Introduction The discrete time model

Insurance: Mathematics and Economics 26 2000 133–149 The discrete-time risk model with correlated classes of business Hélène Cossette, Etienne Marceau ∗ École d’Actuariat, Pavillon Alexandre-Vachon Local 1620, Université Laval, Sainte-Foy, Québec, Canada, G1K 7P4 Received 1 February 1999; received in revised form 1 October 1999; accepted 24 November 1999 Abstract The discrete-time risk model with correlated classes of business is examined. Two different relations of dependence are considered. The impact of the dependence relation on the finite-time ruin probabilities and on the adjustment coefficient is also studied. Numerical examples are presented. © 2000 Elsevier Science B.V. All rights reserved. MSC: IM11; IM12; IM13; IM30 Keywords: Correlated aggregate claims; Shock models; Ruin probability; Adjustment coefficient

1. Introduction

In most actuarial literature related to risk theory, the assumption of independence between classes of business in an insurance book of business is made. In practice, however, there are situations in which this assumption is not verified. In the case of a catastrophe such as an earthquake for example, the damages covered by homeowners and private passenger automobile insurance cannot be considered independent. Papers that treat of a relation of dependence between classes of business include Ambagaspitiya 1998, Cummins and Wiltbank 1983 and Wang 1998. In the present paper, we study the probability of ruin in the discrete-time risk model proposed by Bühlmann 1970 and also presented in Bowers et al. 1997, Klugman et al. 1998 and Rolski et al. 1999. We first give a brief description of the discrete-time model and we define the probability of ruin over finite and infinite-time within this model. Then, we use a Poisson common shock model and a negative binomial component model, proposed by Wang 1998, to introduce a relation of dependence between classes of business. We present numerical examples to illustrate the impact of the introduction of a relation of dependence on the probability of ruin. We also examine its influence on the adjustment coefficient. ∗ Corresponding author. Tel.: +1-418-656-3639; fax: +1-418-656-7790. E-mail addresses: hcossettact.ulaval.ca H. Cossette, etienne.marceauact.ulaval.ca E. Marceau 0167-668700 – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 6 8 7 9 9 0 0 0 5 7 - 8 134 H. Cossette, E. Marceau Insurance: Mathematics and Economics 26 2000 133–149

2. The discrete time model

Assume the discrete-time process {U n , n = 0, 1, 2, . . . } where U n is the surplus for a book of business of an insurer at time n n = 0, 1, 2, . . . which is defined as U n = u + cn − S n , 1 where u is the initial surplus, c the premium income received during each period and S n the total claim amounts over the first n periods. It is also assumed that S n = W 1 + W 2 + · · · + W n , 2 where W i represents the total claim amounts for the book of business in the period i and {W i , i = 1, 2, . . . } is a sequence of independent and identically distributed random variables with E[W i ] = µ W c . The probability distribution and density function of W i i = 1, 2, . . . are denoted by F W w and f W w , respectively. Given 2, we can rewrite 1 as follows: U n = u + c − W 1 + c − W 2 + · · · + c − W n . 3 Let T be the time of ruin defined as T = infn, U n assuming that T = ∞ if U n ≥ 0 for all n = 1, 2, . . . . Let ψu, 1, n be the finite-time ruin probability over the periods 1 to n ψu, 1, n = P T ≤ n. When n → ∞ in ψu, 1, n, we have ψu = P T ∞ which is the infinite-time ruin probability. Also, ϕu = 1 − ψu, ϕu, 1, n = 1 − ψu, 1, n are, respectively, the infinite-and finite-time horizon non-ruin probabilities. Given 3, we have ϕu, 1, n = P U 1 ≥ 0, U 2 ≥ 0, . . . , U n ≥ = P W 1 ≤ u + c, W 1 + W 2 ≤ u + 2c, . . . , W 1 + W 2 + · · · + W n ≤ u + cn. The analytical expression of ϕu, 1, n is given in the following theorem. Theorem 1. Let {W i , i = 1, 2, . . . } be a sequence of i.i.d. random variables and c the annual premium income constant over each period. Then, ϕu, 1, n = Z u+c ϕu + c − w, 1, n − 1 dF W w. 4 Proof. By the theorem of total probabilities, we have ϕu, 1, n = Z u+c ϕu + c − w, 2, n dF W w, where ϕy, j, n = P W j ≤ y + c, W j + W j + 1 ≤ y + 2c, . . . , W j + W j + 1 + · · · + W n−j ≤ y + cn − j . H. Cossette, E. Marceau Insurance: Mathematics and Economics 26 2000 133–149 135 Since W 1 , . . . , W n are i.i.d., we have ϕy, 2, n = ϕy, 1, n − 1 from which the result follows. The calculation of exact values of ϕu, 1, n by the direct application of 4 is rarely possible. Therefore, an algorithm which approximates ϕu, 1, n is presented in Theorem 2. The application of the algorithm requires the discretization of the distribution function F W . Methods of discretization are proposed in Klugman et al. 1998 and Panjer and Willmot 1992. Assume that F ˜ W is the discrete distribution function obtained by one of these methods and ˜ W the corresponding discrete random variable. If P ˜ W = k = f k k = 0, 1, . . . , M, then F ˜ W k = P ˜ W ≤ k = P k j = f j where f k are the mass probabilities. We suppose that the premium income is a constant integer p and that the surplus process takes only integer values. We denote by ψ k, 1,n and ϕ k, 1,n the finite-time ruin and non-ruin probabilities calculated with F ˜ W over the periods 1 to n with an initial surplus k an integer. Theorem 2. Let k, p, j be integers. Then, ϕ k, 1,n = mink+p,M X j = ϕ k+p−j, 1,n−1 f j n = 2, 3, . . . , 5 where ϕ k, 1,1 = F mink+p,M = mink+p,M X j = f j . 6 Proof. The result follows from Theorem 1. The non-ruin probability can therefore be recursively calculated by first applying 6 and then 5. This method is used in De Vylder and Goovaerts 1988 and Dickson and Waters 1991. In the application of the algorithm, the support of F W is discretized by intervals of length r. Setting the parameters k and p such that k ≤ u r k + 1, p ≤ c r p + 1, the following inequalities are obtained ϕ k, 1,n p ≤ ϕu, 1, n ϕ k+ 1,1,n p + 1. Therefore, we have ϕu, 1, n ≃ 1 2 {ϕ k, 1,n p + ϕ k+ 1,1,n p + 1}. In Section 5, special attention is given to the infinite-time ruin probability ψu.

3. Aggregation of dependent classes of business