To find the value of the stock, we will first calculate the required rate of return using the Capital Asset Pricing Model (CAPM), then we will estimate the expected future dividends over the growth periods, and finally, we will find the present value of those expected dividends.
Step 1: Calculate the Required Rate of Return
The required rate of return (r) can be calculated using the CAPM formula:
\[ r = \text{Risk-free rate} + \beta \times \text{Market Risk Premium} \]
Substituting the values:
\[ r = 2.93% + 0.91 \times 5.36% \] \[ r = 2.93% + 4.8676% \] \[ r = 7.7976% \approx 7.80% \]
Step 2: Calculate Future Dividends
The stock just paid a dividend of $1.54, and we need to forecast the dividends for the first three years, growing at 22.07%. We will also calculate the dividend for year 4, which will grow at a constant rate of 3.09% thereafter.
\[ D_0 = 1.54 \] \[ D_1 = D_0 \times (1 + g_1) = 1.54 \times (1 + 0.2207) = 1.54 \times 1.2207 = 1.88278 \approx 1.88 \] \[ D_2 = D_1 \times (1 + g_1) = 1.88 \times 1.2207 = 2.2967 \approx 2.30 \] \[ D_3 = D_2 \times (1 + g_1) = 2.30 \times 1.2207 = 2.80361 \approx 2.80 \] \[ D_4 = D_3 \times (1 + g_2) = 2.80 \times (1 + 0.0309) = 2.80 \times 1.0309 = 2.89152 \approx 2.89 \]
Step 3: Calculate Present Value of Future Dividends
Now we will calculate the present value of the dividends for the first three years, and also the present value of the price at year 3 (which can be calculated using Gordon growth model for perpetuity for dividends starting at year 4).
Present Value of Dividends:
\[ PV(D_1) = \frac{D_1}{(1 + r)^1} = \frac{1.88}{(1 + 0.078)^1} \approx \frac{1.88}{1.078} \approx 1.743 \]
\[ PV(D_2) = \frac{D_2}{(1 + r)^2} = \frac{2.30}{(1 + 0.078)^2} \approx \frac{2.30}{1.165684} \approx 1.9743 \]
\[ PV(D_3) = \frac{D_3}{(1 + r)^3} = \frac{2.80}{(1 + 0.078)^3} \approx \frac{2.80}{1.259712} \approx 2.224 \]
Calculation for Year 4 Dividend Present Value: Using the Gordon Growth Model for the price at year 3 (where the dividend grows at a constant rate from year 4 forward):
\[ P_3 = \frac{D_4}{r - g_2} = \frac{2.89}{0.078 - 0.0309} = \frac{2.89}{0.0471} \approx 61.38 \]
Now we find the present value of that price:
\[ PV(P_3) = \frac{P_3}{(1 + r)^3} = \frac{61.38}{1.259712} \approx 48.70 \]
Total Present Value of Stock:
Now, we sum up all present values:
\[ PV = PV(D_1) + PV(D_2) + PV(D_3) + PV(P_3) \] \[ PV \approx 1.743 + 1.9743 + 2.224 + 48.70 \approx 54.64 \]
Conclusion:
Thus, the estimated value of the stock today is approximately $54.64.