1. finance
2. investing
3. stocks

Chapter Six
How to Value Bonds
and Stocks
Prepared by
Professor Wei Wang
Queen’s University
© 2011 McGraw–Hill Ryerson Limited
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Chapter Outline
6.1
6.2
6.3
6.4
6.5
Definition and Example of a Bond
How to Value Bonds
Bond Concepts
The Present Value of Common Stocks
Estimates of Parameters in the DividendDiscount Model
6.6 Growth Opportunities
6.7 The Dividend Growth Model and the NPVGO
Model (Advanced)
6.8 Price-Earnings Ratio
6.9 Stock Market Reporting
6.10 Summary and Conclusions
© 2011 McGraw–Hill Ryerson Limited
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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.
© 2011 McGraw–Hill Ryerson Limited
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Definition and Example of a Bond LO6.1
• A bond is a legally binding agreement between a
borrower (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)
© 2011 McGraw–Hill Ryerson Limited
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Definition and Example of a Bond LO6.1
• Consider a Government of Canada bond listed as 5.000
December 2014.
– 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 5.000 the payment is $25.000.
– On January 1, 2011 the size and timing of cash flows are:
$25
$25
$25
$1,025
6 / 30 / 14
12 / 31 / 14
1 / 1 / 11
6 / 30 / 11
12 / 31 / 11
© 2011 McGraw–Hill Ryerson Limited
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How to Value Bonds
LO6.2
• Bond value is determined by the present
value of the coupon payments and par value.
• We have to identify the size and timing of
cash flows.
• Discount at the correct discount rate.
– Discount rates are inversely related to present
(i.e., bond) values.
– If you know the price of a bond and the size and
timing of cash flows, the yield to maturity is the
discount rate.
© 2011 McGraw–Hill Ryerson Limited
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Pure Discount Bonds
LO6.2
• Make no periodic interest payments (coupon rate =
0%)
• The entire yield to maturity comes from the
difference between the purchase price and the par
value.
• Cannot sell for more than par value
• Sometimes called zeroes, deep discount bonds, or
original issue discount bonds (OIDs)
• Treasury Bills and principal-only Treasury strips are
good examples of zeroes.
© 2011 McGraw–Hill Ryerson Limited
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Pure Discount Bonds
LO6.2
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 )
© 2011 McGraw–Hill Ryerson Limited
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Pure Discount Bonds: Example
LO6.2
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
$,0$
1
$00
0
20
9
23
1
0
1
2
30
29
F
$1,000
PV =
=
= $174.11
T
30
(1 + r )
(1.06)
© 2011 McGraw–Hill Ryerson Limited
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Pure Discount Bonds: Example
LO6.2
Find the value of a 30-year zero-coupon bond
with a $1,000 par value and a YTM of 6%.
N
I/Y
PV
30
6
– 174.11
PMT
FV
1,000
© 2011 McGraw–Hill Ryerson Limited
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Level Coupon Bonds
LO6.2
• Make periodic coupon payments in addition to the
maturity value
• The payments are equal each period. Therefore, the
bond is just a combination of an annuity and a
terminal (maturity) value.
• Coupon payments are typically semiannual.
© 2011 McGraw–Hill Ryerson Limited
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Level-Coupon Bonds
LO6.2
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
© 2011 McGraw–Hill Ryerson Limited
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Level-Coupon Bonds: Example
LO6.2
• Find the present value (as of January 1, 2011), of a 6-3/8
coupon Government of Canada bond with semi-annual
payments, and a maturity date of December 31, 2016 if the
YTM is 5-percent.
– 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, 2011 the size and timing of cash flows are:
$31.875
$31.875
$31.875
$1,031.875
6 / 30 / 16
12 / 31 / 16
1 / 1 / 11
6 / 30 / 11
12 / 31 / 11
© 2011 McGraw–Hill Ryerson Limited
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Level Coupon Bond: Example
LO6.2
• On January 1, 2011, the required annual yield
is 5%.
• The present value of the bond is:
 $1,000
$31.875 
1
PV =
1−
+
= $1,070.52

