Probability, Finance And Insurance: Proceedings Of The Workshop, The University of Hong Kong 15-17 July 2002

=
rT

+ E[e- V(S2{T),S2(T)\
= S2(T))}

rT

= S [ e - 5 2 ( T ) / ( 5 1 ( T ) = tpS2(T)j\ + Ele-^&mnSriT)

= S2(T))]V (1,1;?)

(27)

by (25) with a = S2(T). Here /(•) denotes the indicator function of an event. With the definitions A(ai, s2; ip) = E [ e - r r 5 2 ( r ) J ( 5 i ( r ) = 52(T))]

(28)

and B(Sl,s2;ip) = E^S^T^S^T)

= S2(T))} ,

(29)

formula (27) becomes V{8i,32;
+ B(8i,82',tp)V(l,l\ip)

.

(30)

To get closed-form expressions for A(si,s2;
(31)

becomes a martingale. Equivalently, we seek 9 so that r e -rt+ex l (t)+(i-e)x 2 (0l

(32)

is a martingale, which means e-rE[eexiW+(i-e)xa(i)]

= 1

_

(33)

Condition (33) gives rise to the quadratic equation 0 = -r + E[9X1(l)

+ (1 - 0)X2(1)] + ^Var[0Xi(l) + (1 - 9)X2(1)] 2

= - r + ^9 + n2(l -9) + ^92

2

+ ^ ( 1 - 9f + pa1(r29(l - 9) , (34)

with n\ and \L2 given by (16). Note that, because of (16), the expression on right-hand side of (34) equals - £ 2 for 9 = 0 and equals - £ i for 9 = 1. Hence, one solution of (34) is negative, and the other is greater than one,

201

say 9i <0,92> 1. With 6 — 9j, the stochastic process (31) is a martingale; if we stop it at time T and apply the optional sampling theorem, we obtain E[e~'TS2{T)[S1{T)/S2(T)]ei]

3* si-"' =

= A(s1,82]
j" = 1,2 .

(35)

These are two linear equations for the functions A and B. Their solutions are Jx

l-ei _ J2

^.« 3 ;
i-e2 a

.

(36)

^ n ^

•

(37)

Substituting (36) and (37) in the right-hand side of (30) yields V(si,s2;tp) (£,01 _ (£02

(38) To determine V(l, 1;
= 0.

(39)

Sl=S2

(An intuitive explanation of this condition is that when s2 is "close" to si, the guarantee or protection will be used instantaneously, and so the value of V is unaffected by marginal changes in s 2 . For a rigorous derivation of a similar condition, see Goldman, Sosin and Gatto [6].) Differentiating (38) with respect to s2 and applying (39), we obtain the equation

o = [(i - ex) - (i - e2)} + [(i - o2)
v(i,i;
Thus, with the definition h(z) = (02 - I)*' 1 + (1 - 6i)z02 ,

z >0,

(40)

we have

Substituting (41) in (38) and simplifying, we obtain

visi s

> ^=nwS2-

(42)

202

Because the option price is V(si, s2) = supV(si, s2; oo for z —> oo and for z - > 0 . Also, h"(z) = (92 - 1)(1 - 0i)( - OxJ*-2 + 92ze>-2) > 0 . Hence the graph of the function h(z), z > 0, is {/-shaped, and the function h{z) has a unique minimum. Let ip be the unique value satisfying h'{ip) = 0 .

(43)

Then

The number (p is between 0 and 1 because both —#i/(l —0i) and (92 — l)/92 are between 0 and 1. The optimal exercise strategy is to exercise the option the first time when Si(t) = ^ll(t), if si >
u

si

W

#2 , (45) if 0 < — < ip \ s2 with h(-) given by (40). In the next section, we shall give an alternative expression for the option. The payoff of the option is path dependent, but with a simple structure: If the option has not been exercised by time t, t > 0, then

/ l (5 1 (i)/n(i)) 1 h($)

-n(f).

(46)

5. An Alternative Pricing Formula We now derive an alternative expression for the option price V(si, s2). The first-order condition (43) is the same as (1 - 9l)62Cpe* = -{92 - 1)9^

,

(47)

203

applying which to (40) yields the formulas h^)=9-^(62-l)
(48)

and

/#) = -\^(l-W 2 -

(49)

It follows from (48) that (fl2 - l)z &1

e2-el\(p)

(50)

"

Similarly, it follows from (49) that (1 - 6»i)z02

- 0 ! (z\<>*

-ar® •

(51

»

Define ,

z

v

92x91 - OixH

g( > =

e2-e\

'

,N

x>0

-

(52)

By (40) the sum of the left-hand sides of (50) and (51) is h(z)/h(
= $(4),

^>0.

(53)

Prom (53) and (45) we obtain the alternative expression for the price of the perpetual dynamic protection option, V(s1,s2)

f 5 K( ^J> 2

= \

^

s2 *•

if 0 < ^ < 1 #2 ^ . if 0 < — < $

(54)

s2

In the special case where S2(t) = m, a positive constant, which can be interpreted as the maximal price of stock 1 in the past, formula (54) was first given by Shepp and Shiryaev [7]. 6. Connection with the Perpetual Maximum Option Price By comparing (2) with (3), we see that the maximum option is cheaper than the dynamic protection option. If both options are perpetual, there is a direct connection between the prices of the two options. See (60) below.

204

Prom Gerber and Shiu [3; 4], a formula for pricing the perpetual maximum option is:

W(Sl,s2)={

s2

if — < b

S29

[il<sS

(t) ^bs '

2

i^^ s

2

S\

2

w

if — > C S2

where g is denned by (52), si = Si(0), s2 = ^2(0),

and Note that

(58)

jj = 0 by (44), and that 0 < 6 < 1< c.

(59)

Observe that the condition

<1

is equivalent to b < cs\/s2 < c by (58). Hence, with this condition, it follows from (54), (58) and (55) that V(si,s2) = s2g(-^-) = s2g(~) = W(csi,s2) , (60) x \ips2J bs2' which gives a simple connection between the two prices. The discussion by Kwok and Chu on [5] provides an elegant explanation for this result in terms of optimal resets. Remark Section 5 of [5] has shown that, for exercise strategy Tv, each time when II(i) falls to the level of S\(t), the replicating portfolio consists of exactly V(l, 1; ip) units(s) of stock 1. Now, V(l,l;$) = V{l,l) = W{&,l) = c.

(61)

by (60) and (55). Thus, for the optimal exercise strategy of the perpetual dynamic protection option, each time when U(t) falls to the level of S\(t), the replicating portfolio consists of exactly c units of stock 1.

205

Acknowledgment Elias Shiu gratefully acknowledges the generous support from the Principal Financial Group Foundation and Robert J. Myers, F.C.A., F.C.A.S., F.S A .

References 1. H. U. Gerber and G. Pafumi. Pricing Dynamic Investment Fund Protection. North American Actuarial Journal 4(2), 28-37 (2000); Discussion 5(1), 153157 (2001). 2. H. U. Gerber and E. S. W. Shiu. Pricing Financial Contracts with Indexed Homogeneous Payoff. Bulletin of the Swiss Association of Actuaries 94 143166 (1994). 3. H. U. Gerber and E. S. W. Shiu. Martingale Approach to Pricing Perpetual American Options on Two Stocks. Mathematical Finance 6 303-322 (1996). 4. H. U. Gerber and E. S. W. Shiu. Pricing Perpetual Options on Two Stocks. Derivatives and Financial Mathematics, edited by John F. Price, Nova Science Publishers, Commack, NY, 91-117 (1997). 5. H. U. Gerber and E. S. W. Shiu. Pricing Perpetual Fund Protection with Withdrawal Option. North American Actuarial Journal 7(2) 60-77; Discussions 77-92 (2003). 6. M. B. Goldman, H. B. Sosin and M. A. Gatto. Path Dependent Options: 'Buy at the Low, Sell at the High'. Journal of Finance 34 1111-1127 (1979). 7. L. Shepp and A. N. Shiryaev. The Russian Option: Reduced Regret. Annals of Applied Probability 3 631-640 (1993).

