Is $\beta$ sometimes Bessel-corrected? I was reading about $\beta$ in finance, and I found the formula  pictured below on this website.  Is the n-1 a typo or do some people actually define $\beta$ with a Bessel correction?    
 A: The formula is correct.
The general unbiased estimator for the variance-covariance matrix is with the Bessel correction.
When the data is known to be normally distributed, the MLE is with the $n$ at the denominator.
It seems that $R_m$ is assumed to be Normal, while $(R_e, R_m)$ clearly is not a bivariate Normal random variable (so the Bessel correction applies).
You can find a derivation here: https://en.wikipedia.org/wiki/Estimation_of_covariance_matrices#Maximum-likelihood_estimation_for_the_multivariate_normal_distribution
Beta can be defined in three equivalent ways:

*

*OLS estimate $\hat{\beta}_1$ for the slope of the linear regression $$
y_i=\beta_0 + \beta_1 x_i + \epsilon_i, \quad i \in \{1, 2, \ldots, N\}
$$


*using the expression $\frac{\text{cov}[X,Y]}{\text{var}[X]}$. This is equivalent to (1) for OLS.
The least squared estimate for $\beta_1$ is
$$
\hat{\beta}_1=\frac{\sum_i x_i y_i - \frac{1}{N}\sum_i x_i \sum_j y_j}{\sum_i x_i^2-\frac{1}{N} \left(\sum_i x_i\right)^2} = \frac{\text{cov}[X,Y]}{\text{var}[X]}
$$


*Using the Pearson's correlation $\rho=\frac{\text{cov}[X,Y]}{\sqrt{\text{var}[X]\text{var}[Y]}}$
$$
\hat{\beta}_1=\frac{\text{cov}[X,Y]}{\text{var}[X]}=\rho\frac{\sqrt{\text{var}[Y]}}{\sqrt{\text{var}[X]}}
$$
