# Fama/Macbeth Regression - negative estimate for market premium

I just conducted a Fama-Macbeth regression to estimate the risk premia of Mkt-Rf, HML and SMB. As a result, I got a negative risk premium for Mkt-Rf which makes no sense in my opinion. As I couldn't find any mistakes in the regression I did it again with the specification of no constant resulting in risk premia as I would expect them. As nice as these results are I don't think holding the constant at zero is correct, so does anyone of you have an idea what went wrong?

Thanks!

By negative risk premium, I am assuming you are referring to a negative $$\beta_i$$, the slope parameter for $$r_{m,t}-r_f$$.

For simplicity, I am going to use the simple CAPM model without the augmented Fame-French three factors here. The interpretation will be the same for the augmented model. The simple CAPM model is as follows, $$r_{i,t} = \alpha_i+\beta_i (r_{m,t}-r_f) +\varepsilon_{i,t}.$$

By the simple linear regression formula, it can be shown that $$\hat{\beta_i}=\hat\rho_{r_i,r_m}\frac{\hat\sigma_{r_i}}{\hat\sigma_{r_m}}.$$

Since $$\hat\sigma_{r_i},\,\hat\sigma_{r_m}>0$$, a negative $$\hat{\beta_i}$$ implies a negative $$\hat\rho_{r_i,r_m}$$, i.e. a negative correlation between the asset and the market return.

Does it make sense to have a negative correlation, and how can one interpret this?

A negative correlation means the asset return tends to be higher (lower) when the market return is doing poorly (well). For example, an investor may hold this asset as an insurance against a recession.

Finally, you should not impose an intercept term of zero for a linear regression, unless all of your variables ($$y$$ and $$X$$) are already demeaned, otherwise, the regression will return misleading results.

• Hi QuantStats, thanks for your answer. Unfortunately, I really meant the premium for rm-rf, not the beta/slope. The negative value for rm-rf was the result of a Fama/Macbeth regression to estimate the values of the different risk premia (rm-rf, hml, smb, mom in my case). Like mentioned earlier, the estimated premium for rm-rf for my selection of companies was negative what makes no sense in my opinion. Maybe you have another idea? – user43224 Nov 25 '19 at 12:31
• In that case, a negative expected rm-rf does not much sense. In a theoretical (CAPM) context, a risk premium for the market return is the reward to an investor for investing in the risky market portfolio rather than the risk-free asset. Outside of the market portfolio, an investor will hold a negative risk-premium asset if it helps to reduce the market portfolio risk. Or for risk lovers, they pay to hold risk, e.g. casino games. In an applied context, it is possible to get negative expected risk premium if the market return data are from a recession e.g. (cont below) – QuantStats Nov 26 '19 at 7:54
• (cont above) Maybe you should reinspect the choice of your proxy to the market return, risk-free rate, period of the data, if your end goal is to have a positive expected risk premium to reconcile and to be able to interpret within the theoretical framework of your augmented CAPM model. I hope it helps. – QuantStats Nov 26 '19 at 7:58