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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.

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    $\begingroup$ 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 $\beta = 1$ $\endgroup$ – Paul Brennan Dec 13 '20 at 4:17
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The problem is, what do you define as the market?

You can easily construct the efficient frontier using the covariance matrix. You can easily show the efficient frontier contains the solution, maybe with or without leverage.

Without the definition of the market (e.g. the cap-weighted index, tangency portfolio of the 2 assets), the question is however not too meaningful.

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