To calculate the value of the stock, we can use the Dividend Discount Model (DDM) which accounts for different growth rates in dividends. Since the dividend grows at two different rates, we will split the calculations into two parts:
- Calculate the required rate of return \( r \) using the Capital Asset Pricing Model (CAPM):
\[ r = r_f + \beta \times (r_m - r_f) \]
Where:
- \( r_f = 3.37% = 0.0337 \) (risk-free rate)
- \( \beta = 1.37 \)
- \( r_m - r_f = 8.88% = 0.0888 \) (market risk premium)
Calculating \( r \):
\[ r = 0.0337 + 1.37 \times 0.0888 \] \[ r = 0.0337 + 0.121056 \] \[ r \approx 0.154756 = 15.48% \]
- Calculate the expected dividends for the first three years. The most recent dividend \( D_0 \) is $2.36, and it is expected to grow at \( 24.05% \) for three years.
Calculating dividends:
- \( D_1 = D_0 \times (1 + g_1) = 2.36 \times (1 + 0.2405) = 2.36 \times 1.2405 \approx 2.92 \)
- \( D_2 = D_1 \times (1 + g_1) = 2.92 \times (1 + 0.2405) \approx 2.92 \times 1.2405 \approx 3.62 \)
- \( D_3 = D_2 \times (1 + g_1) = 3.62 \times (1 + 0.2405) \approx 3.62 \times 1.2405 \approx 4.49 \)
- Calculate the present value of the dividends for the first three years:
\[ PV(D_1) = \frac{D_1}{(1 + r)^1} = \frac{2.92}{(1 + 0.154756)^1} = \frac{2.92}{1.154756} \approx 2.53 \] \[ PV(D_2) = \frac{D_2}{(1 + r)^2} = \frac{3.62}{(1 + 0.154756)^2} = \frac{3.62}{1.333459} \approx 2.71 \] \[ PV(D_3) = \frac{D_3}{(1 + r)^3} = \frac{4.49}{(1 + 0.154756)^3} = \frac{4.49}{1.537650} \approx 2.92 \]
Now summing these present values gives us \( PV(D_1) + PV(D_2) + PV(D_3) \):
\[ PV(D_1) + PV(D_2) + PV(D_3) \approx 2.53 + 2.71 + 2.92 \approx 8.16 \]
- Calculate the present value of the stock after year 3, where the dividends will grow at a constant rate \( g_2 = 3.77% \).
The dividend at year 4 (the first year of constant growth):
\[ D_4 = D_3 \times (1 + g_2) = 4.49 \times (1 + 0.0377) = 4.49 \times 1.0377 \approx 4.66 \]
Using the Gordon Growth Model to find the present value of all future dividends starting from year 4:
\[ PV = \frac{D_4}{r - g_2} = \frac{4.66}{0.154756 - 0.0377} = \frac{4.66}{0.117056} \approx 39.85 \]
Now, we need to find the present value of this amount, discounted back to today (3 years):
\[ PV(\text{forever dividends}) = \frac{39.85}{(1 + r)^3} = \frac{39.85}{1.537650} \approx 25.95 \]
- Sum the present values:
\[ \text{Total PV} = PV(D_1) + PV(D_2) + PV(D_3) + PV(\text{forever dividends}) \approx 8.16 + 25.95 \approx 34.11 \]
Thus, the estimated value of the stock is approximately $34.11.
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