Ruin Probabilities of a Discrete-time Multi-risk Model

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1 Ruin Probabilities of a Discrete-time Multi-risk Model Andrius Grigutis, Agneška Korvel, Jonas Šiaulys Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 4, Vilnius LT-035, Lithuania andrius.grigutis@mif.vu.lt, agneska.korvel@mif.stud.vu.lt, jonas.siaulys@mif.vu.lt May 16, 016 Abstract. In this work, we investigate a multi-risk model describing insurance business with two or more independent series of claim amounts. Each series of claim amounts consists of independent nonnegative random variables. Claims of each series occur periodically with some fixed inter-arrival time. Claim amounts occur until they can be compensated by a common premium rate and the initial insurer s surplus. In this article, we derive a recursive formula for calculation of finite-time ruin probabilities. In the case of bi-risk model, we present a procedure to calculate the ultimate ruin probability. We add several numerical examples illustrating application of the derived formulas. Keywords: multi-risk model; discrete-time risk model; ruin probability; recursive formula; net profit condition 010 Mathematics Subject Classification: 91B30, 60G40. 1 Introduction The main classical collective risk model, used to evaluate insurance business, is the so-called renewal risk model. According to the renewal risk model, the insurer s surplus at each moment of time t 0 satisfies the following equation: Θt W u t u + ct Z i, 1 where u 0 is the initial insurer s surplus, and c > 0 is a premium rate. The claim amounts Z 1, Z,... are assumed to be independent copies of a nonnegative random variable r.v. Z. The number of claims in the interval [0, t] is a counting renewal process Θt n1 1I θ1 +θ +...+θ n t}, where θ 1, θ,..., θ n are independent copies of a nonnegative r.v. θ, which is not concentrated at zero, i.e., Pθ 0 < 1. In addition, the claim amounts Z 1, Z,... and inter-arrival times θ 1, θ,... are supposed to be mutually independent. It is clear that the r.v.s Z and θ, the initial surplus u, and the premium rate c generate the renewal risk model. In the special The firs author was supported by grant No MIP-049/014 from the Research Council of Lithuania, the third author is supported by a grant No MIP from the Research Council of Lithuania. 1

2 case where θ has an exponential distribution, the renewal counting process Θt becomes a homogeneous Poisson process, and the obtained risk model is called the classical risk model. If u N 0, c 1, θ 1, and Z is an integervalued, then we call the model defined by 1 a discrete-time risk model. In such a case, from 1 we get that Usually, W u t u + t t Z i, t N 0. each insurance company works with several series of various claims. Each series of claims Z j1, Z j,...}, j 1,,..., K}, can be driven by a specific initial surplus u j, a specific premium rate c j, and a specific series of inter-arrival times θ j1, θ j,...}. There are two different ways to consider insurance business in this situation. The first way is to create the so-called multidimensional renewal risk model. In this case, we suppose that the insurer s surplus at each moment of time t 0 is a random vector where W 1,u1 t, W,u t,..., W K,uK t, Θ j t W j,uj t u j + c j t Z ji, t 0, j 1,,..., K, and Θ j t is the renewal process generated by the r.v. θ j1. Several problems related to the multidimensional renewal risk model were investigated by Collamore [5], Sundt [], Vernic [3], Denuit et al. [8], Picard et al. [0], Hult et al. [16], Yuen et al. [6], Li et al. [17], Avram et al. [1], Dang et al. [7], Chen et al. [4] and He et al. [15]. The second way to consider insurance business with several series of claims is related to the multi-risk model. We say that the insurer s surplus varies according to the multi-risk model if W u t u + ct K Θ j t Z ji 3 for all time moments t 0. Here we suppose that Z j1, Z j,...} are independent and identically distributed i.i.d. r.v.s for each fixed j 1,,..., K. The r.v.s θ j1, θ j,...} generating counting renewal processes Θ j t are also i.i.d. The random claim amounts Z j1, Z j,...} K and random inter-arrival times θ j1, θ j,...} K are mutually independent. We note that the r.v.s Z 11, Z 1,..., Z K1 and θ 11, θ 1,..., θ K1 in 3 may have different distributions. In the multidimensional risk model, each series of claim amounts has its own dimension, whereas in the multi-risk model, all series of claims are placed in one basket. The multirisk model was investigated by Wang and Wang [4, 5] and by Lu [18], [19], where problems related to large deviations of the sum in 3 where considered. When all renewal counting processes Θ j t in 3 are generated by degenerate r.v.s, the multi-risk model becomes a discrete-time multi-risk model. For instance, if K 3, c 1, θ 11 1, θ 1, and θ 31 3, then from 3 it follows that t t/3 W u t u + t Z 1i Z i for all time moments t 0. Z 3i In this paper, we consider a discrete-time multi-risk model see 4. We derive the recursion formulas to calculate finite-time ruin probabilities and ultimate ruin probabilities. Note that a procedure for calculation of ultimate ruin probabilities is obtained only in the case of two different series of claim amounts. To prove all recurrent relations, we use methods developed in [9], [14], [10], [11], [13], [1] for model and methods developed in [3], [], [6] for so-called multi-seasonal model. The rest of the paper is organized as follows. In Section, we describe a discrete-time

