Reinsurance under the LCR and ECOMOR Treaties with Emphasis on Light-tailed Claims

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1 Reinsurance under the LCR and ECOMOR Treaties with Emphasis on Light-tailed Claims Jun Jiang 1, 2, Qihe Tang 2, 1 Department of Statistics and Finance Universit of Science and Technolog of China Hefei 2326, Anhui, P.R. China 2 Department of Statistics and Actuarial Science The Universit of Iowa 241 Schaeffer Hall, Iowa Cit, IA 52242, USA August 16, 28 Abstract Suppose that, over a fixed time interval of interest, an insurance portfolio generates a random number of independent and identicall distributed claims. Under the LCR treat the reinsurance covers the first l largest claims, while under the ECOMOR treat it covers the first l 1 largest claims in excess of the lth largest one. Assuming that the claim sizes follow an exponential distribution or a distribution with a convolutionequivalent tail, we derive some precise asmptotic estimates for the tail probabilities of the reinsured amounts under both treaties. Kewords: Asmptotics; Convolution-equivalence; Exponential distribution; LCR and ECOMOR treaties; Reinsurance; Tail probabilit 1 Introduction Let {X 1, X 2,...} be a sequence of i.i.d. independent and identicall distributed positive random variables, representing successive claim sizes, with common continuous distribution F on,. Assume that the claims arrive according to a counting process {Nt; t } independent of {X 1, X 2,...}; that is to sa, the random variable Nt counts the number of claims up to time t. Let X1 < X2 < < XNt be the order statistics of the claim sizes occurring in the time interval [, t]. In this paper, we are interested in two large claims reinsurance treaties, LCR largest claims reinsurance and ECOMOR excédent du coût moen relatif, which were introduced Corresponding author. s: junjiang@mail.ustc.edu.cn J. Jiang, qtang@stat.uiowa.edu Q. Tang; Tel.: ; Fax:

2 to the actuarial literature b Ammeter 1964 and Thépaut 195, respectivel. Under the LCR treat, the reinsurer pas the sum of the first l largest claims, L l t l XNt l+i1 {Nt l}, l 1, 1.1 while under the ECOMOR treat, the reinsurer pas the sum of the parts of the first l 1 largest claims in excess of the lth largest one, E l t l X Nt l+i XNt l+1 1{Nt l}, l 2, 1.2 where 1 A denotes the indicator function of a set A. Through a series of papers since 198 s, Kremer made numerous efforts on general formulas or upper bounds for the expectations of reinsured amounts, interpreted as net reinsurance premiums, under some general reinsurance treaties including the present two; see Kremer 1985, 1998 and references therein. Assuming that the number l of the order statistics in 1.1 and 1.2 is fixed or increases in t at a certain rate and that the claim-size distribution F belongs to the maximum domain of attraction of certain extremal value distributions, Beirlant and Teugels 1992 as well as Ladoucette and Teugels 26 obtained limiting distributions for the quantities L l t and E l t as t. Hashorva 27 extended the scenario to the bivariate case. Embrechts et al gave a short review of this stud using extreme value theor. A comprehensive review of the two reinsurance treaties LCR and ECOMOR was made b Ladoucette and Teugels 26. See also Teugels 23 for an extended review and a sstematical treatment of these reinsurance treaties. In recent ears, several researchers started to investigate the asmptotic behavior of the tail probabilities of L l t and E l t. For the subexponential case, Ladoucette and Teugels 26 obtained a precise asmptotic estimate for the tail probabilit of E l t with l and t fixed; see the second relation of 2.7 below. Asimit and Jones 28 used copulas belonging to the maximum domain of attraction of an extreme value copula to describe the dependence among the claim sizes and the derived some precise asmptotic estimates for the tail probabilities of L l t and E l t with l fixed and Nt l nonrandom and fixed. These works initiate a new direction of the mainstream stud of the two reinsurance treaties. A direct application of such asmptotic results is to approximate risk measures of L l t and E l t such as Value at Risk, Expected Shortfall, Conditional Tail Expectation, and so on. The purpose of this paper is to establish precise asmptotic estimates for the tail probabilities of L l t and E l t, with l and t fixed. Our results show that, when F is an exponential distribution, these tail probabilities are both asmptotic to a multiple of the tail of a gamma distribution with suitable parameters, while when F has a convolution-equivalent tail, the are both asmptotic to a multiple of the tail of F. The prefactors involved are completel explicit and transparent. Specificall, for the subexponential case, one of our results coincides with a result first obtained b Ladoucette and Teugels 26. 2