12 
12
.05 2  (1.025)  (1.025)
© 2011 McGraw–Hill Ryerson Limited
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Level-Coupon Bonds: Example
LO6.2
Find the present value (as of January 1, 2011), of a 6-3/8 coupon
Government of Canada bond with semi-annual payments, and a
maturity date of December 31, 2016 if the YTM is 5-percent.
N
I/Y
PV
PMT
12
5
– 1,070.52
31.875 =
1,000×0.06375
2
FV
1,000
© 2011 McGraw–Hill Ryerson Limited
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Bond Pricing with a Spreadsheet
LO6.2
• There are specific formulas for finding bond prices
and yields on a spreadsheet.
– PRICE(Settlement,Maturity,Rate,Yld,Redemption,
Frequency,Basis)
– YIELD(Settlement,Maturity,Rate,Pr,Redemption,
Frequency,Basis)
– Settlement and maturity need to be actual dates
– The redemption and Pr need to be given as % of par value
• Click on the first Excel icon for pricing the bond in
the 6-3/8 December 2016 bond.
• Click on the second Excel icon for another
example.
© 2011 McGraw–Hill Ryerson Limited
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Bond Market Reporting
LO6.2
CANADA
Canada
Coupon
6.375
Mat. Date
Dec 31/16
The Government
of Canada issued
this bond
The bond pays
an annual
coupon rate of
6.375%
Bid $
107.05
Yld%
5.00
The bond’s
quoted annual
yield to
maturity is 5%
The bond
matures on
December 31,
2016
The bond is selling
at 107.05% of the
face value of
$1,000
© 2011 McGraw–Hill Ryerson Limited
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Consols
LO6.2
• Not all bonds have a final maturity.
• British consols pay a set amount (i.e., coupon) every
period forever.
• These are examples of a perpetuity.
C
PV =
R
© 2011 McGraw–Hill Ryerson Limited
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Bond Concepts
LO6.3a
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.
© 2011 McGraw–Hill Ryerson Limited
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YTM and Bond Value
LO6.3a
Bond Value
$1400
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.
© 2011 McGraw–Hill Ryerson Limited
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Bond Example Revisited
LO6.3a
• Using our previous example, now assume that the
required yield is 11%.
• How does this change the bond’s price?
$31.875
$31.875
$31.875
$1,031.875
6 / 30 / 16
12 / 31 / 16
1 / 1 / 11
6 / 30 / 11
12 / 31 / 11
 $1,000
$31.875 
1
PV =
1−
+
= $800.70

12 
12
.11 2  (1.055)  (1.055)
• The bond trades at a discount.
© 2011 McGraw–Hill Ryerson Limited
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Bond Value
Maturity and Bond Price Volatility
LO6.3a
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
Discount Rate
Long Maturity
Bond
© 2011 McGraw–Hill Ryerson Limited
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Bond Value
Coupon Rate and Bond Price Volatility
LO6.3a
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
Low Coupon Bond
Discount Rate
© 2011 McGraw–Hill Ryerson Limited
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Holding Period Return
LO6.3b
• Suppose that on January 1, 2011, you purchased 6.375
coupon Government of Canada bond with semi-annual
payments, and a maturity date of December 31, 2018.
• At that time the YTM was 5-percent, and you paid $1,089.75
(the PV of the bond).
• Six months later (July 1, 2011), You sold the bond when the
YTM was 4-percent. The size and timing of cash flows (as
of July 1, 2011 ) were:
$31.875
$31.875
$31.875
$1,031.875
6 / 30 / 18
12 / 31 / 18
7 / 1 / 11 12 / 31 / 11
6 / 30 / 11
© 2011 McGraw–Hill Ryerson Limited
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Holding Period Return (cont.)
LO6.3b
• Given that the YTM at that time was 4-percent, you sold the
bond for:
$31.875 
1  $1,000
PV =
1−
+
= $1,152.59
15 
15

