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The future earnings, dividends, and common stock price of Carpetto Technologies Inc. are expected to grow \(7 \%\) per year. Carpetto's common stock currently sells for \(\$ 23.00\) per share; its last dividend was \(\$ 2.00 ;\) and it will pay a \(\$ 2.14\) dividend at the end of the current year. a. Using the DCF approach, what is its cost of common equity? b. If the firm's beta is \(1.6,\) the risk-free rate is \(9 \%\), and the average return on the market is \(13 \%,\) what will be the firm's cost of common equity using the CAPM approach? c. If the firm's bonds earn a return of \(12 \%\), based on the bond-yield-plus- risk-premium approach, what will be \(\mathrm{r}_{\mathrm{s}}\) ? Use the midpoint of the risk premium range discussed in Section \(10-5\) in your calculations. d. If you have equal confidence in the inputs used for the three approaches, what is your estimate of Carpetto's cost of common equity?

Step by step solution

01

DCF Approach - Find Dividend Growth Rate

The dividend growth rate (\(g\)) is estimated to be \(7\%\). We have this information directly from the problem statement.

02

DCF Approach - Calculate Cost of Common Equity

In the Dividend Discount Model (DCF approach), the cost of common equity \( (r_s) \) can be calculated using the formula:\[r_s = \frac{D_1}{P_0} + g\]where \(D_1\) is the expected dividend at the end of the year, \(P_0\) is the current stock price, and \(g\) is the growth rate. Plugging in the values:- \(D_1 = 2.14\)- \(P_0 = 23.00\)- \(g = 0.07\)\[r_s = \frac{2.14}{23.00} + 0.07 = 0.0922 + 0.07 = 0.1622 = 16.22\%\]

03

CAPM Approach - Calculate Cost of Common Equity

We use the Capital Asset Pricing Model (CAPM) to find the cost of equity:\[r_s = r_f + \beta (r_m - r_f)\]where \(r_f\) is the risk-free rate, \(\beta\) is the beta of the stock, and \(r_m\) is the expected return of the market. The given values:- \(r_f = 0.09\)- \(\beta = 1.6\)- \(r_m = 0.13\)\[r_s = 0.09 + 1.6(0.13 - 0.09) = 0.09 + 1.6 \times 0.04 = 0.154 = 15.4\%\]

04

Bond-Yield-Plus-Risk-Premium Approach

This approach estimates the cost of common equity by adding a risk premium to the yield of the firm's bonds. Assuming a risk premium of 4% as a midpoint:\[\text{Bond yield = } 12\%\]\[r_s = \text{Bond yield} + \text{Risk Premium} = 12\% + 4\% = 16\%\]

05

Weighted Average of Three Approaches

Since equal confidence is assumed in all three methods, calculate the average of the three cost estimates:\[r_s = \frac{16.22\% + 15.4\% + 16\%}{3} = \frac{47.62}{3} \approx 15.87\%\]

06

Final Cost of Common Equity Estimate

The final estimate of Carpetto's cost of common equity, combining all three approaches, is approximately \(15.87\%\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dividend Discount Model

The Dividend Discount Model (DDM) is a popular tool to assess the cost of common equity for a firm. It is particularly useful when a company consistently pays dividends. The basic premise is that the price of a stock is essentially the present value of its projected future dividends. This is especially true when a company has stable growth rates.

The relevant formula for the DDM is given by:

• \( r_s = \frac{D_1}{P_0} + g \)
• \( D_1 \) is the dividend expected at the end of the year.
• \( P_0 \) is the current stock price.
• \( g \) is the growth rate of the dividends.

To apply this, take Carpetto Technologies Inc. as an example. Suppose the last dividend paid was \(2.00, but the company is expected to pay \)2.14 next year, with a consistent growth rate of 7%. Given the current stock price of $23, the cost of equity can be calculated. Plug these values into the formula and solve for \( r_s \).

This model provides an intuitive understanding because it centers on the company's future prospects via dividends, vital for long-term investors.

Capital Asset Pricing Model

The Capital Asset Pricing Model (CAPM) is a cornerstone in finance that establishes a linear relationship between the risk of an asset and its expected return. It is based primarily on the systematic risk, also known as market risk, which cannot be diversified away.

The CAPM formula is:

• \( r_s = r_f + \beta (r_m - r_f) \)
• \( r_f \) is the risk-free rate.
• \( \beta \) is the beta of the stock, indicating its volatility relative to the market.
• \( r_m \) is the expected market return.

In Carpetto's scenario, the beta is 1.6, reflecting a higher sensitivity to market movements. With a risk-free rate of 9% and a market return of 13%, the CAPM equation shows how the risk of the investment impacts the cost of equity. By analyzing these factors, CAPM provides insight into how much return investors need to take on a certain level of systematic risk. This makes it particularly useful for portfolio managers and investors comparing different investments.

Bond-Yield-Plus-Risk-Premium Approach

This approach provides a straightforward method to estimate the cost of common equity. It does so by adding a specific risk premium to the yield of a company's bonds. This model makes intuitive sense because it recognizes that equity typically carries more risk than debt, thus deserves a higher return.

To use this approach, calculate:

• Consider the firm's bond yield, in Carpetto's case, it's 12%.
• Add a risk premium, typically within a range found in financial literature, here assumed as 4%.

Applying these values for Carpetto Technologies Inc., the cost of common equity would be 12% + 4% = 16%.

This method is particularly pragmatic in practice as it offers a direct linkage between existing debt costs and equity returns, while accounting for additional risks associated with equity investments.