Orthogonal polynomial expansions to evaluate stop-loss premiums

Abstract

A numerical method is proposed to evaluate the survival function of a compound distribution and the stop-loss premiums associated with a non-proportional global reinsurance treaty. The method relies on a representation of the probability density function in terms of Laguerre polynomials and the gamma density. We compare the method against a well established Laplace transform inversion technique at the end of the paper.MSC 2010: 60G55, 60G40, 12E10. This paper concerns approximations of f S N and F S N , though we begin with a discussion of how S N is used in actuarial science.Frequently, S N models the aggregated losses of a non-life insurance portfolio over a given period of time-here N represents the number of claims and U k the claim sizes-yet other applications also exist. Actuaries and risk managers typically want to quantify the risk of large losses by a single comprehensible number, a risk measure.One popular risk measure is the Value-at-Risk (VaR). In actuarial contexts, the VaR at level α ∈ (0, 1) is defined such that the probability of (aggregated) losses exceeding the level VaR is at most 1 − α. We denote this α-quantile asFollowing the European recommendation of the Solvency II directive, the standard value for α is 0.995, see [18]. It is used by risk managers in banks, insurance companies, and other financial institutions to allocate risk reserves and to determine solvency margins. Also, we have stop-loss premiums (slp's) which are risk measures that are commonly used in reinsurance agreements.A reinsurance agreement is a common risk management contract between insurance companies, one called the "cedant" and the other the "reinsurer". Its aim is to keep the cedant's long-term earnings stable by protecting the cedant against large losses. The reinsurer absorbs part of the cedant's loss, say f (S N ) where 0 ≤ f (S N ) ≤ S N , leaving the cedant with I f (S N ) = S N − f (S N ). In return, the cedant pays a premium linked tounder the expected value premium principle.

Approximating the first passage time density from data using generalized Laguerre polynomials

“…Unfortunately in [37] it is shown that the usual method of differentiating the density does not lead to an orthogonal polynomial system, and starting from the Laguerre polynomials leads to a system of orthogonal functions which is not complete. The only way to get a complete system of polynomials is by using the Gram-Schmidt orthogonalisation procedure, but the resulting polynomials are not easy to use (see also [22]). In [25] the authors propose a method to derive the polynomials that involves the so-called bi-orthogonality property but they do not discuss whether this construction leads to a basis.…”