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Valuation and Rates of Return
Author's Overview
The student can clearly see that the material covered in the previous chapter on time value of
money is now being applied. The recurring theme throughout the chapter is that valuation is
based on the present value of benefits to be received in the future. The instructor should
establish this point at the outset and then repeatedly demonstrate it in the evaluation of bonds,
preferred stock, and common stock. The instructor should also emphasize the relationship of
the discount rate in present value analysis to the required rate of return demanded by security
holders. The authors suggest that the instructor go through the process of defining the
investor's required return in terms of a real rate of return, an inflation premium, and a risk
premium. The instructor can then vary one of these components and show the impact on overall
required return and valuation.
Not all professors wish to use an example of supernormal growth, so we have included
Appendix 10A to cover this topic. This model is good at demonstrating that Ke (the required
rate of return on equity) has to be greater than g (the expected growth rate) in order to use the
constant divided growth model. When growth is greater than the required rate of return, the
supernormal growth model is appropriate. This appendix will help solidify the concepts of
uneven cash flow before getting into the capital budgeting section.
Appendix 10B is an optional discussion on using calculators in financial analysis. Both a Texas
Instruments algebraic calculator and the Hewlett Packard HP12C are used to demonstrate how
to use the calculators to calculate the present value, the future value, the bond values, present
value of an annuity, the present value of an uneven cash flow and the internal rate of return.
We suggest that you refer your students to this appendix. It will help reinforce the concepts in
Chapter 9 and Chapter 10 as well as being useful for the chapters that follow.
Chapter Concepts
LO1. The valuation of a financial asset is based on the present value of future cash flows.
LO2. The required rate of return in valuing an asset is based on the risk involved.
LO3. Bond valuation is based on the process of determining the present value of interest
payments plus the present value of the principal payment at maturity.
LO4. Preferred stock valuation is based on the dividend paid and the market required return.
LO5. Stock valuation is based on determining the present value of the future benefits of equity
ownership.
Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill
Education.
3-1
10
Annotated Outline and Strategy
PPT Relationship between Time Value of Money, Required Return, Cost of
Financing, and Investment Decisions (Figure 10-1)
I. Valuation Concepts
A. The value of an asset is the present value of the expected cash flows associated
with the asset. In order to compute the present value of an asset, an investor must
know or estimate the amount of expected cash flows, the timing of expected
cash flows and the risk characteristics of the expected flows.
B. The price (present value) of an asset will be based on the collective assessment
of the asset's cash flow characteristics by the many capital market participants.
II. Valuation of Bonds
A. The value of a bond is derived from cash flows composed of periodic interest
payments and a principal payment at maturity.
B. The present value (price) of a bond is equal to the present value of the interest
payments plus the present value of the principal (Face Value or Par Value)
payment. It is given in Formula 10-1 as follows:
where:
Pb= the market price of the bond
It= the periodic interest payments
Pn= the principal payment at maturity
t= the period from 1 to n
n= the total number of periods
Y= the yield to maturity (required rate of return)
1. The present value of interest payments can be calculated using Formula
9-6 from the previous chapter.
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Education.
3-2
1
(1 ) (1 )
n
t n
bt n
t
I P
PY Y
=
= +
+ +
å
2. The present value of the principal payment (par value) at maturity may be
computed by applying Formula 9-2 from the previous chapter.
3. The present value (price) of the bond will be the sum of the present value
of the interest payments plus the present value of the principal.
Perspective 10-1: Sometimes it is instructive to have the students calculate the present value
of the interest payments and principal independently rather than just the final bond price. This
can be valuable in showing how a change in required return or maturity can cause the two cash
flows to change. This is most easily demonstrated using a calculator or an Excel spreadsheet.
C. Bond valuation using a financial calculator or an Excel spreadsheet is presented
on page 299 of the text. The Excel spreadsheet is a very instructive way to do
sensitivity testing by varying the years to maturity and the required rate of
return.
