On the evaluation of finite-time ruin probabilities in a dependent risk model
Introduction
Research on ruin probability beyond the classical risk model has intensified in recent years. More general ruin probability models assuming dependence between either claim amounts or claim arrivals, or cross-dependence between both arrivals and sizes of claims, and non-linear aggregate premium income have been considered in the actuarial and applied probability literature. Such models are better suited to reflect the dependence in the arrival and severity of losses generated by portfolios of insurance policies. Exploring ruin probability theoretically and numerically, under these more general dependence assumptions, is of utmost importance within the Solvency II framework of internal insolvency-risk model building.
Albrecher and Boxma [2], [3] have considered a collective infinite-horizon ruin model of (semi-)Markovian type where the dependence structure assures that both the consecutive claim inter-arrival times and claim sizes are respectively correlated, and there could also be a cross-correlation between them, and, as in the classical case, premiums accumulate linearly in time. The model considered by Albrecher and Boxma [3] is reasonably general and embeds the classical compound Poisson, and the Sparre-Andersen model with phase type distributed claim inter-arrival times as special cases. However, as the authors note, “in concrete cases, it is sometimes not possible to evaluate the occurring expressions”. It has to be noted also that these expressions relate to the infinite-time ruin case which is not particularly relevant to finite-time applications such as modeling insurance solvency.
Under the classical constant premium rate assumption, Albrecher and Teugels [1] consider a random walk model in which the waiting time for a claim and the claim size are dependent. Asymptotic exponential estimates for both finite and infinite time ruin probabilities are then obtained for light-tail claims, using the Laplace transform. Boudreault et al. [7] also assume that the current claim (amount) is dependent on the inter-occurrence time preceding it, and more precisely that the corresponding conditional density is defined as a mixture of two arbitrary densities with weights defined by exponentials whose powers are proportional to the preceding inter-occurrence time. In [8] the dependence between the claim amount and its corresponding inter-arrival time is modeled by a generalized Farlie–Gumbel–Morgenstern copula. The Laplace transform of the Gerber–Shiu discounted penalty function is derived, and for exponential claims, an explicit formula for the Laplace transform of the ruin time is provided. In a recent paper [26] consider dependence in a risk model under the assumption that the claims arrive according to an order statistics process.
A collective finite-horizon ruin probability model with Poisson claim arrivals, dependent discrete claim amounts having any joint distribution but independent of the claim arrival times, and aggregate premium income represented by any non-decreasing positive, real valued function, has been considered by Ignatov and Kaishev [13]. They give an explicit finite-horizon ruin probability formula in terms of infinite sums of determinants which are shown by the authors to admit representation as classical Appell polynomials. Some useful properties of Appell polynomials, including a recurrence formula are given in the Appendix of that paper. An improved explicit and exact version of the ruin probability formula of Ignatov and Kaishev [13], involving finite summation, is given in [15]. In [14], the same ruin model is considered but assuming the claim amounts have arbitrary continuous (possibly dependent) joint distribution. The finite-time ruin probability formula in that case is obtained explicitly in terms of classical Appell polynomials.
Our goal in this paper is two-fold. First, we summarize the explicit ruin probability formulas which appear in the papers by Ignatov and Kaishev [13], [14] and Ignatov et al. [15], deduce new alternative expressions, and establish some enlightening connections between these formulas. The latter allow for a unified treatment and a fair comparison of their numerical efficiency. Thus, we also study the numerical properties of these formulas and propose an algorithm for their efficient evaluation with a preliminary prescribed accuracy. Based on a series of examples, we demonstrate that these formulas are useful not only theoretically but also for computing ruin probabilities in various risk models with dependence. The latter is important in practical applications. For example, as recently pointed out by Das and Kratz [9], the need to evaluate the Ignatov–Kaishev ruin probability formulas naturally arises in the context of designing early warning systems against ruin of insurance companies. This need also arises in the context of reserving and risk capital allocation in particular, for operational risk, see [17].
This paper is organized as follows. In Section 2, we introduce our main model and give the formulas obtained by Ignatov et al. [15] and Ignatov and Kaishev [14] for both discrete and continuous claim amounts, and also demonstrate the interconnection between these formulas. The latter incorporates classical Appell polynomials and thus, Section 3 introduces various recurrence expressions for computing classical Appell polynomials. Section 4 provides a method of computing survival probability with a prescribed accuracy and a simulation method employing order statistics proposed by Dimitrova and Kaishev [10] is introduced in Section 5. In Section 6, we study the numerical properties of all theoretical results and provide several numerical examples for both discrete and continuous, dependent and independent claim severities. Section 7 concludes the paper.
Section snippets
On non-ruin probability formulas and relations between them
Let us first recall the model which we will be concerned with, which has first been considered in [13], [14], [15]. Let the random variables denote claim severities, and let denote their partial sums, i.e. ,. If claim severities are considered continuous random variables, then will denote their joint density and will denote the joint density of . Clearly, and
On Appell polynomials
As noted, the classical Appell polynomials naturally arise in the ruin probability formulas presented in Section 2. Therefore, in order to evaluate these efficiently, it is necessary to provide efficient means of computing Appell polynomials. In this section, we summarize six recurrence expressions of Appell polynomials and comment on their properties. For instance, we are interested in whether these allow for a recursive implementation only (cf. (15)–(17)), or for both iterative and recursive
A method for computing P(T > x) with a prescribed accuracy
In this section, we introduce a method for computing the survival probability with a pre-specified accuracy by truncating the number of summands in the formulas. As can be observed, formula (4), for instance, involves infinite summation which needs to be appropriately truncated from above in order to evaluate P(T > x) for continuous claim severities. Furthermore, a truncation of the summation from below can also be applied since, depending on the values of the parameters, the first few summands
A simulation-based method for computing the high dimensional integrals/sums in P(T > x) with order statistics
As mentioned above, another aspect which presents a challenge when implementing formulas (4) and (6) numerically is that they incorporate multivariate integration/summation with an increasing dimension k, which could be very computationally intensive in high dimensions. Note that the challenge in the case of discrete claim amounts is related to the fact that for a fixed dimension k and value of the number of terms in formula (6) which have to be evaluated and summed up is . The
Numerical study
This section is devoted to studying the numerical properties of the non-ruin probability formulas presented in Section 2. In particular, we explore how the different recurrence expressions for Appell polynomials given in Section 3 and the numerical algorithms described in Sections 4 and 5 affect the efficiency of computing P(T > x) using formulas (4) and (6), (8) and (9). For the purpose, we have used Mathematica system and a standard PC with 2.93 GHz Intel(R) Core(TM) i7 CPU and 8.00 GB RAM.
Conclusion
We have shown that the survival probability formulas derived by Ignatov and Kaishev [13], [14] and Ignatov et al. [15] can be derived from one another both in their discrete and continuous versions and thus, can all be expressed in terms of the classical Appell polynomials. Various recurrence expressions for computing these polynomials have been presented, and their numerical properties have been investigated. Furthermore, the numerical efficiency of formulas (9) and (8), and (6) and (4) have
Acknowledgments
The authors would like to thank Zvetan Ignatov for many constructive discussions which helped improve the paper. The third author also acknowledges the financial support received by Cass Business School, City University London during the course of his Ph.D.
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