4-1
Chapter 4
The Valuation of
Long-Term
Securities
4-2
The Valuation of
Long-Term Securities
? Distinctions Among Valuation
Concepts
? Bond Valuation
? Preferred Stock Valuation
? Common Stock Valuation
? Rates of Return (or Yields)
4-3
What is Value?
?Going-concern value represents the
amount a firm could be sold for as a
continuing operating business.
?Liquidation value represents the
amount of money that could be
realized if an asset or group of
assets is sold separately from its
operating organization.
4-4
What is Value?
(2) a firm,total assets minus
liabilities and preferred stock as
listed on the balance sheet.
?Book value represents either
(1) an asset,the accounting value
of an asset -- the asset‘s cost
minus its accumulated
depreciation;
4-5
What is Value?
?Intrinsic value represents the
price a security ought to
have?based on all factors
bearing on valuation.
?Market value represents the
market price at which an asset
trades.
4-6
Bond Valuation
? Important Terms
? Types of Bonds
? Valuation of Bonds
? Handling Semiannual
Compounding
4-7
Important Bond Terms
? The maturity value (MV) [or face
value] of a bond is the stated
value,In the case of a U.S,bond,
the face value is usually $1,000,
? A bond is a long-term debt
instrument issued by a
corporation or government.
4-8
Important Bond Terms
? The discount rate (capitalization rate)
is dependent on the risk of the bond
and is composed of the risk-free rate
plus a premium for risk.
? The bond’s coupon rate is the stated
rate of interest; the annual interest
payment divided by the bond’s face
value.
4-9
Different Types of Bonds
A perpetual bond is a bond that never
matures,It has an infinite life.
(1 + kd)1 (1 + kd)2 (1 + kd)?V = + +,.,+
I II
= ??
t=1 (1 + kd)
t
I or I (PVIFA
kd,? )
= I / kd [Reduced Form]
4-10
Perpetual Bond Example
Bond P has a $1,000 face value and
provides an 8% coupon,The appropriate
discount rate is 10%,What is the value of
the perpetual bond?
I = $1,000 ( 8%) = $80.
kd = 10%.
V = I / kd [Reduced Form]
= $80 / 10% = $800.
4-11
Different Types of Bonds
A non-zero coupon bond is a coupon
paying bond with a finite life.
(1 + kd)1 (1 + kd)2 (1 + kd)nV = + +,.,+
I I + MVI
= ?n
t=1 (1 + kd)
t
I
= I (PVIFA kd,n) + MV (PVIF kd,n)
(1 + kd)n+
MV
4-12
Bond C has a $1,000 face value and provides
an 8% annual coupon for 30 years,The
appropriate discount rate is 10%,What is the
value of the coupon bond?
V = $80 (PVIFA10%,30) + $1,000 (PVIF10%,30)
= $80 (9.427) + $1,000 (.057)
[Table IV] [Table II]
= $754.16 + $57.00
= $811.16.
Coupon Bond Example
4-13
Different Types of Bonds
A zero coupon bond is a bond that
pays no interest but sells at a deep
discount from its face value; it provides
compensation to investors in the form
of price appreciation.
(1 + kd)nV =
MV = MV (PVIF
kd,n)
4-14
V = $1,000 (PVIF10%,30)
= $1,000 (.057)
= $57.00
Zero-Coupon
Bond Example
Bond Z has a $1,000 face value and
a 30 year life,The appropriate
discount rate is 10%,What is the
value of the zero-coupon bond?
4-15
Semiannual Compounding
(1) Divide kd by 2
(2) Multiply n by 2
(3) Divide I by 2
Most bonds in the U.S,pay interest
twice a year (1/2 of the annual
coupon).
Adjustments needed:
4-16
(1 + kd/2 ) 2*n(1 + kd/2 )1
Semiannual Compounding
A non-zero coupon bond adjusted for
semiannual compounding.
V = + +,.,+I / 2 I / 2 + MV
= ?2*n
t=1 (1 + kd /2 )
t
I / 2
= I/2 (PVIFAkd /2,2*n) + MV (PVIFkd /2,2*n)
(1 + kd /2 ) 2*n+
MV
I / 2
(1 + kd/2 )2
4-17
V = $40 (PVIFA5%,30) + $1,000 (PVIF5%,30)
= $40 (15.373) + $1,000 (.231)
[Table IV] [Table II]
= $614.92 + $231.00
= $845.92
Semiannual Coupon
Bond Example
Bond C has a $1,000 face value and provides
an 8% semiannual coupon for 15 years,The
appropriate discount rate is 10% (annual rate),
What is the value of the coupon bond?
