Insurance In The Optimal Decision

This paper is divided up two parts. In the first part, we consider a dynamic model for a defined-benefit pension fund in a continuous, certain fixed time period of time. We suppose that the plan is built with the contributions and investment earnings. The sponsoring employer controls the contribution rate and the amount invested in a portfolio composed of a risky assets and a risk-free security. No shortselling is allowed but the manager have the possibility of borrowing. The main purpose of this note is to find the optimal supplementary cost and the optimal asset-allocation strategy. As the supplementary cost is constrained at t — T, the methodological approach we use to solve the optimal supplementary cost and the optimal asset-allocation problem is the stochastic dynamic programming assisted with the Lagrange multiplier method. Under certain regular condition, this note find out the optimal solutions of this problem. As shown in the sequel of this note, the proportion of the optimal supplementary cost on the unfunded actuarial liability is an explicit function of time t and the optimal fund allocated for the risky assets is fixed proportional to the unfunded actuarial liabilities. Finally, a numerical simulation is presented.In the second part, we derive total optimal quota-share reinsurance strategy of n dependent risks-insured with dependent claim numbers under the expectation principle of premium calculation.