CHAPTER 18: EQUITY VALUATION MODELS
1.
Choice (a):
2.
(c)
3.
a.
P0 = D1/(k – g) = $2.10/(.11 – 0) = $19.09
k = D1/P0 + g
.16 = 2/50 + g
g = .12 = 12%
b.
P0 = D1/(k – g) = $2/(.16 – .05) = $18.18
The price falls in response to the more pessimistic dividend forecast. The forecast
for current year earnings, however, is unchanged. Therefore, the P/E ratio must
fall. The lower P/E ratio is evidence of the diminished optimism concerning the
firm's growth prospects.
4.
a.
g = ROE  b = 16%  .5 = 8%
D1 = $2(1 – b) = $2(1 – .5) = $1
P0 = D1/(k – g) = $1/(.12 – .08) = $25
5.
6.
b.
P3 = P0(1 + g)3 = $25(1.08)3 = $31.49
a.
This director is confused. In the context of the constant growth model that P0 =
D1/(k – g), it is true that price is higher when dividends are higher holding
everything else including dividend growth constant. But everything else will
not be constant. If the firm raises the dividend payout rate, the growth rate g
will fall, and stock price will not necessarily rise. In fact if ROE > k, price will
fall.
b.
An increase in dividend payout will reduce the sustainable growth rate as less
funds are reinvested in the firm. The sustainable growth rate is (ROE
plowback), which will fall as plowback ratio falls. The increased dividend
payout rate will reduce the growth rate of book value for the same reason -- less
funds are reinvested in the firm.
a.
k = 6% + 1.25(14% – 6%) = 16%; g = 2/3 9% = 6%
D1 = E0 (1 + g) (1 – b) = 3(1.06) (1/3) = $1.06
18-1
D1
$1.06
P0 = k – g = .16 – .06 = $10.60
b.
Leading P0/E1 = 10.60/3.18 = 3.33
Trailing P0/E0 = 10.60/3 = 3.53
c.
E1
PVGO = P0 – k
= $10.60 –
$3.18
.16 = – $9.275
The low P/E ratios and negative PVGO are due to a poor ROE, 9%, that is less
than the market capitalization rate, 16%.
d.
Now, you revise b to 1/3, g to 1/3  .09 = .03, and D1 to [E0 (1.03)  (2/3)] =
$2.06. Thus, V0 = $2.06/(.16 – .03) = $15.85. V0 increases because the firm
pays out more earnings instead of reinvesting them at a poor ROE. This
information is not yet known to the rest of the market.
7.
Because beta = 1.0, k = market return, 15%. Therefore 15% = D1/P0 + g = 4% + g.
Therefore g = 11%.
8.
FI Corporation.
a.
g = 5%; D1 = $8; k = 10%
D1
$8
P0 = k – g = .10 – .05 = $160
9.
b.
The dividend payout ratio is 8/12 = 2/3, so the plowback ratio is b = 1/3. The
implied value of ROE on future investments is found by solving: g = b  ROE with
g = 5% and b = 1/3. ROE = 15%.
c.
The price assuming ROE = k is just E1/k. P0 = $12/.10 = $120. Therefore, the
market is paying $40 per share ($160 – $120) for growth opportunities.
a. k = 4% + 1.15  (10% – 4%) = 10.9%
b. Using Emma's short term growth projections of 25%, we obtain a two-stage
DDM value as follows:
18-2
D1
P0 = 1 + k
+
D2
(1 + k)2
+
D3
(1 + k)3
.287
= 1.109
+
.359
1.1092
+
.449
1.1093
+
D4 + D5/(k – g)
(1 + k)4
+
.561 + .701/(.109 – .093)
1.1094
= .259 + .292 + .329 + 29.336
= $30.216
c. With these new assumptions, Disney stock has an intrinsic value below its market
price of $37.75. This analysis indicates a sell recommendation. Even though
Disney's 5-year growth rate increases so does its beta and risk premium. The
intrinsic value falls.
10. High-Flyer stock.
a.
k = rf + (rM) – rf ] = 10 + 1.5(15 – 10) = 10% + 7.5% = 17.5%, and g = 5%.
Dl
$2.50
$2.50
Therefore, P0 = k – g = .175 – .05 = .125 = $20.
11.
Stock
Market capitalization rate, k
Expected return on equity, ROE
Estimated earnings per share, E1
Estimated dividends per share, D1
Current market price per share, P0
A
10%
14%
$2.00
$1.00
$27
B
10%
12%
$1.65
$1.00
$25
a.
Dividend payout ratio, 1 – b
.5
.606
b.
Growth rate, g = ROE  b
7%
4.728%
c.
Intrinsic value, V0
$33.33
$18.97
d.
You would invest in Stock A since its intrinsic value exceeds its price. You
might want to sell short stock B.
