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a pool of SBA loans. A factor affecting prepayments is the maturity date
of the loan. It has been found that the fastest speeds on SBA loans and
pools occur for shorter maturities. The purpose of the loan also affects
prepayments. There are loans for working capital purposes and loans to
¬nance real estate construction or acquisition. It has been observed that
SBA pools with maturities of 10 years or less made for working capital
purposes tend to prepay at the fastest speed. In contrast, loans backed
by real estate that have long maturities tend to prepay at a slow speed.
All other factors constant, pools that have capped loans tend to prepay
more slowly than pools of uncapped loans.

EXHIBIT 10.14 Bloomberg Screen of Historical CPR for SBA Pools




Source: Bloomberg Financial Markets
11
CHAPTER

Futures and Forward Rate
Agreements



his chapter is the ¬rst of two devoted to derivative instruments used by
T money market participants. The focus of this chapter is on interest rate
futures and forward rate agreements while in the next we discuss swaps
and caps/¬‚oors. In essence, a derivative instrument is one that derives its
value from some underlying variable or variables. The underlying vari-
ables could be the price of a ¬nancial asset, an interest rate, the spread
between two interest rates, or the amount of snowfall in Aspen, Colo-
rado. Indeed, the possibilities of variables underlying a derivative contract
are limitless. We will discuss the forward contracts ¬rst and then proceed
quickly to a discussion of interest rate futures. In the last section of the
chapter, we discuss forward rate agreements.



FORWARD CONTRACTS
A forward contract is an over-the-counter agreement between two parties
for the future delivery of the underlying at a speci¬ed price at the end of a
designated time period. The party that assumes the long (short) position is
obligated to buy (sell) the underlying at the speci¬ed price. The terms of the
contract are the product of negotiation between the two parties. As such, a
forward contract is speci¬c to the two parties. Although we commonly
refer to taking a long position as “buying a forward contract” and con-
versely taking a short position as “selling a forward contract,” this is a mis-
nomer. No money changes hands between the parties at the time the
forward contract is established. Both sides are making a promise to engage
in a transaction in the future according to terms negotiated upfront.

209
210 THE GLOBAL MONEY MARKETS



At expiration, the party with the long position pays the speci¬ed
price called the forward price in exchange for delivery of the underlying
from the party with the short position. The payoff of the forward con-
tract for the long position on the expiration date is simply the difference
between the price of the underlying minus the forward price. Con-
versely, the payoff of the forward contract for the short position on the
expiration date is the difference between the forward price minus the
price of the underlying. Clearly, a forward contract is a zero-sum game.
Now that we have introduced forward contracts, it is a short walk to
futures contracts.



FUTURES CONTRACTS
A futures contract is a legal agreement between a buyer (seller) and an
established exchange or its clearinghouse in which the buyer (seller)
agrees to take (make) delivery of something at a speci¬ed price at the end
of designated period. The price at which the parties agree to transact in
the future is called the futures price. The designated date at which the
parties must transact is called the settlement or delivery date. When a
market participant takes a position by buying a futures contract, the indi-
vidual is said to be in a long futures position or to be long futures. If,
instead, the market participant™s opening position is the sale of a futures
contract, the investor is said to be in a short position or short futures.
As can be seen from the description, a futures contract is quite simi-
lar to a forward contract. They differ on four dimensions. First, futures
contracts are standardized agreements as to the delivery date (or month)
and quality of the deliverable. Moreover, because these contracts are
standardized, they are traded on organized exchanges. In contrast, for-
ward contracts are usually negotiated individually between buyer and
seller and the secondary markets are often nonexistent or extremely
thin. Second, an intermediary called a clearinghouse (whose function is
discussed shortly) stands between the two counterparties to a futures
contract and guarantees their performance. Both parties to a forward
contract are subject to counterparty risk. Counterparty risk is the risk
that the other party to the contract will fail to perform. Third, a futures
contract is marked-to-market (discussed shortly) while a forward con-
tract may or may not be marked-to-market. Last, although both a
futures and forward contract set forth terms of delivery, futures con-
tracts are not intended to be settled by delivery. In fact, generally less
than 2% of outstanding contracts are settled by delivery. Forward con-
tracts, on the other hand, are intended for delivery.
211
Futures and Forward Rate Agreements



Role of the Clearinghouse
Associated with every futures exchange is a clearinghouse, which per-
forms several functions. One of these functions is guaranteeing that the
two parties to the transaction will perform. When a market participant
takes a position in the futures market, the clearinghouse takes the oppo-
site position and agrees to satisfy the terms set forth in the contract.
Because of the clearinghouse, the user need not worry about the ¬nan-
cial strength and integrity of the counterparty to the contract. After the
initial execution of an order, the relationship between the two parties
ends. The clearinghouse interposes itself as the buyer for every sale and
the seller for every purchase. Thus, users are free to liquidate their posi-
tions without involving the other party in the original contract and
without concern that the other party may default. This is the reason
why we de¬ne a futures contract as an agreement between a party and a
clearinghouse associated with an exchange. In addition to its guarantee
function, the clearinghouse makes it simple for parties to a futures con-
tract to unwind their positions prior to the settlement date.

Margin Requirements
When a position is established in a futures contract, each party must
deposit a minimum dollar amount per contract as speci¬ed by the
exchange in the terms of the contract. This amount, which is called the
initial margin, is required as deposit by the exchange.1
The initial margin may be in the form of an interest-bearing security
such as a Treasury bill. In some futures exchanges around the world, other
forms of margin are accepted such as common stock, corporate bonds or
even letters of credit. As the price of the futures contract ¬‚uctuates, the
value of the user™s equity in the position changes. At the end of each trad-
ing day, the exchange determines the settlement price of the futures con-
tract which is an average of the prices of the last few trades of the day.
This price is used to mark-to-market the user™s position, so that any gain
or loss from the position is re¬‚ected in the investor™s margin account.
Maintenance margin is the minimum level (speci¬ed by the
exchange) to which a user™s margin account may fall as a result of an
unfavorable price change before the user is required to deposit addi-
tional margin. The additional margin deposited is called variation mar-
gin and it is an amount necessary to bring the margin in the account
balance back to its initial margin level. Unlike initial margin, variation
margin must be in cash, not interest-bearing instruments. If a party to a

1
Individual brokerage firms are free to set margin requirements above the minimum
established by the exchange.
212 THE GLOBAL MONEY MARKETS



futures contract receives a margin call and is required to deposit varia-
tion margin fails to do so within 24 hours, the futures position is closed
out. Conversely, any excess margin may be withdrawn by the user.
Although there are initial and maintenance margin requirements for
buying securities on margin, the concept of margin differs for securities
and futures. When securities are acquired on margin, the difference
between the security™s price and the initial margin is borrowed from the
broker. The security purchased serves as collateral for the loan and the
investor pays interest. For futures contracts, the initial margin, in effect,
serves as a performance bond, an indication that the user will be able to
satisfy the obligation of the contract. Normally, no money is borrowed.



SHORT-TERM INTEREST RATE FUTURES CONTRACTS
The more actively traded short-term interest futures contracts in the
United States and the United Kingdom are described below.

