Unformatted text preview: S vvaps The first swap contracts were negotiated in the early 1980s. Since then the market has
seen phenomenal growth. Swaps now occupy a position of central importance in the
over-the-counter derivatives market.
A swap is an agreement between two companies to exchange cash flows in the future.
The agreement defines the dates when the cash flows are to be paid and the way in
which they are to be calculated. Usually the calculation of the cash flows involves the
future value of an interest rate, an exchange rate, or other market variable.
A forward contract can be viewed as a simple example of a swap. Suppose it is
March 1,2006, and a company enters into a forward contract to buy 100 ounces of gold
for $400 per ounce in 1 year. The company can sell the gold in 1 year as soon as it is
received. The forward contract is therefore equivalent to a swap where the company
agrees that on March 1, 2007, it will pay $40,000 and receive 100S, where S is the
market price of 1 ounce of gold on that date.
Whereas a forward contract is equivalent to the exchange of cash flows on just one
future date, swaps typically lead to cash flow exchanges taking place on several future
dates. In this chapter we examine how swaps are used and how they are valued. Our
discussion centers on two popular swaps: plain vanilla interest rate swaps and fixed-forfixed currency swaps. Other types of swaps are discussed in Chapter 30. 7.1 MECHANICS OF INTEREST RATE SWAPS
The most common type of swap is a "plain vanilla" interest rate swap. With this swap a
company agrees to pay cash flows equal to interest at a predetermined fixed rate on a
notional principal for a number of years. In return, it receives interest at a floating rate
on the same notional principal for the same period of time. lIBOR
The floating rate in most interest rate swap agreements is the London Interbank Offer
Rate (LIBOR). We introduced this in Chapter 4. It is the rate of interest at which a bank
is prepared to deposit money with other banks in the Eurocurrency market. Typically,
I-month, 3-month, 6-month, and 12-month LIBOR are quoted in all major currencies.
1Lf.Q 150 CHAPTER 7
Just as prime is often the reference rate of interest for floating-rate loans in the
domestic financial market, LIBOR is a reference rate of interest for loans in international financial markets. To understand how it is used, consider a 5-year bond with a
rate of interest specified as 6-month LIBOR plus 0.5% per annum. The life of the bond
is divided into 10 periods, each 6 months in length. For each period, the rate of interest
is set at 0.5% per annum above the 6-month LIBOR rate at the beginning of the period.
Interest is paid at the end of the period. Illustration
Consider a hypothetical 3-year swap initiated on March 5, 2004, between Microsoft and
Intel. We suppose Microsoft agrees to pay to Intel an interest rate of 5% per annum on
a notional principal of $100 million, and in return Intel agrees to pay Microsoft the
6-month LIBOR rate on the same notional principal. Microsoft is the fixed-rate payer;
Intel is the floating-rate payer. We assume the agreement specifies that payments are to
be exchanged every 6 months and that the 5% interest rate is quoted with semiannual
compounding. This swap is represented diagrammatically in Figure 7.1.
The first exchange of payments would take place on September 5, 2004, 6 months
after the initiation of the agreement. Microsoft would pay Intel $2.5 million. This is the
interest on the $100 million principal for 6 months at 5%. Intel would pay Microsoft
interest on the $100 million principal at the 6-month LIBOR rate prevailing 6 months
prior to September 5, 2004-that is, on March 5, 2004. Suppose that the 6-month
LIBOR rate on March 5, 2004, is 4.2%. Intel pays Microsoft 0.5 x 0.042 x $100 =
$2.1 million. 1 Note that there is no uncertainty about this first exchange of payments
because it is determined by the LIBOR rate at the time the contract is entered into.
The second exchange of payments would take place on March 5, 2005, a year after the
initiation of the agreement. Microsoft would pay $2.5 million to Intel. Intel would pay
interest on the $100 million principal to Microsoft at the 6-month LIBOR rate prevailing
6 months prior to March 5, 2005-that is, on September 5, 2004. Suppose that the
6-month LIBOR rate on September 5, 2004, is 4.8%. Intel pays 0.5 x 0.048 x $100 =
$2.4 million to Microsoft.
In total, there are six exchanges of payment on the swap. The fixed payments are
always $2.5 million. The floating-rate payments on a payment date are calculated
using the 6-month LIBOR rate prevailing 6 months before the payment date. An
interest rate swap is generally structured so that one side remits the difference between
the two payments to the other side. In our example, Microsoft would pay Intel
$0.4 million (= $2.5 million - $2.1 million) on September 5, 2004, and $0.1 million
(= $2.5 million - $2.4 million) on March 5, 2005.
Figure 7.1 Interest rate swap between Microsoft and Intel.
5.0% Intel Microsoft
LIBOR The calculations here are simplified in that they ignore day count conventions. This point is discussed in
more detail later in the chapter. .I 151 S waps Table 7.1 Cash flows (millions' of dollars) to Microsoft in a $100 million 3-year
interest rate swap when a fixed rate of 5% is paid and LIBOR is received.
Mar. Six-month LIBDR Floating cash flow Fixed cash flow
5, 2007 4.20
-2.50 Net cash flow -0.40
+0.45 Table 7.1 provides a complete example of the payments made under the swap for one
particular set of 6-month LIBOR rates. The table shows the swap cash flows from the
perspective of Microsoft. Note that the $100 million principal is used only for the
calculation of interest payments. The principal itself is not exchanged. This is why it is
termed the notional principal.
If the principal were exchanged at the end of the life of the swap, the nature of the
deal would not be changed in any way. The principal is the same for both the fixed and
floating payments. Exchanging $100 million for $100 million at the end of the life of the
swap is a transaction that would have no financial value to either Microsoft or Intel.
Table 7.2 shows the cash flows in Table 7.1 with a final exchange of principal added in.
This provides an interesting way of viewing the swap. The cash flows in the third
column of this table are the cash flows from a long position in a floating-rate bond. The
cash flows in the fourth column of the table are the cash flows from a short position in a
fixed-rate bond. The table shows that the swap can be regarded as the exchange of a
fixed-rate bond for -a floating-rate bond. Microsoft, whose position is described by
Table 7.2, is long a floating-rate bond and short a fixed-rate bond. Intel is long a fixedrate bond and short a floating-rate bond. Table 7.2 Cash flows (millions of dollars) from Table 7.1 when there is a final
exchange of principal.
Six-month LIBD R
rate (%) Date
5.90 Floating cash flow
received Fixed cash flow
paid Net cash flow +2.10
+0.45 152 CHAPTER 7 This characterization of the cash flows in the swap helps to explain why the floating
rate in the swap is set 6 months before it is paid. On a floating-rate bond, interest is
generally set at the beginning of the period to which it will apply and is paid at the end
of the period. The calculation of the floating-rate payments in a "plain vanilla" interest
rate swap such as the one in Table 7.2 reflects this. Using the Swap to Transform a Liability
For Microsoft, the swap could be used to transform a floating-rate loan into a fixed-rate
loan. Suppose that Microsoft has arranged to borrow $100 million at LIBOR plus
10 basis points. (One basis point is one-hundredth of 1%, so the rate is LIBOR
plus 0.1 %.) After Microsoft has entered into the swap, it has the following three sets
of cash flows:
1. It pays LIBOR plus 0.1 % to its outside lenders.
2. It receives LIBOR under the terms of the swap.
3. It pays 5% under the terms of the swap. These three sets of cash flows net out to an interest rate payment of 5.1 %. Thus, for
Microsoft, the swap could have the effect of transforming borrowings at a floating rate
of LIBOR plus 10 basis points into borrowings at a fixed rate of 5.1 %.
For Intel, the swap could have the effect of transforming a fixed-rate loan into a
floating-rate loan. Suppose that Intel has a 3-year $100 million loan outstanding on
which it pays 5.2%. After it has entered into the swap, it has the following three sets of
1. It pays 5.2% to its outside lenders.
2. It pays LIBOR under the terms of the swap.
3. It receives 5% under the terms of the swap. These three sets of cash flows net out to an interest rate payment of LIBOR plus 0.2%
(or LIBOR plus 20 basis points). Thus, for Intel, the swap could have the effect of
transforming borrowings at a fixed rate of 5.2% into borrowings at a floating rate of
LIBOR plus 20 basis points. These potential uses of the swap by Intel and Microsoft
are illustrated in Figure 7.2. Using the Swap to Transform an Asset
Swaps can also be used to transform the nature of an asset. Consider Microsoft in our
example. The swap could have the effect of transforming an asset earning a fixed rate of
interest into an asset earning a floating rate of interest. Suppose that Microsoft owns
$100 million in bonds that will provide interest at 4.7% per annum over the next 3 years.
Figure 7.2 Microsoft and Intel use the swap to transform a liability. 5 .2% 5% Intel Microsoft LIBOR LIB OR + 0.1% 153 Swaps
Figure 7.3 Microsoft and Intel use the swap to transform an asset.
