Estimating Many Fund Alphas with One Fama–French Regression
Summary
The document explains how to estimate alpha for many funds at once using the Fama–French three-factor model. It arranges each fund’s returns in a separate column, subtracts the risk-free rate to form excess returns, and constructs a shared factor matrix with an intercept column. A single matrix least-squares calculation then produces coefficients for every fund; the intercept row contains the estimated alphas.
The method also describes deriving residuals, estimating each fund’s residual variance, and calculating coefficient standard errors from the diagonal of the inverse factor cross-product matrix. This allows uncertainty estimates to be computed alongside alpha and factor exposures. The answer notes that spreadsheet matrix functions can perform the calculation, though a programming language may be more practical. Its formulas assume the same periods and factor data are available across funds, and the stated variance formula uses residual degrees of freedom based on the three factors plus an intercept.
Key ideas
- Subtract the risk-free rate from each fund return before fitting the factor model.
- Place funds in columns and time periods in rows to estimate their coefficients together.
- Add a constant column to the factor matrix so its first coefficient gives each fund’s alpha.
- Use the fitted coefficients to calculate residuals and estimate fund-specific error variances.
- Coefficient standard errors combine each residual variance estimate with the relevant diagonal element of the factor matrix covariance term.
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# Calculating fund alpha using Fama-French 3 factor model?
# Calculating fund alpha using Fama-French 3 factor model?
My dissertation requires me to evaluate fund performance, and for that I need to find the alpha for each fund. I have 173 funds total. I have all the inputs for the 3-factor model, and I realise running a regression and finding the intercept is the fund's alpha - however, is there a faster way of doing this due to the number of funds I have? I do not want to run a regression for every single fund, so is there a quicker way of doing this in Excel?
Thanks in advance.
## Answer by Matthew Gunn (score 4)
https://quant.stackexchange.com/a/37926
In the long run, you'd probably be better off learning a real programming language like Python, R, or MATLAB. While you can do this in Excel using `mmult`, `transpose`, and `minverse`, it's rather horrible.
In any case, you should know about the mathematical idea of a matrix, matrix multiplication, and the inverse of a matrix. (Multiplying by the inverse of a matrix is solving a linear system.)
In my notation, capital letters (eg. A) will be matrices, bold letters (eg. $\mathbf{b}$) will denote column vectors, and lower case letters will denote scalars (i.e. regular numbers). $A_{ij}$ will refer to the $i$th row of column $j$ of matrix $A$.
#### The brief recipe:
- Let $R$ be a $n$ by $k$ matrix of returns where we have $n$ time periods and $k$ assets.
- An excess return is a return in excess of the risk free rate. First you need to compute a matrix $Y$ of excess returns. Let $r^f_t$ denote the risk free rate at time $t$. The excess return of asset $i$ at time $t$ is the return minus the risk free rate: $$ Y_{ti} = R_{ti} - r^f_t$$
- You also need a $n$ by $3$ matrix $F$ of the three Fama-French factors. (Note these are already zero cost portfolios since the risk free rate or other portfolio return has been sbutracted off.) Form a matrix $X$ by pre-appending a column of 1s. $$ X = \begin{bmatrix} \mathbf{1} & F \end{bmatrix}$$
Then your solution to running those 173 regressions is given by: $$ \hat{B} = (X'X)^{-1} X'Y$$ The alphas will be in the first row of $\hat{B}$. The residuals are given by: $$ \hat{U} = Y - X \hat{B}$$ For each column of $\hat{U}$, calculate estimate of the variance of the error terms as:
$$ \hat{\sigma}^2_i = \frac{1}{n - k - 1} \sum_{t=1}^n \hat{U}_{ti}^2 $$
Let $\hat{\mathbf{d}} = \operatorname{diag}((X'X)^{-1})$ be the diagonal elements of $(X'X)^{-1}$. Your standard error for the alpha estimate for security $i$ would be:
$$ \hat{s}_{i1} = \sqrt{\hat{\sigma}^2_i \hat{d}_1}$$ Standard error for estimate of beta would be (assuming the market excess return is in the 2nd column of X): $$ \hat{s}_{i2} = \sqrt{\hat{\sigma}^2_i \hat{d}_2}$$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.