To determine the value of the stock, we can use the Dividend Discount Model (DDM) with the two-stage growth method since the dividend is expected to grow at two different rates over time.
Step 1: Calculate the Required Rate of Return
First, we need to calculate the required rate of return using the Capital Asset Pricing Model (CAPM):
\[ \text{Required rate of return} (r) = \text{Risk-free rate} + \beta \times \text{Market risk premium} \]
Given:
- Risk-free rate = 1.51% or 0.0151
- Market risk premium = 7.01% or 0.0701
- Beta (β) = 1.33
\[ r = 0.0151 + 1.33 \times 0.0701 \] \[ r = 0.0151 + 0.0934783 \] \[ r = 0.1085783 \approx 10.86% \]
Step 2: Calculate Dividends for the First Three Years
The dividend just paid (D0) is $1.24, and it will grow at 24.47% for the next three years. We need to calculate the expected dividends (D1, D2, D3):
\[ D1 = D0 \times (1 + g_1) = 1.24 \times (1 + 0.2447) = 1.24 \times 1.2447 \approx 1.542 \] \[ D2 = D1 \times (1 + g_1) = 1.542 \times (1 + 0.2447) \approx 1.542 \times 1.2447 \approx 1.920 \] \[ D3 = D2 \times (1 + g_1) = 1.920 \times (1 + 0.2447) \approx 1.920 \times 1.2447 \approx 2.392 \]
Step 3: Calculate the Present Value of Dividends for the First Three Years
Now, we will find the present value of D1, D2, and D3 using the required rate of return (10.86%):
\[ PV(D1) = \frac{D1}{(1 + r)^1} = \frac{1.542}{(1 + 0.1085783)^1} \approx \frac{1.542}{1.1085783} \approx 1.39 \]
\[ PV(D2) = \frac{D2}{(1 + r)^2} = \frac{1.920}{(1 + 0.1085783)^2} \approx \frac{1.920}{1.229207} \approx 1.56 \]
\[ PV(D3) = \frac{D3}{(1 + r)^3} = \frac{2.392}{(1 + 0.1085783)^3} \approx \frac{2.392}{1.366173} \approx 1.75 \]
Step 4: Calculate Total Present Value for the First Three Years
Now we sum the present values of D1, D2, and D3:
\[ PV_{\text{total}} = PV(D1) + PV(D2) + PV(D3) \approx 1.39 + 1.56 + 1.75 \approx 4.70 \]
Step 5: Calculate the Present Value of Dividends from Year 4 Onwards
From year 4, the dividend is expected to grow at a constant rate of 4.01%. The dividend in year 4 (D4) is:
\[ D4 = D3 \times (1 + g_2) = 2.392 \times (1 + 0.0401) \approx 2.392 \times 1.0401 \approx 2.487 \]
Using the Gordon Growth Model for the present value of all future dividends starting from year 4, we calculate the present value of the perpetuity:
\[ PV_{\text{perpetuity}} = \frac{D4}{r - g_2} = \frac{2.487}{0.1085783 - 0.0401} \approx \frac{2.487}{0.0684783} \approx 36.39 \]
Step 6: Discount the Present Value of Perpetuity Back to Today
Now we need to discount this value back to present value at year 3:
\[ PV_{\text{perpetuity, discounted}} = \frac{PV_{\text{perpetuity}}}{(1 + r)^3} \approx \frac{36.39}{1.366173} \approx 26.65 \]
Step 7: Calculate Total Value of the Stock
Finally, we can sum the present values calculated earlier to get the total value of the stock:
\[ \text{Stock Value} = PV_{\text{total}} + PV_{\text{perpetuity, discounted}} \approx 4.70 + 26.65 \approx 31.35 \]
Conclusion
The value of the stock is approximately $31.35.
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