The risk contribution helps to quantify the impact of an asset on the portfolio volatility. It considers 3 elements:

• the asset’s volatility

• the asset’s weight

• the asset’s correlation with other assets

\[\text{marginal contribution} = \frac{\partial \sigma_p}{\partial \omega}\]

The marginal contribution is by how much the portfolio variance changes if we slightly increase one holding. For more clarity, we can use the following relationship:

\[\Delta \sigma_p = \frac{\partial \sigma_p}{\partial \omega} \Delta \omega\]

Numerical example:

$\sigma_p^{(1)} = 8.3\%$

$\frac{\partial \sigma_p}{\partial \omega} = 7.8\%$

$\omega_1^{(1)} = 33\%$

If $\omega_1^{(2)} = 83\%$, then $\Delta \sigma_p = 7.8\% * (83\%-33\%) = 3.9\%$.

Thus, $\sigma_p^{(2)} = \sigma_p^{(1)} + \Delta \sigma_p \approx 12.1\%$.

Derivation:

$\frac{\partial \sigma_p}{\partial \omega} = \frac{\partial \sqrt{\omega^T \Sigma \omega}}{\partial \omega}$

We know that: $(\sqrt u)’ = \frac{u’}{2 \sqrt{u}}$

Say there are 3 assets,

\[\begin{align*} x^TAx &= ... \\ &= x_1^2a_{11} + x_1x_2(a_{12}+a_{21}) + x_2^2 a_{22} + x_1x_3(a_{13}+ a_{31}) + x_3^2a_{33} + x_2x_3(a_{23}+a_{32}) \\ &= x_1^2a_{11} + x_2^2 a_{22} + x_3^2a_{33} + 2x_1x_2cov_{1,2} + 2x_1x_3cov_{1,3} + 2x_2x_3cov_{2,3} \\ & (\text{because } A \text{ is symmetric (}a_{ij}=a_{ji}\text{)}) \\ \end{align*}\]

$\frac{\partial \sigma_p}{\partial x_1} = 2x_1a_{11} + 2x_2cov_{1,2} + 2x_3 cov_{1,3}$

In general, $\frac{\partial \sigma_p}{\partial x_k} = 2 \sum_{i=1}^n x_i a_{k_i} = 2Ax$

Hence,

\[\frac{\partial \omega^T \Sigma \omega}{\partial \omega} = 2 \Sigma \omega\]

Finally,

\[\frac{\partial \sqrt{\omega^T \Sigma \omega}}{\partial \omega} = \frac{2 \Sigma \omega}{2 \sqrt{\omega^T \Sigma \omega}} = \frac{1}{\sigma_p} \Sigma \omega\]

Note: $\frac{1}{\sigma_p} \Sigma \omega$ is a vector where each component is the risk contribution of the holding.

One can also weight the risk contributions:

\[\omega \odot \frac{1}{\sigma_p} \Sigma \omega\]

That way, when we sum over the vector’s components, \(\sum_i \big( \omega \odot \frac{1}{\sigma_p} \Sigma \omega \big)_i = \sigma_p\)

In general, one asset would strongly contribute to the portfolio volatility if one of the following statement is true:

• it has a high weight

• it has a high volatility

• it is highly correlated to other assets with a high volatility

See this notebook for examples with synthetic data.