1. The document provides examples and explanations of stock valuation models, including the discounted dividend model and the constant growth model.
2. The constant growth model assumes dividends and stock prices will grow at a constant rate indefinitely. It provides a simplified equation to value a stock based on the next expected dividend, required rate of return, and long-term growth rate.
3. An example applies the constant growth model to value a stock with a $1.15 dividend growing at 8% annually and a required return of 13.4%, calculating the stock price as $23.
1. Nicks Enchiladas Incorporated has preferred stock outstand.docx
1. 1. Nick's Enchiladas Incorporated has preferred stock
outstanding that pays a dividend of $5 at
the end of each year. The preferred stock sells for $35 a share.
What is the stock's required rate of
return? Round the answer to two decimal places.
%
2. A stock is expected to pay a year-end dividend of $2.00, i.e.,
D1 = $2.00. The dividend is
expected to decline at a rate of 5% a year forever (g = −5%). If
the company is in equilibrium and
its expected and required rate of return is 15%, which of the
following statements is CORRECT?
a. The company's dividend yield 5 years from now is expected
to be 10%.
b. The company's current stock price is $20.
c. The company's expected capital gains yield is 5%.
d. The constant growth model cannot be used because the
growth rate is negative.
e. The company's expected stock price at the beginning of next
year is $9.50.
2. 3. The expected return on Natter Corporation's stock is 14%.
The stock's dividend is expected to
grow at a constant rate of 8%, and it currently sells for $50 a
share. Which of the following
statements is CORRECT?
a. The stock's dividend yield is 7%.
b. The current dividend per share is $4.00.
c. The stock's dividend yield is 8%.
d. The stock price is expected to be $54 a share one year from
now.
e. The stock price is expected to be $57 a share one year from
now.
4. Investors require a 16% rate of return on Brooks Sisters'
stock (rs = 16%).
a. What would the value of Brooks's stock be if the previous
dividend was D0 = $2.75 and if
investors expect dividends to grow at a constant compound
annual rate of (1) - 4%, (2)
0%, (3) 7%, or (4) 10%? Round your answers to the nearest
cent.
1. $
2. $
3. 3. $
4. $
b. Using data from part a, what is the Gordon (constant growth)
model's value for Brooks
Sisters's stock if the required rate of return is 16% and the
expected growth rate is (1)
16% or (2) 24%? Are these reasonable results? Explain.
1. (Yes or No)
2. (Yes or No)
c. Is it reasonable to expect that a constant growth stock would
have g > rs? ( Yes or No)
5. Brushy Mountain Mining Company's ore reserves are being
depleted, so its sales are falling.
Also, its pit is getting deeper each year, so its costs are rising.
As a result, the company's earnings
and dividends are declining at the constant rate of 6% per year.
If D0 = $3 and rs = 13%, what is
the value of Brushy Mountain Mining's stock? Round your
answer to the nearest cent.
$
6. Boehm Incorporated is expected to pay a $3.70 per share
dividend at the end of this year (i.e.,
D1 = $3.70). The dividend is expected to grow at a constant rate
of 10% a year. The required rate
of return on the stock, rs, is 18%. What is the value per share of
4. the company's stock? Round your
answer to the nearest cent.
$
7. A company currently pays a dividend of $1.5 per share, D0 =
1.5. It is estimated that the
company's dividend will grow at a rate of 19% percent per year
for the next 2 years, then the
dividend will grow at a constant rate of 6% thereafter. The
company's stock has a beta equal to
1.15, the risk-free rate is 7.5 percent, and the market risk
premium is 3 percent. What is your
estimate is the stock's current price? Round your answer to the
nearest cent.
$
8. The beta coefficient for Stock C is bC = 0.3, and that for
Stock D is bD = - 0.4. (Stock D's beta
is negative, indicating that its rate of return rises whenever
returns on most other stocks fall.
There are very few negative-beta stocks, although collection
agency and gold mining stocks are
sometimes cited as examples.)
a. If the risk-free rate is 7%and the expected rate of return on an
average stock is 12%, what
are the required rates of return on Stocks C and D? Round the
answers to two decimal
places.
1. rC = %
2. rD = %
b. For Stock C, suppose the current price, P0, is $25; the next
expected dividend, D1, is
$1.50; and the stock's expected constant growth rate is 4%. Is
5. the stock in equilibrium?
Explain, and describe what would happen if the stock is not in
equilibrium. (Choose
from the answers below: I, II, III, IV, or V)
I. In this situation, the expected rate of return = 8.50%.
However, the required rate of return is
10%. Investors will seek to buy the stock, raising its price to
$33.33. At this price, the stock will
be in equilibrium.
