## Thursday, June 05, 2008

### What About Residual Variance?

When I was doing my dissertation of variance and stock returns, I remember there was presumably a big difference between idiosyncratic variance and systematic variance. Systematic variance is basically the beta squared, times the market volatility for the period the beta was calculated. residual, or idiosyncratic variance, is the difference between total variance and systematic variance.

This supposedly made a great deal of difference in testing variance, because we all knew the problems with beta. It could be measured with error for a host of reasons, conditional volatility issues, using the right index (S&P500, Market Weighted CRSP index, Equal Weighted Indices, etc). So, if you found a correlation between the total, or systematic variance, nad returns, good luck with that. The beta was so poisoned, so debatable, that you couldn't say anything about it. Thus, the recent papers by Ang, Hodrick, Xing and Zhang (US and international), on volatility and returns is on 'idiosyncratic volatility' and cross sectional returns.

But I remember when I read an interesting paper by Bruce Lehman, Residual Risk Revisted. He noted that idiosyncratic variance should pick up mismeasured factor loadings and mismeasured factor loadings should help explain the poor performance of factor models such as the APT. For example, Stephen Ross and Richard Roll (1994) point out it is possible—though not probable—inefficient estimates of the market portfolio are uncorrelated with returns, yet then residuals should still show a positive correlation with returns. That is, if some factor is misestimated, the measure of beta is not perfect because of imperfections in the risk proxy. But then the residual variance in such an equation should then be positively correlated with returns.

I pulled together a lot of data on betas, total volatility, and residual volatility recently. You get pretty much the same results any way you slice it. They are all negatively correlated with returns, cross sectionally. It doesn't matter, in terms of expected returns. Hmmm. I wonder why? They never tell you in grad school that as a practical matter, risk is very very subtle.