# 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?

• So, you want to calculate the variance of a specific portfolio subset (= sector), then why would you need the correlations of assets in the other sectors? All you need is the same inputs as if you calculate portfolio variance (asset weights (need to sum up to one within the sector), asset variance, and pairwise asset correlations within the same sector). – Matt Aug 17 '13 at 12:23
• This question is not clear to me. If you want to calculate the volatility of the sector portfolio why is the correlation of the portfolio's constituents with other sector's constituents relevant? – Jase Aug 18 '13 at 12:35
• I bring up correlation only in case it was relevant within someone's idea of an answer. I ask the question because I don't know. – strimp099 Aug 20 '13 at 19:11

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.

• Yes, still working on the risk budgets problem... this makes a lot of sense within the construct of the various contribution metrics we have. I'll give it a go, thanks. – strimp099 Aug 20 '13 at 19:08