When Merton Portfolio Weights Match Kelly Bet Sizing

Summary

The document asks how the Merton portfolio choice problem relates to Kelly sizing, which selects a bet size to maximize expected logarithmic wealth. It presents the familiar Merton risky-asset fraction in terms of excess expected return, return variance, and risk aversion, then asks whether setting the risk-free rate to zero and using log utility makes the continuous-time portfolio solution agree with Kelly’s sizing rule.

This is a conceptual prompt rather than a worked derivation: it supplies no answer, numerical example, or evidence of convergence. The comparison is useful because it highlights shared ingredients—growth optimization and exposure scaled by expected return relative to risk—but equivalence depends on matching assumptions about asset dynamics, time horizon, available investments, and how the bet’s outcomes are modeled. The document does not resolve those conditions, so readers would need a separate derivation before applying either formula to a particular trading strategy.

Key ideas

• Kelly sizing chooses exposure to maximize expected logarithmic wealth.
• The Merton fraction scales risky investment with excess return and inversely with variance and risk aversion.
• The prompt asks whether log utility and a zero risk-free rate connect the two sizing rules.
• Any claimed equivalence requires compatible assumptions about returns, horizons, and investment opportunities.

Tags

# Connections between Merton Fraction & Kelly Criterion


# Connections between Merton Fraction & Kelly Criterion












What are the theoretical and practical connections between the solution to the Merton's Portfolio problem and a bet-sizing tool such as optimizing the expected value of the logarithm of wealth (aka the Kelly Criterion)?

Will these two solutions converge under certain conditions?

For example, the Kelly Criterion considers a two-outcome bet scenario and determines the optimal betting level depending on your bankroll. The Merton formula

$$ \pi(W,t) = \frac {\mu - r } {\sigma ^ 2 \gamma} $$

which is a combination of the mean-variance expected return and a risk aversion. If $r=0$ as it does in the Kelly formula, and a log-utility that the Kelly criterion assumes, do these results agree?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.