Two lines of arithmetic, five choices hidden inside them, and a worked mandate that wins on return and loses on risk-adjusted return.
The Sharpe ratio is the average excess return over cash divided by the standard deviation of those excess returns. On five illustrative years, a mandate that returned 6.16 per cent a year against 5.54 for its index scores 0.41 against the index's 0.45: it won on return and lost on risk-adjusted return, because its excess returns swung with a standard deviation of 8.50 against 6.32.
Worked in full in Asset Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The formula fits on one line, which is why it is quoted so often and checked so rarely. Inside it sit five choices: which risk-free rate, arithmetic or geometric mean, excess or total returns in the denominator, sample or population standard deviation, and how to annualise. Each one moves the answer, and a manager and a client who make them differently can both be right and still disagree by a full tenth.
| Year | Fund | Benchmark | Cash | Fund excess | Benchmark excess |
|---|---|---|---|---|---|
| 1 | 11.6 | 9.4 | 0.5 | 11.1 | 8.9 |
| 2 | −6.2 | −3.8 | 1.5 | −7.7 | −5.3 |
| 3 | 15.1 | 12.2 | 3.0 | 12.1 | 9.2 |
| 4 | 2.4 | 3.3 | 4.5 | −2.1 | −1.2 |
| 5 | 7.9 | 6.6 | 4.0 | 3.9 | 2.6 |
| Arithmetic mean | 6.16 | 5.54 | 2.70 | 3.46 | 2.84 |
The fund is the more aggressive of the two: it beats the index in the three good years and falls further in the bad one. That pattern is exactly what the Sharpe ratio is built to price.
Subtract the cash rate from each year's return, year by year, so that a year of 4.5 per cent cash is not treated like a year of 0.5. Then take the mean and the standard deviation of the excess column.
Sharpe = mean(Rt − Rf,t) ÷ stdev(Rt − Rf,t)
=AVERAGE(E2:E6)/STDEV.S(E2:E6)The benchmark, on the same steps, has a mean excess of 2.84 and a standard deviation of 6.32, so its Sharpe ratio is 0.449, or 0.45.
| Measure | Fund | Benchmark | Difference |
|---|---|---|---|
| Arithmetic mean return | 6.16 | 5.54 | 0.62 |
| Annualised (geometric) return | 5.89 | 5.39 | 0.50 |
| Mean excess over cash | 3.46 | 2.84 | 0.62 |
| Standard deviation of excess | 8.50 | 6.32 | |
| Sharpe ratio | 0.41 | 0.45 | −0.04 |
The client was paid 0.62 a year more than the index, and carried a third more volatility to get it. To match the index's ratio at its own volatility the fund would have needed 0.36 a year more return; to match it at its own return, a standard deviation of 7.70 rather than 8.50. Neither is a verdict on skill: five annual observations are far too few for a Sharpe difference of 0.04 to be statistically distinguishable from zero. What the number does establish is that the outperformance came with leverage to the market, not free of it.
A manager's report and a consultant's report can both state the return correctly and still tell opposite stories. The return table says the mandate won. The Sharpe ratio says the client would have done as well, per unit of risk, holding the index with a little more equity in it.
| Convention | Fund | Benchmark |
|---|---|---|
| Base case: arithmetic mean, excess returns, sample deviation | 0.41 | 0.45 |
| Population standard deviation (divide by n) | 0.46 | 0.50 |
| Deviation of total returns, not excess | 0.41 | 0.46 |
| Geometric return less average cash in the numerator | 0.38 | 0.43 |
| Risk-free rate set to zero | 0.74 | 0.90 |
The ranking survives every convention here, but the level does not: the same fund reads anywhere from 0.38 to 0.74. That is why a Sharpe ratio quoted without its conventions cannot be compared with another one. Setting the risk-free rate to zero is the most flattering and the most common shortcut in marketing material; it turns a ratio of excess return to risk into a ratio of total return to risk, which is a different statistic.
Frequency is the sixth choice. Monthly data give more observations and a better estimate of volatility, and the monthly ratio is annualised by multiplying by the square root of 12: a monthly Sharpe of 0.15 becomes 0.15 × 3.464 = 0.52. Do not annualise the mean and the deviation separately and then forget one of them; the mean scales by 12 and the deviation by the square root of 12, so the ratio scales by the square root of 12 only.
The most frequent error is to subtract one average cash rate from the average return, then divide by the standard deviation of total returns. Here that lands close to the right answer, because cash moved slowly. In a period when the cash rate climbs from near zero to several per cent, it does not: the year-by-year subtraction removes the rise in cash from the volatility, and the shortcut leaves it in.
The second mistake is comparing a Sharpe ratio built from monthly data with one built from annual data, or a gross ratio with a net one. Fees reduce the numerator and leave the denominator almost untouched, so a 0.6 per cent fee on a mandate with an 8.50 deviation costs about 0.07 of Sharpe. Always compare like with like: same period, same frequency, same fee basis, same cash rate.
The third is to treat the ratio as a test of the manager. The Sharpe ratio measures the reward for total risk, which is the right question for a client holding this one portfolio. For a manager judged against an index, the question is the reward for active risk, which is the information ratio. On these numbers the information ratio is positive while the Sharpe comparison is negative, and both are correct.
The free workbook on the Asset Management companion page builds both Sharpe ratios and both information ratios for the book’s own five-year mandate, a different case from the one above, where the manager also beats the index on return and loses to it on Sharpe. For the active-risk view of the same question, see how to calculate tracking error and the information ratio.
There is no universal threshold, because the ratio depends on the period, the asset class and the frequency of the data. The useful test is relative: compare the fund with its own benchmark over the same years, computed the same way. In the worked case the fund's 0.41 trails the index's 0.45, so the manager's extra return of 0.62 a year did not pay for its extra volatility.
Multiply the monthly Sharpe ratio by the square root of 12, which is 3.464. A monthly ratio of 0.15 becomes 0.52 a year. The rule assumes monthly returns are independent; if they are autocorrelated, as smoothed or appraisal-based returns usually are, the scaled figure overstates the true annual ratio.
Sample standard deviation, STDEV.S in Excel, which divides by n minus 1. With five years the choice matters: population standard deviation lowers the fund's volatility from 8.50 to 7.60 and lifts its Sharpe ratio from 0.41 to 0.46. Whichever you use, use the same for the benchmark.
This article is one calculation from Asset Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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