Five years of monthly returns put a Sharpe ratio of one somewhere between 0.1 and 1.9

No number in finance does more work than the Sharpe ratio. It sits on fund factsheets, decides which managers are hired and fired, and settles arguments about fees. It is also an estimate calculated from a sample of past returns, almost always published without any indication of how wrong it might be. An estimate without an error bar is an opinion with decimal places. The interesting thing is not that the ratio is imperfect, which everyone concedes, but that its three genuine defects all push the reported figure the same way, and that the defect most complained about is the least of them.

William Sharpe proposed the measure in 1966 to compare mutual funds: the return a fund earned above the risk-free rate, divided by the standard deviation of its returns. The standard deviation is the typical distance between a return and that fund's own average, so the ratio answers a reasonable question: how much excess return per unit of jumpiness. Sharpe wanted to call it the reward-to-variability ratio; the measure became popular and the name did not.¹ Revisiting it in 1994 he entered a caution easy to read past: neither version incorporates information about the correlation of a fund with other assets, with liabilities, or with previous realisations of its own return.¹ That last clause is the subject of this paper.

Start with sampling error, the least glamorous and most damaging. Andrew Lo set out the statistics of the ratio in 2002, and for returns that are independent from one period to the next the standard error of an estimated Sharpe ratio is the square root of one plus half the squared ratio, divided by the number of observations.² Put five years of monthly returns through that. Sixty observations; an annual ratio of 1.0, which is a respectable record, corresponds to about 0.289 a month; the standard error works out at about 0.132 a month, so the ninety-five per cent interval runs from roughly 0.03 to 0.55 monthly, which annualises to roughly 0.1 to 1.9.³ A manager reporting 1.0 over five years and one reporting 0.4 are, on this evidence, indistinguishable. Five years is longer than many allocators wait before deciding.

The second defect is in the denominator, and it is not a statistical problem but an accounting one. The standard deviation is computed from reported returns, and reported returns depend on how often and how independently the assets are priced. Lo showed that a monthly Sharpe ratio cannot be annualised by multiplying by the square root of twelve unless returns are serially uncorrelated, and that serial correlation can overstate the annual ratio of a hedge fund by as much as sixty-five per cent in his illustration.² Serial correlation here means that good months tend to follow good months, which makes a year's variance larger than twelve times a month's. Mila Getmansky, Lo and Igor Makarov traced its source in 908 funds from the TASS database: the serial correlation is most plausibly the product of illiquid holdings and stale prices, and the estimated smoothing varies enough across styles to proxy for illiquidity exposure.⁴ An asset valued by appraisal rather than by trade therefore reports a smaller denominator than it has earned. Cliff Asness gave the practice a name that has stuck, volatility laundering.⁵ The ratio pays a premium for not being priced.

The third defect lies not in the number but in the choosing of it. Goetzmann, Ingersoll, Spiegel and Welch showed that the gaming of popular performance measures can substantially affect them even when transaction costs are high, and characterised the conditions under which a manipulation-proof measure exists.⁶ The standard manoeuvre is to sell options far out of the money: the premium lifts the average return, the variance barely moves, and the risk is pushed into the skewness and fat tails the ratio does not look at.⁷ A subtler version needs no derivatives. When a reported ratio is the best of many computed, its size reflects the search as much as the strategy. David Bailey, Jonathan Borwein, Marcos López de Prado and Qiji Zhu argue that an investor who does not know how many configurations were tried cannot judge how much of a backtest is overfitting, and derive a minimum backtest length as a function of the number of trials.⁸ The deflated Sharpe ratio exists because the maximum of many draws is not an unbiased estimate of any of them.⁹

All three defects run one way. Sampling error is symmetric in principle, but the ratios that get published are the ones that came out high; stale pricing shrinks the denominator; selection picks the top of the distribution. A reported Sharpe ratio is therefore not a noisy measurement of skill but an upward-biased one, and the bias grows with short records, infrequently traded assets and large families of strategies.

The usual response is to build a better ratio, and the field has built several. That addresses the shape of the return distribution, which is the smallest of the three problems. The deeper error is treating the Sharpe ratio as a descriptive statistic when it behaves like a test statistic, and the fix is disclosure rather than mathematics. Three lines would do most of the work: the standard error beside the point estimate, the share of the portfolio that no arm's-length transaction priced during the period, and the number of candidate strategies, share classes or track records the reported one was selected from. A drug trial is judged on its registration and confidence interval rather than on a flattering measure of effect, and the parallel is exact: both are claims about a population inferred from a sample.

Three objections are worth stating. The ratio was designed for a diversified, daily-priced fund measured against an index over decades, and for that job it works; the complaint is about its extension to three-year records and unpriced books. That is right, and it concedes the point, because the extension is where most of its modern use lives. Next, standard errors would be ignored, or gamed by stretching the sample to include a founder's best years, which argues for a standardised window rather than for suppressing the interval. The strongest objection is that not all the volatility reduction in private assets is laundered, since holding periods and capital structures genuinely differ from those of listed equity.⁵ Granted, and it is a reason to disclose the unpriced share rather than to de-smooth every series by assumption.

