READING 3 STATISTICAL MEASURES OF ASSET RETURNS

Question 1

Which measure of central tendency is LEAST affected by outliers in a dataset?
A. Arithmetic mean
B. Median
C. Mode

Answer 1

Correct answer: B

The median is least affected by outliers because it represents the middle value of an ordered dataset, making it robust against extreme values.
A is incorrect the arithmetic mean is heavily influenced by outliers as it incorporates all values in its calculation, including extreme ones.
C is incorrect the mode can be affected by outliers if they create a new frequency pattern in the dataset.

Question 2

A dataset with two values that occur with equal highest frequency is best described as:
A. Unimodal
B. Bimodal
C. Trimodal

Answer 2

Correct Answer: B

A bimodal distribution has two values that occur most frequently.
A is incorrect a unimodal distribution has only one value that appears most frequently.
C is incorrect a trimodal distribution has three values that occur most frequently.

Question 3

The key advantage of using a trimmed mean rather than a simple arithmetic mean is that it:
A. Increases the importance of extreme values
B. Eliminates a stated percentage of extreme observations
C. Substitutes the median for extreme values

Answer 3

Correct Answer: B

A trimmed mean excludes a stated percentage of extreme observations (both high and low), making it more robust.
A is incorrect trimmed means do not increase the importance of extreme values; they do the opposite by removing them.
C is incorrect trimmed means eliminate extreme values rather than substituting them with the median (which would be a winsorized mean).

Question 4

A winsorized mean differs from a trimmed mean in that it:
A. Discards extreme observations entirely
B. Substitutes a value for extreme observations
C. Only considers the median value

Answer 4

B is correct. A winsorized mean substitutes extreme values with less extreme ones (typically using percentiles).
A is incorrect discarding extreme observations describes a trimmed mean, not a winsorized mean.
C is incorrect a winsorized mean still uses all observations, not just the median value.

Question 5

Which of the following BEST describes a percentile in a distribution?
A. The percentage of observations that fall at a specific value
B. The value below which a stated proportion of observations fall
C. The percentage difference between consecutive observations

Answer 5

B is correct. A percentile is the value at or below which a specified percentage of observations fall.
A is incorrect. This describes frequency rather than a percentile.
C is incorrect. This describes a measure of relative differences, not a percentile.

Question 6

The interquartile range is calculated as:
A. The difference between the 25th and 75th percentiles
B. The difference between the highest and lowest values
C. The average of the 25th and 75th percentiles

Answer 6

A is correct. The interquartile range is the difference between the third quartile (75th percentile) and the first quartile (25th percentile).
B is incorrect. This describes the range, not the interquartile range.
C is incorrect. The interquartile range is calculated as a difference, not an average.

Question 7

When comparing dispersion across different datasets with varying means, the most appropriate measure to use is:
A. Standard deviation
B. Variance
C. Coefficient of variation (CV)

Answer 7

C is correct. The coefficient of variation allows for meaningful comparison across datasets with different means as it expresses standard deviation relative to the mean.
A is incorrect. Standard deviation is expressed in the same units as the data and doesn’t account for differences in means, making comparisons difficult.
B is incorrect. Variance is expressed in squared units and doesn’t facilitate easy comparison across different datasets with varying means.

Question 8

In investment analysis, a lower coefficient of variation indicates:
A. Higher risk per unit of expected return
B. Lower risk per unit of expected return
C. No relationship to risk-return characteristics

Answer 8

B is correct. A lower CV indicates less dispersion (risk) per unit of expected return (mean).
A is incorrect. A higher CV, not a lower one, would indicate higher risk per unit of return.
C is incorrect. CV is directly related to risk-return characteristics in investment analysis.

Question 9

The standard deviation differs from variance in that it:
A. Is expressed in the original units of the data
B. Is not affected by outliers
C. Measures central tendency rather than dispersion

Answer 9

A is correct. Standard deviation is the square root of variance, expressed in the same units as the original data.
B is incorrect. Both standard deviation and variance are affected by outliers.
C is incorrect. Standard deviation, like variance, measures dispersion, not central tendency.

Question 10

The primary purpose of calculating the mode for investment returns data is to:
A. Determine the exact middle value of returns
B. Identify the most common return value
C. Calculate the average expected return

Answer 10

B is correct. The mode identifies the most frequently occurring value in a dataset.
A is incorrect. The median determines the middle value, not the mode.
C is incorrect. The arithmetic mean calculates the average expected return, not the mode.

