# Jensen alpha estimation

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?

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*}