Chapter 12: Portfolio Selection and Diversification Objective To understand the theory of personal portfolio selection in theory and in practice Copyright © Prentice Hall Inc 2000 Author: Nick Bagley, bdellaSoft, Inc Chapter 12 Contents • 12.1 The process of personal portfolio selection • 12.2 The trade-off between expected return and risk • 12.3 Efficient diversification with many risky assets Objectives • To understand the process of personal portfolio selection in theory and practice Security Prices 100000 Stock Bond Stock_Mu Bond_Mu Value (Log) 10000 1000 100 10 10 15 20 25 Years 30 35 40 Security Prices 100000 Stock Bond Stock_Mu Bond_Mu Value (Log) 10000 1000 100 10 10 15 20 25 Years 30 35 40 Security Prices 100000 Stock Bond Stock_Mu Bond_Mu Value (Log) 10000 1000 100 10 10 15 20 25 Years 30 35 40 …and Lots More! Security Prices Security Prices 100000 100000 Stock Bond Stock_Mu Bond_Mu Stock Bond Stock_Mu Bond_Mu 10000 Value (Log) Value (Log) 10000 1000 100 1000 100 10 10 Security 15 20Prices 25 10 30 35 40 10 Security 15 20Prices 25 Years 100000 100000 Stock Bond Stock_Mu Bond_Mu 1000 100 35 40 30 35 40 Stock Bond Stock_Mu Bond_Mu 10000 Value (Log) Value (Log) 10000 30 Years 1000 100 10 10 10 15 20 Years 25 30 35 40 10 15 20 Years 25 Security Prices 100000 Stock Bond Stock_Mu Bond_Mu Value (Log) 10000 1000 100 10 10 15 20 Years 25 30 35 40 Probability of Future Price 0.035 Prob_Stock_2 Prob_Bond_2 Prob_Stock_5 Prob_Bond_5 Prob_Stock_10 Prob_Bond_10 Prob_Stock_40 Prob_Bond_40 Probability Density 0.030 0.025 0.020 0.015 0.010 0.005 0.000 50 100 150 Value 200 250 300 Probabilistic Stock Price Changes Over Time 0.020 Stock_Year_1 Stock_Year_2 Stock_Year_3 Stock_Year_4 Stock_Year_5 Stock_Year_6 Stock_Year_7 Stock_Year_8 Stock_Year_9 Stock_Year_10 0.018 Probability Density 0.016 0.014 0.012 0.010 0.008 0.006 0.004 0.002 0.000 200 400 Price 600 10 800 Mnemonic • There 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 summing 34 Variance with Securities W1*Sig1 W1*Sig1 W2*Sig2 Rho(1,2) W2*Sig2 Rho(2,1) σ = w σ + w σ + 2w1w2σ 1σ ρ1, 2 p 2 2 2 35 Variance with Securities W1*Sig1 W2*Sig2 W3*Sig3 W1*Sig1 Rho(1,2) Rho(1,3) W2*Sig2 Rho(2,1) Rho(2,3) W3*Sig3 Rho(3,1) Rho(3,2) σ = w σ + w σ + w σ + 2w1w2σ 1σ ρ1, + p 2 2 2 3 2w1w3σ 1σ ρ1,3 + 2w2 w3σ 2σ ρ 2,3 36 Correlated Common Stock • The next slide shows statistics of two common stock with these statistics: – – – – – – – mean return = 0.15 mean return = 0.10 standard deviation = 0.20 standard deviation = 0.25 correlation of returns = 0.90 initial price = $57.25 Initial price = $72.625 37 2-Shares: Is One "Better?" 