1. Chapter 2 Valuation, Risk, Return, and Uncertainty

2. Introduction Introduction Safe Dollars and Risky Dollars
Relationship Between Risk and Return
The Concept of Return
Some Statistical Facts of Life

3. Safe Dollars and Risky Dollars
A safe dollar is worth more than a risky dollar
Investing in the stock market is exchanging bird-in-the-hand safe dollars for a chance at a higher number of dollars in the future

4. Safe Dollars and Risky Dollars (cont’d)
Most investors are risk averse
People will take a risk only if they expect to be adequately rewarded for taking it
People have different degrees of risk aversion
Some people are more willing to take a chance than others

5. Choosing Among Risky Alternatives
Example
You have won the right to spin a lottery wheel one time. The wheel contains numbers 1 through 100, and a pointer selects one number when the wheel stops. The payoff alternatives are on the next slide.
Which alternative would you choose?

6. Choosing Among Risky Alternatives (cont’d)
$100
Average
payoff
–$89,000
[100]
$550
[91–100] $0
[51–100]
$90
$1,000
[1–99]
$50
[1–90]
$200
[1–50]
$110 D C B A
Number on lottery wheel appears in brackets.

7. Choosing Among Risky Alternatives (cont’d)
Example (cont’d)
Solution:
Most people would think Choice A is “safe.”
Choice B has an opportunity cost of $90 relative to Choice A.
People who get utility from playing a game pick Choice C.
People who cannot tolerate the chance of any loss would avoid Choice D.

8. Choosing Among Risky Alternatives (cont’d)
Example (cont’d)
Solution (cont’d):
Choice A is like buying shares of a utility stock.
Choice B is like purchasing a stock option.
Choice C is like a convertible bond.
Choice D is like writing out-of-the-money call options.

9. Risk Versus Uncertainty
Uncertainty involves a doubtful outcome
What birthday gift you will receive
If a particular horse will win at the track
Risk involves the chance of loss
If a particular horse will win at the track if you made a bet

10. Dispersion and Chance of Loss
There are two material factors we use in judging risk:
The average outcome
The scattering of the other possibilities around the average

11. Dispersion and Chance of Loss (cont’d)
Investment value
Investment A
Investment B
Time

12. Dispersion and Chance of Loss (cont’d)
Investments A and B have the same arithmetic mean
Investment B is riskier than Investment A

13. Concept of Utility
Utility measures the satisfaction people get out of something
Different individuals get different amounts of utility from the same source
Casino gambling
Pizza parties
CDs
Etc.

14. Diminishing Marginal Utility of Money
Rational people prefer more money to less
Money provides utility
Diminishing marginal utility of money
The relationship between more money and added utility is not linear
“I hate to lose more than I like to win”

15. Diminishing Marginal Utility of Money (cont’d)$

16. St. Petersburg Paradox Assume the following game:
A coin is flipped until a head appears
The payoff is based on the number of tails observed (n) before the first head
The payoff is calculated as $2n
What is the expected payoff?

17. St. Petersburg Paradox (cont’d)
Number of Tails Before First
Head
Probability
Payoff
× Payoff
(1/2) = 1/2 $1
$0.50 1
(1/2)2 = 1/4 $2 2
(1/2)3 = 1/8 $4 3
(1/2)4 = 1/16 $8 4
(1/2)5 = 1/32
$16 n
(1/2)n + 1
$2n
Total
1.00 

18. St. Petersburg Paradox (cont’d)
In the limit, the expected payoff is infinite
How much would you be willing to play the game?
Most people would only pay a couple of dollars
The marginal utility for each additional $0.50 declines

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

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

21. 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%

22. 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

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

24. 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

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

26. 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

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

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

29. 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 = %

30. 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

31. 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

32. 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

33. 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

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

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

36. 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

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

38. Definitions
Constants
Variables
Populations
Samples
Sample statistics

39. 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

40. 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.,

41. 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

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

43. 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)

44. 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

45. 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

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

47. Populations (cont’d)
A binomial distribution contains only two random variables
E.g., the toss of a coin
A finite population is one in which each possible outcome is known
E.g., a card drawn from a deck of cards

48. 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

49. 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

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

51. 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

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

53. 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%

54. 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

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

56. 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

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

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

59. 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

60. 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

61. 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:

62. 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:

63. 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

64. Correlations and Covariance (cont’d)

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

66. 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

70. 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

71. 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

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

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

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

75. 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

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

78. 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

79. Runs Test
A runs test allows the statistical testing of whether a series of price movements occurred by chance.
A run is defined as an uninterrupted sequence of the same observation. Ex: if the stock price increases 10 times in a row, then decreases 3 times, and then increases 4 times, we then say that we have three runs.

80. Notation R = number of runs (3 in this example)
n1 = number of observations in the first category. For instance, here we have a total of 14 “ups”, so n1=14.
n2 = number of observations in the second category. For instance, here we have a total of 3 “downs”, so n2=3.
Note that n1 and n2 could be the number of “Heads” and “Tails” in the case of a coin toss.

81. Statistical Test

82. Example Let the number of runs R=23 Let the number of ups n1=20
Let the number of downs n2=30

83. About 2.5% of the area under the normal curve is below a z score of -1.96.

84. Interpretation
Since our z-score is not in the lower tail (nor is it in the upper tail), the runs we have witnessed are purely the product of chance.
If, on the other hand, we had obtained a z-score in the upper (2.5%) or lower (2.5%) tail, we would then be 95% certain that this specific occurrence of runs didn’t happen by chance. (Or that we just witnessed an extremely rare event)