Bond Carry and Rolldown: Reconciling Yield and Return Measures

# question regarding carry & roll of a bond


# question regarding carry & roll of a bond












I have a simple (and might be a dumb) question regarding the calculation of a bond's carry. If someone doesn't take into account cost of financing (e.g. the repo rate) then the bond's approximate return over a short time period is carry (coupon return + pull to par) plus roll-down return:

$$ r\approx C\delta t +(y-C)\delta t -D\delta y $$

But on bloomberg and on several forums I frequently stumbled into the following expression for carry:

$$ \text{carry} = \text{forward yield} - \text{spot yield} $$

Could somebody please clarify or derive what's the logic behind this?

Thanks

## Answer by Helin (score 6, accepted)

https://quant.stackexchange.com/a/25330

The formula you quote (forward minus spot) is the yield carry for a financed position.

The problem is that different people use the word carry to mean different things. The most commonly used convention, at least when we prepare analytical reports and quote sheets, is to use the word "Carry" to refer to the breakeven measure – it tells us how much yield can increase before a financed position starts to lose money. And of course, if spot yield rises to the forward yield, that's when it happens. (If you write out the math, you'll also see this is basically coupon income + pull-to-par - financing cost, in yield terms).

"Rolldown" is typically tabulated separately, and the sum of Carry and Rolldown (usually written as "RD&C") is the complete measure of how much I expect to make from a financed position, assuming an unchanged yield curve.

## Answer by Riccardo (score 6)

https://quant.stackexchange.com/a/37789

Carry and roll-down are two different measures.

The formula you mention [carry = fwd_yield - spot_yield] is the pure carry. There is no roll-down consideration there.

The formula is a consequence of the arbitrage-free assumption: the forward is unobservable in the market and is calculated exactly as fwd_yield = spot_yield + carry. And the carry itself is calculated using something like your approach.

This post discusses what carry and roll are: http://swapsball.net/how-to-calculate-carry-and-roll-down-for-a-bond-futures-asset-swap/

## Answer by xkeecs (score 1)

https://quant.stackexchange.com/a/53243

The first formula is right while the second formula doesn't include the pull to par effect. It's essentially just Cpn - repo. But the (y-C) term is also part of the (unrealized) carry.

## Answer by NikBon (score 0)

https://quant.stackexchange.com/a/69852

Forward(1Yx1Y) = (1+S2)/(1+S1)-1

where S1 and S2 are the Spot rates for 1Y and 2Y

If carry is positive then Forward rate > Spot rate by rewriting the Forward in terms of ratio of spot rates as above you get:

- (1+S2)/(1+S1)-1 > S1

- 1+S2 > (1+S1)^2

This means that you can can borrow money for 1Y (or sell a 1Y bond) at S1 and with that money buy a 2Y bond yielding S2. After 1Y you will have to pay back and borrow again for the remaining year at S1. This assuming the short term rate S1 remained the same, if meanwhile cost of financing went up the trade might turn out to be unprofitable