Reading 7 Statistical Concepts and Market Returns

Question 1

Explain what kurtosis measures, and differentiate between a leptokurtic distribution, a platykurtic distribution, and a mesokurtic distribution.

Answer 1

Kurtosis measures a distribution peak compared to the normal distribution kurtosis of 3.

Leptokurtic: more peaked, fatter tails, kurtosis greater than three (excess kurtosis > 0).

Platykurtic: less peaked, thinner tails, kurtosis less than 3 (excess kurtosis < 0).

A mesokurtic (normal) distribution has kurtosis of 3 (excess kurtosis of 0).

Question 2

Give the formula used to compare the relative dispersion of returns on the two investments using the coefficient of variation.

Answer 2

CV is the ratio of standard deviation of the data set to its mean

CV = s / X
¯¯¯

where:

s = sample standard deviation

¯¯¯
X = the sample mean

Note: CV measures the risk per unit of return.

Question 3

Explain Chebyshev’s inequality, and describe the advantage of using it.

Answer 3

A method of approximating the minimum proportion of observations that lie within k standard deviations from the mean of a data set.

The advantage of using Chebyshev’s inequality is that it holds for samples and populations as well as for discrete and continuous data, regardless of the shape of the distribution.

Question 4

Describe when arithmetic mean and geometric mean should be used.

Answer 4

Arithmetic mean – single period; arithmetic mean is the average of one-period returns.

Geometric mean – multiple periods; geometric mean links investment performance over time.

Question 5

How is the weighted mean calculated?

Answer 5

Where an arithmetic mean assigns equal weight to each observation in the data set, a weighted mean assigns different weights to different observations. (Example: the effect of asset class weight on portfolio mean return.)

Question 6

List the properties of the arithmetic mean, and identify a potential problem with it.

Answer 6

All observations are used in the computation of the arithmetic mean, and a data set has only one arithmetic mean.

All interval and ratio data sets have an arithmetic mean.

The sum of the deviations from the arithmetic mean is always 0.

A potential problem with the arithmetic mean occurs with extreme high or low values, which can disproportionately alter it.

Note: The arithmetic mean is the most frequently used measure of central tendency.

Question 7

Give the formula used to calculate sample skewness.

Answer 7

SK=[n(n−1)(
n−2)]∑ni=1(Xi−––X)3s3

n = number of observations in the sample

s = sample standard deviation

Note: when a distribution is nonsymmetrical, it is considered skewed.

Question 8

Give the formula for the position of a percentile in a data set with n observations sorted in ascending order.

Answer 8

Ly=y 100 (n+1)

where:

y = percentage point at which we are dividing the distribution

Ly = location (L) of the percentile (Py) in the data set sorted in ascending order

Question 9

How is the harmonic mean used, and what is the formula used to calculate it?

Answer 9

To determine the average cost of shares in dollar cost averaging. The dollar amount periodically invested can be divided by the share price to determine the number of shares at each purchase. The total amount invested is then divided by the total number of shares to determine the average cost of shares.

XH=¯n∑ni=1(1/Xi)

Question 10

How are the sample and population variance calculated?

Answer 10

σ2=n∑i=1(Xi−––X)2

where:

Xi = observation i; μ = population mean; and N = size of the population

s2=∑ni=1(Xi−––X)2n−1

where:

n = sample size;
¯¯¯
X, the sample mean, is used in place of μ

Question 11

Define mean absolute deviation (MAD) and give the calculation.

Answer 11

The mean absolute deviation is the average of the absolute values of deviations of observations from the mean.

MAD
=∑ni=1∣∣
∣∣Xi−––X∣∣∣∣n

where:

n = number of items in the data set

¯¯¯
X
=
the arithmetic mean of a sample

Question 12

Identify the important relationships between the arithmetic mean and geometric mean of a data set. And identify when a geometric mean is frequently used.

Answer 12

The geometric mean is always less than or equal to the arithmetic mean.

The geometric mean equals the arithmetic mean only when all the observations are identical.

The difference between the geometric and arithmetic mean increases as the dispersion of observed values increases.

A geometric mean is frequently used when calculating average rates of return over multiple periods or to compute the growth rate of a variable.

Question 13

Describe nominal, ordinal, interval, and ratio scales.

Answer 13

Nominal: weakest level of measurement; categorize or count data but do not rank them

Ordinal: stronger level of measurement than nominal scales; sort data in categories that are ranked according to a certain characteristic

Interval: rank observations such that the differences between scale values are equal so that values can be added and subtracted meaningfully

Ratio: have all the characteristics of interval scales and true zero point as the origin; strongest measurement level

Question 14

Define median and mode.

Answer 14

The median is the value of the middle item of a data set once it has been arranged in ascending or descending order.

The mode is used to identify the most frequently occurring value of a data set.

Question 15

Distinguished between a return distribution with positive skewness and one with negative skewness.

Answer 15

A return distribution with positive skewness has frequent small losses and few large gains, and rmode < rmedian < rmean.

A return distribution with negative skewness has frequent small gains and few large losses, and rmean < rmedian < rmode.

Question 16

Explain why the geometric mean is more appropriate for reporting historical returns.

Answer 16

Because it equals the rate of growth an investor would have to earn each year to match actual cumulative investment performance. The geometric mean is an excellent measure of past performance.

Question 17

Describe relative frequency for an interval.

Answer 17

The proportion or fraction of total observations in an interval. Each interval’s relative frequency is calculated by dividing its absolute frequency by the total number of observations.

Question 18

What is a frequency distribution?

Answer 18

A tabular illustration of data categorized into a small number of intervals or classes. Each observation must fall into only one interval (mutually exclusive), and the set of intervals must cover the entire range of values represented in the data (all-inclusive). The data presented in a frequency distribution may be measured using any type of measurement scale.

Question 19

When is a distribution symmetrical?

Answer 19

When the mean, median, and mode are equal.

Question 20

Give the formula used to calculate the Sharpe ratio, and identify what the ratio measures.

Answer 20

S=–––rp−rfsp

where:

¯
rp = mean portfolio return

rf = risk-free return

sp = standard deviation of portfolio returns

The Sharpe ratio measures the excess return per unit of risk.

Question 21

Give the formula used to calculate sample excess kurtosis.

Answer 21

K
E=
⎛⎜
⎜
⎜
⎝n(n+1)(n−1)(n−2)(n−3)∑ni=1(Xi−––X)4s4
⎞
⎟
⎟
⎟
⎠−3(n−1)2(n−2)(n−3)

Kurtosis itself is the first term.

Question 22

In what way is a histogram used, and what is the advantage of using it?

Answer 22

A histogram graphically represents the data contained in a frequency distribution.

The advantage of a histogram is that users can quickly recognize where most of the data is concentrated.