The Effect of the Real Option to Transfer on the Value of Guaranteed Minimum Death Benefits

ABSTRACT Variable annuity contracts frequently have many options and option-like features embedded in the contracts. Some are obvious, such as guaranteed minimum death benefits (GMDBs), while others are less obviously option-ike. In this article, we consider the effect of the real option to transfer funds between fixed and variable accounts. If a GMDB rider is considered in isolation, it is sometimes in the policyholder's interest to transfer to the fixed fund if the fixed fund earns less than the variable fund in a risk-neutral world. On the other hand, the option to transfer will not be used if the entire annuity and rider are considered together. INTRODUCTION There have been a number of papers in recent years dealing with the valuation of guaranteed minimum death benefits (GMDBs) embedded in annuity contracts. These riders guarantee some minimum benefit at death, which is frequently the return of premiums paid or a fixed percentage rollup on that premium. They resemble a sequence of European put options, as the annuitant cannot choose the time of his own death. A few papers discuss option pricing from an actuarial standpoint. Gerber and Shiu (1994) discuss methods of obtaining closed-form option prices using Esscher transforms. Tiong (2000) uses this approach to value equity-indexed annuities. Milevsky and Promislow (2001) and Milevsky and Posner (2001) extend the discussion to cases more specifically related to insurance and annuity contracts by looking at the option to annuitize and the valuation of GMDBs. There has recently been recognition of the fact that even GMDBs have features of American options. Milevsky and Salisbury (2001) note that policyholders have a real option to lapse their policy and reinstate the death benefit when their policy is out of the money and this option should be priced for. They assume that policyholders exercise their option optimally in that they would lapse and reinstate immediately if there were no surrender charges to prevent it. With this assumption, they calculate the surrender charge necessary to fully compensate the insurance company for this loss of income. This surrender charge is one where the individual owning the policy would be indifferent between maintaining the policy and lapsing it. A second option that is frequently available in these contracts is the option to transfer funds between a variable account and an attached fixed account that earns a guaranteed rate. A particular worry is that individual policyholders will choose to transfer their money from the variable account to the fixed account when the GMDB is in-the-money and thereby lock in a positive GMDB for a substantial period into the future. This option to transfer would behave very much like an American option to lock in a GMDB at a time of the policyholder's own choosing. Unlike an American option, however, the decision to transfer is not irreversible. Another American-type option exists, which allows the policyholder to transfer back to variable account. Clearly, this option has value if the fixed fund were to rise to the value of the GMDB strike, as the value would be zero if the funds were in the fixed account and positive if the funds were in the variable account. The transfer option in these contracts bears some similarity to passport options that have attracted attention recently. They are even more similar to vacation put options introduced by Shreve and Vecer (2000). These options are special cases of call options on a traded account. The individual owning the account starts with an initial wealth [[pi].sub.0] and is allowed to invest a value [q.sub.t] in an asset that follows a geometric Brownian motion process. At maturity time T, the investor receives the maximum of 0 and the value [[pi].sub.T]. The value [q.sub.t] is constrained to be in the range [less than or equal to] [q.sub.t] [less than or equal to] [beta]. The case = [beta] = 1 corresponds to a European call option, and a = [beta] = -1 corresponds to a European put option. …