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Is it quantitatively sound to say that if I have assets $x, y,$ and $z$ in a portfolio, and that the total variance of the portfolio is defined as

$\sigma_p ^2 = w_x^2\sigma_x^2 + w_y^2\sigma_y^2 +w_y^2w\sigma_y^2 + 2w_xw_y\sigma_{xy} + 2w_yw_z\sigma_{yz} + 2w_xw_z\sigma_{xz}$

that the individual risk contribution of each individual asset is:

${\sigma_p}_x^2 = w_x^2\sigma_x^2 + \sigma_{xy} + \sigma_{xz}$

${\sigma_p}_y^2 = w_y^2\sigma_y^2 + \sigma_{xy} + \sigma_{yz}$

${\sigma_p}_z^2 = w_z^2\sigma_z^2 + \sigma_{xz} + \sigma_{yz}$

Is it mathematically sound to assume that the risk of the portfolio is the sum of the risk of each asset with respect to the other assets? Or is the portfolio risk not something that can broken down in a defined way?

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  • $\begingroup$ You have a typo $\sigma_p ^2 = w_x^2\sigma_x^2 + w_y^2\sigma_y^2 +w_y^2\sigma_y^2 + 2 w_x w_y \sigma_{xy} + 2 w_y w_z \sigma_{yz} + 2w_x w_z \sigma_{xz}$ $\endgroup$ – noob2 Nov 2 '16 at 23:06
  • $\begingroup$ When this correction is carried through your risk decomposition is just fine, Sometimes it is written $\frac{\sigma_{px}}{\sigma}+\frac{\sigma_{py}}{\sigma}+\frac{\sigma_{pz}}{\sigma}=1$ sometimes called "risks add up" (Qian). $\endgroup$ – noob2 Nov 3 '16 at 12:42
  • $\begingroup$ Got it. So the risk of the individual asset would rather be $\sigma_{px}^2 = w_x^2 \sigma_x^2 + w_x w_y \sigma_{xy} + w_x w_z \sigma_{xz}$? $\endgroup$ – milkmotel Nov 4 '16 at 14:24
  • $\begingroup$ Yes. Then $\sigma^2_{px}+\sigma^2_{py}+\sigma^2_{pz}=\sigma^2_{p}$ [correcting my own typo]. $\endgroup$ – noob2 Nov 4 '16 at 15:04
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It depends. I don't believe there is any "right" way to allocate risk to asstets in a portfolio, though there are several "good" ways to do so. One I'm partial to is the Euler's method which essentially defines the risk contribution of an asset to the portfolio is the derivative of the portfolio with respect to a change in the asset, as described in Tasche's paper. In the case of variance then this becomes a fairly straightforward calculation. BTW, you have some typos in your portfolio variance calculation.

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Defining the risk of the portfolio as its variance given by $$ \sigma_p^2 = \sum_{ij}w_i\sigma_{ij}w_j $$ where $w_i$ is the portfolio weight of asset $i$ and $\sigma_{ij}$ is the covariance between asset $i$ and $j$. The risk contribution of asset $k$ to the portfolio variance is $$ \sigma_{pk}^2 = \frac{w_k}{2}\frac{\partial \sigma_p^2}{\partial w_k} = w_k\sum_j\sigma_{kj}w_j $$ where we have used the fact that $\sigma_{ij} = \sigma_{ji}$. Note that $\sigma_{kk} = \sigma_k^2$. It is easy to see that the sum of the individual risk contributions add up to the portfolio variance as $$ \sum_k \sigma_{pk}^2 = \sum_{kj}w_k\sigma_{kj}w_j = \sigma_p^2 $$

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