Solutions to Chapter 5
Solutions to Chapter 6
Valuing Stocks
1. No. The dividend discount model allows for the fact that firms may not currently pay dividends. As the market matures, and Rogers Wireless Communication’s growth opportunities moderate, investors may justifiably believe that Rogers Wireless will enjoy high future earnings and will pay dividends then. The stock price today can still reflect the present value of the expected per share stream of dividends.
2. Dividend yield = Expected dividend/Price = DIV1/P0
So: P0 = DIV1/dividend yield
P0 = $2.4/.08 = $30
3. a. The typical preferred stock pays a level perpetuity of dividends. The expected dividend next year is the same as this year’s dividend, $7. Thus the dividend growth rate is zero and the price today is:
P0 = D1/r = 7/.12 = $58.33
b. The expected dividend in two years is this year’s dividend, $7.
P1= D2/r = 7/.12 = $58.33
c. Dividend yield = $7/$58.33 = .12 =12%
Expected capital gains = 0
Expected rate of return = 12%
4. r = DIV1/P0 + g = 8% + 5% = 13%
5. The value of a common stock equals the present value of dividends received out to the investment horizon, plus the present value of the forecast stock price at the horizon. But the stock price at the horizon date depends on expectations of dividends from that date forward. So even if an investor plans to hold a stock for only a year for two, the price ultimately received from another investor depends on dividends to be paid after the date of purchase. Therefore, the stock’s present value is the same for investors with different time horizons.
6. a. P0 = ( r = + g
r = + .04 = .14 = 14%
b. P0 = 2.50/(.165 ( .04) = $20
7. The dividend yield is defined as the annual dividend (or the annualized current dividend) divided by the current price. The current annual dividend is ($2 ( 4) = $8 and the dividend yield is:
DIV1/P0 = .048 ( $8/ P0 = .048 ( P0 = $8/.048 = $166.7
To work with the quarterly dividend, divide the dividend yield by 4 and repeat the above steps:
Quarterly DIV/P0 = .048/4 = .012 ( $2/ P0 = .012 ( P0 = $2/.012 = $166.7
8. Weak, semi-strong, strong, fundamental, technical
9. True. The search for information and insightful analysis is what makes investor assessments of stock values as reliable as possible. Since the rewards accrue to the investors who uncover relevant information before it is reflected in stock prices, competition among these investors means that there is always an active search on for mispriced stocks.
10. a. DIV1 = $1 ( 1.04 = $1.04
DIV2 = $1 ( 1.042 = $1.0816
DIV3 = $1 ( 1.043 = $1.1249
b. P0 = DIV1/(r ( g) = = $13
c. P3 = DIV4/(r ( g) = = $14.6237
d. Your payments are:
Year 1 Year 2 Year 3
DIV 1.04 1.0816 1.1249
Sales Price 14.6237
Total cash flow 1.04 1.0816 15.7486
PV of cash flow .9286 .8622 11.2095
Sum of PV = $13.00, the same as your answer to (b).
11. Dividend growth rate, g = return on equity × plowback ratio:
g = .15 ( .40 = .06
r = + g = + .06 = .16 = 16%
12. a. P0 = = = $21
P0 = = $30
The lower discount rate makes the present value of future dividends higher, raising the value of the stock.
13. r = + g ( g = r - = .14 – = .04 = 4%
14. a. r = + g = + .03 = .0926 = 9.26%.
b. If r = .10, then .10 = 1.64(1.03)/27 + g. So g = .0374 = 3.74%
c. g = Return on equity ( plowback ratio
5% = Return on equity ( .4
Return on equity = = = 12.5%
15. P0 = DIV1/(r ( g)
= $2/(.12 – .06) = $33.33
16. a. P0 = DIV1/(r ( g) = 3/[.15 – ((.10)] = 3/.25 = $12
b. P1 = DIV2/(r ( g) = 3(1 ( .10)/.25 = $10.80
c. return = = = .150 = 15.0%
d. “Bad” companies may be declining, but if the stock price already reflects this fact, the investor still can earn a fair rate of return, as we saw in part c.
