Chapter 2 Valuation, Risk, Return, and Uncertainty

1. Chapter 2Valuation, 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. A B C D [1–50] $110 [1–50] $200 [1–90] $50 [1–99] $1,000 [51–100] $90 [51–100] $0 [91–100] $550 [100] –$89,000 Average payoff $100 $100 $100 $100 Number on lottery wheel appears in brackets. Choosing Among Risky Alternatives (cont’d)

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 • Riskinvolves 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) Utility $

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)

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 $108.33 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:

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 = -13.40%

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