To determine the value of the stock, we can use the Dividend Discount Model (DDM), taking into account two stages of growth: a high-growth period for the first five years, followed by a perpetual growth period.
Step 1: Determine the required rate of return using the Capital Asset Pricing Model (CAPM)
\[ \text{Required Rate of Return (k)} = \text{Risk-free Rate} + \beta \times \text{Market Risk Premium} \] \[ k = 2.71% + 1.48 \times 9.93% \] \[ k = 2.71% + 14.67% \] \[ k = 17.38% \]
Step 2: Calculate the expected dividends for the first 5 years
Dividend at Year 1: \[ D_1 = D_0 \times (1 + g_1) = 1.55 \times (1 + 0.2207) = 1.55 \times 1.2207 = 1.895085 \]
Dividend at Year 2: \[ D_2 = D_1 \times (1 + g_1) = 1.895085 \times 1.2207 = 2.315093 \]
Dividend at Year 3: \[ D_3 = D_2 \times (1 + g_1) = 2.315093 \times 1.2207 = 2.828528 \]
Dividend at Year 4: \[ D_4 = D_3 \times (1 + g_1) = 2.828528 \times 1.2207 = 3.449597 \]
Dividend at Year 5: \[ D_5 = D_4 \times (1 + g_1) = 3.449597 \times 1.2207 = 4.218857 \]
Step 3: Calculate the present value of the dividends for the first 5 years
Using the formula for the present value of future cash flows: \[ PV = \frac{D_t}{(1 + k)^t} \]
- For \( D_1 \): \[ PV_1 = \frac{1.895085}{(1 + 0.1738)^1} = \frac{1.895085}{1.1738} = 1.616027 \]
- For \( D_2 \): \[ PV_2 = \frac{2.315093}{(1 + 0.1738)^2} = \frac{2.315093}{1.378631} = 1.679188 \]
- For \( D_3 \): \[ PV_3 = \frac{2.828528}{(1 + 0.1738)^3} = \frac{2.828528}{1.617098} = 1.746694 \]
- For \( D_4 \): \[ PV_4 = \frac{3.449597}{(1 + 0.1738)^4} = \frac{3.449597}{1.895349} = 1.818868 \]
- For \( D_5 \): \[ PV_5 = \frac{4.218857}{(1 + 0.1738)^5} = \frac{4.218857}{2.223564} = 1.897943 \]
Total Present Value of Dividends for the First 5 Years
\[ PV_{\text{first 5 years}} = PV_1 + PV_2 + PV_3 + PV_4 + PV_5 \] \[ PV_{\text{first 5 years}} = 1.616027 + 1.679188 + 1.746694 + 1.818868 + 1.897943 = 8.95872 \]
Step 4: Calculate the stock price at the end of Year 5
At Year 5, the dividend will grow at a constant rate \( g_2 = 4.01% \):
Dividend at Year 6: \[ D_6 = D_5 \times (1 + g_2) = 4.218857 \times (1 + 0.0401) = 4.218857 \times 1.0401 = 4.392525 \]
Price at Year 5: Using the Gordon Growth Model (for constant growth): \[ P_5 = \frac{D_6}{k - g_2} = \frac{4.392525}{0.1738 - 0.0401} = \frac{4.392525}{0.1337} = 32.89 \]
Step 5: Calculate the present value of the stock price at Year 5
\[ PV_{P_5} = \frac{P_5}{(1 + k)^5} = \frac{32.89}{(1 + 0.1738)^5} = \frac{32.89}{2.223564} = 14.77974 \]
Step 6: Calculate the total present value of the stock
\[ PV_{\text{total}} = PV_{\text{first 5 years}} + PV_{P_5} \] \[ PV_{\text{total}} = 8.95872 + 14.77974 = 23.73846 \]
Conclusion: Value of the stock
The value of the stock is approximately $23.74.
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