1. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 1 Chapter 12: Portfolio Opportunities and Choice Objective To understand the theory of personal portfolio selection in theory and in practice

2. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 2 Chapter 12 Contents 12.1 The process of personal portfolio selection12.1 The process of personal portfolio selection 12.2 The trade-off between expected return and risk12.2 The trade-off between expected return and risk 12.3 Efficient diversification with many risky assets12.3 Efficient diversification with many risky assets

3. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 3 Objectives To understand the process of personal portfolio selection in theory and practiceTo understand the process of personal portfolio selection in theory and practice

4. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 4 Introduction How should you invest your wealth optimally?How should you invest your wealth optimally? –Portfolio selection Your wealth portfolio containsYour wealth portfolio contains –Stock, bonds, shares of unincorporated businesses, houses, pension benefits, insurance policies, and all liabilities

5. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 5 Portfolio Selection Strategy There are general principles to guide you, but the implementation will depend such factors as your (and your spouse’s)There are general principles to guide you, but the implementation will depend such factors as your (and your spouse’s) –age, existing wealth, existing and target level of education, health, future earnings potential, consumption preferences, risk preferences, life goals, your children’s educational needs, obligations to older family members, and a host of other factors

6. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 6 12.1 The Process of Personal Portfolio Selection Portfolio selectionPortfolio selection –the study of how people should invest their wealth –process of trading off risk & expected return to find the best portfolio of assets & liabilities Narrower dfn: consider only securitiesNarrower dfn: consider only securities Wider dfn: house purchase, insurance, debtWider dfn: house purchase, insurance, debt Broad dfn: human capital, educationBroad dfn: human capital, education

7. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 7 The Life Cycle The risk exposure you should accept depends upon your ageThe risk exposure you should accept depends upon your age Consider two investments (rho=0.2)Consider two investments (rho=0.2) –Security 1 has a volatility of 20% and an expected return of 12% –Security 2 has a volatility of 8% and an expected return of 5%

8. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 8 Price Trajectories The following graph show the the price of the two securities generated by a bivariate normal distribution for returnsThe following graph show the the price of the two securities generated by a bivariate normal distribution for returns –The more risky security may be thought of as a share of common stock or a stock mutual fund –The less risky security may be thought of as a bond or a bond mutual fund

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10. 10 Interpretation of the Graph The graph is plotted on a log scale in so that you can see the important featuresThe graph is plotted on a log scale in so that you can see the important features The magenta bond trajectory is clearly less risky than the navy-blue stock trajectoryThe magenta bond trajectory is clearly less risky than the navy-blue stock trajectory The expected prices of the bond and the stock are straight lines on a log scaleThe expected prices of the bond and the stock are straight lines on a log scale

11. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 11 Interpretation of the Graph Recall the log scale: the volatility increases with the length of the investmentRecall the log scale: the volatility increases with the length of the investment You begin to form the conjecture that the chances of the stock price being less than the price bond is higher in earlier yearsYou begin to form the conjecture that the chances of the stock price being less than the price bond is higher in earlier years

12. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 12 Generating More Trajectories This was just one of an infinite number of trajectories generated by the same 2 means, 2 volatilities, and the correlationThis was just one of an infinite number of trajectories generated by the same 2 means, 2 volatilities, and the correlation –I have not cheated you, this was indeed the first trajectory generated by the statistics –the following trajectories are not reordered nor edited Instructor: On slower computers there may be a delayInstructor: On slower computers there may be a delay

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15. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 15 …and Lots More!

16. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 16 From Conjecture to Hypothesis You are probably ready to make the hypothesis thatYou are probably ready to make the hypothesis that –the probability of the high-risk, high-return security will out-perform the low-risk, low- return increases with time

17. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 17 But: I promised to be perfectly frank and honest (pfah) with you about the ordering of the simulated trajectoriesI promised to be perfectly frank and honest (pfah) with you about the ordering of the simulated trajectories The next trajectory truly was the next trajectory in the sequence, honest!The next trajectory truly was the next trajectory in the sequence, honest!

