Principles of Managerial
Finance
9th Edition
Chapter 7
Bond & Stock Valuation
Learning Objectives
• Describe the key inputs and basic model used in the
valuation process.
• Apply the basic bond valuation model to bonds and
describe the impact of required return and time to
maturity on bond values.
• Explain yield to maturity (YTM), its calculation, and the
procedure used to value bonds that pay interest
semiannually.
Learning Objectives
• Understand the concept of market efficiency and basic
common stock valuation under each of three cases:
zero growth, constant growth, and variable growth.
• Discuss the use of book value, liquidation value, and
price/earnings (PE) multiples to estimate common
stock values.
• Understand the relationships among financial
decisions, return, risk, and the firm’s value.
Valuation Fundamentals
• The (market) value of any investment asset is simply
the present value of expected cash flows.
• The interest rate that these cash flows are discounted
at is called the asset’s required return.
• The required return is a function of the expected rate
of inflation and the perceived risk of the asset.
• Higher perceived risk results in a higher required
return and lower asset market values.
Basic Valuation Model
V0 =
CF1
(1 + k)1
+
CF2 + … +
(1 + k)2
CFn
(1 + k)n
Where:
V0 = value of the asset at time zero
CFt = cash flow expected at the end of year t
k = appropriate required return (discount rate)
n = relevant time period
What is a Bond?
A bond is a long-term debt instrument that
pays the bondholder a specified amount of
periodic interest over a specified period of
time.
(note that a bond = debt)
General Features of Debt Instruments
• The bond’s principal is the amount borrowed by the
company and the amount owed to the bond holder on
the maturity date.
• The bond’s maturity date is the time at which a bond
becomes due and the principal must be repaid.
• The bond’s coupon rate is the specified interest rate
(or $ amount) that must be periodically paid.
• The bond’s current yield is the annual interest
(income) divided by the current price of the security.
General Features of Debt Instruments
• The bond’s yield to maturity is the yield (expressed as
a compound rate of return) earned on a bond from the
time it is acquired until the maturity date of the bond.
• A yield curve graphically shows the relationship
between the time to maturity and yields for debt in a
given risk class.
Bonds with Maturity Dates
Annual Compounding
B0 =
I1
I2 + … +
+
(1+i)1
(1+i)2
(In + Pn)
(1+i)n
For example, find the price of a 10% coupon bond
with three years to maturity if market interest rates
are currently 10%.
B0 =
100
+
(1+.10)1
100 + (100+1,000)
(1+i)2
(1+.10)3
Bonds with Maturity Dates
Annual Compounding
Using Excel
For example, find the price of a 10% coupon bond
with three years to maturity if market interest rates
are currently 10%.
Finding the Value of a Bond
Coupon Interest ($)
$
100
Maturity (periods)
3
Face Value ($)
$ 1,000
Market Interest Rate (%)
10%
Market Price ($)
($1,000.00)
Note: the equation
for calculating
price is
=PV(rate,nper,pmt,fv)
Bonds with Maturity Dates
Annual Compounding
Using Excel
For example, find the price of a 10% coupon bond
with three years to maturity if market interest rates
are currently 10%.
Finding the Value of a Bond
Coupon Interest ($)
$
100
Maturity (periods)
3
Face Value ($)
$ 1,000
Market Interest Rate (%)
10%
Market Price ($)
($1,000.00)
When the coupon
rate matches the
discount rate, the
bond always sells
for its par value.
Bonds with Maturity Dates
Annual Compounding
Using Excel
What would happen to the bond’s price if interest
rates increased from 10% to 15%?
Finding the Value of a Bond
Coupon Interest ($)
$
100
Maturity (periods)
3
Face Value ($)
$ 1,000
Market Interest Rate (%)
15%
Market Price ($)
($885.84)
When the interest
rate goes up, the
bond price will
always go down.
Bonds with Maturity Dates
Annual Compounding
Using Excel
What would happen to the bond’s price it had a 15
year maturity rather than a 3 year maturity?
Finding the Value of a Bond
Coupon Interest ($)
$
100
Maturity (periods)
15
Face Value ($)
$ 1,000
Market Interest Rate (%)
15%
Market Price ($)
($707.63)
And the longer the
maturity, the greater
the price decline.
Bonds with Maturity Dates
Annual Compounding
Using Excel
What would happen to the original 3 year bond’s
price if interest rates dropped from 10% to 5%?
