# CAPM model formula for zero cost portfolio?

Let's say I want to run a series of regressions for zero-cost portfolio Y that goes long on stocks based on high variable x and short stocks with a low variable of x. How do I run the regression, for example, for CAPM? Is it:

$$Y - r_f = a + b(market-r_f) + e$$

or

$$Y = a + b(market-r_f) + e$$

Is it the first one only if we assume that the portfolio actually has zero beta/systematic risk?

• Your time series is already a tradable excess return. You do not need to subtract $r_f$ again. You can directly estmate $Y_t=\alpha+\beta (MKT_t-r_{f,t})+\varepsilon_t$. Feb 16 at 7:47
• @Kevin, why exactly is the time series an excess return? Feb 16 at 8:05
• @RichardHardy Because it goes long one set of assets and short another set of assets. You sell short one asset with low x and get \$1 from that and then you invest that \$1 in the asset with high x. It's a zero cost portfolio. It's an excess return. Feb 16 at 8:41
• @Kevin, I was thinking about being in excess of the risk-free rate. But now I realize the risk-free rate can be considered as 0 if you are investing 0 in the first place. Feb 16 at 8:43
• @RichardHardy Excess returns really means only that it's a zero cost portfolio. We normally do in excess of the risk-free rate if consider a standard portfolio of (long) stocks. For example the 25 portfolios on B/M and size. There I'd look at their returns in excess of $r_f$ because I need to fund my investment in any of these portfolios by borrowing from the money market. But for a high-minus-low portfolio, I don't need to include $r_f$. Check out the first table in Cochrane's (2005) first chapter. It lists prices and returns, including excess returns. Feb 16 at 11:43