Fixed-rate payments

An asset swap is a synthetically created structure combining a fixed coupon bond with a fixed-floating IRS, which then transforms the bond s swap fixed rate payments to floating rate. The investor retains the original credit exposure to the fixed... 

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). 

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... 

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... 

Exhibit 19.4 shows the fixed-rate payments based on different assumed values for the swap rate. The first three columns of the exhibit show the same information as in Exhibit 19.3—the beginning and end of the quarter and the number of days in the quarter. Column (4) simply uses the notation for the period. That is, period 1 means the end of the first quarter, period 2 means the end of the second qnarter, and so on. The other columns of the exhibit show the payments for each assumed swap rate. 

EXHIBIT 19.4 Fixed-Rate Payments for Several Assumed Swap Rates... 

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... 

DayS( = Number of days in the payment period t then the fixed-rate payment for period t is equal to... 

The present value of the fixed-rate payment for period t is found by multiplying the previous expression by the forward discount factor. If we let PDF, denote the forward discount factor for period t, then the present value of the fixed-rate payment for period t is equal to... 

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... 

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. 

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... 

In contrast, the fixed-rate receiver must make payments with a present value of 11,459,495, but will only receive fixed-rate payments with a present value equal to 9,473,390. Thus, the value of the swap for the fixed-rate receiver is - 1,986,105. Again, as explained earlier, the fixed-rate receiver is adversely affected by a rise in interest rates because it results in a decline in the value of a swap. 

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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