A Sharpe ratio from 26 weekly returns in a spreadsheet, and how wide its error is

[Fairman]

A Sharpe ratio from 26 weekly returns in a spreadsheet, and how wide its error is

Put 26 weekly returns, in percent, into a column. Suppose the average is 0.45% and the sample standard deviation is 1.8%.

Weekly Sharpe (ignoring the risk free rate): 0.45 / 1.8 = 0.25.
Annualised: multiply by the square root of 52, which is 7.21, so 0.25 x 7.21 = 1.80.

In cells: =AVERAGE(A1:A26)/STDEV.S(A1:A26)*SQRT(52).

1.8 sounds like a very good ratio. But how precise is it with 26 observations? A standard approximation for the error of a Sharpe ratio is the square root of (1 + 0.5 x SR squared) divided by the number of observations, in the same units. Weekly: sqrt((1 + 0.5 x 0.0625) / 26) = 0.199. Annualised: 0.199 x 7.21 = 1.44.

A 95% range is therefore 1.80 plus or minus 1.96 x 1.44, which runs from -1.0 to +4.6.

So the data cannot distinguish this account from a losing one, and equally cannot rule out an excellent one. Half a year of results is too short for the ratio to say much.

The risk free rate matters a little as well. Subtract an illustrative 4% a year, which is 0.077% a week, and the weekly mean becomes 0.373%, the weekly ratio 0.207 and the annual one 1.49.

[FXS HongKong Scalper]

A great reminder about the error. With only 26 observations, the confidence interval is wide, and a Sharpe of 1.8 could easily be closer to zero in reality. Patience and more data are needed before drawing strong conclusions.

[Fairman]

How many weeks does it take? To bring the 95% half width down to 0.5 on an annualised Sharpe near 1.8, the standard error must be about 0.255, which is 0.0354 in weekly units. Then n = (1 + 0.5 x 0.0625) / 0.0354 squared, roughly 824 weeks, or about 16 years. Even for a half width of 1.0 you need around 206 weeks, nearly 4 years. That is for returns that behave like independent draws, which they rarely do. The practical lesson is that Sharpe ratios from a few months of a retail journal are weak evidence either way, and a ratio from a longer record deserves more trust than a higher one from a shorter record.

[Fairman]

The ratio can look fine for the wrong reasons. Take a strategy that makes +0.6% in 95% of weeks and -6% in 5% of weeks. Its average is +0.27% a week and its standard deviation 1.44%, giving an annualised Sharpe near 1.35. That looks respectable. But there is a 0.95 to the 26th power, 26%, chance that a 26 week window contains no losing week at all. In such a window the standard deviation is zero and the Sharpe is undefined or enormous. Strategies that sell rare risk for steady small gains look best just before the event. The ratio uses only the first two moments of the returns, and misses what is in the tail.

[Fairman]

Calmar or the simple ratio of return to maximum drawdown makes a useful companion. Suppose the annual return is 18% and the maximum drawdown during the year was 9%. Then return over drawdown is 2.0. It uses a different part of the record, the worst stretch instead of the average swing, and it is easier to explain: how much did the account make for each unit of pain. Like the Sharpe ratio it needs a long record to mean much, because the maximum drawdown depends on the path and gets larger as the record gets longer. Report both, and look at the drawdown date in the journal.

[Fairman]

If the journal records trades and not weekly returns, build the weekly series first. Sum the trade results by week, divide by the equity at the start of the week to get a percentage, and skip weeks with no trades only if the strategy genuinely sits out, otherwise include them as zeros. A pivot table by week number does the sum in a few clicks. Use the week of the trade's close for consistency. For a daily series use the square root of 252 for annualising, which gives a bigger multiplier of 15.9, so the same account shows a different looking ratio depending on the frequency, unless returns are independent.