Chapter 04 - the Value of Common Stocks

Chapter 04 - The Value of Common Stocks

CHAPTER 4

The Value of Common Stocks

The values shown in the solutions may be rounded for display purposes. However, the answers were derived using a spreadsheet without any intermediate rounding.

Answers to Problem Sets

1.a. True. This is a condition for equilibrium in well-functioning capital markets. All stocks in a particular risk class must offer the same rate of return. If a certain stock, for example, is priced above others in the equivalent risk class, investors would sell their shares to buy cheaper shares from companies in the same risk class. This would force the price of that higher priced stock down to the equilibrium price. The same happens for a stock priced below equilibrium—investors would rush to buy the stock, sending its price back up.

b. True. When shareholders buy a particular stock, they receive cash from the company in the form of future dividends. The rate of return that investors expect is the expected dividend per share plus the expected price appreciation for the stock: r = [DIV1 + (P1 – P0)] / P0.

Est. Time: 01-05

2.Investors who buy stocks may get their return from capital gains as well as dividends. But the future stock price always depends on subsequent dividends. There is no inconsistency.

Est. Time: 01-05

3.P0= (Div1 + P1) / (1 + r)

P0= ($5 + 110)/1.08

P0= $106.48

Chapter 04 - The Value of Common Stocks

Est. Time: 01-05

4.r = DIV1 / P0

r = $5 / $40

r= .125, or 12.5%

Est. Time: 01-05

5.P0 = DIV1 / (r – g)

P0 = $10/(.08 − .05)

P0= $333.33

Est. Time: 01-05

6.P4 = EPS5 / r

P4 = [EPS1 × (1 + g1)3× (1 + g2)] / r

P4 = [$15 × (1 + .05)3 × (1 + 0)] / .08

P4 = $217.05

Note that $15 is the EPS for year 1. The 5 percent growth rate stops after year 4,so the exponent for the first growth rate must be 3, (Year 4 – Year 1). There is no growth in year 5.

P0 = DIV1 / (1 + r) + [DIV1 × (1 + g)] / (1 + r)2 + [DIV1 × (1 + g)2] / (1 + r)3 +

[DIV1 × (1 + g)3] / (1 + r)4 + P4 / (1 + r)4

P0 = $10 / 1.08 + ($10 × 1.05) / 1.082 + ($10 × 1.052) / 1.083 + ($10 × 1.053) /

1.084 + $217.05 / 1.084

P0= $195.06

Est. Time: 01-05

7.P0 = DIV1 / (r – g)

P0 = $10 / (.08 – .05)

P0 = $333.33

P0 = EPS1 / r + PVGO

PVGO = $333.33 – $15 / .08

PVGO = $145.83

Est. Time: 01-05

8.DIV1 = $10

DIV2 = DIV1 × (1 + g) = $10 × 1.05 = $10.50

DIV3= DIV2 × (1 + g) = $10.50 × 1.05 = $11.03

P0 = DIV1 / (r – g) = $10 / (.08 – .05) = $333.33

P1= P0 × (1 + g) = $333.33 × 1.05 = $350.00

P2= P1 × (1 + g) = $350.00 × 1.05 = $367.50

P3= P2 × (1 + g) = $367.50 × 1.05 = $385.88

r1 = (DIV1 + P1 –P0) / P0= ($10 + 350.00 – 333.33) / $333.33 = .08, or 8%

r2 = (DIV2 + P2–P1) / P1 = ($10.50 + 367.50 – 350.00) / $350.00 = .08, or 8%

r3 = (DIV3 + P3–P2) / P2= ($11.03 + 385.88 – 367.50) / $367.50 = .08, or 8%

Since the rate of return each year is 8 percent, each investor should expect to earn 8%.

Est. Time: 06-10

9.a. False.The value of a share equals the present value of the expected future dividends per share. Earnings per share are not used to calculate share price because a portion of the earnings is used to reinvest in plant, equipment, and working capital.

b. True.The expected return is equal to the yearly dividend divided by the share price. If the firm does not grow and all earnings are paid out as dividends, then the expected return is also equal to the EPS/share price. Therefore, P0=DIV1/r=EPS1/r. We must still account for the present value of the growth opportunities, however, so P0 = EPS1/r + PVGO.

Est. Time: 01-05

10.A stock’s capitalization rate equals EPS1/P0 when PVGO = 0, that is when the firm pays out all of its earnings and is not growing.

