The tail probability of discounted sums of Pareto-like losses in insurance

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1 Scandinavian Actuarial Journal, 2005, 6, 446 /461 Original Article The tail probability of discounted sums of Pareto-like losses in insurance MARC J. GOOVAERTS $,%, ROB KAAS $, ROGER J. A. LAEVEN $, *, QIHE TANG $, and RALUCA VERNIC $, $ University of Amsterdam, Department of Quantitative Economics, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands % Catholic University of Leuven, Center for Risk and Insurance Studies, Naamsestraat 69, B-3000 Leuven, Belgium Concordia University, Department of Mathematics and Statistics, 7141 Sherbrooke Street West, Montreal, Quebec, H4B 1R6, Canada Ovidius University of Constanta, Faculty of Mathematics and Computer Science, 124 Mamaia Blvd, Constanta, Romania (Accepted 28 June 2005) In an insurance contet, the discounted sum of losses within a finite or infinite time period can be described as a randomly weighted sum of a sequence of independent random variables. These independent random variables represent the amounts of losses in successive development years, while the weights represent the stochastic discount factors. In this paper, we investigate the problem of approimating the tail probability of this weighted sum in the case when the losses have Pareto-like distributions and the discount factors are mutually dependent. We also give some simulation results. Keywords: Asymptotics; (Log)elliptical distribution; (Log)normal variance-mean mied distribution; Pareto-like distribution; Tail probability. Mathematics Subject Classification: 62P05; 62E20; 91B Introduction Motivated by the work of Resnick & Willekens [1], we investigate the tail probabilities of the randomly weighted sums u k k ; n1; 2;... ; (1:1) *Corresponding author: R.J.A.Laeven@uva.nl Scandinavian Actuarial Journal ISSN print/issn online # 2005 Taylor & Francis DOI: /

2 The tail probability of discounted sums 447 and their maima. Here {, n/1,2,...} is a sequence of independent and identically distributed (i.i.d.) random variables with generic random variable and common distribution function F 1F; while {u n, n/1,2,...} is a sequence of dependent nonnegative random variables, independent of the sequence {, n/1,2,...}. The randomly weighted sums (1.1) and their maima are often encountered in actuarial and economic situations. See the following eamples: EAMPLE 1.1. Just as in Nyrhinen [2] and Tang & Tsitsiashvili [3,4], consider a discrete time risk model, in which the surplus of the insurance company is invested into a risky asset that generates a random, possibly negative, return rate in each year. Denote by A n / (/,) the net income (the total premium income minus the total claim amount) within year n and by R n / (/1,) the random return rate in year n, n/1,2,... Let the initial surplus be ]/0. Hence, if we assume that the net income A n is calculated at the end of year n, then the surplus, denoted by U n, accumulated till the end of year n satisfies the recurrence equation U 0 ]0; U n (1R n )U n1 A n ; n1; 2;...: (1:2) We define the finite time ruin probability as c(; n) min U k B0 05k5n j U 0 and the infinite time ruin probability as c() lim n0 c(; n) min U k B0 05kB j U 0 (1:3) : (1:4) Now write A n ; Y n 1 1 R n ; n1; 2;...: (1:5) The random variable is the net payout within year n and the random variable Y n is the discount factor from year n to year n/1, n/1,2,... In the terminology of Norberg [5] and Tang & Tsitsiashvili [3,4], we call, n/1,2,..., the insurance risks and Y n, n/1,2,..., the financial risks. The discounted value of the surplus process U n, denoted by Ũ n ; is defined by Ũ 0 ; Ũ n Yn Y i U n ; n1; 2;...: By repeatedly substituting (1.2) in the above epression, we find that Ũ n can also be epressed as Ũ 0 ; Ũ n n k Y k Y i W n ; n1; 2;...: One sees that the W n introduced above (with W 0 /0), which denotes the total discounted amount of losses by the end of year n, is of the form (1.1) with u k Q k Y i ; which is a

