# 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 fixed interest rate, which do not change during the term of the loan, and variable 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 interest rate swap.
Definition. (Interest rate swap) Interest rate swap is an agreement between two counterparties to exchanges payments during a specified period (called swap term or swap tenor) in the future based on interest rates applied to a specified principal amount (called notional (principal) amount).
Remark.
* The principal amount is notional in the sense that this amount is never paid by either of the two counterparties.
* The fixed interest rate for the interest rate swap is called swap rate, denoted by $R$.

## 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 payer and another is receiver.
Definition. (Payer) Payer under the interest rate swap is the counterparty who pays payment based on fixed interest rate.
Remark.
* As a result, payer receives payment based on variable interest rate.
Definition. (Receiver) Receiver under the interest rate swap is the counterparty who receives payment based on fixed interest rate.
Remark.
* As a result, receiver pays payment based on variable interest rate.
There are some terminologies related to the exchanges of payments during the swap term.
Definition. (Settlement dates) Settlement dates is the dates specified in the interest rate swap at which the exchanges of payments occur.
Definition. (Settlement period) Settlement period is the time between settlement dates.
Definition. (Net swap payment) Net swap payment is the amount of money paid by one counterparty to another counterparty on each settlement date.
Remark.
* On each settlement date, only one counterparty pays a certain amount of money to another counterparty, depending on the difference between the payment based on fixed interest rate and that based on variable interest rate.
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.
Definition. (Net interest payment) Net interest payment is the total net payment (interest paid on the loan plus net swap payment (if needed)) made by the counterparty with a loan.
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.
Definition. (Accreting swap) Accreting swap is an interest rate swap for which the notional amount increases over the swap term.
Definition. (Amortizing swap) Amortizing swap is an interest rate swap for which the notional amount decreases 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.
Definition. (Deferred swap) Deferred swap is an interest rate swap for which the first settlement period starts at a time that is later than time zero.

## 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 net payments paid (or received) by each counterparty should be zero.
Definition. (Determining swap rate) Under the swap rate, the present value of the payment based on the fixed interest rate equals that based on the variable interest rate.
Remark.
* With a suitable equation, we can solve for the swap rate.
Example. For an interest rate swap, the (level) notional amount is 10000, and the one-year and two-year spot rate is 3% and 5% respectively. The one-year deferred one-year forward rate $f_{1\to 2}=1.05^{2}/1.03-1\approx 0.0704$ (the payment for the second year based on the variable interest rate is based on this forward rate). Then, the swap rate $R$ is determined by $10000R(1.03^{-1})+10000R(1.05^{-2})=10000(0.05)(1.05^{-1})+10000(0.0704)(1.06^{-2})\Rightarrow R={\frac {0.05/1.05+0.0704/1.06^{2}}{1.03^{-1}+1.05^{-2}}}\approx 5.87\%.$
Remark.
* The amount of the level notional amount does not matter for calculating the swap rate.
* For each year, the payment based on the fixed interest rate is $10000R$ for each of the fist and second year, while the payment based on the variable interest rate is $10000(0.03)$ for the first year, and $10000f_{1\to 2}$ for the second year.

## 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.
Definition. (Market value of interest rate swap) The market value of interest rate swap is the present value of the expected future cash flows under the swap, calculated using the spot rates at the time of determining the market value.
Remark.
* The market value can be positive, negative or zero.
* Since the swap is essentially exchange of payment between two counterparties, the market value of the swap for one counterparty is negative of that for another counterparty, at the same time point.
* The future cash flows are the future net swap payments.
* ↑ It is the average interest rate estimated by leading banks in London that the average leading bank would be charged from borrowing from other banks.
* ↑ It is the rate at which banks in the United States will lend money to their most favoured costumers.
* ↑ A basis point is 0.01%, or 0.0001.