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)

Zero Rate(% cont comp)

0.5 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 two-year bond providing a 6% coupon semiannually is

3 3 3

103 98 39

0 05 0 5 0 058 1 0 0 064 1 5

0 068 2 0

e e e

e

. . . . . .

. . .

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

to get y=0.0676 or 6.76%.

3 3 3 103 98 390 5 1 0 1 5 2 0e e e ey y y y . . . . .

2.11

Forward Rates

The forward rate is the future zero rate implied by today’s term structure of interest rates

2.12Calculation 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

2.13

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

R T R T

T T2 2 1 1

2 1

2.14

• Duration of a bond that provides cash flow c i at time t i is

where B is its price & y is its yield (continuously compounded)

• This leads to

tceBi

i

ni

yti

1

BB

D y

Duration

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

2.17An 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

2.18

---------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

Cash Flows to Company B

2.19

Typical Uses of anInterest 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

2.20A and B Transform a Liability

A B

LIBOR

5%

LIBOR+0.8%

5.2%

A: from 5.2 fixed to floating ---> pays Libor+0.2%

B: from floating Libor+0.8% to fixed ---> pays 5%+0.8%

2.21

A and B Transform an Asset

A B

LIBOR

5%

LIBOR-0.25%

4.7%

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

A B

LIBOR

LIBOR+1%

9.95%

10%

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 fixed-rate 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

2.26Swapping a BTP

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