Determine the face amount of the offsetting position that a regression hedge calls for.
The idea
The regression hedge amount is the DV01-neutral amount multiplied by beta. Write the face of the hedge as Hedge, the face of the target as Target, and their DV01s as DV01T and DV01H, both measured per the same face amount, for instance per 100 of face. The hedge face is the target face times the DV01 ratio times beta. Hedge = Target × (DV01T ÷ DV01H) × β. Take the opposite side from the target: sell the hedge if you are long the target. Example: a 100 million target with DV01 0.08 per 100, hedged with an instrument whose DV01 is 0.04 per 100, needs 200 million for DV01 neutrality. With β of 0.7 the regression hedge is 140 million. Equivalently, the hedge DV01 should equal beta times the target DV01. Typical errors are inverting the DV01 ratio, dividing by beta rather than multiplying, and ignoring beta altogether.
Formulas
Face amount of a regression hedge
Hedge = Target × (DV01T ÷ DV01H) × β
- Target
- the face amount of the position being hedged
- DV01T
- the DV01 of the position being hedged, per the same face amount as the hedge's
- DV01H
- the DV01 of the hedging instrument
- β
- the slope of target yield changes regressed on hedge yield changes
Take the opposite side from the target. Multiply by β, never divide, and keep the DV01 ratio target over hedge; the regression intercept does not affect the amount.
A practice question
Which expression gives the face amount of the hedging instrument in a regression hedge?
- Face of target x (DV01 of hedge / DV01 of target) x beta.Why not: Inverts the DV01 ratio; a hedge with a smaller DV01 per unit of face needs more face, not less.
- Face of target x beta.Why not: Leaves out the DV01 ratio and so ignores that the two securities have different price sensitivity per unit of face.
- Face of target x (DV01 of target / DV01 of hedge) / beta.Why not: Divides by beta; the DV01-neutral amount is scaled up or down by multiplying by beta.
- Face of target x (DV01 of target / DV01 of hedge) x beta. - correct
The DV01-neutral face is F_T x DV01_T/DV01_H; the regression hedge multiplies this by beta, so that the hedge DV01 is beta x the target DV01.
A harder one
A desk is long USD 50 million face of a 30-year bond with a DV01 of 0.165 per 100 face. It hedges with a 10-year bond whose DV01 is 0.078 per 100 face. The regression of 30-year yield changes on 10-year yield changes has slope 0.82. The face amount of 10-year bonds to sell is closest to:
- USD 105.8 million.Why not: The DV01-neutral amount; it omits beta.
- USD 86.7 million. - correct
- USD 19.4 million.Why not: Inverts the DV01 ratio (0.078/0.165) before applying beta.
- USD 129.0 million.Why not: Divides the DV01-neutral amount by beta (105.8/0.82) instead of multiplying.
DV01-neutral face = 50 x 0.165/0.078 = 105.77 million. Regression hedge = 0.82 x 105.77 = USD 86.7 million, sold because the position is long.
The hardest tier
A position has a DV01 of USD 38,500 per bp. The hedging bond has a DV01 of USD 640 per bp per USD 1 million face. Daily yield changes have standard deviations of 4.6 bp (target) and 6.4 bp (hedge), with correlation 0.78. The face amount of the hedging bond is closest to:
- USD 60.2 million.Why not: This is the DV01-neutral amount, 38,500/640, with no adjustment for beta.
- USD 65.3 million.Why not: Uses the inverted volatility ratio in beta (0.78 x 6.4/4.6 = 1.085) instead of 0.78 x 4.6/6.4.
- USD 46.9 million.Why not: Uses the correlation 0.78 as beta, forgetting the volatility ratio.
- USD 33.7 million. - correct
Beta = 0.78 x 4.6/6.4 = 0.5606. DV01-neutral face = 38,500/640 = 60.16 million. Regression hedge = 0.5606 x 60.16 = USD 33.7 million.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Regression Hedging and Principal Component Analysis
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