Chapters 6 And 7 Questions: 6-4, 6-5; 7-3. Topics Clarified ✓ Solved

Chapters 6 and 7 Questions: 6-4, 6-5; 7-3. topics clarified.

Answer the following finance problems from Chapters 6 and 7: 6-4, 6-5, and 7-3.

6-4: If investors’ aversion to risk increases, would the risk premium on a high-beta stock increase by more or less than that on a low-beta stock? Explain.

6-5: If a company’s beta doubled, would its expected return double?

7-3: Woidtke Manufacturing’s stock currently sells for $22 a share. The stock just paid a dividend of $1.20 a share (i.e., D0 = $1.20), and the dividend is expected to grow forever at a constant rate of 10% a year. What stock price is expected 1 year from now? What is the estimated required rate of return on Woidtke’s stock (assume the market is in equilibrium with the required return equal to the expected return)?

PROBLEMS: 6-3, 6-5, 7-4: Suppose that the risk-free rate is 5% and that the market risk premium is 7%. What is the required return on (1) the market, (2) a stock with a beta of 1.0, and (3) a stock with a beta of 1.7?

6-5: A stock’s return has the following distribution: State and probability information (as provided in the prompt) with corresponding returns. Use the discrete distribution to compute the expected return and risk metrics as appropriate. If the data in the prompt is incomplete, apply the standard approach to calculating expected return and standard deviation from a discrete state distribution and note any data gaps.

Nick’s Enchiladas Incorporated has preferred stock outstanding that pays a dividend of $5 at the end of each year. The preferred sells for $50 a share. What is the stock’s required rate of return (assume the market is in equilibrium with the required return equal to the expected return)?

7-9: Crisp Cookware’s common stock is expected to pay a dividend of $3 a share at the end of this year (D1 = $3.00); its beta is 0.8; the risk-free rate is 5.2%; and the market risk premium is 6%. The dividend is expected to grow at some constant rate g, and the stock currently sells for $40 a share. Assuming the market is in equilibrium, what does the market believe will be the stock’s price at the end of 3 years (i.e., what is P3)?

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Overview and assumptions

The problems listed hinge on core finance concepts: (a) how risk aversion affects risk premia under CAPM-like pricing, (b) how beta relates to expected returns, (c) perpetual-growth dividend valuation (Gordon growth model), and (d) the use of CAPM to compute required returns for markets and individual stocks. Throughout, I adopt: risk-free rate rf = 5% and market risk premium ERP = 7% for the general CAPM cases, unless problem-specific data differ (e.g., Crisp Cookware uses rf = 5.2% and ERP = 6%). Where applicable, I also use the Gordon growth framework to link dividends, growth, price, and return. These approaches and relationships are standard in modern finance theory and are documented in widely used texts such as Principles of Corporate Finance, Fundamentals of Corporate Finance, and Investments (Brealey et al.; Ross et al.; Bodie et al.) and financial valuation references (Damodaran; Copeland et al.).

6-4: Risk aversion and risk premia across betas

Answer: Under a CAPM-like framework, the required return on a stock can be expressed as r = rf + beta × ERP. An increase in risk aversion typically translates into a higher ERP. Because ERP multiplies by beta, the absolute change in the risk premium is larger for higher-beta stocks than for lower-beta stocks. In other words, when ERP rises, the incremental premium associated with high-beta stocks grows by more, all else equal, than that for low-beta stocks. This follows from the linear relationship between ERP and beta in the standard CAPM formulation. The intuition is that risk-averse investors demand disproportionately higher compensation for the systematic risk carried by high-beta stocks, so their risk premia rise more in absolute terms as risk aversion increases. This is consistent with CAPM-based intuition discussed in introductory finance texts (Brealey, Myers, & Allen, 2019; Ross, Westerfield, & Jordan, 2013).

6-5: If beta doubles, does the expected return double?

Answer: No. Using the CAPM equation r = rf + beta × ERP, doubling beta doubles only the beta × ERP portion of expected return, not the entire return. The risk-free component rf remains unchanged. For example, with rf = 5% and ERP = 7%, a stock with beta β has r = 5% + β×7%. If β doubles (e.g., from 0.8 to 1.6), r changes from 5% + 0.8×7% = 10.6% to 5% + 1.6×7% = 16.2%. The increase is 5.6 percentage points, not a doubling of the original return. The doubling effect applies only to the systematic risk component, not to the fixed rf term. This illustrates the linear, not proportional, relationship between beta and expected return under CAPM (Brealey et al., 2019; Damodaran, 2012).

7-3: Woidtke Manufacturing—price next year and required return

Given: P0 = $22, D0 = $1.20, g = 10%. We compute D1 = D0 × (1+g) = 1.20 × 1.10 = $1.32.

