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I refer here to the paper "Betting against beta" by Pedersen and Frazini.

In the model, they construct the following factor, on page 5.

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I don't quite understand how this portfolio is being constructed. What do they mean by "leveraging the long side to a beta of 1"? Or deleveraging the short side to a beta of 1?

How exactly does that work in practice? What does leveraging have to do with beta?

As I understand leveraging, we just borrow to finance more investment, right? So what does that have to do with changing beta to become 1?

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    $\begingroup$ Let's say you have a return such that $R_i - R_f = \alpha_i + \beta_i \left( R_m - R_f \right) + \epsilon_i$. Now imagine you have a return $R_j = 3 R_i - 2 R_f$. Is this a return? If so, what is the market beta for $R_j$? $\endgroup$ – Matthew Gunn Jun 5 '18 at 17:27
  • $\begingroup$ ........not sure ..................... $\endgroup$ – Donatello Jun 5 '18 at 17:38
  • $\begingroup$ You would have $\beta_j = 3 \beta_i$. If you $3\times$ leverage something, you're $3\times$ your covariances and $3\times$ your betas (whether they're market betas or something else). $\endgroup$ – Matthew Gunn Jun 5 '18 at 17:45
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  • An excess return is the payoff of a zero cost portfolio. For example:
    • $R_i - R_f$ is an excess return.
    • $c \left( R_i - R_f \right) $ is an excess return for any $c \in \mathbb{R}$,.
    • More generally, $R_i - R_j$ is an excess return for any returns $R_i$ and $R_j$.

Excess returns are nice to work with because you cans simply scale them up or scale them down and they're still excess returns. Let's imagine excess return $R_i - R_f$ has a market beta of $\beta_i$.

$$ R_i - R_f = \alpha_i + \beta_i \left( R_m - R_f \right) + \epsilon_i $$

Then excess return $\frac{1}{\beta_i} (R_i - R_f)$ has a market beta of $1$. $$\frac{1}{\beta_i} \left( R_i - R_f\right) = \frac{\alpha_i}{\beta_i} + \left( R_m - R_f \right) + \frac{\epsilon_i}{\beta_i} $$

Excess return $\frac{1}{\beta_i} (R_i - R_f) -\frac{1}{\beta_j} (R_j - R_f) $ will have a market beta of 0.

Since $\beta_H > 1$, multiplying by $\frac{1}{\beta_H}$ to obtain $\frac{1}{\beta_H} (R_H - R_f)$ is deleveraging the excess return $R_H - R_f$. Since $\beta_L < 1$, multiplying by $\frac{1}{\beta_L}$ to obtain $\frac{1}{\beta_L} (R_L - R_f)$ is leveraging the excess return $R_L - R_f$

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  • $\begingroup$ So is this correct: Say we have an asset which has beta 0.5. Say we invest two dollars in this asset, and we borrow two dollars to finance it. Then, we have a portfolio which next period will give a return of $2*R_i$ from the asset but take away $2*R_f$ from the borrowing. So our excess return is $2(R_i - R_f)$. But since $R_i - R_f$ was an excess return corresponding to beta 0.5, $2(R_i - R_f)$ has beta corresponding to $1$. $\endgroup$ – Donatello Jun 5 '18 at 17:47
  • $\begingroup$ @Donatello I think you're on the right track. $\endgroup$ – Matthew Gunn Jun 5 '18 at 18:11

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