fabozzi ch29 bmas 7thed

8/9/2019 Fabozzi Ch29 BMAS 7thEd

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Copyright © 2010 29-1

Chapter 29 Interest-ate !"aps, Caps,

and #loors


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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


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Copyright © 2010 29-$

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 %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

"n 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 !he 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

!he default risk in a swap agreement is called

counterparty risk


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Copyright © 2010 29-)

Interest-Rate Swaps %continued& Interpreting a Swap Position

!here 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 "lthough an interest-rate swap may be nothing more than a package

of forward contracts, it is not a redundant contract, for severalreasons

i %aturities for forward or futures contracts do not extend out as far as

those of an interest-rate swap

ii "n interest-rate swap is a more transactionally efficient instrument

because in one transaction an entity can effectively establish a payoffequivalent to a package of forward contracts

iii Interest-rate swaps now provide more liquidity than forward

contracts, particularly long-dated (ie, long-term) forward contracts


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Interest-Rate Swaps %continued& Interpreting a Swap Position !o 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 &uy ' million par of a five-year floating-rate bond that pays six-month

*I&+ every six months# finance the purchase by borrowing ' million for

five years at . annual interest rate paid every six months !he cash flows for this transaction are shown in xhibit 01- ( see Overhead 29-

8) !he second column shows the cash flow from purchasing the five-yearfloating-rate bond !here is a ' million cash outlay and then cash inflows

!he amount of the cash inflows is uncertain because they depend on future

*I&+ !he next column shows the cash flow from borrowing ' million on a

fixed-rate basis !he last column shows the net cash flow from the entire

transaction "s the last column indicates, there is no initial cash flow (no cashinflow or cash outlay) In all six-month periods, the net position results in a

cash inflow of *I&+ and a cash outlay of '0 million !his 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 or the !"rchase o a Five-#ear Floating-Rate

$on% Finance% b& $orrowing on a Fixe%-Rate $asis

Transaction: 2urchase for ' million a five-year floating-rate bond$ floating rate 3 *I&+,

semiannual pay# borrow ' million for five years$ fixed rate 3 ., semiannual payments

Cash Flow (millions of dollars From!4ix-%onth

2eriod 5loating-ate &onda &orrowing

6ost 7et

8' 9' '

9(*I&+ /0): 80 9 (*I&+ /0):80

0 9(*I&+ 0/0): 80 9 (*I&+ 0/0):80

; 9(*I&+ ;/0): 80 9 (*I&+ ;/0):80

< 9(*I&+ /0): 80 9 (*I&+ >/0):80

? 9(*I&+ ?/0): 80 9 (*I&+ ?/0):801 9(*I&+ 1/0): 80 9 (*I&+ 1/0):80

9(*I&+ /0):9 80 9 (*I&+ /0):80

a !he subscript for *I&+ indicates the six-month *I&+ as per the terms of the floating-rate bond

at time t


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Interest-Rate Swaps %continued& "erminology, Con#entions, and $arket %uotes !he date that the counterparties commit to the swap is called the

trade date

!he 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

!he convention that has evolved for quoting swaps levels is that aswap dealer sets the floating rate equal to the index and then

quotes the fixed-rate that will applyo !he offer price that the dealer would quote the fixed-rate payer

would be to pay ??. and receive *I&+ @flatA (@flatA meaning

with no spread to *I&+)o !he bid price that the dealer would quote the floating-rate payer

would be to pay *I&+ flat and receive ?>.o !he bid-offer spread is basis points


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Interest-Rate Swaps %continued& "erminology, Con#entions, and $arket %uotes

"nother way to describe the position of the counterparties to aswap is in terms of our discussion of the interpretation of a swap

as a package of cash market instrumentso ixed-rate payer: " position that is exposed to the price

sensitivities of a longer-term liability and a floating-rate bondo loating-rate payer: " position that is exposed to the price

sensitivities of a fixed-rate bond and a floating-rate liability !he 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 illustrate this convention, consider a -year swap offered by a

dealer to market participants shown in xhibit 01-0 ( see Overhead

29-!2)


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Interest-Rate Swaps %continued& "erminology, Con#entions, and $arket %uotes