II. In this situation, the expected rate of return = 10%.
However, the required rate of return is
8.50%. Investors will seek to buy the stock, raising its price to
$33.33. At this price, the stock will
be in equilibrium.
III. In this situation, the expected rate of return = 10%.
However, the required rate of return is
8.50%. Investors will seek to sell the stock, raising its price to
$33.33. At this price, the stock will
be in equilibrium.
IV. In this situation, the expected rate of return = 8.50%.
However, the required rate of return is
10%. Investors will seek to sell the stock, raising its price to
$33.33. At this price, the stock will
be in equilibrium.
V. In this situation, both the expected rate of return and the
required rate of return are equal.
Therefore, the stock is in equilibrium at its current price.
6. 9. The risk-free rate of return, rRF , is 12%; the required rate of
return on the market, rM, 15%; and
Schuler Company's stock has a beta coefficient of 1.5.
a. If the dividend expected during the coming year, D1, is
$2.25, and if g is a constant
2.25%, then at what price should Schuler's stock sell? Round
your answer to the nearest
cent.
$
b. Now, suppose the Federal Reserve Board increases the money
supply, causing a fall in
the risk-free rate to 6% and rM to 13%. How would this affect
the price of the stock?
Round your answer to the nearest cent.
$
c. In addition to the change in part b, suppose investors' risk
aversion declines; this fact,
combined with the decline in rRF, causes rM to fall to 9%. At
what price would Schuler's
stock sell? Round your answer to the nearest cent.
$
d. Suppose Schuler has a change in management. The new group
institutes policies that
increase the expected constant growth rate to 7%. Also, the new
management stabilizes
sales and profits, and thus causes the beta coefficient to decline
from 1.5 to 1.0. Assume
that rRF and rM are equal to the values in part c. After all these
changes, what is Schuler's
new equilibrium price? (Note: D1 goes to $2.35.) Round your
answer to the nearest cent.
$
7. 10.
A share of common stock just paid a dividend of $1.00. If the
expected long-run growth rate
for this stock is 5.4%, and if investors' required rate of return is
11.4%, what is the stock price?
a. $18.01
b. $17.57
c. $16.28
d. $16.70
e. $17.13
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Ch07 P20 Build a Model Spring 1, 20137/22/12Chapter 7. Ch
07 P20 Build a ModelExcept for charts and answers that must
be written, only Excel formulas that use cell references or
functions will be accepted for credit. Numeric answers in cells
will not be accepted.Rework Problem 7-19. Taussig
Technologies Corporation (TTC) has been growing at a rate of
20% per year in recent years. This same growth rate is expected
to last for another 2 years (g1 = g2 = 20%).a. If D0 = $1.60, rs
= 10%, and gn = 5%, what is TTC's stock worth today? What
are its expected dividend yield and capital gains yield at this
time?1. Find the price today.D0$1.60rs10.0%gs20%Short-run
g; for Years 1-2 only.gL5%Long-run g; for Year 3 and all
following years.20%5%Year0123DividendPV of dividends=
D3= Terminal value = P2 == rs – gL = P0 2. Find the
expected dividend yield.Recall that the expected dividend yield
is equal to the next expected annual dividend divided by the
9. price at the beginning of the period.Dividend yield
=D1/P0Dividend yield =/Dividend yield =3. Find the expected
capital gains yield.The capital gains yield can be calculated by
simply subtracting the dividend yield from the total expected
return.Cap. Gain yield=Expected return–Dividend yieldCap.
Gain yield=–Cap. Gain yield=Alternatively, we can recognize
that the capital gains yield measures capital appreciation, hence
solve for the price in one year, then divide the change in price
from today to one year from now by the current price. To find
the price one year from now, we will have to find the present
values of the terminal value and second year dividend to time
period one.P1=P2+D2(1 + rs)P1=+P1=Cap. Gain yield=(P1 –
P0)/P0Cap. Gain yield=/Cap. Gain yield=b. Now assume that
TTC's period of supernormal growth is to last for 5 years rather
than 2 years. How would this affect its price, dividend yield,
and capital gains yield?1. Find the price
today.D0$1.60rs10.0%gS20%Short-run g; for Years 1-5
only.gL6%Long-run g; for Year 6 and all following
years.20%6%Year0123456DividendPV of dividends= D6
Kenneth D. Jackson: Discounted 5 years
Kenneth D. Jackson: Discounted two years
Horizon value = P5 == = P0= rs – gLPart 2. Finding the
expected dividend yield.Dividend yield =D1/P0Dividend yield
=/Dividend yield =Part 3. Finding the expected capital gains
yield.Cap. Gain yield=Expected return–Dividend yieldCap. Gain
yield=–Cap. Gain yield=c. What will TTC's dividend yield and
capital gains yield be once its period of supernormal growth
ends? (Hint: These values will be the same regardless of
whether you examine the case of 2 or 5 years of supernormal
growth, and the calculations are very easy.)We used the 5-year
supernormal growth scenario for this calculation, but ultimately
it does not matter which example you use, as they both yield the
same result.Dividend yield =Dn+1/PnDividend yield =/Dividend
10. yield =Cap. Gain yield=Expected return–Dividend yieldCap.
