Testing CAPM Alpha Differences with a Return-Spread Regression
Summary
The document considers whether two funds have significantly different CAPM intercepts. Its proposed test forms a time series of the funds’ return difference and regresses that spread on the same market excess-return variable. The reply says this is a reasonable way to examine the relative intercept: when the funds have similar market betas, their market exposure largely cancels in the spread, so the spread regression can have a very low R-squared even if each fund separately has a high R-squared.
The answer distinguishes explanatory power from the alpha comparison. A low R-squared says the market factor explains little of the variation in the difference between the funds; it does not itself establish whether their intercepts differ. The spread regression’s intercept test addresses that question, and the cited result is described as providing little evidence that either fund reliably outperforms the other. This interpretation is limited to the CAPM specification and the stated funds and sample. The response does not develop alternative inference methods or discuss other model assumptions.
Key ideas
- Regressing the return difference between two funds on the market factor can test their relative CAPM intercept.
- Similar fund betas can make market exposure cancel in the spread, producing a very low R-squared.
- R-squared measures market explanatory power for the spread, not the significance of its intercept.
- An insignificant spread intercept offers little evidence that one fund reliably outperforms the other.
- The conclusion is conditional on the CAPM specification and the sample being analyzed.
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Full text
# 0.0006 r-squared after trying to test whether the intercepts differ significantly. Did I do it wrong? # 0.0006 r-squared after trying to test whether the intercepts differ significantly. Did I do it wrong? I am trying to test if the intercepts of two linear regression (CAPM) differ significantly or not. I have 2 fund's monthly return in the same period and regress them on the same market variable (MKT = Rm-Rf). The regression results seem pretty normal. Now, I have 2 different intercepts. I want to test whether the intercepts differ significantly, so I create a time-series of differences between fund C and fund S (Rd = Rc - Rs) for each time period and then regress it against a market variable MKT. But the result of the regression seems strange. The R-squared is only 0.0006 compared to the 0.8 - 0.9 of two funds above. So, I wonder if the test method is correct and the result is reliable or not. If this method is wrong, could you recommend the right method and some related-paper for me to study? Thanks ## Answer by demully (score 2) https://quant.stackexchange.com/a/49370 No, that's right! One fund has a beta of 81.6% and the other 82.4%, each with a ~90% R^2 to market. Therefore, it makes total sense that the spread between the two should have almost no correlation to the market (which is your very low mkt co-efficient value, and your very low R^2). The >90% P-value on the intercept (of the spread) co-efficients does suggest very little confidence either fund would be biased to outperform (in a flat market). That test is fair - you just have to be mindful of what you're testing here. The very low regressiokn R^2 just says that the market does very little to explain the difference in the two fund's performance. The intercept t/p-tests suggest that neither is reliably better or worse than the other. So what differences you do see between the two are just random noise. Or possibly some weird convoluted non-linear distribution (for which a whole different set of suites exist, but are probably not relevant if these are vanilla funds).
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