The Merton Share: Sizing the Equity Sleeve

Kelly tells you how much to risk on a positive-EV bet. Robert Merton extended the same machinery to the canonical personal-finance question: what fraction of your portfolio should be in stocks versus bonds? His 1969 paper derives a closed-form answer,⁸⁷ now called the Merton share, that collapses the entire stock-bond allocation question to three estimable inputs:

where $w^{\ast }$ is the optimal equity weight, ERP is the equity risk premium (the expected stock return above the risk-free rate), ${\sigma }^2$ is the variance of stock returns, and RRA is your relative risk aversion — a personal parameter that measures how much utility you lose from a given drawdown. The framework is more useful than it looks, because all three inputs are knowable to first order and the formula degenerates gracefully if you get them wrong.

The inputs, in working numbers. Plug in long-run estimates: an ERP of 5%, a stock standard deviation of 20% (so ${\sigma }^2=0.04$), and an RRA of 2 for a typical risk-averse investor. The Merton share is $0.05∕(0.04\times 2)=62.5\%$ in equities. That is within shouting distance of the traditional 60/40 portfolio, which means the allocation the textbooks have been recommending for decades is not a folk heuristic — it is the Merton share evaluated at moderate risk aversion. Push RRA to 1 (someone who genuinely does not flinch at a 100% stock exposure) and the formula returns 125% equities, which is to say moderately leveraged. Push RRA to 4 (someone who feels physically ill at a 10% drawdown) and the formula returns roughly 31% equities — the deeply conservative allocation that behavior, not lifecycle stage, actually calls for. The framework treats risk aversion as a real parameter rather than a euphemism.

The Kelly–Merton synthesis. The half-Kelly recommendation that recurs in the trading literature (section “Beyond the Questionnaire: Kelly Sizing and the Barbell”) is not folk wisdom either. It falls directly out of the Merton share with RRA $\approx 2$: half-Kelly equity weight is Merton-share equity weight for a moderately risk-averse investor. The two frameworks agree because they are solving the same problem — maximize geometric growth subject to a utility function that punishes drawdowns — in two different formulations.⁷⁹ The convergence is one reason to take the half-Kelly heuristic seriously rather than as an arbitrary haircut.

Calibrating your own RRA. The honest way to estimate your RRA is to recall your behavior in the most recent severe drawdown you actually lived through. If you sat through 2008 or March 2020 without altering your allocation and slept fine, your RRA is in the 1–2 range. If you de-risked partway through the decline and re-entered after the recovery, your RRA is at least 3 — and the trades you actually made revealed that, not the answers you gave a broker’s questionnaire at the peak. If you have not lived through a severe drawdown, assume RRA $=2$ and plan to revisit it the first time you do. The questionnaire’s “moderate” label and a Merton-share RRA of 2 produce roughly the same allocation; the difference is that one is a label and the other is a parameter that goes into a formula.

The caveat that breaks the framework: RRA is not constant. The Merton derivation assumes a stable RRA across all market states. Empirically, that is false. RRA rises sharply in panics — the same investor who tolerated 100% equity exposure at the peak finds themselves unable to sleep at 60% during a 30% drawdown. The Merton share you compute at the peak, plugging in the RRA you had at the peak, is a calibration to the version of you that does not exist when it matters. Fred Schwed put this better than any modern textbook:

The practical correction is to size the equity sleeve to the RRA you have in the trough, not the RRA you have at the peak. For most readers that means computing the Merton share with RRA one notch higher than introspection suggests, and accepting that the resulting allocation will look conservative when stocks are running. The cost of getting this wrong is not a slightly suboptimal return curve. It is the version of you in the next drawdown selling the bottom and moving permanently to cash, which is the real-world failure mode no calibration at the peak will ever fix.