Est. Time: 01-05

11.Free cash flow is the amount of cash leftover and available to pay out to investors after all investments necessary for growth. In our simple examples, free cash flow equals operating cash flow minus capital expenditures. Free cash flow can be negative if investments are large.

Est. Time: 01-05

12.Horizon value is the value of a firm at the end of a forecast period. Horizon value can be estimatedusing the constant-growth DCF formula or by using price-earnings or market-book ratios for similar companies.

Est. Time: 01-05

13. If PVGO = 0 at the horizon date,H, then:

Horizon value = Earnings forecastedH + 1 /r

Est. Time: 06-10

14.Internet exercise; answers will vary.

Est. Time: 01-05

15.Internet exercise; answers will vary.

Est. Time: 01-05

16. Internet exercise; answers will vary.

Est. Time: 06-10

17. Internet exercise; answers will vary.

Est. Time: 06-10

18.10 percent capitalization rate:

P0 Stock A = DIV1 / r = $10 / .1 = $100

P0 Stock B = DIV1 / (r – g) = $5 / (.1– .04) = $83.33

P0Stock C= $5 / 1.1 + ($5 × 1.2) / 1.12 + ($5 × 1.22) / 1.13 + ($5 × 1.23) / 1.14 +

($5 × 1.24) / 1.15 + ($5 × 1.25) / 1.16 + {[$5 × 1.25 × (1 + 0)] / .1} / 1.16

P0Stock C= $104.51

At a 10%capitalization rate, Stock C has the largest present value.

Using the same formulas as above with a 7% capitalization rate, the values are:

P0 Stock A = $10 / .07 = $142.86

P0 Stock B = $5 / (.07 - .04) = $166.67

P0Stock C = $5 / 1.07 + ($5 × 1.2) / 1.072 + ($5 × 1.22) / 1.073 + ($5 × 1.23) / 1.074 +

($5 × 1.24) / 1.075 + ($5 × 1.25) /1.076 + {[$5 × 1.25 × (1 + 0)] / .07} / 1.076

P0Stock C= $156.50

At a 7% capitalization rate, Stock B has the largest present value.

Est. Time: 06-10

19.a.P0 = [DIV0 × (1 + g)] / (r – g)

P0 = [($1.35 × (1 + .0275)]/ (.095 – .0275)

P0 = $20.55

b.r = (1 + R) / (1 + h)– 1

r = (1 + .095) / (1 + .0275)– 1

r = .0657, or 6.57%

In real terms, g equals 0, so DIV1 equals DIV0.

P0 = $1.35/ .0657

P0= $20.55

Est. Time: 06-10

20.a.Plowback ratio = 1 – payout ratio

Plowback ratio = 1 – .5

Plowback ratio = .5

gYears 1-4= plowback ratio × ROE

gYears 1-4 =.5 × .14

gYears 1-4 = .07

EPS0 = ROE × book equity per share

EPS0 =.14 × $50

EPS0 = $7.00

DIV0 = payout ratio × EPS0

DIV0 = .5 × $7.00

DIV0= $3.50

gYear 5 and later = plowback ratio × ROE

gYear 5 and later = (1 – .8) × .115

gYear 5 and later = .023, or 2.3%

The annual EPS and DIV are as follows:

Year
0
1 / EPS
$7.00
$7.00 × 1.07 = $7.49 / DIV
$7.49 × .5 = $3.75
2 / $7.00 × 1.072 = $8.01 / $8.01 × .5 = $4.01
3 / $7.00 × 1.073= $8.58 / $8.58 × .5 = $4.29
4
5 / $7.00 × 1.074= $9.18
$7.00 × 1.074 × 1.023 = $9.39 / $9.18 × .5 = $4.59
$9.39 × .8 = $7.51

b.PH= [DIV5 × (1 + g2)] / (r – g2)

PH = ($7.51 × 1.023) / (.115 - .023)

PH = $83.50

P0 = DIV1 / (1 + r) + DIV2 / (1 + r)2+ DIV3 / (1 + r)3 + DIV4 / (1 + r)4

+ DIV5 / (1 + r)5 + PH/ (1 + r)5

P0 = $3.75 / 1.115 + $4.01 / 1.1152 + $4.29 / 1.1153 + $4.59 / 1.1154

+ $7.51 / 1.1155 + $83.50/ 1.1155

P0= $65.45

The last term in the above calculation is dependent on the payout ratio and the growth rate after year 4.

