Risk-Constrained Kelly Gambling

TL;DR

This paper introduces a convex optimization approach to Kelly gambling that incorporates a risk constraint on drawdown probability, balancing growth and risk effectively.

Contribution

It develops a convex bound on drawdown probability that ensures risk constraints are met, linking Kelly gambling with mean-variance portfolio optimization.

Findings

The method outperforms fractional-Kelly bets at the same risk level.

The drawdown probability bound closely matches Monte Carlo estimates.

The approach provides a systematic risk-growth trade-off.

Abstract

We consider the classic Kelly gambling problem with general distribution of outcomes, and an additional risk constraint that limits the probability of a drawdown of wealth to a given undesirable level. We develop a bound on the drawdown probability; using this bound instead of the original risk constraint yields a convex optimization problem that guarantees the drawdown risk constraint holds. Numerical experiments show that our bound on drawdown probability is reasonably close to the actual drawdown risk, as computed by Monte Carlo simulation. Our method is parametrized by a single parameter that has a natural interpretation as a risk-aversion parameter, allowing us to systematically trade off asymptotic growth rate and drawdown risk. Simulations show that this method yields bets that out perform fractional-Kelly bets for the same drawdown risk level or growth rate. Finally, we show that a natural quadratic approximation of our convex problem is closely connected to the classical mean-variance Markowitz portfolio selection problem.