betas /volatility

[ballyb11]

I am using betas to adjust trade sizes in a Pairs trade (stocks x and y). So beta of x in relation to y is the volatility of x compared to y. Mathematically it’s
Covariance of (x, y) / Variance of y. If we happened to look at the reverse - the beta of y in relation to x is the volatility of y compared to x, it would be Covariance of (y, x) / Variance of x.


Simply, my question is I would expect each of the above betas to be the Inverse of the other (start with a trade size TS1 and adjust by the other's beta to get TS2. Now, if you went the other way, starting with TS2, in order to get back to TS1 the 2nd beta has to be the inverse of the 1st) .

But when you calculate the inividual betaas as they relate to one another, they are not inverses.

Why not.

 

[Rhody Trader]

Am I correct in understanding you're expecting the Cov(x, y) to be the inverse of the Cov(y, x)?

 

[ballyb11]

No. they are =. And I guess if that's true, it implies Variance of y = 1 / Variance of x, which I understand doesn't hold.

But what's wrong with the arguement below which is where my thought process is fixed.

1. TSx (Trade Size of x) = TSy / Bx (Beta of x) and

2. TSy = TSx / By

so substituting for TSx in line 2

TSy = (TSy / Bx) / By

Or

By / Bx = 1

so they are inverses.

 

[Rhody Trader]

Your algebra is incorrect.

TSy=(TSy/Bx)/By does not reduce to By/Bx = 1.

It reduces to By x Bx = 1.
By x TSy = TSy/Bx
Bx x By x TSy = TSy

 

[ballyb11]

You are right. I meant to write By = 1 / Bx which is the same as your By x Bx = 1. It's the inverse.

Something in this approach is wrong. Not sure what. Statistically, they are not inverses. But logically they should be? Again,

start with a trade size TSx and adjust by the other's beta to get TSy. Now, if you went the other way, starting with this TSy, in order to get back to TSx the 2nd beta has to be the inverse of the first .

 

[Rhody Trader]

You should end up with an inverse pairs ratio. You have to.

Think of it in broader terms with standard betas (what you're calling beta isn't technically beta as academically defined, but no big deal). Say GOOG has a beta of 1.4 vs. the S&P. That means for every $100 in GOOG you own the index offset would be $140 ($100 x 1.4). Flipping that arround, for every $100 in the index you have the GOOG offset would be about $71.43 ($100/1.4). In other words, the beta of the S&P relative to GOOG is about 0.7143.

The problem you've got is in your Var(x) and Var you've got missmatched terms unless those Vars are inversions. Since your Cov is the same for both beta calculations you're pairing ratio implies 1/Var(x) = Var, but as you've noted, that's not going to be the case. What you're looking for is something like Bx=Var(x)/Var and By=Var/Var(x).

 

[amit1986]

Nice answer John!

 

[ballyb11]

You should end up with an inverse pairs ratio. You have to.

Think of it in broader terms with standard betas (what you're calling beta isn't technically beta as academically defined, but no big deal). Say GOOG has a beta of 1.4 vs. the S&P. That means for every $100 in GOOG you own the index offset would be $140 ($100 x 1.4). Flipping that arround, for every $100 in the index you have the GOOG offset would be about $71.43 ($100/1.4). In other words, the beta of the S&P relative to GOOG is about 0.7143.

The problem you've got is in your Var(x) and Var you've got missmatched terms unless those Vars are inversions. Since your Cov is the same for both beta calculations you're pairing ratio implies 1/Var(x) = Var, but as you've noted, that's not going to be the case. What you're looking for is something like Bx=Var(x)/Var and By=Var/Var(x).

Thanks, but you pointed out my confusion better than I could describe. You say

"In other words, the beta of the S&P relative to GOOG is about 0.7143" (the inverse). That's how I look at it. But IF the statistical calculation of the beta of GOOG is 1.4 vs. the S&P, the statistical calculation of the beta of the S&P vs. GOOG IS NOT 0.7143.


Here's my real life example and why I'm confused. I calculate the beta for Silver (SI) in relation to Gold (GC) and get ~ 1.8. Or I calculate the beta for Gold (GC) in relation to Silver (SI) and get ~ .39. Not inverses. They will result in different trades. So which do I use in determining relative trade sizes?

More importantly what's the error in the movement from "In other words, the beta of the S&P relative to GOOG is about 0.7143" to the statistical calculation of the beta of the S&P relative to GOOG?

 

[amit1986]

How did you calculate beta for the commodities? Did you do the CAPM regression?

 

[Rhody Trader]

Here's my real life example and why I'm confused. I calculate the beta for Silver (SI) in relation to Gold (GC) and get ~ 1.8. Or I calculate the beta for Gold (GC) in relation to Silver (SI) and get ~ .39. Not inverses. They will result in different trades. So which do I use in determining relative trade sizes?