RUIN PROBABILITY FOR A MODEL U N D E R MARKOVIAN SWITCHING REGIME

H. YANG* Department of Statistics and Actuarial Science The University of Hong Kong Pokfulam Road, Hong Kong E-mail: [email protected]

G.YIN1 Department of Mathematics Wayne State University Detroit, MI 4.8202

This work is concerned with insurance risk models, in which continuous dynamics are intertwined with discrete events. We assume t h a t the system under consideration is allowed to have jumps in the regimes and the jump process is modeled as a continuous-time Markov chain. Under suitable conditions, Lunderberg type upper bounds and non-exponential upper bounds for the ruin probability are obtained. In addition, renewal type system of equations for ruin probability and the exponential claim sizes case are discussed.

1. Introduction In the classical insurance risk model, compound Poisson processes are used to model the surplus process. The premium is assumed to be a constant and the claim is assumed to follow a compound Poisson process where the claim sizes are i.i.d. (independent and identically distributed) random variables and the number of claims is assumed to be a Poisson process. This model captures certain basic properties of the insurance portfolio, but it is far from realistic. There is a huge amount of literature devoted to the generalization of the classical model in different ways. For more detailed discussions on the "The research of this author was supported in part by Research Grants Council of HKSAR (Project no: HKU 7139/01H). t T h e research of this author was supported in part by the National Science Foundation under grants DMS-9877090. 206

207

ruin probability under classical models and various extensions, see Gerber (1979), Grandell (1991, 1997), Klugman et al. (1998), Rolski et al. (1999), Asmussen (2000) and the references therein. Recently, people in finance and actuarial science have also started paying attention to regime switching models. Di Masi et al. (1994) considered the European options under the Black-Scholes formulation of the market in which the underlying economy switches among a finite number of states. Buffington and Elliott (2001) discussed the American options under this set-up. Hardy (2001) used monthly data from the Standard and Poor's 500 and the Toronto Stock Exchange 300 indices to fit a regime-switching lognormal model. The fit of the regime-switching model to the data is compared with other econometric models. In this paper, we consider a Markovian regime switching formulation to model the insurance surplus process. This model can capture the feature of insurance policies may need to change if the economical or political environment changes. The model under consideration is a hybrid system, in which continuous dynamics are intertwined with discrete events. The systems are subject to jumps or switches in regime-episodes across which the behavior of the corresponding dynamic systems are markedly different. To model such a regime change, we use a continuous-time Markov chain. Our interest is to figure out the ruin probability of the underlying systems. This model is a special case of the Markov modulate risk model considered in Asmussen (1989) and Janssen (1980). In Asmussen (1989) and Janssen (1980) (see also Asmussen (2000)), the exponential upper bound was obtained for the ruin probability. Under the proposed regime switching model, we also obtain the Lundberg type bound for the ruin probability. Compare to the work of Asmussen (1989) and Janssen (1980), our method used here is simple and has the advantage that the same method can be used to obtain non-exponential upper bounds. We also obtain the integro-differential equation system satisfied by the ruin probability. The rest of the paper is arranged as follows. Section 2 begins with the precise problem formulation. Section 3 proceeds with the derivation of Lunderberg type inequality. Section 4 generalizes the results to nonexponential upper bounds. Section 5 derives a coupled system of integrodifferential equation satisfied by the ruin probability. Finally, Section 6 deals with exponential claim sizes.

208

2. The Model Let £(£) represent a homogeneous continuous-time markov chain taking values in a finite set M = { 1 , 2 , . . . , m}. We assume that P t = ( P / , . . . , P t m ) with PI = Pr{£ ( = i}, i = l , . . . , m , satisfies the Kolmogorov forward equation, i.e. -£- = ptQ,

0
Po = P

(1)

where P is the initial probability of the process {£t} and Q = (g^) is the stationary transition rate matrix of the process {£*,£ G [0,T]}, with qij denotes the transition probability rate from state i to state j and satisfies «« > 0

(2)

9» = -Qu =

2J

9

(3)

*J

For more discussion on Markov chains, see Chung (1967), Yin and Zhang (1998), and the references therein. In this paper, given an i € M, we assume that the sizes of the claims, at a given state i, Xi(i),X2{i), • • • are i.i.d. and nonnegative. The number of claims up to time t, Nt is a Poisson process satisfying P{Nt = n} = ^fe~xt , n = 0,l,2,... . n! Suppose that Nt and Xk(i) are independent. Denote the distribution function, the mean, and the moment generating function by Fx{i)(x)

= P{X(i)<x}

x>0,

I* = E(X(i)), Mx(r)

= E[erX],

respectively. At each state i, we assume that the premium is payable at rate c(i) continuously. Let U(t, i) be the surplus process given the initial suplus x and initial state i: U(x,i) = x+ / c(£(s))ds - St, Jo where N(t)

St=J2 X{£{Ti)),

209

Ti is the ith claim time. We further assume that the following net profit condition is satisfied. E

[ c(tts))ds

Jo

= (l + 0)E[St],

where 9 > 0 such that min.,{c(j)} > Amaxj{/z.,}. Here 6 is called relative security loading. If U(t, i) < 0, we will say that ruin occurs. Let T = inf{t : U(x,i) < 0} be the time of ruin (T = oo if U(t,i) > 0 for all t > 0)._Let 4>{x, i) = P{T = oo\U(Q, ^(0)) = x, £(0) = i) be the ultimate survival probability, and ip{x,i) = 1 — 4>{x,i) be the ultimate ruin probability. 3. Lundberg Type Inequality Let R{t) be the smallest positive solution of the following equation: /•OO

e-Xt-^o
\E[Mxm)){r)

(4)

Jo Denote R = inftg(0]OO){i?(t)}, which is referred to as the adjustment coefficient. We have the following Lemma. L e m m a 3.1. Equation (4) has a positive solution and /•OO

XE[Mx{m)(R)

e-M-RJic(as))dsdt^l

^

Jo for all t S (0,oo). Remark: When m = 1, equation (4) becomes the classical adjustment coefficient equation under the compound Poisson model. Proof: Let /•OO

/(r, t) = XE [Mxm)

(r) / Jo

e" Xt ~ r K c^s»ds}

dt.

210

It is obvious that /(0, t) = 1, therefore 0 is a solution of (4). Since /•OO

f'(0,t)

= \E[M'XU)(0)

e~Xtdt] Jo /•OO

-XE[Mxm))(Q)

ft

e~xt / c{i{s))dsdt] Jo

/ Jo /•OO

< maxf/ij} — A / e~xttmin{c(j)}dt Jo , , minjcfj)} A for all t G (0, oo), /(r,£) is decreasing at 0. Notice that, a s r ^ oo, /•OO

f(r,t)

> XE[Mx{m)(r) > 1

/ ./o

e^-max^)}*^

^ T T T E ^ K W ) ] -

oo.

A+rmax{c(j)} The lemma is thus proved. The following result is a Lundberg type inequality. Theorem 3.1. For x > 0 iP(x,i)<e~Rx,

(6)

where R is the adjustment coefficient defined above, x is the initial surplus, and for i £ M, £(0) = i is the initial state of the Markov chain.

Proof: Define ipn{x, i) to be the probability of ruin on or before the n claim for n = 0,1,2,... We claim that tpn{x, i) < e~Rx for each nonnegative integer n. We prove this by induction. First,

tpo(x,i) = 0 < e

-Rx

211

Suppose ipn(x,i) < e ipn+i(x,i)

Rx

. We proceed to estimate

ipn+i(x,i).

= P{ruin on or before the (n + l ) t n claim} = P{ruin on the 1 s t claim} + P{ruin not on the 1 s t claim, but on or before the next n t n claim} = E J™ X e" A * 11 - Fxm)) /

(x + j T c(£(s))d.

i>n[x+ poo

<E

I

(

Xe-^i

c(£(s))ds -y)

dFxm)(y)

I •/*+;„««

c(«(s))d.

f

i+J 0 'c(f(s))d.

!

