## Definition

In an interest rate swap, party A agrees to pay to party B cash flows equal to interest at a predetermined fixed rate on a notional principal for a predetermined number of years. In return, party A receives interest at a floating rate on the same notional principal for the same period of time from party B.

## Definition

A swap is an over-the-counter derivatives agreement between two counterparties to exchange a series of cashflows at agreed dates.

## A Little Bit More

The swap agreement defines the dates when the cash flows are to be paid and the way in which they are to be calculated. Usually the calculation of the cash flows involves the future value of an interest rate, an exchange rate, or other market variable.

A forward contract can be viewed as a simple example of a swap. Suppose it is March 1, 2017, and a company enters into a forward contract to buy 100 ounces of gold for $1,200 per ounce in one year. The company can sell the gold in one year as soon as it is received. The forward contract is therefore equivalent to a swap where the company agrees that on March 1, 2018, it will pay$120,000 and receive 100S, where S is the market price of one ounce of gold on that date.

Whereas a forward contract is equivalent to the exchange of cash flows on just one future date, swaps typically lead to cash-flow exchanges taking place on several future dates.

## Basics

There are three common types of swaps: interest rate swap, currency swap and equity swap.

## History

The birth of the over-the-counter swap market can be traced to a currency swap negotiated between IBM and the World Bank in 1981. The World Bank had borrowings denominated in U.S. dollars while IBM had borrowings denominated in German deutsche marks and Swiss francs. The World Bank (which was restricted in the deutsche mark and Swiss franc borrowing it could do directly) agreed to make interest payments
on IBM’s borrowings while IBM in return agreed to make interest payments on the World Bank’s borrowings. Since that first transaction in 1981, the swap market has seen phenomenal growth.

## Valuating Swaps

When valuing swaps, we require a ‘‘risk-free’’ discount rate for cash flows. Prior to the 2008 crisis, LIBOR was used as a proxy for the risk-free discount rate. Since the 2008 credit crisis, the market has switched to using the OIS rate for discounting.

## Definition

A forward rate agreement (FRA) is a forward contract between parties that determines the rate of interest, or the currency exchange rate, to be paid or received on an obligation beginning at a future start date.

## Basics

• The long position in an FRA is the party that would borrow money (long the loan, with the contract price being the interest rate on the loan).
• If LIBOR at expiration is above the rate specified in the forward agreement, the long position in the contract can be viewed as the right to borrow at below market rates, and the long will receive a payment.
• If LIBOR at the expiration date is below the FRA rate, the short will receive a cash payment from the long. (The right to lend at above market rates has a positive value.)
• The notation for FRAs is unique. For example, a 2×3 FRA is a contract that expires in two months (60 days), and the underlying loan is settled in three months (90 days). The underlying rate is 1-month (30-day) LIBOR on a 30-day loan in 60 days. A timeline for a 2×3 FRA is shown below.

## Pricing an FRA

The “price” of the FRA is actually the forward interest rate implied by the spot rates consistent with the FRA. For example, the “price” of the 2×3 FRA is the 30-day forward rate in 60 days implied by the 60- and 90-day spot rates.

## Valuing an FRA

The value of an FRA to the long or short position comes from the interest savings on a loan to be made at the settlement date. This value is to be received at the end of the loan, so the value of an FRA after initiation is the present value of these savings. Remember, if the rate in the future is less than the FRA rate, the long is “obligated to borrow” at above-market rates and will have to make a payment to the short. If the market interest rate is greater than the FRA rate, the long will receive a payment from the short.

Lets outline the general steps for valuing a 2×3 FRA (a 30-day loan in 60 days) 40 days after initiation (which means there are 20 days remaining until the FRA expires).

1. Step 1: Calculate the implied 30-day forward rate at the settlement date, 20 days from now, using the current 20-day spot rate and the current 50-day spot rate. Basically, this calculates a new FRA rate (same loan term as the original one) if it is imitated today.
2. Step 2: Calculate the value of the FRA at maturity as the notional principal times the difference between the forward rate from Step 1 and the original FRA “price.” Make sure to convert from an annual rate to a 30-day rate. If the current forward rate is greater than the original FRA price, the long position has positive value. If the current forward rate is less than the original FRA price, the short position has positive value.
3. Step 3: Calculate the value of the FRA today by discounting the value at maturity from Step 2 at the 50-day spot rate.

## Definition

A forward contract is a non-standardized contract between two parties to buy or to sell an asset at a specified future time at a price agreed upon today.

## Key Points

The price of a forward contract is the price specified in the contract at which the long and short sides have agreed to trade the underlying asset when the contract expires.

The value of a forward contract to each side is the amount of money the counterparty would be willing to pay (or receive) to terminate the contract. Its a zero-sum game, so the value of the long position is equal to the negative of the value of the short position.

The no-arbitrage price of the forward contract (with a maturity of T years) is the price at which the value of the long side and the value of the short side are both equal to zero.

$latex FP = S_0 \times (1+R_f)^T$

The value of the short position at any point in time is the negative of the long position.