.04 2  (1.02)  (1.02)
• Your holding period return was:
$31.875 + $1,152.59 − $1,089.75
= 8.69%
$1,089.75
• This annualizes to an effective rate of:
2
(1.0869) − 1 = 18.14%
© 2011 McGraw–Hill Ryerson Limited
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Computing Yield to Maturity
LO6.3b
• Yield to maturity is the rate implied by the
current bond price.
• Finding the YTM requires trial and error if
you do not have a financial calculator and is
similar to the process for finding R with an
annuity.
• 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).
© 2011 McGraw–Hill Ryerson Limited
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YTM with Annual Coupons
LO6.3b
• Find YTM for 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.
N
15
I/Y
11.00
PV
– 928.09
PMT
100
FV
1,000
© 2011 McGraw–Hill Ryerson Limited
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YTM with Semiannual Coupons
LO6.3b
• Find TM for a bond with a 10% coupon rate
and semiannual coupons has a face value of
$1,000, 20 years to maturity, and is selling
for $1,197.93.
– What is the semiannual coupon payment?
– How many periods are there?
© 2011 McGraw–Hill Ryerson Limited
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YTM with Semiannual Coupons (cont.)
LO6.3b
• Find TM for a bond with a 10% coupon rate
and semiannual coupons has a face value of
$1,000, 20 years to maturity, and is selling
for $1,197.93.
N
40
I/Y
4.00
PV
– 1,197.93
PMT
FV
=> YTM = 4%*2 = 8%
50
1,000
© 2011 McGraw–Hill Ryerson Limited
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The Present Value of Common Stocks
LO6.4a
• The value of any asset is the present value of
its expected future cash flows.
• Stock ownership produces cash flows from:
– Dividends
– Capital Gains
• Dividends versus Capital Gains
• Valuation of Different Types of Stocks
– Zero Growth
– Constant Growth
– Differential Growth
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Case 1: Zero Growth
LO6.4b
• 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:
Div1
Div 2
Div 3
P0 =
+
+
+
1
2
3
(1 + r ) (1 + r ) (1 + r )
Div
P0 =
r
© 2011 McGraw–Hill Ryerson Limited
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A Zero Growth Example
LO6.4b
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 the
perpetual stream of dividends:
$0.75
$0.75
$0.75
$0.75
…
0
1
2
3
4
P0 = Div1 / r
= 0.75/0.12 = $6.25
© 2011 McGraw–Hill Ryerson Limited
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Case 2: Constant Growth
LO6.4C
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
© 2011 McGraw–Hill Ryerson Limited
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A Constant Growth Example
LO6.4C
XYZ Corp. has a common stock that paid its
annual dividend this morning. It is expected to
pay a $3.60 dividend one year from now, and
following dividends are expected to grow at a
rate of 4% per year forever.
If stocks of similar risk earn 16% effective
annual return, what is the price of a share of
XYZ stock?
© 2011 McGraw–Hill Ryerson Limited
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A Constant Growth Example (cont.)
LO6.4C
The stock price is given by the the present value
of the perpetual stream of growing dividends:
$3.60
$3.60´1.04
$3.60´1.042
$3.60´1.043
…
0
1
2
3
4
P0 = Div1 / (r-g)
= 3.60/(0.16-0.04)
= $30.00
© 2011 McGraw–Hill Ryerson Limited
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Case 3: Differential Growth
LO6.4d
• 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.
© 2011 McGraw–Hill Ryerson Limited
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Case 3: Differential Growth
LO6.4d
• 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 )
..
.
© 2011 McGraw–Hill Ryerson Limited
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Case 3: Differential Growth
LO6.4d
• 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
N+1
© 2011 McGraw–Hill Ryerson Limited
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Case 3: Differential Growth
LO6.4d
We can value this as the sum of: an N-year
annuity growing at rate g1
Div1  (1 + g1 ) N 
PA =
1 −
N 
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 )
© 2011 McGraw–Hill Ryerson Limited
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Case 3: Differential Growth
LO6.4d
To value a Differential Growth Stock, we can use
 Div N +1 




N
Div1  (1 + g1 )   r − g 2 
P=
+
1 −
N 
N
r − g1 
(1 + r )  (1 + r )
Or we can cash flow it out.
© 2011 McGraw–Hill Ryerson Limited
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A Differential Growth Example
LO6.4d
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?
© 2011 McGraw–Hill Ryerson Limited
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With the Formula
 Div N +1 




T
C  (1 + g1 )   r − g 2 
P=
+
1 −
T 
N
r − g1 
(1 + r )  (1 + r )
LO6.4d
 $2(1.08) 3 (1.04) 


3
.12 − .04
$2 × (1.08)  (1.08)  

