On excess-of-loss reinsurance

Abstract

Abstract. We discuss a unified framework to analyze the distribution of the number of claims and the aggregate claim sizes in an excess-of-loss reinsurance contract based upon the use of point processes and work out several examples explicitly. We first deal with a single excess-of-loss situation with an extra upper bound on the coverage of individual claims. Subsequently the results are extended to a reinsurance chain with k partners. IntroductionTraditionally, an excess-of-loss reinsurance form covers the overshoot over a certain retention level M for all claims whether or not they are considered to be large. Among the insurance branches where it is used, we mention in particular general liability, and to a lesser extent motor liability ([25]) and windstorm reinsurance ([18]). Because of its transparency, an excess-of-loss treaty has been one of the main research objects in the reinsurance literature right from the beginning (see for instance [3,4]). It is clear that excess-of-loss reinsurance limits the liability of the first line insurer but that he himself will cover all claims below the retention M . In this form, the excess-of-loss reinsurance treaty cover has a number of desirable theoretical properties as explained by Bühlmann in [5] and by Asmussen et al. in [2]. For a combination with quota-share reinsurance, see [6].We will look at a more general form where each claim will be considered between two boundaries: we will call the lower retention u while the upper will be taken to be u + v, indicating by v the range for which the treaty is used. As illustrated below, the notation allows us to deal with any partner in an excess-loss reinsurance chain. If u and v depend on the claim orderings, then this reinsurance form is commonly called drop-down-excessof-loss reinsurance. For a study of such a reinsurance form, we refer to Ladoucette et al. [12].In the following we need some basic notation, first for the original portfolio.• The epochs of the claims are denoted by T 0 = 0, T 1 , T 2 , . . . . Apart from the fact that the epochs form a nondecreasing sequence, we in general do not assume anything specific about their interdependence. The random variables defined by W 0 = 0 and {W i+1 := T i+1 − T i ; i = 0, 1, . . .} are called the waiting times in between successive claims. In some particular cases it might be useful to assume that the sequence {W i ; i ≥ 1} consists of independent random variables all with a common distribution V as a random variable T , i.e. P(T ≤ x) = V (x). In that2000 Mathematics Subject Classification. Primary 62P05, 62H20.