How long to trust a Sharpe of 1
A strategy's daily returns are independent and identically distributed, and its true annualised Sharpe ratio is 1 (mean return divided by volatility, both annualised). A year has 252 trading days.
You test whether its mean daily return is positive with a t-statistic computed from years of daily returns. Ignoring the noise in the estimated mean and volatility, how many years of data make the t-statistic equal to 2?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
With observations of mean and standard deviation , the t-statistic of the mean is about .
The annualised Sharpe ratio is the daily one times .
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. The t-statistic of a mean grows with the square root of the number of observations, and so does the annualised Sharpe ratio. Put together, the t-statistic is the annual Sharpe ratio times the square root of the number of years.
Step 1. With daily returns of mean and standard deviation ,
Step 2. The annualised Sharpe ratio is , so
Step 3. With , needs , so years.
The lesson. A Sharpe ratio of 1 is good for a strategy, yet it takes four years of data just to reach a t-statistic of 2. A Sharpe ratio of 0.5 would need 16 years. This is why backtests are so easy to fool yourself with: a few years of data cannot tell a modest real edge from luck.