1. © 2004 by Nelson, a division of Thomson Canada Limited Contemporary Financial Management Chapter 7: Analysis of Risk and Return

2. © 2004 by Nelson, a division of Thomson Canada Limited 2 Introduction  This chapter develops the risk-return relationship for both individual projects (investments) and portfolios of projects

3. © 2004 by Nelson, a division of Thomson Canada Limited 3 Risk and Return  Risk is usually defined as the actual or potential variability of returns from a project or portfolio  Risk-free returns are known with certainty Federal Government Treasury Bills are often considered the risk-free security. The risk-free rate of return sets a floor under all other returns in the market.

4. © 2004 by Nelson, a division of Thomson Canada Limited 4 Holding Period Return HPR = Holding Period Return P 1 = Ending Price P 0 = Beginning Price  Return for holding an investment for one period (i.e. period of days, months, years, etc.)  When there is no cash flow during the holding period, then:

5. © 2004 by Nelson, a division of Thomson Canada Limited 5 Holding Period Return  When there is a cash flow in addition to the ending price (such as the payment of a dividend), the Holding Period Return formula is:

6. © 2004 by Nelson, a division of Thomson Canada Limited 6 Holding Period Return: Example  You bought a stock one year ago for $10. Today, it is worth $12. Yesterday, you received a $1 dividend. What is your holding period return?

7. © 2004 by Nelson, a division of Thomson Canada Limited 7 Returns  Ex Post Returns (After the fact) Return that an investor actually realizes  Ex Ante Returns (Before the fact) Return that an investor expects to earn

8. © 2004 by Nelson, a division of Thomson Canada Limited 8 Analyzing Return  Expected Return ( ): When returns are not known with certainty, there will often exist a probability distribution of possible returns with an associated probability of occurrence. Expected return is a weighted average of the individual possible returns (r j ), with weights being the probability of occurrence (p j ).

9. © 2004 by Nelson, a division of Thomson Canada Limited 9 Expected Return: Example Possible Return Probability of Occurrence -10%5% 0%10% +5%25% +15%50% +25%10%

10. © 2004 by Nelson, a division of Thomson Canada Limited 10 Expected Return: Solution

11. © 2004 by Nelson, a division of Thomson Canada Limited 11 Analyzing Risk  Standard Deviation ( ): a statistical measure of the dispersion, or variability, of outcomes around the mean or expected value ( ). Low standard deviation means that returns are tightly clustered around the mean High standard deviation means that returns are widely dispersed around the mean

12. © 2004 by Nelson, a division of Thomson Canada Limited 12 Calculating Standard Deviation  Three common ways of calculating standard deviation: Returns are known with certainty Standard deviation of a population Standard deviation of a sample Returns are not known with certainty

13. © 2004 by Nelson, a division of Thomson Canada Limited 13 Standard Deviation of a Population The standard deviation ( ) is the square root of the variance: First, calculate the variance ( 2 ):

14. © 2004 by Nelson, a division of Thomson Canada Limited 14 Standard Deviation of a Sample First, calculate the variance (S 2 ): The standard deviation (s) is the square root of the variance:

15. © 2004 by Nelson, a division of Thomson Canada Limited 15 Standard Deviation: Example  You have been given the following sample of stock returns, for which you would like to calculate the standard deviation: {12%, -4%, 0%, 22%, 5%}

16. © 2004 by Nelson, a division of Thomson Canada Limited 16 Standard Deviation: Example  You have been given the following sample of stock returns, for which you would like to calculate the standard deviation: {12%, -4%, 0%, 22%, 5%}  Step 1: Calculate Arithmetic Return

17. © 2004 by Nelson, a division of Thomson Canada Limited 17 Standard Deviation: Example  You have been given the following sample of stock returns, for which you would like to calculate the standard deviation: {12%, -4%, 0%, 22%, 5%}  Step 2: Calculate Variance

