Reading 7: Statistical Concepts and Market Returns-LOS i 习题 2011, face, interpret, variation, center

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Session 2: Quantitative Methods: Basic Concepts
Reading 7: Statistical Concepts and Market Returns

LOS i: Define, calculate, and interpret the coefficient of variation and the Sharpe ratio.

 

 

Given a population of 200, 100, and 300, the coefficient of variation is closest to:

A) 30%.

B) 100%.

C) 40%.

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CV = (σ/mean)
mean = (200 + 100 + 300)/3 = 200
σ = √[(200 - 200)² + (100 - 200)² + (300 - 200)² / 3] = √6666.67 = 81.65
(81.65/200) = 40.82%

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The mean monthly return on U.S. Treasury bills (T-bills) is 0.42%. The mean monthly return for an index of small stocks is 4.56%, with a standard deviation of 3.56%. What is the Sharpe measure for the index of small stocks?

A) 16.56%.

B) 10.60%.

C) 1.16%.

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The Sharpe ratio measures excess return per unit of risk. (4.56 – 0.42) / 3.56 = 1.16%.

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Which of the following statements regarding the Sharpe ratio is most accurate? The Sharpe ratio measures:

A) peakedness of a return distrubtion.

B) excess return per unit of risk.

C) total return per unit of risk.

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The Sharpe ratio measures excess return per unit of risk. Remember that the numerator of the Sharpe ratio is (portfolio return ? risk free rate), hence the importance of excess return. Note that peakedness of a return distribution is measured by kurtosis.

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Portfolio A earned an annual return of 15% with a standard deviation of 28%. If the mean return on Treasury bills (T-bills) is 4%, the Sharpe ratio for the portfolio is:

A) 0.39.

B) 0.54.

C) 1.87.

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(15 ? 4) / 28 = 0.39

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Johnson Inc. manages a growth portfolio of equity securities that has had a mean monthly return of 1.4% and a standard deviation of returns of 10.8%. Smith Inc. manages a blended equity and fixed income portfolio that has had a mean monthly return of 1.2% and a standard deviation of returns of 6.8%. The mean monthly return on Treasury bills has been 0.3%. Based on the Sharpe ratio, the:

A) Johnson and Smith portfolios have exhibited the same risk-adjusted performance.

B) performance of the Johnson portfolio is preferable to the performance of the Smith portfolio.

C) performance of the Smith portfolio is preferable to the performance of the Johnson portfolio.

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The Sharpe ratio for the Johnson portfolio is (1.4 - 0.3)/10.8 = 0.1019.

The Sharpe ratio for the Smith portfolio is (1.2 - 0.3)/6.8 = 0.1324.

The Smith portfolio has the higher Sharpe ratio, or greater excess return per unit of risk.

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A higher Sharpe ratio indicates:

A) lower volatility of returns.

B) a higher excess return per unit of risk.

C) a lower risk per unit of return.

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The Sharpe ratio is excess return (return ? R_f) per unit of risk (defined as the standard deviation of returns).

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A portfolio has a return of 14.2% and a Sharpe’s measure of 3.52. If the risk-free rate is 4.7%, what is the standard deviation of returns?

A) 2.7%.

B) 3.9%.

C) 2.6%.

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Standard Deviation of Returns = (14.2% – 4.7%) / 3.52 = 2.6988.

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Portfolio A earned a return of 10.23% and had a standard deviation of returns of 6.22%. If the return over the same period on Treasury bills (T-bills) was 0.52% and the return to Treasury bonds (T-bonds) was 4.56%, what is the Sharpe ratio of the portfolio?

A) 1.56.

B) 0.56.

C) 0.91.

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Sharpe ratio = (R_p – R_f) / σ_p, where (R_p – R_f) is the difference between the portfolio return and the risk free rate, and σ_p is the standard deviation of portfolio returns. Thus, the Sharpe ratio is: (10.23 – 0.52) / 6.22 = 1.56. Note, the T-bill rate is used for the risk free rate.

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If stock X's expected return is 30% and its expected standard deviation is 5%, Stock X's expected coefficient of variation is:

A) 6.0.

B) 0.167.

C) 1.20.

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The coefficient of variation is the standard deviation divided by the mean: 5 / 30 = 0.167.

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What is the coefficient of variation for a distribution with a mean of 10 and a variance of 4?

A) 40%.

B) 25%.

C) 20%.

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Coefficient of variation, CV = standard deviation / mean. The standard deviation is the square root of the variance, or 4^? = 2. So, CV = 2 / 10 = 20%.