The square-root rule of diversification

Question

Spreading capital across many independent bets famously reduces risk. But diversification has limits once the bets move together.

Why does averaging $n$ independent bets reduce risk by a factor of $\sqrt{n}$ , and what happens to that benefit when the bets are correlated?

Accepted Answer

The independent case. Suppose you split evenly across n bets, each with return standard deviation \sigma and each pair uncorrelated. The portfolio return is the average \bar{R} = \frac1n\sum Ri, and the variance of an average of independent terms is \operatorname{Var}(\bar{R}) = \frac{\sigma^2}{n} \implies \operatorname{SD}(\bar{R}) = \frac{\sigma}{\sqrt{n}}. Risk falls like 1/\sqrt{n}, the same rate as the CLT: the up-and-down luck of individual bets partially cancels, and cancellation improves with the square root of the count. Ten independent bets cut volatility by about \sqrt{10} \approx …

The square-root rule of diversification

Your answer

More Statistics questions