Portfolio diversification

Spreading a portfolio across assets can reduce risk that is not shared by them. For centered asset risks with covariance matrix $\Gamma$, the variance of the weighted risk is $w^T\Gamma w$. Equal weights across $n$ independent risks with common variance $s^2$ give variance $s^2/n$, whereas equal weights across risks that are all the same random variable leave variance $s^2$. Thus small individual weights do not by themselves remove common risk; the covariance criterion for diversification of factor residuals identifies sufficient dependence and weight conditions.