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 . We, thereby, make more objective the goal of maximizing expected utility of death, first considered in a continuous-time framework by Merton (1969). Specifically, instead of requiring the individual tochoose a utility function, we only require the individual to choose a bequest goal b. We learn that, for wealth lying between 0 and b , the optimal investment strategy is {\it independent } of b , a surprising result. Therefore, if the individual were to revise her bequest goal, her investment strategy would not change if her wealth is less than the new goal .
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 .
D
. We, thereby, make more objective the goal of maximizing expected utility of death, first considered in a continuous-time framework by Merton (1969). Specifically, instead of requiring the individual tochoose a utility function, we only require the individual to choose a bequest goal b. We learn that, for wealth lying between 0 and b
R
the optimal investment strategy is
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
R
independent
mathematically
R
of b , a surprising result. Therefore, if the individual were to revise her bequest goal, her investment strategy would not change if her wealth is less than the new goal
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