Option-Adjusted Equilibrium Valuation of Guaranteed Minimum Death Benefits in Variable Annuities

Abstract

In the United States, variable annuity contracts are mutual funds with tax-deferred investment gains containing an additional embedded put option with a stochastic maturity date. At death, the beneficiary - or estate - of the policy owner is assured to receive the greater of the account market value and the original investment premium, perhaps with a minimally guaranteed roll rate. Payment for this life-contingent claim, is not made up front, but rather deducted continuously from the account in the form of a constant insurance expense ratio. This proportional expense stands in contrast to a fixed periodic - or initial - insurance premium and is what differentiates our research from previous analysis of similar equity-linked life products. In this paper we value the guaranteed minimum death benefit (GMDB) of a variable annuity by melding the tools of continuous-time financial economics and the actuarial theory of mortality functions. Specifically, we solve for the equilibrium insurance expense ratio that funds the embedded put option. We demonstrate that the equilibrium expense ratio solves a type of fixed point equation that relates the risk-neutral expected present value of costs and benefits. From an analytic perspective, with an assumed constant force of mortality, we show that the present value of the guaranteed minimum death benefit is the Laplace transform - in time - of the classical Black-Scholes/Merton equation adjusted for dividends, where the dividend is the insurance expense ratio. Remarkably, the final expression can be obtained in closed form and does not even involve the normal density. On the other hand, when the mortality is given a more realistic continuous time Gompertz structure, we obtain quite robust expressions and numerical values for the equilibrium insurance expense ratio. Finally, using a cross section of average insurance expense ratios (known as Mortality and Expense charges) provided by Morningstar Data - and reasonable estimates of market volatility - we conclude that in most cases the insurance industry is charging variable annuity holders approximately five to ten times the economic value of the guarantee.