞ ⎞⎟ ⎟ + 2 ⋅ Y1 ⋅ Y2 ⋅ ∑ ⎜⎜ FRi−1 (π*) ⋅ 1 2 ⎟⎟ ( X1 − 1) ⋅ ( X 2 − 1) ⋅ ( X 2 − X1 ) i =2 ⎝ ⎠⎠
Since due to the assumption that ρ > τ both X1 < 1 and X2 < 1, the infinite sum is always finite and can be simplified as follows: ⎡ ⎤ Z2 Add 3 = γ 2 ⋅ (1 − FR ( π*) ) ⋅ ⎢ ⎥ ⎣ (1 − FR (π*) ) ⋅ (1 − X 2 ⋅ FR ( π*) ) ⎦ ⎡ ⎤ 2 ⋅ Y1 ⋅ Y2 ⋅ FR ( π*) + γ 2 ⋅ (1 − FR ( π*) ) ⋅ ⎢ ⎥ 1 F ( *) 1 X F ( *) 1 X F ( *) ( ) ( ) ( ) − π ⋅ − ⋅ π ⋅ − ⋅ π R 1 R 2 R ⎣ ⎦ = =
2 ⋅ γ 2 ⋅ Y1 ⋅ Y2 ⋅ FR ( π*) γ 2 ⋅ Z2 + (1 − X 2 ⋅ FR (π*)) (1 − X1 ⋅ FR (π*)) ⋅ (1 − X 2 ⋅ FR (π*))
(A.49)
γ 2 ⋅ (Z 2 ⋅ (1 − X1 ⋅ FR ( π*) ) + 2 ⋅ Y1 ⋅ Y2 ⋅ FR ( π*) ) (1 − X1 ⋅ FR (π*)) ⋅ (1 − X 2 ⋅ FR (π*))
Summing up the three sums yields: E (Γ 2 ) = +
E(Π 2 Π > π *) ⋅ X 2 ⋅ (1 − FR (π*)) σ 2A ⋅ E(Π 2 Π > π *) ⋅ Z 2 ⋅ (1 − FR (π*)) + (1 − X 2 ⋅ FR (π*)) (1 − X 2 ⋅ FR (π*))2
2 ⋅ σ 2A ⋅ E(Π 2 Π > π *) ⋅ Y22 ⋅ (1 − FR (π*)) ⋅ FR (π*)
(1 − X 2 ⋅ FR (π*))3 2 ⋅ E(Π Π > π *) ⋅ (1 − FR (π*)) ⋅ γ ⋅ Y2 + (1 − X1 ⋅ FR (π*))(1 − X 2 ⋅ FR (π*)) 2 γ ⋅ (Z 2 ⋅ (1 − X1 ⋅ FR (π*)) + 2 ⋅ Y1 ⋅ Y2 ⋅ FR (π*)) + (1 − X1 ⋅ FR (π*)) ⋅ (1 − X 2 ⋅ FR (π*))
Finally, the following variance formula yields:
(A.50)
Appendix A V (Γ ) = +
381
E (Π 2 Π > π *) ⋅ X 2 ⋅ (1 − FR (π*)) σ 2A ⋅ E (Π 2 Π > π *) ⋅ Z 2 ⋅ (1 − FR (π*)) + (1 − X 2 ⋅ FR (π*)) (1 − X 2 ⋅ FR (π*))2
2 ⋅ E (A 2 ) ⋅ E (Π 2 Π > π *) ⋅ Y22 ⋅ (1 − FR (π*)) ⋅ FR (π*)
(1 − X 2 ⋅ FR (π*))3 2 ⋅ E (Π Π > π *) ⋅ (1 − FR (π*)) ⋅ γ ⋅ Y2 + (1 − X1 ⋅ FR (π*))(1 − X 2 ⋅ FR (π*)) 2 γ ⋅ (Z 2 ⋅ (1 − X1 ⋅ FR ( π*)) + 2 ⋅ Y1 ⋅ Y2 ⋅ FR ( π*)) + (1 − X1 ⋅ FR (π*)) ⋅ (1 − X 2 ⋅ FR (π*)) 2 ⎛ E (Π Π > π *) ⋅ X1 ⋅ (1 − FR ( π*)) + Y1 ⋅ γ ⎞ ⎜ ⎟ −⎜ ⎝
(A.51)
⎟ ⎠
1 − X1 ⋅ FR ( π*)
The variance of the expense at purchase can be computed analogically yielding:
V (Ξ ) = +
E(Π 2 Π < π *) ⋅ FR (π*) ⋅ X 2 σ 2A ⋅ E(Π 2 Π < π *) ⋅ Z 2 ⋅ FR (π*) + (1 − X 2 ⋅ (1 − FR (π*))) (1 − X 2 ⋅ (1 − FR (π*)))2
2 ⋅ σ 2A ⋅ E (Π 2 Π < π *) ⋅ Y22 ⋅ (1 − FR (π*)) ⋅ FR (π*)
+
(1 − X 2 ⋅ (1 − FR (π*)))3
2 ⋅ E (Π Π < π *) ⋅ FR (π*) ⋅ γ ⋅ Y2
(1 − X1 ⋅ (1 − FR (π*))) ⋅ (1 − X 2 ⋅ (1 − FR (π*))) γ 2 ⋅ (Z 2 ⋅ (1 − X1 ⋅ (1 − FR (π*))) + 2 ⋅ Y1 ⋅ Y2 ⋅ FR (π*)) + (1 − X1 ⋅ (1 − FR (π*))) ⋅ (1 − X 2 ⋅ (1 − FR (π*))) ⎛ E (Π Π < π *) ⋅ X1 ⋅ FR (π*) + Y1 ⋅ γ ⎞ ⎟⎟ − ⎜⎜ 1 − X1 ⋅ (1 − FR (π*)) ⎝ ⎠ A.6.
