Ae Van Heerwaarden

Optimal reinsurance in relation to ordering of risks

Abstract Many criteria for choosing optimal reinsurance treaties have the common property that a treaty is considered preferable to another if it is lower in stop-loss order, which means that it generates uniformly lower stop-loss premiums. Such criteria include maximizing utility of wealth, minimizing ruin probability and many others. If the set of feasible reinsurance treaties contains a minimum element in stop-loss order, this element is the solution to the optimization problem. The retained loss under a stop-loss treaty is such a minimum in the set of all treaties with equal mean, implying that stop-loss reinsurance is optimal with respect to many criteria. This extends the result of Borch (1960) and Kahn (1961) on maximizing the utility of wealth.

The Dutch premium principle

A premium principle is derived, in which the loading for a risk is the reinsurance loading for an excess-of-loss cover. It is shown that the principle is well-behaved in the sense that it results in larger premiums for risks that are larger in stop-loss order or in stochastic dominance.

Properties of the Esscher premium calculation principle

Abstract Order preserving and order inducing properties of the Escher premium calculation principle are investigated. It is found that using this principle higher premiums might be asked for smaller risks, and also for less variable risks.

Stop-loss order, unequal means, and more dangerous distributions

Using a sequence of transformations of subsequent cumulative distribution functions, the connections between the following three relations between risks are established: stop-loss order, stop-loss order with equal means, and being more dangerous. By a related technique, stop-loss order is verified between members of some two-parameter families of distributions.

Between Individual and Collective Model for the Total Claims

This article studies random variables whose stop-loss rank falls between a certain risk (assumed to be integer-valued and non-negative, but not necessarily of life-insurance type) and the compound Poisson approximation to this risk. They consist of a compound Poisson part to which some independent Bernoulli-type variables are added.Replacing each term in an individual model with such a random variable leads to an approximating model for the total claims on a portfolio of contracts that is computationally almost as attractive as the compound Poisson approximation used in the standard collective model. The resulting stop-loss premiums are much closer to the real values.

Ordering of risks and ruin probabilities

Abstract The concept of stop-loss order is often restricted to the case that the means of the risks to be ordered are equal, so risks with unequal means are incomparable. In this short note we prove that a risk with higher stop-loss premiums than another can be written as the sum of a risk with higher stop-loss premiums, too, but equal mean, and one that is non-negative with probability one. Using this decomposition it is proven that the risk with higher stop-loss premiums generates a higher ruin probability in a compound Poisson process.

Combining Panjer's recursion with convolution

Abstract In this note we present a model for the total claims on a non-life portfolio that is almost as attractive from the computational point of view as the compound Poisson model. Its stop-loss premiums are much better upper bounds for the exact stop-loss premiums.

New upper-bounds for stop-loss premiums for the individual model

Abstract An analytical expression for an upper bound of the stop-loss premium for the individual model is given.