3 448 M.J. Goovaerts et al. product of positive random variables. We rewrite the ruin probabilities in terms of W k, k/0,1,2,..., as c(; n) ma W k 05k5n j U 0 and c() ma W k 05kB j U 0 : I EAMPLE 1.2. In Nyrhinen [2] and Tang & Tsitsiashvili [3,4], it was assumed that the net incomes A n, n/1, 2,..., constitute a sequence of i.i.d. random variables, that the return rates R n, n/1, 2,..., also constitute a sequence of i.i.d. random variables, and that the two sequences {A n, n/1, 2,...} and {R n, n/1, 2,...} are independent. A particular case is the well-known Black-Scholes-Merton model, in which the financial risks Y n, n/1, 2,..., are assumed to be i.i.d. and lognormally distributed. I EAMPLE 1.3. Since the assumption of independent return rates made in Eample 1.2 is generally considered unrealistic, it is desirable to incorporate some dependence structure in the financial risks. A natural etension is to assume that the log returns follow a multivariate normal distribution, or, more precisely, that there are sequences {m n, n/1, 2,...} and {s ij, i, j/1, 2,...} such that for each n, the vector (Z 1 ; Z 2 ;... ; Z n )(logy 1 ;logy 2 ;... ;logy n ) (1:6) has a multivariate normal distribution with mean vector m n (m 1 ; m 2 ;... ; m n ) and covariance matri 0 1 s 11 s s 1n s S n 21 s s 2n B A : s n1 s n2... s nn Clearly, this assumption is very convenient in calculation because of the attractive properties of the multivariate normal distribution. However, this assumption has also been questioned recently by Bingham et al. [6], because the empirical evidence shows that most financial data ehibit both pronounced asymmetry and much heavier tail behaviour than is consistent with normality. Following the work of Bingham et al. [6], we shall circumvent the limitations of the Black-Scholes- Merton framework by assuming that vector (1.6) either has a multivariate normal variance-mean miture with some miing law, or follows a multivariate elliptical distribution. I Keeping these eamples in mind, we shall investigate the tail probability of the randomly weighted sums (1.1) and their maima, under the assumptions that the distribution function F is Pareto-like and that the random weights {u n, n/1, 2,...} satisfy some conditions.

4 The tail probability of discounted sums 449 The remaining part of the paper is organized as follows: Section 2 gives the main results and some remarks; Section 3 considers two special cases when the quantities involved in the asymptotic results can be handled; and Section 4 proves the main results, after recalling several known results. 2. Main results Throughout this paper, all limit relationships are for 0/ unless stated otherwise. For two positive functions a() and b(), we write a()+/b() if lim sup a()/b()5/1, write a()h/b() if lim inf a()/b()]/1, and write a()/b() if both. Recall the randomly weighted sums (1.1). We assume that the right tail of F is regularly varying in the sense that there eist a constant a ]/0 and a slowly varying function L( /) such that F() a L(); 0: (2:1) We designate the fact (2.1) by F R a : This class contains the famous Pareto distributions. By the well-known representation theorem for slowly varying functions (see Theorem of Bingham et al. [7]), we have that for a distribution function F R a and any b /a, b o(f()): (2:2) More generally, the class R is the union of all R a over the range 05/aB/. For more details on the class R; we refer the reader to Bingham et al. [7]. Now we state the main contributions of this paper. The first result deals with the case of randomly weighted sums of finite summands. THEOREM Wehave Consider the randomly weighted sums (1.1) and let F R a for some a/ m ma 15m5n if there eists some d /0 such that u k k u k k F() n Eu a k (2:3) (1) Eu k ad B/ for each 15/k5/n. I/ By the result of Theorem 2.1, we have under the same assumptions that for all n/1, 2,... lim inf 0 1 F() m ma 15mB u k k ] n Eu a k : (2:4) Hence, by the arbitrariness of n it follows that

