Minimizing the Probability of Lifetime Drawdown under Constant Consumption

TL;DR

This paper derives an optimal investment strategy to minimize the probability of lifetime wealth drawdown under constant consumption in a Black-Scholes market, balancing risk and growth to prevent wealth from falling below a fixed proportion of maximum wealth.

Contribution

It introduces a novel strategy for minimizing lifetime drawdown probability, including a threshold-based approach depending on current maximum wealth.

Findings

Optimal investment strategy depends on current maximum wealth relative to a threshold.

When maximum wealth is below a certain level, the strategy prevents further wealth increase.

If maximum wealth exceeds the threshold but is below the safe level, the strategy allows reaching the safe level.

Abstract

We assume that an individual invests in a financial market with one riskless and one risky asset, with the latter's price following geometric Brownian motion as in the Black-Scholes model. Under a constant rate of consumption, we find the optimal investment strategy for the individual who wishes to minimize the probability that her wealth drops below some fixed proportion of her maximum wealth to date, the so-called probability of {\it lifetime drawdown}. If maximum wealth is less than a particular value, $m^*$, then the individual optimally invests in such a way that maximum wealth never increases above its current value. By contrast, if maximum wealth is greater than $m^*$ but less than the safe level, then the individual optimally allows the maximum to increase to the safe level.