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Corporate FinanceRoss Westerfield Jaffe
Sixth Edition
5
Chapter Five
How to Value Bonds
and Stocks
Prepared by
Gady Jacoby
University of Manitoba
and
Sebouh Aintablian
American University of
Beirut
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Chapter Outline
5.1 Definition and Example of a Bond
5.2 How to Value Bonds
5.3 Bond Concepts
5.4 The Present Value of Common Stocks
5.5 Estimates of Parameters in the Dividend-Discount Model
5.6 Growth Opportunities
5.7 The Dividend Growth Model and the NPVGO
Model (Advanced)5.8 Price Earnings Ratio
5.9 Stock Market Reporting
5.10 Summary and Conclusions
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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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5.1 Definition and Example of a Bond
A bond is a legally binding agreement between aborrower (bond issuer) and a lender (bondholder):
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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5.1 Definition and Example of a Bond
Consider a Government of Canada bond listed as
6.375 of December 2009.
ThePar Valueof the bond is $1,000.
Coupon paymentsare made semi-annually (June 30 and
December 31 for this particular bond). Since the coupon rateis 6.375 the payment is $31.875.
On January 1, 2002 the size and timing of cash flows are:
02/1/1
875.31$
02/30/6
875.31$
02/31/12
875.31$
09/30/6
875.031,1$
09/31/12
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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, theyield to maturityis thediscount rate.
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Pure Discount Bonds
Information needed for valuing pure discount bonds:
Time to maturity (T) = Maturity date - todays date
Face value (F)
Discount rate (r)
Tr
FPV
)1(
Present value of a pure discount bond at time 0:
0
0$
1
0$
2
0$
1T
F$
T
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Pure Discount Bonds: Example
Find the value of a 30-year zero-coupon bondwith a $1,000 par value and a YTM of 6%.
11.174$)06.1(
000,1$
)1( 30
Tr
FPV
0$0$0$
29
0001$
0
0$
1
0$
2
0$
29
000,1$
30
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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
TTr
F
rr
CPV
)1()1(
11
Value of a Level-coupon bond= PV of coupon payment annuity + PV of face value
0
C$
1
C$
2
C$
1T
FC $$
T
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Level-Coupon Bonds: Example
Find the present value (as of January 1, 2002), of a 6.375
coupon Government of Canada bond with semi-annual
payments, and a maturity date of December 31, 2009 if the
YTM is 5-percent.
On January 1, 2002 the size and timing of cash flows are:
02/1/1
875.31$
02/30/6
875.31$
02/31/12
875.31$
09/30/6
875.031,1$
09/31/12
75.089,1$)025.1(
000,1$
)025.1(
11
205.
875.31$1616
PV
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Bond Market Reporting
CANADA
Coupon Mat. Date Bid $ Yld%Canada 6.375 Dec 31/09 108.98 5.00
The Governmentof Canada issued
this bond
The bond paysan annual
coupon rate of
6.375%
The bondmatures on
December 31,
2009
The bond is selling
at 108.98% of the
face value of
$1,000
The bondsquoted annual
yield to
maturity is 5%
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5.3 Bond Concepts
1. Bond prices and market interest rates move in oppositedirections.
2. When coupon rate = YTM, price = par value.When coupon rate > YTM, price > par value (premium
bond)When coupon rate < YTM, price < par value (discountbond)
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 changethan a higher coupon bond when YTM changes. All other
features are identical.
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YTM and Bond Value
800
1000
1100
1200
1300
$1400
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Discount Rate
BondValue
6 3/8
When the YTM < coupon, the bond
trades at a premium.
When the YTM = coupon, the
bond trades at par.
When the YTM > coupon, the bond trades at a discount.
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Maturity and Bond Price Volatility
C
Consider two otherwise identical bonds.
The long-maturity bond will have much more
volatility with respect to changes in the
discount rate
Discount Rate
BondValue
Par
Short Maturity Bond
Long Maturity
Bond
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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
Discount Rate
BondValue
High Coupon Bond
Low Coupon Bond
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Holding Period Return
Suppose that on January 1, 2002, you bought the above6.375 coupon Government of Canada bond with semi-annual
payments, and a maturity date of December 31, 2009.