+

\ dt

poo

I

Jo

dFx(m{y)

"I e-R(x+H

°M>))*-v)dFxm(y)

/ { Jx+JZ

c(Us))ds

\ dt

Jo

<E

+

/0

Ae_At

I

= E

e-*(*+Si°M»)*-v)dFxm)(y)\dt

'£ Xe~xt{ f

= E T

e R{x+I c{i(s))ds y)

~

Xe-Xte-R*-R

°

~ ^^m)(y))d{

/o = « W ) * | r /•OO

e~HXE

xm))\

Jo

<e -Rx

Therefore, i)n+1(x,i)

<e~Rx,

and we have ^n{x, i) < e~Rx

for all n = 0 , 1 , 2 , . . .

Thus, we conclude that i/>(x,i) = lim ipn(x) < e~Rx. n—>tx

The expectation above is with respect to £.

eRydFx{m(y)\dt\

212

4. Non-exponential Upper Bounds In this section, we provide non-exponential upper bounds for the ruin probability. Henceforth, we say a distribution B(X) is a new worse than used (NWU) distribution if B(x) is a distribution function of a nonnegative random variable such that for x > 0 and y > 0, B(x) = 1 - B(x), B(x)B(y) 0 and y > 0, B{x)B(y)>B(x

+ y).

Let B{x) be a NBU distribution function. Let R(t) be the smallest positive solution of the following equation: XE[Mx{m)(r)

J™ e'XtB

Q f c(£(s))ds\ dt] = 1

(7)

Denote R = mf{R(t)}. Then R is called the adjustment coefficient again. Similar as before, we have the following lemma. Lemma 4.1. Equation (7) has a positive solution and XE[Mxm)(R)

J™ e~xtB (J

c(^s))ds\

dt] < 1

(8)

for all t 6 (0,oo). The following theorem provides us with non-exponential upper bounds for the ruin probability. Theorem 4.1. Suppose that B(x) is a NBU distribution function, and B(y-x)
y>x

(9)

Then ip{x,i)
(10)

where x is the initial surplus, and £(0) = i, for i G M is the initial state of the Markov chain. Proof: Define ipn(x, i) to be the probability of ruin on or before the n*11 claim, n — 0,1,2,... Similar as before, we use induction to carry out the proof. First, ipQ(x,i) = 0
.

213

Suppose ipn{x,i) < B(x) is true. For the NBU distribution function B(x), we assume that B(0) = 1 and if we extend the domain of B(x) to x < 0, we have B(x) > 1 for all x < 0. Xe-X4

iPn+1(x,i) = E[J~

1 - Fxm)

+ / f

k/o

dFxm)(y) { Jx+ti

c(i(a))da

rx+S^c(i(s))ds _ / I B[x+ /'OO

r

<E

/

~ \

ft I

\ \ c{Z{s))ds-yjdFx{m{y)\dt

/»00

J

Xe Xt

/

l

dFx«(t))(w)

t

J0 [ Jx+J* c«(s))(fa ,*+/„« c(Z(s))ds _ / ft

<

E[J°°

e

\dt

/*O0

Xe-Xt\

/

+

c(£(s))ds^

Vn ( x + j o c(S(s))ds - yj dFx{m(y) /•OO

<£

(x + jf

~i/0 4

A e - A t | J™

B(X

\

)

+ J* c^{ ))ds\ +

= B(x)^[M x ( e ( t ) ) (i?) ^ ° ° Xe~xtB

S

(J

eRydFx{m(y)

•

\ dt]

c(t;(s))ds ) dt

< B(x). From this we conclude that ip(x,i) = lim ipn(x,i) < B{x). 5. A Coupled System of Integro-differential Equations for Ruin Probability In this section, we show that the ruin probability ip(x, j) satisfies a coupled system of integro-differential equations. Consider the ultimate survival probability cf>(x, i) = 1 — tj)(x, i), we have the following system of integro-differential equations for (j>(x,i). T h e o r e m 5.1. 4>(x,i) satisfies c(i)(p'(x,i) + ^2qi:j(t>(x,j)=X(t>{x,i)-X

(p(x-y,i)dFx{i){y), Jo

(11)

214

where x is the initial surplus and £(0) = i. Proof: By the property of the Poisson process, we have 4>(x,i) = (l-XAt)E[4>(x

+j At

fx+{0

*c(£(a))cfa,£(At)j]

c(Z(s))ds

I

,At

x +

+ XAtE[J^

[

\

c

J

s

(£( ))^-y,£(A£)J

^Fx(S(At))(j/)] o(At)

+

Rearranging the terms, we have E[(x + J XAtE

c(Z(s))ds, e(At) J - {x, a At))} + E[(x, £(A*))] -
4>(x + J

c(£(S))
/•x+/ 0 A t c(i(s))ds

- XAtE [J

I

,At

\

I a; + jf

c(g(s))ds - y, £( At) 1 dF X ( C ( A t ) ) (y)

+ o{At). Therefore, /•At

/ JO

,-At

c(£(s))ds £ [ # z + / Jo

c(e(s))ds, e(At)) - 0 Or, £(At))]

At

+E

Jo

c(S(s))ds

-(j>(x,Z{At))-(t>{x,i) At pAt

= XE[4>(x + J^

c(£{s))ds,ttAt)\]

rx+jf* c(S(s))ds

-XE[j

( t> X +

' \

rAt

J0

\

C s))ds

^

y

At)

~ > ^ J <***<«**» (?/)]

+ o(l). Letting At —> 0, we obtain equation (11). It is not easy to solve the system of equations (11) in general. We now look at the Laplace transform of the survival probability. Define /•OO

0(t, — /

Jo

e~tx<j>(x,i)dx.

215

Then we have the following result: Theorem 5.2. 4>(x,i) satisfies the following linear system [c(i)t - A + A/x W (t)]4>(t, t) + £

Qakt, J) = c(t)0(O. t)

i = 1,2,

...,m.

Proof: Note that /*oo

/ Jo

roo

e-tx'{x, i)dx = I Jo

e-txd(j>(x,i)

= 0(O,i) + t0(M) The theorem can be proved by taking Laplace transform of both sides of equation (11). By virtue of this theorem, if we know the value of <j)(0, i), then we can obtain the Laplace transform of the (j>(x, i) by solving a linear system. We use the method proposed in Dickson and Hipp (1998) to obtain the 0(0, i). Let Qn(t) 921

Qi2

•••

9in

92r,

922 (*) • • •

Mt) 9ml

9m2

...q^m(t)

where

qUt)=c(i)t-\

+ \fx(i)(t)+qil

=

l,...,m,

and let to be the positive solution of |-A0(£)| = 0. Since the Laplace transformation cannot equal to 0, we must have |-Aj(£0)| = 0, i — l , 2 , . . . m , where Ai(t) is the matrix Ao(t) with the ith column replaced by (c(l)<£(0,1), c(2)<j>{0,2),..., c(m)0(O, m))1. Therefore we have obtained a linear system |^(to)|=0

l,...,m.

Prom this linear system, if the above equation has m different positive roots, we can obtain (0, i), i = 1 , . . . , m.

216

6. Exponential Claim Sizes In this section, we assume that the claim size random variables follow exponential distributions. That is, X(i) is exponentially distributed with mean Pi-

Prom equation (11), we have

c(i)(j)'(x, i) + ^21a4>{^, J) i

r i _JL ~ X(j)(x,i) — X I
= X<j>(x, i)

_J£_

/

_£_

e n / Mi

(p(z,i)e"idz.

Jo

Differentiating both sides of the equation above with respect to x, we have c(i)<j>"{x,i) +

Y^QijfiixJ) i A

i

fx

z

= \<j>'(x,i) H — 2 e ~ ^ / Pi Jo = \4>'{x,i) Pi

=

<j>{z,i)e^dz

X


Pi

{x,i) + — Pi \

X<j>(x,i) - c(i)<j)'(x,i) i

-Y]qi:j(j)(x,j) J

(x-^)'(x,i)-±-^qij(x,j). \

Pi J

Mi •

Therefore we obtain the following system of second-order linear ordinary differential equations satisfied by <j)(x,i):

c(i)4>"(x,i) + J2'(x,j) 3

Pi

Pi

7. Concluding Remarks In this paper, we have provided a simple method to deal with the upper bounds of ruin probability under a Markovian regime switching model. Our method enables us to obtain both exponential and non-exponential upper bounds for the ruin probability. We have also derived a system of integro-differential equations satisfied by the ruin probability. The Markovian regime switching insurance risk model is of both theoretical interest

217

and practical importance. This is a relatively new model and there are many research problems in the context of risk theory under this model. We will further investigate this model in future research.

