Merton Optimal Share
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Definition
The optimal fraction of wealth to hold in a risky asset for a CRRA (constant relative risk aversion) investor: w* = (mu - r) / (gamma * sigma^2), where mu is expected return, r the risk-free rate, sigma^2 the return variance, and gamma the coefficient of relative risk aversion.
How to read it
The Merton solution is the continuous-time answer to 'how much should I put at risk given my risk aversion?' It scales the Sharpe-like edge (mu - r)/sigma^2 down by risk aversion gamma. With log utility (gamma = 1) it reduces EXACTLY to full Kelly, so Kelly is the special case of a maximally-growth-seeking, minimally-risk-averse investor. Higher gamma means more conservative sizing; gamma between 2 and 5 is typical for real investors, which naturally produces fractional-Kelly-like allocations without invoking estimation error at all.
How practitioners use it
Used as context among multiple indicators — never as a standalone signal to act.
Less common professional uses
With STOCHASTIC investment opportunities (time-varying mu or sigma), the full Merton solution adds an intertemporal HEDGING DEMAND term beyond the myopic (mu-r)/(gamma sigma^2): investors tilt toward assets that hedge future changes in the opportunity set, which the static formula omits entirely. The Merton share and fractional Kelly converge from two different premises - risk aversion (Merton) vs estimation error (fractional Kelly) - and in practice they compound: a risk-averse investor facing parameter uncertainty should size below BOTH, effectively raising the denominator via a larger gamma AND a Bayesian-shrunk mu. Under fat-tailed or jump-diffusion dynamics the variance term understates true risk, so the Gaussian Merton share over-allocates; robust versions replace sigma^2 with a downside/tail risk measure. Transaction costs turn the point solution into a NO-TRADE REGION around w*: you only rebalance when the weight drifts outside a band whose width grows with cost and shrinks with risk aversion (Davis-Norman).
Sources & provenance
Merton (1969), 'Lifetime Portfolio Selection under Uncertainty', Review of Economics and Statistics; Merton (1971), Journal of Economic Theory
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