Optimal active lifetime investment

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.