Finding the Value of a Bond
Coupon Interest ($)
$
100
Maturity (periods)
3
Face Value ($)
$ 1,000
Market Interest Rate (%)
5%
Market Price ($)
($1,136.16)
When interest rates
go down, bond
prices will always
go up.
Bonds with Maturity Dates
Annual Compounding
Using Excel
What if we considered a similar bond, but with a 15
year maturity rather than a 3 year maturity?
Finding the Value of a Bond
Coupon Interest ($)
$
100
Maturity (periods)
15
Face Value ($)
$ 1,000
Market Interest Rate (%)
5%
Market Price ($)
($1,518.98)
And the longer the
maturity, the greater
the price increase will
be.
Graphically
Effect of Changes in Interest Rates on Price
$1,600
$1,400
$1,200
$1,000
3 yr bond
$800
$600
$400
$200
$-
15 yr bond
As interest rates go up
5%
10%
15%
Bonds with Maturity Dates
Semi-Annual Compounding
Using Excel
If we had the same bond, but with semi-annual coupon
payments, we would have to divide the 10% coupon rate by
two, divided the discount rate by two, and multiply n by two.
Finding the Value of a Bond
Coupon Interest ($)
$
50
Maturity (periods)
6
Face Value ($)
$ 1,000
Market Interest Rate (%)
3%
Market Price ($)
($1,137.70)
For the original
example, divide the 10%
coupon by 2, divide
the 15% discount rate
by 2, and multiply
3 years by 2.
Bonds with Maturity Dates
Semi-Annual Compounding
Using Excel
If we had the same bond, but with semi-annual coupon
payments, we would have to divide the 10% coupon rate by
two, divided the discount rate by two, and multiply n by two.
Finding the Value of a Bond
Coupon Interest ($)
$
50
Maturity (periods)
6
Face Value ($)
$ 1,000
Market Interest Rate (%)
3%
Market Price ($)
($1,137.70)
Thus, the value is
slightly larger than the
price of the annual
coupon bond (1,136.16)
because the investor
receives payments
sooner.
Coupon Effects on Price Volatility
• The amount of bond price volatility depends on three
basic factors:
– length of time to maturity
– risk
– amount of coupon interest paid by the bond
• First, we already have seen that the longer the term to
maturity, the greater is a bond’s volatility
• Second, the riskier a bond, the more variable the
required return will be, resulting in greater price
volatility.
Coupon Effects on Price Volatility
• The amount of bond price volatility depends on three
basic factors:
– length of time to maturity
– risk
– amount of coupon interest paid by the bond
• Finally, the amount of coupon interest also impacts a
bond’s price volatility.
• Specifically, the lower the coupon, the greater will be
the bond’s volatility, because it will be longer before
the investor receives a significant portion of the cash
flow from his or her investment.
Coupon Effects on Price Volatility
10 Year Bond
Interest
Price
Price
Rate
5% Coupon 15% Coupon
0% $
1,500 $
2,500
10% $
693 $
1,307
20% $
371 $
790
Effect of Changes in Interest Rates on Price
$3,000
$2,500
$2,000
5% Coupon
$1,500
15% Coupon
$1,000
$500
$0%
10%
20%
Price Converges on Par at Maturity
• It is also important to note that a bond’s price will
approach par value as it approaches the maturity date,
regardless of the interest rate and regardless of the
coupon rate.
10% Coupon Bond
Interest
Price
Price
Rate
20 Years 1 Year
0% $ 3,000 $ 1,100
10% $ 1,000 $ 1,000
20% $ 513 $ 917
Price Converges on Par at Maturity
• It is also important to note that a bond’s price will
approach par value as it approaches the maturity date,
regardless of the interest rate and regardless of the
coupon rate.
Effect of Changes in Interest Rates on Price
$3,500
$3,000
$2,500
$2,000
20 Years
$1,500
1 Year
$1,000
$500
$0%
10%
20%
Yields
• The Current Yield measures the annual return to an
investor based on the current price.
Current =
Yield
Annual Coupon Interest
Current Market Price
For example, a 10% coupon bond which is currently
selling at $1,150 would have a current yield of:
Current = $100
Yield
$1,150
=
8.7%
Yields
• The yield to maturity measures the compound annual
return to an investor and considers all bond cash
flows. It is essentially the bond’s IRR based on the
current price.
