# Correlation of asset to portfolio, given certain variables

Ultimately I'm trying to calculate stdev contribution, but I've hit a hurdle.

What I have:

20x20 correlation matrix for various assets

Standard deviations for each asset

Returns for each asset

Weights corresponding to various portfolios

What I've derived:

Covariance Matrix

Variance/Stdev for each of the portfolios

What I want:

Risk contributions for each asset in each portfolio, but that requires correlation of each asset to each individual portfolio.

So that's my hang up. I can't seem to figure out how to calculate Covar(asset,portfolio) or correl(asset,portfolio).

If $\Sigma$ is the covariance matrix of all assets and $w$ is the column vector of weightings of the asset in a certain portfolio. Then $$w^T \Sigma w = VAR$$ is the variance of the portfolio. The contribution to volatility of asset $i$ is given by $$w_i (\Sigma w)_i/\sqrt{VAR},$$ where $(\Sigma w)_i$ is the $i_{th}$ entry in the vector $\Sigma w$.

Note that $(\Sigma w)_i$ is the covariance of the asset $i$ to the porfolio with weights $w$.

You can read more details in the following working paper and the references therein: http://arxiv.org/abs/1009.3638

Proof: Write $r_p = \sum_{j=1}^n w_j r_j$, where $r_p$ is the portfolio return, then $$cov(r_i,\sum_{j=1}^n w_j r_j) = \sum_{j=1}^n w_j cov(r_i,r_j) = (\Sigma w)_i.$$

• I think thats only true for $\rho=0$ or independent returns, because your formula is issuming that $VAR=\sum VAR_i$. Jul 9 '14 at 16:08
• Yes, in this case there is correlation. So the portfolio variance is the weight vector * covariance matrix * transpose of weight vector
– Matt
Jul 9 '14 at 17:38
• This answer seems right to me. Jul 9 '14 at 22:14
• No guys. I use the whole covariance matrix. Nowhere I assume anything about its shape. Read the paper or text books about risk contributions to volatility or the Euler allocation principle.
– Ric
Jul 10 '14 at 7:03
• @emcor this is wrong, I don't assume this. I use the linearity of covariance which is correct. See the proof.
– Ric
Jul 10 '14 at 7:03