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Chapter 29
Interest-Rate Swaps, Caps,
and Floors
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Learning Objectives
After reading this chapter, you will understand
what an interest-rate swap is the relationship
between an interest-rate swap and forward contracts
how interest-rate swap terms are quoted in the
market
how the swap rate is calculated
how the value of a swap is determined
the primary determinants of the swap rate
how a swap can be used by institutional investors
for asset/liability management
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Learning Objectives (continued)
After reading this chapter, you will understand
how a structured note is created using an interest-rate
swap
what a swaption is and how it can be used by
institutional investors
what a rate cap and floor are, and how these
agreements can be used by institutional investors
the relationship between a cap and floor and options
how to value caps and floors
how an interest-rate collar can be created
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Interest-Rate Swaps
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 a
predetermined dollar principal, which is called the notional principal
amount.
The dollar amount that each counterparty pays to the other is the
agreed-upon periodic interest rate times the notional principal
amount.
The only dollars that are exchanged between the parties are the
interest payments, not the notional principal amount.
This party is referred to as the fixed-rate payer or the floating-rate
receiver.
The other party, who agrees to make interest rate payments that float
with some reference rate, is referred to as the floating-rate payer or
fixed-rate receiver.
The frequency with which the interest rate that the floating-rate
payer must pay is called the reset frequency.
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Interest-Rate Swaps (continued)
Entering into a Swap and Counterparty Risk
Interest-rate swaps are over-the-counter instruments,
which 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 firm or a
commercial bank that transacts in swaps.
The risks that parties take on when they enter into a
swap are that the other party will fail to fulfill 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.
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Interest-Rate Swaps (continued)
Interpreting a Swap Position
There are two ways that a swap position can be interpreted:
i. as a package of forward/futures contracts
ii. as a package of cash flows from buying and selling cash market
instruments
Although an interest-rate swap may be nothing more than a package
of forward contracts, it is not a redundant contract, for several
reasons.
i. Maturities for forward or futures contracts do not extend out as far as
those of an interest-rate swap.
ii. An interest-rate swap is a more transactionally efficient instrument
because in one transaction an entity can effectively establish a payoff
equivalent to a package of forward contracts.
iii. Interest-rate swaps now provide more liquidity than forward
contracts, particularly long-dated (i.e., long-term) forward contracts.
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Interest-Rate Swaps (continued)
Interpreting a Swap Position
To understand why a swap can also be interpreted as a package of cash market
instruments, consider an investor who enters into the following transaction:
o Buy $50 million par of a five-year floating-rate bond that pays six-month
LIBOR every six months; finance the purchase by borrowing $50 million for
five years at 10% annual interest rate paid every six months.
The cash flows for this transaction are shown in Exhibit 29-1 (see Overhead 29-
8). The second column shows the cash flow from purchasing the five-year
floating-rate bond. There is a $50 million cash outlay and then 10 cash inflows.
The amount of the cash inflows is uncertain because they depend on future
LIBOR. The next column shows the cash flow from borrowing $50 million on a
fixed-rate basis. The last column shows the net cash flow from the entire
transaction. As the last column indicates, there is no initial cash flow (no cash
inflow or cash outlay). In all 10 six-month periods, the net position results in a
cash inflow of LIBOR and a cash outlay of $2.5 million. This net position,
however, is identical to the position of a fixed-rate payer/floating-rate receiver.
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Exhibit 29-1 Cash Flow for the Purchase of a Five-Year Floating-
Rate Bond Financed by Borrowing on a Fixed-Rate Basis
Transaction: Purchase for $50 million a five-year floating-rate bond: floating rate = LIBOR,
semiannual pay; borrow $50 million for five years: fixed rate = 10%, semiannual payments
Cash Flow (millions of dollars) From:
Six-Month
Period Floating-Rate Bond a Borrowing
Cost
Net
0 –$50.0 +$50 $0
1 +(LIBOR1/2)×50 –2.5 + (LIBOR1/2)×50–2.5
2 +(LIBOR2/2)×50 –2.5 + (LIBOR2/2)×50–2.5
3 +(LIBOR3/2)×50 –2.5 + (LIBOR3/2)×50–2.5
4 +(LIBOR4/2)×50 –2.5 + (LIBOR4/2)×50–2.5
5 +(LIBOR5/2)×50 –2.5 + (LIBOR5/2)×50–2.5
6 +(LIBOR6/2)×50 –2.5 + (LIBOR6/2)×50–2.5
7 +(LIBOR7/2)×50 –2.5 + (LIBOR7/2)×50–2.5
8 +(LIBOR8/2)×50 –2.5 + (LIBOR8/2)×50–2.5
9 +(LIBOR9/2)×50 –2.5 + (LIBOR9/2)×50–2.5
10 +(LIBOR10/2)×50+50 –52.5 + (LIBOR10/2)×50–2.5
a
The subscript for LIBOR indicates the six-month LIBOR as per the terms of the floating-rate bond
at time t.
