6.1
Swaps
6.2
Nature of Swaps
A swap is an agreement to exchange
cash flows at specified future times
according to certain specified rules
An Example of a “Plain Vanilla”
Interest Rate Swap
• 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
• Next slide illustrates cash flows
6.3
Cash Flows to Microsoft
6.4
(See Table 6.1, page 127)
---------Millions of Dollars--------LIBOR FLOATING
FIXED
Net
Date
Rate
Cash Flow Cash Flow Cash Flow
Mar.1, 1998
4.2%
Sept. 1, 1998
4.8%
+2.10
–2.50
–0.40
Mar.1, 1999
5.3%
+2.40
–2.50
–0.10
Sept. 1, 1999
5.5%
+2.65
–2.50
+0.15
Mar.1, 2000
5.6%
+2.75
–2.50
+0.25
Sept. 1, 2000
5.9%
+2.80
–2.50
+0.30
Mar.1, 2001
6.4%
+2.95
–2.50
+0.45
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
Intel and Microsoft (MS)
Transform a Liability
(Figure 6.2, page 128)
5%
5.2%
Intel
MS
LIBOR+0.1%
LIBOR
6.6
6.7
Financial Institution is Involved
(Figure 6.4, page 129)
4.985%
5.015%
5.2%
F.I.
Intel
MS
LIBOR+0.1%
LIBOR
LIBOR
Dealer spread = .03% evenly split
Intel and Microsoft (MS)
Transform an Asset
(Figure 6.3, page 128)
5%
4.7%
Intel
MS
LIBOR-0.25%
LIBOR
6.8
6.9
Financial Institution is Involved
(See Figure 6.5, page 129)
4.985%
5.015%
4.7%
F.I.
Intel
MS
LIBOR-0.25%
LIBOR
Dealer spread = .03 %
LIBOR
6.10
The Comparative Advantage
Argument (Table 6.4, page 132)
• 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%
6.11
The Comparative Advantage
Argument
• AAACorp has absolute advantage in both
markets
• But a comparative advantage in fixed
• BBBCorp has comparative advantage in
floating
• If AAA borrows fixed, the gain is 1.2%
• If BBB borrows floating, the gain is
reduced by .7%
• Therefore, we have a net gain of
1.2 - .7 = .5%
• If the gain is split evenly, we have a gain
per party of:
G = (1.2 - .7)/2 = .25%
6.12
Swap Design
• Design the swap so AAA’s
borrowing rate equals the
comparative disadvantage (CD)
rate minus the gain:
•
LIBOR + .3 - .25
• Do the same thing for BBB
• BBB’s rate with swap:
•
11.2 - .25
• Now, draw the diagram
6.13
The Swap (Figure 6.6, page 132)
9.95%
10%
AAA
BBB
LIBOR+1%
LIBOR
The floating rate leg should be LIBOR
6.14
Swap Design with FI
•
•
•
•
•
•
•
•
•
Adjust swap gain for dealer spread
Suppose dealer spread = .04%
Then gain:
G = (1.2 - .7 - .04)/2 = .23%
AAA’s rate with swap:
LIBOR + .3 - .23 = LIBOR + .07
BBB’s rate with swap:
11.2 - .23 = 10.97%
Draw swap diagram
The Swap when a Financial
Institution is Involved
6.15
(Figure 6.7, page 133)
9.93%
9.97%
10%
AAA
F.I.