5% 4.7% Intel
LIB OR After Microsoft has entered into the swap, it has the following three sets of cash flows:
1. It receives 4.7% on the bonds.
2. It receives LIBOR under the terms of the swap.
3. It pays 5% under the terms of the swap. These three sets of cash flows net out to an interest rate inflow of LIBOR minus 30 basis
points. Thus, one possible use of the swap for Microsoft is to transform an asset
earning 4.7% into an asset earning LIBOR minus 30 basis points.
Next, consider Intel. The swap could have the effect of transforming an asset earning
a floating rate of interest into an asset earning a fixed rate of interest. Suppose that Intel
has an investment of $100 million that yields LIBOR minus 20 basis points. After it has
entered into the swap, it has the following three sets of cash flows:
1. It receives LIBOR minus 20 basis points on its investment.
2. It pays LIB OR under the terms of the swap.
3. It receives 5% under the terms of the swap. These three sets of cash flows net out to an interest rate inflow of 4.8%. Thus, one
possible use of the swap for Intel is to transform an asset earning LIBOR minus 20 basis
points into an asset earning 4.8%. Thesepotential uses of the swap by Intel and
Microsoft are illustrated in Figure 7.3. Role of Financial Intermediary
Usually two nonfinancial companies such as Intel and Microsoft do not get in touch
directly to arrange a swap in the way indicated in Figures 7.2 and 7.3. They each deal
with a financial intermediary such as a bank or other financial institution. "Plain
vanilla" fixed-for-floating swaps on US interest rates are usually structured so that
the financial institution earns about 3 or 4 basis points (0.03% or 0.04%) on a pair of
Figure 7.4 shows what the role of the financial institution might be in the situation in
Figure 7.2. The financial institution enters into two offsetting swap transactions with
Figu re 7.4 Interest rate swap from Figure 7.2 when financial institution is involved.
Intel LIBOR 5.015%
LIB OR LIBOR+O.I% 154 CHAPTER 7
Figure 7.5 Interest rate swap from Figure 7.3 when financial institution is involved.
Intel LIBOR-0.2% LIBOR 5.015%
Microsoft LIB OR Intel and Microsoft. Assuming that both companies honor their obligations, the
financial institution is certain to make a profit of 0.03% (3 basis points) per year
multiplied by the notional principal of $100 million. This amounts to $30,000 per year
for the 3-year period. Microsoft ends up borrowing at 5.115% (instead of 5.1 %, as in
Figure 7.2), and Intel ends up borrowing at LIBOR plus 21.5 basis points (instead of at
LIB OR plus 20 basis points, as in Figure 7.2).
Figure 7.5 illustrates the role of the financial institution in the situation in Figure 7.3.
The swap is the same as before and the financial institution is certain to make a profit
of 3 basis points if neither company defaults. Microsoft ends up earning LIBOR minus
31.5 basis points (instead of LIB OR minus 30 basis points, as in Figure 7.3), and Intel
ends up earning 4.785% (instead of 4.8%, as in Figure 7.3).
Note that in each case the financial institution has two separate contracts: one with
Intel and the other with Microsoft. In most instances, Intel will not even know that the
financial institution has entered into an offsetting swap with Microsoft, and vice versa.
If. one of the companies defaults, the -financial institution still has to honor its
agreement with the other company. The 3-basis-point spread earned by the financial
institution is partly to compensate it for the risk that one of the two companies will
default on the swap payments. Market Makers
In practice, it is unlikely that two companies will contact a financial institution at the
same time and want to take qpposite positions in exactly the same swap. For tIlis
reason, many large financial institutions act as market makers for swaps. This means
that they are prepared to enter into a swap without having an offsetting swap with
another counterparty.2 Market makers must carefully quantify and hedge the risks they
are taking. Bonds, forward rate agreements, and interest rate futures are examples of the
instruments that can be used for hedging by swap market makers. Table 7.3 shows
quotes for plain vanilla US dollar swaps that might be posted by a market maker. 3 As
mentioned earlier, the bid-offer spread is 3 to 4 basis points. The average of the bid and
offer fixed rates is known as the swap rate. This is shown in the final column of
Consider a new swap where the fixed rate equals the current swap rate. We can
reasonably assume that the value of this swap is zero. (Why else would a market maker
choose bid-offer quotes centered on the swap rate?) In Table 7.2 we saw that a swap can
2 This is sometimes referred to as lI'are/lOl/sing swaps. The standard swap in the United States is one where fixed payments made every 6 months are exchanged
for floating 'LIBOR payments made every 3 months. In Table 7.1 we assumed that fixed and floating
pa:yments are exchanged every 6 months. As we shall see later, the fixed rate should in theory be the same,
regardless of whether floating payments are made every 3 or every 6 months. 3 155 Szuaps Table 7.3 Bid and offer fixed rates in the swap market and swap
rates (percent per annum).
Maturity (years) Bid Offer Swap rate 2
6.850 be characterized as the difference between a fixed-rate bond and a floating-rate bond.
B fix : Value of fixed-rate bond underlying the swap we are considering
Bft: Value of floating-rate bond underlying the swap we are considering Since the swap is worth zero, it follows that
B fix = Bft (7.1) We will use this result later in the chapter when discussing how the LIBOR/swap zero
curve is determined. 7.2 DAY COUNT ISSUES
We discussed day count conventions in Section 6.1. The day count conventions affect
payments on a swap, and some of the numbers calculated in the examples we have given
do not exactly reflect these day count conventions. Consider, for example, the 6-month
LIBOR payments in Table 7.1. Because it is a money market rate, 6-month LIBOR is
quoted on an actual/360 basis. The first floating payment in Table 7.1, based on the
LIBOR rate of 4.2%, is shown as $2.10 million. Because there are 184 days between
March 5, 2004, and September 5, 2004, it should be
100 x 0.042 x ~:~ = $2.1467 million In general, a LIBOR-based floating-rate cash flow on a swap payment date is calculated
as LRn/360, where L is the principal, R is the relevant LIBOR rate, and 1l is the number
of days since the last payment date.
The fixed rate that is paid in a swap transaction is similarly quoted with a particular
day count basis being specified. As a result, the fixed payments may not be exactly equal
on each payment date. The fixed rate is usually quoted as actual/365 or 30/360. It is not
therefore directly comparable with LIBOR because it applies to a full year. To make the
rates comparable, either the 6-month LIBOR rate must be multiplied by 365/360 or the
fixed rate must be multiplied by 360/365.
For ease of exposition, we will ignore day count issues in the calculations in the rest
of this chapter. 156 CHAPTER 7 Business Snapshot 7.1 Extract from Hypothetical Swap Confirmation Trade date:
Business day convention (all dates):
Termination date: 27-February-2004
Following business day
5-March- 2007 Fixed amounts Fixed-rate payer:
Fixed-rate notional principal:
Fixed-rate day count convention:
Fixed-rate payment "dates: Floating amounts
Floating-rate notional principal:
Floating-rate day count convention:
Floating-rate payment dates: 7.3 Microsoft
USD 100 million
5.015% per annum
Each 5-March and 5-September,
up to and including 5-March-2007
USD 100 million
USD 6-month LIBOR
Each 5-March and 5-September,
up to and including 5-March-2007 CONFIRMATIONS
A confirmation is the legal agreement underlying a swap and is signed by representatives
of the two parties. The drafting of confirmations has been facilitated by the work of the
International Swaps and Derivatives Association (ISDA) in New York. This organization has produced a number of Master Agreements that consist of clauses defining in
some detail the tenllinology used in swap agreements, what happens in the event of
default by either side, and so on. In Business Snapshot 7.1, we show a possible extract
from the confirmation for the swap shown in Figure 7.4 between Microsoft and a
financial institution (assumed here to be Goldman Sachs). Almost certainly, the full
confirmation would state that the provisions of an ISDA Master Agreement apply to
The confirmation specifies that the following business day convention is to be used
and that the US calendar detennines which days are business days and which days are
holidays. This means that, if a payment date falls on a weekend or a US holiday, the
payment is made on the next business day.4 September 5, 2004, is a Sunday. The first
4 Another business day convention that is sometimes specified is the modified following business day
convention. which is the same as the following business day convention except that, when the next business
-day falls in a different month from the specified day. the payment is made on the immediately preceding
business day. Preceding and modified preceding business day conventions are defined analogously. 157
exchange of payments in the swap ,between Microsoft and Goldman Sachs is therefore
on Monday September 6, 2004. 7.4 THE COMPARATIVE-ADVANTAGE ARGUMENT
An explanation commonly put forward to explain the popularity of swaps concerns
comparative advantages. Consider the use of an interest rate swap to transform a
liability. Some companies, it is argued, have a comparative advantage when borrowing
in fixed-rate markets, whereas other companies have a comparative advantage in
floating-rate markets. To obtain a new loan, it makes sense for a company to go to
the market where it has a comparative advantage. As a result, the company may borrow
fixed when it wants floating, or borrow floating when it wants fixed. The swap is used to
transform a fixed-rate loan into a floating-rate loan, and vice versa. Illustration
Suppose that two companies, AAACorp and BBBCorp, both wish to borrow $10 million for 5 years and have been offered the rates shown in Table 7.4. AAACorp has a
AAA credit rating; BBBCorp has a BBB credit rating. 5 We assume that BBBCorp wants
to borrow at a fixed rate of interest, whereas AAACorp wants to borrow at a floating
rate of interest linked to 6-month LIBOR. Because it has a worse credit rating than
AAACorp, BBBCorp pays a higher rate of interest than AAACorp in both fixed and
A key feature of the rates offered to AAACorp and BBBCorp is that the difference
between the two fixed rates is greater than the difference between the two floating rates.