References 1. Asmussen, S. Risk Theory in a Markovian environment. Scand. Act. J., 69100 (1989). 2. Asmussen, S. Ruin Probabilities. World Scientific, Singapore, (2000). 3. Chung, K. L. Markov Chains with Stationary Transition Probabilities, Second Edition, Springer-Verlag, New York, (1967). 4. Di Masi, G. B., Kabanov, Y. M. and Runggaldier, W. J. Mean Variance Hedging of Options on Stocks with Markov Volatility, Theory of Probability and Applications, 39, 173-181 (1994). 5. Dickson, D. C. M. and Hipp, C. Ruin probabilities for Erlang(2) risk process. Insurance: Math, and Econom., 22, 251-262 (1998). 6. Buffington, J. and Elliott, R. J. American Options with Regime Switching. QMF Conference 2001, Sydney, Australia, (2001). 7. ardy, M. R. A Regime-Switching Model of Long-Term Stock Returns. North American Actuarial Journal, 5(2), 41-53 (2001). 8. Gerber, H. U. An Introduction to Mathematical Risk Theory. S.S. Huebner Foundation Monograph Series No.8, R.Irwin, Homewood, IL, (1979). 9. Grandell, J. Aspects of Risk Theory. Springer-Verlag, New York (1991). 10. Grandell, J. Mixed Poisson processes. Chapman and Hall, London, (1997). 11. Janssen, J. (1980). Some transient results on the M/SM/1 Special SemiMarkov Model in Risk and Queneing Theories, ASTIN Bulletin, 11, 41-51 (1980). 12. Klugman, S. A., Panjer, H. H. and Willmot, G. E. Loss Models Prom Data to Decision. Wiley, New York, (1998). 13. Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. StochasticProcesses for Insurance and Finance. Wiley and Sons, New York, (1999). 14. Yin, G. and Zhang, Q. Continuous-Time Markov Chains and Applications. Applications of Mathematics, Vol. 37, Springer, New York, (1998).

HEAVY-TAILED D I S T R I B U T I O N S A N D T H E I R APPLICATIONS *

CHUN SU and QIHE TANG Department of Statistics and Finance, University of Science and Technology of China, Hefei 230026, Anhui, China E-mail: [email protected] Department of Quantitative Economics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands E-mail: [email protected]

Based on some recent investigations by the authors, this paper gives a short review on the concept of heavy-tailedness. Some commonly used and newly introduced classes of heavy-tailed distributions are examined. Applications to precise large deviations and to finite time ruin probabilities are proposed.

1. Heavy-tailed Distributions Let X be a random variable (r.v.) with a distribution function (d.f.) F(x) = V(X < x). Throughout, we denote by F(x) = 1 - F(x) the tail probability of the r.v. X and write by Fe(x) = dx+y1

[*F(z)dz Jo the equilibrium distribution of F provided 0 < /i+ -

/ Jo

(1.1)

~F(z)dz < oo.

Hereafter, we will be using the notation n~p silently. We say that the r.v. X or its d.f. F is heavy tailed to the right if its moment generating function Eexp{tX} = oo

Vt>0;

(1.2)

"This work was supported by the national science foundation of china (no.10371117) and the dutch organization for scientific research (no. nwo 42511013). 218

219

otherwise, we say that it is light tailed to the right. The most important class of heavy-tailed distributions is the subexponential class, denoted by S. This class was first introduced by Chistyakov (1964) in a more general case. Let {Xk, k > 1} be a sequence of independent and identically distributed (i.i.d.) r.v.'s with common d.f. F supported on (0,oo). By definition, the d.f. F belongs to the class S if and only if for each n > 2, the tail probabilities of the sum Sn and the maximum X^n^ of the first n r.v.'s of {Xk, k > 1} are asymptotically of the same order, i.e., P ( 5 n > x ) ~ P ( X ( n ) >x)

asx->oo.

.

(1.3)

In the recent literature, the d.f. F supported on (—oo, oo) is still said to belong to S if the d.f. of the r.v. X+ = .XI(x>o) belongs to S. Some other important classes of heavy-tailed distributions are listed as follows, where the d.f. F involved is always assumed to be supported on (—00,00):

1. the class C (long-tailed): F is said to belong to the class C if x->oo

F(x)

holds for any z > 0 (or equivalently for some z > 0). 2. the class V (of dominated variation): F is said to belong to the class V if hm sup — x—00

< 00

F(x)

holds for any 0 < z < 1 (or equivalently for some 0 < z < 1). 3. the class <S*: F is said to belong to the class S* if 0 < ji J < 00 and

lim r*bAT{z)dz

x

^°° Jo

=

F(x) F(x)

2ti.

4. the class C (of consistent variation): F is said to belong to the class Cif lim lim sup 3 ^ 2 = 1.

(1.4)

m x-^oo F(x) 5. the class ERV (with extended regular variation): F is said to belong to the class ERV(—a, —/?) for some 0 < a < / 3 < o o i f «-/3
x-*oo

F(x)

~

x^oo

F(x)

~

Vw > 1.

(1.5)

220

Especially, if a = fi > 0 then relation (1.5) describes the class Tl-a, which is a well-known subclass of subexponential distributions. The regularity property in (1.4) was first introduced and called "intermediate regular varying" property by Cline (1994). This class has been used in different studies of applied probability such as queueing system and ruin theory; see, for example, Schlegel (1998), Jelenkovic and Lazar (1999), Ng et al. (2004), and references therein. Specifically, the class C covers the class ERV. Most recently Cai and Tang (2004) constructed a simple, but non-trivial, example to illustrate that the inclusion ERV C C is strict; another similar example can be found in Cline and Samorodnitsky (1994). For the classes above, we have, for any 0 < a < 7 < / 3 < o o , H.1cEJ?y(-a,^)cCcDn£c5c£,

S
V£S.

(1.6)

Furthermore, if 0 < yip < oo, then FeX>n£=*F(E<S*C<S.

(1.7)

For more details and related discussions, we refer to Bingham et al. (1987), Embrechts et al. (1997), and Goldie and Kliippelberg (1998), among others. Recently, we have introduced some other classes of heavy-tailed distributions. 6. the class A* (Konstantinides et al. (2002)): F is said to belong to the class A* if 0 < yip < oo, Fe £ S and

liminf .ofg*}, >0.

(1.8)

7. the class M (Tang (2001), Su et al. (2002) and Su and Tang (2003)): F is said to belong to the class M if 0 < jip < oo and

z-»°° J

F(u)au

8. the class M* (Tang (2001), Su et al. (2002) and Su and Tang (2003)): F is said to belong to the class M* if 0 < fx£ < oo and lim sup

xF{x) -

f00

< oo.

221

2. Some asymptotics and bounds 2.1. Applications

of the classes C and S to

asymptotics

Let {Xk,k > 1} be a sequence of independent r.v.'s, each Xk with a d.f. Fk supported on (—oo, oo), k > 1. With XQ — 0, we write n X

(n) ~ 3?$ xk, v '

0
s

n= V ^ , *—' A:=0

S(ln) = max Sk, '

0
n > 1.

The following result shows that, for a large class of heavy-tailed distributions, the tail probability of the maxima of the partial sum is asymptotically equal to the tail probability of the sum of the individual random variables: Theorem 2.1. (Ng et al. (2002)) If Fk G C, k> 1, then for any n G N, F(S{n)>x)~F(Sn>x).

