Risk-Constrained Kelly Gambling

Abstract

The authors 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. They 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 this bound on drawdown probability is reasonably close to the actual drawdown risk, as computed by Monte Carlo simulation. This method is parametrized by a single parameter that has a natural interpretation as a risk-aversion parameter, allowing the authors to systematically trade-off asymptotic growth rate and drawdown risk. Simulations show that this method yields bets that outperform fractional Kelly bets for the same drawdown risk level or growth rate. Finally, they show that a natural quadratic approximation of the convex problem is closely connected to the classical mean–variance Markowitz portfolio selection problem.