3 multi-risk model and present the main results on the finite-time and ultimate ruin probabilities. In Sections 3 and 4, we prove the basic recursive relations for the finite-time and ultimate ruin probabilities. In Section 5, we show the meaning of the net profit condition which is described before Theorem 4 in the discretetime multi-risk model. The procedure for calculating the initial values of the ultimate ruin probability is presented in Section 6. In Section 7, we illustrate the obtained procedures numerically. Model and main results In this section, we describe the discrete-time multi-risk model and define its main critical characteristics. We present the main obtained recursive relations to calculate these characteristics of the model. We say that the insurer s surplus W u varies according to the discrete-time multi-risk model if, for all time moments t N 0, W u t u + t K t/i Z ij, 4 where K is a fixed natural number, u N 0 is the insurer s initial surplus, and Z i1, Z i,... are independent copies of an integer valued nonnegative r.v. Z i for each i 1,,..., K}. In addition, the series of r.v.s Z i1, Z i,...} K are mutually independent. In Eq. 4, time t can be a nonnegative real number. The above conditions imply that W u t W u t + t t. So, in order to describe the behavior of a discrete-time risk model, it suffices to consider t N 0. Obviously, every discrete-time multi-risk model is generated by the insurer s initial surplus u and collection of r.v.s Z 1, Z,..., Z K. The claim amount Z 1 occurs at every time moment, Z occurs at every second time moment, and so on. The time of ruin, finite-time ruin probability, and ultimate ruin probability are the main critical characteristics of the discretetime risk model. We define the time of ruin T u as the first time the insurer s surplus becomes non-positive, that is, inft N : W u t 0}, T u if W u t > 0 for t N. 5 It is obvious that T u is an extended r.v. because, in general, PT u > 0. We call the probability ψu, T PT u T, where T N, the finite-time ruin probability. The finite-time ruin probability depends on the insurer s initial surplus u, time T until the surplus evolution is observed, and the r.v.s Z 1, Z,..., Z K generating the multi-risk model. Definitions 4 and 5 imply that T ψu, T P W u t 0 }. 6 t1 We call the probability ψu PT u < the ultimate ruin probability. Similarly to 6, we have that ψu P W u t 0 }. 7 t1 The nonnegative integer-valued r.v.s Z 1, Z,..., Z K generating the multi-risk model can be described by the local probabilities h ik PZ i k, k N 0, i 1,,..., K, or by their distribution functions d.f. H i x k x h ik, x R, i 1,,..., K. If K, then 4 implies that W u t u + t t k1 X k Y l, t N 0, 8 l1 3

