Estimating CAPM Alpha with Monthly Excess Returns
Summary
The document explains a standard regression approach for estimating a portfolio’s CAPM alpha. It recommends calculating monthly excess returns for the portfolio and market, then regressing portfolio excess returns on market excess returns. The regression intercept is alpha, while the slope estimates market beta; the residual captures variation not explained by the market model. It distinguishes this statistical regression from the CAPM as an economic theory, under which alpha should be zero when the model assumptions hold.
For a rough annualized alpha, the response suggests multiplying a small monthly estimate by twelve. It also recommends heteroscedasticity-robust standard errors for t-statistics. The discussion cautions that a 12-month sample is too short for sensible alpha estimates and argues against using CAPM to forecast expected returns. It points to multifactor models as alternatives, but gives no dataset, regression output, or empirical comparison to assess their performance.
Key ideas
- Estimate alpha by regressing monthly portfolio excess returns on monthly market excess returns.
- The regression intercept is alpha, and its slope is beta.
- For small monthly alpha estimates, multiplying by twelve gives a rough annualized figure.
- Use heteroscedasticity-robust standard errors when calculating t-statistics.
- A 12-month sample is too short for reliable alpha estimation.
- Multifactor models can add return drivers beyond market exposure.
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Full text
# CAPM Calculations
# 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?
> 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?
> 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 by Matthew Gunn (score 4)
https://quant.stackexchange.com/a/34058
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.
- 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.
- 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.
- 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
- The CAPM does not work! It would be a mistake to use the CAPM to forecast expected returns.
- 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.
- 12 months is way too short a period to obtain sensible estimates of alpha.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.