Narrow Results

A quadratic discrete time probabilistic model, for optimal portfolio selection, under risk constraint, is introduced in the context of (re-) insurance and finance. The portfolio is composed of contracts with arbitrary underwriting and maturity times. For positive values of underwriting levels, the expected value of the accumulated final result is optimized under constraints on its variance and on annual Returns On Equity. Existence of a unique solution is proved and a Lagrangian formalism is given. An effective method for solving the Euler-Lagrange equations is developed. The approximate determination of the multipliers is discussed. This basic model, which can include both assets and liabilities, is an important building block for more general models, with constraints also on non-solvency probabilities, market-shares, short-fall distributions and Values at Risk.

In this paper, we study the valuation of variable annuities for an insurer. We concentrate on two types of these contracts, namely guaranteed minimum death benefits and guaranteed minimum living benefits that allow the insured to withdraw money from the associated account. Here, the price of variable annuities corresponds to a fee, fixed at the beginning of the contract, that is continuously taken from the associated account. We use a utility indifference approach to determine the indifference fee rate. We focus on the worst case for the insurer, assuming that the insured makes the withdrawals that minimize the expected utility of the insurer. To compute this indifference fee rate, we link the utility maximization in the worst case for the insurer to a sequence of maximization and minimization problems that can be computed recursively. This allows to provide an optimal investment strategy for the insurer when the insured follows the worst withdrawal strategy and to compute the indifference fee. We finally explain how to approximate these quantities via the previous results and give numerical illustrations of parameter sensitivity.