(2.1)

Now we reduce the class C to the class S and obtain Theorem 2.2. (Ng et al. (2002)) Assume that \imx->00F~k'(x)/F~(x) = ck holds for some constants Cfc > 0 and some d.f. F G <S, k > 1. Then for any n G N, we have Hm

x->oo

F(S^>x) F(x)

= Um

*-»<»

P(5„>.) F(x)

=

^ x

->°°

F(Xln)>x) F(x)

=£ck

~

(2.2) Let F be a d.f. with 0 < /xj < oo. Recall its equilibrium distribution function defined by (1.1). Clearly, Fe is absolutely continuous and supported on (0, oo) with a hazard rate function re(x) = -ll,Te{x)

=1J^t,

z>0,

(2.3)

where the latter equality holds almost everywhere with respect to the Lebesgue measure. The following result extends (2.2) to the random case: Theorem 2.3. (Ng and Tang (2004)) Let {Xk,k > 1} be a sequence of i.i.d. r.v. 's with common d.f. F and finite expectation HF- Assume that r is a nonnegative and integer-valued r.v. with finite expectation and independent of {Xk, k > 1}. / / one of the following three conditions holds: (1) F £ <S and for some 6 > 0 we have E exp{5r} < oo (2) F G V n C, u < 0 and P(r > x) = o (r e (x)) (3) F e ERV(~a, -/3), 1 < a < (3 < oo, and uF < 0

222

then P (5 ( T ) > x) ~ P (ST > x) ~ P ( X ( T ) > x) ~ ErF(a;).

(2.4)

Compared with (1.3), the asymptotic relations (2.2) and (2.4) describe some very natural properties of subexponential distributions. Theorem 2.2 partly improves the related result in Sgibnev (1996). The second sufficient condition of Theorem 2.3 weakens the conditions of Theorem 2.3 in Ng et al. (2002). We refer the reader to Kaas and Tang (2003) and Foss and Zachary (2003) for closely related discussions in more general probabilistic models. The asymptotic relation P (ST > x) ~ ETF(X) in (2.4) describes the tail asymptotic property of the compound sum oo

W(ST < x) = J ^ P ( r =

n)F{-n\x),

n=0

where and throughout F^ (x) denotes the n-fold convolution of the d.f. F whereas F^(x) denotes a d.f. degenerate at 0. Related discussions can be found in Chover et al. (1973), Embrechts et al. (1979), Embrechts and Goldie (1982), Cline (1987), among others. 2.2. Sums of random variables

with dominated

variation

A classical inequality in the literature says that if F £ S, then for any e > 0, there exists a constant C€ > 0 such that + e)nF(x)

P {Sn >x)
(2.5)

holds for all n £ N and x > 0. In the original work by Chistyakov (1964) and Athreya and Ney (1972), the d.f. F is assumed to be supported on (0, oo), but the extension of this result to the general case of F supported on (—oo, oo) is immediate. We remark that the bound given in (2.5) is too rough since it grows exponentially fast with respect to n. Now we present some more precise bounds as follows: Theorem 2.4. (Tang and Yan (2002)) Assume that d.f. F £ V has a finite expectation /J,. Then (1) for any 7 > 0, there exists C(j) > 0 such that FM(X)

> C(j)nF(x)

holds for any n > 1 and x > jn;

(2.6)

223

(2) for any 7 > max{/z, 0}, there exists D(j) > 0 such that jM{x)

< D{j)nF(x)

(2.7)

holds for any n > 1 and x > jn. A closely related reference in this field is Cline and Hsing (1991), who considered more general x-regions Tn. We think that the most important z-region for applications is Tn — [yn, 00), 7 > 0. By Theorem 2.4, for any 7 > max {/j,, 0}, we have 0 < liminf inf n->oo x>jn

_ nF(x)

< limsup sup n-*oo x>-yn

_

< 00.

(2.8)

nF{X)

i.e. the relation F(™)(x) x nF{x) holds uniformly for x > 771 as n —> 00. The following corollary is immediate: Corollary 2.1. (Tang and Yan (2002)) Assume that d.f. F e V has a finite expectation fi. Then for any 7 > /x, there exist two positive constants C(7) and D(j) such that C(j)nF(x

- nn) < F~^){x) < D{-y)nF(x - n/i)

(2.9)

holds for any n > 1 and x > jn. 2.3. Applications

of the class S* to local

asymptotics

For notational convenience we introduce Qh(X;x)

= Qh(F;x) = ^ (F(x) - F(x + h)) = ±-F{x,x + h],

h > 0.

This section is devoted to describing the local asymptotic behavior of the sums Sn and ST and the maxima 5(n) and 5( r ). Recent investigations in this field can be found in Kluppelberg (1989), Bertoin and Doney (1994), Sgibnev (1996), Tang (2002), and Asmussen et al. (20026, 2003). Hereafter, the d.f. F is always assumed to satisfy that Qh(F; x) ~ Cf(x)

for some C > 0 and any h > 0,

(2.10)

where the function f(x) : M+ —>R+ is proportional to the tail of some d.f. from the class S*. Without loss of generality, we assume that f{x) is non-increasing in x G (0,00).

224

Asmussen et al. (2002^, Lemma 1) obtained that if i<\ and F 2 are supported on (0, oo) and satisfy condition (2.10) with the coefficients Ci, Ci > 0, then it holds for any h > 0 that Qh{F1*F2;x)~(C1

+ C2)f{x).

(2.11)

This indicates that the operator Qh(-', x) is additive in the asymptotic sense. The extension of (2.11) to the general case where Ft are supported on (—00,00) is immediate. By induction, we can derive from (2.11) that Qh(Sn;x) ~ nQh(X;x), n > 1. On the other hand, based on (2.11), going along the same line as the proof of Theorem in Sgibnev (1996), we can also obtain that Qh (S(ny,x) ~ nQh{X;x), n > 1. Furthermore, the proof for Qh (X( n );x) ~ nQh (X;x), n > 1, is straightforward. Hence, we summarize: Theorem 2.5. (Ng and Tang (2004)) If the d.f. F satisfies (2.10), then it holds for any n 6 N and h > 0 that Qh{S{ny,x) ~ Q h ( 5 n ; x ) ~Qh{X(n);x)

~n®h(X;x).

(2.12)

Now we aim to extend Theorem 2.5 to the randomized case. By copying the proof of Lemma 2 in Asmussen et al. (2002^) with some adjustments we can obtain that if F supported on (—00,00) satisfies condition (2.10), then for any e > 0, there exists some C(e, h) > 0 such that Qh

(F^]X)

< C(e, /i)(l + £)"/(*)

(2.13)

holds for all n £ N and x > 0. Since Q h (S ( n ) ; 1) < 111=1 Qh{Sk\x), from (2.13) we immediately obtain that for any e > 0, there exists some D(e, h) > 0 such that Q„ (5 ( n ) ;x) < D(e,h)(l

+ e)nf(x)

(2.14)

holds for all n £ N and all x > 0. Hence, if we assume Eexp{<Sr} < 00 for some S > 0, then by the dominated convergence theorem, it follows from (2.12) that Qh(Sr;x) ~ ETh{X;x) holds. Similarly, we can obtain Qh (5( T );x) ~ ¥.TQh(X;x). Finally, the proof for the relationship Qh (X(Ty,x) ~ ErQh (X; x) is also immediate. Hence, we summarize again: Theorem 2.6. (Ng and Tang (2004)) If the d.f. F satisfies (2.10) and Eexp{5r} < 00 for some 5 > 0, then it holds for any h > 0 that Qh{SM;x)~Qh{ST;x)~Qh{X(Ty,x)~ET
(2.15)

225

3. The classes A l and A\* 3.1.

Introduction

In the literature, heavy-tailed distributions are often assumed to be long tailed. The following example, however, indicates that even the union VUC still fails to contain some heavy-tailed distributions of interest: Example 3.1. (Su and Tang (2003)) Assume that the r.v. X is geometrically distributed with a parameter p £ (0,1). Then for any r > 1, the d.f. Fr, say, of the r.v. XT is heavy tailed but Fr (j£ V U C In our recent investigation on heavy-tailed distributions we introduced two other classes, M and M*\ see Tang (2001), Su et al. (2002), Su and Tang (2003). Let F be a d.f. with 0 < jip < oo. Recall the equilibrium hazard rate re(x) of F defined by (2.3). We see that the d.f. F belongs to the class M if and only if lim re(x) = 0

(3.1)

x—>oo

and that the d.f. F belongs to the class M* if and only if Umsupxr e (x) < oo.