(A.52)
2
Covariance between et Sale Receipts
The formula for the covariance between expected net receipts from the sales of two properties is to derive. Two variants are considered separately. In the Variant A, the correlation between unexpected market changes (AX and AY) is independent of the time horizon to which they refer. In the Variant B, the correlation between AX and AY refers only to the time period in which both assets remain unsold. A.6.1. Variant A Receipts from the sale of assets X and Y are defined as follows:
382
Appendix A k
i
ΓX ,i = Π X,i ⋅ (1 + A X ∑ TX ,k ) ⋅ e
−( ρ− τY ) ∑ TY , n
i
k =1
−( ρ− τX ) ∑ TX , n n =1
(A.53)
k =1
k
j
ΓY , j = Π Y, j ⋅ (1 + A Y ∑ TY ,k ) ⋅ e
k
+ ∑ Tk γ X ⋅ e
n =1
−( ρ− τY ) ∑ TY , n n =1
k =1
k
j
+ ∑ TY ,k γ Y ⋅ e
−( ρ− τY ) ∑ TY , n n =1
(A.54)
k =1
Covariance can be computed with a simplified formula as follows: cov(ΓX , ΓY ) = E (ΓX ⋅ ΓY ) − E(ΓX ) ⋅ E (ΓY )
(A.55)
The sale of X takes place when the respective offer exceeds the reservation price πX and the sale of Y takes place when the respective offer exceeds the reservation price πY. Hence, the covariance can be formulated as follows (one expectation operator is used for all random variables): k k i i ⎡⎛ −( ρ−τY ) ∑ TY ,n −( ρ−τX ) ∑ TX ,n ⎞ n =1 n =1 ⎟ + ∑ Tk γ X ⋅ e cov(Γx , ΓY ) = E ⎢⎜ Π X,i ⋅ (1 + A X ∑ TX ,k ) ⋅ e ⎟ ⎢⎣⎜⎝ k =1 k =1 ⎠ k k j j ⎤ ⎛ −( ρ−τY ) ∑ TY , n −( ρ−τY ) ∑ TY ,n ⎞ n =1 n =1 ⎟ Π > π∗ , Π > π ∗ ⎥ ⋅ ⎜ Π Y, j ⋅ (1 + A Y ∑ TY ,k ) ⋅ e + ∑ TY ,k γ Y ⋅ e X X Y Y ⎜ ⎟ k =1 k =1 ⎝ ⎠ ⎦⎥
(A.56)
⎛ ⎞ −( ρ−τY ) ∑ TY ,n −( ρ−τX ) ∑ TX ,n n =1 n =1 − E⎜ Π X,i ⋅ (1 + A X ∑ TX ,k ) ⋅ e + ∑ Tk γ X ⋅ e Π X > π∗X ⎟ ⎜ ⎟ k =1 k =1 ⎝ ⎠ k k j j ⎛ ⎞ −( ρ−τY ) ∑ TY ,n −( ρ−τY ) ∑ TY , n n =1 n =1 ⋅ E⎜ Π Y, j ⋅ (1 + A Y ∑ TY ,k ) ⋅ e + ∑ TY ,k γ Y ⋅ e Π Y > π∗Y ⎟ ⎜ ⎟ k =1 k =1 ⎝ ⎠ k
i
i
k
For better tractability, the conditions of ΠX and ΠY exceeding the respective reservation prices πX and πY are omitted in the following derivations. Furthermore, following substitutions are applied: j i ~ ~ TX ,i = ∑ TX ,k and TY , j = ∑ TY ,k k =1
k
εX = e
−( ρ−τX ) ∑ TX , n n =1
i
H X = ∑ Tk γ X ⋅ e k =1
(A.57)
k =1
k
and ε Y = e k
−( ρ− τX ) ∑ TX , n n =1
−( ρ−τY ) ∑ TY , n n =1
j
and H Y = ∑ TY ,k γ Y ⋅ e k =1
(A.58) k
−( ρ− τY ) ∑ TY , n n =1
(A.59)
Appendix A
383
Applying these simplifications allows for the following presentation for the covariance formula:
[(
)(
)]
~ ~ cov(Γx , ΓY ) = E Π X ⋅ (1 + A X TX ) ⋅ ε X + H X ⋅ Π Y ⋅ (1 + A Y TY ) ⋅ ε Y + H Y ~ ~ − E Π X ⋅ (1 + A X TX ) ⋅ ε X + H X ⋅ E Π Y ⋅ (1 + A Y TY ) ⋅ ε Y + H Y ~ ~ = E Π X ⋅ ε X + Π X ⋅ A X ⋅ TX ⋅ ε X + H X ⋅ Π Y ⋅ ε Y + Π Y ⋅ A Y ⋅ TY ⋅ ε Y + H Y ~ ~ − E Π X ⋅ ε X + A X ⋅ Π X ⋅ TX ⋅ ε X + H X ⋅ E Π Y ⋅ ε Y + Π Y ⋅ A Y ⋅ TY ⋅ ε Y + H Y ~ ´= E Π X ⋅ ε X ⋅ Π Y ⋅ ε Y + Π X ⋅ ε X ⋅ Π Y ⋅ A Y ⋅ TY ⋅ ε X + Π X ⋅ ε X ⋅ H Y ~ + Π X ⋅ A X ⋅ TX ⋅ ε X ⋅ Π Y ⋅ ε Y ~ ~ ~ + Π X ⋅ A X ⋅ TX ⋅ ε X ⋅ Π Y ⋅ A Y ⋅ TY ⋅ ε Y + Π X ⋅ A X ⋅ TX ⋅ ε X ⋅ H Y ~ + H X ⋅ Π Y ⋅ ε Y + H X ⋅ Π Y ⋅ A Y ⋅ TY ⋅ ε Y + H X ⋅ H Y ~ ~ − E Π X ⋅ ε X + Π X ⋅ A X ⋅ TX ⋅ ε X + H X ⋅ E Π Y ⋅ ε Y + Π Y ⋅ A Y ⋅ TY ⋅ ε Y + H X
( (( ( (
) (
(
)
)( ) (
)) )
)
) (
(A.60)
)
Utilizing the fact that random variables referring to X and to Y are independent except from AX and AY, which are correlated with each other but independent of all other variables, the covariance of receipts can be rewritten as: cov(Γx , ΓY ) =
(
)
(
)
~ = E (Π X ⋅ ε X ) ⋅ E (Π Y ⋅ ε Y ) + E (Π X ⋅ ε X ) ⋅ E (A Y ) ⋅ E Π Y ⋅ TY ⋅ ε Y ~ + E (Π X ⋅ ε X ) ⋅ E(H Y ) + E(A X ) ⋅ E Π X ⋅ TX ⋅ ε X ⋅ E (Π Y ⋅ ε Y ) ~ ~ + E(A X ⋅ A Y ) ⋅ E Π X ⋅ TX ⋅ ε X ⋅ E Π Y ⋅ TY ⋅ ε Y ~ + E(A X ) ⋅ E Π X ⋅ TX ⋅ ε X ⋅ E (H Y ) + E(H X ) ⋅ E (Π Y ⋅ ε Y ) ~ + E(H X ) ⋅ E (A Y ) ⋅ E Π Y ⋅ TY ⋅ ε Y + E(H X ) ⋅ E (H Y ) ~ − E (Π X ⋅ ε X ) ⋅ E (Π Y ⋅ ε Y ) − E (Π X ⋅ ε X ) ⋅ E(A Y ) ⋅ E Π Y ⋅ TY ⋅ ε Y ~ − E (Π X ⋅ ε X ) ⋅ E(H Y ) − E(A X ) ⋅ E Π X ⋅ TX ⋅ ε X ⋅ E (Π Y ⋅ ε Y ) ~ ~ − E (A X ) ⋅ E Π X ⋅ TX ⋅ ε X ⋅ E(A Y ) ⋅ E Π Y ⋅ TY ⋅ ε Y ~ − E (A X ) ⋅ E Π X ⋅ TX ⋅ ε X ⋅ E(H Y ) − E(H X ) ⋅ E(Π Y ⋅ ε Y ) ~ − E (H X ) ⋅ E(A Y ) ⋅ E Π Y ⋅ TY ⋅ ε Y − E(H X ) ⋅ E(H Y )
