Finance Quiz: Beta And Cost Of Equity
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Beta And Cost Of EquityQuestion 1 of 20

A publicly traded comparable company in the specialty chemicals industry has an equity beta of 1.4, a debt-to-equity ratio of 0.5, and a corporate tax rate of 25%. An analyst wants to determine the asset beta (unlevered beta) for this company to use as a benchmark. The company's asset beta is closest to:

0.93
1.02
1.93
2.24
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Finance Quiz: Beta And Cost Of Equity

Practice Beta And Cost Of Equity in Finance with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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Question 1

A publicly traded comparable company in the specialty chemicals industry has an equity beta of 1.4, a debt-to-equity ratio of 0.5, and a corporate tax rate of 25%. An analyst wants to determine the asset beta (unlevered beta) for this company to use as a benchmark. The company's asset beta is closest to:

  1. 0.93
  2. 1.02 (correct answer)
  3. 1.93
  4. 2.24
Explanation: The formula to unlever beta is βU=βL1+(1−t)(D/E)\beta_U = \frac{\beta_L}{1 + (1-t)(D/E)}. Plugging in the given values: βU=1.41+(1−0.25)(0.5)=1.41+(0.75)(0.5)=1.41.375≈1.018\beta_U = \frac{1.4}{1 + (1-0.25)(0.5)} = \frac{1.4}{1 + (0.75)(0.5)} = \frac{1.4}{1.375} \approx 1.018, which is closest to 1.02.

Question 2

An analyst is tasked with estimating the cost of equity for a private company. The analyst identifies a publicly traded comparable company with an equity beta of 1.5, a debt-to-equity ratio of 0.6, and a tax rate of 20%. The private company is targeting a debt-to-equity ratio of 1.0 and has a tax rate of 25%. The current risk-free rate is 4.0% and the market risk premium is 5.0%.

Based on the passage, the estimated cost of equity for the private company is closest to:

  1. 9.1%
  2. 11.5%
  3. 12.9% (correct answer)
  4. 13.4%
Explanation: This is a three-step process. First, unlever the comparable company's beta: βU=1.51+(1−0.20)(0.6)=1.51.48≈1.0135\beta_U = \frac{1.5}{1 + (1-0.20)(0.6)} = \frac{1.5}{1.48} \approx 1.0135. Second, relever this asset beta using the private company's target capital structure: βL=1.0135[1+(1−0.25)(1.0)]=1.0135×1.75≈1.7737\beta_L = 1.0135 [1 + (1-0.25)(1.0)] = 1.0135 \times 1.75 \approx 1.7737. Third, use the CAPM to find the cost of equity: ke=4.0%+1.7737×5.0%=4.0%+8.87%≈12.87%k_e = 4.0\% + 1.7737 \times 5.0\% = 4.0\% + 8.87\% \approx 12.87\%, which is closest to 12.9%.

Question 3

An analyst is estimating the cost of equity for a U.S.-based utility company. The company's cash flows are stable, and it is valued as a long-term going concern. Which of the following rates is the most appropriate proxy for the risk-free rate in the CAPM calculation?

  1. The current yield on a 3-month U.S. Treasury bill.
  2. The historical average yield on 10-year U.S. Treasury bonds.
  3. The current yield on a 20-year U.S. Treasury bond. (correct answer)
  4. The current yield on a 20-year AAA-rated corporate bond.
Explanation: The risk-free rate should match the duration of the cash flows being discounted. Since the utility is valued as a long-term going concern, a long-term government bond yield is the most appropriate proxy. The 20-year U.S. Treasury bond yield is a suitable choice. A 3-month T-bill is too short-term. A historical average is not a reflection of current rates. A corporate bond includes a credit spread and is therefore not risk-free.