D. Concept of Yield to Maturity
1. Three factors influence an investor's required rate of return on a bond.
a. The required real rate of return: the rate of return demanded for
giving up current use of funds on a no-risk, non-inflation adjusted
basis.
b. An inflation premium: a premium to compensate the investor for
the effect of inflation on the value of the dollar.
c. Risk premium: all financial decisions are made within a risk-return
framework. An astute investor will require compensation for risk
exposure. There are two types of risk associated with the required
rate of return (yield to maturity) on a bond:
(1) Business risk is the possibility of a firm not being able to
sustain its competitive position and growth in earnings.
(2) Financial risk is the possibility that a firm will be unable
to meet its debt obligations as they come due.
E. Changing the Yield to Maturity and the Impact on Bond Valuation
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Education.
3-3
1
1(1 )
PV
n
A
i
Ai
-+
= ´
1
PV FV (1 )n
i
= ´ +
1. Bond prices are inversely related to required rates of return. A change in
the required rate of return will cause a change in the bond price in the
opposite direction.
PPT Bond Price Table (Table 10-1)
Perspective 10-2: Table 10-1 illustrates the impact of differences between yield to maturity
and coupon rates on bond prices and reinforces the inverse relationship between bond prices
and required yield to maturity.
2. The impact of the change in required rate of return on the bond price is
dependent upon the remaining time to maturity. The impact will be
greater as the time to maturity increases.
PPT Impact of Time to Maturity on Bond Prices (Table 10-2)
Perspective 10-3: Table 10-2 shows the critical effect of time to maturity on bond price
sensitivity. The longer the time to maturity the bigger the change in bond price with a change in
yield to maturity.
PPT Relationship between Time to Maturity and Bond Price (Figure 10-2)
Perspective 10-4: The critical effect of time to maturity on bond price sensitivity is further
supported by this figure.
F. Determining Yield to Maturity from the Bond Price
1. If the bond price, coupon rate, and number of years to maturity are
known, the yield to maturity (market determined required rate of return)
can be computed.
2. The yield to maturity can most easily be found using a financial
calculator. The keystrokes are as follows:
N Number of years to maturity
PV Present value of the bond (current bond price)
PMT Interest payment (annuity)
FV Future value (par value at maturity)
Function
CPT
I/Y The example in the text gives a yield of 12 percent
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Education.
3-4
3. The yield to maturity can also be found by using the Goal Seek function
in Excel. The Goal Seek function can be found in the most recent version
of Excel on the Data tab, in the Data Tools group under What-If Analysis.
Examples of this spreadsheet are found on page 306 in the text.
F. Semiannual Interest and Bond Prices: Often interest payments are made
more frequently than once a year. Semiannual interest payments are common.
To compute the price of such a bond, we divide the annual amount of interest
and the yield to maturity by two and multiply the number of years to maturity by
two. For example, 10 years would be 20 semi-annual periods. Annual coupon
payments of $60 would be semi-annual payments of $30, and an annual yield to
maturity of 8% would become a semi-annual yield to maturity of 4%.
Perspective 10-5: Students have the opportunity to use both the annual and semiannual
approaches in working problems at the back of the chapter.
III. Valuation of Preferred Stock
A. Preferred stock is usually valued as a perpetual stream of fixed dividend
payments
where:
Pp= the price of preferred stock
Dp= the annual dividend for preferred stock
Kp= the required rate of return (discount rate) applied to preferred stock
dividends
B. Since the dividend stream is a perpetuity, the preferred stock valuation formula
can be reduced to a more usable form
C. If Kp changes after preferred stock is issued, Pp will change in an inverse fashion.
Since preferred stock theoretically has a perpetual life, it is highly sensitive to
changes in the required rate of return (Kp).
D. If the market price of preferred stock and the annual dividend are known, the
market determined required rate of return can be computed by using the
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Education.
3-5
1 2 3
....
(1 ) (1 ) (1 ) (1 )
p p p p
pn
p p p
D D D D
PK
K K K
= + + + +
+ + + +
p
p
p
D
PK
=
valuation equation and solving for Kp.