(
( (
(
(
(
( ) (
)
) )
)
(
) )
)
(
(A.61)
)
)
Applying the fact that the expected values of both AX and AY are zero and simplifying yields:
(
) (
~ ~ cov(Γx , ΓY ) = E(A X ⋅ A Y ) ⋅ E Π X ⋅ TX ⋅ ε X ⋅ E Π Y ⋅ TY ε X
)
(A.62)
384
Appendix A
For further simplification consider the following expression: ( τ−ρ ) ∑ Tj ⎛ ⎞ ~ j E Π ⋅ T ⋅ ε = E ⎜ Π ⋅ ( ∑ Tj )e Π > π *⎟ ⎜ ⎟ j ⎝ ⎠ i ∞ ⎡∞∞ ∞ ⎛ i ( τ −ρ ) ∑ t j ⎞ j=1 ⎟ = ∑ ⎢ ∫ ∫ ...∫ ⎜ πi ⋅ ∑ t j ⋅ e ⎜ ⎟ = i =1 ⎢ −∞ 0 j 1 0 ⎝ ⎠ ⎣
(
)
(A.63)
i ⎤ ⋅ Pr(Π i = πi Π i > π *) ⋅ (1 − FR (π*)) ⋅ FRi−1 (π*) ⋅ ∏ dFT ( t j ) ⋅ dπi ⎥ j=1 ⎦
Integrating over πi yields: ∞ ⎡ ~ E Π ⋅ T ⋅ ε = ∑ ⎢ ∫ ...∫ i =1 ⎢ 0 0 ⎣
(
)
∞
∞
i i ⎛ ( τ −ρ ) ∑ t j ⎞ j=1 ⎟ ⎜ E (Π Π > π *) ⋅ t ⋅ e ∑j ⎟ ⎜ j=1 ⎝ ⎠
(A.64)
⎤ ⋅ (1 − FR (π*)) ⋅ FRi−1 (π*) ⋅ ∏ dFT ( t j )⎥ j=1 ⎦ i
Assuming exponential distribution of t1 to ti and substituting for FT(.) yields: ∞ ⎡∞ ∞ ~ E Π ⋅ T ⋅ ε = ∑ ⎢ ∫ ...∫ i =1 ⎣ ⎢0 0
(
)
i i ⎛ ⎞ ⎜ E (Π Π > π *) ⋅ ∑ t j ⋅ ∏ e t j ( τ−ρ) ⎟ ⎜ ⎟ j=1 j=1 ⎝ ⎠
i ⎤ tλ ⋅ (1 − FR (π*)) ⋅ FRi−1 (π*) ⋅ ∏ λ ⋅ e j dt j ⎥ j=1 ⎦
∞ ⎡∞ ∞ = ∑ ⎢ ∫ ...∫ i =1 ⎣ ⎢0 0
i i ⎛ ⎞ ⎜ E (Π Π > π *) ⋅ ∑ t j ⋅ ∏ λ ⋅ e t j ( τ−ρ+λ ) ⎟ ⎜ ⎟ j=1 j=1 ⎝ ⎠
i ⎤ ⋅ (1 − FR (π*)) ⋅ FRi−1 (π*) ⋅ ∏ dt j ⎥ j=1 ⎦
∞ ⎡ ⎛ λ = ∑ ⎢E (Π Π > π *) ⋅ ⎜ i ⋅ ⎜ (ρ − τ + λ ) 2 i =1 ⎢ ⎝ ⎣
⎛ ⎞ λ ⋅ ⎜⎜ ⎟⎟ ⎝ρ−τ+λ⎠
i −1
⎞ ⎟ ⋅ (1 − F (π*)) ⋅ Fi−1 (π*) R R ⎟ ⎠
1 1 − FR ( π*) ∞ ⎛ λ ⋅ FR ( π*) ⎞ ⎟ = E (Π Π > π *) ⋅ ⋅ ⋅∑i ⋅⎜ ρ − τ + λ FR (π*) i=1 ⎜⎝ (λ + ρ − τ) ⎟⎠
]
i
Finally, computing the sum, which is finite if λ ⋅ FR (π*) < λ + ρ − τ , yields:
(A.65)
Appendix A
(
)
385
~ E Π ⋅T⋅ε = = E (Π Π > π *) ⋅ =
1 − FR ( π*) λ ⋅ FR (π*) ⋅ (λ + ρ − τ) 1 ⋅ ⋅ λ + ρ − τ FR (π*) (ρ − τ + λ ⋅ (1 − FR (π*)))2
(A.66)
E (Π Π > π *) ⋅ (1 − FR (π*)) ⋅ λ
(ρ − τ + λ ⋅ (1 − FR (π*)))2
The result can be alternatively expressed as: E (Π Π > π *) ⋅ (1 − FR (π*)) ⋅ Y1 ~ E Π ⋅T⋅ε = (1 − X1 ⋅ FR (π*)) 2
(
)
with: X1 =
λ (ρ − τ) + λ
and
Y1 =
(A.67) λ
((ρ − τ) + λ )2
Applying this result to (A.62) yields and assuming E (A X ⋅ A Y ) to be known and equal σ XY yields the following covariance formula:
cov(ΓX ΓY ) = σ XY ⋅ ⋅
(
)
E (Π X Π X > π X *) ⋅ 1 − FR X ( π X *) ⋅ Y1,X
(
(1 − X1,X ⋅ FR X (π X *))
2
)
E (Π Y Π Y > π Y *) ⋅ 1 − FR Y ( π Y *) ⋅ Y1,Y
(A.68)
(1 − X1,Y ⋅ FR Y ( π Y *)) 2
The assumption of normally distributed offers allows further simplification: cov(Γx , ΓY ) ⎛⎛ ⎛ π* − 1 ⎞ ⎞ ⎛ π* − 1 ⎞ ⎞⎟ ⎛ π* − 1 ⎞ ⎞ ⎛ π* − 1 ⎞ ⎞ ⎛ ⎛ ⎟⎟ ⋅ ⎟⎟ ⎟ + ν X ⋅ ϕ⎜⎜ X ⎟⎟ ⎟ ⋅ ⎜ ⎜1 − Φ⎜⎜ Y ⎟⎟ ⎟ + ν Y ⋅ ϕ⎜⎜ Y = σ XY ⋅ ⎜ ⎜⎜1 − Φ⎜⎜ X ⎜ ⎟ ⎟ ⎜ ⎝ νY ⎠⎠ ⎝ ν Y ⎠ ⎟⎠ ⎝ νX ⎠⎠ ⎝ ν X ⎠ ⎟⎠ ⎜⎝ ⎝ ⎝⎝ Y1,X Y1,Y ⋅ ⋅ 2 2 ⎛ ⎛ * ⎞⎞ ⎛ ⎛ * ⎞⎞ ⎜1 − X1,X ⋅ Φ⎜ π X − 1 ⎟ ⎟ ⎜1 − X1,Y ⋅ Φ⎜ π Y − 1 ⎟ ⎟ ⎜ ν ⎟⎟ ⎜ ⎜ ν ⎟⎟ ⎜ ⎝ X ⎠⎠ ⎝ ⎝ Y ⎠⎠ ⎝
(A.69)
A.6.2. Variant B ~ If the TX is chosen as the (shorter) reference time horizon, the market change factor ~ ~ for the asset Y, (1 + A Y TY ) can be split in two parts: the part referring to TX and the part referring the time before or after the sale of X. Thus:
386
Appendix A
(
~ ~ ~ ~ (1 + A Y TY ) = (1 + A Y TX ) ⋅ 1 + A Y (TY - TX )
)
(A.70)
~ ~ Respectively, if TY is the shorter reference horizon, the (1 + A X TX ) can then be split as follows:
(
~ ~ ~ ~ (1 + A X TX ) = (1 + A X TY ) ⋅ 1 + A X (TX - TY )
)
(A.71)
Only A Y is correlated to AX and A X is correlated to Ay, A Y and A X are independent of all other variables. The relation between AX and A Y and between A X and Ay is assumed to be defined by the covariance coefficient of σXY and σXY respectively. According to the definition of A within the model as unexpected changes, also A X , A X , A Y , and A Y have expected values of zero.