In our illustration, suppose that the -year !reasury yield is ?;. !hen the offer price that the dealer would quote to the fixed-rate

payer is the -year !reasury rate plus basis points versus

receiving *I&+ flat

5or the floating-rate payer, the bid price quoted would be *I&+ flat

versus the -year !reasury rate plus


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Exhibit 29-2 'eaning o a ()*+,* ."ote or

a 1*-#ear Swap /hen 0reas"ries #iel% 3,4

5$i%-Oer Sprea% o 1* $asis !oints6

Floating&RatePayer

Fi'ed&RatePayer

2ay 5loating rate ofsix-month

*I&+

5ixed rate of??.

eceive 5ixed rate of?>.

5loating rate ofsix-month

*I&+


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Interest-Rate Swaps %continued& Calculation of the Swap Rate "t the initiation of an interest-rate swap, the counterparties are agreeing

to exchange future interest-rate payments and no upfront payments byeither party are made

Bhile the payments of the fixed-rate payer are known, the floating-rate

payments are not known

!his is because they depend on the value of the reference rate at the

reset dates 5or a *I&+-based swap, the urodollar 6C futures contract can be

used to establish the forward (or future) rate for three-month *I&+

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 !he 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 xhibit 01-< ( see Overhead 29-!") shows the fixed-rate payments

based on an assumed swap rate of .o !he first three columns of the exhibit show the beginning and end of the

quarter and the number of days in the quarter 6olumn (


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Exhibit 29-) Fixe%-Rate !a&7ents

8ss"7ing a Swap Rate o )9,4

%uarterStarts

%uarter Endsays in%uarter

Period ) Endof %uarter

Fi'ed&Rate Payment if SwapRate Is Assumed to *e +-./01

Dan year %ar ; year 1 ,0

"pr year Dune ; year 1 0 ,0=,>01

Duly year 4ept ; year 10 ; ,0>01

Duly year ; 4ept ; year ; 10 ,0>


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Interest-Rate Swaps %continued&Calculation of the Swap Rate

Eiven the swap payments, we can show how tocompute the swap rate "t the initiation of an interest-rate swap, the counterparties are

agreeing to exchange future payments and no upfront payments by either party are made

!his means that the present value of the payments to be made by the counterparties must be at least equal to the present valueof the payments that will be received

!o eliminate arbitrage opportunities, the present value of the

payments made by a party will be equal to the present value ofthe payments received by that same party

!he 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 !he present value of ' 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 !hese are the same forward rates that are used to compute the floating-

rate paymentsFthose obtained from the urodollar 6C futures contracto Be must make Gust one more adGustmento Be must adGust the forward rates used in the formula for the number of

days in the period (ie, the quarter in our illustrations) in the same way

that we made this adGustment to obtain the paymentso 4pecifically, 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 Eiven the payment for a period and the forward discount factor

for the period, the present value of the payment can becomputed

!he forward discount factor is used to compute the present value

of the both the fixed-rate payments and floating-rate payments

&eginning with the basic relationship for no arbitrage to exist$ PV of floatingrate payments ) PV of fixedrate payments

!he formula for the swap rate is derived as follows Be begin

with$

fixedrate payment for period tdays in period

notional amount swap rate360

=

× ×


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Calculation of the Swap Rate !he 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

Be have$

4umming up the present value of the fixed-rate payment for each

period gives the present value of the fixed-rate payments *etting $ be

the number of periods in the swap, we have$

present !alue of the fixedrate payment for period t

days in period t notional amount swap rate forward dis"ount fa"tor for period t 360

=

× × ×

present !alue of the fixedrate payment

days in period t swap rate notional amount forward dis"ount fa"tor for period t

360

=

× × ×∑


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Interest-Rate Swaps %continued& Calculation of the Swap Rate 4olving for the swap rate gives

2aluing a Swap +nce the swap transaction is completed, changes in market

interest rates will change the payments of the floating-rate side

of the swap !he value of an interest-rate swap is the difference between the

present value of the payments of the two sides of the swap

-

#

t

$

present !alue of floatingrate payments

days in period t notional amount forward dis"ount fa"tor for period t

360

swap rate

=

× ×∑


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Interest-Rate Swaps %continued& uration of a Swap "s with any fixed-income contract, the value of a swap will change

as interest rates change Collar duration is a measure of the interest-rate sensitivity of a

fixed-income contract 5rom the perspective of the party who pays floating and receives

fixed, the interest-rate swap position can be viewed as follows$long a fixedrate bond 3 short a floatingrate bond

