burning0spear

In a skewed distribution, what is the minimum proportion of observations between +/- two standard deviations from the mean?

A) 95%.

B) 84%.

C) 75%.

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For any distribution we can use Chebyshev’s Inequality, which states that the proportion of observations within k standard deviations of the mean is at least 1 – (1 / k2).
1 – (1 / 22) = 0.75, or 75%.
Note that for a normal distribution, 95% of observations will fall between +/- 2 standard deviations of the mean.

burning0spear

Regardless of the shape of a distribution, according to Chebyshev’s Inequality, what is the minimum percentage of observations that will lie within +/– two standard deviations of the mean?

A) 68%.

B) 89%.

C) 75%.

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According to Chebyshev’s Inequality, for any distribution, the minimum percentage of observations that lie within k standard deviations of the distribution mean is equal to:
1 – (1 / k2), with k equal to the number of standard deviations. If k = 2, then the percentage of distributions is equal to 1 – (1 / 4) = 75%.

burning0spear

In a skewed distribution, what is the minimum amount of observations that will fall between +/- 1.5 standard deviations from the mean?

A) 44%.

B) 56%.

C) 95%.

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Because the distribution is skewed, we must use Chebyshev’s Inequality, which states that the proportion of observations within k standard deviations of the mean is at least 1 – (1 / k2).
1 – (1 / 1.52) = 0.5555, or 56%.

burning0spear

According to Chebyshev’s Inequality, for any distribution, what is the minimum percentage of observations that lie within three standard deviations of the mean?

A) 94%.

B) 89%.

C) 75%.

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According to Chebyshev’s Inequality, for any distribution, the minimum percentage of observations that lie within k standard deviations of the distribution mean is equal to: 1 – (1 / k2). If k = 3, then the percentage of distributions is equal to 1 – (1 / 9) = 89%.

burning0spear

Which of the following statements about statistical concepts is least accurate?

A) The coefficient of variation is useful when comparing dispersion of data measured in different units or having large differences in their means.

B) For a normal distribution, only 95% of the observations lie within ±3 standard deviations from the mean.

C) For any distribution, based on Chebyshev’s Inequality, 75% of the observations lie within ±2 standard deviations from the mean.

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For a normal distribution, 95% of the observations lie within ±2 standard deviations of the mean while 99% of the observations lie within plus or minus three standard deviations of the mean. Both remaining statements are true. Note that 75% of observations for any distribution lie within ±2 standard deviations of the mean using Chebyshev’s inequality.

burning0spear

A higher Sharpe ratio indicates:

A) a higher excess return per unit of risk.

B) lower volatility of returns.

C) a lower risk per unit of return.

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

burning0spear

A portfolio of options had a return of 22% with a standard deviation of 20%. If the risk-free rate is 7.5%, what is the Sharpe ratio for the portfolio?

A) 0.725.

B) 0.568.

C) 0.147.

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Sharpe ratio = (22% – 7.50%) / 20% = 0.725.

burning0spear

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

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

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

burning0spear

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) 0.54.

B) 1.87.

C) 0.39.

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(15 − 4) / 28 = 0.39

burning0spear

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

A) excess return per unit of risk.

B) peakedness of a return distrubtion.

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.