At that time the YTM was 5-percent, and you paid $1,089.75
(the PV of the bond). Six months later (July 1, 2002), You sold the bond when the
YTM was 4-percent. The size and timing of cash flows (as of
July 1, 2002 ) were:
02/1/7
875.31$
02/31/12
875.31$
03/30/6
875.31$
09/30/6
875.031,1$
09/31/12
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Your holding period returnwas:
This annualizes to an effective rate of:
Given that the YTM at that time was 4-percent, you sold thebond for:
Holding Period Return (continued)
59.152,1$
)02.1(
000,1$
)02.1(
11
204.
875.31$1515
PV
%77.5$1,089.75
1,089.75$59.152,1$
%87.111)0577.1( 2
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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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Case 1: Zero Growth
Assume that dividends will remain at the same levelforever
rP
rrrP
Div
)1(
Div
)1(
Div
)1(
Div
0
3
3
2
2
1
1
0
321 DivDivDiv
Since future cash flows are constant, the value of a zerogrowth stock is the present value of a perpetuity:
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P0= Div1/ r
= 0.75/0.12 = $6.25
ABC Corp. is expected to pay $0.75 dividend per annum,starting a year from now, in perpetuity. If stocks of similar
risk earn 12% annual return, what is the price of a share of
ABC stock?
The stock price is given by the present value of theperpetual stream of dividends:
A Zero Growth Example
0 1 2 3 4
$0.75 $0.75 $0.75 $0.75
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Case 2: Constant Growth
)1(DivDiv 01 g
Since future cash flows grow at a constant rate forever,
the value of a constant growth stock is the presentvalue of a growing perpetuity:
gr
P
10Div
Assume that dividends will grow at a constant rate,g,forever. i.e.
2
012 )1(Div)1(DivDiv gg
3
023 )1(Div)1(DivDiv gg ...
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A Constant Growth Example
XYZ Corp. has a common stock that paid itsannual dividend this morning. It is expected to
pay a $3.60 dividend one year from now, and
following dividends are expected to grow at arate of 4% per year forever.
If stocks of similar risk earn 16% effectiveannual return, what is the price of a share of
XYZ stock?
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A Constant Growth Example (continued)
The stock price is given by the the present valueof the perpetual stream of growing dividends:
$3.60 $3.601.04 $3.601.042 $3.601.043
P0= Div1/ (r-g)
= 3.60/(0.16-0.04)
= $30.00
0 1 2 3 4
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Case 3: Differential Growth
Assume that dividends will grow at differentrates in the foreseeable future and then willgrow at a constant rate thereafter.
To value a Differential Growth Stock, we needto:
Estimate future dividends in the foreseeablefuture.
Estimate the future stock price when the stockbecomes a Constant Growth Stock (case 2).
Compute the total present value of the estimatedfuture dividends and future stock price at the
appropriate discount rate.
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Case 3: Differential Growth
)(1DivDiv 101 g
Assume that dividends will grow at rateg1forNyears and grow at rateg2thereafter
2
10112 )(1Div)(1DivDiv gg
N
NN gg )(1Div)(1DivDiv 1011
)(1)(1Div)(1DivDiv 21021 ggg N
NN
.
.
.
.
.
.
2
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Case 3: Differential Growth
)(1Div 10 g
Dividends will grow at rateg1forNyears andgrow at rateg2thereafter
2
10 )(1Div g
Ng)(1Div 10 )(1)(1Div
)(1Div
210
2
gg
g
NN
0 1 2
N N+1
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Case 3: Differential Growth
We can value this as the sum of:
anN-year annuity growing at rateg1
T
T
A
r
g
gr
CP
)1(
)1(1 1
1
plus the discounted value of a perpetuity growing at rateg2that starts in yearN+1
NB r
grP
)1(
Div
2
1N
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Case 3: Differential Growth
To value a Differential Growth Stock, we can use
NT
T
rgr
rg
grCP
)1(
Div
)1()1(1 2
1N
1
1
Or we can cash flow it out.