18. © 2004 by Nelson, a division of Thomson Canada Limited 18 Standard Deviation: Example  You have been given the following sample of stock returns, for which you would like to calculate the standard deviation: {12%, -4%, 0%, 22%, 5%}  Step 3: Calculate Standard Deviation

19. © 2004 by Nelson, a division of Thomson Canada Limited 19 Standard Deviation – Returns Not Certain r j = return at time period j = expected return p j = probability of return j occurring ˆ r

20. © 2004 by Nelson, a division of Thomson Canada Limited 20 Standard Deviation: Example  You have been provided with the following possible returns and their associated probabilities. Calculate the expected return and the standard deviation of return. State of EconomyReturnProbability Boom30%15% Normal15%60% Recession0%25%

21. © 2004 by Nelson, a division of Thomson Canada Limited 21 Standard Deviation: Solution  Step #1: Calculate the Expected Return

22. © 2004 by Nelson, a division of Thomson Canada Limited 22 Standard Deviation: Solution  Step #2: Calculate the Variance

23. © 2004 by Nelson, a division of Thomson Canada Limited 23 Standard Deviation: Solution  Step #3: Calculate the Standard Deviation

24. © 2004 by Nelson, a division of Thomson Canada Limited 24 Normal Probability Distribution  A symmetrical, bell-like curve where 50% of possible outcomes are greater than the expected value and 50% are less than the expected value.  A normal distribution is fully described by just two statistics: Mean Standard deviation

25. © 2004 by Nelson, a division of Thomson Canada Limited 25 Normal Probability Distribution Mean- 1 σ 50% Probability - 3 σ- 2 σ 1 σ 2 σ3 σ 50% Probability 68.26%95.44% 99.74%

26. © 2004 by Nelson, a division of Thomson Canada Limited 26 Normal Distribution Example 10% 50% Probability 5% 15% 0% 20% - 5% 25% 50% Probability 68.26%95.44% 99.74% Mean = 10%; Standard Deviation = 5%

27. © 2004 by Nelson, a division of Thomson Canada Limited 27 Standard Normal Probability  Problem: Standard deviation is correlated with size of the mean  Solution: To allow for easy comparison among distributions with different means, standardize using a Z score  Z score measures the number of standard deviations ( ) a particular rate of return (r) is from the mean or expected value ( ).

28. © 2004 by Nelson, a division of Thomson Canada Limited 28 Standard Normal: Example  What is the probability of a loss on an investment with an expected return of 20% and a standard deviation of 17%?  Step #1: Calculate the Z Score for the number of standard deviations from the mean for a 0% return

29. © 2004 by Nelson, a division of Thomson Canada Limited 29 Standard Normal: Example  What is the probability of a loss on an investment with an expected return of 20% and a standard deviation of 17%?  Step #2: Consult Table V on Page 712. Find the row with 1.10 in the left hand column. Then find the column with 0.08 in the top row. The cell where the row & column intersect is the probability of obtaining a value less than 1.18 standard deviations from the mean. The answer is 0.1190 or 11.90%

30. © 2004 by Nelson, a division of Thomson Canada Limited 30 Probability of Earning Less then 0% 20%3% - 31%-14% 37% 54%71% 1.18 st. dev. from the mean 11.9% Probability

31. © 2004 by Nelson, a division of Thomson Canada Limited 31 Concept of Efficient Portfolios  Has the highest possible expected return for a given level of risk (or standard deviation)  Has the lowest possible level of risk for a given expected return A B C A dominates B because it has the same expected return for a given risk. C dominates B because it has a higher expected return for a given risk.

32. © 2004 by Nelson, a division of Thomson Canada Limited 32 Coefficient of Variation (v)  The ratio of the standard deviation ( ) to the expected value ( ).  Tells us the risk per unit of return.  An appropriate measure of total risk when comparing two investment projects of different size.