PV =
I1
(1+i)1
+
I2 + … +
(1+i)2
(In + Pn)
(1+i)n
Notice that this is the same equation we saw earlier
when we solved for price. The only difference then is
that we are solving for a different unknown. In this
case, we know the market price but are solving for
return.
Yields
• The yield to maturity measures the compound annual
return to an investor and considers all bond cash
flows. It is essentially the bond’s IRR based on the
current price.
Using Excel
For Example, suppose we wished to determine the YTM
on the following bond.
Finding Yield to Maturity
Market Price ($)
($1,000.00)
Coupon Interest ($)
$
100
Maturity (periods)
10
Face Value ($)
$
1,000
Market Interest Rate (%)
?
Yields
• The yield to maturity measures the compound annual
return to an investor and considers all bond cash
flows. It is essentially the bond’s IRR based on the
current price.
Using Excel
Finding Yield to Maturity
To compute the
yield on this bond
we simply listed
all of the bond
cash flows in a
column and
computed the
IRR
Period Cash Flow
0
($1,000.00)
Market Price ($)
($1,000.00)
1
$
100
Coupon Interest ($)
$
100
2
$
100
Maturity (periods)
10
3
$
100
Face Value ($)
$
1,000
4
$
100
Market Interest Rate (%)
10%
5
$
100
6
$
100
7
$
100
8
$
100
=IRR(d10:d20)
9
$
100
10
$
1,100
Yields
• The yield to maturity measures the compound annual
return to an investor and considers all bond cash
flows. It is essentially the bond’s IRR based on the
current price.
• Note that the yield to maturity will only be equal if the
bond is selling for its face value ($1,000).
• And that rate will be the same as the bond’s coupon
rate.
• For premium bonds, the current yield > YTM.
• For discount bonds, the current yield < YTM.
Yields
• The yield to call is the yield earned on a callable bond.
• To calculate the yield to call, simply substitute the call
date for the maturity date plus the call premium if there
is one.
For Example, suppose we wished to determine the yield
to call (YTC) on the following bond where the call
premium is equal to one year extra coupon interest.
Finding Yield to Call
Market Price ($)
Coupon Interest ($)
Maturity (periods)
Face Value ($)
Call Premium
Market Interest Rate (%)
($1,000.00)
$
100
10
$
1,000
$
100
?
Yields
• The yield to call is the yield earned on a callable bond.
• To calculate the yield to call, simply substitute the call
date for the maturity date plus the call premium if there
is one.
Finding Yield to Call
Market Price ($)
Coupon Interest ($)
Maturity (periods)
Face Value ($)
Call Premium
Market Interest Rate (%)
($1,000.00)
$
100
10
$
1,000
$
100
11%
Period
0
1
2
3
4
5
6
7
8
9
10
Cash Flow
($1,000.00)
$
100
$
100
$
100
$
100
$
100
$
100
$
100
$
100
$
100
$
1,200
Risk and Yield Fluctuations
Bond Ratings and Yields for Bonds Maturing in 2007
Bond Issue
AT&T, 7 3/4, 07
Seagram & Sons, 8 5/8, 07
Long Island Lighting, 7 1/2, 07
Inland Steel, 9.9, 07
S&P Bond
Rating
AA
AA
BB+
BB-
Yield to
Maturity
7.08%
7.16%
9.23%
9.64%
Risk and Yield Fluctuations
Moody's Aaa versus Baa Yield Spread
17%
15%
13%
Aaa
Baa
11%
9%
7%
5%
1965
1970
1975
1980
1985
1990
1995
2000
Year
The Reinvestment Rate Assumption
• It is important to note that the computation of the YTM
implicitly assumes that interest rates are reinvested at
the YTM.
• In other words, if the bond pays a $100 coupon and
the YTM is 8%, the calculation assumes that all of the
$100 coupons are invested at that rate.
• If market interest rates fall, however, the investor may
be forced to reinvest at something less than 8%,
resulting a a realized YTM which is less than
promised.
• Of course, if rates rise, coupons may be reinvested at
a higher rate resulting in a higher realized YTM.
Common Stock Valuation
Stock Returns are derived from both dividends and
capital gains, where the capital gain results from the
appreciation of the stock’s market price.due to the
growth in the firm’s earnings. Mathematically, the
expected return may be expressed as follows:
E(r) = D/P + g
For example, if the firm’s $1 dividend on a $25
stock is expected to grow at 7%, the expected
return is:
E(r) = 1/25 + .07 = 11%
Stock Valuation Models
The Basic Stock Valuation Equation
D1
D2
D
PO 

 ... 