!his 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 fixedrate bond

8 dollar duration of a floatingrate bond


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Application of a Swap to Asset45ia6ility

$anagement "n interest-rate swap can be used to alter the cash flow

characteristics of an institutionHs assets so as to provide a

better match between assets and liabilities

"n interest-rate swap allows each party to accomplish itsasset/liability obGective of locking in a spread

"n 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 " lia%ility swap permits two institutions to change the

cash flow nature of their liabilities


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Creation of Structured 7otes 8sing Swaps

6orporations can customie medium-term notes forinstitutional investors who want to make a market play

on interest rate, currency, and/or stock market

movements

!hat is, the coupon rate on the issue will be based onthe movements of these financial variables

" corporation can do so in such a way that it can still

synthetically fix the coupon rate

!his can be accomplished by issuing an %!7 andentering into a swap simultaneously

%!7s created in this way are called structured &T$s


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Interest-Rate Swaps %continued& Primary eterminants of Swap Spreads

!he swap spread is determined by the same factors that influence thespread over !reasuries on financial instruments (futures / forward

contracts or cash) that produce a similar return or funding profile Eiven 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 " urodollar 6C futures contract is a swap where a fixed dollar

payment (ie, the futures price) is exchanged for three-month

*I&+ " market participant can synthesie a (synthetic) fixed-rate security

or a fixed-rate funding vehicle of up to five years by taking a position in a strip of urodollar 6C futures contracts (ie, a position

in every three-month urodollar 6C up to the desired maturity date)


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Interest-Rate Swaps %continued& Primary eterminants of Swap Spreads

5or swaps with maturities longer than five years, the spread isdetermined primarily by the credit spreads in the corporate bond

market

&ecause 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 keydeterminant of the swap spread

&oundary conditions for swap spreads based on prices for fixed-

rate and floating-rate corporate bonds can be determined

4everal 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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e#elopment of the Interest&Rate Swap $arket !he initial motivation for the interest-rate-swap market was borrower

exploitation of what was perceived to be @credit arbitrageA

opportunities

o !hese opportunities resulted from differences in the quality spread

between lower- and higher-rated credits in the J4 and urodollar bond

fixed-rate market and the same spread in these two floating-rate markets

&asically, the argument for swaps was based on a well-known economic principle of comparative advantage in international economics

o !he 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 (ie, have an absolute advantage in both), it will have a

comparative advantage relative to a lower credit-rated issuer in one ofthe markets (and a comparative disadvantage in the other)


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Role of the Intermediary

!he role of the intermediary in an interest-rate swap shedssome 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 swapso !hat is, they found in their client bases those entities that needed

the swap to accomplish a funding or investing obGective, and they

matched the two entities

o In essence, the intermediary in this type of transaction performed

the function of a brokero !he only time that the intermediary would take the opposite side

of a swap (ie, would act as a principal) was to balance out the

transaction

S


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Interest-Rate Swaps %continued& *eyond the Plain 2anilla Swap In a generic or plain vanilla swap, the notional principal amount does

not vary over the life of the swap !hus it is sometimes referred to as

a %ullet swap In contrast, for amortiing, accreting, and roller

coaster swaps, the notional principal amount varies over the life of

the swap "n amorti'ing swap is one in which the notional principal amount

decreases in a predetermined way over the life of the swapo 4uch a swap would be used where the principal of the asset that is being

hedged with the swap amorties over time *ess common than the

amortiing swap are the accreting swap and the roller coaster swap "n 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

I R S


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*eyond the Plain 2anilla Swap

!he terms of a generic interest-rate swap call for theexchange of fixed- and floating-rate payments

In a %asis rate swap, both parties exchange floating-rate

payments based on a different reference rate

o !he risk is that the spread between the prime rate and*I&+ will change !his is referred to as %asis risk

