6.1
Swaps
Chapter 7
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.2
Nature of Swaps
A swap is an agreement to exchange
cash flows at specified future times
according to certain specified rules
(traded on OTC)
• Notional Principal
• Counterparties: companies and swap
dealers (market makers)
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.3
An Example of a “Plain Vanilla”
Interest Rate Swap
• An agreement by Microsoft to receive 6month LIBOR & pay a fixed rate of 5% per
annum every 6 months for 3 years on a
notional principal of $100 million
• Next slide illustrates cash flows
• Note that in practice Fixed Rates are on
actual/365 or 30/360 basis whereas Floating
Rates are on actual/360 basis.
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.4
Cash Flows to Microsoft
---------Millions of Dollars--------LIBOR FLOATING
FIXED
Net
Date
Rate
Cash Flow Cash Flow Cash Flow
Mar.5, 2001
4.2%
Sept. 5, 2001
4.8%
+2.10
–2.50
–0.40
Mar.5, 2002
5.3%
+2.40
–2.50
–0.10
Sept. 5, 2002
5.5%
+2.65
–2.50
+0.15
Mar.5, 2003
5.6%
+2.75
–2.50
+0.25
Sept. 5, 2003
5.9%
+2.80
–2.50
+0.30
Mar.5, 2004
6.4%
+2.95
–2.50
+0.45
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.5
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
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.6
Intel and Microsoft (MS)
Transform a Liability
5%
5.2%
Intel
MS
LIBOR+0.1%
LIBOR
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.7
Financial Institution is
Involved
4.985%
5.015%
5.2%
Intel
F.I.
MS
LIBOR+0.1%
LIBOR
LIBOR
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
Financial Institution is
Involved
6.8
• The fixed rate paid by FI is called its bid, the fixed
rate received by FI is its offer/ask rate.
• In our example Bid = 4.985% and Ask = 5.015%
• Swap rate (quote) is the average of the bid and
ask. Swap = 5% here.
4.985%
5.015%
5.2%
Intel
F.I.
MS
LIBOR+0.1%
LIBOR
LIBOR
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.9
Intel and Microsoft (MS)
Transform an Asset
5%
4.7%
Intel
MS
LIBOR-0.25%
LIBOR
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.10
Financial Institution is
Involved
4.985%
5.015%
4.7%
F.I.
Intel
MS
LIBOR-0.25%
LIBOR
LIBOR
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.11
The Comparative Advantage
Argument
• AAACorp wants to borrow
floating
• BBBCorp wants to borrow
fixed
Fixed
Floating
AAACorp
10.00%
6-month LIBOR + 0.30%
BBBCorp
11.20%
6-month LIBOR + 1.00%
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.12
The Comparative Advantage
Argument
• We are not comparing companies, rather
different markets from a single
company’s point of view:
– AAACorp has a comparative advantage in
fixed-rate markets
– BBBCorp has a comparative advantage in
floating-rate markets
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
The Comparative Advantage
Argument
6.13
• No Swap:
– Total cost = R+0.3+11.2 = R+11.5%
• Swap:
– Total cost = R+1+10 = R+11%
• Gain from using a swap is 0.5%
• Assume that they split it 50/50. So, each
company gets a reduction in borrowing costs of
0.5/2 = .25% or 25 basis points.
• AAA will be able to borrow at R+0.3-0.25 =
R+.05%
• BBBwill be able to borrow at 11.2-0.25 = 10.95%
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.14
The Swap
9.95%
10%
AAA
BBB
LIBOR+1%
LIBOR
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.15
The Swap when a Financial
Institution is Involved
9.93%
9.97%
10%
AAA
F.I.
BBB
LIBOR+1%
LIBOR
LIBOR
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
Criticism of the Comparative
Advantage Argument
6.16
• The 10.0% and 11.2% rates available to AAACorp
and BBBCorp in fixed rate markets are 5-year
rates
• The LIBOR+0.3% and LIBOR+1% rates available
in the floating rate market are six-month rates
• Floating Spread smaller than Fixed Spread may
just reflect that default probabilities of the BBB
company grow faster than those of AAA
• BBBCorp’s fixed rate depends on the spread
above LIBOR it borrows at in the future
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.17
Valuation of an Interest Rate
Swap
• 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)
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.18
Valuation in Terms of FRAs
• Each exchange of payments in an
interest rate swap is an FRA
• The FRA’s can be valued on the
assumption that today’s forward rates
are realized
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.19
Example
• Problem 7.3
• NP = $100 mil; T = 10 months; Fixed
Rate = 12% (semiannual)
• Term structure of zero rates is flat at
10% (continuous).