Est. Time: 11-15

21.a.An Incorrect Application. Hotshot Semiconductor’s earnings and dividends have grown by 30% per year since the firm’s founding 10 years ago. Current stock price is $100, and next year’s dividend is projected at $1.25. Thus:

r = DIV1 / P0 + g = $1.25 / $100 + .30 = .3125, or 31.25%

This is wrong because the formula assumes perpetual growth; it is not possible for Hotshot to grow at 30% per year forever.

A Correct Application. The formula might be correctly applied to the Old Faithful Railroad, which has been growing at a steady 5% rate for decades. Its EPS1=$10, DIV1 = $5, and P0 = $100. Thus:

r = Div1 / P0 + g = $5 / $100 + .05 = .10, or 10%

Even here, you should be careful not to blindly project past growth into the future. If Old Faithful hauls coal, an energy crisis could turn it into a growth stock.

b.An Incorrect Application. Hotshot has current earnings of $5 per share. Thus:

r = EPS1 / P0 = $5 / $100 = .05, or 5%

This is too low to be realistic. The reason P0 is so high relative to earnings is not that r is low, but rather that Hotshot is endowed with valuable growth opportunities. Suppose PVGO = $60:

P0 = EPS1 / r + PVGO

$100 = $5 / r + $60

r = 12.5%

A Correct Application. Unfortunately, Old Faithful has run out of valuable growth opportunities. Since PVGO = 0:

P0 = EPS1 / r + PVGO

$100 = $10 / r + $0

r = 10%

Est. Time: 06-10

22.

Therefore:

The statement in the question implies the following:

Rearranging, we have:

NPV < NPV; everything else equal
(r– .15) > (r – .08); everything else equal

; everything else equal
;everything else equal

Est. Time: 06-10

23.a.P0= Div1 / (1 + r) + Div2 / (1 + r)2 + Div3 / (1 + r)3 + (Div4 / (r – g) / (1 + r)3

P0= $.50 / 1.12 + $.60 / 1.122 + $1.15 / 1.123 + [$1.24 / (.12 – .08)] / 1.123

P0= $23.81

The horizon value P3contributes:

P0 = [$1.24 / (.12 – .08)] / 1.123

P0 = $22.07

Without PVGO, P3 would equal earnings for year 4 capitalized at 12%, so PVGO3 is valued as:

PVGO3 = [DIV4 / (r – g)] – EPS4 / r

PVGO3= $1.24 / (.12 – .08) – $2.48 / .12

PVGO3= $10.33

The PVGO of $10.33 is lost at Year 3. Therefore, the current stock price of $23.81 will decrease by the present value of PVGO:

P0 No-growth = P0 – PVGO3 / (1 + r)3

P0 No-growth= $23.81 – $10.33 / 1.123

P0 No-growth= $16.45

Est. Time: 11-15

24.a.r = DIV1 / P0+ g

r = $4/ $100 + .04

r = .08, or 8%

EPS1 = Div1 / (1 – reinvestment rate)

EPS1 = $4 / (1 –.40)

EPS1 = $6.67

P0 = EPS1 / r + PVGO

PVGO = P0 – EPS1 / r

PVGO = $100 – $6.67 / .08

PVGO = $16.67

DIV1 will decrease to: .20 $6.67 = $1.33.

By plowing back 80% of earnings, CSI will grow by 8% per year for five years before returning to its long-run growth rate of 4%.The dividend will be 20% of earnings for years 1-5 and 60% of earnings in year 6 and beyond.

Year / 1 / 2 / 3 / 4 / 5 / 6
EPSt / $6.67 / $7.20 / $7.78 / $8.41 / $9.07 / $9.80
DIVt / 1.33 / 1.44 / 1.56 / 1.68 / 1.81 / 5.88

P5 = DIV6 / (r – g)

P5 = $5.88 / (.08 – .04)

P5 = $146.93

P0 = DIV1 / (1 + r) + DIV2 / (1 + r)2 + DIV3 / (1 + r)3 + DIV4 / (1 + r)4 +

DIV5 / (1 + r)5 + P5 / (1 + r)5

P0 = $1.33 / 1.08 + $1.44 / 1.082 + $1.56 / 1.083 + $1.68 / 1.084 +

$1.81 / 1.085 + $146.93 / 1.085

P0 = $106.17

Est. Time: 11-15

25.a.First, compute the dividends, which equal net income, for 2016 through 2020:

