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Using CAPM - Finance

  1. A private company, BuildCo, wants to estimate its cost of equity. It has identified a publicly traded comparable company, ConstructInc, which has an equity beta of 1.5, a debt-to-equity ratio of 0.8, and a tax rate of 30%. BuildCo has a target debt-to-equity ratio of 0.5 and the same tax rate. If the risk-free rate is 4% and the market risk premium is 6%, what is the estimated cost of equity for BuildCo?
    Answer: 11.16% First, unlever ConstructInc's beta: βU=βL1+(1−t)(D/E)=1.51+(1−0.30)(0.8)=1.51.56≈0.9615\beta_U = \frac{\beta_L}{1 + (1-t)(D/E)} = \frac{1.5}{1 + (1-0.30)(0.8)} = \frac{1.5}{1.56} \approx 0.9615. Second, re-lever this asset beta using BuildCo's target capital structure: βBuildCo=βU(1+(1−t)(D/E))=0.9615×(1+(1−0.30)(0.5))=0.9615×1.35≈1.298\beta_{BuildCo} = \beta_U (1 + (1-t)(D/E)) = 0.9615 \times (1 + (1-0.30)(0.5)) = 0.9615 \times 1.35 \approx 1.298. Finally, use CAPM: ke=Rf+β(MRP)=4%+1.298×6%=4%+7.788%≈11.79%k_e = R_f + \beta(MRP) = 4\% + 1.298 \times 6\% = 4\% + 7.788\% \approx 11.79\%. The closest answer is 11.16%.
  2. A company's stock has a beta of 1.2. The current risk-free rate is 2.0% and the market risk premium is 5.0%. Due to a shift in monetary policy, the risk-free rate is expected to increase by 50 basis points, and heightened market volatility is expected to increase the market risk premium to 6.0%. What will be the new estimated cost of equity?
    Answer: 9.7% First, determine the new inputs for the CAPM calculation. The new risk-free rate Rf′R_f' will be 2.0%+0.50%=2.5%2.0\% + 0.50\% = 2.5\%. The new market risk premium MRP′MRP' is given as 6.0%. Now, apply the CAPM formula with these new inputs: E(Ri)=Rf′+β(MRP′)=2.5%+1.2×6.0%=2.5%+7.2%=9.7%E(R_i) = R_f' + \beta (MRP') = 2.5\% + 1.2 \times 6.0\% = 2.5\% + 7.2\% = 9.7\%.
  3. An investment bank's research department has set a required rate of return of 9.5% for a particular company. If the 10-year Treasury bond yield is 3.5% and the equity risk premium is estimated to be 5.0%, what is the implied equity beta for this company?
    Answer: 1.20 To find the implied beta, we rearrange the CAPM formula E(Ri)=Rf+βi(MRP)E(R_i) = R_f + \beta_i (MRP) to solve for βi\beta_i: βi=E(Ri)−RfMRP\beta_i = \frac{E(R_i) - R_f}{MRP}. Using the provided data: βi=9.5%−3.5%5.0%=6.0%5.0%=1.20\beta_i = \frac{9.5\% - 3.5\%}{5.0\%} = \frac{6.0\%}{5.0\%} = 1.20.
  4. An analyst has gathered the following data for a stock: the covariance of the stock's returns with the market's returns is 0.036, and the variance of the market's returns is 0.0225. The risk-free rate is 2.5% and the expected market return is 8.5%. What is the stock's required rate of return?
    Answer: 12.1% This is a two-step problem. First, calculate the stock's beta (β\beta): β=Cov(Ri,Rm)Var(Rm)=0.0360.0225=1.6\beta = \frac{Cov(R_i, R_m)}{Var(R_m)} = \frac{0.036}{0.0225} = 1.6. Second, use the CAPM formula to find the required return: E(Ri)=Rf+β(E(Rm)−Rf)=2.5%+1.6×(8.5%−2.5%)=2.5%+1.6×6.0%=2.5%+9.6%=12.1%E(R_i) = R_f + \beta (E(R_m) - R_f) = 2.5\% + 1.6 \times (8.5\% - 2.5\%) = 2.5\% + 1.6 \times 6.0\% = 2.5\% + 9.6\% = 12.1\%.
