Swaps
Chapter 7
Options, Futures, and Other Derivatives, 7th International Edition,
Copyright © John C. Hull 2008
1
Nature of Swaps
A swap is an agreement btw two parties to
exchange cash flows at specified future times
according to certain specified rules.
- A forward contract is equivalent to the
exchange of CFs on just one future date, swaps
lead to CF exchanges taking a place on several
future dates.
The most common swap contracts are;
- plain vanilla interest rate swap
- fixed-for-fixed currency swap
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2008
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Mechanics of Interest Rate
Swaps
Plain vanilla swap: Company agrees to pay
CFs equal to interest at a predetermined fixed
rate on a notional principal for a number of
years. In return, it receives interest at a
floating rate on the same notional principal for
the same period of time.
LIBOR is used as the floating rate in most
interest rate swaps.
Notional: Principal is not exchanged
An Example of a “Plain Vanilla”
Interest Rate Swap
An interst rate swap btw Microsoft and
Intel. An agreement by Microsoft to
receive 6-month LIBOR & pay a fixed
rate of 5% per annum every 6 months
for 3 years on a notional principal of
$100 million.
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Exp. Cont.
Microsoft is long a floating rate bond and
short a fixed rate bond.
Intel is long a fixed rate bond and short a
floating rate bond.
On a floating rate bond, interest is generally
set at the beginning of the period to which it
will apply and is paid at the end of the period.
Cash Flows to Microsoft
---------Millions of Dollars--------LIBOR FLOATING
FIXED
Net
Date
Rate
Cash Flow Cash Flow Cash Flow
Mar.5, 2010
4.2%
Sept. 5, 2010
4.8%
+2.10
–2.50
–0.40
Mar.5, 2011
5.3%
+2.40
–2.50
–0.10
Sept. 5, 2011
5.5%
+2.65
–2.50
+0.15
Mar.5, 2012
5.6%
+2.75
–2.50
+0.25
Sept. 5, 2012
5.9%
+2.80
–2.50
+0.30
Mar.5, 2013
6.4%
+2.95
–2.50
+0.45
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Typical Uses of an
Interest Rate Swap
Converting a liability from
fixed rate to floating rate
floating rate to fixed rate
Converting an investment from
fixed rate to floating rate
floating rate to fixed rate
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Using a swap to transform a
liability
Microsoft has agrees to borrow $100 million
at LIBOR+0.1%, wants to transform a floating
rate loan into a fixed rate loan. After it has
entered into a swap:
Loan payment:
LIBOR+0.1%
Add:Paid under swap
+ 5%
Less:Received under swap
- LIBOR
Net Payment
5.1%
For Intel swap is used to transform fixed rate
into floating rate. Suppose it has 3-year $100
million loan on which it pays 5.2%
Loan payment
5.2%
Add: Paid under swap
+LIBOR
Less: Received under swap
- 5%
Net payment
LIBOR+0.2%
Intel and Microsoft (MS)
Transform a Liability
5%
5.2%
Intel
MS
LIBOR+0.1%
LIBOR
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Using the swap to transform
an asset (nature of an asset)
Suppose MS owns $100 million in bonds that
will provide 4.7% per annum over the next 3
years. MS enters into a swap, wants to switch
its assets from fixed to floating rate.
Investment income
4.7%
Less: Paid under swap
-5%
Add: Received under swap
+LIBOR
Net income
LIBOR-0.3%
Intel is transforming an asset earning floating
to fixed. Suppose Intel has an investment
$100 million that yields LIBOR-0.20. After it
has entered into the swap:
Investment income
LIBOR-0.20
Less: Paid under swap
- LIBOR
Add: Received under swap
+ 5%
Net Investment income
4.8%
Intel and Microsoft (MS)
Transform an Asset (Figure 7.3, page 151)
5%
4.7%
Intel
MS
LIBOR-0.2%
LIBOR
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Role of Financial Intermediary
Suppose fin. intermediary enters into two
offsetting swap transactions with Intel and MS
and receives 0.03% fee per year:
0.03% x 1 million = $30,000
Transform a Liability (Fin int.
İnvolved)
M.S
Intel
Loan payment
LIBOR + 0.1%
5.2%
Add: Paid under
swap
Less: Received
under swap
+ 5.015
+ LIBOR
- LIBOR
-4.985%
Net Payment
(Without fin. int.)
5.115%
(5.1%)
LIBOR+0.215
(LIBOR+ 0.2%)
Financial Institution is Involved
(Figure 7.4, page 151)
4.985%
5.015%
5.2%
Intel
F.I.
LIBOR
MS
LIBOR
LIBOR+0.1
%
Financial Institution has two offsetting
swaps
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Transform an Asset (Fin. Inter.
involved)
In. Income
M.S
Intel
4.7%
LIBOR-0.2%
Less: Paid under -5.015
swap
Add: Received
+ LIBOR
under swap
-LIBOR
Net Income
LIBOR-0.315%
(Without a fin. int) (LIBOR-0.3%)
4.785%
(4.8%)
+ 4.985%
Financial Institution is Involved
4.985%
5.015%
4.7%
Intel
F.I.