4 where u N 0, X 1, X,... are independent copies of a nonnegative integer valued r.v. X Z 1, and Y 1, Y,... are independent copies of an integer valued r.v. Y Z. We call the model defined by 8 a bi-risk discrete-time risk model. It is clear that such a model is generated by the insurer s surplus u and two random claim amounts X and Y, where X occurs at every time increment, and Y occurs at every double time increment. In such a case, we use the following notation for the local probabilities and d.f.s of X and Y : a k PZ k, k N 0 ; b l PY l, l N 0 ; Ax a k, x R; Bx 0 k x 0 l x b l, x R. Our first assertion gives a recursive procedure to calculate the finite-time ruin probabilities for a general discrete-time multi-risk model. Theorem 1 Suppose that r.v.s Z 1, Z,..., Z K, K 1, generate the discrete-time multi-risk model. For all u, l N 0, let where B Kl D K lu k ij N 0 : i 1,,..., K}, j N, B Kl u + l}, K Then, for all u N 0, we have: l+1/i ψu, 1 k 11 >u l+1 k ij h 1k11, ψu, ψu, 1 + ψu, T ψu, T 1 + l+1/ k 1j + k 11 u k 11 +k 1 +k 1 >u+1 l+1/k k j + + h 1k11 h 1k1 h k1, D D K 0u DK 1u DK T u DK T 1u k ij D for all T 3, 4,..., M}, where M is the least common multiple of numbers 1,,..., K. If u N 0 and T M + 1, then ψu, T ψu, M + D D0u K DK 1u DK M u DK k ij D M 1u k Kj. h ikij h ikij ψu + M B KM 1, T M. For a larger K, the obtained recursive formulas are quite complex, and numerical application of these formulas requires much resources. Otherwise, when K is relatively small, the formulas of Theorem 1 imply a sufficiently simple algorithm to calculate finite-time ruin probabilities. For example, in the bi-risk model, for each u N 0, we have that 4

5 ψu, 1 a k, k>u ψu, ψu, 1 + ψu, T ψu, + k+ l+ m>u+1 k+ l+ m u+1 Theorem 1 allows us to calculate the values of ψu, T, u N 0, T N, for an arbitrary discrete-time multi-risk model. Similar recursive procedures for ultimate ruin probabilities, ψu + k l m, T, T 3. 9 are a bit different. The following result allows us to calculate the values of ψu only in the case of a discrete-time bi-risk model. Theorem Let us consider a discrete-time bi-risk model with generating r.v.s X and Y. Then, for all u N 0, ψu k>u a k + k+l+m>u+1 + k+l+m u+1 ψu + k l m. We see from the last theorem that we can calculate the values of ψu for u if we know ψ0 and ψ1. Theorems 3 and 4 provide an algorithm for finding ψ0 and ψ1. For every u N 0, we denote: A A Bu, k+l+m u A A Bu 1 A A Bu. Theorem 3 Let us consider a discrete-time bi-risk model with generating r.v.s X and Y for finite means EX and EY. i If µ x,y : EX + EY/ 1 and the r.v.s X, Y are non-degenerate, then ψu 1 for all u N 0. ii If µ x,y < 1 and b 0 0, then we have: ψ0 µ x,y 1, ψ1 1 A A B1 1 µ x,y, ψu for all u, 3,...}. 1 A A B 1 u 1 ψva A Bu + 1 v + v1 vu+1 A A Bv As usual, we call condition µ x,y < 1 the net profit condition. It states that, in every two units of time, the insurer s premium income exceeds the insurer s expected aggregate claim amount. The following theorem provides a recursive procedure to calculate ψ0 and ψ1 under the net profit condition. 5

6 Theorem 4 Let us consider a discrete-time bi-risk model with generating r.v.s X and Y. Suppose that a 0 0, b 0 0, and µ x,y EX + EY/ < 1. Then γ n+1 γ n ψ0 1 µ x,y 1 lim, n β n+1 β n ψ1 1 µ x,y 1 + a 0 b 0 ψ0, a 0 b 0 where β n } and γ n } are two recurrent sequences defined as follows: β 0 1, β 1 1, β n 1 n 1 β n α i β n i a n 1, n, 3,...}, a 0 b 0 α 0 γ 0 0, γ 1 1, γ n 1 n 1 γ n α i γ n i + a n 1, n, 3,...}, a 0 b 0 α 0 and α r k+l+mr for r N 0. 3 Proof of Theorem 1 We prove the assertion only in the particular case K. In fact, we prove only the equations given in 9. The proof of the general case is similar. By 6 we have that ψu, 1 PW u 1 0 Pu + 1 X 1 0 PX > u a k. k>u Consequently, the first two equalities of 9 hold. It remains to prove the third one. If T 3, then equalities 4, 6 and the law of total probability imply that ψu, T T P t1 u + t t } X i Y j 0 Similarly, ψu, P W u 1 0} W u 0} PW u 1 0} + PW u 0} W u 1 > 0} ψu, 1 u + PX 1 + X + Y 1 u +, X 1 k by the law of total probability. It is obvious that the second term of the last equality is u Pk + X + Y 1 u + a k k+l+m>u+1. P u + 1 X a k P X l, Y 1 m, l0 m0 T t } u + t k X i Y j 0 t ψu, a k l,m: k+l+m u+1 i a k l,m: k+l+m>u+1 k l m t i3 T a l b m P t3 a l b m u + t X i Y j 0}. j The random variables X 1, X,... are independent and identically distributed i.i.d. as well 6

