The carry on a swap, as with that of any financial instrument (e.g. a bond), is the difference between a fixed coupon received by the investor and a floating financing rate paid by that investor over a specific horizon. This implies that the carry can be locked in at the trade date of the swap by paying forward on the swap. The forward price of the swap, in this sense, reflects the effects of carry:
Forward price = sport price – carry
It follows that the carry is equal to the difference between the spot price of the swap (it is zero for an at-the-money swap) and the “locked-in” forward price of this swap.
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