Transcript TITLE HERE - Texas Tech University

How to Value Bonds
and Stocks
What is a Bond?

A bond is a legally binding agreement between
a borrower and a lender
 IOU
2
Bond Terminology

Face value (F) or Principal
 For
a corporate bond this is generally $1,000
Zero- coupon bond
 Coupon Rate

 This
is a Stated Annual Rate
 Determines the coupon payment
Coupon payment (C )
 Rating

3
Bond Pricing Terminology

Par
 The

Premium
 The

price of the bond is greater than its face value
Discount
 The

price of the bond equals its face value
price of the bond is less than its face value
Yield to Maturity
4
Yield to Maturity



YTM is the return that the bond is offering if
you bought it today and held it till maturity
The YTM is determined by the riskiness of
the bond
Risk comes from:
1.
Risk of default

Risk is often measured with bond ratings

2.
Investment Grade / Junk
Time to maturity

Longer term bonds are riskier
5
Pure Discount Bonds

Have no coupon
 Sometimes
called zeroes, deep discount bonds, or original
issue discount bonds (OIDs)



Example: T-Bill
Yield to Maturity comes only from the difference
between the purchase price and principal repayment
A pure discount bond cannot sell at a premium
WHY?
6
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
Present value of a pure discount bond at time 0:
FV
PV 
(1  R ) T
7
Pure Discount Bond: 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
$ 1,0 0 0
29
30
1,000/(1.0630) = 174.11
8
Coupon Bonds
 Make
periodic coupon payments in addition
to repaying the principal
 Coupon payments are the same each period
Typically
 An
occur semi-annual
investor’s return is comprised of:
Difference between the purchase price & face value
2. Coupon payments
1.
9
Valuing a Coupon Bond


The value of a bond is simply the present
value of it’s future cash flows
We value a bond is a package of two
investments:
Present value of the coupon payments
2. Present value of the principal repayment
1.
10
Determining Coupon Payments
Coupon ($)= (Principal * Coupon Rate) / Frequency
 Ex:
 8%

semi-annual
(1,000 * 0.08) / 2 = 40
 12%

(1,000 * 0.12) / 12 = 10
 20%

monthly
annual
(1,000 * 0.20) / 1 = 200
11
Coupon Bond Pricing Equation
C
1 
FV
Bond Value  1 

T
r  (1 R)  (1 R)T
Annuity
Coupon Payments
Lump Sum
Principal
Repayment
12
Coupon Bond Pricing: BA II plus
N = The number of coupon payments
 I/Y= The rate corresponding to the coupon
frequency
 PV = The price of the bond today
 PMT= The amount of the coupon payment
 FV = The principal that will be repaid

13
Coupon Example 2
What is the yield to maturity of a 9% 15 year,
bond that sells for $1,200%?
 N = 30 = 15 * 2
3.42%
 I/Y = ??
 PV = -1,200
 PMT = 45 = (1000 * 0.08)/2
 FV = 1,000

The 3.42% is a 6 month rate, the YTM = 6.84%
14
Coupon Example 1
What is the present value of a 8% 10 year,
bond with the yield to maturity is 12%?
 N = 20 = 10 * 2
 I/Y = 12
402.44
 PV = ??
 PMT = 40 = (1000 * 0.08)/2
 FV = 1,000

15
Valuing a Corporate Bond

DuPont issued a 30 year bonds with a coupon rate of
7.95%.
 Interest is




paid semi-annually
These bonds currently have 28 years remaining to
maturity and are rated AA.
The bonds have a par value of $1,000
Newly issued AA bonds with maturities greater than
10 years are currently yielding 7.73%
What is the value of DuPont bond today?
16
DuPont example (continued)
Annual interest ($) =
 Semiannual coupon payment =
 Semiannual discount rate =
 Number of semiannual periods=
 PV=

17
DuPont example (continued)
Annual interest ($) = 0.0795*1000 =79.50
 Semiannual coupon payment =
 Semiannual discount rate =
 Number of semiannual periods=
 PV=

18
DuPont example (continued)
Annual interest ($) = 0.0795*1000 =79.50
 Semiannual coupon payment = 79.5/2= 39.75
 Semiannual discount rate =
 Number of semiannual periods=
 PV=

