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Learning Objectives
After reading this chapter, you will understand
ď what an interest-rate swap is the relationship
between an interest-rate swap and forward contracts
ď how interest-rate swap terms are quoted in the
market
ď how the swap rate is calculated
ď how the value of a swap is determined
ď the primary determinants of the swap rate
ď 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
ď how a structured note is created using an interest-rate
swap
ď what a swaption is and how it can be used by
institutional investors
ď what a rate cap and floor are, and how these
agreements can be used by institutional investors
ď the relationship between a cap and floor and options
ď how to value caps and floors
ď how an interest-rate collar can be created
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Interest-Rate Swaps
ď In an interest-rate swap, two parties (called counterparties) agree to
exchange periodic interest payments.
ď The dollar amount of the interest payments exchanged is based on a
predetermined dollar principal, which is called the notional principal
amount.
ď The dollar amount that each counterparty pays to the other is the
agreed-upon periodic interest rate times the notional principal
amount.
ď The only dollars that are exchanged between the parties are the
interest payments, not the notional principal amount.
ď This party is referred to as the fixed-rate payer or the floating-rate
receiver.
ď 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.
ď The frequency with which the interest rate that the floating-rate
payer must pay is called the reset frequency.
5. www.StudsPlanet.com 29-5
Interest-Rate Swaps (continued)
ď Entering into a Swap and Counterparty Risk
ďź Interest-rate swaps are over-the-counter instruments,
which means that they are not traded on an exchange.
ďź 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.
ďź 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.
ďź The default risk in a swap agreement is called
counterparty risk.
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Interest-Rate Swaps (continued)
ď Interpreting a Swap Position
ďź 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
ďź 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.
7. www.StudsPlanet.com 29-7
Interest-Rate Swaps (continued)
ď Interpreting a Swap Position
ďź 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.
ďź 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.
8. www.StudsPlanet.com 29-8
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 â$50.0 +$50 $0
1 +(LIBOR1/2)Ă50 â2.5 + (LIBOR1/2)Ă50â2.5
2 +(LIBOR2/2)Ă50 â2.5 + (LIBOR2/2)Ă50â2.5
3 +(LIBOR3/2)Ă50 â2.5 + (LIBOR3/2)Ă50â2.5
4 +(LIBOR4/2)Ă50 â2.5 + (LIBOR4/2)Ă50â2.5
5 +(LIBOR5/2)Ă50 â2.5 + (LIBOR5/2)Ă50â2.5
6 +(LIBOR6/2)Ă50 â2.5 + (LIBOR6/2)Ă50â2.5
7 +(LIBOR7/2)Ă50 â2.5 + (LIBOR7/2)Ă50â2.5
8 +(LIBOR8/2)Ă50 â2.5 + (LIBOR8/2)Ă50â2.5
9 +(LIBOR9/2)Ă50 â2.5 + (LIBOR9/2)Ă50â2.5
10 +(LIBOR10/2)Ă50+50 â52.5 + (LIBOR10/2)Ă50â2.5
a
The subscript for LIBOR indicates the six-month LIBOR as per the terms of the floating-rate bond
at time t.
9. www.StudsPlanet.com 29-9
Interest-Rate Swaps (continued)
ď Terminology, Conventions, and Market Quotes
ďź The date that the counterparties commit to the swap is called the
trade date.
ďź 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.
ďź 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 âflatâ (âflatâ 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.
10. www.StudsPlanet.com 29-10
Interest-Rate Swaps (continued)
ď Terminology, Conventions, and Market Quotes
ďź 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.
ďź 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.
ďź 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)
ď Terminology, Conventions, and Market Quotes
ď¨ In our illustration, suppose that the 10-year Treasury yield is 8.35%.
ď¨ 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.
ď¨ For the floating-rate payer, the bid price quoted would be LIBOR flat
versus the 10-year Treasury rate plus 40 basis points.
ď¨ The dealer would quote such a swap as 40â50, 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.
ď¨ The difference between the Treasury rate paid and received is the
bid-offer spread.
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Exhibit 29-2 Meaning of a â40â50â 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
13. www.StudsPlanet.com 29-13
Interest-Rate Swaps (continued)
ď Calculation of the Swap Rate
ďź 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.
ďź While the payments of the fixed-rate payer are known, the floating-rate
payments are not known.
ďź This is because they depend on the value of the reference rate at the
reset dates.
ďź For a LIBOR-based swap, the Eurodollar CD futures contract can be
used to establish the forward (or future) rate for three-month LIBOR.
ďź In general, the floating-rate payment is determined as follows:
floating rate payment
number of days in period
notional amount three month LIBOR
360
â =
Ă â Ă
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Interest-Rate Swaps (continued)
ď Calculation of the Swap Rate
ďź The equation for determining the dollar amount of the fixed-rate
payment for the period is:
ďź It is the same equation as for determining the floating-rate payment
except that the swap rate is used instead of the reference rate.
ďź 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
â =
Ă Ă
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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
16. www.StudsPlanet.com 29-16
Interest-Rate Swaps (continued)
ď Calculation of the Swap Rate
ďź Given the swap payments, we can show how to
compute the swap rate.
ď¨ 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.
ď¨ 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.
ď¨ 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.
ď¨ The equivalence of the present value of the payments is the key
principle in calculating the swap rate.
