How do you compute a Sharpe ratio without fooling yourself?

The Sharpe ratio is the most quoted number in performance measurement and the most frequently miscomputed. The formula takes one line; the traps take a page.

The formula

Sharpe ratio = (portfolio return − risk-free rate) ÷ volatility of the portfolio

In words: how much return you earned above cash, per unit of volatility endured. The subtraction matters. If cash paid 5%, a portfolio returning 5% earned nothing for its trouble, and its Sharpe should be zero regardless of how smooth the ride was.

The simple annual version

A portfolio returned 11% over a year. Cash paid 3%. Its annualised volatility was 16%.

(11% − 3%) ÷ 16% = 8 ÷ 16 = 0.50

Half a point of excess return for every point of volatility.

Doing it properly from a return series

Real computation starts from daily or monthly returns, and annualisation is where people go wrong. The rule: mean excess return scales with time, volatility scales with the square root of time.

From monthly data:

• Annualised excess return = mean monthly excess return × 12
• Annualised volatility = monthly standard deviation × √12
• Sharpe = the first ÷ the second — which is identical to monthly Sharpe × √12

Worked example. Sixty monthly returns with a mean of 0.9%, a mean monthly cash rate of 0.2%, and a monthly standard deviation of 3.5%.

• Monthly excess return: 0.9% − 0.2% = 0.7%
• Monthly Sharpe: 0.7 ÷ 3.5 = 0.20
• Annualised: 0.20 × √12 = 0.20 × 3.464 = 0.69

Cross-check the long way: 0.7% × 12 = 8.4% annualised excess; 3.5% × √12 = 12.1% annualised volatility; 8.4 ÷ 12.1 = 0.69. Same answer, as it must be.

From daily data the constant is 252 (trading days), so multiply the daily Sharpe by √252 ≈ 15.87. Annualising a daily volatility by multiplying by 252 instead of √252 inflates it roughly sixteen-fold and is the single most common arithmetic error in this whole course.

State which return the numerator is. Mean monthly excess return times 12 is an arithmetic annual figure. A compound annual growth rate is geometric and sits lower by about half the variance. At 16% volatility the gap is 0.16 squared over 2, about 1.3 points a year, which moves a Sharpe ratio by roughly 0.08. Two reports on one fund, built each way, rank it differently against the same peer. Academic papers and most risk vendors use the arithmetic form; fund factsheets mostly quote a CAGR. Read the footnote and use one convention across every fund in a comparison. A period shorter than a year is never annualised: six months at +6% is reported as +6%.

Four things Sharpe cannot see

1. It treats upside and downside alike. A portfolio that occasionally leaps 15% is punished exactly as much as one that occasionally falls 15%. The next lesson exists because of this.
2. It assumes volatility captures risk. Strategies that quietly sell insurance — writing options, carry trades, illiquid credit — produce long, calm stretches of small gains and a rare enormous loss. Their Sharpe looks magnificent right up until the event it was blind to.
3. It is period-dependent and noisy. Computed over one or two years, the number is dominated by sampling error. Over such windows, ranking managers by Sharpe ranks luck.
4. Smoothed prices flatter it. Assets marked infrequently or by appraisal show artificially low volatility, which lands directly in the denominator.

For orientation, broad equity indices have historically shown long-run Sharpe ratios roughly in the 0.3–0.5 region. A sustained figure far above 1 is a prompt to understand how it was produced, not a reason to admire it — and never, in any case, a signal to buy.

In the data

The risk-free leg is usually a Treasury bill rate, and every bill is quoted two ways. The latest day for the 13-week and the 52-week bill:

The discount rate is quoted on a bank-discount basis and is not a yield that can be subtracted from a compounded return; it always sits a little below the coupon-equivalent figure, which is the yield. Use the second. Both are annual percentages, so the frequency conversion above still has to happen after the right one is chosen.

Try it now

1. The window is the S&P 500 fund from 3 January 2023, below. Nine hundred daily returns are not a hand calculation, so here is the arithmetic done for you on 28 September 2026: daily percentage returns from adjusted closes, then mean × 252 and standard deviation × √252, for the whole window and for its last twelve months.

2. Check the annualisation on the first row: recover the daily standard deviation from 14.95%. Then say what the volatility would have read had someone multiplied the daily figure by 252 instead of √252.
3. Subtract a cash rate from each annualised return — the coupon-equivalent bill rate above, not the discount rate, for the reason given there — then divide by annualised volatility, and you have the Sharpe ratio for each window. Note how much it moves between the two — that movement is the noise problem, made visible. Say too which way using one recent bill rate biases the longer window, which began when bills paid more than they do now.