4-18
Preferred Stock is a type of stock
that promises a (usually) fixed
dividend,but at the discretion of the
board of directors.
Preferred Stock has preference over
common stock in the payment of
dividends and claims on assets.
Preferred Stock Valuation
4-19
Preferred Stock Valuation
This reduces to a perpetuity!
(1 + kP)1 (1 + kP)2 (1 + kP)?V = + +,.,+
DivP DivPDivP
= ??
t=1 (1 + kP)t
DivP or Div
P(PVIFA kP,? )
V = DivP / kP
4-20
Preferred Stock Example
DivP = $100 ( 8% ) = $8.00.
kP = 10%.
V = DivP / kP = $8.00 / 10%
= $80
Stock PS has an 8%,$100 par value
issue outstanding,The appropriate
discount rate is 10%,What is the value
of the preferred stock?
4-21
Common Stock Valuation
? Prorata share of future earnings
after all other obligations of the
firm (if any remain).
? Dividends may be paid out of
the prorata share of earnings.
Common stock represents a
residual ownership position in the
corporation.
4-22
Common Stock Valuation
(1) Future dividends
(2) Future sale of the common
stock shares
What cash flows will a shareholder
receive when owning shares of
common stock?
4-23
Dividend Valuation Model
Basic dividend valuation model accounts
for the PV of all future dividends.
(1 + ke)1 (1 + ke)2 (1 + ke)?V = + +,.,+
Div1 Div?Div2
= ??
t=1 (1 + ke)
t
Divt Divt,Cash Dividend
at time t
ke,Equity investor’s
required return
4-24
Adjusted Dividend
Valuation Model
The basic dividend valuation model
adjusted for the future stock sale.
(1 + ke)1 (1 + ke)2 (1 + ke)nV = + +,.,+
Div1 Divn + PricenDiv2
n,The year in which the firm’s
shares are expected to be sold.
Pricen,The expected share price in
year n,
4-25
Dividend Growth
Pattern Assumptions
The dividend valuation model requires the
forecast of all future dividends,The
following dividend growth rate assumptions
simplify the valuation process.
Constant Growth
No Growth
Growth Phases
4-26
Constant Growth Model
The constant growth model assumes that
dividends will grow forever at the rate g.
(1 + ke)1 (1 + ke)2 (1 + ke)?V = + +,.,+
D0(1+g) D0(1+g)?
= (k
e - g)
D1 D1,Dividend paid at time 1.g, The constant growth rate.
ke,Investor’s required return.
D0(1+g)2
4-27
Constant Growth
Model Example
Stock CG has an expected growth rate of
8%,Each share of stock just received an
annual $3.24 dividend per share,The
appropriate discount rate is 15%,What
is the value of the common stock?
D1 = $3.24 ( 1 +,08 ) = $3.50
VCG = D1 / ( ke - g ) = $3.50 / (,15 -,08 )
= $50
4-28
Zero Growth Model
The zero growth model assumes that
dividends will grow forever at the rate g = 0.
(1 + ke)1 (1 + ke)2 (1 + ke)?V = + +,.,+
D1 D?
= k
e
D1 D1,Dividend paid at time 1.
ke,Investor’s required return.
D2
4-29
Zero Growth
Model Example
Stock ZG has an expected growth rate of
0%,Each share of stock just received an
annual $3.24 dividend per share,The
appropriate discount rate is 15%,What
is the value of the common stock?
D1 = $3.24 ( 1 + 0 ) = $3.24
VZG = D1 / ( ke - 0 ) = $3.24 / (,15 - 0 )
= $21.60
4-30
D0(1+g1)t Dn(1+g2)t
Growth Phases Model
The growth phases model assumes
that dividends for each share will grow
at two or more different growth rates.
(1 + ke)t (1 + ke)tV =?t=1
n ?
t=n+1
?