MS
LIBOR-0.2%
LIBOR
LIBOR
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Market Makers
It is unlikely that two companies will contact a
fin. intermediary at the same time and want to
take opposite positions in exactly the same
swap. For this reason many large fin.
intermediaries act as market makers for
swaps.
Swap Rate: is the average the bid and offer
fixed rates.
Quotes By a Swap Market
Maker
Maturity
Bid (%)
Offer (%)
Swap Rate (%)
2 years
6.03
6.06
6.045
3 years
6.21
6.24
6.225
4 years
6.35
6.39
6.370
5 years
6.47
6.51
6.490
7 years
6.65
6.68
6.665
10 years
6.83
6.87
6.850
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2008
20
The Comparative Advantage Argument
AAACorp and BBBCorp wish to
borrow $10 million for 5 years
• AAACorp wants to borrow floating
• BBBCorp wants to borrow fixed
Fixed
Floating
AAACorp
4.0%
6-month LIBOR − 0.10%
BBBCorp
5.2%
6-month LIBOR + 0.6%
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BBBCorp has a comparative advantage in
floating market and AAACorp has a
comparative advantage in fixed rate market.
They can enter into a swap that AAACorp
ends up with floating rate funds and BBBCorp
ends up with fixed rate funds.
Suppose AAACorp agrees to pay interest at 6
month LIBOR on $10 million and BBBCorp
agrees to pay 4.35% per annum on $10
million.
AAACorp
BBBCorp
Loan payment
4%
LIBOR+0.6%
Add: Paid under
swap
Less: Received
under swap
+ LIBOR
+ 4.35%
-4.35%
-LIBOR
Net Payment
(Before)
LIBOR-0.35%
(LIBOR-0.10%)
(0.25% gain)
4.95%
(5.2%)
(0.25% gain)
The swap agreement appears to improve the position
of both company.
Total gain is: 0.25+0.25=0.50%
If a is the difference btw the interest rates in fixed
market and if b is the difference between the interest
rates in floating market.
Total gain is; a-b
a = 1.2% (5.2%-5%)
b= 0.7% ( LIBOR+0.6%-(LIBOR-0.1%))
a-b = 1.2-0.7=0.5%
The Swap agreement btw two
parties
4.35%
4%
AAACorp
BBBCorp
LIBOR+0.6%
LIBOR
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The Swap when a Financial
Institution is Involved
4.33%
4.37%
4%
AAACorp
LIBOR
F.I
.
BBBCorp
LIBOR+0.6%
LIBOR
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Criticism of the Comparative
Advantage Argument
The 4.0% and 5.2% rates available to
AAACorp and BBBCorp in fixed rate
markets are 5-year rates
The LIBOR−0.1% and LIBOR+0.6% rates
available in the floating rate market are sixmonth rates
BBBCorp’s fixed rate depends on the
spread above LIBOR it borrows at in the
future
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The Nature of Swap Rates
Six-month LIBOR is a short-term AA
borrowing rate
The 5-year swap rate has a risk
corresponding to the situation where 10 sixmonth loans are made to AA borrowers at
LIBOR
This is because the lender can enter into a
swap where income from the LIBOR loans
is exchanged for the 5-year swap rate
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Using Swap Rates to Bootstrap the
LIBOR/Swap Zero Curve
Consider a new swap where the fixed rate is the
swap rate
When principals are added to both sides on the final
payment date the swap is the exchange of a fixed
rate bond for a floating rate bond
The floating-rate rate bond is worth par. The swap is
worth zero. The fixed-rate bond must therefore also
be worth par
This shows that swap rates define par yield bonds
that can be used to bootstrap the LIBOR (or
LIBOR/swap) zero curve
Traders
use swap rates for longer-term zero rates.
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Example 7.2
Suppose that the 6-month, 12-month and 18 month
LIBOR/swap zero rates have been determined as
4%, 4.5%, and 4.8% with cont.compounding and that
the 2-year swap rate (for a swap where payments are
made semiannually) is 5%.
Note: This 5% swap rate means that a bond with a
principal of $100 million and semiannual coupon of
5% per annum sells for par.
Find the 2-year zero rate.
Example 7.3
The 400-day LIBOR zero rate has been
calculated as 4.8% with cont.compounding
and, from a Eurodollar futures quote, it has
been calculated that the forward rate for a 91day period beginning in 400 days is 5.3% with
cont. compounding. Find the 491-day rate.
