Calculating a correlation from a combination of other correlations?

**vasco** · March 19th 2009, 07:56 AM

My task:

I have an (n,n) correlation matrix. I want to add another variable (which is a combination of two existing variables) and end up with an (n+1,n+1) matrix.

I have estimates of all expected returns and variances.

However, I do not know how to calculate correlation coefficients or covariances for the new variable (or if this is possible).

I am not working from a sample. These figures are estimates of long-term asset returns. The new variable could be thought of as a portfolio containing x% of asset i and (100-x)% of asset j - so I have already calculated it's expected return and variance.

Any help would be much appreciated.

**meymathis** · March 21st 2009, 01:55 PM

I am assuming that we are defining the new variable as a linear combination of 2 "old" ones.

So let's say we have $X_1,\ldots,X_n$ , and we define

$X_{n+1}=AX_a+BX_b$ where A,B are constants, and a,b are the indexes of two of the X's that you are going to combine.

So the correlation coef of this new one versus any old one is:

$\rho_{X_i,X_{n+1}} = \frac{E[(X_i-\mu_{X_i})(X_{n+1}-\mu_{X_{n+1}})]}{ \sigma_{X_i}\sigma_{X_{n+1}}}$

Let's reduce this to a function of the known variables.

$= \frac{E[(X_i-\mu_{X_i})(AX_a+BX_b- A\mu_{X_a}-B\mu_{X_b)}]}{ \sigma_{X_i}\sigma_{X_{n+1}}}$

$= \frac{E[(X_i-\mu_{X_i})(AX_a-A\mu_{X_a})]+ E[(X_i-\mu_{X_i})(BX_b-B\mu_{X_b)}]}{ \sigma_{X_i}\sigma_{X_{n+1}}}$

$= \frac{AE[(X_i-\mu_{X_i})(X_a-\mu_{X_a})]+ BE[(X_i-\mu_{X_i})(X_b-\mu_{X_b)}]}{ \sigma_{X_i}\sigma_{X_{n+1}}}$

$= \frac{A\rho_{X_i,X_a}\sigma_{X_i}\sigma_{X_a}+ B\rho_{X_i,X_b}\sigma_{X_i}\sigma_{X_b}}{ \sigma_{X_i}\sigma_{X_{n+1}}}$

$= \frac{A\rho_{X_i,X_a}\sigma_{X_a}+ B\rho_{X_i,X_b}\sigma_{X_b}}{ \sigma_{X_{n+1}}}$

Finally we know $\sigma_{X_{n+1}}= \sqrt{A^2\sigma^2_{X_a}+B^2\sigma^2_{X_b} + 2AB\rho_{X_a,X_b}\sigma_{X_a}\sigma_{X_b}}$

And so we get:

$\rho_{X_i,X_{n+1}} = \frac{A\rho_{X_i,X_a}\sigma_{X_a}+ B\rho_{X_i,X_b}\sigma_{X_b}}{ \sqrt{A^2\sigma^2_{X_a}+B^2\sigma^2_{X_b}+ 2AB\rho_{X_a,X_b}\sigma_{X_a}\sigma_{X_b}}}$

Now you said you have estimates of all these parameters. I can't say how good your corr corf will be, if you stick your estimates into the formula. And looking back at this, maybe I should have started with the sample correlation coefficient formula, rather than with the population. So maybe you might want to rederive what I did using the same version. I think just trying to do the arithmetic in my head that you will get the same formula except that the population parameters will be replaced with the sample ones (the sample corr coeff just has an (n-1) instead of an nin the denominator which will cancel out ).

**vasco** · March 23rd 2009, 08:17 AM

Many thanks for this.

I just want to check one thing:

Finally we know

Does it matter if there is a covariance term here, so that (the square root should obviously cover my additions):

+ 2. A.B . ρ(A,B) . σ(A) . σ(B)

**meymathis** · March 23rd 2009, 12:39 PM

Yes you are correct. Good catch. Thank you.

I'll fix my post.

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