Swaps

Below are links to the following topics:

• Types of Swaps
• The Term Structure of Interest Rates
• Stripped Treasury Securities and Forward Rate Agreements
• Swaps
• Plain Vanilla Swaps
• Plain Vanilla Currency Swap
• Motivations for Swaps
• Converting a Fixed Rate Asset into a Floating Rate Asset
• Creating Hybrid Fixed Floating Debt
• Pricing of Swaps
• The Economic Analysis of Swaps
• Plain Vanilla Receive Fixed Interest Rate Swap
• Plain Vanilla Pay Fixed Interest Rate Swap
• Plain Vanilla Currency Swap II
• Seasonal Interest Rate Swap
• An Interest Rate Swap as a Portfolio of Forward Rate Agreements
• Interest Swaps and FRA's
• Interest Rate Swap Pricing
• Interest Rate Parity
• Swaps: The Parallel Loan and How Swaps Began
• Creating Synthetic Securities with Swaps
• Pricing Flavored Interest Rate and Currency Swaps
• Swaps the All-In Cost
• The Seasonal Swap
• The Rate Differential Swap
• Day Count Conventions
• The Currency Annuity Swap

Types of Swaps

Types of Swaps
Floating Rate Note (FRN)	Floating Rate Agreement (FRA)
Plain Vanilla Swap	Yield Curve Swap
Amortizing Swap	Constant Maturity Swap
Accreting Swap	Rate-Differential Swap
Seasonal Swap	Corridor Swap
Roller Swap	All-In Swap
Off-Market Swap	CIRCUS Swap
Forward Swap	Equity Swap
Extension Swap	Credit Swap
Cancelable Swap	Total Return Swap
Basis Swap	Swaption
Commodity Swap	Equity Swap
Pay Fixed currency swap	Plain Vanilla Pay Fixed Interest Rate Swap
Receive Fixed currency swap	Plain Vanilla Received Fixed Interest Rate Swap

The Term Structure of Interest Rates [viewable here in Excel]

In this section, we explore the term structure of interest rates - the relationship between term-to-maturity of bonds and their respective yields. Analyzing the term structure of interest rates requires focusing on bonds that are as similar as possible, except with respect to maturity. Consequently, term structure analysis avoids attention to bonds that differ in their tax status, default risk, callability, and coupon levels. Table 19.2 (not shown) shows data for certain period in tabular form, based on bonds selling at par. Therefore, these data express the par yield curve - the relationship between yield and maturity for bonds selling at par.

Starting with data based on the term structure of par bonds which are published daily in sources such as The Wall Street Journal, it is possible to derive other yield measures that will be important in understanding interest rate options. We are particularly interested in the zero-coupon yield curve and in the implied forward yield curve. The zero-coupon yield curve expresses the relationship between yield and maturity for bonds paying no coupon payments over the life of the bond. A forward rate of interest or forward rate is a rate of interest for a period that begins at some date in the future and extends to a more distant date. For example, a forward rate might cover a period beginning two years from today and extending for three years to a time five years from today. The implied forward yield curve expresses the relationship between term to maturity and rates for forward rates as implied by the par yield curve. All of these interest rate relationships, the par yield curve, the zero-coupon yield curve, and the implied forward yield curve, form a single system of mutually consistent interest rates. Given one set of rates, it is possible to find the others, as we now show.

We begin by considering the zero-curve. The zero-coupon yield curve, or zero curve, captures the relationship between zero-coupon spot yields and term to maturity as of a particular date. The zero curve, as opposed to other kinds of yield curves, represents a set of yields unencumbered by complicating assumptions about reinvestment rates for coupons received before bond maturity. As a result, practitioners can be confident that the discount factors and expected forward rates derived from the zero curve do not depend on any reinvestment rate assumptions. Because of the desirable properties that it possesses, the zero curve has become a key ingredient to the valuation of many financial instruments, including swaps.

A zero curve can be based on any class of interest rates and denominated in any type of currency. Therefore, there are many different types of zero curves. However, we focus on the US dollar LIBOR zero curve. The zero curve must be constructed - as opposed to observed - because directly observable, and reliable, zero rates are available for only a limited number of maturities. For other maturities, zero rates must be constructed using a combination of boot-strapping and interpolation techniques.

Any coupon bond may be regarded as a portfolio of zero-coupon bonds. Each cash flow on the coupon bond is considered to be the payoff from a zero-coupon bond. Therefore, we can express the value of a coupon bond as follows:

Equation (19.2) P_o = Σ_t=im C_t/Z_o,t

Where P_o is the price of the bond at t = 0, C_t is the cash flow from the bond occurring at time t, and Z_x,y is the zero-coupon factor for a payment to be received at time y measured from time x. Equation 19.2 is the same as the familiar bond pricing formula, except that it allows a different interest rate to be applied to each payment on the bond. Table 19.3 presents US Treasury bonds from the par yield curve, all with an assumed par value of 100. Table 19.4 shows the cash flows from each of the bonds in Table 19.3. From these, we show how to compute the zero-coupon yield curve and the implied forward yield curve. Based on Table 19.4, we have the following:

Table 19.3 Illustrative Treasury instruments, par = 100
Instrument	Maturity	Annual Coupon	Price (% of par)
A	6 mo	5.80%	100
B	1 yr	6.00%	100
C	1.5 yrs	6.40%	100
D	2 year	6.80%	100
E	3 year	7.00%	100

Table 19.4 Cash-flows from Treasury bonds		6 months	12 months	18 months	24 months	30 months	36 months
	Bond Term	0.5	1.00	1.5	2	2.5	3
2.90%	A	102.9
3.00%	B	3.0	103.0
3.20%	C	3.2	3.2	103.2
3.40%	D	3.4	3.4	3.4	103.4
3.50%	E	3.5	3.5	3.5	3.5	3.5	103.5	100.00644	7.12%

				Bootstrap Zero-rate Coupon Factor	Forward-rate 6 mo Factor (FRF)	Forward-rate 1 yr Factor (FRF)	Forward-rate 1.5 yr Factor (FRF)	Zero Annual Yield w CC	Annual Coupon Rate (% of Par)		Trial and error
6 months	PA = 102.9/Z0,0.5	100=102.9/Z0,0.5	100.00	1.02900				5.6380%	5.80%	96.00	10.68%
1 year	PB = 3.0/Z0,0.5+103.0/Z0,1.0	100 = 3.0/1.029+103.0/Z0,1.0	100.00	1.060931	1.03103			5.9147%	6.00%
1.5 years	PC = 3.2/Z0,0.5+3.2/Z0,1.0 +103.2/Z0,1.5	100 = 3.2/1.029+3.2/1.060931+103.2/Z0,1.5	100.00	1.099346	1.03621	1.0683638		6.4130%	6.40%
2.0 years	PD = 3.4/Z0,0.5+3.4/Z0,1.0 +3.4/Z0,1.5+103.4/Z0,2.0	100 = 3.4/1.029+3.4/1.060931+3.4/1.099346+103.4/Z 0,2.0	100.00	1.143826	1.04046	1.0781346	1.11159018	6.9445%	6.80%
3.0 years	PE = 3.5/Z0,0.5+3.5/Z0,1.0 +3.5/Z0,1.5+3.5/Z0,2.0+3.5/Z 0,2.5+103.5/Z0,3.0	100 = 3.5/1.029+3.5/1.060931+3.5/1.099346+3.5/1.143826+103.5/Z 0,2.5	100.00	1.188889				6.1060%	7.00%

where the subscripts A-E on the price indicates the bond, since we are finding all prices at time zero and the subscripts on Z are expressed in years.

Our first task is to find the six-month zero-coupon factor Z 0,0.5		Zero-Coupon	Zero-Coupon Annual w C/C	How do you compute the implied annual interest rate?
100=102.9/Z0,0.5	100.00	1.02900	5.64%	5.88%
100 = 3.0/1.029+103.0/Z0,1.0	100.00	1.060931
100 = 3.2/1.029+3.2/1.060931+103.2/Z0,1.5	100.00	1.099346
100 = 3.4/1.029+3.4/1.060931+3.4/1.099346+103.4/Z 0,2.0	100.00	1.143826
100 = 3.5/1.029+3.5/1.060931+3.5/1.099346+3.5/1.143826+103.5/Z 0,2.5	100.00	1.188889

With a six-month zero-coupon factor of 1.029, the implied annualized yield, assuming semiannual compounding, is 5.88 percent.

We next find the one-year zero-coupon factor by treating the one-year coupon bond as a portfolio of zero-coupon instruments. The one-year bond consists of a payment in six months of $3 (half the annual coupon rate) plus a second payment in one year of 103 (the par value, plus the final semiannual coupon payment):

Po = 3.0/Z_0,0.5+103.0/Z0,1.0
100 = 3.0/1.029+103.0/Z_0,1.0
Z_0,1.0 = 103/(100-(3/1.029)) = 1.06093

Is essence, we have begun with a short-term zero-coupon instrument, found the zero-coupon factor, and used this initial factor to find the factor for the next shortest term. This process is called bootstrapping - the sequential process of using a short-term rate to find a longer-term rate. Continuing the process of bootstrapping gives Z_0,1.5 = 1.099346 and Z_0,2 = 1.14386. Notice that we are stymied at this point with respect to instruments in Tables 19.3 and 19.4. Our next desired factor would be Z_0,2.5. However, we do not have the price of a bond maturing at t = 2.5. Also, we cannot use the bond maturing at t = 3, because it has an intervening payment at t = 2.5. Therefore, bootstrapping requires an instrument maturing at each date up to and including the longest maturity date for which we wish to compute the zero-coupon factor. Note also that all of the bonds in Tables 19.3 and 19.4 traded at par. The bootstrapping technique could be applied to bonds and that are not at par, due to a diversity of coupon rates. However, differing coupon rates can affect bond yields, so it is best to use par bonds only.

The set of bond yields we have been considering also implies a set of forward rates on interest. Let FR_x,y indicate a forward rate of interest to cover the period that begins at future time x and ends at a later time y. Also, let FR_x,y be the forward rate zero-coupon factor to cover the period that begins at future time x and ends at a later time y. Given the zero-coupon factors, any FRF can be found as

FRF_x,y = Z_0,y / Z_0,x

Using the data of Table 19.3 and Table 19.4, along with the zero-coupon factors already computed, we have the following:

FRF0.5,1 = Z0,1 / Z0,.5 = 1.060931/1.029 = 1.031031	forward interest rate for the period yr.5 - yr 1
FRF1.0,1.5 = Z0,1.5 / Z1,1.5 = 1.099346/1.060931 = 1.036209	forward interest rate for the period yr 1 - yr 1.5
FRF1.5,2 = Z0,2 / Z0,2 = 1.143826/1.099346 = 1.040460	forward interest rate for the period yr 1.5 - yr 2.0
FRF0.5,1.5 = Z0,1.5 / Z0,0.5 = 1.060931/1.029 = 1.068363	forward interest rate for the period yr.5 - yr 1.5
FRF0.5,2 = Z0,2 / Z0,0.5 = 1.143826/1.029 = 1.111590	forward interest rate for the period yr.5 - yr 2.0
FRF1,2 = Z0,2 / Z0,1 = 1.143826/1.060931 = 1.078134	forward interest rate for the period yr1 - yr 2

Given the various FRFs, the forward rates, the FRs, are the interest rates implied by the forward rate factors. By using FRFs, we can make computations based strictly on cash flows and then move to forward rates under various compounding assumptions.

In the example data of Tables 19.3 and 19.4, the yield curve is upward sloping, with longer maturity bonds having higher yields than short maturity bonds. In this situation, the zero-coupon rates are high than the rates on a two year zero-coupon bonds. For example assuming annual compounding, the zero-coupon interest rate on a two-year zero-coupon bond is 6.9498 percent, compared to the yield of 6.8 percent on the coupon bond. Forward rates map onto these spot yield measures only approximately, but for the same example, the forward interest rate to cover from year 1 to year 2 is 7.8134 percent. This lead to the following rule:

For upward-sloping yield curves: forward rate > zero-coupon > coupon bond rate

For downward-sloping yield curves: forward rate < zero-coupon rate < coupon bond rate

We will use the bootstrapping technique and the various yield measures in pricing interest rate options. As we will see in later sections, bootstrapping is also important in pricing swap agreements.

Stripped Treasury Securities and Forward Rate Agreements (FRAs) [viewable here in Excel]

In the previous section, we saw how to compute zero-coupon factors and rates along with forward rate factors and rates from the par yield curve. Explicit markets for zero-coupon instruments and forward rates also exist.

Treasury Strips

A stripped T-bond is created when a normal T-bond is decomposed into a series of zero-coupon bonds corresponding to the various coupon and principal payments that constitute the bond. For example, a 30-year semiannual coupon T-bond could be stripped to give 60 zero-coupon instruments that pay the original coupon payment of the bond, plus one zero-coupon bond that corresponds to the principal repayment on the bond.

Forward Rate Agreement (FRAs)

There is also an explicit market for forward rates. A contract based on the forward rate is known as a forward rate agreement (FRA). Forward rate agreements are typically based on LIBOR, which stands for "London Interbank Offering Rate," a rate at which large international banks lend funds to each other. Typically, an FRA calls for the exchange of a payment based on LIBOR at a future date in return for a payment based on a fixed rate of interest agreed on the contracting date. One can contract to either pay-fixed and receive floating or to receive-fixed and pay floating. Typically, an FRA market maker will have a spread on the fixed side of the deal to provide a profit margin.

For example, consider an FRA market maker who agrees today to pay six-month LIBOR in six months in exchange for a fixed interest payment at an annual rate of 5 percent and a notional principal of $20 million. (If the market maker is willing to take the receive-fixed side of the deal at 5.00 percent might be willing to take the pay-fixed side at only 4.96 percent. Spreads of four basis points are typical in this market.) When the FRA expires in six months, assume that six-month LIBOR stands at 5.8 percent. Payments on the FRA would be the interest rate times the fraction of the year times the notional principal.

Receive-fixed: 0.050 x 0.5 x $20,000,000 = $500,000
Pay-floating: -0.058 x 0.5 x $20,000,000 = -$580,000
Net payment: -$80,000

In this case, the market maker would be obligated to make a net payment of $80,000. In the FRA market, it is customary for settlement amounts to be "determined in advance and paid in arrears." The payment based on the FRA in our example would be due six months after the date of determination, because six months was the maturity of the interest rate being used in the FRA. Thus, FRAs based on three-month LIBOR are paid three months after the determination date, FRAs based on six-month LIBOR are paid six months after the determination date, and so on.

FRAs are Quoted in the Following Manner:

term to expiration in months x term to end of period covered by agreement rate

For example, the FRA we considered above, with six months to expiration for six-month LIBOR and a fixed rate of 5 percent, would be quoted as:

6 x 12 5.00%

The first number indicates the months until the FRA expires, the second indicates the number of months until the instrument presumed to underlie the FRA matures, while the differences between the two numbers shows the maturity of the presumed underlying instrument. There is no actual instrument that is delivered; instead, the presumed underlying instrument is simply an instrument of the underlying maturity that pays LIBOR. Instead of exchanging instruments, the profit or loss is settled in cash, as in our preceding example. The 5 percent in the question is the fixed rate of interest to be paid in exchange for LIBOR.

The rates on FRA agreements also tie in with the zero-coupon yield curve. Assume that the current six-month spot rate is 4.95 percent, and consider now the following FRA quotations that imply a rising yield curve:

0 x 6 4.95%
6 x 12 5.0%
12 x 18 5.1%
18 x 24 5.2%

We adopt the following notation:

FRA_x,j is the rate of interest on an FRA for a period beginning at time x and ending at time y

FRA_0,y indicates a spot rate from time zero to time y

On the basis of the notation we have the following:

	Forward Rates
6 months 0 x 6 or FRA0,6	4.95%
6 months 6 x 12 or FRA6,12	5.00%
6 months 12 x 18 or FRA12,18	5.10%
6 months 18 x 24 or FRA18,24	5.20%

Find the Forward Rate if you have the Zero rate

Solve for y where 1.02475 x (1 + .5 * y) = 1.0506388	Solve for y (1 + .5 * y) = 1.02475	Solve for y where 1.05036875 x (1 + .5 * y) = 1.077153
1.02475 x (1 + .5y) = 1.0506388	(1 + .5 y) = 1.02475	1.05036875 x (1 + .5y) = 1.077153
1.02475 x (1 + .5y)/1.02475 = 1.0506388/1.02475	1 + .5y -1 = 1.02475 - 1	1.05036875 x (1 + .5y)/1.05036875 = 1.077153/1.05036875
1.02475 x (1 + .5y) /1.02475 = 1.0506388/1.02475	1 + .5y - 1 = 1.02475 - 1	1.05036875 x (1 + .5y)/1.05036875 = 1.077153/1.05036875
1 + .5y = 1.0506388/1.02475	1 + .5y = 1.02475	1 + .5y = 1.077153/1.05036875
Subtract 1 from both sides	Subtract 1 from both sides	Subtract 1 from both sides
1 + .5y -1 = (1.0506388/1.02475) -1	1 + .5y -1 = 1.02475 -1	1 + .5y -1 = (1.077153/1.05036875) -1
1 + .5y -1 = (1.0506388/1.02475) -1	1 + .5y -1 = 1.02475 -1	1 + .5y -1 = (1.077153/1.05036875) -1
.5y = (1.0506388/1.02475) -1	.5y = .02475	.5y = (1.077153/1.05036875) -1
Convert 1 into fraction: 1 * 1.02475/1.02475	y = .02475/.5	Convert 1 into fraction: 1 * 1.05036875/1.05036875
-(1 * 1.02475)/1.02475 + (1.0506388/1.02475)	y = .0495	-(1 * 1.05036875)/1.05036875 + (1.077153/1.05036875)
Since the denominators are equal, combine the fractions a/c +- b/c = a+-b/c		Since the denominators are equal, combine the fractions a/c +- b/c = a+-b/c
(-1 * 1.02475 + 1.0506388)/1.02475)		(-1 * 1.05036875 + 1.077153)/1.05036875)
(-1 * 1.02475 + 1.0506388) = .025888		(-1 * 1.05036875 + 1.077153) = .0267844
.5y = .025888/1.02475		.5y = .027844/1.05036875
y =(.025888/1.02475)/.5		y =(.0267844/1.05036875)/.5 = 0.0255
y = .0253/.5 = .0506		y = .0255/.5 = .0510
y = .0506	y = .0506	y = .05100
1.02475 x (1 + .5 * y) = 1.0506388	1.02475 x (1 + .5 * y) = 1.0506388	1.050369 x (1 + .5 * y) = 1.077153
1.0506761750	1.050676175	1.077153

Note the following equivalences:

FRA_x,y = FR_x,y = FRF_x,y - 1

We will use FR_x,y to refer to forward rates generically and FRA_x,y to refer to a rate on a forward rate agreement. To avoid arbitrage the two must be equal. We also define the following:

FRAC is the fraction of the year covered by the FRA
NP is the notional principal

We now want to integrate the quotation and payment mechanism of the FRA market with our technique of bootstrapping to find zero-coupon factors. In particular, we need to take into account the fact that payments occur in arrears and that the dollar amount of the payment depends on the part of the year covered by the FRA multiplied times the notional principal.