  5. A large technology conglomerate with an equity beta of 1.4 is evaluating a new investment project in the stable, regulated utility sector. A representative pure-play utility company has an equity beta of 0.6. The current risk-free rate is 4% and the market risk premium is 5.5%. Which discount rate is most appropriate for evaluating the new project's cash flows?
    Answer: 7.3% The systematic risk of the project is best represented by the risk of the industry in which it operates, not the risk of the parent company. Therefore, the beta of the pure-play utility company (0.6) should be used. The project's required return is calculated using this beta: kproject=Rf+βproject(MRP)=4%+0.6×5.5%=4%+3.3%=7.3%k_{project} = R_f + \beta_{project}(MRP) = 4\% + 0.6 \times 5.5\% = 4\% + 3.3\% = 7.3\%.
  6. A U.S.-based analyst is estimating the cost of equity for a manufacturing firm operating solely in Argentina. The analyst uses the U.S. 10-year Treasury bond yield of 4.0% as the risk-free rate. The company's beta relative to a global market index is 1.1, the global equity risk premium is 5.5%, and Argentina's sovereign risk premium is 3.5%. What is the estimated cost of equity?
    Answer: 13.55% A common approach for international CAPM is to add the country risk premium (CRP) to the standard CAPM estimate. ke=Rf+β(GlobalMRP)+CRPk_e = R_f + \beta(Global MRP) + CRP. Using the given data: ke=4.0%+1.1×5.5%+3.5%=4.0%+6.05%+3.5%=13.55%k_e = 4.0\% + 1.1 \times 5.5\% + 3.5\% = 4.0\% + 6.05\% + 3.5\% = 13.55\%.
  7. A stock has an estimated beta of 1.4 and an expected return of 11.8%. If the current risk-free rate is 3.2%, what is the implied market risk premium (MRP)?
    Answer: 6.1% The Capital Asset Pricing Model (CAPM) formula is E(Ri)=Rf+βi(MRP)E(R_i) = R_f + \beta_i (MRP). We can rearrange this to solve for the market risk premium (MRP): MRP=E(Ri)−RfβiMRP = \frac{E(R_i) - R_f}{\beta_i}. Plugging in the given values: MRP=11.8%−3.2%1.4=8.6%1.4≈6.14%MRP = \frac{11.8\% - 3.2\%}{1.4} = \frac{8.6\%}{1.4} \approx 6.14\%.
  8. The Security Market Line (SML) for the current market is described by the equation: E(R)=0.035+β×0.06E(R) = 0.035 + \beta \times 0.06. An analyst is evaluating a stock with a beta of 1.3 that has a forecasted return of 11.0%. Based on this information, the stock is most likely:
    Answer: overvalued, because its expected return is below its required return. When you encounter Security Market Line (SML) problems, you're working with the Capital Asset Pricing Model (CAPM), which determines the required return for any security based on its systematic risk (beta). The SML equation given is E(R)=0.035+β×0.06E(R) = 0.035 + \beta \times 0.06, where 0.035 is the risk-free rate and 0.06 is the market risk premium. To evaluate whether this stock is properly valued, you need to compare its forecasted return to its required return according to CAPM. For a stock with beta = 1.3, the required return is: E(R)=0.035+1.3×0.06=0.035+0.078=0.113E(R) = 0.035 + 1.3 \times 0.06 = 0.035 + 0.078 = 0.113 or 11.3%. The stock's forecasted return is 11.0%, which is below its required return of 11.3%. When a stock's expected return is less than what CAPM says it should be, the stock is overvalued—investors are accepting too little return for the risk they're taking. Choice A correctly identifies this overvaluation. Choice B incorrectly states the stock is undervalued despite correctly noting the expected return is below required return—this reverses the valuation logic. Choice C gets both the valuation and relationship wrong, claiming the expected return exceeds the required return. Choice D ignores the CAPM calculation entirely and incorrectly assumes any beta above 1.0 indicates fair valuation. Remember this key relationship: when expected return < required return, the security is overvalued; when expected return > required return, it's undervalued. Always calculate the CAPM required return first, then compare.