Gain yield=–Cap. Gain yield=Upon reflection, we see that these
calculations were unnecessary because the constant growth
assumption holds that the long-term growth rate is the dividend
growth rate and the capital gains yield, hence we could have
simply subtracted the long-run growth rate from the required
return to find the dividend yield.
Sheet27/22/12
Chapter4/11/10Chapter 7 Tool Kit for Stock Valuation
VALUING COMMON STOCKS (Section 7.5)Stocks can be
evaluated in two ways: (1) find the stock price directly by
calculating the present value of the expected future dividends,
or (2) find the stock price indirectly by first calculating the
value of the entire corporation, which is the the present value of
the firm's expected future free cash flows, and then subtracting
the value of the debt and preferred stock to find the total value
of the common equity. Only the first approach (the dividend
model) is discussed in this chapter. The second approach (the
corporate valuation model) is described in Chapter 13.THE
DISCOUNTED DIVIDEND APPROACHThe value of any
financial asset is the present value of the future cash flows
provided by the asset. When an investor buys a share of stock,
he or she typically expects to receive cash in the form of
dividends and then, eventually, to sell the stock and to receive
cash from the sale. However, the price the first investor
receives is dependent upon the dividends the next investor
expects to earn, and so on for different generations of investors.
Thus, the stock's value ultimately depends on the cash dividends
the company is expected to provide and the discount rate used
to find the present value of those dividends.Here is the basic
dividend valuation equation:P0 =D1+D2+. . . .DN( 1 + rs )( 1
+ rs ) 2( 1 + rs ) NThe dividend stream theoretically extends on
out forever, i.e., to n = infinity. Obviously, it would not be
feasible to deal with an infinite stream of dividends, but
fortunately, a relatively simple equation has been developed
11. that can be used to find the PV of the dividend stream, provided
it is growing at a constant rate.VALUING A CONSTANT
GROWTH STOCK (Section 7.6)In the constant growth model,
we assume that the dividend and stock will grow forever at a
constant growth rate. Naturally, assuming a constant growth
rate for the rest of eternity is a rather bold assumption.
However, considering the implications of imperfect information,
information asymmetry, and general uncertainty, the assumption
of constant growth is often reasonable. It is reasonable to guess
that a given stock will experience ups and downs throughout its
life. By assuming constant growth, we are trying to find the
average of the good times and the bad times, and we assume
that we will see both scenarios over the firm's life. In addition
to a constant growth rate, we also need the estimated long-term
required return for the stock, and it too must be constant. If
these variables are constant, our price equation for common
stock simplifies to the following expression:P0 =D1( rs – g )In
this equation, the long-run growth rate (g) can be approximated
by multiplying the firm's return on assets by the retention ratio.
Generally speaking, the long-run growth rate of a firm is likely
to fall between 5% and 8% a year.EXAMPLE: CONSTANT
GROWTHA firm just paid a $1.15 dividend and its dividend is
expected to grow at a constant rate of 8%. What is its stock
price, assuming it has a required return of 13.4%?D0 =$1.15g
=8%rs =13.4%P0 =D1=D0 (1 + g)=$1.2420( rs – g )( rs – g
)0.0540P0 =$23.00How sensitive is the stock's price to changes
in the dividend, the growth rate, and rs? We can construct a
series of data tables and a graph to examine this
question.Resulting % ChangeLastPricein D0Dividend,
D0$23.00-30%$0.81$16.10-
15%$0.98$19.550%$1.15$23.0015%$1.32$26.4530%$1.50$29.9
0% ChangeReq'd Return$23.00-30%9.38%$90.00-
15%11.39%$36.640%13.40%$23.0015%15.41%$16.7630%17.42
%$13.18% ChangeGrowth Rate$23.00-30%5.60%$15.57-
15%6.80%$18.610%8.00%$23.0015%9.20%$29.9030%10.40%$
42.32This chart shows that the stock price has a positive
12. relationship with the dividend and the growth rate, and a
negative relationship with the required return. Furthermore, we
see that the dividend has a linear relationship with price, while
the growth rate seems to have a quadratic relationship. The
relationship between required return and stock price is not only
negative, but it is a quadratic relationship with greater
convexity than the growth rate. This indicates that the required
return is the factor that more directly influences the stock price.