Based on the first FRA, FRA_0,6 = 0.0495, to cover the period from the present to six months, the fixed-side payment would be:

_FRA0,6 x FRAC x NO = 0.0495 x 0.5 x notional principal

We have not defined the notional principal for this example because the notional principal does not affect the rates on FRAs, just the dollar amount of the payment. Therefore, in general, we may ignore the notional principal in our computations of FRA rates. We can compute the value for the first zero-coupon factor Z_0,6, as follows:

Z_0,6 = 1 + 0.5 x 0.0495 = 1.024750

where FRAC = 0.5 reflects the half-year between payments and FRA_0,6 = 0.495 is the current LIBOR six-month spot rate. Note that the payment actually occurs at the end of the period covered by the FRA, but is based on the observation of LIBOR at the beginning of the period covered by the FRA.

The zero-coupon factor for the second-payment covers 12 months, and given the value of Z0,6, we can compute the value of Z_0,12 as

_Z6,12 = Z_0,6 x (1 + 0.5 x 0.0500) = 1.02475 *2 = 1.050369

where 0.5 reflects the half-year between payments and 0.0500 is the rate on the 6 x 12 FRA. Values for Z_0,18 and Z_0,24 follow similarly:

Z_0,18 = Z_0,12 x (1 + 0.5 x 0.0510) = 1.050369 x 1.0255 = 1.077153
Z_0,24 = Z_0,18 x (1 + 0.5 x 0.0520) = 1.077153 x 1.0260 = 1.105159

Starting with Z_0,6 and using the bootstrapping technique, we have found all of the zero-coupon factors that we need. As we will see later, FRA agreements tie in closely with interest rate options, as do the pricing conventions for the FRA market that we have just discussed. These zero-coupon factors and FRA rates are also important for understanding the swaps market, which is explored later.

Swaps

A swap is an agreement between two or more parties to exchange a sequence of cash flows over a period in the future. For example, Party A might agree to pay a fixed rate of interest on $1 million each year for five years to Party B. In return, Party B might pay a floating rate of interest on $1 million each year for five years. The parties that agree to the swap are known as counterparties. The cash flows that the counterparties make are generally tied to the value of debt instruments or to the value of foreign currencies. Therefore, the two basic kinds of swaps are interest rate swaps and currency swaps.

The origins of the swap market can be traced to the late 1970s when currency traders developed currency swaps as a technique to evade British controls on the movement of foreign currency. The first interest rate swap occurred in 1981, in an agreement between IBM and the World Bank. Since that time, the market has grown rapidly. By 2005, interest rate swaps with $172 trillion in underlying value where outstanding, and currency swaps totaled another $8.5 trillion. The total swaps market exceeded a notional amount of $180 trillion, with about 95 percent of the swaps being interest rate swaps and the remaining 5 percent primarily currency swaps. The growth in this market has been phenomenal; in fact, it has been the most rapid for any financial product in history.

The Swaps Market

In this section, we consider the special features of the swaps market. For purposes of comparison, we begin by summarizing some of the key features of futures and options markets. Against this background, we focus on the most important features of the swap product. The section concludes with a brief summary of the development of the swaps market.

A Review of Futures and Options Market Features

Previously, we have explored the futures and options markets. We noted that futures contracts trade exclusively in markets operated by futures exchanges and regulated by the Commodity Futures Trade Commissions (CFTC). In our discussion of options we focused primarily on exchanged-traded options. This portion of the options market regulated by the Securities and Exchange Commission (SEC), is highly formalized with the options exchanges playing a major role in the market.

Futures markets trade highly standardized contracts and the options traded on exchange also have specified contract terms that cannot be altered. For example the S&P 500 futures contract is based on a particular set of stocks for a particular dollar amount, with only four fixed maturity dates per year. In addition futures and exchange-traded options generally have a fairly short time horizon. In many cases, futures contracts are listed only about one to two years before they expire. Even when it is possible to trade futures for expiration in three years or more, the markets do not become liquid until the contract comes much closer to expiration. For exchange-traded stock options, the longest time to maturity is generally less than one year. These futures and options cannot provide a means of dealing with risks that extend farther into the future than the expiration of the contract that are traded. For example, if a firm faces associated with a major building project, the futures market allows risk management only for the horizon of futures contracts currently being traded, which is about three years. In recent years, over-the-counter (OTC) markets for options have become more important.

The Characteristics of the Swaps Market

On futures and options exchanges, major financial institutions are readily identifiable. For example, in a futures pit, traders can discern the activity of particular firms, because traders know who represents which firm. Therefore, exchange trading necessarily involves a certain loss of privacy. In the swaps market, by contrast, only the counterparties know that the swap takes place. Thus, the swaps market affords a privacy that cannot be obtained in exchange trading.

We have noted that futures and options exchanges are subject to considerable government regulation. By contrast, the swaps market has virtually no government regulations. As we will see later, swaps are similar to futures. The Commodity Futures Modernization Act of 2000 (CFMA) excluded swaps on financial products from regulation under the Commodity Exchange Act, the law governing futures trading in the United States. The CFMA exempted by statute swaps authority in these markets. The CFMA provided legal certainty about the enforceability of swap agreements. The International Swaps and Derivatives Association, Inc (ISDA) is an industry organization that provides standard documentation for swap agreements and keeps records of swap activity.

The swap market also has some inherent limitations. First, to consummate a swap transaction, one potential counterparty must find another counterparty that is willing to take the opposite side of a transaction. If one party needs a specific maturity, or a certain pattern of cash flows, it could be difficult to find a willing counter party. In the early days of the swap market, counterparties generally faced this problem as end-users of the swaps market transacted directly with each other. The swap market is now served by swap dealers who make markets in swaps. Second, because a swap agreement is a contract between two counterparties, the swap cannot be altered or terminated early without the agreement of both parties. Third, for futures and exchange-traded options, the exchanges effectively guarantee performance on the contracts for all parties. By its very nature, the swaps market has no such guarantor. As a consequence parties to the swap must be certain of the creditworthiness of their counterparties.

Plain Vanilla Swaps

In this section, we analyze the two basic kinds of swaps that are available. The basic swap is known as a plain vanilla swap, which can be an interest rate swap or a foreign currency swap. We begin by considering the mechanics of these plain vanilla swaps. Later in this chapter, we consider more complicated swaps called flavored swaps.

Interest Rate Swaps

In a plain vanilla interest rate swap, one counterparty agrees to a pay a sequence of fixed-rate interest payments and to receive a sequence of floating-rate interest payments. This counter-party is said to have the pay-fixed side of the deal. The opposing counterparty agrees to receive a sequence of fixed-rate interest payments and to pay a sequence of floating-rate payments. This counterparty has the receive-fixed side of the deal.

Pay fixed - receive floating (short) Pay floating - receive fixes (long)

The swap agreement specifies a time over which the periodic interest payments will be made, which is the tenor of the swap. The amount of the periodic interest payments is a fraction of a dollar amount specified in the swap agreement, which is called the notional principal. The notional principal is a nominal quantity used as a scale factor to determine the size of the interest payments in the swap agreement. Later, we explore the motivation that these counterparties might have for taking their respective positions. First, however, we need to understand the transactions.

To see the nature of the plain vanilla interest rate swap most clearly, we use an example. We assume that the swap covers a five-year period (a five-year tenor) and involves annual payments on a $1 million notional principal amount. Let us assume that Party A is the pay-fixed counterparty and agrees to pay a fixed rate of 9 percent to Party B. In return, Party B, the receive-fixed counterparty agrees to pay a floating rate of LIBOR to Party A. As we have seen, LIBOR, is the London International Offering Rate and represents the rate at which large international banks lend to each other. LIBOR-based loans are essentially privately negotiated business loans that may have a variety of maturities. Most of the maturities range from one month to a year. Quotations of LIBOR rates appear daily in the "Money Rates" column of The Wall Street Journal. Floating rates in the swaps market are most often set as equaling LIBOR, which is also called LIBOR flat.

In our example swap agreement, Party A pays 9 percent of $1 million or $90,000, each year to Party B. Party B makes payments to Party A in return, but the actual amount of the payments depends on movement in LIBOR. The LIBOR maturity for this plain vanilla swap will be the one-year LIBOR rate, because there is one year between each of the payments on the swap. We assume that one-year LIBOR stands at 8.75 percent at the time the swap agreement is negotiated, a rate may differ substantially from the fixed rate on the swap.

Conceptually, the two parties also exchange the principal amount of $1 million. However, actually making the transaction of sending each other $1 million would not make practical sense. As a consequence, principal amounts are generally not exchanged. Instead, the notional principal is used to determine the amount of the interest payments. Because the principal is not actually exchanged it is only a notional principal, an amount used as a base for computations, but not an amount that is actually transferred from one party to another. In our example, the notional principal is $1 million, and knowing that amount lets us compute the actual dollar amount of the cash flows that the two parties make to each other each year.

Generally, the determination of LIBOR occurs at one settlement date, with payment occurring at the next settlement date. The payment is said to be determined in advance and paid in arrears. Table 20.2 shows what is known about the cash flows on this swap agreement at t = 0, the time t which the swap is negotiated. As mentioned above, LIBOR at the time of negotiation is 8.75 percent, but the future course of LIBOR is unknown. In each period, the fixed-rate payment will be $90,000. In general, the floating-rate payment at a given time t depends on the level of LIBOR at period t-1. Thus, at the inception of the swap agreement, LIBOR is observed and this determines the floating-rate payment that occurs in the first period. In our example, with LIBOR at 8.75 percent when the swap is negotiated, the first floating-rate payment will be $87,500. The second payment, which occurs at t = 2, depends on LIBOR at t = 1, and this is unknown when the swap is negotiated. The table shows LIBOR at the end of the swap's tenor as "N/A." This rate is nonapplicable, because it does not determine any of the cash flows associated with the swap. This convention of determining the floating-rate payment in advance and actually making the payment in arrears matches the conventions that prevail in the market for floating-rate notes and bank loans.

	Table 20.2 Cash flows for a plain vanilla interest rate swap
Year	LIBOR	Floating-rate obligation: Party B pay Party A	Fixed-rate obligation: Party A pays Party B
0	8.75%	0	0
1	LIBOR1 = ?	LIBOR0 x $1.0 Million	$90,000
		= 0.0875 x $1.0 Million = $87,000
2	LIBOR2 = ?	LIBOR1 x $1.0 Million	$90,000
3	LIBOR3 = ?	LIBOR2 x $1.0 Million	$90,000
4	LIBOR4 = ?	LIBOR3 x $1.0 Million	$90,000
5	N/A	LIBOR4 x $1.0 Million	$90,000

Figure 20.1 parallels Table 20.2 and shows the cash flows on the swap from the perspective of each party. In this time line, an up arrow indicates a cash inflow, while a down-arrow indicates a cash outflow. The upper panel of Figure 20.1 pertains to Party A, the pay-fixed counter-party. In each year, Party A will pay a fixed amount of $90,000. In return, Party A receives a payment that depends on LIBOR. At year 1, Party A will receive $87,500, because LIBOR stands at 8.75 percent when the swap is negotiated, because those payments depend on the unknown future of LIBOR. The lower panel shows the same swap from the perspective of Party B, the receive-fixed counterparty. Each year, Party B will receive a fixed payment of $90,000, and make a payment based on LIBOR. From the figure, it is clear that Parties A and B have "mirror-image" cash flows. The receipt for one are the payments for the other. This emphasizes the fact that a swap is a zero-sum game - one party's gain is the other's loss. In general, we can show everything about a swap by just looking at it from the perspective of a single party.

Continuing with our example of a plain vanilla interest rate swap, let us now assume that LIBOR is 10 percent at t = 1. This rate will determine the floating-rate payments to occur at t = 2, assuming that the swap is paid in arrears. This means that Party A will be obligated to pay $90,000 to Party B, while Party B will owe $100,000 to Party A. Offsetting the two mutual obligations, Party B owes $10,000 to Party A. Generally, only the net payment, the difference between the two obligations, takes place in interest rate swaps. Again, this practice avoids unnecessary payments.

For our example swap, the payments are annual and the tenor is five years. Consequently, the swap has five determination dates, the dates at which the amount of payments are due from each party are determined. There are also five payment dates, the actual dates on which each net payment is made. The determination dates occur at years 0, 1, 2, 3, and 4. The corresponding payment dates occur one year later than their respective determination dates. In general, the payment date occurs one period after the determination date. For example on a swap with quarterly payments, the payment date follows its corresponding determination date by one quarter.

While most swaps are "determined in advance and paid in arrears" some swap agreements specify payments in advance. These swaps are called in-advance swaps. For a swap with payments in advance the payment due is the present value of the in-arrears obligation, discounted at LIBOR. This is the marketwide convention for in-advance swaps. If our sample plain vanilla swap had been an in-advance swap the first payment would occur at T = 0. The fixed-rate payment would be the present value of $90,000, while the floating-rate payment would be the present value of $87,500, each discounted at the LIBOR of 8.75 percent. Thus, the floating-rate payment would be $80,459.77, and the fixed-rate payment would be $82,758.62. Since only net payments are made, the actual payment would be from the fixed-rate to the floating-rate payer (from Party A to Party B) for $82,758.62 - $80,459.77 = $2,295.85 [=90000/1.0875=$82,759 and =87500/1.0875 = $80,460].

For the same plain vanilla interest rate swap, which we now interpret as an in-advance swap, let us now assume that the time is t = 1 and that one-year LIBOR now stands at 10 percent. For this in-advanced swap, the payments at t = 1 will be as follows. The floating-rate payer must pay the present value of its obligations (0.10 x $1,000,000 = $100,000) discounted at the prevailing LIBOR of 10 percent. This equals $100,000/1.10 = $90,900.09. The fixed-rate payment is the present value of $90,000/1.10 = $81,818.18. This means that the floating-rate payer, Party B will make a net payment of $9,090.91 to Party A.

Plain Vanilla Currency Swap

In a plain vanilla currency swap, one party typically holds one currency and desires a different currency. The swap arises when one party provides a certain principal in one currency to its counterparty in exchange for an equivalent amount of a different currency. Each party will then pay interest on the currency it receives in the swap, and this interest payment can be made at either a fixed or floating rate.

(1) For example, Party C pays a fixed rate on dollars received, and Party D pays a fixed rate on euros received.
(2) Party C pays a floating rate on dollars received, and Party D pays a fixed rate on euros received.
(3) Party C pays a fixed rate on dollars received, and Party D pays a floating rate on euros received.
(4) Party C pays a floating rate on dollars received, and Party D pays a floating rate on euros received.

Although all four patterns of interest payments are observed in the market, the predominant quotation is of the second type: pay floating on dollars/pay fixed on the foreign currency, and this is known as a plain vanilla currency swap.

Before analyzing the cash flows on the plain vanilla currency swap (floating rate on dollars/fixed rate on a foreign currency), we begin with a simple case. The simplest kind of currency swap arises when each party pays a fixed rate of interest on the currency it receives. The fixed-for-fixed currency swap involves three different sets of cash flows. First, at the initiation of the swap, the two parties actually exchange cash. Typically, the motivation for the currency swap is the actual need for funds denominated in a different currency. This differs from the interest rate swap in which both parties deal in a single currency and can pay the net amount. Second, the parties make periodic interest payments to each other during the life of the swap agreement, and these payments are made in full without netting. Third, at the termination of the swap, the parties again exchange the principal.

As an example of the fixed-for-fixed currency swap, let us assume that the current spot exchange rate between euros and US dollars is €0.8 per dollar. Thus, the euro is worth $1.25 (1/.8= 1.25). We assume that the US interest rate is 10 percent and the EU interest rate is 8 percent. Party C holds €25 million and wishes to exchange those euros for dollars. In return for the euros, Party D would pay $31.25 million to Party C at the initiation of the swap. We also assume that the tenor of the swap is seven years and the parties will make annual interest payments. With the interest rates in our example, Party D will pay 8 percent interest on €25 million it received, so the annual payment from Party D to Party C will be €2 million. Party C receives $31.25 (€25/.8) million and pays interest at 10 percent, so Party C will pay $3.125 million each year to Party D. As the payments are made in different currencies, netting is not a typical practice. Instead, each party makes the full interest payment.

At the end of seven years, the two parties again exchange principal. In our example, Party C would pay $31.25 million and Party D would pay €25 million. This final payment terminates the currency swap. Figure 20.2 shows the cash flows on this fixed-for-fixed currency swap from the perspective of each party. At time t = 0, the principal amounts are exchanged. At the end of each of the seven years, the fixed interest payments are exchanged. Finally, at the end of the swap, t = 7, the principal amounts are again exchanged.

When this fixed-for-fixed currency swap is negotiated at t = 0, the entire sequence of cash flows is known for the entire tenor of the swap. Which set of cash flows is more desirable, only time will tell, because interest rates for the dollar and the euro will fluctuate, as will the dollar/euro exchange rate. Like the plain vanilla interest rate swap we have considered, the fixed-for-fix currency swap is a zero-sum game. One set of cash flows will turn out to be better than the other. The party that gains does so at the other party's expense.

As noted above, the fixed-for-floating currency swap is the prevalent type of currency swap, and is considered to be the plain vanilla currency swap. In this type of swap, parties typically exchange principal at the outset of the swap, but one party pays a fixed rate of interest on the foreign currency it receives, while the other pays a floating rate on the currency it receives.

As an example of a fixed-for-floating (plain vanilla) currency swap, consider a swap arranged between a US and a Japanese firm assuming that $1 is worth ¥120 when the swap is negotiated. Let the notional amounts be $10 million and ¥1.2 billion, with a tenor of four years based on annual payments. The Japanese four-year fixed interest rate is 7 percent, and the US firm promises to pay this fixed rate. For its part, the Japanese firm promises to pay one year LIBOR flat, which is currently 5 percent. Table 20.3 shows the anticipated cash flows. Customarily, foreign currency swaps are determined in advance and paid in arrears, just as we have seen with interest rate swaps. However, foreign currency swaps can sometimes be in-advance swaps as well. Figure 20.3 parallels Table 20.3 and shows the cash flows on the plain vanilla currency swaps from the perspective of each counterparty.

	Table 20.3 Cash flows for a plain vanilla currency swap
Year	LIBOR	Floating-rate obligation: Party B pay Party A	Fixed-rate obligation: Party A pays Party B
0	5.00%	¥1,200,000,000	$10,000,000
1	LIBOR1 = ?	LIBOR0 x $10 Million	$90,000
		= 0.05 x $10 Million = $500,000
2	LIBOR2 = ?	LIBOR1 x $10.0 Million	¥84,000,000
3	LIBOR3 = ?	LIBOR2 x $10.0 Million	¥84,000,000
4	LIBOR4 = ?	LIBOR3 x $10.0 Million	¥84,000,000
5	N/A	LIBOR4 x $10.0 Million	¥84,000,000 + ¥1,200,000,000

Motivations for Swaps

Today, the swaps market is a mature market that is well understood by many sophisticated practitioners. Therefore, there are likely to be few, if any, arbitrage opportunities available. Instead, the swaps market has succeeded because it offers more operationally efficient and flexible means of packaging and transforming cash flows than other instruments, such as exchange-traded options and futures. These uses of the swaps market are not motivated by perceived informational inefficiencies. Instead, the motivation turns on reducing transaction costs, lowering hedging costs, avoiding costly regulations, and maintaining privacy. These business applications in an informationally efficient market will be the primary focus of our discussion of swaps.

