An asymptotic formula for the net premium of some reinsurance treaties

Abstract

In the following a rather general class of reinsurance treaties is defined, including the excess-of-Ioss, largest claims and ECOMOR reinsurance treaties. Based on asymptotic considerations a simple premium formula is developed for the general treaty. The result is applied to two special types of reinsurance treaties. As a byproduct the paper indicates that there are intimate connections between reinsurance and the field of robust statistics.

Tail asymptotic expansions for L-statistics

“…Most common distributions satisfy the smoothness varying condition in Theorem 3.1, e.g., Burr, Pareto, absolute student t, etc (see Examples in Section 5). In the literature, E{S n−1 (c)} is so-called the net premium, see Kremer [18]. In our simulation study, we use empirical estimators to replace E{S n−1 (c)}.…”

Largest Claims Reinsurance Premiums under Possible Claims Dependence

“…is subject of many mathematical investigations (see e.g. Kremer (1984) For continuous F one knows under (A.I) -(A.3) that (see Kremer (1985)):…”

“…De Vylder & Goovaerts (1983), Kremer (1990 c)) and a comprehensive new theory was developed for reinsurance treaties of the largest claims type (see e.g. Kremer (1984), (1985), (1986), (1988), (1990 a), (1990 b), (1992), (1994 a)).…”

A General Bound for the Net Premium of the Largest Claims Reinsurance Covers

“…and in the case that Prob(TV< p) = 0, we get the approximate bound These formulas are correct for each finite collective, whereas the formulas in KREMER (1982KREMER ( , 1983KREMER ( , 1984 are derived for growing collectives with asymptotic considerations.…”

“…The starting point was from papers of AMMETER (1964), BENKTANDER (1978), BERLINER (1972) and KUPPER (1971), treating two special cases under very special model assumptions, and a more general paper of CIMINELLI (1976). In 1984 the author developed general formulas for the net premium of the generalized reinsurance treaties under asymptotic considerations, in 1985 corresponding non-asymptotic formulas (see KREMER 1984KREMER , 1985 and a handy recursive rating method in 1986 (see KREMER, 1986b). Special cases were given by the author in 1982 and 1986 (see KREMER, 1982KREMER, , 1986a.…”