Quant - Stat. Measures of Asset Returns

Question 1

How do you calculate the arithmetic mean?

Answer 1

The arithmetic mean is the sum of the observations divided by the number of observations.

Question 1

What are three “measures of central tendency” and what is the purpose of a “measure of central tendency”?

Answer 1

Measures of central tendency include the mean, the median and the mode, and specify where data are centered.

Question 2

How do you calculate the median value?

Answer 2

The median is the value of the middle item of observations, or the mean of the values of the two middle items, when the items in a set are sorted into ascending or descending order.

Question 3

When is the median more useful than the mean?

Answer 3

Since the median is not influenced by extreme values, it is most useful in the case of skewed distributions.

Question 4

How do you calculate the mode?

Answer 4

The mode is the most frequently observed value and is the only measure of central tendency that can be used with nominal or categorical data. A distribution may be unimodal (one mode), bimodal (two modes), trimodal (three modes), or have even more modes.

Question 5

What are quartiles?

Answer 5

Quantiles, as the median, quartiles, quintiles, deciles, and percentiles, are location parameters that divide a distribution into halves, quarters, fifths, tenths, and hundredths, respectively.

Question 6

How do you calculate the first, second, and third quartiles given a data set?

Answer 6

To calculate the quartiles, you first arrange your data in ascending order.

Q1 can be calculated by finding the median of the first half of your dataset.

Q2 is the median of the dataset.

Q3 is found by calculating the median of the second half of your dataset.

If the dataset has an odd number of observations, the median itself is typically excluded from the halves when finding Q1 and Q3.

Question 7

What does the “Box” in a “Box and Whiskers Plot” represent?

Answer 7

A box and whiskers plot illustrates the distribution of a set of observations. The “box” depicts the interquartile range, the difference between the first and the third quartile. The “whiskers” outside of the “box” indicate the others measures of dispersion.

Question 8

What is “dispersion”?

Answer 8

In statistics, dispersion (or variability) refers to how spread out or scattered the values in a data set are. It’s a measure of how much the data points differ from each other and from the central tendency (mean, median, or mode) of the distribution.

Question 9

What are some measures of dispersion?

Answer 9

range
variance
standard deviation
coefficient of variation

Question 10

What is “range” and how do you calculate it?

Answer 10

Range: The simplest measure of dispersion, which is the difference between the maximum and minimum values in a data set. It gives a quick sense of the breadth of the values but doesn’t account for how the data is distributed between these extremes.

Question 11

What is “inter quartile range” and how do you calculate it?

Answer 11

Interquartile Range (IQR): This measure focuses on the middle 50% of the data, calculated as the difference between the third quartile (Q3) and the first quartile (Q1). It helps mitigate the effect of outliers and provides a clearer picture of the central spread of data.

Question 12

What is “variance” and how do you calculate it?

Answer 12

Variance: A more comprehensive measure that describes the average squared deviations from the mean. By squaring the differences, variance weighs outliers more heavily, making it sensitive to extreme values.

Question 13

What is the benefit of “variance” and what is the issue with “variance’?

Answer 13

Benefit: can handle negative numbers.
Issue: by squaring the differences, variance weighs outliers more heavily, making it sensitive to extreme values.

Question 14

What is “standard deviation” and how is it calculated?

Answer 14

Standard Deviation: The square root of the variance, which brings the measure of dispersion back into the units of the original data.

Question 15

Which is more commonly used and why?

• Standard Deviation
• Mean

Answer 15

Standard Deviation: It’s one of the most widely used statistics for measuring dispersion because it is interpretable in the context of the mean.

Question 16

What is the “MAD”?

Answer 16

MAD, or the Mean Absolute Deviation, is a statistical measure that quantifies the average absolute dispersion of a set of data points around their mean. It is one of the ways to measure the variability or spread in a data set.

Question 17

How do you calculate “MAD”?

Answer 17

the average absolute dispersion of a set of data points around their mean.

Question 18

Which is less sensitive to outliers, “MAD” or “Standard Deviation”?

Answer 18

MAD is less sensitive to outliers than SD. Because it calculates absolute differences without squaring them, extreme values have less impact on the overall measure.

Question 19

Why is “Standard Deviation” more common than “MAD”?

