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I have three assets and a covariance matrix.

How do I calculate the correlation of asset X with a portfolio that includes assets Y and Z?

For example, assume I want to calculate the correlation of high yield bonds to a portfolio that includes 60% equities and 40% U.S. Treasuries.

Thanks.

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closed as off-topic by LocalVolatility, noob2, lehalle, zer0hedge, amdopt Jun 12 '17 at 15:29

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  • $\begingroup$ Try to expand $\rho(x,\lambda y + (1-\lambda) z)$ using the definition and the known properties of correlation. $\endgroup$ – noob2 Jun 9 '17 at 15:00
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This is fairly basic but anyway...

Let us define:

  • $X$, $Y$ and $Z$ your assets $-$ without any loss of generality, assume $X$ designates high yield bonds, $Y$ equities and $Z$ US Treasuries $-$ and $P$ your portfolio;
  • $\sigma_X$, $\sigma_Y$ and $\sigma_Z$ their respective standard deviation, and $\sigma_P$ the standard deviation of the portfolio;
  • $\sigma_{X,Y}$, $\sigma_{X,Z}$ and $\sigma_{Y,Z}$ their pairwise covariances;
  • $w$ the proportion of equities in your portfolio $-$ so $60\%$ in your example.

Your portfolio $P$ can be written as:

$$ P = wY+(1-w)Z $$

You are interested in computing the correlation $\mathbb{Corr}(X,P)$ between your portfolio and high yield bonds. By the properties of variance and covariance, we obtain:

$$ \begin{align} \mathbb{Corr}(X,P) & = \frac{\mathbb{Cov}(X,wY+(1-w)Z)}{\sigma_X\sigma_P} \\[9pt] & = \frac{w\,\sigma_{X,Y}+(1-w)\sigma_{X,Z}}{\sigma_X\sqrt{(w^2\sigma_Y^2+(1-w)^2\sigma_Z^2+2w(1-w)\sigma_{Y,Z})}} \end{align} $$

Your are left with $\sigma_X$, $\sigma_Y$, $\sigma_Z$, $\sigma_{X,Y}$, $\sigma_{X,Z}$ and $\sigma_{Y,Z}$, which are the elements of your variance-covariance matrix, and $w$ which you set yourself.

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    $\begingroup$ Thanks! The only comment/edit would be to make the Y and Z standard deviation terms in the lower denominator variance terms instead, correct? $\endgroup$ – Parallax Jun 9 '17 at 17:37
  • $\begingroup$ Indeed, my mistake. $\endgroup$ – Daneel Olivaw Jun 9 '17 at 18:46

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