33. © 2004 by Nelson, a division of Thomson Canada Limited 33 Coefficient of Variation: Example  You are asked to rank the following set of investments according to their risk per unit of return. SecurityReturnStandard Deviation A6%7% B10%13% C18%21%

34. © 2004 by Nelson, a division of Thomson Canada Limited 34 Coefficient of Variation: Solution SecurityReturnStandard Deviation Coefficient of Variation A6%7% B10%13% C18%20% Most Risk Least Risk

35. © 2004 by Nelson, a division of Thomson Canada Limited 35 Relationship Between Risk and Return  Required Rate of Return Discount rate used to present value a stream of expected cash flows from an asset.  Risk-Return Relationship The riskier, or the more variable, the expected cash flow stream, the higher the required rate of return.

36. © 2004 by Nelson, a division of Thomson Canada Limited 36 Relationship Between Risk and Return  Required Rate of Return = Risk-free Rate of Return + Risk Premium  Risk-free Rate: rate of return on securities that are free of default risk, such as T-bills.  Risk Premium: expected “reward” the investor expects to earn for assuming risk

37. © 2004 by Nelson, a division of Thomson Canada Limited 37 Risk-Free Rate of Return  Risk-free Rate of Return (r f ) = Real Rate of Return + Exp. Inflation Premium  Real Rate of Return: the reward for deferring consumption  Expected Inflation Premium: compensates investors for the loss of purchasing power due to inflation

38. © 2004 by Nelson, a division of Thomson Canada Limited 38 Types of Risk Premiums  Maturity risk premium  Default risk premium  Seniority risk premium  Marketability risk premium  Business risk  Financial risk

39. © 2004 by Nelson, a division of Thomson Canada Limited 39 Term Structure of Interest Rates  Term structure is a plot of the yield on securities with similar risk but different maturities  Term structure is used to explain the maturity risk premium (why long securities tend to have higher yields than short maturity securities)  Three theories of the Term Structure: Expectations theory Liquidity premium theory Market segmentation theory

40. © 2004 by Nelson, a division of Thomson Canada Limited 40 Canada’s Term Structure 2.65% 2.69% 2.76% 2.70% 2.91% 3.28% 3.57% 4.13% 4.88% 5.39% 0% 1% 2% 3% 4% 5% 6% 1 Month 2 3 6 12 Months 2 year 3 year 5 year 10 year Long term November 13, 2003

41. © 2004 by Nelson, a division of Thomson Canada Limited 41 Expectations Theory  The long interest rate is the geometric average of expected future short interest rates.  If the term structure is sloping up, future short interest rates are expected to be higher than current short interest rates.  If the term structure is sloping down, future short interest rates are expected to be lower than current short interest rates.

42. © 2004 by Nelson, a division of Thomson Canada Limited 42 Liquidity Premium Theory  Investors prefer liquidity (the ability to convert to cash at or near face value)  Long securities are less liquid than short securities  Therefore, to induce investors to hold long securities, must pay a “liquidity premium”

43. © 2004 by Nelson, a division of Thomson Canada Limited 43 Market Segmentation Theory  The yield in each segment of the yield curve is determined by the supply and demand for funds in that maturity zone  Supply & demand driven by firms which deal primarily in a specific maturity zone Chartered banks – short maturities Trust companies – medium maturities Pension funds – long maturities

44. © 2004 by Nelson, a division of Thomson Canada Limited 44 Modern Portfolio Theory Harry MarkowitzWilliam F. SharpeMerton Miller Modern portfolio theory was introduced by Harry Markowitz in 1952. Markowitz, Sharpe & Miller were co-recipients of the Nobel Prize in Economics in 1990 for their pioneering work in portfolio theory

45. © 2004 by Nelson, a division of Thomson Canada Limited 45 Expected Return  The expected return on a portfolio is the weighted average of the returns of each asset within the portfolio Example: A portfolio is comprised of three securities with the following returns: SecurityReturn% of Portfolio A5%30% B10%45% C15%25%

46. © 2004 by Nelson, a division of Thomson Canada Limited 46 Expected Return  The expected return of the portfolio is the weighted average: r j = return at time period j = expected return w j = proportion of the portfolio comprised of asset j

47. © 2004 by Nelson, a division of Thomson Canada Limited 47 Portfolio Risk: Two Risky Assets  Standard deviation of a two-asset portfolio is calculated as follows:

48. © 2004 by Nelson, a division of Thomson Canada Limited 48 Portfolio Risk: Two Risky Assets Portfolio risk is driven mainly by the correlation between the assets!!