BBBCorp pays 1.2% more than AAACorpin fixed-rate markets and only 0.7% more
than AAACorp in floating-rate markets. BBBCorp appears to have a comparative
advantage in floating-rate markets, whereas AAACorp appears to have a comparative
advantage in fixed-rate markets. 6 It is this apparent anomaly that can lead to a swap
being negotiated. AAACorp borrows fixed-rate funds at 4% per annum. BBBCorp
borrows floating-rate funds at LIBOR plus 1% per annum. They then enter into a swap
Table 7.4 Borrowing rates that provide a basis for the
BBBCorp Floating 4.0%
5.2% 6~month LIBOR
6-month LIBOR + 0.3% + 1.0% The credit ratings assigned to companies by S&P (in order of decreasing creditworthiness) are AAA, AA,
A, BBB, BB, B, and CCc. The corresponding ratings assigned by Moody's are Aaa, Aa, A, Baa, Ba, B, and
Caa, respectively. 5 Note that BBBCorp's comparative advantage in floating-rate markets does not imply that BBBCorp pays
less than AAACorp in this market. It means that the extra amount that BBBCorp pays over the amount paid
by AAACorp is less in this market. One of my students summarized the situation as follows: "AAACorp pays
more less in fixed-rate markets; BBBCorp pays less more in floating-rate markets." 6 158 CHAPTER 7
Figure 7.6 Swap agreement between AAACorp and BBBCorp when rates in Table 7.4 apply.
4% LIB OR LIB OR + 1% agreerp.ent to ensure that AAACorp ends up with floating-rate funds and BBBCorp
ends up with fixed-rate funds.
To understand how this swap might _
work, we first assume that AAACorp and
BBBCorp get in touch with each other directly. The sort of swap they might negotiate
is shown in Figure 7.6. This is similar to our example in Figure 7.2. AAACorp agrees to
pay BBBCorp interest at 6-month LIBOR on $10 million. In return, BBBCorp agrees to
pay AAACorp interest at a fixed rate of 3.95% per annum on $10 million.
AAACorp has three sets of interest rate cash flows:
1. It pays 4% per annum to outside lenders.
2. It receives 3.95% per annum from BBBCorp.
3. It pays LIBOR to BBBCorp. The net effect of the three cash flows is that AAACorp pays LIBOR plus 0.05% per
annum. This is 0.25% per annum less than it would pay if it went directly to floatingrate markets. BBBCorp also has three sets of interest rate cash flows:
1. It pays LIBOR + 1% per annum to outside lenders.
2. It receives LIBOR from AAACorp.
3. It pays 3.95% per annum to AAACorp. The net effect of the three cash flows is that BBBCorp pays 4.95% per annum. TIns is
0.25% per annum less than it would pay if it went directly to fixed-rate markets.
In this example, the swap has been structured so that the net gain to both sides is the
same, 0.25%. This need not be the case. However, the total apparent gain from this
type of interest rate swap arrangement is always a - b, where a is the difference between
the interest rates facing the two companies in fixed-rate markets, and b is the difference
between the interest rates facing the two companies in floating-rate markets. In this
case, a = 1.2% and b = 0.7%, so that the total gain is 0.5%.
If AAACorp and BBBCorp did not deal directly with each other and used a financial
institution, an arrangement such as that shown in Figure 7.7 might result. (This is similar
Swap agreement between AAACorp and BBBCorp when rates in Table 7.4
apply and a financial intermediary is involved. Figure 7.7 3.93% 4%
AAACorp LIB OR 3.97%
LIB OR LIB OR + 1% 159 S waps to the example in Figure 7.4.) In this case, AAACorp ends up borrowing at LIBOR +
0.07%, BBBCorp ends up borrowing at 4.97%, and the finaI1cial institution earns a
spread of 4 basis points per year. The gain to AAACorp is 0.23%; the gain to BBBCorp is
0.23%; and the gain to the financial institution is 0.04%. The total gain to all three
parties is 0.50% as before. Criticism of the Comparative-Advantage Argument
The comparative-advantage argument we have just outlined for explaining the attractiveness of interest rate swaps is open to question. Why in Table 7.4 should the spreads
between the rates offered to AAACorp and BBBCorp be different in fixed and floating
markets? Now that the swap market has been in existence for some time, we might
reasonably expect these types of differences to have been arbitraged away.
The reason that spread differentials appear to exist is due to the nature of the
contracts available to companies in fixed and floating markets. The 4.0% and 5.2%
rates available to AAACorp and BBBCorp in fixed-rate markets are 5-year rates (e.g.,
the rates at which the companies can issue 5-year fixed-rate bonds). The LIBOR +
0.3% and LIBOR + 1.0% rates available to AAACorp and BBBCorp in floating-rate
markets are 6-month rates. In the floating-rate market, the lender usually has the
opportunity to review the floating rates every 6 months. If. the creditworthiness of
AAACorp or BBBCorp has declined, the lender has the option of increasing the spread
over LIBOR that is charged. In extreme circumstances, the lender can refuse to roll over
the loan at alL The providers of fixed-rate financing do not have the option to change
the terms of the loan in this way.7
The spreads between the rates offered to AAACorp and BBBCorp are a reflection of
the extent to which BBBCorp is more likely than AAACorp to default. During the next
6 months, there is very little chance that either AAACorp or BBBCorp will default. As
we look further ahead, default statistics show that on average the probability of a
default by a company with a relatively low credit rating (such as BBBCorp) increases
faster than the probability of a default by a company with a relatively high credit rating
(such as AAACorp). This is why the spread between the 5-year rates is greater than the
spread between the 6-month rates.
After negotiating a floating-rate loan at LIBOR + 1.0% and entering into the swap
shown in Figure 7.7, BBBCorp appears to obtain a fixed-rate loan at 4.97%. The
arguments just presented show that this is not really the case. In practice, the rate paid
is 4.97% only if BBBCorp can continue to borrow floating-rate funds at a spread of
1.0% over LIBOR. If, for example, the credit rating of BBBCorp declines so that the
floating-rate loan is rolled over at LIBOR + 2.0%, the rate paid by BBBCorp increases
to 5.97%. The market expects that BBBCorp's spread over 6-month LIB OR will on
average rise during the swap's life. BBBCorp's expected average borrowing rate when it
enters into the swap is therefore greater than 4.97%.
The swap in Figure 7.7 locks in LIBOR + 0.07% for AAACorp for the whole of the
next 5 years, not just for the next 6 months. Tllis appears to be a good deal for AAACorp.
The downside is that it is bearing the risk of a default by the financial institution. If it
borrowed floating-rate funds in the usual way, it would not be bearing this risk.
If the floating-rate loans are structured so that the spread over LIBOR is guaranteed in advance regardless
of changes in credit rating, there is in practice little or no comparative advantage. 7 160 7.5 CHAPTER 7 THE NATURE OF SWAP RATES
At this stage it is appropriate to examine the nature of swap rates and the relationship
between swap and LlBOR markets. We explained in Section 4.1 that LIBOR is the rate of
interest at which AA-rated banks borrow for periods between 1 and 12 months from
other banks. As shown in Table 7.3, a swap rate is the average of (a) the fixed rate that a
swap market maker is prepared to pay in exchange for receiving LIBOR (its bid rate) and
(b) the fixed rate that it is prepared to receive in return for paying LIBOR (its offer rate).
Like LIBOR rates, swap rates are not risk-free lending rates. However, they are close
to risk-free. A financial institution can earn the 5-year swap rate on a certain principal
by doing the following:
1. Lend the principal for the first 6 months to a AA borrower and then re1end it for successive 6 month periods to other AA borrowers; and
2. Enter into a swap to exchange the LIBOR income for the 5-year swap rate.
This shows that the 5-year swap rate is an interest rate with a credit risk corresponding
to the situation where 10 consecutive 6-month LIBOR loans to AA companies are
made. Similarly the 7-year swap rate is an interest rate with a credit risk corresponding
to the situation where 14 consecutive 6-month LIBOR loans to AA companies are
made. Swap rates of other maturities can be interpreted analogously.