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Interest-Rate Swaps (continued)
Terminology, Conventions, and Market Quotes
The date that the counterparties commit to the swap is called the
trade date.
The date that the swap begins accruing interest is called the
effective date, and the date that the swap stops accruing interest is
called the maturity date.
The convention that has evolved for quoting swaps levels is that a
swap dealer sets the floating rate equal to the index and then
quotes the fixed-rate that will apply.
o The offer price that the dealer would quote the fixed-rate payer
would be to pay 8.85% and receive LIBOR “flat” (“flat” meaning
with no spread to LIBOR).
o The bid price that the dealer would quote the floating-rate payer
would be to pay LIBOR flat and receive 8.75%.
o The bid-offer spread is 10 basis points.
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Interest-Rate Swaps (continued)
Terminology, Conventions, and Market Quotes
Another way to describe the position of the counterparties to a
swap is in terms of our discussion of the interpretation of a swap
as a package of cash market instruments.
o Fixed-rate payer: A position that is exposed to the price
sensitivities of a longer-term liability and a floating-rate bond.
o Floating-rate payer: A position that is exposed to the price
sensitivities of a fixed-rate bond and a floating-rate liability.
The convention that has evolved for quoting swaps levels is that a
swap dealer sets the floating rate equal to the index and then
quotes the fixed rate that will apply.
To illustrate this convention, consider a 10-year swap offered by a
dealer to market participants shown in Exhibit 29-2 (see
Overhead 29-12).
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Interest-Rate Swaps (continued)
Terminology, Conventions, and Market Quotes
In our illustration, suppose that the 10-year Treasury yield is 8.35%.
Then the offer price that the dealer would quote to the fixed-rate
payer is the 10-year Treasury rate plus 50 basis points versus
receiving LIBOR flat.
For the floating-rate payer, the bid price quoted would be LIBOR flat
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
fixed rate equal to the 10-year Treasury rate plus 40 basis points, and
it would be willing to enter into a swap to pay LIBOR and receive a
fixed rate equal to the 10-year Treasury rate plus 50 basis points.
The difference between the Treasury rate paid and received is the
bid-offer spread.
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Exhibit 29-2 Meaning of a “40–50” Quote for
a 10-Year Swap When Treasuries Yield 8.35%
(Bid-Offer Spread of 10 Basis Points)
Floating-Rate
Payer
Fixed-Rate
Payer
Pay Floating rate of
six-month
LIBOR
Fixed rate of
8.85%
Receive Fixed rate of
8.75%
Floating rate of
six-month
LIBOR
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Interest-Rate Swaps (continued)
Calculation of the Swap Rate
At the initiation of an interest-rate swap, the counterparties are agreeing
to exchange future interest-rate payments and no upfront payments by
either party are made.
While the payments of the fixed-rate payer are known, the floating-rate
payments are not known.
This is because they depend on the value of the reference rate at the
reset dates.
For a LIBOR-based swap, the Eurodollar CD futures contract can be
used to establish the forward (or future) rate for three-month LIBOR.
In general, the floating-rate payment is determined as follows:
floating rate payment
number of days in period
notional amount three month LIBOR
360
− =
× − ×
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Interest-Rate Swaps (continued)
Calculation of the Swap Rate
The equation for determining the dollar amount of the fixed-rate
payment for the period is:
It is the same equation as for determining the floating-rate payment
except that the swap rate is used instead of the reference rate.
