Question
A stock just paid a dividend of $2.74. The dividend is expected to grow at 29.02% for three years and then grow at 3.01% thereafter. The required return on the stock is 10.36%. What is the value of the stock?
Answers
GPT-4o mini
To find the value of the stock, we need to calculate the present value of the expected future dividends.
### Step 1: Calculate the expected dividends for the first three years
1. **Year 1 Dividend (D1):**
\[
D_1 = D_0 \times (1 + g_1) = 2.74 \times (1 + 0.2902) = 2.74 \times 1.2902 \approx 3.53
\]
2. **Year 2 Dividend (D2):**
\[
D_2 = D_1 \times (1 + g_1) = 3.53 \times (1 + 0.2902) \approx 3.53 \times 1.2902 \approx 4.55
\]
3. **Year 3 Dividend (D3):**
\[
D_3 = D_2 \times (1 + g_1) = 4.55 \times (1 + 0.2902) \approx 4.55 \times 1.2902 \approx 5.86
\]
### Step 2: Calculate the present value of the first three dividends
Using the required return (discount rate) of 10.36%:
\[
PV_{D1} = \frac{D_1}{(1 + r)^1} = \frac{3.53}{(1 + 0.1036)^1} \approx \frac{3.53}{1.1036} \approx 3.20
\]
\[
PV_{D2} = \frac{D_2}{(1 + r)^2} = \frac{4.55}{(1 + 0.1036)^2} \approx \frac{4.55}{1.2180} \approx 3.73
\]
\[
PV_{D3} = \frac{D_3}{(1 + r)^3} = \frac{5.86}{(1 + 0.1036)^3} \approx \frac{5.86}{1.3437} \approx 4.35
\]
### Step 3: Sum the present values of the first three dividends
\[
PV_{\text{first 3 dividends}} = PV_{D1} + PV_{D2} + PV_{D3} \approx 3.20 + 3.73 + 4.35 \approx 11.28
\]
### Step 4: Calculate the present value of dividends from Year 4 onwards
The dividend in Year 4 (D4) will grow at a rate of 3.01%:
\[
D_4 = D_3 \times (1 + g_2) = 5.86 \times (1 + 0.0301) \approx 5.86 \times 1.0301 \approx 6.04
\]
### Step 5: Calculate the terminal value at Year 3
The terminal value at Year 3 (the present value of all dividends from Year 4 onwards) can be calculated using the Gordon Growth Model:
\[
TV_3 = \frac{D_4}{r - g_2} = \frac{6.04}{0.1036 - 0.0301} = \frac{6.04}{0.0735} \approx 82.16
\]
### Step 6: Calculate the present value of the terminal value
\[
PV_{TV} = \frac{TV_3}{(1 + r)^3} = \frac{82.16}{1.3437} \approx 61.00
\]
### Step 7: Calculate the total present value (value of the stock)
\[
\text{Total PV} = PV_{\text{first 3 dividends}} + PV_{TV} \approx 11.28 + 61.00 = 72.28
\]
Therefore, the estimated value of the stock is approximately **$72.28**.
### Step 1: Calculate the expected dividends for the first three years
1. **Year 1 Dividend (D1):**
\[
D_1 = D_0 \times (1 + g_1) = 2.74 \times (1 + 0.2902) = 2.74 \times 1.2902 \approx 3.53
\]
2. **Year 2 Dividend (D2):**
\[
D_2 = D_1 \times (1 + g_1) = 3.53 \times (1 + 0.2902) \approx 3.53 \times 1.2902 \approx 4.55
\]
3. **Year 3 Dividend (D3):**
\[
D_3 = D_2 \times (1 + g_1) = 4.55 \times (1 + 0.2902) \approx 4.55 \times 1.2902 \approx 5.86
\]
### Step 2: Calculate the present value of the first three dividends
Using the required return (discount rate) of 10.36%:
\[
PV_{D1} = \frac{D_1}{(1 + r)^1} = \frac{3.53}{(1 + 0.1036)^1} \approx \frac{3.53}{1.1036} \approx 3.20
\]
\[
PV_{D2} = \frac{D_2}{(1 + r)^2} = \frac{4.55}{(1 + 0.1036)^2} \approx \frac{4.55}{1.2180} \approx 3.73
\]
\[
PV_{D3} = \frac{D_3}{(1 + r)^3} = \frac{5.86}{(1 + 0.1036)^3} \approx \frac{5.86}{1.3437} \approx 4.35
\]
### Step 3: Sum the present values of the first three dividends
\[
PV_{\text{first 3 dividends}} = PV_{D1} + PV_{D2} + PV_{D3} \approx 3.20 + 3.73 + 4.35 \approx 11.28
\]
### Step 4: Calculate the present value of dividends from Year 4 onwards
The dividend in Year 4 (D4) will grow at a rate of 3.01%:
\[
D_4 = D_3 \times (1 + g_2) = 5.86 \times (1 + 0.0301) \approx 5.86 \times 1.0301 \approx 6.04
\]
### Step 5: Calculate the terminal value at Year 3
The terminal value at Year 3 (the present value of all dividends from Year 4 onwards) can be calculated using the Gordon Growth Model:
\[
TV_3 = \frac{D_4}{r - g_2} = \frac{6.04}{0.1036 - 0.0301} = \frac{6.04}{0.0735} \approx 82.16
\]
### Step 6: Calculate the present value of the terminal value
\[
PV_{TV} = \frac{TV_3}{(1 + r)^3} = \frac{82.16}{1.3437} \approx 61.00
\]
### Step 7: Calculate the total present value (value of the stock)
\[
\text{Total PV} = PV_{\text{first 3 dividends}} + PV_{TV} \approx 11.28 + 61.00 = 72.28
\]
Therefore, the estimated value of the stock is approximately **$72.28**.