# Interest Rate Swaps

An interest rate swap is an over-the-counter derivative contract in which counterparties exchange cash flows based on two different fixed or floating interest rates. The swap contract in which one party pays cash flows at the fixed rate and receives cash flows at the floating rate is the most widely used interest rate swap and is called the plain-vanilla swap or just vanilla swap.

You can think of an interest rate swap as a series of forward contracts. Because an interest rate swap is a tailor-made contract purchased over the counter, it is subject to credit risk. Just like a forward contract, the swap has zero value at inception and hence no cash changes hand at initiation. However, a swap must have a notional amount which represent the amount to which interest rates are applied to calculate periodic cash flows.

Let’s say you have a 5-years $100 million loan at a variable interest rate which equals LIBOR plus a spread of 100 basis points. You would prefer to pay a fixed interest rate to be able to better forecast your cash flow requirements. You can’t go back to the bank and change the loan to a fixed-rate loan. However, you can manage your risk by entering a fixed-for-floating interest rate swap which requires you to pay an amount determined based on a fixed rate and receive an amount calculated at the floating rate. You can match the amount and timing of the swap cash flows you receive with the cash flows you are required to make on the loan.

## Pricing of interest rate swap

You can think of a pay fixed, receive floating swap as a combination of a long position in a fixed rate bond and a short position in a floating rate bond. This is because you will receive cash flows equal to the periodic cash flows on a fixed-coupon bond and you must pay cash flows which can be replicated as equivalent to coupons on floating rate bond.

The value the swap thus equals the difference between the present value of fixed cash flows and present value of variable cash flows:

### Swap Rate

The swap rate is the rate that applies to the fixed payment leg of a swap. It can be worked out using the following equation:

$$ c=\frac{1\ -\ {\rm PVF}_n}{\sum_{i}^{n}{\rm PVF}_i} $$

It means that the fixed rate on the swap (let's call it c) equals 1 minus the present value factor that applies to the last cash flow date of the swap divided by the sum of all the present value factors corresponding to all the swap dates.

For a fixed-for-floating interest rate swap, the rate is determined and locked at initiation. However, at any point in the swap tenor, it changes with change in floating rates. The new fixed rate corresponding to the new floating rates can be termed as the equilibrium swap rate or equilibrium fixed rate. The value of a swap to the party that pays fixed and receives floating is the difference between the (new) equilibrium fixed rate (let’s say f) minus the fixed rate negotiated and locked at initiation (referred to as c above) at the valuation date multiplied by the sum of all present value factors multiplied by the notional amount N. This can be expressed mathematically as follows:

$$ V_{pay\ fixed}=N\times(f-c)\times\sum_{i}^{n}{\rm PVF}_i $$

## Example

Let’s say you negotiate a 2% pay fixed, receive floating swap to convert your 5-years $100 million loan to a fixed loan. Calculate the value of your swap one year down the road, given the following floating rates present value factor schedule:

Year | PVF |
---|---|

1 | 0.99 |

2 | 0.97 |

3 | 0.95 |

4 | 0.94 |

The equilibrium fixed swap rate one year down the road is 1.56%:

$$ F=\frac{1\ -\ 0.94}{0.99+0.97+0.95+0.94}=\frac{0.06}{3.85}=1.56\% $$

Because you have locked a 2% fixed rate on the loan, the value of the swap is -$1.7 million:

$$ V_{pay\ fixed}=$100\ million\times(1.56\%-2\%)\times3.85=-1.7\ million $$

Because the current equilibrium fixed rate is lower than the rate that you negotiated for the whole life of your swap, the current value of swap is negative for you. This is because you have committed to pay 2% for the life of the swap but the current floating rates structure corresponds to 1.56% fixed rate.

The value to the counterparty, i.e. the party paying floating and receiving fixed is the exact opposite of the value above.