With Leverage, the Gambler is Not the One Who Gets Ruined

In this note we explore the following simple question: How does the classic \emph{gambler’s ruin} phenomenon change if we allow the gambler \emph{financial leverage}? We show that, with a moderate amount of leverage, it is possible for an (idealized) gambler to sustain seemingly improbable profits over a remarkably long time horizon without ever going bankrupt, \emph{even} if the gambler makes only fair bets. With a leverage ratio of $L=10$ (less than what Lehman Brothers Holdings had from 1995–2008), the probability of multiplying one’s money by $1000\times$ without going bankrupt is roughly $50\mathrm{\%}$. More generally, given $L$-fold leverage, and given the goal of multiplying one’s money by a factor of $r>1$, the optimal gambling strategy has probability roughly $r^{-1\mathrm{/}L}$ of success. The results in this note are designed as part of the course development for the CMU randomized algorithms course.