P=
+
1 −
3
3
.12 − .08  (1.12) 
(1.12)
(
$32.75)
P = $54 × [1 − .8966] +
3
(1.12)
P = $5.58 + $23.31
P = $28.89
© 2011 McGraw–Hill Ryerson Limited
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A Differential Growth Example (cont) LO6.4d
$2(1.08)
$2(1.08)
2
3
3
$2(1.08) $2(1.08) (1.04)
…
0
1
$2.16
0
1
2
3
4
$2.33
$2.62
$2.52 +
.08
2
3
The constant
growth phase
beginning in year 4
can be valued as a
growing perpetuity
at time 3.
$2.62
P3 =
= $32.75
.08
$2.16 $2.33 $2.52 + $32.75
P0 =
+
+
= $28.89
2
3
1.12 (1.12)
(1.12)
© 2011 McGraw–Hill Ryerson Limited
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A Differential Growth Example (cont) LO6.4d
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?
$28.89 = 5.58 + 23.31
First find the PV of the supernormal dividend stream then
find the PV of the steady-state dividend stream.
N
I/Y
3.70 =
PV
– 5.58
PMT
FV
$2 =
0
3
N
3
1.12
1.08
–1 ×100
2×1.08
1.08
I/Y
12
PV
– 23.31
PMT
0
FV
2×(1.08)3 ×(1.04)
32.75 =
.08
© 2011 McGraw–Hill Ryerson Limited
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Estimates of Parameters in the DividendDiscount Model
LO6.5
• The value of a firm depends upon its growth
rate, g, and its discount rate, r.
– Where does g come from?
– Where does r come from?
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Where does g come from?
LO6.5
Formula for Firm’s 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
Earnings
=
+
next Year this Year
Retained Return on
earnings ´ retained
this Year earnings
© 2011 McGraw–Hill Ryerson Limited
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Where does g come from? (con.)
LO6.5
• Dividing both sides by this year’s earnings, we get:
Earnings next Year
= 1+
Earnings this Year
1+g
Retained earnings
Return on
this Year
´ retained
Earnings this Year
earnings
Retention ratio
• This leads to the formula for the firm’s growth rate:
g = Retention ratio × Return on retained earnings
• The return on retained earnings can be estimated
using the firm’s historical return on equity (ROE)
© 2011 McGraw–Hill Ryerson Limited
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Where does g come from? An Example
Ontario Book Publishers (OBP) just reported earnings
of $1.6 million, and it plans to retain 28-percent of its
earnings.
If OBP’s historical ROE was 12-percent, what is the
expected growth rate for OBP’s earnings?
With the above formula:
g = 0.28 × 0.12 = 0.0336 = 3.36%
Or:
(0.28×$1.6 million)×0.12
Change in earnings
=
= 0.0336
$1.6 million
Total earnings
© 2011 McGraw–Hill Ryerson Limited
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Where does r come from?
LO6.5
• From the constant growth cas, we can write:
Div1
r=
+g
P0
• The discount rate can be broken into two
parts.
– The dividend yield
– The growth rate (in dividends)
• In practice, there is a great deal of estimation
error involved in estimating r.
© 2011 McGraw–Hill Ryerson Limited
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Where does r come from? An Example
Manitoba Shipping Co. (MSC) is expected to pay a
dividend 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%
© 2011 McGraw–Hill Ryerson Limited
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Growth Opportunities
LO6.6
• Growth opportunities are opportunities to
invest in positive NPV projects.
• The value of a firm can be conceptualized as
the sum of the value of a firm that pays out
100-percent of its earnings as dividends and
the net present value of the growth
opportunities.
EPS
P=
+ NPVGO
r
© 2011 McGraw–Hill Ryerson Limited
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Growth Opportunities
LO6.6
• Two conditions must be met in order to
increase value:
– Earnings must be retained so that projects can be
funded.
– The projects must have positive NPV.
• A firm’s value increases when it invests in
growth opportunities with positive NPVGOs.
© 2011 McGraw–Hill Ryerson Limited
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The No-Dividend Firm
LO6.6
• Firms that pay no dividends sell at positive
prices (e.g. Amazon.com and RIM)
• These firms have many growth opportunities
and investors believe these firms will pay
dividends at some point.
• The model for constant growth of dividends
does not apply.
© 2011 McGraw–Hill Ryerson Limited
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The Dividend Growth Model and the
NPVGO Model (Advanced)
LO6.7
• 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 pershare value of its growth opportunities.
© 2011 McGraw–Hill Ryerson Limited
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The Dividend Growth Model and the
NPVGO Model
LO6.7
Consider a firm that has EPS of $5 at the end of the
first 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:
Div1
$1.50
P0 =
=
= $75
r − g .16 − .14
© 2011 McGraw–Hill Ryerson Limited
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The NPVGO Model
LO6.7
First, we must calculate the value of the firm as a
cash cow.
EPS $5
=
= $31.25
r
.16
Second, we must calculate the value of the growth
opportunities.
3.50 × .20 