U.S. Treasury Bill Futures
The Treasury bill futures market, which is traded on the International
Monetary Market (IMM) of the Chicago Mercantile Exchange, is based
on a 13-week (3-month) Treasury bill with a face value of $1 million.
More speci¬cally, the seller of a Treasury bill futures contract agrees to
deliver to the buyer on the settlement date a Treasury bill with 13 weeks
remaining to maturity and a face value of $1 million. The Treasury bill
delivered can be newly issued or seasoned. The futures price is the price
at which the Treasury bill will be sold by the short and purchased by the
long. For example, a Treasury bill futures contract that settles in 3
months requires that 3 months from now the short deliver to the long
$1 million face value of a Treasury bill with 13 weeks remaining to
maturity. The Treasury bill delivered could be a newly issued 13-week
Treasury bill or a seasoned 26-week Treasury bill that has only 13
weeks remaining until maturity.
As explained in Chapter 3, the convention for quoting bids and
offers in the secondary market is different for Treasury bills and Trea-
sury coupon securities. Bids/offers on bills are quoted in a special way.
Unlike bonds that pay coupon interest, Treasury bill values are quoted
on a bank discount basis, not on a price basis. The yield on a bank dis-
count basis is computed as follows:

Y d = D — 360
---- ---------
- -
F t
213
Futures and Forward Rate Agreements



where:

Yd = annualized yield on a bank discount basis (expressed as a
decimal)
D = dollar discount, which is equal to the difference between the
face value and the price
F = face value
t = number of days remaining to maturity

Given the yield on a bank discount basis, the price of a Treasury bill
is found by ¬rst solving the formula for the dollar discount (D), as fol-
lows:

D = Yd — F — (t /360)

The price is then

price = F ’ D

In contrast, the Treasury bill futures contract is quoted not directly
in terms of yield, but instead on an index basis that is related to the
yield on a bank discount basis as follows:

Index price = 100 ’ (Yd — 100)

For example, if Yd is 1.77%, the index price is

100 ’ (0.0177 — 100) = 98.23

Given the index price of the futures contract, the yield on a bank
discount basis for the futures contract is determined as follows:

100 “ Index price
Y d = --------------------------------------------
-
100

To illustrate how this works, let™s use Bloomberg™s Futures Contract
Description screen presented in Exhibit 11.1. This 3-month U.S. Trea-
sury bill futures contract began trading on June 19, 2001 and expires on
March 18, 2002. On December 19, 2001, the index price was 98.230,
which is labeled as “Contract Price” and is located on the left-hand side
of the screen. The yield on a bank discount basis for this Treasury bill
futures contract is:
214 THE GLOBAL MONEY MARKETS




Y d = 100 “ 98.230 = 0.0177 or 1.77%
---------------------------------
-
100

The invoice price that the buyer of $1 million face value of 13-week
Treasury bills must pay at settlement is found by ¬rst computing the
dollar discount, as follows:

D = Yd — $1,000,000 — t /360

where t is either 90 or 91 days.
Typically, the number of days to maturity of a 13-week Treasury bill
is 91 days. The invoice price is then:

Invoice price = $1,000,000 ’ D

For example, if the index price is 98.230 (and a yield on a bank dis-
count basis of 1.77%), the dollar discount for the 13-week Treasury bill
to be delivered with 91 days to maturity is:

D = 0.0177 — $1,000,000 — 91/360 = $4,474.167

EXHIBIT 11.1 Bloomberg Futures Contract Description Screen for a
U.S. Treasury Bill Futures Contract




Source: Bloomberg Financial Markets
215
Futures and Forward Rate Agreements



The invoice price is:

Invoice price = $1,000,000 ’ $4,474.167 = $995,525.833

The minimum index price ¬‚uctuation or “tick” for this futures con-
tract is 0.005. A change of 0.005 for the minimum index price translates
into a change in the yield on a bank discount basis of one-half of a basis
point (0.00005). A one-half basis point change results in a change in the
invoice price as follows:

0.00005 — $1,000,000 — t/360

For a 13-week Treasury bill with 91 days to maturity, the change in
the dollar discount is:

0.00005 — $1,000,000 — 91/360 = $12.639

For a 13-week Treasury bill with 90 days to maturity, the change in
the dollar discount would be $12.50. Despite the fact that a 13-week
Treasury bill usually has 91 days to maturity, market participants com-
monly refer to the value of a tick for this futures contract as $12.50. As
evidence of this, on the left side of Exhibit 11.1, the “Tick Value” is
$12.50.

Eurodollar CD Futures
As discussed in Chapter 6, Eurodollar certi¬cates of deposit (CDs) are
denominated in dollars but represent the liabilities of banks outside the
United States. The contracts are traded on the International Monetary
Market of the Chicago Mercantile Exchange and the London Interna-
tional Financial Futures Exchange (LIFFE). As noted several times in the
book, the rate paid on Eurodollar CDs is the London interbank offered
rate (LIBOR).
The 3-month (90 day) Eurodollar CD is the underlying instrument
for the Eurodollar CD futures contract. Exhibit 11.2 presents the
Bloomberg Futures Contract Description screen for the April 2002 con-
tract. As with the Treasury bill futures contract, this contract is for $1
million of face value and is traded on an index price basis. The index
price basis in which the contract is quoted is equal to 100 minus the
annualized futures LIBOR. For example, a Eurodollar CD futures price
of 98.00 means a futures 3-month LIBOR of 2%.
The minimum price ¬‚uctuation (tick) for this contract is 0.005 or ¹ ‚‚
basis point. This means that the tick value for this contract is $12.50,
which is determined as follows:
216 THE GLOBAL MONEY MARKETS



tick value = $1,000,000 — (0.005 — 90/360)

tick value = $12.50

This expression appears in the lower right-hand corner of Exhibit 11.2.
The Eurodollar CD futures contract is a cash settlement contract.
Speci¬cally, the parties settle in cash for the value of a Eurodollar CD
based on LIBOR at the settlement date. The Eurodollar CD futures con-
tract is one of the most heavily traded futures contracts in the world.
Exhibit 11.3 presents Bloomberg™s Contract Table screen for the active
90-day Eurodollar CD futures contracts on January 22, 2002. Note the
very large open interest for March, June, September, and December
2002 contracts.2
The Eurodollar CD futures contract is used frequently to trade the
short end of the yield curve and many hedgers believe this contract to be
the best hedging vehicle for a wide range of hedging situations.

EXHIBIT 11.2 Bloomberg Futures Contract Description Screen for a
Eurodollar CD Futures Contract




Source: Bloomberg Financial Markets

2
Open interest is the number of futures contracts established that have yet to be off-
set.
217
Futures and Forward Rate Agreements



EXHIBIT 11.3 Bloomberg Contract Table for a Eurodollar CD Futures Contract




Source: Bloomberg Financial Markets

The 90-day sterling Libor interest rate futures contract trades on the
main London futures exchange, LIFFE. The contract is structured simi-
larly to the Eurodollar futures contract described above. The Bloomberg
Futures Contract Description screen for the March 2002 contract is pre-
sented in Exhibit 11.4. Prices are quoted as 100 minus the interest rate
and the delivery months are March, June, September, and December.
The contract size is £500,000. A tick is 0.01 or one basis point and the
tick value is £12.5. Exhibit 11.5 presents a Bloomberg screen with set-
tlement prices of the near-term 90-day sterling LIBOR contract on Janu-
ary 22, 2002.
The LIFFE exchange also trades short-term interest rate futures for
other major currencies including euros, yen, and Swiss franc. For exam-
ple, Exhibit 11.6 presents a Bloomberg Futures Contract Description
screen for the June 2002 90-day Euro Euribor contract. Short-term
interest rate contracts in other currencies are similar to the 90-day ster-
ling Libor contract and trade on exchanges such as Deutsche Termin-
bourse in Frankfort and MATIF in Paris.
218 THE GLOBAL MONEY MARKETS



EXHIBIT 11.4 Bloomberg Futures Contract Description Screen for a
90-Day Sterling Libor Contract




Source: Bloomberg Financial Markets

EXHIBIT 11.5 Bloomberg Contract Table for the 90-Day Sterling Libor Contracts




Source: Bloomberg Financial Markets
219
Futures and Forward Rate Agreements



EXHIBIT 11.6 Bloomberg Futures Contract Description Screen for the
90-Day Euro Euribor Contract