0.16 0.14 Expected Return 0.12 0.1 0.08 0.06 0.04 0.02 0 0.05 0.1 0.15 0.2 Standard Deviation 38 0.25 0.3 Share Prices 350 Value (adjusted for Splits) 300 250 200 ShareP_1 ShareP_2 150 100 50 0 Years 39 10 Portfolio of Two Securities 0.25 Efficient Share Expected Return 0.20 Share 0.15 Minimum Variance 0.10 0.05 Suboptima l 0.00 0.15 0.17 0.19 0.21 0.23 Standard Deviation 40 0.25 0.27 0.29 Fragments of the Output Table -0.30 -0.20 Data For two securities -0.10 This data has been constructed to produce the mean-varience paradox 0.00 0.10 mu_1 15.00% 0.20 mu_2 10.00% 0.30 sig_1 20.00% sig_2 25.00% 1.30 rho 90.00% 1.40 1.50 1.60 w_1 w_2 Port_Sig Port_Mu 1.70 -2.50 3.50 0.4776 -0.0250 1.80 -2.40 3.40 0.4674 -0.0200 -2.30 3.30 0.4573 -0.0150 1.90 -2.20 3.20 0.4472 -0.0100 2.00 2.10 -2.10 3.10 0.4372 -0.0050 -2.00 3.00 0.4272 0.0000 2.20 -1.90 2.90 0.4173 0.0050 2.30 -1.80 2.80 0.4074 0.0100 2.40 -1.70 2.70 0.3976 0.0150 2.50 41 1.30 1.20 1.10 1.00 0.90 0.80 0.70 0.2723 0.2646 0.2571 0.2500 0.2432 0.2366 0.2305 0.0850 0.0900 0.0950 0.1000 0.1050 0.1100 0.1150 -0.30 -0.40 -0.50 -0.60 -0.70 -0.80 -0.90 -1.00 -1.10 -1.20 -1.30 -1.40 -1.50 0.1953 0.1949 0.1953 0.1962 0.1978 0.2000 0.2028 0.2062 0.2101 0.2145 0.2194 0.2247 0.2305 0.1650 0.1700 0.1750 0.1800 0.1850 0.1900 0.1950 0.2000 0.2050 0.2100 0.2150 0.2200 0.2250 Sample of the Excel Formulae =w_1*mu_1 + w_2*mu_2 w_1 -2.5 =A14+0.1 =A15+0.1 =A16+0.1 w_2 =1-A14 =1-A15 =1-A16 =1-A17 Port_Sig =SQRT(w_1^2*sig_1^2 + =SQRT(w_1^2*sig_1^2 + =SQRT(w_1^2*sig_1^2 + =SQRT(w_1^2*sig_1^2 + 2*w_1*w_2*sig_1*sig_2*rho + 2*w_1*w_2*sig_1*sig_2*rho + 2*w_1*w_2*sig_1*sig_2*rho + 2*w_1*w_2*sig_1*sig_2*rho + w_2^2*sig_2^2) w_2^2*sig_2^2) w_2^2*sig_2^2) w_2^2*sig_2^2) Port_Mu =w_1*mu_1 + =w_1*mu_1 + =w_1*mu_1 + =w_1*mu_1 + w_2*mu_2 w_2*mu_2 w_2*mu_2 w_2*mu_2 =SQRT(w_1^2*sig_1^2 + 2*w_1*w_2*sig_1*sig_2*rho + w_2^2*sig_2^2) 42 Formulae for Minimum Variance Portfolio σ − ρ1, 2σ 1σ * w1 = 2 σ − ρ1, 2σ 1σ + σ σ − ρ1, 2σ 1σ w = 2 σ − ρ1, 2σ 1σ + σ 2 * = 1− w * 43 Formulae for Tangent Portfolio w1tan = (µ (µ ) − r σ f − ( µ − r f ) ρ1, 2σ 1σ 2 ) ( ) ( ) − r σ − µ − r + µ − r ρ σ σ + µ − r σ f 1 f f 1, 2 f w2tan = − w1 ( ) 10 * 25 − ( 0.05) * 0.90 * 0.20 * 0.25 tan w1 = ( 0.05) * 0.20 − ( 0.10 + 0.05) * 0.90 * 0.20 * 0.25 + ( 0.10) * 0.252 w1tan = 23 w2tan = −1 23 44 Example: What’s the Best Return given a 10% SD? µ tan = w1tan µ1 + w2tan µ µ tan = 0.15 − 0.10 3 µ tan = 0.2333 σ tan = (w ) σ + (w ) σ tan 2 tan 2 2 + 2w1tan w2tanσ 1σ ρ1, 2 8 5 σ tan = 0.20 + − 0.252 + 2 − * 0.2 * 0.25 * 0.90 3 3 σ tan = 0.2409 µ tan − rf 0.2333 − 0.05 µ= σ + rf = 0.10 + 0.05 = 0.1261 σ tan 0.2409 45 Achieving the Target Expected Return (2): Weights • Assume that the investment criterion is to generate a 30% return µ criterion = µ tangent w1 + rf (1 − w1 ) µ criterion − rf 0.30 − 0.05 w1 = = = 1.3636 µ tangent − rf 0.2333 − 0.05 • This is the weight of the risky portfolio on the CML 46 Achieving the Target Expected Return (2):Volatility • Now determine the volatility associated with this portfolio σ = w1σ tangent = 1.3636 * 0.2409 = 0.3285 • This is the volatility of the portfolio we seek 47 Achieving the Target Expected Return (2): Portfolio Weights COMPUTATION WEIGHT RISKLESS -0.3636 -0.3636 ASSET 1.3636*2.6667 ASSET 1.3636*(-1.6667) 3.6363 TOTAL -2.2727 1.0000 48 [...]