1
2
(1  k ) (1  k )
(1  k ) 
Stock Valuation Models
The Zero Growth Model
• The zero dividend growth model assumes that the
stock will pay the same dividend each year, year after
year.
• For assistance and illustration purposes, I have
developed a spreadsheet tutorial on Excel.
• A non-functional excerpt from the spreadsheet
appears on the following slide.
Stock Valuation Models
The Zero Growth Model
Using Excel
1. Zero Growth (Constant Dividend) Model
A. Solving for Price:
V = D/k, where D = dividend and k = required return
What would an investor be willing to pay for a stock if she expected to receive
a dividend of $2.50 each year indefinitely and her required return is 15%?
D
k
V?
$
2.50
15.00%
$ 16.67
Stock Valuation Models
The Zero Growth Model
Using Excel
B. Solving for Return: k = D/V
What rate of return would an investor expect if the current price of a stock
is $119 and she expected the firm to pay a constant dividend of $4/year?
V
D
k?
$ 119.00
$ 4.00
3.4%
Stock Valuation Models
The Constant Growth Model
• The constant dividend growth model assumes that the
stock will pay dividends that grow at a constant rate
each year -- year after year.
• For assistance and illustration purposes, I have
developed a spreadsheet tutorial using Excel
• A non-functional excerpt from the spreadsheet
appears on the following slide.
Stock Valuation Models
The Constant Growth Model
Using Excel
Valuation
(Note: The tables below have been w ritten using formulas w hich allow you to alter the information or assumptions.)
1. Constant Growth Model
A. Solving for Price: V = D0(1+g)/k-g = D1/(k-g) , where D0 = current dividend, k = required return,
and g = growth rate
What would an investor be willing to pay for a stock if she just received a
dividend of $2.50, her required return is 15%, and she expected dividneds
to grow at a rate of 5% per year.
D0
k
g
V?
$
2.50
15.00%
5.00%
$ 26.25
Stock Valuation Models
The Constant Growth Model
Using Excel
B. Solving for Return: k = D0(1+g)/V + g = D1/V + g
What is my expected return on a stock that costs $26.50, just paid a
dividend of $2.50, and has an expected growth rate of 5%?
D0
V
g
k?
$ 2.50
$ 26.25
5.00%
15.00%
Stock Valuation Models
Variable Growth Model
• The non-constant dividend growth model assumes
that the stock will pay dividends that grow at one rate
during one period, and at another rate in another year
or thereafter.
• For assistance and illustration purposes, I have
developed a spreadsheet tutorial available under the
heading “Course Materials” on Course Web-Page.
• A non-functional excerpt from the spreadsheet
appears on the following slide.
Stock Valuation Models
Variable Growth Model
Using Excel
Valuation
(Note: The tables below have been w ritten using formulas
w hich allow you to alter the informatins or assumptions.)
1. Non-Constant Growth Model
A. Solving for Price: This model involves the computation of year-to-year dividends which
are then dicounted at the investors required rate of return.
What would an investor be willing to pay for a stock if she just received a
dividend of $2.50, her required return is 15%, and she expected dividneds
to grow at a rate of 10% per year for the first two years, and then at a rate of
5% thereafter.
Stock Valuation Models
Variable Growth Model
What would an investor be willing to pay for a stock if she just received a
dividend of $2.50, her required return is 15%, and she expected dividneds
to grow at a rate of 10% per year for the first two years, and then at a rate of
5% thereafter.
Step 1: Compute the expected dividends during the first growth period.
g
D0
10.0%
$ 2.50
D1
$
2.75
D2
$
3.03
Stock Valuation Models
Variable Growth Model
What would an investor be willing to pay for a stock if she just received a
dividend of $2.50, her required return is 15%, and she expected dividneds
to grow at a rate of 10% per year for the first two years, and then at a rate of
5% thereafter.
Step 2: Compute the Estimated Value of the stock at the end of year 2
using the Constant Growth Model
D2
k
g
V2?
