Volatility of investment: Joint-Volatility of mutually independent variables
The variances of the sum of independent random variables are additive. Therefore, if X and Y are independent, Var(X+Y)= Var(X) + Var(Y). The standard deviation is defined as $\sigma_z = \sqrt{Var(z)}$. Therefore, if X and Y are independent, $\sigma_{X+Y} = \sqrt{Var(X+Y)} = \sqrt{Var(X) + Var(Y)}$. This gives the result above.