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 fixedrate payer and a fixedrate 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 fixedrate payer and fixedrate 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 fixedrate 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 floatingrate 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 fixedrate 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 floatingrate payments is equal to the present value of the fixedrate 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 fixedrate 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 fixedrate 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 floatingrate 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 floatingrate 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 floatingrate payments and the present value of the floatingrate 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 fixedrate payments equal to the present value of the floatingrate payments. Finally, we will see how the value of swap is determined after the inception of a swap.
Calculating the FloatingRate Payments
For the first floatingrate payment, the amount is known. For all subsequent payments, the floatingrate payment depends on the value of the reference rate when the floating rate is determined. To illustrate the issues associated with calculating the floatingrate payment, we will assume that
■ a swap starts today, January 1 of year 1(swap settlement date)
■ the floatingrate payments are made quarterly based on “actual/360”
■ the reference rate is 3month LIBOR
■ the notional amount of the swap is $100 million
■ the term of the swap is three years
The quarterly floatingrate 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 floatingrate payment is set at the beginning of the quarter but paid at the end of the quarter— that is, the floatingrate payments are made in arrears.
Suppose that today 3month LIBOR is 4.05%. Let’s look at what the fixedrate 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 floatingrate payment will be. In general, the floatingrate payment is determined as follows:
notional amount x (3month Libor) x no.of days in period / 360
In our illustration, assuming a nonleap year, the number of days from January 1 of year 1 to March 31 of year 1 (the first quarter) is 90. If 3month LIBOR is 4.05%, then the fixedrate 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 floatingrate payment after the first quarterly payment. That is, for the 3year swap there will be 12 quarterly floatingrate payments. So, while the first quarterly payment is known, the next 11 are not. However, there is a way to hedge the next 11 floatingrate payments by using a futures contract. Specifically, the futures contract used to hedge the future floatingrate payments in a swap whose reference rate is 3month LIBOR is the Eurodollar CD futures contract.
Determining Future FloatingRate Payments
Now let’s determine the future floatingrate payments.These payments can be locked in over the life of the swap using the Eurodollar CD futures contract. We will show how these floatingrate 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 floatingrate payment will be determined by 3month LIBOR on April 1 of year 1 and paid on June 30 of year 1. Where might the fixedrate payer look to today (January 1 of year 1) to project what 3month LIBOR will be on April 1 of year 1? One possibility is the Eurodollar CD futures market. There is a 3month 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 3month Eurodollar CD futures contract that settles on June 30 of year 1 is 95.85, then as explained above, the 3month Eurodollar futures rate is 4.15%. We will refer to that rate for 3month LIBOR as the “forward rate.” Therefore, if the fixedrate 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 floatingrate 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 floatingrate payment is not made until the end of period t.
Exhibit shows this for the 3year 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 floatingrate 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 3month 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 FixedRate Payments
The swap will specify the frequency of settlement for the fixedrate payments. The frequency need not be the same as the floatingrate payments. For example, in the 3year swap we have been using to illustrate the calculation of the floatingrate payments, the frequency is quarterly. The frequency of the fixedrate payments could be semiannual rather than quarterly.
In our illustration we will assume that the frequency of settlement is quarterly for the fixedrate payments, the same as with the floatingrate payments. The day count convention is the same as for the floatingrate payment, “actual/360”. The equation for determining the dollar amount of the fixedrate 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 floatingrate payment except that the swap rate is used instead of the reference rate (3month LIBOR in our illustration).
For example, suppose that the swap rate is 4.98% and the quarter has 90 days. Then the fixedrate 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 fixedrate 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 fixedrate and floatingrate sides of a swap where the reference rate is 3month LIBOR given (1) the current value for 3month LIBOR, (2) the expected 3month 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 FloatingRate 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:
Fixedrate 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 3year 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 floatingrate 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 3year swap, the convention is to use the 3year onthe 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 fixedrate payments to that of the floatingrate payments.
Valuing a Swap
Once the swap transaction is completed, changes in market interest rates will change the payments of the floatingrate 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 3month LIBOR forward rates from the current Eurodollar CD futures contracts are used to (1) calculate the floatingrate 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


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