Statistical Concepts and Market Returns

Question 1

Statistics used to summarize important characteristics of large data sets

Answer 1

Descriptive Statistics

Question 2

The procedures used to make forecasts, estimates or judgments about large data sets on the basis of the statistical characteristic of a smaller set (sample).

Answer 2

Inferential Statistics

Question 3

The set of all possible members of a stated group

Answer 3

Population

Question 4

A subset of the population of interest

Answer 4

Sample

Question 5

A measurement scale that contains the least information. Observations are classified or counted with no particular order. (i.e. binary)

Answer 5

Nominal Scale

Question 6

A measurement scale that has every observation assigned to one of several categories. Then the categories are ordered with respect to certain characteristics. (I.e. Top 100 stocks of SP500)

Answer 6

Ordinal Scale

Question 7

A measurement scale that provides relative ranking, plus the assurance that the differences between scale values are equal. (i.e. temperature )

Answer 7

Interval Scale

Question 8

A measurement scale that provides ranking and equal differences between scale values and a true zero point at the origin. (i.e. money purchasing power)

Answer 8

Ratio Scale

Question 9

A measure used to describe a characteristic of a population

Answer 9

parameter

Question 10

used to measure a characteristic of a sample

Answer 10

Sample Statistic

Question 11

A tabular presentation of statistical data that aids the analysis of large data sets. (For frequency distributions, the interval with the greatest frequency is the modal interval)

Answer 11

Frequency Distribution

Question 12

Calculated by dividing the absolute frequency of each return interval by the total number of observations

Answer 12

Relative Frequency

*The percentage of total observation that fall within each interval

Question 13

Summing the absolute frequencies starting at the lowest interval and progressing through the highest

Answer 13

Cumulative Absolute Frequencies

Question 14

Calculation for Population/Sample Mean

Answer 14

SUM(Xi) / N

Question 15

Calculation for Sum of Mean Deviations

Answer 15

Sum(Xi-Xbar) = 0

Question 16

Calculation for Weight Mean

Answer 16

Sum(wi*Xi) = 1
wi == weight for each Xi

• example is a portfolio weight by %stock, bond, cash

Question 17

Calculation for Geometric Mean

Answer 17

G = (X1X2…*Xn)^(1/n)

*Always less than arithmetic mean

Question 18

Calculation for Geometric Mean Return

Answer 18

1 + Rg = ((1+Rg1)(1+Rg2)…*(1+Rgn))^(1/n)

Question 19

Calculation for Harmonic Mean

Answer 19

N / SUM(1/Xi)

*ex. average cost per share
harmonic mean < geometric mean < arithmetic mean

Question 20

Percentile Calculation

Answer 20

Ly = (n+1) *(y/100) == the number below which observation is the quartile

Question 21

The variability around the Central Tendency

Answer 21

Dispersion

Question 22

Mean Absolute Deviation

Answer 22

SUM(abs(Xi-Xbar)) / N

Question 23

Population Variance Calculation

Answer 23

sigma^2 = SUM(Xi- u)^2 / N

Question 24

Calculation for Population Standard Deviation

Answer 24

sigma = ((SUM(X-u)^2) / N)^(1/2)

Question 25

Calculation for Sample Variance

Answer 25

s^2 = SUM(Xi-Xbar)^2 / (n-1)

Question 26

If n is used, not (n-1), the sigma squared is systematically underestimated

Answer 26

biased estimator

Question 27

Sample Standard Deviation

Answer 27

s = (SUM(Xi-Xbar)^2/(n-1))^(1/2)

Question 28

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/k^2)) for all k > 1

Answer 28

Chebyshev’s Inequality

Question 29

36% of all observations lie within +- 1.25 sdev's of mean
56% "" +/- 1.5 sdev's of mean 
75%  " " +/- 2 sdev's of mean 
89% "" +/- 3 sdev's of mean 
94% "" +/- 4 sdev's of mean

Answer 29

Using Chebyshev’s Inequality

Question 30

The amount of variability in a distribution relative to a reference point or benchmark. It is measured with coefficient of variation

Answer 30

Relative Dispersion

Question 31

Coefficient of Variation Calculation

• measures the amount of dispersion in a distribution relative to its mean
• • risk per unit of return

Answer 31

CV = sx/ Xbar = (Std. Dev of x) / (avg. value of x)

Question 32

Sharpe Ratio Calculation

*measures excess return per unit

Answer 32

s = (Rpbar - Rf) / sigmap

Rpbar = portfolio return 
Rf = risk-free return
sigmap = std. dev. of portfolio returns

*T-bills are convention for risk free return

Question 33

2 limitations of Sharpe Ratio

Answer 33

1. If 2 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.
2. Investment strategies with the option characteristics produce Sharpe Ratios to high and underestimate risk because of asymmetric return distributions

Question 34

Skewness refers to the extent to which a distribution is not symmetrical

Answer 34

• positively skewed == many outliers in the upper region, or right tail
• *negatively skewed == many outliers in the lower region, or left tail.

Question 35

In positively skewed distribution, (organize mean, median and mode)

Answer 35

mode < median < mean

Question 36

In negatively skewed distribution, (organize mean, median and mode)

Answer 36

mean < median < mode

Question 37

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

Answer 37

Kurtosis

Question 38

Leptokurtic

Answer 38

More peaked

Question 39

Platykurtic

Answer 39

less peaked

Question 40

Mesokurtic

Answer 40

equal or same as normal

Question 41

This distribution will have more returns clustered around the mean and more return with large deviation from the mean (fatter tails). This is perceived as risk increasing.

Answer 41

Leptokurtic Distribution

Question 42

If it has more or less kurtosis than the normal distribution

Answer 42

excess kurtosis

Question 43

Kurtosis for normal kurtosis = 3
Excess Kurtosis = kurtosis - 3

*Generally, more positive kurtosis and more negative skew signify increased risk.

Answer 43

excess kurtosis:
leptokurtic if >0
platykurtic if < 0

Question 44

Sample Skewness Calculation

Answer 44

Sk = (1/n) * (SUM(Xj-Xbar)^3 / s^3)

s = sample standard deviation

Question 45

Sample Kurtosis Calculation

Answer 45

== (1/n) * (SUM(Xi-Xbar)^4 / s^4)

Question 46

Excess kurtosis > +/- 1 is considered large

Answer 46

””

Question 47

Geometric Mean when measuring the past

Answer 47

Arithmetic Mean when measuring the future.