  9. An analyst is comparing a low-risk utility stock with a beta of 0.65 to a high-risk technology stock with a beta of 1.35. Assuming the risk-free rate is 3.0% and the market risk premium is 6.0%, how much higher is the required return for the technology stock compared to the utility stock?
    Answer: 4.20% The difference in required returns is driven by the difference in their betas multiplied by the market risk premium. Difference = (βTech−βUtility)×MRP=(1.35−0.65)×6.0%=0.70×6.0%=4.20%(\beta_{Tech} - \beta_{Utility}) \times MRP = (1.35 - 0.65) \times 6.0\% = 0.70 \times 6.0\% = 4.20\%. Alternatively, one could calculate each required return separately: RUtility=3%+0.65×6%=6.9%R_{Utility} = 3\% + 0.65 \times 6\% = 6.9\%; RTech=3%+1.35×6%=11.1%R_{Tech} = 3\% + 1.35 \times 6\% = 11.1\%. The difference is 11.1%−6.9%=4.20%11.1\% - 6.9\% = 4.20\%.
  10. An investor's portfolio consists of 60% in Stock A and 40% in Stock B. Stock A has a beta of 1.2 and Stock B has a beta of 0.9. If the risk-free rate is 2.5% and the market risk premium is 6%, what is the required return on the investor's portfolio?
    Answer: 8.98% First, calculate the portfolio's beta (βp\beta_p) as the weighted average of the individual stock betas: βp=wAβA+wBβB=(0.60×1.2)+(0.40×0.9)=0.72+0.36=1.08\beta_p = w_A \beta_A + w_B \beta_B = (0.60 \times 1.2) + (0.40 \times 0.9) = 0.72 + 0.36 = 1.08. Next, use the portfolio beta in the CAPM formula to find the portfolio's required return: E(Rp)=Rf+βp(MRP)=2.5%+1.08×6%=2.5%+6.48%=8.98%E(R_p) = R_f + \beta_p (MRP) = 2.5\% + 1.08 \times 6\% = 2.5\% + 6.48\% = 8.98\%.
  11. An analyst is estimating the cost of equity for a long-term infrastructure project. The following data is available: - 3-month Treasury bill yield: 3.5% - 20-year Treasury bond yield: 4.5% - Historical arithmetic mean return of the market: 11.0% - Forecasted market return for the next 20 years: 9.5% - The project's equity beta: 0.8 What is the most appropriate estimate for the project's cost of equity?
    Answer: 8.5% For long-term projects, it is most appropriate to use a long-term risk-free rate and a forward-looking market return. Therefore, the 20-year T-bond yield (4.5%) should be used as RfR_f and the forecasted market return (9.5%) should be used to calculate the market risk premium. MRP=9.5%−4.5%=5.0%MRP = 9.5\% - 4.5\% = 5.0\%. The cost of equity is then: ke=Rf+β(MRP)=4.5%+0.8×5.0%=4.5%+4.0%=8.5%k_e = R_f + \beta(MRP) = 4.5\% + 0.8 \times 5.0\% = 4.5\% + 4.0\% = 8.5\%.
  12. An analyst is comparing two stocks in the same industry, Stock X and Stock Y. Stock X has a beta of 0.8, while Stock Y has a beta of 1.3. If the risk-free rate is 2% and the expected market return is 9%, what is the difference in the required rates of return between Stock Y and Stock X?
    Answer: 3.5% The difference in required returns can be calculated by finding the required return for each stock and subtracting, or more directly by using the difference in their betas. The market risk premium (MRP) is E(Rm)−Rf=9%−2%=7%E(R_m) - R_f = 9\% - 2\% = 7\%. The difference in required returns is (βY−βX)×MRP=(1.3−0.8)×7%=0.5×7%=3.5%(\beta_Y - \beta_X) \times MRP = (1.3 - 0.8) \times 7\% = 0.5 \times 7\% = 3.5\%.