However, we begin by considering a swap transaction designed to exploit a market inefficiency, which turns on the competitive advantage in borrowing costs between two firms. We then turn to understanding the motivation for swaps for risk management purposes in an operationally efficient market. We consider two simple examples of how firms might use the swaps market to manage internal risk. The motivation for these latter swap transactions is operational efficiency and cost-effectiveness, rather than an effort to exploit an informational inefficiency.

The Comparative Advantage

In some situations, one firm may have better access to the capital market than another firm. For example, a US firm may be able to borrow easily in the Unites States, but it might not have such favorable access to the capital market in Germany. Similarly, a German firm may have good borrowing opportunities domestically but poor opportunities in the Unites States. Notice that this comparative advantage implies an inefficiency in the financial market, because the differential access to the markets implies that lenders evaluate the firms differently in different countries.

Table 20.4 presents borrowing rates for Parties C and D, the firms of our fixed-for-fixed currency swap example. In the previous example, we assumed that, for each currency both parties faced the same rate. We now assume that Party C is a German firm with access to euros at a rate of 7 percent, while the US firm, party D, must pay 8 percent to borrow euros. On the other hand, Party D can borrow dollars at 9 percent, while the German Party C must pay 10 percent for its borrowings.

	Table 20.4 Borrowing rates for two firms in two currencies
Firm	US dollar rate	Euro rate
Party C	10%	7%
Party D	9%	8%

As the table shows, Party C enjoys a comparative advantage in borrowing euros and Party D has a comparative advantage in borrowing dollars. (Again, notice the market inefficiency that these rates imply: in one currency, Party C is regarded as a better credit risk; in the other currency Party D is the better credit risk.) These rates raise the possibility that each firm can exploit its comparative advantage and share the gains by reducing overall borrowing costs. This possibility is shown in Figure 20.4, which parallels Figure 20.2, but focuses just on the exchange of currencies at the initiation of the swap.

Figure 20.4 shows that Party C borrows €25 million from a third-party lender at its borrowing rate of 7 percent, while Party D borrows $31.25 million from a fourth party at 9 percent. After these borrowings, both parties have the funds to engage in the fixed-for-fixed currency swap that we have already analyzed. To initiate the swap, Party C forwards the €25 million it has just borrowed to Party D, which reciprocates with the $31.25 million it has borrowed. In effect, the two parties have made independent borrowings and then exchanged the proceeds. For this reason, a currency swap is also known as an exchange of borrowings.

Figure 20.5 shows the annual interest cash flows for the loans and the fixed-for-fixed currency swap. Party C annually receives €2 million (€25x.08) from Party D and pays interest of €1.75 million (€25x.07) on its loan. This gives a net inflow of €250,000 per year. Valuing these euros at the exchange rate of $1 = €0.8, the net flow has a value of $312,500 (250,000/.8). Party C also pays $3.125 million ($31.25x.10) annually to Party D, giving Party C a net annual cash outflow of $2,812,500 ($31.25x.09) for the use of $31.25 million, for an effective interest rate of 9 percent. This compares favorably with savings of 1 percent financing cost to Party C.

Assuming that $1 = €0.80

Party C borrows dollars at an effective interest rate of 9 percent as follows, with all values expressed ultimately in dollars. Interest payments = $3,125,000 - €2,000,000 x 1.25 + €1,750,000 x 1.25 = $2,812,500 on $31.25 million.

Party D borrows euros at an effective interest rate of 7 percent as follows, with all values expressed ultimately in euros. Interest payments = €2,000,000 - $3,125,000 x 0.80 + $2,812,500 x .80 = €2,000,000 - €2,500,000 + €2,500,000 €1,750,000 on €25 million.

Each year, Party D receives $3.125 million from Party C and pays $2,812,500 on its loan, for an annual inflow of $312,500. At the exchange rate of $1 = €0.8, this inflow is worth €250,000. Party D also pays €2 million to Party C, for a net annual cash outflow of €1.75 million (€2 million-€250,000 = €1.75 million). This outflow pays for the use of €25 million, for an effective interest rate of 7 percent. This is better than the EU interest rate that is available to Party D of 8 percent, as shown in Table 20.4. Thus, party D also saves 1 percent on its financing costs. By using the swap, both parties achieve an effective borrowing rate that is much lower than they could have obtained by borrowing the currency they needed directly. Parties C and D share equally in this example. By engaging in the swap, both firms can use the comparative advantage of the other to reduce their borrowing costs. Figure 20.6 shows the termination cash flows for the swap, when both parties repay the principal.

Converting a Fixed-Rate Asset into a Floating-Rate Asset

As we noted above, the comparative advantage fixed-for-fixed swap was predicated on a market imperfection - conflicting credit risk assessments of the two counterparties in two countries. We now focus on two examples of using swaps that do not rely on the presence of market imperfections. These examples are more closely related to current market conditions and the actual business practices of firms.

As an example of a prime candidate for an interest rate swap, consider a typical savings and loan association. Savings and loan associations accept deposits and lend those funds for long-term mortgages. Because most deposits are short term, deposit rates must adjust to changing interest rate conditions. Most mortgagors wish to borrow at a fixed rate for a long time. As a result, the savings and loan association can be left with floating-rate liabilities and fixed-rate assets. This means that the savings and loan is vulnerable to rising rates. If rates rise, the savings and loan will be forced to increase the rate it pays on deposits, but it cannot increase the interest rate it charges on the mortgages that have already been issued.

To escape this interest rate risk, the savings and loan might use the swaps market to transform its fixed-rate assets into floating-rate assets or transform its floating-rate liabilities into fixed-rate liabilities. Let us assume that the savings and loan wishes to transform a fixed-rate mortgage into an asset that pays a floating rate of interest. In terms of our plain vanilla interest rate swap example, the savings and loan association is like Party A - in exchange for the fixed-rate mortgage that it holds, it wants to pay a fixed rate of interest and receive a floating rate of interest. Thus, the savings and loan wants to be a pay-fixed counterparty in a swap. Engaging in a swap as Party A did will help the association to resolve its interest rate risk.

To make the discussion more concrete, we extend our example of the plain vanilla interest rate swap. We assume that the savings and loan association has just loaned $1 million for five years at 9 percent with annual payments, and we assume that the saving and loan pays a deposit rate that equals LIBOR minus 1 percent. With these rates, the association will lose money if LIBOR exceeds 10 percent, and it is this danger that prompts the association to consider a interest rate swap.

Figure 20.7 shows our original plain vanilla interest rate swap with the additional information about the savings and loan that we have just elaborated. In the figure, Party A is the saving and loan association, and it receives payments at a fixed rate of 9 percent on the mortgage. After it enters the swap, the association also pays a fixed rate of 9 percent on a notional principal of $1 million. In effect, it receives mortgage payments and passes them through to Party B under the swap agreement. Under the swap agreement, Party A receives a floating rate of LIBOR flat. From this cash inflow, the association pays its depositors LIBOR minus 1 percent. This leaves a periodic inflow to the association of 1 percent, which is the spread that it makes on the loan.

In our example, the association now has a fixed-rate inflow of 1 percent, and it has succeeded in avoiding its exposure to interest rate risk. No matter what happens to the level of interest rates, the association will enjoy a net cash inflow of 1 percent on $1 million. This example clarifies how the savings association has a strong motivation to enter the swap market. From the very nature of the savings and loan industry, the association finds itself with a risk exposure to rising interest rates. However, by engaging in an interest rate swap, the association can secure a fixed-rate position.

Notice that the savings and loan could have achieved the same result in other ways if it were free from regulatory constraints and were willing to radically alter its business operations. For example, the savings and loan could achieve the same reduction in interest rate risk by paying off all of its depositors and issuing a fixed-rate bond to fund its mortgage lending. Obviously, this approach is not available to a savings and loan institution, because such a course of action would mean that the firm would cease to be a depository institution altogether. For the firm in this example, the swaps market is attractive because it provides a means of altering its interest rate risk without changing its business operations. The motivation is business efficiency, not the pursuit of an arbitrage profit.

Creating Hybrid Fixed/Floating Debt

Considering an industrial firm with an outstanding FRN (floating-rate note) paying LIBOR plus 2 percent semiannually, a remaining term to maturity of six years, and a par value of $30 million. The issuer has decided that it would like to fix its financing cost for the first three years of the remaining maturity, while allowing the rate to float for the remaining three years. A "brute force" approach to this need would be to purchase all of the existing bonds in the open market, and issue a new bond that has the desired characteristics of a fixed rate for three years followed by a floating rate for three years. This course of action would be quite expensive and difficult. First, a bond buyback is an expensive undertaking in itself. Second, the firm would have to register and issue the new bond, incurring substantial registration fees and flotation costs. Third, the firm might have difficulty finding investors who would want a hybrid fixed/floating bond. Through the swap market, however, the firm can realize its desire efficiently and at low cost.

The firm can change the structure of its debt by leaving its existing FRN intact and entering a pay-fixed interest rate swap. Specifically, the firm initiates a swap to pay a fixed rate on $30 million with semiannual payments with a tenor of three years and to receive LIBOR flat. Assuming the fixed rate for such a swap is 6 percent, the firm will have fixed its financing cost at 8 percent for the first three years while allowing the rate it pays on the bond to float for the last three years of the FRN's life. The firm's fixed-rate financing cost is 8 percent, because with the swap agreement, the firm pays LIBOR plus 2 percent on its FRN, receives LIBOR flat on the swap agreement, and pays 6 percent fixed on the swap agreement. We illustrate this transaction by focusing on a single payment, since all six of the semiannual payments have the same structure:

Initial position:
Semiannual cash flow on outstanding FRN:	-(LIBOR + 2%) x 1/2 year x $30,000,000
Pay-fixed swap semiannual cash flows:	- 6% x 1/2 year x $30,000,000 + LIBOR x 1/2 year x $30,000,000
Net semiannual cash flow (swap flows plus outstanding FRN):	-(LIBOR + 2%) x 1/2 year x $30,000,000 - 6% x 1/2 year x $30,000,000 + + LIBOR x 1/2 year x $30,000,000 = -8% x 1/2 year x $30,000,000 = -$1,200,000

Compared with the difficulty and expense of a bond buyback and a reissuance, the swap agreement can be arranged quickly and cheaply to achieve the same financial results for the firm. This example illustrates the real current motivation for the swaps market. Swaps provide a cost-effective and operationally efficient means of altering a financial position that could probably also be achieved in a more expensive and cumbersome manner. Today, the popularity of the swaps market depends much more on the operational efficiencies that it offers, rather than on attempts to exploit informational inefficiencies.

Pricing of Swaps

This section illustrates the bare basic of swap pricing by focusing on the intuition behind the pricing of plain vanilla interest rate swaps. The information below explores the pricing of interest rate and currency swaps in detail. For now, we seek to convey the basic principal. Consider the three yield curves of Figure 20.9. Each curve has a two-year spot rate of 5.40 percent. One curve slopes modestly upward to 6.20 percent by year 10. One curve is flat at 5.40 percent, and one slopes modestly downward to 4.6 percent at year 10. We want to consider how a plain vanilla interest rate swap would be priced in each environment. In every case, the plain vanilla interest rate swap requires the pay-floating party to pay LIBOR each period in return for a fixed-rate payment.

If we compare the upward- and downward-sloping yield curves, we can ask which environment would justify a higher fixed-payment in exchange for LIBOR for a vanilla swap with a ten-year tenor. To answer this question, consider the forward rates of interest that prevail in each environment. The forward rates are clearly higher in the upward-sloping yield curve environment.

When the forward rates are higher, the fixed rate on the swap should be higher. Without attempting to prove this proposition here, there are two intuitive justifications for this claim. First, forward rates are often taken as a forecast of future expected spot interest rates. Thus, one would expect higher spot rates over the ten-year horizon in the upward-sloping term structure environment. If those higher rates were to materialize, the LIBOR payments would rise over time and this would require a higher fixed-rate payment on the swap to avoid arbitrage. With the ready availability of forward rate agreements (FRAs), one could hedge the various payments on the swap at rates consistent with the shape of the term structure. Therefore, the fixed rate on the swap must reflect both the level and shape of the term structure. This is the essential factor that determines swap pricing.

The no-arbitrage fixed rate on an interest rate swap agreement depends principally on the level of interest rates and the shape of the term structure.

Below illustrates this principal in detail and illustrates the mathematics of interest rate swap pricing. By analogy with our intuitive considerations of the term structure for interest rate swap pricing, we state an analogous principle for foreign currency swap pricing without attempting proof at this point.

The no-arbitrage fixed rate on a foreign currency swap agreement depends principally on the term structure of interest rates for the two currencies in the swap (which together also define the term structure of foreign exchange rates between the two currencies).

We also demonstrate this principle of pricing for foreign currency swaps and illustrate it with several examples. For the present, we accept as given that the yield curve principally determines interest rate swap pricing, and go on to explore how the dealer sets prices to be consistent with the existing term structure of interest rates in a way to yield a profit.

The Economic Analysis of Swaps

In this section we consider swaps from a variety of economic viewpoints. By understanding how to analyze swaps in terms of more familiar financial instruments, we can deepen our understanding of the swaps market in general and lay a foundation for understanding how prices of swap contracts are determined.

We begin by analyzing interest rate and foreign currency swaps in terms of bonds. As we will see, swaps can be interpreted as a combination of buying and selling a pair of bonds. We will see that it is possible to view an interest rate swap as a collection of forward or futures contracts. The analysis of swaps as a portfolio of forwards is critical in the marketplace for the pricing of swaps. As a final type of analysis, we show that a swap agreement can be interpreted as a portfolio of option contracts.

If a swap is shown to have identical cash flows to another portfolio of securities, this information can be helpful in pricing a swap. If two instruments, or portfolios of instruments, have identical cash flows, they must have the same price. In some cases, it is easy to find a price for one pack of securities but not the other. Being able to price one portfolio reveals the price of the other equivalent package of cash flows. For example, we will see that an interest rate swap is equivalent to a portfolio of bonds, in the sense that the swap and the portfolio of bonds have identical cash flows. If we know the price of the bonds in the portfolio, then we know the price of the swap.

An interest rate swap as a combination of capital market instruments In this section, we show how four different types of interest rate and foreign currency swaps can be interpreted as a pair of bond transactions. In each case, a swap is equivalent to the simultaneous purchase of one bond and the sale of another. Key to this analysis is a bond that pays a floating rate of interest, know as a floating-rate note (FRN).

Plain Vanilla Receive-Fixed Interest Rate Swap

A plain vanilla receive-fixed interest rate swap may be constructed from a long position in a bond coupled with the issuance of an FRN, as the following example illustrates. Consider a 6 percent corporate bond with an annual coupon payment, a remaining maturity of four years, and a market value of $40 million principal that pays LIBOR annually and has a four-year maturity. Figure 21.1 shows the cash flows associated with buying the corporate bond and issuing the FRN.

The net flows from the pair of bond transaction are as follows. At the outset, the firm buys a bond for $40 million and issues an FRN with a principal balance of $40 million, for a net zero cash flow. Similarly, at the end of the four-year period, both bonds will mature. At maturity, the firm will be repaid its $40 million principal on the corporate bond, and it will repay the $40 million on the FRN, for a net zero cash flow on the principal amounts. This leaves the four annual coupons to consider on both the bond and the FRN. Each annual coupon payment net cash flow will consist of a $2.4 million inflow on the corporate bond and an outflow on the FRN equal to LIBOR time $40 million:

$2.4 million - LIBOR x $40 million

Whether this net flow will be positive or negative depends on movement in interest rates. The important point to notice about the net cash flows is that they are identical to a receive-fixed plain vanilla interest rate swap with annual payments and a four-year tenor. Thus, the bond portfolio is financially equivalent to an interest rate swap.

Plain Vanilla Pay-Fixed Interest Rate Swap

From figure 21.1, it is also clear that a similar strategy can be used to create a plain vanilla pay-fixed interest rate swap. To create the pay-fixed swap, one would issue a corporate fixed-coupon bond and buy an FRN. Using the same bond and FRN described in Figure 21.1, issuing a fixed-coupon bond and buying an FRN would result in no net principal cash flows and four annual flows equal to:

LIBOR x $40 million - $2.4 million

Other more complex interest rate swaps can be interpreted as more complex bond portfolios by following the same basic strategy.

Fixed-for-fixed currency swap

To create a fixed-for-fixed currency swap, one can buy a bond denominated in one currency and issue a bond denominated in a second currency. For example, assume that one wishes to create a fixed-for-fixed currency swap with a notional principal of €50 million, and a tenor of five years with annual payments to pay US dollars and receive euros. Assume that the spot exchange rate is $1 = €0.8, the prevailing euro interest rate (Euribor) is 7 percent, and the US dollar rate is 6 percent.

In the example, one wishes to receive euros and pay dollars, so the fixed-for-fixed swap is created by buying the euro-denominated bond and issuing a dollar-denominated bond. With a desired notional amount of €50 million and an exchange rate of €1.0 = $0.8, the issuance will be for a dollar-denominated bond with a principal amount of $40 million. The upper panel of Figure 21.2 shows the separate cash flows from buying the euro-denominated bond and issuing the dollar-denominated bond. The lower panel shows the overall cash flows from the combined purchase and sale. In the lower panel at the outset, the net flow is to receive $40 million and pay €50 million. At the end of five years, the principal payments will be to receive €50 million and pay $40 million. In addition, there will be five annual coupon payments of $2.4 million and a coupon receipt of €3.5 million. The cash flows in the lower panel that we have been describing are the same cash flows as a fixed-for-fixed currency swap to receive euros and pay US dollars with a notional principal of €50 million and a tenor of five years with annual payments. Thus, we may analyze a fixed-for-fixed currency swap as being equivalent to purchasing a bond in one currency and issuing a bond in another currency.

Plain Vanilla Currency Swap II

A plain vanilla currency swap may be analyzed also in terms of a two-bond portfolio. To create a plain vanilla currency swap (pay floating US dollars and receive-fixed foreign currency), one would issue a dollar-denominated FRN and buy a foreign bond. Consider a party that issues at par an annual coupon FRN for $20 million with a maturity of three years to pay LIBOR. With an exchange rate of $1 = ¥120, this same party also buys a yen-denominated bond at par with a market value of ¥2.4 billion, a maturity of three years, and annual coupon payments of 5 percent. The upper panel of Figure 21.3 shows the cash flows associated with each bond, while the lower panel shows the combined flows from the two bonds. In combination, these two bonds have the same cash flows as a plain vanilla currency swap to pay LIBOR and receive-fixed yen with a notional principal of $20 million, annual payments, and a tenor of three years.

Forward Interest Rate Swap

In addition to the simple swaps considered already in this section, many more complicated swap structures can be analyzed in terms of bonds. Consider the following four bond transactions:

1 Purchase an eight-year annual coupon FRN based on one-year LIBOR at par with a face value of $30 million.
2 Issue and eight-year 8 percent annual coupon bond at par with a face value of $30 million.
3 Issue a three-year annual coupon FRN based on one-year LIBOR at par with a face value of $30 million
4 Purchase a three-year 8 percent annual coupon bond at par with a face value of $30 million.

