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

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asked May 24

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 Answer

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

Best answer
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.

commented May 25 by (760 points)

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