# CAPM Calculations

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?

1. 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)

• How the fund has performed is measured by a single number called the (monthly) Alpha. It is equal to the average of the things you called 'alphas' in Step 1, but are more commonly called the 'monthly residual returns'. Other than this Method 1 looks OK. It is also possible to compute the Alpha and Beta in a single step if you have an OLS regression routine, and that is how I do it. – noob2 May 3 '17 at 17:52
• Once you have the 'monthly Alpha' you can report it as is (many people do) or you can annualize it if you wish. – noob2 May 3 '17 at 17:57
• @noob2 thank you for your answer. So you are saying that calculating beta with the Cov/Var formula and finding Alpha. Will give the same Alpha as using Rt-Rf as my "y" variable and (Rm-Rf) as my "x" variable and using single regression to find the intercept? – PeterNiklas May 3 '17 at 17:59
• @noob2, The thing is. If i use the regression approach, with monthly data then i will have to do a regression to find a Alpha for every single month. If I'm not mistaken isn't it like this: The regression approach of finding Alpha works if i use yearly data on returns and so on in the regression to find the intercept. The other approach of calculating beta with Cov/var works when using monthly data, because then i can calculate a monthly value of Alpha and then annualising it? – PeterNiklas May 3 '17 at 18:09
• I am very confused. – noob2 May 3 '17 at 18:18

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 $R^m_t - R^f_t$
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.