Question 4

An analyst reviewing a portfolio of stocks identifies a company with a beta of -0.5. According to the Capital Asset Pricing Model (CAPM), this finding implies that the company's required rate of return is:

  1. equal to the risk-free rate, as negative beta is theoretically impossible.
  2. negative, indicating that investors expect the stock's price to decline.
  3. less than the risk-free rate, but still likely positive. (correct answer)
  4. greater than the risk-free rate, but less than the market return.
Explanation: The CAPM formula is ke=Rf+β(MRP)k_e = R_f + \beta(MRP). If beta is negative, the second term, β(MRP)\beta(MRP), becomes negative (assuming a positive market risk premium). This means the required return will be the risk-free rate minus some amount. For example, if Rf=3%R_f=3\% and MRP=6%MRP=6\%, ke=3%+(−0.5)(6%)=3%−3%=0%k_e = 3\% + (-0.5)(6\%) = 3\% - 3\% = 0\%. If Rf=5%R_f=5\% and MRP=6%MRP=6\%, ke=5%−3%=2%k_e = 5\% - 3\% = 2\%. The required return is less than the risk-free rate but is not necessarily negative.

Question 5

An analyst calculates a raw beta of 0.70 for a stable utility company based on historical data. The analyst decides to use the Blume adjustment technique to account for beta's tendency to revert toward the mean of 1.0. The adjustment formula is: Adjusted β=(2/3)×Raw β+(1/3)×1.0\beta = (2/3) \times \text{Raw } \beta + (1/3) \times 1.0. What is the adjusted beta?

  1. 0.70
  2. 0.80 (correct answer)
  3. 0.87
  4. 1.05
Explanation: Plugging the raw beta into the formula gives: Adjusted β=(2/3)×0.70+(1/3)×1.0=0.4667+0.3333=0.80\beta = (2/3) \times 0.70 + (1/3) \times 1.0 = 0.4667 + 0.3333 = 0.80. This adjustment pulls the raw beta of 0.70 closer to the market average beta of 1.0.

Question 6

A country's government announces a surprise, permanent increase in the corporate tax rate. For a levered company with a positive asset beta, what is the most likely immediate impact on its equity beta (βL\beta_L) and its cost of equity (kek_e), assuming all else is held constant?

  1. βL\beta_L will decrease and kek_e will decrease. (correct answer)
  2. βL\beta_L will increase and kek_e will increase.
  3. βL\beta_L will decrease and kek_e will increase.
  4. There will be no change to βL\beta_L or kek_e.
Explanation: The equity beta is given by βL=βU[1+(1−t)(D/E)]\beta_L = \beta_U [1 + (1-t)(D/E)]. When the tax rate (t) increases, the term (1−t)(1-t) decreases. This reduces the effect of leverage on the equity beta, causing βL\beta_L to decrease. Since the cost of equity, ke=Rf+βL(MRP)k_e = R_f + \beta_L(MRP), is a direct function of βL\beta_L, a lower βL\beta_L will lead to a lower estimated cost of equity, all else being equal.

Question 7

A company's stock currently trades for $40 per share. It just paid an annual dividend of $1.50 per share, and this dividend is expected to grow at a constant rate of 6% per year indefinitely. The current risk-free rate is 2.5% and the market risk premium is 5.0%. What is the implied equity beta that equates the cost of equity from the CAPM with the cost of equity from the constant-growth Dividend Discount Model (DDM)?

  1. 1.48 (correct answer)
  2. 1.59
  3. 1.88
  4. 2.00
Explanation: First, calculate the cost of equity using the DDM (Gordon Growth Model): ke=D1P0+gk_e = \frac{D_1}{P_0} + g. D_1 = D_0(1+g) = \1.50(1.06) = $1.59.So,. So, k_e = \frac{$1.59}{$40} + 0.06 = 0.03975 + 0.06 = 9.975%.Next,setthisequaltotheCAPMformulaandsolveforbeta:. Next, set this equal to the CAPM formula and solve for beta: 9.975% = 2.5% + \beta(5.0%).Thissimplifiesto. This simplifies to 7.475% = \beta(5.0%).Solvingforbeta:. Solving for beta: \beta = \frac{7.475%}{5.0%} \approx 1.495$, which is closest to 1.48.

