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Descriptive Statistics and Risk-Adjusted Returns - Series 66

  1. A client's portfolio is allocated 60% to an equity fund with an expected return of 12% and a standard deviation of 15%, and 40% to Treasury bills earning 2%. The T-bill return is risk-free, and the correlation between the T-bill holding and the equity fund is zero. What is the Sharpe ratio of the client's overall portfolio?
    Answer: 0.67 Sharpe ratio = (portfolio return - risk-free rate) / portfolio standard deviation. The portfolio's expected return is 60% × 12% + 40% × 2% = 8%. Because the T-bill return is risk-free, its standard deviation is zero, so the portfolio's standard deviation is 60% × 15% = 9%. The Sharpe ratio is (8% - 2%) / 9% = 0.67, the same as the equity fund's own Sharpe ratio because combining the risky fund with the risk-free asset moves the client along the capital allocation line. The 0.40 figure weights the fund's Sharpe ratio by the 60% allocation (0.6 × 0.67), or equivalently divides the portfolio's 6% excess return by the fund's 15% standard deviation. The 1.11 figure uses the equity fund's 10% excess return (12% - 2%) over the portfolio's 9% standard deviation, mixing the fund's risk premium with the portfolio's risk. The 0.58 figure omits the T-bill interest, using 7.2% (60% × 12%) as the portfolio return before subtracting the 2% risk-free rate.
  2. Which of the following statements regarding the coefficient of variation (CV) is correct?
    Answer: The CV is not a meaningful risk measure when the expected return is negative because the ratio becomes negative and loses interpretive value. The coefficient of variation is the standard deviation divided by the arithmetic mean, and it measures risk per unit of return. When the mean or expected return is negative, the CV is negative and loses its usual interpretive value, so it is not a meaningful risk-adjusted comparison. The statement that a lower CV means lower total risk is incorrect: the CV is a relative measure of risk per unit of return, not a measure of total risk. The statement using the geometric mean is incorrect because the CV uses the arithmetic mean. The statement that reverses the ratio and adds a market-relative element is also incorrect; the CV does not compare an investment with the market.
  3. An IAR is comparing two equity funds for a client's portfolio. Fund X has an expected return of 15%, a standard deviation of 18%, and a beta of 1.4. Fund Y has an expected return of 12%, a standard deviation of 12%, and a beta of 0.9. The risk-free rate is 3%. Based solely on the Sharpe ratio, which fund offers the greater return per unit of total risk, and what is that fund's Sharpe ratio?
    Answer: Fund Y, with a Sharpe ratio of 0.75 Sharpe ratio = (expected return − risk-free rate) ÷ standard deviation. For Fund X: (15% − 3%) ÷ 18% = 0.67. For Fund Y: (12% − 3%) ÷ 12% = 0.75. Fund Y offers the greater return per unit of total risk; the two betas are not used in the Sharpe ratio and are a red herring. The 0.83 for Fund X divides the gross 15% return by 18% without subtracting the risk-free rate, and the 1.00 for Fund Y makes the same error (12% ÷ 12%). Choosing Fund X with a Sharpe ratio of 0.67 selects the fund with the higher expected return even though its Sharpe ratio is lower than Fund Y's.
  4. An investment adviser is evaluating the performance of two large-cap equity funds for a client. The risk-free rate is 2%. The market return is 10%. The following statistics have been calculated for each fund over the past five years: Fund A: - Average annual return: 13.5% - Standard deviation of returns: 18% - Beta: 1.2 Fund B: - Average annual return: 12.0% - Standard deviation of returns: 15% - Beta: 0.9
    Which of the following statements correctly compares the risk-adjusted performance of Fund A and Fund B using both the Treynor ratio and the Sharpe ratio?
