# Chapter 2 Valuation, Risk, Return, and Uncertainty

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Chapter 2 Valuation, Risk, Return, and Uncertainty

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

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

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

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?

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.

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.

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.

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

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

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

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

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.

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”

Diminishing Marginal Utility of Money (cont’d)
\$

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?

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

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

The Concept of Return Measurable return Expected return
Return on investment

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

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%

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

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

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

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

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

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

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

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 = %

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

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

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

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

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

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

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

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

Definitions Constants Variables Populations Samples Sample statistics

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

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.,

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

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

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)

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

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

Populations (cont’d) Positive Skewness Negative Skewness

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

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

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

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

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

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

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%

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

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

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

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

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

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

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

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:

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:

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

Correlations and Covariance (cont’d)

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

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

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

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

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

R Squared and Standard Errors
Application R squared Standard Errors

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

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

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

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

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.

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.

Statistical Test

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

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

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)