Approximate Solutions to Retirement Spending Problems and the Optimality of Ruin

Abstract

Milevsky and Huang (2011) investigated the optimal retirement spending policy for a utility-maximizing retiree facing a stochastic lifetime but assuming deterministic investment returns. They solved the problem using techniques from the calculus of variations and derived analytic expressions for the optimal spending rate and wealth depletion time under the Gompertz law of mortality. Of course, in the real world financial returns are stochastic as well as lifetimes, raising the question of whether their qualitative insights and approximations are generalizable or practical. We solve the retirement income problem when investment returns are indeed stochastic using numerical PDE methods, assuming the principles of stochastic control theory and dynamic programming. But then – and this is key – we compare the proper optimal spending rates to the analytic approach presented in Milevsky and Huang (2011) by updating the portfolio wealth inputs to current market values. Our main practical conclusion is that this simplistic approximation when calibrated properly and frequently can indeed be used as an accurate guide for rational retirement spending policy. As a by-product of our PDE-based methodology, our results indicate that even though the wealth depletion time is no longer a certainty under stochastic returns, the expected age at which liquid wealth is exhausted (i.) takes place well before the maximum lifetime and (ii.) is also well approximated by our analytical solution.