Spread of a 25-day cumulative return

Question

Daily returns are independent and identically distributed with a standard deviation of 2% per day (variance 4 in percent-squared units). Consider the 25-day cumulative return, which is the sum of 25 daily returns.

What is the standard deviation of the 25-day cumulative return?

Accepted Answer

For a sum of n independent pieces each with variance \sigma^2, the variances add: \operatorname{Var}(	ext{sum}) = n\sigma^2, \qquad 	ext{sd(sum)} = \sigma\sqrt{n}. Spread grows with the square root of the number of days, not linearly. Here: \operatorname{Var} = 25 	imes 4 = 100, \qquad 	ext{sd} = \sqrt{100} = 10\%. Contrast this with the average daily return over the 25 days, whose spread shrinks: 	ext{sd} = \sigma/\sqrt{n} = 2/\sqrt{25} = 0.4\%. Summing makes spread grow like \sqrt{n}; averaging makes it shrink like 1/\sqrt{n}. Takeaway. Independent pieces add variances, so a sum's sd s…

Spread of a 25-day cumulative return

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