+
4-31
D0(1+g1)t Dn+1
Growth Phases Model
Note that the second phase of the
growth phases model assumes that
dividends will grow at a constant rate g2,
We can rewrite the formula as:
(1 + ke)t (ke - g2)V =?t=1
n
+ 1(1 + k
e)n
4-32
Growth Phases
Model Example
Stock GP has an expected growth
rate of 16% for the first 3 years and
8% thereafter,Each share of stock
just received an annual $3.24
dividend per share,The appropriate
discount rate is 15%,What is the
value of the common stock under
this scenario?
4-33
Growth Phases
Model Example
First,determine the annual dividend.
D0 = $3.24
D1 = D0(1+g1)1 = $3.24(1.16)1 =$3.76
D2 = D0(1+g1)2 = $3.24(1.16)2 =$4.36
D3 = D0(1+g1)3 = $3.24(1.16)3 =$5.06
D4 = D3(1+g2)1 = $5.06(1.08)1 =$5.46
4-34
Growth Phases
Model Example
Second,determine the PV of cash flows.
PV(D1) = D1(PVIF15%,1) = $3.76 (.870) = $3.27
PV(D2) = D2(PVIF15%,2) = $4.36 (.756) = $3.30
PV(D3) = D3(PVIF15%,3) = $5.06 (.658) = $3.33
P3 = $5.46 / (.15 -,08) = $78 [CG Model]
PV(P3) = P3(PVIF15%,3) = $78 (.658) = $51.32
4-35
D0(1+.16)t D4
Growth Phases
Model Example
Third,calculate the intrinsic value by
summing all of cash flow present values.
(1 +,15)t (.15-.08)V = ?t=1
3
+ 1(1+.15)n
V = $3.27 + $3.30 + $3.33 + $51.32
V = $61.22
4-36
Calculating Rates of
Return (or Yields)
1,Determine the expected cash flows.
2,Replace the intrinsic value (V) with
the market price (P0).
3,Solve for the market required rate of
return that equates the discounted
cash flows to the market price,
Steps to calculate the rate of
return (or Yield).
4-37
Determining Bond YTM
Determine the Yield-to-Maturity
(YTM) for the coupon paying bond
with a finite life.
P0 = ?
n
t=1 (1 + kd )
t
I
= I (PVIFA kd,n) + MV (PVIF kd,n)
(1 + kd )n+
MV
kd = YTM
4-38
Determining the YTM
Julie Miller want to determine the YTM
for an issue of outstanding bonds at
Basket Wonders (BW),BW has an
issue of 10% annual coupon bonds
with 15 years left to maturity,The
bonds have a current market value of
$1,250.
What is the YTM?
4-39
YTM Solution (Try 9%)
$1,250 = $100(PVIFA9%,15) +
$1,000(PVIF9%,15)
$1,250 = $100(8.061) +
$1,000(.275)
$1,250 = $806.10 + $275.00
= $1,081.10
[Rate is too high!]
4-40
YTM Solution (Try 7%)
$1,250 = $100(PVIFA7%,15) +
$1,000(PVIF7%,15)
$1,250 = $100(9.108) +
$1,000(.362)
$1,250 = $910.80 + $362.00
= $1,272.80
[Rate is too low!]
4-41
.07 $1,273
.02 IRR $1,250 $192
.09 $1,081
X $23
.02 $192
YTM Solution (Interpolate)
$23X
=
4-42
.07 $1,273
.02 IRR $1,250 $192
.09 $1,081
X $23
.02 $192
YTM Solution (Interpolate)
$23X
=
4-43
.07 $1273
.02 YTM $1250 $192
.09 $1081
($23)(0.02)
$192
YTM Solution (Interpolate)
$23X
X = X =,0024
YTM =,07 +,0024 =,0724 or 7.24%
4-44
Determining Semiannual
Coupon Bond YTM
P0 = ?
2n
t=1 (1 + kd /2 )t
I / 2
= (I/2)(PVIFAkd /2,2n) + MV(PVIFkd /2,2n)
+ MV
[ 1 + (kd / 2)2 ] -1 = YTM
Determine the Yield-to-Maturity
(YTM) for the semiannual coupon
paying bond with a finite life.
(1 + kd /2 )2n
4-45
Bond Price - Yield
Relationship
Discount Bond -- The market required
rate of return exceeds the coupon rate
(Par > P0 ).
Premium Bond -- The coupon rate
exceeds the market required rate of
return (P0 > Par).
Par Bond -- The coupon rate equals the
market required rate of return (P0 = Par).