In other words, required return is the value driver in this
valuation technique. However, the final effects also depend on
the amount of change in each of the three variables. If the
required return and dividend are expected to be stable, but the
dividend growth rate is expected to change significantly, then
the growth rate will be the primary determinant of the stock
price.DO STOCK PRICES REFLECT LONG-TERM OR
SHORT-TERM CASH FLOWS?Managers often claim that stock
prices are "short-term" in nature in the sense that they reflect
what is happening in the near-term and ignore the long-term.
We can use the results for the constant growth model to shed
light on this claim.The first step is to forecast the dividends for
the next 5 years. Then we find the present value of these
dividends and compare that PV with the current stock price,
which reflects the PV of all future dividends.P0 =$23.00D0
=$1.15g =8%rs
=13.4%Year012345Dividend$1.15$1.24$1.34$1.45$1.56$1.69P
V of dividends in Years 1 through 5 =$4.98Current stock price
=$23.00Percent of current stock price due to
dividends in Years 1 through 5 =21.6%Percent of current stock
price due to
dividends beyond Year 5 =78.4%For most stock, the percentage
of the current price that is due to long-term cash flows is over
80%.EXPECTED RATE OF RETURN ON A CONSTANT
GROWTH STOCK (Section 7.7)Using the constant growth
equation introduced earlier, we can re-work the equation to
solve for rs. In doing so, we are now solving for an expected
return. The expression we are left is:rs =D1+gP0This
13. expression tells us that the expected return on a stock comprises
two components. First, it consists of the expected dividend
yield, which is simply the next expected dividend divided by the
current price. The second component of the expected return is
the expected capital gains yield. The expected capital gains
yield is the expected annual price appreciation of the stock, and
is given by g. This shows us the dual role of g in the constant
growth rate model. Not only does g indicate expected dividend
growth, but it is also the expected stock price growth
rate.EXAMPLE: EXPECTED RATE OF RETURN ON A
CONSTANT GROWTH STOCKYou buy a stock for $23, and
you expect the next annual dividend to be $1.242. Furthermore,
you expect the dividend to grow at a constant rate of 8%. What
is the expected rate of return on the stock, and what is the
dividend yield of the stock?P0
$23.00D1$1.242g8%rs13.40%Dividend Yield + Capital Gains
Yield = Expected Rate of ReturnDividend yield5.40%
5.40% + 8% = 13.40%EXAMPLE: EXPECTED PRICE IN THE
FUTUREWhat is the expected price of this stock in five
years?N =5Using the growth rate we find that:P5
=$33.79=B152*(1+B154)^B163VALUING NONCONSTANT
GROWTH STOCKS (Section 7.8)For many companies, it is
unreasonable to assume that they grow at a constant growth
rate. Hence, valuation for these companies proves a little more
complicated. The valuation process, in this case, requires us to
estimate the short-run nonconstant growth rate and predict
future dividends. Then, we must estimate a constant long-term
growth rate at which the firm is expected to grow. Generally,
we assume that after a certain point of time, all firms begin to
grow at a rather constant rate. Of course, the difficulty in this
framework is estimating the short-term growth rate, how long
the short-term growth will hold, and the long-term growth
rate.Specifically, we will predict as many future dividends as
we can and discount them back to the present. Then we will
treat all dividends to be received after the convention of
constant growth rate with the Gordon constant growth model
14. described above. The point in time when the dividend begins to
grow constantly is called the horizon date. When we calculate
the constant growth dividends, we solve for a terminal value (or
a continuing value) as of the horizon date. The terminal value
can be summarized as:TVN =PN =DN+1=DN (1 + g)( rs –
g )( rs – g )This condition holds true, where N is the terminal
