# VALUING INTEREST RATE SWAPS - Global Money Markets

In an interest rate swap, the counterparties agree to exchange periodic interest payments. The dollar amount of the interest payments exchanged is based on the notional principal. In the most common type of swap, there is a fixed-rate payer and a fixed-rate receiver. The convention for quoting swap rates is that a swap dealer sets the floating rate equal to the reference rate and then quotes the fixed rate that will apply.

Computing the Payments for a Swap

In the previous section we described in general terms the payments by the fixed-rate payer and fixed-rate receiver but we did not give any details. That is, we explained that if the swap rate is 6% and the notional amount is $100 million, then the fixed-rate payment will be$6 million for the year and the payment is then adjusted based on the frequency of settlement. So, if settlement is semiannual, the payment is $3 million. If it is quarterly, it is$1.5 million. Similarly, the floating-rate payment would be found by multiplying the reference rate by the notional amount and then scaling based on the frequency of settlement.

It was useful to illustrate the basic features of an interest rate swap with simple calculations for the payments such as described above and then explain how the parties to a swap either benefit or hurt when interest rates change. However, we will show how to value a swap in this section. To value a swap, it is necessary to determine both the present value of the fixed-rate payments and the present value of the floating rate payments. The difference between these two present values is the value of a swap. As will be explained below, whether the value is positive (i.e., an asset) or negative (i.e., a liability) will depend on the party.

At the inception of the swap, the terms of the swap will be such that the present value of the floating-rate payments is equal to the present value of the fixed-rate payments.That is, the value of the swap is equal to zero at its inception. This is the fundamental principle in determining the swap rate (i.e., the fixed rate that the fixed-rate payer will make).

2 A question that commonly arises is why is the fixed rate of a swap is quoted as a fixed spread above a Treasury rate when Treasury rates are not used directly in swap valuation? Because of the timing difference between the quote and settlement, quoting the fixed-rate side as a spread above a Treasury rate allows the swap dealer to hedge against changing interest rates.

Here is a road map of the presentation. First we will look at how to compute the floating-rate payments.We will see how the future values of the reference rate are determined to obtain the floating rate for the period. From the future values of the reference rate we will then see how to compute the floating-rate payments taking into account the number of days in the payment period. Next we will see how to calculate the fixed rate payments given the swap rate. Before we look at how to calculate the value of a swap, we will see how to calculate the swap rate. This will require an explanation of how the present value of any cash flow in an interest rate swap is computed. Given the floating-rate payments and the present value of the floating-rate payments, the swap rate can be determined by using the principle that the swap rate is the fixed rate that will make the present value of the fixed-rate payments equal to the present value of the floating-rate payments. Finally, we will see how the value of swap is determined after the inception of a swap.

Calculating the Floating-Rate Payments

For the first floating-rate payment, the amount is known. For all subsequent payments, the floating-rate payment depends on the value of the reference rate when the floating rate is determined. To illustrate the issues associated with calculating the floating-rate payment, we will assume that

■ a swap starts today, January 1 of year 1(swap settlement date)

■ the floating-rate payments are made quarterly based on “actual/360”

