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# Optimally Investing to Reach a Bequest Goal

24 Pages Posted: 5 Mar 2015 Last revised: 26 May 2016

## Erhan Bayraktar

University of Michigan at Ann Arbor - Department of Mathematics

## V.R. Young

University of Michigan at Ann Arbor - Department of Mathematics

Date Written: May 20, 2016

### 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 $c/r$.

Keywords: Bequest motive, consumption, optimal investment, stochastic control

JEL Classification: C61, G02, G11

Suggested Citation

Bayraktar, Erhan and Young, V.R., Optimally Investing to Reach a Bequest Goal (May 20, 2016). Available at SSRN: https://ssrn.com/abstract=2573421 or http://dx.doi.org/10.2139/ssrn.2573421