Stochastic optimization of retirement portfolio asset allocations and withdrawals

Abstract Stochastic optimization identifies the asset allocation that minimizes the probability of exhausting the retirement portfolio, thereby minimizing risk, from unmanaged (constant) and optimally managed withdrawals over the retirement life span. Optimal equity compositions and minimized probabilities of prematurely exhausting the portfolio increase with higher withdrawal rates and earlier retirements with both managed and unmanaged withdrawals. However, optimal withdrawal management from optimally managed portfolios reduces the sensitivity of premature portfolio exhaustion to higher initial withdrawal rates or earlier retirements, thereby reducing the increase in the risk of exhausting the portfolio necessary to support the improved lifestyles from higher withdrawals, longer retirements, or both. © 2008 Academy of Financial Services. All rights reserved. Jel classification: G11 Keywords: Retirement portfolio allocation; Withdrawal management; Stochastic optimization 1. Introduction The success of a retirement portfolio is one of a set of problems in which the objective function cannot be evaluated precisely because it depends upon uncertain independent variables. In such problems, Monte Carlo simulation is widely accepted as a means of estimating the objective function by randomly generating values for uncertain outcomes from known or hypothesized distributions of input variables. The outputs from Monte Carlo simulation are statistical distributions of one or more output variables, and although based upon thousands of randomly generated values of input variables, the outputs retain their random characterization because they are a function of the randomness in the distribution sampling process. Stochastic optimization refers to the optimization of a function in the presence of randomness in the input data or the optimization algorithm. Stimulated by the acceptance of Monte Carlo simulation, some simulation packages now include stochastic optimization procedures that recognize simulated outcomes as random inputs to an optimization algorithm. The technique is commonly applied in the physical and natural sciences, but it has not been applied in retirement portfolio and withdrawal management. Some retirement withdrawal planning models or distribution planning models have employed Monte Carlo simulation to examine the sustainability of a constant stream of inflation-adjusted withdrawals from a retirement portfolio subject to uncertain portfolio returns and uncertain remaining retiree lifetimes, and more recently, Monte Carlo simulation has been used to examine the effects of withdrawal management decision rules in variable withdrawal models. Outputs from these simulations include the distribution of the probability of running out of money before death, the distribution of ending portfolio values, and, in variable withdrawal rate models, the distribution of withdrawal rates. This paper applies stochastic optimization in a Monte Carlo simulation model to identify the optimal portfolio allocations for minimizing the probabilities of depleting the retirement portfolio before death, oftentimes termed portfolio ruin or shortfall risk, from constant inflation-adjusted withdrawals. Optimal portfolio allocations and minimum probabilities of portfolio ruin are regressed on the withdrawal rate and age at retirement to more clearly quantify the impacts of earlier retirement, higher withdrawal rates, or both, on optimum portfolio allocations and the minimum probability of ruin. Monte Carlo stochastic optimization is also applied in a model of variable withdrawal rates to identify both the optimal portfolio allocations and the optimal withdrawal rate decision rule parameters for minimizing the probabilities of portfolio ruin. Optimal portfolio allocations and minimized probabilities of portfolio ruin are regressed on the initial withdrawal rate and age at retirement to quantify the impacts of earlier retirement, higher initial withdrawal rates, or both, on the minimum probability of ruin from optimally managed withdrawals from optimum portfolios. …