49. © 2004 by Nelson, a division of Thomson Canada Limited 49 Correlation  Correlation is a measure of the linear relationship between two assets  Correlation varies between perfect negative (-1) to perfect positive (+1)  Perfect negative correlation: when the return on asset A rises, the return on Asset B falls and vice versa  Perfect positive correlation: the returns on asset A and Asset B move in perfect unison

50. © 2004 by Nelson, a division of Thomson Canada Limited 50 Correlation & Risk Reduction B A RARA σAσA σBσB RBRB Perfect Negative Correlation Perfect Positive Correlation Less than Perfect Correlation To minimize portfolio risk, choose assets that have very low correlations with each other.

51. © 2004 by Nelson, a division of Thomson Canada Limited 51 Moving Toward Many Risky Assets  When the portfolio consists of many risky assets, they form a plot similar to a broken egg shell shape  Each dot within the broken egg shell shape represents the risk/return profile for a single risky asset or portfolio of risky assets  To maximize return per unit of risk assumed, an investor would always choose an asset or portfolio that plots along the efficient frontier

52. © 2004 by Nelson, a division of Thomson Canada Limited 52 Portfolios: Many Risky Assets You would never choose Asset A, as you can earn a higher return with similar risk by choosing the asset that plots along the Efficient Frontier. RARA σAσA Return Standard Deviation A

53. © 2004 by Nelson, a division of Thomson Canada Limited 53 Choosing a Portfolio: So Far  The investor first decides how much risk to assume  The investor then chooses the portfolio that plots along the efficient frontier with that amount of risk

54. © 2004 by Nelson, a division of Thomson Canada Limited 54 Introducing the Risk Free Security  When a risk-free asset (Treasury Bill) is introduced into the set of risky assets, a new efficient frontier emerges  This new efficient frontier is known as the Capital Market Line (CML)  The CML represents all possible portfolios comprised of Treasury Bills and the Market Portfolio

55. © 2004 by Nelson, a division of Thomson Canada Limited 55 Adding the Risk-Free Asset RMRM σMσM Return Standard Deviation A RfRf Capital Market Line

56. © 2004 by Nelson, a division of Thomson Canada Limited 56 Capital Market Line (CML)  To maximize return for an amount of risk, investors should hold a portion of their assets in T-bills and a portion in the market portfolio.  Linear relationship between risk and return  To earn an expected return greater than the return on the market portfolio, invest more than 100% of one’s own wealth in the market portfolio.

57. © 2004 by Nelson, a division of Thomson Canada Limited 57 What are we Missing?  We know: Investors should split their assets between Treasury bills and the market portfolio To reduce risk, invest a greater proportion of assets in Treasury bills To enhance expected return, invest a greater proportion of assets in the market portfolio  We do not know how to calculate the expected return (and hence the price) for a single risky asset The Capital Asset Pricing Model is needed

58. © 2004 by Nelson, a division of Thomson Canada Limited 58 The Missing Link  We need to measure the Market risk that cannot be diversified away Unique or Non-systematic Risk Market or Systematic Risk DiversifiableNon- Diversifiable Total Standard Deviation (or Risk)

59. © 2004 by Nelson, a division of Thomson Canada Limited 59 The Missing Link Unique Risk Market Risk (measured with Beta) The market will not compensate us for risk that can be diversified away. The market will compensate us for market risk – the risk that cannot be diversified away

60. © 2004 by Nelson, a division of Thomson Canada Limited 60 CAPM: Systematic Risk is Relevant  Systematic, or non-diversifiable, risk is caused by factors affecting the entire market interest rate changes changes in purchasing power change in business outlook  Unsystematic, or diversifiable, risk is caused by factors unique to the firm strikes regulations management’s capabilities

