Statistical Concepts and Market Returns

Question 1

Measures of Central Tendency

Answer 1

Measures of central tendency provide an indication of an investment’s expected return.
1. arithmetic mean 2. geometric mean 3. weighted mean 4. median 5. mode
Measures of central tendency identify the center, or average, of a data set.

Question 2

Measures of Dispersion

Answer 2

Measures of dispersion indicate the riskiness of an investment.
1. range 2. mean absolute deviation 3. variance

Question 3

Two Categories of statistics

Answer 3

1. descriptive statistics 2. inferential statistics

Question 4

Descriptive statistics

Answer 4

used to summarize the important characteristics of large data sets.

Question 5

Inferential statistics

Answer 5

Inferential statistics pertain to the procedures used to make forecasts, estimates, or judgments about a large set of data on the basis of the statistical characteristics of a smaller set (a sample).

Question 6

Population

Answer 6

The set of all possible members of a stated group. A cross­ section of the returns of all of the stocks traded on the New York Stock Exchange (NYSE) is an example of a population.

Question 7

Sample

Answer 7

a subset of the population of interest.

Question 8

Types of Measurement Scales

Answer 8

1. Nominal scales 2.Ordinal scales 3. Interval scale 4. Ratio scales

Question 9

Nominal scales

Answer 9

Nominal scales are the level of measurement that contains the least information. Observations are classified or counted with no particular order. An example would be assigning the number 1 to a municipal bond fund, the number 2 to a corporate bond fund, and so on for each fund style.

Question 10

Ordinal scales

Answer 10

Ordinal scales represent a higher level of measurement than nominal scales. When working with an ordinal scale, every observation is assigned to one of several categories. Then these categories are ordered with respect to a specified characteristics. For example, the ranking of 1,000 small cap growth stocks by performance may be done by assigning the number 1 to the 100 best performing stocks, the number 2 to the next 1 00 best performing stocks, and so on, assigning the number 1 0 to the 100 worst performing stocks. Based on this type of measurement, it can be concluded that a stock ranked 3 is better than a stock ranked 4, but the scale reveals nothing about performance differences or whether the difference between a 3 and a 4 is the same as the difference between a 4 and a 5.

Question 11

Interval scale

Answer 11

Interval scale measurements provide relative ranking, like ordinal scales, plus the assurance that differences between scale values are equal. Temperature measurement in degrees is a prime example. Certainly, 49°C is hotter than 32°C, and the temperature difference between 49°C and 32°C is the same as the difference between 67°C and 50°C. The weakness of the interval scale is that a measurement of zero does not necessarily indicate the total absence of what we are measuring. This means that interval-scale-based ratios are meaningless. For example, 30°F is not three times as hot as 1 0°F

Question 12

Ratio scales

Answer 12

Ratio scales represent the most refined level of measurement. Ratio scales provide ranking and equal differences between scale values, and they also have a true zero point as the origin. Order, intervals, and ratios all make sense with a ratio scale. The measurement of money is a good example. If you have zero dollars,
you have no purchasing power, but if you have $4.00, you have twice as much purchasing power as a person with $2.00.

Question 13

frequency distribution

Answer 13

A frequency distribution groups observations into classes, or intervals. An interval is a range of values. Frequency distributions summarize statistical data by assigning it to specified groups, or intervals. Also, the data employed with a frequency distribution may be measured using any type of measurement scale.

Question 14

How to construct a frequency distribution

Answer 14

1. Define the intervals.
2. Tally the observations.
3. Count the observations.

Question 15

Modal interval

Answer 15

For any frequency distribution, the interval with the greatest frequency is referred to as the modal interval.

Question 16

Relative Frequency

Answer 16

Relative frequency is the percentage of total observations falling within an interval. It is calculated by dividing the absolute frequency of each return interval by the total number of observations.

Question 17

cumulative absolute frequency

Answer 17

Calculated by adding the frequency of all observations at or below that point.

Question 18

cumulative relative frequency

Answer 18

Cumulative relative frequency for an interval is the sum of the relative frequencies for all values less than or equal to that interval’s maximum value. Sum of relative frequency percentages

Question 19

Histogram

Answer 19

The graphical presentation of the absolute frequency distribution.

Question 20

Populations Mean Formula

Answer 20

Sum all observed values in the population and divide by # of observations in the population

Question 21

Sample Mean Formula

Answer 21

Sum of all values in a sample population divided by the # of observations in the sample

Question 22

mode

Answer 22

the value that occurs most frequently in a data set.

Question 23

Harmonic Mean

Answer 23

A harmonic mean is used for certain computations, such as the average cost of shares purchased over time

Question 24

Geometric Mean

Answer 24

geometric mean is often used when calculating investment returns over multiple periods or when measuring compound growth rates.
G= (1+X1 * 1+X2 * 1+X3….)^1/n

Question 25

Properties of arithmetic mean

Answer 25

All interval and ratio data sets have an arithmetic mean.
All data values are considered and included in the arithmetic mean computation.
A data set has only one arithmetic mean (i.e., the arithmetic mean is unique).
The sum of the deviations of each observation in the data set from the mean is always zero.