Exhibit 29-4 (see Overhead 29-15) shows the fixed-rate payments
based on an assumed swap rate of 4.9875%.
o The first three columns of the exhibit show the beginning and end of the
quarter and the number of days in the quarter. Column (4) simply uses
the notation for the period.
o That is, period 1 means the end of the first quarter, period 2 means the
end of the second quarter, and so on.
o Column (5) shows the fixed-rate payments for each period based on a
swap rate of 4.9875%.
fixed rate payment
number of days in period
notional amount swap rate
360
− =
× ×
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Exhibit 29-4 Fixed-Rate Payments
Assuming a Swap Rate of 4.9875%
Quarter
Starts
Quarter Ends
Days in
Quarter
Period = End
of Quarter
Fixed-Rate Payment if Swap
Rate Is Assumed to Be 4.9875%
Jan 1 year 1 Mar 31 year 1 90 1 1,246,875
Apr 1 year 1 June 30 year 1 91 2 1,260,729
July 1 year 1 Sept 30 year 1 92 3 1,274,583
Oct 1 year 1 Dec 31 year 1 92 4 1,274,583
Jan 1 year 2 Mar 31 year 2 90 5 1,246,875
Apr 1 year 2 June 30 year 2 91 6 1,260,729
July 1 year 2 Sept 30 year 2 92 7 1,274,583
Oct 1 year 2 Dec 31 year 2 92 8 1,274,583
Jan 1 year 3 Mar 31 year 3 90 9 1,246,875
Apr 1 year 3 June 30 year 3 91 10 1,260,729
July 1 year 3 Sept 30 year 3 92 11 1,274,583
Oct 1 year 3 Dec 31 year 3 92 12 1,274,583
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Interest-Rate Swaps (continued)
Calculation of the Swap Rate
Given the swap payments, we can show how to
compute the swap rate.
At the initiation of an interest-rate swap, the counterparties are
agreeing to exchange future payments and no upfront payments
by either party are made.
This means 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.
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 of the present value of the payments is the key
principle in calculating the swap rate.
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Interest-Rate Swaps (continued)
Calculation of the Swap Rate
The present value of $1 to be received in period t is the forward discount
factor.
In calculations involving swaps, we compute the forward discount factor
for a period using the forward rates.
These are the same forward rates that are used to compute the floating-
rate payments—those obtained from the Eurodollar CD futures contract.
o We must make just one more adjustment.
o We must adjust the forward rates used in the formula for the number of
days in the period (i.e., the quarter in our illustrations) in the same way
that we made this adjustment to obtain the payments.
o Specifically, the forward rate for a period, which we will refer to as the
period forward rate, is computed using the following equation:
days in period
period forward rate annual forward rate
360
= ×
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Interest-Rate Swaps (continued)
Calculation of the Swap Rate
Given the payment for a period and the forward discount factor
for the period, the present value of the payment can be
computed.
The forward discount factor is used to compute the present value
of the both the fixed-rate payments and floating-rate payments.
Beginning with the basic relationship for no arbitrage to exist:
PV of floating-rate payments = PV of fixed-rate payments
The formula for the swap rate is derived as follows. We begin
with:
fixed-rate payment for period t
days in period
notional amount swap rate
360
=
× ×
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Interest-Rate Swaps (continued)
Calculation of the Swap Rate
The present value of the fixed-rate payment for period t is found by
multiplying the previous expression by the forward discount factor for
period t.
We have:
Summing up the present value of the fixed-rate payment for each
period gives the present value of the fixed-rate payments. Letting N be
the number of periods in the swap, we have:
present value of the fixed-rate payment for period t
days in period t
notional amount swap rate forward discount factor for period t
360
=
× × ×
present value of the fixed-rate payment
days in period t
swap rate notional amount forward discount factor for period t
360
=
× × ×∑
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Interest-Rate Swaps (continued)
Calculation of the Swap Rate
Solving for the swap rate gives
Valuing a Swap
Once the swap transaction is completed, changes in market
interest rates will change the payments of the floating-rate side
of the swap.
The value of an interest-rate swap is the difference between the
present value of the payments of the two sides of the swap.
1
N
t
=
present value of floating-rate payments
days in period t
notional amount forward discount factor for period t
360
swap rate
=
× ×∑
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Interest-Rate Swaps (continued)
Duration of a Swap
As with any fixed-income contract, the value of a swap will change
as interest rates change.
Dollar duration is a measure of the interest-rate sensitivity of a
fixed-income contract.
From the perspective of the party who pays floating and receives
fixed, the interest-rate swap position can be viewed as follows:
long a fixed-rate bond + short a floating-rate bond
This means that the dollar duration of an interest-rate swap from the
perspective of a floating-rate payer is simply the difference between
the dollar duration of the two bond positions that make up the swap;
that is,
dollar duration of a swap = dollar duration of a fixed-rate bond
– dollar duration of a floating-rate bond
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Interest-Rate Swaps (continued)
Application of a Swap to Asset/Liability
Management
An interest-rate swap can be used to alter the cash flow
characteristics of an institution’s assets so as to provide a
better match between assets and liabilities.
An interest-rate swap allows each party to accomplish its
asset/liability objective of locking in a spread.
An asset swap permits the two financial institutions to
alter the cash flow characteristics of its assets: from fixed
to floating or from floating to fixed.
A liability swap permits two institutions to change the
cash flow nature of their liabilities.
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Interest-Rate Swaps (continued)
Creation of Structured Notes Using Swaps
Corporations can customize medium-term notes for
institutional investors who want to make a market play
on interest rate, currency, and/or stock market
movements.
That is, the coupon rate on the issue will be based on
the movements of these financial variables.
A corporation can do so in such a way that it can still
synthetically fix the coupon rate.
This can be accomplished by issuing an MTN and
entering into a swap simultaneously.
MTNs created in this way are called structured MTNs.
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Interest-Rate Swaps (continued)
Primary Determinants of Swap Spreads
The swap spread is determined by the same factors that influence the
spread over Treasuries on financial instruments (futures / forward
contracts or cash) that produce a similar return or funding profile.
Given that a swap is a package of futures/forward contracts, the swap
spread can be determined by looking for futures/forward contracts
with the same risk/return profile.
A Eurodollar CD futures contract is a swap where a fixed dollar
payment (i.e., the futures price) is exchanged for three-month
LIBOR.
A market participant can synthesize a (synthetic) fixed-rate security
or a fixed-rate funding vehicle of up to five years by taking a
position in a strip of Eurodollar CD futures contracts (i.e., a position
in every three-month Eurodollar CD up to the desired maturity date).
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Interest-Rate Swaps (continued)
Primary Determinants of Swap Spreads
For swaps with maturities longer than five years, the spread is
determined primarily by the credit spreads in the corporate bond
market.
Because a swap can be interpreted as a package of long and short
positions in a fixed-rate bond and a floating-rate bond, it is the
credit spreads in those two market sectors that will be the key
determinant of the swap spread.
Boundary conditions for swap spreads based on prices for fixed-
rate and floating-rate corporate bonds can be determined.
Several technical factors, such as the relative supply of fixed-rate
and floating-rate corporate bonds and the cost to dealers of
hedging their inventory position of swaps, influence where
between the boundaries the actual swap spread will be
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Interest-Rate Swaps (continued)
Development of the Interest-Rate Swap Market
The initial motivation for the interest-rate-swap market was borrower
exploitation of what was perceived to be “credit arbitrage”
opportunities.
o These opportunities resulted from differences in the quality spread
between lower- and higher-rated credits in the U.S. and Eurodollar bond
fixed-rate market and the same spread in these two floating-rate markets.
Basically, the argument for swaps was based on a well-known economic
principle of comparative advantage in international economics.
o The argument in the case of swaps is that even though a high credit-
rated issuer could borrow at a lower cost in both the fixed- and floating-
rate markets (i.e., have an absolute advantage in both), it will have a
comparative advantage relative to a lower credit-rated issuer in one of
the markets (and a comparative disadvantage in the other).