Question 11

For continuous data like investment returns, analysts typically use which of the following instead of a single mode?
A. Trimmed mean
B. Modal interval
C. Median value

Answer 11

B is correct. For continuous data, a modal interval (the interval containing the largest number of observations) is used.
A is incorrect. A trimmed mean addresses outliers but doesn’t identify frequency patterns.
C is incorrect. The median identifies the middle value but not frequency patterns.

Question 12

When the arithmetic mean is greater than the median in a distribution, this typically indicates:
A. A left-skewed (negatively skewed) distribution
B. A right-skewed (positively skewed) distribution
C. A perfectly symmetric distribution

Answer 12

B is correct. When the mean exceeds the median, the distribution typically has a longer right tail (positive skew).
A is incorrect. Left-skewed distributions have means less than medians, not greater.
C is incorrect. In symmetric distributions, the mean equals the median.

Question 13

The primary benefit of using the median rather than the mean when analyzing investment returns is that:
A. It always provides a higher value
B. It is not affected by extreme values or outliers
C. It accounts for compounding effects

Answer 13

B is correct. The median is not affected by outliers, making it useful when analyzing returns with extreme values.
A is incorrect. The median is not consistently higher or lower than the mean.
C is incorrect. The median does not account for compounding effects in returns.

Question 14

Which of these measures is most appropriate for making inferences about the population mean?
A. Sample median
B. Sample mode
C. Sample mean

Answer 14

C is correct. The sample mean is used to make inferences about the population mean.
A is incorrect. The sample median is less commonly used for inferring the population mean.
B is incorrect. The sample mode typically isn’t used for making inferences about population parameters.

Question 15

The relationship between variance and standard deviation is best described as:
A. Standard deviation is the cube root of variance
B. Standard deviation is the square root of variance
C. Standard deviation is the logarithm of variance

Answer 15

B is correct. Standard deviation is calculated as the square root of variance.
A is incorrect. Standard deviation is not the cube root of variance.
C is incorrect. Standard deviation is not the logarithm of variance.

Question 16

A box and whisker plot is primarily used to visualize:
A. The mean and standard deviation
B. Quartiles and potential outliers
C. Probability distributions

Answer 16

B is correct. Box and whisker plots display quartiles (the “box”) and extreme values (the “whiskers”), helping identify outliers.
A is incorrect. They don’t specifically highlight mean and standard deviation.
C is incorrect. While related to distributions, they focus on displaying data spread rather than probability functions.

Question 17

In a dataset of investment returns, the modal interval is defined as:
A. The interval containing the median return
B. The interval containing the largest number of observations
C. The interval between the highest and lowest returns

Answer 17

B is correct. The modal interval contains the largest number of observations.
A is incorrect. The median interval would contain the middle value, not necessarily the most frequent one.
C is incorrect. The range describes the interval between highest and lowest values.

Question 18

The coefficient of variation (CV) is calculated by:
A. Dividing the standard deviation by the mean
B. Dividing the mean by the standard deviation
C. Multiplying the standard deviation by the mean

Answer 18

A is correct. CV = Standard Deviation ÷ Mean
B is incorrect. This is the inverse of the CV formula.
C is incorrect. Multiplying standard deviation by mean does not produce the CV.

Question 19

When comparing the risk-adjusted returns of two investments, which statement is correct based on their coefficients of variation?
A. The investment with the higher CV offers better risk-adjusted returns
B. The investment with the lower CV offers better risk-adjusted returns
C. CV cannot be used to compare risk-adjusted returns

Answer 19

B is correct. A lower CV indicates less risk (standard deviation) per unit of return (mean).
A is incorrect. A higher CV indicates more risk per unit of return, which is less desirable.
C is incorrect. CV is specifically used to compare risk-adjusted performance across investments.

Question 20

Which of the following statements about a symmetrical distribution is MOST accurate?
A. The mean is greater than both the median and the mode
B. The mean, median, and mode are all equal
C. Outliers occur with equal frequency in both tails, but the mean is affected more by positive outliers

Answer 20

Correct Answer: B

In a symmetrical distribution, the mean, median, and mode are all equal as the distribution is shaped identically on both sides of its central point.
Option A describes a positively skewed distribution where outliers pull the mean to the right of the median and mode.
Option C is incorrect because while outliers do occur with equal frequency in both tails of a symmetrical distribution, they affect the mean equally in both directions, not more in the positive direction.

