Betting Around the Clock: Time Change and Long Term Model Risk

We investigate the performance of the Kelly rule in a setting in which the dynamics of the return is represented by a time change process. We find that in this general semi-martingale setting the Kelly rule does not maximize the average growth rate, unless the log-return is normally distributed. Namely, the investment position proposed by the Kelly rule is too large, and the investor could achieve a higher average growth rate by investing less aggressively. The higher the variance of the stochastic clock, the more material the failure of the Kelly rule. The ruin threshold proposed by Thorp (1969) is closer, even though examples based on stochastic clock variance estimates taken from the literature show that Kelly rule investment remains safely in the ruin-free region. Finally, the goal of keeping the investment below the ruin threshold for a family of stochastic clock distributions generates a long term investment problem that parallels the "acceptable investment" theory.