3 The rest of this paper is organized as follows: Section 2 presents the main results after briefl introducing our conditions on the claim-size distribution, Section 3 prepares some lemmas, and Sections 4 and 5 prove the main results. 2 Preliminaries and Main Results Throughout the paper, all limit relationships are for x unless stated otherwise; for two positive functions a and b, we write ax bx if lim ax/bx On the Claim-size Distribution Since claim sizes are alwas nonnegative, we onl consider distributions on [,. Due to its memorless propert, the exponential distribution with parameter γ >, F x 1 F x e γx, x >, 2.1 is often an ideal candidate for claim-size distributions in the actuarial literature. It is well known that, for each k 1, 2,..., its k-fold convolution F k is a gamma distribution with probabilit densit function Hence, f x; k, 1/γ F k x γ k k 1! xk 1 e γx, x >. 2.2 γk 1 k 1! xk 1 e γx. 2.3 As a natural generalization, a distribution F is said to belong to the class Lγ for some γ if F x > for all x and the relation F x lim x F x e γ 2.4 holds for all. When γ >, we usuall sa that F has an exponential tail. In particular, if a distribution F belonging to the class Lγ is such that the limit F lim 2 x x F x 2c 2.5 exists and is finite, then we sa that F has a convolution-equivalent tail, written as F Sγ. As we go along we shall often suppress the phrase γ, but it remains in place. Since it was introduced b Chistakov 1964 and Chover et al. 1973a,b, the class Sγ has been extensivel investigated b man researchers and applied to various fields. Recent studies of this class can be found in Pakes 24, Tang 26, Foss and Korshunov 27, and Watanabe 28, among man others. This class is often used to model claim-size 3

4 distributions; see, for example, Embrechts and Veraverbeke 1982, Klüppelberg 1989a, and Tang and Tsitsiashvili 24. It is well known that the constant c in relation 2.5 is equal to the moment generating function of F at γ, defined to be m F γ e γx F dx; see Rogozin 2 and references therein. Therefore, for F Sγ it is necessar that m F γ <. This unfortunatel excludes the exponential distribution as defined in 2.1. Examples and criteria for membership of the class Sγ for γ > can be found in the Theorem of Embrechts 1983 and Theorems 2-4 of Cline When γ it reduces to the well-known subexponential class S, which contains Pareto, lognormal, and heavtailed Weibull distributions. See Embrechts et al for a review of applications of the class S to insurance and finance. In this paper, we shall assume that F either is an exponential distribution or belongs to the class Sγ. 2.2 Theorems Denote b Q t z Ez Nt the probabilit generating function of Nt. If it is analtic at z > then Q r n! t z Pr Nt n n r! zn r, r 1, 2, nr Note that, when z 1, the condition E Nt r < suffices for the series on the right-hand side of equalit 2.6 to converge. In the sequel, we shall borrow the notation Q r t 1 to represent this series for z 1, but we do not require that Q t z is analtic at z 1. It is clear that, for all t for which < ENt <, the relation Pr L 1 t > x ENtF x holds as long as F x > for all x; see Lemma 1 of Ladoucette and Teugels 26. Hence in this paper, we onl deal with the case l 2 for both reinsurance treaties. Theorem 2.1. Recall equalit 1.1 with l 2 and t fixed such that Pr Nt l >. Let the claim sizes X 1, X 2,... be i.i.d. with common continuous distribution F on, and independent of Nt. Assume E Nt l <. i If F is an exponential distribution with parameter γ as in 2.1, then ii If F Sγ, then Pr L l t > x γl 1 x l 1 e γx Q l t 1; l! l 1! Pr L l t > x F x l 2! e γ 4 l 2 e γu F du Q l 1 t F d.

