Modelos de Avaliação de Ações

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18-1 
CHAPTER 18: EQUITY VALUATION MODELS 
 
1. Choice (a): P0 = D1/(k – g) = $2.10/(.11 – 0) = $19.09 
2. (c) 
3. a. 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 
 b. P3 = P0(1 + g)3 = $25(1.08)3 = $31.49 
 
5. 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. 
 
 
6. 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 
 
 P0 = 
D1
k – g = 
$1.06
.16 – .06 = $10.60 
 18-2 
 
 b. Leading P0/E1 = 10.60/3.18 = 3.33 
 Trailing P0/E0 = 10.60/3 = 3.53 
 
 c. PVGO = P0 – 
E1
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% 
 
 P0 = 
D1
k – g = 
$8
.10 – .05 = $160 
 
 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. 
 
 
9. 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: 
 
 18-3 
 P0 = 
D1
1 + k + 
D2
(1 + k)2
 + 
D3
(1 + k)3
 + 
D4 + D5/(k – g)
(1 + k)4
 
 
 = 
.287
1.109 + 
.359
1.1092
 + 
.449
1.1093
 + 
.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 + β [Ε(rM) – rf ] = 10 + 1.5(15 – 10) = 10% + 7.5% = 17.5%, and g = 5%. 
 
 Therefore, P0 = 
Dl
k – g = 
$2.50
.175 – .05 = 
$2.50
.125 = $20. 
 
 
11. Stock 
 A B 
 Market capitalization rate, k 10% 10% 
 Expected return on equity, ROE 14% 12% 
 Estimated earnings per share, E1 $2.00 $1.65 
 Estimated dividends per share, D1 $1.00 $1.00 
 Current market price per share, P0 $27 $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. 
 
 
 
 18-4 
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. • 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. 
 
 b. 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. 
 
 18-5 
 
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: 
 
 PVGO = P0 – 
E1
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% 
 
 P0 = 
D1
k – g = 
D0(1 + g)
k – g = 
$.50 × 1.10
.15 – .10 = $11 
 
 b. time EPS Dividend Comment 
 0 1.00 .50 This data is given in the question 
 1 1.10 .55 g = 10%, plowback = .50 
 
 2 1.21 .726 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 
 3 1.2826 .7696 EPS grows by .4 × 15% = 6% and payout ratio = .60 
 
 
 At time 2, P2 = 
D3
k – g = 
.7696
.15 – .06 = $8.551 
 
 18-6 
 At time 0, V0 = 
.55
1.15 + 
.726 + 8.551
(1.15)2
 = $7.493 
 
 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
11 = .150 = 15.0% 
 
 2 
(8.551 – 12.10) + .726
12.10 = – .233 = −23.3% 
 
 3 
(9.064 – 8.551) + .7696
8.551 = .150 = 15.0% 
 
 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 + β[E(rM) – rf] = 8% + 1.2(15% – 8%) = 16.4% 
 g = b × ROE = .6 × 20% = 12% 
 
 V0 = 
D0(1 + g)
k – g = 
$4 × 1.12
.164 – .12 = $101.82 
 b. P1 = V1 = V0(1 + g) = $101.82 × 1.12 = $114.04 
 
 E(r) = 
D1 + P1 – P0
P0
 = 
$4.48 + $114.04 – $100
$100 = .1852 = 18.52% 
 
 18-7 
17. a. k = rf + β [Ε(rM) – rf ] = .045 + 1.15(.145 − .045) = .16 = 16% 
 
 b. Year Dividends 
 1999 $1.72 
 2000 1.72 × 1.12 = $1.93 
 2001 1.72 × 1.122 = $2.16 
 2002 1.72 × 1.123 = $2.42 
 2003 1.72 × 1.123 × 1.09 = $2.64 
 
 Present value of dividends in 2000 – 2002: 
 
 Year Dividends 
 2000 1.93 / 1.161 = $1.66 
 2001 2.16 / 1.162 = $1.61 
 2002 2.42 / 1.163 = $1.55 
 Total: $4.82 
 
 Price at year-end 2002 = 
D2003
k − g
 = 
$2.64
.16 − .09
 = $37.71 
 
 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. 
 
