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Essentials of Investments: Chapter 6 - Efficient Diversification includes Two-Security Portfolio, Correlation Coefficients, Three Security Portfolio, Minimum Variance Combination, Extending Concepts to All Securities, Optimal Risky Portfolios.
1 Bodie • Kane • Marcus Essentials of Investments Fourth Edition Chapter Efficient Diversification Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved Bodie • Kane • Marcus Essentials of Investments Fourth Edition Two-Security Portfolio: Return rp = W1r1 + W2r2 W1 = Proportion of funds in Security W2 = Proportion of funds in Security r1 = Expected return on Security r2 = Expected return on Security n S Wi = i=1 Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved Bodie • Kane • Marcus Essentials of Investments Fourth Edition Two-Security Portfolio: Risk sp2 = w12s12 + w22s22 + 2W1W2 Cov(r1r2) s12 = Variance of Security s22 = Variance of Security Cov(r1r2) = Covariance of returns for Security and Security Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved Bodie • Kane • Marcus Essentials of Investments Fourth Edition Covariance Cov(r1r2) = r1,2s1s2 r1,2 = Correlation coefficient of returns s1 = Standard deviation of returns for Security s2 = Standard deviation of returns for Security Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved Bodie • Kane • Marcus Essentials of Investments Fourth Edition Correlation Coefficients: Possible Values Range of values for r 1,2 -1.0 < r < 1.0 If r = 1.0, the securities would be perfectly positively correlated If r = - 1.0, the securities would be perfectly negatively correlated Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved Bodie • Kane • Marcus Essentials of Investments Fourth Edition Three-Security Portfolio rp = W1r1 + W2r2 + W3r3 s2p = W12s12+ W22s22 + W32s32 + 2W1W2 Cov(r1r2) + 2W1W3 Cov(r1r3) + 2W2W3 Cov(r2r3) Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved Bodie • Kane • Marcus Essentials of Investments Fourth Edition In General, For an n-Security Portfolio: rp = Weighted average of the n securities sp2 = (Consider all pair-wise covariance measures) Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved Bodie • Kane • Marcus Essentials of Investments Fourth Edition Two-Security Portfolio E(rp) = W1r1 + W2r2 sp2 = w12s12 + w22s22 + 2W1W2 Cov(r1r2) sp = [w12s12 + w22s22 + 2W1W2 Cov(r1r2)]1/2 Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved Bodie • Kane • Marcus E(r) Essentials of Investments Fourth Edition TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS 13% r = -1 r=0 8% r = -1 r=1 12% Irwin / McGraw-Hill r = 20% St Dev © 2001 The McGraw-Hill Companies, Inc All rights reserved 10 Bodie • Kane • Marcus Essentials of Investments Fourth Edition Portfolio Risk/Return Two Securities: Correlation Effects • Relationship depends on correlation coefficient • -1.0 < r < +1.0 • The smaller the correlation, the greater the risk reduction potential • If r = +1.0, no risk reduction is possible Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved 22 Bodie • Kane • Marcus Essentials of Investments Fourth Edition Case 2: c =0.3 s2p= 0.017187 sp = 0.1311, rp = 0.105 Return 0.105 0.08 0.12 Irwin / McGraw-Hill 0.13 0.2 stand dev © 2001 The McGraw-Hill Companies, Inc All rights reserved 23 Bodie • Kane • Marcus Essentials of Investments Fourth Edition Portfolio Return/Risk Return c=-1 c=0.3 c=-1 c=1 Stand Dev Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved 24 Bodie • Kane • Marcus Essentials of Investments Capital Allocation for Two Risky Assets Fourth Edition Return rf Sp Max (rp -rf)/sp {w} w* =f(r1, r2, s1, s2, cov(1,2)) then, we get: rp, sp Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved 25 Bodie • Kane • Marcus Essentials of Investments Fourth Edition Example of optimal portfolio The optimal weight in the less risky asset will be: w1= (r1-rf)s22-(r2-rf)cov(1,2) (r1-rf)s22+(r2-rf)s21-(r1-rf+r2-rf)cov(1,2) w2 =1-w1 Given: r1=0.1, s1=0.2 r2=0.3, s2=0.6 c(coeff of corr)=-0.2 Then: cov=-0.24 w1=0.68 w2=1-w1=0.32 Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved 26 Bodie • Kane • Marcus Essentials of Investments Fourth Edition Lending v.s Borrowing Return U p rf Lending Sp Assume two portfolios (p, rf), weight in portfolio, y, will be: y = (rp -rf)/0.01As2p Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved 27 Bodie • Kane • Marcus Essentials of Investments Fourth Edition Markowitz Portfolio Selection • Three assets case return and variance formula for the portfolio • N-assets case Return and variance formula for the portfolio Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved 28 Bodie • Kane • Marcus E(r) Essentials of Investments Fourth Edition The minimum-variance frontier of risky assets Efficient frontier Global minimum variance portfolio Individual assets Minimum variance frontier St Dev Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved 29 Bodie • Kane • Marcus Essentials of Investments Fourth Edition Extending to Include Riskless Asset • The optimal combination becomes linear • A single combination of risky and riskless assets will dominate Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved 30 Bodie • Kane • Marcus E(r) Essentials of Investments ALTERNATIVE CALS CAL (P) Fourth Edition CAL (A) M M P P A CAL (Global minimum variance) A G F P Irwin / McGraw-Hill P&F M A&F s © 2001 The McGraw-Hill Companies, Inc All rights reserved 31 Bodie • Kane • Marcus Essentials of Investments Fourth Edition Dominant CAL with a Risk-Free Investment (F) CAL(P) dominates other lines it has the best risk/return or the largest slope Slope = (E(R) - Rf) / s [ E(RP) - Rf) / s P ] > [E(RA) - Rf) / sA] Regardless of risk preferences combinations of P & F dominate Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved 32 Bodie • Kane • Marcus Essentials of Investments Fourth Edition Single Factor Model - CAPM ri = E(Ri) + ßiF + e ßi = index of a securities’ particular return to the factor F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns Assumption: a broad market index like the S&P500 is the common factor Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved 33 Bodie • Kane • Marcus Essentials of Investments Fourth Edition Single Index Model r r r r e i f Risk Prem i i i m f i Market Risk Prem or Index Risk Prem = the stock’s expected return if the market’s excess return is zero (rm - rf) = ßi(rm - rf) = the component of return due to movements in the market index ei = firm specific component, not due to market movements Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved 34 Bodie • Kane • Marcus Essentials of Investments Fourth Edition run OLS in the estimation window, get ˆ i and ˆ i then compute the abnormal returns ARit Rit ˆ i ˆi Rmt ARit is the abnormal for security i at time t - it is the error term of the market model calculated on an out of sample basis - it is basically the forecast error - the difference between the actual Rit and the forecast Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved 35 Bodie • Kane • Marcus Essentials of Investments Fourth Edition Risk Premium Format Let: Ri = (ri - rf) Rm = (rm - rf) Risk premium format Ri = i + ßi(Rm) + ei Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved 36 Bodie • Kane • Marcus Essentials of Investments Fourth Edition Estimating the Index Model Excess Returns (i) Irwin / McGraw-Hill Security Characteristic Line Excess returns on market index Ri = i + òiRm + ei â 2001 The McGraw-Hill Companies, Inc All rights reserved ... (r1-rf)s2 2-( r2-rf)cov(1,2) (r1-rf)s22+(r2-rf)s2 1-( r1-rf+r2-rf)cov(1,2) w2 =1-w1 Given: r1=0.1, s1=0.2 r2=0.3, s2=0 .6 c(coeff of corr) =-0 .2 Then: cov =-0 .24 w1=0 .68 w2=1-w1=0.32 Irwin / McGraw-Hill... (.15)2 + (.2)2 - 2(.2)(.15) (-. 3) W1 = 60 87 W2 = (1 - 60 87) = 3913 Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved 15 Bodie • Kane • Marcus Essentials of Investments... - (.2)(.15)(.2) W1 = (.15)2 + (.2)2 - 2(.2)(.15)(.2) W1 = 67 33 W2 = (1 - 67 33) = 3 267 Irwin / McGraw-Hill © 2001 The McGraw-Hill Companies, Inc All rights reserved 13 Bodie • Kane • Marcus Essentials