On Optimal Reinsurance

In this paper an attempt is made to find an answer to the question, “What is the most advantageous size for the retention limit of a risk portfolio, given the fact that a certain stability requirement is to be satisfied?†This problem will be approached from the viewpoint of an insurer who wishes to obtain a certain degree of stability at lowest cost.It is assumed that in his choice of reinsurance methods, the insurer restricts himself to either a surplus treaty, a stop loss treaty or a combination of both these types.Moreover it is assumed that “stability†can be adequately measured by the variance of the risks retained for own account.We start to consider a reinsurance policy based on the surplus system where the amount of risk in excess of a retention limit u is ceded.By thus limiting the potential loss on each risk individually, the variance is kept at a certain level, but at the expense of an amount of premium payable to a reinsurer.The insurer could, of course, reduce the reinsurance cost by increasing his retention but he then is bound to incur a higher variance in his portfolio, which would mean a loss of stability.One might ask, however, whether a suitably chosen stop loss coverage could bring the variance down again to the proper level at lesser cost than the profit obtained by taking a higher retention. A reduction in reinsurance cost would then have been effected.The question leads to an optimization problem, which in a more general setting, has been discussed by K. Borch.