BBB
LIBOR+1%
LIBOR
LIBOR
Check that dealer spread = .04%
6.16
Criticism of the Comparative
Advantage Argument
• 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 sixmonth rates
• BBBCorp’s fixed rate depends on the spread
above LIBOR it borrows at in the future
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
6.18
Swap Valuation
• Fixed Receive: Vswap = Vfixed – Vfloating
• Fixed Pay: Vswap = Vfloating - Vfixed
• The fixed rate stream is valued as an
annuity
• The floating rate stream is valued by
noting that it is worth par immediately
after the next payment date
6.19
Floating Rate Perpetuity
• To create a floating rate perpetuity, invest
principal value in 6-month LIBOR
• At the end of 6-months, remove the interest
and reinvest the principal value at the new 6month LIBOR
• Therefore, the cost of a floating rate
perpetuity is principal value
• It always sells for its par value immediately
after interest payment
6.20
Floating Rate Instrument with
Maturity T
• At the end of T years, floating rate
perpetuity is worth the principal value M
• Therefore, the floating rate instrument is
worth: M – MdT
• Or M - M/(1+y/2)2T, where is the rate on
a zero-coupon bond maturing in T years
6.21
Swap Valuation
• The floating rate instrument is worth:
Vfloating = M – M/(1+y/2)2T
• The fixed rate stream is worth:
Vfixed = (C/2)(Annuity Factor)
• So for Fixed Receive Swap:
Vswap = (C/2)(Annuity Factor) - M
+M/(1+y/2)2T
6.22
Swap Valuation
• The swap is structured such that initial value is zero
to either party
• Set Vswap = 0
• Rearrange terms:
M=(C/2)(Annuity Factor) + M/(1+y/2)2T
• The left-hand side is the present value of a bond at y
• Since the bond is selling at par, CR = C/M = y
• For the swap to have zero value the fixed rate must
equal the yield to maturity on a par bond
• The swap rate is the coupon rate on a LIBOR bond
that causes it to be worth par
6.23
Example
• Zero coupon LIBOR curve is 5%, 6%,
and 7% for one, two, and three years
• What is the swap rate on a three year
interest rate swap?
• Assume payments are annual and
yields are compounded annually
• Solve for LIBOR par yield
• M = CRxM(d1 + d2 + d3) + Md3
6.24
Example Continued
• Solution:
1
1
3
1
.
07
CR 
 6.91%
1
1
1


2
1.05 1.06 1.073
6.25
Interest Rate Risk
• Receive Fixed: Vswap = Vfixed – Vfloating
• Pay Fixed: Vswap = Vfloating – Vfixed
6.26
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
6.27
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
6.28
Three Cash Flow Components
• t = 0: exchange principal based
upon current exchange rates
Pay: $15 M
Rcv: £ 10 M
• t = 1, 2, 3, 4, 5:
Pay: .11x10 = £1.1 M
Rcv: .08x15 = $1.2 M
• t = 5:
Pay: £ 10 M
Rcv: $ 15 M
6.29
The Cash Flows (Table 6.6, page 140)
Dollars Pounds
$
£
Years ------millions-----0
–15.00 +10.00
+1.20 –1.10
1
2
+1.20 –1.10
3
+1.20 –1.10
4
+1.20 –1.10
5
+16.20 -11.10
6.30
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
6.31
Comparative Advantage Arguments
for Currency Swaps (Table 6.7, page 141)
General Motors wants to borrow AUD
Qantas wants to borrow USD
USD
AUD
General Motors 5.0%
12.6%
Qantas
13.0%
7.0%
6.32
Comparative Advantage
• GM has absolute advantage in both markets
• But GM has comparative advantage in dollars
• Qantas has comparative advantage in
Australian dollars
• So GM should borrow dollars and Qantas
Australian dollars
• Then swap cash flows to earn gain from
comparative advantage
6.33
Comparative Advantage
• Gain per party:
G = (2 - .4)/2 = .8%
• GM’s rate with swap:
12. 6 - .8 = AUD 11.8%
• Qantas’ rate with swap:
7 - .8 = USD 6.2%
6.34
Qantas Assumes Exchange Rate Risk
USD 5%
USD 5%
AUD 13%
GM
Qantas
AUD 11.8%
6.35
GM Assumes Exchange Rate Risk
USD 6.2%
USD 5%
AUD 13%
GM
Qantas
AUD 13.0%
6.36
FI Assumes Exchange Rate Risk
• Adjust swap gain for dealer spread
• Suppose dealer spread = .2%
• Then gain:
• Gain per party:
G = (2 - .4 - .2)/2 = .7%
• GM’s rate with swap:
12. 6 - .7 = AUD 11.9%
• Qantas’ rate with swap:
7 - .7 = USD 6.3%
6.37
FI Assumes Exchange Rate Risk
USD 5%
USD 6.3%
USD 5%
GM
F.I.
Q
AUD 13%
AUD11.9%
AUD 13%
Check that dealer spread = .2%
Pay: 13.0 – 11.9 = AUD 1.1%
Rcv: 6.3 – 5.0 = USD 1.3%
6.38
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
6.39
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 6 Fraps
• The “fixed for fixed” currency swap in
our example consisted of a cash
transaction & 5 forward contracts
6.40
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
6.41
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