$
3.03
15.00%
5.00%
$ 31.76
Stock Valuation Models
Variable Growth Model
What would an investor be willing to pay for a stock if she just received a
dividend of $2.50, her required return is 15%, and she expected dividneds
to grow at a rate of 10% per year for the first two years, and then at a rate of
5% thereafter.
Step 3: Compute the Present Value of all expected cash flows
to find the price of the stock today.
1 D1
Cash
Flow
$
2.75
PV at
15%
$
2.39
2 D2
$
3.03
$
2.29
3 V2?
$
31.76
$
20.88
?
$
25.56
V0
Other Approaches to Stock Valuation
Book Value
• Book value per share is the amount per share that
would be received if all the firm’s assets were sold for
their exact book value and if the proceeds remaining
after paying all liabilities were divided among common
stockholders.
• This method lacks sophistication and its reliance on
historical balance sheet data ignores the firm’s
earnings potential and lacks any true relationship to
the firm’s value in the marketplace.
Other Approaches to Stock Valuation
Liquidation Value
• Liquidation value per share is the actual amount per
share of common stock to be received if al of the firm’s
assets were sold for their market values, liabilities
were paid, and any remaining funds were divided
among common stockholders.
• This measure is more realistic than book value
because it is based on current market values of the
firm’s assets.
• However, it still fails to consider the earning power of
those assets.
Other Approaches to Stock Valuation
Valuation Using P/E Ratios
• Some stocks pay no dividends. Using P/E ratios are
one way to evaluate a stock under these
circumstances.
• The model may be written as:
– P = (m)(EPS)
– where m = the estimated P/E multiple.
For example, if the estimated P/E is 15, and a
stock’s earnings are $5.00/share, the estimated
value of the stock would be P = 15*5 =
$75/share.
Other Approaches to Stock Valuation
Weaknesses of Using P/E Ratios
• Determining the appropriate P/E ratio.
– Possible Solution: use the industry average P/E
ratio
• Determining the appropriate definition of earnings.
– Possible Solution: adjust EPS for extraordinary
items
• Determining estimated future earnings
– forecasting future earnings is extremely difficult
Decision Making and Common Stock Value
• Valuation equations measure the stock value at a
point in time based on expected return and risk.
• Any decisions of the financial manager that affect
these variables can cause the value of the firm to
change as shown in Figure 8.3 below.
Decision Making and Common Stock Value
Changes in Dividends or Dividend Growth
• Changes in expected dividends or dividend growth
can have a profound impact on the value of a stock.
Price Sensitivity to Changes in Dividends and Dividend Growth
(Using the Constant Growth Model)
D0
g
D1
$
kS
P
10.0%
10.0%
10.0%
10.0%
10.0%
10.0%
$ 29.43 $ 36.79 $ 44.14 $ 29.43 $ 53.00 $ 218.00
$
2.00 $
3.0%
2.06 $
2.50 $
3.0%
2.58 $
3.00 $
3.0%
3.09 $
2.00 $
3.0%
2.06 $
2.00 $
6.0%
2.12 $
2.00
9.0%
2.18
Decision Making and Common Stock Value
Changes in Risk and Required Return
• Changes in expected dividends or dividend growth
can have a profound impact on the value of a stock.
Price Sensitivity to Changes Risk (Required Return)
(Using the Constant Growth Model)
D0
g
D1
$
kS
P
5.0%
7.5%
10.0%
12.5%
15.0%
17.5%
$ 103.00 $ 45.78 $ 29.43 $ 21.68 $ 17.17 $ 14.21
$
2.00 $
3.0%
2.06 $
2.00 $
3.0%
2.06 $
2.00 $
3.0%
2.06 $
2.00 $
3.0%
2.06 $
2.00 $
3.0%
2.06 $
2.00
3.0%
2.06
Decision Making and Common Stock Value
Changes in Risk and Required Return
• Changes in expected dividends or dividend growth
can have a profound impact on the value of a stock.
Price Sensitivity to Changes in Both Dividends and Required Return
(Using the Constant Growth Model)
D0
g
D1
$
kS
P
5.0%
7.5%
10.0%
12.5%
15.0%
17.5%
$ 103.00 $ 176.67 $ 327.00 $ 21.68 $ 29.44 $ 38.47
$
2.00 $
3.0%
2.06 $
2.50 $
6.0%
2.65 $
3.00 $
9.0%
3.27 $
2.00 $
3.0%
2.06 $
2.50 $
6.0%
2.65 $
3.00
9.0%
3.27