17. a. (i) reinvest 0% of earnings: g = 0 and DIV1 = $5:
P0 = = = $33.33
(ii) reinvest 40%: g = 15% ( .40 = 6% and DIV1 = $5 ( (1 – .40) = $3:
P0 = $3/(.15 – .06) = $33.33
(iii) reinvest 60%: g = 15% ( .60 = 9% and DIV1 = $5 ( (1 – .60) = $2:
P0 = 2/(.15 – .09) = $33.33
b. (i) reinvest 0%: P0 = 5/(.15 – 0) = $33.33
PVGO = $0
(ii) reinvest 40%: P0 = = $42.86
PVGO = $42.86 – $33.33 = $9.53
(iii) reinvest 60%: P0 = = $66.67
PVGO = $66.67 – $33.33 = $33.34
c. In part (a), the return on reinvested earnings is equal to the discount rate. Therefore, the NPV of the firm’s new projects is zero, and PVGO is zero in all cases, regardless of the reinvestment rate. While higher reinvestment results in higher growth rates, it does not result in a higher value of growth opportunities. This example illustrates that there is a difference between growth and growth opportunities.
In part (b), the return on reinvested earnings is greater than the discount rate. Therefore, the NPV of the firm’s new projects is positive, and PVGO is positive. PVGO is higher when the reinvestment rate is higher in this case, since the firm is taking greater advantage of its opportunities to invest in positive NPV projects.
20. a. P0 = + + = 18.10
b. DIV1/P0 = $1/18.10 = .0552 = 5.52%
21. Stock A Stock B
a. Payout ratio $1/$2 = .50 $1/$1.50 = .67
b. g = ROE ( plowback 15% ( .5 = 7.5% 10% ( .333 = 3.33%
c. Price = DIV1/(r ( g) = $14.33 = $8.85
Note: We interpret “recent” to mean in the past. The current stock price
depends on future dividends – so the next dividend must be 1 + g times higher.
22. a. ROE ( plowback ratio = 20% ( .3 = 6%
b. E = $2, plowback ratio = .3, r = .12, g = .06 ( P0 = = $23.33
c. No-growth value = E/r = $2/.12 = $16.67
PVGO = P0 ( no-growth value = $23.33 ( $16.67 = $6.66
d. P/E = 23.33/2 = 11.665
e. If all earnings were paid as dividends, price would equal the no-growth value, $16.67, and P/E would be 16.67/2 = 8.335.
f. High P/E ratios reflect expectations of high PVGO.
23. a. =$60
b. No-growth value = E/r = $6.20/.12 = $51.67
PVGO = P0 ( no-growth value = $60 ( 51.67 = $8.33
24. a. Earnings = DIV1 = $4. Growth rate g = 0.
P0 = = $33.33
P/E = 33.33/4 = 8.33
b. If r = .10, P0 = = 40, and P/E increases to 40/4 = 10
A decrease in the required rate of return, holding dividends constant, raises the stock price and the P/E ratio.
25. a. Plowback ratio = 0 implies DIV1 = $3 and g = 0.
Therefore, P0 = = $30
and the P/E ratio is 30/3 = 10.
b. Plowback ratio = .40 implies DIV1 = $3(1 – .40) = $1.80, and g = 10% ( .40 = 4%.
Therefore P0 = $1.80/(.10 – .04) = $30
and the P/E ratio is 30/3 = 10.
c. Plowback ratio = .80 implies DIV1 = $3(1 – .80) = $.60, and g = 10% ( .80 = 8%.
Therefore P0 = $.60/(.10 – .08) = $30
and the P/E ratio is 30/3 = 10.
Regardless of the plowback ratio, the stock price = $30 because all projects
offer return on equity just equal to the opportunity cost of capital.
26. a. P0 = DIV1/(r ( g) = $5/(.10 – .06) = $125
b. If Trendline followed a zero-plowback strategy, it could pay a perpetual dividend of $8. Its value would be $8/.10 = $80, and therefore, the value of assets in place is $80. The remainder of its value must be due to growth opportunities, so PVGO = $125 – $80 = $45.