IV. Valuation of Common Stock
A. The value of a share of common stock is the present value of an expected stream
of dividends
where:
P0= price of the stock at time zero (today)
D= dividend for each year
Ke= the required rate of return for common stock
B. Unlike dividends on most preferred stock, common stock dividends may vary.
The valuation formula may be applied, with modification, to three different
circumstances: no growth in dividends, constant growth in dividends, and
variable growth in dividends.
1. No Growth in Dividends. Common stock with constant (no growth)
dividends is valued in the same manner as preferred stock.
where:
P0= price of common stock
D0= current annual dividend on common stock = D1
(Expected to remain the same in the future)
Ke= required rate of return for common stock
2. Constant Growth in Dividends. The price of common stock with
constant growth in dividends is the present value of an infinite stream of
growing dividends. Fortunately, in this circumstance the basic valuation
equation can be reduced to the more usable form below if the discount
rate (Ke) is assumed to be greater than the growth rate.
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Education.
3-6
p
p
p
D
PK
=
then
p
p
p
D
KP
=
1 2 3
02 3
....
( 1 ) ( 1 ) ( 1 ) ( 1 )
n
n
ee e e
D D D D
PKK K K
= + + + +
++ + +
0
0
e
D
PK
=
where:
D1= dividend expected at the end of the first year = D0(1 + g)
g= constant growth rate in dividends
P0= price of common stock
Ke= required rate of return for common stock
a. The above formula can also be thought to represent the present
value of dividends for a period of time (such as n = 3) plus the
present value of the stock price after a period of time (such as P3).
Since P3 represents the present value of dividends from D4 through
Dn, P0 will still represent the present value of all future dividends.
b. The value of P0 is quite sensitive to any change in the required rate
of return (Ke) and the growth rate (g).
3. Rearrangement of the constant growth equation allows the calculation of
the required rate of return, Ke, when P0, D1, and g are given.
The first term represents the dividend yield that the stockholder expects
to receive and the second term represents the anticipated growth in
dividends, earnings and stock price.
4. The Price Earnings Ratio Concept and Valuation. Stock valuation may
also be linked to the concept of price-earnings ratios discussed in Chapter
2. Although this is a less theoretical, more pragmatic approach than the
dividend valuation models, the end results may be similar because of the
common emphasis on risk and growth under either approach.
Perspective 10-6: Table 10-4 from Barron’s illustrates how P/E ratios are shown in the
financial press. IBM can be used as an example, and dividend yield and price changes can also
be mentioned to stimulate interest.
5. Variable Growth in Dividends. The most likely variable growth case is
one of supernormal growth followed by constant growth.
a. Value can be found through taking the present value of the
dividends during the supernormal growth period plus the price of
the stock at end of the supernormal growth period. Since growth
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Education.
3-7
1
0( )
e
D
Pg
K
=-
1
0
e
Dg
KP
= +
is then constant, Formula 10-8 can be used for the terminal value.
Finance in Action: An Important Question – What’s a Small Business Really Worth?
This box presents some of the practical issues faced in valuing a small business and serves as a
contrast to the formulas presented in the text.
b. Another type of variable growth is where
the firm is assumed to pay no dividends for a period of time and
then begins paying dividends. In this case, the present value of
deferred dividends can be computed as a representation of value.
c. If no dividends are ever intended, then valuation may rest solely
on the present value of future earnings and the present value of a
future stock price.
Perspective 10-7: Appendix10A Valuation of a Supernormal Growth Firm is optional
found on pages 330-332.
Perspective 10-8: Appendix10B Using Calculators for Financial Analysis is an often
overlooked presentation of comprehensive calculator solutions for both the Texas Instruments
BAII Plus and the Hewlett-Packard 12C. Found on pages 332-340.
Other Chapter Supplements
Cases for Use with Foundations of Financial Management
Case 13, Gilbert Enterprises (Stock Valuation)
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Education.
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