Attributing Portfolio Volatility to Sectors with Risk Contributions

# How to compute a sector's volatility within a portfolio?


# How to compute a sector's volatility within a portfolio?












Assume I have a large portfolio of equities spread across three sectors.

I am attempting to compute the volatility of these sectors within the portfolio considering the cross correlations among the assets in the other sectors.

Further, how might I compute the relevant cross correlations for use in the volatility measure?

Any resources or papers on the topic? Or perhaps this is an easy problem that I'm just unaware of?

## Answer by SRKX (score 2, accepted)

https://quant.stackexchange.com/a/8784

I perform this kind of analysis using the risk contribution concept.

I understand from this post that your already know about the contributions, but let's just restate the idea here for the sake of completeness.

We have a portfolio of $n$ assets with allocation $w \in \mathbb{R}^n$ and volatility $\sigma_P(w)$.

The marginal risk contribution of asset $i$ is defined as:

$$MRC_i= \frac{\partial \sigma_P(w)}{\partial w_i}$$

The total risk contribution of asset $i$ is defined as:

$$\sigma_i(w) = w_i \cdot MRC_i$$

Finally, note that:

$$\sigma_P(w) = \sum_{i=1}^n \sigma_i(w) $$

(See this canonical paper for the details).

In order to compute how much of the volatility is coming from some sector S, you can just sum the total risk contributions of all assets you consider in the sector.

$$\sigma_S(w)=\sum_{i \in S} \sigma_i(w)$$

This would give you an absolute value.

An interesting way of looking at this is also to compute its relative version by dividing the total risk contribution of the sector by the total volatility of the portfolio which enables you to discuss the percentage of the risk to be attributed to this given sector.