Source: Bloomberg Financial Markets

Fed Funds Futures Contract
When the Federal Reserve formulates and executes monetary policy, the
federal funds rate is frequently a signi¬cant operating target. Accord-
ingly, the federal funds rate is a key short-term interest rate. The fed
funds futures contract is designed for hedgers who have exposure to this
rate or speculators who want to make a bet on the direction of U.S.
monetary policy. Underlying this contract is the simple average over-
night federal funds rate (i.e., the effective rate) for the delivery month.
As such, this contract is settled in cash.
Exhibit 11.7 presents the Bloomberg Futures Contract Description
screen for the May 2002 fed funds futures contract. The contract size is
$5,000,000 and the tick size is 0.005 or ¹„‚‚ basis point. Accordingly, the
tick value is 20.835. Just as the other short-term interest futures con-
tracts discussed above, prices are quoted as 100”the interest rate.
Exhibit 11.8 presents the Bloomberg Contract Table screen for the
active fed funds futures contracts on January 22, 2002.
220 THE GLOBAL MONEY MARKETS



EXHIBIT 11.7 Bloomberg Futures Contract Description Screen for the
Federal Funds Futures Contract




Source: Bloomberg Financial Markets

EXHIBIT 11.8 Bloomberg Contract Table for the Federal Funds Futures Contract




Source: Bloomberg Financial Markets
221
Futures and Forward Rate Agreements



FORWARD RATE AGREEMENTS
A forward rate agreement (FRA) is an over-the-counter derivative
instrument that trades as part of the money markets. In essence, an FRA
is a forward-starting loan, but with no exchange of principal, so the
cash exchanged between the counterparties depend only on the differ-
ence in interest rates. While the FRA market is truly global, much busi-
ness is transacted in London. Trading in FRAs began in the early 1980s
and the market now is large and liquid. According to the British Bank-
ers Association, turnover in London exceeds $5 billion each day.
In effect an FRA is a forward dated loan, transacted at a ¬xed rate, but
with no exchange of principal”only the interest applicable on the notional
amount between the rate agreed to when the contract is established and the
actual rate prevailing at the time of settlement changes hands. For this rea-
son, FRAs are off-balance sheet instruments. By trading today at an interest
rate that is effective at some point in the future, FRAs enable banks and
corporations to hedge forward interest rate exposure. Naturally, they are
also used to speculate on the level of future interest rates.

FRA Basics
An FRA is an agreement to borrow or lend a notional cash sum for a
period of time lasting up to 12 months, starting at any point over the
next 12 months, at an agreed rate of interest (the FRA rate). The
“buyer” of a FRA is borrowing a notional sum of money while the
“seller” is lending this cash sum. Note how this differs from all other
money market instruments. In the cash market, the party buying a CD,
Treasury bill, or bidding for bond in the repo market, is the lender of
funds. In the FRA market, to “buy” is to “borrow.” Of course, we use
the term “notional” because with an FRA no borrowing or lending of
cash actually takes place. The notional sum is simply the amount on
which interest payment is calculated (i.e., a scale factor).
Accordingly, when a FRA is traded, the buyer is borrowing (and the
seller is lending) a speci¬ed notional sum at a ¬xed rate of interest for a
speci¬ed period, the “loan” to commence at an agreed date in the future.
The buyer is the notional borrower, and so if there is a rise in interest rates
between the date that the FRA is traded and the date that the FRA comes
into effect, she will be protected. If there is a fall in interest rates, the buyer
must pay the difference between the rate at which the FRA was traded and
the actual rate, as a percentage of the notional sum. The buyer may be
using the FRA to hedge an actual exposure, that is an actual borrowing of
money, or simply speculating on a rise in interest rates. The counterparty to
the transaction, the seller of the FRA, is the notional lender of funds, and
has ¬xed the rate for lending funds. If there is a fall in interest rates, the
222 THE GLOBAL MONEY MARKETS



seller will gain, and if there is a rise in rates, the seller will pay. Again, the
seller may have an actual loan of cash to hedge or is acting as a speculator.
In FRA trading, only the payment that arises because of the differ-
ence in interest rates changes hands. There is no exchange of cash at the
time of the trade. The cash payment that does arise is the difference in
interest rates between that at which the FRA was traded and the actual
rate prevailing when the FRA matures, as a percentage of the notional
amount. FRAs are traded by both banks and corporations. The FRA
market is liquid in all major currencies and rates are readily quoted on
screens by both banks and brokers. Dealing is over the telephone or
over a dealing system such as Reuters.
The terminology quoting FRAs refers to the borrowing time period
and the time at which the FRA comes into effect (or matures). Hence if a
buyer of a FRA wished to hedge against a rise in rates to cover a 3-
month loan starting in three months™ time, she would transact a “3-
against-6 month” FRA, or more usually denoted as a 3—6 or 3v6 FRA.
This is referred to in the market as a “threes-sixes” FRA, and means a
3-month loan beginning in three months™ time. So correspondingly, a
“ones-fours” FRA (1v4) is a 3-month loan in one month™s time, and a
“three-nines” FRA (3v9) is a 6-month loan in three months™ time.
As an illustration, suppose a corporation anticipates it will need to
borrow in six months time for a 6-month period. It can borrow today at
6-month LIBOR plus 50 basis points. Assume that 6-month LIBOR rates
are 4.0425% but the corporation™s treasurer expects rates to go up to
about 4.50% over the next several weeks. If the treasurer™s suspicion is
correct, the corporation will be forced to borrow at higher rates unless
some sort of hedge is put in place to protect the borrowing requirement.
The treasurer elects to buy a 6v12 FRA to cover the 6-month period begin-
ning six months from now. A bank quotes 4.3105% for the FRA, which
the corporation buys for a £1,000,000 notional principal. Suppose that six
months from now, 6-month LIBOR has indeed backed-up to 4.50%, so
the treasurer must borrow funds at 5% (LIBOR plus the 50 basis point
spread). However, offsetting this rise in rates, the corporation will receive
a settlement amount which will be the difference between the rate at
which the FRA was bought (4.3105%) and today™s 6-month LIBOR rate
(4.50%) as a percentage of the notional principal of £1,000,000. This
payment will compensate for some of the increased borrowing costs.

FRA Mechanics
In virtually every market, FRAs trade under a set of terms and conven-
tions that are identical. The British Bankers Association (BBA) has com-
piled standard legal documentation to cover FRA trading. The following
standard terms are used in the market.
223
Futures and Forward Rate Agreements



– Notional sum: The amount for which the FRA is traded.
– Trade date: The date on which the FRA is transacted.
– Settlement date: The date on which the notional loan or deposit of
funds becomes effective, that is, is said to begin. This date is used, in
conjunction with the notional sum, for calculation purposes only as no
actual loan or deposit takes place.
– Fixing date: This is the date on which the reference rate is determined,
that is, the rate to which the FRA rate is compared.
– Maturity date: The date on which the notional loan or deposit expires.
– Contract period: The time between the settlement date and maturity
date.
– FRA rate: The interest rate at which the FRA is traded.
– Reference rate: This is the rate used as part of the calculation of the set-
tlement amount, usually the Libor rate on the ¬xing date for the con-
tract period in question.
– Settlement sum: The amount calculated as the difference between the
FRA rate and the reference rate as a percentage of the notional sum,
paid by one party to the other on the settlement date.

These key dates are illustrated in Exhibit 11.9.
The spot date is usually two business days after the trade date, how-
ever it can by agreement be sooner or later than this. The settlement date
will be the time period after the spot date referred to by the FRA terms:
for example a 1—4 FRA will have a settlement date one calendar month
after the spot date. The ¬xing date is usually two business days before
the settlement date. The settlement sum is paid on the settlement date,
and as it refers to an amount over a period of time that is paid up front
(i.e., at the start of the contract period), the calculated sum is a dis-
counted present value. This is because a normal payment of interest on a
loan/deposit is paid at the end of the time period to which it relates;
because an FRA makes this payment at the start of the relevant period,
the settlement amount is a discounted present value sum. With most FRA
trades, the reference rate is the LIBOR setting on the ¬xing date.