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A Differential Growth Example
A common stock just paid a dividend of $2.The dividend is expected to grow at 8% for 3
years, then it will grow at 4% in perpetuity.
If stocks of similar risk earn 12% effective
annual return, what is the stock worth?
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With the Formula
NT
T
r
gr
r
g
gr
CP
)1(
Div
)1(
)1(1 2
1N
1
1
3
3
3
3
)12.1(
04.12.
)04.1()08.1(2$
)12.1(
)08.1(1
08.12.
)08.1(2$
P
3)12.1(75.32$8966.154$ P
31.23$58.5$ P 89.28$P
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A Differential Growth Example (continued)
08).2(1$ 2
08).2(1$
0 1 2 3 4
3
08).2(1$ )04.1(08).2(1$ 3
16.2$ 33.2$
0 1 2 3
08.
62.2$52.2$
89.28$
)12.1(
75.32$52.2$
)12.1(
33.2$
12.1
16.2$320
P
75.32$08.
62.2$3 P
The constant
growth phasebeginning in year 4
can be valued as a
growing perpetuity
at time 3.
5 31 i f i h
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Dividend-Discount Model
The value of a firm depends upon its growth
rate,g, and its discount rate, r.
Where doesg come from?Where does rcome from?
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Where doesgcome from?
Formula for Firms Growth Rate (g):
The firm will experience earnings growth if its net
investment (total investment-depreciation) is
positive.
To grow, the firm must retain some of its earnings.
This leads to:
Earnings
next Year
Earnings
this Year
Retained
earnings
this Year
Return on
retained
earnings= +
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This leads to the formula for the firms growth rate:
g= Retention ratio Return on retained earnings
The return on retained earnings can be estimatedusing the firms historical return on equity (ROE)
Where doesgcome from?
Dividing both sides by this years earnings, we get:
Earnings this Year
Earnings next Year1
Retained earnings
this YearReturn on
retained
earnings= +
Earnings this Year
1 +g Retention ratio
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Where doesgcome from? An Example
Ontario Book Publishers (OBP) just reportedearnings of $1.6 million, and it plans to retain 28-
percent of its earnings.
If OBPs historical ROE was 12-percent, what is the
expected growth rate for OBPs earnings?
With the above formula:
g= 0.28 0.12 = 0.0336 = 3.36%
Or:
Total earnings
Change in earnings=
$1.6 million
(0.28$1.6 million)0.12= 0.0336
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In practice, there is a great deal of estimationerror involved in estimating r.
The discount rate can be broken into twoparts.
The dividend yield
The growth rate (in dividends)
From the constant growth cas, we can write:
Where does rcome from?
gP
r 0
1Div
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Where does rcome from? An Example
Manitoba Shipping Co. (MSC) is expected to pay adividend next year of $8.06 per share. Future
Dividends for MSC are expected to grow at a rate of
2% per year indefinitely.
If an investor is currently willing to pay $62.00 per
one MSC share, what is her required return for this
investment?
With the above formula:
r = (8.06/62) + 0.02 = 0.15 = 15%
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5.6 Growth Opportunities
Growth opportunities are opportunities toinvest in positive NPV projects.
The value of a firm can be conceptualized as
the sum of the value of a firm that pays out100-percent of its earnings as dividends and
the net present value of the growth
opportunities.
NPVGOr
EPSP
5-38 5 7 Th Di id d G th M d l d th
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5 38 5.7 The Dividend Growth Model and the
NPVGO Model (Advanced)
We have two ways to value a stock:
The dividend discount model.
The price of a share of stock can be calculated as
the sum of its price as a cash cow plus the per-share value of its growth opportunities.
5-39 Th Di id d G th M d l d th
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5 39 The Dividend Growth Model and the
NPVGO Model
Consider a firm that has EPS of $5 at the end of thefirst year, a dividend-payout ratio of 30-percent, a
discount rate of 16-percent, and a return on retained
earnings of 20-percent.