    Answer: Fund B has a higher Treynor ratio and a higher Sharpe ratio. The Sharpe ratio measures excess return per unit of total risk (standard deviation): (RpR_p - RfR_f) / σ. Fund A: (13.5% - 2%) / 18% = 11.5 / 18 = 0.639. Fund B: (12% - 2%) / 15% = 10 / 15 = 0.667. The Treynor ratio measures excess return per unit of systematic risk (beta): (RpR_p - RfR_f) / β. Fund A: (13.5% - 2%) / 1.2 = 11.5 / 1.2 = 9.58%. Fund B: (12% - 2%) / 0.9 = 10 / 0.9 = 11.11%. Fund B has the higher Sharpe ratio and the higher Treynor ratio. The option stating Fund A has a higher Treynor ratio but Fund B has a higher Sharpe ratio is wrong because Fund A's Treynor ratio is lower. The option stating Fund B has a higher Treynor ratio but Fund A has a higher Sharpe ratio is wrong because Fund A's Sharpe ratio is also lower. The option stating Fund A has both higher ratios is wrong on both counts. The correct answer is Fund B on both measures.
  5. A portfolio has a Sharpe ratio of 0.45 and a Sortino ratio of 0.60. Which of the following statements is most likely true about the portfolio's risk?
    Answer: The portfolio's downside deviation is lower than its standard deviation. The Sharpe ratio uses standard deviation as the denominator, while the Sortino ratio uses downside deviation. Because both ratios have the same numerator (excess return), a higher Sortino ratio (0.60) than Sharpe ratio (0.45) implies that the denominator of the Sortino ratio is smaller. Therefore, downside deviation is lower than standard deviation. The statement that standard deviation is lower than downside deviation has the relationship reversed. The positive-skewness statement is not implied; skewness does not directly determine the relationship between downside deviation and standard deviation. The negative-excess-return statement is false because a positive Sharpe ratio requires a positive excess return.
  6. An analyst is evaluating two funds using Sortino ratios. Fund A has an average annual return of 9% and a downside deviation of 5%. Fund B has an average annual return of 8% and a downside deviation of 3%. The minimum acceptable return (MAR) is 5%. Both funds are being considered for a client who is very concerned about losses below the MAR.
    Which of the following statements correctly compares the Sortino ratios of Fund A and Fund B, and which fund is more appropriate for this client?
    Answer: Fund B has the higher Sortino ratio, making it the more appropriate fund for this client's needs. The Sortino ratio measures excess return over the minimum acceptable return per unit of downside deviation. Fund A: (9% - 5%) / 5% = 4% / 5% = 0.80. Fund B: (8% - 5%) / 3% = 3% / 3% = 1.00. Fund B therefore has the higher Sortino ratio, indicating better return per unit of downside risk. Because the client is specifically concerned about losses below the MAR, Fund B is the more appropriate choice. The statements claiming Fund A has the higher Sortino ratio misstate the comparison, since 0.80 is less than 1.00; Fund B also has the lower downside deviation. The statement that Fund B has the higher ratio but Fund A is more appropriate incorrectly favors raw return over the client's stated downside-risk concern.
  7. A portfolio's expected returns are approximately normally distributed with a mean of 10% and a standard deviation of 4%. Which statement is correct?
    Answer: The probability of an annual return below 2% is approximately 2.5%. The empirical rule for a normal distribution states that about 68% of observations fall within one standard deviation of the mean and about 95% within two standard deviations. The mean is 10% and the standard deviation is 4%, so 2% is two standard deviations below the mean (10% − 2 × 4%). Because about 5% of observations fall outside two standard deviations, roughly 2.5% fall in each tail, making the probability of a return below 2% approximately 2.5%. The 6%–14% range is the one-standard-deviation band, so labeling it 95% misstates the rule; the 2%–18% range is the two-standard-deviation band, where the correct probability is 95%, not 68%. The 16% figure is the one-standard-deviation tail probability and would apply to returns above 14%, not above 18%.
  8. Which of the following statements regarding the use of standard deviation and beta as risk measures for a well-diversified portfolio is correct?
    Answer: For a well-diversified portfolio, beta is more relevant than standard deviation because unsystematic risk has been eliminated. In a well-diversified portfolio, unsystematic (company-specific) risk is effectively eliminated, leaving predominantly systematic (market) risk. Beta measures sensitivity to systematic risk, so it is the more relevant risk measure in this context. Standard deviation measures total risk; once unsystematic risk is diversified away, beta is preferred. The statement that standard deviation is the best measure because it captures both systematic and unsystematic risk is incorrect: for a diversified portfolio, unsystematic risk is no longer a significant component. The statement that standard deviation measures systematic risk while beta measures total risk reverses the two measures. The statement that beta is a measure of total risk and is not affected by diversification is incorrect because beta measures systematic risk, not total risk; diversification does not reduce beta, but the total-risk characterization is wrong.
