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?

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.