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19. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 19 Explanation The bond and the stock end up at about the same price, when the expected prices are more than a magnitude apartThe bond and the stock end up at about the same price, when the expected prices are more than a magnitude apart There is either a very good explanation for this, or there is a very high probability that I have been much less than perfectly frank and honest with youThere is either a very good explanation for this, or there is a very high probability that I have been much less than perfectly frank and honest with you

20. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 20 Another View of the Model A little mathematics, and we are able to generate the following price distributions for the stock and the bond for 2, 5, 10, and 40 years into the futureA little mathematics, and we are able to generate the following price distributions for the stock and the bond for 2, 5, 10, and 40 years into the future

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22. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 22 There is a lot going on here, so we will further constrain our viewThere is a lot going on here, so we will further constrain our view First look at stock prices over a period of 10 yearsFirst look at stock prices over a period of 10 years The prices are distributed according to the lognormal distributionThe prices are distributed according to the lognormal distribution

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24. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 24 Note –the scale is $0 to $800 –the distribution diffuses and drifts towards higher prices with time –the diffusion is more pronounced in the earlier years than in the later years –you may see that the mode, median, and mean appear to drift apart with time

25. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 25 Bond in Time You will recall that if you invest in a 5- year default-free pure discount bond for 5 years, the return is known with certaintyYou will recall that if you invest in a 5- year default-free pure discount bond for 5 years, the return is known with certainty To avoid this effect, assume we invest in short term bonds, and roll them over as they matureTo avoid this effect, assume we invest in short term bonds, and roll them over as they mature

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27. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 27 Note –the scale is now $0 to $400 (not $0 to $800 as in the case of the stock) –we observe the same kind of diffusion and drift behavior, and there is less of each (remember to adjust for the scale)(remember to adjust for the scale)

28. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 28 Contrast of Trajectories and Distributions The price distributions and the trajectories were generated from the same distribution. ButThe price distributions and the trajectories were generated from the same distribution. But They do not seem to agreeThey do not seem to agree –The distributions appear to produce much lower averages (expected returns) than the trajectories

29. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 29 Meaty Tails The resolution is that the distributions have much meatier tails than your intuition allows, pushing the median and mean further and further from the mode with timeThe resolution is that the distributions have much meatier tails than your intuition allows, pushing the median and mean further and further from the mode with time The region where the left tail appears to have drifted into insignificance has a profound affect on the meanThe region where the left tail appears to have drifted into insignificance has a profound affect on the mean

30. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 30 Stock and Bonds Distributions Compared at the Same Times The next sequence of slides contrasts the distribution of stock and bond prices at 1, 2, 5, 10, and 40 into the futureThe next sequence of slides contrasts the distribution of stock and bond prices at 1, 2, 5, 10, and 40 into the future Some of the slides have different measures of central tendency indicatedSome of the slides have different measures of central tendency indicated Note the behavior of these statistics as time increasesNote the behavior of these statistics as time increases

31. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 31 Mode =104Mode =106Median=104Mean =104Median=111Mean = 113

32. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 32

33. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 33 Mode = 122Mode = 135Median= 126 Mean = 128Median= 165 Mean = 182

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35. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 35 Mode =503Mode =1,102Median=650Mean =739Median=5,460Mean =12,151

36. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 36 Slide Sequence Summary The next table summarizes the drifts of the measures of central tendencyThe next table summarizes the drifts of the measures of central tendency Note that the means do in fact tie back to the trajectoriesNote that the means do in fact tie back to the trajectories The last (anomalous?) trajectory not an uncommon occurrence, and I was pfah with youThe last (anomalous?) trajectory not an uncommon occurrence, and I was pfah with you

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38. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 38 Implication for Investors –If you are older, the average remaining life of the investment is relatively short, and there is a larger probability that an investment in the risky security will result in a loss –This is not serious if you have substantial assets, in which case you can afford to take the risk, and enjoy higher expected returns

39. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 39 Implication for Investors –If you are younger, the average remaining life of retirement investment is longer, and there is only a small probability that an investment in the risky security will be less than the “safer” one –Investing in the less risky security will almost always result in a significantly smaller retirement income

40. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 40 Implication for Investors –Relatively early during a typical life cycle, there may be a need to liquidate some invested funds, perhaps for a house deposit, a child’s education, or an uninsured medical emergency –In the case where liquidating an investment early may damage long-term goals, some precautionary funds should be kept in lower- risk securities

41. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 41 Time Horizons –Planning horizon The total length of time for which one plansThe total length of time for which one plans –Decision horizon The length of time between decisions to revise a portfolioThe length of time between decisions to revise a portfolio –Trading horizon The shortest possible time interval over which investors may revise their portfoliosThe shortest possible time interval over which investors may revise their portfolios

42. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 42 Computing Life Expectancy Mortality tables may be organized as three columns: actuary age, deaths/year per 1000 live births, and remaining life expectation. Note:Mortality tables may be organized as three columns: actuary age, deaths/year per 1000 live births, and remaining life expectation. Note: if you survive from 60 to 65, for example, the expected date of your death advances by 3 to 4 yearsif you survive from 60 to 65, for example, the expected date of your death advances by 3 to 4 years young women have a higher life expectation than men, but this is lost with advancing ageyoung women have a higher life expectation than men, but this is lost with advancing age

43. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 43 Useful Internet Address The Society of Actuaries maintain a web site that provides detailed mortality tables, interactive computer models, mortgage experiences, career information, and current research papersThe Society of Actuaries maintain a web site that provides detailed mortality tables, interactive computer models, mortgage experiences, career information, and current research papers www.soa.orgwww.soa.org

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45. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 45

46. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 46 Life Expectancy 0 5 10 15 20 25 6065707580859095 Age Remaining Expected Life MExLife FExLife

47. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 47 Risk Tolerance Your tolerance for bearing risk is a major determinant of portfolio choicesYour tolerance for bearing risk is a major determinant of portfolio choices –It is the mirror image of risk aversion –Whatever its cause, we do not distinguish between capacity to bear risk and attitude towards risk

48. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 48 Role of Professional Asset Managers Most people have neither the time nor the skill necessary to optimize a portfolio for risk and returnMost people have neither the time nor the skill necessary to optimize a portfolio for risk and return –Professional fund managers provide this service as individually designed solutions to the precise needs of a customer ($$$$)individually designed solutions to the precise needs of a customer ($$$$) a set of financial products which may be used together to satisfy most customer goals ($$)a set of financial products which may be used together to satisfy most customer goals ($$)

49. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 49 12.2 Trade-Off between Expected Return and Risk Assume a world with a single risky asset and a single riskless assetAssume a world with a single risky asset and a single riskless asset The risky asset is, in the real world, a portfolio of risky assetsThe risky asset is, in the real world, a portfolio of risky assets The risk-free asset is a default-free bond with the same maturity as the investor’s decision (or possibly the trading) horizonThe risk-free asset is a default-free bond with the same maturity as the investor’s decision (or possibly the trading) horizon

50. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 50 Trade-Off between Expected Return and Risk The assumption of a risky and riskless security simplifies the analysisThe assumption of a risky and riskless security simplifies the analysis

51. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 51 The Risk-Reward Trade-Off Line

52. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 52 Combining the Riskless Asset and a Single Risky Asset Assume that you invest W1 proportion of your wealth in security 1 and proportion W2 of your wealth in security 2Assume that you invest W1 proportion of your wealth in security 1 and proportion W2 of your wealth in security 2 You must invest in either 1 or 2, so W1+W2 = 1You must invest in either 1 or 2, so W1+W2 = 1 Let 2 be the riskless asset, and 1 be the risky asset (portfolio)Let 2 be the riskless asset, and 1 be the risky asset (portfolio)

53. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 53 Combining the Riskless Asset and a Single Risky Asset Your statistics background tells you how to determine the expected return and volatility of any two-security portfolioYour statistics background tells you how to determine the expected return and volatility of any two-security portfolio –1. Form a new random variable, the return of the portfolio,RP, from the two given random variables, R1 and R2 RP = W1*R1 + W2*R2 RP = W1*R1 + W2*R2

54. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 54 Combining the Riskless Asset and a Single Risky Asset –The expected return of the portfolio is the weighted average of the component returns  p = W 1*  1 + W 2*  2  p = W 1*  1 + (1- W 1 ) *  2

55. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 55 Combining the Riskless Asset and a Single Risky Asset –The volatility of the portfolio is not quite as simple:  p = (( W 1*  1) 2 + 2 W 1*  1* W 2*  2 + ( W 2*  2) 2 ) 1/2

56. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 56 Combining the Riskless Asset and a Single Risky Asset –We know something special about the portfolio, namely that security 2 is riskless, so  2 = 0, and  p becomes:  p = (( W 1*  1) 2 + 2 W 1*  1* W 2* 0 + ( W 2* 0 ) 2 ) 1/2  p = | W 1| *  1

57. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 57 Combining the Riskless Asset and a Single Risky Asset –In summary  p = | W 1| *  1, And:  p = W 1*  1 + (1- W 1 ) * r f, So: If W 1>0,  p = [( r f -  1)/  1]*  p + r f If W 1>0,  p = [( r f -  1)/  1]*  p + r f Else  p = [(  1- r f )/  1]*  p + r f Else  p = [(  1- r f )/  1]*  p + r f

58. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 58 Reflection The risk-free rate, r f, the risky security’s expected rate of return,  1, and volatility,  1, are constants, so we have a “ray” that “reflects” from the expected return axes at  p = r fThe risk-free rate, r f, the risky security’s expected rate of return,  1, and volatility,  1, are constants, so we have a “ray” that “reflects” from the expected return axes at  p = r f

59. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 59 Illustration Consider the set of all portfolios that may be formed by investing (long and or short) inConsider the set of all portfolios that may be formed by investing (long and or short) in –a risky security with a volatility of 20% and an expected return of 15% –a riskless security with a volatility of 0% and a known return of 5%

60. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 60

61. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 61 Sub-Optimal Investments Investments on the higher part of the line are always preferred (by normal folk) to investments on the lower part of the line, so for our current purposes we may ignore the lower lineInvestments on the higher part of the line are always preferred (by normal folk) to investments on the lower part of the line, so for our current purposes we may ignore the lower line That is, we will not sell the risky asset short and invest the proceeds in the riskless securityThat is, we will not sell the risky asset short and invest the proceeds in the riskless security

62. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 62 Long risky and short risk-free Long both risky and risk-free 100% Risky 100% Risk- less

63. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 63 Observations –An investor with a low risk tolerance may invest in a portfolio containing a small % of risky securities, and a correspondingly higher % of riskless securities –An investor with a high tolerance for risk may sell risk-free securities he does not own, and invest the proceeding in the risky investment –They both use the same two securities

64. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 64 Observations –The graph has been labeled the “capital market line” a little prematurely We will soon discover that ifWe will soon discover that if –the risky security is the market portfolio of risky securities –investors have similar expectations and time horizons All investors will invest (long or short) in the market portfolio and risk-free securityAll investors will invest (long or short) in the market portfolio and risk-free security –The line joins the capital markets for risky and risk-less securities

65. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 65 Achieving a Target Expected Return (1) Your boss has just read an ad’ that included the data for the Janus Twenty Fund (Scientific American, Sept 1998, page 6)Your boss has just read an ad’ that included the data for the Janus Twenty Fund (Scientific American, Sept 1998, page 6) “You beat them, or I’ll find another portfolio manager”, she quips“You beat them, or I’ll find another portfolio manager”, she quips “Wrong way to compute return?” you venture, as you rush for the door“Wrong way to compute return?” you venture, as you rush for the door

66. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 66 Mutual Fund Average % Total Returns

67. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 67 To obtain a 20% Return You settle on a 20% return, and decide not to pursue on the computational issueYou settle on a 20% return, and decide not to pursue on the computational issue –Recall:  p = W 1*  1 + (1- W 1 ) * r f –Your portfolio:  = 20%,  = 15%, rf = 5% –So: W 1 = (  p - r f )/(  1 - r f ) = (0.20 - 0.05)/(0.15 - 0.05) = 150% = (0.20 - 0.05)/(0.15 - 0.05) = 150%

68. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 68 To obtain a 20% Return Assume that you manage a $50,000,000 portfolioAssume that you manage a $50,000,000 portfolio A W1 of 1.5 or 150% means you invest (go long) $75,000,000, and borrow (short) $25,000,000 to finance the differenceA W1 of 1.5 or 150% means you invest (go long) $75,000,000, and borrow (short) $25,000,000 to finance the difference Borrowing at the risk-free rate is mootBorrowing at the risk-free rate is moot

69. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 69 To obtain a 20% Return How risky is this strategy?How risky is this strategy?  p = | W 1| *  1 = 1.5 * 0.20 = 0.30 The portfolio has a volatility of 30%The portfolio has a volatility of 30%

70. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 70 Important Observation It doesn’t require much skill to leverage a portfolio; stockbrokers will let most investors trade “on margin”It doesn’t require much skill to leverage a portfolio; stockbrokers will let most investors trade “on margin” When evaluating an investment’s performance, you must examine both the risk and the expected returnWhen evaluating an investment’s performance, you must examine both the risk and the expected return

71. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 71 Returning to the Example Advertisements for mutual funds do not generally disclose a quantifiable measure of risk, and Janus is no exceptionAdvertisements for mutual funds do not generally disclose a quantifiable measure of risk, and Janus is no exception –The advertised “Janus Twenty Fund” returns are completely meaningless from a financial point of view –More information is needed

72. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 72 Returning to the Example You can leverage the funds expected returns up or downYou can leverage the funds expected returns up or down If you want an expected returns of 10%, or, 20%, 30%, 40%, 50%, 60%… you can have it (under the condition you can continue to borrow at the risk-free rate)If you want an expected returns of 10%, or, 20%, 30%, 40%, 50%, 60%… you can have it (under the condition you can continue to borrow at the risk-free rate)

73. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 73 How Should my Boss Judge my Fund’s Performance? It is a little early to answer this questionIt is a little early to answer this question –If the risky security is the market portfolio, then given your portfolio’s risk, consistent returns above the CML line may appear appealing

74. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 74 Portfolio Efficiency

75. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 75 Portfolio Efficiency An efficient portfolio is defined as the portfolio that offers the investor the highest possible expected rate of return at a specific riskAn efficient portfolio is defined as the portfolio that offers the investor the highest possible expected rate of return at a specific risk We now investigate more than one risky asset in a portfolioWe now investigate more than one risky asset in a portfolio

76. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 76 12.3 Efficient Diversification with Many Risky Assets We have consideredWe have considered –Investments with a single risky, and a single riskless, security –Investments where each security shares the same underlying return statistics We will now investigate investments with more than one (heterogeneous) stockWe will now investigate investments with more than one (heterogeneous) stock

77. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 77 Portfolio of Two Risky Assets Recall from statistics, that two random variables, such as two security returns, may be combined to form a new random variableRecall from statistics, that two random variables, such as two security returns, may be combined to form a new random variable A reasonable assumption for returns on different securities is the linear model:A reasonable assumption for returns on different securities is the linear model:

78. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 78 The Risk-Reward Trade-Off Curve: Risky Assets Only

79. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 79 Equations for Two Shares The sum of the weights w1 and w2 being 1 is not necessary for the validity of the following equations, for portfolios it happens to be trueThe sum of the weights w1 and w2 being 1 is not necessary for the validity of the following equations, for portfolios it happens to be true The expected return on the portfolio is the sum of its weighted expectationsThe expected return on the portfolio is the sum of its weighted expectations

80. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 80 Equations for Two Shares Ideally, we would like to have a similar result for riskIdeally, we would like to have a similar result for risk –Later we discover a measure of risk with this property, but for standard deviation:

81. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 81 Mnemonic There is a mnemonic that will help you remember the volatility equations for two or more securitiesThere is a mnemonic that will help you remember the volatility equations for two or more securities To obtain the formula, move through each cell in the table, multiplying it by the row heading by the column heading, and summingTo obtain the formula, move through each cell in the table, multiplying it by the row heading by the column heading, and summing

82. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 82 Variance with 2 Securities

83. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 83 Variance with 3 Securities

84. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 84 Note: The correlation of a with b is equal to the correlation of b with aThe correlation of a with b is equal to the correlation of b with a For every element in the upper triangle, there is an element in the lower triangleFor every element in the upper triangle, there is an element in the lower triangle – so compute each upper triangle element once, and just double it This generalizes in the expected mannerThis generalizes in the expected manner

85. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 85 Correlated Common Stock The next slide shows statistics of two common stock with these statistics:The next slide shows statistics of two common stock with these statistics: –mean return 1 = 0.15 –mean return 2 = 0.10 –standard deviation 1 = 0.20 –standard deviation 2 = 0.25 –correlation of returns = 0.90 –initial price 1 = $57.25 –initial price 2 = $72.625

86. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 86

87. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 87 Observation The statistics indicate that one security appears to totally dominate the otherThe statistics indicate that one security appears to totally dominate the other –Security 1 has a lower risk and higher return than security 2 –In an efficient market: Wouldn’t everybody short 2, and buy 1?Wouldn’t everybody short 2, and buy 1? Wouldn’t supply and demand then cause the relative expected returns to “flip”?Wouldn’t supply and demand then cause the relative expected returns to “flip”?

88. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 88 Does it Happen? The purpose of selecting two shares with this paradoxical form is to illustrate an important point laterThe purpose of selecting two shares with this paradoxical form is to illustrate an important point later This kind of relationship does occur in the real worldThis kind of relationship does occur in the real world

89. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 89 A Pair of Price Trajectories The next graph shows a trajectory of two share prices with the statistics we have selectedThe next graph shows a trajectory of two share prices with the statistics we have selected

90. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 90

91. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 91 Observation If you were to “cut a piece” from one trajectory, re-scale it for relative price differences, and slide it over the other, you would observe that both trajectories behave in a broadly similar manner, but each has independent behavior as wellIf you were to “cut a piece” from one trajectory, re-scale it for relative price differences, and slide it over the other, you would observe that both trajectories behave in a broadly similar manner, but each has independent behavior as well Quick confirmation is seen in the region 1 to 4 years where prices are closeQuick confirmation is seen in the region 1 to 4 years where prices are close

92. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 92 Correlation The two shares are highly correlatedThe two shares are highly correlated –They track each other closely, but even adjusting for the different average returns, they have some individual behavior

93. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 93

94. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 94 Observation Shorting the high-risk, low-return stock, and re-investing in the low-risk, high- return stock, creates efficient portfoliosShorting the high-risk, low-return stock, and re-investing in the low-risk, high- return stock, creates efficient portfolios –Shorting high-risk by 80% of the net wealth crates a portfolio with a volatility of 20% and a return of 19% (c.f. 15% on security 1) –Shorting by 180% gives a volatility of 25%, and a return of 24% (c.f. 10% on security 2)

95. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 95 Observation In order to generate a portfolio that generates the same risk, but with a higher returnIn order to generate a portfolio that generates the same risk, but with a higher return –Compute the weights of the minimum portfolio, W 1 (min-var), W 2 (min-var) (Formulae given later)(Formulae given later) –Use the relationship W i (sub-opt) +W i (opt) = 2 * W i (min-var)W i (sub-opt) +W i (opt) = 2 * W i (min-var)

96. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 96 Observation –Another way to generate the two securities is to form two portfolios consisting of a risky and a riskless security that each meet the efficient frontier –Result: two portfolios that are long the risky security, and short the riskless security –Short one of the portfolios and invest in the other to generate one of the desired efficient portfolios –Repeat to generate the other Prove that the riskless security becomes irrelevantProve that the riskless security becomes irrelevant

97. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 97 Optimal Combination of Risky Assets The following slides are samples of the computations used to generate the graphsThe following slides are samples of the computations used to generate the graphs

98. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 98 Fragments of the Output Table

99. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 99 Sample of the Excel Formulae

100. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 100 Formulae for Minimum Variance Portfolio

101. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 101 Selection of the Preferred Portfolio

102. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 102 Formulae for Tangent Portfolio

103. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 103 Example: What’s the Best Return given a 10% SD?

104. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 104 Selecting the Preferred Portfolio The procedure is as followsThe procedure is as follows –Find the portfolio weights of the tangent portfolio of the line (CML) through (0, rf) –Determine the standard deviation and expectation of this point –Construct the equation of the CML –Apply investment criterion

105. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 105 Achieving the Target Expected Return (2): Weights Assume that the investment criterion is to generate a 30% returnAssume that the investment criterion is to generate a 30% return This is the weight of the risky portfolio on the CMLThis is the weight of the risky portfolio on the CML

106. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 106 Achieving the Target Expected Return (2):Volatility Now determine the volatility associated with this portfolioNow determine the volatility associated with this portfolio This is the volatility of the portfolio we seekThis is the volatility of the portfolio we seek

107. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 107 Achieving the Target Expected Return (2): Portfolio Weights

108. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 108 Investment Strategies We have examined two strategies in detail whenWe have examined two strategies in detail when –the volatility is specified –the return is specified Additionally, one of the graphs indicated an approach to take when presented with investor’s risk/return preferencesAdditionally, one of the graphs indicated an approach to take when presented with investor’s risk/return preferences

109. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 109 Portfolio of Many Risky Assets In order to solve problems with more than two securities requires tools such as quadratic programmingIn order to solve problems with more than two securities requires tools such as quadratic programming The “Solve” function in Excel may be used to obtain solutions, but it is generally better to use a software package such as the one that came with this bookThe “Solve” function in Excel may be used to obtain solutions, but it is generally better to use a software package such as the one that came with this book

110. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 110 Chapter Assumptions The theory underlying this chapter is essentially just probability theory, but there are financial assumptionsThe theory underlying this chapter is essentially just probability theory, but there are financial assumptions

111. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 111 –We do not have to assume that the generating process of returns is normal, but we do assume that the process has a mean and a variance. This is may not be the case in real life

112. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 112 –We assumed that the process was generated without inter-temporal correlations. Some investors believe that there is valuable information in old price data that has not been incorporated into the current price--this runs counter to many rigorous empirical studies.

113. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 113 –There are no “hidden variables” that explain some of the noise

114. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 114 –Investors make decisions based on mean- variances alone statistics such as skewness & kurtosis have been ignoredstatistics such as skewness & kurtosis have been ignored

115. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 115 We have made the assumption the we can lend at the risk-free rate, and that we can “short” common stock aggressivelyWe have made the assumption the we can lend at the risk-free rate, and that we can “short” common stock aggressively

116. Copyright © 2009 Pearson Education, Inc. Publishing as Prentice Hall 116 Summary There is no single investment strategy that is suitable for all investors; nor for a single investor for his whole lifeThere is no single investment strategy that is suitable for all investors; nor for a single investor for his whole life Time makes risky investments more attractive than safer investmentsTime makes risky investments more attractive than safer investments In practice, diversification has somewhat limited power to reduce riskIn practice, diversification has somewhat limited power to reduce risk