interest rate swaps, caps

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 29-1

Chapter 29 Interest-Rate Swaps, Caps,

and Floors


Learning ObjectivesAfter 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


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


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.


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.


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 contractsii. 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.


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.


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 aBorrowing

CostNet

0 –$50.0 +$50 $01 +(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.


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.


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).


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.


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


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 periodnotional amount three month LIBOR

360


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 periodnotional amount swap rate

360


Exhibit 29-4 Fixed-Rate Payments Assuming a Swap Rate of 4.9875%

Quarter Starts

Quarter EndsDays 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 1,260,729July 1 year 3 Sept 30 year 3 92 11 1,274,583Oct 1 year 3 Dec 31 year 3 92 12 1,274,583


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.


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 periodperiod forward rate annual forward rate

360


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 periodnotional amount swap rate

360


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 tnotional amount swap rate forward discount factor for period t

360

present value of the fixed-rate payment

days in period tswap rate notional amount forward discount factor for period t

360


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 paymentsdays in period t

notional amount forward discount factor for period t360

swap rate


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



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.



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.



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).


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



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).



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.


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.



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.


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.


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.


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.


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.



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.



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.



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.


Exhibit 29-11 Binomial Interest Rate Tree with Dates and Years Identified

N3.500%

NL

NHH

NLL

NHL

7.0053%

5.7354%

4.6958%

5.4289%

4.4448%

NH

Dates: 0 1 2 3

Years: One Two Threes


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.



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.


Exhibit 29-13 Valuing a Cap Using a Single Binomial Tree

N

$72,7533.500%

NL

$180,530

$53,540

$104,026$21,711

$125,7375.4289%

$24,241$ 0

$24,2414.4448%

NH

$168,7117.0053%

$50,6365.7354%

$04.6958%

Years: One Two ThreesDates: 0 1 2 3

$0



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.



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.


All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America.