The Sharpe Ratio, and What It Quietly Assumes

One division, and what the denominator is really counting

The ratio is (Rp − Rf) ÷ σp. The numerator is the return you earned above what a risk-free instrument would have paid — the part attributable to taking risk at all. The denominator is the standard deviation of the same return series over the same period. The quotient has a clear reading: units of excess return per unit of volatility. A fund returning 1.2167 percent a month against a 0.15 percent monthly risk-free rate, with a monthly standard deviation of 2.1595 percent, scores (1.2167 − 0.15) ÷ 2.1595 = 0.4939.

Everything that follows comes from the choice of denominator. Standard deviation measures dispersion about the mean without regard to sign: an observation 4 points above the average and an observation 4 points below contribute exactly the same squared term. A manager who beats her own average by a lot is penalised for it as heavily as one who misses it by the same amount. That is not a flaw in the arithmetic — it is what standard deviation is — but it means the ratio answers a narrower question than the one most readers think they are asking. It does not measure risk. It measures variability, and treats variability as a proxy for risk.

William Sharpe himself was careful about this. His 1966 paper called the measure a reward-to-variability ratio, and his 1994 revisit spells out that the denominator is the standard deviation of the differential return and that the whole construction is an ex-ante concept applied, out of necessity, to ex-post data. The name that stuck is less careful than the papers were.

Two funds, one Sharpe ratio: the mirror construction

Here is a twelve-month series for Fund A, in percent: 0.4, 0.3, 0.5, −0.2, 6.2, 0.3, 0.5, 0.4, −0.1, 5.4, 0.4, 0.5. Its mean is 1.2167 and its sample standard deviation is 2.1595. Now build Fund B by reflecting every observation about A's mean — Bᵢ = 2μ − Aᵢ — which gives 2.03, 2.13, 1.93, 2.63, −3.77, 2.13, 1.93, 2.03, 2.53, −2.97, 2.03, 1.93. Reflection about the mean leaves the mean where it was and leaves every deviation the same size, so B has mean 1.2167 and standard deviation 2.1595 too. The Sharpe ratio is therefore identical: 0.4939 for both, not approximately but exactly, and it would stay identical at any risk-free rate you cared to use.

The two funds are not remotely equivalent. A's worst month is −0.2 percent and its deepest drawdown is 0.2 percent; B's worst month is −3.77 percent and its deepest drawdown is 3.77 percent, nineteen times as large. A's dispersion sits almost entirely on the upside — two outstanding months of 6.2 and 5.4 — while B's sits on the downside. Both compound to about the same place over the year, 15.34 percent for A and 15.32 for B. Anyone offered the choice takes A, and no calculation is needed to see why.

The Sortino ratio separates them, because it changes only the denominator. Keep the same numerator — mean return minus a minimum acceptable return, here the same 0.15 percent risk-free rate — and replace the standard deviation with the downside deviation: the root mean square of min(rᵢ − MAR, 0), which is zero for every month at or above the threshold. Fund A's downside deviation is 0.1242 and its Sortino ratio is 8.59. Fund B's downside deviation is 1.4449 and its Sortino ratio is 0.74. The ranking that the Sharpe ratio could not express — one fund is 11.6 times better — is exactly what the change of denominator recovers.

One caution on Sortino: there is no single convention for the divisor inside the downside deviation. Dividing the sum of squared shortfalls by the total number of observations, as done here, is the usual choice and keeps the measure comparable across series; dividing only by the number of losing periods gives a different, larger number and a different ranking. A Sortino ratio without its convention stated is as slippery as a Sharpe ratio without its frequency stated.

The square root of twelve is a choice, not arithmetic

Sharpe ratios are quoted annually, but computed from monthly or daily observations. The bridge everyone uses is to multiply the period ratio by the square root of the number of periods in a year: √12 = 3.4641 for monthly data, √252 ≈ 15.87 for daily. That factor comes out of the variance of a sum. If twelve monthly returns are independent, the variance of their sum is twelve times the variance of one, so the standard deviation grows by √12 while the mean grows by 12 — and the ratio of the two grows by 12 ÷ √12 = √12. The whole convention rests on that first word: independent.

Returns are frequently not independent. Illiquid or infrequently priced holdings — private credit, property, structured positions, anything marked with a lag — produce return series that are smoothed, in which a good month tends to be followed by another good month. That is positive serial correlation, and it deflates the measured standard deviation of the short-period returns without deflating the true annual one. Andrew Lo's 2002 paper gives the correction: the honest annualisation factor for q periods is q ÷ √(q + 2·Σₖ₌₁^(q−1)(q − k)·ρₖ), which collapses to √q only when every autocorrelation ρₖ is zero.

Put a plausible number in. Assume a first-order autocorrelation of 0.2 with the usual geometric decay — a level of smoothing that is unremarkable in a fund holding hard-to-price assets. The corrected factor for twelve months is 2.879 instead of 3.464. Our monthly Sharpe ratio of 0.4939 annualises to 1.422 rather than the 1.711 that goes on the factsheet: the published figure is 20.3 percent too high. At an autocorrelation of 0.3 the overstatement is 32.5 percent, and at 0.5 it is 63.3 percent. Note that the correction runs the other way for mean-reverting returns: at ρ = −0.2 the √12 convention understates by 16.9 percent. The table above sweeps the whole range.

Selling insurance produces a beautiful Sharpe ratio

Consider a strategy whose payoff is structurally that of an insurer: it collects a small, reliable premium most of the time and pays a large claim rarely. Selling far out-of-the-money options, carry trades, and many strategies described as absolute return share that shape. Model it as 35 months returning about 0.90 percent each, with tiny variation, and then one month at −22 percent.

Over the first 35 months the mean monthly return is 0.90 percent and the monthly standard deviation is 0.0574 percent, because nothing ever happens. The monthly Sharpe ratio is (0.90 − 0.15) ÷ 0.0574 = 13.07, which annualises under the √12 convention to 45.3. That number is absurd and it is arithmetically correct. It is the exact output of the exact formula on the exact data, and a three-year track record is long enough for most allocators to take a serious look.

Add the 36th month. The mean falls to 0.2639 percent, the standard deviation jumps to 3.8171 percent, and the monthly Sharpe ratio becomes 0.030 — an annualised 0.10. One observation moved the score from 45.3 to 0.10. The distribution now has a skewness of −5.75 and an excess kurtosis of 31.0, and standard deviation is simply not designed to describe a shape like that: it summarises a distribution with two numbers, and this one is not remotely captured by two numbers. Notice also what the cumulative return did — 36.83 percent over 35 months, 6.73 percent over 36 — and that recovering the −22 percent month at 0.90 percent a month takes another 28 months.

The general lesson is not that the ratio lies. It is that any measure built on the first two moments of a sample cannot see a risk that has not yet shown up in the sample. If the loss is rare enough, a track record of any realistic length will usually not contain one, and the ratio will read the absence of the loss as evidence of safety. Ask what the strategy's worst plausible month looks like, not what its historical standard deviation was.

What a Sharpe ratio must state before it means anything

A bare number is uninterpretable. Three things have to travel with it. First, the period: 0.4939 over twelve months and 0.4939 over ten years are different claims, and a short window is dominated by luck — Lo's paper is largely about how wide the confidence interval around an estimated Sharpe ratio is. Second, the risk-free rate used: the same series scores differently against a 0.15 percent monthly rate and a 0.35 percent one, and the gap widens as the strategy's own return falls. Third, the observation frequency and the annualisation method, because as the table shows the same underlying performance can be published as 1.048 or as 2.060 depending on what was assumed.

Two further conditions are structural rather than presentational. The ratio is only comparable across two things measured over the same window, because market conditions dominate the numerator; and it is only comparable at all between strategies whose return distributions have broadly the same shape. Ranking a long-only equity fund against an options-selling programme on Sharpe alone compares a roughly symmetric distribution with a heavily left-skewed one, and the measure has no way to register the difference.

What the ratio is still genuinely good for

None of this makes the Sharpe ratio useless. It does one job extremely well: comparing two portfolios of similar character over the same window, where the question is whether the extra return one of them produced was worth the extra variability. That is the question Sharpe posed in 1966, and for that question the ratio is the right tool. It also has a clean theoretical role — it is the slope of the line from the risk-free rate to a portfolio in mean-standard-deviation space, so the highest Sharpe ratio identifies the portfolio you would want to lever up or down to reach any risk level you like.