Valuation of an Interest Rate Swap
That Is Not New
Interest rate swaps can be valued as the
difference between the value of a fixed-rate
bond and the value of a floating-rate bond
Alternatively, they can be valued as a
portfolio of forward rate agreements (FRAs)
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Valuation in Terms of Bonds
The fixed rate bond is valued in the usual way
The floating rate bond is valued by noting that
it is worth par immediately after the next
payment date
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Example
Pay six-month LIBOR, receive 8% (s.a.
compounding) on a principal of $100 million
Remaining life 1.25 years
LIBOR rates for 3-months, 9-months and 15months are 10%, 10.5%, and 11% (cont
comp)
6-month LIBOR on last payment date was
10.2% (semiann compounding)
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Valuation Using Bonds
Time Bfix cash Bfl cash
flow
flow
Disc
factor
PV
Bfix
PV
Bfl
0.25
4.0
0.9753
3.901
102.505
0.75
4.0
0.9243
3.697
1.25
104.0
0.8715
90.640
Total
105.100
98.238
102.505
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Valuation Using Bonds
Vswap= 98.238-102.505= -4.267
Valuation in Terms of FRAs
Each exchange of payments in an interest
rate swap is an FRA
The FRAs can be valued on the
assumption that today’s forward rates are
realized
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Valuation of Example Using FRAs
(page 162)
Time
Fixed
cash
flow
Floating
cash
flow
Net Cash Disc
Flow
factor
PV
Bfl
0.25
4.0
-5.100
-1.100
0.9753
-1.073
0.75
4.0
-5.522
-1.522
0.9243
-1.407
1.25
4.0
-6.051
-2.051
0.8715
-1.787
Total
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-4.267
38
Currency Swaps
Exchanging principal and interest payments
in one currency for principal and interest
payments in another currency.
An Example of a Currency Swap
An agreement to pay 5% on a sterling
principal of £10,000,000 & receive 6% on
a US$ principal of $18,000,000 every year
for 5 years.
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Exchange of Principal
In an interest rate swap the principal is
not exchanged
In a currency swap the principal is usually
exchanged at the beginning and the end
of the swap’s life
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The Cash Flows (Table 7.7, page 164)
Year
2004
2005
2006
2007
2008
2009
Dollars Pounds
$
£
------millions-----–18.00 +10.00
+1.08 –0.50
+1.08 –0.50
+1.08 –0.50
+1.08 –0.50
+19.08 −10.50
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Typical Uses of a Currency Swap
Conversion from a liability in one
currency to a liability in another currency
Conversion from an investment in one
currency to an investment in another
currency
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Comparative Advantage Arguments for
Currency Swaps
General Electric wants to borrow 20 miilion
AUD and Qantas wants to borrow 15
million USD (Exchange rate is $0.75 per
AUD)
USD
AUD
General Motors
5.0%
7.6%
Qantas
7.0%
8.0%
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GE has a comparative advantage in USD and
Qantas has a comparative advantage in AUD.
So that GE borrows USD and Qantas
borrows AUS then they enter into a currency
swap to transform GE’s loan into a AUD and
Qantas loan into a USD.
Total gain to all parties: 2%-0.4% = 1.6%
Valuation of Currency Swaps
Like interest rate swaps, currency swaps
can be valued either as the difference
between 2 bonds or as a portfolio of
forward contracts
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Example
All Japanese LIBOR/swap rates are 4%
All USD LIBOR/swap rates are 9%
5% is received in yen; 8% is paid in dollars.
Payments are made annually
Principals are $10 million and 1,200 million
yen
Swap will last for 3 more years
Current exchange rate is 110 yen per dollar
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Valuation in Terms of Bonds
Time
Cash Flows
($)
PV ($)
Cash flows
(yen)
PV (yen)
1
0.8
0.7311
60
57.65
2
0.8
0.6682
60
55.39
3
0.8
0.6107
60
53.22
3
10.0
7.6338
1,200
1,064.30
Total
9.6439
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1,230.55
48
Valuation in Terms of Forwards
Time $ cash
flow
Yen cash Forward
flow
Exch
rate
Yen cash
flow in $
Net
Cash
Flow
Present
value
1
-0.8
60
0.009557
0.5734
-0.2266
-0.2071
2
-0.8
60
0.010047
0.6028
-0.1972
-0.1647
3
-0.8
60
0.010562
0.6337
-0.1663
-0.1269
3
-10.0
1200
0.010562
12.6746
+2.674
6
2.0417
Total
1.5430
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Swaps & Forwards
A swap can be regarded as a convenient
way of packaging forward contracts
Although the swap contract is usually
worth zero at the outset, each of the
underlying forward contracts are not
worth zero
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Credit Risk
A swap is worth zero to a company initially
At a future time its value is liable to be either positive
or negative
The company has credit risk exposure only when its
value is positive
Some swaps are more likely to lead to credit risk
exposure than others
What is the situation if early forward rates have a
positive value?
What is the situation when the early forward rates
have a negative value?
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Other Types of Swaps
Floating-for-floating interest rate swaps,
amortizing swaps, step up swaps, forward
swaps, constant maturity swaps,
compounding swaps, LIBOR-in-arrears
swaps, accrual swaps, diff swaps, cross
currency interest rate swaps, equity swaps,
extendable swaps, puttable swaps,
swaptions, commodity swaps, volatility
swaps……..
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