• Two months ago 6-m LIBOR was 9.6%
(semiannual).
• What is the swap value to the payfloating party?
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.20
Example
• Swap is equivalent to a portfolio of a 4-m
FRA and a 10-m FRA with values V4 and
V10, respectively.
• Forward rates are all 10% (continuous) or 2x
(e0.1/2 – 1) = 10.254% (semiannual)
• V4 = 0.5 x 100 x (.12-.096)e-0.1x1/3 = 1.16066
mil
• V10 = 0.5 x 100 x (.12-.10254)e-0.1x10/12 =
0.803 mil
• Vswap = V4 + V10 = 1.964 mil (current value)
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.21
An Example of a Currency
Swap
An agreement to pay 11% on a
sterling principal of £10,000,000 &
receive 8% on a US$ principal of
$15,000,000 every year for 5 years
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.22
Exchange of Principal
• In an interest rate swap the
principal is not exchanged
• In a currency swap the
principal is exchanged at the
beginning and the end of the
swap
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.23
The Cash Flows
IBM pays fixed 11% in sterling and receives 8% in USD
Year
2001
2002
2003
2004
2005
2006
Dollars Pounds
$
£
------millions-----–15.00 +10.00
+1.20 –1.10
+1.20 –1.10
+1.20 –1.10
+1.20 –1.10
+16.20 -11.10
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.24
Typical Uses of a
Currency Swap
• Conversion from • Conversion from
a liability in one
an investment in
currency to a
one currency to
liability in
an investment in
another currency
another currency
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.25
Comparative Advantage
Arguments for Currency Swaps
General Motors wants to borrow AUD
Qantas wants to borrow USD
USD
AUD
General Motors 5.0%
12.6%
Qantas
13.0%
7.0%
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.26
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
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.27
Example
• Suppose the term structure of interest
rates is flat both in US at 9% and in Japan
at 4% (both continuous).
• A FI enters a swap in which it receives 5%
in yen and pays 8% in USD once a year.
• T = 3 years (remaining)
• The principals are $10 mil and 1,200 mil
yen. Current Spot exchange rate is 110
yen = $1.
• What is the swap value to the FI?
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.28
Example
• Remember that forwards are valued (under no
arbitrage) as though the forward price on the
underlying (in this case, yen) is realized:
f  ( F0  K )e
 rT
• In this example the swap is equivalent to 3 forward
contracts (1, 2 and 3 years)
• The corresponding forward rates:
• 1/110 x e.05x1 = .009557
• 1/110 x e.05x2 = .010047
( r  rf ) T
F

S
e
.05x3
0
0
• 1/110 x e
= .010562
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
Example
6.29
• Exchange of interest involves receiving 5% x 1,200
mil = 60 mil yen and paying 8% x 10 = $0.8 mil.
Exchange of principals at the end involves receiving
1,200 mil yen and paying $10 mil. The values of the
forwards are as follows:
• f1 = (60x .009557 - 0.8)e-.09x1 = -$0.2071
• f2 = (60x .010047 - 0.8)e-.09x2 = -$0.1647
• f3 = (60x .010562 - 0.8)e-.09x3 = -$0.1269
• f4 = (1,200x .010562 - 10)e-.09x3 = $2.0416
• Thus, the total value is $1.543 mil, if entered into in
the past.
• If the swap was initiated today, the value would be
$1.543 mil – $0.9091mil = $633,909
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.30
Swaps & Forwards
• A swap can be regarded as a
convenient way of packaging forward
contracts
• The “plain vanilla” interest rate swap in
our example consisted of several FRAs
• The “fixed for fixed” currency swap in
our example consisted of a cash
transaction and 4 forward contracts
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.31
Swaps & Forwards
(continued)
• The value of the swap is the sum of the
values of the forward contracts underlying
the swap
• Swaps are normally “at the money” initially
– This means that it costs nothing to enter
into a swap
– It does not mean that each forward
contract underlying a swap is “at the
money” initially
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull
6.32
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
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull