Hedge ratio (beta)

The number that turns a view into a position

Suppose you think two assets have drifted apart and will come back together. To act on that without also betting on the direction of the whole market, you need to be long one leg and short the other in proportions that cancel out the shared movement.

The hedge ratio is that proportion. In the regression

log(A) = α + β·log(B) + ε

β is the slope, and it says: when B moves 1%, A has historically moved β%. If β is 1.42, then a 1% move in B came with a 1.42% move in A on average over the fitted window.

Turn that into sizing. Hold $10,000 of A. To balance it, the second leg needs enough size that a shared 1% move produces the same dollar change on both sides:

second leg = 10,000 / 1.42 = $7,042

Or, sizing from the other direction, $10,000 of B is balanced by $14,200 of A. Either arithmetic gives the same position. What matters is that the legs are weighted by their measured responsiveness, not held at equal notional.

Equal notional — $10,000 against $10,000 — is the position a plain price ratio implies. When β is meaningfully different from 1, that position is not neutral. It carries leftover directional exposure equal to the mismatch, and in a market where most assets are correlated with the largest one, that leftover exposure is usually exposure to the market itself.

Why beta is fitted on log prices

The slope of a regression on raw prices depends on the units. Fit XMR against DOGE in dollars and you get a slope in the hundreds, because one asset happens to be priced far higher than the other. That number is arithmetically correct and useless for sizing: it conflates the price level with the responsiveness.

On log prices, both sides are percentage changes, so β comes out as a pure ratio of responsiveness. It is unitless and comparable across pairs. A β of 1.4 means the same thing whether the assets trade at five dollars or at fifty thousand.

Reading the value

β near 1 — the legs move roughly one-for-one. Near-equal notional on each side is close to neutral, and the pair behaves much like its price ratio.

β above 1 — the first leg is the more responsive one. It amplifies whatever the second leg does, so it needs the smaller position to balance.

β well below 1 but positive — the first leg is the quieter one. Most of its movement is unexplained by the second, which often means the two are not tightly linked to begin with. A low β and a low correlation together usually mean the fit is describing noise.

β at or below 0 — the legs moved in opposite directions over the window. There is no hedge to construct here. A position sized on a negative β would be long both directions of the same trade, and both legs can move against you at once. Pairs with β ≤ 0 are shown on this site but excluded from the tradable ranking, and the paper journal refuses to open a position on them.

The part that is easy to forget

Beta is an estimate from a finite window, not a property of the pair.

Fit the same pair on 30 days and on 180 days and you will usually get two different slopes. Which one is right depends on the horizon you care about, and neither is a forecast. A hedge ratio fitted on the last quarter describes the last quarter.

Two practical consequences follow.

The first is that a position sized today drifts out of balance as the relationship shifts. What was a neutral weighting at entry accumulates directional exposure over weeks, quietly, without anything visibly going wrong.

The second is that instability is itself a diagnostic. If β swings widely depending on the window you fit, the pair does not have a stable relationship to trade — the regression will still return a number, because it always does, but the number is describing a relationship that is not there. Correlation is the usual first check on this, and correlation versus cointegration covers why a high correlation still is not sufficient.

Every pair page on this site shows β alongside the window it was fitted on and the correlation over the same window, so you can see whether the slope rests on a relationship or on noise. The screener can be filtered to exclude pairs with β at or below zero.