~
Consider the case when TX is the reference horizon. For better tractability rental revenues are omitted and the simplified notation from Version A is applied.
(
)
~ ~ cov ΓX , ΓY TX < TY ~ ~ ~ ~ ~ = E Π X ⋅ (1 + A X TX + A Y TX + A X A Y TX2 ) ⋅ ε X ⋅ Π Y ⋅ 1 + A Y (TY - TX ) ⋅ ε Y ~ ~ − E Π X ⋅ (1 + A X TX ) ⋅ ε X ⋅ E Π Y ⋅ (1 + A Y TY ) ⋅ ε Y
[ (
) (
)
(
) ]
(A.72)
Recognizing that only AX and A Y are correlated and conducting a simplification analogical to the one in Variant B leads to the following covariance formula:
(
) [
~ ~ ~ ~ ~ cov T~ ΓX , ΓY TX < TY = E Π X ⋅ Π Y ⋅ (A X A Y TX2 ) ⋅ e −(ρ−τX ) TX ⋅ e −(ρ−τY ) TY X ~ ~ ~ = E (A X A Y ) ⋅ E(Π X ⋅ TX2 ⋅ e −( ρ−τX ) TX ) ⋅ E(Π Y ⋅ e −( ρ−τY ) TY )
Resolving the expectation operator yields:
]
(A.73)
Appendix A
387
(
)
~ ~ cov T~ ΓX , ΓY TX < TY = X
∞
∞
(
)
= σ X Y ⋅ ∑∑ E (Π X Π X > π*X ) ⋅ 1 − FR ,X ( π*X ) ⋅ FRi−,1X (π*X ) i =1 j=1
(
)
⋅ E (Π Y Π Y > π*Y ) ⋅ 1 − FR ,Y (π*Y ) ⋅ FRj−,1Y (π*Y ) ⎛ 2 ⋅ λX ⋅ ⎜i ⋅ ⎜ ((ρ − τ X ) + λ X )3 ⎝ + i ⋅ (i − 1) ⋅
⎛ ⎞ λX ⎟⎟ ⋅ ⎜⎜ ρ − τ + λ ( ) ⎝ X X ⎠
λ2X ((ρ − τ X ) + λ X )4
(
i −1
⎛ ⎞ λX ⎟⎟ ⋅ ⎜⎜ ρ − τ + λ ( ) ⎝ X X ⎠
i−2
)
j ⎞ ⎛ ⎞ λY ⎟⋅⎜ ⎟ ⎟ ⎜⎝ (ρ − τ Y ) + λ Y ⎟⎠ ⎠
= σ X Y ⋅ E (Π X Π X > π*X ) ⋅ 1 − FR ,X (π*X ) ⋅ E (Π Y Π Y > π*Y )
(
)
⋅ 1 − FR ,Y (π*Y ) ⋅
1 ((ρ − τ X ) + λ X )2 i
∞ ∞ ⎛ ⎛ ⎞ ⎞ λX λY ⎟⎟ ⋅ FRj−,1Y ( π*Y ) ⋅ ⎜⎜ ⎟⎟ ⋅ ∑∑ FRi−,1X (π*X ) ⋅ i ⋅ (i − 1) ⋅ ⎜⎜ ρ − τ + λ ρ − τ + λ ( ) ( ) i =1 j=1 ⎝ ⎝ X X ⎠ Y Y ⎠
= ⋅
( (1 − F
) ))
E (Π Y Π Y > π*Y ) ⋅ 1 − FR ,Y (π*Y ) ⋅ λ Y (ρ − τ Y ) + λ Y
R ,Y
(π
* Y
j
(A.74)
(
2 ⋅ σ X Y ⋅ FR ,X (π*X ) ⋅ λ2X ⋅ E(Π X Π X > π*X ) ⋅ 1 − FR ,X (π*X )
((ρ − τ X ) + λ X ) ⋅ ((ρ − τ X ) + λ X (1 − FR ,X (π*X ) ))
)
3
~
Analogically, the covariance in the case of TY being the shorter reference horizon equals:
(
~ ~ cov ΓX , ΓY TX > TY = ⋅
)
( (1 − F
) ))
E(Π X Π X > π*X ) ⋅ 1 − FR ,X (π*X ) ⋅ λ X (ρ − τ X ) + λ X
R ,X
(π
* X
(A.75)
(
2 ⋅ σ XY ⋅ FR ,Y (π*Y ) ⋅ λ2Y ⋅ E(Π Y Π Y > π*Y ) ⋅ 1 − FR ,Y (π*Y )
((ρ − τ Y ) + λ Y ) ⋅ ((ρ − τ Y ) + λ Y (1 − FR ,Y (π
* Y
)
))
)
3
Analytical derivation of the unconditional covariance requires the computation of probabilities of one asset being sold earlier that the other. This proves, however, to be very difficult, if at all possible, when continuous time is assumed. Due to the lack of a satisfactory solution of this problem no closed form formula for the covariance of net
388
Appendix A
receipts can be provided here. A numerical solution can be applied when precision of the covariance estimation with the method presented in the Variant A is not satisfactory.
Appendix B
B.1.
Statistics of the GAA Data on Condominium Transactions in Selected German Urban Areas
All condominium prices and their statistics (means and standard deviations) are in Euro per square meter; transaction volumes are in thousands of Euros. Two-digit precision is used in most cases.
