# Portfolio volatility

Problem True or fale? The stock of a firm has an expected return of 10%, and a volatility of 10%. The weight of the stock in a portfolio is 5%, and the correlation of the stock’s return with the portfolio is 0.5. In that case, the contribution of the stock to the volatility of the portfolio is 0.25%.

Attempt $Var(Portfolio)=Var(aX,bY)=a^2 VarX + b^2 VarY+2ab StDev(X) StDev(Y) Corr(X,Y)$

so $Var(Portfolio)_{contributedByX}=a^2 VarX + +2aStDev(X)Corr(X,Y)$

and we have Corr(X,Y)=0.5 ; VarX=10% ; a=5%.

So I get $Var(Portfolio)_{contributedByX} = 0.050025$ so $Volatility=\sqrt{0.050025}=0.22366...$

Solution True, is the right answer. So 0.22366 must be wrong..

• added _{contributedByX} for clarification – jacob May 22 '15 at 12:44

For a portfolio you have that the variance is: $$\sigma^2 = w \Sigma w$$ Thus the volatility is $\sigma^2/\sigma = w \Sigma w/\sigma$.
Just focusing on one asset with weight $w_i$ and return $r_i$ we get $$\sigma^2 = covar(\sum_{i=1}^n w_i r_i, r_P) = \sum_{i=1}^n w_i covar(r_i, r_P),$$ where $r_p$ is the return of the portfolio, and thus $$\sigma = \sum_{i=1}^n w_i covar(r_i, r_P)/\sigma.$$ Thus the risk contribution to volatility by asset $i$ can be formulated as $$w_i covar(r_i, r_P)/\sigma.$$ Pluggin in your numbers we get (using $covar(r_i,r_p) = \sigma_i \cdot cor \cdot \sigma$ and simplifying $\sigma$ from the above equation ) $$5\% \cdot 10\% \cdot 0.5 = 0.25\%.$$