Unbiased estimation of expected return using CAPM
Jan Bartholdy, Paula Peare – 2003
Interesting paper that uses some fairly simple math to show why the kind of traditional time series regression approach to estimating expected returns leads to a downward bias in the estimates.
The idea is that when you think about the CAPM model the market risk premium that you multiply by the stock’s beta needs to be estimated, and we don’t have access to the actual world market portfolio so you need to use some proxy for that (let’s say the S&P 500 or something similar) and these proxies are imperfectly correlated with the actual world market portfolio. That imputes a bias in the estimation of expected returns unless you’re careful to undo that bias. Now undoing that bias is non-trivial because it turns out that it requires estimating the correlation between the proxy returns and the true market returns, but the true market is by definition unobservable and so you can’t calculate the adjustment factor.
The fix is to use the Fama-MacBeth procedure:
- Run time-series regressions as you normally would to get estimated betas
- Run a cross-sectional regression in each time period that relates the excess returns of each stock to the estimated betas for stocks in that particular time period
- Run that regression over multiple time periods (e.g., 5 years)
- Take the average of the coefficients in each of those regressions over that period of time
That coefficient is an unbiased estimate for the true market risk premium, which is the degree to which investors are being paid to take market risk over and above the risk-free rate. That can then be multiplied by the estimated beta for any given stock in order to get its expected return. This will be an unbiased estimate of the stock’s expected return.
The issue here is quite analogous to that in regression of the problem of measurement error in the regressors. When you regress y on x, if there is measurement error in x that will cause attenuation bias in the estimated coefficients. Similarly, here the x’s are essentially the market risk premium which, because we are using a proxy for the market, have some sort of measurement error in them. And so the betas and then therefore the expected returns end up being downward biased.