The Grossman and Zhou investment strategy is not always optimal

Abstract

Grossman and Zhou [1993. Optimal investment strategies for controlling drawdowns. Math. Finance 3, 241–276] proposed a strategy to maximize the asymptotic long-run growth rate of one's fortune F t subject to its never falling below λ sup 0 ⩽ t ′ ⩽ t F t ′ e r ( t - t ′ ) , where 0 ⩽ λ ⩽ 1 is a fixed constant chosen by the investor and r is a fixed, known, non-negative, continuously compounded interest rate on invested capital. In this paper we show that the strategy proposed in Grossman and Zhou does not retain its optimal long-run growth property when generalized to the discrete-time setting.