61. © 2004 by Nelson, a division of Thomson Canada Limited 61 Portfolio Diversification  When assets are put into a well-diversified portfolio, some of the unique or nonsystematic risk is diversified away  The number of assets required to diversify away most of the unique risk varies with the correlation between the assets Canada’s capital markets are more highly correlated with the natural resource sector than are the US capital markets Require more securities in Canada to diversify away most of the unique risk

62. © 2004 by Nelson, a division of Thomson Canada Limited 62 Diversifying Unique Risk Risk Unique Risk Number of Securities Portfolio Risk Market Risk

63. © 2004 by Nelson, a division of Thomson Canada Limited 63 Systematic Risk is Measured by Beta  Beta is a measure of the volatility of a security’s return compared to the volatility of the return on the Market Portfolio

64. © 2004 by Nelson, a division of Thomson Canada Limited 64 Concept of Beta To calculate Beta, use historical monthly rates of return on both the security and the market index. Return on Stock A Return on TSX Index Slope equals Beta

65. © 2004 by Nelson, a division of Thomson Canada Limited 65 Security Market Line (SML)  Shows the relationship between required rate of return and beta (ß). rfrf ß Security Market Line Required Rate of Return ßjßj k j

66. © 2004 by Nelson, a division of Thomson Canada Limited 66 Required Rate of Return  The required return for any security j may be defined in terms of systematic risk,  j, the expected market return, r m, and the expected risk free rate, r f. ^ ^

67. © 2004 by Nelson, a division of Thomson Canada Limited 67 SML: Example  A security has a Beta of 1.25. If the yield on Treasury Bills is 5% and the return on the market portfolio is 11%, what is the expected return for holding the security? An investor expects a return of 12.5% to hold the security.

68. © 2004 by Nelson, a division of Thomson Canada Limited 68 Market Risk Premium  The reward for bearing risk Equal to (r m – r f ) Equal to the slope of security market line (SML)  Will increase or decrease with uncertainties about the future economic outlook the degree of risk aversion of investors

69. © 2004 by Nelson, a division of Thomson Canada Limited 69 Security Market Line (Again)  SML Return MM RMRM A B A – return is too high; price is too low B – return is too low; price is too high

70. © 2004 by Nelson, a division of Thomson Canada Limited 70 CAPM Assumptions  Investors hold well-diversified portfolios  Competitive markets  Borrow and lend at the risk-free rate  Investors are risk averse  No taxes  Investors are influenced by systematic risk  Freely available information  Investors have homogeneous expectations  No brokerage charges

71. © 2004 by Nelson, a division of Thomson Canada Limited 71 CAPM Drawbacks  Estimating expected future market returns on historic returns.  Determining an appropriate r f  Determining the best estimate of   Investors don’t totally ignore unsystematic risk  Betas are frequently unstable over time  Required returns are determined by macroeconomic factors

72. © 2004 by Nelson, a division of Thomson Canada Limited 72 Market Efficiency  Capital markets are efficient if prices adjust fully and instantaneously to new information affecting a security’s prospective return.

73. © 2004 by Nelson, a division of Thomson Canada Limited 73 Three Degrees of Market Efficiency  Weak form  Semi-strong form  Strong form

74. © 2004 by Nelson, a division of Thomson Canada Limited 74 Weak Form Market Efficiency  Security prices capture all of the information contained in the record of past prices and volumes  Implication: No investor can earn excess returns using historical price or volume information. Technical analysis should have no marginal value.

75. © 2004 by Nelson, a division of Thomson Canada Limited 75 Semi-Strong Form Market Efficiency  Security prices capture all of the information contained in the public domain.  Implication: No investor can earn excess returns using publicly available information. Fundamental analysis should have no marginal value.

76. © 2004 by Nelson, a division of Thomson Canada Limited 76 Strong Form Market Efficiency  Security prices capture all information, both public and private.  Markets are quite efficient (but it is illegal to use private information for personal gain, when trading securities)!