Chapter 3Principles of

Corporate FinanceTenth Edition

Valuing Bonds

Slides by

Matthew Will

McGraw-Hill/Irwin Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.

3- 2

Topics Covered

Using The Present Value Formula to Value Bonds

How Bond Prices Vary With Interest RatesThe Term Structure of Interest RatesExplaining the Term StructureReal and Nominal Rates of InterestCorporate Bonds and the Risk of Default

3- 3

Valuing a Bond

NN

r

C

r

C

r

CPV

)1(

000,1...

)1()1( 22

11

3- 4

Valuing a Bond

Example If today is October 1, 2010, what is the value of the

following bond? An IBM Bond pays $115 every September 30 for 5 years. In September 2015 it pays an additional $1000 and retires the bond. The bond is rated AAA (WSJ AAA YTM is 7.5%)

Cash Flows

Sept 1112 13 14 15

115 115 115 115 1115

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Valuing a Bond

Example continued If today is October 1, 2010, what is the value of the following bond? An

IBM Bond pays $115 every September 30 for 5 years. In September 2015 it pays an additional $1000 and retires the bond. The bond is rated AAA (WSJ AAA YTM is 7.5%)

84.161,1$

075.1

115,1

075.1

115

075.1

115

075.1

115

075.1

1155432

PV

3- 6

Valuing a Bond

Example - France In December 2008 you purchase 100 Euros of bonds in France which

pay a 8.5% coupon every year. If the bond matures in 2012 and the YTM is 3.0%, what is the value of the bond?

Euros 44.120

03.1

5.108

03.1

5.8

03.1

5.8

03.1

5.8432

PV

3- 7

Valuing a Bond

Another Example - Japan In July 2010 you purchase 200 Yen of bonds in Japan which pay a 8%

coupon every year. If the bond matures in 2015 and the YTM is 4.5%, what is the value of the bond?

Yen 57.243

045.1

216

045.1

16

045.1

16

045.1

16

045.1

165432

PV

3- 8

Valuing a Bond

Example - USA In February 2009 you purchase a 3 year US Government bond. The

bond has an annual coupon rate of 4.875%, paid semi-annually. If investors demand a 0.6003% semiannual return, what is the price of the bond?

95.107,1$

006003.1

375.1024

006003.1

375.24

006003.1

375.24

006003.1

375.24

006003.1

375.24

006003.1

375.2465432

PV

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Valuing a Bond

Example continued - USA Take the same 3 year US Government bond. If investors demand a

4.0% semiannual return, what is the new price of the bond?

09.918$

04.1

375.1024

04.1

375.24

04.1

375.24

04.1

375.24

04.1

375.24

04.1

375.2465432

PV

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Interest Rate on 10yr Treasuries

Year

Yie

ld ,

%

3- 11

Bond Prices and Yields

Interest Rates, %

Bon

d P

rice

, %

80.00

85.00

90.00

95.00

100.00

105.00

110.00

115.00

3- 12

Maturity and Prices

Interest Rates, %

Bon

d P

rice

, ($) 30 yr bond

3 yr bond

When the interest rateequals the 5% coupon,both bonds sell forface value

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Duration Formula

PV

CPVT

PV

CPV

PV

CPV

PV

CPV T )(...

)(3)(2)(1Duration 321

yield1

duration(%) volatilityDuration Modified

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Duration Calculation

Year Ct PV(Ct) at 5.0%Proportion of Total Value

[PV(Ct)/V]Proportion of Total

Value Time

1 100 95.24 0.084 0.0842 100 90.7 0.08 0.163 1100 950.22 0.836 2.509

V = 1136.16 1 Duration= 2.753 years

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Duration

Year CF PV@YTM % of Total PV % x Year

1 68.75 65.54 .060 0.060

2 68.75 62.48 .058 0.115

3 68.75 59.56 .055 0.165

4 68.75 56.78 .052 0.209

5 68.75 841.39 .775 3.875

1085.74 1.00 Duration 4.424

Example (Bond 1)

Calculate the duration of our 6 7/8 % bond @ 4.9 % YTM

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Duration

Year CF PV@YTM % of Total PV % x Year

1 90 82.95 .081 0.081

2 90 76.45 .075 0.150

3 90 70.46 .069 0.207

4 90 64.94 .064 0.256

5 1090 724.90 .711 3.555

1019.70 1.00 Duration= 4.249

Example (Bond 2)Given a 5 year, 9.0%, $1000 bond, with a 8.5% YTM, what is this bond’s

duration?

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Duration & Bond Prices

Interest rate, percent

Bon

d P

rice

, per

cent

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Interest Rates

Short- and long-term interest rates do not always move in parallel. Between September 1992 and April 2000 U.S. short-term rates rose sharply while long term rates declined.