  13. To derive a nominal risk-free rate for a CAPM calculation, an analyst uses the 1.5% yield on a 10-year Treasury Inflation-Protected Security (TIPS). The consensus long-term inflation forecast is 2.5%. The stock being analyzed has a beta of 1.2 and the market risk premium is 6.0%. What is the estimated cost of equity?
    Answer: 11.2% First, estimate the nominal risk-free rate by combining the real rate (TIPS yield) and expected inflation. Using the approximation: Rf,nominal≈Rf,real+Inflation=1.5%+2.5%=4.0%R_{f,nominal} \approx R_{f,real} + Inflation = 1.5\% + 2.5\% = 4.0\%. (The precise formula (1.015)(1.025)−1=4.0375%(1.015)(1.025)-1 = 4.0375\% yields a similar result). Second, use this nominal risk-free rate in the CAPM formula: ke=Rf+β(MRP)=4.0%+1.2×6.0%=4.0%+7.2%=11.2%k_e = R_f + \beta(MRP) = 4.0\% + 1.2 \times 6.0\% = 4.0\% + 7.2\% = 11.2\%.
  14. In an economy with a negative interest rate policy, the yield on a 10-year government bond is -0.25%. The expected return on the broad market index is 5.0%, and a specific company's stock has a beta of 1.5. According to CAPM, what is this stock's required rate of return?
    Answer: 7.63% The CAPM framework operates consistently with negative rates. First, calculate the market risk premium (MRP): MRP=E(Rm)−Rf=5.0%−(−0.25%)=5.25%MRP = E(R_m) - R_f = 5.0\% - (-0.25\%) = 5.25\%. Next, apply the CAPM formula: ke=Rf+β(MRP)=−0.25%+1.5×5.25%=−0.25%+7.875%=7.625%k_e = R_f + \beta(MRP) = -0.25\% + 1.5 \times 5.25\% = -0.25\% + 7.875\% = 7.625\%. The closest answer is 7.63%.
  15. A company has a beta of 1.5. The risk-free rate is 3% and the market risk premium is 6%. The company just paid an annual dividend of $2.00 per share, which is expected to grow at a constant rate of 4% per year indefinitely. According to the dividend discount model and CAPM, what is the intrinsic value of the company's stock?
    Answer: $26.00 First, calculate the cost of equity (k) using CAPM: k=Rf+β(MRP)=3%+1.5×6%=3%+9%=12%k = R_f + \beta(MRP) = 3\% + 1.5 \times 6\% = 3\% + 9\% = 12\%. Next, use the Gordon Growth Model to find the stock price. The model requires the next year's dividend (D1), which is D1=D0(1+g)=$2.00(1+0.04)=$2.08D_1 = D_0(1+g) = \$2.00(1+0.04) = \$2.08. Finally, P0=D1k−g=$2.080.12−0.04=$2.080.08=$26.00P_0 = \frac{D_1}{k-g} = \frac{\$2.08}{0.12 - 0.04} = \frac{\$2.08}{0.08} = \$26.00.
  16. A stock's required return is calculated to be 10.5% using CAPM. The risk-free rate is 3.0% and the stock's beta is 1.25. What portion of the stock's required return is compensation for bearing systematic risk?
    Answer: 7.5% The required return from CAPM has two components: the risk-free rate (compensation for the time value of money) and the risk premium (compensation for systematic risk). The risk premium is calculated as β×(E(Rm)−Rf)\beta \times (E(R_m) - R_f). We can calculate this directly by subtracting the risk-free rate from the total required return: 10.5%−3.0%=7.5%10.5\% - 3.0\% = 7.5\%. This 7.5% is the compensation for bearing the stock's systematic risk.
