CFA 7: Statistical Concepts and Market Returns

Question 1

statistics

Some Fundamental Concepts

Answer 1

The term statistics can have two broad meanings, one referring to data, and the other to method.

A company’s average earnings per share (EPS) for the last 20 quarters, or its average returns for the past 10 years, are statistics. We may also analyze historical EPS to forecast future EPS, or use the company’s past returns to infer its risk. The totality of methods we employ to collect and analyze data is also called statistics.

Question 2

descripitive statistics

Some Fundamental Concepts

Answer 2

The study how data can be summarized effectively to describe the important aspects of large data sets.

By consolidating a mass of numerical details, descriptive statistics turns data into information.

Question 3

statistical inference

Some Fundamental Concepts

Answer 3

Involves making forecasts, estimates, or judgments about a larger group from the smaller group actually observed. The foundation for statistical inference is probability theory.

Question 4

population

Some Fundamental Concepts

Answer 4

All members of a specified group.

Question 5

parameter

Some Fundamental Concepts

Answer 5

Any descriptive measure of a population characteristic. Although a population can have many parameters, investment analysts are usually concerned with only a few, such as the mean value, the rate of investment returns, and the variance.

Question 6

sample statistic

Some Fundamental Concepts

Answer 6

A quantity computed from or used to describe a sample.

Question 7

measurement scales

Some Fundamental Concepts

Answer 7

All data measurements are taken on one of four major scales: nominal, ordinal, interval, or ratio.

Question 8

nominal scales

Some Fundamental Concepts

Answer 8

Nominal scales represent the weakest level of measurement: They categorize data but do not rank them.

If we assigned integers to mutual funds that follow different investment strategies, the number 1 might refer to a small-cap value fund, the number 2 to a large-cap value fund, and so on for each possible style.

This nominal scale categorizes the funds according to their style but does not rank them.

Question 9

ordinal scales

Some Fundamental Concepts

Answer 9

Ordinal scales sort data into categories that are ordered with respect to some characteristic.

For example, the Morningstar and Standard & Poor’s star ratings for mutual funds represent an ordinal scale in which one star represents a group of funds judged to have had relatively the worse performance, with two, three, four, and five stars representing groups with increasingly better performance, as evaluated by those services.

Question 10

interval scales

Some Fundamental Concepts

Answer 10

Interval scales provide not only ranking but also assurance that the differences between scale values are equal. As a result scale values can be added and subtracted meaningfully.

The Celsius and Fahrenheit scales are interval measurement scales. the difference in temperature between 10c and 11c is the same amount as the difference between 40 and 41C. We can state accurately that 12C = 9C + 3C, for example.

Nevertheless, the zero point of an interval scale does not reflect complete absence of what is being measured; it is not a true zero point or natural zero.

Zero degrees Celsius corresponds to the freezing point of water, not the absence of temperature. As a consequence of the absence of a true zero point, we cannot meaningfully for ratios on interval scales.

As an example, 50C, although five times as large a number of 10C, does not represent five times as much temperature. Also, questionnaire scales are often treated as interval scale. If an investor is asked to rank his risk aversion on a scale from 1 (extremely risk-averse) to 7 (extremely risk-loving), the difference between a response of 1 and a response of 2 is sometimes assumed to represent the same difference in risk aversion as the difference between a response of 6 and a response of 7. When that assumption can be justified, the data are measured on an interval scale.

Question 11

ratio scales

Some Fundamental Concepts

Answer 11

Ratio scales represent the strongest level of measurement. They have all the characteristics of interval measurement scales as well as a true zero point as the origin. With ratio scales, we can meaningfully compute ratios as well as meaningfully add and subtract amounts within the scale.

As a result, we can apply the widest range of statistical tools to data measured on a ratio scale. Rates of return are measured on a ratio scale, as is money. If we have twice as much money, then we have twice the purchasing power. Note that the scale has a natural zero - zero means no money

Question 12

Construction of a Frequency Distribution

Summarizing Data Using Frequency Distributions

Answer 12

1) Sort the data in ascending order
2) Calculate the range of the data, defined as Range = Max value - Min value
3) Decide on the number of intervals in the frequency distribution, k
4) Determine the interval width as Range/k
5) Determine the intervals by successively adding the interval width to the minimum value, to determine the ending points of intervals, stopping after reaching an interval that includes the maximum value.
6) Count the number of observations falling in each interval
7) Construct a table of the intervals listed from smalles to largest that shows the number of observations falling in each interval

Question 13

frequency distribution

Summarizing Data Using Frequency Distributions

Answer 13

A tabular display of data summarized in a relatively small number of intervals.