But you're not actually calculating a beta of SI relative to GC if you're using the Cov/Var formulae mentioned above. Instead you're getting a "beta" which is a ratio of each market's Var relative to the Cov. That's not a direct SI to GC comparisson, but instead a comparisson of volatility to covariance. Not the same thing.

As amit1986 suggested, beta is actually a function of regression.

This isn't to say you have to use that type of ratio. As noted above, you could use Var(x)/Var as your ratio, or a number of other things based on what you're trying to achieve.

 

[ballyb11]

I used Covariance of (x, y) / Variance of y.

 

[ballyb11]

But you're not actually calculating a beta of SI relative to GC if you're using the Cov/Var formulae mentioned above. Instead you're getting a "beta" which is a ratio of each market's Var relative to the Cov. That's not a direct SI to GC comparisson, but instead a comparisson of volatility to covariance. Not the same thing.

As amit1986 suggested, beta is actually a function of regression.

This isn't to say you have to use that type of ratio. As noted above, you could use Var(x)/Var as your ratio, or a number of other things based on what you're trying to achieve.

OK, that makes sense. My stats proficiency is mediocre at best. I guess implied in your answer is that if I calculated the true betas using a regression model, they would be inverses?

 

[amit1986]

It won't be 100% inverse, but it should be close. You are using a theoretical model and fitting it to empirical data, so by definition, the inverse relationship should hold.

Here are the steps you should take:

1) Get daily prices of the each commodity. Compute the daily return of each as per the traditional return formula:

Rt = (P2t - P1t)/P1t

2) Perform a simple linear regression twice on your returns. First regress the returns of gold on silver, then regress the returns of silver on gold. You will then end up with two beta coefficients from your regression.

3) Paste the results here for us to see

 

[ballyb11]

I want to thank you both - Amit & Rhody Trader. You've been a big help.

 

[SanMiguel]

I am using betas to adjust trade sizes in a Pairs trade (stocks x and y). So beta of x in relation to y is the volatility of x compared to y. Mathematically it’s
Covariance of (x, y) / Variance of y. If we happened to look at the reverse - the beta of y in relation to x is the volatility of y compared to x, it would be Covariance of (y, x) / Variance of x.


Simply, my question is I would expect each of the above betas to be the Inverse of the other (start with a trade size TS1 and adjust by the other's beta to get TS2. Now, if you went the other way, starting with TS2, in order to get back to TS1 the 2nd beta has to be the inverse of the 1st) .

But when you calculate the inividual betaas as they relate to one another, they are not inverses.

Why not.

Why adjust it at all? Why not just be market neutral with the same amount of money in each part of the pair?

 

[ballyb11]

Why adjust it at all? Why not just be market neutral with the same amount of money in each part of the pair?

"What if" one is relatively stable and the second is insanely volatile? That's an extreme, but there is a continuum of different volatilities and in a pairs trade I like to try to equalize the risk.

 

[SanMiguel]

"What if" one is relatively stable and the second is insanely volatile? That's an extreme, but there is a continuum of different volatilities and in a pairs trade I like to try to equalize the risk.

If you adjust the trade size, it will no longer be market neutral regardless of the volatility.
You are calculating how much company A moves in % results in a % move in company B but ignoring the ratio of the stock price between the 2 companies aren't you?

 

[ballyb11]

Essentially you start w/ equal $, and adjust one by the others beta.

 

[SanMiguel]

Essentially you start w/ equal $, and adjust one by the others beta.

But what are you targeting? A pairs trade usually targets a return to the mean so should be market neutral based upon the ratio between the shares.
Let's say the normal difference between company A and company B is a ratio of 2 but currently Company A is priced at $100 and Company B is $25. You want to place trades such that you make a profit when it returns to a ratio of 2 instead of the current 4. There is an exact price ratio of shares you should use for that. If you adjust for volatility then it is no longer a neutral strategy as you will be over or underweight in one of the shares. The volatility does not affect the historic mean ratio of the 2 pairs even though it may go wildly out of sync in between.

 

[ballyb11]

But what are you targeting? A pairs trade usually targets a return to the mean so should be market neutral based upon the ratio between the shares.
Let's say the normal difference between company A and company B is a ratio of 2 but currently Company A is priced at $100 and Company B is $25. You want to place trades such that you make a profit when it returns to a ratio of 2 instead of the current 4. There is an exact price ratio of shares you should use for that. If you adjust for volatility then it is no longer a neutral strategy as you will be over or underweight in one of the shares. The volatility does not affect the historic mean ratio of the 2 pairs even though it may go wildly out of sync in between.

Point well taken. I guess I used the term 'pairs trade" too loosely. I don't do long term trading. These trades are for several days, 7 - 10 at the most. And I'm trading commodities. And I don't like pain, so I'm trying to minimize the volatility in a trade. Equal $ in gold and silver doesn't work for me, just way too volatile. And I'm using these betas in my analysis. If I'm trading beta adjusted, then my signals should be generated using beta adjusted data (I use a fancy data / analysis tool, LIM's MIM). So I think this is the right approach for me.