A company has an existing ten-year loan from a bank on which it pays a floating rate of interest...
A company has an existing ten-year loan from a bank on which it pays a floating rate of interest (linked to an interbank lending rate, such as, say, Libor). It is now worried that interest rates will rise but does not want the hassle and expense of cancelling the loan.
So instead it enters into a separate swap agreement with another bank. It agrees to pay a fixed rate of interest to that bank over the same term as the original loan and on the same 'notional' amount of borrowing (notional because the swap does not involve a new loan). It receives a floating rate in return (also linked to Libor).
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Since it is now paying a variable rate on its original borrowing and receiving a variable rate under the swap, the firm's effective interest-rate exposure is the swap fixed rate. That is the 'swap rate'.