4-46
Bond Price - Yield
Relationship
Coupon Rate
MARKET REQUIRED RATE OF RETURN (%)
BO
ND
PR
ICE
($
)
1000
Par
1600
1400
1200
600
0
0 2 4 6 8 10 12 14 16 18
5 Year
15 Year
4-47
Bond Price-Yield
Relationship
Assume that the required rate of
return on a 15 year,10% coupon
paying bond rises from 10% to 12%,
What happens to the bond price?
When interest rates rise,then the
market required rates of return rise
and bond prices will fall.
4-48
Bond Price - Yield
Relationship
Coupon Rate
MARKET REQUIRED RATE OF RETURN (%)
BO
ND
PR
ICE
($
)
1000
Par
1600
1400
1200
600
0
0 2 4 6 8 10 12 14 16 18
15 Year
5 Year
4-49
Bond Price-Yield
Relationship (Rising Rates)
Therefore,the bond price has
fallen from $1,000 to $864.10.
The required rate of return on a 15
year,10% coupon paying bond
has risen from 10% to 12%.
4-50
Bond Price-Yield
Relationship
Assume that the required rate of
return on a 15 year,10% coupon
paying bond falls from 10% to 8%,
What happens to the bond price?
When interest rates fall,then the
market required rates of return fall
and bond prices will rise,
4-51
Bond Price - Yield
Relationship
Coupon Rate
MARKET REQUIRED RATE OF RETURN (%)
BO
ND
PR
ICE
($
)
1000
Par
1600
1400
1200
600
0
0 2 4 6 8 10 12 14 16 18
15 Year
5 Year
4-52
Bond Price-Yield Relationship
(Declining Rates)
Therefore,the bond price has
risen from $1000 to $1171.
The required rate of return on a 15
year,10% coupon paying bond
has fallen from 10% to 8%.
4-53
The Role of Bond Maturity
Assume that the required rate of return
on both the 5 and 15 year,10% coupon
paying bonds fall from 10% to 8%,What
happens to the changes in bond prices?
The longer the bond maturity,the
greater the change in bond price for a
given change in the market required rate
of return.
4-54
Bond Price - Yield
Relationship
Coupon Rate
MARKET REQUIRED RATE OF RETURN (%)
BO
ND
PR
ICE
($
)
1000
Par
1600
1400
1200
600
0
0 2 4 6 8 10 12 14 16 18
15 Year
5 Year
4-55
The Role of Bond Maturity
The 5 year bond price has risen from
$1,000 to $1,080.30 for the 5 year bond
(+8.0%).
The 15 year bond price has risen from
$1,000 to $1,171 (+17.1%),Twice as fast!
The required rate of return on both the
5 and 15 year,10% coupon paying
bonds has fallen from 10% to 8%.
4-56
The Role of the
Coupon Rate
For a given change in the
market required rate of return,
the price of a bond will change
by proportionally more,the
lower the coupon rate.
4-57
Example of the Role of
the Coupon Rate
Assume that the market required rate
of return on two equally risky 15 year
bonds is 10%,The coupon rate for
Bond H is 10% and Bond L is 8%,
What is the rate of change in each of
the bond prices if market required
rates fall to 8%?
4-58
Example of the Role of the
Coupon Rate
The price for Bond H will rise from $1,000
to $1,171 (+17.1%).
The price for Bond L will rise from
$847.88 to $1,000 (+17.9%),Faster Rise!
The price on Bond H and L prior to the
change in the market required rate of
return is $1,000 and $847.88
respectively.
4-59
Determining the Yield on
Preferred Stock
Determine the yield for preferred
stock with an infinite life.
P0 = DivP / kP
Solving for kP such that
kP = DivP / P0
4-60
Preferred Stock Yield
Example
kP = $10 / $100.
kP = 10%.
Assume that the annual dividend on
each share of preferred stock is $10,
Each share of preferred stock is
currently trading at $100,What is
the yield on preferred stock?
4-61
Determining the Yield on
Common Stock
Assume the constant growth model
is appropriate,Determine the yield
on the common stock.
P0 = D1 / ( ke - g )
Solving for ke such that
ke = ( D1 / P0 ) + g
4-62
Common Stock Yield
Example
ke = ( $3 / $30 ) + 5%
ke = 15%
Assume that the expected dividend
(D1) on each share of common stock
is $3,Each share of common stock
is currently trading at $30 and has an
expected growth rate of 5%,What is
the yield on common stock?
