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

Uses of Statistics

Statistics is one of the tools used to make decisions in business

We apply statistical concepts in our lives As a student of business or economics,

basic knowledge and skills to organize, analyze, and transform data and to present the information.

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Why Study Statistics?

1. Numerical information is everywhere 2. Statistical techniques are used to make decisions that

affect our daily lives 3. The knowledge of statistical methods will help you

understand how decisions are made and give you a better understanding of how they affect you.

No matter what line of work you select, you will find

yourself faced with decisions where an understanding of data analysis is helpful.

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Who Uses Statistics?

Statistical techniques are used extensively by marketing, accounting, quality control, consumers, professional sports people, hospital administrators, educators, politicians, physicians, etc...

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Types of Statistics – Descriptive Statistics and Inferential Statistics

Descriptive Statistics - methods of organizing, summarizing, and presenting data in an informative way.

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Inferential Statistics: A decision, estimate, prediction, or generalization about a population, based on a sample.

Why take a sample instead of studying every member of the population?

1. Prohibitive cost of census 2. Destruction of item being studied may

be required 3. Not possible to test or inspect all

members of a population being studied

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Central Tendency - Measures of Location The purpose of a measure of location is to

pinpoint the center of a distribution of data. There are many measures of location. We

will consider five: 1. The arithmetic mean, 2. The weighted mean, 3. The median, 4. The mode, and 5. The geometric mean

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Properties of the Arithmetic Mean 1. Every set of interval-level and ratio-level data has a

mean. 2. All the values are included in computing the mean. 3. The mean is unique. 4. The sum of the deviations of each value from the mean is

zero.

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Population Mean For ungrouped data, the population mean is the

sum of all the population values divided by the total number of population values:

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Sample Mean

For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values:

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Weighted Mean

The weighted mean of a set of numbers X1, X2, ..., Xn, with corresponding weights w1, w2, ...,wn, is computed from the following formula:

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Example Weighted Mean

Stock Return Value of Investment

Madrid Barca

Liverpool

15% 17% 12%

$5,000,000 $8,000,000 $3,000,000

The Median

PROPERTIES OF THE MEDIAN 1. There is a unique median for each data set. 2. It is not affected by extremely large or small values and is

therefore a valuable measure of central tendency when such values occur.

3. It can be computed for ratio-level, interval-level, and ordinal-level data.

4. It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class.

MEDIAN The midpoint of the values after they have been ordered from the smallest to the largest, or the largest to the smallest.

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The Mode

MODE The value of the observation that appears most frequently.

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The Geometric Mean

Useful in finding the average change of percentages, ratios, indexes, or growth rates over time.

It has a wide application in business and economics because we are often interested in finding the percentage changes in sales, salaries, or economic figures, such as the GDP, which compound or build on each other.

The geometric mean will always be less than or equal to the arithmetic mean. The formula for the geometric mean is written:

EXAMPLE: The return on investment earned by Atkins Construction Company for four successive years was: 30 percent, 20 percent, -40 percent, and 200 percent. What is the geometric mean rate of return on investment?

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The Geometric Mean – Finding an Average Percent Change Over Time

EXAMPLE During the decade of the 1990s, and into the 2000s, Las

Vegas, Nevada, was the fastest-growing city in the United States. The population increased from 258,295 in 1990 to 607,876 in 2009. This is an increase of 349,581 people, or a 135.3 percent increase over the period. The population has more than doubled.

What is the average annual increase?

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Mean Deviation

A shortcoming of the range is that it is based on only two values, the highest and the lowest; it does not take into consideration all of the values.

The mean deviation does. It measures the mean amount by which the values in a population, or sample, vary from their mean

MEAN DEVIATION The arithmetic mean of the absolute values of the deviations from the arithmetic mean.

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Variance and Standard Deviation

VARIANCE The arithmetic mean of the squared deviations from the mean.

STANDARD DEVIATION The square root of the variance.

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Variance – Formula and Computation

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Sample Variance

sample the in nsobservatio of number the is sample the of mean the is

sample the in nobservatio each of value the is variance sample the is

:Where2

nX

Xs

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Sample Standard Deviation

sample the in nsobservatio of number the is sample the of mean the is

sample the in nobservatio each of value the is variance sample the is

:Where2

nX

Xs

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Coefficient of Variation

Indicates the relative magnitude of the standard deviation as compared with the mean of the distribution of measurements

Example

2005 2006 2007 2008 2009 2010 2011 2012 2013 13 11 15 12 17 12 13 11 17

Data of Stock Return in Percentage

Find: Mode, Median, Mean, Geometric Mean, Standard Deviation

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))((σ

)(RECV σ

=Risk-return trade0ff

Example Suppose there are data about stock return

on different investment conditions Investment Condition

Probability Return

Bullish Average Bearish

0.5 0.4 0.1

24% 15% 9%

Find: Expected Return, Risk, and CV