(3.2)

x—»oo

It is easy to check that the d.f. FT in Example 3.1 belongs to the class M and that any d.f. F g P U £ with 0 < //p < oo should be an element of the class M.. Recently, we extensively investigated the classes M. and M* and discovered the following implications and inclusions; see Tang (2001), Su et al. (2002), Su and Tang (2003) and Su and Hu (2003): (1) F e M ^ F e £ £ ; (2) F e M* «=> Fe £ C Fe € V; (3) F € A* n M* <=$ Fe e ERV(-aF, 0 < aF =: liminf

x~F(x) -

f00

-/3F),

where x~F(x)

< limsup

f00—

=: 0F < oo. (3.3)

(4) A* n M* C V. 3.2. Tails of distributions

from the classes Ad and Ai*

A mistake often encountered in the literature is the assertion that relation (1.2) implies lim etxF(x) = oo,

Vt > 0.

(3.4)

226

In fact counterexamples to this assertion can easily be provided. For any r.v. X with a d.f. F supported on (—00,00), we define its moment index by I F = l(X) = sup{u : E ^ < 00}.

(3.5)

This index indeed describes a characteristic of the right tail of the r.v. X. We refer to Daley (2001) and Tang and Tsitsiashvili (2003) for some interesting discussions. Our further investigation discovers the following fact: T h e o r e m 3.1. (Su and Hu (2003)) I/FGM has a moment indexlp > 1, then (3-4) holds. But there exist d.f. 's in the class M. with only finite 1st moment such that (3.4) doesn't hold. Hence, those distributions with only finite 1st moment possesses some abnormal properties, which may have not been discovered so far. Moreover, we have the following result: T h e o r e m 3.2. (Su and Hu (2003)) If F € M* has a moment indexlp > 1, then we have lim x 7 F(x) = 0 0 ,

V7 > J f ^ r

x—>oo

(3.6)

Ip — 1

where the quantity ftp is defined by (3.3). But there exist d.f. 's in the class M* with only finite 1st moment such that (3.6) doesn't hold. The theorem shows that, as an extension of V, the class M* inherits some properties of the class V. For the equilibrium distribution of a d.f. F in the class M*, a more interesting property is discovered: T h e o r e m 3.3. (Su and Hu (2003)) For F £ M*, we have lim xWe{x)

= 00,

V 7 > (3F.

(3.7)

x—»oo

3.3. Closure of the classes .M and

Ai*

The following theorem indicates the closure of the classes M and M* under convolution: T h e o r e m 3.4. (Su et al. (2002)) Let F = F1*F2. then F&M*

<=^FX&M*.

IfT2{x)

=

0{F\{x)), (3.8)

227

IfF2(x) = o(f™F1(t)dt), then FiEMt^FeM.

(3.9)

By Theorem 3.4 we know that, for any n £ N , F G M*
F ( n ) e M.

FEM&

Now we consider the difference X = Z - Y,

(3.10)

where Z and V are two independent and nonnegative r.v.'s which are not degenerate at 0, and Z is distributed by G. Then the d.f. F, say, of the difference X is therefore concentrated on (—00,00). The following result describes the intimate relationship between the right tails of the r.v.'s X and Z. Theorem 3.5. (Su et al. (2002)) Consider the independent difference (3.10). IfEZ^EY, then GEM*^-FEM*,

G&M<==>FEM.

4. Precise large deviations Throughout this section, {X, Xk,k > 1} denotes a sequence of i.i.d. and nonnegative r.v.'s with common d.f. F and finite expectation fi. As before, we denote by Sn the nth partial sum, n > 1, of the sequence {Xk, k > 1}. All limit relationships are for n —> 00 or t —* 00 unless stated otherwise. The main stream of research on precise large deviation probabilities has been concentrated on the study of the asymptotics V(Sn - nfi > x) ~ nF(x)

(4.1)

uniformly for some a;-region Tn. The uniformity in (4.1) is understood in the following sense: F{Sn-n/j,>x) lim sup 0. nF(x) - °° xeTn Some classical work on precise large deviations can be found in Nagaev (1969a, &), Heyde (1967a, b, 1968). See also Nagaev (1973, 1979), who studied the asymptotic relation (4.1) for r.v.'s with d.f. F £ ~R-a for some a > 1. A nice review of recent research on precise large deviations is Mikosch and Nagaev (1998). Other reviews can be found in Nagaev n >

228

(1979) and Rozovski (1993). See also the monographs Vinogradov (1994), Gnedenko and Korolev (1996) and Meerschaert and Schemer (2001). Recently, the precise large deviations for the randomly indexed sum (random sum), JV(t)

^w(t) = 2 ^ xk,

t > o,

fc=l

have been investigated by many researchers such as Cline and Hsing (1991), Kluppelberg and Mikosch (1997), Mikosch and Nagaev (1998) and Tang et al. (2001), among others. Here the random index N(t) is a nonnegative and integer-valued process, independent of the sequence {X, Xk,k > 1}.. We always suppose that the mean function \(t) —» oo as t —+ oo. Most recently, Ng et al. (2003) established the large deviations result for a socalled prospective-loss process under a very mild condition on the counting process N(t). We cite here the following result of Theorem 3.1 in Kluppelberg and Mikosch (1997); see also Proposition 7.1 in Mikosch and Nagaev (1998) for a further extension: Proposition 4.1. (Kluppelberg and Mikosch (1997)) Let the common d.f. F S ERV(—a, — (3) for some 1 < a < /? < oo. If the process N(t) satisfies that N(t)/X(t)

—>p 1

Assumption

A

and that for some e > 0 and any S > 0, Y^

(1 + e)k¥{N(t)

> k) = o(l),

Assumption

B

fc>(l+(5)A(t)

then for any 7 > 0, we have F (SN{t) - n\(t) > x) ~ X(t)F(x)

uniformly for x > 7A(t).

(4.2)

As verified in Kluppelberg and Mikosch (1997), homogenous Poisson process satisfies both assumptions A and B above. Some applications of result (4.2) to insurance and finance can be found in Chapter 8 of Embrechts et al. (1997) and some of the above-mentioned references. Lately, Su et al. (2001) and Tang et al. (2001) weakened Assumptions A and B into one as E ((N(t)f+e

V(t)>(i+«)A(t))) = O (A(i)).

Assumption C

229

for some e > 0 and any S > 0. This is a much weaker condition, which can be satisfied, for example, by the general renewal process and the compound renewal process. Recently, we have made some progress on precise large deviations. Theorem 4.1. (Ng et al. (2004)) If F GC has a finite expectation /x, then for any fixed 7 > 0, it holds that P (Sn - n/j, > x) ~ nF(x)

uniformly for x > jn.

(4.3)

Using the method of Kliippelberg and Mikosch (1997), we can extend Theorem 4.1 to the random case. To this end, we introduce the following notation. Let X be a r.v. with a d.f. F supported on (—00,00). For any y > 0 we set F.{y) = liminf ^ M x—00 F(x) and then define

F*(y) = limsup ^ M i-»oo F(x)

n = M{J!£M-.V>l\--to,*£M,

(4.4)

(4.5)

j- = su P (-^M :y > a = _ lim is^M. (4.6) [

logy

J

y^oo

logy

In the terminology of Bingham et al. (1987), here the quantities j £ and J ^ are the upper and lower Matuszewska indices of the function f(x) — (F(x)) , x > 0. Without any confusion, we simply call JJ^ the upper/lower Matuszewska index of the d.f. F. The latter equalities in (4.5) and (4.6) are due to Theorem 2.1.5 in Bingham et al. (1987). Trivially, for any d.f. F it holds that 0 < J ^ < j j < 00. But from inequality (2.1.9) in Theorem 2.1.8 in Bingham et al. (1987) we see that F € V if and only if J j < 00. Moreover, if F € TZ-a for some a > 0, then J ^ = a. For more details of the Matuszewska indices, see Chapter 2.1 of Bingham et al. (1987), Cline and Samorodnitsky (1994) and Tang and Tsitsiashvili (2003). Theorem 4.2. (Ng et al. (2004)) If F eC has a finite expectation \i and {N(t),t>0} satisfies EAT p (i)I ( N ( t ) > ( 1 + 5 ) A ( t ) ) = O (A(i))

Assumption D

for some p > j £ and any 5 > 0, then, the large deviations result (4-2) holds.

