In the CAPM section of the notes they give a practical example of why the beta of a portfolio should be the weighted sum of the betas of its constituents. They are basically trying to justify why in reality for a portfolio, \(E_P = \sum_{I=1}^N x_iE_i \Leftrightarrow\beta_P =\sum_{I=1}^{N}x_i\beta_i \).

They say to consider a simple portfolio of equal parts asset 1 and 2 with \(\beta_1\) and \(\beta_2\) being their betas. They then say if the expected return on the portfolio is perhaps higher than \(\frac{1}{2}(E_1+E_2)\) but the beta of the portfolio, \(\beta_p =\frac{1}{2}(\beta_1+\beta_2)\), then you could sell the portfolio and buy equal parts of asset 1 and asset 2 and end up with a positive expected value and and hence a risk-free profit.

To me this does not make any sense. This is because all it is doing is giving a net **expected** profit which doesn't mean it is risk free at all.

I understand that if the market has mispriced an index fund or something like that then you could technically make a risk free profit by instantaneously buying all the constituents and selling the index fund or vice versa but that's not what this is saying, is it?