Estimating Ex Ante Correlation Between Active Portfolios
Summary
The document considers two equity portfolios with the same benchmark and investable universe, and asks how to estimate the correlation of their ex ante tracking errors. It describes forming each portfolio’s active weights by subtracting benchmark weights, then using the asset covariance matrix to calculate the covariance matrix of active portfolio returns. For two portfolios, the resulting matrix contains each portfolio’s active-return variance on the diagonal and their covariance off the diagonal. Scaling by the diagonal standard deviations converts it to a correlation matrix.
This calculation gives a model-based estimate from current weights and an estimated covariance matrix; it is not the same as measuring realized tracking error from historical fund net asset values. The answers disagree about framing: one questions the meaning of ex ante correlation, while another points out that fixed weights and forecast covariances produce an ex ante risk estimate that changes as weights change. The calculation therefore depends on aligned active-weight definitions, a suitable covariance estimator, and the intended horizon. It estimates correlation of active returns, rather than an allocation measure by itself.
Key ideas
- Active weights are portfolio weights minus benchmark weights.
- The covariance matrix of active returns can be calculated from portfolio active weights and the asset covariance matrix.
- The off-diagonal element gives covariance between two portfolios’ active returns.
- Standardizing that covariance matrix by its marginal standard deviations yields correlation.
- An ex ante estimate uses current weights and forecast covariance, while ex post analysis uses realized fund returns.
Tags
Full text
# ex ante tracking error correlation between funds
# 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.
## Answer by user40411 (score 1)
https://quant.stackexchange.com/a/25136
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)
## Answer by Richi Wa (score 0)
https://quant.stackexchange.com/a/11106
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.