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Estimating Portfolio Beta and Hedging with Multiple Benchmarks

Article Quant Q&A · Author: inf
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Summary

The document asks how to define a stock’s beta relative to a market proxy such as SPY, and whether beta as a covariance-to-variance ratio equals the slope from a regression of stock returns on benchmark returns. It then considers hedging a long-short stock portfolio with two benchmark assets. One proposed method adds each stock’s individual beta exposures to each hedger; another estimates both hedge coefficients in a multiple regression of portfolio returns on the two hedgers.

It asks which method is appropriate, whether the portfolio coefficients should be scaled by the sum of absolute notionals, and how long-short weights affect the hedge. The text does not provide an answer or numerical evidence. The comparison highlights that portfolio construction, return weighting, benchmark correlation, and the units used for hedge notionals matter; it does not establish that either proposed approach is generally correct.

Key ideas

  • In a simple regression with an intercept, the beta slope equals covariance with the benchmark divided by benchmark variance.
  • A multi-hedger regression estimates portfolio exposures jointly rather than summing separate single-hedger estimates.
  • Long-short portfolio weights and notional scaling affect the conversion from return exposures to hedge positions.
  • The document poses these questions but does not resolve them or provide a worked hedge calculation.

Tags

StatisticsRisk managementPortfolio constructionEquities
Full text
# Some questions about beta hedging


# Some questions about beta hedging












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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.