Note that swap rates are less than AA borrowing rates. It is much more attractive to
.lend money for successive 6-month periods to borrowers who are always AA at the
beginning of the periods than to lend it to one borrower for the whole 5 years when all
we can be sure of is that the borrower is AA at the beginning of the 5 years. 7.6 DETERMINING L1BOR/SWAP ZERO RATES
We explained in Section 4.1 that derivative traders tend to use LIBOR rates as a proxies
for risk-free rates when valuing derivatives. One problem with LIBOR rates is that
direct observations are possible only for maturities out to 12 months. As described in
Section 6.4, one way of extending the LIBOR zero curve beyond 12 months is to use
Eurodollar futures. Typically Eurodollar futures are used to produce a LIBOR zero
curve out to 2 years-and sometimes out to as far as 5 years. Traders then use swap
rates to extend the LIBOR zero curve further. The resulting zero curve is sometimes
referred to as the LIBOR zero curve and sometimes as the swap zero curve. To avoid
any confusion, we will refer to it as the LIBORjswap zero curve. We will now describe
how swap rates are used in the determination of the LIBORjswap zero curve.
The first point to note is that the value of a newly issued floating-rate bond that
pays 6-month LIBOR is always equal to its principal value (or par value) when the
LIBORjswap zero curve is used for discounting. s The reason is that the bond provides
a rate of interest of LIBOR, and LIBOR is the discount rate. The interest on the bond
exactly matches the discount rate, and as a result the bond is fairly priced at par.
In equation (7.1), we showed that for a newly issued swap where the fixed rate equals
the swap rate, Bfix = Bft. We have just shown that Bft equals the notional principal: It
follows that Bfix also equals the swap's notional principal. Swap rates therefore define a
s The same is true of a newly issued bond that pays I-month, 3-month, or I2-month LIBOR. 161 Swaps set of par yield bonds. For example? from the swap rates in Table 7.3, we can deduce
that the 2-year LIBOR/swap par yield is 6.045%, the 3-year LIBOR/swap par yield is
6.225%, and so on. 9
. The usual method for determining the LIBOR/swap zero curve is the bootstrap
method which we used to determine the Treasury zero curve in Section 4.5. LIBOR
rates define the zero curve out to I year. Swap rates define par yield bonds that are used
to determine longer-term rates.
Example 7.1 Suppose that the 6-month, 12-month, and 18-month LIBOR/swap zero rates have
been determined as 4%, 4.5%, and 4.8% with continuous compounding and that
the 2-year swap rate (for a swap where payments are made semiannually) is 5%.
This 5% swap rate means that a bond with a principal of $100 and a semiannual
coupon of 5% per annum sells for par. It follows that, if R is the 2-year zero rate,
2.5e-o.04xO.5 + 2.5e-o.045xl.O + 2.5e-o.048x1.5 + 102.5e2R = 100
Solving this, we obtain R = 4.953%. (Note that this calculation is simplified in
that it does not take the swap's day count conventions and holiday calendars into
account. See Section 7.2.) 7.7 VALUATION OF INTEREST RATE SWAPS
We now move on to discuss the valuation of interest rate swaps. An interest rate swap is
worth zero, or close to zero, when it is first initiated. After it has been in existence for
some time, its value may become positive or negative. There are two valuation
approaches. The first regards the swap as the difference between two bonds; the second
regards it as a portfolio of FRAs. Valuation in Terms of Bond Prices
Principal payments are not exchanged in an interest rate swap. However, as illustrated
in Table 7.2, we can assume that principal payments are both received and paid at the
end of the swap without changing its value. By doing this, we find that, from the point
of view of the floating-rate payer, a swap can be regarded as a long position in a fixedrate bond and a short position in a floating-rate bond, so that where V
swap is the value of the swap, Bfl is the value of the floating-rate bond (corresponding to payments that are made), and Bfix is the value of the fixed-rate bond
(corresponding to payments that are received). Similarly, from the point of view of
the fixed-rate payer, a swap is a long position in a floating-rate bond and a short
Analysts frequently interpolate between swap rates before calculating the zero curve, so that they have swap
rates for maturities at 6-month intervals. For example, for the data in Table 7.3 the 2.5-year swap rate would
be assumed to be 6.135%; the 7.5-year swap rate would be assumed to be 6.696%; and so on. 9 162 CHAPTER 7
position in a fixed-rate bond, so that the value of the swap is The value of the fixed rate bond, Bfix ' can be determined as described in Section 4.4. To
value the floating-rate bond, we note that the bond is worth the notional principal
immediately after an interest payment. This is because at this time the bond is a "fair
deal" where the borrower pays LIBOR for each subsequent accrual period.
Suppose that the notional principal is L, the next exchange of payments is at time t*,
and the floating payment that will be made at time t* (which was determined at the last
payment date) is k*. Immediately after the payment B n = L as just explained. It follows
that immediately before the payment B n.= L + k*. The floating-rate bond can therefore
be regarded as an instrument providing a single cash flow of L --)- k* at time t*. Discounting this, the value of the floating-rate bond today is
(L + k*)e-r*r* where r* is the LIBOR/swap zero rate for a maturity of t* .
. Example 7.2 Suppose that a financial institution has agreed to pay 6-month LIBOR and
receive 8% per annum (with semiannual compounding) on a notional principal
of $100 million. The swap has a remaining life of 1.25 years. The LIBOR rates
with continuous compounding for 3-month, 9-month, and l5-month maturities
are 10%, 10.5%, and 11 %, respectively. The 6-month LIBOR rate at the last
payment date was 10.2% (with semiannual compounding).
The calculations for valuing the swap in terms of bonds are summarized in
Table 7.5. The fixed-rate bond has cash flows of 4, 4, and 104 on the three
payment dates. The discount factors for these cash flows are, respectively,
e-O.lxO.25, e-O.105xO.75, e-O.llxI.25 and are shown in the fourth column of Table 7.5. The table shows that the value
of the fixed-rate bond (in millions of dollars) is 98.238.
In this example, k* = 0.5 x 0.102 x 100 = $5.1 million and t* = 0.25, so that
the floating-rate bond can be valued as though it produces a cash flow of
$105.1 million in 3 months. The table shows that the value of the floating bond
(in millions of dollars) is 102.505.
Table 7.5 Valuing a swap in terms of bonds ($ millions). Here, Bfix is fixed-rate
bond underlying the swap, and Bft is floating-rate bond underlying the swap. Time Bfix
cash floll' Bn
cash floll' Discount
factor Present value
Bfix cash floll' Present value
Bft cashfloll' 0.25
. 1.25 4.0
104.0 105.100 0.9753
90.640 102.505 98.238 102.505 Total: 163 Swaps The value of the swap is tl?-e difference between the two bond prices:
swap = 98.238 - 102.505 = -4.267 or -4.267 million dollars.
If the financial institution had been in the opposite position of paying fixed
and receiving floating, the value of the swap would be +$4.267 million. Note
that our calculations do not take account of day count conventions and holiday
calendars. Valuation in Terms of FRAs
A swap can be characterized as a portfolio of forward rate agreements. Consider the
swap between Microsoft and Intel in Figure 7.1. The swap is a 3-year deal entered into
on March 5, 2004, with semiannual payments. The first exchange of payments is known
at the time the swap is negotiated. The other five exchanges can be regarded as FRAs.
The exchange on March 5, 2005, is an FRA where interest at 5% is exchanged for
interest at the 6-month rate observed in the market on September 5, 2004; the exchange
on September 5, 2005, is an FRA where interest at 5% is exchanged for interest at the
6-month rate observed in the market on March 5, 2005; and so on.
As shown at the end of Section 4.7, an FRA can be valued by assuming that forward
interest rates are realized. Because it is nothing more than a portfolio of forward rate
agreements, a plain vanilla interest rate swap can also be valued by making the
assumption that forward interest rates are realized. The procedure is as follows:
1. Use the LIBOR/swap zero curve to calculate forward rates for each of the LIBOR rates that will determine swap cash flows.
2. Calculate swap cash flows on the assumption that the LIBOR rates will equal the
3. Discount these"swap cash flows (using the LIBOR/swap zero curve) to obtain the
Example 7.3 Consider again the situation in Example 7.2. Under the terms of the swap, a
financial institution has agreed to pay 6-month LIBOR and receive 8% per annum
(with semiannual compounding) on a notional principal of$100 million. The swap
has a remaining life of 1.25 years. The LIB OR rates with continuous compounding for 3-month, 9-month, and 15-month maturities are 10%, 10.5%, and 11 %,
respectively. The 6-monthLiBOR rate at the last payment date was 10.2% (with
The calculations are summarized in Table 7.6. The first row of the table shows the
cash flows that will be exchanged in 3 months. These have already been determined.
