# ex ante tracking error correlation between funds

I have two portfolio's called Comb & Global. They both have the same investable universe lets says 3000 stocks & are measured against the same benchmark. So it is possible that both funds hold the same stocks. I would like to examine the correlation of the ex-ante between the two funds.

I know I can calculate the ex-ante tracking error as below,

te = sqrt((port_wgt - bm_wgt)' * cov_matrix * (port_wgt - bm_wgt))


I also know the correlation is calculated by

 p = cov(x,y) / stdev(x) * stdev(y)


I was wondering the best way to calculate the ex ante correlation between the two funds? Is there a relationship between the two funds weights that I can make use of?

Update

I should have mentioned that the two portfolios are sub portfolios and are combined into one portfolio. So I wanted to see the correlation of the ex ante tracking error between the two sub portfolio's.

I realised I can do the following,

port_wgts - number_of_companies x 2 matrix
cov_matrix - number_of_companies x number_of_companies matrix


so the below line will return a 2x2 covariance matrix.

port_wgts' * cov_matrix * prt_wgts


So I have the variances of both sub portfolios - taking the square root of this gives me the tracking error for both.

Convert the 2 X 2 covariance matrix to a correlation matrix by the following

  D = Diag(cov_matrix)^(1/2)
corr_matrix = D^-1 * cov_matrix * D^-1


So I now have the correlation between the two sub portfolios just using the weights.

## 2 Answers

Correlation is not an asset allocation measure and thus should have nothing to do with the weights in the portfolio. What you want to do is to figure out the correlation between the two portfolios using: p = cov(x,y) / stdev(x) * stdev(y)

and depending on the results, you can then run a solver function to find out the weights in each sub portfolio that minimizes the correlation of said portfolio to a benchmark (i.e. S&P)

It is unclear to me what you ask. You have the covariance matrix and today's weights - then you get an ex-ante TE. Why do you need ex-ante correlation? Ex-ante means that weights are fixed and you take an estimator of the future covariance matrix. What you get is an ex-ante TE (if you scale by $\sqrt{T}$ and you have $T$ periods in a year). If weights change tomorrow then you have and new ex-ante TE.

If you observe the funds for a while and you calculate the TE between the two funds NAVs then you get the ex-post (i.e. realized) TE.