Frequency distributions help in the analysis of large amounts of statistical data, and they work with all types of measurement scales.

The frequency distribution is the list of intervals together with the corresponding measures of frequency.

Gives us a sense of not only where most of the observations lie, but also whether the distribution is evenly distributed, lopsided, or peaked.

Question 14

interval

Summarizing Data Using Frequency Distributions

Answer 14

A set of values within which an observation falls. Each observation falls into only one interval, and the total number of intervals covers all the values represented in the data.

Question 15

absolute frequency

Summarizing Data Using Frequency Distributions

Answer 15

the actual number of observations in a given interval.

Question 16

relative frequency

Summarizing Data Using Frequency Distributions

Answer 16

The absolute frequency of each interval divided by the total number of observations.

Question 17

cumulative relative frequency

Summarizing Data Using Frequency Distributions

Answer 17

Cumulates the relative frequencies as we move from the first to the last interval. It tells us the fraction of observations that are less than the upper limit on each interval.

Question 18

histogram

The Graphic Presentation of Data

Answer 18

A bar chart of data that have been grouped into a frequency distribution.

Question 19

frequency polygon

The Graphic Presentation of Data

Answer 19

Plot the midpoint of each interval on the x-axis and the absolute frequency for that interval on the y-axis.

Question 20

measure of central tendency

Measures of Central Tendency

Answer 20

Specifies where the data are centered. Measures of central tendency are probably more widely used than any other statistical measure becase they can be computed and applied easily.

Question 21

measures of location

Measures of Central Tendency

Answer 21

Include not only measures of central tendency but other measures that illustrate the location or distribution of data.

Question 22

arithmetic mean

Measures of Central Tendency

Answer 22

The sum of the observations divided by the number of observations.

Question 23

population mean (u)

Measures of Central Tendency

Answer 23

The arithmetic mean value of a population.

The population mean is an example of a parameter. The population mean is unique; that is, a given population has only one mean.

Question 24

sample mean (x bar)

Measures of Central Tendency

Answer 24

The sum of the sample observations, divided by the sample size.

Question 25

cross-sectional data

Measures of Central Tendency

Answer 25

Observations over individual units at a point in time, as opposed to time-series data.

Question 26

time-series data

Measures of Central Tendency

Answer 26

Observations of a variable over time.

Question 27

median

Measures of Central Tendency

Answer 27

The value of the middle item of a set of items that has been sorted into ascending or descending order.

In an odd-numbered sample of n items, the median occupies the (n+1)/2 position. In an even-numbered sample, we define the median as the mean of the values of items occupying the n/2 and (n+2)/2 positions (the two middle items).

A potential advantage of the median is that, unlike the mean, extreme values do not affect it. The median, however, does not use all the information about the size and magnitude of the observations; it focuses only on the relative position of the ranked observations.

Question 28

mode

Measures of Central Tendency

Answer 28

The most frequently occurring value in a distribution.

A distribution can have more than one mode, or even no mode. When a distribution has one most frequently occurring value, the distribution is said to be unimodal.

If a distribution has two most frequently occurring values, the nit has two modes and we say it is bimodal.

If the distribution has three most frequently occurring values, then it is trimodal.

When all the values in a data set are different, the distribution has no mode because no value occurs more frequently than any other value.

The modal interval always has the highest bar in the histogram.

The mode is the only measure of central tendency that can be used with NOMINAL data. For example, when we categorize mutual funds into different styles and assign a number to each style, the mode of these categorized data is the most frequent mutual fund style.

Question 29

weighted mean

Measures of Central Tendency

Answer 29

An average in which each observation is weighted by an idex of its relative importance.

Question 30

expected value

Measures of Central Tendency

Answer 30

The probability-weighted average of the possible outcomes of a random variable.

Question 31

geometric mean (G)

Measures of Central Tendency

Answer 31

The geometric mean is most frequently used to average rates of change over time or to compute the growth rate of a variable. In invstments, we frequently use the geoemetric mean to average a time series of rate of retun on an asset or a portfolio or to ocmpute th growth rate of a financial variable such as earnings or sales.

