Floating-rate payments

Normally, the fixed interest payments are paid on the basis of a 30/ 360 day count floating-rate payments are paid on the basis of an actual/ 360 day count. Accordingly, the fixed interest payments will differ slightly owing to the differences in the lengths of successive coupon periods. The floating payments will differ owing to day counts as well as movements in the reference rate. 

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 scaled based on the frequency of settlement. 

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). 

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... 

The floating-rate payments are made quarterly based on actual/360. ... 

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... 

Suppose that today 3-month EURIBOR is 4.05%. Let s look at what the fixed-rate payer will receive on 31 March 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 ... 

In our illustration, assuming a nonleap year, the number of days from 1 January of year 1 to 31 March of year 1 (the first quarter) is 90. If 3-month EURIBOR is 4.05%, then the fixed-rate payer will receive a floating-rate payment on March 31 of year 1 equal to... 

Now let s return to our objective of determining the future floating-rate payments. These payments can be locked in over the life of the swap using the EURIBOR futures contract. We will show how these floating-rate payments are computed using this contract. 

We will begin with the next quarterly payment—from 1 April of year 1 to 30 June of year 1. This quarter has 91 days. The floating-rate payment will be determined by 3-month EURIBOR on 1 April of year 1 and paid on 30 June of year 1. Where might the fixed-rate payer look to today... 

Note that each futures contract is for 1 million and hence 100 contracts have a notional amount of 100 million.) Similarly, the EURI-BOR futures contract can be used to lock in a floating-rate payment for each of the next 10 quarters. 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 19.3 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 EURIBOR (4.05%). The payment is shown in the last column. Column (8). 

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 euro amount of the fixed-rate payment for the period is... 

EXHIBIT 19.3 Floating-Rate Payments Based on Initial EURIBOR and EURIBOR Eutures... 

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 EURIBOR in our illustration). 

Calculating the Present Value of the Floating-Rate Payments... 

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 EURIBOR 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 ... 

Given the floating-rate payment for a period and the forward discount factor for the period, the present value of the payment can be computed. For example, from Exhibit 19.3 we see that the floating-rate payment for period 4 is 1,206,222. From Exhibit 19.5, the forward discount factor for period 4 is 0.95689609. Therefore, the present value of the payment is... 

Exhibit 19.6 shows the present value for each payment. The total present value of the 12 floating-rate payments is 14,052,917. Thus, the present value of the payments that the fixed-rate payer will receive is 14,052,917 and the present value of the payments that the fixed-rate receiver will make is 14,052,917. 

The fixed-rate payer will require that the present value of the fixed-rate payments that must be made based on the swap rate not exceed the 14,052,917 payments to be received from the floating-rate payments. The fixed-rate receiver will require that the present value of the fixed-rate payments to be received is at least as great as the 14,052,917 that must be paid. This means that both parties will require a present value for the fixed-rate payments to be 14,052,917. If that is the case, the present value of the fixed-rate payments is equal to the present value of the floating-rate payments and therefore the value of the swap is zero for both parties at the inception of the swap. The interest rates that should be used to compute the present value of the fixed-rate payments are the same interest rates as those used to discount the floating-rate payments. 

PV of floating-rate payments = PV of fixed-rate payments... 

We can now sum up the present value of the fixed-rate payment for each period to get the present value of the floating-rate payments. Using the Greek symbol sigma, X, to denote summation and letting N be the number of periods in the swap, then the present value of the fixed-rate payments can be expressed as... 

Let s apply the formula to determine the swap rate for our 3-year swap. Exhibit 19.7 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 19.6. The sum of the last column in Exhibit 19.7 shows that the denominator of the swap rate formula is 281,764,282. We know from Exhibit 19.6 that the present value of the floating-rate payments is 14,052,917. Therefore, the swap rate is... 

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. 

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 EURIBOR forward rates from the current EURIBOR 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. 

To illustrate this, consider the 3-year swap used to demonstrate how to calculate the swap rate. Suppose that one year later, interest rates change as shown in Columns (4) and (6) in Exhibit 19.8. In Colnmn (4) shows the current 3-month EURIBOR. In Column (5) are the EURIBOR futures price for each period. These rates are used to compute the forward rates in Column (6). Note that the interest rates have increased one year later since the rates in Exhibit 19.8 are greater than those in Exhibit 19.3. As in Exhibit 19.3, the current 3-month EURIBOR and the forward rates are used to compute the floating-rate payments. These payments are shown in Column (8) of Exhibit 19.8. 

In Exhibit 19.10 the forward discount factor (from Exhibit 19.9) and the floating-rate payments (from Exhibit 19.8) are shown. The fixed-rate payments need not be recomputed. They are the payments shown in Column (8) of Exhibit 19.4. This is the fixed-rate payments for the swap rate of 4.9875% and is reproduced in Exhibit 19.10. Now the two payment streams must be discounted using the new forward discount factors. As shown at the bottom of Exhibit 19.10, the two present values are as follows ... 

Present value of floating-rate payments 11,459,495 Present value of fixed-rate payments 9,473,390... 

EXHIBIT 19.8 Rates and Floating-Rate Payments One Year Later if Rates Increase... 

The fixed-rate payer will receive the floating-rate payments. And these payments have a present value of 11,459,495. The present value of the payments that must be made by the fixed-rate payer is 9,473,390. Thus, the swap has a positive value for the fixed-rate payer equal to the difference in the two present values of 1,986,105. This is the value of the swap to the fixed-rate payer. Notice, when interest rates increase (as they did in the illustration analyzed), the fixed-rate payer benefits because the value of the swap increases. 

The present value at time 0 of the floating-rate payment is given by equation (7.4). 

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