Valuation and Optimal Surrender of Variable Annuities with Guaranteed Minimum Benefits and Periodic Fees

Abstract

This work studies the valuation and optimal surrender of variable (equity-linked) annuities under a L\'evy-driven equity market with mortality risk. We consider a practical periodic fee structure which can vary over time, and is assessed as a proportion of the fund value. At maturity, the fund value is returned to the policyholder according to a guaranteed minimum accumulation benefit (GMAB). Mortality risk is also modeled discretely, and the contract offers a guaranteed minimum death benefit (GMBD) prior to maturity. The benefits accommodate caps on the growth of funds (in addition to the rising floor) to reduce the fee level, and as a disincentive to early surrender. Interest rates are modeled via a deterministic discounting term structure, which can be calibrated (bootstrapped) to the rates market, according to market convention.

An efficient and accurate valuation framework is developed, along with closed form pricing formulas in the case where policy surrender is not permitted. Numerous experiments are conducted to illustrate the interplay between contract parameters and the decision to surrender, and we provide an extensive analysis that investigates how to structure contracts to disincentivize early surrender.

Keywords: Surrender, equity-linked, variable annuity, GMDB, GMAB, Guaranteed Minimum Death Benefit, GMxB, Levy, variable annuity, mortality, pricing, valuation, lapse

JEL Classification: G13, G22, G23, C15