A note on the Taylor series expansions for multivariate characteristics of classical risk processes.

The series expansion introduced by Frey and Schmidt (1996) [Taylor Series expansion for multivariate characteristics of classical risk processes. Insurance: Mathematics and Economics 18, 1–12.] constitutes an original approach in approximating multivariate characteristics of classical ruin processes, specially ruin probabilities within finit time with certain surplus prior to ruin and severity of ruin. This approach can be considered alternative to inversion of Laplace transforms for particular claim size distributions [Gerber, H., Goovaerts, M., Kaas, R., 1987. On the probability and severity of ruin. ASTIN Bulletin 17(2), 151–163; Dufresne, F., Gerber, H., 1988a. The probability and severity of ruin for combinations of exponential claim amount distributions and their translations. Insurance: Mathematics and Economics 7, 75–80; Dufresne, F., Gerber, H., 1988b. The surpluses immediately before and at ruin, and the amount of the claim causing ruin. Insurance: Mathematics and Economics 7, 193–199.] or discretization of the claim size and time [Dickson, C., 1989. Recursive calculation of the probability and severity of ruin. Insurance: Mathematics and Economics 8, 145–148; Dickson, C., Waters, H., 1992. The probability and severity of ruin in finit and infinit time. ASTIN Bulletin 22(2), 177–190; Dickson, C., 1993. On the distribution of the claim causing ruin. Insurance: Mathematics and Economics 12, 143–154.] applying the so-called Panjer’s recursive algorithm [Panjer, H.H., 1981. Recursive calculation of a family of compound distributions. ASTIN Bulletin 12, 22–26.]. We will prove that the recursive relation involved in the calculations of the the nth derivative with respect to – average number of claims in the time unit – of the multivariate finit time ruin probability (developed in the original paper by Frey and Schmidt (1996) can be simplified The cited simplificatio leads to a substantial reduction in the number of multiple integrals used in the calculations and makes the series expansion approach more appealing for practical implementationFinite time ruin probability; Surplus prior to ruin; Severity of ruin; Series expansion; Recursive methods;

Robustness analysis and convergence of empirical finite-time ruin probabilities and estimation risk solvency margin.

We consider the classical risk model and carry out a sensitivity and robustness analysis of finite-time ruin probabilities. We provide algorithms to compute the related influence functions. We also prove the weak convergence of a sequence of empirical finite-time ruin probabilities starting from zero initial reserve toward a Gaussian random variable. We define the concepts of reliable finite-time ruin probability as a Value-at-Risk of the estimator of the finite-time ruin probability. To control this robust risk measure, an additional initial reserve is needed and called Estimation Risk Solvency Margin (ERSM). We apply our results to show how portfolio experience could be rewarded by cut-offs in solvency capital requirements. An application to catastrophe contamination and numerical examples are also developed.Finite-time ruin probability; robustness; Solvency II; reliable ruin probability; asymptotic Normality; influence function; Estimation Risk Solvency Margin (ERSM)

Reinsurance, ruin and solvency issues: some pitfalls

In this paper, we consider optimal reinsurance from an insurer's point of view. Given a (low) ruin probability target, insurers want to find the optimal risk transfer mechanism, i.e. either a proportional or a nonproportional reinsurance treaty. Since it is usually admitted that reinsurance should lower ruin probabilities, it should be easy to derive an efficient Monte Carlo algorithm to link ruin probability and reinsurance parameter. Unfortunately, if it is possible for proportional reinsurance, this is no longer the case in nonproportional reinsurance. Some examples where reinsurance might increase ruin probabilities are given at the end, when claim arrival and claim size are not independent.Dependence; Reinsurance; Ruin probability; Solvency requirements

Finite time ruin probabilities with one Laplace inversion.

In this work we present an explicit formula for the Laplace transform in time of the finite time ruin probabilities of a classical Levy model with phase-type claims. Our result generalizes the ultimate ruin probability formula of Asmussen and Rolski [IME 10 (1991) 259]—see also the analog queuing formula for the stationary waiting time of the M/Ph/1 queue in Neuts [Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach. Johns Hopkins University Press, Baltimore, MD, 1981]—and it considers the deficit at ruin as wellFinite-time ruin probability; Phase-type distribution; Deficit at ruin; Lundberg’s equation; Laplace transform;

On the accuracy of phase-type approximations of heavy-tailed risk models

Numerical evaluation of ruin probabilities in the classical risk model is an important problem. If claim sizes are heavy-tailed, then such evaluations are challenging. To overcome this, an attractive way is to approximate the claim sizes with a phase-type distribution. What is not clear though is how many phases are enough in order to achieve a specific accuracy in the approximation of the ruin probability. The goals of this paper are to investigate the number of phases required so that we can achieve a pre-specified accuracy for the ruin probability and to provide error bounds. Also, in the special case of a completely monotone claim size distribution we develop an algorithm to estimate the ruin probability by approximating the excess claim size distribution with a hyperexponential one. Finally, we compare our approximation with the heavy traffic and heavy tail approximations.Comment: 24 pages, 13 figures, 8 tables, 38 reference

Convergence and asymptotic variance of bootstrapped finite-time ruin probabilities with partly shifted risk processes.

The classical risk model is considered and a sensitivity analysis of finite-time ruin probabilities is carried out. We prove the weak convergence of a sequence of empirical finite-time ruin probabilities. So-called partly shifted risk processes are introduced, and used to derive an explicit expression of the asymptotic variance of the considered estimator. This provides a clear representation of the influence function associated with finite time ruin probabilities, giving a useful tool to quantify estimation risk according to new regulations.Finite-time ruin probability; robustness; Solvency II; reliable ruin probability; asymptotic normality; influence function; partly shifted risk process; Estimation Risk Solvency Margin. (ERSM).

The Effect of a Threshold Proportional Reinsurance Strategy on Ruin Probabilities

In the context of a compound Poisson risk model, we define a threshold proportional reinsurance strategy: A retention level k1 is applied whenever the reserves are less than a determinate threshold b, and a retention level k2 is applied in the other case. We obtain the integro-differential equation for the Gerber-Shiu function (defined in Gerber and Shiu (1998)) in this model, which allows us to obtain the expressions for ruin probability and Laplace transforms of time of ruin for several distributions of the claim sizes. Finally, we present some numerical results.time of ruin, threshold proportional reinsurance strategy, ruin probability, gerber-shiu function

Ruin probabilities in the Cram\'er-Lundberg model with temporarily negative capital

We study the asymptotics of the ruin probability in the Cram\'er-Lundberg model with a modified notion of ruin. The modification is as follows. If the portfolio becomes negative, the asset is not immediately declared ruined but may survive due to certain mechanisms. Under a rather general assumption on the mechanism - satisfied by most such modified models from the literature - we study the relation of the asymptotics of the modified ruin probability to the classical ruin probability. This is done under the Cram\'er condition as well as for subexponential integrated claim sizes.Comment: 9 page