Question

Suppose the risk-free rate is 1.92% and an analyst assumes a market risk premium of 5.49%. Firm A just paid a dividend of $1.27 per share. The analyst estimates the β of Firm A to be 1.33 and estimates the dividend growth rate to be 4.61% forever. Firm A has 258.00 million shares outstanding. Firm B just paid a dividend of $1.91 per share. The analyst estimates the β of Firm B to be 0.72 and believes that dividends will grow at 2.87% forever. Firm B has 195.00 million shares outstanding. What is the value of Firm B?

Answer 1

To value Firm B using the Gordon Growth Model (also known as the Dividend Discount Model), we can use the following formula:

\[ P_0 = \frac{D_1}{r - g} \]

where:

• \( P_0 \) is the current price of the stock,
• \( D_1 \) 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 Growth Rate and Required Rate of Return

For Firm B:

1. Growth Rate \( g \): \[ g = 2.87% = 0.0287 \]

2. Calculate the Required Rate of Return \( r \) using the Capital Asset Pricing Model (CAPM): \[ r = R_f + \beta \times (R_m - R_f) \] where:

   • \( R_f \) = risk-free rate = 1.92% = 0.0192
   • \( \beta \) = 0.72
   • Market Risk Premium \( (R_m - R_f) \) = 5.49% = 0.0549

   Substituting values: \[ r = 0.0192 + 0.72 \times 0.0549 \] \[ r = 0.0192 + 0.039525 \] \[ r = 0.058725 \text{ or } 5.87% \]

Step 2: Calculate the Expected Dividend Next Year \( D_1 \)

Firm B's last paid dividend \( D_0 \) is $1.91. The expected dividend next year is calculated as: \[ D_1 = D_0 \times (1 + g) = 1.91 \times (1 + 0.0287) = 1.91 \times 1.0287 \approx 1.9642 \]

Step 3: Calculate the Price of the Stock \( P_0 \)

Now, substitute \( D_1 \), \( r \), and \( g \) into the Gordon Growth Model: \[ P_0 = \frac{1.9642}{0.058725 - 0.0287} \] Calculate \( r - g \): \[ r - g = 0.058725 - 0.0287 = 0.030025 \] Now, substitute back: \[ P_0 = \frac{1.9642}{0.030025} \approx 65.3 \]

Step 4: Calculate the Value of Firm B

To find the total value of Firm B, multiply the price per share by the number of shares outstanding: \[ \text{Value of Firm B} = P_0 \times \text{Number of Shares} = 65.3 \times 195,000,000 \] \[ \text{Value of Firm B} \approx 12,749,500,000 \text{ or } $12.75 \text{ billion} \]

Final Answer

The estimated value of Firm B is approximately $12.75 billion.