Figure 21.4 presents a cash flow diagram for each bond transaction in the upper panel. The final time line of the figure shows the net cash flows. The cash flows on bond 3 and 4 together exactly cancel out the cash flows on bonds 1 and 2 for the first three years. The resulting cash flow pattern is that of a pay-fixed forward swap to begin in three years, to have annual payments, and to have a tenor of five years. Therefore, the four-bond portfolio and the forward swap are equivalent.

Seasonal Interest Rate Swap

A typical seasonal interest rate swap might have quarterly payments with one payment each year being based on a substantially larger notional principal. The paradigm here is a retailer with the last quarterly flow of the year being larger to reflect the Christmas retailing surge. Consider the following portfolio of bonds:

(1) Issue a semiannual coupon FRN based on six-month LIBOR at par with a face value of $10 mill-ion with payment dates in May and November and a maturity of seven years.

(2) Issue a semiannual coupon FRN based on six-month LIBOR at par with a face value of $10 mill-ion with payment dates in February and August and a maturity of seven years.

(3) Purchase a semiannual 6.5 percent coupon bond at par with payment dates in May and November and a maturity of seven years, with a face value of $10 million.

(4) Purchase a semiannual 6.5 percent coupon bond at par with payment dates in February and August and a maturity of seven years, with a face value of $10 million.

(5) Issue an annual coupon seven-year FRN at par based on one-year LIBOR with a face value of $20 million, with each annual payment date in November.

(6) Purchase a 6.5 percent annual coupon bond at par with a face value of $20 million, with each annual payment date in November and a maturity of seven years.

Together, these six bonds create a bond portfolio that is equivalent to a quarterly payment receive-fixed seasonal interest rate swap with a tenor of seven years. The February, May and August payments in the swap will be based on a notional principal of $10 million, while the November payment will be based on a notional principal of $30 million.

At the initiation of the bond portfolio, the net cash flow is zero, as the bond issuances offset the bond purchases. The same is true when the bonds mature: the repayment of principal on a bond that was issued if funded by the maturity of a bond that was purchased. Figure 21.5 shows the cash flows that would result for a single year of the bond portfolio. The cash flows have the same form as those of a swap.

In a typical quarterly payment swap, the floating-rate payments would normally be based on three-month LIBOR. In our bond portfolio, they are based on six-month LIBOR for the first four bonds, and the additional November payment from bonds 5 and 6 is based on one-year LIBOR.

Comparing the bond portfolio to a seasonal wrap, we see that the bond portfolio is cumbersome relative to the swap. The bond portfolio requires six bonds to replicate (or almost replicate) a swap that can be described fairly completely as:

Receive-fixed at 6.5 percent quarterly with a notional principal of $10 million in February, May, and August, and with a notional principal of $30 million in November for a tenor of seven years, with floating payments based on three-month LIBOR.

Notice that the bond portfolio is not exactly equivalent to the swap just described, because the floating payments are based on either six-month or one-year LIBOR in the bond portfolio, and based on three-month LIBOR in the swap description. As a swap becomes slightly more complicated, the replicating bond portfolio quickly becomes extremely complicated. Constructing a bond portfolio that matches the swap cash flows and interest calculations exactly would be difficult. One would need the semiannual payment FRNs in the portfolio to be based on three-month LIBOR, which would be unusual. Alternatively, one might seek a quarterly payment FRN with payments based on three-month LIBOR. Either of these strategies would still leave the larger annual November payment, which would need to be based on three-month LIBOR as well. This kind of bond also would be rare.

Even ignoring the problem with the quarterly payments in the bond portfolio being based on nonquarterly LIBOR, the bond portfolio is a cumbersome and costly way of securing the cash flow obligations that can be achieved inexpensively and easily through the swap directly. The operational efficiency of this kind of swap relative to the bond portfolio illustrates one of the reasons for the swap market's stunning success.

An Interest Rate Swap as s Portfolio of Forward Rate Agreements

In this section, we analyze an interest rate swap in terms of a portfolio of interest rate forward contracts. In the interest rate market, these forward contracts are known as forward rate agreements (FRA). We briefly review the key features of FRA's here, before showing how an interest rate swap can be analyzed as a portfolio of FRAs.

Key Features of FRAs

Typically, an FRA calls for the exchange of LIBOR at a future date in return for a payment based on a fixed rate of interest agreed on the contracting date. For example, consider an FRA market maker who agrees today to pay six-month LIBOR in six months in exchange for a fixed interest payment at an annual rate of 5 percent and a notional principal of $20 million. When the determination date of the FRA arrives, assume that LIBOR stand at 5.8 percent. Payments on the FRA would be the interest rate times the fraction of the year times the notional principal:

Receive-fixed: 0.050 x 0.5 x $20,000,000 = $500,000
Pay-floating: 0.058 x 0.5 x $20,000,000 = -$580,000
Net payment: 0.058 x 0.5 x $20,000,000 = -$580,000

In this case, the market maker would be obligated to make a net payment of $80,000. In essence, this FRA agreement is a one-date swap agreement. Therefore, we may analyze an interest rate swap agreement as a sequence of FRAs.

FRAs are quoted in the following manner:

term to expiration in months x term to end of period covered by agreement

For example, the FRA we considered above, with six months to expiration for six-month LIBOR, would be quoted as:

6 x 12 5%

The first number indicates the months until the FRA expires, the second indicates the number of months until the instrument presumed to underlie the FRA matures, while the differences between the two numbers shows the maturity of the presumed underlying instrument. There is no actual instrument that is delivered; instead, the presumed underlying instrument is simply an instrument of the underlying maturity that pays LIBOR. Instead of exchanging instruments, the profit or loss is settled in cash as in our preceding example. The 5 percent is the fixed rate of interest or loss to be paid in exchange for LIBOR. FRAs are normally "determined in advance and settled in arrears." In our example of the 6 x 12 FRA, the determination date would be at month 6 and the actual payment would occur at month 12.

On-market and Off-Market FRAs

As we have seen, FRA market makers offer to make a market in FRAs for a variety of maturities at stated rates. An FRA agreement entered at the prevailing market-determined rate is an on-market FRA. For the example we have just been considering, the on-market rate for a six-month FRA with a determination date in six months and a payment date in 12 months would be 5 percent. Entering an on-market FRA is costless, as it is simply a forward contract initiated at the prevailing rate. An off-market FRA is an FRA entered at a rate that differs from the prevailing market-determined rate. Because the terms for an off-market FRA differ from those prevailing in the market, a payment is required to enter the FRA. For example, let us assume that the prevailing rate on an FRA is:

6 x 12 5%

An off-market receive-fixed FRA for this period is entered at a rate of 7 percent with a notional principal of $10 million. For this agreement, the on-market rate is 5 percent, so the on-market fixed payment would be:

0.05 x 1/2 x $10,000,000 = $25,000,000

while the off-market fixed payment would be

0.07 x 1/2 x $10,000,000 = $350,000

0.07 - 0.05 x 1/2% x $10,000,000 = $100,000

Both payments would occur in one year at month 12, and both FRAs call for the same dollar payment on the floating side. From this example, it is clear that entering the off-market receive-fixed FRA will pay $100,000 more than the on-market FRA in one year. Therefore, this receive-fixed off-market FRA will require the payment of the present value of the $100,000 when the FRA is initiated or the payment of $100,000 at month 12. As we now show, off-market FRAs play a critical role in analyzing an interest rate swap as a portfolio of FRAs.

Interest Swaps and FRAs

With this background, we can now see how a plain vanilla interest rate swap can be analyzed as a portfolio of FRAs. Consider the following FRA quotations:

0 x 6 5.00%
6 x 12 5.00%
12 x 18 5.00%
18 x 24 5.00%

These quotations offer a fixed rate of interest for six-month LIBOR 6, 12 and 18 months from now, and these rates are consistent with a flat yield curve. Faced with these rates, a firm enters all three FRAs to receive-fixed and pay six-month LIBOR.

A sequence of evenly spaced instruments with the same notional principal is called a strip. By entering these three FRA agreements at a notional principal of $20 million, the firm has entered a strip of FRAs and is obligated to receive a fixed rate of 5 percent and pay six-month LIBOR on a notional principal of $20 million each six months over the next two years. The determination dates will be at months 6, 12, and 18, and the corresponding payment dates will be at months 12, 18, and 24.

This sequence of three FRAs is equivalent to a receive-fixed interest rate swap at 5 percent with semiannual payments on a notional principal of $20 million and a tenor of two years. From the point of view of the receive-fixed party, the periodic cash flow will be

0.05 x 1/2 year x $20 million - LIBOR x 1/2 year x $20 million

Therefore, when the yield curve is flat, a plain vanilla interest rate swap is equivalent to a sequence of FRAs.

We now consider the more realistic case of a yield curve with shape. Assume that the current six-month spot rate is 4.95 percent, and consider now the following FRA quotations that imply a rising yield curve:

0 x 6 4.95%
6 x 12 5.00%
12 x 18 5.10%
18 x 24 5.20%

These spot and FRA yields coverage range from 4.95 percent to 5.20 percent. A plain vanilla interest rate swap call for a single fixed rate for the entire tenor of the swap. With these varying FRA rates, it is not clear how the equivalence between the strip of FRAs and the plain vanilla swap can be maintained.

Intuitively, the fixed rate on the swap must be greater than 4.9 percent, but less than 5.20 percent. However, the fixed swap rate is unlikely to match any of the four rates. Instead, the fixed rate on the swap must be a function of the spot rate and the three rates on the FRA, such that the fixed rate on the swap implies payments with the same present value as the sequence of FRAs. If this pricing rule were not maintained, arbitrage would be possible. The arbitrage would result because the cash flows on an interest rate swap can be replicated by a sequence of FRAs. This no-arbitrage principle enforces an equality between the present value of the fixed payments and the present value of the floating payments viewed from the initiation of the swap. It also provides a way to find the no-arbitrage fixed rate for the swap given the FRA quotations.

Later, we show exactly how to find the fixed rate for this swap. For now, let us simply assume that the no-arbitrage fixed rate for this swap is 5.075 percent. We select this rate as it is the midpoint between the high and low rates. Notice that it does not match any of the four FRA rates actually available. This will generally be the case in any term structure environment when the yield curve has shape. Thus, for a swap in this situation with a fixed rate of 5.075 percent, the swap may be replicated by a strip of off-market FRAs:

Quarter	FRA rate (%)	Fixed Rate (%)	Fixed-rate bias
1	4.950	5.075	Too high
2	5.000	5.075	Too high
3	5.100	5.075	Too low
4	5.200	5.075	Too low
	5.063

In this rising yield environment of our example, the fixed rate on the swap is first too high, for periods 1 and 2, and then too low for periods 3 and 4. In general, we would expect the fixed rate on the swap to be about the average of the FRA rates covering the tenor of the swap. The first two payments on this swap have a present value that is beneficial to the receive-fixed counterparty while the last two payments benefit the pay-fixed party, assuming that the FRA rates materialize as the actual rates to cover the various periods.

In summary, we may interpret an interest rate swap as a strip of FRAs. In the rare event of a flat yield curve, all FRA rates will be the same and the fixed rate for the swap will be the same as the common FRA note. In the more usual case, when the yield curve has shape, the fixed rate on the swap will not equal any of the FRA rates. Instead, it will be the unique fixed rate that makes the present values of all the individual payments sum to zero. This is the one fair fixed rate for the swap that does not disadvantage either party. The strip of FRAs will then be a strip of off-market FRAs, and it is this strip of off-market FRAs that is equivalent to a plain vanilla interest rate swap.

Interest Rate Swap Pricing [viewable here in Excel]

The term structure of interest rates is the key to pricing interest rate swaps; the term structure of interest rates and the term structure of foreign exchange rates are the keys to pricing currency swaps. We have seen that bonds, FRNs, FRAs, call and puts on LIBOR, foreign exchange forward contracts, interest rate futures, and swaps are all related instruments. If they are not priced properly relative to each other and relative to the existing term structures, arbitrage will be possible. Thus, we approach swap pricing in a no-arbitrage environment that focuses on the term structure of interest and currency rates

The Term Structure of Interest Rates and Foreign Exchange Rates

There are three related term structures or yield curves: the par yield curve, the zero-coupon yield curve, and the implied forward yield curve. The par curve expresses the relationship between the yield on a coupon bond selling at par and bond maturity. The zero-coupon yield curve shows the relationship between yield and maturity for single future payments. The implied yield curve shows the relationship between forward rates of interest for various future periods as implied by the par yield curve and the zero-coupon yield curve. These three term structures form an integrated system - one set of rates implies another. For example, the zero-coupon yield curve can be found by using the data of the par yield curve in conjunction with the technique of bootstrapping, as discussed above. Below are a few key notations related to bootstrapping:

Z_x,y = the zero-coupon factor for an investment initiated at time x and extended until time y
FR_x,y = the forward rate of interest for a period beginning at time x and extending until time y
FRA_x,y = the interest rate on an FRA for a period beginning at time x and extending until time y
FRF_x,y = the forward rate factor for a period beginning at time x and extending until time y

We also note the following
FR_x,y = FRA_x,y = FRF_x,y - 1

FRF_x,y = Z_0,y / Z_0,x

Table 19.3 Illustrative Treasury instruments, par = 100
Instrument	Maturity	Annual Coupon	Price (% of par)
A	6 mo	5.80%	100
B	1 yr	6.00%	100
C	1.5 yrs	6.40%	100
D	2 year	6.80%	100
E	3 year	7.00%	100

Table 19.4 Cash-flows from Treasury bonds		6 months	12 months	18 months	24 months	30 months	36 months
	Bond Term	0.5	1.00	1.5	2	2.5	3
2.90%	A	102.9
3.00%	B	3.0	103.0
3.20%	C	3.2	3.2	103.2
3.40%	D	3.4	3.4	3.4	103.4
3.50%	E	3.5	3.5	3.5	3.5	3.5	103.5	100.00644	7.12%

				Bootstrap Zero-rate Coupon Factor	Forward-rate 6 mo Factor (FRF)	Forward-rate 1 yr Factor (FRF)	Forward-rate 1.5 yr Factor (FRF)	Zero Annual Yield w CC	Annual Coupon Rate (% of Par)		Trial and error
6 months	PA = 102.9/Z0,0.5	100=102.9/Z0,0.5	100.00	1.02900				5.6380%	5.80%	96.00	10.68%
1 year	PB = 3.0/Z0,0.5+103.0/Z0,1.0	100 = 3.0/1.029+103.0/Z0,1.0	100.00	1.060931	1.03103			5.9147%	6.00%
1.5 years	PC = 3.2/Z0,0.5+3.2/Z0,1.0 +103.2/Z0,1.5	100 = 3.2/1.029+3.2/1.060931+103.2/Z0,1.5	100.00	1.099346	1.03621	1.0683638		6.4130%	6.40%
2.0 years	PD = 3.4/Z0,0.5+3.4/Z0,1.0 +3.4/Z0,1.5+103.4/Z0,2.0	100 = 3.4/1.029+3.4/1.060931+3.4/1.099346+103.4/Z 0,2.0	100.00	1.143826	1.04046	1.0781346	1.11159018	6.9445%	6.80%
3.0 years	PE = 3.5/Z0,0.5+3.5/Z0,1.0 +3.5/Z0,1.5+3.5/Z0,2.0+3.5/Z 0,2.5+103.5/Z0,3.0	100 = 3.5/1.029+3.5/1.060931+3.5/1.099346+3.5/1.143826+103.5/Z 0,2.5	100.00	1.188889				6.1060%	7.00%

Bootstrap Zero-rate Coupon Factor	Forward-rate 6mo Factor (FRF)	Forward-rate 1 yr Factor (FRF)	Forward-rate 1.5 yr Factor (FRF)	Zero Annual Yield w CC	Annual Coupon Rate (% of Par)
1.02900				5.6380%	5.80%
1.060931	1.03103			5.9147%	6.00%
1.099346	1.03621	1.06836378		6.4130%	6.40%
1.143826	1.04046	1.07813456	1.111590187	6.9445%	6.80%
1.188889				6.1060%	7.00%
	SFR Rate		Forward Rate	Swap Fixed Rate (Average)	6.2349%
	2,305,855	=	2,305,855	Geometric Average (Forward Rate)	6.2383%

	Forward Rates
6 months 0 x 6	4.95%
6 months 6 x 12	5.00%
6 months 12 x 18	5.10%
6 months 18 x 24	5.20%

Find the Forward Rate if you have the Zero rate
Solve for y where 1.02475 x (1 + .5 * y) = 1.0506388	Solve for y (1 + .5 * y) = 1.02475	Solve for y where 1.05036875 x (1 + .5 * y) = 1.077153
1.02475 x (1 + .5y) = 1.0506388	(1 + .5 y) = 1.02475	1.05036875 x (1 + .5y) = 1.077153
1.02475 x (1 + .5y)/1.02475 = 1.0506388/1.02475	1 + .5y -1 = 1.02475 - 1	1.05036875 x (1 + .5y)/1.05036875 = 1.077153/1.05036875
1.02475 x (1 + .5y) /1.02475 = 1.0506388/1.02475	1 + .5y - 1 = 1.02475 - 1	1.05036875 x (1 + .5y)/1.05036875 = 1.077153/1.05036875
1 + .5y = 1.0506388/1.02475	1 + .5y = 1.02475	1 + .5y = 1.077153/1.05036875
Subtract 1 from both sides	Subtract 1 from both sides	Subtract 1 from both sides
1 + .5y -1 = (1.0506388/1.02475) -1	1 + .5y -1 = 1.02475 -1	1 + .5y -1 = (1.077153/1.05036875) -1
1 + .5y -1 = (1.0506388/1.02475) -1	1 + .5y -1 = 1.02475 -1	1 + .5y -1 = (1.077153/1.05036875) -1
.5y = (1.0506388/1.02475) -1	.5y = .02475	.5y = (1.077153/1.05036875) -1
Convert 1 into fraction: 1 * 1.02475/1.02475	y = .02475/.5	Convert 1 into fraction: 1 * 1.05036875/1.05036875
-(1 * 1.02475)/1.02475 + (1.0506388/1.02475)	y = .0495	-(1 * 1.05036875)/1.05036875 + (1.077153/1.05036875)
Since the denominators are equal, combine the fractions a/c +- b/c = a+-b/c		Since the denominators are equal, combine the fractions a/c +- b/c = a+-b/c
(-1 * 1.02475 + 1.0506388)/1.02475)		(-1 * 1.05036875 + 1.077153)/1.05036875)
(-1 * 1.02475 + 1.0506388) = .025888		(-1 * 1.05036875 + 1.077153) = 0.0267844
.5y = .025888/1.02475		.5y = .027844/1.05036875
y =(.025888/1.02475)/.5		y =(.0267844/1.05036875)/.5 = 0.0255
y = .0253/.5 = 0.0506		y = .0255/.5 = 0.0510
y = .0506	y = .0506	y = .05100
1.02475 x (1 + .5 * y) = 1.0506388	1.02475 x (1 + .5 * y) = 1.0506388	1.050369 x (1 + .5 * y) = 1.077153

Pricing a Plain Vanilla Interest Rate Swap

In a plain vanilla interest rate swap, the receive-fixed party will pay a floating rate equal to LIBOR flat in each period. Determining a price for the plan vanilla interest rate swap requires finding the fixed rate that will be received in exchange for making the sequence of floating-rate payments. This is called the swap fixed rate or SFR. At the time the swap is negotiated, the agreed SFR must not give an arbitrage opportunity to either side. This no-arbitrage condition means that the agreed SFR must be consistent with the term structure of interest rates. One way of determining the SFR is to use FRA quotations as representative of the term structure. Based on these FRA quotations. We use the no-arbitrage equivalence of the present value of the strip of FRAs and the present value of the fixed payments to find the no-arbitrage fixed rate for the swap that is consistent with the FRA quotations. Actual market pricing of swaps relies on money market yield and day counting conventions. We can abstract from these technicalities.

