Question
A stock just paid a dividend of $2.51. The dividend is expected to grow at 27.28% for three years and then grow at 4.92% thereafter. The required return on the stock is 11.24%. What is the value of the stock?
Answers
GPT-4o mini
To calculate the value of the stock, we will follow the steps below:
1. **Calculate the dividends for the first three years when the dividend grows at 27.28%**.
2. **Calculate the stock price at the end of Year 3** based on the constant growth of 4.92%.
3. **Discount all future cash flows (dividends and terminal value) back to the present value** to find the stock's value today.
### Step 1: Calculate Dividends for the First Three Years
The dividend just paid is \( D_0 = 2.51 \).
#### Year 1:
\[
D_1 = D_0 \times (1 + g_1) = 2.51 \times (1 + 0.2728) = 2.51 \times 1.2728 \approx 3.197
\]
#### Year 2:
\[
D_2 = D_1 \times (1 + g_1) = 3.197 \times (1 + 0.2728) = 3.197 \times 1.2728 \approx 4.067
\]
#### Year 3:
\[
D_3 = D_2 \times (1 + g_1) = 4.067 \times (1 + 0.2728) = 4.067 \times 1.2728 \approx 5.177
\]
### Step 2: Calculate the Terminal Value at Year 3
Starting from Year 4, the dividend will grow at a constant rate of 4.92%. We need to calculate the dividend for Year 4:
\[
D_4 = D_3 \times (1 + g_2) = 5.177 \times (1 + 0.0492) = 5.177 \times 1.0492 \approx 5.437
\]
Next, we calculate the terminal value at the end of Year 3, which is the present value of all dividends from Year 4 onward:
\[
TV_3 = \frac{D_4}{r - g_2} = \frac{5.437}{0.1124 - 0.0492} = \frac{5.437}{0.0632} \approx 86.051
\]
### Step 3: Discount Future Cash Flows Back to Present Value
Now, we will discount the dividends and the terminal value back to the present value using the required return of 11.24% (r = 0.1124).
#### Present Value of Dividends
\[
PV(D_1) = \frac{D_1}{(1 + r)^1} = \frac{3.197}{(1 + 0.1124)^1} \approx \frac{3.197}{1.1124} \approx 2.872
\]
\[
PV(D_2) = \frac{D_2}{(1 + r)^2} = \frac{4.067}{(1 + 0.1124)^2} \approx \frac{4.067}{1.2437} \approx 3.269
\]
\[
PV(D_3) = \frac{D_3}{(1 + r)^3} = \frac{5.177}{(1 + 0.1124)^3} \approx \frac{5.177}{1.3845} \approx 3.740
\]
#### Present Value of Terminal Value
\[
PV(TV_3) = \frac{TV_3}{(1 + r)^3} = \frac{86.051}{(1 + 0.1124)^3} \approx \frac{86.051}{1.3845} \approx 62.141
\]
### Total Present Value
Now, we sum all the present values:
\[
PV_{Total} = PV(D_1) + PV(D_2) + PV(D_3) + PV(TV_3) \approx 2.872 + 3.269 + 3.740 + 62.141 \approx 72.022
\]
Thus, the estimated value of the stock is approximately **$72.02**.
1. **Calculate the dividends for the first three years when the dividend grows at 27.28%**.
2. **Calculate the stock price at the end of Year 3** based on the constant growth of 4.92%.
3. **Discount all future cash flows (dividends and terminal value) back to the present value** to find the stock's value today.
### Step 1: Calculate Dividends for the First Three Years
The dividend just paid is \( D_0 = 2.51 \).
#### Year 1:
\[
D_1 = D_0 \times (1 + g_1) = 2.51 \times (1 + 0.2728) = 2.51 \times 1.2728 \approx 3.197
\]
#### Year 2:
\[
D_2 = D_1 \times (1 + g_1) = 3.197 \times (1 + 0.2728) = 3.197 \times 1.2728 \approx 4.067
\]
#### Year 3:
\[
D_3 = D_2 \times (1 + g_1) = 4.067 \times (1 + 0.2728) = 4.067 \times 1.2728 \approx 5.177
\]
### Step 2: Calculate the Terminal Value at Year 3
Starting from Year 4, the dividend will grow at a constant rate of 4.92%. We need to calculate the dividend for Year 4:
\[
D_4 = D_3 \times (1 + g_2) = 5.177 \times (1 + 0.0492) = 5.177 \times 1.0492 \approx 5.437
\]
Next, we calculate the terminal value at the end of Year 3, which is the present value of all dividends from Year 4 onward:
\[
TV_3 = \frac{D_4}{r - g_2} = \frac{5.437}{0.1124 - 0.0492} = \frac{5.437}{0.0632} \approx 86.051
\]
### Step 3: Discount Future Cash Flows Back to Present Value
Now, we will discount the dividends and the terminal value back to the present value using the required return of 11.24% (r = 0.1124).
#### Present Value of Dividends
\[
PV(D_1) = \frac{D_1}{(1 + r)^1} = \frac{3.197}{(1 + 0.1124)^1} \approx \frac{3.197}{1.1124} \approx 2.872
\]
\[
PV(D_2) = \frac{D_2}{(1 + r)^2} = \frac{4.067}{(1 + 0.1124)^2} \approx \frac{4.067}{1.2437} \approx 3.269
\]
\[
PV(D_3) = \frac{D_3}{(1 + r)^3} = \frac{5.177}{(1 + 0.1124)^3} \approx \frac{5.177}{1.3845} \approx 3.740
\]
#### Present Value of Terminal Value
\[
PV(TV_3) = \frac{TV_3}{(1 + r)^3} = \frac{86.051}{(1 + 0.1124)^3} \approx \frac{86.051}{1.3845} \approx 62.141
\]
### Total Present Value
Now, we sum all the present values:
\[
PV_{Total} = PV(D_1) + PV(D_2) + PV(D_3) + PV(TV_3) \approx 2.872 + 3.269 + 3.740 + 62.141 \approx 72.022
\]
Thus, the estimated value of the stock is approximately **$72.02**.