Interpreting Alpha in CAPM Regressions With and Without Excess Returns

Summary

The document compares two CAPM regression forms: one uses the asset's return in excess of the risk-free rate, while the other regresses the raw asset return on the market's excess return. It explains that, when the same observations and regressors are used, the market beta is unchanged by adding the risk-free rate back to the dependent variable. The intercept shifts by that rate.

In the excess-return regression, alpha represents the asset's estimated abnormal return relative to the CAPM relation. In the raw-return formulation, the intercept combines that excess-return alpha with the risk-free rate, so its economic interpretation differs. The response offers a simple algebraic explanation but no literature references, empirical example, or discussion of cases where the risk-free rate varies over time or the regression specification changes.

Key ideas

• The standard CAPM regression models asset excess returns against market excess returns.
• Using raw asset returns instead leaves the estimated market beta unchanged when the regressors and observations are otherwise identical.
• The intercept in the raw-return regression equals the excess-return alpha plus the risk-free rate.
• The two regression forms therefore assign different economic meanings to the intercept.

Tags

# CAPM estimation model alternatives


# CAPM estimation model alternatives












Let's take a look at the standard CAPM: $$ r_{i} -r_F = \alpha+\beta(r_{MKT}-r_F) + \varepsilon $$ I would like to consider the alternative formulation: $$ r_{i} = \alpha+\beta(r_{MKT}-r_F) + \varepsilon $$ here the return of the asset $i$ is not corrected for the risk free rate. Is there any literature that refers to this formulation? What is the difference in the economic interpretation of the estimates?

## Answer by Roronoa (score 1, accepted)

https://quant.stackexchange.com/a/71413

I think that if you do a regression with these two equations, the beta will be the same, but what will change is of course the "alpha". In first equation the alpha is the well-know alpha in the industry, which means excess-return or abnormal return. In the second equation the alpha is not anymore only the excess-return, but the excess-return + the risk-free rate (rf). The proof is simple as you only have to do + rf to the first equation.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.