STOCK VALUATION
22. Here we have a stock paying a constant dividend for a fixed period, and an increasing dividend after
We need to find the present value of the two different cash flows using the appropriate quarterly interest rate. The constant dividend is an annuity, so the present value of these dividends is:
PVA = C(PVIFAR,t) PVA = ₪1(PVIFA2.5%,12) PVA = ₪10.26
Now we can find the present value of the dividends beyond the constant dividend phase. Using the present value of a growing annuity equation, we find:
P12 = D13 / (R – g)
P12 = ₪1(1 + .005) / (.025 – .005) P12 = ₪50.25
This is the price of the stock immediately after it has paid the last constant dividend. So, the present value of the future price is:
PV = ₪50.25 / (1 + .025)12 PV = ₪37.36
The price today is the sum of the present value of the two cash flows, so:
P0 = ₪10.26 + 37.36 P0 = ₪47.62
23. We can find the price of the stock in Year 4 when it begins a constant increase in dividends using the growing perpetuity equation. So, the price of the stock in Year 4, immediately after the dividend payment, is:
P4 = D4(1 + g) / (R – g) P4 = 元 2(1 + .07) / (.16 – .07) P4 = 元 23.78
The stock price today is the sum of the present value of the two fixed dividends plus the present value of the future price, so:
P0 = 元 2 / (1 + .16)3 + 元 2 / (1 + .16)4 + 元 23.78 / (1 + .16)4 P0 = 元 15.51
24. Here we need to find the dividend next year for a stock with nonconstant growth. We know the stock price, the dividend growth rates, and the required return, but not the dividend. First, we need to realize that the dividend in Year 3 is the constant dividend times the FVIF. The dividend in Year 3 will be:
D3 = D(1.04)
The equation for the stock price will be the present value of the constant dividends, plus the present value of the future stock price, or:
P0 = D / 1.12 + D/1.122 + D(1.04)/(.12 – .04)]/1.122
£30 = D / 1.12 + D/1.122 + D(1.04)/(.12 – .04)]/1.122
We can factor out D0 in the equation, and combine the last two terms. Doing so, we get:
£30 = D{1/1.12 + 1/1.122 + [(1.04)/(.12 – .04)] / 1.122}
Reducing the equation even further by solving all of the terms in the braces, we get:
£30 = D(12.0536)
D = £30 / 12.0536 = £2.49
This is the dividend today, so the projected dividend for the next year will be:
D1 = £2.49(1.30) = £2.49
25. The required return of a stock consists of two components, the capital gains yield and the dividend yield. In the constant dividend growth model (growing perpetuity equation), the capital gains yield is the same as the dividend growth rate, or algebraically:
R = D1/P0 + g
We can find the dividend growth rate by the growth rate equation, or:
g = ROE × b g = .11 × .75 g = .0825 or 8.25%
This is also the growth rate in dividends. To find the current dividend, we can use the information provided about the net income, shares outstanding, and payout ratio. The total dividends paid is the net income times the payout ratio. To find the dividend per share, we can divide the total dividends paid by the number of shares outstanding. So:
Dividend per share = (Net income × Payout ratio) / Shares outstanding Dividend per share = ($10,000,000 × .25) / 1,250,000
Dividend per share = $2.00
Now we can use the initial equation for the required return. We must remember that the equation uses the dividend in one year, so:
R = D1/P0 + g
R = $2(1 + .0825)/$40 + .0825 R = .1366 or 13.66%
26. First, we need to find the annual dividend growth rate over the past four years. To do this, we can use the future value of a lump sum equation, and solve for the interest rate. Doing so, we find the dividend growth rate over the past four years was:
FV = PV(1 + R)t
$1.66 = $0.90(1 + R)4 R = .1654 or 16.54%
We know the dividend will grow at this rate for five years before slowing to a constant rate indefinitely. So, the dividend amount in seven years will be:
D7 = D0(1 + g1)5(1 + g2)2
D7 = $1.66(1 + .1654)5(1 + .08)2 D7 = $4.16
27. a. We can find the price of the all the outstanding company stock by using the dividends the same way we would value an individual share. Since earnings are equal to dividends, and there is no growth, the value of the company’s stock today is the present value of a perpetuity, so:
P = D / R
P = £800,000 / .15 P = £5,333,333.33
The price-earnings ratio is the stock price divided by the current earnings, so the price-earnings ratio of each company with no growth is:
P/E = Price / Earnings
P/E = £5,333,333.33 / £800,000 P/E = 6.67 times
b. Since the earnings have increased, the price of the stock will increase. The new price of the all the outstanding company stock is:
P = D / R
P = (£800,000 + 100,000) / .15 P = £6,000,000.00
The price-earnings ratio is the stock price divided by the current earnings, so the price-earnings with the increased earnings is:
P/E = Price / Earnings P/E = £6,000,000 / £800,000 P/E = 7.50 times
c. Since the earnings have increased, the price of the stock will increase. The new price of the all the outstanding company stock is:
P = D / R
P = (£800,000 + 200,000) / .15 P = £6,666,666.67
The price-earnings ratio is the stock price divided by the current earnings, so the price-earnings with the increased earnings is:
P/E = Price / Earnings
P/E = £6,666,666.67 / £800,000 P/E = 8.33 times
28. a. If the company does not make any new investments, the stock price will be the present value of the constant perpetual dividends. In this case, all earnings are paid dividends, so, applying the perpetuity equation, we get: NPVGO is simply the present value of the investment plus the present value of the increases in EPS. SO, the NPVGO will be:
NPVGO = C1 / (1 + R) + C2 / (1 + R)2 + C3 / (1 + R)3
NPVGO = –€1.75 / 1.12 + €1.90 / 1.122 + €2.10 / 1.123 NPVGO = €1.45
So, the price of the stock if the company undertakes the investment opportunity will be:
P = €58.33 + 1.45 P = €59.78
c. After the project is over, and the earnings increase no longer exists, the price of the stock will revert back to €58.33, the value of the company as a cash cow.