5 450 M.J. Goovaerts et al. 1 lim inf 0 F() m ma 15mB u k k ] Eu a k (2:5) regardless of whether/a Eua k converges. For any real number, we write its positive part by / /ma{,0}. The following result etends Theorem 2.1 to the case of infinite sums. THEOREM 2.2. have For the randomly weighted sums (1.1) with F R a for some a /0, we ma 15nB u k k if one of the following assumptions holds: (2) 0B/aB/1 and u k k F() Eu a k (2:6) Eu ad k B and Eu ad k B for some d0; (2:7) (3) 15/a5/ and (Eu ad 1 ad k ) B and (Eu ad 1 k ) ad B for some d0: (2:8) I/ REMARK 2.1. Both Theorems 2.1 and 2.2 do not require any information about the dependence structure of the sequence {u n, n/1,2,...}./ I REMARK 2.2. Recall Eample 1.2, where the random variables u k in (1.1) are interpreted as discount factors from time k to time 0 and are epressed as u k Yk j1 Y j ; ; 2;... ; (2:9) with i.i.d. nonnegative random variables {Y n, n/1,2,...}. Clearly, in this standard case, assumption (1) of Theorem 2.1 is equivalent to (4) EY 1 ad B/ for some d /0, and assumptions (2) and (3) of Theorem 2.2 are equivalent to (5) EY a9d 1 B/1 for some d /0. Under these assumptions, it follows from Theorems 2.1 and 2.2 that c(; n)f() EY 1 a(1 (EY 1 a)n ) 1 EY1 a

6 The tail probability of discounted sums 451 and that c()f() : 1 EY1 a The latter result etends Theorem 5.2(3) of Tang & Tsitsiashvili [3] to the case of ultimate ruin. I REMARK 2.3. Since the asymptotic relations given by Theorems 2.1 and 2.2 are completely eplicit, the evaluation of some actuarial quantities becomes quite easy. As an eample, we consider the evaluation of stop-loss premiums of the randomly weighted sums (1.1). Under the conditions of Theorem 2.1 with the additional restriction that a / 1, we have for each n/1, 2,... that, as d 0/, E u k k d g d E[ 1 d] EY a 1 u k k d n Eu a k : Eukg a F()d In particular, if the random variables u k are given by (2.9) with i.i.d. random variables {Y n, n/1, 2,...}, then for each n/1, 2,..., as d 0/, E u k k d EY1 a E[ 1 d] (1 (EY 1 a)n ) : 1 EY1 a Furthermore, if assumption (5) holds, then by Theorem 2.2, it also holds that, as d 0/, E u k k d EY1 a E[ 1 d] : I 1 EY1 a d 3. Some specific cases In order to apply Theorems 2.1 and 2.2, we need to calculate the epectations Eu a k for k/1, 2,.... In this section, we give some concrete eamples in which such calculation can be performed Logelliptically discounted process It has been pointed out in a large number of papers that the normality assumption regarding log returns of a risky investment is often not realistic. This rejection of normality has led researchers to investigate alternative models for the investment returns, including the family of elliptical distributions. In this direction, we refer the reader to Owen & Rabinovitch [8] and Vorkink [9], among others. In multivariate statistical analysis, elliptical distributions have provided an alternative to the normal model. Being an etension of the multivariate normal distribution, the class

