Fixed-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). 

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. 

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). 

In the accounting practice the operating and capital costs must be expressed on a common basis. The first are normally reported on yearly basis. With the capital the situation is different. We may consider that the capital is borrowed over a period of time, usually between five and ten years, the repayment will be done following a fixed sum called annuity. Suppose that the present value of the invested capital is P. If invested at a fixed interest rate i over n years, this will produce an equivalent amount F, as given by the equation (15.6). The reimbursement by yearly payments A can be... 

Strictly speaking, the FIAT 1 transaction does not generate excess spread. This explains the high level of credit enhancement from the unrated class M notes (usually, unrated tranches are either privately sold or kept as an equity tranche by the originator). On the closing date, an amount of notes was issued which was equal to the net present value of all future cash payments due from the collateral (as opposed to the principal balance of the collateral). The discount rate used was the fixed rate payable to the swap counterparty (swap rate plus coupon on the class A notes and all fees associated with the transaction). Structured this way, the receivables always yield the discount rate, leaving no excess spread in the transaction. However, losses on the FIAT 1 portfolio can be covered to a certain degree from interest collections because the structure provides for delinquent principal and defaults to be covered before interest is paid on the class M notes. 

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 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. 

Consider a plain vanilla interest rate swap with a notional principal of M that pays n interest payments through its maturity date, T. Payments are made on dates 4 where i = 1,. ..n. The present value today of a future payment made at time r, is denoted as PV 0, t,). If the swap rate is r, the present value of the fixed-leg payments, PVfi s is given by equation (7.2). 

Level-payment fixed-rate mortgages, the conventional form in the United States, have fixed terms to maturity and specify monthly payments of fixed-rate interest. The monthly interest payment on a conventional fixed-rate mortgage is given by formula (14.3), which is derived from the conventional present-value equation via the steps shown in (14.1) and (14.2). 

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