Bài giảng Essentials of Investments - Chapter 6 Efficient Diversification

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus1 Chapter 6 Efficient Diversification Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus2 rp = W1r1 + W2r2 W1 = Proportion of funds in Security 1 W2 = Proportion of funds in Security 2 r1 = Expected return on Security 1 r2 = Expected return on Security 2 Two-Security Portfolio: Return WiS i=1 n = 1 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus3 sp 2 = w1 2s1 2 + w2 2s2 2 + 2W1W2 Cov(r1r2) s1 2 = Variance of Security 1 s2 2 = Variance of Security 2 Cov(r1r2) = Covariance of returns for Security 1 and Security 2 Two-Security Portfolio: Risk Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus4 Covariance r1,2 = Correlation coefficient of returns Cov(r1r2) = r1,2s1s2 s1 = Standard deviation of returns for Security 1 s2 = Standard deviation of returns for Security 2 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus5 Correlation Coefficients: Possible Values If r = 1.0, the securities would be perfectly positively correlated If r = - 1.0, the securities would be perfectly negatively correlated Range of values for r 1,2 -1.0 < r < 1.0 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus6 s2p = W1 2s1 2+ W2 2s2 2 + 2W1W2 rp = W1r1 + W2r2 + W3r3 Cov(r1r2) + W3 2s3 2 Cov(r1r3)+ 2W1W3 Cov(r2r3)+ 2W2W3 Three-Security Portfolio Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus7 rp = Weighted average of the n securities sp 2 = (Consider all pair-wise covariance measures) In General, For an n-Security Portfolio: Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus8 E(rp) = W1r1 + W2r2 Two-Security Portfolio sp 2 = w1 2s1 2 + w2 2s2 2 + 2W1W2 Cov(r1r2) sp = [w1 2s1 2 + w2 2s2 2 + 2W1W2 Cov(r1r2)] 1/2 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus9 r = 0 E(r) r = 1 r = -1 r = -1 r = .3 13% 8% 12% 20% St. Dev TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus10 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 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus11 1 1 2 - Cov(r1r2) W1 = + - 2Cov(r1r2) 2 W2 = (1 - W1) Minimum Variance Combination s 2 s 2 2E(r2) = .14 = .20Sec 2 12 = .2 E(r1) = .10 = .15Sec 1 s s r s 2 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus12 W1 = (.2)2 - (.2)(.15)(.2) (.15)2 + (.2)2 - 2(.2)(.15)(.2) W1 = .6733 W2 = (1 - .6733) = .3267 Minimum Variance Combination: r = .2 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus13 rp = .6733(.10) + .3267(.14) = .1131 p = [(.6733)2(.15)2 + (.3267)2(.2)2 + 2(.6733)(.3267)(.2)(.15)(.2)] 1/2 p= [.0171] 1/2 = .1308 Minimum Variance: Return and Risk with r = .2 s s Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus14 W1 = (.2)2 - (.2)(.15)(.2) (.15)2 + (.2)2 - 2(.2)(.15)(-.3) W1 = .6087 W2 = (1 - .6087) = .3913 Minimum Variance Combination: r = -.3 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus15 rp = .6087(.10) + .3913(.14) = .1157 p = [(.6087)2(.15)2 + (.3913)2(.2)2 + 2(.6087)(.3913)(.2)(.15)(-.3)]1/2 p = [.0102] 1/2 = .1009 Minimum Variance: Return and Risk with r = -.3 s s Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus16 Extending Concepts to All Securities • The optimal combinations result in lowest level of risk for a given return • The optimal trade-off is described as the efficient frontier • These portfolios are dominant Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus17 Optimal Risky Portfolios • Portfolio Diversification • Portfolios of Two Risky Assets • Asset Allocation • Markowitz Portfolio Model Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus18 Portfolio Diversification Return Time Stock x Stock y x + y Implications: combination of stocks can reduce overall risk (variance). Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus19 Portfolio Risk behavior Variance # Assets systematic risk Unsystematic risk After a certain number of securities, portfolio variance can no longer be reduced Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus20 Portfolios of Two Risky Assets • Given r1 =0.08, s1=0.12 r2 =0.13, s2=0.20 w1 =w2 = 0.5 (assumption) • rp = w1r1 + w2r2 =0.5(0.08) + 0.5(0.13) = 0.105 s2p=w 2 1s 2 1 + w 2 2s 2 2 + 2w1w2cov12 =0.25(0.0144)+0.25(0.04) + 2(0.5)(0.5)cs1s2 • case (1): Assume c=1.0 s2p = 0.0256 sp = 0.16 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus21 return stand. dev 0.12 0.16 0.08 0.2 1 2 0.105 0.13 Portfolio Return/Risk If more weight is invested in security 1, the tradeoff line will move downward. Otherwise, it will move upward. Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus22 Case 2: c =0.3 s2p= 0.017187 sp = 0.1311, rp = 0.105 0.08 0.105 Return stand. dev 0.12 0.13 0.2 1 2 . Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus23 Return Stand. Dev. Portfolio Return/Risk c=-1 c=0.3 c=-1 c=1 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus24 Capital Allocation for Two Risky Assets rf 1 2 Return Sp Max (rp -rf)/sp {w} w* =f(r1, r2, s1, s2, cov(1,2)) then, we get: rp, sp Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus25 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)s 2 2+(r2-rf)s 2 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 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus26 Lending v.s Borrowing rf 1 2 Return Sp U Assume two portfolios (p, rf), weight in portfolio, y, will be: y = (rp -rf)/0.01As 2 p Lending p Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus27 Markowitz Portfolio Selection • Three assets case return and variance formula for the portfolio • N-assets case Return and variance formula for the portfolio Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus28 E(r) The minimum-variance frontier of risky assets Efficient frontier Global minimum variance portfolio Minimum variance frontier Individual assets St. Dev. Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus29 Extending to Include Riskless Asset • The optimal combination becomes linear • A single combination of risky and riskless assets will dominate Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus30 E(r) CAL (Global minimum variance) CAL (A)CAL (P) M P A F P P&F A&FM A G P M s ALTERNATIVE CALS Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus31 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 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus32 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 Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus33 Single Index Model Risk Prem Market Risk Prem or Index Risk Prem i = the stock’s expected return if the market’s excess return is zero ßi(rm - rf) = the component of return due to movements in the market index (rm - rf) = 0 ei = firm specific component, not due to market movements      errrr ifmiifi   Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus34 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 returns abnormal thecomputethen ˆ and ˆget window,estimation in the OLSrun ii  mtiiitit RRAR  ˆˆ  Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus35 Let: Ri = (ri - rf) Rm = (rm - rf) Risk premium format Ri = i + ßi(Rm) + ei Risk Premium Format Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie • Kane • Marcus36 Estimating the Index Model Excess Returns (i) Security Characteristic Line . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . . ... . . .. . Excess returns on market index Ri =  i + ßiRm + ei