A last objection says uncertainty does not allocate capital, and that an investor facing wide intervals still has to choose. It does allocate. If the headline number cannot separate a good manager from a mediocre one over the period anybody observes, the rational response is to decide on what can be measured without a long sample: fees, liquidity terms, the pricing of the book and the discipline of the process. The Sharpe ratio is not useless. It is a sample statistic read as a fact, and the cure is to make it answer the three questions every sample statistic should: how wide is the interval, what priced the assets, and how many times did you look.

Notes

1William F. Sharpe, "The Sharpe Ratio," The Journal of Portfolio Management (Fall 1994), for the 1966 origin of the measure in his study of mutual fund performance, for his statement that he proposed the term reward-to-variability ratio while the measure gained popularity and the name did not, and for his caution that neither the ex ante nor the ex post version incorporates information about the correlation of a fund or strategy with other assets, liabilities, or previous realisations of its own return. The author's own reprint is the copy consulted; the environment's network policy blocked every attempt to open it, so the quoted wording comes from search results quoting that page. Sharpe's own copy of the article and a reference account of the measure

2Andrew W. Lo, "The Statistics of Sharpe Ratios," Financial Analysts Journal 58, no. 4 (2002), pp. 36-52, for the finding that monthly Sharpe ratios cannot be annualised by multiplying by the square root of twelve except under special conditions, for the illustration in which serial correlation in monthly returns overstates a hedge fund's annual Sharpe ratio by as much as 65 per cent, and for the standard error of the estimated ratio under independent and identically distributed returns. Not opened. The 65 per cent figure is from an illustrative example rather than a general estimate. CFA Institute record and a copy hosted by a Berkeley student organisation

3The confidence interval in the title and in the text is my own arithmetic, not a figure taken from a source. Using Lo's standard error for independent and identically distributed returns, the square root of one plus half the squared ratio divided by the number of observations: an annual ratio of 1.0 is 0.2887 monthly; with 60 observations the standard error is 0.1318; 1.96 standard errors either side gives 0.030 to 0.547 monthly, which multiplied by the square root of twelve gives 0.105 to 1.894. Returns that are serially correlated would widen it further. The formula is restated, with the same algebra, in a statistical package's documentation. Steven Pav, "Notes on the Sharpe ratio" (SharpeR package vignette, 2024) and Lo's article

4Mila Getmansky, Andrew W. Lo and Igor Makarov, "An Econometric Model of Serial Correlation and Illiquidity in Hedge Fund Returns," Journal of Financial Economics 74 (2004), pp. 529-609, for the argument that serial correlation in hedge fund returns is most plausibly the result of illiquid holdings and stale prices, for the estimation on 908 funds from the TASS database, for the finding that estimated smoothing coefficients differ considerably across style categories and could proxy for illiquidity exposure, and for the derivation of a smoothing-adjusted Sharpe ratio. Not opened. NBER working paper record and the authors' page at MIT

5For volatility laundering, the term Cliff Asness of AQR gave to the understatement of risk in private asset returns caused by stale marks and the autocorrelation they produce, for his argument that the move into private assets rests on the belief that private equity is meaningfully less volatile than public equity, and for the countervailing point that some of the volatility reduction in private vehicles is structural rather than an artefact of valuation. The term was coined in a social media post whose date the available sources do not agree on, so no date is given here. A research summary of the argument, a report on return smoothing in private funds and an interview with Asness in the Journal of Private Markets Investing, Spring 2026

6William Goetzmann, Jonathan Ingersoll, Matthew Spiegel and Ivo Welch, "Portfolio Performance Manipulation and Manipulation-proof Performance Measures," Review of Financial Studies 20, no. 5 (2007), pp. 1503-1546, for the finding that gaming of popular performance measures can substantially affect them even when transaction costs are high, for the characterisation of a manipulation-proof measure resembling the average of a power utility function over a return history, and for the authors' view that the case for such a measure is strongest for hedge funds, where derivative use is unconstrained and manager pay is itself nonlinear. Some citations render the title in the singular. Not opened. The authors' working paper at Yale

7Didier Maillard, on the manipulation of the Sharpe ratio by selling out-of-the-money put options, which raises the average return through premium collection without much raising variance while changing the skewness and kurtosis of the distribution. Cited for the mechanism rather than for a figure; the paper is a 2017 conference submission and was not opened. Conference paper

8David H. Bailey, Jonathan M. Borwein, Marcos López de Prado and Qiji Jim Zhu, "Pseudo-Mathematics and Financial Charlatanism: The Effects of Backtest Overfitting on Out-of-Sample Performance," Notices of the American Mathematical Society (2014), for the argument that without knowing the number of configurations tried for a backtest an investor cannot evaluate the degree of overfitting, for the use of the Sharpe ratio as the measure of strategy attractiveness, and for the derivation of a minimum backtest length given the number of configurations tried. Not opened. A copy on one author's site and a second copy

9David H. Bailey and Marcos López de Prado, "The Probability of Backtest Overfitting," Journal of Computational Finance (2015), for the estimation of the probability of backtest overfitting by combinatorially symmetric cross-validation, and for the deflated Sharpe ratio as the corrective statistic for the multiple-testing case. The description of the deflated ratio comes from teaching notes citing the authors rather than from their own text, which was not opened. SSRN record and lecture notes setting out the deflated Sharpe ratio