 18-8 
 
 18-9 
18. DEQS Corporation. 
 Time: 0 1 5 6 . . . 
 Et $10 $12 $24.883 $29.860 
 Dt $0 $0 $0 $11.944 
 b 1 1 1 .60 
 g .20 .20 .20 .09 
 
 a. V5 = 
D6
k – g = 
$11.944
.15 – .09 = $199.07 
 
 V0 = 
V5
(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: 
V0 = 
D1
(1 + k)1
 + 
D2
(1 + k)2
 + 
D3
(1 + k)3
 + 
D4
(1 + k)4
 + 
D5
(1 + k)5
 + 
D6
(k – g)
(1 + k)5
 
 = 
$2.29
1.101
 + 
$2.75
1.102
 + 
$3.30
1.103
 + 
$3.96
1.104
 + 
$4.75
1.105
 + 
$5.09
(0.10 – 0.07)
1.105
 
 = $117.84 
 
 where: D1 = D0 (1 + 0.20) = 2.29 
 D2 = D1 (1 + 0.20) = 2.75 
 D3 = D2 (1 + 0.20) = 3.30 
 D4 = D3 (1 + 0.20) = 3.96 
 D5 = D4 (1 + 0.20) = 4.75 
 D6 = D5 (1 + 0.07) = 5.09 
 k = Equity required return = 0.10 
 g = Growth in second stage = 0.07 
 18-10 
 
 18-11 
b. 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 
 
 
c. 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 
 
 
 
20. a. Multistage Dividend Discount Model 
 
Advantages Disadvantages 
 
1. Excellent for comparing greatly different 
companies. 
1. Need to forecast well into the future. 
2. Solid theoretical framework. 2. Problem with non-dividend paying companies. 
3. Ease in adjusting for risk levels. 3. Problem with high growth companies (g>k). 
 
4. Dividends relatively easy to project. 4. Problems projecting “forever after” ROE and 
payout ratio. 
5. Dividends not subject to distortions from 
arbitrary accounting rules. 
5. Small changes in assumptions can have big 
impact. 
6. Flexibility in use and more realistic than 
constant growth model. 
6. Need technology for more advanced models. 
 
 
 18-12 
 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 
companies in specific industries. 
2. Need to determine what is a “normal” 
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 Disadvantages 
 
1. Incorporates some concept of asset 
values. 
1. Subject to differing accounting rules. 
2. Easy to compute even for companies with 
volatile or negative earnings. 
2. Affected by non-recurring items. 
3. Easy to compare with market and specific 
industries. 
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. 
 18-13 
 
 
 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 2 3 4 5 . . . 
 Dt 1.00 1.25 1.5625 1.953125 
 g .25 .25 .25 .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
1.20 + 
1.5625
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% 
 
 18-14 
 The implied capital gains rate and the expecteddividend yield sum to the market 
 capitalization rate. This is consistent with the DDM. 
 
 
22. Generic Genetics (GG) Corporation. 
 0 1 . . . 4 5 
 Et 5 6 10.368 12.4416 
 Dt 0 0 0 12.4416 
 
 k = 15%, and we are told that dividends for the next four years = 0, so that b = 1.0 (100% 
plowback ratio). 
 
 a. P4 = 
D5
k = 
12.4416
.15 = $82.944 
 
 V0 = 
P4
(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 $2,100,000 
 Depreciation 210,000 
 Taxable Income 1,890,000 
 Taxes (@ 34%) 642,600 
 After-tax unleveraged income 1,247,400 
 After-tax cash flow from operations 1,457,400 
 (After-tax unleveraged income + depreciation) 
 New investment (20% of cash flow from operations) 420,000 
 Free cash flow 
 (After-tax cash flow from operations – new investment) 1,037,400 
 
 k = 12% Debt = $4 million 
 The value of the whole firm, debt plus equity, is 
 
 18-15 
 V0 = 
C1
k – g = 
$1037400
.12 – .05 = $14,820,000 
 
 Since the value of the debt is $4 million, the value of the equity is $10,820,000. 
 18-16 
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]