7 452 M.J. Goovaerts et al. of elliptical distributions shares many of its nice statistical properties, though it contains many other non-normal multivariate distributions such as the multivariate Student s t, Cauchy, logistic, and so on. For details of the class of elliptical distributions, we refer the reader to Fang et al. [10] and Gupta et al. [11]. There are several equivalent ways to define elliptical distributions. We shall use the definition based on the characteristic function. DEFINITION 3.1. A random vector Z(Z 1 ;... ; Z n ) is said to have an elliptical distribution with parameter vector m n and parameter matri S n BB T for some n/m-matri B, if its characteristic function is of the form E[ep(it?Z)]ep(it?m n )f(t?s n t); for some function f( ):R 0 R: We write Z d E n (m n ; S n ; f). The function f( /) is called the characteristic generator. The matri S n is symmetric, positive definite and has positive elements on its diagonal. The characteristic generator may eplicitly depend on n, the dimension of Z. Hence, we denote by F n the family of all possible characteristic generators for a given n/1,2,..., that is, F n ff( ):f(t t2 n ) is an n dimensional characteristic functiong: Clearly, Let F 1 F 2 F 3 : F \ F n: n1 From Theorem 2.21 of Fang et al. [10], we know that a function f belongs to the class F if and only if f() g 0 e r2 F (dr) (3:1) with F a distribution function over (0,). DEFINITION 3.2. Let Y be a random vector with positive components. We say that Y has a logelliptical distribution with parameters m n, S n and f, written as Y d LE n (m n ; S n ; f); if log Y(logY 1 ;... ; logy n ) d E n (m n ; S n ; f): Let us go back to the eamples given in Section 1. Recall relation (1.6). We assume that for each n/1,2,..., Z(Z 1 ;... ; Z n ) d E n (m n ; S n ; f); hence that Y(Y 1 ;... ; Y n ) d LE n (m n ; S n ; f): Let s ij ; i; j 1;... ; n; denote the entry in row i and column j of the matri S n. From Theorem 2.16 of Fang et al. [10], we know

8 The tail probability of discounted sums 453 that the marginal distributions and any linear combination of an elliptically distributed random vector are also elliptically distributed with the same generator f. Therefore, with m (k) a k m i and s (k) a 15i; j5k s ij ; we have that Z 1 Z k d E 1 (m (k) ; s (k) ; f); hence that e Z 1Z k d LE 1 (m (k) ; s (k) ; f): Theorem 2.26 of Fang et al. [10] gives an eplicit epression for the moments Eu a k for a / 0 and k/1,2,..., namely Eu a k Eea(Z 1Z ) k ep(am (k) )f(a 2 s (k) ): To consider the infinite dimensional case, we must assume that f is of the form (3.1). In that case, we have that Eu a k ep(am (k) ) g 0 e a2 s (k) r 2 F (dr): (3:2) The lognormally discounted process results when f()e =2, that is, when the p distribution function F in (3.1) is degenerated at 1= ffiffiffi 2 : Because the lognormal case possesses many attractive properties and is easy to calculate, we restrict ourselves to this case in the remainder of this section. For this case, from (3.2), we have that Eu a k ep am (k) 1 2 a2 s (k) : (3:3) Therefore, under the conditions of Theorems 2.1 and 2.2, we have that for n/1,2,... u k k F() n ep am (k) 12 a2 s (k) : (3:4) Now we give some numerical results for relation (3.4). We assume that the random variables {, n/1, 2,...} are i.i.d. with common Pareto(a,b) distribution for some a /0 and b /0, with density function f () aba a1 ; b; and that for each n/1,2,..., thevector (Y 1,...Y n ) follows an n-dimensional lognormal distribution with parameters /m n, S n. We take the dimension n/10, the mean vector m 10 /(0.1,0.1,...,0.1) and the covariance matri

