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Let's say I have 2 time-series, how would I "standard beta hedge" them against each other? For example, what if the position in 1 timeseries is 100 shares at 16 USD per share. Another time-series is 25 USD per share. Covariance of the two is 50 and the standard deviation of time-series 1 is 5 and of time-series 2 is 7. So I know that:

beta = Cov(ts_1, ts_2)/(SD(ts_1)*SD(ts_2))
beta = 50/(7*5)
beta = 1.428

How would I determine the position of ts_2 that I should short?

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  • $\begingroup$ Covariance over product of standard deviations like you have is a $\rho$ (correlation), not a $\beta$. And also Beta is usually defined with respect to a third security. such as a market portfolio; so you would have two Betas, not one. $\endgroup$ – noob2 Apr 2 '17 at 20:02
  • $\begingroup$ Beta is of the from $Covar(r_i,r_m)/Var(r_m)$ . One of the securities plays a special role: it is in the denominator by itself. That is usually the market portfolio. $\endgroup$ – noob2 Apr 2 '17 at 20:18
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To "standard beta hedge" you would make your positions dollar values inversely proportional to their Betas. So if your standard position is 1000 USD long vs 1000 USD short when the Betas are 1, then you would have 909 long vs 1000 short when the betas are 1.1 and 1. In general $1000/\beta_1$ long of security 1 vs $1000/\beta_2$ short of security 2..

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  • $\begingroup$ Ok, so I need to get 2 positions. To get the first position I would do $position ts_1 = 1000 / (Covar(ts_1, ts_2) / Var(ts_2))$ and to get the second position I would do $position ts_2 = -1000 / (Covar(ts_1, ts_2) / Var(ts_1))$. Is that right? $\endgroup$ – user1367204 Apr 2 '17 at 21:30
  • $\begingroup$ I think my above comment is wrong. I think the correct thing to do is to buy as much as you want of $ts_1$, then find out the $\beta$ of $ts_2$ as it relates to $ts_1$ by doing $Cov(ts_1, ts_2)/Var(ts_1)$, and then shorting $ts_2$ so that $USDposition\_ts_2 = USDposition\_ts_1 * \beta * -1$ $\endgroup$ – user1367204 Apr 3 '17 at 2:54

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