Statistics tools for Risk

Statistics...

A set of mathematical procedures for

describing, synthesizing, analyzing, and

interpreting quantitative data

…the selection of an appropriate

statistical technique is determined by

the research design, hypothesis, and

the data collected

Descriptive statistics are

numerical estimates that organize

and sum up or present the data.

Inferential statistics is the process

of inferring from a sample to the

population.

Preparing data for analysis...

Data must be accurately scored and

systematically organized to facilitate

data analysis:

tabulating: organizing the data in a

systematic manner

coding: assigning numerals (e.g., ID)

to data

scoring: assigning a total to each

participant’s instrument

descriptive statistics...

…permit the researcher to describe many

pieces of data with a few indices

statistics...

…indices calculated by the researcher for a

sample drawn from a population

parameters...

…indices calculated by the researcher for an

entire population

Types of descriptive statistics…

1. graphs

2. measures of central tendency

3. measures of variability

graphs...

…representations of data enabling the

researcher to see what the distribution

of scores look like

1. Graphs…

frequency polygon

pie chart

boxplot

stem-and-leaf chart

measures of central tendency...

…indices enabling the researcher to

determine the typical or average score

of a group of scores

2. Measures of central tendency…

mode

median

mean

mode...

…the score attained by more participants than

any other score

median...

…the point in a distribution above and below

which are 50% of the scores

mean...

…the arithmetic average of the scores

Mean (arithmetic mean)

Example

2003 2004 2005 2006 2007 2008 2009 2010 2012

12 10 16 11 16 10 11 12 16

Data of Stock Return in Percentage

Find: Mode, Median, Mean, Geometric

Mean, Standard Deviation

Geometric Mean

The average of a set of products,

the calculation of which is commonly

used to determine the performance

results of an investment or portfolio.

)1()1()1( 21)1(

1

XXX ngeomean n

measures of variability...

…indices enabling the researcher to

indicate how spread out a group of

scores are

3. Measures of variability…

range

quartile deviation

variance

standard deviation

range...

…the difference between the highest and

lowest score in a distribution

quartile deviation...

…one half of the difference between the

upper quartile (the 75%’ile) and the

lower quartile (the 25%’ile) in a

distribution

variance...

…a summary statistic indicating the

degree of variability among participants

for a given variable

standard deviation...

…the square root of variance providing an

index of variability in the distribution of

scores

Statistics in Finance

Geometric mean

Expected Return

The average of a probability distribution of

possible returns

Risk (standard deviation)

Coefficient of Variation (CV)

n

iiPRERi

1

2

))((

)(RECV

Example

Suppose there are data about stock return on different investment conditions

Investment Condition Probability Return

Very bullish

Bullish

Average

Bearish

Very bearish

0.1

0.2

0.4

0.2

0.1

27%

18%

14%

8%

4%

Find: Expected Return, Risk, and CV