CAPM Calculations

Question

Im trying to calculate Alpha using CAPM & I have data on everything necessary.

$$R_t-R_f={\alpha}+{\beta}\times(R_m-R_f)$$

i.e.

$${\alpha}=R_t-R_t-{\beta}\times(R_m-R_f)$$

In more detail, I have monthly data on returns, market returns and the risk free rate. Now lets say I'm interested in how a fund has performed in 12 months, which one of the following two methods is correct?

1. Alpha = monthly returns - monthly risk free rate - beta(monthly market returns - monthly risk free rate)

This will yield a monthly Alpha and then calculating the yearly Alpha per fund by the formula $(a+{\alpha}_1)(a+{\alpha}_2)...-1$ and beta here is calculated by Covariance(Monthly fund return, Monthly market return)/Variance(Monthly Market return)

Or do I first convert the returns to yearly and then calculate Alpha?

2. Alpha = Yearly returns - Yearly risk free - beta ( yearly market return - yearly risk free rate)

and beta here is calculated by Covariance(Yearly fund return, Yearly market return)/Variance(Yearly market return)

Answer 1

The question above looks somewhat confused. Where's the error term?

A recipe for a standard calculation

It's customary to work with monthly returns.

1. For each portfolio $i$, calculate monthly excess returns $R^x_{i,t} = R_{i,t} - R^f_t$ where $R^f_t$ denotes the 1-month risk free rate.
2. Calculate or download the monthly excess return of the market $R^m_t - R^f_t$
   • Eg. go to the data library section of Ken French's website.
3. Regress excess returns of portfolio $i$ on excess returns of the market:

$$ R^x_{i,t} = \alpha_i + \beta_i \left( R^m_t - R^f_t \right) + \epsilon_{i,t} $$

This regression estimates what Eugene Fama calls the market model. It is a statistical model. The CAPM in turn is an economics based, asset pricing theory that says the $\alpha_i$ in the market model regression should be zero if you're using excess returns (like I did in this example).

For small values, you can annualize $\alpha_i$ by multiplying by twelve.

For calculation of t-stats, make sure to use heteroscedasticity robust standard errors.

Some related results

1. The CAPM does not work! It would be a mistake to use the CAPM to forecast expected returns.
2. More sensible asset pricing models might be the Fama-French 3 Factor Model, the Carhart 4 Factor Model, or the Fama-French 5 Factor Model. Basically, you add right hand side variables to the regression that actually do explain cross-sectional variation in average returns.
3. 12 months is way too short a period to obtain sensible estimates of alpha.