Kelly Betting Can Be Too Conservative

TL;DR

This paper challenges the common view that Kelly betting is overly aggressive, showing instead that it can be overly conservative when based on theoretical models versus empirical data, leading to missed opportunities.

Contribution

The paper introduces the Restricted Betting Theorem, demonstrating situations where theoretical Kelly bets are more conservative than empirical ones, highlighting potential pitfalls of relying solely on theory.

Findings

Theoretical Kelly bets can be smaller than empirical bets, missing profitable opportunities.

Empirical data can reveal 'golden' bets that theoretical models reject.

In cases with unbounded support, Kelly betting may suggest no bets at all.

Abstract

Kelly betting is a prescription for optimal resource allocation among a set of gambles which are typically repeated in an independent and identically distributed manner. In this setting, there is a large body of literature which includes arguments that the theory often leads to bets which are "too aggressive" with respect to various risk metrics. To remedy this problem, many papers include prescriptions for scaling down the bet size. Such schemes are referred to as Fractional Kelly Betting. In this paper, we take the opposite tack. That is, we show that in many cases, the theoretical Kelly-based results may lead to bets which are "too conservative" rather than too aggressive. To make this argument, we consider a random vector X with its assumed probability distribution and draw m samples to obtain an empirically-derived counterpart Xhat. Subsequently, we derive and compare the resulting Kelly bets for both X and Xhat with consideration of sample size m as part of the analysis. This leads to identification of many cases which have the following salient feature: The resulting bet size using the true theoretical distribution for X is much smaller than that for Xhat. If instead the bet is based on empirical data, "golden" opportunities are identified which are essentially rejected when the purely theoretical model is used. To formalize these ideas, we provide a result which we call the Restricted Betting Theorem. An extreme case of the theorem is obtained when X has unbounded support. In this situation, using X, the Kelly theory can lead to no betting at all.