Why the Sharpe Ratio Rewards Volatility When You Are Losing

Bitcoin closed the Tuesday 1 September 2026 session at 77,425.10 on Phemex spot, down 1.47% from the Monday 31 August close of 78,581.30. One session cannot produce a Sharpe ratio at all, because a single observation has no standard deviation to divide by. Stretch the window back to 251 daily closes ending on that same Tuesday bar and something far stranger falls out of the arithmetic. Ether lost 16.72% over those 250 sessions against Bitcoin's 11.24%, carried 38% more daily volatility, and finished with the better Sharpe ratio anyway.

That is not a data error and it is not a quirk of this particular window. The formula does it by construction every time the numerator turns negative, which means most of the risk-adjusted return tables published during a drawdown are ranking the wrong strategy first.

What the Sharpe Ratio Actually Measures

William Sharpe, who shared the 1990 Nobel Memorial Prize in Economic Sciences, restated his own measure in a 1994 Journal of Portfolio Management paper, and the version he wrote is not the version most calculators run. In Sharpe's own 1994 restatement of the ratio, the ratio is the mean of the differential return divided by the standard deviation of that same differential return. Differential means the fund's return minus the return on what he calls a benchmark portfolio or security. He is explicit that the benchmark does not have to be a risk-free rate, and the instruction he gives for computing it is a single spreadsheet line, the average of the differential column divided by the standard deviation of that column.

Read that denominator again. The standard deviation belongs to the differential, not to the portfolio.

The popular version, including the standard Investopedia reference, uses the portfolio's own standard deviation instead and gets away with it, because when the benchmark is a constant the shortcut is arithmetically identical. Subtracting a fixed cash rate from every return shifts the mean and leaves the spread untouched, so the two denominators agree. That agreement collapses the moment your benchmark moves. Measured against Bitcoin over those same 250 sessions, Ether's differential return has a standard deviation of 1.5073% per session against Ether's own 3.3281%, because the two streams are 0.9111 correlated and most of the movement cancels. The correct denominator is less than half of what a portfolio-volatility calculator hands you.

If you are pulling standard deviations of realized returns, the Phemex historical volatility guide covers how that backward-looking figure gets built. It is a separate quantity from the forward-looking implied volatility priced into options, and the two are not interchangeable.

Why a Losing Strategy Scores Better When It Gets Wilder

Take the two largest assets on the tape and run the calculation properly. The series below comes from Phemex spot daily closes, 251 bars producing 250 daily returns, ending on the Tuesday 1 September 2026 close. The benchmark is a flat zero, which is what an idle USDT balance earns, so the differential return equals the asset return and the shortcut denominator is legitimate here.

The mechanism sits in two of those rows. Both numerators land within 0.0003 percentage points of each other, while Ether's denominator is 38% larger, and dividing a negative number by a bigger positive number pushes the result toward zero rather than away from it. Zero is the good end of this scale. So the asset that gave back half again as much capital collects a score that reads 29% healthier, purely because it shook harder on the way down.

And the effect is not an artefact of where the window starts. Running the same calculation at every window length from 220 to 300 sessions, the inversion holds at every one of them except 260. Swap the zero benchmark for a 4% annual cash rate and the gap widens rather than narrows, to -0.2354 for Bitcoin against -0.1685 for Ether. The Phemex explainer on Bitcoin volatility covers why both assets sat in negative territory across that stretch. What matters here is that the scoreboard ranked the worse outcome first.

The Israelsen Correction and How to Run It

Craig Israelsen published the fix in the Journal of Asset Management, volume 5 issue 6, in 2005, and it is one line of arithmetic. Raise the denominator to the power of the excess return divided by its own absolute value. When excess return is positive that exponent equals 1 and the standard formula is untouched. When excess return is negative the exponent equals -1, the denominator becomes the reciprocal of the standard deviation, and dividing by a reciprocal is the same as multiplying. You multiply by volatility instead of dividing by it.

Run it on the same two columns and the ordering flips back. Bitcoin scores -6.83% multiplied by 45.99%, which is -0.0314. Ether scores -6.72% multiplied by 63.58%, which is -0.0427. Bitcoin ranks ahead under the corrected form, and that matches what actually happened to the capital.

One caveat that most write-ups of the correction skip. The modified output no longer sits on the same scale as a conventional Sharpe ratio, so a -0.0314 and a +0.85 are not comparable quantities. The corrected form ranks a set of losers against each other correctly, and that is all it claims to do. If your table holds both positive and negative scores, rank the two groups separately rather than sorting the combined column.

Why a Crypto Sharpe Ratio and an Equity Sharpe Ratio Are Different Numbers

Every annualized Sharpe ratio you read is a daily or monthly figure scaled up by the square root of the number of periods, and Sharpe attaches a condition to that scaling in the same 1994 paper. The differential returns have to carry zero serial correlation. He notes that even when the underlying process is clean, a specific historical sample can fail the test on its own.