29. a. The price of the stock is the present value of the dividends. Since earnings are equal to dividends, we can find the present value of the earnings to calculate the stock price. Also, since we are excluding taxes, the earnings will be the revenues minus the costs. We simply need to find the present value of all future earnings to find the price of the stock. The present value of the revenues is:
PVRevenue = C1 / (R – g)
PVRevenue = $3,000,000(1 + .05) / (.15 – .05) PVRevenue = $31,500,000
And the present value of the costs will be:
PVCosts = C1 / (R – g)
PVCosts = $1,500,000(1 + .05) / (.15 – .05) PVCosts = $15,750,000
So, the present value of the company’s earnings and dividends will be:
PVDividends = $31,500,000 – 15,750,000 PVDividends = $15,750,000
Note that since revenues and costs increase at the same rate, we could have found the present value of future dividends as the present value of current dividends. Doing so, we find:
D0 = Revenue0 – Costs0
D0 = $3,000,000 – 1,500,000 D0 = $1,500,000
Now, applying the growing perpetuity equation, we find:
PVDividends = C1 / (R – g)
PVDividends = $1,500,000(1 + .05) / (.15 – .05) PVDividends = $15,750,000
This is the same answer we found previously. The price per share of stock is the total value of the company’s stock divided by the shares outstanding, or:
P = Value of all stock / Shares outstanding P = $15,750,000 / 1,000,000
P = $15.75
b. The value of a share of stock in a company is the present value of its current operations, plus the present value of growth opportunities. To find the present value of the growth opportunities, we need to discount the cash outlay in Year 1 back to the present, and find the value today of the increase in earnings. The increase in earnings is a perpetuity, which we must discount back to today. So, the value of the growth opportunity is:
NPVGO = C0 + C1 / (1 + R) + (C2 / R) / (1 + R)
NPVGO = –$15,000,000 – $5,000,000 / (1 + .15) + ($6,000,000 / .15) / (1 + .15) NPVGO = $15,434,782.61
To find the value of the growth opportunity on a per share basis, we must divide this amount by the number of shares outstanding, which gives us:
NPVGOPer share = $15,434,782.61 / $1,000,000 NPVGOPer share = $15.43
The stock price will increase by $15.43 per share. The new stock price will be:
New stock price = $15.75 + 15.43 New stock price = $31.18
30. a. If the company continues its current operations, it will not grow, so we can value the company as a cash cow. The total value of the company as a cash cow is the present value of the future earnings, which are a perpetuity, so:
Cash cow value of company = C / R
Cash cow value of company = 110,000,000 / .15
Cash cow value of company = 733,333,333.33
The value per share is the total value of the company divided by the shares outstanding, so:
Share price = 733,333,333.33 / 20,000,000 Share price = 36.67
b. To find the value of the investment, we need to find the NPV of the growth opportunities. The initial cash flow occurs today, so it does not need to be discounted. The earnings growth is a perpetuity. Using the present value of a perpetuity equation will give us the value of the earnings growth one period from today, so we need to discount this back to today. The NPVGO of the investment opportunity is:
NPVGO per share = 39,884,057.97 / 20,000,000 NPVGO per share = 1.99
This means the per share stock price if the company undertakes the project is:
Share price = Cash cow price + NPVGO per share
Share price = 36.67 + 1.99 Share price = 38.66
31. a. If the company does not make any new investments, the stock price will be the present value of the constant perpetual dividends. In this case, all earnings are paid dividends, so, applying the perpetuity equation, we get:
P = Dividend / R P = £5 / .14
P = £35.71
b. The investment occurs every year in the growth opportunity, so the opportunity is a growing perpetuity. So, we first need to find the growth rate. The growth rate is:
g = Retention Ratio × Return on Retained Earnings g = 0.25 × 0.40
g = 0.10 or 10%
Next, we need to calculate the NPV of the investment. During year 3, twenty-five percent of the earnings will be reinvested. Therefore, £1.25 is invested (£5 × .25). One year later, the shareholders receive a 40 percent return on the investment, or £0.50 (£1.25 × .40), in perpetuity.
The perpetuity formula values that stream as of year 3. Since the investment opportunity will continue indefinitely and grows at 10 percent, apply the growing perpetuity formula to calculate the NPV of the investment as of year 2. Discount that value back two years to today.
NPVGO = [(Investment + Return / R) / (R – g)] / (1 + R)2
NPVGO = [(–£1.25 + £0.50 / .14) / (0.14 – 0.1)] / (1.14)2 NPVGO = £44.66
The value of the stock is the PV of the firm without making the investment plus the NPV of the investment, or:
P = PV(EPS) + NPVGO
P = £35.71 + £44.66 P = £80.37 Challenge
32. We are asked to find the dividend yield and capital gains yield for each of the stocks. All of the