DV01: dollars per basis point, and hedging with them
Traders think in dollars per basis point, not in duration, because dollars add across a book. Match the DV01s of two bonds and a parallel move in yields cancels. What is left is a bet on the shape of the curve.
- Compute a position's DV01 and the P&L of a parallel yield move
- Size a DV01-neutral hedge in a shorter bond, and the hedge ratio
- Turn a DV01 limit into a maximum face, and read the duration a risk system is using
- Work out what a DV01-matched position makes when the curve does not move in parallel
The intuition
You are long $25 million of a ten-year bond with a modified duration of 7.5, priced at par. A one-basis-point rise in yield costs 0.075% of $25 million: $18,750. That is the DV01, and it is the number that matters on the risk report, because DV01s add. A hundred positions of different sizes and maturities, longs against shorts, net to one dollar figure; durations are percentages of different bases and do not.
It is also what makes hedging work. A two-year note with a modified duration of 1.9 moves about a quarter as much per dollar of face, so shorting $25 million of it against the ten-year leaves three quarters of the risk on. To be neutral you short enough note to match the DV01: $18,750 ÷ ($1.9 per $1,000 of face) is about $98.7 million, almost four times the face. Then a parallel move in yields cancels. A move that is not parallel does not: if ten-year yields rise 10 bp and two-year yields only 5, the long bond loses $187,500 and the short note makes back only $93,750. You have swapped outright rate risk for curve risk.
DV01 per $1 of face = modified duration × price ÷ 100 × 0.0001; position DV01 = that × face. P&L ≈ −DV01 × Δy in bp. Hedge face = position DV01 ÷ the hedge's DV01 per $1. Hedged P&L = −DV01 × (Δ long yield − Δ hedge yield). Face for a limit = limit ÷ DV01 per $1.
Why it works
- The conventions here: annual-coupon bonds, price per 100. DV01 = modified duration × price ÷ 100 × face × 0.0001, positive for a long position (the dollar loss for a 1 bp rise). Prices and modified durations are printed to two decimals and every dollar answer comes from the printed figures. The hedge is a short position in a shorter bond sized so the two DV01s match. No carry, no convexity.
- Why dollars add and percentages do not. A percentage change needs a base, and every position has a different one. Dollars per basis point are already in the same unit, so the book's rate risk is one sum.
- Equal face is not a hedge. The note's DV01 per dollar of face is a fraction of the bond's, the cover ratio. Equal face offsets only that fraction of the risk; the rest is an outright long.
- What a matched hedge leaves. With equal DV01s, the P&L is −DV01 × (Δ long yield − Δ hedge yield). A parallel move cancels. A steepening, long yields rising more than short, loses; a flattening makes money. Long the long end and short the short end is a flattener.
- Run it backwards, twice. A DV01 limit divided by the DV01 per dollar of face is the most face you may hold unhedged. A DV01 divided by market value and by 0.0001 is the modified duration the risk system is using.
- What is still missing. Funding the two legs costs carry, and for big moves convexity differs between the legs. Chapter five covers carry; chapter four convexity.
| Bond DV01: $25,000,000 × 7.5 × 0.0001 | $18,750 a basis point |
| Note DV01 per $1 of face: 1.9 × 0.0001 | $0.00019 |
| Face to short: 18,750 ÷ 0.00019 | $98.7M, a hedge ratio of 3.95x |
| Share of the risk that equal face would hedge: 1.9 ÷ 7.5 | 25% |
| 10-year yields +10 bp, 2-year +5 bp, unhedged | −$187,500 |
| The same move, DV01-hedged: −18,750 × (10 − 5) | −$93,750: the curve steepened |
The formulas
What one basis point costs per dollar of face.
Dollars per basis point, times the move. A long loses when yields rise.
Match the dollars per basis point, not the face.
A parallel move cancels; the shape of the curve is what is left.
The backwards reading.
Worked example
Each bond's DV01 per dollar of face, the position's DV01, then divide to get the face of the note to short. The follow-up asks what the hedge still leaves you exposed to.
See it move
Same bond and hedge. Change the face you hold, the long bond's yield and maturity, and how each end of the curve moves.
- Hold more face. The DV01 and the hedge face rise in proportion; the hedge ratio and the cover do not move, because they are per dollar of face.
- Lengthen the long bond. Its DV01 and the hedge face rise, the hedge ratio rises, and equal face would cover less of the risk.
- Raise the long bond's yield. Its price and duration fall, so its DV01, the hedge face and the hedge ratio fall.
- Raise the move in the long yield. Both P&Ls fall. Raise the move in the hedge yield instead: the hedged P&L rises and the unhedged one does not move.
Run it backwards
The desk's DV01 limit, the bond's price and its modified duration are given. What is the most face you can hold unhedged?
DV01 per dollar of face is modified duration × price ÷ 100 × 0.0001. Divide the limit by it. The same limit allows far more face of a short note, in proportion to how much less risk each dollar of it carries.
The other inverse reads a risk report: a DV01 divided by the market value and by 0.0001 is the modified duration the system is using, and it must sit below the bond's maturity.
Traps
Say it in the interview
“How would you hedge $25 million of a ten-year bond with two-year notes?”
Check yourself
4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- DV01 = modified duration × price ÷ 100 × face × 0.0001: dollars per basis point, and they add.
- Hedge face = position DV01 ÷ the hedge's DV01 per dollar of face. Equal face is not a hedge.
- A DV01-matched position is neutral to parallel moves and long or short the shape of the curve.
- Limit ÷ DV01 per $1 is the most face you may hold; DV01 ÷ (market value × 0.0001) is the duration in use.