The Merton Share: Can a Formula Beat 100% Stocks, or Just Size Risk Better?

In 2016, Victor Haghani and Richard Dewey handed $25 each to 61 people trained in finance and economics, showed them a digital coin rigged to land heads 60% of the time, and let them bet on it for 30 minutes. Despite a large, known edge, about a third finished with less money than they started with, 28% went completely bust, and 18 of the 61 staked everything on a single flip at some point. Only five participants had heard of the formula for sizing such bets.¹ Knowing an investment is attractive and knowing how much of it to hold are different skills, and most people, including professionals, lack the second one. The Merton share is the formula for the second skill. This guide explains what it says, whether the dynamic version beats 100% stocks, and how a DIY investor should use it.

The formula, briefly

In 1969, Paul Samuelson and Robert Merton published companion papers in the same issue of the Review of Economics and Statistics, solving lifetime portfolio choice in discrete and continuous time respectively.^2,3 Merton’s closed-form answer for the optimal fraction of wealth in the risky asset is:

where $\mu$ is the expected return of stocks in excess of the safe asset, $\sigma$ is stock volatility, and $\gamma$ is your coefficient of relative risk aversion: how fast the pain of losses grows relative to the pleasure of gains. Plug in a 4% expected excess return, 18% volatility, and moderate risk aversion of 3, and the formula says hold about 41% in stocks. Our lifecycle asset allocation guide walks the formula and a full worked example, including the human capital adjustment that typically pushes young investors far above this; Elm Wealth’s derivations note is the best free source for the math itself.⁴

The most useful way to read the formula multiplies both sides by volatility:

Your optimal portfolio volatility equals the market’s reward-to-risk ratio divided by your risk aversion. It is a risk budget: take more total risk when compensation per unit of risk is high, less when your aversion to loss is high. Everything else in this guide is about what happens when the three inputs meet the real world.

The original Merton share is constant

The most common misreading of this literature treats the Merton share as a market-timing rule. It is the opposite. Merton’s 1969 result shows that for an investor with constant relative risk aversion facing constant expected returns and volatility, the optimal allocation is the same fraction of wealth at every age, every wealth level, and every horizon.² You rebalance to maintain the fraction, but the target never moves. Samuelson’s companion paper made the same point against the idea that long horizons justify more risk by themselves.³

What Elm Wealth and others market as dynamic asset allocation comes from re-estimating the inputs, mainly the expected excess return, and re-running the formula as those estimates change. When expected returns genuinely vary over time, theory does support adjusting, and Merton’s own later work shows the full answer adds a hedging demand on top of the simple formula.⁵ But the dynamic version stands or falls on a forecasting problem the original theorem never had to solve. The formula is only as good as the $\mu$ you feed it.

Does it beat 100% stocks?

The question needs to be split apart, because “beat” means six different things and the answer differs across them.

The compound-growth row deserves the detail, because it is where the real case against 100% stocks lives. For a continuously rebalanced portfolio, expected log growth is approximately:

Growth rises with allocation up to a peak and then falls, because volatility drag grows with the square of exposure. The peak is the full Kelly allocation, which is the Merton share with $\gamma=1$:

At a 4% expected excess return and 18% volatility, Kelly is about 123%, so an unlevered all-stock portfolio sits below the growth peak and a lower allocation gives up expected growth. Shrink the premium estimate to 3% and Kelly falls to about 93%, and 100% stocks is now overbet: reducing stock exposure would raise expected compound growth while cutting risk. Whether 100% stocks is a growth-maximizing bet or an overbet flips on one percentage point of an unobservable input. Our position sizing guide covers Kelly and why practitioners bet fractions of it. And a growth-optimal bettor is not the same as a utility-optimal one: an investor with $\gamma=3$ deliberately gives up some expected compound wealth to shrink the range of outcomes their family has to live through.

Auditing the headline numbers

The case for dynamic Merton allocation reached most readers through two Economist articles drawing on work by Victor Haghani and James White of Elm Wealth, whose book The Missing Billionaires argues that America’s Gilded Age fortunes disappeared mainly through oversized risk and overspending rather than bad luck.⁶ The numbers are striking and worth stating precisely, because the two articles report different comparisons and are easy to conflate.