We begin by using an example of a plain vanilla interest rate swap with a notational principal of $20 million, a tenor of two years, and payments every six months. The following table shows the relevant quotations for a strip of FRAs and their associated payments with a notional principal of $20 million:

Forward Rates

6 months .0495 x 0.50 x $20,000,000 = $495,000
12 months .0500 x 0.50 x $20,000,000 = $500,000
18 months .0510 x 0.50 x $20,000,000 = $510,000
24 months .0520 x 0.50 x $20,000,000 = $520,000

Each payment on the fixed side would be the unknown swap fixed rate, SFR, times the half-year between payments times the notional principal of $20 million. Each fixed payment would be:

SFR x 0.5 x $20,000,000

According to our no-arbitrage principle, the sequences of fixed and floating payments must have the same present value at the initiation of the swap. We see that these rates range from 4.95 percent to 5.20 percent, so it seems intuitively reasonable that the fixed rate must be in that range and that is should be some kind if average of these various floating rates.

Recalling that Z_0,t is the zero-coupon factor for a horizon starting at the present, time zero, and ending at time t, we will express time in months, so Z_0,18 is the zero-coupon factor to cover the period from the present until the end if the eighteenth month. In our example, the no-arbitrage condition become:

.0495 x 0.50 x $20,000,000/Z_0,6 + .0500 x 0.50 x $20,000,000/Z_0,12 + .0510 x 0.50 x $20,000,000/Z_0,18 + .0520 x 0.50 x $20,000,000/Z_0,24

Equation (21.9)

SFR x 0.50 x $20,000,000/Z_0,6 + SFR x 0.50 x $20,000,000/Z_0,12 + SFR x 0.50 x $20,000,000/Z_0,18 + SFR x 0.50 x $20,000,000/_Z0,24

In this no-arbitrage condition, it appears that we have four unknown zero-coupon factors (Z_0,6 + Z_0,12 + Z_0,18 + Z_0,24) plus the unknown swap fixed rate (SFR). However, we actually do have the information to find Z_0,6, and we can use Z_0,6 to find the other factors through a technique known as bootstrapping, as shown above.

Applying the bootstrapping technique, we can use an initially known zero-coupon factor and forward rate to find successive factors. We illustrate the technique for our example by noting that we can compute the value for the first factor, Z_0,6 as follows:

Z_0,6 = 1 + 0.5 x 0.0495 = 1.02475

where 0.5 reflects the half-year between payments and 0.0495 is the current LIBOR spot rate. The factor for the second payment covers the first 12 months of the swap. Given the value of Z_0,6. We can compute the value of Z_0,12 as follows:

Z_0,6 = Z_{0,6 x (1 + 0.5 x 0.0500) = 1.050369}

where 0.5 reflects the half-year between payments and 0.0500 is the forward rate from the 6 x 12 FRA. Values for Z_0,18 and Z_0,24 follow similarly:

Z_0,18 = Z_0,12 x (1 + 0.5 x 0.0510) = 1.050369 x 1.0255 = 1.077153

Z_0,24 = Z_0,18 x (1 + 0.5 x 0.0520) = 1.077153 x 1.026 = 1.105159

Starting with Z_0,6 and using the bootstrapping technique, we have found all of zero-coupon factors that we need.

We can now use these factors in Equation 21.9 to find the swap rate, SFR:

.0495 x 0.50 x $20,000,000/1.02475 + .0500 x 0.50 x $20,000,000/1.050369 + .0510 x 0.50 x $20,000,000/1.077153 + .0520 x 0.50 x $20,000,000/1.105159

SFR = (.0495/1.02475 + .0500/1.050369 + .0510/1.077153 + .0520/1.105159) / (1/1.02475 + 1/1.050369 + 1/1.077153 + 1/1.105159) = 0.050598

Thus, this swap fixed rate of 5.0598% is the rate on the fixed side if the swap that prevents arbitrage.

We can generalize this example:

SFR = (FRA_0,6/Z_0,6 + FRA_0,6/Z_0,12 + FRA_0,6/Z0,18 + FRA_0,6/Z_0,24) / (1/Z_0,6 + 1/Z_0,12 + 1/Z_0,18 + 1/Z_0,24)

Forward Rate Agreement (FRAs)
Off-Market Forward Rates		Zero Rate Calculation		Zero-rate	Spot rate	Forward
FRA0,6	4.95%	Z0,6	= 1 + FRAC X FRA0,6	1.02475	4.950%	4.95%
FRA6,12	5.00%	Z0,12	=Z0,6 X (1 + FRAC X FRA6,12)	1.05037	5.001%	5.05%
FRA12,18	5.10%	Z12,18	=Z0,12 x ( 1 + FRAC X FRA12,18)	1.07715	5.104%	5.31%
FRA18,24	5.20%	Z18,24	= Z0,18 x (1 + FRAC X FRA18,24)	1.10516	5.207%	5.52%
	5.0616%	SFR Rate		Forward Rate	Geometric Average (FRAs)
	5.0598%	1,903,059	=	1,903,059	Swap Fixed Rate (SFR)

Interest Rate Parity [viewable here in Excel]

Consider two countries with debt instruments issued in the currencies of each country and the spot and forward currency exchange rates in the two currencies. The interest rate parity theorem asserts that the interest rates in the two countries and the exchange rates between the two currencies form an integrated system. There must be parity in this system of interest and exchange rates to avoid arbitrage. Specifically, interest rate parity asserts that investment in one currency must yield the same proceeds as the following alternative strategy over a given investment horizon:

(1) Convert funds into a foreign currency at the spot rate (today's rate) at the outset
(2) Invest in the foreign currency
(3) Reconvert the proceeds to the original currency via a foreign exchange forward contract initiated at the outset of the investment horizon to convert the investment proceeds into the original currency at the investment horizon.

Let

_x,yF_x,y = for a foreign exchange forward contract initiated at time t with delivery at time T, the value of one unit of the x-currency in terms of the y-currency

Thus

_$,euroFX_0,3 = the forward exchange value of $1 in terms of euros for a contract initiated at time
t = 0, with delivery at time T = 3, and
_$,¥FX_0,0 = the spot price of $1 in terms of Japanese yen for immediate delivery, because t = T

Previously, we introduced the notation of zero-coupon factors such that Z_t,T is the factor for a payment to be received at time T measured from time t. Where necessary, we will indicate currencies by a prescript for the factor.

In terms of this notation, we can express the interest rate parity theorem as follows:

Interest rate parity
_xZ_t,T = _x,yFX_{t,T x y}Z_t,T x _y,xFX_t,T

Again, in English this theorem says that:

Investments in currency x from time t until time T must have the same proceeds as converting funds from currency x to currency y at the foreign exchange spot rate, investing in currency y from time t until time T, and converting the proceeds of the investment in currency y back into currency x via a forward contract initiated at time t with payoff at time T.

To illustrate this parity condition, assume that the foreign exchange spot rate between the dollar and the euro is $1 = €0.8, the one-year US interest rate is 9%, and the one-year European EMU interest rate is 12%, all with annual compounding. Interest rate parity asserts that the one-year foreign exchange forward rate must be 1 euro = $1.216518 or $1 - €0.822018 (=(1.12 x .8)/1.09))

_xZ_t,T = _x,yFX_t,T x _yZ_t,T x _y,xFX_t,T

_$Z_0,1 = _$,€FX_0,0 x _€Z_0,1, x _€,$FX_0,1

	If rates are:	If rates are:	If rates are:
EMU Interest Rate	12%	5%	5%
US Interest Rate	9%	8%	8%
Foreign exchange forward rate are:		Foreign exchange forward rate are:		Foreign exchange forward rate are:
1.216518	=1.09/(0.8*1.12)	1.28571	=1.08/(0.8*1.05)	1.21528	=1.05/(0.8*1.08)
1.090000	=1.216517857*(0.8*1.12)	0.77778	=(0.8*1.05)/1.08	0.82286	=(0.8*1.08)/1.05	Forward exchange rate
0.822018	=(1.12*0.8)/1.09	1.08000	=1.28571*(1.05*0.8)	1.05000	=1.21528*(1.08*0.8)	Euro Zero Coupon Factor
		1.05000	=(1.08/0.8)*0.77778	1.15800	=(1.05/0.8)*0.882286	$ Zero Coupon Factor

If interest rate parity did not hold, arbitrage would be possible. For example, if _€,RFX_0,1 < 1.216518 in this situation, investments in dollars would be clearly superior. If _€,RFX_0,1 > 1.216518, investment in euros would yield a higher return. Arbitrage would proceed by borrowing in the currency in which funds are relatively cheap, and investing in the currency where the return is relatively high, with a forward contract to convert funds to the currency that was borrowed originally. For example, assume that the forward exchange rate is €1 = $1.30 and the other variables are as given. One would transact as follows:

t = 0
Borrow $1,000,000 million in the United States for one year at 9 percent
Exchange $1,000,000 million for €800,000 at the spot exchange rate
Invest €800,000 one year forward at €1 = $1.30 for a total of $1,164,800
Net cash flow = 0

t = 1 year
Collect proceeds of European investment: €800,000 x 1.12 = €896,000
Deliver €896,000 against forward contract: collect $1,164,800
Repay US loan: $1,000,000 x 1.09 = $1,090,000
Net cash flow = +-74,800

This is a clear example of arbitrage, as it produces a riskless profit without investment. It should be borne in mind that interest rate parity expresses a relationship among interest rates in two countries, the spot exchange rate and the forward exchange rate. All four elements must be mutually consistent to avoid arbitrage.

Consistent with interest rate parity, there is a term structure of foreign exchange rates. The term structure of interest rates in two countries and the term structure of exchange rates between two countries form an integrated system that must be consistent with interest rate parity to avoid arbitrage. For example, assume that the term structure of interest rates in the United States is strongly upward sloping, while the term structure of interest rates in Europe is more gently upward sloping at a lower level. Table 21.1 shows this situation in a simplified form, along with the term structure of dollar/euro exchange rates that is consistent with interest rate parity. Because the US term structure of interest rates lies above the European term structure of interest rates, the value of the mark in terms of dollars must rise as maturities lengthen.

Table 21.1 Term structure of dollar and European interest rates and the dollar/euro exchange rate

Maturity	US Dollar interest rate par yields	US $ interest rate zero-coupon factor	Euro interest rate par yields	European EMU interest rate zero-coupon factor	Forward exchange rate euro value of $1	Forward exchange rate $ value of euro
0	N/A	N/A	N/A	N/A	0.8000000	1.20000
1	0.080	1.080000000	0.050	1.0500000000	0.7777778	1.28571
2	0.085	1.177688442	0.052	1.1068136273	0.7518550	1.28921
3	0.088	1.289411384	0.054	1.1713939028	0.7267775	1.29032
4	0.091	1.420765515	0.055	1.2397555033	0.6980775	1.29265
5	0.093	1.567391306	0.056	1.3149137717	0.6711349	1.29380

A fixed-for-fixed currency swap as a strip of foreign exchange forward contracts.

With this background on interest rate parity, we can now see how to interpret a fixed-for-fixed currency swap as a strip of foreign exchange forward contracts. Consider a five-year fixed-for-fixed currency swap with annual payments negotiated in the context of the interest rate and exchange rate environment shown in table 21.1, and assume that determination dates occur at years 0, 1, 2,3, and 4, with payment dates falling one year later. Assume that the notional principle is $100 million, equivalent to €80 million, and that the dollar payer promises to pay a fixed rate of 8.5 percent, while the European payer promises to pay a fixed rate of 5.3 percent. (These rates are arbitrarily set at roughly the midpoints of the two yield curves for the tenor of the swap. Later, we show how to find the no-arbitrage rates for this kind of swap.) The first time line in Figure 21.6 shows the cash flows from the perspective of the dollar payer. In each year, 1-5, the dollar payer pays $8.5 million and receives €4,240,000. The time line also shows the exchanges of principal at the inception of the swap and at year 5.

The cash flows shown in the upper time line of Figure 21.6 essentially express the cash flows from the following portfolio of foreign exchange contracts:

Sell €80,000,000 spot at a rate of $1 = €.80 for $100,000,000
Sell €8,500,000 one year forward at a rate of $1 = €.498824 for €4,240,000 (.498824=4,240,000/8,500,000)
Sell €8,500,000 two year forward at a rate of $1 = €.498824 for €4,240,000
Sell €8,500,000 three year forward at a rate of $1 = €.498824 for €4,240,000
Sell €8,500,000 four year forward at a rate of $1 = €.498824 for €4,240,000
Sell €8,500,000 five year forward at a rate of $1 = €.498824 for €4,240,000
Sell $100,000,000 five year forward at a rate of $1 = €.80 for €80,000,000

The cash flows for the fixed-for-fixed currency swap in the upper time line of Figure 21.6 are equivalent to this portfolio of one spot foreign exchange transaction and six forward exchange transactions. Notice that the foreign exchange rates for each year are the same (except for the last exchange of principal) and that they do not equal any of the forward exchange rates in Table 21.1. Therefore, we see that a fixed-for-fixed currency swap may be viewed as a portfolio of off-market foreign exchange transactions.

This perspective also provides guidance to the pricing of this kind of currency swap. The second time line of Figure 21.6 shows the dollar equivalents of the cash flows in the upper time lines. All of the euro cash flows have been translated into dollars using the respective forward rates. For example, the euro inflow at year 3 is €4,200,000, and the three-year forward currency rate is 0.726778 from Table 21.1. Therefore, the dollar value of that flow is €4,200,000/.726778 = $5,833,972. In the second time line, the exchange of principal is a wash. The dollar payer loses by at least $2,000,000 on each of the annual coupon payments in years 1-5. The reexchange of principal at year 5 (€80 million for $100 million) implies an exchange rate of $1 = €0.80, which was the original spot rate. On the reexchange of principal, the dollar payer recoups much of the losses on the individual coupon payments. For this fixed-for-fixed currency swap to be fairly priced, the present value of all of the cash flows in the second time line must be zero, when these cash flows are discounted based on the US dollar zero-coupon term structure. Otherwise, one side gains at the expense of the other.

Similarly we can view the entire transaction from the perspective of one concerned with euros. To do so, we convert all of the dollar flows into euros at the respective exchange rates. For example, at year four, the euro payer receives $8.5 million. The four-year forward rate is $1 = €0.698078, so the euro value of that payment would be €5,933,633. The cash flows in the third time line must have a zero present value when discounted according to the European zero-coupon discount rates (Euribor). Otherwise, one party gains at the other's expense. As we show later, this equivalence between a fixed-for-fixed currency swap and a portfolio of off-market foreign exchange transactions provides a valuable guide to pricing currency swaps.

A plain vanilla currency swap as a fixed-for-fixed currency swap plus a plain vanilla interest rate swap In this section, we explore the equivalences between a plain vanilla currency swap, on the one hand, and a fixed-for-fixed currency swap plus a plain vanilla interest rate swap, on the other. The plain vanilla currency swap calls for the dollar payer to pay LIBOR against a fixed-rate on a foreign currency. Consider a plain vanilla currency swap in the rate environment of Table 21.1. Assume that the swap has a five-year tenor, and annual payments determined in advance and paid in arrears based on one-year LIBOR. The notional principal is $100 million, equivalent to €80 million. The euro fixed rate is 5.3 percent. Determination dates occur at years 0, 1, 2, 3, and 4, with settlement dates one year later. LIBOR, indicates the one-year LIBOR rate prevailing at time t. This swap is the plain vanilla analog of the fixed-for-fixed swap that we explored in the previous section. The first time line of Figure 21.7 shows the cash flows for the plain vanilla currency swap just described from the perspective of the receive-fixed party.

We now want to show that this plain vanilla currency swap is equivalent to a fixed-for-fixed currency swap plus a plain vanilla interest rate swap. The second time line of Figure 21.7 shows the cash flows for the fixed-for-fixed currency swap analyzed in the preceeding section. It consists of the same notional principal of $100 million, equivalent to €80 million, with the dollar payer paying 8.5 percent fixed and the euro payer paying 5.3 percent. (The second time line of Figure 21.7 is identical to the first time line of Figure 21.6.) The third time line of Figure 21.7 shows the cash flows for a plain vanilla interest rate swap with a notional principal of $100 million, a five-year tenor, annual payments, and a fixed rate of 8.5 percent. If we add the cash flows for the dollar payer fixed-for-fixed currency swap and the cash flows for the receive-fixed plain vanilla interest rate swap the combined cash flows are exactly the same as the cash flows for the receive-fixed plain vanilla currency swap. Therefore, we can conclude as follows:

Equation (21.2) receive-fixed plain vanilla currency swap
= dollar payer fixed-for-fixed currency swap
+ receive-fixed plain vanilla interest rate swap

pay-fixed plain vanilla currency swap
= FOREX payer fixed-for-fixed currency swap
+ receive-fixed plain vanilla interest rate swap

We have already seen that a plain vanilla interest swap can be analyzed as a pair of bonds or as a strip of FRAs. Similarly, we noted that a fixed-for-fixed currency swap can be analyzed as a portfolio of foreign exchange contracts. Therefore, it is possible to decompose the plain vanilla currency swap into these more basic elements. Previously, we discussed a CIRCUS swap, a combined interest rate and currency swap. A CIRCUS swap is a fixed-for-fixed currency swap created by combining a plain vanilla currency swap and a plain vanilla interest rate swap.