9 454 M.J. Goovaerts et al :05 0:01 0: :01 0:1 0:01 0: :01 0:01 0:1 0:01 0: :02 0:01 0:05 0:05 0: :02 0:05 0:1 0:01 0: S :01 0:01 0:1 0:02 0: : :01 0:02 0:05 0:01 0: :01 0:01 0:02 0:01 0:01 B :01 0:01 0:1 0:05A :01 0:05 0:05 Some numerical results are given in Table 1. The number of simulations is 5,000,000. The considered values of a are 1.2 and 1.5 as is reasonable, for eample, in fire insurance; see Beirlant et al. [12]. Apart from the values of and the simulated and asymptotic tail probabilities in (3.4), we also display the values of 1 asymptotic : Theoretically, these values simulated must tend to 0 when 0/. This seems to be the case from the table. Table 2 presents some numerical results for the case of n/50, i.e., when more time periods are considered. The values of the elements of m 50 and S 50 are similar to the ones for n/10. The number of simulations is again 5,000,000. In the notes of the tables, we display the simulated values of several quantiles of the discounted sums under consideration Lognormal variance-mean mied discounted process As announced in Eample 1.3, we now concentrate on the situation when the vector (1.6) is a normal variance-mean miture with some miing law. Some distributions from this Table 1. Simulated versus asymptotic values of the tail probability for Pareto claims with lognormal discount factors (n/10) a/1.2, Simulated b/2 1 asymptotic Asymptotic / simulated a/1.5, Simulated b/2 1 asymptotic Asymptotic / simulated Notes: Simulated quantiles at level p for a/1.2 are as follows: (p/0.950), (p/0.975), (p/ 0.990), (p/0.995), (p/0.999). Simulated quantiles at level p for a/1.5 are as follows: (p/ 0.950), (p/0.975), (p/0.990), (p/0.995), (p/0.999).

10 The tail probability of discounted sums 455 Table 2. Simulated versus asymptotic values of the tail probability for Pareto claims with lognormal discount factors (n/50) a/1.2, Simulated b/2 1 asymptotic Asymptotic / simulated a/1.5, Simulated b/2 1 asymptotic Asymptotic / simulated Notes: Simulated quantiles at level p for a/1.2 are as follows: (p/0.950), (p/0.975), (p/ 0.990), (p/0.995), (p/0.999). Simulated quantiles at level p for a/1.5 are as follows: (p/ 0.950), (p/0.975), (p/0.990), (p/0.995), (p/0.999). class have already been studied in the financial literature; see Eberlein & Keller [13], Barndorff-Nielsen [14] and Bingham et al. [6]. DEFINITION 3.3. A random vector Z(Z 1 ;...; Z n ) is said to be a normal variance-mean miture with position m n, drift b n, structure matri S n and miing distribution G on [0,) if for some random variable U sampled from G, the conditional distribution of Z given (U/ u) is N n (m n ub n ; us n ). Here the structure matri S n is symmetric and positive definite with js n j1. We write Z d NVMM n (m n ; b n ; S n ; G). The characteristic function of Z is then given by 1 8 Z (t)ep(it?m n )F 2 t?s ntit?b n ; where F is the Laplace-Stieltjes transform of G, that is, F(s) g 0 e sr G(dr); s0: DEFINITION 3.4. If b n /0 in the above definition, then we obtain the class of normal variance mitures, denoted NVM (m n, S n, G). In fact, in this case we have Z d E n (m n ; S n ; f) with f(s)f(s=2); so that NVM n ƒe n ; see Bingham et al. [6]. DEFINITION 3.5. Let Y be a random vector with positive components. We say that Y has a lognormal variance-mean mied distribution with parameters m n, b n, S n and G, denoted LNVMM n, if

11 456 M.J. Goovaerts et al. logy(logy 1 ;... ; logy n ) d NVMM n (m n ; b n ; S n ; G): Let us go back again to the eamples given in Section 1. We assume that Z(Z 1 ; :::; Z n ) d NVMM n (m n ; b n ; S n ; G): From Definition 3.3, for k/1, 2,... and u / 0, we have (Z 1 :::Z k )j(u u) d N 1 k (m i ub i ); us (k) ; where s (k) a 15i;j5k s ij as before. It follows that for a /0, Eu a k E[E(ea(Z 1:::Z ) k ju)] E ep a k (m i b i U) a2 s (k) 2 ep a k m i F a k b i a2 s (k) 2 U This gives an eplicit epression for the moments Eu a k for a /0 and k/1,2,... Therefore, under the conditions of Theorems 2.1 and 2.2, we have in this case that for n/1,2,...,, u k k F() n ep a k m i F a k : b i a2 s (k) : (3:5) 2 In particular, when b n /0 and F(s)/e s, we are again in the lognormal setting and relation (3.5) coincides with relation (3.4). In the following, we consider the particular case when Y is a lognormal variance-mean miture with the inverse Gaussian distribution as the miing distribution. This miing distribution was also considered e.g., by Barndorff-Nielsen [14] and Bingham et al. [6]. The inverse Gaussian density is sffiffiffiffiffiffiffiffiffiffi l g() ep l 2p 3 2n 2 (n)2 ; 0; with l,n /0, and its Laplace-Stieltjes transform is s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!! F(s)ep l 1 1 2sn2 ; s0: n l Hence, from (3.5), in this case u k k F() n 0 0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiff1 ep am (k) l u 1 1 2an2 k b i as B B t AA: n l 2 To assess the quality of this asymptotic relation by simulation, we assume, as in the previous section, that the random variables {, n/1, 2,...} are i.i.d. with common

