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

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

3. Valuation Valuation may be the most important part of the study of investments – Security analysts make a career of estimating “what you get” for “what you pay” – The time value of money is one of the two key concepts in finance and is very useful in valuation 3

4. Growing Income Streams A growing stream is one in which each successive cash flow is larger than the previous one – A common problem is one in which the cash flows grow by some fixed percentage 4

5. Growing Annuity A growing annuity is an annuity in which the cash flows grow at a constant rate g: 5

6. Growing Perpetuity A growing perpetuity is an annuity where the cash flows continue indefinitely: 6

7. 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 7

8. 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 8

9. 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? 9

10. Choosing Among Risky Alternatives (cont’d) 10 $100 Average payoff –$89,000[100]$550[91–100]$0[51–100]$90[51–100] $1,000[1–99]$50[1–90]$200[1–50]$110[1–50] DCBA Number on lottery wheel appears in brackets.

11. 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. 11

12. 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. 12

13. 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 13

14. 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 14

15. Dispersion and Chance of Loss (cont’d) 15 Investment A Investment B Time Investment value

16. Dispersion and Chance of Loss (cont’d) Investments A and B have the same arithmetic mean Investment B is riskier than Investment A 16

17. Types of Risk Total risk refers to the overall variability of the returns of financial assets Undiversifiable risk is risk that must be borne by virtue of being in the market – Arises from systematic factors that affect all securities of a particular type 17

18. Types of Risk (cont’d) Diversifiable risk can be removed by proper portfolio diversification – The ups and down of individual securities due to company-specific events will cancel each other out – The only return variability that remains will be due to economic events affecting all stocks 18

19. Relationship Between Risk and Return Direct Relationship Concept of Utility Diminishing Marginal Utility of Money St. Petersburg Paradox Fair Bets The Consumption Decision Other Considerations 19

20. Direct Relationship The more risk someone bears, the higher the expected return The appropriate discount rate depends on the risk level of the investment The riskless rate of interest can be earned without bearing any risk 20

21. Direct Relationship (cont’d) 21 Risk Expected return RfRf 0

22. Direct Relationship (cont’d) The expected return is the weighted average of all possible returns – The weights reflect the relative likelihood of each possible return The risk is undiversifiable risk – A person is not rewarded for bearing risk that could have been diversified away 22

23. 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. 23

24. 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” 24

25. Diminishing Marginal Utility of Money (cont’d) 25 $ Utility

26. 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 $2 n What is the expected payoff? 26

27. St. Petersburg Paradox (cont’d) 27 Number of Tails Before First HeadProbabilityPayoff Probability × Payoff 0(1/2) = 1/2$1$0.50 1(1/2) 2 = 1/4$2$0.50 2(1/2) 3 = 1/8$4$0.50 3(1/2) 4 = 1/16$8$0.50 4(1/2) 5 = 1/32$16$0.50 n(1/2) n + 1 $2 n $0.50 Total1.00 

28. 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 28

29. Fair Bets A fair bet is a lottery in which the expected payoff is equal to the cost of playing – e.g., matching quarters – e.g., matching serial numbers on $100 bills Most people will not take a fair bet unless the dollar amount involved is small – Utility lost is greater than utility gained 29

30. The Consumption Decision The consumption decision is the choice to save or to borrow – If interest rates are high, we are inclined to save e.g., open a new savings account – If interest rates are low, borrowing looks attractive e.g., a bigger home mortgage 30

31. The Consumption Decision (cont’d) The equilibrium interest rate causes savers to deposit a sufficient amount of money to satisfy the borrowing needs of the economy 31

32. Other Considerations Psychic Return Price Risk versus Convenience Risk 32

33. Psychic Return Psychic return comes from an individual disposition about something – People get utility from more expensive things, even if the quality is not higher than cheaper alternatives e.g., Rolex watches, designer jeans 33

34. Price Risk versus Convenience Risk Price risk refers to the possibility of adverse changes in the value of an investment due to: – A change in market conditions – A change in the financial situation – A change in public attitude e.g., rising interest rates influence stock prices, and a change in the price of gold can affect the value of the dollar 34

