Financial leverage can reduce gambler ruin risk significantly
With Leverage, the Gambler is Not the One Who Gets Ruined
Data Structures and Algorithms
Summary
This paper looks at how adding financial leverage changes the usual risk of losing everything when gambling. The authors found that with moderate leverage, a gambler can often avoid bankruptcy and even make very large profits over time without needing to win more than fair bets. For example, with ten times leverage, there is about a 50% chance to multiply money by 1000 without going broke. The authors provide a formula for the chance of success based on leverage and desired profit. Their work is part of a course on randomized algorithms at CMU.
What this means in practice
- •For financial traders: Develop risk management strategies that incorporate leverage to sustain trading without bankruptcy over long time periods.
- •For casino game designers: Assess how leverage mechanisms might influence player behavior and game outcomes under fair betting conditions.
A theory result. No direct application yet.
Authors
William Kuszmaul
Abstract
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 $50\%$. 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 $r^{-1/L}$ of success. The results in this note are designed as part of the course development for the CMU randomized algorithms course.