19
DuPont example (continued)
Annual interest ($) = 0.0795*1000 =79.50
 Semiannual coupon payment = 79.5/2= 39.75
 Semiannual discount rate = 0.0773/2 =0.03865
 Number of semiannual periods=
 PV=

20
DuPont example (continued)
Annual interest ($) = 0.0795*1000 =79.50
 Semiannual coupon payment = 79.5/2= 39.75
 Semiannual discount rate = 0.0773/2 =0.03865
 Number of semiannual periods= 28*2 = 56
 PV=

N = ??, I/Y = ??, PV= ????, PMT =??, FV=??
21
DuPont example (continued)
Annual interest ($) = 0.0795*1000 =79.50
 Semiannual coupon payment = 79.5/2= 39.75
 Semiannual discount rate = 0.0773/2 =0.03865
 Number of semiannual periods= 28*2 = 56
 PV=

 N=
56, I/Y = ??, PV= ????, PMT =??, FV=??
22
DuPont example (continued)
Annual interest ($) = 0.0795*1000 =79.50
 Semiannual coupon payment = 79.5/2= 39.75
 Semiannual discount rate = 0.0773/2 =0.03865
 Number of semiannual periods= 28*2 = 56
 PV=

N = 56, I/Y = 3.865, PV= ????, PMT = ??, FV= ??
23
DuPont example (continued)
Annual interest ($) = 0.0795*1000 =79.50
 Semiannual coupon payment = 79.5/2= 39.75
 Semiannual discount rate = 0.0773/2 =0.03865
 Number of semiannual periods= 28*2 = 56
 PV=

N = 56, I/Y = 3.865, PV= ????, PMT = 39.75, FV= ??
24
DuPont example (continued)
Annual interest ($) = 0.0795*1000 =79.50
 Semiannual coupon payment = 79.5/2= 39.75
 Semiannual discount rate = 0.0773/2 =0.03865
 Number of semiannual periods= 28*2 = 56
 PV=

N = 56, I/Y = 3.865, PV= ????, PMT = 39.75, FV = 1,000
25
DuPont example (continued)





Annual interest ($) = 0.0795*1000 =79.50
Semiannual coupon payment = 79.5/2= 39.75
Semiannual discount rate = 0.0773/2 =0.03865
Number of semiannual periods= 28*2 = 56
PV= 1,025.06
 The
bond is currently selling for 1,025.06
N = 56, I/Y = 3.865, PV= ????, PMT = 39.75, FV= 1,000
26
Level Coupon Bond: Example (Given)

Consider a U.S. government bond with a 6 3/8%
coupon that expires in December 2010.
 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, 2006 the size and timing of cash flows are:
 The
1 /1 / 06
require annual rate is 5%
$ 3 1 .8 7 5
$ 3 1 .8 7 5
$ 3 1 .8 7 5
$ 1,0 3 1 .8 7 5
6 /30 /06
12 /31/06
6 / 30 /10
12 / 31 /10
27
Level Coupon Bond: Example (Given)

Coupon Rate 6 3/8%, pay semi-annually
 10




Semi-Annual Payments of $31.875.
Maturity December 2010, Start Jan. 2006
The Par Value of the bond is $1,000.
The require annual rate is 5%
N = 10, I/Y = 2.5, PV=???, PMT = 31.875,
FV=1,000::: PV = $1,060.17

31.875
1
1,000
Bond Value 
1


10 
10
0.025  (1.025)  (1.025)
28
Valuing a Corporate Bond (Given)

Value a bond with the following
characteristics (calculator):
 Face
value: $1,000
 Coupon rate (C ): 8%
 Time to maturity: 4 years
 Discount rate: 9%
 Present Value: $967.02

You should know how to get any one of
these numbers given the other 4.
29
YTM and Bond Prices

How are prices and YTM related?
 Inversely,
as one goes up the other goes down
 As you pay more for the bond you earn a lower
return
30
Coupon Rate and YTM



Coupon rate = YTM
 Price = Face, Bond is selling at Par
 Coupons provide all the required return
Coupon rate > YTM
 Price > Face, Bond is selling at a Premium
 Coupons provide more than the required return
Coupon rate < YTM
 Price < Face, Bond is selling at a Discount
 Coupons do not provide the required return need
to increased the return by paying less
31
YTM and Bond Value
When the YTM < coupon, the bond
trades at a premium.
Bond Value
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
Coupon Rate
0.08
0.09
0.1
Discount Rate
When the YTM > coupon, the bond trades at a discount.
32
Computing Yield to Maturity
Finding the YTM requires trial and error if you
do not have a financial calculator
 If you have a financial calculator, enter N, PV,
PMT, and FV,