Rearranging terms in Equation 21.2 gives the following:

dollar payer fixed-for-fixed currency swap
= receive-fixed plain vanilla currency swap
- receive-fixed plain vanilla interest rate swap
= receive-fixed plain vanilla currency swap
+ pay-fixed plain vanilla interest rate swap

and a similar rearrangement of Equation 21.3 gives

FOREX Payer Fixed-For-Fixed Currency Swap

FOREX payer fixed-for-fixed plain vanilla currency swap
= pay-fixed plain vanilla currency swap
- pay-fixed plain vanilla interest rate swap
= pay-fixed plain vanilla currency swap
+ receive-fixed plain vanilla interest rate swap

Table 21.3 Cash flow analysis for fixed-for-fixed currency swap
	Dollar SFR	Euro FRA
	9.30%	5.600%
US Dollar Payer - Euro Perspective Settlement date (year)	Euro cash flow	Dollar cash flow	FOREX rate (euro per dollar)	Euro value of dollar cash flow ($ cf x euro FOREX)	Net euro cash flow ($US cf + Euro cf), (euro value of $ cf + euro cf)	Euro Zero-coupon factor	Present value of euro cash flow
0	(80,000,000)	100,000,000	0.800000	80,000,000	0	1.000000
1	4,480,000	(9,300,000)	0.7777778	(7,233,333)	(2,753,333)	1.050000	(2,622,222)
2	4,480,000	(9,300,000)	0.7518550	(6,992,251)	(2,512,251)	1.106814	(2,269,805)
3	4,480,000	(9,300,000)	0.7267775	(6,759,030)	(2,279,030)	1.171394	(1,945,571)
4	4,480,000	(9,300,000)	0.6980775	(6,492,121)	(2,012,121)	1.239756	(1,622,998)
5	84,480,000	(109,300,000)	0.6711349	(73,355,045)	11,124,955	1.314914	8,460,596
						NPV =	0.0
Euro payer - US dollar Perspective Settlement date (year)	Euro cash flow	Dollar cash flow	FOREX rate (euro per dollar)	Dollar value of euro cash flow (Euro cf/euro FOREX)	Net dollar flow Euro cf + $US cf ($ value of euro cf + dollar cf)	US dollar Zero-coupon factor	Present value of dollar cash flow
0	80,000,000	(100,000,000)	0.800000	100,000,000	0	1.000000
1	(4,480,000)	9,300,000	0.7777778	(5,760,000)	3,540,000	1.080000	3,277,778
2	(4,480,000)	9,300,000	0.7518550	(5,958,596)	3,341,404	1.177688	2,837,256
3	(4,480,000)	9,300,000	0.7267775	(6,164,198)	3,135,802	1.289411	2,431,964
4	(4,480,000)	9,300,000	0.6980775	(6,417,626)	2,882,374	1.420766	2,028,747
5	(84,480,000)	109,300,000	0.6711349	(125,876,332)	(16,576,332)	1.567391	(10,575,746)
						NPV =	(0.0)

Table 21.4 Cash flow analysis for plain vanilla currency swap

US Dollar Payer - Euro Perspective Settlement date (year)	Euro cash flow	US one-year FRA	Dollar cash flow	FOREX rate (euro per dollar)	Euro value of dollar cash flow ($ cf x euro FOREX)	Net euro cash flow $US cf + Euro cf (euro value of $ cf + euro cf)	Euro Zero-coupon factor	Present value of euro cash flow
0	(80,000,000)	0.080000000	100,000,000	0.800000	80,000,000	0	1.000000	0
1	4,480,000	0.09045226	(8,000,000)	0.7777778	(6,222,222)	(1,742,222)	1.050000	(1,659,259)
2	4,480,000	0.0948663	(9,045,226)	0.7518550	(6,800,698)	(2,320,698)	1.106814	(2,096,738)
3	4,480,000	0.10187139	(9,486,630)	0.7267775	(6,894,669)	(2,414,669)	1.171394	(2,061,363)
4	4,480,000	0.10320196	(10,187,139)	0.6980775	(7,111,412)	(2,631,412)	1.239756	(2,122,525)
5	84,480,000	N/A	(110,320,196)	0.6711349	(74,039,735)	10,440,265	1.314914	7,939,886
							NPV =	0.0
Euro payer - US dollar Perspective Settlement date (year)	Euro cash flow	US one-year FRA	Dollar cash flow	FOREX rate (euro per dollar)	Dollar value of euro cash flow (Euro cf/euro FOREX)	Net dollar flow Euro cf + $US cf ($ value of euro cf + dollar cf)	US dollar Zero-coupon factor	Present value of dollar cash flow
0	(80,000,000)	0.080000	(100,000,000)	0.800000	100,000,000	0	1.000000	0
1	(4,480,000)	0.090452	8,000,000	0.7777778	(5,760,000)	2,240,000	1.080000	2,074,074
2	(4,480,000)	0.094866	9,045,226	0.7518550	(5,958,596)	3,086,630	1.177688	2,620,922
3	(4,480,000)	0.101871	9,486,630	0.7267775	(6,164,198)	3,322,432	1.289411	2,576,704
4	(4,480,000)	0.103202	10,187,139	0.6980775	(6,417,626)	3,769,513	1.420766	2,653,157
5	84,480,000	N/A	110,320,196	0.6711349	(125,876,332)	(15,556,135)	1.567391	(9,924,857)
							NPV =	0.0

Table 21.1 Term structures of dollar and European interest rates and the dollar/euro exchange rate
Instrument	Maturity	US dollar interest rate par yields	Price	Zero-Rate
A	0	N/A	100
B	1	8.00%	100
C	2	8.50%	100
D	3	8.80%	100
E	4	9.10%	100
F	5	9.30%	100

	Term	0	1	2	3	4	5
0.00%	A	0
8.00%	B	108
8.50%	C	8.5	108.5
8.80%	D	8.8	8.8	108.8
9.10%	E	9.1	9.1	9.1	109.1
9.30%	F	9.3	9.3	9.3	9.3	9.3	109.3	128

This forward calculation can be found here:

Year/Maturity
1	PA = 108/Z0,0.1	100=108/Z0,1.0	100.00	=108/1.08
2	PB = 8.5/Z0,0.1+108.5/Z0,2.0	100 = 8.5/1.08+108.5/Z0,1.0	100.00	= 8.5/1.08+108.5/1.177688
3	PC = 8.8/Z0,.0+8.8/Z0,2.0 +108.8/Z0,3.0	100 = 8.8/1.08+8.8/1.77688+108.8/Z0,1.5	100.00	= 8.8/1.08+8.8/1.177688+108.8/1.289411
4	PD = 9.1/Z0,1.0+9.1/Z0,2.0 +9.1/Z0,3.0+109.1/Z0,4.0	100 = 9.1/1.08+9.1/1.177688+9.1/1.289411+109.1/Z 0,4.0	100.00	= 9.1/1.08+9.1/1.177688+9.1/1.28941+109.1/1.420766
5	PE = 9.3/Z0,1.0+9.3/Z0,2.0 +9.3/Z0,3.0+9.3/Z0,4.0+109.3/Z 0,5.0	100 = 9.3/1.08+9.3/1.177688+9.3/1.289411+9.3/1.420766+109.3/Z 0,5.0	100.00	= 9.3/1.08+9.3/1.177688+9.3/1.289411+9.3/1.420766+109.3/1.567391

Bootstrap Zero-rate Coupon Factor	Forward-rate 1 yr Factor (FRF)	Forward-rate 2-yr Factor (FRF)	Forward-rate 3-yr Factor (FRF)	Forward-rate 4-yr Factor (FRF)	Zero Annual Yield w CC	Annual Coupon Rate (% of Par)
1.08000	1.08000				7.6961%	8.00%
1.177688	1.09045				8.5117%	8.50%
1.289411	1.09487	1.19389943			9.2096%	8.80%
1.420766	1.10187	1.206401849	1.3155236		10.0019%	9.10%
1.567391	1.10320	1.215586682	1.3309049	1.4513	10.7489%	9.30%
	SFR Rate		Forward Rate		Swap Fixed Rate Average	8.6805%
	33,788,179	=	33,788,179		Geometric Average (Forward Rate)	8.7278%
					Forward Average	9.408%
					Forward Geometric Average	9.405%

The first task is to find the one-year zero-coupon factor Z 0,1.0:
P0 = C1.0/Z0,1.0
100=108/Z0,1.0	100=108.0/Z0,1.0 = 108.0/100=1.08
Find the Forward Rate if you have the Zero rate
Z0,1 = 1 + 1.0 x 0.08 = 1.08		Z1,2 =(1.1718)*( 1 + 1 x 0.088) = 1.274914
Solve for y where 1.08 x (1 + 1 * y) = 1.1718		Solve for y where (1.1718)*( 1 + 1 x 0.088) = 1.274914
=(1.08/(1+(1.0*0.085)))		=(1.1718)*( 1 +1 * 0.088)
=1.08*(1+(1.0*0.085)) = 1.1718		=(1.1718)*( 1 +1 * 0.088) = 1.2749184
Solve for y where 1.08 x (1 + 1 * y) = 1.1718	Solve for y (1 + 1 * y) = 1.08	Solve for y where 1.1718 x (1 + 1.0 * y) = 1.2749184
1.08 x (1 + 1.0y) = 1.1718	(1 + 1.0 y) = 1.08	1.1718 x (1 + 1.0y) = 1.2749184
1.08 x (1 + 1.0y)/1.08 = 1.1718/1.08	1 + 1.0y -1 = 1.08 - 1	1.1718 x (1 + 1.0y)/1.1718 = 1.2749184/1.1718
1.08 x (1 + 1.0y)/ 1.08 = 1.1718/1.08	1 + 1.0y -1 = 1.08 - 1	1.1718 x (1 + 1.0y)/ 1.1718 = 1.2749184/1.1718
1 + 1.0y = 1.1718/1.08	1 + 1.0y = 1.08	1 + 1.0y = 1.2749184/1.1718
Subtract 1 from both sides	Subtract 1 from both sides	Subtract 1 from both sides
1 + 1.0y -1 = (1.1718/1.08) -1	1 + 1.0y -1 = 1.08 -1	1 + 1.0y -1 = (1.2749184/1.1718) -1
1 + 1.0y - 1 = (1.1718/1.08) -1	1 + 1.0y - 1 = 1.08 -1	1 + 1.0y - 1 = (1.2749184/1.1718) -1
1.0y = (1.1718/1.08) -1	1.0y = .08	1.0y = (1.2749184/1.1718) -1
Convert 1 into fraction: 1 * 1.08/1.08	y = .08/1.0	Convert 1 into fraction: 1 * 1.1718/1.1718
-(1 * 1.08)/1.08 + (1.1718/1.08)	y = .08	-(1 * 1.1718)/1.1718 + (1.2749184/1.1718)
Since the denominators are equal, combine the fractions a/c +- b/c = a+-b/c		Since the denominators are equal, combine the fractions a/c +- b/c = a+-b/c
(-1 * 1.08 + 1.1718)/1.08)		(-1 * 1.1718 + 1.2749184)/1.1718)
(-1 * 1.08 + 1.1718) = .0918		(-1 * 1.1718 + 1.2749184) = .1031884
1.0y = .0918/1.08		1.0y = .1031184/1.1718
y =(.0918/1.08)/1		y =(.103115/1.1718)/1.0 = .0880
y = .085/1 = 0.085		y = .0880/1.0 = .0880
y = .085/1 = 8.50%	y = .08	y = .088
1.08 x (1 + 1 * y) = 1.1718	1.08 x (1 + 1.0 * y) = 1.08	1.718 x (1 + 1 * y) = 1.2749184
1.1718000000		1.274918

Swaps: The Parallel Loan: How Swaps Began

Today, there are no officially imposed restrictions on the movement of most major currencies. In the not too recent past, central banks in many industrialized countries imposed active restrictions on the flow of currency. The parallel loan market developed to circumvent restrictions imposed by the Bank of England on the free flow of British pounds. British firms wishing to invest abroad generally needed to convert pounds into US dollars. The Bank of England required these firms to buy dollars at an exchange rate above the market price. The purpose of this policy was to defend the value of the pound in terms of other currencies. Firms, naturally, were not interested in subsidizing the Bank of England by paying the above-market rate for dollars required by the Bank of England's policies. Attempts to evade these currency controls led directly to the development of the market for currency swaps.

Consider two similar firms, one British and one American, each with operating subsidiaries in both countries. Assume that the free-market value of the pound is £1 = $1.60 and that the officially required exchange rate for British firms to acquire dollars is £1 = $1.44. In this environment, the British firm would like to exploit an investment opportunity in the United States that requires an outlay of $100 million. The free-market value of the needed $100 million is £62.5 million (100,000,000/1.60). If the British firm complies with the Bank of England's regulations, it will have to pay £69,444,4444 for the needed dollars (100,000,000/1.44). From the firm's point of view, this regulation would require the firm to pay a subsidy of almost £1 million.

By cooperating with a US firm that has operations in England, the firm can evade the currency controls. The British firm lends pounds to the US subsidiary operating in England, while the US firm lends a similar amount to the British subsidary operating in the United States. This is a parallel loan - two multinational firms lend each other equivalent amounts of two different currencies on equivalent terms in two countries. A parallel loan is also know as a back-to-back loan.

Let us suppose that the US subsidiary of the British firm borrows $100 million from the parent of the US firm for five years at one-year US LIBOR, while the British subsidiary of the US firm borrows £62.5 million from the British parent for five years at a fixed rate of 7 percent interest. Figure 22.1 shows the cash flows from the perspective of the British firm, integrating the cash flows of the British parent and subsidiary. These are exactly the cash flows from a plain vanilla receive-fixed currency swap. In creating this parallel loan, the British firm would be careful to record these loans as two unrelated transactions, keeping the Bank of England in the dark. In this example, the British firm got the free-market exchange rate. In practice, the US and British firms might share the gains from evading the currency controls by an exchange rate value of the pound between $1.44 and $1.60. Alternatively, the two firms might share the gains from currency evasion by having the US firm pay a fixed rate that is somewhat lower.

The development of swaps stemmed directly from these incentives to create parallel loans. Although the projected cash flows from the parallel loan and the plain vanilla currency swap are identical, there are still some subtle but important differences. In the parallel loan, both parties need to pretend that the transaction are completely distinct. If so, default on one of the loans would not justify default by the other party. In an interest rate swap agreement, there are cross-default clauses. (The parallel loan cannot contain those cross-default clauses, because the two parties are trying to pretend that the parallel loans are distinct.) Also, a swap agreement would have lower transaction costs than arranging two separate loans.

Creating Synthetic Securities with Swaps

In other sections, we saw that certain securities could be synthesized from combinations of derivatives. For example, "Forward put-call parity" in Interest Rate Option, we explored the put-call parity relationship, which shows how any three of four instruments (a put, a call, the underlying good, a risk-free bond) can synthesize the fourth instrument. In "Swaps", we saw that swaps could be analyzed as a portfolio of other instruments, such as a portfolio of bonds, a portfolio of floating-rate agreements (FRAs), or a portfolio of options.

Exploring the creation of synthetic securities has two motivations. First, it provides a deeper analysis of swaps and an understanding of how to use swaps to change the form of existing instruments. In "Swaps", we showed how to deconstruct swaps in terms of other instruments; now we show how to construct other instruments using swaps. Second, by seeing that an existing security is equivalent to a synthetic security, we learn how to compare financing and investment alternatives in the search for cheaper financing costs and higher investment returns. The synthetic securities discussed in this section are quite simple in character. As we will see later in this section, they are akin to fundamental building blocks that can be combined to create more complex and flexible financial structures.

Synthetic Fixed-Rate Debt

Consider a firm with an existing floating-rate debt obligation that wishes to eliminate the uncertainty inherent in floating debt. A firm in this position could create a synthetic fixed-rate debt instrument by combining its existing floating-rate obligations with an interest rate swap. Assume that a firm has an outstanding issue of $50 million on which it pays a floating-rate annual coupon and the debt matures in six years. The firm wished to transform this obligation into a fixed-rate instrument with the same maturity.

Figure 22.2 shows the firm's existing obligation in the upper time line. To transform this existing obligation into a fixed-rate instrument, the firm can engage in a swap agreement to receive-floating/pay-fixed, with a tenor and payment timing to match its existing debt, as shown in the bottom time line of Figure 22.2. The combination of the existing debt and the pay-fixed/receive-floating interest rate swap gives the firm a synthetic fixed-rate obligation instead of its current floating-rate debt.

Synthetic Floating-Rate Debt

An existing fixed-rate obligation can be transformed into floating-rate debt by reversing the technique used to create synthetic fixed-rate debt. Assume that a firm has an existing fixed-rate debt obligation with a maturity of six years that requires annual interest rate payments. The upper time line of Figure 22.3 shows the cash flows associated with this obligation. (This example parallels that of Figure 22.2, except that the initial obligation has fixed-rate coupons.)

By combining this fixed-rate obligation with a receive-fixed/pay-floating interest rate swap, the instrument can be transformed from a fixed-rate to a synthetic floating-rate obligation. The lower time line of Figure 22.3 shows the cash flows on a receive-fixed/pay-floating interest rate swap. By combining this swap with the existing obligation, the firm transforms its existing fixed-rate obligation into a synthetic floating-rate debt with the same maturity.

Synthetic Callable Debt

Consider a firm with an outstanding fixed-rate obligation that possesses no call feature. The issuing firm would like to be able to call this debt in three, but does not want the obligation to retire the issue. In essence, the firm wishes that the existing noncallable debt had a call provision allowing a call in three years.

When a firm calls an existing debt instrument, it repays the debt. We may view that repayment as creating a new financing need that the firm will meet from floating-rate obligations. After all, in calling the debt, it retired the existing fixed-rate obligation. From this perspective, we may see that the decision to call an existing fixed-rate obligation is like creating a synthetic floating-rate debt obligation using a call. As we have just seen, a fixed-rate debt obligation, combined with an interest rate swap to receive fixed payments and pay floating, transforms the fixed-rate instrument into a floating-rate obligation

However, in the present instance, the issuer wishes to have the option, but not the obligation to make this transformation. Therefore, the firm can create a synthetic callable bond by using a swaption. Because the firms wants to possess the call option feature of a callable bond, we know that the firm must purchase a swaption, because only buying a swaption - an option on a swap - gives that flexibility. The swap that underlies the desired swaption must allow the firm to receive-fixed/pay-floating. Therefore, the firm needs to purchase a receiver swaption with an expiration of three years.

At the expiration of the swaption, the firm can simulate a callable bond by exercising its swaption. Upon exercise, the firm will continue to pay fixed on its existing bond (which is not callable), receive fixed on the swap, and pay floating on the swap. Netting out the fixed payment and fixed receipt, the firm is left with pay-floating obligation, which is analogous to having been able to call the bond.

Synthetic Noncallable Debt

A firm with outstanding callable debt can use interest rate swaps to eliminate the call feature and capture its value. When it issued the callable debt, the firm essentially purchased a call option from the bondholders. If the firm is sure that it will not wish to call the debt, it may wish to recapture the value represented by that call option. We consider this situation from the perspective of corporate financial management in our discussion of swaptions in Section "Swaption". In essence, we now regard the example firm of the swaption discussion in "Swaption" as having created synthetic noncallable debt by using swaptions. Table 22.1 shows how the firm maintains fixed-rate noncallable debt whether or not the swaption is exercised.

Table 22.1 Transforming callable into noncallable debt

Call date scenario	Swap	Issuer	Result
Interest rates higher	Swaption not exercised	Does not call the bond	Issuer has fixed-rate financing
Interest rates lower	Swaption exercised; issuer pays fixed and receives floating for remainder of bond's life	Calls the bond and funds with floating for remainder of	Issuer has fixed-rate financing

Synthetic Dual-Currency Debt

A dual-currency bond has principal payments denominated in one country, with coupon payments denominated in a second currency. For example, an issuing firm might borrow dollars and pay coupon payments on the instrument in euros. When the bond expires, the firm would repay its principal obligation in dollars. This dual-currency bond can be synthesized from a regular single-currency bond with all payments in dollars (a dollar-pay bond) combined with a fixed-for-fixed currency swap.

