Tuesday, March 23, 2010

Introducing Betamax: A new measure of covariance stationarity (Part 1)

Setting the Stage

Risk managers are recognizing the importance of correlation, but there are precious few quantitative tools around to address correlation risk. While the adage that "during a sharp market correction, correlations go to one" is widely repeated, this is usually a qualitative statement rarely accompanied by any real quantitative measure. In this post, I will introduce a new quantitative measure of covariance stationarity (also known as weak form stationarity), which is an important topic that is generally misunderstood. In a finance context, covariance stationarity simply means that the covariance (or, alternatively, correlation) structure of asset returns is constant through time. This assumption is made by some equity return factor models, structured credit models, portfolio optimizers, and return attribution models and has implications for tracking error. As such, it is implicitly incorporated into many of the software packages popular among risk professionals. Discussion of this new measure begins with a visit to an old friend: Beta.

Beta is perhaps the oldest and most well known financial risk measure. Defined as the slope of the security market line (SML) in the original Capital Asset Pricing Model of Sharpe and Lintner, Beta is the classical measure of non-diversifiable risk. As far as risk metrics go, Beta had a good run. Born in the '60s, Beta was instantly loved by rational economics and a decade of adoration culminated in its coronation as king in the early-'70s by Fama-MacBeth. As with any monarch, Beta’s reign was almost immediately besieged by challengers (such as the APT) and faced its fair share of assassination attempts (Roll’s critic, Fisher Black’s commentary on market noise). Yet it managed to survive, gain influence, and have a bit of a golden age through the following two decades. Sadly, Beta was rather violently overthrown in the early-'90s by Fama-French, who loudly declared in 1992 that “Beta is dead” and at the same time advanced their new three factor model. The king is dead. Long live the king!

There are many criticisms of the CAPM and, by extension, Beta. The first might be that it is paradoxically on the one hand a Nobel Prize worthy contribution to finance, yet on the other hand has as one of its core tenets, indeed its very mechanism of action, the assumption that all investors know the Beta of each stock fully and are all too willing to engage in statistical arbitrage to return the expected return of all assets back to their security market lines. It assumes that all investors know, and agree upon, the marginal distribution for every asset and, more importantly, the joint distribution of returns for all assets. In particular, it is assumed that investors have full knowledge of the return covariance structure of all assets within the market. That empirical tests have so far shown that if it ever was a valid theory, it worked better prior to its discovery than after two decades of brow beating MBA students and the investing public with it. That is a little something I like to call irony.

Putting all other assumptions and criticisms aside, our aim in this post is to examine one particular assumption that not only applies to the simple two factor model proposed by the CAPM, but of a very common assumption made by a host of statistical factor models. In this post we will examine the assumption of covariance stationarity and will limit our analysis to the U.S. equity market. We do this because equities have the deepest record of return data.

As pointed out by Chan-Lakonishok (Are the Reports of Beta's Death Premature?), in their rebuttal to Fama-French:


“Even if there were no compensation for Beta risk, this does not mean that Betas serve no use for investment decision-making. As long as Beta is a stable measure of exposure to market movements, investors should still consider the 'Beta factor' of a stock.”
Chan and Lakonishok go on to show that, at least for the 10 largest down months, high Beta stocks perform worse than low Beta stocks. Of course, if the general adage holds true, that when the market undergoes sharp corrections correlations go to one, then a similar result would be true of stocks with high variance relative to stocks with low variance. The important assertion of the CAPM is that it is not individual stock variance that investors should demand compensation for, but rather the individual contribution to the variance of the market return that determines expected return. This individual contribution is a stock’s Beta. Why all this discussion of Beta?

In part two, I’ll make some critically important observations about Beta that lead directly to a natural measure of covariance stationarity.

Continue to part two.

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