Using the Gordon growth model at price today, P0 = D1 / (r − g). Solve for r:

r = D1 / P0 + g = 1.32 / 22 + 0.10 = 0.06 + 0.10 = 0.16 or 16%.

To find P1 (the expected price one year from now), note that P1 = D2 / (r − g), where D2 = D1 × (1+g) = 1.32 × 1.10 = 1.452. With r − g = 0.16 − 0.10 = 0.06, P1 = 1.452 / 0.06 = $24.20.

Thus, P1 ≈ $24.20 and the estimated required return on Woidtke’s stock is 16%. These results align with the Gordon growth framework and the assumption that the market is in equilibrium (Brealey et al., 2019; Bodie, Kane, & Marcus, 2014).

CAPM base-case problems (6-3, 6-4, 7-4)

Given rf = 5% and ERP = 7% (the market risk premium), the CAPM implies:

• Market return: rM = rf + ERP = 5% + 7% = 12%.
• Stock with beta β = 1.0: r = 12% (since r = rf + βERP).
• Stock with beta β = 1.7: r = 5% + 1.7×7% = 5% + 11.9% = 16.9%.

These results illustrate the linear relationship between beta and expected return under CAPM. They also reflect the standard assumptions in asset pricing that the market risk premium is the same for all assets, with beta translating market risk into expected return (Brealey et al., 2019; Sharpe, 1964; Luenberger, 1998).

6-5: Discrete-state return distribution (conceptual approach)

When a stock’s return is described by a discrete-state distribution (states with probabilities and associated returns), the expected return is the probability-weighted average of the state returns: E[R] = Σ p_i × R_i. The risk (variance/standard deviation) is computed as Var(R) = Σ p_i × (R_i − E[R])^2, with SD = √Var(R). If the prompt data for states and probabilities is incomplete, compute using the available states and acknowledge any remaining degrees of freedom (e.g., missing final state probability or return). This methodology is standard in portfolio and investment analysis (Elton, Gruber, Brown, & Goetzmann, 2014; Bodie, Kane, & Marcus, 2014).

Nick’s Enchiladas: preferred stock return

Preferred stock pays a perpetual, fixed dividend. With D = $5 and P0 = $50, the required rate of return is: r = D / P0 = 5 / 50 = 0.10 or 10%. In equilibrium, the expected return on preferred stock equals its required return (Brealey et al., 2019; Copeland, Koller, & Murrin, 2000).

Crisp Cookware: P3 under Gordon growth with CAPM inputs

Given: D1 = $3; β = 0.8; rf = 5.2%; ERP = 6% => r = 0.052 + 0.8 × 0.06 = 0.052 + 0.048 = 0.100 = 10%.

Assuming a constant growth rate g, and current price P0 = $40, we find g from P0 = D1 / (r − g): 40 = 3 / (0.10 − g) => 0.10 − g = 3/40 = 0.075 => g = 0.10 − 0.075 = 0.025 or 2.5%.

Then D2 = D1 × (1+g) = 3 × 1.025 = 3.075; D3 = D2 × (1+g) = 3.075 × 1.025 ≈ 3.1519.

P3 = D4 / (r − g) where D4 = D3 × (1+g) = 3.1519 × 1.025 ≈ 3.2267 and r − g = 0.10 − 0.025 = 0.075. Therefore P3 ≈ 3.2267 / 0.075 ≈ 42.96.

Thus, P3 ≈ $42.96, consistent with the assumed growth and discount rate (Brealey et al., 2019; Damodaran, 2012).

References

• Brealey, R. A., Myers, S. C., & Allen, F. (2019). Principles of Corporate Finance (13th ed.). McGraw-Hill Education.
• Ross, S. A., Westerfield, R. W., & Jordan, B. D. (2013). Fundamentals of Corporate Finance (10th ed.). McGraw-Hill Education.
• Bodie, Z., Kane, A., & Marcus, A. J. (2014). Investments (10th ed.). McGraw-Hill Education.
• Damodaran, A. (2012). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset (3rd ed.). Wiley.
• Damodaran, A. (2010). The Little Book of Valuation. Wiley.
• Hull, J. C. (2018). Options, Futures, and Other Derivatives (10th ed.). Pearson Education.
• Elton, E. J., Gruber, M. J., Brown, S. J., & Goetzmann, W. N. (2014). Modern Portfolio Theory and Investment Analysis (9th ed.). Wiley.
• Berk, J., & DeMarzo, P. (2017). Corporate Finance (4th ed.). Pearson.
• Copeland, T., Koller, J., & Murrin, J. (2000). Valuation: Measuring and Managing the Value of Companies. Wiley.
• Luenberger, D. G. (1998). Investment Science. Oxford University Press.