Introduc corporate finance ch5

5-1

Valuation of Bonds and Stock
• First Principles:

– Value of financial securities = PV of expected
future cash flows
• To value bonds and stocks we need to:

– Estimate future cash flows:
• Size (how much) and
• Timing (when)

– Discount future cash flows at an appropriate rate:
• The rate should be appropriate to the risk presented by
the security.
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5-2

5.1 Definition and Example of a Bond
• A bond is a legally binding agreement between a
borrower and a lender:
– Specifies the principal amount of the loan.
– Specifies the size and timing of the cash flows:
• In dollar terms (fixed-rate borrowing)
• As a formula (adjustable-rate borrowing)

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5-3

5.1 Definition and Example of a Bond
• Consider a U.S. government bond listed as 6 3/8 of
December 2009.
– The Par Value of the bond is $1,000.
– Coupon payments are made semi-annually (June 30 and
December 31 for this particular bond).
– Since the coupon rate is 6 3/8 the payment is $31.875.
– On January 1, 2002 the size and timing of cash flows are:

$31.875

$31.875

$31.875

$1,031.875

6 / 30 / 09

12 / 31 / 09


1 / 1 / 02

6 / 30 / 02

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12 / 31 / 02

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5-4

5.2 How to Value Bonds
• Identify the size and timing of cash flows.
• Discount at the correct discount rate.

– If you know the price of a bond and the size and
timing of cash flows, the yield to maturity is the
discount rate.

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5-5

Pure Discount Bonds
Information needed for valuing pure discount bonds:
– Time to maturity (T) = Maturity date - today’s date

– Face value (F)
– Discount rate (r)

$0

$0

$0

$F

T −1

T


0

1

2

Present value of a pure discount bond at time 0:

F
PV =
T
(1 + r )
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5-6

Pure Discount Bonds: Example
Find the value of a 30-year zero-coupon bond with a $1,000
par value and a YTM of 6%.

$0

$0

$0

$1,000
0$ 0$0,1$

01 22930


0

1

2

29

30

F
$1,000
PV =
=
= $174.11
T
30
(1 + r )
(1.06)
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5-7

Level-Coupon Bonds
Information needed to value level-coupon bonds:
– Coupon payment dates and time to maturity (T)
– Coupon payment (C) per period and Face value (F)
– Discount rate

$C

$C

$C

$C + $ F

T −1

T


0

1

2

Value of a Level-coupon bond
= PV of coupon payment annuity + PV of face value

C
1 
F
PV = 1 −
+
T 
r  (1 + r )  (1 + r )T
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5-8

Level-Coupon Bonds: Example

Find the present value (as of January 1, 2002), of a 6-3/8
coupon T-bond with semi-annual payments, and a maturity
date of December 2009 if the YTM is 5-percent.
– On January 1, 2002 the size and timing of cash flows are:

$31.875

$31.875

$31.875

$1,031.875

6 / 30 / 09

12 / 31 / 09


1 / 1 / 02

6 / 30 / 02

12 / 31 / 02

 $1,000
$31.875 
1
PV =
1−
+

= $1,049.30

16 
16
.05 2  (1.025)  (1.025)
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5-9

Bond Rates and Yields
 Suppose a $1,000 face value bond currently sells for

$932.90, pays an annual coupon of $70, and matures
in 10 years.


The coupon rate is the annual dollar coupon expressed
as a percentage of the face value:
Coupon rate = $___ /$_____ = 7.0%



The current yield is the annual coupon divided by the
price:
Current yield = $___ /_____ = 7.5%

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5-10

Bond Rates and Yields
 The yield to maturity is the rate that makes the price of

the bond just equal to the present value of its future cash
flows.

 How to find yield to maturity?
–
–
–

Trial and error
Approximation formula
Financial calculator

McGraw-Hill/Irwin

YTM = 8%

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5-11

5.3 Bond Concepts
1.

Bond prices and market interest rates move in opposite
directions.

2.

When coupon rate = YTM, price = par value.
When coupon rate > YTM, price > par value (premium
bond)
When coupon rate < YTM, price < par value (discount
bond)

3.

A bond with longer maturity has higher relative (%) price
change than one with shorter maturity when interest rate
(YTM) changes. All other features are identical.

4.

A lower coupon bond has a higher relative price change
than a higher coupon bond when YTM changes. All other
features are identical.

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5-12

YTM and Bond Value
$1400

Bond Value

When the YTM < coupon, the bond
trades at a premium.

1300

1200

When the YTM = coupon, the
bond trades at par.

