Swaps discount factors

We assume we have constructed a market curve of Libor discount factors where Df(t) is the price today of 1 to be paid at time t. From the perspective of the asset swap seller, it sells the bond for par plus accrued interest. The net up-front payment has a value 100 F where P is the market price of the bond. If we assume both parties to the swap are interbank credit quality, we can price the cash flows off the Libor curve. 

The traditional approach to yield curve fitting involves the calculation of a set of discount factors from market interest rates. From this, a spot yield curve can be estimated. The market data can be money market interest rates, futures and swap rates and bond yields. In general, though this approach tends to produce ragged spot rates and a forward rate curve with pronounced jagged knot points, due to the scarcity of data along the maturity structure. A refinement of this technique is to use polynomial approximation to the yield curve. 

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 ... 

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... 

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 ... 

For each moment in the life of the default swap we can sum up all of the instantaneous chances of actually receiving a default payout of 1 - R by integrating the above equation from time 0 to the maturity date, T. However, we need to make one adjustment, and that is to weight each chance of default by its present value. It is only appropriate that receiving a payout of 1 - R is worth more if that payout occurs tomorrow rather than next year, so after weighting each potential payout by its appropriate discount factor the value of the default protection becomes... 

Expression (7.9) formalizes the hootstrapping process described in chapter 3. Essentially, the -year discount factor is computed using the discount factors for years one to n- and the -year swap or zero-coupon rate. Given the discount factor for any period, that period s zero-coupon, or spot, rate can be derived using (7.9) rearranged as (7.10). 

It is not surprising that the net present value is zero. The zero-coupon curve is used to derive the discount factors that are then used to derive the forward rates that are used to determine the swap rate. As with any financial instrument, the fair value is its break-even price or hedge cost. The bank that is pricing this swap could hedge it with a series of FRAs transacted at the forward rates shown. This method is used to price any interest rate swap, even exotic ones. 

To understand this principal, consider figure 7.5, which shows the present value of both legs of the 5-year swap to be 2,870,137. The same result is obtained by using the 5-year discount factor, as shown in (7.18). 

Our starting point is a set of zero curve tenors (or discount factors) obtained from a collection of market instruments such as cash deposits, futures, swaps, or coupon bonds. We therefore have a set of tenor points and their respective zero rates (or discount factors). The mathematics of cubic splines is straightforward, but we assume a basic understanding of calculus and a familiarity with solving simultaneous linear equations by substitution. An account of the methods analyzed in this section is given in Burden and Faires (1997), which has very accessible text on cubic spline interpolation. ... 

Equation (7.16) captures the ins ht that an interest rate swap can be considered as a strip of futures. Since this strip covers the same period as the swap, it makes sense that, as (7.16) states, the swap rate can be computed as the average of the forward rates from tfo to rf weighted according to the discount factor for each period. 

The floating-leg payments of an interest rate swap can be valued using just the discount factor for the final maturity period and the notional principal. This short-cut method is based on the fact that the value of the floating-leg interest payments is conceptually the same as that of a strategy... 

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