2.1
Swaps
Lecture 2
2.2
Types of Rates
• Treasury rates
• LIBOR rates
• Euribor rates
2.3
2.4
2.5
2.6
2.7
Zero Rates
A zero rate (or spot rate), for maturity T,
is the rate of interest earned on an
investment that provides a payoff only
at time T
2.8
Example
Maturity
(years)
0.5
Zero Rate
(% cont comp)
5.0
1.0
5.8
1.5
6.4
2.0
6.8
2.9
Bond Pricing
• To calculate the cash price of a bond we
discount each cash flow at the appropriate
zero rate
• In our example, the theoretical price of a twoyear bond providing a 6% coupon
semiannually is
3e
0.05 0.5
 3e
0.0581.0
 3e
 103e 0.0682.0  98.39
0.064 1.5
2.10
Bond Yield
• The bond yield is the discount rate that
makes the present value of the cash flows on
the bond equal to the market price of the
bond
• Suppose that the market price of the bond in
our example equals its theoretical price of
98.39
• The bond yield is given by solving
3e  y 0.5  3e  y 1.0  3e  y 1.5  103e  y 2.0  98.39
to get y=0.0676 or 6.76%.
2.11
Forward Rates
The forward rate is the future zero rate
implied by today’s term structure of
interest rates
Calculation of Forward Rates
Zero Rate for
Forward Rate
an n -year Investment for n th Year
Year (n )
(% per annum)
(% per annum)
1
10.0
2
10.5
11.0
3
10.8
11.4
4
11.0
11.6
5
11.1
11.5
2.12
2.13
Formula for Forward Rates
• Suppose that the zero rates for time
periods T1 and T2 are R1 and R2 with
both rates continuously compounded.
• The forward rate for the period between
times T1 and T2 is
R2 T2  R1 T1
T2  T1
2.14
Duration
• Duration of a bond that provides cash flow c i at time t i is
 c i e  yt i 
ti 


i 1
 B 
n
where B is its price & y is its yield (continuously
compounded)
• This leads to
B
B
  Dy
2.15
Duration Matching
• This involves hedging against interest
rate risk by matching the durations of
assets and liabilities
• It provides protection against small
parallel shifts in the zero curve
2.16
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 “Company B” 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
2.17
Cash Flows to Company B
---------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
2.18
2.19
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
A and B Transform a Liability
5%
5.2%
A
B
LIBOR+0.8%
LIBOR
A: from 5.2 fixed to floating ---> pays Libor+0.2%
B: from floating Libor+0.8% to fixed ---> pays 5%+0.8%
2.20
2.21
A and B Transform an Asset
5%
4.7%
A
B
LIBOR-0.25%
LIBOR
2.22
The Comparative Advantage
Argument
• Company A wants to borrow
floating
• Company B wants to borrow
fixed
Fixed
Floating
Company A 10.00%
6-month LIBOR + 0.30%
Company B 11.20%
6-month LIBOR + 1.00%
2.23
The Swap
9.95%
10%
A
B
LIBOR+1%
LIBOR
A: from 10% fixed to floating ---> pays Libor+0.05%
B: from floating Libor+1% to fixed ---> pays 9.95%+1%
2.24
Valuation of an Interest Rate
Swap
• Interest rate swaps can be
valued as the difference
between the value of a fixedrate bond & the value of a
floating-rate bond
2.25
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
Swapping a
BTP
2.26
2.27
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
2.28
Examples of Other Types of
Swaps
•
•
•
•
•
•
Amortizing & step-up swaps
Extendible & puttable swaps
Index amortizing swaps
Equity swaps
Commodity swaps
Differential swaps