Excess of Loss Reinsurance with Reinstatements Revisited

Abstract

The classical evaluation of pure premiums for excess of loss reinsurance with reinstatements requires the knowldege of the claim size distribution of the insurance risk. In the situation of incomplete information, where only a few characteristics of the aggregate claims to an excess of loss layer can be estimated, the method of stop-loss ordered bounds yields a simple analytical distribution-free approximation to pure premiums of excess of loss reinsurance with reinstatements. It is shown that the obtained approximation is enough accurate for practical purposes and improves the analytical approximations obtained using either a gamma, translated gamma, translated inverse Gaussian or a mixture of the last two distributions.

What do distortion risk measures tell us on excess of loss reinsurance with reinstatements?

“…Even if their analysis is conducted by a numerical recursion, the choice of the PH-transform risk measure as a particular concave distortion risk measure strengthens our interest. Furthermore, in an excess of loss reinsurance with reinstatements the computation of premiums requires the knowledge of the claim size distribution of the insurance risk: with reference to the expected value equation of the XL reinsurance with reinstatements (8), Sundt [5] based the computation on the Panjer recursion numerical method and Hürlimann [3] provided distribution-free approximations to pure premiums.…”

“…The problem, previously studied by Sundt [5] and, more recently, by Mata [4] and Hürlimann [3], requires the evaluation of pure premiums given the knowledge of the claim size distribution of the insurance risk: in order to face this question, different approaches have been followed in the actuarial literature. Sundt [5] based the computation on the Panjer recursion numerical method and Hürlimann [3] provided distribution-free approximations to pure premiums.…”

Optimal retention for a stop-loss reinsurance with incomplete information

“…the stochastic dominance order (stop-loss order). Assuming that the first two moments and support of X are known, Hürlimann (2005) uses the stop-loss ordered random variables to develop the analytical lower and upper bounds of X, and approximates pure premiums for excess of loss reinsurance with reinstatements. These papers have shown that the obtained approximations are accurate enough for practical purpose, especially when one agrees to calculate some risk measures, not based on the actual loss function, but based on stochastic bounds of the loss.…”

Ruin theory with excess of loss reinsurance and reinstatements

“…For the discrete case, Walhin and Paris [13] derived a multivariate Panjer-type algorithm in order to approximate the resulting compound claim distribution for the cedent. A distribution-free approximation in a market with incomplete information was given by Hürlimann [7]. Hess and Schmidt [6] determine an optimal premium plan for reinsurance contracts with reinstatements by minimizing the expected squared difference between the loss and the premium income of the reinsurer.…”