Question
A stock just paid a dividend of $2.10. The dividend is expected to grow at 24.62% for two years and then grow at 3.31% thereafter. The required return on the stock is 11.86%. What is the value of the stock?
Answers
GPT-4o mini
To find the value of the stock based on its dividends, we can use the Dividend Discount Model (DDM), which accounts for the expected dividends and their growth.
### Step 1: Calculate the Expected Dividends
1. **D0:** The most recent dividend paid is $2.10.
2. **Growth Rate for Years 1 and 2:** 24.62%
The expected dividends for the next two years (D1 and D2) are calculated as follows:
- **D1 (dividend for Year 1)**:
\[
D1 = D0 \times (1 + \text{growth rate for Year 1}) = 2.10 \times (1 + 0.2462) \approx 2.10 \times 1.2462 \approx 2.6150
\]
- **D2 (dividend for Year 2)**:
\[
D2 = D1 \times (1 + \text{growth rate for Year 2}) = D1 \times (1 + 0.2462) = 2.6150 \times 1.2462 \approx 3.2597
\]
3. **Growth Rate for Year 3 and beyond:** 3.31%
Now we calculate **D3**:
- **D3 (dividend for Year 3)**:
\[
D3 = D2 \times (1 + \text{growth rate for Year 3}) = 3.2597 \times (1 + 0.0331) = 3.2597 \times 1.0331 \approx 3.3645
\]
### Step 2: Discount the Expected Dividends to Present Value
The required return on the stock is 11.86%, or 0.1186.
Using this rate, we can discount the expected dividends back to present value:
- **PV of D1**:
\[
PV(D1) = \frac{D1}{(1 + r)^1} = \frac{2.6150}{(1 + 0.1186)^1} \approx \frac{2.6150}{1.1186} \approx 2.3398
\]
- **PV of D2**:
\[
PV(D2) = \frac{D2}{(1 + r)^2} = \frac{3.2597}{(1 + 0.1186)^2} \approx \frac{3.2597}{1.2519} \approx 2.6013
\]
### Step 3: Calculate the Present Value of D3 and Future Dividends Using Gordon Growth Model
For the dividends from Year 3 onwards (starting with D3), we use the Gordon Growth Model (GGM):
\[
P2 = \frac{D3}{r - g} = \frac{3.3645}{0.1186 - 0.0331} = \frac{3.3645}{0.0855} \approx 39.3142
\]
Now, we need to discount \( P2 \) back to present value:
- **PV of P2**:
\[
PV(P2) = \frac{P2}{(1 + r)^2} = \frac{39.3142}{1.2519} \approx 31.4071
\]
### Step 4: Total Present Value
Now we sum up all the present values:
\[
\text{Total Present Value} = PV(D1) + PV(D2) + PV(P2) = 2.3398 + 2.6013 + 31.4071 \approx 36.3482
\]
### Final Stock Value
Thus, the estimated total value of the stock is approximately:
\[
\text{Value of the Stock} \approx 36.35
\]
The value of the stock is approximately **$36.35**.
### Step 1: Calculate the Expected Dividends
1. **D0:** The most recent dividend paid is $2.10.
2. **Growth Rate for Years 1 and 2:** 24.62%
The expected dividends for the next two years (D1 and D2) are calculated as follows:
- **D1 (dividend for Year 1)**:
\[
D1 = D0 \times (1 + \text{growth rate for Year 1}) = 2.10 \times (1 + 0.2462) \approx 2.10 \times 1.2462 \approx 2.6150
\]
- **D2 (dividend for Year 2)**:
\[
D2 = D1 \times (1 + \text{growth rate for Year 2}) = D1 \times (1 + 0.2462) = 2.6150 \times 1.2462 \approx 3.2597
\]
3. **Growth Rate for Year 3 and beyond:** 3.31%
Now we calculate **D3**:
- **D3 (dividend for Year 3)**:
\[
D3 = D2 \times (1 + \text{growth rate for Year 3}) = 3.2597 \times (1 + 0.0331) = 3.2597 \times 1.0331 \approx 3.3645
\]
### Step 2: Discount the Expected Dividends to Present Value
The required return on the stock is 11.86%, or 0.1186.
Using this rate, we can discount the expected dividends back to present value:
- **PV of D1**:
\[
PV(D1) = \frac{D1}{(1 + r)^1} = \frac{2.6150}{(1 + 0.1186)^1} \approx \frac{2.6150}{1.1186} \approx 2.3398
\]
- **PV of D2**:
\[
PV(D2) = \frac{D2}{(1 + r)^2} = \frac{3.2597}{(1 + 0.1186)^2} \approx \frac{3.2597}{1.2519} \approx 2.6013
\]
### Step 3: Calculate the Present Value of D3 and Future Dividends Using Gordon Growth Model
For the dividends from Year 3 onwards (starting with D3), we use the Gordon Growth Model (GGM):
\[
P2 = \frac{D3}{r - g} = \frac{3.3645}{0.1186 - 0.0331} = \frac{3.3645}{0.0855} \approx 39.3142
\]
Now, we need to discount \( P2 \) back to present value:
- **PV of P2**:
\[
PV(P2) = \frac{P2}{(1 + r)^2} = \frac{39.3142}{1.2519} \approx 31.4071
\]
### Step 4: Total Present Value
Now we sum up all the present values:
\[
\text{Total Present Value} = PV(D1) + PV(D2) + PV(P2) = 2.3398 + 2.6013 + 31.4071 \approx 36.3482
\]
### Final Stock Value
Thus, the estimated total value of the stock is approximately:
\[
\text{Value of the Stock} \approx 36.35
\]
The value of the stock is approximately **$36.35**.