Question 8

A company is considering two mutually exclusive projects. Project A involves manufacturing basic consumer necessities and has low operating leverage. Project B involves developing and selling luxury goods and requires a factory with high fixed costs, resulting in high operating leverage. If each project were a standalone firm, which of the following statements about their betas is most likely true?

  1. Project A would have a higher beta due to more predictable revenue streams.
  2. Project B would have a higher beta due to higher revenue cyclicality and operating leverage. (correct answer)
  3. Both projects would have the same beta, as project-specific risk is diversified away.
  4. Project B would have a lower beta because luxury goods have higher profit margins.
Explanation: A firm's asset beta is determined by the cyclicality of its revenues and its operating leverage. Luxury goods (Project B) have highly cyclical revenues that are sensitive to the state of the economy. High operating leverage (high fixed costs) further amplifies the effects of revenue fluctuations on earnings. Both factors lead to higher systematic risk and therefore a higher beta compared to a firm selling non-cyclical necessities with low operating leverage (Project A).

Question 9

An investor's portfolio consists of two assets. 70% of the portfolio is invested in Stock A, which has a beta of 1.4. The remaining 30% is invested in Stock B, which has a beta of 0.7. What is the beta of the investor's portfolio?

  1. 1.00
  2. 1.05
  3. 1.19 (correct answer)
  4. 2.10
Explanation: The beta of a portfolio is the weighted average of the betas of the individual assets in the portfolio. The formula is βp=wAβA+wBβB\beta_p = w_A \beta_A + w_B \beta_B. Plugging in the values: βp=(0.70×1.4)+(0.30×0.7)=0.98+0.21=1.19\beta_p = (0.70 \times 1.4) + (0.30 \times 0.7) = 0.98 + 0.21 = 1.19.

Question 10

An analyst is calculating the cost of equity for a firm using the Capital Asset Pricing Model (CAPM). The analyst has determined the following inputs: the risk-free rate is 3.0%, the firm's equity beta is 1.2, and the expected return on the market is 8.0%. What is the firm's estimated cost of equity?

  1. 9.0% (correct answer)
  2. 9.6%
  3. 12.6%
  4. 6.0%
Explanation: The cost of equity is calculated using the CAPM formula: ke=Rf+β(E[Rm]−Rf)k_e = R_f + \beta (E[R_m] - R_f). First, calculate the market risk premium: E[Rm]−Rf=8.0%−3.0%=5.0%E[R_m] - R_f = 8.0\% - 3.0\% = 5.0\%. Then, apply the CAPM formula: ke=3.0%+1.2×5.0%=3.0%+6.0%=9.0%k_e = 3.0\% + 1.2 \times 5.0\% = 3.0\% + 6.0\% = 9.0\%.

Question 11

When estimating beta for a publicly traded company, analysts often prefer using monthly stock returns over a five-year period rather than daily stock returns over a one-year period. The primary reason for this preference is that monthly returns:

  1. always produce a higher beta estimate, which is more conservative for valuation.
  2. are required by regulatory bodies for financial reporting and valuation disclosures.
  3. result in a higher regression R-squared, guaranteeing a more precise beta estimate.
  4. reduce the statistical noise and potential biases from non-synchronous trading. (correct answer)
Explanation: Daily returns can be subject to significant 'noise' from short-term market movements and microstructure effects like bid-ask bounce and non-synchronous trading (when a stock doesn't trade at the exact same time the market index is measured). Using a longer interval like monthly returns helps to smooth out this noise and capture the more fundamental, underlying relationship between the stock and the market, leading to a more stable and reliable beta estimate.

Question 12

A company with a stable capital structure undertakes a large, debt-financed share repurchase program. This action significantly increases its debt-to-equity ratio. Assuming the company's underlying business operations and earnings potential are unaffected, what is the most likely immediate consequence for its asset beta (βU\beta_U) and equity beta (βL\beta_L)?