35. Price Risk versus Convenience Risk (cont’d) Convenience risk refers to a loss of managerial time rather than a loss of dollars – e.g., a bond’s call provision Allows the issuer to call in the debt early, meaning the investor has to look for other investments 35

36. The Concept of Return “Return” can mean various things, and it is important to be clear when discussing an investment 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 percent, compounded continuously is worth $108.33 after one year The return is $8.33, or 8.33 percent 36

37. 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 percent The 11.58 percent could have been earned over one year or one week 37

38. Arithmetic Mean Return The arithmetic mean return is the arithmetic average of several holding period returns measured over the same holding period: 38

39. Arithmetic Mean Return (cont’d) Arithmetic means are a useful proxy for expected returns Arithmetic means are not especially useful for describing historical return data – It is unclear what the number means once it is determined 39

40. Geometric Mean Return The geometric mean return is the nth root of the product of n returns: 40

41. Arithmetic and Geometric Mean Returns Example Assume the following sample of weekly stock returns: 41 WeekReturnReturn Relative 10.00841.0084 2–0.00450.9955 30.00211.0021 40.00001.000

42. Arithmetic and Geometric Mean Returns (cont’d) Example (cont’d) What is the arithmetic mean return? Solution: 42

43. Arithmetic and Geometric Mean Returns (cont’d) Example (cont’d) What is the geometric mean return? Solution: 43

44. Comparison of Arithmetic and Geometric Mean Returns The geometric mean reduces the likelihood of nonsense answers – Assume a $100 investment falls by 50 percent in period 1 and rises by 50 percent in period 2 – The investor has $75 at the end of period 2 Arithmetic mean = [(0.50) + (–0.50)]/2 = 0% Geometric mean = (0.50 × 1.50) 1/2 – 1 = –13.40% 44

45. Comparison of Arithmetic and Geometric Mean Returns (Cont’d) 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 mean and geometric mean 45

46. 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 46

47. Return on Investment Return on investment (ROI) is a term that must be clearly defined – Return on assets (ROA) Return ÷ Total Assets – Return on equity (ROE) Return ÷ Total stockholder’s Equity ROE is a leveraged version of ROA 47

48. 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 48

49. Standard Deviation and Variance (cont’d) General equation for variance: If all outcomes are equally likely: 49

50. 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 50

51. Example Assume the following monthly stock returns for Stocks A and B: 51 MonthStock AStock B 12%3% 2–1%0% 34%5% 41%4%

52. 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 52

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

54. Dispersion Investors are interested in the variation of actual values around the average A common measure of dispersion is the variance or standard deviation 54

55. Example (cont’d) The variance and standard deviation for Stock A are: 55

56. Example (cont’d) The variance and standard deviation for Stock B are: 56

57. 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 relative Logarithms make other statistical tools more appropriate – e.g., linear regression 57

58. 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 58

59. 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 59

60. Correlations and Covariance (cont’d) 60

61. Example (cont’d) The covariance and correlation for Stocks A and B are: 61

62. Correlations and Covariance (cont’d) 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 62

63. 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 Linear regression finds the equation of a line through the points that gives the best possible fit 63

64. Linear Regression (cont’d) Example Assume the following sample of weekly stock and stock index returns: 64 WeekStock ReturnIndex Return 10.00840.0088 2–0.0045–0.0048 30.00210.0019 40.00000.0005

65. Linear Regression (cont’d) Example (cont’d) 65 Intercept = 0 Slope = 0.96 R squared = 0.99

66. R Squared and Standard Errors R squared and the standard error are used to assess the accuracy of calculated securities 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 66

67. Standard Errors The standard error is equal to the standard deviation divided by the square root of the number of observations: 67

68. Standard Errors (cont’d) The standard error enables us to determine the likelihood that the coefficient is statistically different from zero – About 68 percent of the elements of the distribution lie within one standard error of the mean – About 95 percent lie within 1.96 standard errors – About 99 percent lie within 3.00 standard errors 68