A policyholder's indifference valuation for the guaranteed annuity option

Guaranteed annuity options belong to the class of long term guarantees that insurance companies may offer to their policy- holders. They were very common in U.S. tax-sheltered plans and U.K. retirement savings. These options may represent a very valuable liability for the insurer, being exposed to two dif- ferent sources of randomness: the future interest rates and the future mortality (hazard) rates. Both financial and actuarial ap- proaches have been used to evaluate and describe the nature of such options. In the present work, we present an indifference valuation method for the guaranteed annuity option, from the policyholder’s point of view. In a setting where interest rates are constant, we find an explicit solution for the indifference problem, where the individual is described by a power (instan- taneous) utility function. In this setting, we compare two strate- gies at the time of conversion, and two strategies at the moment when the policy is purchased. In the former, we assume that if the annuitant does not exercise the option, first she withdraws her policy’s accumulated funds, and then seeks to solve a stan- dard Merton’s problem, under an infinite time horizon setting. In the latter strategy, we compare the agent’s expected utility associated to a policy that embeds a guaranteed annuity option, and a policy that does not embed such an option. In order to accumulate the retirement funds, we assume in both cases a pure premium paid at a constant continuous stream. Regard- ing the optimal strategy, we are able to derive explicit solutions for a class of problems where finite horizon, bequest motive and power consumption utility are considered. We conclude the present framework by allowing the agent to earn a constant labor income. As expected, since the income is non-random, we find that the indifference valuation of implicit guaranteed annuity option is not influenced by this richer setting.