  9. A portfolio manager is evaluating a fund using the information ratio. The fund's benchmark is the S&P 500 Index. Over the past year, the fund returned 12.5% while the benchmark returned 10.0%. The standard deviation of the fund's excess returns (tracking error) was computed to be 4.0%. The fund's beta relative to the benchmark is 1.10 and its standard deviation of returns is 18.0%. The risk-free rate is 2.0%.
    Which of the following statements about the fund's information ratio and its interpretation is most accurate?
    Answer: The information ratio is 0.625, indicating the fund generated 0.625% of active return for each 1% of active risk taken. The information ratio is calculated as active return divided by tracking error. Active return is the fund return minus the benchmark return: 12.5% - 10.0% = 2.5%. Tracking error is the standard deviation of active returns: 4.0%. Therefore, the information ratio is 2.5% / 4.0% = 0.625, meaning the fund earned 0.625% of active return for each 1% of active risk taken. The distractor stating 0.625 but describing it as excess return over the risk-free rate per unit of total risk describes the Sharpe ratio; the Sharpe ratio here would be (12.5% - 2%) / 18% = 0.583, so that is not 0.625. The distractor that states 0.139 and interprets it as active return per unit of total risk arises from dividing active return (2.5%) by the fund's total standard deviation (18%) = 0.139, incorrectly using total risk instead of active risk. The distractor that states 0.139 and interprets it as excess return over the risk-free rate per unit of active risk uses an incorrect numerator: (12.5% - 2%) / 4% = 2.625, not 0.139.
  10. An investment produced annual returns of 20%, -10%, and 30% over a three-year period. Which of the following statements regarding the investment's historical performance is correct?
    Answer: The geometric mean return of 11.98% is the appropriate measure because it reflects the compound growth rate. The geometric mean is the appropriate measure of historical performance because it reflects compound growth. The total growth factor is (1.20 × 0.90 × 1.30) = 1.404; the cube root is approximately 1.11975, so the geometric mean return is approximately 11.98%. The arithmetic mean is (20% − 10% + 30%) / 3 = 13.33%, which does not reflect compounding. The statement treating the 13.33% arithmetic mean as a compound-growth measure is incorrect; the 12.00% arithmetic mean figure is incorrect; and the 13.33% geometric mean/simple-average figure is incorrect.
  11. An IAR is evaluating the performance of a client's equity portfolio. Over the past year, the portfolio returned 14.2%. Its beta is 1.20. The risk-free rate was 2.5%, and the market returned 10.5%. Based on the capital asset pricing model, what is the portfolio's alpha?
    Answer: 2.1% Alpha measures actual return against the capital asset pricing model's required return: alpha = actual return − [risk-free rate + beta × (market return − risk-free rate)]. Here the required return is 2.5% + 1.20 × (10.5% − 2.5%) = 2.5% + 9.6% = 12.1%, so alpha = 14.2% − 12.1% = 2.1%. The 3.7% figure simply subtracts the market's 10.5% return from the portfolio's 14.2% return, ignoring beta and the risk-free rate. The 1.6% figure applies beta to the full market return (1.20 × 10.5% = 12.6%) but never adds the risk-free rate. The −0.9% figure treats the market's 10.5% return as if it were the equity risk premium, computing 2.5% + 1.20 × 10.5% = 15.1% as the required return.
  12. A portfolio produced annual returns of 3%, 5%, 7%, 9%, and 30% over the past five years. Which statement correctly describes the distribution of these returns?