EXHIBIT 11.9 Key Dates in a FRA Trade
224 THE GLOBAL MONEY MARKETS



The settlement sum is calculated after the ¬xing date, for payment
on the settlement date. We can illustrate this with a hypothetical exam-
ple. Consider a case where a corporation has bought £1 million notional
of a 1—4 FRA, and transacted at 5.75%, and that the market rate is
6.50% on the ¬xing date. The contract period is 90 days. In the cash
market the extra interest charge that the corporate would pay is a sim-
ple interest calculation, and is:

6.50 “ 5.75
extra interest charge = ---------------------------- — £1,000,000 — ( 91 „ 365 ) = £1,869.86
-
100

Note that in the U.S. money market, a 360 day year is assumed rather
than the 365 day year used in the UK money market.
This extra interest that the corporation is facing would be payable
with the interest payment for the loan, which (as it is a money market
loan) is paid when the loan matures. Under a FRA then, the settlement
sum payable should, if it was paid on the same day as the cash market
interest charge, be exactly equal to this. This would make it a perfect
hedge. As we noted above though, FRA settlement value is paid at the
start of the contract period, that is, the beginning of the underlying loan
and not the end. Therefore, the settlement sum has to be adjusted to
account for this, and the amount of the adjustment is the value of the
interest that would be earned if the unadjusted cash value were invested
for the contract period in the money market. The settlement value is
given by the following expression:

( r ref “ r FRA ) — M — ( n „ B )
settlement value = ---------------------------------------------------------------
-
1 + [ r ref — ( n „ B ) ]


where
rref = the reference interest ¬xing rate
rFRA = the FRA rate or contract rate
M = the notional value
n = the number of days in the contract period
B = the day-count basis (360 or 365).

The expression for the settlement value above simply calculates the
extra interest payable in the cash market, resulting from the difference
between the two interest rates, and then discounts the amount because it
is payable at the start of the period and not, as would happen in the
cash market, at the end of the period.
225
Futures and Forward Rate Agreements



In our hypothetical illustration, as the ¬xing rate is higher than the
contract rate, the buyer of the FRA receives the settlement sum from the
seller. This payment compensates the buyer for the higher borrowing
costs that they would have to pay in the cash market. If the ¬xing rate
had been lower than 5.75%, the buyer would pay the difference to the
seller, because the cash market rates will mean that they are subject to a
lower interest rate in the cash market. What the FRA has done is hedge
the interest rate exposure, so that whatever happens in the market, the
buyer will pay 5.75% on its borrowing.
A market maker in FRAs is trading short-term interest rates. The
settlement sum is the value of the FRA. The concept is exactly as with
trading short-term interest-rate futures; a trader who buys a FRA is run-
ning a long position, so that if on the ¬xing date the reference rate is
greater than the contract rate then the settlement sum is positive and the
trader realizes a pro¬t. What has happened is that the trader, by buying
the FRA, “borrowed” money at the FRA rate, which subsequently rose.
This is a gain, exactly like a short position in an interest rate futures
contract, where if the price goes down (that is, interest rates go up), the
trader realizes a gain. Conversely, a “short” position in a FRA which is
accomplished by selling a FRA realizes a gain if on the ¬xing date the
reference rate is less than the FRA rate.

FRA Pricing
FRAs are forward rate instruments and are priced using standard for-
ward rate principles.3 Consider an investor who has two alternatives,
either a 6-month investment at 5% or a 1-year investment at 6%. If the
investor wishes to invest for six months and then rollover the invest-
ment for a further six months, what rate is required for the rollover
period such that the ¬nal return equals the 6% available from the 1-year
investment? If we view a FRA rate as the break-even forward rate
between the two periods, we simply solve for this forward rate and that
is our approximate FRA rate.
In practice, FRAs are priced off the exchange-traded short-term
interest rate futures for that currency. For this reason, the contract rates
(FRA rates) for FRAs are possibly the most liquid and transparent of
any non-exchange-traded derivative instrument. To illustrate the pricing
of FRAs, we will assume that



3
For a discussion of these principles, see Frank J. Fabozzi and Steven V. Mann, In-
troduction to Fixed-Income Analytics (New Hope, PA: Frank J. Fabozzi Associates,
2001)
226 THE GLOBAL MONEY MARKETS



– the FRAs start today, January 1 of year 1 (FRA settlement date)
– the reference rate is LIBOR
– today 3-month LIBOR is 4.05%

Exhibit 11.10 presents the information that we will utilize in the FRA
pricing. We will in an analogous manner as when we determined the
future ¬‚oating-rate payments in a swap contract in the next chapter.
Shown in Column (1) is when the quarter begins and in Column (2) when
the quarter ends in year 1. Column (3) lists the number of days in each
quarter. Column (4) shows the current value of 3-month LIBOR. Column
(5) contains the prices of 3-month Eurodollar CD futures contracts used
to determine the implied 3-LIBOR forward rates in Column (6). Lastly,
Column (7) contains the forward rate for the period that we will refer to
as the period forward rate. The period forward rate is computed using the
following formula:

period forward rate = annual forward rate — (days in period/360)

For example, the annual forward rate for the second quarter is
4.15%. The period forward rate for quarter 2 is:

period forward rate = 4.15% — (91/360) = 1.0490%

Using the information presented above, let™s illustrate the pricing of a
3v9 FRA. Simply put, using the forward rates implied by the Eurodollar
CD futures contracts, we are asking what is the annualized implied 6-
month LIBOR forward rate three months hence. Accordingly, the 3v9
FRA price is calculated as follows:

[(1.010490)(1.011628) ’ 1](360/183) = 0.043751 = 4.3751%

EXHIBIT 11.10 Calculating the Implied Forward Rates

(1) (2) (3) (4) (5) (6) (7)
Number of Current Eurodollar Period
Quarter Quarter days in 3-month CD futures Forward forward
starts ends quarter LIBOR price rate rate

Jan 1 year 1 Mar 1 year 1 90 4.05% ” 1.0125%
Apr 1 year 1 June 30 year1 91 95.85 4.15% 1.0490%
July 1 year 1 Sept 30 year 1 92 95.45 4.55% 1.1628%
Oct 1 year 1 Dec 31 year 1 92 95.28 4.72% 1.2062%
227
Futures and Forward Rate Agreements



A couple of points should be noted here. First, in the U.S. money
markets an Actual/360, day count convention is used but in the UK the
day count convention is Actual/365. Second, in the calculation, the 183
days is the length of the 6-month period beginning three months from
now.
By the same reasoning, we can price a 3v12 FRA. In this illustra-
tion, we are calculating the implied 9-month forward rate (annualized)
three months hence. The price of a 3v12 is calculate as follows:

[(1.010490)(1.011628)(1.012062) ’ 1](360/275)
= 0.045256 = 4.5256%

Exhibit 11.11, Panels A, B, and C present three Bloomberg screens
of bid/ask rates for FRAs for various maturities and currencies. These
data are supplied to Bloomberg by Tullett and Tokyo Forex Interna-
tional. The currencies are U.S. dollars, sterling, and euros, respectively.