 Remembering

the sign convention
PMT and FV need to have the same sign, PV the
opposite sign
33
YTM with Semiannual Coupons

A bond has a 10% coupon rate, 20yrs to maturity,
makes coupon payments semi-annually, a $1,000
face, and is selling at $1,197.93
 Is

the YTM more or less than 10%?
LESS
 What

(1,000 * 0.10) / 2 = $50
 How

is the semi-annual coupon payment?
many periods are there?
20 * 2 = 40
 What
is the YTM?
N= 40,I/Y = ?, PV= -1197.93, PMT = 50, FV= 1,000→ 3.99%
 YTM
= 7.99998011%
34
YTM with Annual Coupons (Given)

Consider a bond with a 10% annual coupon rate, 15
years to maturity, and a par value of $1,000. The
current price is $928.09.
 Will the YTM be more
 MORE
 What is the YTM?
 N
= 15
 I/Y
= ???? = 11%
 PV
= 928.09
 PMT = 100
 FV
= 1000
or less than 10%?
35
Rate Changes and Bond Prices
Known as interest rate risk
 Consider two identical 8% coupon bonds except
that one matures in 4 years, the other matures in
10 years
 Calculate the change in the price of each bond
if:

 Interest
rates fall from 8% to 6%
 Interest rates rise from 8% to 10%
36
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=_, I/Y = _, PV=_, PMT = _, FV = _

4 years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_ PMT = _, FV = _

10 years @ 6%, 8% Coupon
 N=_, I/Y = _, PV=_ PMT = _, FV = _

10 Years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_ PMT = _, FV = _
37
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=8, I/Y = _, PV=_,PMT = _, FV = _

4 years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_ PMT = _, FV = _

10 years @ 6%, 8% Coupon
 N=_, I/Y = _, PV=_ PMT = _, FV = _

10 Years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_ PMT = _, FV = _
38
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=8, I/Y = 3, PV=_, PMT = _, FV = _

4 years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_ PMT = _, FV = _

10 years @ 6%, 8% Coupon
 N=_, I/Y = _, PV=_ PMT = _, FV = _

10 Years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_ PMT = _, FV = _
39
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=8, I/Y = 3, PV=?, PMT = _, FV = _

4 years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_ PMT = _, FV = _

10 years @ 6%, 8% Coupon
 N=_, I/Y = _, PV=_ PMT = _, FV = _

10 Years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_ PMT = _, FV = _
40
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=8, I/Y = 3, PV=?, PMT = 40, FV = _

4 years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_ PMT = _, FV = _

10 years @ 6%, 8% Coupon
 N=_, I/Y = _, PV=_ PMT = _, FV = _

10 Years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_ PMT = _, FV = _
41
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=8, I/Y = 3, PV=?, PMT = 40, FV = 1,000

4 years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_ PMT = _, FV = _

10 years @ 6%, 8% Coupon
 N=_, I/Y = _, PV=_ PMT = _, FV = _

10 Years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_ PMT = _, FV = _
42
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000


PV =$1,070.20
4 years @ 10%, 8% Coupon
 N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% Coupon
 N=_, I/Y = _, PV=_, PMT = _, FV = _

10 Years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_, PMT = _, FV = _
49
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000


PV =$1,070.20
4 years @ 10%, 8% Coupon
 N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% Coupon
 N=20, I/Y = _, PV=_ PMT = _, FV = _

10 Years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_, PMT = _, FV = _
50
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000


PV =$1,070.20
4 years @ 10%, 8% Coupon
 N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% Coupon
 N=20, I/Y = 3, PV=_ PMT = _, FV = _

10 Years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_, PMT = _, FV = _
51
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000


PV =$1,070.20
4 years @ 10%, 8% Coupon
 N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% Coupon
 N=20, I/Y = 3, PV=? PMT = _, FV = _

10 Years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_, PMT = _, FV = _
52
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000


PV =$1,070.20
4 years @ 10%, 8% Coupon
 N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% Coupon
 N=20, I/Y = 3, PV=? PMT = 40, FV = _

