Capitalization Coupling: Major Indices at Correlation Highs

Last summer, I often covered the difference in short-term performance between the Russell 2000 and S&P 500. I suggested that the VIX, as a measure of implied volatility, was a good predictor for this capitalization premium, and that claim often held up. I even went so far as to analyze the high-amplitude periods of this relationship. However, as the actual volatility of volatility has increased dramatically since last fall, my suggestions have been more and more difficult to implement.

I wanted to explore why this relationship had changed, and so I've taken a look at the Dow 30 (DIA), S&P 500 (SPY), and Russell 2000 (IWM) since 2002. The figure below shows the cumulative return of each index ETF in the top pane. The bottom pane shows the trailing 100-session percentage-correlation between each pair of indexes.

click to enlarge

One of the most striking features of these plots is that all three indexes are trading at or above their highest historical correlations on this range. The only timespan of comparable length was during late 2002 and early 2003 as a short bear market held sway.

The other relationship that caught my eye was that the trend in correlation between the indexes was inversely related to the overall market performance. In other words, as the correlation between indexes fell, the markets rose on average. Furthermore, during these falling correlation periods, the Russell often outperformed its counterparts, and vice versa in rising correlation periods. This relationship likely reflects the fact that the capitalization premium and discount on small caps and large caps is very much a function of the strength of the economy and credit market.

In the future, I'll be watching closely for a decline in the correlation between these indexes as a confirmation of overall market uptrend.

Comments

[Rudi]

"In other words, as the correlation between indexes fell, the markets rose on average."
This is more likely to be explained by a higher correlation when markets crash. Your interpretation does not bear in mind the temporal causality, the interpretation should be the opposite, it is therefore worse than tassology.
The increasing correlation has been well established in conjunction with copulas.
en.wikipedia.org/wiki/Copula_%28statistics%29

[Michael Bommarito]

Rudi, are you talking about correlations between assets *within* an index or correlation *between* these specific capitalization-oriented indexes?

Both rank and Pearson-product correlations indicate that:
i) daily R^2 between the average correlation change and return is -1%
ii) weekly R^2 between the average correlation change and return is 2%
iii) monthly R^2 between the average correlation change and return is 4%
iv) quarterly R^2 between the average correlation change and return is 6%

Which seems to contradict your statement.

[Michael Bommarito]

By the way, I'd appreciate links to studies or your own analysis in the future instead of suggestions about tea leaves.

[Rudi]

Michael,

I replicated a part of your study. I selected DIA/IWM since it has the highest volatility and correlation. I used the past 100 days to calculate the correlations. In another replication of your study, I also used weekly data und based the correlations on the past 20 weeks.
On both occassions, there wasn't any significant relationship.
If there would have been one, I would have done the following (and suggest you try that for the set of data where you indeed found something): Lag the correlation-variable (differentiated) and do a regression analysis with the differences in returns.(Thus regarding the temporal structure.) Btw: A regression is not the best way in this case. A VECM would be a better choice, but it means a lot of work. Then try it vice versa. Lag the variable containing the returns and regress it on the future period of correlation. This may work. It should be even more significant if you shorten the period, to calculate the correlation. (Like 30 days e.g.)

Your R² statistics are random. You did not post any p-values and the negative (-1%) number for daily-based analysis suggests, that you are not able to reject the null-hypothesis. R² also increases with a shrinking N, so your increasing percentages cannot form the basis of any serious argument.

[Michael Bommarito]

Yes, I accept that my suggestion about the trend was not significant under this model, but what I meant to show in my reply was that the opposite was not true either, as you argued. My p-values were all significant at least to the 0.025 level, though I did not record them.

I would also note that your samples were contained strictly within the local phenomenon that I am trying to explain, and thus not only do they have no chance of demonstrating a difference from prior relationships, but you have an N smaller than mine by an order of magnitude wrt power.

[d_teller]

I believe that much of the variance in the correlations and regressions has to be due to the magnitude of differences, on a daily basis. If you want an unweighted, ordinal test, use a Kappa-like function, eg., Tau-beta, which is a measure of concordance, with +1.00 if all move in the same direction; -1.000 if they are totally inverse and randomly moving. Then the relative magnitude of changes is removed, and only directionality is counted.

[Rudi]

As you asked me for some links to studies of copulas:

www3.interscience.wiley.com/cgi-bin/abst...

This here is about what is called tail-dependency (stock crash
together but in positive return times, they are less correlated
(indexes as well).

db.riskwaters.com/public/showPage.html?p...
This is interesting as well but german.

www.imw.tuwien.ac.at/fc/Teaching/Schwaig...
This is an introduction to the topic, stating that conjunct negative
returns are more likely than conjunct positive returns. Also german,
sorry.

papers.ssrn.com/sol3/papers.cfm?abstract...
This is a good study about copulas, but not within stocks, although interesting.

Googeling "tail-dependency" seems promising if you want to find more

Nevertheless: All this stuff is based on Embrechts (1999)
Embrechts P., McNeil A. and D. Straumann (1999), Correlation: Pitfalls
and alternatives, RISK

Link: www.ma.hw.ac.uk /~mcneil/ftp/risk.pdf !!


To d_teller:

This seems an interesting idea, as a non-parametric approach may deal
with the problem of non-normality of return distributions.
Have you had any success with that or any of the links to the studies?


In spite of that I have to repeat, that no approach is worth spending
time on, as long as it does not generate cash!

For this purpose, one should always lag the independent variables AND
split the sample in a test-sample and validation-sample.
Fit your model with lagged variables and apply it to the unused data
of the validation sample. It is only worth money if you find some
relation in the validation sample.
Keep in mind, that you bias the study if you use knowledge from the
validation-sample to alter your model. Best is to keep a (third) final
validation-sample until you are really sure with your model.

Michael, have you tried adding more variables and maybe
interaction terms to your model? E.g. include the lagged returns in
interaction with the correlation. I also would use a smaller time
frame than 100 days to calculate the correlation, to have a more
sensitive variable.


As an impetus: To generate cash, you could spend time to build models
that predict future correlations. One could use that to better
minimize the portfoliorisk and therefore absorb a higher lever. There
you get your free lunch. GARCH Models and Kalman filters are suitable for this purpose.

best regards
Rudi

[Michael Bommarito]

Middle of exams here so I don't have time for any stochastic volatility or adaptive models, but feel free to provide any results you find doing so. Here are some more extensive results just under this simple model with non-parametrics for comparison.

Colums:
Pearson/Sign Pearson/Spearman Rank/Sign Spearman Rank

Rows:
1st: Pearson correlation between return & each correlation
2nd: Spearman rank correlation between return & each correlation

Results:

0-Day (no lag)
ans =
-0.0279 -0.0283 -0.0377 -0.0268
-0.0006 -0.0555 -0.0439 -0.0555

1-Day
ans =
0.0115 0.0147 0.0236 0.0145
0.0261 0.0224 0.0368 0.0269

2-Day
ans =
0.0010 0.0035 0.0004 0.0029
0.0015 0.0074 -0.0087 0.0057

[Michael Bommarito]

These are all from June 2001 to yesterday, and significant to at least p=0.025 again.