The fixed rate of 8% will lead to a cash inflow of 100 x 0.08 x 0.5 = 4 million. The
floating rate of 10.2% (which was set 3 months ago) will lead to a cash outflow of
100 x 0.1 02 x 0.5 = 5.1 million. The second row of the table shows the cash flows
that will be exchanged in 9 months assuming that forward rates are realized. The
cash inflow is 4.0 million as before. To calculate the cash outflow, we we must first
calculate the forward rate corresponding to the period between 3 and 9 months. 164 CHAPTER 7
Valuing swap in terms of FRAs ($ millions). Floating cash flows are
calculated by assuming that forward rates will be realized. Table 7.6
cash flow Floating
cash flow Net
cash flow Discol/nt
factor Presen t vallie
of net cash flow 0.25
-4.267 Total: From equation (4.5), this is
0.105 x 0.7~~ 0.10 x 0.25 = 0.1075
or 10.75% with continuous compounding. From equation (4.4), the forward rate
becomes 11.044% with semiannual compounding. The cash outflow is therefore
100 x 0.11044 x 0.5 = 5.522 million. The third row similarly shows the cash flows
that will be exchanged in 15 months assuming that forward rates are realized. The
discount factors for the three payment dates are, respectively,
e-O.lxO.25, e~O.I05xO.75, e-O.1 Ix 1.25 The present value of the exchange in three months is - 1.073 million. The values
of the FRAs corresponding to the exchanges in 9 months and 15 months are
-1.407 and -1.787, respectively. The total value of the swap is -$4.267 million.
This is in agreement with the value we calculated in Example 7.2 by decomposing
the swap into bonds.
The fixed rate in an interest rate swap is chosen so that the swap is worth zero initially.
This means that at the outset of a swap the sum of the values of the FRAs underlying
the swap is zero. It does not mean that the value of each individual FRA is zero. In
general, some FRAs will have positive values whereas others have negative values.
Consider the FRAs underlying the swap between Microsoft and Intel in Figure 7.1:
Value of FRA to Microsoft> 0 when forward interest rate> 5.0%
Value of FRA to Microsoft = 0 when forward interest rate = 5.0%
Value of FRA to Microsoft < 0 when forward interest rate < 5.0%
Suppose that the term structure of interest rates is upward-sloping at the time the swap
is negotiated. This means that the forward interest rates increase as the maturity of the
FRA increases. Since the sum of the values of the FRAs is zero, the forward interest
rate must be less than 5.0% for the early payment dates and greater than 5.0% for the
later payment dates. The value to Microsoft of the FRAs corresponding to early
payment dates is therefore negative, whereas the value of the FRAs corresponding to
later payment dates is positive. If the term structure of interest rates is downwardsJoping at the time the swap is negotiated, the reverse is true. The impact of the shape of
the term structure of interest rates on the values of the forward contracts underlying a
swap is summarized in Figure 7.8. S waps 165
Valuing of forward rate agreements underlying a swap as a function of
maturity. In (a) the term structure of interest rates is upward..:sloping and we receive
fixed, or it is downward-sloping and we receive floating; in (b) the term structure of
interest rates is upward-sloping and we receive floating, or it is downward-sloping and
we receive fixed. Figure 7.8 Value of forward
contract Maturity (a) Value of forward
contract Maturity (b) 7.8 CURRENCY SWAPS
Another popular type of swap is known as a currency swap. In its simplest form, tlus
involves exchanging principal and interest payments in one currency for principal and
interest payments in another.
A currency swap agreement requires the principal to be specified in each of the two
currencies. The principal amounts in each currency are usually exchanged at the
beginning and at the end of the life of the swap. Usually the principal amounts are
chosen to be approximately equivalent using the exchange rate at the swap's initiation.
When they are exchanged at the end of the life of the swap, their values may be quite
different. CHAPTER·7 166
A currency swap. Figure 7.9 Dollars 4%
Petroleum IBM Sterling 7% lIIustfiition
Consider a hypothetical 5-year currency swap agreement between IBM and British
Petroleum entered into on February 1, 2004. We suppose that IBM pays a fixed rate of
interest of7% in sterling and receives a fixed rate of interest of 4% in dollars from British
Petroleum. Interest rate payments are made once a year and the principal amounts are
- $15 million and £10 million. This is termed afixed-for-fixed currency swap because the
interest rate in both currencies is fixed. The swap is shown in Figure 7.9. Initially, the
principal amounts flow in the opposite direction to the arrows in Figure 7.9. The interest
p~yments during the life of the swap and the final principal payment flow in the same
direction as the arrows. Thus, at the outset of the swap, IBM pays $15 million and
receives £10 million. Each year during the life of the swap contract, IBM receives
$0.60 million (= 4% of$15 million) and pays £0.70 million (= 7% of £10 million). At the
end of the life of the swap, it pays a prin~ipal of £10 million and receives a principal of
$15 million. These cash flows are shown in Table 7.7. Use of a Currency Swap to Transform loans and Assets
A swap such as the one just considered can be used to transform borrowings in one currency to borrowings in another. Suppose that IBM can issue $15 million of USdollar-denominated bonds at 4% interest. The swap has the effect of transforming tIlis
transaction into one where IBM has borrowed £10 million at 7% interest. The initial
exchange of principal converts the proceeds of the bond issue from US dollars to
sterling. The subsequent exchanges in the swap have the effect of swapping the interest
and principal payments from dollars to sterling.
The swap can also be used to transform the nature of assets. Suppose that IBM can
invest £10 million in the UK to yield 7% per annum for the next 5 years, but feels that Table 7.7 Cash flows to IBM in currency swap. Date February
Eebruary Dollar cash flow
1,2009 Sterling cash flow
-10.70 Swaps 167
the US dollar will strengthen again~t sterling and prefers a US-dollar-denominated
investment. The swap has the effect of transforming the U~ investment into a
$15 million investment in the US yielding 4%. Comparative Advantage
Currency swaps can be motivated by comparative advantage~ To illustrate this, we
consider another hypothetical example. Suppose the 5-year fixed-rate borrowing costs
to General Motors and Qantas Airways in US dollars (USD) and Australian dollars
(AUD) are as shown in Table 7.8. The data in the table suggest that Australian rates are
higher than USD interest rates, and also that General Motors is more creditworthy than
Qantas Airways, because it is offered a more favorable rate of interest in both currencies.
From the viewpoint of a swap trader, the interesting aspect of Table 7.8 is that the
spreads between the rates paid by General Motors and Qantas Airways in the two
markets are not the same. Qantas Airways pays 2% more than General Motors in the
US dollar market and only 0.4% more than General Motors in the AUD market.
This situation is analogous to that in Table 7.4. General Motors has a comparative
advantage in the USD market, whereas Qantas Airways has a comparative advantage
in the AUD market. In Table 7.4, where a plain vanilla interest rate swap was
considered, we argued that comparative advantages are largely illusory. Here we are
comparing the rates offered in two different currencies, and it is more likely that the
comparative advantages are genuine. One possible source of comparative advantage is
tax. General Motors' position might be such that USD borrowings lead to lower taxes
on its worldwide income than AUD borrowings. Qantas Airways' position might be the
reverse. (Note that we assume that the interest rates in Table 7.8 have been adjusted to
reflect these types of tax advantages.)
We suppose that General Motors wants to borrow 20 million AUD and Qantas
Airways wants to borrow 12 million USD and that the current exchange rate (USD per
AUD) is 0.6000. This creates a perfect situation for a currency swap. General Motors
and Qantas Airways .each borrow in the market where they have a comparative
advantage; that is, General Motors borrows USD whereas Qantas Airways borrows
AUD. They then use a currency swap to transform General Motors' loan into an AUD
loan and Qantas Airways' loan into a USD loan.
As already mentioned, the difference between the USD interest rates is 2%, whereas
the difference between the AUD interest rates is 0.4%. By analogy with the interest rate
swap case, we expect the total gain to all parties to be 2.0 - 0.4 = 1.6% per annum.
There are many ways in which the swap can be arranged. Figure 7.10 shows one way
swaps might be entered into with a financial institution. General Motors borrows USD
and Qantas Airways borrows AUD. The effect of the swap is to transform the USD
Table 7.8 Borrowing rates providing basis for currency swap.
USD* General Motors
Qantas Airways AUD* 5.0%
13.0% * Quoted rates have been adjusted to reflect the differential impact of taxes. 168 CHAPTER 7
Figure 7.10 A currency swap motivated by comparative advantage.
USD 5.0% USD5.0% General
Motors AUD 11.9% USD 6.3%
institution AUD 13.0% Qantas
Airways AUD 13.0% interest rate of 5% per annum to an A UD interest rate of 11.9% per annum for
General Motors. As a result, General Motors is 0.7% per annum better off than it
would be if it went directly to AUD markets. Similarly, Qantas exchanges an AUD
loan at 13% per annum for a USD loari at 6.3% per annum and ends up 0.7% per
annum better off than it would be if it went directly to USD markets. The financial
institution gains 1.3 % per annum on its USD cash flows and loses 1.1 % per annum on
its AUD flows. If we ignore the difference between the two currencies, the financial
institution makes a net gain of 0.2% per annum. As predicted, the total gain to all
parties is 1.6% per annum.