Duisburg Mean 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 Ø
1359.15 1350.35 1381.68 1518.60 1346.11 1368.61 1396.00 1585.96 1539.45 1304.64 1415.06
Good Quality Stand. Volume Dev. 259.61 3,344.41 311.97 4,742.45 362.25 4,518.57 300.39 4,169.84 292.19 3,798.93 287.30 2,248.35 322.83 4,295.80 455.36 1,801.39 352.18 1,416.00 373.84 2,884.90 331.79 3,322.06
Number
Mean
36 49 45 38 39 23 41 15 12 30 32.80
1179.02 1232.88 1214.68 1208.70 1191.98 1172.80 1021.32 1085.71 1094.62 1167.03 1179.02
Number
Mean
71 81 95 79 77 79 80 80 93 107 84
1,724.19 1,650.15 1687.17
Medium Quality Stand. Volume Dev. 359.78 19,267.56 306.29 15,572.95 345.78 20,204.08 348.80 14,657.22 323.96 9,126.65 345.93 12,018.55 383.07 11,557.41 455.56 16,791.58 441.09 16,435.08 406.84 7,511.60 359.78 14,314.27
Number 232 176 232 170 106 142 157 219 207 92 232.00
Frankfurt Mean 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Ø
2,412.13 2,277.79 2344.96
Good Quality Stand. Volume Dev. 757.41 15,702.97 701.56 17,060.65 729.48 16,381.81
Medium Quality Stand. Volume Dev. 606.91 47,311.77 623.76 55,214.02 615.34 51,262.90
Number 388 357 405 403 376 431 366 332 392 478 393
390
Appendix B
Hannover
1997
1,573.57
Good Quality Stand. Volume Dev. 275.57 12,378.19
Number
Mean
119
1,434.65
Medium Quality Stand. Volume Dev. 239.93 55,899.92
1998
1,479.92
317.29
13,615.07
130
1,439.60
301.97
61,485.66
589
1999
1,543.24
2000
1,433.87
431.34
14,519.59
141
1,360.53
263.69
65,570.47
635
304.79
10,199.91
103
1,266.33
280.20
46,062.78
2001
466
1,489.45
308.06
10,531.92
107
1,257.28
288.74
47,562.17
484
2002
1,465.41
325.81
11,266.83
119
1,178.43
252.27
50,880.98
538
2003
1,422.81
379.34
10,445.67
111
1,147.25
294.88
47,172.66
502
2004
1,240.96
168.64
9,168.22
101
1,130.49
299.80
41,403.69
455
2005
1,413.18
360.61
13,056.34
149
1,131.19
377.79
58,962.44
671
Mean
Number 537
2006
1,343.94
327.55
10,332.95
120
1,062.17
417.75
46,663.60
540
Ø
1,440.64
319.90
11,551.47
119.98
1,440.64
319.90
11,551.47
119.98
Köln Mean Ø
1,930.44
Good Quality Stand. Volume Dev. 630.50 -
Number
Mean
4,050
1,673.51
Medium Quality Stand. Volume Dev. 437.03 -
Number 8,133
Stuttgart Good Quality
Medium Quality
1995
1,666.77
Stand. Dev. 358.14
1996
1,658.73
356.11
15,631.26
142
1,807.50
343.62
38,746.84
311
1997
1,532.71
263.15
15,077.82
147
1,799.33
308.22
35,763.08
292
1998
1,523.45
272.68
21,194.14
202
1,751.89
319.95
41,709.65
348
1999
1,484.56
260.33
19,078.68
190
1,748.48
375.14
44,614.06
375
2000
1,467.38
222.08
15,223.83
151
1,674.18
340.47
35,353.69
306
2001
1,466.67
248.44
14,086.92
145
1,718.26
358.67
38,479.66
324
2002
1,481.77
259.82
17,750.05
175
1,718.19
354.00
44,470.38
376
2003
1,512.40
311.02
19,155.59
188
1,732.10
385.22
38,562.29
317
2004
1,459.77
266.62
15,367.12
158
1,718.80
351.32
41,058.73
345
2005
1,513.25
261.91
22,824.01
220
1,758.03
329.75
46,716.61
382
Ø
1,524.31
280.03
17,184.38
167.27
1,761.62
345.12
40,331.31
333.09
Mean
Volume
Number
1,951.02
Stand. Dev. 329.93
38,169.38
288
Volume
Number
Mean
13,638.71
122
Appendix B
Duisburg M
Frankfurt G
Frankfurt M
Hanover G
Hanover M
Stuttgart G
Stuttgart M
Mean
1.62
1.55
1.34
1.65
1.44
1.84
1.40
1.59
1.36
1.41
Standard Deviation
0.13
0.09
0.11
0.12
0.14
0.13
0.09
0.08
0.07
0.07
Skewness
1.47
1.48
1.25
1.44
1.27
1.52
1.41
1.44
1.51
1.51
Kurtosis
2.67
2.73
1.72
2.67
1.82
2.89
2.52
2.44
2.93
3.08
1.40
Cologne G
Duisburg G
Summary Statistics of Sale Receipts generated in a MC Simulation Cologne M
B.2.
