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Chapter 3

A Review of Statistical Principles Useful in Finance

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Statistical thinking will one day be as necessary for effective citizenship as the

ability to read and write.

- H.G. Wells

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Outline Introduction The concept of return Some statistical facts of life

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Introduction Statistical principles are useful in:

• The theory of finance

• Understanding how portfolios work

• Why diversifying portfolios is a good idea

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The Concept of Return Measurable return Expected return Return on investment

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Measurable Return Definition Holding period return Arithmetic mean return Geometric mean return Comparison of arithmetic and geometric

mean returns

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Definition A general definition of return is the benefit

associated with an investment• In most cases, return is measurable• E.g., a $100 investment at 8%, compounded

continuously is worth $108.33 after one year– The return is $8.33, or 8.33%

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Holding Period Return The calculation of a holding period return

is independent of the passage of time• E.g., you buy a bond for $950, receive $80 in

interest, and later sell the bond for $980– The return is ($80 + $30)/$950 = 11.58%– The 11.58% could have been earned over one year

or one week

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Arithmetic Mean Return The arithmetic mean return is the

arithmetic average of several holding period returns measured over the same holding period:

1

Arithmetic mean

the rate of return in period

ni

i

i

Rn

R i

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Arithmetic Mean Return (cont’d)

Arithmetic means are a useful proxy for expected returns

Arithmetic means are not especially useful for describing historical returns• It is unclear what the number means once it is

determined

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Geometric Mean Return The geometric mean return is the nth root

of the product of n values:

1/

1

Geometric mean (1 ) 1nn

ii

R

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Arithmetic and Geometric Mean Returns

Example

Assume the following sample of weekly stock returns:

Week Return Return Relative

1 0.0084 1.00842 -0.0045 0.99553 0.0021 1.00214 0.0000 1.000

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Arithmetic and Geometric Mean Returns (cont’d)

Example (cont’d)

What is the arithmetic mean return?

Solution:

1

Arithmetic mean

0.0084 0.0045 0.0021 0.00004

0.0015

ni

i

Rn

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Arithmetic and Geometric Mean Returns (cont’d)

Example (cont’d)

What is the geometric mean return?

Solution:

1/

1

1/ 4

Geometric mean (1 ) 1

1.0084 0.9955 1.0021 1.0000 1

0.001489

nn

ii

R

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Comparison of Arithmetic &Geometric Mean Returns

The geometric mean reduces the likelihood of nonsense answers• Assume a $100 investment falls by 50% in

period 1 and rises by 50% in period 2

• The investor has $75 at the end of period 2– Arithmetic mean = (-50% + 50%)/2 = 0%– Geometric mean = (0.50 x 1.50)1/2 –1 = -13.40%

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Comparison of Arithmetic &Geometric Mean Returns

The geometric mean must be used to determine the rate of return that equates a present value with a series of future values

The greater the dispersion in a series of numbers, the wider the gap between the arithmetic and geometric mean

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Expected Return Expected return refers to the future

• In finance, what happened in the past is not as important as what happens in the future

• We can use past information to make estimates about the future

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Return on Investment (ROI) Definition Measuring total risk

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Definition Return on investment (ROI) is a term that

must be clearly defined• Return on assets (ROA)

• Return on equity (ROE)– ROE is a leveraged version of ROA

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Measuring Total Risk Standard deviation and variance Semi-variance

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Standard Deviation and Variance

Standard deviation and variance are the most common measures of total risk

They measure the dispersion of a set of observations around the mean observation

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Standard Deviation and Variance (cont’d)

General equation for variance:

If all outcomes are equally likely:

2

2

1

Variance prob( )n

i ii

x x x

2

2

1

1 n

ii

x xn

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Standard Deviation and Variance (cont’d)

Equation for standard deviation:

2

2

1

Standard deviation prob( )n

i ii

x x x

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Semi-Variance Semi-variance considers the dispersion only

on the adverse side• Ignores all observations greater than the mean• Calculates variance using only “bad” returns

that are less than average• Since risk means “chance of loss” positive

dispersion can distort the variance or standard deviation statistic as a measure of risk

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Some Statistical Facts of Life Definitions Properties of random variables Linear regression R squared and standard errors

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Definitions Constants Variables Populations Samples Sample statistics

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Constants A constant is a value that does not change

• E.g., the number of sides of a cube• E.g., the sum of the interior angles of a triangle

A constant can be represented by a numeral or by a symbol

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Variables A variable has no fixed value

• It is useful only when it is considered in the context of other possible values it might assume

In finance, variables are called random variables• Designated by a tilde

– E.g., x

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Variables (cont’d) Discrete random variables are countable

• E.g., the number of trout you catch

Continuous random variables are measurable• E.g., the length of a trout

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Variables (cont’d) Quantitative variables are measured by real

numbers• E.g., numerical measurement

Qualitative variables are categorical• E.g., hair color

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Variables (cont’d) Independent variables are measured directly

• E.g., the height of a box

Dependent variables can only be measured once other independent variables are measured• E.g., the volume of a box (requires length,

width, and height)

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Populations A population is the entire collection of a

particular set of random variables The nature of a population is described by

its distribution• The median of a distribution is the point where

half the observations lie on either side• The mode is the value in a distribution that

occurs most frequently

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Populations (cont’d) A distribution can have skewness