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Term Structure of Interest Rates

Spot Rate - The actual interest rate today (t=0)

Forward Rate - The interest rate, fixed today, on a loan made in the future at a fixed time.

Future Rate - The spot rate that is expected in the future

Yield To Maturity (YTM) - The IRR on an interest bearing instrument

YTM (r)

Year

1981

1987 & Normal

1976

1 5 10 20 30

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Yield Curve

Maturity

U.S. Treasury Strip Spot Rates as of February 2009

Spo

t rat

es (

%)

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Law of One Price

All interest bearing instruments are priced to fit the term structure

This is accomplished by modifying the asset price

The modified price creates a New Yield, which fits the Term Structure

The new yield is called the Yield To Maturity (YTM)

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Yield to Maturity

ExampleA $1000 treasury bond expires in 5 years.

It pays a coupon rate of 10.5%. If the market price of this bond is 107.88, what is the YTM?

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Yield to Maturity

ExampleA $1000 treasury bond expires in 5 years. It pays

a coupon rate of 10.5%. If the market price of this bond is 107.88, what is the YTM?

C0 C1 C2 C3 C4 C5

-1078.80 105 105 105 105 1105

Calculate IRR = 8.5%

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Term Structure

What Determines the Shape of the Term Structure?

Expectations Theory

Term Structure & Capital BudgetingCF should be discounted using Term Structure infoSince the spot rate incorporates all forward rates, then you

should use the spot rate that equals the term of your project.

If you believe in other theories take advantage of the arbitrage.

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Debt & Interest Rates

Classical Theory of Interest Rates (Economics)developed by Irving Fisher

Nominal Interest Rate = The rate you actually pay when you borrow money

Real Interest Rate = The theoretical rate you pay when you borrow money, as determined by supply and demand

Supply

Demand

$ Qty

r

Real r

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Inflation Rates

Annual rates of inflation in the United States from 1900–2008.A

nnua

l Inf

lati

on (

%)

3- 27

Global Inflation Rates

0.00

2.00

4.00

6.00

8.00

10.00

12.00

Ave

rag

e In

flat

ion

, %

Switzer

land

Nether

lands

USA

Canada

Sweden

Norway

Austra

lia

Denmar

kUK

Irelan

d

South

Afri

ca

Avera

ge

Germ

any (

ex 192

2/23)

Belgium

Spain

Franc

e

Japa

nIta

ly

Averages from 1900-2006

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Debt & Interest Rates

Nominal r = Real r + expected inflation (approximation)

Real r is theoretically somewhat stable

Inflation is a large variable

Q: Why do we care?A: This theory allows us to understand the Term Structure of

Interest Rates.

Q: So What?A: The Term Structure tells us the cost of debt.

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Debt & Interest Rates

Actual formula

)1()1(1 realnominal irr

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02468101214161820

1-Jan-84

1-Apr-85

1-Jul-8

6

1-Oct-87

1-Jan-89

1-Apr-90

1-Jul-9

1

1-Oct-92

1-Jan-94

1-Apr-95

1-Jul-9

6

1-Oct-97

1-Jan-99

1-Apr-00

1-Jul-0

1

1-Oct-02

1-Jan-04

1-Apr-05

1-Jul-0

6

1-Oct-07

1-Jan-09

UK Bond Yields

10 year nominal interest rate

10 year real interest rate

Inte

rest

rat

e (%

)

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Govt. Bills vs. Inflation (’53-’08)%

United Kingdom

Inflation

T-Bill Returns

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Govt. Bills vs. Inflation (’53-’08)%

United States

Inflation

T-Bill Returns

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Govt. Bills vs. Inflation (’53-’08)%

Germany

Inflation

T-Bill Returns

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Bond Ratings

Key to bond ratings. The highest-quality bonds are rated triple A. Bonds rated triple B or above are investment grade. Lower-rated bonds are called high-yield, or junk, bonds.

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Yield Spread

Yie

ld s

prea

d be

twee

n co

rpor

ate

and

gove

rnm

ent b

onds

, %Yield spreads between corporate and 10-year

Treasury bonds.

Years

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Prices and Yields

Prices and yields of a sample of corporate bonds, December 2008.

Source: Bond transactions reported on FINRA’s TRACE service: http://cxa.marketwatch.com/finra/BondCenter

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Web Resources

Click to access web sitesClick to access web sites

Internet connection requiredInternet connection required

http://cxa.marketwatch.com/finra/BondCenter

www.ft.com

www.smartmoney.com

www.wsj.com

www.finpipe.com

www.investinginbonds.com

www.investorguide.com

http://money.cnn.com/markets/bondcenter

www.federalreserve.gov

www.stls.frb.org

www.ustreas.gov