Each year the financial institution makes a gain of USD 156,000 (= 1.3% of
12 million) and incurs a loss of AUD 220,000 (= 1.1 % of 20 million). The financial
institution can avoid any foreign exchange risk by buying AUD 220,000 per annum in
the forward market for each year of the life of the swap, thus locking in a net gain
, It is possible to redesign the swap so that the financial institution makes a 0.2%
spread in USD. Figures 7.11 and 7.12 present two alternatives. These alternatives are
unlikely to be used in practice because they do not lead to General Motors and Qantas
being free of foreign exchange risk. 1O In Figure 7.11, Qantas bears some foreign
exchange risk because it pays 1.1 % per annum in AUD and pays 5.2% per annum
in USD. In Figure 7.12, General Motors bears some foreign exchange risk because it
receives 1.1 % per annum in USD and pays 13% per annum in AUD. 7.9 VALUATION OF CURRENCY SWAPS
Like interest rate swaps, fixed-far-fixed currency swaps can be decomposed into either
the difference between two bonds or a portfolio of forward foreign exchange contracts. Alternative arrangement for currency swap: Qantas Airways bears some
foreign exchange risk. Figure 7.11 USD 5.0%
U5D5.0% -'0 General
Motors USD 5.2%
institution AUD 11.9% Qantas
AUD 11.9% AUD 13.0% Usually it makes sense for the financial institution to bear the foreign exchange risk, because it is in the
best position to hedge the risk. 169 Swaps Figure 7.12 Alternative arrangement for currency swap: General Motors bears some
foreign exchange risk.
Motors AUD 13.0% USD 6.3%
institution AUD 13.0% Qantas
. Airways AUD 13.0% Valuation in Terms of Bond Prices
If we define V
swap as the value in US dollars of an outstanding swap where dollars are
received and a foreign currency is paid, then where B F is the value, measured in the foreign currency, of the bond defined by the
foreign cash flows on the swap and B D is the value of the bond defined by the domestic
cash flows on the swap, and So is the spot exchange rate (expressed as number of dollars
per unit of foreign currency). The value of a swap can therefore be determined from
LIBOR rates in the two currencies, the term structure of interest rates in the domestic
currency, and the spot exchange rate.
Similarly, the value of a swap where the foreign currency is received and dollars are
Example 7.4 Suppose that the term structure of LIBOR/swap interest rates is flat in both Japan
and the United States. The Japanese rate is 4% per annum and the US rate is
9% per annum (both with continuous compounding). A financial institution has
entered into a currency swap in which it receives 5% per annum in yen and pays
8% per annum in dollars once a year. The principals in the two currencies are
$10 million and 1,200 million yen. The swap will last for another 3 years, and the
current exchange rate is 110 yen = $1.
The calculations are summarized in Table 7.9. In this case the cash flows from
the dollar bond underlying the swap are as shown in the second column. The
Table 7.9 Valuation of currency swap in terms of bonds. (All amounts in
Time Cash flows
on dollar bond ($) Present value
($) Cash flows forward
on yen bond (yen) Present value
1,064.30 Total: 9.6439 1,230.55 170 CHAPTER 7
present value of the cash flows using the dollar discount rate of 9% are shown in
the third column. The cash flows from the yen bond underlying the swap are
shown in the fourth column of the table. The present value of the cash flows using
the yen discount rate of 4% are shown in the final column of the table.
The value of the dollar bond, B D, is 9.6439 million dollars. The value of the yen
bond is 1230.55 million yen. The value of the swap in dollars is therefore
110 - 9.6439 = 1.5430 millIon Valuation as Portfolio of Forward Contracts
Each exchange of payments in a fixed-for-fixed currency swap is a forward contract. As
shown in Section 5.7, forward foreign exchange contracts can be valued by assuming
_ that forward exchange rates are realized. The forward exchange rates themselves can be
calculated from equation (5.9).
Example 7.5 Consider again the situation in Example 7.4. The LIBOR/swap term structure of
interest rates is flat in both Japan and the United States. The Japanese rate is 4%
per annum and the US rate is 9% per annum (both with continuous compounding).
A financial institution has entered into a currency swap in which it receives 5% per
annum in yen and pays 8% per annJlm in dollars once a year. The principals in the
two currencies are $10 million and 1,200 million yen. The swap will last for another
3 years, and the current exchange rate is 110 yen = $1.
The calculations are summarized in Table 7.10. The financial institution pays
0.08 x 10 = $0.8 million dollars and receives 1,200 x 0.05 = 60 million yen each
year. In addition, the dollar principal of $10 million is paid and the yen principal
of 1,200 is received at the end of year 3. The current spot rate is 0.009091 dollar
per yen. In this case r = 4% and IJ = 9%, so that, from equation (5.9), the I-year
forward rate is
0.009091 e(O.09-0.04) x I = 0.009557
The 2- and 3-year forward rates in Table 7.10 are calculated similarly. The forward contracts underlying the swap can be valued by assuming that the forward
rates are realized. If the I-year forward rate is realized, the yen cash flow in year 1
Valuation of currency swap as a portfolio of forward contracts.
(All amounts in millions.) Table 7.10 Time 1
cash flow cash flow -0.8
1200 Forward Dollar value of Net cash flow
yell cash flow 0.009557
1.5430 171 S waps is worth 60 x 0.009557 = 0.5734 million dollars and the net cash flow at the end
of year 1 is 0.8 - 0.5734 = -0.2266 million dollars. This has a present value of
_0.2266e-o.09xl = -0.2071
million dollars. This is the value of forward contract corresponding to the exchange
of cash flows at the end of year 1. The value of the other forward contracts are
calculated similarly. As shown in Table 7.10, the total value of the forward contracts is $1.5430 million. This agrees with the value calculated for the swap in
Example 7.4 by decomposing it into bonds.
The value of a currency swap is normally zero when it is first negotiated. If the two
principals are worth exactly the same using the exchange rate at the start of the swap,
the value of the swap is also zero immediately after the initial exchange of principal.
However, as in the case of interest rate swaps, this does not mean that each of the
individual forward contracts underlying the swap has zero value. It can be shown that,
when interest rates in two currencies are significantly different, the payer of the
currency with the high interest rate is in the position where the forward contracts
corresponding to the early exchanges of cash flows have negative values, and the
forward contract corresponding to final exchange of principals has a positive value.
(This is the situation in our example in Table 7.10.) The payer of the currency with the
low interest rate is likely to be in the opposite position; that is, the early exchanges of
cash flows have positive values and the final exchange has a negative value.
For the payer of the low-interest currency, the swap will tend to have a negative value
during most of its life. The forward contracts corresponding to the early exchanges of
payments have positive values, and once these exchanges have taken place, there is a
tendency for the remaining forward contracts to have, in total, a negative value. For the
payer of the high-interest currency, the reverse.is true. The value of the swap will tend to
be positive during most of its life. These results are important when the credit risk in the
swap is being evaluated. 7.10 CREDIT RISK
Contracts such as swaps that are private arrangements between two companies entail
credit risks. Consider a financial institution that has entered into offsetting contracts
with two companies (see Figure 7.4, 7.5, or 7.7). If neither party defaults, the financial
institution remains fully hedged. A decline in the value of one contract will always be
offset by an increase in the value of the other contract. However, there is a chance that
one party will get into financial difficulties and default. The financial institution then
still has to honor the contract it has with the other party.
Suppose that, some time after the initiation of the contracts in Figure 7.4, the
contract with Microsoft has a positive value to the financial institution, whereas the
contract with Intel has a negative value. If Microsoft defaults, the financial institution is
liable to lose the whole of the positive value it has in this contract. To maintain a
hedged position, it would have to find a third party willing to take Microsoft's position.
To induce the third party to take the position, the financial institution would have to
pay the third party an amount roughly equal to the value of its contract with Microsoft
prior to the default. 172 CHAPTER 7
A financial institution has credit-risk exposure from a swap only when the value of
the swap to the financial institution is positive. What happens when this value is
negative and the counterparty gets into financial difficulties? In theory, the financial
institution could realize a windfall gain, because a default would lead to it getting rid of
a liability. In practice, it is likely that the counterparty would choose to sell the contract
to a third party or rearrange its affairs in some way so that its positive value in the
contract is not lost. The most realistic assumption for the financial institution is
therefore as follows. If the counterparty goes bankrupt, there will be a loss if the value
of the swap to the financial institution is positive, and there will be no effect on the
financial institution's position if the value of the swap to the financial institution is
negative. This situation is summarized in Figure 7.13.
Potential losses from defaults on a swap are much less than the potential losses from
defaults on a loan with the same principal. This is because the value of the swap is
usually only a small fraction of the value of the loan. Potential losses from defaults on a
currency swap are greater than on an interest rate swap. The reason is that, because
principal amounts in two different currencies are exchanged at the end of the life of a
currency swap, a currency swap is liable to have a greater value at the time of a default
than an interest rate swap.
It is important to distinguish between the credit ,risk and market risk to a financial
institution in any contract. As discussed earlier, the credit risk arises from the
possibility of a default by the counterparty when the value of the contract to the
.financial institution is positive. The market risk arises from the possibility that market
variables such as interest rates and exchange rates will move in such a way that the value
of a contract to the financial institution becomes negative. .l'4arket risks can be hedged
by entering into offsetting contracts; credit risks are less easy to hedge.