2016 / 2017 / 2018 / 2019 / 2020
Production (million barrels) / 1.8000 / 1.6740 / 1.5568 / 1.4478 / 1.3465
Price of oil/barrel ($) / 65 / 60 / 55 / 50 / 52.5
Costs per barrel ($) / 25 / 25 / 25 / 25 / 25
Revenue / 117,000,000 / 100,440,000 / 85,625,100 / 72,392,130 / 70,690,915
Expenses / 45,000,000 / 41,850,000 / 38,920,500 / 36,196,065 / 33,662,340
Net Income (= Dividends) / 72,000,000 / 58,590,000 / 46,704,600 / 36,196,065 / 37,028,574

Horizon growth rates:

gRevenue = (1+ sales price growth rate) × (1 + production growth rate) – 1

gRevenue = (1 + .05) × [1 + (–.07)] – 1

gRevenue = –.0235, or –2.35%

gCosts = (1+ costs growth rate) × (1 + production growth rate) – 1

gCosts= (1 + 0) × [1 + (–.07)] – 1

gCosts = –.07, or –7%

PV2019 Revenues= revenue2020 / (r – g)

PV2019 Revenues= $70,690,915 / [.09 – (–.0235)]

PV2019 Revenues = $622,827,444

PV2019 Costs = costs2020 / (r – g)

PV2019 Costs= $33,662,340 / [.09 – (–.07)]

PV2019 Costs= $210,389,628

PV2019 =PV2019 revenues – PV2019 costs

PV2019 = $622,827,444 – 210,389,628

PV2019 = $412,437,817

PV2016 = DIV2017 / (1 + r) + DIV2018 / (1 + r)2 + (DIV2019 + P2019) / (1 + r)3

PV2016= $58,590,000 / 1.09 + $46,704,600 / 1.092 + ($36,196,065 +

412,437,817) / 1.093

PV2016 = $439,490,293

Price per share2016 = PV2016 / number of shares

Price per share2016 = $439,490,293 / 7,000,000

Price per share2016 = $62.78

b.EPS2016= net income2016 / number of shares

EPS2016= $72,000,000/7,000,000

EPS2016= $10.29

EPS/P = $10.29/$62.78

EPS/P = .164, or 16.4%

The EPS/P is greater than the cost of capital because production and earnings are declining.

Est. Time: 15-20

26.The free cash flow for years 1 through 8 is computed in the following table:

Year
($ in millions) / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8
Asset value / 10 / 11.2 / 12.54 / 14.05 / 15.31 / 16.69 / 18.19 / 19.29
Earnings / 1.2 / 1.34 / 1.51 / 1.69 / 1.84 / 1.92 / 2.00 / 2.03
Investment / 1.20 / 1.34 / 1.51 / 1.26 / 1.38 / 1.50 / 1.09 / 1.16
Free Cash Flow / .00 / .00 / .00 / .43 / .46 / .42 / .91 / .87

The present value of the business is:

PV = FCF1 / (1 + r) + FCF2 / (1 + r)2+ FCF3 / (1 + r)3 + FCF4 / (1 + r)4 +

FCF5 / (1 + r)5 + FCF6 / (1 + r)6 + FCF7 / (1 + r)7 + [FCF8 / (r – g)] / (1 + r)7

PV= $0 / 1.1 + $0 / 1.12 + $0 / 1.13 + $.43 / 1.14 + $.46 / 1.15 + $.42 / 1.16 +

$.91 / 1.17 + [$.87 / (.1 – .08)] / 1.17

PV= $23.47 million

Est. Time: 15-20

27. Currency amounts are in millions of pesos.

a. r = DIV1/ P0 + g

r= 8.5 / 200 + .075

r = .1175, or 11.75%

b. g = ROE × (1 – reinvestment rate)

g = .12 × (1 – .50)

g = .06, or 6%

r = DIV1/ P0 + g

r = 8.5 / 200 + .06

r = .1025, or 10.25%

Est. Time: 15-20

28.a. PV2016= DIV2017 / (1 + r) + DIV2018 / (1 + r)2 + DIV2019 / (1 + r)3 +

DIV2020 / (1 + r)4 + DIV2021 / (1 + r)5 + (DIV2021 / r) / (1 + r)5

PV2016 = $0 / 1.09 + $1 / 1.092 + $2 / 1.093 + $2.3 / 1.094 + $2.6 / 1.095

+ ($2.6 / .09) / 1.095

PV2016 = $24.48 million

b.Price per share2016= PV2016 / number of shares

Price per share2016= $24.48 / 12

Price per share2016= $2.04

c. Based on $1million of net income for 2016:

P/E 2016 = $24.48 / $1 = 24.48

The PV of the cash flows at various points in time are as follows:

PV2017=$1 / 1.09 + $2 / 1.092 + $2.3 / 1.093 + $2.6 / 1.094 +

($2.6 / .09) / 1.094c.