Question 21

A distribution with a negative skewness is BEST characterized by:
A. The mean being less than the median, which is less than the mode
B. A relatively long upper (right) tail with positive outliers
C. The median being exactly in the middle between the mean and mode

Answer 21

Correct Answer: A

In a negatively skewed distribution, the mean is less than the median, which is less than the mode, as negative outliers pull the mean downward (to the left).
Option B describes a positively skewed distribution with a long right tail.
Option C is incorrect because while the median is between the mean and mode in skewed distributions, it is not necessarily exactly in the middle.

Question 22

In the context of evaluating investment returns, which measure is MOST important to consider when a distribution has fat tails?
A. Standard deviation
B. Kurtosis
C. Median

Answer 22

Correct Answer: B

Kurtosis measures the “tailedness” of a distribution and is most important when evaluating distributions with fat tails, as it specifically addresses the likelihood of extreme values.
Standard deviation, while important, does not specifically address the frequency of extreme outcomes in the tails.
The median, as a measure of central tendency, does not provide information about the tails of a distribution.

Question 23

What does a leptokurtic distribution indicate about investment risk compared to a normal distribution?
A. Lower risk due to fewer extreme values
B. Higher risk due to more returns clustered around the mean
C. Higher risk due to greater probability of extreme deviations

Answer 23

Correct Answer: C

A leptokurtic distribution has fatter tails than a normal distribution, indicating a higher probability of extreme deviations from the mean, which translates to higher investment risk.
Option A is incorrect because leptokurtic distributions have more extreme values, not fewer.
Option B is partially correct in describing the clustering around the mean, but this alone would suggest lower risk, not higher risk. The higher risk comes from the greater frequency of extreme values.

Question 24

In a positively skewed distribution of investment returns, which of the following is MOST likely to be true?
A. Most returns are above the mean
B. Large negative outliers tend to pull the mean downward
C. The mode is less than the median, which is less than the mean

Answer 24

Correct Answer: C

In a positively skewed distribution, the mode is less than the median, which is less than the mean. This occurs because large positive outliers pull the mean upward (to the right).
Option A is incorrect because in a positively skewed distribution, most returns are actually below the mean.
Option B describes a negatively skewed distribution, where large negative outliers pull the mean downward.

Question 25

The property that correlation ranges from -1 to +1 is important primarily because it: A. Allows comparisons between relationships of different variables regardless of their units B. Ensures that covariance is always positive C. Guarantees that variables with high correlation have causal relationships

Answer 25

Correct Answer: A Correlation being bounded between -1 and +1 standardizes the measure of relationship strength, allowing meaningful comparisons between different pairs of variables regardless of their units. Option B is incorrect because covariance can be negative, zero, or positive. Option C is incorrect because correlation does not imply causation, regardless of its strength.

Question 26

What does a platykurtic distribution indicate about the concentration of observations relative to a normal distribution? A. More observations are concentrated around the mean B. More observations are in the tails C. More observations are in the shoulders of the distribution

Answer 26

Correct Answer: C A platykurtic distribution has more observations in the "shoulders" of the distribution and fewer in both the tails and center compared to a normal distribution. Option A describes a leptokurtic distribution, which has more observations concentrated around the mean. Option B is incorrect because platykurtic distributions have fewer observations in the tails than a normal distribution, not more.

Question 27

An analyst notes that a returns distribution has excess kurtosis of -0.5. This observation MOST likely indicates: A. The distribution is positively skewed B. The distribution is platykurtic C. The distribution has a higher peak than a normal distribution

Answer 27

Correct Answer: B Excess kurtosis of -0.5 (less than zero) indicates a platykurtic distribution, which has flatter shoulders and thinner tails than a normal distribution. Option A is incorrect because kurtosis and skewness are separate measures; negative excess kurtosis does not indicate positive skew. Option C is incorrect because a negative excess kurtosis indicates a lower peak, not a higher one, compared to a normal distribution.

Question 28

Which of the following combinations of skewness and kurtosis would indicate the HIGHEST potential risk for an investment? A. Positive skewness and leptokurtic distribution B. Negative skewness and leptokurtic distribution C. Zero skewness and mesokurtic distribution

Answer 28

Correct Answer: B Negative skewness combined with a leptokurtic distribution indicates higher risk because there are more extreme negative outcomes (from negative skew) and a higher probability of extreme values in general (from leptokurtosis). Option A presents less risk from a downside perspective because the positive skew indicates more extreme positive outcomes. Option C describes a normal distribution (zero skewness and mesokurtic), which has less risk than the other options.

