Estimating a Two-Asset Hedge Ratio from Return Covariance
Summary
The document outlines a basic covariance hedge for two securities believed to move together. It first converts each price series into daily percentage returns, then estimates each return variance and their covariance. The hedge ratio is the covariance of the proposed hedge security with the chosen reference security divided by the reference security’s variance.
To size the position, the example starts with a dollar allocation to the security being hedged, multiplies that amount by the estimated ratio to obtain the hedge’s dollar exposure, and divides by the hedge security’s price to get units. The hedge is taken in the opposite direction from the original position. This is a simple historical return-based method: the document does not discuss estimation windows, rebalancing, costs, shorting constraints, or whether the pair has a stable relationship. The choice of which asset goes in the denominator also determines the interpretation of the hedge ratio.
Key ideas
- Estimate the hedge ratio from covariance and variance of asset returns.
- Use percentage changes rather than price levels for the covariance calculation.
- Multiply the hedged position’s dollar value by the ratio to obtain hedge dollar exposure.
- Convert hedge dollars into units by dividing by the hedge security’s price.
- Take the hedge in the opposite direction to the position being hedged.
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# Is this the correct way to hedge two securities against each other?
# Is this the correct way to hedge two securities against each other?
Let's say I believe that $ts_1$ and $ts_2$ move together and I would like to pairs trade them. Am I correct in understanding that to hedge them against each other I would get their $Var_1$, $Var_2$, and $Cov_1,_2$ all in USD, then I would buy 1000 USD worth of $ts_1$ and then find how much money to put into shorting $ts_2$ by doing $\beta = Cov_1,_2$/$Var_1$, and then doing $\$\_position\_ts_2 = $\$1000* $\beta$?
## Answer by user1367204 (score 1)
https://quant.stackexchange.com/a/33460
This is (one way) how to hedge two securities against each other. I am synthesizing the material from here and here.
Let's call one security the $security_{market}$, and another $security_{unique}$. I'm taking for granted that you have done your research and believe that you found a good pair of securities for hedging.
- Convert each security time-series from a price to a daily percent change. Call this $pct\_change_{market}$ and $pct\_change_{unique}$.
- Get $Var(pct\_change_{market}$), $Var(pct\_change_{unique})$ and $Cov(pct\_change_{market}, pct\_change_{unique}$).
- $\beta$ = $Cov(pct\_change_{market}, pct\_change_{unique}$)/$Var(pct\_change_{market}$).
- Buy \$100,000 (or whatever amount) of $security_{unique}$. Now you need to know how many dollars to sink into $security_{market}$.
- Multiply \$100,000 by the $\beta$, this will be the amount of dollars to spend on your hedge.
- Divide the number of dollars you came up with in step 5 by the price of $security_{market}$. Just to be clear, numerator is number of dollars, denominator is price of $security_{market}$. This will be your position for the hedge.
Note: Obviously, if you want to long one security then you will short the other security.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.