1. 1 Chapter 2 Valuation, Risk, Return, and Uncertainty Portfolio Construction, Management, & Protection, 5e, Robert A. Strong Copyright ©2009 by South-Western, a division of Thomson Business & Economics. All rights reserved.

2. 2 It’s what we learn after we think we know it all that counts. Kin Hubbard

3. 3 Outline u Introduction u Valuation u Safe Dollars and Risky Dollars u Relationship Between Risk and Return u The Concept of Return u Some Statistical Facts of Life

4. 4 Introduction u The occasional reading of basic material in your chosen field is an excellent philosophical exercise Do not be tempted to conclude that you “know it all” –e.g., what is the present value of a growing perpetuity that begins payments in five years?

5. 5 Valuation u 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

6. 6 Growing Income Streams u 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

7. 7 Growing Annuity u A growing annuity is an annuity in which the cash flows grow at a constant rate g:

8. 8 Growing Perpetuity u A growing perpetuity is an annuity where the cash flows continue indefinitely:

9. 9 Safe Dollars and Risky Dollars u 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

10. 10 Safe Dollars and Risky Dollars (cont’d) u Most investors are risk averse People will take a risk only if they expect to be adequately rewarded for taking it u People have different degrees of risk aversion Some people are more willing to take a chance than others

11. 11 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?

12. 12 Choosing Among Risky Alternatives (cont’d) $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.

13. 13 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.

14. 14 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.

15. 15 Risk Versus Uncertainty u Uncertainty involves a doubtful outcome What birthday gift you will receive If a particular horse will win at the track u Risk involves the chance of loss If a particular horse will win at the track if you made a bet

16. 16 Dispersion and Chance of Loss u There are two material factors we use in judging risk: The average outcome The scattering of the other possibilities around the average

17. 17 Dispersion and Chance of Loss (cont’d) Investment A Investment B Time Investment value

18. 18 Dispersion and Chance of Loss (cont’d) u Investments A and B have the same arithmetic mean u Investment B is riskier than Investment A

19. 19 Types of Risk u Total risk refers to the overall variability of the returns of financial assets u 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

20. 20 Types of Risk (cont’d) u 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

21. 21 Relationship Between Risk and Return u Direct Relationship u Concept of Utility u Diminishing Marginal Utility of Money u St. Petersburg Paradox u Fair Bets u The Consumption Decision u Other Considerations

22. 22 Direct Relationship u The more risk someone bears, the higher the expected return u The appropriate discount rate depends on the risk level of the investment u The riskless rate of interest can be earned without bearing any risk

23. 23 Direct Relationship (cont’d) Risk Expected return RfRf 0

24. 24 Direct Relationship (cont’d) u The expected return is the weighted average of all possible returns The weights reflect the relative likelihood of each possible return u The risk is undiversifiable risk A person is not rewarded for bearing risk that could have been diversified away

25. 25 Concept of Utility u 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.

26. 26 Diminishing Marginal Utility of Money u 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”

27. 27 Diminishing Marginal Utility of Money (cont’d) $ Utility

28. 28 St. Petersburg Paradox u 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 u What is the expected payoff?

29. 29 St. Petersburg Paradox (cont’d) 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 

30. 30 St. Petersburg Paradox (cont’d) u In the limit, the expected payoff is infinite u 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

31. 31 Fair Bets u 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 u Most people will not take a fair bet unless the dollar amount involved is small Utility lost is greater than utility gained

32. 32 The Consumption Decision u 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

33. 33 The Consumption Decision (cont’d) u The equilibrium interest rate causes savers to deposit a sufficient amount of money to satisfy the borrowing needs of the economy

34. 34 Other Considerations u Psychic Return u Price Risk versus Convenience Risk

35. 35 Psychic Return u 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

36. 36 Price Risk versus Convenience Risk u 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 u e.g., rising interest rates influence stock prices, and a change in the price of gold can affect the value of the dollar

37. 37 Price Risk versus Convenience Risk (cont’d) u 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

38. 38 The Concept of Return u “Return” can mean various things, and it is important to be clear when discussing an investment u 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

39. 39 Holding Period Return u 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

40. 40 Arithmetic Mean Return u The arithmetic mean return is the arithmetic average of several holding period returns measured over the same holding period:

41. 41 Arithmetic Mean Return (cont’d) u Arithmetic means are a useful proxy for expected returns u Arithmetic means are not especially useful for describing historical return data It is unclear what the number means once it is determined

42. 42 Geometric Mean Return u The geometric mean return is the nth root of the product of n values:

43. 43 Arithmetic and Geometric Mean Returns Example Assume the following sample of weekly stock returns: WeekReturnReturn Relative 10.00841.0084 2–0.00450.9955 30.00211.0021 40.00001.000

44. 44 Arithmetic and Geometric Mean Returns (cont’d) Example (cont’d) What is the arithmetic mean return? Solution:

45. 45 Arithmetic and Geometric Mean Returns (cont’d) Example (cont’d) What is the geometric mean return? Solution:

46. 46 Comparison of Arithmetic and Geometric Mean Returns u 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%

47. 47 Comparison of Arithmetic and Geometric Mean Returns (Cont’d) u The geometric mean must be used to determine the rate of return that equates a present value with a series of future values u The greater the dispersion in a series of numbers, the wider the gap between the arithmetic mean and geometric mean

48. 48 Expected Return u 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

49. 49 Return on Investment u 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

50. 50 Standard Deviation and Variance u Standard deviation and variance are the most common measures of total risk u They measure the dispersion of a set of observations around the mean observation

51. 51 Standard Deviation and Variance (cont’d) u General equation for variance: u If all outcomes are equally likely:

52. 52 Standard Deviation and Variance (cont’d) u Equation for standard deviation:

53. 53 Semi-Variance u 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

54. 54 Some Statistical Facts of Life u One has to understand key terms: “Constants,” “Variables”, “Populations,” “Samples,” and “Sample statistics” u Properties of Random Variables u Linear Regression u R Squared and Standard Errors

55. 55 Constants u 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 u A constant can be represented by a numeral or by a symbol

56. 56 Variables u A variable has no fixed value It is useful only when it is considered in the context of other possible values it might assume u In finance, variables are called random variables Designated by a tilde –e.g.,

57. 57 Variables (cont’d) u Discrete random variables are countable e.g., the number of trout you catch u Continuous random variables are measurable e.g., the length of a trout

58. 58 Variables (cont’d) u Quantitative variables are measured by real numbers e.g., numerical measurement u Qualitative variables are categorical e.g., hair color

59. 59 Variables (cont’d) u Independent variables are measured directly e.g., the height of a box u Dependent variables can only be measured once other independent variables are measured e.g., the volume of a box (requires length, width, and height)

60. 60 Populations u A population is the entire collection of a particular set of random variables u 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

61. 61 Populations (cont’d) u 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

62. 62 Populations (cont’d) Positive Skewness Negative Skewness

63. 63 Populations (cont’d) u A binomial distribution contains only two random variables e.g., the toss of a coin (heads or tails) u A finite population is one in which each possible outcome is known e.g., a card drawn from a deck of cards

64. 64 Populations (cont’d) u An infinite population is one where not all observations can be counted e.g., the microorganisms in a cubic mile of ocean water u A univariate population has one variable of interest

65. 65 Populations (cont’d) u A bivariate population has two variables of interest e.g., weight and size u A multivariate population has more than two variables of interest e.g., weight, size, and color

66. 66 Samples u A sample is any subset of a population e.g., a sample of past monthly stock returns of a particular stock

67. 67 Sample Statistics u Sample statistics are characteristics of samples A true population statistic is usually unobservable and must be estimated with a sample statistic –Expensive –Statistically unnecessary

68. 68 Properties of Random Variables u Example u Central Tendency u Dispersion u Logarithms u Expectations u Correlation and Covariance

69. 69 Example Assume the following monthly stock returns for Stocks A and B: MonthStock AStock B 12%3% 2–1%0% 34%5% 41%4%

70. 70 Central Tendency u Central tendency is what a random variable looks like, on average u The usual measure of central tendency is the population’s expected value (the mean) The average value of all elements of the population

71. 71 Example (cont’d) The expected returns for Stocks A and B are:

72. 72 Dispersion u Investors are interested in the variation of actual values around the average u A common measure of dispersion is the variance or standard deviation

73. 73 Example (cont’d) The variance and standard deviation for Stock A are:

74. 74 Example (cont’d) The variance and standard deviation for Stock B are:

75. 75 Logarithms u 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 u Logarithms make other statistical tools more appropriate e.g., linear regression

76. 76 Logarithms (cont’d) u 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

77. 77 Expectations u The expected value of a constant is a constant: u The expected value of a constant times a random variable is the constant times the expected value of the random variable:

78. 78 Expectations (cont’d) u The expected value of a combination of random variables is equal to the sum of the expected value of each element of the combination:

79. 79 Correlations and Covariance u Correlation is the degree of association between two variables u Covariance is the product moment of two random variables about their means u Correlation and covariance are related and generally measure the same phenomenon

80. 80 Correlations and Covariance (cont’d)

81. 81 Example (cont’d) The covariance and correlation for Stocks A and B are:

82. 82 Correlations and Covariance (cont’d) u 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

83. 83 Linear Regression u 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 u Linear regression finds the equation of a line through the points that gives the best possible fit

84. 84 Linear Regression (cont’d) Example Assume the following sample of weekly stock and stock index returns: WeekStock ReturnIndex Return 10.00840.0088 2–0.0045–0.0048 30.00210.0019 40.00000.0005

85. 85 Linear Regression (cont’d) Example (cont’d) Intercept = 0 Slope = 0.96 R squared = 0.99

86. 86 R Squared and Standard Errors u R squared and the standard error are used to assess the accuracy of calculated securities u 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% u 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

87. 87 Standard Errors u The standard error is equal to the standard deviation divided by the square root of the number of observations:

88. 88 Standard Errors (cont’d) u 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