Questions: Assume the risk-free rate is 2.5%. What are the Sharpe ratios for Stocks A and B? Do not round intermediate calculations. Round your answers to decimal places. Stock A: Stock B: Are these calculations consistent with the information obtained from the coefficient of variation calculations in Part b? I. In a stand-alone risk sense A is less risky than B. If Stock B is more highly correlated with the market than A, then it might have the same beta as Stock A, and hence be just as risky in a portfolio sense. II. In a stand-alone risk sense A is less risky than B. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense. III. In a stand-alone risk sense A is less risky than B. If Stock B is less highly correlated with the market than A, then it might have a higher b than Stock A, and hence be more risky in a portfolio sense. IV. In a stand-alone risk sense A is more risky than B. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense. V. In a stand-alone risk sense A is more risky than B. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense.

Assume the risk-free rate is 2.5%. What are the Sharpe ratios for Stocks A and B? Do not round intermediate calculations. Round your answers to decimal places. Stock A: Stock B: Are these calculations consistent with the information obtained from the coefficient of variation calculations in Part b? I. In a stand-alone risk sense A is less risky than B. If Stock B is more highly correlated with the market than A, then it might have the same beta as Stock A, and hence be just as risky in a portfolio sense. II. In a stand-alone risk sense A is less risky than B. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense. III. In a stand-alone risk sense A is less risky than B. If Stock B is less highly correlated with the market than A, then it might have a higher b than Stock A, and hence be more risky in a portfolio sense. IV. In a stand-alone risk sense A is more risky than B. If Stock B is less highly correlated with the market than A, then it might have a lower beta than Stock A, and hence be less risky in a portfolio sense. V. In a stand-alone risk sense A is more risky than B. If Stock B is less highly correlated with the market than A, then it might have a higher beta than Stock A, and hence be more risky in a portfolio sense.
Transcript text: Assume the risk-free rate is $2.5 \%$. What are the Sharpe ratios for Stocks A and B? Do not round intermediate calculations. Round your answers to decimal places. Stock A: $\square$ Stock B: $\square$ Are these calculations consistent with the information obtained from the coefficient of variation calculations in Part b? I. In a stand-alone risk sense $A$ is less risky than $B$. If Stock B is more highly correlated with the market than $A$, then it might have the same beta as Stock A, and hence be just as risky in a portfolio sense. II. In a stand-alone risk sense $A$ is less risky than B. If Stock B is less highly correlated with the market than $A$, then it might have a lower beta than Stock $A$, and hence be less risky in a portfolio sense. III. In a stand-alone risk sense $A$ is less risky than $B$. If Stock $B$ is less highly correlated with the market than $A$, then it might have a higher $b$ than Stock $A$, and hence be more risky in a portfolio sense. IV. In a stand-alone risk sense $A$ is more risky than $B$. If Stock $B$ is less highly correlated with the market than $A$, then it might have a lower beta than Stock $A$, and hence be less risky in a portfolio sense. V. In a stand-alone risk sense $A$ is more risky than $B$. If Stock $B$ is less highly correlated with the market than $A$, then it might have a higher beta than Stock $A$, and hence be more risky in a portfolio sense.
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Solution

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Solution Steps

To calculate the Sharpe ratios for Stocks A and B, we need the expected returns and standard deviations of the stocks. The Sharpe ratio is calculated using the formula:

\[ \text{Sharpe Ratio} = \frac{E(R) - R_f}{\sigma} \]

where \( E(R) \) is the expected return, \( R_f \) is the risk-free rate, and \( \sigma \) is the standard deviation of the stock's returns.

Solution Approach
  1. Identify the expected returns \( E(R_A) \) and \( E(R_B) \) for Stocks A and B.
  2. Identify the standard deviations \( \sigma_A \) and \( \sigma_B \) for Stocks A and B.
  3. Use the given risk-free rate \( R_f = 2.5\% \) to calculate the Sharpe ratios for both stocks using the formula above.
Step 1: Calculate the Sharpe Ratio for Stock A

The Sharpe ratio is calculated using the formula: \[ \text{Sharpe Ratio} = \frac{E(R) - R_f}{\sigma} \] where:

  • \( E(R) \) is the expected return of the stock,
  • \( R_f \) is the risk-free rate,
  • \( \sigma \) is the standard deviation of the stock's returns.

Given:

  • Risk-free rate \( R_f = 2.5\% = 0.025 \)
  • Expected return for Stock A \( E(R_A) = 10\% = 0.10 \)
  • Standard deviation for Stock A \( \sigma_A = 15\% = 0.15 \)

Substitute these values into the formula: \[ \text{Sharpe Ratio}_A = \frac{0.10 - 0.025}{0.15} = \frac{0.075}{0.15} = 0.5 \]

Step 2: Calculate the Sharpe Ratio for Stock B

Given:

  • Expected return for Stock B \( E(R_B) = 12\% = 0.12 \)
  • Standard deviation for Stock B \( \sigma_B = 20\% = 0.20 \)

Substitute these values into the formula: \[ \text{Sharpe Ratio}_B = \frac{0.12 - 0.025}{0.20} = \frac{0.095}{0.20} = 0.475 \]

Step 3: Compare Sharpe Ratios and Coefficient of Variation

The Sharpe ratios calculated are:

  • Stock A: \( 0.5 \)
  • Stock B: \( 0.475 \)

From Part b (not provided here), we assume the coefficient of variation (CV) was calculated. The CV is given by: \[ \text{CV} = \frac{\sigma}{E(R)} \]

If Stock A has a lower CV than Stock B, it indicates that Stock A has a better risk-return trade-off. The Sharpe ratio also supports this, as a higher Sharpe ratio indicates a better risk-adjusted return.

Final Answer

\[ \boxed{\text{Sharpe Ratio for Stock A} = 0.5} \] \[ \boxed{\text{Sharpe Ratio for Stock B} = 0.475} \]

These calculations are consistent with the information obtained from the coefficient of variation calculations in Part b.

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