5 Theorem 2.2. Recall equalit 1.2 with l 2 and t fixed such that Pr Nt l >. Let the claim sizes X 1, X 2,... be i.i.d. with common continuous distribution F on, and independent of Nt. Assume E Nt l 1 <. i If F is an exponential distribution with parameter γ as in 2.1, then ii If F Sγ, then Pr E l t > x Pr E l t > x γl 2 x l 2 e γx l 2! F x l 2! e l 1γ Pr Nt l ; l 2 e γu F du Q l 1 t F d. Note that the probabilit generating function Q t z is automaticall analtical at an z, 1. Also recall our convention for Q r t 1. Therefore, both Q l t 1 and Q l 1 t F in Theorems 2.1 and 2.2 are well defined. Theorems 2.1i and 2.2i show that the tail probabilities of L l t and E l t are both asmptoticall proportional to the tail of a gamma distribution with suitable parameters; recall relation 2.3. Note also that, when F S, Theorems 2.1ii and 2.2ii reduce to the relations Pr L l t > x Pr E l t > x ENt1 {Nt l} F x. 2.7 Ladoucette and Teugels 26 obtained the latter relation of 2.7 for E l t under the stronger condition that Q t z is analtic at z 1. Finall, we remark that all formulas obtained in Theorems 2.1 and 2.2 well capture the impact of all stochastic factors including the claim-size distribution F, the total number of claims Nt, as well as the number of claims l covered b reinsurance. Hence, these formulas should work fine for a relativel large value of x when either Nt has a non-degenerate distribution or γ >. However, for the case where γ and Nt is degenerate at some constant the can poorl perform unless x is extremel large because, as seen from 2.7, for this case the even fail to capture the impact of the number l. To keep the paper short, we shall not pursue numerical studies of these formulas. 3 Lemmas In this section we prepare several lemmas for later use. Lemma 3.1. Assume F Sγ. i For each function h such that hx and x hx, it holds that F hx F x hx o F x ; ii If, for each i 1,..., n, the distribution F i satisfies F i x c i F x for some c i >, then F 1 F n Sγ and F 1 F n x O F x. 5

6 Proof. The two items are direct consequences of Lemma 2 and Corollar 1 of Cline 1986, respectivel. Lemma 3.2. Let X and Y be two independent positive random variables with distributions F and G, respectivel. Assume F Sγ and Gx cf x for some c >. Then, Pr X + Y > x, X > Y m G γf x. 3.1 Proof. Since the class Sγ is closed under tail equivalence see page 26 of Klüppelberg 1989b, we have G Sγ. Moreover, as x uniforml for all >, Pr X + Y > x, X > Y F x Gd 1 c G x Gd, which is further equal to c 1 Pr Y 1 + Y 2 > x, Y 1 > Y 2 for Y 1 and Y 2 being i.i.d. copies of Y. B Lemma 3.1i, Pr Y 1 Y 2 > x/2 Gx/2 2 o Gx. Hence, Pr X + Y > x, X > Y 1 2c Pr Y 1 + Y 2 > x + o Gx m G γf x. This proves relation 3.1. The following lemma will pla a ke role in proving Theorems 2.1ii and 2.2ii: Lemma 3.3. Let X 1, X 2,... be a sequence of independent positive random variables such that X 1 follows a distribution F and X 2, X 3,... follow a common distribution F. Assume F Sγ and F x cf x for some c >. Write S k k X i for k 1, 2,.... Then, for each integer k 2 and each constant a, Pr S k > x + ka, X k > > X 1 > a F xe kγa e γ Pr S k 1 d, X k 1 > > X 1 > a. 3.2 Proof. For k 1, 2,... and a fixed, we write A k Pr X k > > X 1 > a and write G k as the conditional distribution of S k on X k > > X 1 > a. Thus b 2.4, relation 3.2 amounts to the relation Pr S k > x + ka, X k > > X 1 > a A k 1 m Gk 1 γf x + ka. 3.3 We are to prove that relation 3.3 holds for each integer k 2 and each a. As a b-product, our proof also shows that G k 1 Sγ. First, we claim that relation 3.3 holds for k 2 with A 1 Pr X 1 > a and G 1 Pr X 1 X 1 > a, which obviousl belongs to the class Sγ. Actuall, b Lemma 3.2, Pr S 2 > x + 2a, X 2 > X 1 m F γf x + 2a, 6

