Calculating an Asset’s Contribution to Portfolio Volatility

Summary

The document tests whether a stock contributes a stated amount to portfolio volatility when its weight, volatility, and correlation with the portfolio are specified. The questioner starts from the portfolio variance expansion but applies it incorrectly, mixing variance terms and arriving at a portfolio volatility figure that does not answer the contribution question.

The accepted explanation derives an asset’s contribution to portfolio volatility as its portfolio weight multiplied by its covariance with portfolio returns, divided by total portfolio volatility. Since covariance can be expressed using the asset’s volatility, its correlation with the portfolio, and portfolio volatility, the latter cancels. Applying the stated inputs gives the claimed contribution. This is a marginal risk contribution concept, distinct from the asset’s standalone volatility or its share of total variance; the excerpt does not provide the portfolio’s full covariance matrix or explain other risk attribution conventions.

Key ideas

• An asset’s volatility contribution depends on its weight and covariance with the portfolio.
• The contribution formula divides weighted covariance with portfolio returns by portfolio volatility.
• Using correlation and asset volatility makes the portfolio volatility term cancel in the expression.
• The resulting contribution differs from the asset’s standalone volatility and from its direct variance contribution.

Tags

# Portfolio volatility


# 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..

## Answer by Richi Wa (score 2)

https://quant.stackexchange.com/a/17946

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 ) $$ 5\% \cdot 10\% \cdot 0.5 = 0.25\%. $$

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