1. **Stating the problem:** We are given a dividend growth model with multiple growth rates and need to find the price of the stock at time 0, denoted as $Price_0$. 2. **Given data:** - Initial dividend $D_0 = 10$ - Growth rates: - $g_1 = 10\%$ for years 1 to 1 - $g_2 = 5\%$ for years 2 to 5 - $g_3 = 2\%$ from year 6 onwards - Required rate of return $r = 0.2$ - Time points: 0, 1, 5, 6, 25, 26 (used to segment growth periods) 3. **Formula used:** The price of a stock with multiple growth rates is the present value of all future dividends: $$ Price_0 = \sum_{t=1}^{n} \frac{D_t}{(1+r)^t} + \frac{P_n}{(1+r)^n} $$ where $P_n$ is the price at the start of the last growth period, calculated using the Gordon Growth Model: $$ P_n = \frac{D_{n+1}}{r - g_3} $$ 4. **Calculate dividends for each period:** - Dividend at year 1: $$ D_1 = D_0 \times (1 + g_1) = 10 \times 1.10 = 11 $$ - Dividend at year 2: $$ D_2 = D_1 \times (1 + g_2) = 11 \times 1.05 = 11.55 $$ - Dividend at year 5: $$ D_5 = D_1 \times (1 + g_2)^4 = 11 \times 1.05^4 = 11 \times 1.21550625 = 13.37056875 $$ - Dividend at year 6: $$ D_6 = D_5 \times (1 + g_3) = 13.37056875 \times 1.02 = 13.637980125 $$ 5. **Calculate price at year 5 using Gordon Growth Model:** $$ P_5 = \frac{D_6}{r - g_3} = \frac{13.637980125}{0.2 - 0.02} = \frac{13.637980125}{0.18} = 75.76655625 $$ 6. **Calculate present value of dividends from year 1 to 5:** We discount each dividend and the price at year 5 back to time 0: $$ Price_0 = \sum_{t=1}^5 \frac{D_t}{(1+r)^t} + \frac{P_5}{(1+r)^5} $$ Calculate each term: - $\frac{D_1}{(1+r)^1} = \frac{11}{1.2} = 9.1667$ - $\frac{D_2}{(1+r)^2} = \frac{11.55}{1.2^2} = \frac{11.55}{1.44} = 8.0208$ - $\frac{D_3}{(1+r)^3} = \frac{D_2 \times 1.05}{1.2^3} = \frac{11.55 \times 1.05}{1.728} = \frac{12.1275}{1.728} = 7.0177$ - $\frac{D_4}{(1+r)^4} = \frac{D_3 \times 1.05}{1.2^4} = \frac{12.1275 \times 1.05}{2.0736} = \frac{12.7339}{2.0736} = 6.1433$ - $\frac{D_5}{(1+r)^5} = \frac{13.3706}{2.48832} = 5.3733$ - $\frac{P_5}{(1+r)^5} = \frac{75.7666}{2.48832} = 30.4413$ Sum all: $$ Price_0 = 9.1667 + 8.0208 + 7.0177 + 6.1433 + 5.3733 + 30.4413 = 66.1631 $$ 7. **Final answer:** $$ \boxed{Price_0 \approx 66.16} $$