EXHIBIT 11.11 FRA Rates for Various Maturities and Currencies
Panel A: U.S. Dollar FRAs
228 THE GLOBAL MONEY MARKETS



EXHIBIT 11.11 (Continued)
Panel B: Sterling FRAs




Panel C: Euro FRAs




Source: Bloomberg Financial Markets
12
CHAPTER

Swaps and Caps/Floors



n addition to interest rate futures and FRAs, there are two additional
I derivative instruments used by money market participants to control
their exposure to interest rate risk”swaps and caps/¬‚oors. These instru-
ments have an important feature in common. Namely, both swaps and
caps/¬‚oors are combinations of more basic derivative instruments. A
swap is a portfolio of forward contracts; caps/¬‚oors are portfolios of
options on interest rates.
The most prevalent swap contract is an interest rate swap. An interest
rate swap contract provides a vehicle for market participants to transform
the nature of cash ¬‚ows and the interest rate exposure of a portfolio or
balance sheet. In this chapter, we explain how to analyze interest rate
swaps. We will describe a generic interest rate swap, the parties to a swap,
the risk and return of a swap, and the economic interpretation of a swap.
Then we look at how to compute the ¬‚oating-rate payments and calculate
the present value of these payments. Next we will see how to calculate the
¬xed-rate payments given the swap rate. Before we look at how to calcu-
late the value of a swap, we will see how to calculate the swap rate. Given
the swap rate, we will then see how the value of a swap is determined
after the inception of a swap. We will also discuss other types of swaps,
options on swaps called swaptions, and swap futures contracts. The ¬nal
section of the chapter introduces caps and ¬‚oors.



DESCRIPTION OF AN INTEREST RATE SWAP
In an interest rate swap, two parties (called counterparties) agree to
exchange periodic interest payments. The dollar amount of the interest
payments exchanged is based on some predetermined dollar principal,

229
230 THE GLOBAL MONEY MARKETS



which is called the notional amount. The dollar amount each counter-
party pays to the other is the agreed-upon periodic interest rate times the
notional amount. The only dollars that are exchanged between the parties
are the interest payments, not the notional amount. Accordingly, the
notional principal serves only as a scale factor to translate an interest rate
into a cash ¬‚ow. In the most common type of swap, one party agrees to
pay the other party ¬xed interest payments at designated dates for the life
of the contract. This party is referred to as the ¬xed-rate payer. The other
party, who agrees to make interest rate payments that ¬‚oat with some ref-
erence rate, is referred to as the ¬‚oating-rate payer.
The reference rates that have been used for the ¬‚oating rate in an
interest rate swap are various money market rates: Treasury bill rate, the
London interbank offered rate, commercial paper rate, bankers acceptan-
ces rate, certi¬cates of deposit rate, the federal funds rate, and the prime
rate. The most common is the London interbank offered rate (LIBOR).
LIBOR is the rate at which prime banks offer to pay on Eurodollar depos-
its available to other prime banks for a given maturity. There is not just
one rate but a rate for different maturities. For example, there is a 1-
month LIBOR, 3-month LIBOR, and 6-month LIBOR.
To illustrate an interest rate swap, suppose that for the next ¬ve years
party X agrees to pay party Y 10% per year, while party Y agrees to pay
party X 6-month LIBOR (the reference rate). Party X is a ¬xed-rate
payer/¬‚oating-rate receiver, while party Y is a ¬‚oating-rate payer/¬xed-
rate receiver. Assume that the notional amount is $50 million, and that
payments are exchanged every six months for the next ¬ve years. This
means that every six months, party X (the ¬xed-rate payer/¬‚oating-rate
receiver) will pay party Y $2.5 million (10% times $50 million divided by
2). The amount that party Y (the ¬‚oating-rate payer/¬xed-rate receiver)
will pay party X will be 6-month LIBOR times $50 million divided by 2.
If 6-month LIBOR is 7%, party Y will pay party X $1.75 million (7%
times $50 million divided by 2). Note that we divide by two because one-
half year™s interest is being paid.
Interest rate swaps are over-the-counter instruments. This means that
they are not traded on an exchange. An institutional investor wishing to
enter into a swap transaction can do so through either a securities ¬rm or
a commercial bank that transacts in swaps.1 These entities can do one of
the following. First, they can arrange or broker a swap between two par-


1
Do not get confused here about the role of commercial banks. A bank can use a
swap in its asset/liability management. Or, a bank can transact (buy and sell) swaps
to clients to generate fee income. It is in the latter sense that we are discussing the
role of a commercial bank in the swap market here.
231
Swaps and Caps/Floors



ties that want to enter into an interest rate swap. In this case, the securi-
ties ¬rm or commercial bank is acting in a brokerage capacity.
The second way in which a securities ¬rm or commercial bank can get
an institutional investor into a swap position is by taking the other side of
the swap. This means that the securities ¬rm or the commercial bank is a
dealer rather than a broker in the transaction. Acting as a dealer, the secu-
rities ¬rm or the commercial bank must hedge its swap position in the
same way that it hedges its position in other securities. Also it means that
the swap dealer is the counterparty to the transaction.
The risks that the two parties take on when they enter into a swap is
that the other party will fail to ful¬ll its obligations as set forth in the
swap agreement. That is, each party faces default risk. The default risk in
a swap agreement is called counterparty risk. In any agreement between
two parties that must perform according to the terms of a contract, coun-
terparty risk is the risk that the other party will default. With futures and
exchange-traded options the counterparty risk is the risk that the clear-
inghouse will default. Market participants view this risk as small. In con-
trast, counterparty risk in a swap can be signi¬cant.
Because of counterparty risk, not all securities ¬rms and commercial
banks can be swap dealers. Several securities ¬rms have established sub-
sidiaries that are separately capitalized so that they have a high credit rat-
ing which permit them to enter into swap transactions as a dealer.
Thus, it is imperative to keep in mind that any party who enters into
a swap is subject to counterparty risk.



INTERPRETING A SWAP POSITION
There are two ways that a swap position can be interpreted: (1) a package
of forward/futures contracts and (2) a package of cash ¬‚ows from buying
and selling cash market instruments.

Package of Forward Contracts
Consider the hypothetical interest rate swap used earlier to illustrate a
swap. Let™s look at party X™s position. Party X has agreed to pay 10%
and receive 6-month LIBOR. More speci¬cally, assuming a $50 million
notional amount, X has agreed to buy a commodity called “6-month
LIBOR” for $2.5 million. This is effectively a 6-month forward contract
where X agrees to pay $2.5 million in exchange for delivery of 6-month
LIBOR. The ¬xed-rate payer is effectively long a 6-month forward con-
tract on 6-month LIBOR. The ¬‚oating-rate payer is effectively short a 6-
232 THE GLOBAL MONEY MARKETS



month forward contract on 6-month LIBOR. There is therefore an
implicit forward contract corresponding to each exchange date.
Consequently, interest rate swaps can be viewed as a package of more
basic interest rate derivative instruments”forwards. The pricing of an
interest rate swap will then depend on the price of a package of forward
contracts with the same settlement dates in which the underlying for the
forward contract is the same reference rate.
While an interest rate swap may be nothing more than a package of
forward contracts, it is not a redundant contract for several reasons. First,
maturities for forward or futures contracts do not extend out as far as
those of an interest rate swap; an interest rate swap with a term of 15
years or longer can be obtained. Second, an interest rate swap is a more
transactionally ef¬cient instrument. By this we mean that in one transac-
tion an entity can effectively establish a payoff equivalent to a package of
forward contracts. The forward contracts would each have to be negoti-
ated separately. Third, the interest rate swap market has grown in liquid-
ity since its establishment in 1981; interest rate swaps now provide more
liquidity than forward contracts, particularly long-dated (i.e., long-term)
forward contracts.

Package of Cash Market Instruments
To understand why a swap can also be interpreted as a package of cash
market instruments, consider an investor who enters into the transaction
below:

– buy $50 million par value of a 5-year ¬‚oating-rate bond that pays 6-
month LIBOR every six months
– ¬nance the purchase by borrowing $50 million for ¬ve years at a 10%
annual interest rate paid every six months.