10 Years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_, PMT = _, FV = _
53
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000


PV =$1,070.20
4 years @ 10%, 8% Coupon
 N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $935.37

10 years @ 6%, 8% Coupon
 N=20, I/Y = 3, PV=? PMT = 40, FV = 1,000

10 Years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_, PMT = _, FV = _
54
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000


4 years @ 10%, 8% Coupon
 N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000


PV = $935.37
10 years @ 6%, 8% Coupon
 N=20, I/Y = 3, PV=? PMT = 40, FV = 1,000


PV =$1,070.20
PV = $1,148.77
10 Years @ 10%, 8% Coupon
 N=_, I/Y = _, PV=_, PMT = _, FV = _
55
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000


4 years @ 10%, 8% Coupon
 N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000


PV = $935.37
10 years @ 6%, 8% Coupon
 N=20, I/Y = 3, PV=? PMT = 40, FV = 1,000


PV =$1,070.20
PV = $1,148.77
10 Years @ 10%, 8% Coupon
 N=20, I/Y = _, PV=_ PMT = _, FV = _
56
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000


4 years @ 10%, 8% Coupon
 N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000


PV = $935.37
10 years @ 6%, 8% Coupon
 N=20, I/Y = 3, PV=? PMT = 40, FV = 1,000


PV =$1,070.20
PV = $1,148.77
10 Years @ 10%, 8% Coupon
 N=20, I/Y = 5, PV=_ PMT = _, FV = _
57
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000


4 years @ 10%, 8% Coupon
 N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000


PV = $935.37
10 years @ 6%, 8% Coupon
 N=20, I/Y = 3, PV=? PMT = 40, FV = 1,000


PV =$1,070.20
PV = $1,148.77
10 Years @ 10%, 8% Coupon
 N=20, I/Y = 5, PV=? PMT = _, FV = _
58
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000


4 years @ 10%, 8% Coupon
 N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000


PV = $935.37
10 years @ 6%, 8% Coupon
 N=20, I/Y = 3, PV=? PMT = 40, FV = 1,000


PV =$1,070.20
PV = $1,148.77
10 Years @ 10%, 8% Coupon
 N=20, I/Y = 5, PV=? PMT = 40, FV = _
59
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000


4 years @ 10%, 8% Coupon
 N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000


PV = $935.37
10 years @ 6%, 8% Coupon
 N=20, I/Y = 3, PV=? PMT = 40, FV = 1,000


PV =$1,070.20
PV = $1,148.77
10 Years @ 10%, 8% Coupon
 N=20, I/Y = 5, PV=? PMT = 40, FV = 1,000
60
Rate Change and Bond Pricing

4 years @ 6%, 8% Coupon
 N=8, I/Y = 3, PV=? PMT = 40, FV = 1,000


4 years @ 10%, 8% Coupon
 N=8, I/Y = 5, PV=? PMT = 40, FV = 1,000


PV = $935.37
10 years @ 6%, 8% Coupon
 N=20, I/Y = 3, PV=? PMT = 40, FV = 1,000


PV =$1,070.20
PV = $1,148.77
10 Years @ 10%, 8% Coupon
 N=20, I/Y = 5, PV=? PMT = 40, FV = 1,000

PV = $875.38
61
Interest Rates and Time to Maturity
The longer a bond has till maturity, the greater
the price impact of a change in interest rates
 WHY?
 Longer maturity bond have more payments
affected by the rate change, and the
principal repayment is further away so it
will be more heavily discounted

62
Interest Rates and Bond Prices

Bond Prices and Interest Rates have an Inverse
Relationship
63
Pricing Stocks
Remember: The value of any asset is the
present value of its expected future cash flows.
 Bond cash flows are: Coupon & Principal
 Stock produces cash flows from:
Dividends
Capital Gains

64
Stock Valuation Terminology
Dt or Divt –dividend expected at time t
 P0 – market price of stock at time 0
 Pt – expected mkt price of stock at time t
 g- expected growth rate of dividends
 rs or re- required rate of return on equity
 D1 / P0 – expected one-year dividend yield
 (P1 - P0)/ P0 – expected one year capital gain