• The December 2024 Christmas special reports that a Merton-share strategy using US stocks and inflation-protected Treasuries from 1900 to 2022 returned about 10% annualized versus 8.5% for a static 65/35 mix, and beat 100% stocks with roughly 40% less risk.⁷
• The June 2025 inheritance article reports about 10% annualized versus 9.8% for all-stocks since 1900, with roughly 30% less volatility, and separately estimates that a portfolio with a 4.1% expected return supports a sustainable spending ratio of only about 2.4%.⁸

Elm’s own research states the backtested edge as about 2.5 percentage points of compound return per year over a similar-risk static allocation, using an expected equity premium built from the cyclically adjusted earnings yield, risk aversion of 2, and monthly rebalancing. Its forward-looking estimate is far more modest: about 1.5% per year before tax and roughly 1% after tax.⁹ Three caveats matter before anyone trades on this.

• The results are provider-produced. They come from the firm selling the strategy, simulated on one country’s single historical path. Elm itself cautions readers against adopting the approach on the backtest alone, and notes many 10-year stretches where the dynamic strategy did worse than static.⁹
• The result is estimator-dependent. In a separate Elm study, a strategy that varied allocation using the earnings yield alone, without also adjusting for volatility, failed to beat a static allocation over the same 120 years.¹⁰ Change the estimator and the headline changes.
• The broader timing literature is sobering. Goyal and Welch found the standard predictive variables generally failed to beat the simple historical average out of sample, a result their 2024 update largely reaffirms.^11,12 Campbell and Thompson showed economically sensible restrictions help somewhat, so the literature is not one-sided, but the gains are small.¹³ AQR’s study of valuation-based timing concluded investors should at most “sin a little,” and found that pairing valuation with momentum improves the modest timing case that valuation alone fails to make.¹⁴

To Elm’s credit, it describes static and dynamic index investing as “nearly on top of each other way over on the far right, sensible end of the continuum” of investment options.⁹ The honest reading of the evidence: expected returns probably do vary, exploiting that variation in real time is much harder than backtests suggest, and the after-tax prize for a taxable US investor is closer to Elm’s 1% estimate than to the 2.5-point backtest, with real tracking-error risk attached.

Optimization amplifies input error

The allocation moves one-for-one with the expected excess return estimate: at $\gamma=3$ and 18% volatility, each percentage point of assumed premium moves the recommended allocation by about 10 points. Estimation error passes straight through to your portfolio. DeMiguel, Garlappi, and Uppal tested 14 optimized portfolio models across seven datasets and found none consistently beat naive equal weighting out of sample once estimation error and turnover were counted.¹⁵

Crashes expose a second problem: the formula’s numerator and denominator disagree. After a severe decline, valuations improve, raising the estimated premium, while realized and implied volatility spike, raising the denominator. Whether the formula says buy or sell depends entirely on the estimators, lookback windows, smoothing, and update frequency you chose in advance. A system without those choices predefined is discretion disguised by a formula.

Which safe asset? It depends on the liability

The model assumes a risk-free asset. Nothing is risk-free for every purpose. Treasury bills are stable in nominal terms but leave a 30-year retirement exposed to inflation and reinvestment risk. TIPS held to maturity lock a real payout to a date; a long-duration TIPS fund swings hard when real yields move. The safe asset for next year’s tuition is different from the safe asset for spending in 2055. Define the liability first, then pick the asset that is safe relative to it; our TIPS ladder and asset-liability matching guides cover this step. The Economist simulations used inflation-protected bonds as the safe leg for exactly this reason.⁷

Taxes, trading costs, and the no-trade band

The strongest practical result in this whole literature is that precision is cheap to skip. The welfare loss from holding an allocation near the optimum instead of at it is quadratic:

Sit 10 points from the optimum at $\gamma=3$ and 18% volatility and the model-implied cost is about 5 basis points a year. Selling appreciated stock to close that gap in a taxable account can cost 100 times more in realized capital gains. Classic transaction-cost research reaches the matching conclusion: with trading frictions, the optimal policy is a no-trade band around the target, trading only to the band’s edge when you drift outside it.^16,17 A tax-aware implementation order follows directly: steer new contributions and withdrawals first, rebalance inside tax-advantaged accounts second, harvest losses, sell highest-basis lots, keep wide bands, and realize gains deliberately only when the risk reduction is worth the tax bill.

