Optimal active lifetime investment

Abstract

Optimal decision-making regarding investments is important. In this study, we examine how an individual aims to find optimal investment policies to overcome a benchmarked opponent during their lifetime. The individual evaluates the trade-off between the achievement of a performance goal and the risk of a shortfall. The dynamic programming method is applied in this study. When the individual's lifetime follows an exponential distribution, the associated Hamilton–Jacobi–Bellman (HJB) equation is an ordinary differential equation, and the explicit optimal policies for several important performance functions are derived. When the lifetime follows a general distribution, the HJB equation is a parabolic partial differential equation, and a Markov chain approximating numerical scheme is presented to estimate the value function. We further illustrate the numerical method when the random lifetime follows the Gompertz–Makeham distribution.