Optimal Insurance Strategy in the Individual Risk Model under a Stochastic Constraint on the Value of the Final Capital

Abstract

We solve a problem of optimal risk control in the static model by choosing an admissible insurance policy, where the objective functional is the so-called Markowitz utility functional, i.e., a functional that depends only on the mean value and standard deviation of the insurer’s final capital after an insurance transaction. Interests of the insurer are taken into account by introducing probabilistic or, more precisely, quantile constraints (value at risk constraint) on the final capital of the insurer, using a normal distribution to model the distribution of total damage. Additionally, we impose a restriction with probability one on the risk taken from an individual policy holder. Optimal from the point of view of the insurer is the so-called stop-loss insurance. We find explicit forms of conditions for refusing an insurance transaction. We give an example that illustrates the proven results in case of an exponential distribution of claim size.