7 and b the local uniformit of the convergence of relation 2.4, Pr S 2 > x + 2a, X 2 > a X 1 Therefore, a Pr S 2 > x + 2a, X 2 > X 1 > a F x + 2a F d F x + 2a Pr S 2 > x + 2a, X 2 > X 1 Pr S 2 > x + 2a, X 2 > a X 1 F x + 2a a e γ F d. a e γ F d. Now we inductivel assume that relation 3.3 holds for k j for some integer j 2 with G j 1 Sγ. Straightforwardl, the distribution G j also belongs to the class Sγ since G j x Pr S j > x X j > > X 1 > a A j 1 m Gj 1 γf x. A j We need to prove that relation 3.3 still holds for k j + 1. For notational convenience, we write x x + j + 1 a, which tends to infinit uniforml for all j 2, 3,... and a. B comparing X j+1 and S j with j x/ j + 1 and x/ j + 1, respectivel, we split the probabilit of the left-hand side of relation 3.3 with k j + 1 into four parts as 4 Pr S j+1 > x, X j+1 > > X 1 > a B s s1 4 I s x, 3.4 s1 where the events B 1 -B 4 are defined to be B 1 {X j+1 x/ j + 1}, B 2 { x/ j + 1 < X j+1 j x/ j + 1}, B 3 {X j+1 > j x/ j + 1, S j > x/ j + 1}, and B 4 {X j+1 > j x/ j + 1, S j x/ j + 1}. Triviall, I 1 x. B Lemma 3.1ii, I 2 x jex/j+1 ex/j+1 Pr S j > x u F du O1 jex/j+1 ex/j+1 F x u F du o F x, where the last step follows from Lemma 5.5 of Pakes 24. Moreover, b items ii and i of Lemma 3.1, in turn, I 3 x Pr X j+1 > j x Pr S j > x j x x O1F F o F x. j + 1 j + 1 j + 1 j + 1 Therefore, plugging these estimates into 3.4 ields that Pr S j+1 > x, X j+1 > > X 1 > a I 4 x + o F x

8 Now we focus on I 4 x. Introduce a positive random variable Y j independent of X j+1 and distributed b G j, so that I 4 x Pr S j+1 > x, S j x j + 1, X j > > X 1 > a A j Pr X j+1 + Y j > x, Y j x. 3.6 j + 1 On the one hand, recalling G j Sγ, b Lemma 3.2 we have I 4 x A j Pr X j+1 + Y j > x, X j+1 > Y j A j m Gj γf x. 3.7 On the other hand, b the local uniformit of the convergence of relation 2.4, for arbitraril fixed M > and all large x, I 4 x A j M M F x G j d A j F x e γ G j d. 3.8 It follows from relations 3.7 and 3.8 and the arbitrariness of M > that I 4 x A j m Gj γf x. 3.9 We conclude from relations 3.5 and 3.9 that relation 3.3 holds for k j + 1. This completes the proof of Lemma Proof of Theorem 2.1 Hereafter, for notational convenience, we write p n t Pr Nt n for t and n, 1,.... Since F is continuous on,, it is clear that the random variables Xn l+1,..., X n have a joint probabilit densit function Hence, l Pr X n l+i dx i Pr L l t > x n! n l! F n l x 1 l F dx i, x l > > x 1 >. l p n t Pr Xn l+i > x n! p n t n l! L x F n l x 1 l F dx i, { where L x x 1,..., x l : } l x i > x, x l > > x 1 >. Introduce a positive random variable X 1 independent of X 2,..., X l and distributed b F n l+1, so that Pr X 1 > x 8

9 n l + 1 F x. Then, l n! Pr L l t > x p n t n l + 1! Pr X i + X 1 > x, X l > > X 2 > X 1 i2 n! p n t n l + 1! P Lx, n. 4.1 i Let F be an exponential distribution as given in 2.1. B comparing X 1 with x/l, we split P L x, n in 4.1 into two parts as P L x, n Pr X l > > X 2 > X 1 > x l + Pr X i + X 1 > x, X l > > X 2 > X l 1, X 1 x l P L,1 x, n + P L,2 x, n. i2 4.2 Clearl, P L,1 x, n Pr X l,..., X 2, X 1 > x n l + 1 e γx. 4.3 l B the memorless propert of the exponential distribution, 1 x/l l P L,2 x, n Pr X i > x l, X l,..., X 2 > F n l+1 d l 1! i2 1 x/l l F l 1 Pr X i > x l F n l+1 d. l 1! i2 Recall that the sum l i2 X i has the densit function f u; l 1, 1/γ as given in 2.2. Plugging it into the above and using change of variables, x/l n l + 1 P L,2 x, n l 1! l 2! γl x l 2 e γx u + x l l 2 e γu 1 e γ n l dud x l 2 n l + 1 l 1! l 2! γl x l 2 e γx Jx. 4.4 On the one hand, Jx x/l u x + 1 l l 2 e γu ddu x u x l l 1 x l l 1 γ. x + 1 l 1 u x l 1 e γu du 9

