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in BUS 4028F - Financial Economics by (820 points)

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? 

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The notes are saying exactly what you outline in your last paragraph. The portfolio itself should be viewed as a tradable security like the other individual securities.

If \(\beta_{\mathrm{portfolio}} = \sum_{i = 1}^N x_i\beta_i\) the risks are matched exactly and the arbitrage strategy mentioned, buying the portfolio (which itself is now a security) and selling the underlying securities, leads to an arbitrage profit of:
$$(1+E_{\mathrm{portfolio}}) - \sum_{i = 1}^N x_i (1 + E_i)  > 0 $$ 
Note that I assumed the investment period to be 1 year and \(E\) is be a nominal annual compounded annual rate for demonstration purposes.


by (820 points)

Thanks, makes perfect sense. Seeing the portfolio as a tradable asset itself like an ETF or something makes sense. 

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