The practical protocol is short. Compute it, and then compute the Sortino ratio alongside it — the two together tell you whether the volatility being penalised is the kind you mind. Look at the worst single period and the deepest drawdown, which no ratio contains. State the window, the risk-free rate and the frequency whenever you quote a figure, and treat any Sharpe ratio above about 2 on a live strategy as a question rather than an answer.

Frequently asked questions

What counts as a good Sharpe ratio?

There is no threshold that means anything on its own, because the number depends on the window, the risk-free rate, the observation frequency and the annualisation assumption. The rules of thumb you will see — 1 is acceptable, 2 is good, 3 is excellent — are conventions about annualised figures from broadly symmetric return distributions, and they break immediately outside those conditions. Our worked example makes the point: the same monthly Sharpe ratio of 0.4939 publishes as anything from 1.048 to 2.060 depending only on what was assumed about serial correlation. And a very high number is more often a warning than a compliment: the insurance-selling example in this article scored an annualised 45.3 over 35 months and 0.10 over 36.

Should I use the Sortino ratio instead?

Alongside, not instead. Sortino replaces the standard deviation with the downside deviation, so it stops penalising upside dispersion — which is exactly the fix the mirror example needs: 8.59 against 0.74 where Sharpe saw 0.4939 against 0.4939. But it inherits every other weakness, including the annualisation problem and the blindness to a loss that has not happened yet, and it adds one of its own: the divisor convention inside the downside deviation is not standardised, so two providers can publish different Sortino ratios for the same series. Compute both, state your conventions, and read the drawdown as well.

Which risk-free rate should go in the numerator?

One whose maturity and currency match the measurement period and the portfolio's currency — a short government bill yield for a monthly series, converted to a monthly figure. Sharpe's own 1994 note is stricter than most practice: the correct construction is a differential return, the portfolio minus the risk-free asset period by period, and the standard deviation belongs to that difference rather than to the portfolio alone. Where the risk-free rate is essentially constant the two coincide; where it moves a lot, they do not. The practical consequences are: never mix currencies, never use a long bond yield for a short measurement window, and always state which rate you used, since the same series scores differently at 0.15 percent a month and at 0.35 percent.

Can I compare a Sharpe ratio computed on daily data with one computed on monthly data?

Not safely, even after both have been annualised. The annualisation only makes them comparable if returns are independent at both frequencies, and for most real series they are not: daily data picks up short-horizon noise and any day-to-day mean reversion, while monthly data smooths both away. Whichever frequency produces the lower measured standard deviation produces the higher Sharpe ratio, and there is no rule saying which that will be. Compare like with like — same frequency, same window, same risk-free series, same annualisation method — or do not compare. If a factsheet does not say which frequency was used, the ratio on it cannot be checked.

Why does a smoothed, illiquid fund show a higher Sharpe ratio?

Because the denominator is measured on reported returns, and reported returns for hard-to-price assets are stale. When a holding is marked with a lag or by appraisal rather than by trade, a move in its true value is spread across several reporting periods. That spreading raises the correlation between consecutive periods and lowers the measured period-by-period standard deviation, while the true annual variability is unchanged. The √12 convention then treats the artificially small monthly figure as if it came from independent draws, and multiplies the error into the annual number. At a monthly autocorrelation of 0.2 the published ratio is 20.3 percent too high; at 0.3, 32.5 percent. Positive serial correlation in a monthly return series is worth checking before comparing that fund with a daily-priced one.

Does a negative Sharpe ratio mean anything?

It means the portfolio returned less than the risk-free rate over the window, and that is all it means. The magnitude is not interpretable, because dividing a negative numerator by the standard deviation inverts the usual ranking: of two losing portfolios with the same shortfall, the more volatile one scores closer to zero and therefore looks better. A fund that lost 2 percent against the risk-free rate with a 4 percent standard deviation scores −0.50, while one that lost the same 2 percent with an 8 percent standard deviation scores −0.25 and appears superior. It is not. When the numerator is negative, stop reading the ratio and read the return and the drawdown directly.