CRRA Expected-Utility Sizing
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Definition
Sizing positions by maximizing the expected value of a CRRA (power) utility function, U(W) = (W^(1-gamma) - 1)/(1-gamma) for gamma != 1 and U(W) = ln(W) for gamma = 1, where gamma is the coefficient of relative risk aversion.
How to read it
CRRA utility formalizes the idea that a fixed PERCENTAGE gain or loss feels the same regardless of wealth level (constant relative risk aversion), which is why optimal fractions of wealth are wealth-independent. Maximizing expected CRRA utility over the return distribution yields the Merton share as its closed form under Gaussian returns and full Kelly at gamma = 1. Its real power is that you can plug in the ACTUAL (non-Gaussian) return distribution and let the concave utility automatically penalize fat left tails - something Sharpe and mean-variance cannot do.
How practitioners use it
Used as context among multiple indicators — never as a standalone signal to act.
Less common professional uses
Expected-utility maximization over the EMPIRICAL return distribution (not a Gaussian fit) is the rigorous way to size fat-tailed strategies: the concavity of power utility applies a heavier penalty to the realized left tail than variance does, so it endogenously produces the fat-tail Kelly discount without an ad-hoc haircut. CRRA delivers a Taylor expansion linking utility to moments: E[U] ~ mean - (gamma/2)*variance + (gamma(gamma+1)/6)*skew-term - ..., so it rewards positive skew and penalizes kurtosis, formalizing why negative-skew carry trades should be sized smaller than their Sharpe suggests. Parameter uncertainty is handled by maximizing utility over the POSTERIOR predictive distribution of returns (Bayesian), which shrinks the optimal fraction below the plug-in estimate - the utility-theoretic justification for fractional Kelly. CRRA is time-consistent and additive over horizons for i.i.d. returns, which is why it underpins the Merton dynamic program; with regime switching the optimal policy becomes state-contingent.
Sources & provenance
Pratt (1964); Arrow (1965) on risk aversion; Merton (1969/1971); Campbell & Viceira (2002), 'Strategic Asset Allocation'
This page is educational content published by Pachira Aquatica Global LLC. It is not investment advice and not a recommendation.