391
Maximal Reservation Price, ρ = 5%
Maximal Reservation Price, ρ = 25% Mean
1.60
1.53
1.28
1.60
1.40
1.78
1.38
1.55
1.34
Standard Deviation
0.13
0.10
0.12
0.13
0.15
0.13
0.09
0.09
0.07
0.08
Skewness
1.37
1.51
1.13
1.43
1.24
1.44
1.37
1.56
1.42
1.46
Kurtosis
2.22
2.95
1.35
2.39
1.91
2.51
2.31
3.23
2.51
2.78
1.36
Maximal Reservation Price, ρ = 50% Mean
1.55
1.49
1.21
1.53
1.34
1.70
1.34
1.50
1.30
Standard Deviation
0.13
0.10
0.13
0.14
0.16
0.14
0.10
0.09
0.08
0.08
Skewness
1.36
1.33
1.06
1.38
1.16
1.37
1.28
1.47
1.34
1.43
Kurtosis
2.42
1.92
1.20
2.28
1.54
2.15
1.94
2.71
2.12
2.46
1.70
Minimal Reservation Price, ρ = 5% Mean
2.01
1.90
1.60
1.96
1.89
2.27
1.66
1.85
1.60
Standard Deviation
0.37
0.44
0.35
0.23
1.23
0.46
0.23
0.18
0.29
0.57
Skewness
-0.19
-0.13
-0.44
-0.33
-0.18
-0.39
-0.29
-0.25
-0.15
-0.16
Kurtosis
8.80
8.68
7.97
7.67
6.28
9.88
9.67
8.68
9.53
9.25
1.49
Minimal Reservation Price. ρ = 25% Mean
1.74
1.65
1.35
1.72
1.52
1.90
1.48
1.64
1.41
Standard Deviation
0.11
0.09
0.11
0.12
0.14
0.12
0.08
0.08
0.06
0.07
Skewness
1.13
1.37
0.85
1.22
0.70
1.47
1.10
1.39
1.22
1.29
Kurtosis
2.38
2.93
1.78
2.05
2.24
3.27
2.36
2.59
2.43
3.05
1.44
Minimal Reservation Price. ρ = 50% Mean
1.66
1.58
1.27
1.63
1.43
1.81
1.41
1.58
1.36
Standard Deviation
0.12
0.09
0.12
0.12
0.14
0.13
0.09
0.08
0.07
0.07
Skewness
1.36
1.37
0.84
1.23
0.92
1.43
1.24
1.43
1.38
1.40
Kurtosis
2.86
2.57
1.23
2.01
1.99
2.83
2.29
2.99
3.11
2.80
392
Stuttgart G
Stuttgart M
1.84
1.55
1.89
1.83
2.19
1.60
1.79
1.55
1.65
16.9
16.5
11.7
15.3
16.3
19.7
12.7
15.1
12.4
15.0
Return Volatility (%)
23.5
24.5
12.8
14.9
32.2
19.4
20.1
14.1
17.4
31.7
Cologne G
1.00
0.52
0.37
0.44
0.27
0.35
0.30
0.09
0.25
0.17
Cologne M
0.52
1.00
0.25
0.23
0.14
0.46
0.12
0.23
0.32
0.31
Duisburg G
0.37
0.25
1.00
0.53
0.11
0.26
0.30
0.07
0.43
0.29
Duisburg M
0.44
0.23
0.53
1.00
0.19
0.33
0.11
0.10
0.54
0.38
Frankfurt G
0.27
0.14
0.11
0.19
1.00
0.16
0.57
0.07
0.21
0.08
Frankfurt M
0.35
0.46
0.26
0.33
0.16
1.00
-0.02
0.35
0.57
0.53
Hanover G
0.30
0.12
0.30
0.11
0.57
-0.02
1.00
0.08
0.05
-0.06
Hanover M
0.09
0.23
0.07
0.10
0.07
0.35
0.08
1.00
0.36
0.24
Stuttgart G
0.25
0.32
0.43
0.54
0.21
0.57
0.05
0.36
1.00
0.64
0.17
0.31
0.29
0.38
0.08
0.53
-0.06
0.24
0.64
1.00
0.24
Hanover G
1.94
Expected Return (%)
Cologne M
Reservation Price
Cologne G
Hanover M
Frankfurt M
Frankfurt G
Duisburg M
Summary Statistics of Returns Corrected for Search Effects in a Planned Liquidation Duisburg G
B.3.
Appendix B
Liquidation Strategies Maximizing Expected Returns
Correlations
Stuttgart M
Liquidation Strategies Minimizing Return Volatilities Reservation Price
0.36
0.46
0.78
0.69
0.36
0.88
0.38
0.72
0.63
Expected Return (%)
7.5
7.9
6.8
7.3
7.8
9.7
6.6
7.7
6.8
8.3
Return Volatility (%)
17.5
16.2
10.0
12.0
17.6
10.1
17.6
11.4
13.2
21.2
Cologne G
1.00
0.60
0.41
0.49
0.33
0.43
0.33
0.10
0.29
0.21
Cologne M
0.60
1.00
0.29
0.27
0.17
0.57
0.14
0.27
0.37
0.37
Duisburg G
0.41
0.29
1.00
0.57
0.13
0.32
0.33
0.08
0.48
0.34
Duisburg M
0.49
0.27
0.57
1.00
0.23
0.42
0.12
0.11
0.60
0.44
Frankfurt G
0.33
0.17
0.13
0.23
1.00
0.19
0.75
0.09
0.26
0.10
Frankfurt M
0.43
0.57
0.32
0.42
0.19
1.00
-0.03
0.44
0.71
0.66
Hanover G
0.33
0.14
0.33
0.12
0.75
-0.03
1.00
0.08
0.05
-0.07
Hanover M
0.10
0.27
0.08
0.11
0.09
0.44
0.08
1.00
0.40
0.28
Stuttgart G
0.29
0.37
0.48
0.60
0.26
0.71
0.05
0.40
1.00
0.75
Stuttgart M
0.21
0.37
0.34
0.44
0.10
0.66
-0.07
0.28
0.75
1.00
Correlations
Appendix B
B.4.
393
Exemplary Efficient Portfolios with a Liquidity Decision Criterion
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.00 0.00 0.04 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.24 0.00 0.19 0.00 0.20 0.00 0.21 0.00 0.07
Stuttgart G
Hanover G
Frankfurt G
Implicit Spread
P16 P32 P48 P64 P80 P96 P112 P128 P144 P160 P176 P192 P208 P224 P240 P256 P272 P288 P304 P320 P336 P352 P368 P384 P400 P416 P432 P448 P464 P480 P496 P512 P528 P544 P560 P576
1.28 1.43 1.41 0.18 0.15 -8.06 Asset Weights 0.86 0.02 0.04 0.92 0.06 0.00 0.89 0.09 0.00 0.81 0.09 0.00 0.71 0.13 0.00 0.61 0.12 0.00 0.56 0.20 0.00 0.52 0.26 0.00 0.64 0.35 0.00 0.57 0.36 0.00 0.44 0.31 0.00 0.55 0.40 0.00 0.40 0.35 0.00 0.52 0.44 0.00 0.36 0.39 0.00 0.50 0.47 0.00 0.32 0.41 0.00 0.47 0.50 0.00 0.28 0.44 0.00 0.43 0.54 0.00 0.24 0.46 0.00 0.40 0.57 0.00 0.20 0.49 0.00 0.38 0.61 0.00 0.18 0.54 0.00 0.35 0.64 0.00 0.16 0.58 0.00 0.04 0.42 0.00 0.15 0.62 0.00 0.00 0.48 0.00 0.11 0.65 0.00 0.00 0.52 0.00 0.07 0.68 0.00 0.00 0.57 0.00 0.04 0.71 0.00 0.00 0.70 0.00
