Forward discount factor

The starting point is that we set discount curves in all the main currencies, which are the relevant OIS curve. We can extract the discount factors for each currency from these curves, which we call DfccY for general discount factor and Dfojs for the relevant discount factor for the OIS in that currency. If we assume that EX rates are not correlated to interest rates (a big assumption, but necessary in this analysis), this implies that forward FX rates - which are a deposit product, as forward EX rates are simply spot FX rate adjusted for the deposit interest rate in each currency - are not a function of the discounting level in each currency. This further implies that the ratio of forward discount factors is constant. 

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 ... 

Also shown in Exhibit 19.5 is the forward disconnt factor for all 12 periods. These values are shown in the last colnmn. Let s show how the forward discount factor is compnted for periods 1, 2, and 3. For period 1, the forward discount factor is... 

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... 

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... 

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... 

In Exhibit 19.9, the forward discount factor is computed for each period. The calculation is the same as in Exhibit 19.5 to obtain the forward discount factor for each period. The forward discount factor for each period is shown in the last column of Exhibit 19.9. 

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 ... 

EXHIBIT 19.9 Period Forward Rates and Forward Discount Factors One Year Later if Rates Increase... 

If money is borrowed, interest must be paid over the time period if money is loaned out, interest income is expected to accumulate. In other words, there is a time value associated with the money. Before money flows from different years can be combined, a compound interest factor must be employed to translate all of the flows to a common present time. The present is arbitrarily assumed often it is either the beginning of the venture or start of production. If future flows are translated backward toward the present, the discount factor is of the form (1 + i) , where i is the annual discount rate in decimal form (10% = 0.10) and n is the number of years involved in the translation. If past flows are translated in a forward direction, a factor of the same form is used, except that the exponent is positive. Discounting of the cash flows gives equivalent flows at a common time point and provides for the cost of capital. 

Equation (3.22) describes the bond price as a function of the spot rate only, as opposed to the multiple processes that apply for aU the forward rates from t to T. As the bond has a nominal value of 1, the value given by Equation (3.22) is the discount factor for that term the range of zero-coupon bond prices would give us the discount function. 

From an elementary understanding of the markets, we know that there is a relationship between a set of discount factors, and the discount function, the par yield curve, the zero-coupon yield curve and the forward yield curve. If we know one of these functions, we may readily compute the other three. In practice, although the zero-coupon yield curve is directly observable from the yields of zero-coupon... 

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. 

Expression (7.1) states that the price of a zero-coupon bond is equal to the discount factor from time t to its maturity date or the average of the discount factors under all interest-rate scenarios, weighted by their probabilities. It can be shown that the T-maturity forward rate at time t is given by... 

The effect of weighting using discount factors is to make the lower level interest-rate scenario more significant because the discount factors are higher under these scenarios. This means that a lower interest-rate scenario has more influence on the forward rate than a higher rate scenario, and this influence steadily increases as the forward rate term grows in maturity, since the difference between the discount factors increases. This is an important result. 

The expected future short rate at any point in the future will be 8%, given the probabilities however, the forward rate will be lower than 8% because it is calculated by weighting each interest-rate scenario by the relevant discount factors. This is illustrated in Figure 7.1. 

Coupon Period Quoted Day Forward Margin Count Rate (%) (%) Cash Flow ( ) Required Margin (%) Discount Factor PVof Cash Flow ( )... 

It is from the assumed values of 3-month LIBOR (i.e., the current spot rate and the implied forward rates) and the required margin in Column (6) that the discount rate that will be used to determine the present value of the cash flows will be calculated. The discount factor is found as follows ... 

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. 

Carleton and Cooper (1976) describes an approach to estimating tetm structure that assumes default-free bond cash flows, payable on specified discrete dates, to each of which a set of unrelated discount factors are applied. These discount factors are estimated as regression coefficients, with the bond cash flows beir the independent variables and the bond price at each payment date the dependent variable. This type of simple linear regression produces a discrete discount fimction, not a continuous one. The forward-rate curves estimated from this fimction are accordir ly very ja ed. 

Calculating the Forward Rate from Spot-Rate Discount Factors... 

The relationship between discount factors and the spot rates for the same periods can be used to calculate forward rates. Say the spot rate for period 1 is known. The corresponding discount rate can be derived using (7.9), which reduces to (7.11). 

This can be generalized to form an expression, (713), that calculates the discount factor for any period, +1, given the discount rate for the previous period, n, and the forward rate, rf jot the period n to n+1. Expression (7-13) can then be rearranged as (7.14), to solve for the forward rate... 

Equation (7.17) states that the zero-coupon rate is the geometric average of one plus the forward rates. The w-period forward rate is obtained using the discount factors for periods n and n-. The discount factor for the complete period is obtained by multiplying the individual discount factors together. Exactly the same result would be obtained using the zero-coupon interest rate for the whole period to derive the discount factor. ... 

PERIOD ZERO-COUPON RATE % 1 5.5 DISCOUNT FACTOR 0.947867298 FORWARD RATE % 5.5... 

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. 

PERIOD ZERO-COUPON RATE % DISCOUNT FACTOR FORWARD RATE %... 

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. 

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