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I am a bit confused about how to calculate Jensen's alpha, having encountered a variety of methodologies.

Based on his 1967 paper, Jensen's equation for the estimation of alpha is:

$\tilde{R_{jt}} - R_{ft} = \alpha_j + \beta_j[\tilde{R_{mt}} - R_{ft}] + \tilde{u_{jt}}$

Accordingly, please correct me if I am wrong, $\alpha_j$ would be estimated by OLS, as the intercept of a linear regression of the fund's excess returns on the market's excess returns and beta would be:

$\hat\beta_j = Cov(\tilde{R_{jt}} - R_{ft}, \tilde{R_{mt}} - R_{ft})/Var(\tilde{R_{mt}} - R_{ft})$

However, I have stumbled upon another way to compute alpha, based on estimating beta as:

$\hat\beta_j = Cov(\tilde{R_{jt}}, \tilde{R_{mt}})/Var(\tilde{R_{mt}})$

And then computing the fund's expected return as:

$E(\tilde{R_{jt}}) = R_{ft} + \beta_j[E(\tilde{R_{mt}}) - R_{ft}]$

And, finally, calculating alpha as:

$\hat\alpha_j = R_{jt} - E(\tilde{R_{jt}})$

Where $R_{jt}$ would be the observed return of fund j at time t.

What is the difference between the two approaches and which would should I use if I am interested in estimating fund alphas?

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Note that both formulae for $\beta_j$ are the same since $\mathbb{V}\text{ar}[\tilde{X}+c]=\mathbb{V}\text{ar}[\tilde{X}]$ and $\mathbb{C}\text{ov}(\tilde{X}+c,\tilde{Y}+c)=\mathbb{C}\text{ov}(\tilde{X},\tilde{Y})$ for all $c\in\mathbb{R}$.

Regarding the rest (note that I will surpress the $t$ subscript), if you have the Single Index model (which is just a statistical tool, basically an OLS regression) \begin{align*} \tilde{R_j} - R_f = \alpha_j + \beta_j (\tilde{R_m}-R_f) + \tilde{u_j}, \end{align*} where $\tilde{u_j}\sim (0,\sigma^2_j)$ model i.i.d. idiosyncratic effects. Here, $\alpha_j$ is indeed the intercept of an OLS regression (asset $j$ excess returns against market excess returns).

In the CAPM, which is an economic theory, fairly priced assets lie on the security market line (SML) with \begin{align*} \mathbb{E}[\tilde{R_j}] = R_f + \beta_j (\mathbb{E}[\tilde{R_m}]-R_f) \end{align*} There is no alpha as they are neither over- or underpriced. Furthermore note that only the exposure to systematic risk measured by $\beta_j$ matters and not the overall total variance of $\tilde{R_j}$.

Now, if you have observed the return of an asset $i$, then you can compare this return, $R_j$ to its expected return from the CAPM, $E[R_j]$ and call this difference alpha, i.e. \begin{align*} \alpha_j &= R_j - \mathbb{E}[\tilde{R_j}]. \end{align*}

Finally, you may want to have a look at this post.

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