Question 26

arithmetic mean

Answer 26

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

Question 27

Quantile

Answer 27

Quantile is the general term for a value at or below which a stated proportion of the data in a distribution lies.

Question 28

Quartile

Answer 28

Quartiles- the distribution is divided into quarters.

Question 29

Quintile

Answer 29

Quintile- the distribution is divided into fifths.

Question 30

Decile

Answer 30

Decile- the distribution is divided into tenths.

Question 31

Percentile

Answer 31

Percentile- the distribution is divided into hundredths (percents)

Question 32

Formula for the position of any observation at any given percentile

Answer 32

Ly = (n + 1)* y/100

n= # of observations
y= percentile

Question 33

Dispersion

Answer 33

Dispersion is defined as the variability around the central tendency.

Question 34

Range

Answer 34

range = maximum value - minimum value

Question 35

Mean Absolute Deviation (MAD)

Answer 35

The mean absolute deviation (MAD) is the average of the absolute values of the deviations of individual observations from the arithmetic mean.

Question 36

Population Variance

Answer 36

Population variance is defined as the average of the squared deviations from the mean.

Question 37

Population Standard Deviation

Answer 37

The population standard deviation is the square root of the population variance

Question 38

Sample Variance

Answer 38

The sample variance is the measure of dispersion that applies when we are evaluating a sample of n observations from a population.

Question 39

Sample Standard Deviation

Answer 39

sample standard deviation can be calculated by taking the square root of the sample variance.

Question 40

Chebyshev’s inequality

Answer 40

Chebyshev’s inequality states that for any set of observations, whether sample or population data and regardless of the shape of the distribution, the percentage of the observations that lie within k standard deviations of the mean is at least 1 - 1/k2 for all k > l.

Question 41

Percentage of observation within standard deviations of the mean

Answer 41

According to Chebyshev’s inequality, the following relationships hold for any distribution. At least:
36% ofobservations lie within ±1.25 standard deviations ofthe mean.
56% ofobservations lie within ± 1.50 standard deviations ofthe mean.
75% of observations lie within ±2 standard deviations of the mean.
89% of observations lie within ±3 standard deviations of the mean.
94% of observations lie within ±4 standard deviations of the mean.

Question 42

Relative dispersion

Answer 42

Relative dispersion is the amount of variability in a distribution relative to a reference point or benchmark. Relative dispersion is commonly measured with the coefficient of variation

Question 43

Coefficient of variation

Answer 43

Standard deviation of X / avg value of X

Relative dispersion is commonly measured with the coefficient of variation

Question 44

Sharpe Ratio

Answer 44

Portfolio Return - Risk free rate of return / Stdev of portfolio returns

Question 45

Limitations of the Sharpe ratio

Answer 45

(1) If two portfolios have negative Sharpe ratios, it is not necessarily true that the higher Sharpe ratio implies superior risk-adjusted performance. Increasing risk moves a negative Sharpe ratio closer to zero (i.e., higher). (2) The Sharpe ratio is useful when standard deviation is an appropriate measure of risk. However, investment strategies with option characteristics have asymmetric return distributions, reflecting a large probability of small gains coupled with a small probability of large losses. In such cases, standard deviation may underestimate risk and produce Sharpe ratios that are too high.

Question 46

Skewness affects the location of the mean, median, and mode of a distribution.

Answer 46

positively skewed, unimodal distribution: mode < median < mean.

negatively skewed, unimodal distribution:
mean < median < mode.

The key to remembering how measures ofcentral tendency are affected by skewed data is to recognize that skew affects the mean more than the median and mode, and the mean is “pulled” in the direction of the skew.

Question 47

Kurtosis

Answer 47

Kurtosis is a measure of the degree to which a distribution is more or less “peaked” than a normal distribution.

Question 48

Leptokurtic

Answer 48

Leptokurtic describes a distribution that is more peaked than a normal distribution,

Question 49

Platykurtic

Answer 49

Platykurtic refers to a distribution that is less peaked, or flatter than a normal distribution.

Question 50

mesokurtic

Answer 50

A distribution is mesokurtic if it has the same kurtosis as a normal distribution.

Question 51

Sample Skewness

Answer 51

Sample skewness is equal to the (sum of the cubed deviations from the mean) divided by (the cubed standard deviation) and by (the number of observations.)

Question 52

Sample kurtosis

Answer 52

Sample Kurtosis is equal to the (sum of the deviations from the mean raised to the 4th) divided by (the standard deviation to the 4th) and by (the number of observations.)

Question 53

Geometric mean return

Answer 53

= (1 + annual return)*(1+annual return)…… ^1/n -1

Question 54

Arithmetic mean return

Answer 54

= (1 + annual return)*(1+annual return)…… / n

Question 55

Parameter

Answer 55

Any measurable characteristic of a population