Closed Form Solutions of a Re-Insurer’s Surplus, Stochastic and Time-Dependent Investment Returns with Random Parameters

Abstract

One of the challenges faced by insurance companies is the ability to manage the risk of their client (policyholders) and pay their claims whenever the need arises; hence there is a need to study the portfolio of a reinsurer which is a combination of his surplus and invested funds in the financial market. To achieve this, we consider cases where the expected rates of return of the risky asset are linear and quadratic. Ito’s lemma, maximum principle and variable separation technique were used in solving for the closed-form solutions of the prices of the risky assets for all considered cases. Similarly, the closed-form solutions of the reinsurer’s surplus were obtained for both time-dependent and stochastic cases. Finally, some numerical simulations were presented to study the effect of some sensitive parameters on the price process of the risky asset and also the behaviour of the reinsurer’s surpluses with time. It was observed that the price process of the risky asset is an increasing function of the expected rate of return and a decreasing function of the instantaneous volatilities while the surplus process does not necessarily depend on the expected rate of returns but on contribution rate and the number claims to be serviced at any given time.