Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics

1. 1
Chapter 3
A Review of Statistical Principles
Useful in Finance

2. 2
Statistical thinking will one day be as
necessary for effective citizenship as the
ability to read and write.
- H.G. Wells

3. 3
Outline
Introduction
The concept of return
Some statistical facts of life

4. 4
Introduction
Statistical principles are useful in:
• The theory of finance
• Understanding how portfolios work
• Why diversifying portfolios is a good idea

5. 5
The Concept of Return
Measurable return
Expected return
Return on investment

6. 6
Measurable Return
Definition
Holding period return
Arithmetic mean return
Geometric mean return
Comparison of arithmetic and geometric
mean returns

7. 7
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%

8. 8
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

9. 9
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
n
i
i
i
R
n
R i
=
=
=
∑

10. 10
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

11. 11
Geometric Mean Return
The geometric mean return is the nth root
of the product of n values:
°
1/
1
Geometric mean (1 ) 1
nn
i
i
R
=
 
= + − 
 
∏

12. 12
Arithmetic and
Geometric Mean Returns
Example
Assume the following sample of weekly stock returns:
Week Return Return Relative
1 0.0084 1.0084
2 -0.0045 0.9955
3 0.0021 1.0021
4 0.0000 1.000

13. 13
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 0.0000
4
0.0015
n
i
i
R
n=
=
− + +
=
=
∑

14. 14
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
i
i
R
=
 
= + − 
 
= × × × −
=
∏

15. 15
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%

16. 16
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

17. 17
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

18. 18
Return on Investment (ROI)
Definition
Measuring total risk

19. 19
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

20. 20
Measuring Total Risk
Standard deviation and variance
Semi-variance

21. 21
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

22. 22
Standard Deviation and
Variance (cont’d)
General equation for variance:
If all outcomes are equally likely:
[ ]
2
2
1
Variance prob( )
n
i i
i
x x xσ
=
= = −∑
[ ]
2
2
1
1 n
i
i
x x
n
σ
=
= −∑

23. 23
Standard Deviation and
Variance (cont’d)
Equation for standard deviation:
[ ]
2
2
1
Standard deviation prob( )
n
i i
i
x x xσ σ
=
= = = −∑

24. 24
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

25. 25
Some Statistical Facts of Life
Definitions
Properties of random variables
Linear regression
R squared and standard errors

26. 26
Definitions
Constants
Variables
Populations
Samples
Sample statistics

27. 27
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

28. 28
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%

29. 29
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

30. 30
Variables (cont’d)
Quantitative variables are measured by real
numbers
• E.g., numerical measurement
Qualitative variables are categorical
• E.g., hair color

31. 31
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)

32. 32
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

33. 33
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

34. 34
Populations (cont’d)
Positive Skewness Negative Skewness

35. 35
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

36. 36
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

37. 37
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

38. 38
Samples
A sample is any subset of a population
• E.g., a sample of past monthly stock returns of
a particular stock

39. 39
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

40. 40
Properties of
Random Variables
Example
Central tendency
Dispersion
Logarithms
Expectations
Correlation and covariance

41. 41
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%

42. 42
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 i
i
E R R
n =
= ∑% %

43. 43
Example (cont’d)
The expected returns for Stocks A and B are:
1
1 1
( ) (2% 1% 4% 1%) 1.50%
4
n
A i
i
E R R
n =
= = − + + =∑% %
1
1 1
( ) (3% 0% 5% 4%) 3.00%
4
n
B i
i
E R R
n =
= = + + + =∑% %

44. 44
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
( )
( )
22
22
i
i
E x x
E x x
σ
σ σ
 = −
 
 = = −
 
%
%

45. 45
Example (cont’d)
The variance ad standard deviation for Stock A are:
( )
22
2 2 2 2
2
1
(2% 1.5%) ( 1% 1.5%) (4% 1.5%) (1% 1.5%)
4
1
(0.0013) 0.000325
4
0.000325 0.018 1.8%
iE x xσ
σ σ
 = −
 
 = − + − − + − + − 
= =
= = = =
%

46. 46
Example (cont’d)
The variance ad standard deviation for Stock B are:
( )
22
2 2 2 2
2
1
(3% 3.0%) (0% 3.0%) (5% 3.0%) (4% 3.0%)
4
1
(0.0014) 0.00035
4
0.00035 0.0187 1.87%
iE x xσ
σ σ
 = −
 
 = − + − + − + − 
= =
= = = =
%

47. 47
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

48. 48
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

49. 49
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=% %

50. 50
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+ = +% % % %

51. 51
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

52. 52
Correlations and Covariance
(cont’d)
( , ) ( )( )ABCOV A B E A A B Bσ  = = − − 
% %% %
( , )
AB
A B
COV A B
ρ
σ σ
=
% %

53. 53
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%)
4
1
(0.001225)
4
0.000306
ABσ = × + − ×− + × + − ×
=
=
( , ) 0.000306
0.909
(0.018)(0.0187)
AB
A B
COV A B
ρ
σ σ
= = =
% %

54. 54
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

55. 55
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

56. 56
Linear Regression (cont’d)
Example
Assume the following sample of weekly stock and stock
index returns:
Week Stock Return Index Return
1 0.0084 0.0088
2 -0.0045 -0.0048
3 0.0021 0.0019
4 0.0000 0.0005

57. 57
Linear Regression (cont’d)
Example (cont’d)
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
-0.01 -0.005 0 0.005 0.01
Return (Market)
Return(Stock)
Intercept = 0
Slope = 0.96
R squared = 0.99

58. 58
R Squared and
Standard Errors
Application
R squared
Standard Errors

59. 59
Application
R-squared and the standard error are used
to assess the accuracy of calculated
statistics

60. 60
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

61. 61
Standard Errors
The standard error is the standard deviation
divided by the square root of the number of
observations:
Standard error
n
σ
=

62. 62
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