One of the more bizarre stories in swap markets is outlined in Business Snapshot 7.2.
It concerns the British Local Authority, Hammersmith and Fulham and shows that, in
Figure 7.13 The credit exposure in a swap.
Exposure Swap value 173 S waps Business Snapshot 7.2 The Hammersmith and Fulham Star Between 1987 to 1989 the London Borough of Ha
u am In Great
Britain entered into about 600 interest rate swaps and related instruments with a total
notional principal of about 6 billion pounds. The transactions appear to have been
entered into for speculative rather than hedging purposes::The two em 10 ees of
Hammersmith and fulliam that were responsible for the trades had 0
understanding of the risks they were taking and how the product
By 1989, because of movements in sterling intere
Fulham had lost several hundred million pounds on the swaps:
the other side of the transactions, the swaps were worth severaL
pounds. The banks were concerned about credit risk. They ha.d entered into offsetting swaps to hedge their interest rate risks. If Hammersmith and, Fulham
defaulted, the banks would stilI have to honor their oblicr
swaps and would take a huge loss.
What happened was something a little different fr
Fulham's auditor asked to have the transactions declared void
and Fulham did not have the authority to enter into the transac 10
courts agreed. The case was appealed and went all the way to the H
Britain's highest court. The final decision was that HamI
not have the authority to enter into the swaps, but th
authority to do so in the future for risk-management purp
banks were furious that their contracts were overturned in this
addition to bearing market risk and credit risk, banks trading swaps also sometimes
bear legal risk. 7.11 OTHER TYPES OF SWAPS
In this chapter we have covered interest rate swaps where LIBOR is exchanged for a fixed
rate of interest and currency swaps where a fixed rate of interest in one currency is
exchanged for a fixed rate of interest in another currency. Many other types of swaps are
traded. We will discuss many of them in detail in later chapters, such as in Chapters 21,
26, and 30. At this stage, we will provide an overview. Variations on the Standard Interest Rate Swap
In fixed-for-floating interest rate swaps, LIBOR is the most common reference floating
interest rate. In the examples in this chapter, the tenor (i.e., payment frequency) of
LIBOR has been 6 months, but swaps where the tenor of LIBOR is 1 month, 3 months,
and 12 months trade regularly. The tenor on the floating side does not have to match
the tenor on the fixed side. (Indeed, as pointed out in footnote 3, the standard interest
rate swap in the United States is one where there are quarterly LIBOR payments and
semiannual fixed payments.) LIBOR is the most common floating rate, but others such
as the commercial paper (CP) rate are occasionally used. Sometimes floating-for- 174 CHAPTER 7
floating interest rates swaps are negotiated. For example, the 3-month CP rate plus
10 basis points might be exchanged for 3-month LIBOR with both being applied to the
same principal. (TIlls deal would allow a company to hedge its exposure when assets
and liabilities are subject to different floating rates.)
The principal in a swap agreement can be varied throughout the term of the swap to
meet the needs of a counterparty. In an amortizing swap, the principal reduces in a
predetermined way. (This might be designed to correspond to the amortization schedule
on a loan.) In a step-up swap, the principal increases in a predetermined way. (This
might be designed to correspond to drawdowns on a loan agreement.) Deferred swaps
or forward swaps, where the parties do not begin to exchange interest payments until
some future date, are also sometimes arranged. Sometimes swaps are negotiated where
the principal to which the fixed payments are applied is different from the principal to
which the floating payments are applied.
A constant maturity swap (eMS swap) is an agreement to exchange a LIBOR rate for
a swap rate. An example would be an agreement to exchange 6-month LIBOR applied
to a certain principal for the 10-year swap rate applied to the same principal every
6 months for the next 5 years. A constant maturity Treasury swap (CMT swap) is a
sinlilar agreement to exchange a LIBOR rate for a particular Treasury rate (e.g., the
1O-year Treasury rate).
In a compounding swap, interest on one or both sides is compounded forward to the
end of the life of the swap according to preagreed rules arid there is only one payment
date at the end of the life of the swap.-In a LIBOR-in arrears swap, the LIBOR rate
observed on a payment date is used to calculate the payment on that date. (As explained
in Section 7.1, in a standard deal the LIBOR rate observed on one payment date is used
to determine the payment on the next payment date.) In an accrual swap, the interest on
one side of the swap accrues only when the floating reference rate is in a certain range. Other Currency Swaps
In this chapter we have considered fixed-for-fixed currency swaps. Another type of swap
is a fixed-for-floating currency swap, whereby a floating rate (usually LIBOR) in one
currency is exchanged for a fixed rate in another currency. This is a combination of a
fixed-for-floating interest rate swap and a fixed-for-fixed currency swap and is known as
a cross-currency interest rate swap. A further type of currency swap is a floating-forfloating currency swap, where a floating rate in one currency is exchanged for a floating
rate in another currency.
Sometimes a rate observed in one currency is applied to a principal amount in
another currency. One such deal nlight be where 3-month LIBOR observed in the
United States is exchanged for 3-month LIBOR in Britain, with both principals being
applied to a principal of 10 million British pounds. This type of swap is referred to as a
dtff swap or a quanta. Equity Swaps
An equity swap is an agreement to exchange the total return (dividends and capital
gains) realized on an equity index for either a fixed or a floating rate of interest. For
-example, the total return on the S&P 500 in successive 6-month periods might be
exchanged for LIBOR, with both being applied to the same principal. Equity swaps can 175 Swaps be used by portfolio managers to convert returns from a fixed or floating investment to
the returns from investing in an equity index, and vice versa. Options
Sometimes there are options embedded in a swap agreement. For example, in an
extendable swap, one party has the option to extend the life ·of the swap beyond the
specified period. In a· puttable swap, one party has the option to terminate the swap
early. Options on swaps, or swaptions, are also available. These provide one party with
the right at a future time to enter into a swap where a predetermined fixed rate is
exchanged for floating. Commodity Swaps, Volatility Swaps, and Other Exotic Instuments
Commodity swaps are in essence a series of forward contracts on a commodity with
different maturity dates and the same delivery prices. In a l'olatility sll'ap there are a
series of time periods. At the end of each period, one side pays a preagreed volatility,
while the other side pays the historical volatility realized during the period. Both
volatilities are multiplied by the same notional principal in calculating payments.
Swaps are limited only by the imagination of financial engineers and the desire of
corporate treasurers and fund mangers for exotic structures. In Chapter 30, we will
describe the famous 5/30 swap entered into between Procter and Gamble and Bankers
Trust, where payments depended in a complex way on the 3D-day commercial paper
rate, a 3D-year Treasury bond price, and the yield on a 5-year Treasury bond. SUMMARY
The two most common types of swaps are interest rate swaps and currency swaps. In an
interest rate swap, one party agrees to pay the other party interest at a fixed rate on a
notional principal for a number of years. In return, it receives interest at a floating rate
on the same notional principal for the same period of time. In a currency swap, one
party agrees to pay interest on a principal amount in one currency. In return, it receives
interest on a principal amount in another currency.
Principal amounts are not usually exchanged in an interest rate swap. In a currency
swap, principal amounts are usually exchanged at both the beginning and the end of the
life of the swap. For a party paying interest in the foreign currency, the foreign principal
is received, and the domestic principal is paid at the beginning of the life of the swap. At
the end of the life of the swap, the foreign principal is paid and the domestic principal is
An interest rate swap can be used to transform a floating-rate loan into a fixed-rate
loan, or vice versa. It can also be used to transform a floating-rate investment to a fixedrate investment, or vice versa. A currency swap can be used to transform a loan in one
currency into a loan in another currency. It can also be used to transform an investment
denominated in one currency into an investment denominated in another currency.
There are two ways of valuing interest rate and currency swaps. In the first, the swap
is decomposed into a long position in one bond and a short position in another bond.
In the second it is regarded as a portfolio of forward contracts. 176 CHAPTER 7
When a financial institution enters into a pair of offsetting swaps with different
counterparties, it is exposed to credit risk. If one of the counterparties defaults when
the financial institution has positive value in its swap with that counterparty, the
financial institution loses money because it still has to honor its swap agreement with
the other counterparty. FURTHER READING
Baz, J., ,and M. Pascutti. "Alternative Swap Contracts Analysis and Pricing," Journal of
Derivatives, (Winter 1996): 7-21.
Brown, K. c., and D. J. Smith. Interest Rate and Currency Swaps: A Tutorial. Association for
Investment Management and Research, 1996.Cooper, 1., and A. Mello. "The Default Risk in Interest Rate Swaps," Journal of Finance, 46, 2
- Dattatreya, R. E., and K. Hotta. Advanced Interest Rate and Currency Swaps: State-of-the-Art
Products. Strategies, and Risk Management Applications. Irwin, 1993.
Flavell, R. Swaps and Other Instruments. Chichester: Wiley, 2002.
