# Hedge ratio with non-whole betas

Pardon me if this is a simple question but it has been a while since I dealt with this. Last time was in my quantitative investment class.

Let's suppose I have a couple highly correlated instruments $$X$$ and $$Y$$. I would like to hedge these. The simplest way would be to run a linear regression on them. Suppose the result is:

$$X = .35Y + \epsilon$$

To hedge this, I would need to buy $$1$$ $$X$$ and short $$.35$$ $$Y$$. I dont know any brokers that will allow me to do this!

If I take the floor of $$\beta$$ I get $$0$$. So that won't work. If I take the ceiling I get $$X = Y$$ which will not be hedged correctly. In fact, it will be off quite a bit (though this may be the answer for something with a $$\beta$$ closer to a whole number).

Mathematically I could also long $$1/.35 = ~3.8$$ $$X$$ and get the same result. This time taking the ceiling of $$3.8$$ gives me $$4$$. Not perfect, but it doesn't allow much to slip.

Is there a hard and fast rule to this? I vaguely remember my professor telling us to "just get the nearest whole number" but I don't exactly remember the entire discussion around it.

## 1 Answer

This is only a problem with small numbers and even small (smart) retail investors would not buy one share or one index tracking ETF (because of fixed transaction costs). This problem disappears almost entirely if you buy 10.000 of something.

• I dont entirely understand. If I did $10X = 4Y + \epsilon$ I'd still run into the same rounding problem? Wouldn't this be the same as doing $(1/\beta)*X = Y + \epsilon$ without the need for a 10 share outlay? – CL40 Oct 29 '18 at 20:10
• You always need to round and thus you’ll be of by at most half a share. The good news is that percentage you’re off will be smaller with more shares. – Bob Jansen Oct 29 '18 at 21:03