10 On the other hand, with M > arbitraril fixed, Jx 1 x/l 1 l l 2 1 e γ n l d γ M x 1 x/l 1 e γm n l 1 l l 2 d γ M x 1 e γm n l x l l 1 γ. Therefore, Jx We conclude from relations that x l l 1 γ. 4.5 P L x, n n l + 1 l!l 1! γl 1 x l 1 e γx. 4.6 Now we return to equalit 4.1. B relation 2.3 with k l, there exists some constant C 1 > such that, for all n l, l + 1,... and all large x, l n! P L x, n n! Pr X i > x C n l + 1! x l 1 e γx n l! x l 1 e γx 1 n l. Therefore, appling the dominated convergence theorem to 4.1 and using relations 4.6 and 2.6, Pr L l t > x γl 1 x l 1 e γx n! p n t l! l 1! n l! γl 1 x l 1 e γx Q l t 1. l! l 1! ii For F Sγ, we begin with equalit 4.1. B Lemma 3.1ii, there exists some constant C 2 > such that, for all n l, l + 1,... and all large x, l n! P L x, n n! Pr X i > x C 2 n l. n l + 1! F x n l! F x Hence, appling the dominated convergence theorem to 4.1 and using Lemma 3.3 with k l, a, and F F n l+1, n! Pr L l t > x F x p n t n l + 1! K Ln, 4.7 where K L n e γu Pr 1 l 2! 1 l 2! l 1 X i + X 1 du, X l 1 > > X 2 > X 1 i2 l 1 e γ i2 E e γx i 1 {Xi >} F n l+1 d l 2 e γ e γu F du F n l+1 d

11 Therefore, b relations 4.7, 4.8, and 2.6 we have Pr L l t > x F x l 2! F x l 2! This completes the proof of Theorem 2.1. n! l 2 p n t e γ e γu F du F n l+1 d n l + 1! l 2 e γ e γu F du Q l 1 t F d. 5 Proof of Theorem 2.2 Similarl as in the proof of Theorem 2.1, l Pr E l t > x p n t Pr X n l+i Xn l+1 > x n! p n t n l! E x l 1 F n l x F dx i, where in the integral we suitabl changed the subscripts and { } l 1 E x x,..., x l 1 : x i x > x, x l 1 > > x >. i Introduce a positive random variable Y independent of X 1,..., X l 1 and distributed b F n l+1. Thus, n! Pr E l t > x p n t n l + 1! P Ex, n, 5.1 where P E x, n Pr l 1 i X i l 1 Y > x, X l 1 > > X 1 > Y l 1 Pr X i > x + l 1, X l 1 > > X 1 > F n l+1 d. i Let F be an exponential distribution as given in 2.1. Then b its memorless propert, 1 l 1 P E x, n Pr X i > x, X l 1,..., X 1 > F n l+1 d l 1! n l + 1 l 1 l 1! γ Pr X i > x e lγ 1 e γ n l d. 11

12 Hence, b relation 2.3 with k l 1, P E x, n n l + 1! n! l 2! γl 2 x l 2 e γx, 5.2 and there exists some constant C 3 > such that, for all n l, l + 1,... and all large x, l 1 n! P E x, n n l + 1! x l 2 e n! Pr X i > x C γx n l + 1! x l 2 e γx 3 n l 1. Therefore, appling the dominated convergence theorem to 5.1 and using relation 5.2, Pr E l t > x γl 2 x l 2 e γx l 2! Pr Nt l. ii For F Sγ, b Lemma 3.1ii, there exists some constant C 4 > such that, for all n l, l + 1,... and all large x, l 1 n! P E x, n n! Pr X i > x C 4 n l 1. n l + 1! F x n l + 1! F x Hence, appling the dominated convergence theorem to 5.1 and using Lemma 3.3 with k l 1, a, and F F, Pr E l t > x n! lim p n t x F x n l + 1! lim P E x, n x F x n! p n t K E e l 1γ F n l+1 d, 5.3 n l + 1! where K E e γu Pr l 2 X i du, X l 2 > > X 1 > 1 l 2 E e γx i 1 {Xi >} l 2! 1 l 2! Therefore, b relations 5.3, 5.4, and 2.6 we have l 2 e γu F du. 5.4 Pr E l t > x F x n! l 2 p n t e l 1γ e γu F du F n l+1 d l 2! n l + 1! F x l 2 e l 1γ e γu F du Q l 1 t F d. l 2! This completes the proof of Theorem 2.2. Acknowledgments. The authors wish to thank an anonmous referee for his/her helpful comments and suggestions. 12

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