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

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

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

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

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

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

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 $ after one year
The return is $8.33, or 8.33%

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. Arithmetic Mean Return
The arithmetic mean return is the arithmetic average of several holding period returns measured over the same holding period:

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. Geometric Mean Return
The geometric mean return is the nth root of the product of n values:

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
0.9955 3
0.0021
1.0021 4
0.0000
1.000

13. Arithmetic and Geometric Mean Returns (cont’d)
Example (cont’d)
What is the arithmetic mean return?
Solution:

14. Arithmetic and Geometric Mean Returns (cont’d)
Example (cont’d)
What is the geometric mean return?
Solution:

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 = %

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

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. Measuring Total Risk
Standard deviation and variance
Semi-variance

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. Standard Deviation and Variance (cont’d)
General equation for variance:
If all outcomes are equally likely:

23. Standard Deviation and Variance (cont’d)
Equation for standard deviation:

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

26. Definitions
Constants
Variables
Populations
Samples
Sample statistics

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. 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.,

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

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

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

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

41. Example Assume the following monthly stock returns for Stocks A and B:
Stock A
Stock B 1 2% 3% 2
-1% 0% 3 4% 5% 4 1%

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

43. Example (cont’d)
The expected returns for Stocks A and B are:

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

45. Example (cont’d)
The variance ad standard deviation for Stock A are:

46. Example (cont’d)
The variance ad standard deviation for Stock B are:

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. 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. 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:

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:

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. Correlations and Covariance (cont’d)

53. Example (cont’d)
The covariance and correlation for Stocks A and B are:

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. 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. 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 3
0.0021
0.0019 4
0.0000
0.0005

57. Linear Regression (cont’d)
Example (cont’d)
Intercept = 0
Slope = 0.96
R squared = 0.99

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

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

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

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