Chapter 4
The Valuation of
Long-Term
Securities
4-2
The Valuation of
Long-Term Securities
? Distinctions Among Valuation
Concepts
? Bond Valuation
? Preferred Stock Valuation
? Common Stock Valuation
? Rates of Return (or Yields)
4-3
What is Value?
?Going-concern value represents the
amount a firm could be sold for as a
continuing operating business.
?Liquidation value represents the
amount of money that could be
realized if an asset or group of
assets is sold separately from its
operating organization.
4-4
What is Value?
(2) a firm,total assets minus
liabilities and preferred stock as
listed on the balance sheet.
?Book value represents either
(1) an asset,the accounting value
of an asset -- the asset‘s cost
minus its accumulated
depreciation;
4-5
What is Value?
?Intrinsic value represents the
price a security ought to
have?based on all factors
bearing on valuation.
?Market value represents the
market price at which an asset
trades.
4-6
Bond Valuation
? Important Terms
? Types of Bonds
? Valuation of Bonds
? Handling Semiannual
Compounding
4-7
Important Bond Terms
? The maturity value (MV) [or face
value] of a bond is the stated
value,In the case of a U.S,bond,
the face value is usually $1,000,
? A bond is a long-term debt
instrument issued by a
corporation or government.
4-8
Important Bond Terms
? The discount rate (capitalization rate)
is dependent on the risk of the bond
and is composed of the risk-free rate
plus a premium for risk.
? The bond’s coupon rate is the stated
rate of interest; the annual interest
payment divided by the bond’s face
value.
4-9
Different Types of Bonds
A perpetual bond is a bond that never
matures,It has an infinite life.
(1 + kd)1 (1 + kd)2 (1 + kd)?V = + +,.,+
I II
= ??
t=1 (1 + kd)
t
I or I (PVIFA
kd,? )
= I / kd [Reduced Form]
4-10
Perpetual Bond Example
Bond P has a $1,000 face value and
provides an 8% coupon,The appropriate
discount rate is 10%,What is the value of
the perpetual bond?
I = $1,000 ( 8%) = $80.
kd = 10%.
V = I / kd [Reduced Form]
= $80 / 10% = $800.
4-11
Different Types of Bonds
A non-zero coupon bond is a coupon
paying bond with a finite life.
(1 + kd)1 (1 + kd)2 (1 + kd)nV = + +,.,+
I I + MVI
= ?n
t=1 (1 + kd)
t
I
= I (PVIFA kd,n) + MV (PVIF kd,n)
(1 + kd)n+
MV
4-12
Bond C has a $1,000 face value and provides
an 8% annual coupon for 30 years,The
appropriate discount rate is 10%,What is the
value of the coupon bond?
V = $80 (PVIFA10%,30) + $1,000 (PVIF10%,30)
= $80 (9.427) + $1,000 (.057)
[Table IV] [Table II]
= $754.16 + $57.00
= $811.16.
Coupon Bond Example
4-13
Different Types of Bonds
A zero coupon bond is a bond that
pays no interest but sells at a deep
discount from its face value; it provides
compensation to investors in the form
of price appreciation.
(1 + kd)nV =
MV = MV (PVIF
kd,n)
4-14
V = $1,000 (PVIF10%,30)
= $1,000 (.057)
= $57.00
Zero-Coupon
Bond Example
Bond Z has a $1,000 face value and
a 30 year life,The appropriate
discount rate is 10%,What is the
value of the zero-coupon bond?
4-15
Semiannual Compounding
(1) Divide kd by 2
(2) Multiply n by 2
(3) Divide I by 2
Most bonds in the U.S,pay interest
twice a year (1/2 of the annual
coupon).
Adjustments needed:
4-16
(1 + kd/2 ) 2*n(1 + kd/2 )1
Semiannual Compounding
A non-zero coupon bond adjusted for
semiannual compounding.
V = + +,.,+I / 2 I / 2 + MV
= ?2*n
t=1 (1 + kd /2 )
t
I / 2
= I/2 (PVIFAkd /2,2*n) + MV (PVIFkd /2,2*n)
(1 + kd /2 ) 2*n+
MV
I / 2
(1 + kd/2 )2
4-17
V = $40 (PVIFA5%,30) + $1,000 (PVIF5%,30)
= $40 (15.373) + $1,000 (.231)
[Table IV] [Table II]
= $614.92 + $231.00
= $845.92
Semiannual Coupon
Bond Example
Bond C has a $1,000 face value and provides
an 8% semiannual coupon for 15 years,The
appropriate discount rate is 10% (annual rate),
What is the value of the coupon bond?
