# How do I measure the "dispersions" of a group of stock returns

I have $$n$$ stock return time series $$X_1, X_2, ... X_n$$. I want to measure how much they have "dispersed" over time. i.e. are they moving "more together" in 2023, comparing to 2022. $$n$$ is about 50. Time series are measured in the same points in time (in fact, daily, so no more than 365 data points a year).

If there are just two time series, then I just measure the correlation of $$X_1$$ and $$X_2$$ in two different years. What do I do with $$n$$ of them?

One thing I can think of is to run a PCA and compare if the first principal component takes up more variance, but it can get messy. For example why don't we look at first two principal components? How do we compare two PCs? I'd like to know if there is a systematic, existing measure.

Arguably there is not a single, objective measure that quantifies how much assets disperse over time. There are multiple that I've come and they all require some form of creativity and subjectivity (e.g. rolling window or non-overlapping samples, high or low frequency, raw or preprocessed data). Here's a selection of choices:

• Realized Cross-sectional Return Dispersion: this is the most common way to approach the problem. I'm pretty sure cross-sectional dispersion (or cross-sectional standard deviation, CSSD) has been around for age, perhaps it dates back to Christie and Huang (1995) but I don't have that paper available. More recently, Stivers et al. (2009) use it to analyse common factor strategies. At each monthly cross-section they calculate the universe dispersion as the standard deviation of monthly returns: $$RD_t=\sqrt{\frac{1}{n-1}\sum_{i=1}^n(R_{i,t}-R_{\mu,t})^2}$$. This is also used by the likes of MSCI.

• Value-weighted dispersion measure: Fei et al. (2019) list variations of the first measure by including market cap weights $$w_{i,t}$$ to the CSSD. Similarly, one can also use absolute instead of squared deviations. It is calculated as $$CSSD_{vw,t}=\sqrt{\sum_{i=1}^n w_{i,t}(r_{i,t}-r_{\mu,t})^2}$$. To make the estimate less noisy, you could also use portfolio or factor returns instead of stock returns. The authors also introduce a heterogeneous autoregressive (HAR) version of CSSD, which is an interesting idea.

• Average Pairwise Correlation: this is not the same as dispersion but still gives you an idea about how assets move together. Tierens and Anadu (2004) calculate this as $$\rho_{av(1)}=\frac{2\sum_{i=1}^N \sum_{j>1}^N w_i w_j \rho_{i,j}}{1-\sum_{i=1}^N w_i^2}$$ where $$w_i$$ are the portfolio or cap weights. It excludes the diagonal elements of the correlation matrix. I don't have the original Goldman Sachs paper at hand but refer to this summary.

• PCA: you've mentioned this already but the way I've seen this used in research notes is by solely looking at the first PC. Using the first PC is somewhat of an arbitrary choice and doesn't necessarily have the same kind of interpretation in equities as it does in fixed income. However, many people do use PC1 as the "global" market factor.

• Machine Learning Approaches: I haven't given this too much thought (nor applied it in practice) but I'm sure if you dive into the ML literature you will find some interesting approaches. You could for example use a clustering algorithm (e.g. Ward) and then calculate the dispersion of clusters.

Hope this gives you a few ideas and inspiration on how to approach the problem.

## References

Christie, William G., and Roger D. Huang. ‘Following the Pied Piper: Do Individual Returns Herd around the Market?’ Financial Analysts Journal 51, no. 4 (July 1995): 31–37. https://doi.org/10.2469/faj.v51.n4.1918.

Fei, Tianlun, Xiaoquan Liu, and Conghua Wen. ‘Cross-Sectional Return Dispersion and Volatility Prediction’. Pacific-Basin Finance Journal 58 (December 2019): 101218. https://doi.org/10.1016/j.pacfin.2019.101218.

Stivers, Chris T., and Licheng Sun. ‘Cross-Sectional Return Dispersion and Time-Variation in Value and Momentum Premiums’. SSRN Scholarly Paper. Rochester, NY, 29 January 2009. https://papers.ssrn.com/abstract=1064101.

Tierens, I., and M. Anadu. "Does it matter which methodology you use to measure average correlation across stocks." Goldman Sachs Equity Derivatives Strategy (2004).

• Arguably there is not a single, objective measure that quantifies how much assets disperse over time. I think this is because there is no broadly accepted (?) definition of dispersion in this context. If we could agree on a definition, a measure would be implied immediately. Commented Feb 12 at 17:04