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Interest-Rate Swaps (continued)
Role of the Intermediary
The role of the intermediary in an interest-rate swap sheds
some light on the evolution of the market.
o Intermediaries in these transactions have been commercial banks
and investment banks, who in the early stages of the market
sought out end users of swaps.
o That is, they found in their client bases those entities that needed
the swap to accomplish a funding or investing objective, and they
matched the two entities.
o In essence, the intermediary in this type of transaction performed
the function of a broker.
o The only time that the intermediary would take the opposite side
of a swap (i.e., would act as a principal) was to balance out the
transaction.
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Interest-Rate Swaps (continued)
Beyond the Plain Vanilla Swap
In a generic or plain vanilla swap, the notional principal amount does
not vary over the life of the swap. Thus it is sometimes referred to as
a bullet swap. In contrast, for amortizing, accreting, and roller
coaster swaps, the notional principal amount varies over the life of
the swap.
An amortizing swap is one in which the notional principal amount
decreases in a predetermined way over the life of the swap.
o Such a swap would be used where the principal of the asset that is being
hedged with the swap amortizes over time. Less common than the
amortizing swap are the accreting swap and the roller coaster swap.
An accreting swap is one in which the notional principal amount
increases in a predetermined way over time.
In a roller coaster swap, the notional principal amount can rise or
fall from period to period.
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Interest-Rate Swaps (continued)
Beyond the Plain Vanilla Swap
The terms of a generic interest-rate swap call for the
exchange of fixed- and floating-rate payments.
In a basis rate swap, both parties exchange floating-rate
payments based on a different reference rate.
o The risk is that the spread between the prime rate and
LIBOR will change. This is referred to as basis risk.
Another popular swap is to have the floating leg tied to
a longer-term rate such as the two-year Treasury note
rather than a money market rate.
o Such a swap is called a constant maturity swap.
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Interest-Rate Swaps (continued)
Beyond the Plain Vanilla Swap
There are options on interest-rate swaps.
o These swap structures are called swaptions and grant the
option buyer the right to enter into an interest-rate swap at a
future date.
o There are two types of swaptions – a payer swaption and a
receiver swaption.
i. A payer swaption entitles the option buyer to enter into an
interest-rate swap in which the buyer of the option pays a
fixed-rate and receives a floating rate.
ii. In a receiver swaption the buyer of the swaption has the
right to enter into an interest-rate swap that requires paying
a floating rate and receiving a fixed-rate.
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Interest-Rate Swaps (continued)
Forward Start Swap
A forward start swap is a swap wherein the
swap does not begin until some future date that
is specified in the swap agreement.
Thus, there is a beginning date for the swap at
some time in the future and a maturity date for
the swap.
A forward start swap will also specify the swap
rate at which the counterparties agree to
exchange payments commencing at the start
date.
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Interest-Rate Caps and Floors
An interest-rate agreement is an agreement between two
parties whereby one party, for an upfront premium,
agrees to compensate the other at specific time periods if
a designated interest rate, called the reference rate, is
different from a predetermined level.
When one party agrees to pay the other when the
reference rate exceeds a predetermined level, the
agreement is referred to as an interest-rate cap or
ceiling.
The agreement is referred to as an interest-rate floor
when one party agrees to pay the other when the
reference rate falls below a predetermined level.
The predetermined interest-rate level is called the strike
rate.
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Interest-Rate Caps and Floors
(continued)
Interest-rate caps and floors can be
combined to create an interest-rate collar.
This is done by buying an interest-rate cap
and selling an interest-rate floor.
Some commercial banks and investment
banking firms write options on interest-rate
agreements for customers.
Options on caps are captions; options on
floors are called flotions.
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Interest-Rate Caps and Floors
(continued)
Risk/Return Characteristics
In an interest-rate agreement, the buyer pays an upfront
fee representing the maximum amount that the buyer can
lose and the maximum amount that the writer of the
agreement can gain.
The only party that is required to perform is the writer of
the interest-rate agreement.
The buyer of an interest-rate cap benefits if the underlying
interest rate rises above the strike rate because the seller
(writer) must compensate the buyer.
The buyer of an interest rate floor benefits if the interest
rate falls below the strike rate, because the seller (writer)
must compensate the buyer.
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Interest-Rate Caps and Floors
(continued)
Valuing Caps and Floors
The arbitrage-free binomial model can be used to value a cap and a
floor.
