Chance a 100-day average return looks great

Question

A trading strategy has a true mean daily return of $0$ with daily standard deviation $\sigma = 2\%$ . Daily returns are independent, and over $100$ days the central limit theorem makes the average roughly normal.

Over $n = 100$ days, what is the probability the average daily return exceeds $+0.4\%$ ?

Accepted Answer

By the central limit theorem, the average of many independent daily returns is approximately normal, centered at the true mean (0) but with a shrunken spread. The spread of the average is the standard error: 	ext{SE} = \frac{\sigma}{\sqrt{n}} = \frac{2\%}{\sqrt{100}} = \frac{2\%}{10} = 0.2\%. Averaging over 100 days pulls the spread in by a factor of \sqrt{100} = 10. Now measure the +0.4\% threshold in units of that standard error. This is the z-score: z = \frac{0.4\% - 0}{0.2\%} = 2. From the normal table, the chance of landing more than 2 standard errors above the mean is P(Z 2) \approx 0.…

Chance a 100-day average return looks great

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