18-3
12. a.
It is true that NewSoft sells at higher multiples of earnings and book value than
does Capital Corp. But this difference may be justified by NewSoft's higher
expected growth rate of earnings and dividends. NewSoft is in a growing
market with abundant profit and growth opportunities. Capital Corp. is in a
mature industry with fewer growth prospects. Both the price-to-earnings and
price-to-book ratios will reflect the prospect of growth opportunities, implying
that the ratios for these firms do not necessarily imply mispricing.
b.
The most important weakness of the constant-growth dividend discount model
in this application is that it assumes a perpetual constant growth rate of
dividends. While dividends may be on a steady growth path for Capital Corp.,
which is a more mature firm, that is far less likely to be a realistic assumption
for NewSoft.
c.
NewSoft should be valued using a multi-stage DDM, which allows for rapid
growth in the early years, but recognizes that growth ultimately must slow to a
more sustainable rate.
13. a.
b.

Investors might extrapolate recent performance of growth stocks too far into
the future and thus overestimate the value of growth stocks. The inevitable
correction would make growth stocks underperform value stocks over
extended periods.

Momentum investors might focus on recently growing firms and bid up their
prices. Overpricing of these growth stocks results.

Value stocks may not be as extensively researched if they are not as exciting
or publicized. The resultant “neglect effect” will give them higher average
returns.

During market run-ups, investors may underestimate the risk of growth
stocks, and not remember that just as these stocks perform well in up
markets, they can perform poorly in down markets. Their performance in up
markets is not sustainable “long-run” performance.
In an efficient market, stock prices correctly reflect all available information. If
so, both growth and value stocks will provide the same risk-adjusted returns.
The effects listed in part (a) all rely on some form of investor irrationality or
error, which should not characterize an efficient market.
18-4
14. Nogro Corporation.
a. P0 = $10, El = $2, b = .5, ROE = .20
k = D1/P0 + g
D1 = .5  $2 = $1
g = b  ROE = .5  .2 = .1
Therefore, k = $1/$10 + .1 = .1 + .1 = .2 or 20%
b. Since k = ROE, the NPV of future investment opportunities is zero:
E1
PVGO = P0 – k = 10 – 10 = 0
c. Since k = ROE, the stock price would be unaffected by cutting the dividend and
investing the additional earnings.
15. a. E0 = $1; D0 = $.50; ROE = 20%; k = 15%; b = .5.
Therefore, g = ROE  b = 20%  .5 = 10%
D1
D0(1 + g) $.50  1.10
P0 = k – g = k – g = .15 – .10 = $11
b. time EPS
0
1.00
1
1.10
Dividend
.50
.55
2
1.21
.726
3
1.2826 .7696
Comment
This data is given in the question
g = 10%, plowback = .50
EPS has grown by 10% based on last year's earnings
plowback, and ROE; this year's earnings plowback
ratio now falls to .40 and payout ratio = .60
EPS grows by .4  15% = 6% and payout ratio = .60
D3
.7696
At time 2, P2 = k – g = .15 – .06 = $8.551
.55
.726 + 8.551
At time 0, V0 = 1.15 +
= $7.493
(1.15)2
18-5
c. P0 = $11 and P1 = P0(1 + g) = $12.10. (Because the market is unaware of the
changed competitive situation, it believes the stock price should grow at 10% per
year.)
P2 = $8.551 after the market becomes aware of the changed competitive situation.
P3 = $8.551  1.06 = $9.064. The new growth rate is 6%.
Year
Return
1
(12.10 – 11) + .55
= .150 = 15.0%
11
2
(8.551 – 12.10) + .726
= – .233 = 23.3%
12.10
3
(9.064 – 8.551) + .7696
= .150 = 15.0%
8.551
Moral: In "normal periods" when there is no special information, the stock return =
k = .15. When special information arrives, all the abnormal return accrues in that
period, as one would expect in an efficient market.
16. Xyrong Corporation
a. k = rf +[E(rM) – rf] = 8% + 1.2(15% – 8%) = 16.4%
g = b  ROE = .6  20% = 12%
V0 =
D0(1 + g) $4  1.12
k – g = .164 – .12 = $101.82
b. P1 = V1 = V0(1 + g) = $101.82  1.12 = $114.04
E(r) =
D1 + P1 – P0
$4.48 + $114.04 – $100
=
= .1852 = 18.52%
P0
$100
18-6
17. a. k = rf + (rM) – rf ] = .045 + 1.15(.145  .045) = .16 = 16%
b. Year
1999
2000
2001
2002
2003
Dividends
$1.72
1.72  1.12
1.72  1.122
1.72  1.123
1.72  1.123  1.09
= $1.93
= $2.16
= $2.42
= $2.64
Present value of dividends in 2000 – 2002:
Year
2000
2001
2002
Dividends
1.93 / 1.161 = $1.66
2.16 / 1.162 = $1.61
2.42 / 1.163 = $1.55
Total:
$4.82
Price at year-end 2002 =
D2003
$2.64
=
= $37.71
kg
.16  .09
PV in 1999 of this stock price = 37.71/1.163 = $24.16
Intrinsic value of stock = $4.82 + $24.16 = $28.98
c. The table presented in the problem indicated that Quick Brush was selling below
intrinsic value, while we’ve just shown that SmileWhite is selling a bit above the
estimate of intrinsic value. Based on this analysis, Quick Brush offers the
potential for considerable abnormal returns, while SmileWhite would offer
slightly below-market risk-adjusted returns.