27. a. g = 20% ( .30 = 6%
DIV1 = $2(1 – .30) = $1.40
P0 = DIV1/(r ( g) = $1.40/(.12 ( .06) = $23.33
P/E = 23.33/2 = 11.665
b. If the plowback ratio is reduced to .20, g = 20% ( .20 = 4%
DIV1 = $2(1 – .20) = $1.60
P0 = DIV1/(r ( g) = $1.60/(.12 – .04) = 20
P/E = 20/2 = 10
P/E falls because the firm’s value of growth opportunities is now lower: It
takes less advantage of its attractive investment opportunities.
c. If the plowback ratio = 0, g = 0, and DIV1 = $2,
P0 = $2/.12 = 16.67 and E/P = 2/16.67 = .12
28. a. DIV1 = 2.00 PV = 2/1.10 = 1.818
DIV2 = 2(1.20) = 2.40 PV = 2.40/1.102 = 1.983
DIV3 = 2(1.20)2 = 2.88 PV = 2.88/1.103 = 2.164
b. This could not continue indefinitely. If it did, the stock would be worth an infinite amount. Another way to think about the feasible perpetual growth rate is to compare the company’s growth rate with the growth rate of the economy. The economy grows about 3% a year. To grow faster than the economy as a whole is feasible when the company is small. However, to continue to grow at 20%, the company must take over other companies and eventually become the entire economy. But in the long run, it still can only grow as quickly as the entire economy. So it is impossible to grow at 20% in perpetuity. Think about Microsoft – it has had phenomenal growth partly by acquiring other companies and partly by growing its own businesses. However, even if it were to own all of the companies in the world, eventually its growth rate would fall to the growth rate of the world economy. We are assuming that Bill Gates is not able to successfully market his software to still to be discovered alien worlds!! Finally, note too that the constant dividend growth model fails when the assumed perpetual growth rate is greater than the discount rate.
29. a. Book value = $100 million
First year earnings = $100 million ( .24 = $24 million
Dividends = Earnings ( (1 – plowback ratio) = $12 million
g = return on equity ( plowback ratio = .24 ( .50 = .12
Market value = = $400 million
Market-to-book ratio = $400/$100 = 4
b. Now g falls to .10 ( .50 = .05, first year earnings decline to $10 million (=$100 million × .1), and dividends decline to $5 million (=$10 million × .5).
Market value = = $50 million
Market-to-book ratio = ½
This makes sense, because the firm now earns less than the required rate of return on its investments. Its project is worth less than it costs.
30. P0 = + + = $16.59
31. a. DIV1 = $2 ( 1.20 = $2.40
b. DIV1 = $2.40 DIV2 = $2.88 DIV3 = $3.456
P3 = = $32.675
P0 = + + = $28.02
c. P1 = + = $29.825
d. Capital gain = P1 ( P0 = $29.825 ( $28.02 = 1.805
r = = .15 = 15%
32. a. Note: If students carry at least 4 decimal places, the results will be clearer.
Also, it is easier to solve the prices in reverse order.
DIV1 = $.5 DIV2 = $.5 DIV3 = $.5
DIV4 = $.5 × 1.04 = $.52 DIV5 = $.5 × 1.042 = $.5408
P4 = = = $7.7257
P3 = = = $7.4286
P2 = = = $7.1429
P1 = = = $6.8855
P0 = = = $6.6536
b. Year 0
Dividend yield = = = .07515
Capital gains yield = = = .03485
Dividend yield + capital gains yield = .07515 + .03485 = .11
Year 1
Dividend yield = = = .07262
Capital gains yield = = = .03738
Dividend yield + capital gains yield = .07262 + .03738 = .11
Year 2
Dividend yield = = = .0700
Capital gains yield = = = .0400
Dividend yield + capital gains yield = .07 + .04 = .11
Year 3
Dividend yield = = = .0700
Capital gains yield = = = .0400
Dividend yield + capital gains yield = .07 + .04 = .11
Yes, each year the sum of the dividend yield and the capital gains yield equal 11 percent, the required rate of return. Once the company hits constant growth rate of 4 percent, both the dividend yield and the capital gains yield also become constant.
