To find the value of Firm B, we can use the Gordon Growth Model (Dividend Discount Model), which is given by the formula:
\[ P_0 = \frac{D_1}{r - g} \]
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 dividends.
Step 1: Calculate the Required Rate of Return (r) for Firm B
Using the Capital Asset Pricing Model (CAPM):
\[ r = r_f + \beta (r_m - r_f) \]
where:
- \(r_f\) = risk-free rate = 3.94% = 0.0394
- \(\beta\) for Firm B = 0.73
- Market risk premium (\(r_m - r_f\)) = 6.66% = 0.0666
Now substituting the values:
\[ r = 0.0394 + 0.73 \times 0.0666 \] \[ r = 0.0394 + 0.048658 \] \[ r = 0.088058 \text{ or } 8.81% \]
Step 2: Calculate \(D_1\) for Firm B
Firm B just paid a dividend of $1.64. We need to calculate the dividend expected next year:
\[ D_1 = D_0 \times (1 + g) \] where:
- \(D_0\) = last paid dividend = $1.64
- \(g\) = growth rate = 2.14% = 0.0214
Now substituting the values:
\[ D_1 = 1.64 \times (1 + 0.0214) \] \[ D_1 = 1.64 \times 1.0214 \] \[ D_1 = 1.673416 \]
Step 3: Calculate the value of Firm B (P_0)
Now we can substitute \(D_1\), \(g\), and \(r\) into the Gordon Growth Model formula:
\[ P_0 = \frac{D_1}{r - g} \]
Substituting the known values:
\[ P_0 = \frac{1.673416}{0.088058 - 0.0214} \] \[ = \frac{1.673416}{0.066658} \] \[ = 25.11 \]
Step 4: Calculate the total market value of Firm B
The total value of Firm B is calculated as:
\[ \text{Market Value} = P_0 \times \text{Shares Outstanding} \]
Where the number of shares outstanding of Firm B is 199.00 million:
\[ \text{Market Value} = 25.11 \times 199,000,000 \] \[ \text{Market Value} \approx 4,999,890,000 \text{ or } \approx 5.00 \text{ billion} \]
Thus, the value of Firm B is approximately $5.00 billion.
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