7 as the r.v.s Y 1, Y,.... Therefore, t t d X i X i, 10 i3 j for t 3, 4,...}. 1 d Y j second equality of 9 imply that ψu, T ψu, + k+l+m u+1 P l m Y j 11 The last relations and the T τ τ1 u + + τ k τ/ X i Y j 0}. Now we see that the last equality of 9 follows from expression 6, and the particular case of Theorem 1 is proved. 4 Proof of Theorem The proof is similar to the proof of Theorem 1. Indeed, by 7 and the law of total probability we have that ψu P W u t 0} t1 PW u PW u 1 > 0, W u 0 + P W u 1 > 0, W u > 0, W u t 0} k>u a k + + k+l+m u+1 k+l+m>u+1 m P t i3 t3 t3 u + t k l X i Y j 0}. j To complete the proof, it suffices to observe that the last sum equals P k+l+m u+1 m k+l+m u+1 τ τ1 u + + τ k l τ/ } X i Y j 0 ψu + k l m due to Eqs. 10, 11 and definition 7. Theorem is proved. 5 Proof of Theorem 3 Proof of part i. Let S t : t X i + Y j t, t N. 1 It follows from 7 that, for every u N, However, ψu PS t u for some t N S n PS n u for some n N Plim sup S n u. 13 n n X i + n Y j n n for every n N, where ξ 1, ξ,...} are independent copies of the r.v. ξ X 1 + X + Y 1. that Since Eξ 0 and Pξ 0 < 1, we have Plim sup S n 1 14 n see, for instance, Proposition 7..3 in [1]. The obtained relations 13 and 14 imply part i of Theorem 3. steps. implies Proof of part ii consists of several First, we prove that the condition µ x,y < 1 ξ i lim ψu u 7

8 According to definition 7 we have that, for every u N, ψu PS t u for some t N P S n u + P S n+1 u, 16 sup n N sup n N where S t is defined in 1. It is clear that S n n 1 n n ξ i, where r.v.s ξ 1, ξ,... are described in the proof of part i. Hence, by the strong law of large numbers, S n n a.s. n n m 1 Eξ µ x,y 1 : < 0. Therefore, P sup S n n + If N and u is positive, then P P P P sup n N N 1 n1 N 1 n1 N 1 n1 S n < u Sn u }, Sn u } + P nn 1 Sn u } + P sup n N nn m Sn u } Sn u } 1 S n n + 1. This inequality and relation 17 imply that lim P sup S n < u u n N On the other hand, for all n N, S n+1 n + 1 n n n n ξ i + X n+1 1 n + 1. Due to the strong law of large numbers, 1 n n ξ i Therefore, and we obtain a.s. n µ x,y 1, S n+1 n + 1 n N X n+1 1 n + 1 a.s. n, a.s. n 0. lim P sup S n+1 < u 1 19 u using the same procedure as for the sums S n, n N. Equality 15 follows now from estimate 16 and Eqs. 18, 19. In this step, we prove that ψ0 + a 0 b 0 Av + 1ψ1 v+1 A A Bu + a 0b 0 ψv + + ψv + 1 u1 v+1 ψu A A Bv + u u1 for all v N 0. + a 0 b 0 Av + 1 a 0 0 From Theorem we have that ψu A A Bu a u+1 a 0 b 0 + ψu + k l m k+l+m u+1 for all u N 0. Therefore, v ψu u0 where v N 0, and S v u0 k+l+m u+1 v A A Bu + 1 u0 v + a 0 b 0 a u+1 + S, 1 u0 ψu + k l m. 8

