Optimal Fee Structure of Variable Annuities

We study the design of fee structures of variable annuities as a stochastic control problem, in which an insurer is allowed to choose the fee structure in any form, and seeks an optimal one to maximize her expected discounted net profit. We obtain a semi-explicit characterization result for the optimal fee structure which is of barrier type with a free boundary. The insurer's optimal strategy is to charge fees if and only if the account value of variable annuities hits the free boundary from below, which differs from the constant fee structure that is commonly used in the industry, and other state- and time-dependent fee structures that have been proposed in the literature.