1100

1000

800
0

0.01

0.02

0.03

0.04

0.05

0.06
0.07
6 3/8

0.08

0.09

0.1

Discount Rate

When the YTM > coupon, the bond trades at a discount.
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5-13

Bond Value

Maturity and Bond Price Volatility
Consider two otherwise identical bonds.
The long-maturity bond will have much more

volatility with respect to changes in the
discount rate

Par
Short Maturity Bond

C

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Discount Rate
Long Maturity
Bond
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5-14

Bond Value

Coupon Rate and Bond Price Volatility
Consider two otherwise identical bonds.
The low-coupon bond will have much more
volatility with respect to changes in the
discount rate

High Coupon Bond
Discount Rate
Low Coupon Bond

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5-15

Bond Example:
 Bond J has a 4% coupon and Bond K a 10% coupon. Both

have 10 years to maturity, make semiannual payments, and
have 9% YTMs. If market rates rise by 2%, what is the
percentage price change of these bonds? What if rates fall
by 2%?

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5-16

 Percentage changes in bond prices

Bond prices and market rates
7%

9%

11%

Bond J
% Chg.

$786.81
(+16.60%)

$674.80

$581.74
(-13.79%)

Bond K
%Chg.

$1,213.19
(+13.9%)

$1,065.04

$940.25
(-11.72%)

The results above demonstrate that, all else equal, the price of the
lower-coupon bond changes more (in percentage terms) than the price
of the higher-coupon bond when market rates change.

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5-17

5.4 The Present Value of Common Stocks
• Dividends versus Capital Gains
• Valuation of Different Types of Stocks

– Zero Growth
– Constant Growth
– Differential Growth

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5-18

Case 1: Zero Growth
• Assume that dividends will remain at the same level
forever

Div1 = Div 2 = Div 3 = 
• Since future cash flows are constant, the value of a zero

growth stock is the present value of a perpetuity:

Div 3
Div1

Div 2
P0 =
+
+
+
1
2
3
(1 + r ) (1 + r ) (1 + r )
Div
P0 =
r
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5-19

Case 2: Constant Growth
Assume that dividends will grow at a constant rate, g,
forever. i.e.
Div1 = Div 0 (1 + g )
Div 2 = Div1 (1 + g ) = Div 0 (1 + g ) 2
Div 3 = Div 2 (1 + g ) = Div 0 (1 + g ) 3
..
.

Since future cash flows grow at a constant rate forever,
the value of a constant growth stock is the present

value of a growing perpetuity:

Div1
P0 =
r−g
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5-20

Case 3: Differential Growth
• Assume that dividends will grow at different
rates in the foreseeable future and then will
grow at a constant rate thereafter.
• To value a Differential Growth Stock, we need
to:
– Estimate future dividends in the foreseeable future.
– Estimate the future stock price when the stock
becomes a Constant Growth Stock (case 2).
– Compute the total present value of the estimated
future dividends and future stock price at the
appropriate discount rate.
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5-21

Case 3: Differential Growth
• Assume that dividends will grow at rate g1 for N

years and grow at rate g2 thereafter
Div1 = Div 0 (1 + g1 )

Div 2 = Div1 (1 + g1 ) = Div 0 (1 + g1 )
..
.

2

Div N = Div N −1 (1 + g1 ) = Div 0 (1 + g1 )

N

Div N +1 = Div N (1 + g 2 ) = Div 0 (1 + g1 ) N (1 + g 2 )
..
.
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5-22

Case 3: Differential Growth
• Dividends will grow at rate g1 for N years and

grow at rate g2 thereafter

Div 0 (1 + g1 ) Div 0 (1 + g1 ) 2

…
0

1

2

Div 0 (1 + g1 ) N

Div N (1 + g 2 )
= Div 0 (1 + g1 ) N (1 + g 2 )

…

…
N

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N+1
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5-23

Case 3: Differential Growth

We can value this as the sum of:
an N-year annuity growing at rate g1

C  (1 + g1 )T 
PA =
1 −
T 
r − g1 
(1 + r ) 
plus the discounted value of a perpetuity growing at rate
g2 that starts in year N+1

 Div N +1 


r − g2 

PB =
N
(1 + r )
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5-24

Case 3: Differential Growth
To value a Differential Growth Stock, we can use

 Div N +1 




T
C  (1 + g1 )   r − g 2 
P=
+
1 −
T 
N
r − g1 
(1 + r )  (1 + r )
• Or we can cash flow it out.

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5-25

A Differential Growth Example
A common stock just paid a dividend of $2. The
dividend is expected to grow at 8% for 3 years, then
it will grow at 4% in perpetuity.
What is the stock worth?

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