"nother popular swap is to have the floating leg tied to

a longer-term rate such as the two-year !reasury note

rather than a money market rate

o 4uch a swap is called a constant maturity swap

I t t R t S


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Copyright © 2010 29-$0

Interest-Rate Swaps %continued& *eyond the Plain 2anilla Swap !here are options on interest-rate swapso !hese swap structures are called swaptions and grant the

option buyer the right to enter into an interest-rate swap at a

future dateo !here are two types of swaptions 8 a payer swaption and a

receiver swaptioni " 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 theright to enter into an interest-rate swap that requires paying

a floating rate and receiving a fixed-rate

I t t R t S


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Interest-Rate Swaps %continued& Forward Start Swap

" forward start swap is a swap wherein theswap does not begin until some future date that

is specified in the swap agreement

!hus, there is a beginning date for the swap atsome time in the future and a maturity date for

the swap

" forward start swap will also specify the swap

rate at which the counterparties agree toexchange payments commencing at the start

date

I t t R t C % Fl


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Interest-Rate Caps an% Floors "n 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 Bhen one party agrees to pay the other when the

reference rate exceeds a predetermined level, theagreement is referred to as an interest-rate cap or

ceiling !he agreement is referred to as an interest-rate floor

when one party agrees to pay the other when thereference rate falls below a predetermined level

!he predetermined interest-rate level is called the strike

rate

I t t R t C % Fl


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Interest-Rate Caps an% Floors%continued&

Interest-rate caps and floors can becombined to create an interest-rate collar

!his is done by buying an interest-rate cap

and selling an interest-rate floor 4ome commercial banks and investment

banking firms write options on interest-rate

agreements for customers +ptions on caps are captions# options on

floors are called flotions

Interest Rate Caps an% Floors


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Risk4Return 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

!he only party that is required to perform is the writer ofthe interest-rate agreement

!he 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 !he 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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2aluing Caps and Floors !he arbitrage-free binomial model can be used to value a cap and a

floor !his is because a cap and a floor are nothing more than a package

or strip of options %ore specifically, they are a strip of uropean options on interest

rates !hus to value a cap the value of each periodHs cap, called a caplet ,

is found and all the caplets are then summed Be refer to this approach to valuing a cap as the caplet method

(!he same approach can be used to value a floor) +nce the caplet

method is demonstrated, we will show an easier way of valuing a

cap 4imilarly, an interest rate floor can be valued !he value for the floor for any year is called a floorlet



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2aluing Caps and Floors

!o illustrate the caplet method, we will use the binomialinterest-rate tree used in 6hapter ? to value an interest rate

option to value a 0., three-year cap with a notional amount

of ' million

!he reference rate is the one-year rates in the binomial tree andthe payoff for the cap is annual

!here 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

!his is due to the fact that settlement for the typical cap andfloor is paid in arrears

E'hi6it 9-&:: (see ;#erhead 9-&


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8sing a Single *inomial "ree to 2alue a Cap !he valuation of a cap can be done by using a single binomialtree

!he procedure is easier only in the sense that the number of

times discounting is required is reduced

!he method is shown in xhibit 01-; ( see Overhead 29-())

!he three values at Cate 0 are obtained by simply computing

the payoff at Cate ; and discounting back to Cate 0

*etHs look at the higher node at Cate (interest rate of


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8sing a Single *inomial "ree to 2alue a Cap

!he number below it, '0,>, is the payoff of the Lear !wocaplet on Cate

!he third number down at the top node at Cate in xhibit 01-

;, 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 !he same procedure is used to get the values shown in the

boxes at the lower node at Cate Eiven the values at the two nodes at Cate , the bolded values

are averaged to obtain ('0,>;> 9 '00,


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Interest-Rate Caps an% Floors %continued&

Applications

!o see how interest-rate agreements can be used forasset/liability management, consider the problems faced by a

commercial bank which needs to lock in an interest-rate spread

over its cost of funds

&ecause the bank borrows short term, its cost of funds is

uncertain !he 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

!he 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

I t t R t C % Fl


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Interest-Rate Caps an% Floors %continued&

Applications

!he 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

!he bank receives a fee for selling the floor, but ithas sold off its opportunity to benefit from a decline

in rates below the strike rate

&y buying a cap and selling a floor the bank creates

a @collarA with a predetermined range for its cost of

funds


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