9 Changing the order of summation in S, we obtain the following expression of S: v u u0 v uk v + v u+1 k v l0 u+1 k l0 u+1 k l m0 v u+1 k l m0 ψu + k l m ψu + k l m u+1 k l l1 uk+l 1 m0 v 0 v l0 uk v ψu + k l m u+1 k l m0 u+1 k l l1 uk+l 1 m0 + a 0 v v v uk l1 + a 0 v v ψu + k l m ψu + k l m u+1 k l m0 l m0 a k b m ψu + k m v uk+l+m 1 ψu + k l m v v + a 0 b 0 a k ψu + k l1 uk m1 + a 0 b 0 v l m0 + a 0 v v uk+m 1 v+ k r v+ k l m m1 ψra k a k b m ψu + k m v+ k m ψr ψra k b m. v v l1 v+ k l v+ + a 0 b 0 r + a 0 v ψr v+ + a 0 b 0 v+1 ψr r v+ k l r ψr m0 v+ r ψr v+ k r v+1 + a 0 ψr v+1 r v+ + a 0 b 0 r a k v+ k r ψr m1 v+ k l r l1 m0 v+ r ψr a k v+ k r v+1 + a 0 ψr v+1 r v+ k r m1 v+ k l r l1 m0 v+ r ψr a k v+1 r v+1 ψra A Bv + r v+ k r m1 a k b m a k b m a k b m a 0 b 0 Av + 1ψ1 + a 0b 0 ψv +. The last expression and Eq. 1 immediately imply relation 0. In this step, we complete the proof of Theorem 3. By Eq. 15 we have lim v u1 v+1 ψua A Bv + u 0. On the other hand, lim v u1 v+1 A A Bu A A Bu u0 A A B0 µ x,y 1 + a 0b 0 Now, changing the order of summation in the opposite direction, we get that S can be written in the following form: because of A A Bu µ x,y. u0 9

10 Therefore, ψ0 + a 0 b 0 ψ1 µ x,y + a 0 b 0 1 as v in both sides of Eq. 0. If b 0 0, then ψ0 µ x,y 1, and the first statement of part ii follows. get If we set v 0 and b 0 0 in 0, then we ψ1 1 1 ψ0 A A B1 1 1 ψ0 a 0 b, 1 and the second equality of ii follows. The third equality of ii also follows from 0 if b 0 0. Theorem 3 is proved. 6 Proof of Theorem 4 Recall that α k PX 1 + X + Y 1 k for all k N 0. Let ϕu 1 ψu be the survival probability of the discrete-time bi-risk model for the initial insurer s surplus u N 0. By definition 7, the law of total probability, and Eqs. 10, 11 we obtain ϕu P t1 P t P t u + t u + t u + t t t t } X i Y j > 0 } X i Y j > 0 } X i Y j > 0, X 1 u + 1 u+1 P X 1 + X + Y 1 k, t3 u + t t i3 } X i Y j k > 0 j P X 1 u + 1, X 0, Y 1 0, t3 u + t t i3 } X i Y j > 0 j u+1 α k ϕu + k a 0 b 0 a u+1 ϕ1. So, for an arbitrary u N 0, we have that u+1 ϕu α u+1 k ϕk + 1 a 0 b 0 a u+1 ϕ1. 3 Let β n and γ n be two recurrent sequences defined in Theorem 4. Let us prove by induction that ϕn β n ϕ0 + 1 µ x,y γ n 4 for all n 0. If n 0, then 4 is evident. If n 1, then relation 4 follows from because ϕ1 1 a 0 b 0 ϕ0 + 1 µ x,y a 0 b 0. We now prove that 4 is true for n N +1 assuming that it holds for n N. Substituting u N 1 into Eq. 3, we get ϕn 1 Therefore, N α N k ϕk + 1 a 0 b 0 a N ϕ1. ϕn α 0 ϕn 1 N k1 α k ϕn k a 0 b 0 a N ϕ1, and by the induction hypothesis we have that ϕn α 0 β N 1 ϕ0 + 1 µ x,y γ N 1 N α k β N k+1 ϕ0 k1 + 1 µ x,y γ N k+1 10