Your brokerage account is a minority of your balance sheet

The formula sizes risk for financial wealth, but the household balance sheet includes future paychecks, Social Security, pensions, housing, and often employer stock. Research on lifecycle portfolio choice shows that stable, bond-like labor income justifies holding more equities, while risky or equity-correlated income argues for less.^18,19 A tech worker with RSUs vesting every quarter already carries substantial equity exposure before buying a single index share; our human capital risk guide works through that case. The right $w$ applies to total economic exposure, computed across every account and income stream, not to one brokerage login.

Estimating your own gamma

Risk aversion is the one input nobody can look up, and the research offers four practical ways to approximate it.

• Permanent-income gambles. Barsky and co-authors asked people whether they would take a new job with a 50/50 chance of doubling lifetime income or cutting it by a third, then by smaller fractions, sorting respondents into risk-tolerance bands.²⁰ Stakes tied to permanent living standards beat any abstract “do you like risk” question.
• Lottery menus. Holt and Laury’s ten-choice menu locates your switch point between safer and riskier gambles, and their key finding is cautionary: measured risk aversion jumped when hypothetical stakes became real and large.²¹ Your questionnaire answers in a calm market are a lower bound on your risk aversion in a crash.
• The single validated question. Dohmen and co-authors showed that one self-rating of general willingness to take risk predicts real behavior, though far too coarsely to set a portfolio weight by itself.²²
• Invert your own allocation. Rearrange the formula: $\gamma=\mu/(w\,\sigma^{2})$. If you hold 80% stocks and believe in a 4% premium at 18% volatility, you are acting like a $\gamma\approx 1.5$ investor, close to a full-Kelly gambler. If a 50% temporary drawdown or a permanent 20% spending cut would horrify you, your allocation and your preferences disagree, and one of them has to move.

Triangulate these into a range, something like “my answers are consistent with gamma between 2 and 5,” and treat any tool that reports your risk aversion to two decimal places as decoration. Keep the three-way distinction from our risk budget guide in view: willingness to take risk is preference, capacity is whether a bad outcome is survivable, and need is the return your plan requires. The Merton share models the first plus the market; a financial plan needs all three.

Does following the formula make you happier?

No study shows that Merton-share investors are happier or more fulfilled. The formula maximizes expected utility, a mathematical representation of preferences under risk, and the number it optimizes cannot be deposited or felt. The happiness research itself is unsettled in instructive ways: Kahneman and Deaton found day-to-day emotional well-being flattening around $75,000 while life evaluation kept rising,²³ Killingsworth later found experienced well-being rising with log income well past that point,²⁴ and their 2023 adversarial collaboration resolved the conflict: flattening is real mainly for the least-happy minority of people, roughly at six-figure incomes, while happiness keeps rising with log income for everyone else.²⁵ Diminishing marginal utility of money is directionally right, and its exact shape varies enormously across people, which is another reason to treat gamma as a personal range rather than a constant of nature.

What the framework honestly offers is protection from catastrophe and from regret-driven decisions: it makes the risk decision explicit before the drawdown, and it caps exposure below the level where a bad sequence permanently damages your standard of living. Those are welfare gains. Continuous re-optimization, benchmark envy in the years stocks run, and false confidence in an estimated optimum are welfare losses, and they are just as real. Our money and happiness guide covers what wealth can and cannot buy.

Try it: the sandbox

The tool below computes the Merton share at your inputs, inverts your current allocation into an implied gamma, and shows the two results that matter more than any single recommendation: the utility hill is nearly flat on top, and the recommended allocation swings wildly across inputs that reasonable people disagree about. Use it to stress-test your allocation, never to time trades.

What we recommend

For most DIY investors, the default remains a diversified global stock portfolio, a high-quality bond or liability-matching sleeve, a stable written allocation, threshold rebalancing, and tax-aware implementation. That approach survives estimation error, costs little, and is easy to hold through the years when any timing model looks foolish. Use the Merton framework on top of it as a diagnostic, in three specific ways.

• Check the Kelly ceiling. Under your own honest premium estimate, confirm your allocation is not past the growth-optimal point where extra risk lowers expected compound growth. Leveraged positions fail this test far more often than unlevered ones.
• Check the implied gamma. If your allocation implies a risk aversion you would never claim in a permanent-income gamble, resolve the contradiction now, in a calm market, through contributions and sheltered accounts.
• Check the risk budget. Portfolio volatility should be near the Sharpe ratio divided by your gamma range. If it is far outside, you are taking meaningfully more or less risk than your preferences justify.