■ the reference rate is 3-month LIBOR

■ the notional amount of the swap is $100 million ■ the term of the swap is three years The quarterly floating-rate payments are based on an “actual/360” day count convention. Recall that this convention means that 360 days are assumed in a year and that in computing the interest for the quarter, the actual number of days in the quarter is used. The floating-rate payment is set at the beginning of the quarter but paid at the end of the quarter— that is, the floating-rate payments are made in arrears. Suppose that today 3-month LIBOR is 4.05%. Let’s look at what the fixed-rate payer will receive on March 31 of year 1—the date when the first quarterly swap payment is made. There is no uncertainty about what the floating-rate payment will be. In general, the floating-rate payment is determined as follows: notional amount x (3-month Libor) x no.of days in period / 360 In our illustration, assuming a non-leap year, the number of days from January 1 of year 1 to March 31 of year 1 (the first quarter) is 90. If 3-month LIBOR is 4.05%, then the fixed-rate payer will receive a floating rate payment on March 31 of year 1 equal to:$100,000,000 x 0.0405 x 90/360 = $1,012,500 Now the difficulty is in determining the floating-rate payment after the first quarterly payment. That is, for the 3-year swap there will be 12 quarterly floating-rate payments. So, while the first quarterly payment is known, the next 11 are not. However, there is a way to hedge the next 11 floating-rate payments by using a futures contract. Specifically, the futures contract used to hedge the future floating-rate payments in a swap whose reference rate is 3-month LIBOR is the Eurodollar CD futures contract. Determining Future Floating-Rate Payments Now let’s determine the future floating-rate payments.These payments can be locked in over the life of the swap using the Eurodollar CD futures contract. We will show how these floating-rate payments are computed using this contract. We will begin with the next quarterly payment—from April 1 of year 1 to June 30 of year 1. This quarter has 91 days. The floating-rate payment will be determined by 3-month LIBOR on April 1 of year 1 and paid on June 30 of year 1. Where might the fixed-rate payer look to today (January 1 of year 1) to project what 3-month LIBOR will be on April 1 of year 1? One possibility is the Eurodollar CD futures market. There is a 3-month Eurodollar CD futures contract for settlement on June 30 of year 1. That futures contract will express the market’s expectation of 3- month LIBOR on April 1 of year 1. For example, if the futures price for the 3-month Eurodollar CD futures contract that settles on June 30 of year 1 is 95.85, then as explained above, the 3-month Eurodollar futures rate is 4.15%. We will refer to that rate for 3-month LIBOR as the “forward rate.” Therefore, if the fixed-rate payer bought 100 of these 3- month Eurodollar CD futures contracts on January 1 of year 1 (the inception of the swap) that settle on June 30 of year 1, then the payment that will be locked in for the quarter (April 1 to June 30 of year 1) is$100,000,000 × 0.0415 91 / 360 =$1,049,028 Floating rate payments based on intial LIBOR and Euro dollar cd features (Note that each futures contract is for$1 million and hence 100 contractshave a notional amount of $100 million.) Similarly, the Eurodollar CD futures contract can be used to lock in a floating-rate payment for each of the next 10 quarters.3 Once again, it is important to emphasize that the reference rate at the beginning of period t determines the floating rate that will be paid for the period. However, the floating-rate payment is not made until the end of period t. Exhibit shows this for the 3-year swap. Shown in Column (1) is when the quarter begins and in Column (2) when the quarter ends. The payment will be received at the end of the first quarter (March 31 of year 1) and is$1,012,500. That is the known floating-rate payment as explained earlier. It is the only payment that is known. The information used to compute the first payment is in Column (4) which shows the current 3-month LIBOR (4.05%). The payment is shown in the last column, Column (8).
Notice that Column (7) numbers the quarters from 1 through 12.

Look at the heading for Column (7). It identifies each quarter in terms of the end of the quarter. This is important because we will eventually be

Calculating the Fixed-Rate Payments

The swap will specify the frequency of settlement for the fixed-rate payments. The frequency need not be the same as the floating-rate payments. For example, in the 3-year swap we have been using to illustrate the calculation of the floating-rate payments, the frequency is quarterly. The frequency of the fixed-rate payments could be semiannual rather than quarterly.
In our illustration we will assume that the frequency of settlement is quarterly for the fixed-rate payments, the same as with the floating-rate payments. The day count convention is the same as for the floating-rate payment, “actual/360”. The equation for determining the dollar amount of the fixed-rate payment for the period is:

notional amount x (swap rate) x no.of days in a period / 360

It is the same equation as for determining the floating-rate payment except that the swap rate is used instead of the reference rate (3-month LIBOR in our illustration).