The upper time line in Figure 22.4 shows the cash flows from owning a typical dollar-pay bond. The purchaser of the bond invests at the outset and then receives coupon inflows and the return of principal upon maturity. The second time line in Figure 22.4 shows the cash flows for a fixed-for-fixed foreign currency swap in which the party receives fixed euro inflows and pays fixed dollar amounts. Notice that there is no exchange of borrowings in this swap. (A currency swap with no exchange of borrowings is known as a currency annuity swap. Later in the section, we show how to price such a swap.) The amounts of the cash flows in the currency swap are constructed to equal the coupon payments.

Figure 22.5 shows the effect of combining the dollar-pay bond with the foreign currency swap. The dollar coupon payments on the dollar-pay bond and the dollar payments on the fixed-for-fixed currency swap perfectly offset each other. This leaves euro inflows from the swap that take the place of the coupon payments. As Figure 22.5 shows, the principal payment and repayment are in dollars; all of the coupon cash flows are in euros. Thus, a dollar-pay bond combined with the appropriate fixed-for-fixed currency swap with no exchange of borrowings produces a dual-currency bond.

Swaps: The All-In Cost

The all-in cost is the internal rate of return (IRR) for a given financing alternative. It is called the all-in-cost because it includes all costs associated with the alternative being evaluated, such as flotation costs and administrative expenses, as well as the actual cash flows for the instrument being evaluated. As such, the all-in-cost represents an effective annual percentage cost and provides a comprehensive basis for comparing different financing alternatives. As we will see in some more comprehensive applications later in this section, the all-in-cost is a useful technique in a variety of situations.

We illustrate the concept of the all-in-cost by comparing two financing alternatives available to the firm that are different in structure but that have very similar actual cash flows. The first instrument is a ten-year semiannual payment bond with a principal amount of $40 million and a coupon rate of 7 percent. This instrument is priced at par:

Equation (22.1) P = M$Sigma;_t=1 C_t / (1 + Y)^t

where M is the maturity date of the bond; C_t is the cash flow from the bond at time t, which could be principal or interest; and y is the yield-to-maturity. In Equation 22.1, y is the yield-to-maturity on the bond, which is also the IRR that equates the price and the present value of the cash flows associated with the bond. Therefore, the yield-to-maturity meets the definition of the all-in costs for this bond. Because the bond pays a coupon of 7 percent and is priced at par, the yield on the bond and the all-in-cost are 7 percent. Therefore, the firm can secure its fixed rate financing at an all-in cost of 7 percent by issuing a straight bond.

As a second financing alternative, the firm can borrow $40 million for ten years at a floating rate of LIBOR plus 30 basis points, with the rate being reset each six months. LIBOR currently stands at 6.5 percent. Because the firm currently faces a floating finance rate of 6.8 percent, it looks attractive compared to the 7 percent fixed-rate financing vehicle. However, the firm has determined to secure fixed-rate financing. As we saw earlier in this section, issuing a floating-rate bond combined with a pay-fixed/receive-floating interest rate swap is equivalent to a fixed-rate bond. Upon inquiry, the firm learns that it can enter a pay-fixed/receive floating interest rate swap, to pay 6.5 percent and receive LIBOR. The fee for arranging the swap and the associated administrative cost is an immediate payment of $400,000.

In summary of this second financing alternative, the firm would borrow $40 million at a floating rate of 6.8 percent, and it would enter a swap agreement to pay-fixed at 6.5 percent and receive LIBOR. The firm must also pay $400,000 fee for the swap, so it will net only $39.6 million of actual financing The two financing alternatives have very similar cash flows and both imply a fixed-rate financing of about $40 million. The choice of financing, therefore, reduces to comparing the all-in-costs of the two deals. Figure 22.6 shows the cash flow line for the second financing alternative, reflecting the effect of the swap. The firm receives $39.6 million at inception, makes 20 semiannual payments of $1.36 million, and repays the principal of $40 million at the end of ten years. The all-in cost for the second alternative is simply the IRR that equates the present cash inflow $39.6 million with all of the cash outflows. For these flows, the IRR is 0.034703 on a semiannual basis, or 0.069406 in annual terms. This is slightly lower than the 7 percent IRR on the straight bond financing, so the firm prefers the floating-rate instrument coupled with the interest rate swap. Being able to compute the IRR on the two deals and compare the all-in costs leads to the correct decision. However, the firm should also be aware that the six basis point differential might simply reflect an additional credit risk on the swap if the market is fully efficient. If so, the firm would be indifferent between the two financing alternatives.

Pricing Flavored Interest Rate and Currency Swaps [viewable here in Excel]

To price a flavored interest rate swap, we apply the same technique used to price plain vanilla interest rate swap: we find the swap fixed rate (SFR) that equates the present value of the cash flows from the pay-fixed and received-fixed sides of the swap. The zero-coupon factors for the present value calculations are derived from the yield curve in effect when the swap is initiated. In using these discount rates, no assumptions is made about the future course of interest rates. Instead, this pricing approach finds the swap rate terms that prevent arbitrage.

We will use the SFR for a plain vanilla swap as follows

In this section, we apply these general principles of swap pricing to four different flavored swaps: a forward interest rate swap, a seasonal swap, a diff swap and a currency annuity swap. In each case, the pricing solution is to find the terms of the swap that make it arbitrage free. This means that the present value for each party must be equal, given the structure of interest rates and exchange rates that prevail when the swap is initiated. The principles illustrated for the swap in this section provide a model for pricing any kind of flavored swap.

The Forward Swap

A forward swap can be replicated as a portfolio of four bonds. The forward swap would have to have the same price as the four-bond portfolio. It is also possible to price the forward swap directly, using the same approach that we applied to find the SFR for a plain vanilla swap interest rate swap. The pay-fixed forward swap synthetically duplicated by the four-bond portfolio shown in "Stripped Treasury Securities…" had a notional principal of $30 million and annual payments, was set to begin in three years, and had a tenor of five years. For that swap the yield curve was flat at 8 percent, Therefore, the SFR for the forward swap also had to be 8 percent. We will now show how to price this kind of forward swap in a more typical yield curve environment.

As the following analysis shows, a forward-swap is really just a plain vanilla interest rate swap with a deferred starting date. Because of this, the principles of plain vanilla pricing apply quite directly to forward swap. The actual LIBOR rates that will prevail are, of course unknown from our perspective at time zero when the forward swap is initiated. However, the no-arbitrage rates for our contracting purposes are the forward rates from the yield curve that correspond to the various one-year LIBOR rates. Given the necessary yield curve information, our pricing problem is to find the SFR that makes the present values of the fixed and floating cash flows equal from the perspective of time zero.

If we look at the time line of Figure 22.12 from the vantage point of a potential counterparty at time 3, the cash flows on the forward swap have exactly the same form as the cash flows on a receive-fixed plain vanilla swap. Therefore, we can price the forward swap from time zero or time 3.

We begin by pricing this forward swap from the perspective of time zero. Table 22.7 presents the necessary information on the yield curve at the time the swap is to be initiated. The zero-coupon factors and forward rate factors have been computed from the par yields by boot-strapping, as shown above. As Table 22.7 shows, the yield curve is humped, with the par yields initially rising and then falling. The floating rates implied by the yield curve, and shown in Table 22.7 are as follows:

Table 22.7 Term structure information for pricing the forward swap

Instrument	Maturity	Par Yield	Zero-coupon factor	Forward rate factor
0	0	N/A	0	0
1	1	5.030000%	1.050300	1.050300
2	2	6.350000%	1.131936	1.077726
3	3	7.040000%	1.229247	1.085969
4	4	7.500000%	1.341535	1.091348
5	5	7.690000%	1.457308	1.086299
6	6	7.610000%	1.560784	1.071005
7	7	7.500000%	1.664352	1.066356
8	8	7.180000%	1.734682	1.042257

Cash-flows from Treasury bonds
	Term	0	1	2	3	4	5	6	7	8
0.00%	0	0
5.0300%	1	1.10313
6.3500%	2	0.072	1.122
7.0400%	3	0.087	0.087	1.218
7.5000%	4	0.101	0.101	0.101	1.330
7.6900%	5	0.112	0.112	0.112	0.112	1.454
7.6100%	6	0.119	0.119	0.119	0.119	0.119	1.576
7.5000%	7	0.125	0.125	0.125	0.125	0.125	0.125	1.686
7.1800%	8	0.125	0.125	0.125	0.125	0.125	0.125	0.125	0.125	1.789	7.542578

	Zero-coupon Factor
100 = 105.03/Z0,1.0	0.050300	100.00
100 = 6.35/Z0,1.0+106.35/Z0,2.0	0.077726	100.00
100 = 7.04/Z0,1.0+7.04/Z0,2.0+107.04/Z 0,3.0	0.085969	100.00
100 = 7.50/Z0,1.0+7.50/Z0,2.0+7.50/Z 0,3.0+107.50/Z0,4.0	0.091348	100.00
100 = 7.69/Z0,1.0+7.69/Z0,2.0+7.69/Z0,3.0+7.69/Z0,4.0+107.69/Z 0,5.0	0.086299	100.00
100 = 7.61/Z0,1.0+7.61/Z0,2.0+7.61/Z0,3.0+7.61/Z0,4.0+7.61/Z0,5+107.61/Z 0,6.0	0.071005	100.00
100 = 7.50/Z0,1.0+7.50/Z0,2.0+7.50/Z 0,3.0+7.50/Z0,4.0+7.50/Z0,5.0 +7.50/Z0,6.0+107.50/Z0,7.0	0.066356	100.00
100 = 7.18/Z0,1.0+7.18/Z0,2.0+7.18/Z 0,3.0+7.18/Z0,4.0+7.18/Z0,5.0 +7.18/Z0,6.0+7.18/Z0,7.0+107.18/Z 0,8.0	0.042257	100.00

Zero-Coupon Factor

Zero-coupon Factor	1-yr Forward Rate Factor (FRF)	2-yr Forward Rate Factor (FRF)	3-yr Forward Rate Factor (FRF)	4-yr Forward Rate Factor (FRF)	5-yr Forward Rate Factor (FRF)	6-yr Forward Rate Factor (FRF)	7-yr Forward Rate Factor (FRF)	Zero Annual Yield w CC	Annual Coupon Rate (% of Par)
1.05030	1.050300							4.9076%	5.030%	5.030%
1.131936	1.077726							6.3883%	6.350%	7.773%
1.229247	1.085969	1.170377						7.3637%	7.040%	8.597%
1.341535	1.091348	1.185169	1.2772878					8.1934%	7.500%	9.135%
1.457308	1.086299	1.18553	1.2874481	1.3875				8.7518%	7.690%	8.630%
1.560784	1.071005	1.163431	1.2697078	1.3789	1.3789			8.9351%	7.610%	7.100%
1.664352	1.066356	1.142073	1.2406322	1.3540	1.3540	1.3540		9.0670%	7.500%	6.636%
1.734682	1.042257	1.111417	1.1903326	1.2931	1.2931	1.2931	1.2931	8.7860%	7.180%	4.226%
	SFR Rate		Forward Rate				Geometric Average	7.6543%	6.9292%	6.9228%
	12,154,384	=	12,154,384				Swap Fixed Rate	6.8684%
							Geometric Average (Forward Rate)	6.92%

The fixed-rate payments will occur at times, 4,5,6,7 and 8 and will equal the SFR times the notional principal of $30 million.

Table 22.8 details the cash flows, zero-coupon factors, and present values for each payment. Consistent with table 22.8, the present value of the floating-rate cash flows is given by the following:

Table 22.8 Cash flows for issuance and transformation of a bear floater to an FRN

	Cash flows for forward swap, time-zero perspective Date (year)	Floating-rate cash flow	Fixed-rate cash flow	Zero-coupon factor	Present value of floating-rate cash flow	Present value of fixed-rate cash flow
PVFloat = (FRA3,4 x NP)/Z0,4 + (FRA4,5 x NP)/Z0,5 + (FRA5,6 x NP)/Z0,6 + (FRA6,7 x NP)/Z0,7 + (FRA7,8 x NP)/Z0,8	1	0	0	1.050300	0	0
	2	0	0	1.131936	0	0
= .019347 x 30,000,000/1.3441535 + .086299 x 30,000,000/1.457308 + .071005 X 30,000,000/1.560784 + .066356 X 30,000,000/1.664352 + .042257 x 30,000,000/1.734682	3	0	0	1.229247	0	0
	4	.091348 x 30000000	SFR x 30,000,000	1.341535	2,042,764	22,362,437
	5	.0862999 x 30000000	SFR x 30,000,000	1.457308	1,776,561	20,585,897
= 2,042,742 + 1,776,541 + 1,364,793 + 1,196,076 + 730,797	6	.071005 x 30000000	SFR x 30,000,000	1.560784	1,364,795	19,221,107
	7	.066356 x 30000000	SFR x 30,000,000	1.664352	1,196,069	18,025,034
= $7,110,949	8	.042257 x 30000000	SFR x 30,000,000	1.734682	730,803	17,294,240
					7,110,991	97,488,715

On the fixed-rate side, the present value of the fixed-rate flows is as follows:

PVFixed = (SFR_3,4 x NP)/Z_0,4 + (SFR_4,5 x NP)/Z_0,5 + (SFR_5,6 x NP)/Z_0,6 + (SFR_6,7 x NP)/Z_0,7 + (SFR_7,8 x NP)/Z_0,8

= SFR x 30,000,000/1.3441535 + SFR x 30,000,000/1.457308 + SFR X 30,000,000/1.560784 + SFR X 30,000,000/1.664352 + SFR x 30,000,000/1.734682

= SFR x 22,362,443 + SFRx20,585,902 + SFR*19,221,109 + SFR*18,025,033 + SFRx17,294,236

= 22,362,443 + 20,585,902 + 19,221,109 + 18,025,033 + 17,294,236

= $97,488,723

We computed the calculation of the SFR by equating the present value of the cash flow on the floating-rate and fixed-rate sides and solving for the SFR:

PV_FLOAT = PV_FIXED
7,110,949 = SFR X 97,488,724
SFR = 0.072941

We now show that the forward swap can also be priced from the standpoint of the time at which the comparable plain vanilla swap would be initiated. In general, this is one period before the first cash flow; for our example, that would be at time 3. The cash flows are not affected by this change in perspective. The discounting does change, however, because we are discounting to time 3 instead of to time zero. The zero-coupon factors for each cash flow are obtained from the forward rate factors or the zero-coupon factors in Table 22.7, as follows.

Z_3,4 = Z₄/Z₃ = 1.241535/1.229247 = 1.091347
Z_3,5 = Z₅/Z₃ = 1.457308/1.229247 = 1.091347
Z_3,6 = Z₆/Z₃ = 1.560784/1.229247 = 1.091347
Z_3,7 = Z₇/Z₃ = 1.664352/1.229247 = 1.091347
Z_3,8 = Z₈/Z₃ = 1.734682/1.229247 = 1.091347

PVFloat = (FRA_3,4 x NP)/Z_0,4 + (FRA_4,5 x NP)/Z_0,5 + (FRA_5,6 x NP)/Z_0,6 + (FRA_6,7 x NP)/Z_0,7 + (FRA_7,8 x NP)/Z_0,8

= .019347 x 30,000,000/1.091347 + .086299 x 30,000,000/1.185529 + .071005 X 30,000,000/1.269707 + .066356 X 30,000,000/1.353961 + .042257 x 30,000,000/1.411174

= 2,511,035 + 2,183,810 + 1,677,671 + 1,470,264 + 898,337

= $8,741,116

On the fixed-rate side, the present value of the fixed-rate flows is as follows:

PVFixed = (SFR_3,4 x NP)/Z_3,4 + (SFR_3,5 x NP)/Z_0,5 + (SFR_3,6 x NP)/Z_0,6 + (SFR_3,7 x NP)/Z_0,7 + (SFR_3,8 x NP)/Z_0,8

= SFR x 30,000,000/1.091347 + SFR x 30,000,000/1.185529 + SFR X 30,000,000/1.269707 + SFR X 30,000,000/1.353961 + SFR x 30,000,000/1.411174

= SFR x 22,362,443 + SFRx20,585,902 + SFR*19,221,109 + SFR*18,025,033 + SFRx17,294,236

= SFR x 27,488,965 + SFR x 25,305,159 + SFR x 23,627,498 + SFR x 22,157,211 + SFR x 21,258,895

= SFR x $119,837,728

We computed the calculation of the SFR by equating the present value of the cash flow on the floating-rate and fixed-rate sides and solving for the SFR:

PV_FLOAT = PV_FIXED
8,741,116 = SFR X 119,837,728
SFR = 0.072941

This is the identical solution that we reached from the time-zero perspective. In sum a forward swap can be priced by finding the SFR that equates the present value of the cash flows on the two sides of the swap from the time of contracting (time zero) or one period before the first cash flow.

The Seasonal Swap

In a seasonal swap, the notional principal varies according to a fixed plan. This kind of swap can be useful in matching the financing needs of retailers. For example, the swap could be structured on a seasonal basis to match the typically heavy fourth-quarter needs of retailing firms. When the notional principal on the swap first increases and then amortizes to zero over the life of the swap, the swap is called a roller coaster swap. Thus, the notional principal can be structured to conform to the notional principal of risk management need.

A seasonal swap could be synthesized by a six-bond portfolio. The resulting seasonal swap previously explored had quarterly payments in February, May, August and November. The notional principal was $10 million for payments in February, May, and August, and $30 million for the November payment. The swap had a tenor of seven years. Pricing this swap uses the same technology that we have explored for plain vanilla swaps; that is, we need to find the SFR that equates the present value of the floating-rate and fixed-rate cash flows. However, we must take into account of the varying notional principal. We can allow the notional principal to vary by including a different potential notional principal in each element of:

Equation 22.15

SFR = (ΣN_n=1 x FRA_{(n-1) x Mann x MON}/Z0,n x MON) / (ΣN_n=1 / Z0,n x MON)

Let N_n be the notional principal for the nth time period. In this case, the formula for the SFR on a seasonal swap is as follows:

Equation 22.16

SFR = (ΣN_n=1 NP_n x FRA_{(n-1) x MON,n x MON}/Z0,n x MON) / (ΣNP_{n= 1} / Z0,n x MON)

The only difference between Equation 22.15, for the SFR of a plain vanilla swap, and Equation 22.16, for the SFR on a seasonal swap, is the inclusion of a varying notional principal when the notional principal is constant in all periods. We can use Equation 22.16 to price our seasonal swap. Figure 22.14 (not shown) shows the gently rising term structure over the 28 quarters that comprise the tenor of the seasonal swap. The yields to the right are par yields. As our swap has quarterly payments, we must take account of quarterly compounding.

Table 22.9 presents information on the term structure environment and the cash flows on this seasonal swap. The second column shows the annualized par yields, while the third column shows the notional principal for each of the 28 quarterly periods covered by the swap. Columns 4-6 show the quarterly par yields, the zero-coupon factor for each quarter and the forward rate for each quarter. The quarterly par yield is just the annualized par yield divided by four. The zero-coupon factors were found by the bootstrapping method, and the forward rate factors were computed from the zero-coupon factors.

