Hey Giuseppe,

I usually think of hedging in terms of beta exposure, and generally this beta exposure is to the market. Lesson 4 in the lecture series talks about beta hedging, and Lesson 13 discusses beta exposure in general.

A long-short strategy with a large number of securities in each leg is very likely going to be fully hedged. My reasoning for this is that if you sample n securities, the chance that the beta of that sample is not 1.0 decreases as n grows larger. So by the time you're sampling 500 securities for one leg, the chance that that leg's beta is not 1.0 is low. This is, of course, assuming that you don't have a sampling bias built into your ranking system.

Because both leg's betas are likely to be 1.0, running a portfolio of both will yield a beta of 1.0 - 1.0 = 0. Again, if your ranking system introduces a weird bias in which the longs have lower or higher beta exposure, then you may need to correct for this to achieve low beta. That said, the ranking system may introduce say a 0.8 beta leg in the top and a 1.0 leg on the bottom, I would be very surprised if you managed to rank such that the beta spread was very large. Consequently, I suspect you'll still end up with low beta (0.8 - 1.0 = -0.2). Lastly, beta exposure can be to anything. And of course your beta exposure to your ranking factor will be 1.0. The important thing is deciding the factors to which you want no exposure. The market is usually one of those, because it's quite volatile.

If you do find your long-short strategy to not be very well hedged in practice, I suspect you have a strong bias in your ranking system. Approaches to correct this might include re-evaluating your ranking system as discussed in Lesson 18, and weighting your legs differently. If you find that in practice the beta of the long leg is b1, and the beta of the short is b2, then you can re-weight your legs by solving a system of equations:

w1 * b1 + w2 * b2 = 0.0
w1 + w2 = 1.0

Of course, this is assuming that you can predict future betas. The historical betas would have to be pretty consistent for this to work. Lessons 7 and 10 go into estimate instability more.

Hope this helps, I'm around if you have further questions.

Thanks,
Delaney

Thank you Delaney.

So the number of stocks (let's say 2n) in both legs has to guarantee that there are probably some more volatile equities (high beta to the market) and some less volatile ones in both legs so that it's reasonable to assume that we have a mean value of volatility almost equal in both legs.

If this is true let's consider for a second a pair trading situation:
Now we cannot assume this compensation because -of course. there is only one stock in each leg, so we do have to hedge the less volatile asset with a multiplyer factor in order to compensate the fact that the money on this less volatile leg move slower.
In other words if we have stock X with a beta to the SPY equal to 1.9 and a stock Y with a beta to the SPY of 0.2 is it enough to allocate just the same amount of money? or do we have to compensate the "laziness" of Y with more money?
In the codes examples i always see an even distributed money allocation for both legs even in a pair trading situation, but i don't understand why volatility (or the beta of a stock to the other) does not influence the amount of capital allocated.
Thank you
-Giuseppe

Actually, generally in pairs trading the leg weights aren't equal. For instance, in the algorithm attached to the pairs trading lecture on the lectures page, the weights are determined by a linear regression between the two security price histories. This is known as computing the hedge ratio, and is distinct from computing a market beta. I will be updating the pairs trading lecture to include a discussion of hedge ratios at some point in the future, but the computation comes from the formal definition of cointegration. Computing the ratio to also include information about the stocks' market exposure is an interesting idea and probably worth some exploration.