12 The tail probability of discounted sums 457 Table 3. Simulated versus asymptotic values of the tail probability for Pareto claims with lognormal variancemean inverse Gaussian mied discount factors a/1.2, Simulated b/2 1 asymptotic Asymptotic / simulated a/1.5, Simulated b/2 1 asymptotic Asymptotic / simulated / / / /0.030 Pareto (a,b) distribution, for some a /0 and b /0. We take again n/10 and consider the parameters /m 10, S 10 of the lognormal variance-mean inverse Gaussian miture to be as given before, while b 1 b 10 1; l1 and n1: The number of simulations is 5,000,000. Some numerical results are given in Table oof of the theorems 4.1. Some lemmas The following lemma is from Breiman [15]; see also Cline & Samorodnitsky [16] for more general discussions. LEMMA 4.1. Let and Y be two independent random variables with cdf s F and G, where Y is nonnegative. If F R a for some a/0 and EY ad B for some d/0, then (Y ) lim 0 ( ) EY a : The following is a restatement of Lemma 2.1 of Davis & Resnick [17]. LEMMA 4.2. For a sequence of nonnegative random variables f 1 ;... ; g and a distribution function F R a for some a0, if lim 0 ( i ) c i F() for ;... ; n and lim 0 ( i ; j ) 0 for 15i"j 5n; F()

13 458 M.J. Goovaerts et al. then Pn i lim 0 F() c i : The following result is the one-dimensional version of Theorem 2.1 of Resnick & Willekens [1]. LEMMA 4.3. Consider the randomly weighted series a n1 u n n with f ; n1; 2;...g and fu n ; n1; 2;...g the same as in (1.1). Then under the conditions of Theorem 2.2, we have n1 u n n F() n1 Eu a n : 4.2. oof of Theorem 2.1 By Lemma 4.1, we have (u k k )F()Eua k ; 15k5n: Now choose some 0Bo B1 such that (1o)(ad)a: Then for all 15k"l 5n; (u k k ; u l l )5(u k 1o )(u k k ; u l l ; u k 5 1o ) 5 (1o)(ad) Eu ad k ( k o )(u l l ) o(f()); where we have used the Markov inequality and the property in (2.2). Thus, applying Lemma 4.2 we obtain that Since u k k F() n Eu a k : (4:1) m u k k 5 ma 15m5n u k k 5 n u k k ; it suffices to prove the relation u k k HF() n Eu a k : (4:2) For an arbitrary set I ƒf1; 2; ; ng; we denote by kik the cardinal number of the set I and introduce two events V 1 (I)( k 0 for all k I); V 2 (I)( k 50 for all kqi):

