Rates · R3 · 8 min
Duration, DV01 and convexity
Turn any move in yields into a profit or loss in pounds, and say why long bonds move more than short ones.
In the news: “The 30-year gilt yield rose 13bp”: a loss for every holder, and this lesson shows how big.
The essentials
- Price and yield
- A bond’s price and its move in opposite directions. Long-dated bonds move further than short-dated ones for the same change in yield.
- Duration
- The percentage a bond’s price moves for a one-point (100bp) change in yield. A of 7.7 means a 1-point rise in yield costs about 7.7% of the price.
- What lengthens duration
- A later maturity and a lower coupon. A has the most duration for its maturity, because all of its value arrives at the end.
- DV01
- The profit or loss on a position for a 1bp move in yield: about notional × duration × 0.0001. £10m of a bond with a duration of 7.7 has a DV01 of about £7,700.
- Long and short
- Owning bonds (long) loses money when yields rise and gains when they fall. Having sold bonds you don’t own (short) is the reverse.
- Convexity
- The price–yield line is a curve, so a bond gains more when yields fall than it loses when they rise by the same amount. It matters most for large moves.
P&L from a yield move
DV01 ≈ notional × D × 0.0001
- DV01
- profit or loss for a 1bp move in yield
- notional
- face value of the bonds held
- D
- modified duration
How it works
1. Long bonds move more
You know that bond prices fall when yields rise. They don’t all fall by the same amount. If yields rise by one percentage point, a 2-year bond loses about 2% of its price, but a 30-year bond about 15%. The further away a bond’s payments, the more a change in the discount rate hits what they are worth today.
2. Duration puts a number on it
A bond’s , usually just called duration, is the percentage its price moves for a one-point (100) move in yield. A 10-year with a 5% coupon has a duration of about 7.7: a 1-point rise in yield costs about 7.7% of its price, and a 10bp rise about 0.77%. The rule: % price change ≈ −duration × change in yield.
3. What gives a bond more duration
Two things lengthen duration: a later maturity, and a lower coupon. A low coupon means more of the bond’s value sits in the final repayment, far away. The extreme is a zero-coupon bond, which pays nothing until the end: a 10-year zero has more duration than any 10-year bond that pays coupons.
4. DV01: the move in money
Traders turn duration into pounds with : the profit or loss for a 1bp move in yield. For a bond priced near 100, DV01 ≈ amount held × duration × 0.0001. £10m of a gilt with a duration of 7.7 has a DV01 of about £7,700: every basis point is worth £7,700.
5. Long or short sets the sign
Owning bonds is being long: you lose when yields rise. With a DV01 of £7,700, a 12bp rise costs 12 × £7,700 = £92,400. Being short, having sold bonds you don’t own and must buy back later, is the opposite: you gain when yields rise and lose when they fall.
6. Convexity: the curve bends
Duration treats the link between price and yield as a straight line, but it is a curve. For the 10-year gilt, a 1-point rise in yield costs 7.4%, not the 7.7% duration predicts, and a 1-point fall gains 8.1%. Gaining more than you lose for the same move is . It barely matters for small moves and matters a lot for big ones.
Try it
Drag the yield. The straight line is the duration estimate and the curve is the true price; the gap between them is convexity.
5% coupon at par · duration 7.72
- Duration says
- −11.58%
- True change
- −10.78%
- Gap (convexity)
- +0.80pts
Worked example
You hold £10m of a 10-year gilt with a 5% coupon, priced at 100. Its duration is 7.72. Gilt yields rise 12bp after a hot inflation figure.
- DV01: £10,000,000 × 7.72 × 0.0001 = £7,720 per bp.
- Size of the move: 12bp × £7,720 = £92,640.
- Sign: you own the bonds, so higher yields are a loss: −£92,640.
- As a share of the position: −7.72 × 0.12 = −0.93%.
About −£92,600: a 12bp rise costs just under 1% of a 10-year position.
Check yourself
6 quick questions on the explanation above.
Practice
New numbers every set. Misses and guesses go into your review deck.
In the interview
Answer out loud in under a minute, then compare.
What’s DV01, and how would you hedge a long gilt position with it?
Why does a zero-coupon bond have more duration than a coupon bond of the same maturity?
Why is positive convexity valuable, and who is short it?
Remember
DV01 ≈ notional × duration × 0.0001
Long bonds lose when yields rise, and the longer the bond, the more each basis point costs.