Gupta, A., and M. G. Subrahmanyam. "An Empirical Examination of the Convexity Bias in the
Pricing of Interest Rate Swaps," Journal of Financial Economics, 55, 2 (2000): 239-79.
Litzenberger, R. H. "Swaps: Plain and Fanciful," Journal of Finance, 47, 3 (1992): 831-50.
Minton, B. A. "An Empirical Examination of the Basic Valuation Models for Interest Rate
Swaps," Journal of Financial Economics, 44~ 2 (1997): 251-77.
Sun, T., S. Sundaresan, and C. Wang. "Interest Rate Swaps: An Empirical Investigation,"
Journal of Financial Economics, 34, 1 (1993): 77-99.
Titman, S. "Interest Rate Swaps and Corporate Financing Choices," Journal of Finance, 47, 4
(1992): 1503-16. Questions and Problems (Answers in Solutions Manual)
7.1. Companies A and B have been offered the following rates per annum on a $20 million
Fixed rate Company A:
Company B: 12.0%
13.4% Floating rate LIBOR
LIBOR + 0.1%
+ 0.6% Company A requires a floating-rate loan; company B requires a fixed-rate loan. Design a
swap that will net a bank, acting as intermediary, 0.1 % per annum and that will appear
equally attractive to both companies.
7.2. Company X wishes to borrow US dollars at a fixed rate of interest. Company Y wishes to
borrow Japanese yen at a fixed rate of interest. The amounts required by the two
companies are roughly the same at the current exchange rate. The companies have been
quoted the following interest rates, which have been adjusted for the impact of taxes:
Yen Company X:
Company Y : Dollars 5.0%
10.0% 177 S waps Design a swap that will net a billlk, a,cting as intermediary, 50 basis points per annum.
Make the swap equally attractive to the two companies and e.!1sure that all foreign
exchange risk is assumed by the bank.
7.3. A $'100 million interest rate swap has a remaining life of 10 months. Under the terms of the
swap, 6-month LIBOR is exchanged for 12% per annum (compounded semiannually). The
average of the bid-offer rate being exchanged for 6-month LIBOR in swaps of all
maturities is currently 10% per annum with continuous compounding. The 6-month
LIBOR rate was 9.6% per annum 2 months ago. What is the current value of the swap to
the party paying floating? What is its value to the party paying fixed?
7.4. Explain what a swap rate is. What is the relationship between swap rates and par
7.5. A currency swap has a remaining life of 15 months. It involves exchanging interest
at 14% on £20 million for interest at 10% on $30 million once a year. The term
structure of interest rates in both the United Kingdom and the United States is currently
flat, and if the swap were negotiated today the interest rates exchanged would be 8% in
dollars and 11 % in sterling. All interest rates are quoted with annual compounding. The
current exchange rate (dollars per pound sterling) is 1.6500. What is the value of the
swap to the party paying sterling? What is the value of the swap to the party paying
7.6. Explain the difference between the credit risk and the market risk in a financial contract.
7.7. A corporate treasurer tells you that he has just negotiated a 5-year loan at a competitive
fixed rate of interest of 5.2%. The treasurer explains that he achieved the 5.2% rate by
borrowing at 6-month LIBOR plus 150 basis points and swapping LIBOR for 3.7%. He
goes on to say that this was possible because his company has a comparative advantage in
the floating-rate market. What has the treasurer overlooked?
7.8. Explain why a bank is subject to credit risk when it enters into two offsetting swap
7.9. Companies X and Y have been offered the following rates per annum on a $5 million
Fixed rate Company X:
Company Y: Floating rate 8.0%
LIBOR Company X requires a fixed-rate investment; company Y requires a floating-rate investment. Design a swap that will net a bank, acting as intermediary, 0.2% per annum and
will appear equally attractive to X and Y.
7.10. A financial institution has entered into an interest rate swap with company X. Under the
terms of the swap, it receives 10% per annum and pays 6-month LIBOR on a principal of
$10 million for 5 years. Payments are made every 6 months. Suppose that company X
defaults on the sixth payment date (at the end of year 3) when the interest rate (with
semiannual compounding) is 8% per annum for all maturities. What is the loss to the
financial institution? Assume that 6-month LIBOR was 9% per annum halfway through
7.11. A financial institution has entered into a lO-year currency swap with company Y. Under
the terms of the swap, the financial institution receives interest at 3% per annum in Swiss 178 CHAPTER 7 francs and pays interest at 8% per annum in US dollars. Interest payments are exchanged
once a year. The principal amounts are 7 million dollars and 10 million francs. Suppose
that company Y declares bankruptcy at the end of year 6, when the exchange rate is $0.80
per franc. What is the cost to the financial institution? Assume that, at the end of year 6,
the interest rate is 3% per annum in Swiss francs and 8% per annum in US dollars for all
maturities. All interest rates are quoted with annual compounding.
7.12. Companies A and B face the following interest rates (adjusted for the differential impact
Company A US dollars (floating rate) :
Canadian dollars (fixed rate): Company B LIBOR + 0.5%
5.0% - LIBOR + 1.0%
6.5% Assume that A wants to borrow US dollars at a floating rate of interest and B wants to
- borrow Canadian dollars at a fixed rate of interest. A financial institution is planning to
arrange a swap and requires a 50-basis-point spread. If the swap is equally attractive to
A and B, what rates of interest will A and B end up paying?
7.13. After it hedges its foreign exchange risk using forward contracts, is the financial
institution's average spread in Figure 7.10 likely to be greater than or less than 20 basis
points? Explain your answer.
7.14. "Companies with high credit risks are the ones that cannot access fixed-rate markets
directly. They are the companies that are most likely to be paying fixed and receiving
floating in an interest rate swap." Assume that this statement is true. Do you think it
increases or decreases the risk of a financial institution's swap portfolio? Assume that
companies are most likely to default when interest rates are high.
7.15. Why is the expected loss from a default on a swap less than the expected loss from the
default on a loan with the same principal?
7.16. A bank finds that its assets are not matched with its liabilities. It is taking floating-rate
deposits and making fixed-rate loans. How can swaps be used to offset the risk?
7.17. Explain how you would value· a swap that is the exchange of a floating rate in one
currency for a fixed rate in another currency.
7.18. The LIBOR zero curve is flat at 5% (continuously compounded) out to 1.5 years. Swap
rates for 2- and 3-year semiannual pay swaps are 5.4% and 5.6%, respectively. Estimate
the LIBOR zero rates for maturities of 2.0, 2.5, and 3.0 years. (Assume that the 2.5-year
swap rate is the average of the 2- and 3-year swap rates.) Assignment Questions
7.19. The I-year LIBOR rate is 10%. A bank trades swaps where a fixed rate of interest is
exchanged for 12-month LIBOR with payments being exchanged annually. The 2- and
3-year swap rates (expressed with annual compounding) are 11 % and 12% per annum.
Estimate the 2- and 3-year LIBOR zero rates.
7.20. Company A, a British manufacturer, wishes to borrow US dollars at a fixed rate of
int.erest. Company B, a US multinational, wishes to borrow sterling at a fixed rate of 179 Szvaps interest. They have been quoted the following rates per annum (adjusted for differential
Sterling Company A
Company B US dollars 11.0%
6.2% Design a swap that will net a bank, acting as intermediary, 10 basis points per annum and
that will produce a gain of 15 basis points per annum for each of the two companies.
7.21. Under the terms of an interest rate swap, a financial institution has agreed to pay 10% per
annum and to receive 3-month LIBOR in return on a notional principal of $100 million
with payments being exchanged every 3 months. The swap has a remaining life of
14 months. The average of the bid and offer fixed rates currently being swapped for
3-month LIBOR is 12% per annum for all maturities. The 3-month LIBOR rate 1 month
ago was 11.8% per annum. All rates are compounded quarterly. What is the value of
7.22. Suppose that the term structure of interest rates is flat in the United States and Australia.
The USD interest rate is 7% per annum and the AUD rate is 9% per annum. The current
value of the AUD is 0.62 USD. Under the terms of a swap agreement, a financial
institution pays 8% per annum in AUD and receives 4% per annum in USD. The
principals in the two currencies are $12 million USD and 20 million AUD. Payments
are exchanged every year, with one exchange having just taken place. The swap will last
2 more years. What is the value of the swap to the financial institution? Assume all
interest rates are continuously compounded.
7.23. Company X is based in the United Kingdom and would like to borrow $50 million at a
fixed rate of interest for 5 years in US funds. Because the company is not well known in
the United States, this has proved to be impossible. However, the company has been
quoted 12% per annum on fixed-rate 5-year sterling funds. Company Y is based in the
United States and would like to borrow the equivalent of $50 million in sterling funds for
5 years at a fixed rate of interest. It has been unable to get a quote but has been offered
US dollar funds at 10.5% per annum. Five-year government bonds currently yield
9.5% per annum in the United States and 10.5% in the United Kingdom. Suggest an
appropriate currency swap that will net the financial intermediary 0.5% per annum. ...
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