To find the value of the stock, we can use the Dividend Discount Model (DDM) which accounts for both the short-term high growth and the long-term stable growth phases of dividends. We will calculate the expected dividends for the first three years when the growth is 26.77%, and then we will use the Gordon Growth Model for the dividends in perpetuity after the third year.
Step 1: Calculate the expected dividends for the first three years
Let \( D_0 = 1.02 \) (the dividend just paid).
Using the growth rate \( g_1 = 26.77% \) for the first three years, we can calculate:
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Year 1 Dividend \( D_1 \): \[ D_1 = D_0 \times (1 + g_1) = 1.02 \times (1 + 0.2677) \] \[ D_1 \approx 1.02 \times 1.2677 \approx 1.294 \]
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Year 2 Dividend \( D_2 \): \[ D_2 = D_1 \times (1 + g_1) \] \[ D_2 \approx 1.294 \times 1.2677 \approx 1.64 \]
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Year 3 Dividend \( D_3 \): \[ D_3 = D_2 \times (1 + g_1) \] \[ D_3 \approx 1.64 \times 1.2677 \approx 2.08 \]
Step 2: Calculate the dividend for Year 4
The growth rate changes to \( g_2 = 4.48% \) from Year 4 onward.
- Year 4 Dividend \( D_4 \): \[ D_4 = D_3 \times (1 + g_2) \] \[ D_4 \approx 2.08 \times (1 + 0.0448) \] \[ D_4 \approx 2.08 \times 1.0448 \approx 2.18 \]
Step 3: Calculate the present value of the dividends for the first three years
We discount each of the dividends by the required return \( r = 12.74% \):
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Present Value of \( D_1 \): \[ PV(D_1) = \frac{D_1}{(1 + r)^1} = \frac{1.294}{(1 + 0.1274)^1} \approx \frac{1.294}{1.1274} \approx 1.1477 \]
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Present Value of \( D_2 \): \[ PV(D_2) = \frac{D_2}{(1 + r)^2} = \frac{1.64}{(1 + 0.1274)^2} \approx \frac{1.64}{1.2718} \approx 1.2876 \]
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Present Value of \( D_3 \): \[ PV(D_3) = \frac{D_3}{(1 + r)^3} = \frac{2.08}{(1 + 0.1274)^3} \approx \frac{2.08}{1.4328} \approx 1.4535 \]
Step 4: Calculate the present value of the dividends from Year 4 onward
To find the value of the stock from Year 4 onward, we can use the Gordon Growth Model: \[ P_3 = \frac{D_4}{r - g_2} = \frac{2.18}{0.1274 - 0.0448} = \frac{2.18}{0.0826} \approx 26.37 \] Now we need to discount this back to present value at Year 3: \[ PV(P_3) = \frac{P_3}{(1 + r)^3} = \frac{26.37}{1.4328} \approx 18.42 \]
Step 5: Add all present values together to find the stock price
\[ PV_{Total} = PV(D_1) + PV(D_2) + PV(D_3) + PV(P_3) \] \[ PV_{Total} \approx 1.1477 + 1.2876 + 1.4535 + 18.42 \approx 22.31 \]
Therefore, the value of the stock is approximately $22.31.