Separating Carry and Roll-Down for a Zero-Coupon Swap

Summary

The discussion compares two ways to describe the one-day profit and loss of a payer in a short-dated zero-coupon interest rate swap. One interpretation defines carry as the predictable accrual component and roll-down as the value change from the curve’s movement as the swap ages. Under that framing, a zero-coupon swap has no coupon accrual, so its return is attributed to roll-down, while financing can be treated as carry.

A second convention assigns carry to the difference between the overnight rate and the swap rate over the day, and roll-down to repricing the remaining swap at the unchanged spot curve after one day. The answers show that the terms depend on desk convention and the chosen decomposition. The example provides a useful conceptual distinction, but it does not establish a single universally accepted definition or resolve details such as discounting and financing treatment.

Key ideas

• Carry is often used for predictable accrual, while roll-down describes repricing as a position ages along the curve.
• A zero-coupon swap has no coupon accrual under the accrual-based definition of carry.
• Another convention measures carry from the overnight rate relative to the swap rate.
• The precise carry and roll-down decomposition depends on the market convention being used.

Tags

# Carry vs Roll-Down on a zero-coupon IRS


# Carry vs Roll-Down on a zero-coupon IRS












I am trying to understand the differences between carry vs roll-down on a zero-coupon interest rate swap.

Lets say we have a 10 day ZC IRS, meaning we will only swap once on maturity. We are a payer of the swap.

- Current 10-day spot rate: 3%

- Current 9-day spot rate: 2.9%

- Current Overnight rate: 3.2%

What is the carry on this trade? What is the roll-down?

## Answer by Dimitri Vulis (score 4)

https://quant.stackexchange.com/a/48828

(This is my opinion; someone is likely to disagee).

I like to think of the carry as the predictable part (e.g. the coupon that accrues daily) and the rolldown as the stochastic part (the curves moved - maybe the forwards realized, maybe not. A good estimate of what it might turn out to be as to reprice for the next day assuming all forwards are realized.

I would therefore view a zero-coupon as having all rolldown and no carry.

You could view the financing cost of the swap as carry.

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/48834

Most people would say: carry = the 1day p/l resulting from overnight rate being different from coupon = (3.2- 3.0)* 1day accrual. Roll down = p/l on remaining swap assuming spot rates remain the same = (2.9-3.0) * 9 days accrual.

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