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Sorry if this is obvious to you. I've got my brain spinning for a while and think I should seek some insights.

Question 1: What's the definition of $\beta$ between a stock and hedger/market portfolio, assuming the hedger is SPY?

I saw two expressions:

  • $\beta = \frac{Cov(r_{s1}, r_{SPY})}{Var(r_{SPY})}$
  • $\beta$ is the coefficient in the regression $r_{S1} = \alpha + \beta r_{SPY}$

Are these two equivalent? If yes, how to get the first one from the second one?

Question 2: For a portfolio of stocks, what's the right way to beta hedge with multiple hedgers?

For example, a long-short portfolio has stocks $S_{i}$ with notional $N_{i}$ and weights $w_{i}$. So $w_{i} = \frac{N_{i}}{\sum{|N_{i}|}}$. Assume $i=3$, i.e. 3 stocks.

Also assume there are two hedgers, say $H_{1}$ and $H_{2}$.

I saw two different beta hedging approaches in practice:

  • Assume we have the following individual betas. $\beta_{(i, j)}$, where $i = 1...3$ and $j=1..2$. For example $\beta_{3,1}$ means the $\beta$ between stock $S_{3}$ and hedger $H_{1}$. Then the final hedging is simply $-(\beta_{(1,1)}N_1 + \beta_{(2,1)}N_2 + \beta_{(3,1)}N_3)$ notional in $H_1$ and $-(\beta_{(1,2)}N_1 + \beta_{(2,2)}N_2 + \beta_{(3,2)}N_3)$ notional in $H_2$.
  • We use $w_i$'s and stock $S_i$'s returns to calculate a portfolio return $r_{p}$. Then we run a linear regression $r_{p} = \alpha + \beta_1 r_{H1} + \beta_2 r_{H2}$ where $r_{H1}$ is returns of hedger $H_1$ and $r_{H2}$ is returns of hedger $H_2$. Then final hedging is $-(\beta_1 \sum{|N_i|})$ notional on $H_{1}$ and $-(\beta_2 \sum{|N_i|})$ on $H_2$.

Which approach is right? Are they equivalent? Is multiplying $\sum|N_i|$ in the second approach right? What should be the right way if none of them are right?

Thanks

inf
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    For Question 1, the two are the same as shown on the wiki of simple linear regression: Relationship with the sample covariance matrix – autoencoder Mar 21 '24 at 05:59
  • Why do you need to define weigths and the factor $\sum | N_i |$, isn't the calculation the same as before i.e. the dollar value ("notional") of the stock times the Beta gives the required notional of the hedging instrument (with a change of sign, i.e. you hedge a long position with a short hedge and vice versa) ? – nbbo2 Mar 21 '24 at 16:31
  • @nbbo2, thanks for your inputs. What you mentioned is the approach 1 in the portfolio case above. One argument I saw is this linear addition doesn't count for correlations between core stocks. Hence it may over hedge. I guess this is why others use approach 2 of the portfolio case above. But I'm not sure if the final regressed betas should be scaled up by $\sum{|N_{i}|}$ for a long/short portfolio. – inf Mar 26 '24 at 14:34
  • Yet another version of portfolio beta hedge I saw goes like below:
    1. based on the weights $w_{i} =\frac{N_{i}}{\sum{|N_{i}|}}$ and individual betas $\beta_{i,j}$ for core positions $i$ and hedger $j$, a weighted overall beta is calculated for each hedger $j$ as $\beta{j} = \sum{w_{i} \beta{i,j}}$.
    2. then each calculated $\beta{j}$ above is scaled up by portfolio notional and flip the sign to get the final hedge notionals.

    First, I failed to unify this approach with any of existing 2 approached I mentioned above. Second, for long/short portfolio, should I scale by $\sum{|N_{i}|}$?

     – inf Mar 26 '24 at 14:42

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