What is the total correlation between assets in a portfolio?

Question

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 1

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.

Answer 2

I just want to add to vonjd's answer some info on the comparison of the 3 methods. This is too big for a comment so I'm posting as a separate answer but please upvote his answer, not mine.

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Do the differences in methodologies matter in practice?

To gauge the practical importance of the biases in methods 2 and 3, we calculate the weighted stock correlation for the stocks in the S&P 500 index during the period January 2002 through March 2004. For each month in our sample, we use the daily total returns of each of the S&P 500 constituents to calculate the pair-wise correlations needed in method 1, and the single stock volatilities needed in methods 2 and 3. In addition, we calculate the volatility of the S&P 500 index based on its daily total returns during the month, and we use the start-of-month index weights to obtain the weighted average stock correlation.

Exhibit 2A shows the resulting weighted average cross-stock correlations for each of the 27 months in the sample based on each of the three calculation methods. The choice of method has a modest impact on the average correlation number for a well-diversified portfolio or index such as the S&P 500. Exhibit 2B makes this point even clearer by plotting the differences between the correlations numbers obtained from each of the methods. The absolute difference in correlation fell below 0.05 during the past 2+ years. Exhibit 2B also visualizes the consistent upward bias in method 3 as compared to method 2, but the overestimation is less than 0.01 in absolute value.

Exhibits 3A and 3B analyze the difference between methods 1 and 2 further, by looking at some crude measures associated with the volatility bias in method 2 identified above. They suggest that larger differences tend to happen more often in periods when average stock volatility is higher or when stock volatility has changed by a larger amount.