  1. Asset beta will increase and equity beta will increase.
  2. Asset beta will remain unchanged and equity beta will increase. (correct answer)
  3. Asset beta will decrease and equity beta will remain unchanged.
  4. Both asset beta and equity beta will remain unchanged.
Explanation: The asset beta (unlevered beta) reflects the systematic risk of the company's assets, independent of its capital structure. Since the underlying business operations are unaffected, the asset beta should remain unchanged. The equity beta reflects both the business risk (via asset beta) and the financial risk from leverage. By increasing the debt-to-equity ratio, the financial risk borne by equity holders increases, which in turn increases the equity beta.

Question 13

A large, diversified conglomerate operates in the aviation and healthcare sectors. The conglomerate's overall equity beta is 1.2. To evaluate a potential acquisition of a new hospital chain, what would be the most appropriate beta to use in estimating the project's cost of equity?

  1. The conglomerate's equity beta of 1.2, as it reflects the firm's overall risk.
  2. A beta of 1.0, because the specific risk of the hospital can be diversified away.
  3. The average unlevered beta of publicly traded 'pure-play' hospital companies, relevered to the conglomerate's target capital structure. (correct answer)
  4. The equity beta of the conglomerate's existing healthcare division, derived from internal accounting data.
Explanation: The most appropriate method for determining the cost of capital for a project in a specific industry is the 'pure-play' method. This involves identifying publicly traded companies that operate solely in that industry (in this case, hospitals), calculating their average asset (unlevered) beta to find the industry's business risk, and then relevering that asset beta based on the capital structure that will be used for the project or the company's target capital structure.

Question 14

When applying the CAPM to estimate the cost of equity for a small, founder-owned private company, a major concern is that the founder has the majority of their personal wealth invested in the business. This situation most directly violates which core assumption of the CAPM?

  1. Investors have homogeneous expectations regarding asset returns.
  2. Investors hold fully diversified portfolios and are therefore only concerned with systematic risk. (correct answer)
  3. There are no taxes or transaction costs that affect investment decisions.
  4. Investors can borrow and lend unlimited amounts at the risk-free rate of interest.
Explanation: The CAPM assumes that all investors hold the 'market portfolio' and are fully diversified. This means they have eliminated all firm-specific (unsystematic) risk and are only compensated for bearing systematic risk, which is measured by beta. A founder with most of their wealth in their own company is highly undiversified. Consequently, they are concerned with the total risk of the firm (both systematic and unsystematic), not just its systematic risk. This violates the assumption and makes beta an incomplete measure of risk from the founder's perspective.

Question 15

The standard deviation of an individual stock's returns is 40%, while the standard deviation of the market portfolio's returns is 25%. The correlation between the stock's returns and the market's returns is 0.8. The beta of the stock is closest to:

  1. 0.50
  2. 0.80
  3. 1.28 (correct answer)
  4. 1.60
Explanation: The formula for beta based on standard deviation and correlation is βi=Cov(Ri,Rm)σm2=ρi,mσiσmσm2=ρi,mσiσm\beta_i = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^2} = \frac{\rho_{i,m} \sigma_i \sigma_m}{\sigma_m^2} = \frac{\rho_{i,m} \sigma_i}{\sigma_m}. Plugging in the given values: β=0.8×40%25%=32%25%=1.28\beta = \frac{0.8 \times 40\%}{25\%} = \frac{32\%}{25\%} = 1.28.

Question 16

The average asset beta for firms in the renewable energy sector is 0.90. A company entering this sector plans to maintain a target debt-to-equity ratio of 0.80 and is subject to a 30% corporate tax rate. What is the company's projected equity beta?