4-18
Preferred Stock is a type of stock
that promises a (usually) fixed
dividend,but at the discretion of the
board of directors.
Preferred Stock has preference over
common stock in the payment of
dividends and claims on assets.
Preferred Stock Valuation
4-19
Preferred Stock Valuation
This reduces to a perpetuity!
(1 + kP)1 (1 + kP)2 (1 + kP)?V = + +,.,+
DivP DivPDivP
= ??
t=1 (1 + kP)t
DivP or Div
P(PVIFA kP,? )
V = DivP / kP
4-20
Preferred Stock Example
DivP = $100 ( 8% ) = $8.00.
kP = 10%.
V = DivP / kP = $8.00 / 10%
= $80
Stock PS has an 8%,$100 par value
issue outstanding,The appropriate
discount rate is 10%,What is the value
of the preferred stock?
4-21
Common Stock Valuation
? Prorata share of future earnings
after all other obligations of the
firm (if any remain).
? Dividends may be paid out of
the prorata share of earnings.
Common stock represents a
residual ownership position in the
corporation.
4-22
Common Stock Valuation
(1) Future dividends
(2) Future sale of the common
stock shares
What cash flows will a shareholder
receive when owning shares of
common stock?
4-23
Dividend Valuation Model
Basic dividend valuation model accounts
for the PV of all future dividends.
(1 + ke)1 (1 + ke)2 (1 + ke)?V = + +,.,+
Div1 Div?Div2
= ??
t=1 (1 + ke)
t
Divt Divt,Cash Dividend
at time t
ke,Equity investor’s
required return
4-24
Adjusted Dividend
Valuation Model
The basic dividend valuation model
adjusted for the future stock sale.
(1 + ke)1 (1 + ke)2 (1 + ke)nV = + +,.,+
Div1 Divn + PricenDiv2
n,The year in which the firm’s
shares are expected to be sold.
Pricen,The expected share price in
year n,
4-25
Dividend Growth
Pattern Assumptions
The dividend valuation model requires the
forecast of all future dividends,The
following dividend growth rate assumptions
simplify the valuation process.
Constant Growth
No Growth
Growth Phases
4-26
Constant Growth Model
The constant growth model assumes that
dividends will grow forever at the rate g.
(1 + ke)1 (1 + ke)2 (1 + ke)?V = + +,.,+
D0(1+g) D0(1+g)?
= (k
e - g)
D1 D1,Dividend paid at time 1.g, The constant growth rate.
ke,Investor’s required return.
D0(1+g)2
4-27
Constant Growth
Model Example
Stock CG has an expected growth rate of
8%,Each share of stock just received an
annual $3.24 dividend per share,The
appropriate discount rate is 15%,What
is the value of the common stock?
D1 = $3.24 ( 1 +,08 ) = $3.50
VCG = D1 / ( ke - g ) = $3.50 / (,15 -,08 )
= $50
4-28
Zero Growth Model
The zero growth model assumes that
dividends will grow forever at the rate g = 0.
(1 + ke)1 (1 + ke)2 (1 + ke)?V = + +,.,+
D1 D?
= k
e
D1 D1,Dividend paid at time 1.
ke,Investor’s required return.
D2
4-29
Zero Growth
Model Example
Stock ZG has an expected growth rate of
0%,Each share of stock just received an
annual $3.24 dividend per share,The
appropriate discount rate is 15%,What
is the value of the common stock?
D1 = $3.24 ( 1 + 0 ) = $3.24
VZG = D1 / ( ke - 0 ) = $3.24 / (,15 - 0 )
= $21.60
4-30
D0(1+g1)t Dn(1+g2)t
Growth Phases Model
The growth phases model assumes
that dividends for each share will grow
at two or more different growth rates.
(1 + ke)t (1 + ke)tV =?t=1
n ?
t=n+1
?