The cash ¬‚ows for this transaction are set forth in Exhibit 12.1. The sec-
ond column of the exhibit shows the cash ¬‚ows from purchasing the 5-
year ¬‚oating-rate bond. There is a $50 million cash outlay and then ten
cash in¬‚ows. The amount of the cash in¬‚ows is uncertain because they
depend on future levels of 6-month LIBOR. The next column shows the
cash ¬‚ows from borrowing $50 million on a ¬xed-rate basis. The last col-
umn shows the net cash ¬‚ows from the entire transaction. As the last col-
umn indicates, there is no initial cash ¬‚ow (the cash in¬‚ow and cash
outlay offset each other). In all ten 6-month periods, the net position
results in a cash in¬‚ow of LIBOR and a cash outlay of $2.5 million. This
net position, however, is identical to the position of a ¬xed-rate payer/
¬‚oating-rate receiver.
233
Swaps and Caps/Floors



EXHIBIT 12.1 Cash Flows for the Purchase of a 5-Year Floating-Rate Bond
Financed by Borrowing on a Fixed-Rate Basis
Transaction:
– Purchase for $50 million a 5-year ¬‚oating-rate bond:
floating rate = LIBOR, semiannual pay
– Borrow $50 million for ¬ve years:
fixed rate = 10%, semiannual payments

Cash Flow (In Millions of Dollars) From:
Six Month
Floating-rate Bond a
Period Borrowing Cost Net

’$50
0 +$50.0 $0
+ (LIBOR1/2) — 50 ’2.5 + (LIBOR1/2) — 50 ’ 2.5
1
+ (LIBOR2/2) — 50 ’2.5 + (LIBOR2/2) — 50 ’ 2.5
2
+ (LIBOR3/2) — 50 ’2.5 + (LIBOR3/2) — 50 ’ 2.5
3
+ (LIBOR4/2) — 50 ’2.5 + (LIBOR4/2) — 50 ’ 2.5
4
+ (LIBOR5/2) — 50 ’2.5 + (LIBOR5/2) — 50 ’ 2.5
5
+ (LIBOR6/2) — 50 ’2.5 + (LIBOR6/2) — 50 ’ 2.5
6
+ (LIBOR7/2) — 50 ’2.5 + (LIBOR7/2) — 50 ’ 2.5
7
+ (LIBOR8/2) — 50 ’2.5 + (LIBOR8/2) — 50 ’ 2.5
8
+ (LIBOR9/2) — 50 ’2.5 + (LIBOR9/2) — 50 ’ 2.5
9
+ (LIBOR10/2) — 50 + 50 ’52.5 + (LIBOR10/2) — 50 ’ 2.5
10
a
The subscript for LIBOR indicates the 6-month LIBOR as per the terms of the float-
ing-rate bond at time t.

It can be seen from the net cash ¬‚ow in Exhibit 12.1 that a ¬xed-rate
payer has a cash market position that is equivalent to a long position in a
¬‚oating-rate bond and a short position in a ¬xed-rate bond”the short
position being the equivalent of borrowing by issuing a ¬xed-rate bond.
What about the position of a ¬‚oating-rate payer? It can be easily
demonstrated that the position of a ¬‚oating-rate payer is equivalent to
purchasing a ¬xed-rate bond and ¬nancing that purchase at a ¬‚oating-
rate, where the ¬‚oating rate is the reference rate for the swap. That is, the
position of a ¬‚oating-rate payer is equivalent to a long position in a ¬xed-
rate bond and a short position in a ¬‚oating-rate bond.



TERMINOLOGY, CONVENTIONS, AND MARKET QUOTES
Here we review some of the terminology used in the swaps market and
explain how swaps are quoted. The trade date for a swap is not surpris-
ingly, the date on which the swap is transacted. The terms of the trade
234 THE GLOBAL MONEY MARKETS



include the ¬xed interest rate, the maturity, the notional amount of the
swap, and the payment bases of both legs of the swap. The date from
which ¬‚oating interest payments are determined is the reset or setting date,
which may also be the trade date. In the same way as for FRAs (discussed
in the previous chapter), the rate is ¬xed two business days before the
interest period begins. The second (and subsequent) reset date will be two
business days before the beginning of the second (and subsequent) swap
periods. The effective date is the date from which interest on the swap is
calculated, and this is typically two business days after the trade date. In a
forward-start swap the effective date will be at some point in the future,
speci¬ed in the swap terms. The ¬‚oating-interest rate for each period is
¬xed at the start of the period, so that the interest payment amount is
known in advance by both parties (the ¬xed rate is known of course,
throughout the swap by both parties).
While our illustrations assume that the timing of the cash ¬‚ows for
both the ¬xed-rate payer and ¬‚oating-rate payer will be the same, this is
rarely the case in a swap. An agreement may call for the ¬xed-rate payer
to make payments annually but the ¬‚oating-rate payer to make payments
more frequently (semiannually or quarterly). Also, the way in which
interest accrues on each leg of the transaction differs. Normally, the ¬xed
interest payments are paid on the basis of a 30/360 day count which is
described in Chapter 2. Floating-rate payments for dollar and euro-
denominated swaps use an Actual/360 day count similar to other money
market instruments in those currencies. Sterling-denominated swaps use
an Actual/365 day count.
Accordingly, the ¬xed interest payments will differ slightly owing to
the differences in the lengths of successive coupon periods. The ¬‚oating
payments will differ owing to day counts as well as movements in the ref-
erence rate.
The terminology used to describe the position of a party in the swap
markets combines cash market jargon and futures market jargon, given
that a swap position can be interpreted either as a position in a package
of cash market instruments or a package of futures/forward positions. As
we have said, the counterparty to an interest rate swap is either a ¬xed-
rate payer or ¬‚oating-rate payer.
The ¬xed-rate payer receives ¬‚oating-rate interest and is said to be
"long" or to have "bought" the swap. The long side has conceptually
purchased a ¬‚oating-rate note (because it receives ¬‚oating-rate interest)
and issued a ¬xed coupon bond (because it pays out ¬xed interest at peri-
odic intervals). In essence, the ¬xed-rate payer is borrowing at ¬xed-rate
and investing in a ¬‚oating-rate asset. The ¬‚oating-rate payer is said to be
"short" or to have "sold" the swap. The short side has conceptually pur-
chased a coupon bond (because it receives ¬xed-rate interest) and issued a
235
Swaps and Caps/Floors



¬‚oating-rate note (because it pays ¬‚oating-rate interest). A ¬‚oating-rate
payer is borrowing at ¬‚oating rate and investing in a ¬xed rate asset.
The convention that has evolved for quoting swaps levels is that a
swap dealer sets the ¬‚oating rate equal to the reference rate and then
quotes the ¬xed rate that will apply. To illustrate this convention, con-
sider the following 10-year swap terms available from a dealer:

– Floating-rate payer:
Pay ¬‚oating rate of 3-month LIBOR quarterly.
Receive ¬xed rate of 8.75% semiannually.
– Fixed-rate payer:
Pay ¬xed rate of 8.85% semiannually
Receive ¬‚oating rate of 3-month LIBOR quarterly.

