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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

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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.

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    $\begingroup$ Thanks, could you please write out the math? It would a help a lot. $\endgroup$
    – Daniel
    Apr 9, 2016 at 17:53
  • $\begingroup$ I think this definition of carry is a bit deceptive, because if you think of carry as how much you earn if the spot yield stays at the same level, then it is exactly the spot yield-repo rate for a financed position instead of the difference between the forward yield and the spot yield. But if you think of carry as a cushion against the change in spot yield before you start loosing money then it's correct. $\endgroup$
    – Daniel
    Apr 10, 2016 at 13:59
  • $\begingroup$ "But if you think of carry as a cushion against the change in spot yield before you start loosing money then it's correct." And you also have to assume that the yield curve is flat, in order to separate the rolldown effect from the carry. $\endgroup$
    – Daniel
    Apr 10, 2016 at 15:10
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    $\begingroup$ Carry is actually the most reliable part of bond returns; it's exactly known on an ex-ante basis and is not contingent on what happens to the yield curve. In dollar terms, carry = (ending accrued interest – starting accrued interest) – (starting price + starting AI) x repo rate x year fraction [or in words, carry = coupon income – financing cost]. Incidentally, forward price = spot price MINUS the quantity above; i.e., forward price = spot price – carry. This is how everything ties together. $\endgroup$
    – Helin
    Apr 10, 2016 at 22:33
  • $\begingroup$ For anyone coming across this answer the related answer here is a cross-reference to the same concept: quant.stackexchange.com/questions/36253/… "LINK" $\endgroup$
    – Attack68 ♦
    Oct 1, 2017 at 20:44
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Carry and roll-down are two different measures.

The carry is the PNL resulting from holding a position. However, even if you don't finance the bond in repo, you can still measure your carry as the yield-to-maturity of maturity of the bond vs the yield of the alternative default investment you would have made with your cash (for example 0% if sitting on your bank account at 0%, but maybe it'd be 1%, etc).

The formula you mention [carry = fwd yield - spot yield] is due to the arbitrage-free assumption: say carry > fwd yield - spot yield, then the fwd yield is priced too low and I could sell the bond fwd, buy it spot, hold it until the fwd delivery date and make a positive PNL.

Roll down is the mark-to-market due to the passage of time assuming that the shape of the curve doesn't change. This is a strong assumption and has a few limitations.

In this post I discuss what carry and roll are, and look at the bond future's asset swap as well: http://swapsball.net/how-to-calculate-carry-and-roll-down-for-a-bond-futures-asset-swap/

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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.

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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. (1+S2)/(1+S1)-1 > S1
  2. 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

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