+
4-31
D0(1+g1)t Dn+1
Growth Phases Model
Note that the second phase of the
growth phases model assumes that
dividends will grow at a constant rate g2,
We can rewrite the formula as:
(1 + ke)t (ke - g2)V =?t=1
n
+ 1(1 + k
e)n
4-32
Growth Phases
Model Example
Stock GP has an expected growth
rate of 16% for the first 3 years and
8% thereafter,Each share of stock
just received an annual $3.24
dividend per share,The appropriate
discount rate is 15%,What is the
value of the common stock under
this scenario?
4-33
Growth Phases
Model Example
First,determine the annual dividend.
D0 = $3.24
D1 = D0(1+g1)1 = $3.24(1.16)1 =$3.76
D2 = D0(1+g1)2 = $3.24(1.16)2 =$4.36
D3 = D0(1+g1)3 = $3.24(1.16)3 =$5.06
D4 = D3(1+g2)1 = $5.06(1.08)1 =$5.46
4-34
Growth Phases
Model Example
Second,determine the PV of cash flows.
PV(D1) = D1(PVIF15%,1) = $3.76 (.870) = $3.27
PV(D2) = D2(PVIF15%,2) = $4.36 (.756) = $3.30
PV(D3) = D3(PVIF15%,3) = $5.06 (.658) = $3.33
P3 = $5.46 / (.15 -,08) = $78 [CG Model]
PV(P3) = P3(PVIF15%,3) = $78 (.658) = $51.32
4-35
D0(1+.16)t D4
Growth Phases
Model Example
Third,calculate the intrinsic value by
summing all of cash flow present values.
(1 +,15)t (.15-.08)V = ?t=1
3
+ 1(1+.15)n
V = $3.27 + $3.30 + $3.33 + $51.32
V = $61.22
4-36
Calculating Rates of
Return (or Yields)
1,Determine the expected cash flows.
2,Replace the intrinsic value (V) with
the market price (P0).
3,Solve for the market required rate of
return that equates the discounted
cash flows to the market price,
Steps to calculate the rate of
return (or Yield).
4-37
Determining Bond YTM
Determine the Yield-to-Maturity
(YTM) for the coupon paying bond
with a finite life.
P0 = ?
n
t=1 (1 + kd )
t
I
= I (PVIFA kd,n) + MV (PVIF kd,n)
(1 + kd )n+
MV
kd = YTM
4-38
Determining the YTM
Julie Miller want to determine the YTM
for an issue of outstanding bonds at
Basket Wonders (BW),BW has an
issue of 10% annual coupon bonds
with 15 years left to maturity,The
bonds have a current market value of
$1,250.
What is the YTM?
4-39
YTM Solution (Try 9%)
$1,250 = $100(PVIFA9%,15) +
$1,000(PVIF9%,15)
$1,250 = $100(8.061) +
$1,000(.275)
$1,250 = $806.10 + $275.00
= $1,081.10
[Rate is too high!]
4-40
YTM Solution (Try 7%)
$1,250 = $100(PVIFA7%,15) +
$1,000(PVIF7%,15)
$1,250 = $100(9.108) +
$1,000(.362)
$1,250 = $910.80 + $362.00
= $1,272.80
[Rate is too low!]
4-41
.07 $1,273
.02 IRR $1,250 $192
.09 $1,081
X $23
.02 $192
YTM Solution (Interpolate)
$23X
=
4-42
.07 $1,273
.02 IRR $1,250 $192
.09 $1,081
X $23
.02 $192
YTM Solution (Interpolate)
$23X
=
4-43
.07 $1273
.02 YTM $1250 $192
.09 $1081
($23)(0.02)
$192
YTM Solution (Interpolate)
$23X
X = X =,0024
YTM =,07 +,0024 =,0724 or 7.24%
4-44
Determining Semiannual
Coupon Bond YTM
P0 = ?
2n
t=1 (1 + kd /2 )t
I / 2
= (I/2)(PVIFAkd /2,2n) + MV(PVIFkd /2,2n)
+ MV
[ 1 + (kd / 2)2 ] -1 = YTM
Determine the Yield-to-Maturity
(YTM) for the semiannual coupon
paying bond with a finite life.
(1 + kd /2 )2n
4-45
Bond Price - Yield
Relationship
Discount Bond -- The market required
rate of return exceeds the coupon rate
(Par > P0 ).
Premium Bond -- The coupon rate
exceeds the market required rate of
return (P0 > Par).
Par Bond -- The coupon rate equals the
market required rate of return (P0 = Par).