... Asset and a Single Risky Asset – The volatility of the portfolio is not quite as simple: σp = ((W1* σ1)2 + 2W1* σ1* W2* σ2 + (W2* σ2)2)1/2 22 Combining the Riskless Asset and a Single Risky Asset – We know something special about the portfolio, namely that security 2 is riskless, so σ2 = 0, and σp becomes: σp = ((W1* σ1)2 + 2W1* σ1* W2* 0 + (W2* 0)2)1/2 σp = |W1| * σ1 23 Combining the Riskless Asset and. .. |W1| * σ1, And: µp = W1*µ1 + (1- W1)*rf , So: If W1>0, µp = [(rf -µ1)/ σ1]*σp + rf Else µp = [(µ1-rf )/ σ1]*σp + rf 24 A Portfolio of a Risky and a Riskless Security 0.30 0.25 0.20 Return 0.15 0.10 0.05 0.00 0.00 -0.05 0.10 0.20 0.30 -0.10 -0.15 -0.20 Volatility 25 0.40 0.50 Capital Market Line 0.30 100% Risky 0.25 Long risky and short risk-free Return 0.20 0.15 100% RiskLong both risky and risk-free... on a 20% return, and decide not to pursue on the computational issue – Recall: µp = W1*µ1 + (1- W1)*rf – Your portfolio: σ = 20%, µ = 15%, rf = 5% – So: W1 = (µp - rf)/(µ1 - rf) = (0.20 - 0.05)/(0.15 - 0.05) = 150% 28 To obtain a 20% Return • Assume 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 difference... 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 variable • A reasonable assumption for returns on different securities is the linear model: rp = w1r1 + w2 r2 ; with w1 + w2 = 1 31 Equations for Two Shares • The sum of the weights w1 and w2 being 1 is not necessary for the validity of the following... 100 120 Price 13 140 160 180 200 5-Years Out Mode = 122 0.020 0.018 Stock_5_Year Bond_5_Year 0.016 Density 0.014 Median= 126 Mean = 128 Mode = 135 0 .012 0.010 Median= 165 Mean = 182 0.008 0.006 0.004 0.002 0.000 0 100 200 300 Price 400 14 500 10-Years Out 0 .012 0.010 Stock_10_Year Density 0.008 Bond_10_Year 0.006 0.004 0.002 0.000 0 200 400 600 800 Value 15 1,000 40 Years Out Mode =503 0.002 Median=650... 4.46 116.1 5.18 90 221.77 3.18 190.75 3.45 95 329.96 1.87 317.32 1.91 18 Deaths Per Thousand M & F 350 300 MDePm Deaths / 1000 250 FDePm 200 150 100 50 0 60 65 70 75 80 85 Age 19 90 95 Life Expection Remaining Expected Life 25 20 MExLife 15 FExLife 10 5 0 60 65 70 75 80 85 Age 20 90 95 Combining the Riskless Asset and a Single Risky Asset – The expected return of the portfolio is the weighted average... measure of risk with this property, but for standard deviation: 2 2 2 2 2 p 1 1 1 2 1 2 1, 2 2 2 σ = w σ + 2w w σ σ ρ + w σ 33 Mnemonic • There 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 summing 34 Variance with 2 Securities W1*Sig1