# Build a portfolio with $\beta=1$ and minimize $\sigma^2$ using CAPM

Suppose there are two stocks A and B:

• expected returns are $$E[R_A]=0.1$$, $$E[R_B]=0.15$$;
• standard deviations are $$\sigma_A=0.1$$, $$\sigma_A=0.2$$;
• correlation is $$corr(A,B)=0.6$$;
• their betas to some index (not the market) are 0.45 and 0.9, respectively.

If we want to construct a portfolio using stock A and B such that portfolio beta to the market is 1 and sigma as small as possible, what would the ratio between weight of stock A and weight of stock B in this portfolio?

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I could go as far as getting the covariance between A and B ($$\sigma_{AB}=0.012$$), and let weight of stock A $$= X$$, express the portfolio's variance in terms of X = $$0.026X^2+0.016X+0.04$$. But I have no idea how to switch from index beta to market beta.

• Are you allowed leverage. $\beta$ is linear so it is just a combination of the portfolio weights. You would need leverage to get the beta to one, so this is a hard constaint. Draw a graph of the weights of A ana B that get you a $\beta = 1$ Dec 13 '20 at 4:17