To find the value of the stock given the growth rates and other information, we can use the Dividend Discount Model (DDM) which accounts for varying growth rates in dividends.
First, we'll calculate the required rate of return using the Capital Asset Pricing Model (CAPM):
\[ \text{Required Rate of Return} (r) = \text{Risk-Free Rate} + \beta \times \text{Market Risk Premium} \]
Plugging in the values:
\[ r = 1.49% + 1.78 \times 5.67% \] \[ r = 1.49 + (1.78 \times 5.67) \] \[ r = 1.49 + 10.08 \] \[ r = 11.57% \text{ or } 0.1157 \]
Step 1: Calculate Dividends for the First 5 Years
The dividend just paid is \(D_0 = 2.79\). The dividends for the next five years with a growth rate of \(g_1 = 21.40%\) will be calculated as follows:
\[ D_1 = D_0 \times (1 + g_1) = 2.79 \times (1 + 0.214) = 2.79 \times 1.214 = 3.39 \] \[ D_2 = D_1 \times (1 + g_1) = 3.39 \times 1.214 = 4.12 \] \[ D_3 = D_2 \times (1 + g_1) = 4.12 \times 1.214 = 5.00 \] \[ D_4 = D_3 \times (1 + g_1) = 5.00 \times 1.214 = 6.07 \] \[ D_5 = D_4 \times (1 + g_1) = 6.07 \times 1.214 = 7.38 \]
Step 2: Calculate the Present Value of the First 5 Dividends
Now we will calculate the present value of these dividends:
\[ PV(D_1) = \frac{D_1}{(1+r)^1} = \frac{3.39}{(1 + 0.1157)^1} = \frac{3.39}{1.1157} = 3.04 \] \[ PV(D_2) = \frac{D_2}{(1+r)^2} = \frac{4.12}{(1 + 0.1157)^2} = \frac{4.12}{1.2486} = 3.30 \] \[ PV(D_3) = \frac{D_3}{(1+r)^3} = \frac{5.00}{(1 + 0.1157)^3} = \frac{5.00}{1.3925} = 3.59 \] \[ PV(D_4) = \frac{D_4}{(1+r)^4} = \frac{6.07}{(1 + 0.1157)^4} = \frac{6.07}{1.5598} = 3.89 \] \[ PV(D_5) = \frac{D_5}{(1+r)^5} = \frac{7.38}{(1 + 0.1157)^5} = \frac{7.38}{1.7539} = 4.20 \]
Step 3: Total Present Value of Dividends for the First 5 Years
Adding these present values together:
\[ PV_{5 \text{ years}} = 3.04 + 3.30 + 3.59 + 3.89 + 4.20 = 17.12 \]
Step 4: Calculate Terminal Value at Year 5
Starting from Year 6, the dividend will grow at a constant rate \(g_2 = 4.50%\). The dividend in Year 6 \(D_6\) will be:
\[ D_6 = D_5 \times (1 + g_2) = 7.38 \times (1 + 0.045) = 7.38 \times 1.045 = 7.71 \]
The Terminal Value (TV) at Year 5 can be calculated using the Gordon Growth Model:
\[ TV = \frac{D_6}{r - g_2} = \frac{7.71}{0.1157 - 0.045} = \frac{7.71}{0.0707} = 109.06 \]
Step 5: Present Value of Terminal Value
Now, we need to find the present value of the terminal value:
\[ PV(TV) = \frac{TV}{(1 + r)^5} = \frac{109.06}{1.7539} = 62.13 \]
Step 6: Calculate Total Value of the Stock
Finally, the total value of the stock (V) is the sum of the present values of the first 5 dividends and the present value of the terminal value:
\[ V = PV_{5 \text{ years}} + PV(TV) = 17.12 + 62.13 = 79.25 \]
So, the estimated value of the stock is approximately $79.25.