Given the DV01 of each bond, calculate how much face amount is required to hedge a position that is sensitive to interest rates.
The idea
To hedge interest rate risk with DV01, choose a hedge position whose total DV01 equals that of the position being hedged, with the opposite sign. When DV01 is quoted per 100 of face, the face amount of the hedge follows from a ratio. Face of hedge = Face of position × DV01 per 100 of the position ÷ DV01 per 100 of the hedge instrument. The bond with the smaller DV01 per 100 of face needs a larger face amount, so the ratio puts the position's DV01 on top. If you are long the original position you sell the hedge, and if you are short you buy it. Matching face amounts or market values is the usual mistake, because bonds with different maturities or coupons have different DV01 per unit of face. The hedge works only for small parallel moves and must be rebalanced as DV01s drift.
Formulas
Face amount for a DV01-weighted hedge or curve trade leg
Fh = Fp × DV01p ÷ DV01h
- Fh
- the face amount of the hedge or second leg
- Fp
- the face amount of the position being matched
- DV01p
- the DV01 per 100 of face of that position
- DV01h
- the DV01 per 100 of face of the hedge instrument
Prices do not enter once DV01 per 100 of face is given. Sell the hedge against a long position and buy it against a short one.
A practice question
A risk manager is long Bond A and will hedge it with a short position in Bond B using DV01s. Which condition defines the hedge?
- The face amount of B equals the face amount of A.Why not: Equal face amounts leave a DV01 mismatch unless the two bonds have the same DV01 per 100 of face.
- The market value of B equals the market value of A.Why not: Matching value ignores differences in duration between bonds, so DV01s would still not offset.
- The face amount of B is chosen so that B's total DV01 equals A's total DV01. - correct
- The convexity of B is chosen to equal the convexity of A.Why not: Convexity is a second-order property; the first-order hedge matches DV01, not convexity.
A DV01 hedge makes the two legs equal and opposite in dollar sensitivity to a 1bp move. Face or value matching does not do this when the bonds differ in maturity or coupon.
A harder one
A long position in a 10-year bond has a DV01 of 0.0780 per 100 of face. A hedge uses a 5-year bond with DV01 of 0.0430 per 100 of face. The long position is USD 40 million face and the bond prices are 96.40 and 101.30 respectively. The face amount of the hedge is closest to:
- USD 22.1 millionWhy not: Inverts the DV01 ratio, using the 5-year DV01 over the 10-year DV01.
- USD 69.0 millionWhy not: Adjusts the DV01 ratio again for the price ratio, which double-counts price because DV01 per 100 of face already includes it.
- USD 72.6 million - correct
- USD 40.0 millionWhy not: Matches face amounts instead of DV01s.
Hedge face = 40 million x 0.0780 / 0.0430 = USD 72.56 million. DV01 per 100 of face already reflects each bond's price.
The hardest tier
A trader is long USD 30 million face of Bond A, with DV01 of 0.145 per 100 of face. She has already sold USD 12 million face of Bond B, with DV01 of 0.062 per 100 of face. To make the position DV01-neutral, how much additional Bond B face should she sell?
- USD 58.2 million - correct
- USD 70.2 millionWhy not: Ignores the existing short in Bond B and hedges the whole of Bond A's DV01.
- USD 24.9 millionWhy not: Divides the net DV01 by Bond A's DV01 per 100 of face instead of Bond B's.
- USD 46.2 millionWhy not: Subtracts the existing 12 million face from the net requirement, counting the existing hedge twice.
Bond A DV01 = 30,000,000 x 0.145 / 100 = 43,500. Existing hedge = 12,000,000 x 0.062 / 100 = 7,440. Net = 36,060. Extra face = 36,060 / 0.062 x 100 = USD 58.16 million.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Applying Duration, Convexity, and DV01
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