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
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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