Solving a savings allocation problem by numerical dynamic programming with shape-preserving interpolation

Abstract

This article introduces a bivariate shape-preserving interpolation algorithm to approximate the value function of a dynamic program. First, we present a savings allocation problem between a pension account and another non-pension one. With the objective of maximizing the present value of utility over a life cycle, the investor can distribute his or her savings, in each account, between stocks and cash funds. Formally, this complex problem involved with various tax rules is in dynamic programming formulation and can only be solved numerically. It is known that the value function of the associated two-dimensional dynamic program inherits monotonicity and convexity of the investor's risk-averse utility function. To preserve these shape characteristics, we apply a bivariate shape-preserving interpolation algorithm in the successive approximation of the value function. Finally, we have computational results for this savings allocation problem, showing that the proposed shape-preserving interpolation method is superior to other dynamic programming methods with less sophisticated interpolation techniques. Scope and purpose The savings allocation problems with several dimensions of continuous states are too complicated and thus can only be solved by numerical dynamic programming. Theory of dynamic programming has shown that the associated value function inherits the shape characteristics – monotonicity and concavity – of a risk-averse investor's utility function. However, there are no numerical methods which guarantee to preserve these shape features in the course of approximation of the value function. In this article, we model a savings allocation problem as a two-dimensional dynamic program and we present a bivariate shape-preserving interpolation method to solve it.