− 3.50 + .16 
$.875
NPVGO =
=
= $43.75
r−g
.16 − .14
Finally, P0 = 31.25 + 43.75 = $75
© 2011 McGraw–Hill Ryerson Limited
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Price Earnings Ratio
LO6.8
• 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 Post uses last 4 quarters’ earnings
Price per share
P/E ratio =
EPS
• Firms whose shares are “in fashion” sell at high
multiples. Growth stocks for example.
• Firms whose shares are out of favour sell at low
multiples. Value stocks for example.
© 2011 McGraw–Hill Ryerson Limited
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Other Price Ratio Analysis
LO6.8
• 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
© 2011 McGraw–Hill Ryerson Limited
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Stock Market Reporting
52W
high
42.10
LO6.9
52W
Yield
Vol
low Stock Ticker Div % P/E 00s High Low
32.25 BCE Inc BCE 1.20 3.5 11.8 19210 34.59 33.80
BCE pays a
BCE has dividend of 1.2
dollars/share
been as
Given the
high as
$42.10 in current price,
the dividend
the last
yield is 3½ %
year.
BCE has
Given the
been as low
current price, the
as $32.25 in
P/E ratio is 11.8
the last year.
times earnings
Close
34.50
Net
chg
-0.47
BCE ended
trading at $34.50,
down $0.47 from
yesterday’s close
1,921,000 shares
traded hands in the
last day’s trading
© 2011 McGraw–Hill Ryerson Limited
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Stock Market Reporting
52W
high
42.10
52W
Yield
Vol
low Stock Ticker Div % P/E 00s High Low
32.25 BCE Inc BCE 1.20 3.5 11.8 19210 34.59 33.80
Close
34.50
Net
chg
-0.47
BCE Incorporated is having a tough year, trading near their 52week low. Imagine how you would feel if within the past year you
had paid $42.10 for a share of BCE and now had a share worth
$34.50! That $1.20 dividend wouldn’t 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
© 2011 McGraw–Hill Ryerson Limited
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Summary and Conclusions
LO6.10
In this chapter, we used the time value of
money formulae from previous chapters to
value bonds and stocks.
1. The value of a zero-coupon bond is
F
PV =
T
(1 + r )
2. The value of a perpetuity is
C
PV =
r
© 2011 McGraw–Hill Ryerson Limited
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Summary and Conclusions (cont.)
LO6.10
3. The value of a coupon bond is the sum of
the PV of the annuity of coupon payments
plus the PV of the par value at maturity.
C
1 
F
PV = 1 −
+
T 
r  (1 + r )  (1 + r )T
4. The yield to maturity (YTM) of a bond is
that single rate that discounts the payments
on the bond to the purchase price.
© 2011 McGraw–Hill Ryerson Limited
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Summary and Conclusions (cont.)
LO6.10
5. A stock can be valued by discounting its
dividends. There are three cases:
Div
1) Zero growth in dividends P0 =
r
Div1
2) Constant growth in dividends P0 =
r−g
 Div N +1 




N
Div1  (1 + g1 )   r − g 2 
P=
+
1 −
N 
N
r − g1 
(1 + r )  (1 + r )
3) Differential growth in dividends
© 2011 McGraw–Hill Ryerson Limited
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Summary and Conclusions (cont.)
LO6.10
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.
EPS
P=
+ NPVGO
r
© 2011 McGraw–Hill Ryerson Limited
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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 Dividend Growth
Model ?
• Decompose a stock’s price into constant growth and
NPVGO values.
• Discuss the importance of the P/E ratio.
© 2011 McGraw–Hill Ryerson Limited