The offer price that the dealer would quote the ¬xed-rate payer
would be to pay 8.85% and receive LIBOR “¬‚at.” (The word ¬‚at means
with no spread.) The bid price that the dealer would quote the ¬‚oating-
rate payer would be to pay LIBOR ¬‚at and receive 8.75%. The bid-offer
spread is 10 basis points.
In order to solidify our intuition, it is useful to think of the swap mar-
ket as a market where two counterparties trade the ¬‚oating reference rate
in a series of exchanges for a ¬xed price. In effect, the swap market is a
market to buy and sell LIBOR. So, buying a swap (pay ¬xed/receive ¬‚oat-
ing) can be thought of as buying LIBOR on each reset date for the ¬xed
rate agreed to on the trade date. Conversely, selling a swap (receive ¬xed/
pay ¬‚oating) is effectively selling LIBOR on each reset date for a ¬xed rate
agreed to on the trade date. In this framework, a dealer™s bid-offer spread
can be easily interpreted. Using the numbers presented above, the bid price
of 8.75% is the price the dealer will pay to the counterparty to receive 3-
month LIBOR. In other words, buy LIBOR at the bid. Similarly, the offer
price of 8.85% is the price the dealer receives from the counterparty in
exchange for 3-month LIBOR. In other words, sell LIBOR at the offer.
The ¬xed rate is some spread above the Treasury yield curve with the
same term to maturity as the swap. In our illustration, suppose that the
10-year Treasury yield is 8.35%. Then the offer price that the dealer
would quote to the ¬xed-rate payer is the 10-year Treasury rate plus 50
basis points versus receiving LIBOR ¬‚at. For the ¬‚oating-rate payer, the
bid price quoted would be LIBOR ¬‚at versus the 10-year Treasury rate
plus 40 basis points. The dealer would quote such a swap as 40-50,
meaning that the dealer is willing to enter into a swap to receive LIBOR
and pay a ¬xed rate equal to the 10-year Treasury rate plus 40 basis
points; and he or she would be willing to enter into a swap to pay LIBOR
and receive a ¬xed rate equal to the 10-year Treasury rate plus 50 basis
236 THE GLOBAL MONEY MARKETS



points. The difference between the Treasury rate paid and received is the
bid-offer spread.2


VALUING INTEREST RATE SWAPS
In an interest rate swap, the counterparties agree to exchange periodic
interest payments. The dollar amount of the interest payments exchanged
is based on the notional principal. In the most common type of swap,
there is a ¬xed-rate payer and a ¬xed-rate receiver. The convention for
quoting swap rates is that a swap dealer sets the ¬‚oating rate equal to the
reference rate and then quotes the ¬xed rate that will apply.

Computing the Payments for a Swap
In the previous section we described in general terms the payments by
the ¬xed-rate payer and ¬xed-rate receiver but we did not give any
details. That is, we explained that if the swap rate is 6% and the
notional amount is $100 million, then the ¬xed-rate payment will be $6
million for the year and the payment is then adjusted based on the fre-
quency of settlement. So, if settlement is semiannual, the payment is $3
million. If it is quarterly, it is $1.5 million. Similarly, the ¬‚oating-rate
payment would be found by multiplying the reference rate by the
notional amount and then scaling based on the frequency of settlement.
It was useful to illustrate the basic features of an interest rate swap
with simple calculations for the payments such as described above and
then explain how the parties to a swap either bene¬t or hurt when inter-
est rates change. However, we will show how to value a swap in this
section. To value a swap, it is necessary to determine both the present
value of the ¬xed-rate payments and the present value of the ¬‚oating-
rate payments. The difference between these two present values is the
value of a swap. As will be explained below, whether the value is posi-
tive (i.e., an asset) or negative (i.e., a liability) will depend on the party.
At the inception of the swap, the terms of the swap will be such that
the present value of the ¬‚oating-rate payments is equal to the present
value of the ¬xed-rate payments. That is, the value of the swap is equal to
zero at its inception. This is the fundamental principle in determining the
swap rate (i.e., the ¬xed rate that the ¬xed-rate payer will make).

2
A question that commonly arises is why is the fixed rate of a swap is quoted as a
fixed spread above a Treasury rate when Treasury rates are not used directly in swap
valuation? Because of the timing difference between the quote and settlement, quot-
ing the fixed-rate side as a spread above a Treasury rate allows the swap dealer to
hedge against changing interest rates.
237
Swaps and Caps/Floors



Here is a roadmap of the presentation. First we will look at how to
compute the ¬‚oating-rate payments. We will see how the future values of
the reference rate are determined to obtain the ¬‚oating rate for the
period. From the future values of the reference rate we will then see how
to compute the ¬‚oating-rate payments taking into account the number of
days in the payment period. Next we will see how to calculate the ¬xed-
rate payments given the swap rate. Before we look at how to calculate the
value of a swap, we will see how to calculate the swap rate. This will
require an explanation of how the present value of any cash ¬‚ow in an
interest rate swap is computed. Given the ¬‚oating-rate payments and the
present value of the ¬‚oating-rate payments, the swap rate can be deter-
mined by using the principle that the swap rate is the ¬xed rate that will
make the present value of the ¬xed-rate payments equal to the present
value of the ¬‚oating-rate payments. Finally, we will see how the value of
swap is determined after the inception of a swap.

Calculating the Floating-Rate Payments
For the ¬rst ¬‚oating-rate payment, the amount is known. For all subse-
quent payments, the ¬‚oating-rate payment depends on the value of the
reference rate when the ¬‚oating rate is determined. To illustrate the issues
associated with calculating the ¬‚oating-rate payment, we will assume that

– a swap starts today, January 1 of year 1(swap settlement date)
– the ¬‚oating-rate payments are made quarterly based on “actual/360”
– the reference rate is 3-month LIBOR
– the notional amount of the swap is $100 million
– the term of the swap is three years

The quarterly ¬‚oating-rate payments are based on an “actual/360”
day count convention. Recall that this convention means that 360 days
are assumed in a year and that in computing the interest for the quarter,
the actual number of days in the quarter is used. The ¬‚oating-rate pay-
ment is set at the beginning of the quarter but paid at the end of the quar-
ter”that is, the ¬‚oating-rate payments are made in arrears.
Suppose that today 3-month LIBOR is 4.05%. Let™s look at what the
¬xed-rate payer will receive on March 31 of year 1”the date when the
¬rst quarterly swap payment is made. There is no uncertainty about what
the ¬‚oating-rate payment will be. In general, the ¬‚oating-rate payment is
determined as follows:

no. of days in period
notional amount — ( 3-month LIBOR ) — ----------------------------------------------------
-
360
238 THE GLOBAL MONEY MARKETS



In our illustration, assuming a non-leap year, the number of days from
January 1 of year 1 to March 31 of year 1 (the ¬rst quarter) is 90. If 3-
month LIBOR is 4.05%, then the ¬xed-rate payer will receive a ¬‚oating-
rate payment on March 31 of year 1 equal to:

90-
$100,000,000 — 0.0405 — --------- = $1,012,500
360

Now the dif¬culty is in determining the ¬‚oating-rate payment after
the ¬rst quarterly payment. That is, for the 3-year swap there will be 12
quarterly ¬‚oating-rate payments. So, while the ¬rst quarterly payment is
known, the next 11 are not. However, there is a way to hedge the next
11 ¬‚oating-rate payments by using a futures contract. Speci¬cally, the
futures contract used to hedge the future ¬‚oating-rate payments in a
swap whose reference rate is 3-month LIBOR is the Eurodollar CD
futures contract.