A constrained dynamic overlay is defensible for a narrower group: investors who understand the assumptions, hold most assets in tax-advantaged accounts, precommit estimators and update rules in a written policy before seeing the next market move, cap allocation changes to modest sizes, use no leverage, and can tolerate years of tracking error without abandoning the rule. If any of those conditions fails, and for beginners, investors with large embedded taxable gains, or anyone drawn mainly by the backtest chart, the static default is the better strategy. The choice between static and constrained-dynamic is a small decision between two prudent options; the decision that dominates outcomes is holding total risk at a level you can survive and sustain.

How Summitward helps

Summit’s allocation tool runs a Merton-style calculation with your actual inputs: age, income, human capital, and risk tolerance, rather than a context-free formula. Portfolio analysis measures your equity exposure across every account, which is the $w$ the formula requires. The retirement Monte Carlo evaluates what matters more than CAGR: whether a given risk level funds your plan across thousands of simulated paths, including bad ones. For hands-on readers, Engineer Investor’s Python tutorial on the Merton share computes it from live market data in a free Colab notebook.

Frequently asked questions

What is the Merton share?

The Merton share is the fraction of wealth an investor should hold in risky assets to maximize expected utility, derived by Robert Merton in 1969: expected excess return divided by risk aversion times variance. At a 4% excess return, 18% volatility, and risk aversion of 3, it recommends about 41% in stocks.

Is the Merton share the same as the Kelly criterion?

Kelly is the special case with risk aversion equal to 1: the allocation that maximizes expected compound growth while ignoring the discomfort of the ride. The Merton share generalizes it, scaling the Kelly bet down by your personal risk aversion. Most investors behave like fractional-Kelly bettors with gamma between 2 and 5.

Does dynamic asset allocation beat buy-and-hold?

In Elm Wealth’s historical simulations, yes, by about 2.5 percentage points a year against a similar-risk static mix. But the result is provider-produced, based on one country’s history, sensitive to the choice of estimator, and Elm’s own forward-looking estimate shrinks to roughly 1% a year after tax. The academic evidence on return forecasting suggests treating any timing edge as small and uncertain.

What gamma should I use?

Estimate a range rather than a number. Combine a permanent-income gamble (would you risk a one-third income cut for a coin-flip chance to double it?), your tolerance for portfolio drawdowns and spending cuts, and the gamma implied by inverting your current allocation. Most people land between 2 and 5. If your methods disagree sharply, that disagreement is the finding: act on the conservative end until you have lived through a real drawdown.

Key takeaways

• The Merton share is a sizing tool. It answers “how much risk should I hold?” with a defensible formula. The original result recommends a constant allocation; the dynamic version adds a forecasting problem nobody has reliably solved.
• “Beats 100% stocks” needs unpacking. A lower allocation never wins on expected return, always wins on volatility, and wins on compound growth only when all-stocks is past the Kelly point, which depends on an unobservable premium.
• The utility hill is flat on top. Being 10 points from the optimum costs single-digit basis points; closing the gap with taxable sales can cost far more. Wide bands and contribution-based adjustment beat precision.
• Estimate gamma as a range. Income gambles, drawdown tolerance, and inverting your current allocation each give a noisy reading; triangulate them and distrust decimal-point precision.
• The framework’s deliverable is discipline. It aligns risk with consequences and guards against overbetting. It does not promise more wealth than 100% stocks, and no evidence shows it delivers more happiness.

Related guides

• Lifecycle Asset Allocation: Why Young Investors Should Hold More Stocks the Merton share with human capital: the derivation and worked examples this guide builds on.
• Position Sizing: The Skill That Matters More Than Picking Winners Kelly, fractional Kelly, and why sizing beats selection.
• 60/40, Target-Date Funds, or 100% Stocks Forever? what each static alternative is solving for.
• Do Stock Valuations Still Matter? the CAPE evidence behind every dynamic-allocation premium estimate.
• Money and Happiness: What the Research Actually Says what rising wealth does and does not buy.