Answer 19

Standard Deviation fully captures variability around the mean, particularly in contexts that assume a normal distribution, or when engaged in more complex statistical modeling that relies on the properties of the normal distribution.

Question 20

How do you calculate “Coefficient of Variation”?

Answer 20

The coefficient of variation (CV) is the ratio of the standard deviation of a set of observations to their mean value.

Question 21

Why is the “Coefficient of Variation” useful?

Answer 21

By expressing the magnitude of variation among observations relative to their average size, the CV allows for the direct comparisons of dispersion across different datasets. Reflecting the correction for scale, the CV is a scale-free measure, that is, it has no units of measurement.

Question 22

What does “Skewness” refer to?

Answer 22

Skewness describes the degree to which a distribution is asymmetric about its mean.

Question 23

Consider two portfolios:

(1) the first has an asset return distribution that is normal
(2) the second has an asset return distribution that displays positive skewness.

How does the return of the second portfolio (i.e., the one with positive skewness) compare to the first portfolio (i.e., the normal one)?

Answer 23

An asset return distribution with positive skewness has frequent small losses and a few extreme gains compared to a normal distribution.

Question 24

Consider two portfolios:

(1) the first has an asset return distribution that is normal
(2) the second has an asset return distribution that displays negative skewness.

How does the return of the second portfolio (i.e., the one with negative skewness) compare to the first portfolio (i.e., the normal one)?

Answer 24

An asset return distribution with negative skewness has frequent small gains and a few extreme losses compared to a normal distribution.

Question 25

Consider a portfolio with zero skewness and the mean return is 0.0%:

What is more frequent, losses or gains?

Answer 25

Zero skewness indicates a symmetric distribution of returns. Therefore the likeliness of losses or gains is identical.

Question 26

What does “Kurtosis” measure?

Answer 26

Kurtosis measures the combined weight of the tails of a distribution relative to the rest of the distribution.

Question 27

A distribution with fat tails can be said to be:

(A) leptokurtic
(B) platykurtic
(C) mesokurtic

Answer 27

leptokurtic

Question 28

A distribution with narrow tails can be said to be:

(A) leptokurtic
(B) platykurtic
(C) mesokurtic

Answer 28

platykurtic

Question 29

A distribution with normally-distributed tails can be said to be:

(A) leptokurtic
(B) platykurtic
(C) mesokurtic

Answer 29

mesokurtic

Question 30

What does the correlation coefficient measure?

Answer 30

The correlation coefficient measures the association between two variables.

Question 31

What does a positive correlation coefficient indicate?

Answer 31

A positive correlation coefficient indicates that the two variables tend to move together

Question 32

What does a negative correlation coefficient indicate?

Answer 32

a negative coefficient indicates that the two variables tend to move in opposite directions.

Question 33

What does the middle line represent in a box-and-whiskers plot?

Answer 33

the median.

Question 34

What does the x represent in a box-and-whiskers plot?

Answer 34

the average

Question 35

What does the width of the box w/in the box-and-whiskers plot represent?

Answer 35

the interquartile range.

Question 36

What does the top line indicate in a box-and-whiskers plot?

Answer 36

the highest value.

Question 37

What does the bottom line indicate in a box-and-whiskers plot?

Answer 37

the lowest value.

Question 38

What is the “Target Downside Deviation”?

Answer 38

the target downside deviation, also referred to as the target semideviation, is a measure of dispersion of the observations (here, returns) below a target

Question 39

How do we calculate the “Target Downside Deviation” (or Target Semideviation)?

Answer 39

to calculate a sample target semideviation, we first specify the target. After identifying observations below the target, we find the sum of those squared negative deviations from the target, divide that sum by the total number of observations in the sample minus 1, and, finally, take the square root.

Question 40

What is “Excess Kurtosis”?

Answer 40

Excess kurtosis thus characterizes kurtosis relative to the normal distribution. A normal distribution has excess kurtosis equal to 0. A fat-tailed distribution has excess kurtosis greater than 0, and a thin-tailed distribution has excess kurtosis less than 0. A return distribution with positive excess kurtosis—a fat-tailed return distribution—has more frequent extremely large deviations from the mean than a normal distribution.

Question 41

What type of plot is useful for identifying the relationship between two variables?

Answer 41

A scatter plot is a useful tool for displaying and understanding potential relationships between two variables.