• There is more dispersion on one side of the distribution

• Positive skewness means the mean is greater than the median

– Stock returns are positively skewed• Negative skewness means the mean is less than

the median

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Populations (cont’d)Positive Skewness Negative Skewness

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Populations (cont’d) A binomial distribution contains only two

random variables• E.g., the toss of a die

A finite population is one in which each possible outcome is known• E.g., a card drawn from a deck of cards

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Populations (cont’d) An infinite population is one where not all

observations can be counted• E.g., the microorganisms in a cubic mile of

ocean water

A univariate population has one variable of interest

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Populations (cont’d) A bivariate population has two variables of

interest• E.g., weight and size

A multivariate population has more than two variables of interest• E.g., weight, size, and color

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Samples A sample is any subset of a population

• E.g., a sample of past monthly stock returns of a particular stock

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Sample Statistics Sample statistics are characteristics of

samples• A true population statistic is usually

unobservable and must be estimated with a sample statistic

– Expensive– Statistically unnecessary

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Properties of Random Variables

Example Central tendency Dispersion Logarithms Expectations Correlation and covariance

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Example

Assume the following monthly stock returns for Stocks A and B:

Month Stock A Stock B

1 2% 3%2 -1% 0%3 4% 5%4 1% 4%

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Central Tendency Central tendency is what a random variable

looks like, on average The usual measure of central tendency is

the population’s expected value (the mean)• The average value of all elements of the

population

1

1( )n

i ii

E R Rn

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Example (cont’d)

The expected returns for Stocks A and B are:

1

1 1( ) (2% 1% 4% 1%) 1.50%4

n

A ii

E R Rn

1

1 1( ) (3% 0% 5% 4%) 3.00%4

n

B ii

E R Rn

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Dispersion Investors are interest in the best and the

worst in addition to the average A common measure of dispersion is the

variance or standard deviation

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22

i

i

E x x

E x x

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Example (cont’d)

The variance ad standard deviation for Stock A are:

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2 2 2 2

2

1 (2% 1.5%) ( 1% 1.5%) (4% 1.5%) (1% 1.5%)41 (0.0013) 0.0003254

0.000325 0.018 1.8%

iE x x

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Example (cont’d)

The variance ad standard deviation for Stock B are:

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2 2 2 2

2

1 (3% 3.0%) (0% 3.0%) (5% 3.0%) (4% 3.0%)41 (0.0014) 0.000354

0.00035 0.0187 1.87%

iE x x

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Logarithms Logarithms reduce the impact of extreme

values• E.g., takeover rumors may cause huge price

swings• A logreturn is the logarithm of a return

Logarithms make other statistical tools more appropriate• E.g., linear regression

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Logarithms (cont’d) Using logreturns on stock return

distributions:• Take the raw returns

• Convert the raw returns to return relatives

• Take the natural logarithm of the return relatives

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Expectations The expected value of a constant is a

constant:

The expected value of a constant times a random variable is the constant times the expected value of the random variable:

( )E a a

( ) ( )E ax aE x

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Expectations (cont’d) The expected value of a combination of

random variables is equal to the sum of the expected value of each element of the combination:

( ) ( ) ( )E x y E x E y

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Correlations and Covariance Correlation is the degree of association

between two variables

Covariance is the product moment of two random variables about their means

Correlation and covariance are related and generally measure the same phenomenon

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Correlations and Covariance (cont’d)

( , ) ( )( )ABCOV A B E A A B B

( , )AB

A B

COV A B

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Example (cont’d)

The covariance and correlation for Stocks A and B are:

1 (0.5% 0.0%) ( 2.5% 3.0%) (2.5% 2.0%) ( 0.5% 1.0%)41 (0.001225)40.000306

AB

( , ) 0.000306 0.909(0.018)(0.0187)AB

A B

COV A B

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Correlations and Covariance Correlation ranges from –1.0 to +1.0.

• Two random variables that are perfectly positively correlated have a correlation coefficient of +1.0

• Two random variables that are perfectly negatively correlated have a correlation coefficient of –1.0

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Linear Regression Linear regression is a mathematical

technique used to predict the value of one variable from a series of values of other variables• E.g., predict the return of an individual stock

using a stock market index Regression finds the equation of a line

through the points that gives the best possible fit

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Linear Regression (cont’d)Example

Assume the following sample of weekly stock and stock index returns:

Week Stock Return Index Return

1 0.0084 0.00882 -0.0045 -0.00483 0.0021 0.00194 0.0000 0.0005

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Linear Regression (cont’d)Example (cont’d)

-0.006

-0.004-0.002

00.002

0.004

0.0060.008

0.01

-0.01 -0.005 0 0.005 0.01

Return (Market)

Ret

urn

(Sto

ck)

Intercept = 0Slope = 0.96R squared = 0.99

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R Squared and Standard Errors

Application R squared Standard Errors

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Application R-squared and the standard error are used

to assess the accuracy of calculated statistics

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R Squared R squared is a measure of how good a fit we get

with the regression line• If every data point lies exactly on the line, R squared is

100%

R squared is the square of the correlation coefficient between the security returns and the market returns• It measures the portion of a security’s variability that is

due to the market variability

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Standard Errors The standard error is the standard deviation

divided by the square root of the number of observations:

Standard errorn

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Standard Errors (cont’d) The standard error enables us to determine

the likelihood that the coefficient is statistically different from zero• About 68% of the elements of the distribution

lie within one standard error of the mean• About 95% lie within 1.96 standard errors• About 99% lie within 3.00 standard errors