A Combined Stochastic Programming and Optimal Control Approach to Personal Finance and Pensions

Abstract

We combine a dynamic programming approach (stochastic optimal control) with a multi-stage stochastic programming approach (MSP) in order to solve various problems in personal finance and pensions. Stochastic optimal control produces an optimal policy that is easy to understand and implement. However, explicit solution may not exist, especially when we want to deal with constraints, such as limits on portfolio composition, limits on the sum insured, an inclusion of transaction costs or taxes on capital gains, which are important issues regularly mentioned in the literature. Both optimization methods are integrated into one MSP formulation, and in a short computational time produce a solution, which takes into account the entire lifetime of an individual with a focus on the practical constraints during the first years of a contract.

Two applications are considered: (A) optimal investment, consumption and sum insured for an individual maximizing the expected utility of consumption and bequest, and (B) optimal investment for a pension saver who wishes to maximize the expected utility of retirement benefits. Numerical results show that among the considered practical constraints, the presence of taxes affects the optimal controls the most. Furthermore, the individual's preferences, such as impatience level and risk aversion, have even a higher impact on the controlled processes than the taxes on capital gains.

Keywords: dynamic programming, multi-period stochastic programming, power utility, personal finance, retirement

JEL Classification: C44, C61, G11, G22, G23