Financial Math FM/Interest Rate Swaps

Learning objectives
The Candidate will understand key concepts concerning interest rate swaps, and how to perform related calculations.

Learning outcomes
The Candidate will be able to:
 * Define and recognize the definitions of the following terms: swap rate, swap term or swap tenor, notional amount, market value of a swap, settlement dates, settlement period, counterparties, deferred swap, amortizing swap, accreting swap, interest rate swap net payments.
 * Given sufficient information, calculate the market value, notional amount, spot rates or swap rate of an interest rate swap, deferred or otherwise, with either constant or varying notional amount.

Introduction and motivation
As suggested by the name "interest rate swap", there is some sort of "swap" between interest rates. Indeed, the interest rates to be swapped are interest rate, which do not change during the term of the loan, and interest rate, which changes based on the market interest rate.

Since the future market interest rate is uncertain, there are some uncertainties involved for the variable interest rate. Since some may want to avoid such uncertainties, they may want to have fixed, rather than variable, interest rates for their loans. This motivates.

Terminologies
For the variable interest rate, it is often linked to a specified index, and common indexes include the London Inter-Bank Offered Rate (LIBOR) and the prime interest rate. By linking to an index, it does not mean the variable interest rate is exactly the same as the rate from that index. There can be some modifications on it. Typically, the variable interest rate equals an index plus some basis point(s) (bp(s)). The basis point(s) added is call "spread". For the two counterparties, one is and another is.

There are some terminologies related to the exchanges of payments during the swap term.

Sometimes, one counterparty has a loan from a third party for which interest needs to be paid to the third party. In this case, in addition to the net swap payment (if needed), additional payment is also needed.

Most interest rate swaps have a level notional amount over the swap term, but this is not a must, and the notional amount can change over the swap term.

Typically, the first settlement period of an interest rate swap starts at time zero, but again, this is not a must, and the first settlement period can start at later time.

Swap rate
The fundamental principle for determining the swap rate here is that the swap rate is set such that there is no cost to either counterparty to enter into an interest rate swap (assume the transaction cost is zero from now on), i.e. neither of the counterparties are better off or worse off from entering into the interest rate swap. To do this, the present value of expected future payments for each counterparty should be the same. As a result, the expected future payments paid (or received) by each counterparty should be zero.

Market value
An interest rate swap can be sold or closed by a counterparty at a time during the swap term. To sold the swap, the seller receives the market value of the swap at the time point, calculated using the spot rates at that time. To close the swap, a possible way is that one counterparty pays the market value to another counterparty at that time.