Sources

1. Haghani, V., & Dewey, R. (2016). Rational Decision-Making Under Uncertainty: Observed Betting Patterns on a Biased Coin. SSRN. 61 participants; 28% went bust; 18 of 61 staked everything on one flip.
2. Merton, R. C. (1969). Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case. Review of Economics and Statistics 51(3), 247–257.
3. Samuelson, P. A. (1969). Lifetime Portfolio Selection by Dynamic Stochastic Programming. Review of Economics and Statistics 51(3), 239–246.
4. Rosenbluth, J. M., & White, J. (2024). Merton Share Derivations: What’s in your denominator? Elm Wealth.
5. Merton, R. C. (1973). An Intertemporal Capital Asset Pricing Model. Econometrica 41(5), 867–887.
6. Haghani, V., & White, J. (2023). The Missing Billionaires: A Guide to Better Financial Decisions. Wiley.
7. The Economist (2024, December 19). How much happiness does money buy? Merton strategy 10% vs. 8.5% for 65/35, 1900–2022.
8. The Economist (2025, June 12). How to invest your enormous inheritance. 10% vs. 9.8% for all-stocks with 30% less volatility.
9. Haghani, V. (2024). Victor Meets the Bogleheads. Elm Wealth. Backtest edge ~2.5%/yr; prospective ~1.5% pre-tax, ~1% after-tax.
10. Elm Wealth. Man Doth Not Invest by Earnings Yield Alone. Earnings-yield-only timing failed to beat static over ~120 years.
11. Goyal, A., & Welch, I. (2008). A Comprehensive Look at the Empirical Performance of Equity Premium Prediction. Review of Financial Studies 21(4), 1455–1508.
12. Goyal, A., Welch, I., & Zafirov, A. (2024). A Comprehensive 2022 Look at the Empirical Performance of Equity Premium Prediction. Review of Financial Studies 37(11), 3490–3557.
13. Campbell, J. Y., & Thompson, S. B. (2008). Predicting Excess Stock Returns Out of Sample: Can Anything Beat the Historical Average? Review of Financial Studies 21(4), 1509–1531.
14. Asness, C., Ilmanen, A., & Maloney, T. (2017). Market Timing: Sin a Little. Journal of Investment Management 15(3).
15. DeMiguel, V., Garlappi, L., & Uppal, R. (2009). Optimal Versus Naive Diversification: How Inefficient Is the 1/N Portfolio Strategy? Review of Financial Studies 22(5), 1915–1953.
16. Constantinides, G. M. (1986). Capital Market Equilibrium with Transaction Costs. Journal of Political Economy 94(4), 842–862.
17. Davis, M. H. A., & Norman, A. R. (1990). Portfolio Selection with Transaction Costs. Mathematics of Operations Research 15(4), 676–713.
18. Bodie, Z., Merton, R. C., & Samuelson, W. F. (1992). Labor Supply Flexibility and Portfolio Choice in a Life-Cycle Model. Journal of Economic Dynamics and Control 16(3–4), 427–449.
19. Cocco, J. F., Gomes, F. J., & Maenhout, P. J. (2005). Consumption and Portfolio Choice over the Life Cycle. Review of Financial Studies 18(2), 491–533.
20. Barsky, R. B., Juster, F. T., Kimball, M. S., & Shapiro, M. D. (1997). Preference Parameters and Behavioral Heterogeneity: An Experimental Approach in the Health and Retirement Study. Quarterly Journal of Economics 112(2), 537–579.
21. Holt, C. A., & Laury, S. K. (2002). Risk Aversion and Incentive Effects. American Economic Review 92(5), 1644–1655.
22. Dohmen, T., Falk, A., Huffman, D., Sunde, U., Schupp, J., & Wagner, G. G. (2011). Individual Risk Attitudes: Measurement, Determinants, and Behavioral Consequences. Journal of the European Economic Association 9(3), 522–550.
23. Kahneman, D., & Deaton, A. (2010). High Income Improves Evaluation of Life but Not Emotional Well-Being. PNAS 107(38), 16489–16493.
24. Killingsworth, M. A. (2021). Experienced Well-Being Rises with Income, Even above $75,000 per Year. PNAS 118(4), e2016976118.
25. Killingsworth, M. A., Kahneman, D., & Mellers, B. (2023). Income and Emotional Well-Being: A Conflict Resolved. PNAS 120(10), e2208661120.

Editor’s note

Educational content, not investment advice. Elm Wealth figures and Economist reporting are cited as published and current as of July 2026; backtested results are hypothetical and provider-produced. The calculator implements the single-period CRRA approximation and is a model explorer, not a recommendation engine.