# 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.

• 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