Allston Inc. just paid an annual dividend of $1.00. The expected dividend next year $1.50, and the year after that, $2.00. After the third year, dividends will grow at a constant rate of 5% per year. If your required rate of return for Allston is 10%, what is the most you should pay for this stock.

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Quizplus · Accepted Answer

To find the value of the stock, we use the dividend discount model for a stock with dividends that grow at a constant rate after a certain period. The formula for the present value of dividends in the first three years is:$$ PV = \frac{D_1}{(1+r)^1} + \frac{D_2}{(1+r)^2} + \frac{D_3}{(1+r)^3} $$Where $D_1, D_2, $ and $D_3$ are the dividends for years 1, 2, and 3, respectively, and $r$ is the required rate of return.Given:- $D_1 = \$1.50$- $D_2 = \$2.00$- $D_3 = \$2.00 \times 1.05 = \$2.10$ (since dividends grow at 5% after the third year)- $r = 10\% = 0.10$$$ PV = \frac{\$1.50}{(1+0.10)^1} + \frac{\$2.00}{(1+0.10)^2} + \frac{\$2.10}{(1+0.10)^3} $$$$ PV = \frac{\$1.50}{1.10} + \frac{\$2.00}{1.21} + \frac{\$2.10}{1.331} $$$$ PV = \$1.36 + \$1.65 + \$1.58 = \$4.59 $$The present value of the dividends for the first three years is \$4.59.Next, we calculate the present value of the stock at the end of year 3, which is when the dividends start growing at a constant rate. We use the Gordon Growth Model (also known as the Dividend Growth Model for a perpetuity):$$ P_3 = \frac{D_4}{r - g} $$Where $D_4$ is the dividend in year 4, $r$ is the required rate of return, and $g$ is the growth rate.Given:- $D_4 = \$2.10 \times 1.05 = \$2.205$- $g = 5\% = 0.05$$$ P_3 = \frac{\$2.205}{0.10 - 0.05} $$$$ P_3 = \frac{\$2.205}{0.05} $$$$ P_3 = \$44.10 $$Now, we discount $P_3$ back to the present value:$$ PV_{P_3} = \frac{\$44.10}{(1+0.10)^3} $$$$ PV_{P_3} = \frac{\$44.10}{1.331} $$$$ PV_{P_3} = \$33.14 $$Finally, we add the present value of the dividends for the first three years to the present value of $P_3$:$$ PV_{total} = PV + PV_{P_3} $$$$ PV_{total} = \$4.59 + \$33.14 = \$37.73 $$Therefore, the most you should pay for this stock is \$37.73.