17. www.StudsPlanet.com 29-17
Interest-Rate Swaps (continued)
ď Calculation of the Swap Rate
ďź The present value of $1 to be received in period t is the forward discount
factor.
ďź In calculations involving swaps, we compute the forward discount factor
for a period using the forward rates.
ďź These are the same forward rates that are used to compute the floating-
rate paymentsâthose 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
= Ă
18. www.StudsPlanet.com 29-18
Interest-Rate Swaps (continued)
ď Calculation of the Swap Rate
ď¨ Given the payment for a period and the forward discount factor
for the period, the present value of the payment can be
computed.
ď¨ The forward discount factor is used to compute the present value
of the both the fixed-rate payments and floating-rate payments.
ď¨ Beginning with the basic relationship for no arbitrage to exist:
PV of floating-rate payments = PV of fixed-rate payments
ď¨ 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
=
Ă Ă
19. www.StudsPlanet.com 29-19
Interest-Rate Swaps (continued)
ď Calculation of the Swap Rate
ď¨ 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.
ď¨ We have:
ď¨ 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
=
Ă Ă Ă
present value of the fixed-rate payment
days in period t
swap rate notional amount forward discount factor for period t
360
=
Ă Ă Ăâ
20. www.StudsPlanet.com 29-20
Interest-Rate Swaps (continued)
ď Calculation of the Swap Rate
ď¨ Solving for the swap rate gives
ď Valuing a Swap
ďź Once the swap transaction is completed, changes in market
interest rates will change the payments of the floating-rate side
of the swap.
ďź 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
=
Ă Ăâ
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Interest-Rate Swaps (continued)
ď Duration of a Swap
ďź As with any fixed-income contract, the value of a swap will change
as interest rates change.
ďź Dollar duration is a measure of the interest-rate sensitivity of a
fixed-income contract.
ďź 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
ďź 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
â dollar duration of a floating-rate bond
22. www.StudsPlanet.com 29-22
Interest-Rate Swaps (continued)
ď Application of a Swap to Asset/Liability
Management
ďź An interest-rate swap can be used to alter the cash flow
characteristics of an institutionâs assets so as to provide a
better match between assets and liabilities.
ďź An interest-rate swap allows each party to accomplish its
asset/liability objective of locking in a spread.
ďź 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.
ďź A liability swap permits two institutions to change the
cash flow nature of their liabilities.
34. www.StudsPlanet.com 29-34
Interest-Rate Caps and Floors
(continued)
ď Risk/Return Characteristics
ďź 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.
ďź The only party that is required to perform is the writer of
the interest-rate agreement.
ďź 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.
ďź 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)
ď Valuing Caps and Floors
ďź The arbitrage-free binomial model can be used to value a cap and a
floor.
ďź This is because a cap and a floor are nothing more than a package
or strip of options.
ďź More specifically, they are a strip of European options on interest
rates.
ďź Thus to value a cap the value of each periodâs cap, called a caplet,
is found and all the caplets are then summed.
ďź 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.
ďź Similarly, an interest rate floor can be valued.
ďź The value for the floor for any year is called a floorlet.
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Interest-Rate Caps and Floors
(continued)
ď Valuing Caps and Floors
ď 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.
ď The reference rate is the one-year rates in the binomial tree and
the payoff for the cap is annual.
ď 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.
ďź This is due to the fact that settlement for the typical cap and
floor is paid in arrears.
ď Exhibit 29-11 (see Overhead 29-37) shows the binomial
interest rate tree with dates and years.
37. www.StudsPlanet.com 29-37
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
38. www.StudsPlanet.com 29-38
Interest-Rate Caps and Floors
(continued)
ď Using a Single Binomial Tree to Value a Cap
ď¨ The valuation of a cap can be done by using a single binomial
tree.
ď¨ The procedure is easier only in the sense that the number of
times discounting is required is reduced.
ď¨ The method is shown in Exhibit 29-13 (see Overhead 29-40).
ď¨ The three values at Date 2 are obtained by simply computing
the payoff at Date 3 and discounting back to Date 2.
ď¨ Letâs look at the higher node at Date 1 (interest rate of
5.4289%).
ď¨ 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)
ď Using a Single Binomial Tree to Value a Cap
ď¨ The number below it, $21,711, is the payoff of the Year Two
caplet on Date 1.
ď¨ 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.
ď¨ The same procedure is used to get the values shown in the
boxes at the lower node at Date 1.
ď¨ Given the values at the two nodes at Date 1, the bolded values
are averaged to obtain ($125,737 + $24,241)/2 = $74,989.
ď¨ Discounting this value at 3.5% gives $72,453.
ď¨ 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
41. www.StudsPlanet.com 29-41
Interest-Rate Caps and Floors (continued)
ď Applications
ďź 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.
ďź Because the bank borrows short term, its cost of funds is
uncertain.
ďź 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.
ďź If short-term rates decline, the bank does not benefit from the
cap, but its cost of funds declines.
ďź 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.
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Interest-Rate Caps and Floors (continued)
ď Applications
ďź The bank can reduce the cost of purchasing the cap
by selling a floor.
ďź In this case the bank agrees to pay the buyer of the
floor if the reference rate falls below the strike rate.
ďź 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.
ďź By buying a cap and selling a floor the bank creates
a âcollarâ with a predetermined range for its cost of
funds.
43. www.StudsPlanet.com 29-43
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