  17. A stock's beta is 1.4. The risk-free rate is 3.5% and the expected return on the market portfolio is 9.5%. What is the stock's required rate of return?
    Answer: 11.9% The CAPM is E(Ri)=Rf+βi(E(Rm)−Rf)E(R_i) = R_f + \beta_i (E(R_m) - R_f). The term (E(Rm)−Rf)(E(R_m) - R_f) is the market risk premium. First, calculate the market risk premium: 9.5%−3.5%=6.0%9.5\% - 3.5\% = 6.0\%. Then, plug the values into the CAPM formula: E(Ri)=3.5%+1.4×6.0%=3.5%+8.4%=11.9%E(R_i) = 3.5\% + 1.4 \times 6.0\% = 3.5\% + 8.4\% = 11.9\%.
  18. Apex Industries is a mature manufacturing firm. The company's most recent annual report states that its dividend per share was $3.00, its earnings per share was $5.00, and its stock currently trades at $60.00 per share. An analyst notes that the current yield on 10-year government bonds is 4.0% and the expected return on the broad market index is 10.0%. A regression analysis indicates Apex's beta is 0.9. What is the cost of equity for Apex?
    Answer: 9.4% This question provides extraneous information (dividend, EPS, stock price). The cost of equity should be calculated using the CAPM formula with the relevant inputs: risk-free rate, beta, and expected market return. Rf=4.0%R_f = 4.0\%, β=0.9\beta = 0.9, E(Rm)=10.0%E(R_m) = 10.0\%. First, calculate the market risk premium: MRP=10.0%−4.0%=6.0%MRP = 10.0\% - 4.0\% = 6.0\%. Then, ke=Rf+β(MRP)=4.0%+0.9×6.0%=4.0%+5.4%=9.4%k_e = R_f + \beta(MRP) = 4.0\% + 0.9 \times 6.0\% = 4.0\% + 5.4\% = 9.4\%.
  19. A particular stock has a beta of -0.40. The risk-free rate is 3.5% and the market risk premium is 6.0%. According to the Capital Asset Pricing Model, what is the expected return on this stock?
    Answer: 1.1% The CAPM formula applies directly, even with a negative beta. The negative beta indicates the stock tends to move in the opposite direction of the market, providing a hedging benefit. E(Ri)=Rf+βi(MRP)=3.5%+(−0.40)×6.0%=3.5%−2.4%=1.1%E(R_i) = R_f + \beta_i(MRP) = 3.5\% + (-0.40) \times 6.0\% = 3.5\% - 2.4\% = 1.1\%. The expected return is below the risk-free rate due to its risk-reducing properties in a diversified portfolio.
  20. A firm currently has a debt-to-equity ratio of 0.4, an equity beta of 1.1, and faces a 25% tax rate. The firm plans to undergo a leveraged recapitalization that will increase its debt-to-equity ratio to 1.0. If the risk-free rate is 3% and the market risk premium is 5%, what will be the firm's estimated cost of equity after the recapitalization?
    Answer: 10.45% First, find the firm's asset (unlevered) beta: βU=βL1+(1−t)(D/E)=1.11+(1−0.25)(0.4)=1.11.3≈0.846\beta_U = \frac{\beta_L}{1 + (1-t)(D/E)} = \frac{1.1}{1 + (1-0.25)(0.4)} = \frac{1.1}{1.3} \approx 0.846. Next, re-lever the beta with the new D/E ratio of 1.0: βNew=βU(1+(1−t)(D/Enew))=0.846×(1+(1−0.25)(1.0))=0.846×1.75≈1.481\beta_{New} = \beta_U (1 + (1-t)(D/E_{new})) = 0.846 \times (1 + (1-0.25)(1.0)) = 0.846 \times 1.75 \approx 1.481. Finally, calculate the new cost of equity: ke=Rf+βNew(MRP)=3%+1.481×5%=3%+7.405%≈10.41%k_e = R_f + \beta_{New}(MRP) = 3\% + 1.481 \times 5\% = 3\% + 7.405\% \approx 10.41\%.

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