1. 1 Chapter 12: Portfolio Selection and Diversification Copyright © Prentice Hall Inc. 2000. Author: Nick Bagley, bdellaSoft, Inc. Objective To understand the theory of personal portfolio selection in theory and in practice

2. 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. 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

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7. 7 …and Lots More!

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12. 12 Mode =104Mode =106Median=104Mean =104Median=111Mean = 113

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14. 14 Mode = 122Mode = 135Median= 126 Mean = 128Median= 165 Mean = 182

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16. 16 Mode =503Mode =1,102Median=650Mean =739Median=5,460Mean =12,151

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21. 21 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

22. 22 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

23. 23 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

24. 24 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

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26. 26 Long risky and short risk-free Long both risky and risk-free 100% Risky 100% Risk- less

27. 27 Mutual Fund Average % Total Returns

28. 28 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%

29. 29 To obtain a 20% Return Assume that your manage a $50,000,000 portfolioAssume that your 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

30. 30 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%

31. 31 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:

32. 32 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

33. 33 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:

34. 34 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

35. 35 Variance with 2 Securities

36. 36 Variance with 3 Securities

37. 37 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

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41. 41 Fragments of the Output Table

42. 42 Sample of the Excel Formulae

43. 43 Formulae for Minimum Variance Portfolio

44. 44 Formulae for Tangent Portfolio

45. 45 Example: What’s the Best Return given a 10% SD?

46. 46 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

47. 47 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

48. 48 Achieving the Target Expected Return (2): Portfolio Weights