Guaranteed Minimum Death Benefits Research Articles

The Titanic Option: Valuation of the Guaranteed Minimum Death Benefit in Variable Annuities and Mutual Funds

ABSTRACT The authors use risk-neutral option pricing theory to value the guaranteed minimum death benefit (GMDB) in variable annuities (VAs) and some recently introduced mutual funds. A variety of death benefits, such as return-of-premium, rising floors, and ratches, are analyzed. Specifically, the authors compute the fair insurance risk fee, charged to assets, that funds the embedded option. The authors derive analytic option prices for a simplified exponential mortality model and robust numerical estimates in the case of a properly calibrated Gompertz model. The authors label this contingent claim a Titanic option because its payoff structure is in between European and American style but is triggered by death. The authors' main objective is to compare theoretical estimates against a cross-section of insurance risk charges, as reported by Morningstar, Inc. The authors' main conclusion is that a simple return-of-premium death benefit is worth between one and ten basis points, depending on gender, purchase age, and asset volatility. In contrast, the median Mortality and Expense risk charge for return-of-premium variable annuities is 115 basis points. Presumably, the remaining markup can be attributed to profits, model imperfections, or, more cynically, to an implicit payment for the tax-deferral privilege. INTRODUCTION In the United States, a variable annuity (VA) policy is a collection of investment subaccounts that are wrapped with a life insurance contract. The raison d'etre of the VA is the convenient deferral of all income and capital gains taxation until the funds are withdrawn or annuitized at retirement. The tax deferral on investment gains--of increasing long-term benefit--make VAs a popular alternative to fully taxable mutual funds, even though they involve higher management fees. In fact, despite the recent reduction in the long-term capital gains tax rate, sales of variable annuities amounted to $121 billion during 1999 and $100 billion during 1998. Interestingly, an estimated 55 percent of these variable annuity purchases were conducted within qualified, i.e., tax-sheltered, IRAs, 401(k)s, 403(b)s, and 457 retirement savings plans for which the tax deferral is redundant. [1] However, in addition to the tax deferral--and in contrast to most mutual funds--the variable annuity provides a unique death benefit, possibly justifying its sale within tax-sheltered plans. The variable annuity death benefit stipulates that at least the original investment will be returned to the estate or beneficiary of the policy, regardless of the performance of the underlying assets in the account. This guaranty will be in-the-money if the VA policyholder (annuitant) dies during a bear market and at the same time the investment is showing a paper loss. Recently, some innovative mutual funds have joined with insurance companies to offer a similar death benefit on their investments. Presumably, this innovation has been partially driven by an attempt to compete with the VA market and its implicit investment protection. Of course, the tax deferral can only be provided by VAs. The coincidence of death and market downturn, as unlikely as it may seem, is the impetus for this study. Specifically, the authors are interested in the economic value and fair cost of this death benefit. Before embarking on this endeavor, it is important to mention that not all VA contracts and death-protected mutual funds (DPMF) are created equal. Many innovative insurance companies sell VA products with death benefits that are more than others. By generous the authors mean that the guaranteed minimum death benefit is higher than a simple return of premium plus minimal interest. For example, some products guarantee a more lucrative death benefit that locks in accrued investment gains during the life of the contract. As such, the VA and death-protected mutual fund market is not homogenous, and one must be careful with any generalizations about these products. …

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

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.