How many years to know a mean return?

Question

A strategy has annual return volatility (standard deviation) of $20\%$ . You want to estimate its true mean annual return to a precision (standard error) of $\pm 2\%$ .

How many years of data do you need, and what does the answer say about estimating expected returns?

Accepted Answer

The standard error of the sample mean is \operatorname{SE}(\bar X) = \frac{\sigma}{\sqrt{n}}. This is not the standard deviation \sigma = 20\%, which describes how much a single year's return varies and never shrinks. The SE describes how precisely you know the average return, and it shrinks like 1/\sqrt{n}. Set the SE to the target 2\% and solve for the number of years n: \frac{20\%}{\sqrt{n}} = 2\% \implies \sqrt{n} = \frac{20}{2} = 10 \implies n = 100 	ext{ years}. You would need a century of returns just to know the mean to within \pm 2\%, and to halve that error to \pm 1\% you would nee…

How many years to know a mean return?

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