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.