For example, suppose that the swap rate is 4.98% and the quarter has 90 days. Then the fixed-rate payment for the quarter is:

$100,000,000 x 0.0498 x 90 / 360 =$1,245,000

If there are 92 days in a quarter, the fixed-rate payment for the quarter is: Note that the rate is fixed for each quarter but the dollar amount of the payment depends on the number of days in the period.

Calculation of the Swap Rate

Now that we know how to calculate the payments for the fixed-rate and floating-rate sides of a swap where the reference rate is 3-month LIBOR given (1) the current value for 3-month LIBOR, (2) the expected 3-month LIBOR from the Eurodollar CD futures contract, and (3) the assumed swap rate, we can demonstrate how to compute the swap rate.

At the initiation of an interest rate swap, the counterparties are agreeing to exchange future payments and no upfront payments are made by either party. This means that the swap terms must be such that the present value of the payments to be made by the counterparties must be at least equal to the present value of the payments that will be received. In fact, to eliminate arbitrage opportunities, the present value of the payments made by a party will be equal to the present value of the payments received by that same party. The equivalence (or no arbitrage) of the present value ofthe payments is the key principle in calculating the swap rate.

Since we will have to calculate the present value of the payments, let’s show how this is done.

Calculating the Present Value of the Floating-Rate Payments

As explained earlier, we must be careful about how we compute the present value of payments. In particular, we must carefully specify (1) the timing of the payment and (2) the interest rates that should be used to discount the payments. We have already addressed the first issue. In constructing the exhibit for the payments, we indicated that the payments are at the end of the quarter. So, we denoted the time periods with respect to the end of the quarter.

Now let’s turn to the interest rates that should be used for discounting.

First, every cash flow should be discounted at its own discount rate using a spot rate. So, if we discounted a cash flow of $1 using the spot rate for period t, the present value would be: Fixed-rate payments for several assumed swap rates Calculating the forward discount factor Second, forward rates are derived from spot rates so that if we discounted a cash flow using forward rates rather than spot rates, we would come up with the same value. That is, the present value of$1 to be received in period t can be rewritten as:

We will refer to the present value of $1 to be received in period t as the forward discount factor. In our calculations involving swaps, we will compute the forward discount factor for a period using the forward rates. These are the same forward rates that are used to compute the floating rate payments—those obtained from the Eurodollar CD futures contract. We must make just one more adjustment. We must adjust the forward rates used in the formula for the number of days in the period (i.e., the quarter in our illustrations) in the same way that we made this adjustment to obtain the payments. Specifically, the forward rate for a period, which we will refer to as the period forward rate, is computed using the following equation: Solving for the swap rate All of the values to compute the swap rate are known. Let’s apply the formula to determine the swap rate for our 3-year swap. Exhibit shows the calculation of the denominator of the formula. The forward discount factor for each period shown in Column (5) is obtained from Column (4) of Exhibit .The sum of the last column in Exhibit shows that the denominator of the swap rate formula is$281,764,282. We know from Exhibit that the present value of the floating-rate payments is \$14,052,917. Therefore, the swap rate is

Given the swap rate, the swap spread can be determined. For example, since this is a 3-year swap, the convention is to use the 3-year on-the run Treasury rate as the benchmark. If the yield on that issue is 4.5875%, the swap spread is 40 basis points (4.9875% − 4.5875%).

The calculation of the swap rate for all swaps follows the same principle: equating the present value of the fixed-rate payments to that of the floating-rate payments.

Valuing a Swap

Once the swap transaction is completed, changes in market interest rates will change the payments of the floating-rate side of the swap. The value of an interest rate swap is the difference between the present value of the payments of the two sides of the swap. The 3-month LIBOR forward rates from the current Eurodollar CD futures contracts are used to (1) calculate the floating-rate payments and (2) determine the discount factors at which to calculate the present value of the payments.

Calculating the denominator for the swap rate formula