The dividend at year one will be $5 .30 = $1.50 per share.
The retention ratio is .70 ( = 1 -.30) implying a growth rate
in dividends of 14% = .70 20%
From the dividend growth model, the price of a share is:
75$14.16.
50.1$Div 10
grP
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The NPVGO Model
First, we must calculate the value of the firm as acash cow.
25.31$16.
5$EPS
r
Second, we must calculate the value of the growthopportunities.
75.43$14.16.
875$.16.
20.50.350.3
grNPVGO
Finally, 75$75.4325.310 P
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5.8 Price Earnings Ratio
Many analysts frequently relate earnings per share to
price.
The price earnings ratio is a.k.a the multiple
Calculated as current stock price divided by annual EPS
The National Postuses last 4 quarters earnings
Firms whose shares are in fashion sell at highmultiples. Growth stocksfor example.
Firms whose shares are out of favour sell at low
multiples. Value stocksfor example.
EPS
shareperPriceratioP/E
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Other Price Ratio Analysis
Many analysts frequently relate earnings per
share to variables other than price, e.g.:
Price/Cash Flow Ratio
cash flow = net income + depreciation = cash flow
from operations or operating cash flow
Price/Sales
current stock price divided by annual sales per share
Price/Book (a.k.a. Market to Book Ratio) price divided by book value of equity, which is
measured as assets - liabilities
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52W 52W Yield Vol Net
high low Stock Ticker Div % P/E 00s High Low Close chg42.10 32.25 BCE Inc BCE 1.20 3.5 11.8 19210 34.59 33.80 34.50 -0.47
5.9 Stock Market Reporting
BCE has
been as
high as
$42.10 in
the last
year.BCE has
been as low
as $32.25 in
the last year.
Given the
current price,
the dividend
yield is 3 %Given the
current price, the
P/E ratio is 11.8
times earnings
1,921,000 shares
traded hands in the
last days trading
BCE ended
trading at $34.50,down $0.47 from
yesterdays close
BCE pays a
dividend of 1.2
dollars/share
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5.9 Stock Market Reporting
BCE Incorporated is having a tough year, trading near their 52-
week low. Imagine how you would feel if within the past yearyou had paid $42.10 for a share of BCE and now had a share
worth $34.50! That $1.20 dividend wouldnt go very far in
making amends.
Yesterday, BCE had another rough day in a rough year. BCE
opened the day down beginning trading at $34.59, which was
down from the previous close of $34.97 = $34.50 + $0.47
52W 52W Yield Vol Net
high low Stock Ticker Div % P/E 00s High Low Close chg
42.10 32.25 BCE Inc BCE 1.20 3.5 11.8 19210 34.59 33.80 34.50 -0.47
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5.10 Summary and Conclusions
In this chapter, we used the time value ofmoney formulae from previous chapters to
value bonds and stocks.
1. The value of a zero-coupon bond is
2. The value of a perpetuity is
Tr
FPV
)1(
r
CPV
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5.10 Summary and Conclusions (continued)
3. The value of a coupon bond is the sum ofthe PV of the annuity of coupon payments
plus the PV of the par value at maturity.
4. The yield to maturity (YTM) of a bond isthat single rate that discounts the payments
on the bond to the purchase price.
TTr
Frr
CPV)1()1(
11
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5.10 Summary and Conclusions (continued)
5. A stock can be valued by discounting its
dividends. There are three cases:
1. Zero growth in dividends
2. Constant growth in dividends
3. Differential growth in dividends
r
P Div
0
grP
10Div
NT
T
r
gr
r
g
gr
CP
)1(
Div
)1(
)1(1 2
1N
1
1
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5.10 Summary and Conclusions (continued)
6. The growth rate can be estimated as:
g= Retention ratio Return on retained earnings
7. An alternative method of valuing a stock
was presented. The NPVGO values a stock
as the sum of its cash cow value plus the
present value of growth opportunities.
NPVGOr
EPSP