Compute the beta of portfolios constructed from the Betting Against Beta strategy [on hold]

http://pages.stern.nyu.edu/~lpederse/papers/BettingAgainstBeta.pdf

I am trying to understand how to apply the Betting Against Beta strategy. Let's for instance consider two stocks A and B with $\beta_A<1$ and $\beta_B>1$ and current prices $P_A$ and $P_B$. With inspiration from Equation 9 I will construct these two portfolios:

In my Low-beta portfolio I will buy $\beta_A^{-1}/P_A$ shares of A and maybe borrow/store some money at the riskfree rate $r$.

In my high-beta portfolio I will short sell $\beta_B^{-1}/P_B$ shares of B and maybe borrow/store some money at the riskfree rate $r$.

Now ... according to the theory from the paper, both of my portfolios should have a beta of $1$. How can I make sure that this is satisfied?

In otherwords the following must hold: $$\beta_L = Cov(r^L,r^m)/var(r^m)=1$$$$r^m:=r^{\text{market}}$$ Does it?

put on hold as unclear what you're asking by Alex C, Helin, LocalVolatility, Attack68, amdopt15 hours ago

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• I do not understand your question. As you yourself said, the way to make each of the portfolio betas equal to 1 is to "borrow or lend some money at the risk free rate r" until this is so. – Alex C Aug 11 at 2:34