Beta of portfolio being weighted sum of betas - Arbitrage opportunity if this does not hold ?

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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?

+1 vote
answered May 25 by (820 points)
selected May 25

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.