Carry and roll-down: the return from standing still
A financed bond position earns money even if nothing happens: the coupon less the repo cost, and the price gain from ageing down an upward-sloping curve. Together they say how far yields can move against you before the trade loses.
- Compute carry on a repo-financed par bond, in dollars
- Compute roll-down by repricing the bond one year older on an unchanged curve
- Find the parallel shift in yields that wipes out a year of carry and roll, and the repo rate at which the trade breaks even
- Say what a flat or inverted curve does to each piece
The intuition
You buy a five-year bond at par with a 4% coupon and finance it in repo at 3% for a year. If nothing happens you collect 4 and pay 3: one point of carry per 100. And something else happens even if the curve does not move: a year later your five-year bond is a four-year bond, and on an upward-sloping curve four-year yields are lower, say 3.80%. A 4% coupon at a 3.80% yield is worth about 100.73, so the bond has gained 0.73 just by ageing. That is roll-down.
Carry plus roll-down, about 1.73 per 100, is the return from standing still, and it is also your cushion. Divide it by what one basis point costs on the bond at the horizon and you have the parallel shift that wipes the year out: roughly 47 bp here. Divide coupon plus roll-down by 100 and you have the repo rate at which the trade breaks even, 4.73%. On a flat curve there is no roll-down; on an inverted curve the bond ages into a higher yield and roll-down turns into a cost; and when repo is above the coupon, carry does too.
Carry per 100 = coupon − repo × 100. Horizon price = the bond priced at today's (n−1)-year yield with n−1 years left; roll-down = horizon price − 100. Breakeven parallel shift ≈ (carry + roll-down) ÷ horizon DV01 per 100. Breakeven repo = (coupon + roll-down) ÷ 100.
Why it works
- The conventions here: an n-year par bond (coupon = today's n-year yield, price 100) financed in full in repo for one year at a simple rate on the price of 100. "Unchanged curve" means the bond is then an (n−1)-year bond priced at today's (n−1)-year yield. The breakeven is a parallel shift over the year, estimated as (carry + roll-down) ÷ the horizon bond's DV01 per 100, which is its modified duration × its price × 0.0001. No coupon reinvestment, no convexity.
- Carry is a slope too. Positive carry means the bond's yield is above repo: the very front of the curve sits below this point, the normal shape. Negative carry means a central bank has pushed the front above it, and holding the bond costs money every day.
- Roll-down is slope times duration. Roughly the yield drop over one year of maturity times the bond's duration. It is biggest where the curve is steepest per year, usually the two- to five-year sector, so a steep slope on a medium bond often beats a flat slope on a long one.
- Why a short bond has the bigger breakeven. The cushion is divided by duration. A two-year bond's small DV01 means each basis point of sell-off costs little, so the same carry absorbs many more basis points. Long bonds can offer more carry and roll in dollars and a thinner cushion in yield.
- Run it backwards. Set the year's return to zero and the repo rate falls out: coupon plus roll-down, per 100. If actual repo is below it, the trade works on an unchanged curve.
- What this is not. A view that yields will fall. It is the return for waiting, and the cushion that view has if it is wrong. On an inverted curve the cushion is negative and buying the bond is purely a bet on cuts.
| Carry: 4 − 3 | 1.00 per 100 |
| Horizon price: a 4% coupon with 4 years left at 3.80% | 100.729 |
| Roll-down: 100.729 − 100 | 0.729 per 100 |
| Total: carry + roll-down | 1.729 per 100 |
| Horizon DV01: modified duration 3.64 × 100.729 × 0.0001 | 0.0366 per 100 per bp |
| Breakeven parallel shift: 1.729 ÷ 0.0366 | about 47 bp |
| Breakeven repo: (4 + 0.729) ÷ 100 | 4.73% |
On a flat curve at 4% the horizon price is 100, roll-down is zero and the year earns carry alone.
The formulas
The coupon you collect less the financing you pay.
The bond repriced one year older at today's yield for that maturity.
The return from standing still.
The cushion divided by what one basis point costs at the horizon.
The financing rate at which the year earns nothing.
Worked example
Price the bond one year older at today's shorter yield, take the gain over par, add the carry. The follow-up asks where on a curve roll-down is usually biggest.
See it move
Same bond and financing. Change the yield, how much the curve drops per year of maturity, the repo rate, the maturity and the face.
- Steepen the curve. The horizon price, the roll-down, the total, the breakeven shift and the breakeven repo all rise; carry does not move, because it depends only on the coupon and repo.
- Raise repo. Carry, the total and the breakeven shift fall; roll-down and the breakeven repo do not move.
- Raise the yield. The coupon and carry rise with it, and so does the total; the roll-down shrinks a little, because a higher-yielding bond has less duration for the same slope to work on.
- Lengthen the maturity. Roll-down and the total rise, but so does the horizon duration, so watch the breakeven shift: more cushion divided by a bigger DV01 can go either way.
Run it backwards
A par bond's coupon and its horizon price on an unchanged curve are given. At what repo rate does financing it for a year exactly break even?
The year earns the coupon plus the roll-down and pays repo on 100. Set them equal: breakeven repo = (coupon + roll-down) ÷ 100. If actual repo is below it, the trade works on an unchanged curve by the difference.
The other inverse divides the same cushion by the horizon DV01 instead, and gives the parallel shift in yields that wipes the year out. Both are the same question: how much can go wrong before the return for waiting is gone.
Traps
Say it in the interview
“What do carry and roll-down mean for a financed bond position?”
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.
- Carry = coupon − repo. Roll-down = the bond repriced one year older at today's shorter yield, minus par.
- Carry + roll-down is the return from standing still and the cushion against a sell-off.
- Breakeven shift = cushion ÷ horizon DV01; breakeven repo = (coupon + roll-down) ÷ 100.
- Flat curve: no roll. Inverted curve: roll is a cost. Repo above the coupon: carry is a cost.