Euler Risk Contributions for Portfolio Variance
Summary
The document explains how to allocate portfolio variance across assets. It first questions a proposed allocation that adds covariance terms directly to each asset’s weighted variance, then presents a derivative-based approach associated with Euler allocation. For asset k, its contribution is its weight multiplied by the weighted sum of its covariances with every asset, including its own variance. The factor of one half in the derivative expression accounts for variance appearing symmetrically in the covariance matrix.
The stated check is that summing these contributions across assets recovers total portfolio variance, making the allocation additive and mathematically consistent. The answer also notes that risk allocation is a choice among several defensible conventions rather than a uniquely correct definition. The exposition is limited to variance and assumes symmetric covariances; it does not compare alternative risk measures or discuss estimation uncertainty, constraints, or practical portfolio implementation.
Key ideas
- Risk allocation is not uniquely defined, though coherent methods can be used.
- Euler allocation defines an asset's variance contribution through the portfolio variance derivative with respect to its weight.
- An asset's contribution equals its weight times its covariance-weighted exposure to the portfolio.
- The contributions sum to total portfolio variance when covariance symmetry holds.
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Full text
# Risk contribution of part of a portfolio
# Risk contribution of part of a portfolio
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?
## Answer by user9403 (score 1)
https://quant.stackexchange.com/a/30846
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.
## Answer by RRG (score 1)
https://quant.stackexchange.com/a/31779
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 $$ \sum_k \sigma_{pk}^2 = \sum_{kj}w_k\sigma_{kj}w_j = \sigma_p^2 $$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.