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1. Nonconstant Dividends. Apocalyptica Corporation is expected to pay the following dividends over the next four years: $6, $12, $17, and $3.25. Afterward, the company pledges to maintain a constant 5 percent growth rate in dividends, forever. If the required return on the stock is 11 percent, what is the current share price?
With nonconstant dividends, we find the price of the stock when the dividends level off at a constant growth rate, and then find the present value of the future stock price, plus the present value of all dividends during the nonconstant growth period. The stock begins constant growth after the fourth dividend is paid, so we can find the price of the stock at Year 4, when the constant dividend growth begins, as:
P4 = D4 (1 + g) / (R – g)
P4 = $3.25(1.05) / (.11 – .05)
P4 = $56.88
The price of the stock today is the present value of the first four dividends, plus the present value of the Year 4 stock price. So, the price of the stock today will be:
P0 = $6 / 1.11 + $12 / 1.112 + $17 / 1.113 + $3.25 / 1.114 + $56.88 / 1.114
P0 = $67.18
2. Supernormal Growth. Burton Corp. is growing quickly. Dividends are expected to grow at a rate of 25 percent for the next three years, with the growth rate falling off to a constant 6 percent thereafter. If the required return is 11.5 percent and the company just paid a dividend of $2.50, what is the current share price?
With nonconstant dividends, we find the price of the stock when the dividends level off at a constant growth rate, and then find the present value of the future stock price, plus the present value of all dividends during the nonconstant growth period. The stock begins constant growth after the third dividend is paid, so we can find the price of the stock in Year 3, when the constant dividend growth begins as:
P3 = D3 (1 + g2) / (R – g2)
P3 = D0 (1 + g1)3 (1 + g2) / (R – g2)
P3 = $2.50(1.25)3(1.06) / (.115 – .06)
P3 = $94.11
The price of the stock today is the present value of the first three dividends, plus the present value of the Year 3 stock price. The price of the stock today will be:
P0 = $2.50(1.25) / 1.115 + $2.50(1.25)2 / 1.1152 + $2.50(1.25)3 / 1.1153 + $94.11 / 1.1153
P0 = $77.35
3. Nonconstant Growth. Metallica Bearings, Inc., is a young start-up company. No dividends will be paid on the stock over the next nine years, because the firm needs to plow back its earnings to fuel growth. The company will then pay a dividend of $19 per share 10 years from today and will increase the dividend by 5 percent per year thereafter. If the required return on this stock is 13 percent, what is the current share price?
Here, we have a stock that pays no dividends for nine years. Once the stock begins paying dividends, it will have a constant growth rate of dividends. We can use the constant growth model at that point. It is important to remember that the general constant dividend growth formula is:
Pt = [Dt × (1 + g)] / (R – g)
This means that since we will use the dividend in Year 10, we will be finding the stock price in Year 9. The dividend growth model is similar to the present value of an annuity and the present value of a perpetuity: The equation gives you the present value one period before the first payment. So, the price of the stock in Year 9 will be:
P9 = D10 / (R – g)
P9 = $19.00 / (.13 – .05)
P9 = $237.50
The price of the stock today is simply the PV of the stock price in the future. We simply discount the future stock price at the required return. The price of the stock today will be:
P0 = $237.50 / 1.139
P0 = $79.06
4. Wesen Corp. will pay a dividend of $3.14 next year. The company has stated that it will maintain a constant growth rate of 4.5 percent a year forever. If you want a return of 12 percent, how much will you pay for the stock? What if you want a return of 8 percent? What does this tell you about the relationship between the required return and the stock price?
Here, we need to value a stock with two different required returns. Using the constant growth model and a required return of 12 percent, the stock price today is:
P0 = D1 / (R – g)
P0 = $3.14 / (.12 – .045)
P0 = $41.87
And the stock price today with a required return of 8 percent will be:
P0 = D1 / (R – g)
P0 = $3.14 / (.08 – .045)
P0 = $89.71
5. Take Time Corporation will pay a dividend of $3.65 per share next year. The company pledges to increase its dividend by 5.1 percent per year, indefinitely. If you require a return of 11 percent on your investment, how much will you pay for the company’s stock today?
The Toy Chest will pay an annual dividend of $2.64 per share next year and currently sells for $48.30 a share based on a market rate of return of 11.67 percent. What is the capital gains yield?
g = .1167- ($2.64/$48.30) = .0620, or 6.20 percent
6. For the past six years, the price of Slippery Rock stock has been increasing at a rate of 8.21 percent a year. Currently, the stock is priced at $43.40 a share and has a required return of 11.65 percent. What is the dividend yield?
Dividend yield = .1165-.0821 = .0344, or 3.44 percent
7. Village East expects to pay an annual dividend of $1.40 per share next year, and $1.68 per share for the following two years. After that, the company plans to increase the dividend by 3.4 percent annually. What is this stock’s current value at a discount rate of 13.7 percent?
P3= ($1.68 × 1.034) / (.137 – .034)= $16.86
P0 = $1.40 / 1.137 + $1.68 / 1.1372 + ($1.68+ $16.86)/ 1.1373
P0= $15.15
8. Toy Mart recently announced that it will pay annual dividends at the end of the next two years of $1.60 and $1.10 per share, respectively. Then, in Year 5 it plans to pay a final dividend of $13.50 a share before closing its doors permanently. At a required return of 13.5 percent, what should this stock sell for today?
P0 = $1.60/1.135 + $1.10/1.1352 + $13.50/1.1355= $9.43
9. Business Solutions is expected to pay its first annual dividend of $.84 per share in Year 3. Starting in Year 6, the company plans to increase the dividend by 2 percent per year. What is the value of this stock today, Year 0, at a required return of 14.4 percent?
P5 = ($.84 ×1.02)/(.144-.02) = $6.91
P0 =($.84/1.1443) + ($.84/1.1444) + [($.84 + 6.91)/1.1445] = $5.01
10. This morning, you purchased a stock that will pay an annual dividend of $1.90 per share next year. You require a 12 percent rate of return and the dividend increases at 3.5 percent annually. What will your capital gain be in dollars on this stock if you sell it three years from now?
P0 = $1.90/(.12 -.035) = $22.35
P3 = [$1.90 x (1.035)3]/(.12 -.035) = $24.78
Capital gain = $24.78 -22.35 = $2.43
11. Dry Dock Marina is expected to pay an annual dividend of $1.58 next year. The stock is selling for $18.53 a share and has a total return of 9.48 percent. What is the dividend growth rate?
g =.0948- ($1.58/$18.53) = .0095, or .95 percent