Floating-rate payments present value

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). 

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. 

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. 

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... 

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). 

The pricing of a floating-rate note at issue does not differ from a conventional bond. In fact, it is the present value of coupon payments and principal repayment and is given by (10.3) ... 

A swap s fixed-rate payments are known in advance, so deriving their present values is a straightforward process. In contrast, the floating rates, by definition, are not known in advance, so the swap bank predicts them using the forward rates applicable at each payment date. The fotward rates are those that are implied from current spot rates. These are calculated using equation (7.6). 

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