14 The tail probability of discounted sums 459 Clearly, u k k I:Iƒf1;2;;ng&I"f u k k ; V 1 (I)SV 2 (I) : (4:3) For any large L /0 and M /0, we further write V 3 (I; L)(LB k 50 for kqi); V 4 (I; M)(u k 5M for kqi): Then, the probability on the right-hand side of (4.3) is not smaller than ] u k k ; V 1 (I)SV 2 (I)SV 3 (I; L)SV 4 (I; M) k I k I u k k (nkik)lm; V 1 (I)SV 3 (I; L)SV 4 (I; M) u k k (nkik)lm j V 1 (I)SV 3 (I; L)SV 4 (I; M) (V 1 (I))(V 3 (I; L))(V 4 (I; M)): (4:4) Similarly to (4.1), applying Lemma 4.2, the conditional probability on the right-hand side of (4.4) is asymptotically equal to E(u a k jv 4 (I; M))( k (nkik)lmj k 0): k I Since F R implies ( k (nkik)lmj k 0)( k j k 0); we have H F() F(0) F() F(0) u k k ; V 1 (I)SV 2 (I)SV 3 (I; L)SV 4 (I; M) k I k I E(u a k jv 4 (I; M))(V 1 (I))(V 3 (I; L))(V 4 (I; M)) Eu a k 1 V 4 (I;M) (V 1 (I))(V 3 (I; L)): Substituting this into (4.3) yields that u k k H F() F(0) I:Iƒf1;2;;ng&I"f k I Eu a k 1 V 4 (I;M) (V 1 (I))(V 3 (I; L)): Since L and M can be arbitrarily large, this proves that u k k H F() F(0) I:Iƒf1;2;;ng&I"f k I Eu a k (F(0))kIk (F(0)) nkik : By interchanging the order of the two sums, we calculate the right-hand side of the above as

15 460 M.J. Goovaerts et al. F() F(0) Eu a k I:Iƒf1;2;;ng&k I (F(0)) kik (F(0)) nkik F() F(0) Eu a k (F(0)F(0))n1 F(0): This proves the announced result (4.2) oof of Theorem 2.2 It is trivial that ma 15nB u k k 5 u k k : In view of this, relation (2.5), and the convergence of the series a Eua k ; it suffices to prove that u k k However, relation (4.5) is given by Lemma 4.3. F() Eu a k : (4:5) Acknowledgement Rob Kaas and Roger Laeven acknowledge the support from the Actuarial Education and Research Fund of the Society of Actuaries. Qihe Tang acknowledges the supports from the Dutch Organization for Scientific Research (no. NWO ) and the Natural Sciences and Engineering Research Council of Canada (no ). Raluca Vernic acknowledges the support from the Dutch Organization for Scientific Research (no. NWO ). References [1] Resnick, S.I. and Willekens, E., 1991, Moving averages with random coefficients and random coefficient autoregressive models. Communications in Statistics / Stochastic Models, 7(4), 511 /525. [2] Nyrhinen, H., 1999, On the ruin probabilities in a general economic environment. Stochastic ocesses and their Applications, 83(2), 319 /330. [3] Tang, Q. and Tsitsiashvili, G., 2003, ecise estimates for the ruin probability in finite horizon in a discretetime model with heavy-tailed insurance and financial risks. Stochastic ocesses and their Applications, 108(2), 299 /325. [4] Tang, Q. and Tsitsiashvili, G., 2004, Finite and infinite time ruin probabilities in the presence of stochastic return on investments. Advances in Applied obability, 36(4), 1278/1299. [5] Norberg, R., 1999, Ruin problems with assets and liabilities of diffusion type. Stochastic ocesses and their Applications, 81(2), 255 /269. [6] Bingham, N.H., Kiesel, R. and Schmidt, R., 2003, A semi-parametric approach to risk management. Quantitative Finance, 3(6), 426 /441. [7] Bingham, N.H., Goldie, C.M. and Teugels, J.L., 1987, Regular Variation (Cambridge: Cambridge University ess). [8] Owen, J. and Rabinovitch, R., 1983, On the class of elliptical distributions and their applications to the theory of portfolio choice. Journal of Finance, 38(3), 745 /752. [9] Vorkink, K., 2003, Return distributions and improved tests of asset pricing models. Review of Financial Studies, 16(3), 845 /874.

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