 The
stocks total return = div yield + cap. gain
65
Valuing Common Stock

The price of a share is simply the present value of
the expected future cash flows
An investor planning on selling his share in a
year is willing to pay:
The investor buying the share next year plans
on selling it a year later so he is only willing to
pay:
66
Valuing Common Stock

The price of a share is simply the present value of
the expected future cash flows
An investor planning on selling his share in a
year is willing to pay: P0=(D1+P1)/(1+R)
The investor buying the share next year plans
on selling it a year later so he is only willing to
pay:
67
Valuing Common Stock

The price of a share is simply the present value of
the expected future cash flows
An investor planning on selling his share in a
year is willing to pay: P0=(D1+P1)/(1+R)
The investor buying the share next year plans
on selling it a year later so he is only willing to
pay: P1=(D2+P2)/(1+R)
Therefore: P0
68
Valuing Common Stock

The price of a share is simply the present value of
the expected future cash flows
An investor planning on selling his share in a
year is willing to pay: P0=(D1+P1)/(1+R)
The investor buying the share next year plans
on selling it a year later so he is only willing to
pay: P1=(D2+P2)/(1+R)
Therefore: P0=(D1+{(D2+P2)/(1+R)})/(1+R)
P0=D1 / (1+R) + (D2 + P2)/(1+R)2
69
Keep Going

This process can be repeated into the future
D iv 1
D iv 2
D iv H  PH
P0 

 ... 
1
2
H
(1  r )
(1  r )
(1  r )
Using summation:
 P0 = H Dh / (1 + r)h + PH / (1 + r)H
 What happens to PH as H approaches infinity?

 The
present value becomes insignificant
70
Dividend Valuation Model

As H approaches infinity PH goes to zero
 Because
of this we only need to be concerned with
the stock’s future dividends

The price of a stock is equal to the present
value of its expected future dividends
71
Constant Dividend

How do you value a stock that will pay a
constant dividend?
 Hint:
what does the cash flow stream look similar
to?
 Firms are a going concern so treat dividends as
a perpetual cash flow
P = D / r
72
Constant Dividend Example

What is the value of a stock that is expected to
pay a constant dividend of $2 per share?
 The
required rate of return is 10%
 P = 2 / 0.1 = 20
73
Growing Dividends
Now we are assuming that the firm’s dividends
will grow at a constant rate, g forever
 This is similar to a: A Growing Perpetuity
 So the price of a share is: P = D1 / (r-g)

D iv 1  D iv 0 ( 1  g )
D iv 2  D iv 1 ( 1  g )  D iv 0 ( 1  g ) 2
D iv 3  D iv 2 ( 1  g )  D iv 0 ( 1  g ) 3
74
Growing Dividend Example
Geneva steel just paid a dividend of $2.10.
Dividend payments are expected to grow at a
constant rate of 6%. The appropriate discount
rate is 12%. What is the price of Geneva
stock?
 Div1 =
 P0 =

75
Growing Dividend Example
Geneva steel just paid a dividend of $2.10.
Dividend payments are expected to grow at a
constant rate of 6%. The appropriate discount
rate is 12%. What is the price of Geneva
stock?
 Div0 = $2.10 so Div1 = 2.10*(1.06)=$ 2.226
 P0 = 2.226 / (0.12- 0.06) = $37.10

76
Valuing Stock with Changing g
1.
2.
3.
4.
Find the PV of dividends during the period of
non-constant growth, PA
Find the price of the stock at the end of the nonconstant growth period, PN
Discount the price found in 2 back to the
present, PB
Add the two present values (1+3) to find the
intrinsic value (price) of the stock P0 = PA + PB
77
Differential Growth Rates
Dividends will grow at g1 for N years and g2
thereafter
Step 1: An N-year annuity growing at rate g1

C
PA 
R  g1

(1  g 1 )
1


(1  R )

{
}
T



Step 2: A growing perpetuity at rate g2
PN = DivN+1 / (R-g2)
Step 3: PB = PN / (1+R)N
Step 4: P0 = PA + PB
78
Non-Constant Growth Example
(Given)

Websurfers Inc, a new internet firm is expected to do
very well during its initial growth period. Investors
expect its dividends to grow at 25% for the next 3
years. Obviously one cannot expect such
extraordinary growth to continue forever, and it is
expected that dividends will grow at 5% after year 3
in perpetuity. Its current dividend is $1/share.
Required rate of return on the stock = 10%. Calculate
what the current price should be.
79
Websurfer Inc, Example (Given)
1
1*1.25
= 1.25
1.25*1.25
=1.56
1.56*1.25
= 1.95
1.95*1.05
= 2.05
0
1
2
3
4
1.PA=[(1.25)/(0.10-0.25)]*[1-{1.25/1.10}3] =