33. DIV1 = dividend payout × earnings1 = .4 × $3 = $1.2
DIV2 = dividend payout × earnings2 = .4 × $3 × 1.1 = $1.32
DIV3 = dividend payout × earnings3 = .4 × $3 × 1.12 = $1.452
DIV4 = dividend payout × earnings4 = .4 × $3 × 1.13 = $1.5972
DIV5 = dividend payout × earnings5 = .4 × $3 × 1.14 = $1.75692
P0 = + + + + × = $13.95
37. Before-tax rate of return:
= = = .078 = 7.8%
After-tax rate of return:
=
= = .0596 = 5.96%
43. 50 =
g = .15 – = .10
g = .10 = return on equity ( plowback ratio = return on equity ( .60
return on equity = .10/.60 = .1667 = 16.67%
44. a. P0 = +
DIV1 = $1
DIV2 = $2
P2 = = = $30
P0 = + = $26.40
b. Next year, P1 = = = $28.57
c. return = = = .12
= + = .12
45. DIV1 = $1
DIV2 = $2
DIV3 = $3
g = .06 Therefore P3 = 3(1.06)/(.14 – .06) = $39.75
P0 = + + = $31.27
46. a. DIV1 = $4, and g = 4%
Expected return = DIV1/P0 + g = 4/100 + 4% = 8%
b. Since DIV1 = Earnings ( (1 – plowback ratio),
Earnings = DIV1/(1 – plowback ratio) = 4/(1 – .4) = $6.667
If the discount rate is 8% (the expected return on the stock), then the no-growth value of the stock is 6.667/.08 = $83.34. Therefore PVGO =$100 – $83.34 = $16.66
c. For the first 5 years, g = 10% ( .8 = 8%. Thereafter, g = 10% ( .4 = 4%
Year 1 2 3 4 5 6 . . . Earnings 6.67 7.20 7.78 8.40 9.07 9.80
plowback .80 .80 .80 .80 .80 .40
DIV 1.33 1.44 1.56 1.68 1.81 5.88
g .08 .08 .08 .08 .08 .04
After year 6, the plowback ratio falls to .4 and the growth rate falls to 4 percent. [We assume g = 8% in year 5 (i.e., from t = 5 to t = 6), since the plowback ratio in year 5 is still high at b = .80. Notice the big jump in the dividend when the plowback ratio falls.] By year 6, the firm enters a steady-growth phase, and the constant-growth dividend discount model can be used to value the stock.
The stock price in year 6 will be
P6 = = = $152.88
P0 = + + + + + = $106.22
47. a. DIV1 = 1.00 ( 1.20 = $1.20
DIV2 = 1.00 ( (1.20)2 = $1.44
DIV3 = 1.00 ( (1.20)3 = $1.728
DIV4 = 1.00 ( (1.20)4 = $2.0736
b. P4 = DIV5/(r ( g) = DIV4(1 + g)/(r ( g)
= 2.0736(1.05)/(.10 – .05) = $43.55
c. P0 = + + + = $34.74
d. DIV1/P0 = 1.20/34.74 = .0345 = 3.45%
e. Next year the price will be:
+ + = $37.01
f. return =
= = .10
The expected return equals the discount rate (as it should if the stock is fairly priced).
48. Before-tax rate of return
= = = .1 = 10%
After-tax dividend:
Grossed up dividend = 1.25 × 2 = 2.50
Gross federal tax = .22 × 2.50 = .55
Federal tax credit = .1333 × 2.50 = .3333
Net federal tax = .55 - .3333 = .2167
Gross provincial tax = .119 × 2.50 = .2975
Provincial tax credit = .066 × 2.50 = .165
Net provincial tax = .2975 - .165 = .1325
Total dividend tax = .2167 + .1325 = .3492
After-tax dividend = 2 – .3492 = 1.6508
Capital gains tax = .5 × (.22 + .119) × (53 – 50) = .5085
After-tax capital gains = 53 – 50 – .5085 = 2.4915
After-tax rate of return
=
= = .0828 = 8.28%
49. Assume taxes do not change. We make the easiest reinvestment assumption: dividends are spent as they are received and do not earn any interest. Thus, the future value of dividends received is 2 × 3 = 6. The selling price is $55.