This is because a cap and a floor are nothing more than a package
or strip of options.
More specifically, they are a strip of European options on interest
rates.
Thus to value a cap the value of each period’s cap, called a caplet,
is found and all the caplets are then summed.
We refer to this approach to valuing a cap as the caplet method.
(The same approach can be used to value a floor.) Once the caplet
method is demonstrated, we will show an easier way of valuing a
cap.
Similarly, an interest rate floor can be valued.
The value for the floor for any year is called a floorlet.
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Interest-Rate Caps and Floors
(continued)
Valuing Caps and Floors
To illustrate the caplet method, we will use the binomial
interest-rate tree used in Chapter 18 to value an interest rate
option to value a 5.2%, three-year cap with a notional amount
of $10 million.
The reference rate is the one-year rates in the binomial tree and
the payoff for the cap is annual.
There is one wrinkle having to do with the timing of the
payments for a cap and floor that requires a modification of the
binomial approach presented to value an interest rate option.
This is due to the fact that settlement for the typical cap and
floor is paid in arrears.
Exhibit 29-11 (see Overhead 29-37) shows the binomial
interest rate tree with dates and years.
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Exhibit 29-11 Binomial Interest Rate
Tree with Dates and Years Identified
N
3.500%
NL
NHH
NLL
NHL
7.0053%
5.7354%
4.6958%
5.4289%
4.4448%
NH
Dates: 0 1 2 3
Years: One Two Threes
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Interest-Rate Caps and Floors
(continued)
Using a Single Binomial Tree to Value a Cap
The valuation of a cap can be done by using a single binomial
tree.
The procedure is easier only in the sense that the number of
times discounting is required is reduced.
The method is shown in Exhibit 29-13 (see Overhead 29-40).
The three values at Date 2 are obtained by simply computing
the payoff at Date 3 and discounting back to Date 2.
Let’s look at the higher node at Date 1 (interest rate of
5.4289%).
The top number, $104,026, is the present value of the two Date
2 values that branch out from that node.
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Interest-Rate Caps and Floors
(continued)
Using a Single Binomial Tree to Value a Cap
The number below it, $21,711, is the payoff of the Year Two
caplet on Date 1.
The third number down at the top node at Date 1 in Exhibit 29-
13, which is in bold, is the sum of the top two values above it.
It is this value that is then used in the backward induction.
The same procedure is used to get the values shown in the
boxes at the lower node at Date 1.
Given the values at the two nodes at Date 1, the bolded values
are averaged to obtain ($125,737 + $24,241)/2 = $74,989.
Discounting this value at 3.5% gives $72,453.
This is the same value obtained from using the caplet
approach.
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Exhibit 29-13
Valuing a Cap Using a Single Binomial Tree
N
$72,753
3.500%
NL
$180,530
$53,540
$104,026
$21,711
$125,737
5.4289%
$24,241
$ 0
$24,241
4.4448%
NH
$168,711
7.0053%
$50,636
5.7354%
$0
4.6958%
Years: One Two Threes
Dates: 0 1 2 3
$0
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Interest-Rate Caps and Floors (continued)
Applications
To see how interest-rate agreements can be used for
asset/liability management, consider the problems faced by a
commercial bank which needs to lock in an interest-rate spread
over its cost of funds.
Because the bank borrows short term, its cost of funds is
uncertain.
The bank may be able to purchase a cap, however, so that the
cap rate plus the cost of purchasing the cap is less than the rate
it is earning on its fixed-rate commercial loans.
If short-term rates decline, the bank does not benefit from the
cap, but its cost of funds declines.
The cap therefore allows the bank to impose a ceiling on its
cost of funds while retaining the opportunity to benefit from a
decline in rates.
42.
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Interest-Rate Caps and Floors (continued)
Applications
The bank can reduce the cost of purchasing the cap
by selling a floor.
In this case the bank agrees to pay the buyer of the
floor if the reference rate falls below the strike rate.
The bank receives a fee for selling the floor, but it
has sold off its opportunity to benefit from a decline
in rates below the strike rate.
By buying a cap and selling a floor the bank creates
a “collar” with a predetermined range for its cost of
funds.
43.
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