2.PN


2.05*1.05
= 2.15
5
3.90
D1 = D0 * (1 + g1) = 1 * 1.25 = 1.25
={2.05}/(0.10-0.05) = 41.00
D4 = D3 * (1 + g2) = 1.95 * 1.05 = 2.05
D4 = D0*(1 + g1)3*(1 + g2)= 1*1.253* 1.05 = 2.05
3.PB =41.00/(1.103)
= 30.80
4.P0 = PA + PB = 3.90+ 30.80 = $34.70
80
A Differential Growth Example


0
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? The discount rate is 12%.
2*1.081
=2.16
2*1.082
=2.33
2*1.083
1
2
3
=2.52
2*1.083 *1.04
=2.62
4
2*1.083*
1.042 =
2.72
5
81
Solution

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. R=12%
PA =
2. PN =
3. PB =
4. P0 = PA + PB =
1.
82
Solution
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. R=12%
PA=[(2*1.08)/(0.12-0.08)]*[1-{1.08/1.12}3]=5.58
2. PN =
3. PB =
4. P0 = PA + PB =
1.
83
Solution
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. R=12%
PA=[(2*1.08)/(0.12-0.08)]*[1-{1.08/1.12}3]=5.58
2. PN ={2*1.083*1.04}/(0.12-0.04) = 32.75
3. PB =
4. P0 = PA + PB =
1.
84
Solution
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. R=12%
PA=[(2*1.08)/(0.12-0.08)]*[1-{1.08/1.12}3]=5.58
2. PN ={2*1.083*1.04}/(0.12-0.04) = 32.75
3. PB =32.75/(1.123) = 23.31
4. P0 = PA + PB =
1.
85
Solution
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. R=12%
PA=[(2*1.08)/(0.12-0.08)]*[1-{1.08/1.12}3]=5.58
2. PN ={2*1.083*1.04}/(0.12-0.04) = 32.75
3. PB =32.75/(1.123) = 23.31
4. P0 = PA + PB = 5.58 + 23.31 = $28.89
1.
86
Important Parameters

The value of a firm depends on the discount
rate, the growth rate, and the initial dividend.
87
The Discount Rate
 The
market consensus of the firm’s required
rate
This
is the Market Capitalization Rate
Return that an investor expects to make
This is similar to what for a bond?
Yield to Maturity
88
Where does “r” come from?


We generally estimate r from one of the dividend
valuation models
Using constant dividend growth model:
D1
P0 
R -g
Rearrange and solve for R:

D1
R
g
P0
In practice, estimates of r have a lot of estimation
error
89
Where does “R” come from?
D1
R
g
P0
What is D1/P0? Dividend Yield
 What is g? The growth rate, or the Capital
Gains Yield
 An investor’s return comes from either the
dividends received or price appreciation

90
Classifying Stocks
 Firms
are often classified based on where
investors expect to earn their return from
“Income/Value stocks”:
have a higher
dividend yield
“Growth stocks”: have a higher growth
component
 As
long as both are equally risky, the
return should be the same
91
Where does “g” come from?

From analysts' estimates
 I/B/E/S,

Google, Yahoo, or WSJ
From earnings re-investment
g
= plowback ratio * ROE

How much does the firm reinvest, and what is the return on
the investment
92
Link between stock prices and earnings
A “new valuation model” :
 Consider a firm with a 100% payout ratio, so
Div = EPS and earnings remain flat.
 P0 = DIV / r
 Because Div = EPS
 P0 = EPS / r

93
Present Value of all Future Growth
Opportunities (PVGO)

The price is composed of the value of the
firm’s current assets (100% payout firm) and
the firm’s growth opportunities
 Growth
opportunities are opportunities to invest in
positive NPV projects.
P0 = EPS / r + PVGO
EPS / r : This is the value of the firm’s current assets
PVGO : This is the value of what the firm can invest in
94
Who cares about PVGO?

For what type of stock is the PVGO more
important?
 Growth
or Value stocks
95
Who cares about PVGO?