4-46
Bond Price - Yield
Relationship
Coupon Rate
MARKET REQUIRED RATE OF RETURN (%)
BO
ND
PR
ICE
($
)
1000
Par
1600
1400
1200
600
0
0 2 4 6 8 10 12 14 16 18
5 Year
15 Year
4-47
Bond Price-Yield
Relationship
Assume that the required rate of
return on a 15 year,10% coupon
paying bond rises from 10% to 12%,
What happens to the bond price?
When interest rates rise,then the
market required rates of return rise
and bond prices will fall.
4-48
Bond Price - Yield
Relationship
Coupon Rate
MARKET REQUIRED RATE OF RETURN (%)
BO
ND
PR
ICE
($
)
1000
Par
1600
1400
1200
600
0
0 2 4 6 8 10 12 14 16 18
15 Year
5 Year
4-49
Bond Price-Yield
Relationship (Rising Rates)
Therefore,the bond price has
fallen from $1,000 to $864.10.
The required rate of return on a 15
year,10% coupon paying bond
has risen from 10% to 12%.
4-50
Bond Price-Yield
Relationship
Assume that the required rate of
return on a 15 year,10% coupon
paying bond falls from 10% to 8%,
What happens to the bond price?
When interest rates fall,then the
market required rates of return fall
and bond prices will rise,
4-51
Bond Price - Yield
Relationship
Coupon Rate
MARKET REQUIRED RATE OF RETURN (%)
BO
ND
PR
ICE
($
)
1000
Par
1600
1400
1200
600
0
0 2 4 6 8 10 12 14 16 18
15 Year
5 Year
4-52
Bond Price-Yield Relationship
(Declining Rates)
Therefore,the bond price has
risen from $1000 to $1171.
The required rate of return on a 15
year,10% coupon paying bond
has fallen from 10% to 8%.
4-53
The Role of Bond Maturity
Assume that the required rate of return
on both the 5 and 15 year,10% coupon
paying bonds fall from 10% to 8%,What
happens to the changes in bond prices?
The longer the bond maturity,the
greater the change in bond price for a
given change in the market required rate
of return.
4-54
Bond Price - Yield
Relationship
Coupon Rate
MARKET REQUIRED RATE OF RETURN (%)
BO
ND
PR
ICE
($
)
1000
Par
1600
1400
1200
600
0
0 2 4 6 8 10 12 14 16 18
15 Year
5 Year
4-55
The Role of Bond Maturity
The 5 year bond price has risen from
$1,000 to $1,080.30 for the 5 year bond
(+8.0%).
The 15 year bond price has risen from
$1,000 to $1,171 (+17.1%),Twice as fast!
The required rate of return on both the
5 and 15 year,10% coupon paying
bonds has fallen from 10% to 8%.
4-56
The Role of the
Coupon Rate
For a given change in the
market required rate of return,
the price of a bond will change
by proportionally more,the
lower the coupon rate.
4-57
Example of the Role of
the Coupon Rate
Assume that the market required rate
of return on two equally risky 15 year
bonds is 10%,The coupon rate for
Bond H is 10% and Bond L is 8%,
What is the rate of change in each of
the bond prices if market required
rates fall to 8%?
4-58
Example of the Role of the
Coupon Rate
The price for Bond H will rise from $1,000
to $1,171 (+17.1%).
The price for Bond L will rise from
$847.88 to $1,000 (+17.9%),Faster Rise!
The price on Bond H and L prior to the
change in the market required rate of
return is $1,000 and $847.88
respectively.
4-59
Determining the Yield on
Preferred Stock
Determine the yield for preferred
stock with an infinite life.
P0 = DivP / kP
Solving for kP such that
kP = DivP / P0
4-60
Preferred Stock Yield
Example
kP = $10 / $100.
kP = 10%.
Assume that the annual dividend on
each share of preferred stock is $10,
Each share of preferred stock is
currently trading at $100,What is
the yield on preferred stock?
4-61
Determining the Yield on
Common Stock
Assume the constant growth model
is appropriate,Determine the yield
on the common stock.
P0 = D1 / ( ke - g )
Solving for ke such that
ke = ( D1 / P0 ) + g
4-62
Common Stock Yield
Example
ke = ( $3 / $30 ) + 5%
ke = 15%
Assume that the expected dividend
(D1) on each share of common stock
is $3,Each share of common stock
is currently trading at $30 and has an
expected growth rate of 5%,What is
the yield on common stock?