    Answer: The distribution is positively skewed; the mean return is 10.8% and the median is 7%. With five observations, the median is the middle value after sorting: 3%, 5%, 7%, 9%, 30%, so the median is 7%. The mean is (3% + 5% + 7% + 9% + 30%) ÷ 5 = 10.8%. The 30% return stretches the right tail, making the distribution positively (right) skewed, and in a positively skewed distribution the mean exceeds the median. The 8% median comes from averaging the two middle observations (7% and 9%), which applies the even-number-of-observations rule to an odd count. The statements calling the distribution negatively skewed reverse the direction of the skew; a negative (left) skew would pull the mean below the median, the opposite of what this data shows.
  13. An investment adviser is comparing two portfolios for a client. Portfolio X has an expected return of 14%, a beta of 1.2, and a standard deviation of 16%. Portfolio Y has an expected return of 11%, a beta of 0.8, and a standard deviation of 12%. The risk-free rate is 3%. A client is considering adding one of these portfolios to a well-diversified portfolio that already contains many stocks.
    Based on the client's situation, which portfolio is more appropriate, and which performance measure is the most relevant for this decision?
    Answer: Portfolio Y, because it has a higher Treynor ratio, and the Treynor ratio is most relevant for a well-diversified portfolio. Because the client already holds a well-diversified portfolio, unsystematic risk has been largely eliminated, so the relevant risk measure is systematic risk (beta). The Treynor ratio uses beta and is therefore the most appropriate measure for evaluating a portfolio to be added to a well-diversified portfolio. Portfolio X's Treynor ratio is (14% - 3%) / 1.2 = 9.17%. Portfolio Y's Treynor ratio is (11% - 3%) / 0.8 = 10.0%. Portfolio Y has the higher Treynor ratio, so it is the more appropriate addition. The Sharpe ratio uses total risk (standard deviation) and is more relevant when evaluating a portfolio as a standalone holding. Portfolio X has the higher Sharpe ratio (11% / 16% = 0.6875 vs. 8% / 12% = 0.6667), but that measure is not the most relevant for this decision. Selecting Portfolio X on the basis of a higher Treynor ratio is incorrect because Portfolio X's Treynor ratio is lower. Selecting Portfolio X on the basis of the Sharpe ratio is incorrect because the Sharpe ratio is not the most relevant measure here, even though X's Sharpe ratio is higher. Selecting Portfolio Y on the basis of the Sharpe ratio is incorrect because Portfolio Y's Sharpe ratio is lower than Portfolio X's and the Sharpe ratio is not the most relevant measure for a well-diversified investor adding a portfolio.
  14. An IAR is reviewing a client's annual returns for the past five years: 5%, 5%, 10%, 15%, and 15%. What is the sample standard deviation of these returns?
    Answer: 5% The sample standard deviation is the square root of the sum of squared deviations from the mean divided by n − 1. The mean of 5%, 5%, 10%, 15%, and 15% is 10%; the deviations are −5%, −5%, 0%, 5%, and 5%, and the squared deviations sum to 100 (25 + 25 + 0 + 25 + 25). Dividing 100 by 4 (n − 1) gives a sample variance of 25, and the square root is 5%. The 25% figure is the variance with the square root not taken. The 4.5% figure divides the sum of squared deviations by 5 rather than 4, applying the population variance formula to a sample (√20 = 4.47). The 10% figure is the range — the highest return minus the lowest — not the standard deviation.
  15. A portfolio has a beta of 1.5 and an expected return of 16%. The market portfolio has an expected return of 11%. The risk-free rate is 4%.
    Which of the following statements about the portfolio's Jensen's alpha and its interpretation is most accurate?
    Answer: The portfolio's alpha is 1.5%, indicating the portfolio outperformed the market on a risk-adjusted basis. Jensen's alpha compares the portfolio's return with the CAPM required return: required return = 4% + 1.5(11% - 4%) = 14.5%. The portfolio's alpha is 16% - 14.5% = 1.5%. Because alpha is positive, the portfolio's expected return exceeds the return required for its systematic risk, so it outperformed on a risk-adjusted basis. The 1.5% underperformance statement misstates the sign convention for alpha. The 5.0% statements come from comparing the portfolio's return directly with the market's 11% expected return (16% - 11% = 5%); that is a raw-return comparison, not Jensen's alpha, because it ignores beta and the risk-free rate. The only accurate statement is a 1.5% alpha with outperformance.

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