Chapter 3
A Review of Statistical Principles
Useful in Finance
1
Statistical thinking will one day be as
necessary for effective citizenship as the
ability to read and write.
- H.G. Wells
2
Outline
 Introduction
 The
concept of return
 Some statistical facts of life
3
Introduction
 Statistical
principles are useful in:
• The theory of finance
• Understanding how portfolios work
• Why diversifying portfolios is a good idea
4
The Concept of Return
 Measurable
return
 Expected return
 Return on investment
5
Measurable Return
 Definition
 Holding
period return
 Arithmetic mean return
 Geometric mean return
 Comparison of arithmetic and geometric
mean returns
6
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%
7
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:
n
Ri
Arithmetic mean  
i 1 n
Ri  the rate of return in period i
9
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
10
Geometric Mean Return
 The
geometric mean return is the nth root
of the product of n values:
1/ n


Geometric mean   (1  Ri ) 
 i 1

n
1
11
Arithmetic and
Geometric Mean Returns
Example
Assume the following sample of weekly stock returns:
Week
Return
Return Relative
1
2
0.0084
-0.0045
1.0084
0.9955
3
0.0021
1.0021
4
0.0000
1.000
12
Arithmetic and Geometric
Mean Returns (cont’d)
Example (cont’d)
What is the arithmetic mean return?
Solution:
n
Ri
Arithmetic mean  
i 1 n
0.0084  0.0045  0.0021  0.0000

4
 0.0015
13
Arithmetic and Geometric
Mean Returns (cont’d)
Example (cont’d)
What is the geometric mean return?
Solution:
1/ n


Geometric mean   (1  Ri ) 
 i 1

n
1
 1.0084  0.9955 1.00211.0000
1/ 4
 0.001489
1
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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%
15
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
16
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
17
Return on Investment (ROI)
 Definition
 Measuring
total risk
18
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
19
Measuring Total Risk
 Standard
deviation and variance
 Semi-variance
20
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
21
Standard Deviation and
Variance (cont’d)
 General
equation for variance:
2
n
Variance     prob( xi )  xi  x 
2
i 1
 If
all outcomes are equally likely:
n
2
1
    xi  x 
n i 1
2
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Standard Deviation and
Variance (cont’d)
 Equation
for standard deviation:
Standard deviation     2 
2
n
 prob( x )  x  x 
i 1
i
i
23
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
24
Some Statistical Facts of Life
 Definitions
 Properties
of random variables
 Linear regression
 R squared and standard errors
25
Definitions
 Constants
 Variables
 Populations
 Samples
 Sample
statistics
26
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
27
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
28
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
29
Variables (cont’d)
 Quantitative
variables are measured by real
numbers
• E.g., numerical measurement
 Qualitative
variables are categorical
• E.g., hair color
30
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
32
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
33
Populations (cont’d)
Positive Skewness
Negative Skewness
34
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
36
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
37
Samples
 A sample
is any subset of a population
• E.g., a sample of past monthly stock returns of
a particular stock
38
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
39
Properties of
Random Variables
 Example
 Central
tendency
 Dispersion
 Logarithms
 Expectations
 Correlation and covariance
40
Example
Assume the following monthly stock returns for Stocks A
and B:
Month
Stock A
Stock B
1
2
3
2%
-1%
4%
3%
0%
5%
4
1%
4%
41
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 n
E ( Ri )   Ri
n i 1
42
Example (cont’d)
The expected returns for Stocks A and B are:
1 n
1
E ( RA )   Ri  (2%  1%  4%  1%)  1.50%
n i 1
4
1 n
1
E ( RB )   Ri  (3%  0%  5%  4%)  3.00%
n i 1
4
43
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
  E  xi  x  
2
2


    E  xi  x  
2
2


44
Example (cont’d)
The variance ad standard deviation for Stock A are:
2
 2  E  xi  x  


1
(2%  1.5%) 2  (1%  1.5%) 2  (4%  1.5%) 2  (1%  1.5%) 2 
4
1
 (0.0013)  0.000325
4

   2  0.000325  0.018  1.8%
45
Example (cont’d)
The variance ad standard deviation for Stock B are:
2
 2  E  xi  x  


1
(3%  3.0%)2  (0%  3.0%)2  (5%  3.0%)2  (4%  3.0%)2 
4
1
 (0.0014)  0.00035
4

   2  0.00035  0.0187  1.87%
46
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
47
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
48
Expectations
 The
expected value of a constant is a
constant:
E (a)  a
 The
expected value of a constant times a
random variable is the constant times the
expected value of the random variable:
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 )
50
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
51
Correlations and Covariance
(cont’d)
COV ( A, B)   AB  E ( A  A)( B  B ) 
 AB 
COV ( A, B)
 A B
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Example (cont’d)
The covariance and correlation for Stocks A and B are:
 AB
1
  (0.5%  0.0%)  (2.5%  3.0%)  (2.5%  2.0%)  (0.5%  1.0%)
4
1
 (0.001225)
4
 0.000306
 AB 
COV ( A, B)
 A B
0.000306

 0.909
(0.018)(0.0187)
53
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
54
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
55
Linear Regression (cont’d)
Example
Assume the following sample of weekly stock and stock
index returns:
Week
Stock Return
Index Return
1
2
0.0084
-0.0045
0.0088
-0.0048
3
4
0.0021
0.0000
0.0019
0.0005
56
Linear Regression (cont’d)
Return (Stock)
Example (cont’d)
0.01
Intercept = 0
0.008
Slope = 0.96
R squared = 0.99 0.006
0.004
0.002
0
-0.01
-0.005 -0.002 0
0.005
0.01
-0.004
-0.006
Return (Market)
57
R Squared and
Standard Errors
 Application
R
squared
 Standard Errors
58
Application
 R-squared
and the standard error are used
to assess the accuracy of calculated
statistics
59
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
60
Standard Errors
 The
standard error is the standard deviation
divided by the square root of the number of
observations:
Standard error 

n
61
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
62