Ruin probabilities with compounding assets

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Abstract

We consider a classical surplus process modified by the action of a constant force of interest. We derive recursive algorithms for the calculation of the probability of ruin in finite time. We also discuss the numerical evaluation of the probability of ultimate ruin using methods proposed by De Vylder [De Vylder, F., 1996. Advanced Risk Theory. Editions de l’Université de Bruxelles.] and Sundt and Teugels [Sundt, B., Teugels, J.L., 1995. Insurance: Mathematics and Economics 16, 7–22]. Finally, we consider the problem of recovery from ruin.

Introduction

In this paper we present and discuss some algorithms to calculate the probability of ruin in both finite and infinite time for a classical surplus process modified by the inclusion of (deterministic) interest on the insurer’s surplus. The inclusion of interest is a natural extension of the classical surplus process but it makes the calculation of the probability of ruin a far more difficult problem. This is evidenced by the relatively small number of published papers on this topic, and by the even smaller number which contain any numerical results. Our aim is to present numerical results.

A useful review of the literature in this area is provided by Paulsen (1998). Delbaen and Haezendonck (1987) use martingale methods to produce an upper bound for the probability of ruin in finite time in cases where the individual claim amount is exponentially bounded. Embrechts and Schmidli (1994) use the theory of piece-wise deterministic Markov processes to study the probability of ruin in infinite time. It is interesting to note that the numerical examples in the latter two papers all assume individual claim amounts are exponentially distributed.

In previous studies Dickson and Waters, 1991, Dickson et al., 1995) algorithms have been presented to calculate ruin probabilities for the classical surplus process. In these studies, the basic idea is that the probability of ruin for a classical surplus process can be approximated by a discrete time ruin probability for an integer-valued surplus process. We shall retain elements of this approach in this paper.

In the case of finite time, we will approximate the probability of ruin in continuous time by a probability of ruin in discrete time. However, due to the fact that income is not a linear function of time in the modified process, we cannot base our algorithm on an integer-valued process. Instead we must construct an algorithm based on integral equations. In the case of infinite time ruin, we discuss the numerical evaluation of the probability of ruin using methods proposed by De Vylder, 1996, Sundt and Teugels, 1995.

In the final section we consider the problem of recovery from ruin when the insurer must borrow funds at some fixed rate of interest until the surplus process returns to zero. We present algorithms to calculate the probability of the surplus process returning to zero without passing below specified levels, and discuss the circumstances under which these algorithms may usefully be applied.

Section snippets

Model assumptions

In this and the following section we will consider a surplus process for which aggregate claims are generated by a compound Poisson process. The Poisson parameter for the number of claims per annum is λ and the individual claim amount distribution function is F(x). We assume the latter distribution is continuous and that F(0)=0. We denote by μ the mean individual claim amount. Premium income is received continuously at constant rate P per annum, where P=(1+θ)λμ for some premium loading factor θ

De Vylder’s non-adapted method

We denote by ψ(U) and ψ(U) the probabilities of ultimate ruin and ultimate survival, respectively, for the surplus process described in Section 2.1. De Vylder proposed the following method for the calculation of ψ(U) (De Vylder, 1996, I.10.4.2, Theorem 18).

For any (integer) value of the initial surplus U, let τ(U) denote the time interval in which, if there were no claims, the surplus would increase to U+1, so thatU+1=Uexpτ(U)}+P(expτ(U)}−1)/δ,and henceτ(U)=1δlogU+1+P/δU+P/δ.The probability

Recovery from ruin

In this section we consider, for the same surplus process as in 2 The probability of ruin in finite time, 3 The probability of ruin in infinite time, the distribution of the maximal negative surplus before recovery once ruin has occurred. More specifically, we derive an algorithm for the approximate calculation of the probability π(−y,−z), whereπ(−y,−z)=Pr[surplusreaches0withoutgoingbelow−z∣currentsurplusis−y],and where y and z are such that −P/δ<−z≤−y<0. The probability π(−y,−z) is the

Acknowledgements

Financial support from the Faculty of Economics and Commerce, University of Melbourne, is gratefully acknowledged.

Much of the work on this paper was completed while the first named author was visiting the Laboratory of Actuarial Mathematics, University of Copenhagen, in the first half of 1997.

This paper has benefitted very much from comments from an anonymous referee.

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