Question 29

The primary limitation of using correlation as a measure of relationship between two variables is that: A. It only measures linear relationships B. It is affected by the units of measurement C. It cannot be negative

Answer 29

Correct Answer: A Correlation only measures the strength of linear relationships between variables and may miss non-linear relationships. Option B is incorrect because correlation is standardized and is not affected by units of measurement (unlike covariance). Option C is incorrect because correlation can range from -1 to +1, including negative values.

Question 30

Spurious correlation is BEST described as: A. A statistical error that occurs when calculating covariance B. A relationship between variables that appears significant but is actually due to chance or an unmeasured third variable C. A correlation coefficient that exceeds +1 or is less than -1

Answer 30

Correct Answer: B Spurious correlation refers to an apparent relationship between variables that is misleading because it is caused by chance or by an unmeasured third variable, not by a direct causal relationship. Option A is incorrect because spurious correlation is not a calculation error. Option C is incorrect because correlation coefficients cannot exceed +1 or fall below -1 by definition.

Question 31

When a distribution of investment returns is described as mesokurtic, this means: A. The distribution has the same kurtosis as a normal distribution B. The mean equals the median C. The distribution has no skewness

Answer 31

Correct Answer: A A mesokurtic distribution has the same kurtosis as a normal distribution, with excess kurtosis equal to zero. Option B is incorrect because "mesokurtic" refers to the peakedness of the distribution, not the relationship between mean and median. Option C is incorrect because kurtosis and skewness are separate concepts; a mesokurtic distribution can still be skewed.

Question 31

In risk management, why are the shape characteristics of return distributions important? A. They determine whether mean-variance analysis is sufficient for portfolio optimization B. They affect the calculation of standard deviation C. They inform decisions about the potential for extreme outcomes not captured by standard deviation alone

Answer 31

Correct Answer: C The shape characteristics of return distributions (skewness and kurtosis) provide information about the likelihood of extreme outcomes that may not be fully captured by standard deviation alone. Option A is partially correct, but the primary importance is the information about extreme risks, not just portfolio optimization techniques. Option B is incorrect because standard deviation calculation does not depend on skewness or kurtosis.

Question 32

The relationship between mean, median, and mode in a symmetrical unimodal distribution is BEST described as: A. Mean > Median > Mode B. Mean = Median = Mode C. Mode > Median > Mean

Answer 32

Correct Answer: B In a symmetrical unimodal distribution, the mean, median, and mode are all equal and located at the center of the distribution. Option A describes a positively skewed distribution. Option C describes a negatively skewed distribution.

Question 32

Which of the following statements about covariance is MOST accurate? A. Covariance is always a value between -1 and +1 B. Covariance values cannot be compared across different pairs of variables with different units C. A positive covariance implies a causal relationship between variables

Answer 32

Correct Answer: B Covariance values depend on the units of the variables being measured, making it difficult to compare covariance across different pairs of variables with different units. Option A is incorrect because correlation, not covariance, is bounded between -1 and +1. Option C is incorrect because covariance (like correlation) measures association but does not imply causation.

Question 33

When interpreting a scatter plot between two variables, a key pattern that would indicate a strong non-linear relationship is: A. Points forming a clear straight line B. Points showing a clear pattern but not along a straight line C. Points randomly dispersed with no discernible pattern

Answer 33

Correct Answer: B A scatter plot showing a clear pattern that is not a straight line indicates a strong non-linear relationship between variables. Option A indicates a strong linear relationship, not a non-linear one. Option C indicates no relationship between the variables.

Question 34

In the context of investment returns, outliers are MOST important to analyze because they: A. Always indicate data collection errors B. Represent potential extreme gains or losses that may significantly impact performance C. Are usually removed before statistical analysis

Answer 34

Correct Answer: B Outliers in investment returns represent potential extreme gains or losses that may significantly impact overall performance and risk assessment. Option A is incorrect because outliers are not always errors; they can represent real but rare events. Option C is incorrect because while outliers may sometimes be examined separately, they should not be routinely removed, especially in investment analysis, as they contain valuable information about potential extreme outcomes.

