statistics - What is the total correlation between assets in a portfolio?

Suppose I have portfolio with 10 assets, each one of them with a weight of 10% from the total portfolio (equally weighted).

It's well known how to measure from historical prices->returns a variance-co-variance matrix. And from here to have portfolio's Variance and STD (and later on this is useful for VaR calculation etc.)

However it's also useful to know the regular correlation (Pearson correlation coefficient) between each pair of assets.

The question is: What is the correct measure to some kind of "Total" or "Average" correlation between all the assets in a portfolio?

Naively because it's equally weighted portfolio I just took simple arithmetic average of all pairwise correlation coefficients...

Answer

This is indeed an interesting question.

According to this website, a paper by Goldman Sachs [Tierens and Anadu (2004)] proposes three alternative methods for estimating average stock correlations:

1. Calculate a full correlation matrix, weighting its elements in line with the weight of the corresponding stocks in the portfolio/index, and excluding correlations between the stock and itself (i.e. the diagonal elements of the correlation matrix)



2. Proxy average correlation using only individual stock volatilities and that of the portfolio/index as a whole


3. Refine 2. by reference to the ratio of index to average stock volatility

You can find more details on the abovementioned website. Unfortunately I haven't found the original paper, but if somebody provides a link in the comments I will update the post.

So to answer your question about a "correct" method: As always there is no "god-given" way how to model statistical phenomena, there are always tradeoffs with certain characteristics which are helpful in some situations but less so in others. Some important characteristics and tradeoffs for the different methods can be found in section 3 (Comments) of the abovementioned website.