Optimally Investing to Reach a Bequest Goal

TL;DR

This paper derives the optimal investment strategy in a Black-Scholes market to maximize the probability of meeting a bequest goal at death, considering continuous consumption and different boundary conditions from prior goal-reaching problems.

Contribution

It introduces a novel approach to maximize the probability of reaching a bequest goal with continuous consumption, differing from previous models by setting the ruin level at zero and analyzing ongoing wealth dynamics.

Findings

Optimal investment strategy derived for maximizing bequest probability.

Comparison with Browne's goal-reaching problems highlights differences in boundary conditions.

Results show how investment strategies vary with consumption rate and bequest level.

Abstract

We determine the optimal strategy for investing in a Black-Scholes market in order to maximize the probability that wealth at death meets a bequest goal $b$, a type of goal-seeking problem, as pioneered by Dubins and Savage (1965, 1976). The individual consumes at a constant rate $c$, so the level of wealth required for risklessly meeting consumption equals $c/r$, in which $r$ is the rate of return of the riskless asset. Our problem is related to, but different from, the goal-reaching problems of Browne (1997). First, Browne (1997, Section 3.1) maximizes the probability that wealth reaches $b < c/r$ before it reaches $a < b$. Browne's game ends when wealth reaches $b$. By contrast, for the problem we consider, the game continues until the individual dies or until wealth reaches 0; reaching $b$ and then falling below it before death does not count. Second, Browne (1997, Section 4.2) maximizes the expected discounted reward of reaching $b > c/r$ before wealth reaches $c/r$. If one interprets his discount rate as a hazard rate, then our two problems are {\it mathematically} equivalent for the special case for which $b > c/r$, with ruin level $c/r$. However, we obtain different results because we set the ruin level at 0, thereby allowing the game to continue when wealth falls below $c/r$.