Question 35

The key difference between skewness and kurtosis in describing return distributions is that: A. Skewness measures asymmetry, while kurtosis measures tailedness B. Skewness affects the mean, while kurtosis affects the standard deviation C. Skewness is always calculated, while kurtosis is only relevant for non-normal distributions

Answer 35

Correct Answer: A Skewness measures the asymmetry of a distribution (whether it's tilted to the left or right), while kurtosis measures the tailedness (whether extreme values are more or less likely than in a normal distribution). Option B is incorrect because while skewness does affect the relationship between mean and median, kurtosis doesn't directly affect standard deviation. Option C is incorrect because both measures are relevant for all distributions, not just non-normal ones.

Question 36

Which of the following statements about measures of central tendency in skewed distributions is MOST accurate? A. The mean is the most representative measure regardless of skewness B. The median is generally more representative than the mean in skewed distributions C. The mode is always the best measure to use in highly skewed distributions

Answer 36

Correct Answer: B The median is generally more representative than the mean in skewed distributions because it is not influenced by extreme values or outliers. Option A is incorrect because the mean is heavily influenced by outliers and can be misleading in skewed distributions. Option C is incorrect because while the mode can be useful, it is not always the best measure, particularly in multimodal distributions.

Question 36

When examining a correlation coefficient of 0.98 between two variables, an analyst should be MOST concerned about: A. The possibility of spurious correlation B. The potential for a non-linear relationship C. The statistical significance of the relationship

Answer 36

Correct Answer: A With an extremely high correlation coefficient of 0.98, an analyst should be concerned about spurious correlation—a relationship that appears strong but may be due to chance or an unmeasured third variable. Option B is incorrect because such a high correlation almost certainly indicates a linear relationship. Option C is less concerning than spurious correlation because such a high correlation value would almost certainly be statistically significant.

Question 36

How does excess kurtosis of securities returns affect risk management strategies? A. Positive excess kurtosis indicates returns are more predictable B. Positive excess kurtosis suggests more emphasis should be placed on tail risk C. Negative excess kurtosis requires more conservative position sizing

Answer 36

Correct Answer: B Positive excess kurtosis (leptokurtic distribution) indicates fatter tails than a normal distribution, suggesting risk managers should place more emphasis on managing tail risk and extreme events. Option A is incorrect because positive excess kurtosis actually indicates less predictability due to more extreme values. Option C is incorrect because negative excess kurtosis (platykurtic) implies fewer extreme values, not necessarily requiring more conservative positioning.

Question 37

When analyzing an investment with both positive skewness and high kurtosis, which risk characteristic is MOST likely to be true? A. The investment has a higher probability of extreme positive returns than would be expected in a normal distribution B. The investment returns are primarily clustered around the mean with few outliers C. The investment has symmetrical risk with equal probabilities of positive and negative extreme returns

Answer 37

Correct Answer: A An investment with positive skewness and high kurtosis has a higher probability of extreme positive returns than would be expected in a normal distribution, as positive skewness indicates more extreme positive values and high kurtosis indicates fatter tails overall. Option B is incorrect because high kurtosis indicates more outliers, not fewer. Option C is incorrect because positive skewness indicates asymmetrical risk, not symmetrical risk.

Question 38

The primary reason that most risk managers focus more on the distribution of returns in the tails rather than the mean and standard deviation is: A. The mean and standard deviation are less accurate measures B. Normal distribution assumptions often fail to account for the frequency of extreme events C. Tail events have no impact on long-term performance

Answer 38

Correct Answer: B Risk managers focus on tail distributions because normal distribution assumptions (which are defined by mean and standard deviation) often fail to account for the actual frequency of extreme events in financial markets. Option A is incorrect because mean and standard deviation are accurate measures, but they're insufficient for fully describing non-normal distributions. Option C is incorrect because tail events can have significant impacts on long-term performance.

Question 39

When analyzing the correlation between two variables, what is the primary concern when outliers are present in the dataset? A. Outliers make the calculation of correlation impossible B. Outliers may significantly reduce the calculated correlation C. Outliers may create the appearance of correlation where none truly exists in the main data

Answer 39

Correct Answer: C The primary concern with outliers when analyzing correlation is that they may create the appearance of correlation where none truly exists in the main body of data, leading to spurious conclusions. Option A is incorrect because correlation can still be calculated when outliers are present. Option B is incorrect because outliers can either increase or decrease correlation, depending on their position relative to the overall pattern.