Revisiting Gambler’s Ruin: Trading Edge, Drawdowns, and Survival

The gambler's ruin problem is a classical probability problem that examines the likelihood of a gambler reaching a specified wealth target or losing all available capital. Reference [1] revisits this problem. Specifically, it seeks to answer the following questions:

Suppose a gambler enters a casino with finite capital,

• What is the probability of reaching a target wealth level or going bankrupt, and
• What is the expected duration of play?

The author first derives analytical formulas to address these questions, then uses Monte Carlo simulations to validate the theoretical results. Finally, the findings are applied to a real-world investment problem using daily S&P 500 data from January 1990 through September 2026.

The author pointed out,

Applied to daily S&P 500 returns, the model estimates a win rate of p = 0.5369 and finds that the sign sequence is statistically distinguishable from an independent one while being, in magnitude, very nearly independent. The more serious failure is elsewhere. When the fixed unit stake is replaced by a bootstrap of the actual return distribution, holding the win rate and the mean absolute move constant, the projected probability of ruin rises from 5.0% to 42.7%. The model’s weakness is not that markets are predictable but that they do not move in equal steps. That is a useful thing to know about a model, and it is the kind of thing that only becomes visible when the mathematics is exact enough that the discrepancy cannot be blamed on the algebra.

A central finding of the paper is that risk models based solely on the frequency of winning/losing days can substantially underestimate the risk of ruin when actual return magnitudes are unequal and heavy-tailed. In other words, the size of losses relative to gains, as well as the presence of fat tails, is critical in determining ruin probabilities.

Another interesting finding is that a relatively small increase in the probability of winning can dramatically reduce the probability of ruin. For example, at a 51% win rate, the probability of ruin is 44.93%, whereas at a 56% win rate, it falls to just 0.8%.

This paper makes an important contribution by formally confirming two principles already familiar to practitioners, namely

• Keeping losses and drawdowns small, and
• Increasing the trading edge whenever possible.

Let us know what you think in the comments below or in the discussion forum.

References

[1] Jasman Mander, Random Walks and the Gambler’s Ruin Problem: Exact Results, Independent Derivations, and an Empirical Application to Equity Index Returns, Working Paper, 2026

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