230

Now we propose a new application for Theorem 4.2. In the context of insurance risk theory, the claim sizes Xk, k > 1, are often assumed to be i.i.d. nonnegative r.v.'s with common d.f. F. Their occurrence times o^, fc > 1, constitute an ordinary renewal counting process N{t)=sup{n:an
t > 0,

(4.7)

with X(t) = EN(t) < oo for any t > 0. With er0 = 0, we can write the interarrival times by Yfc = ak — o~k-i, k > 1, which therefore form a sequence of i.i.d. nonnegative r.v.'s. These are standard assumptions in the ordinary renewal model. Now we consider the model in Denuit et al. (2002). In their model, the fcth insurance policy, k > 1, is associated with a Bernoulli variable Ik, and the variable Ik has a common expectation 0 < q < 1, where q is the claim occurrence probability of the fcth policy, k > 1. Then, the total claim amount up to time t is N(t)

s!{t) = j2XkI^

t

^°-

(4-8)

fc=l

We assume that the three sequences {Xk,k > 1}, {Yk,k > 1}, and {Ik, k > 1} are mutually independent and that the sequence {Ik, k > 1} is negatively associated. Therefore, Si (t) is a non-standard random sum unless the parameter q = l. Generally speaking, a sequence {Ik, k > 1} is said to be negatively associated (NA) if, for any disjoint finite subsets A and B of {1,2, • • •} and any coordinatewise monotonically increasing functions / and g, the inequality Cav\f(Ii]i

£ A), < ? ( J j ; j e J 3 ) ] < 0

holds whenever the moment involved exists. For details about the notation of NA, please refer to Alam and Saxena (1981), Joag-Dev and Proschan (1983), and Shao (2000), among others. We proved the following result: Theorem 4.3. (Ng et al. (2004)) If F & C has a finite expectations \x, then it holds for any 7 > 0 and all x > jX(t) that V(Si(t) - M\{t) > x) ~

q\{t)F(x).

5. The finite time ruin probability The following renewal risk model has been extensively investigated in the literature since it was introduced by Sparre Anderson (1957). As described in Section 4.3, the costs of claims Xi} i > 1, form a sequence of i.i.d.,

231

nonnegative r.v.'s with common d.f. F, and the inter-occurrence times Yi,i > 1, form another sequence of i.i.d. nonnegative r.v.'s which are not degenerate at 0. We assume that the sequences {Xi, i > 1} and {Yi, i > 1} are mutually independent. The locations of the successive claims constitute a renewal process defined by (4.7). The surplus process of the insurance company is then expressed as N(t) x

R(t) =x + ct-^2 i,

* > 0,

i=l

where x > 0 denotes the initial surplus, c > 0 denotes the constant premium rate, and Y^t=i %% — 0 by convention. If Y\ is exponentially distributed, hence N(t) is a homogenous Poisson process, the model above is just the classical risk model, which is also called the compound Poisson risk model. We define, as usual, the time to ruin of this model with initial surplus x > 0 as T(X)

= inf {t > 0 : R(t) < 0 | fl(0) = x} .

Hence, the probability of ruin within finite time t > 0 can be defined as a bivariate function V>(z; t) = P (T(Z) < t | R{0) =x),

(5.1)

and the ultimate ruin probability can be defined as ip(x; oo) = lim ip(x; t) = P ( r ( i ) < oo | R{0) = x). t—*oo

In order for the ultimate ruin not to be certain, it is natural to assume that the safety loading condition /x = cEYi — MX\ > 0 holds. Under the assumption that the equilibrium d.f. Fe defined by (1.1) is subexponential, Veraverbeke (1979) and Embrechts and Veraverbeke (1982) established a celebrated asymptotic relation for the ultimate ruin probability that tf>(x;oo)~-

1 r°° — /

F(u)du.

(5.2)

As universally admitted, the study of the finite time ruin probability is more practical, but much harder, than that of the infinite time ruin probability. The problem of finding accurate approximations to the finite time ruin probability for the renewal model has a long history and during the period many methods have been developed. In this section we aim at a simple asymptotic relation for the finite time ruin probability ip(x;t) as x —> oo with special request that this asymptotic

232

relation should be uniform for t in a relevant infinite time interval. Under some mild assumptions on the distribution functions of the heavy-tailed claim size X\ and of the inter-occurrence time Y\, applying a recent result by Tang (2004) on the tail probability of the maxima over a finite horizon inspired by Korshunov (2002), we obtained the following result: Theorem 5.1. (Tang (2003)) Consider the renewal risk model with the safety loading condition. If F € C with its upper Matuszewska index j £ and EYf < oo for some p > Jf£ + 1, then for arbitrarily given to > 0 such that A(to) > 0, it holds uniformly for t £ [to, oo] that T/>(x;t)~- / F{u)du. M Jx Moreover, relation (5.3) can be rewritten as

(5.3)

tp (x; t) ~ - / ~F(u)du M Jx if the horizon t is restricted to a smaller region [t(x),oo] for arbitrarily given function t(x) —> oo, where A = 1/EYi. The uniformity of the asymptotic relation (5.3) allows us to flexibly vary the horizon t as the initial surplus x increases. For example, under the conditions of Theorem 5.1, from (5.3) we can obtain that the relation ip{x;t(x))~-

F{u)du (5.4) A* Jx holds for any real function t(x) such that to < t(x) < oo. If we specify the d.f. F G 7?._Q for some a > 1, a more explicit formula can be derived immediately, which extends Corollary 1.6(a) of Asmussen and Kluppelberg (1996) to the renewal model. The uniformity of relation (5.3) also makes it possible to change the horizon t into a random variable as long as it is independent of the risk system, as done by Asmussen et al. (2002) and Avram and Usabel (2003). In fact, we have the following consequence of Theorem 5.1: Corollary 5.1. (Tang (2003)) Let the conditions of Theorem 5.1 be valid and let T be an independent r.v. with a d.f. H satisfying E T < oo and P(T > to) = 1 for some 10 > 0 such that A(t0) > 0. Then it holds that •0(x;T)~EA(T)-B(x).

(5.5)

233

It is clear that ip (i; T) = P (

max

Y" (Zj - c9i) > x ) .

(5.6)

i=l

A similar result as (3.3) has recently been established by Foss and Zachary (2003). However, we remark t h a t our Corollary 5.1 is never a special case of theirs since the r.v. N(T) in (5.6) cannot be explained as a stopping time in their sense.

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T H E I N S U R A N C E REGULATORY R E G I M E IN HONG KONG (WITH A N EMPHASIS ON THE A C T U A R I A L ASPECT)

AUGUST CHOW Office of the Commissioner of Insurance 21/F Queensway Government Offices 66 Queensway, Hong Kong E-mail: [email protected]

1. The Insurance Regulatory Regime in Hong Kong In this article, I would like to share with you the insurance regulatory regime in Hong Kong, and in particular some of the actuarial applications for purposes of monitoring the life insurance industry in Hong Kong today. 2. Outline This article will cover 3 areas: First, to enable you to have a better understanding as to what we do, I would like to give you an overview of the life insurance market and our insurance regulatory philosophy in Hong Kong. Then, I would like to cover the Appointed Actuary System that we have adopted in Hong Kong and the roles played by the actuaries in the market place. And finally, I would like to provide you with an example of an application on Reserving for Investment Guarantees with respect to the retirement scheme business sold in Hong Kong. 3. An Overview of the Hong Kong Life Insurance Market Compared with other territories in the region, Hong Kong is a free and open city with many insurers competing in the market, and has the highest number of insurers per capita. However, the penetration ratio, which is defined as 100% times the number of policies per capita, is only 70% in Hong Kong which is relatively 237

238

low as compared to over 200% in the United States and Japan. Therefore, there is still ample room for life insurance to prosper further in Hong Kong. Under our prudential regulatory regime, insurers in Hong Kong have the freedom to set premium rates and design products. 4. Number of Authorized Life Insurers by Place of Incorporation as of 30 June 2002 At the end of June 2002, there were 48 life insurers and 18 composite insurers authorized to write long term business in and from Hong Kong. About 1/3 of them (21) were incorporated locally in Hong Kong, with the remaining 2/3 incorporated in overseas (10 in Bermuda, 8 in UK, 7 in the US and the rest from China, Canada, Switzerland, Isle of Man etc). Figure 1. Number of authorized life insurers by place of incorporation as at 30 June 2002.