Question 32

harmonic mean (X bar h)

Measures of Central Tendency

Answer 32

The harmonic mean may be viewed as a special type of weighted mean in which an observation’s weight is inversely proportional to its magnitude. The harmonic mean is a relatively specialized concept of the mean that is appropriate when averaging ratios (“amount per unit”) when the ratios are repeatedly applied to a fixed quantity to yield a variable number of units.

A mathematical fact concerning the harmonic, geometirc, and arithmetic means is that unless all the observations in a data set have the same value the harmonic mean is less than the geometric mean, which in turn is less than the aritmetic mean.

Question 33

quantile (fractile)

Other Measures of Location: Quantiles

Answer 33

A value at or below which a stated fraction of the data lies.

quartiles divide the distribution into quarters

quintiles divide the distribution into fifths

percentiles divide the distribution into hundredths

Question 34

linear interpolation

Other Measures of Location: Quantiles

Answer 34

Interpolation means estimating an unknown value on the basis of two known values that surround it (lie above and below it); the term “linear” refers to a straight-line stimate.

Question 35

dispersion

Measures of Dispersion

Answer 35

The variability around the central tendency.

If mean return addresses reward, dispersion addresses risk.

Question 36

absolute dispersion

Measures of Dispersion

Answer 36

The amount of vairability present without comparison to any reference point or benchmark.

Question 37

range

Measures of Dispersion

Answer 37

The difference betseen teh maximum and minimum values of a data set.

Question 38

mean absolute deviation (MAD)

Measures of Dispersion

Answer 38

With reference to a sample, the mean of the absolute values of deviations from the sample mean.

Question 39

variance

Measures of Dispersion

Answer 39

The average of the squared deviations around the mean

Question 40

standard deviation

Measures of Dispersion

Answer 40

The positive square root of the variance

Question 41

semivariance

Measures of Dispersion

Answer 41

The average squared deviation below the mean.

Target semivariance is the average squared deviation below a stated target.

Question 42

semideviation

Measures of Dispersion

Answer 42

The positive square root of semi variance

Target semideviation is the positive square root of of semi variance below a state target.

Question 43

Chebyshev’s Inequality

Measures of Dispersion

Answer 43

The inequality gives the proportion of values within k standard deviations of the mean.

For any distribution with finite variance, the proportion of the observations within k standard deviations of the aritmetic mean is at least 1-1/k^2 for all k > 1.

Question 44

coefficient of variation (CV)

Measures of Dispersion

Answer 44

Ratio of the standard deviation of a set of observations to their mean value.

Question 45

Sharpe ratio

Measures of Dispersion

Answer 45

The average return in excess of the risk-free rate divided by the standard deviation of return; a measure of the average excess return earned per unit of standard deviation of return.

Question 46

mean excess return

Measures of Dispersion

Answer 46

The average rate of return in excess of the risk-free rate.

Question 47

skewed

Symmetry and Skewness in Return Distributions

Answer 47

A distribution that is not symmetrical.

A return distribution with a positive skew has frequent small losses and a few extreme gains. A return distribution with a negative skew has frequent small gains and a few extreme losses.

Positively skewed distribution has a long tail on its right side;

Negatively skewed distribution has long tail on its left side.

For the positively skewed unimodal distribution, the mode is less than the median, which is less than the mean.

For the negatively skewed unimodal distribution, the mean is less than the median, which is less than the mode.

Question 48

kurtosis

Kurtosis in Return Distributions

Answer 48

The statistical measure that tells us when a distribution is more or less peaked than a normal distribution.

Question 49

leptokurtic

Kurtosis in Return Distributions

Answer 49

A distribution that is more peaked than normal.

Question 50

platykurtic

Kurtosis in Return Distributions

Answer 50

A distribution that is less peaked than normal.

Question 51

mesokurtic

Kurtosis in Return Distributions

Answer 51

A distribution identical to the normal distribution.

Question 52

excess kurtosis

Kurtosis in Return Distributions

Answer 52

Kurtosis minus 3.

Excess kurtosis characherizes kurtosis relative to the normal distribtuion.

A normal or other mesokurtic distribution has excess kurtosis equal to 0.

A leptokurtic distrbution has excess kurtosis greater than 0.

A platykurtic distribution has excess kurtosis less than 0.

Question 53

semilogarithmic

Answer 53

Describes a scale constructed so that equal intervals on the vertical scale represent equal rates of change, and equal intervals on the horizontal scale represent equal amounts of change.