d. Strengths of 2-stage versus constant growth DDM:
 2-stage model allows for separate valuation of two distinct periods in a
company’s future. This can accommodate life cycle effects. It also can avoid
the difficulties posed by initial growth that is higher than the discount rate.
 2-stage model allows for initial period of above-sustainable growth. It allows
the analyst to make use of her expectations of when growth may shift from
off-trend to a more sustainable level.
A weakness of all DDMs is that they are all very sensitive to input values. Small
changes in k or g can imply big changes in estimated intrinsic value. These
inputs are difficult to measure.
18-7
18. DEQS Corporation.
Time: 0
Et
$10
Dt
$0
b
1
g
.20
1
$12
$0
5
$24.883
$0
1
.20
1
.20
6
$29.860
$11.944
...
.60
.09
D6
$11.944
a. V5 = k – g = .15 – .09 = $199.07
V5
V0 = (1 + k)5 = $98.97
b. The price should rise by 15% per year until year 6: because there is no dividend,
the entire return must be in capital gains.
c. It would have no effect since ROE = k.
19. a. The formula for a multistage DDM model with two distinct growth stages
consisting of a first stage with five years of above-normal constant growth
followed by a second stage of normal constant growth is:
D6
D1
D2
D3
D4
D5
(k – g)
V0 =
+
+
+
+
+
1
2
3
4
5
(1 + k)
(1 + k)
(1 + k)
(1 + k)
(1 + k)
(1 + k)5
$5.09
(0.10
– 0.07)
$2.29
$2.75
$3.30
$3.96
$4.75
=
+
+
+
+
+
1
2
3
4
5
1.10
1.10
1.10
1.10
1.10
1.105
= $117.84
where:
D1
D2
D3
D4
D5
D6
k
g
=
=
=
=
=
=
=
=
D0 (1 + 0.20) = 2.29
D1 (1 + 0.20) = 2.75
D2 (1 + 0.20) = 3.30
D3 (1 + 0.20) = 3.96
D4 (1 + 0.20) = 4.75
D5 (1 + 0.07) = 5.09
Equity required return = 0.10
Growth in second stage = 0.07
18-8
b.
c.
20. a.
Philip Morris P/E (12/31/91) =
$80.25/$4.24
=
18.9
S&P 500 P/E (12/31/91)
=
$417.09/$16.29 =
25.6
Philip Morris relative P/E
=
18.9/25.6
0.74
=
Philip Morris book value (12/31/91) =
$12,512/920 = $13.60 per share
Philip Morris P/B (12/31/91)
=
=
$80.25/$13.60
5.90
S&P 500 P/B (12/31/91)
=
=
$417.09/$161.08
2.59
Philip Morris relative P/B
=
=
5.90/2.59
2.28
Multistage Dividend Discount Model
Advantages
1. Excellent for comparing greatly
different companies.
2. Solid theoretical framework.
3. Ease in adjusting for risk levels.
4. Dividends relatively easy to project.
5. Dividends not subject to distortions
from arbitrary accounting rules.
6. Flexibility in use and more realistic
than constant growth model.
Disadvantages
1. Need to forecast well into the future.
2. Problem with non-dividend paying
companies.
3. Problem with high growth companies (g>k).
4. Problems projecting “forever after” ROE
and payout ratio.
5. Small changes in assumptions can have big
impact.
6. Need technology for more advanced models.
18-9
Absolute and Relative Price/Earnings Ratio
Advantages
Disadvantages
1. Widely used by investors.
1. Difficult with volatile earnings.
2. Easy to compare with market and other 2. Need to determine what is a “normal”
companies in specific industries.
P/E ratio.
3. Difficult to project earnings.
4. Effect of accounting differences.
5. Many factors influence multiples.
6. Can be used only for relative rather
than absolute measurement.
7. Doesn’t address quality of earnings.
8. Problem with companies with no (or
negative) income.
Absolute and Relative Price/Book Ratio
Advantages
1. Incorporates some concept of asset
values.