date. The terminal value can be described as the expected value
of the firm in the time period corresponding to the horizon
date.EXAMPLE: NONCONSTANT GROWTHA company's
stock just paid a $1.15 dividend, which is expected to grow at
30% for the next three years. After three years the dividend is
expected to grow constantly at 8% forever. The stock's required
return is 13.4%, what is the price of the stock today?D0
=$1.15rs =13.4%gs =30%Short-run g; for Years 1-3 only.gL
=8%Long-run gL; for all years after Year 3.Growth
rate30%30%30%8%8%Year01234Dividends$1.15$1.4950$1.943
5$2.5266$2.7287PV of dividends discounted at
rs$1.31831.51131.7326$2.7287$4.5622 = PV of nonconstant
dividends$50.5310= Horizon value =──────$34.6512 = PV
of horizon value5.4%$39.2135 = P0rs – gLPREFERRED
STOCK (Section 7.11)Consider an issue of preferred stock that
pays a $10 dividend and has a required return of 10%. What is
the price of this preferred stock?Vps =Dps÷rps
=$10.00÷10.00% =$100.00Some preferred stock has a
maturity date. Consider a firm whose preferred stock matures
in 50 years, pays a $10 annual dividend, has a par value of
$100, and has a required return of 8%. What is the price of this
preferred stock?Years to Maturity (N):50Annual Dividend
(PMT):$10Par value (FV):$100Required return, rd
(I/YR):8%Vps =$124.47STOCK MARKET EQUILIBRIUM
(Section 7.12)Changes in Equilibrium Stock PricesSmall
changes in the market's expectations can cause large changes in
stock price!OriginalNewRisk-free rate, rRF8%7%Market risk
premium, rM – rRF4%3%Stock i’s beta coefficient,
bi21ri16.00%10.00%Stock i’s expected growth rate,
gi5%6%D0$2.8571$2.8571Price of Stock i$27.27$75.71
22. Changes in Dividends, Req's Return, and Growth: Effect on
Stock Price
Dividend -0.3 -0.15 0.0 0.15 0.3 16.1 19.55 23
26.45 29.9 Req's Return -0.3 -0.15 0.0 0.15
0.3 90.00000000000004 36.63716814159292 23
16.76113360323887 13.18471337579617 Growth -0.3 -
0.15 0.0 0.15 0.3 15.56923076923077 18.60909090909091 23
29.89999999999999 42.32
Percent Change from Base
Stock Price
7.5SECTION 7.5SOLUTIONS TO SELF-TESTIf D1 = $3.00, P0
= $50, and the expected P at t=1 is equal to $52, what are the
stock’s expected dividend yield, capital gains yield, and total
return for the coming year?D1$3.00P0$50.00Expected
P1$52.00Exp. dividend yield6.0%=B6/B7Exp. capital gains
yield4.0%=(B8-B7)/B7Exp. total return10.0%=C10+C11
7.6SECTION 7.6SOLUTIONS TO SELF-TEST A stock is
expected to pay a dividend of $2 at the end of the year. The
required rate of return is rs = 12%. What would the stock’s
price be if the growth rate were 4%? D1$2.00g4%rs12%Stock
price$25.00A stock is expected to pay a dividend of $2 at the
end of the year. The required rate of return is rs = 12%. What
would the stock’s price be if the growth rate were
0%?D1$2.00g0%rs12%Stock price$16.67
7.7SECTION 7.7SOLUTIONS TO SELF-TEST If D0 = $4.00, rs
= 9%, and g = 5% for a constant growth stock, what are the
stock’s expected dividend yield and capital gains yield for the
coming year?D0$4.00g5%rs9%Expected D1$4.20Stock
price$105.00Expected dividend yield4.00%Expected capital
gains yield5.00%Alternatively, you know that the capital gains
yield is equal to the growth rate.Expected capital gains yield =
growth rate = 5.00%Because the total return is rs, the dividend
23. yield is rs minus the capital gains yield:Expected dividend
yield4.00%
7.8SECTION 7.8SOLUTIONS TO SELF-TEST Suppose D0 =
$5.00 and rs = 10%. The expected growth rate from Year 0 to
Year 1
(g0 to 1) = 20%, the expected growth rate from Year 1 to Year
2 (g1 to 2) = 10%, and the constant rate beyond Year 2 is gn =
5%. What are the expected dividends for Year 1 and Year 2?
What is the expected horizon value price at Year 2? What is the
expected P0?D0$5.00g0 to 120%g1 to
210%gn5%rs10%Year12D1D2Expected
dividends$6.00$6.60Expected P2$138.60PV of expected
dividends$10.91PV of expected P2$114.55Expected P0$125.45
7.11SECTION 7.11SOLUTIONS TO SELF-TEST A preferred
stock has an annual dividend of $5. The required return is 8%.
What is the Vps?Dps$5.00rps8%Vps$62.50
