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

  • William Kuszmaul
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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× 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.

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Cite

BibTeXRIS
BibTeX
@misc{kuszmaul2026with,
  title = {{With Leverage, the Gambler is Not the One Who Gets Ruined}},
  author = {William Kuszmaul},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19239},
  eprint = {2610.19239},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19239},
}
RIS
TY  - UNPB
TI  - With Leverage, the Gambler is Not the One Who Gets Ruined
AU  - William Kuszmaul
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.19239
UR  - https://arxiv.org/abs/2610.19239
AB  - 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× 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.
ER  -