For what type of stock is the PVGO more
important?
 Growth
or Value stocks
96
PVGO Example
Assume that a firm has 2 potential projects.
Project A & B with NPV’s of $2m, and $3m,
respectively. The firm pays out all its earnings as
dividends, and paid a dividend of $1/share last
year. It has 200,000 shares outstanding. Assume
the discount rate is 10%.
 What is the share price, if the cash flow from the
firm's existing assets are expected to remain the
same in perpetuity, and the firm takes on Project
A, and B?

97
PVGO Example
P0 = Div / r + PVGO, since Div = EPS
P0 = 1 / 0.1 + PVGO
What are the firm’s growth opportunities? Worth?
Project A & B, $5 million
PVGO per share?
5,000,000 / 200,000 = $25
P0 = 1 / 0.1 + 25 = $35 / share
The firm is worth: 200,000 * 35 = $7million
98
Stock Value Represents:
Present value of expected future dividends,
 Present value of free cash flow,
 Present value of average future earnings under
a no-growth policy plus the present value of
growth opportunities

99
Price-Earnings Ratio

The price-earnings ratio is calculated as the
current stock price divided by annual EPS.
Wall Street Journal uses last 4 quarter’s
earnings
 The
P rice p er share
P /E ratio 
EPS

Many analysts use this to determine how the
market feels about a company
100
Price/Earnings Ratio


Is selling at a high P/E good?
Why might the P/E be high:
1.
2.
3.

r is low (investors think the firm is relatively safe)
Good growth opportunities (high PVGO)
Current EPS is low
Remember, earnings are an accounting measure,
which means P/E is an accounting measure
101
Problem 1 (Given)
A firm is expected to grow at 25% for the next 3
years. Its growth is expected to decline to
15% for the following 4 years. It is then
expected to grow at 5% in perpetuity. Find
the current share price if the current dividend
is $1 and the discount rate is 10%.
102
Problem 1 (Given)
A firm is expected to grow at 25% for the next 3 years. Its growth is expected to
decline to 15% for the following 4 years. It is then expected to grow at 5% in
perpetuity. Find the current share price if the current dividend is $1 and the
discount rate is 10%.
PA=[(1*1.25)/(0.10-0.25)]*[1-{1.25/1.10}3]=3.89
PN1 ={1*1.253*1.15}/(0.10-0.15)]*
[1-{1.15/1.10}4]=8.76
PB =8.76/(1.103) = 6.58
PN2 ={1*1.253*1.154*1.05}/(0.10-0.05)] = 71.80
PC =71.80/(1.107) = 36.83
P0 = PA + PB + Pc = 3.89+6.58+36.83 = $47.30
103
Problem 2 (Given)
Consider a firm whose dividend growth is
expected to decline gradually. For the next two
years, the growth is expected to be 20%. In the
following years, it is expected to grow at 18%,
13% and 10%. From year 6 onwards,
dividends are expected to grow at 5% for
perpetuity. Assume the current dividend is $1
and the required rate of return is 10%. What is
the current price?
104
Problem 2: Phase 1 – Years 1-5
(Given)
DIV1 = 1.00 * 1.20 = 1.20
 DIV2 = 1.20 * 1.20 = 1.44
 DIV3 = 1.44 * 1.18 = 1.70
 DIV4 = 1.70 * 1.13 = 1.92
 DIV5 = 1.92 * 1.10 = 2.11
 PA = 1.2/1.1 + 1.44/1.12 + 1.70/1.13 +
1.92/1.14 + 2.11/1.15 = 6.18

105
Problem 2 Phase 2 – Years 6-
(Given)
DIV6 = 2.11 * 1.05 = 2.22
 PN = DIV6 / (r-g) = 2.22/(0.10-0.05) = 44.4
 PB = 44.4 * 1/1.15 = 27.57
 Current Price = 6.18 + 27.57 = $33.75

106
Quick Quiz






How do you find the value of a bond, and why do
bond prices change?
What is a bond indenture, and what are some of the
important features?
What determines the price of a share of stock?
What determines g and R in the DGM?
Decompose a stock’s price into constant growth and
NPVGO values.
Discuss the importance of the PE ratio.
107
Why We Care
Basic real world application of the time value
of money
 Foundation of Investment/ Financial Analysis

108