11 + a 0 b 0 a N β 1 ϕ0 + 1 µ x,y γ 1 1 ϕ0 β N 1 α 0 N α k β N+1 k k1 + a 0 b 0 a N β µ x,y γ N 1 α 0 N α k γ N+1 k + a 0 b 0 a N γ 1 k1 β N+1 ϕ0 + 1 µ x,y γ N+1. Consequently, Eq. 4 holds for all n 0. Now we derive both equalities of Theorem 4. The sequence ψu, u N 0, is nonincreasing by 7. Therefore, ϕu is nondecreasing with respect to u, and there exists a finite limit lim ϕu. Consequently, u lim ϕn + 1 ϕn 0. n From the last equality and relation 4 we obtain that lim βn+1 β n ϕ0+1 µ x,y γ n+1 γ n 0. n Therefore, γ n+1 γ n ϕ0 µ x,y 1 lim, 5 n β n+1 β n provided that for a positive constant c. inf n N 0 β n+1 β n c 6 We observe that the statement of Theorem 4 follows immediately from 5 and. It remains to show that β k+1 β k 1, β k β k, 7 for all k N because 6 with c follows from 7 by considering odd and even n separately. It is easy to see that 7 is true for k 0. Let us show that it holds for k N + 1 if it does for k 1,,..., N. If k N + 1, then β N+ 1 N+1 β N α i β N+ i a N+1. α 0 By the induction hypothesis, β N+ 1 α 0 β N α β N α 4 β N α N β N α 1 β 1 α 3 β 1 α N+1 β 1 a N+1 1 α 0 β N 1 α α 4 α N β 1 α 1 + α α N+1 a N+1 1 α 0 β N α a 0 b 0 α N+1 a N+1 β N. 8 Similarly, by the induction hypothesis and the proved estimate 8 we have β N+3 1 N+ β N+1 α i β N+3 i a N+ α 0 1 α 0 β N α α 4 α N+ β N+1. 9 Inequalities 8 and 9 imply that 7 holds for all k N. This finishes the proof of Theorem 4. 7 Numerical examples In this section, we present four examples of computing numerical values of finite-time ruin probability and ultimate ruin probability for various discrete-time multi-risk models. All calculations are carried out using software MATH- EMATICA. In the presented tables, the numbers are rounded up to three decimal places. 11

12 Example 1. Let us consider the bi-risk model generated by r.v.s X and Y having the following simple distributions: X 0 1 P 3/4 1/8 1/8 ; Y 0 1 P 1/10 8/10 1/10. Using Theorem 1, we obtain Table 1 of the values of the function ψu, T. The last row of this table shows the values of ψu obtained by Theorems and 4. Note that the net profit condition is satisfied in this example. Example. Let us now consider the bi-risk model generated by a r.v. X as in Example 1 and a r.v. Y such that PY 1 9/10 and PY 1/10. In such a case, we have that b 0 PY 0 0 and µ x,y < 1. We fill Table by applying Theorems 1 and 3. Example 3. We say that a r.v. ξ has the Poisson distribution with parameter λ > 0 ξ Πλ if Pξ k e λ λ k /k!, k N 0. Consider the multi-risk model generated by three r.v.s Z 1 Π1/3, Z Π1/, and Z 3 Π1. We fill Table 3 of numerical values of the function ψu, T using Theorem 1. Note that EZ 1 + EZ / + EZ 3 /3 < 1 in the case under consideration. Example 4. We say that a r.v. ξ has the geometric distribution with parameter p 0, 1 ξ Gp if Pξ k p1 p k, k N 0. Consider the multi-risk model generated by three r.v.s Z 1 G1/3, Z G1/, and Z 3 G/3. Using the formulas of Theorem 1, we fill Table 4 similarly as in Example 3. We observe that EZ 1 + EZ / + EZ 3 /3 > 1 in the last example. In view of Table 4, it seems that lim ψu, T 1 for each fixed u. T Table 1. Values of functions ψu, T and ψu for the model in Example 1. T \ u

13 Table. Values of functions ψu, T and ψu for the model in Example. T \ u Table 3. Values of function ψu, T of the discrete-time multi-risk model generated by three Poisson r.v.s. T \ u

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