Table 22.9 Term structure and cash flow data for the seasonal swap

Quarter	Par yield	Notional principal	Quarterly par yield	Zero-coupon factor	Forward rate factor (FCF)	PV of floating payments	Value of NPt / Zt
1	6.000%	$10,000,000	1.5000%	1.01500	1.015000	147,783	$9,852,217
2	6.010%	$10,000,000	1.5025%	1.030276	1.015050	146,081	$9,706,136
3	6.050%	$10,000,000	1.5125%	1.046070	1.015330	146,545	$9,559,591
4	6.110%	$30,000,000	1.5275%	1.062534	1.015739	444,383	$28,234,390
5	6.190%	$10,000,000	1.5475%	1.079861	1.016307	151,011	$9,260,452
6	6.260%	$10,000,000	1.5650%	1.097752	1.016568	150,927	$9,109,525
7	6.310%	$10,000,000	1.5775%	1.115940	1.016569	148,473	$8,961,052
8	6.410%	$30,000,000	1.6025%	1.135911	1.017895	472,624	$26,410,531
9	6.440%	$10,000,000	1.6100%	1.154933	1.016747	145,002	$8,658,508
10	6.450%	$10,000,000	1.6125%	1.173839	1.016370	139,453	$8,519,056
11	6.530%	$10,000,000	1.6325%	1.195580	1.018521	154,913	$8,364,143
12	6.590%	$30,000,000	1.6475%	1.217465	1.018305	451,059	$24,641,369
13	6.680%	$10,000,000	1.6700%	1.241484	1.019729	158,911	$8,054,879
14	6.750%	$10,000,000	1.6875%	1.265636	1.019455	153,715	$7,901,164
15	6.770%	$10,000,000	1.6925%	1.288071	1.017726	137,617	$7,763,547
16	6.860%	$30,000,000	1.7150%	1.315198	1.021060	480,390	$22,810,251
17	6.930%	$10,000,000	1.7325%	1.342301	1.020608	153,524	$7,449,893
18	6.940%	$10,000,000	1.7350%	1.366265	1.017853	130,668	$7,319,224
19	6.950%	$10,000,000	1.7375%	1.390738	1.017912	128,796	$7,190,428
20	6.970%	$30,000,000	1.7425%	1.416564	1.018570	393,282	$21,178,002
21	7.020%	$10,000,000	1.7550%	1.445745	1.020600	142,486	$6,916,848
22	7.120%	$10,000,000	1.7800%	1.480883	1.024304	164,118	$6,752,730
23	7.140%	$10,000,000	1.7850%	1.509355	1.019227	127,384	$6,625,346
24	7.220%	$30,000,000	1.8050%	1.545419	1.023893	463,824	$19,412,214
25	7.310%	$10,000,000	1.8275%	1.584434	1.025245	159,334	$6,311,403
26	7.320%	$10,000,000	1.8300%	1.614720	1.019115	118,378	$6,193,025
27	7.330%	$10,000,000	1.8325%	1.645692	1.019181	116,552	$6,076,473
28	7.360%	$30,000,000	1.8400%	1.680413	1.021098	376,665	$17,852,754
						6,103,900	327,085,148
						SFR (Quarterly)	1.8662%
						SFR (Annualized)	7.4646%

To complete the calculation of the SFR on this swap, we need to find the present value of the floating-rate cash flows (the numerator of Equation 22.16) and the present value of the notional principals (the denominator of Equation 22.16). In both the numerator and denominator of Equation 22.16, there will be one term corresponding to each period of the swap. As an example, we show how the numerator and denominator elements for the 15th period were found. The notional principal for period 15 is $10 million. The forward rate for this period is 0.017726, and the zero-coupon factor is 1.288071. Therefore, the corresponding element from the present value of the floating cash flow in the numerator is:

$10,000,000 x 0.017726 / 1.288071 = $137,617

The value appears along with the 27 other present values in the seventh column of Table 22.9.

The present value of the 15th notional principal is as follows:

$10,000,000 / 1.288071 = $7,773,547

This element appears in the last column of Table 22.9.

The sum of the present value of all of the floating payments is shown at the bottom of column 7 in the table and equals $6,103,881. This is the value of the numerator of Equation 22.16 as applied to our seasonal swap. The sum of the present value of all of the notional principals appears at the bottom of column 8 and equals $327,085,139. This is the value of the denominator of Equation 22.16.

Given this information, the swap fixed rate on a quarterly basis is as follows:

SFR = $6,103,881 / $327,085,139 = 0.01866144 = 1.87%
The annualized terms, the SFR = 0.074646 = 4 x 0.01866144 = 7.46%

The Rate-Differential (Diff) Swap [viewable here in Excel]

A rate-differential swap, or diff-swap has payments tied to interest rate indexes in two different currencies, but all payments are made in a single currency. For example, a diff swap might be structured with all payments in US dollars, with one party paying three-month US LIBOR and the other party paying three-month Euribor, but with payments in dollars. Assume that both the LIBOR and Euribor yield curves are flat, that the dollar rate is 7 percent, and that the euro rate is 6.75 percent. The payment based on US LIBOR might be LIBOR flat and the payment based on Euribor would also be Euribor flat, but paid in dollars. As we will see, with currency swap pricing, shows both parties would pay LIBOR flat, even though the rates are 7 percent in the US and 6.75 percent in the EU. This kind of swap would exploit changes in US versus EU interest rates over the tenor of the swap.

The Cross-Index Basis Note

In a cross-index basis note, or quanto note, the investor receives a rate of interest that is based on a floating-rate index for a foreign short-term rate, but is paid in the investor's domestic currency. For example, a US investor might buy a note with an interest rate based on European interest rates, but with all payments on the note made in US dollars. From the investor's point of view, the quanto note allows exposure to foreign interest rates without currency exposure. Also, the quanto note allows the investor to speculate on relative changes in the foreign and domestic yield curves. From the point of view of the issuer, the quanto note can offer investors attractive investment opportunities that might not be available elsewhere. Also, the issuer can issue a quanto note, but use swaps to transform its risk exposure to a perhaps more congenial form.

As an example, consider Figure 22.10, which shows yield curves for one-year LIBOR in both US dollars and British pounds sterling. Current rates are 5 percent in dollars and 8 percent in pounds. However, the US yield curve is strongly upward sloping, while the British yield curve is even more strongly downward sloping.

In this environment, a firm might issue a quanto note that pays a floating rate equal to one-year British LIBOR plus or minus a spread, with a maturity of five years, annual payments, a principal of $30 million, and with all payments being made in US dollars. The spread on British LIBOR would reflect the difference between US dollar and British pound term structures. From figure 22.10, it seems clear that British yields are expected to be below US yields for most of the five years, even though the initial yield on the British pound is higher. In this environment, the quanto note will have to pay British LIBOR plus a spread to attract dollar investments.

Given that the quanto note will pay British LIBOR plus a spread, the initial yield on the quanto will be more than 3 percent higher than the floating rate available on a straight US dollar FRN. However, the yield curves in Figure 22.10 suggest that this yield differential is expected to narrow quickly with British rates even becoming lower than US rates. If the future interest rates corresponding to the forward rates implied by the yield curves in Figure 22.10, the US investor will soon receive less on the quanto note than a purely domestic US FRN. Therefore, the investor in this quanto note has an implicit speculation that the interest rates will not converge as quickly or fully as the market seems to expect.

On the other side, the issuer of the quanto note has an implicit speculation that rates will narrow more quickly than the market expects. However, the issuer may wish to offset this risk by engaging in a swap. The first time line of Figure 22.11 shows the cash flows from issuing this quanto note. The principal is paid in dollars. The amount of each interest payment is determined by the level of British LIBOR, but the actual payment is made in US dollars. As it stands, the issuer of the quanto note is exposed to interest rate risk in US and the UK.

A participant in a diff swap might receive British LIBOR, plus or minus a spread, and pay US dollar LIBOR, with all payments being made in US dollars. The spread would reflect the differences in the yield curves for the two currencies.

The issuer of our example quanto note might find a diff swap an attractive hedging vehicle. The second time line of Figure 22.11 shows the cash flows on a diff swap in which the issuer receives British LIBOR plus or minus a spread and pays US LIBOR. The spread in the diff swap and the quanto note are the same. The diff swap has a notional principal of $30 million, annual payments, and a tenor of five years. (The determination of the exact spread on the British pound interest rate requires more information on the exact shape of the two yield curves. Later we show how to price the quanto note and its accompanying diff swap.)

The Rate-Differential (Diff) Swap

As with all swap pricing the key is to find the terms that equate the present value of the two sides of the swap, consistent with the prevailing term structure. In this diff swap, one party will pay British LIBOR plus a spread, while the other will pay US LIBOR, with all payments on both sides of the swap being denominated in dollars. The present value of the cash flows based on British LIBOR and US LIBOR are as follows:

PV_€LIBOR = (Σ⁵_{t=1 €LIBORt + SPRD) * $30,000,000 / Z0,j}

PV$LIBOR = (Σ⁵_{t=1 $LIBORt + SPRD) * $30,000,000 / Z0,j}

The pricing solution is to find the value of SPRD that makes these two present values equal:

(Σ⁵_{t=1 €LIBORt + SPRD) * $30,000,000 / Z0,j = (Σ⁵t=1 $LIBORt + SPRD) * $30,000,000 / Z0,j}

Solving for SPRD gives the following:

SPRD = ((Σ⁵_{t=1 €LIBORt / Z0,j) - (Σ⁵t=1 $LIBORt)) / (Σ⁵t=1 1/Z0,j)}

=(0.33189 - 0.16381) / 4.04745 = 4.1527%

This equation shows that the appropriate spread of the diff-swaps is 415.27 basis points, which is the same as the spread on the quanto note.

This equation also provides a general equation for the appropriate spread on a diff swap.

Day Count Conventions

Throughout our discussion of swaps, we have abstracted from the technicalities of the exact interest calculation that is used in actual market transactions. This choice has allowed us to avoid some tedious arithmetic to focus on the economics of the swap market. This section provides a brief overview of the actual conventions employed. The basic interest rate swap involves a series of fixed-rate payments associated with a long-term instrument against a series of floating-rate payments tied to the money market. The bond market and money market use fundamentally different assumptions about day counts and interest computations.

In the money market, interest payments are generally computed on the assumption that the year has 360 days and that interest accrues each calendar day. For example, a LIBOR rate of 8.30 percent on a notional principal of $10 million, over a quarter period of 91 days, would generate an interest payment of:

$10,000,000 x 0.0830 x 91/360 = $209,805.56

As this example indicates, money market yields are computed according to a day count convention of actual/360.

In the bond market, two different day count conventions are common. One is actual/365, the other is 30/360, assuming that the year has 360 days and that each month has 30 days. For most bonds, the actual/365 convention applies. For example, a $100,000 par value Treasury bond with a yield of 8.30 percent for a semiannual period with 182 days would generate an interest payment of:

$10,000,000 x 0.0830 x 182/365 = $413,863

Because these conventions differ, it is important to take their effects into account in swap contracting and cash flow computations. For example, consider a two-year plain vanilla interest rate swap with semiannual payments. The terms are an unknown SFR against six-month LIBOR. The notional principal is $100 million. The four six-month periods covered by the swap have 182, 183, 181 and 182 days, respectively. The fixed-side cash flows are computed on an actual/365 basis, while the floating-side cash flows are computed on an actual/360. The basic no-arbitrage pricing condition requires equality in the present value of the fixed-side and floating-side cash flows. On this swap, the first fixed cash flow is:

$10,000,000 x SFR x 182/365

and the first floating cash flow is

$10,000,000 x LIBOR x 182/360

Differences in day count conventions affect each cash flow, the present value of the two sides of the swap, and the swap rates.

As an example, we price this four-period swap, in the term structure environment of Table 22.13. First, we price if according to our simplified Equation 22.15. then we price the same swap considering the day count conventions For the simplified Equation 22.15, the numerator and denominator are as follows:

numerator = FRA_0,1/Z_0,1 + FRA_1,2/Z_1,2 + FRA_2,3/Z_0,3 + FRA_3,4/Z_0,4

numerator = 0.030450/1.03045 +0.032889/1.064341 + 0.035376/1.101993 + 0.035801/1.141445 = .123917

denominator = 1/Z_0,1 + 1/Z_1,2 + 1/Z_0,3 + 1/Z_0,4

denominator = 1/1.0345 + 1/1.064341 + 1/1.101993 + 1/1.141445 = 3.693528

The SFR = denominator/numerator = 0.123917/3.693528 = 0.033550

SFR - annual = 3.355 * 2 = 6.710%

This SFR of 0.033550 is expressed in semiannual terms. The corresponding annualized SFR is 6.71%.

We now complete the SFR taking the day count convention into account. The four semiannual periods have 182, 183, 181 and 182 days, respectively. The fixed-rate payments are computed on an actual/365 basis, while the floating payments are based on an actual/360 day assumption. The basic computation is the same as in Equation 22.15 except the fraction of the year cannot be factored out, as it differs for each payment and differs between the numerator (for the floating payment) and the denominator (for the fixed payments). Taking the day count conventions into account, the present value of the floating payments is as follows:

PV_FLOATING = NP x (FRA_0,1 x (182/360)/Z_0,1 + FRA_1,2 x (183/360)/Z_1,2 + FRA_2,3 x (181/360)/Z_0,3 + FRA_3,4 x (182/360)/Z_0,4

= (0.060900 x (182/360))/1.03045 +(0.065778 x (183/360))/1.064341 + (0.070752 x (181/360))/1.101993 + (0.071602 x (182/360))/1.141445 = .125287663

PV_FIXRD = NP x (SFR_0,1 x (182/365)/Z_0,1 + SFR_1,2 x (183/365)/Z_1,2 + SFR_2,3 x (181/3650)/Z_0,3 + SFR_3,4 x (182/365)/Z_0,4

= $100,000,000 x SFR x ((182/365))/1.03045 + ((183/365))/1.064341 + ((181/365))/1.101993 + ((182/365))/1.141445 = 1.841792124

= $100,000,000 x SFR x 1.841792

= SFR x $184,179,200

Because we are not factoring out the expression for the fraction of the year, as it differs for each payment, note that the FRA rates that appear above are the annualized rates.

To compute the SFR, we equate the present value for the fixed and floating payment streams, and solve for the SFR:

PV _FLOATING = _PVFIXED
$12,528,800 SFR x $184,179,200

SFR = $12,528,800/$184,179,200 = 0.068025

This SFR is already in annualized terms. The annualized SFR according to the simplified Equation 22.15 is 0.067100, compared t an SFR = 0.068025 taking the day count convention into account. This is a difference of 9.25 basis points - a significant difference that needs to be considered in actual swap pricing.

The Currency Annuity Swap [viewable here in Excel]

A currency annuity swap is similar to a plain vanilla swap without the exchange of principal at the initiation or the termination of the swap. It is also know as a currency basis swap. For example, one party might make a sequence of payments based on British LIBOR while the other makes a sequence of payments based on US LIBOR. As we will see, the currency annuity swap generally requires one party to pay an additional spread to the other or to make an up-front payment at the time of the swap. Valuations of this structure can be created by allowing one, or both, parties to pay at a fixed rate. In pricing these swaps, the key is to specify a spread or up-front payment that makes the present value of the cash flows incurred by each party equal.

As an example, we consider a swap of British LIBOR versus US LIBOR made in the context of the interest rate environment of Figure 22.10 and Table 22.10, which we have already explored for the diff swap. The spot exchange rate at time zero is £1 = $1.60, and the corresponding forward exchange rates are shown in the final column of Table 22.10. The notional principal is £50 million, equivalent to $80 million. The specific terms of the swap require one party to pay a floating rate equal to one-year British LIBOR on a notional principal of £50 million for five years, plus or minus a spread. Unlike the diff swap, these payments will be made in British pounds. The other party will pay US LIBOR on a notional principal of $80 million.

Table 22.10 Interest rate data for rate-differential (diff) swap (for annual coupon bonds)

	US Dollar	US Dollar	US Dollar	British Pound	British Pound	British Pound	British Pound
Maturity (yrs)	Par yield	Zero-coupon or Future Value	One-year Forward Rate Factor	Par yield	Zero-coupon or Future Value	One-year Forward Rate Factor	Forward Value of LIBOR
1	5.00%	1.05000	1.050000	8.00%	1.08000	1.080000	1.555560
2	6.50%	1.135279	1.081218	6.00%	1.122353	1.039216	1.618427
3	7.20%	1.235012	1.087848	4.80%	1.148130	1.022967	1.721076
4	7.80%	1.357937	1.099534	4.20%	1.174603	1.023057	1.849731
5	8.20%	1.496762	1.102232	3.90%	1.205379	1.026202	1.989678
Geometric Mean	6.840%		8.400%	5.192%		3.807%	8.73447
Zero Coupon -US Dollar $US 1/ZoJ	Zero Coupon -US Dollar LIBOR 1/ZoJ	US Forward/US Zero	LIBOR Forward/US Zero Rate	LIBOR Forward/US Zero Rate
0.95238	0.92593	0.04762	0.076190	0.08
0.88084	0.89099	0.07154	0.034543	0.03
0.80971	0.87098	0.07113	0.018597	0.02
0.73641	0.85135	0.07330	0.016979	0.02
0.66811	0.82961	0.06830	0.017504	0.02
4.04745	4.36886	0.33189	0.16381	0.16381

The time line in Figure 22.15 shows the cash flows on this swap from the point of view of the dollar payer. For the dollar payer, the present value of the outflows is as follows:

PV_OUTFLOWS = $80,000,000 X (Σ⁵_t=1 $LIBOR_t / Z_0,j = $80,000,000 x 0.331891 = $26,551,280

Recall that _€,.5FX_0,t indicates the dollar value of a forward contract for €1 initiated at zero for payment at time t. Then, also from the perspective of the dollar payer, the US dollar value of the British pound inflows is a follows:

PV_INFLOWS = £50,000,000 X (Σ⁵_t=1 £LIBOR_t + SPRD) x _€,5FX_0,t/ Z_0,j

PV_INFLOWS = £50,000,000 X (Σ⁵_t=1 £LIBOR_t + SPRD) x _€,5FX_0,t/ Z_0,j + £50,000,000 x SPRD x Σ⁵_t=1 ^{€,5FX0,t/ Z0,j}

= £50,000,000 x 0.272614 + £50,000,000 x SPRD x 6.990144

= $13,630,700 + $349,507,200 x SPRD

Now equating the values of the inflows and outflows from the point of view of the dollar payer, we have the following:

PV_INFLOWS = PV_OUTFLOWS
= $13,630,700 + $349,507,200 x SPRD = $26,551,280
SPRD = 0.036968

The payer of British pounds should pay British LIBOR plus 3.6968 percent versus US LIBOR flat.

Alternatively, the swap could be structured such that the five annual payments are British LIBOR flat versus US LIBOR flat. In this case, the payer of pounds would have to make a payment to the dollar payer that equals the present value of the spread payments:

PV_SPRD = SPRD x £50,000,000 (Σ⁵_{t=1 €,5}FX_0,t/ Z_0,j

= 0.036968 x £50,000,000

= $12,920,580

Sources: Damodaran, Hull, Basu, Kolb, Brueggeman, Fisher, Bodie, Merton, White, Case, Pratt, Agee.