STOCK VALUATION
13. With supernormal dividends, we find the price of the stock when the dividends level off at a constant growth rate, and then find the PV of the future stock price, plus the PV of all dividends during the
supernormal growth period. The stock begins constant growth in Year 4, so we can find the price of the stock in Year 3, one year before the constant dividend growth begins, as:
P4 = D4 (1 + g) / (R – g) = £2.00(1.05) / (.13 – .05) = £26.25
The price of the stock today is the PV of the first three dividends, plus the PV of the Year 3 stock price. So, the price of the stock today will be:
P0 = £8.00 / 1.13 + £6.00 / 1.132 + £3.00 / 1.133 + £2.00 / 1.134 + £26.25 / 1.134 = £31.18
14. With supernormal dividends, we find the price of the stock when the dividends level off at a constant growth rate, and then find the PV of the future stock price, plus the PV of all dividends during the supernormal growth period. The stock begins constant growth in Year 4, so we can find the price of the stock in Year 3, one year before the constant dividend growth begins as:
P3 = D3 (1 + g) / (R – g) = D0 (1 + g1)3 (1 + g2) / (R – g) = €2.80(1.25)3(1.07) / (.13 – .07) = €97.53 The price of the stock today is the PV of the first three dividends, plus the PV of the Year 3 stock price. The price of the stock today will be:
P0 = 2.80(1.25) / 1.13 + €2.80(1.25)2 / 1.132 + €2.80(1.25)3 / 1.133 + €97.53 / 1.133 P0 = €77.90
15. Here we need to find the dividend next year for a stock experiencing supernormal 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 current dividend times the FVIF. The dividend in Year 3 will be:
D3 = D0 (1.30)3
And the dividend in Year 4 will be the dividend in Year 3 times one plus the growth rate, or:
D4 = D0 (1.30)3 (1.18)
The stock begins constant growth in Year 4, so we can find the price of the stock in Year 4 as the dividend in Year 5, divided by the required return minus the growth rate. The equation for the price of the stock in Year 4 is:
P4 = D4 (1 + g) / (R – g)
Now we can substitute the previous dividend in Year 4 into this equation as follows:
P4 = D0 (1 + g1)3 (1 + g2) (1 + g3) / (R – g)
P4 = D0 (1.30)3 (1.18) (1.08) / (.14 – .08) = 46.66D0
When we solve this equation, we find that the stock price in Year 4 is 46.66 times as large as the dividend today. Now we need to find the equation for the stock price today. The stock price today is the PV of the dividends in Years 1, 2, 3, and 4, plus the PV of the Year 4 price. So:
P0 = D0(1.30)/1.14 + D0(1.30)2/1.142 + D0(1.30)3/1.143+ D0(1.30)3(1.18)/1.144 + 46.66D0/1.144 We can factor out D0 in the equation, and combine the last two terms. Doing so, we get:
P0 = $70.00 = D0{1.30/1.14 + 1.302/1.142 + 1.303/1.143 + [(1.30)3(1.18) + 46.66] / 1.144} Reducing the equation even further by solving all of the terms in the braces, we get:
$70 = $33.04D0
D0 = $70.00 / $33.04 = $2.12
This is the dividend today, so the projected dividend for the next year will be:
D1 = $2.12(1.30) = $2.75
16. The constant growth model can be applied even if the dividends are declining by a constant percentage, just make sure to recognize the negative growth. So, the price of the stock today will be:
P0 = D0 (1 + g) / (R – g) = ZW$215(1 – .08) / [(.11 – (–.08)] = ZW$1,041.05
17. We are given the stock price, the dividend growth rate, and the required return, and are asked to find the dividend. Using the constant dividend growth model, we get:
P0 = €50 = D0 (1 + g) / (R – g)
Solving this equation for the dividend gives us:
D0 = €50(.14 – .08) / (1.08) = €2.78
18. The price of a share of preferred stock is the dividend payment divided by the required return. We know the dividend payment in Year 6, so we can find the price of the stock in Year 5, one year before the first dividend payment. Doing so, we get:
P5 = ¥295 / .07 = ¥4,214.28
The price of the stock today is the PV of the stock price in the future, so the price today will be:
P0 = ¥4,214.28 / (1.07)5 = ¥3,004.72
19. The annual dividend paid to stockholders is $0.15, and the dividend yield is 0.2 percent. Using the equation for the dividend yield:
Dividend yield = Dividend / Stock price
We can plug the numbers in and solve for the stock price:
.002 = $0.15 / P0
P0 = $0.15/.002 = $75.00
The “Net Chg” of the stock shows the stock increased by $2.20 on this day, so the closing stock price yesterday was:
Yesterday’s closing price = $75.00 – 2.20 = $72.80
To find the net income, we need to find the EPS. The stock quote tells us the P/E ratio for the stock is 14. Since we know the stock price as well, we can use the P/E ratio to solve for EPS as follows:
P/E = 14 = Stock price / EPS = $75.00 / EPS EPS = $75.00 / 14 = $5.357
We know that EPS is just the total net income divided by the number of shares outstanding, so:
EPS = NI / Shares = $5.357 = NI / 25,000,000 NI = $5.357(25,000,000) = $133,928,571
20. To find the number of shares owned, we can divide the amount invested by the stock price. The share price of any financial asset is the present value of the cash flows, so, to find the price of the stock we need to find the cash flows. The cash flows are the two dividend payments plus the sale price. We also need to find the aftertax dividends since the assumption is all dividends are taxed at the same rate for all investors. The aftertax dividends are the dividends times one minus the tax rate, so:
Year 1 aftertax dividend = $2.00(1 – .28) Year 1 aftertax dividend = $1.44
Year 2 aftertax dividend = $4.00(1 – .28) Year 2 aftertax dividend = $2.88
We can now discount all cash flows from the stock at the required return. Doing so, we find the price of the stock is:
P = $1.44/1.15 + $2.88/(1.15)2 + $50/(1+.15)3 P = $36.31
The number of shares owned is the total investment divided by the stock price, which is:
Shares owned = $100,000 / $36.31 Shares owned = 2,754
21. If the company’s earnings are declining at a constant rate, the dividends will decline at the same rate since the dividends are assumed to be a constant percentage of income. The dividend next year will less than this year’s dividend, so
P0 = D0 (1 + g) / (R – g) = ₦5.00(1 – .08) / [(.14 – (–.08)] = ₦20.91