To calculate the value of Firm A, we will use the Gordon Growth Model (also known as the Dividend Discount Model for a firm with constant growth). The formula for the present value of a stock is:
\[ P_0 = \frac{D_1}{r - g} \]
Where:
- \( P_0 \) is the price of the stock today,
- \( D_1 \) is the expected dividend next year,
- \( r \) is the required rate of return,
- \( g \) is the growth rate of the dividends.
Step 1: Calculate the required rate of return (r)
To calculate the required rate of return for Firm A, we will use the Capital Asset Pricing Model (CAPM):
\[ r = r_f + \beta \times (r_m - r_f) \]
Where:
- \( r_f = 3.67% \) (risk-free rate),
- \( \beta = 1.49 \) (beta for Firm A),
- \( r_m - r_f = 7.11% \) (market risk premium).
Substituting in the values:
\[ r = 3.67% + 1.49 \times 7.11% \]
Calculating \( \beta \times (r_m - r_f) \):
\[ 1.49 \times 7.11% = 10.58% \]
Now, adding the risk-free rate:
\[ r = 3.67% + 10.58% = 14.25% \]
Step 2: Calculate the expected dividend next year (D1)
The expected dividend next year can be calculated using the growth rate on the dividend just paid:
\[ D_1 = D_0 \times (1 + g) \]
Where:
- \( D_0 = 1.21 \) (the dividend just paid for Firm A),
- \( g = 4.56% \) (dividend growth rate).
Calculating \( D_1 \):
\[ D_1 = 1.21 \times (1 + 0.0456) = 1.21 \times 1.0456 \approx 1.26 \]
Step 3: Calculate the price of Firm A's stock (P0)
Now substituting \( D_1 \), \( r \), and \( g \) into the Gordon Growth Model:
\[ P_0 = \frac{1.26}{0.1425 - 0.0456} = \frac{1.26}{0.0969} \approx 13.00 \]
Step 4: Calculate the total value of Firm A
To find the total value of the firm, we need to multiply the price per share by the number of shares outstanding:
\[ \text{Total Value of Firm A} = P_0 \times \text{Shares Outstanding} \]
Where:
- Shares Outstanding = 278 million.
Calculating the total value:
\[ \text{Total Value of Firm A} = 13.00 \times 278,000,000 \approx 3,614,000,000 \]
Final Answer:
The value of Firm A is approximately $3.614 billion.
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