A Continuous Time Framework for Sequential Goal-Based Wealth Management

Abstract

We develop a continuous time framework for sequential goals-based wealth management. A stochastic factor process drives asset price dynamics and the client’s goal amount and income. We prove the weak dynamic programming principle for the value function of our control problem, which we show to be the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. We develop an equivalent and computationally efficient representation of the Hamiltonian, which yields the optimal portfolio within a factor-dependent opportunity set defined by the maximum and minimum variance hypersurfaces. Our analysis shows that it is optimal to fund an expiring goal up to the level where the marginal benefit of additional fundedness is exceeded by the opportunity cost of diverting wealth from future goals. An all-or-nothing investor is more risk averse toward an approaching goal deadline if well funded, but more risk seeking if not on track with upcoming goals, compared with an investor with flexible goals. This paper was accepted by David Simchi-Levi, finance. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada [Discovery Grant RGPIN-2020-06290] and Fi-Tek.