Return Volatility
1.65 0.07
Mean Return
Sale Res. Price Purch. Res. Price
Duisburg G
Cologne G
Implicit Spread as Liquidity Criterion
6.18% 6.18% 6.23% 6.28% 6.38% 6.44% 6.54% 6.64% 6.64% 6.69% 6.74% 6.74% 6.79% 6.79% 6.85% 6.85% 6.90% 6.90% 6.95% 6.95% 7.00% 7.00% 7.05% 7.05% 7.10% 7.10% 7.15% 7.20% 7.20% 7.26% 7.26% 7.31% 7.31% 7.36% 7.36% 7.41%
0.84% 0.84% 0.81% 0.79% 0.76% 0.78% 0.77% 0.79% 0.84% 0.85% 0.84% 0.90% 0.89% 0.95% 0.94% 1.01% 1.00% 1.08% 1.05% 1.15% 1.12% 1.23% 1.19% 1.32% 1.28% 1.42% 1.37% 1.29% 1.48% 1.38% 1.58% 1.47% 1.68% 1.58% 1.80% 1.82%
1.111 1.109 1.115 1.121 1.133 1.140 1.151 1.171 1.161 1.167 1.188 1.173 1.185 1.178 1.187 1.184 1.192 1.190 1.200 1.196 1.205 1.201 1.210 1.207 1.215 1.213 1.221 1.310 1.226 1.300 1.232 1.308 1.238 1.316 1.244 1.275
1.35 0.18 0.08 0.02 0.03 0.09 0.15 0.27 0.24 0.19 0.01 0.08 0.22 0.05 0.24 0.04 0.25 0.03 0.27 0.03 0.28 0.03 0.30 0.03 0.31 0.02 0.28 0.01 0.26 0.31 0.22 0.33 0.24 0.28 0.25 0.23 0.25 0.23
394
Appendix B
Cologne G
Duisburg G
Frankfurt G
Hanover G
Stuttgart G
Cologne G
Duisburg G
Frankfurt G
Hanover G
Stuttgart G
Mean Return (in %)
Return Volatility (in %)
Liq. Risk Reward
Liquidity Risk Reward as Liquidity Criterion
0.08 0.10 0.12 0.12 0.11 0.15 0.14 0.17 0.16 0.19 0.18 0.20 0.19 0.22 0.21 0.23 0.23 0.25 0.25 0.23 0.28 0.25 0.30 0.29 0.33 0.32 0.35 0.34 0.38 0.38 0.41 0.39 0.30 0.43 0.31 0.45 0.34 0.48 0.37 0.51 0.39 0.51 0.36 0.43
0.45 0.39 0.34 0.37 0.44 0.26 0.33 0.18 0.26 0.14 0.22 0.12 0.19 0.10 0.15 0.07 0.12 0.05 0.07 0.15 0.03 0.10 0.00 0.06 0.00 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.08 0.08 0.08 0.13 0.22 0.09 0.17 0.07 0.16 0.08 0.16 0.09 0.18 0.11 0.19 0.13 0.19 0.14 0.19 0.29 0.19 0.28 0.20 0.27 0.23 0.28 0.25 0.29 0.27 0.31 0.29 0.34 0.43 0.36 0.46 0.38 0.48 0.41 0.51 0.43 0.54 0.47 0.60 0.55
0.11 0.12 0.14 0.09 0.00 0.15 0.06 0.19 0.09 0.21 0.09 0.22 0.09 0.21 0.09 0.21 0.10 0.21 0.12 0.00 0.15 0.02 0.15 0.01 0.14 0.00 0.12 0.05 0.12 0.05 0.11 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.28 0.30 0.33 0.30 0.23 0.35 0.30 0.38 0.33 0.38 0.34 0.37 0.35 0.36 0.36 0.35 0.37 0.34 0.37 0.33 0.36 0.34 0.35 0.36 0.31 0.37 0.27 0.32 0.22 0.27 0.18 0.27 0.27 0.22 0.23 0.17 0.17 0.11 0.12 0.06 0.07 0.02 0.05 0.01
1.58 1.57 1.55 1.56 1.57 1.54 1.55 1.53 1.54 1.52 1.53 1.52 1.53 1.52 1.53 1.52 1.52 1.52 1.52 1.53 1.52 1.53 1.52 1.53 1.52 1.52 1.52 1.52 1.52 1.52 1.52 1.52 1.54 1.52 1.54 1.53 1.54 1.53 1.54 1.53 1.54 1.53 1.55 1.54
1.19 1.19 1.19 1.19 1.19 1.19 1.19 1.19 1.19 1.20 1.19 1.20 1.20 1.21 1.20 1.22 1.21 1.23 1.22 1.21 1.25 1.22 1.13 1.23 1.08 1.25 1.04 1.06 1.01 0.92 1.13 1.00 1.00 1.01 1.00 1.05 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.34 1.34 1.33 1.31 1.31 1.31 1.30 1.32 1.30 1.31 1.29 1.31 1.29 1.30 1.29 1.30 1.29 1.29 1.29 1.28 1.29 1.28 1.29 1.29 1.29 1.29 1.29 1.29 1.29 1.29 1.29 1.29 1.28 1.29 1.28 1.29 1.29 1.29 1.29 1.29 1.29 1.29 1.29 1.29
1.36 1.35 1.34 1.35 1.22 1.33 1.36 1.32 1.35 1.32 1.35 1.32 1.35 1.32 1.35 1.32 1.34 1.32 1.34 1.39 1.34 1.39 1.34 1.06 1.34 -.04 1.35 1.37 1.35 1.38 1.36 1.03 1.00 1.05 1.00 1.10 1.00 0.01 1.00 1.10 1.00 1.00 1.00 1.00
1.30 1.30 1.29 1.30 1.31 1.29 1.30 1.29 1.29 1.29 1.29 1.29 1.29 1.29 1.29 1.29 1.29 1.29 1.29 1.30 1.29 1.30 1.29 1.30 1.30 1.30 1.30 1.30 1.31 1.31 1.31 1.31 1.31 1.31 1.32 1.32 1.32 1.33 1.33 1.34 1.34 1.35 1.34 1.35
6.53 6.57 6.61 6.65 6.69 6.69 6.73 6.73 6.77 6.77 6.81 6.81 6.85 6.85 6.88 6.88 6.92 6.92 6.96 7.00 7.00 7.04 7.04 7.08 7.08 7.12 7.12 7.16 7.16 7.20 7.20 7.24 7.27 7.27 7.31 7.31 7.35 7.35 7.39 7.39 7.43 7.43 7.47 7.47
0.80 0.81 0.84 0.83 0.81 0.88 0.86 0.94 0.90 0.98 0.93 1.01 0.96 1.04 1.00 1.08 1.04 1.12 1.09 1.06 1.16 1.11 1.21 1.17 1.26 1.22 1.32 1.30 1.40 1.37 1.47 1.39 1.41 1.48 1.48 1.56 1.57 1.67 1.67 1.79 1.78 1.88 1.88 1.92