Crypto runs seven days a week and annualizes with 365. Equities run 252 sessions and annualize with 252. The square root of 365 divided by 252 is 1.2035, so the identical strategy carries a score 20.35% larger in magnitude under the crypto convention. Bitcoin's -0.1484 above becomes -0.1233 under the equity convention, and Ether's -0.1056 becomes -0.0878.

Note the direction, because it runs opposite to the way the shorthand is usually stated. The crypto convention inflates magnitude, so it flatters a winning strategy and punishes a losing one. A crypto fund posting a genuine +0.9 would report +0.75 on an equity desk's convention, while the Bitcoin column above looks 20.35% worse under 365 than under 252. Anyone dropping a crypto fund and an equity fund into one column without restating both on a single convention is comparing numbers that were never on the same scale, which is worth carrying into anything on how big-tech equities and crypto volatility feed each other.

The condition does happen to hold on this sample. Lag-one autocorrelation across the 250 sessions is -0.0204 for Bitcoin and +0.0238 for Ether, both close enough to zero that the square-root scaling is defensible. That will not hold for a strategy with monthly marks, a lending book, or anything holding illiquid positions, where the annualized figure is inflated by smoothing rather than by skill.

What the Formula Never Sees

The numerator is an arithmetic mean, and an arithmetic mean knows nothing about compounding. Bitcoin averaged -0.0187% per session and compounded at -0.0477%. Ether averaged -0.0184% and compounded at -0.0732%. The gap between those two figures is volatility drag, it runs at roughly half the variance, and it works out to 0.0290 percentage points a day for Bitcoin against 0.0548 for Ether. That mechanism alone explains how two assets with effectively identical average daily returns finished 5.48 percentage points apart, and the numerator is blind to every bit of it.

Sharpe names the second gap himself in the closing pages of the paper. His wording is that it is essential to remember that the Sharpe Ratio does not take correlations into account, and the correlation he means is the one between the strategy and everything else the investor already holds. A book already long Bitcoin gains almost nothing from adding a second position 0.9111 correlated to it, however the two ratios rank, which is the same reasoning behind checking how Bitcoin tracks the S&P 500 before sizing anything as a diversifier.

Then comes the sample size problem, and Sharpe supplies the tool for it in the same document. The t-statistic of the mean differential return equals the Sharpe ratio multiplied by the square root of the number of observations. Bitcoin's daily ratio over 250 sessions gives a t-statistic of -0.123 and Ether's gives -0.087. Neither is distinguishable from zero at any confidence level a trader would act on, so the entire ranking argument above rests on two numbers that are statistically noise. That is the correct conclusion, and the ratio alone will never hand it to you.

Leverage deserves a closing note, because traders reach for this measure to justify sizing. Doubling exposure on a daily-rebalanced position scales the mean and the standard deviation by the same factor, so the Sharpe ratio comes out identical while the distance to liquidation halves. The measure is deliberately indifferent to how much of the account you are risking, a property worth holding alongside the mechanics in the Phemex guide to perpetual futures contracts.

Frequently Asked Questions

Can a negative Sharpe ratio still be useful?

The measure is only usable there once the sign correction is applied. A raw negative Sharpe ratio ranks strategies backwards, so the ordering it produces is worse than no ordering at all. Apply the Israelsen exponent, or compare drawdown and total return directly, since neither of those can invert.

What counts as a good Sharpe ratio for a crypto strategy?

Above 1.0 annualized is genuinely strong, and anything above 2.0 in crypto usually means the sample is too short, the returns are smoothed, or a tail risk has not printed yet. Check the t-statistic before believing any of them, because a ratio of 2.0 built on 40 observations carries a t-statistic near 1.3 and proves nothing.

Should I annualize a crypto Sharpe ratio with 365 or 252?

Use 365 for crypto, because the asset genuinely trades 365 days a year and the observation count is real. The rule that matters is restating both figures on one convention before any crypto-versus-equity comparison, since the two differ by a factor of 1.2035.

Does the Sortino ratio solve the negative return problem?

No. Sortino swaps total volatility for downside deviation in the denominator, which is a better risk proxy, but that denominator is still positive and the same sign inversion applies the moment the numerator drops below zero. The exponent correction has to be applied to Sortino too.

Bottom Line

Any risk-adjusted table you build before the next sustained uptrend will rank your losers in reverse unless the sign correction goes in, and that covers every strategy that spent the 250 sessions to the Tuesday 1 September 2026 close underwater. Three checks close the gap. Confirm the denominator is the standard deviation of the differential rather than of the portfolio, apply the excess-over-absolute-excess exponent whenever the numerator is negative, and print the annualization factor beside the number so nobody compares a 365 figure to a 252 one. The t-statistic is the fourth check and the one that will quietly kill most of the rankings, because a ratio built on a few hundred sessions of crypto returns almost never clears significance. A measure that flatters the wildest loser in the room is not measuring risk, and knowing precisely where it breaks is worth more than the number it prints.

This article is for informational purposes only and does not constitute financial or investment advice. Cryptocurrency trading involves substantial risk. Always conduct your own research before making trading decisions.