  1. 0.58
  2. 1.40 (correct answer)
  3. 1.46
  4. 1.62
Explanation: To find the levered (equity) beta, the asset beta must be relevered using the company's target capital structure and tax rate. The formula is βL=βU[1+(1−t)(D/E)]\beta_L = \beta_U [1 + (1-t)(D/E)]. Plugging in the values: βL=0.90[1+(1−0.30)(0.80)]=0.90[1+(0.70)(0.80)]=0.90[1+0.56]=0.90×1.56=1.404\beta_L = 0.90 [1 + (1-0.30)(0.80)] = 0.90 [1 + (0.70)(0.80)] = 0.90 [1 + 0.56] = 0.90 \times 1.56 = 1.404.

Question 17

An analyst is estimating the beta for a pharmaceutical company that just received FDA approval for a blockbuster drug expected to triple its revenue over the next two years. Using the company's historical stock returns from the past five years for a regression analysis against a market index will likely produce a beta estimate that is:

  1. unreliable, because the company's systematic risk profile has fundamentally changed. (correct answer)
  2. reliable, because five years of historical data provides a statistically significant sample.
  3. biased high, because pharmaceutical stocks generally become less risky after drug approval.
  4. biased low, because the market has not yet priced in the future growth from the new drug.
Explanation: Beta is a forward-looking measure of risk, but it is typically estimated using historical data. When a company undergoes a major structural change, such as the launch of a product that will fundamentally alter its size and business mix, its historical risk profile is no longer a good representation of its future risk. Therefore, the beta calculated from past returns is unreliable for estimating a forward-looking cost of equity.

Question 18

When estimating the equity risk premium (ERP) for use in the CAPM, an analyst compares a historical premium, calculated as the average excess return of a market index over government bonds for the last 100 years, with an implied premium, derived from a dividend discount model based on current market prices. The implied premium is often preferred for valuation because it:

  1. is statistically more robust due to the long time series of historical data available.
  2. is less volatile than historical premiums, which can change significantly year to year.
  3. reflects the market's current, forward-looking expectations of risk and return. (correct answer)
  4. consistently provides a lower, more conservative estimate for the cost of capital.
Explanation: Valuation is inherently forward-looking. A historical equity risk premium assumes that the future will resemble the past, which may not be true, especially if economic conditions, risk aversion, or market structures have changed. An implied premium is derived from current stock prices and expected future cash flows (dividends or earnings), so it reflects the market's current expectations and required return for bearing equity risk. This makes it a more relevant input for a forward-looking valuation.

Question 19

A company is considering two mutually exclusive projects. Project A involves manufacturing basic consumer necessities and has low operating leverage. Project B involves developing and selling luxury goods and requires a factory with high fixed costs, resulting in high operating leverage. If each project were a standalone firm, which of the following statements about their betas is most likely true?

  1. Project A would have a higher beta due to more predictable revenue streams.
  2. Project B would have a higher beta due to higher revenue cyclicality and operating leverage. (correct answer)
  3. Both projects would have the same beta, as project-specific risk is diversified away.
  4. Project B would have a lower beta because luxury goods have higher profit margins.
Explanation: A firm's asset beta is determined by the cyclicality of its revenues and its operating leverage. Luxury goods (Project B) have highly cyclical revenues that are sensitive to the state of the economy. High operating leverage (high fixed costs) further amplifies the effects of revenue fluctuations on earnings. Both factors lead to higher systematic risk and therefore a higher beta compared to a firm selling non-cyclical necessities with low operating leverage (Project A).

Question 20

An investor's portfolio consists of two assets. 70% of the portfolio is invested in Stock A, which has a beta of 1.4. The remaining 30% is invested in Stock B, which has a beta of 0.7. What is the beta of the investor's portfolio?

  1. 1.00
  2. 1.05
  3. 1.19 (correct answer)
  4. 2.10
Explanation: The beta of a portfolio is the weighted average of the betas of the individual assets in the portfolio. The formula is βp=wAβA+wBβB\beta_p = w_A \beta_A + w_B \beta_B. Plugging in the values: βp=(0.70×1.4)+(0.30×0.7)=0.98+0.21=1.19\beta_p = (0.70 \times 1.4) + (0.30 \times 0.7) = 0.98 + 0.21 = 1.19.