Applying Fractional Regression Betas to Whole-Unit Hedge Positions

Summary

The document asks how to implement a regression hedge when the estimated beta is fractional and a broker may not permit fractional units. In its example, a regression relates the price or value of one instrument to another with a beta of 0.35. A textbook hedge would offset one unit of the first instrument with 0.35 units of the second, but the questioner considers rounding the hedge ratio or scaling the position to obtain whole units.

The reply says the issue is mainly noticeable for small trades: increasing the position size can make fractional-unit differences comparatively less important. This is a practical sizing observation, not a precise rounding rule or proof that the residual hedge error is acceptable. The exchange does not discuss dollar-value or contract multipliers, transaction costs in detail, changing betas, or uncertainty in the regression estimate. In practice, tradable hedge quantities depend on instrument units and exposure values, so rounding should be evaluated against the resulting residual risk and trading costs.

Key ideas

• A regression beta implies a relative exposure, which may not map directly to whole tradable units.
• Scaling both legs can make a fractional hedge ratio easier to approximate with integer quantities.
• Rounding the ratio changes the hedge and leaves residual exposure.
• The exchange offers no universal rounding rule or quantitative assessment of the resulting hedge error.
• Instrument multipliers, trading costs, and beta uncertainty also affect practical hedge sizing.

Tags

# Hedge ratio with non-whole betas


# 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.

## Answer by Bob Jansen (score 1)

https://quant.stackexchange.com/a/42436

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.

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