The before-tax rate of return is
= ()1/3- 1 = ()1/3- 1 = .0685 = 6.85%
Annual after-tax dividends = 2 – .3492 = 1.6508 (from question 48)
Future value of dividends received = 1.6508 × 3 = 4.9524
Capital gain tax = .5 × (.22 + .119) × (55 – 50) = .8475
The after-tax rate of return is
= ()1/3- 1
= ()1/3- 1 = .0573 = 5.73%
50. a. Both of these instruments are perpetuities. Recall the price of a perpetuity is
P0 =
Rearranging the equation to find the required rate of return. The consol’s annual cash flow is its coupon payment and the preferred share’s annual cash flow is its dividend payment.
Consol rate of return = bond coupon/P0 = .04 × 1000/800 = .05 = 5%
Preferred share rate of return = preferred dividend/P0 = 6/120 = .05 = 5%
b. Convert the annual cash flows to their after-tax amounts:
After-tax bond coupon = (1 - .35) × .04 × 1000 = .65 × 40 = 26
After-tax preferred dividend = (1 - .29) × 6 = 4.26
Consol after-tax rate of return = after-tax coupon/P0 = 26/800 = .0325 = 3.25%
Preferred after-tax rate of return = after-tax dividend/P0 = 4.26/120 = .0355 = 3.55%
c. With a 35% corporate tax rate, the after-tax rate of return on the consol is the same as we calculated in (b), 3.25%. However, the corporate tax rate on dividends received from other Canadian corporations is zero. Thus the rate of return on the preferred shares is the before-tax rate from (a), 5%.
d. Corporations with spare cash to invest will prefer to purchase dividend-paying securities of Canadian corporations than corporate bonds to take advantage of preferential dividend tax treatment.
51. The first dividend comes in 3 years from today and thereafter grows at a constant annual rate of 6%. Use the constant dividend growth model to calculate the price two years from today, P2, and then discount that price to today to today’s price, P0:
P2 = = = 12.5
P0 =× P2 = × 12.5 = 10.33
52. The growing annuity formula is × [1 – ()T]
Years 1 – 4
r = 12%, g = 10%, T = 4,
C1 = DIV1 = (1 + g) × DIV0 = 1.1 × 1 = 1.1
Present value of Year 1 – 4 dividends
= × [1 – ()T] = × [1 – ()4]= 3.82
Years 5 – 14
r = 12%, g = 8%, T = 10
To get the Year 5 dividend (which is the first cash flow in the second interval of constant growth), figure out the Year 4 dividend first. Since dividends are expected to grow 10% a year for 4 years and then grow at 8%:
DIV4 = 1.14 × DIV0 = 1.14 × 1 = 1.4641
DIV5 = (1 + g) × DIV4 = 1.08 × 1.4641 = 1.581228
Present value of Years 5 – 14 dividends at the end of Year 4
= × [1 – ()10] = × [1 – ()10]= 12.0523
Present value of Years 5 – 14 dividends today
= × × [1 – ()10] = × 12.0523 = 7.66
Year 15 and on
r = 12%, g = 5%
DIV14 = (1.08)10 × DIV4 = (1.08)10 × 1.4641 = 3.1609
DIV15 = (1.05) × DIV14 = 1.05 × 3.1609 = 3.3190
Present value of Year 15 and on dividends at the end of Year 14
= = = 47.41
Present value of Year 15 and on dividends today
= = × 47.41 = 9.70
Price today = present value of all dividends to be received
P0 = 3.82 + 7.66 + 9.70 = 21.18
53.
|Event |PV of dividends |
|High quality gold |8 × annuity factor(9%, 20 years) = 73.03 |
|Medium quality gold |2 × annuity factor(9%, 20 years) = 18.26 |
|No gold |0 |
| | |
P0 = .4 × 73.03 + .5 × 18.26 = $38.34
54. a. Price on May 1, 2007 = 2.50/(.1 - .06) = 62.50
b. Price on May 1, 2007 = 1.248/(.1 - .04) = 20.8
c. Price on May 1, 2006 = [1.20 + .3 × 62.50 + .7 × 20.8] = $31.37
d. Rate of return if R&D is successful = - 1 = 1.031 = 103.1%
Rate of return if R&D is unsuccessful = - 1 = -0.299 = -29.9%
Expected rate of return = .3 × 1.031 + .7 × (-.299) = .10 = 10%
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