Determining Future Floating-Rate Payments
Now let™s determine the future ¬‚oating-rate payments. These payments
can be locked in over the life of the swap using the Eurodollar CD futures
contract. We will show how these ¬‚oating-rate payments are computed
using this contract.
We will begin with the next quarterly payment”from April 1 of year
1 to June 30 of year 1. This quarter has 91 days. The ¬‚oating-rate pay-
ment will be determined by 3-month LIBOR on April 1 of year 1 and paid
on June 30 of year 1. Where might the ¬xed-rate payer look to today
(January 1 of year 1) to project what 3-month LIBOR will be on April 1
of year 1? One possibility is the Eurodollar CD futures market. There is a
3-month Eurodollar CD futures contract for settlement on June 30 of
year 1. That futures contract will express the market™s expectation of 3-
month LIBOR on April 1 of year 1. For example, if the futures price for
the 3-month Eurodollar CD futures contract that settles on June 30 of
year 1 is 95.85, then as explained above, the 3-month Eurodollar futures
rate is 4.15%. We will refer to that rate for 3-month LIBOR as the “for-
ward rate.” Therefore, if the ¬xed-rate payer bought 100 of these 3-
month Eurodollar CD futures contracts on January 1 of year 1 (the incep-
tion of the swap) that settle on June 30 of year 1, then the payment that
will be locked in for the quarter (April 1 to June 30 of year 1) is

91
$100,000,000 — 0.0415 — --------- = $1,049,028
-
360
239
Swaps and Caps/Floors



EXHIBIT 12.2 Floating-Rate Payments Based on Initial LIBOR and
Eurodollar CD Futures

(1) (2) (3) (4) (5) (6) (7) (8)

Number of Current Eurodollar Period = Floating-rate
Quarter Quarter days in 3-month CD futures Forward End of payment at
starts ends quarter LIBOR price rate quarter end of quarter

Jan 1 year 1 Mar 31 year 1 90 4.05% ” 1 1,012,500
Apr 1 year 1 June 30 year 1 91 95.85 4.15% 2 1,049,028
July 1 year 1 Sept 30 year 1 92 95.45 4.55% 3 1,162,778
Oct 1 year 1 Dec 31 year 1 92 95.28 4.72% 4 1,206,222
Jan 1 year 2 Mar 31 year 2 90 95.10 4.90% 5 1,225,000
Apr 1 year 2 June 30 year 2 91 94.97 5.03% 6 1,271,472
July 1 year 2 Sept 30 year 2 92 94.85 5.15% 7 1,316,111
Oct 1 year 2 Dec 31 year 2 92 94.75 5.25% 8 1,341,667
Jan 1 year 3 Mar 31 year 3 90 94.60 5.40% 9 1,350,000
Apr 1 year 3 June 30 year 3 91 94.50 5.50% 10 1,390,278
July 1 year 3 Sept 30 year 3 92 94.35 5.65% 11 1,443,889
Oct 1 year 3 Dec 31 year 3 92 94.24 5.76% 12 1,472,000


(Note that each futures contract is for $1 million and hence 100 con-
tracts have a notional amount of $100 million.) Similarly, the Eurodollar
CD futures contract can be used to lock in a ¬‚oating-rate payment for
each of the next 10 quarters.3 Once again, it is important to emphasize
that the reference rate at the beginning of period t determines the ¬‚oating-
rate that will be paid for the period. However, the ¬‚oating-rate payment is
not made until the end of period t.
Exhibit 12.2 shows this for the 3-year swap. Shown in Column (1) is
when the quarter begins and in Column (2) when the quarter ends. The
payment will be received at the end of the ¬rst quarter (March 31 of year 1)
and is $1,012,500. That is the known ¬‚oating-rate payment as explained
earlier. It is the only payment that is known. The information used to com-
pute the ¬rst payment is in Column (4) which shows the current 3-month
LIBOR (4.05%). The payment is shown in the last column, Column (8).
Notice that Column (7) numbers the quarters from 1 through 12.
Look at the heading for Column (7). It identi¬es each quarter in terms of
the end of the quarter. This is important because we will eventually be

3
The Chicago Mercantile Exchange offers pre-packaged series of Eurodollar CD fu-
tures contracts that expire on consecutive dates called bundles. Specifically, a bundle
is the simultaneous sale or purchase of one of each of a consecutive series of Euro-
dollar CD futures contracts. So, rather than construct the same positions with indi-
vidual contracts, a series of contracts can be sold or purchased in a single transaction.
240 THE GLOBAL MONEY MARKETS



discounting the payments (cash ¬‚ows). We must take care to understand
when each payment is to be exchanged in order to properly discount. So,
for the ¬rst payment of $1,012,500 it is going to be received at the end of
quarter 1. When we refer to the time period for any payment, the refer-
ence is to the end of quarter. So, the ¬fth payment of $1,225,000 would
be identi¬ed as the payment for period 5, where period 5 means that it
will be exchanged at the end of the ¬fth quarter.

Calculating the Fixed-Rate Payments
The swap will specify the frequency of settlement for the ¬xed-rate pay-
ments. The frequency need not be the same as the ¬‚oating-rate payments.
For example, in the 3-year swap we have been using to illustrate the cal-
culation of the ¬‚oating-rate payments, the frequency is quarterly. The fre-
quency of the ¬xed-rate payments could be semiannual rather than
quarterly.
In our illustration we will assume that the frequency of settlement is
quarterly for the ¬xed-rate payments, the same as with the ¬‚oating-rate
payments. The day count convention is the same as for the ¬‚oating-rate
payment, “actual/360”. The equation for determining the dollar amount
of the ¬xed-rate payment for the period is:

no. of days in period
notional amount — ( swap rate ) — ----------------------------------------------------
-
360

It is the same equation as for determining the ¬‚oating-rate payment
except that the swap rate is used instead of the reference rate (3-month
LIBOR in our illustration).
For example, suppose that the swap rate is 4.98% and the quarter
has 90 days. Then the ¬xed-rate payment for the quarter is:

90
$100,000,000 — 0.0498 — --------- = $1,245,000
-
360

If there are 92 days in a quarter, the ¬xed-rate payment for the quarter is:

92-
$100,000,000 — 0.0498 — --------- = $1,272, 667
360

Note that the rate is ¬xed for each quarter but the dollar amount of the
payment depends on the number of days in the period.
241
Swaps and Caps/Floors



Exhibit 12.3 shows the ¬xed-rate payments based on different assumed
values for the swap rate. The ¬rst three columns of the exhibit show the
same information as in Exhibit 12.2”the beginning and end of the quarter
and the number of days in the quarter. Column (4) simply uses the notation
for the period. That is, period 1 means the end of the ¬rst quarter, period 2
means the end of the second quarter, and so on. The other columns of the
exhibit show the payments for each assumed swap rate.

Calculation of the Swap Rate
Now that we know how to calculate the payments for the ¬xed-rate and
¬‚oating-rate sides of a swap where the reference rate is 3-month LIBOR
given (1) the current value for 3-month LIBOR, (2) the expected 3-month
LIBOR from the Eurodollar CD futures contract, and (3) the assumed
swap rate, we can demonstrate how to compute the swap rate.
At the initiation of an interest rate swap, the counterparties are agree-
ing to exchange future payments and no upfront payments are made by
either party. This means that the swap terms must be such that the present
value of the payments to be made by the counterparties must be at least
equal to the present value of the payments that will be received. In fact, to
eliminate arbitrage opportunities, the present value of the payments made
by a party will be equal to the present value of the payments received by
that same party. The equivalence (or no arbitrage) of the present value of
the payments is the key principle in calculating the swap rate.
Since we will have to calculate the present value of the payments, let™s
show how this is done.

Calculating the Present Value of the Floating-Rate Payments
As explained earlier, we must be careful about how we compute the
present value of payments. In particular, we must carefully specify (1) the
timing of the payment and (2) the interest rates that should be used to dis-
count the payments. We have already addressed the ¬rst issue. In con-
structing the exhibit for the payments, we indicated that the payments are
at the end of the quarter. So, we denoted the time periods with respect to
the end of the quarter.
Now let™s turn to the interest rates that should be used for discount-
ing. First, every cash ¬‚ow should be discounted at its own discount rate
using a spot rate. So, if we discounted a cash ¬‚ow of $1 using the spot
rate for period t, the present value would be:

$1
present value of $1 to be received in period t = ---------------------------------------------------------------------
-
t
( 1 + spot rate for period t )

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