2. Easy to compute even for companies
with volatile or negative earnings.
3. Easy to compare with market and
specific industries.
Disadvantages
1. Subject to differing accounting rules.
2. Affected by non-recurring items.
3. Subject to historical costs.
4. Book may be poor guide to actual asset
values.
5. Ignores future earnings prospects and
growth potential.
b.
Support can be given to either position:
Philip Morris is undervalued because:




DDM indicates intrinsic value above current market price.
Given forecasts of dividends over two stages, DDM is best to use for this
situation and should be given more weight.
P/E below market despite past growth and forecast of superior future growth.
P/E relative below 10-year average.
18-10
Philip Morris is overvalued because:




P/B considerably higher than market.
P/B relative higher than 10-year average.
DDM discount rate used should be higher than market’s 10% due to large
potential risks in cigarette manufacturing business (although whether this
risk is systematic is far from clear).
P/E on Philip Morris should be low relative to market and past growth due
to risks inherent in its business.
21. Duo Growth Co.
0
1
Dt 1.00 1.25
g
.25
.25
2
1.5625
.25
3
4
5
...
1.953125
.05 . . . . . . .
a. The dividend to be paid at the end of year 3 is the first installment of a dividend
stream that will increase indefinitely at the constant growth rate of 5%.
Therefore, we may use the constant growth model as of the end of year 2, and can
calculate intrinsic value by adding the present value of the first two dividends
plus the present value of the sales price of the stock at the end of year 2.
The expected price 2 years from now is:
P2 = D3 / (k – g) = $1.953125 / ( .20 – .05 ) = $13.02
The PV of this expected price is 13.02 / 1.202 = $9.04
The PV of expected dividends in years 1 and 2 is:
1.25 1.5625
1.20 + 1.202 = $2.13
Thus the current price should be $9.04 + $2.13 = $11.17.
b. Expected dividend yield = D1 / P0 = 1.25/11.17 = .112 or 11.2%
c. The expected price one year from now is the PV of P2 and D2.
P1 = ( D2 + P2 ) / 1.20 = (1.5625 + 13.02 ) / 1.20 = $12.15.
The implied capital gain is (P1 – P0) / P0 = (12.15 – 11.17) / 11.17 = .088 = 8.8%
The implied capital gains rate and the expected dividend yield sum to the market
capitalization rate. This is consistent with the DDM.
18-11
22. Generic Genetics (GG) Corporation.
0
5
0
Et
Dt
1
6
0
...
4
10.368
0
5
12.4416
12.4416
k = 15%, and we are told that dividends for the next four years = 0, so that b = 1.0
(100% plowback ratio).
D5 12.4416
a. P4 = k = .15
= $82.944
P4
V0 = (1 + k)4 = $47.42
b. Its price should increase at a rate of 15% over the next year, so that the HPR will
equal k.
23. MoMi Corporation.
Projected Free Cash Flow in Year 1
Before-tax cash flow from operations
Depreciation
Taxable Income
Taxes (@ 34%)
After-tax unleveraged income
After-tax cash flow from operations
(After-tax unleveraged income + depreciation)
New investment (20% of cash flow from operations)
Free cash flow
(After-tax cash flow from operations – new investment)
k = 12%
$2,100,000
210,000
1,890,000
642,600
1,247,400
1,457,400
420,000
1,037,400
Debt = $4 million
The value of the whole firm, debt plus equity, is
C1
$1037400
V0 = k – g = .12 – .05 = $14,820,000
Since the value of the debt is $4 million, the value of the equity is $10,820,000.
18-12
24. CPI Corporation.
a. k* = D1* / P0 + g*
= 1/20 + .04 = .05 + .04 = .09 or 9% per year.
b. Nominal capitalization rate:
k = (1 + k*) (1 + i) – 1
= 1.09  1.06 – 1 = .1554 or 15.54%
Nominal dividend yield:
D1/P0 = (1  1.06)/20 = .053 or 5.3%
Growth rate of nominal dividends:
g = (1 + g*) (1 + i) – 1
= 1.04  1.06 – 1 = .1024 or 10.24%
c. If expected real EPS are $1.80, then the estimate of intrinsic value using the
simple capitalized earnings model is:
V0 = $1.80 / .09 = $20
25. a. (ii)
b. (ii)
c. $23.91
[($1.50 + $26)/1.15 = $23.91]
d. (iii)
e. $11.03
[$10  (1 + .05)2 = 11.03]
f. $53.15
Solution:
g = ( 1  .4)  15% = 9%;
P2 = D3/(k – g) = $2/(.12  .09) = $66.67
$66.67/1.122 = $53.15
g. (iii)
h. (iii)
[from equation 18.8, P/E = (1  b)/(k  g) = .45/(.15  .10) = 9]
18-13