Stop-loss Contracts Research Articles

Reinsurance in arbitrage-free markets

The paper studies dynamic reinsurance policies in the continuous Lundberg model, where claims, premiums and interest rates are stochastic processes. The main purpose of the paper is to study the consequences of arbitrage-free markets for the premium calculation of arbitrary reinsurance contracts like stop loss or excess loss. An explicit formula is derived for the stop-loss contract when the claim process is Compound Poisson. Conditions are given when Pareto-optimal allocations of the total risk in an economy can be achieved through markets for proportional risk sharing.

Pareto-Optimal Risk Exchanges and Related Decision Problems

In various branches of applied mathematics the problem arises of making decisions to reconcile conflicting criteria. One example is the classical statistical problem, where a type 1 error cannot be arbitrarily reduced without increasing the probability for a type 2 error. Another example, quite familiar to actuaries, is graduation, where a compromise between smoothness and fit has to be reached. This motivates the concept of Pareto-optimal decisions, which is discussed in section 2. There is a simple method, maximizing a weighted average of the scores, to obtain certain Pareto-optimal decisions. In section 3 a condition is given, which is satisfied in most applications, that guarantees that all the Pareto-optimal decisions can be found by this method. This is applied in section 4, where the problem of risk exchange between n insurance companies is considered. The original model of Borch is generalized: it is assumed that some of the companies are not willing to contribute more than a certain fixed amount towards the aggregate loss of the other companies. The theorem in section 4 gives a characterization of all the Pareto-optimal risk exchanges. Because of the restrictions, these risk exchanges do not just depend on the combined surplus (which would amount to pooling) in general, and can be found by an algorithm. One benefit of this generalization of Borch's Theorem is that two seemingly unrelated results (optimality of a stop loss contract, and optimality of certain dividend formulas in group insurance) follow from it as special cases.

The optimal reinsurance treaty

1. Some years ago I discussed optimal reinsurance treaties, without trying to give a precise definition of this term [1]. I suggested that a reinsurance contract could be called “most efficient” if it, for a given net premium, maximized the reduction of the variance in the claim distribution of the ceding company. I proved under fairly restricted conditions that the Stop Loss contract was most efficient in this respect.I do not consider this a particularly interesting result. I pointed out at the time that there are two parties to a reinsurance contract, and that an arrangement which is very attractive to one party, may be quite unacceptable to the other.2. In spite of my own reservations, it seems that this result —which I did not think deserved to be called a theorem—has caused some interest. Kahn [4] has proved that the result is valid under far more general conditions, and recently Ohlin [5] has proved that the result holds for a much more general class of measures of dispersion.In view of these generalizations it might be useful to state once more, why I think the original result has relatively little interest. In doing so, it is by no means my purpose to reduce the value of the mathematical generalizations of Kahn and Ohlin. Such work has a value in itself, whether the results are immediately useful or not. I merely want to point out that there are other lines of research, which appear more promising, if our purpose is to develop a realistic theory of insurance.

Minimum Variance Reinsurance

In a paper entitled “An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance” (XVIth Int. Congr. Act. Bruxelles 1960) Karl Borch has shown that, if the reinsurance premium is given, the smallest variance of the cedent's payments is obtained by a stop-loss reinsurance contract. Paul Markham Kahn, in “Some Remarks on a Recent Paper by Borch”, a paper read to the 1961 Astin Colloquium, has given an elegant proof of this theorem which appears to apply also to cases not considered by Borch. In this paper we study the problem from the reinsurer's point of view and it will be seen that, under natural conditions which are also used in the proof of the Borch-Kahn theorem, the minimum variance of the reinsurer's payments is obtained by a quota contract. This focusses attention on a peculiar opposition of interests of the two partners of a reinsurance contract. However, we do not enter any further into the investigation of a possible resolution of this conflict.We study a problem concerning the division of risk between a cedent and his reinsurer. The risk may refer to a whole portfolio (in which case one might consider a Stop-Loss contract), or to a single contract (when an Excess-Loss contract is a possibility). We shall here use the nomenclature of a portfolio reinsurance.Let it be assumed that a function F(x) is known which gives the probability of a total claim not exceeding x. We have then in Stieltjes integral notationThe two partners to a reinsurance arrangement agree that the reinsurer reimburses m(x).x out of a claim of x, where m(x) is a continuous and differentiate function of x and o ≤ m(x) ≤ 1.