5.24 5.76 6.30 5.98 4.91 7.04 6.11 7.57 6.80 7.73 7.02 7.79 7.08 7.77 7.18 7.71 7.21 7.60 7.26 6.19 7.22 6.33 7.06 6.31 6.85 6.23 6.61 6.24 6.37 6.05 6.13 5.75 5.25 5.57 5.04 5.37 4.90 5.19 4.76 5.01 4.58 4.81 4.29 4.48
Asset Weights
P16 P32 P48 P64 P80 P96 P112 P128 P144 P160 P176 P192 P208 P224 P240 P256 P272 P288 P304 P320 P336 P352 P368 P384 P400 P416 P432 P448 P464 P480 P496 P512 P528 P544 P560 P576 P592 P608 P624 P640 P656 P672 P688 P704
Reservation Prices
Appendix B
395
Stuttgart G
Cologne G
Duisburg G
Frankfurt G
Hanover G
Stuttgart G
0.00 0.03 0.07 0.30 0.14 0.06 0.18 0.18 0.00 0.24 0.15 0.13 0.22 0.17 0.07 0.38 0.19 0.15 0.01 0.31 0.17 0.04 0.41 0.22 0.15 0.08 0.01 0.35 0.27 0.10 0.49 0.43 0.36 0.30 0.26 0.22 0.49 0.46 0.44 0.39 0.37 0.34 0.44 0.53 0.51 0.49 0.59
0.22 0.31 0.24 0.00 0.23 0.23 0.11 0.00 0.15 0.00 0.25 0.32 0.13 0.14 0.20 0.00 0.10 0.15 0.17 0.07 0.19 0.19 0.00 0.05 0.00 0.12 0.07 0.12 0.13 0.04 0.00 0.03 0.06 0.00 0.05 0.03 0.00 0.00 0.03 0.00 0.03 0.00 0.00 0.00 0.01 0.01 0.00
0.24 0.28 0.34 0.23 0.37 0.30 0.37 0.22 0.25 0.27 0.28 0.20 0.38 0.33 0.21 0.28 0.30 0.23 0.10 0.33 0.08 0.00 0.31 0.19 0.19 0.00 0.00 0.10 0.03 0.00 0.24 0.16 0.08 0.09 0.00 0.00 0.14 0.11 0.06 0.06 0.01 0.02 0.07 0.06 0.03 0.02 0.01
1.59 1.54 1.53 1.50 1.49 1.55 1.53 1.65 1.65 1.61 1.52 1.53 1.49 1.54 1.56 1.57 1.56 1.55 1.58 1.48 1.54 1.59 1.57 1.56 1.59 1.58 1.60 1.51 1.53 1.55 1.49 1.47 1.52 1.60 1.53 1.53 1.53 1.49 1.50 1.56 1.53 1.54 1.65 1.57 1.48 1.49 1.65
1.24 1.22 1.22 1.21 1.18 1.25 1.21 1.28 1.28 1.26 1.26 1.27 1.25 1.27 1.25 1.19 1.25 1.26 1.21 1.17 1.24 1.29 1.18 1.25 1.25 1.29 1.17 1.13 1.13 1.17 1.13 1.13 1.13 1.20 1.19 1.17 1.13 1.10 1.10 1.13 1.13 1.13 1.12 1.12 1.18 1.21 1.12
1.29 1.39 1.33 1.25 1.25 1.38 1.29 1.43 1.27 1.39 1.31 1.33 1.26 1.33 1.39 1.24 1.35 1.35 1.42 1.24 1.35 1.42 1.28 1.36 1.41 1.41 1.43 1.27 1.32 1.39 1.24 1.24 1.29 1.39 1.33 1.34 1.28 1.25 1.26 1.34 1.30 1.33 1.19 1.32 1.24 1.25 1.17
1.37 1.33 1.33 1.24 1.29 1.35 1.34 1.25 1.41 1.25 1.33 1.33 1.31 1.36 1.37 1.19 1.38 1.37 1.39 1.30 1.36 1.39 1.19 1.40 1.22 1.40 1.40 1.34 1.37 1.40 1.17 1.32 1.38 1.21 1.39 1.40 1.18 1.39 1.37 1.20 1.39 1.18 1.17 1.17 1.19 1.38 1.16
1.33 1.31 1.29 1.26 1.27 1.32 1.29 1.35 1.35 1.34 1.30 1.31 1.27 1.31 1.33 1.23 1.32 1.33 1.35 1.26 1.34 1.18 1.28 1.34 1.35 1.18 1.17 1.32 1.35 1.19 1.26 1.27 1.33 1.35 1.41 1.20 1.32 1.30 1.32 1.35 1.35 1.35 1.14 1.35 1.31 1.33 1.14
Receipts’ Volat. (in %).
Hanover G
0.25 0.14 0.11 0.33 0.08 0.09 0.11 0.24 0.08 0.19 0.02 0.01 0.01 0.01 0.00 0.13 0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Exp. Receipts (in %)
Frankfurt G
0.30 0.24 0.23 0.13 0.18 0.33 0.23 0.37 0.52 0.30 0.30 0.34 0.25 0.35 0.51 0.21 0.41 0.47 0.71 0.29 0.55 0.77 0.24 0.54 0.66 0.79 0.92 0.43 0.57 0.86 0.27 0.38 0.51 0.61 0.69 0.76 0.37 0.43 0.46 0.55 0.58 0.64 0.49 0.41 0.45 0.48 0.40
Return Volatility (in %)
Duisburg G
P64 P96 P160 P192 P256 P288 P352 P384 P448 P480 P544 P576 P640 P672 P736 P768 P832 P864 P928 P960 P1056 P1120 P1152 P1216 P1248 P1312 P1344 P1408 P1440 P1536 P1600 P1632 P1696 P1728 P1792 P1824 P1888 P1920 P1984 P2016 P2080 P2112 P2176 P2208 P2272 P2304 P2400
Reservation Prices
Cologne G
Asset Weights
Mean Return
Expected Receipts and Receipts’ Volatility as Independent Liquidity Criteria
6.69 6.74 6.80 6.86 6.86 6.86 6.92 6.92 6.92 6.97 6.97 6.97 7.03 7.03 7.03 7.09 7.09 7.09 7.09 7.15 7.15 7.15 7.20 7.20 7.20 7.20 7.20 7.26 7.26 7.26 7.32 7.32 7.32 7.32 7.32 7.32 7.38 7.38 7.38 7.38 7.38 7.38 7.39 7.43 7.43 7.43 7.48
1.00 1.06 1.03 0.91 1.06 1.13 1.04 1.08 1.37 1.06 1.24 1.38 1.18 1.23 1.48 1.14 1.31 1.42 1.93 1.28 1.71 2.20 1.30 1.56 1.77 2.24 2.57 1.64 1.83 2.41 1.49 1.57 1.73 1.81 2.05 2.17 1.64 1.68 1.77 1.81 1.92 1.96 1.77 1.80 1.85 1.88 1.94
1.43 1.42 1.41 1.36 1.38 1.45 1.41 1.46 1.51 1.44 1.44 1.46 1.41 1.46 1.51 1.39 1.48 1.50 1.56 1.42 1.52 1.59 1.42 1.52 1.56 1.59 1.62 1.47 1.52 1.59 1.42 1.44 1.50 1.55 1.55 1.57 1.47 1.46 1.48 1.53 1.52 1.54 1.49 1.50 1.46 1.48 1.45
0.07 0.06 0.05 0.07 0.05 0.06 0.06 0.09 0.11 0.08 0.06 0.06 0.06 0.07 0.08 0.07 0.07 0.08 0.11 0.07 0.09 0.12 0.08 0.09 0.11 0.12 0.15 0.08 0.09 0.12 0.09 0.09 0.09 0.11 0.11 0.11 0.10 0.09 0.10 0.11 0.10 0.11 0.12 0.11 0.10 0.10 0.12
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