5. View of the Hong Kong Insurance Regulatory Framework The insurance industry in Hong Kong is regulated by the Insurance Companies Ordinance. It provides a legislative framework for the prudential supervision of the insurance industry in Hong Kong. The objectives of the Insurance Authority are to protect the interests of the policyholders and to maintain the general stability of the market place. In addition to being an insurance regulator, the Insurance Authority is a proactive market enabler that we welcome and stand ready to facilitate

239

entry of newcomers and a referee of insurers. As mentioned earlier, we adopt a prudential supervisory approach rather than a close supervision, with emphasis on self-control mechanism and the appropriate checks and balances. 6. Appointed Actuary System in Hong Kong At the beginning of 2001, we had introduced a fully fledged Appointed Actuary System for the life insurance business in Hong Kong. The adoption of this system is to strengthen the monitoring of the financial condition of life insurers and is also to be in line with the development in other developed countries. Under this system, the appointed actuaries have many other responsibilities apart from the traditional work in product pricing and valuation. Appointed Actuaries now have the responsibility to the company's board of directors and to the regulators. To enable the Appointed Actuary can properly carry out the responsibility, the actuary is given unrestricted access to all of the company's relevant financial and non-financial data that are complete and accurate. 7. Specific Reference to Actuaries in the Insurance Regulation As mentioned earlier, one of the important duties of the appointed actuary is to monitor the financial condition of insurance company and to report them to the board of directors. To fulfill this duty, the Insurance Authority has imposed regulations on the valuation of long term liabilities and the determination of solvency margin. Moreover, the appointed actuary has a whistle-blowing function under our Insurance Companies Ordinance. If the Appointed Actuary is aware of a situation that could threaten the company's financial condition, the Appointed Actuary has an obligation to report this to the company's management and the Board. If the company does not correct the situation, the Appointed Actuary is obligated to report the situation directly to the Insurance Authority to take proper action. In order to fulfill the duties of the Appointed Actuaries, the Insurance Authority has prescribed a professional standard for the Appointed Actuaries to follow. The professional standard was issued by a local actuarial body, the Actuarial Society of Hong Kong (ASHK) which I would further comment later on. Every year, the appointed actuary is required to issue

240

an actuarial certification indicating the compliance with this professional standard or some other standards acceptable by the Insurance Authority. 8. Roles of Actuaries in the Market Place Apart from all the regulatory responsibilities, actuaries also play a very important role in the operation of a life insurance company. Some of them could include identifying and targeting profitable market segments, designing sensible product features, setting the correct premium rate, assessing and conducting the risk control through reinsurance; and managing asset liability. For participating policyholders, the Appointed Actuary is also required to preserve the Policyholders' Reasonable Expectation (PRE). The Policyholders' Reasonable Expectation has often to do with the amount of bonus or dividends expected to be payable to policyholders in the future. 9. Actuarial Professional Body In order to have a strong Appointed Actuary system, it is very important that the actuarial professional body provides a professional code of conduct and a standard of practice to guide its members so that its members can carry out the responsibilities given to them. The Actuarial Society of Hong Kong (the ASHK) has taken up this important role. The ASHK has issued a number of professional standards and guidance notes over the years, such as the PS1 which I have already mentioned, and the GN3, an additional guidance for Appointed Actuary with respect to the determination of reserves and solvency margin. The ASHK also conducts training and professional development meetings such as the Appointed Actuaries Symposium held last summer. 10. Actuarial Application: Reserving for Investment Guarantee Finally, I would like to talk about an actuarial application on reserving standards for investment guarantees. In January 2001, the Insurance Authority issued a guidance note, GN7, which requires life insurers carrying on Class G retirement scheme insurance business with investment guarantees on capital or return to set up an adequate reserve. The Class G business includes the Mandatory Provident Funds (MPF) Scheme which is a community wide scheme providing a com-

241

pulsory retirement saving vehicle for all working population in Hong Kong, except for the very low income group. To better assist the members in complying with the GN7, the ASHK has recently released a draft guideline which further provides guidance on satisfying the requirements of GN7. 11. Details of the Reserving for Investment Guarantees The guidance note detailed the approaches to determine the provision such as the annual projection of assets and policy liabilities cash flow based on the stochastic scenario with 99% level of confidence. If deterministic scenario or factor approach is used, a stochastic adequacy test must be performed at least once a year on the total provision on the adverse scenarios with a 99% level of confidence. The provisions for investment guarantees must be revised at least quarterly to reflect current information. Provisions at month-ends within a quarter may be extrapolated from the previous quarter-end values. However, where there are significant fluctuations in the underlying factors that have been modeled by the stochastic analysis, the scenario testing should be done on a monthly basis. 12. Stochastic Approach Under the stochastic approach, the process for completing a full stochastic investigation includes the modeling and projection of investment returns as well as the in-force business liabilities. In the modeling and projection of investment returns, one needs to determine the relevant economic/market risk factors to be included in the model, the underlying investment policy of the assets and its investment returns, and the use of stochastic model to simulate the evolution of these quantities. For the modeling and projection of in-force business liabilities, one needs to extract the relevant policy in-force data and to determine the assumptions used in the projection such as the mortality, lapse, withdrawal, future contributions, expenses, policy holders behaviors, management strategies. Then the investment and business liabilities models are run with over 1000 economic scenarios. The result of the scenarios, which is the present value of claims less the present value of income, are ranked in the order of severity of the required reserve.

242

Finally, the total provision is established for the investment guarantees which is consistent with a 99% level of confidence. A transitional arrangement is provided for certain Class G insurance policies which are not related to MPF schemes. For these non-MPF policies, the level of reserves for guarantees is required to reach the 99% level of confidence by year 2004. It is also required that the results of stochastic testing must be traceable and reasonably reproducible for audit purposes. 13. Other Actuarial Applications Apart from this, there are other actuarial applications too. Due to the time constraint, I'm not going to cover them in details with you. I would just mention that the Insurance Authority had requested the ASHK to look into the possibility of implementing Risk based capital and the Dynamic solvency testing for purposes of enhancing the financial monitoring of life insurers. Starting from the end of 2001, the Insurance Authority also required general insurers engaging an actuary to certify reserves of certain compulsory lines of business. 14. Conclusion In conclusion, actuaries and actuarial mathematics have played and will continue to play a significant role in the regulatory framework of the insurance market. I hope you would agree with me that the demand for actuarial skills is getting greater in Hong Kong and particularly in China with the new opportunities upon China's entry into the WTO. References 1. GN 7 Guidance Note On Reserving Standards For Investment Guarantees. 2. PS 1 Professional Standard For Fellow Members Of The Actuarial Society Of Hong Kong.

PROBABILITY, FINANCE AND INSURANC; This workshop was the first of its kind in bringing together researchers in probability theory, stochastic processes, insurance and finance from mainland China, Taiwan, Hong Kong, Singapore, Australia and the United States. In particular, as China has joined the WTO, there is a growing demand for expertise in actuarial sciences and quantitative finance. The strong probability research and graduate education programs in many of China's universities can be enriched by their outreach in fields that are of growing importance to the country's expanding economy, and the workshop and its proceedings can be regarded as the first step in this direction. This book presents the most recent developments in probability, finance and actuarial sciences, especially in Chinese probability research. It focuses on the integration of probability theory with applications in finance and insurance. It also brings together academic researchers and those in industry and government. With contributions by leading authorities on probability theory — particularly limit theory and large derivations, valuation of credit derivatives, portfolio selection, dynamic protection and ruin theory — it is an essential source of ideas and information for graduate students and researchers in probability theory, mathematical finance and actuarial sciences, and thus every university should acquire a copy.

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