Question
Suppose the risk-free rate is 3.32% and an analyst assumes a market risk premium of 5.54%. Firm A just paid a dividend of $1.19 per share. The analyst estimates the β of Firm A to be 1.41 and estimates the dividend growth rate to be 4.13% forever. Firm A has 252.00 million shares outstanding. Firm B just paid a dividend of $1.74 per share. The analyst estimates the β of Firm B to be 0.82 and believes that dividends will grow at 2.39% forever. Firm B has 188.00 million shares outstanding. What is the value of Firm B?
Answers
GPT-4o mini
To find the value of Firm B, we can use the Gordon Growth Model (also known as the Dividend Discount Model), which calculates the value of a stock based on its expected dividends and growth rate. The formula is:
\[
P_0 = \frac{D_1}{r - g}
\]
Where:
- \( P_0 \) is the price of the stock today,
- \( D_1 \) is the dividend expected next year,
- \( r \) is the required rate of return (which can be calculated using the Capital Asset Pricing Model, CAPM),
- \( g \) is the growth rate of the dividends.
### Step 1: Calculate \( r \) using CAPM
The required rate of return \( r \) can be calculated using the formula:
\[
r = r_f + \beta \cdot (r_m - r_f)
\]
Where:
- \( r_f \) is the risk-free rate.
- \( \beta \) is the beta of the stock.
- \( r_m - r_f \) is the market risk premium.
#### For Firm B:
- \( r_f = 3.32\% = 0.0332 \)
- \( \beta = 0.82 \)
- Market risk premium \( = 5.54\% = 0.0554 \)
Now, substitute these values into the CAPM formula:
\[
r = 0.0332 + 0.82 \cdot 0.0554
\]
Calculating the market risk component:
\[
0.82 \cdot 0.0554 = 0.045428
\]
Thus:
\[
r = 0.0332 + 0.045428 = 0.078628 \approx 7.86\%
\]
### Step 2: Calculate \( D_1 \)
Firm B just paid a dividend of $1.74, and the growth rate \( g \) is 2.39%. To find \( D_1 \):
\[
D_1 = D_0 \cdot (1 + g) = 1.74 \cdot (1 + 0.0239)
\]
Calculating \( D_1 \):
\[
D_1 = 1.74 \cdot 1.0239 \approx 1.78
\]
### Step 3: Calculate \( P_0 \)
Now we can find the price of Firm B’s stock today \( P_0 \):
\[
P_0 = \frac{D_1}{r - g} = \frac{1.78}{0.078628 - 0.0239}
\]
Calculating \( r - g \):
\[
r - g = 0.078628 - 0.0239 \approx 0.054728
\]
Now calculate \( P_0 \):
\[
P_0 = \frac{1.78}{0.054728} \approx 32.45
\]
### Step 4: Calculate the total value of Firm B
To find the total value of Firm B, multiply the price per share by the number of shares outstanding.
Given that Firm B has 188 million shares:
\[
\text{Total Value of Firm B} = P_0 \cdot \text{Number of Shares}
\]
\[
\text{Total Value of Firm B} = 32.45 \cdot 188,000,000 \approx 6,104,600,000
\]
Thus, the value of Firm B is approximately **$6.10 billion**.
\[
P_0 = \frac{D_1}{r - g}
\]
Where:
- \( P_0 \) is the price of the stock today,
- \( D_1 \) is the dividend expected next year,
- \( r \) is the required rate of return (which can be calculated using the Capital Asset Pricing Model, CAPM),
- \( g \) is the growth rate of the dividends.
### Step 1: Calculate \( r \) using CAPM
The required rate of return \( r \) can be calculated using the formula:
\[
r = r_f + \beta \cdot (r_m - r_f)
\]
Where:
- \( r_f \) is the risk-free rate.
- \( \beta \) is the beta of the stock.
- \( r_m - r_f \) is the market risk premium.
#### For Firm B:
- \( r_f = 3.32\% = 0.0332 \)
- \( \beta = 0.82 \)
- Market risk premium \( = 5.54\% = 0.0554 \)
Now, substitute these values into the CAPM formula:
\[
r = 0.0332 + 0.82 \cdot 0.0554
\]
Calculating the market risk component:
\[
0.82 \cdot 0.0554 = 0.045428
\]
Thus:
\[
r = 0.0332 + 0.045428 = 0.078628 \approx 7.86\%
\]
### Step 2: Calculate \( D_1 \)
Firm B just paid a dividend of $1.74, and the growth rate \( g \) is 2.39%. To find \( D_1 \):
\[
D_1 = D_0 \cdot (1 + g) = 1.74 \cdot (1 + 0.0239)
\]
Calculating \( D_1 \):
\[
D_1 = 1.74 \cdot 1.0239 \approx 1.78
\]
### Step 3: Calculate \( P_0 \)
Now we can find the price of Firm B’s stock today \( P_0 \):
\[
P_0 = \frac{D_1}{r - g} = \frac{1.78}{0.078628 - 0.0239}
\]
Calculating \( r - g \):
\[
r - g = 0.078628 - 0.0239 \approx 0.054728
\]
Now calculate \( P_0 \):
\[
P_0 = \frac{1.78}{0.054728} \approx 32.45
\]
### Step 4: Calculate the total value of Firm B
To find the total value of Firm B, multiply the price per share by the number of shares outstanding.
Given that Firm B has 188 million shares:
\[
\text{Total Value of Firm B} = P_0 \cdot \text{Number of Shares}
\]
\[
\text{Total Value of Firm B} = 32.45 \cdot 188,000,000 \approx 6,104,600,000
\]
Thus, the value of Firm B is approximately **$6.10 billion**.