PV2017= $26.68

PV2018 = $2 / 1.09 + $2.3 / 1.092 + $2.6 / 1.093 + ($2.6 / .09) / 1.093

PV2018= $28.09

PV2019 = $2.3 / 1.09 + $2.6 / 1.092 + ($2.6 / .09) / 1.092

PV2019= $28.61

PV2020 = $2.6 / 1.09 + ($2.6 / .09) / 1.092

PV2020= $28.89

PV2021 = $2.6 + ($2.6 / .09) / 1.09

PV2021= $28.89

Thus, the future PE ratios are estimated as:

PE2017 = $26.68 / $1 = 26.68

PE2018 = $28.09 / $2 = 14.04

PE2019 = $28.61 / $3.2 = 8.94

PE2020 = $28.89 / $3.7 = 7.81

PE2021 = $28.89 / $4 = 7.22

d.Using the formula, r0 = (DIV1 + P1 – P0) / P0, the annual rates of return are:

Rate of return2018 = ($1 + 28.09 – 26.68 / $26.68 = .09, or 9%

Rate of return2019 = ($2 + 28.61 – 28.09 / $28.09 = .09, or 9%

Rate of return2020 = ($2.3 + 28.89 – 28.61) / $28.61 = .09, or 9%

Rate of return2021 = ($2.6 + 28.89 – 28.89) / $28.89 = .09, or 9%

Est. Time: 15-20

29.P0 = [ROE(1 – b)BVPS] / (r – bROE)

P0 / BVPS = [ROE(1 – b)] / (r – bROE)

P0 / BVPS = (1 – b) / [(r / ROE) –b]

Consider three cases:

ROE < r (P0/BVPS) < 1

ROE = r (P0/BVPS) = 1

ROE > r (P0/BVPS) > 1

Thus, as ROE increases, the price-to-book ratio also increases, and, when ROE = r, price-to-book equals one.

Est. Time: 11-15

30.Value at a dividend yield of 5 percent:

r = dividend yield +g

r – g = dividend yield

r – g=.05, or 5%

Value = (annual fee × portfolio value) / (r – g)

Value = (.005 × $100 million) / .05

Value= $10 million

Value at a dividend yield of 4 percent using the same formulas as above:

Value = (.005 × $100 million) / .04

Value= $12.5 million

Est. Time: 6-10

31. a. Under the new faster growth assumption, Table 4.7 is reproduced here.

Year
($ millions) / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10
Asset value / 10.00 / 12.00 / 14.40 / 17.28 / 19.35 / 21.68 / 24.28 / 25.73 / 27.28 / 28.91
Earnings / 1.20 / 1.44 / 1.73 / 2.07 / 2.32 / 2.60 / 2.91 / 3.09 / 3.27 / 3.47
Net investment / 2.00 / 2.40 / 2.88 / 2.07 / 2.32 / 2.60 / 1.46 / 1.54 / 1.64 / 1.73
Free cash flow / –.80 / –.96 / –1.15 / .00 / .00 / .00 / 1.46 / 1.54 / 1.64 / 1.73
Return on equity / .12 / .12 / .12 / .12 / .12 / .12 / .12 / .12 / .12 / .12
Asset growth rate / .20 / .20 / .20 / .12 / .12 / .12 / .06 / .06 / .06
Earnings growth rate / .20 / .20 / .20 / .12 / .12 / .12 / .06 / .06 / .06

The present value of the near-term cash flows for the first six years is computed using the FCF for years 1 to 3 as the FCF is zero for years 4 to 6.

PV = FCF1 / (1 + r) + FCF2 / (1 + r)2 + FCF3 / (1 + r)3

PV = –$.80 / 1.1 + (–$.96) / 1.12 + (–$1.15) / 1.13

PV= –$2.39 million

The present value at time 0 of the horizon value, computed as of year 6, is:

PVH = [$1.46 / (.1 – .06)] / 1.16

PVH = $20.56 million

PV(business) = PV(free cash flow) + PV(horizon value)

PV(business) = –$2.39 + 20.56

PV(business)= $18.17million

Under the original growth assumptions, the PV of free cash flow was $.9 million, the PV of the horizon value was $15.4 million, and the PV of the business was $16.3 million. All three present values increased as a result of the increased rate of growth.

b. Issuing new shares does not affect the overall value of the company as that value is dependent only on the free cash flows. However, if new shares are issued to fund the negative present values of the new cash flows, the value of the existing shares would be diluted.

Est. Time: 15-20