Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 1 Chapter 7 Efficient Diversification

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 2 r p = W 1 = W 2 = r 1 = r 2 = Two-Security Portfolio: Return W i  i=1n = 1

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 3 p2 =p2 = p2 =p2 =  1 2 = Variance of Security 1  2 2 = Variance of Security 2 Cov(r 1 r 2 ) = Covariance of returns for Security 1 and Security 2 Cov(r 1 r 2 ) = 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 Marcus 4 Covariance  1,2 = Correlation coefficient of returns  1,2 = Correlation coefficient of returns Cov(r 1 r 2 ) =  1 = Standard deviation of returns for Security 1  2 = Standard deviation of returns for Security 2  1 = Standard deviation of returns for Security 1  2 = 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 Marcus 5 Correlation Coefficients: Possible Values If  = 1.0, If  = - 1.0, Range of values for  1,2

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 6  2 p = r p = Three-Security Portfolio

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 7 r p = Weighted average of the n securities r p = Weighted average of the n securities  p 2 = (Consider all pair-wise covariance measures)  p 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 Marcus 8 E(r p ) = Two-Security Portfolio  p 2 =  p =

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 9  = 0 E(r)  = 1  = -1  =.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 Marcus 10 Portfolio Risk/Return Two Securities: Correlation Effects Relationship depends on correlation coefficient -1.0 <  < +1.0

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus W1W1 W1W1 = = W2W2 W2W2 = (1 - W 1 ) Minimum Variance Combination     2 2 E(r 2 ) =.14 =.20 Sec 2 12 =.2 E(r 1 ) =.10 =.15 Sec 1        

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 12 W1W1W1W1 = (.2) 2 - (.2)(.15)(.2) (.15) 2 + (.2) 2 - 2(.2)(.15)(.2) W1W1W1W1= W2W2W2W2= Minimum Variance Combination:  =.2

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 13 rp =rp =rp =rp = p = p = Minimum Variance: Return and Risk with  =.2    

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 14 W1W1W1W1 = (.2) 2 - (-.3)(.15)(.2) (.15) 2 + (.2) 2 - 2(.2)(.15)(-.3) W1W1W1W1= W2W2W2W2= Minimum Variance Combination:  = -.3

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 15 rp =rp =rp =rp = p = p = Minimum Variance: Return and Risk with  = -.3    

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 16 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 Marcus 17 E(r) The minimum-variance frontier of risky assets Efficientfrontier Globalminimumvarianceportfolio Minimumvariancefrontier Individualassets St. Dev.

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 18 Extending to Include Riskless Asset

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 19 E(r) CAL (Global minimum variance) CAL (A) CAL (P) M P A F PP&FA&F M A G P M  ALTERNATIVE CALS

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 20 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 P ) - R f ) /  P   E(R A ) - R f ) /    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 Marcus 21 Single Factor Model r i = ß i = 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 Marcus 22 Single Index Model Risk Prem Market Risk Prem or Index Risk Prem or Index Risk Prem i = the stock’s expected return if the market’s excess return is zero market’s excess return is zero ß i (r m - r f ) = the component of return due to movements in the market index movements in the market index (r m - r f ) = 0 e i = firm specific component, not due to market movements movements 

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 23Let: Risk premium format R i =  i + ß i (R m ) + e i Risk Premium Format

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 24 Estimating the Index Model Excess Returns (i) SecurityCharacteristicLine Excess returns on market index R i =  i + ß i R m + e i

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 25 Components of Risk Market or systematic risk: risk related to the macro economic factor or market index Unsystematic or firm specific risk: risk not related to the macro factor or market index Total risk =

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 26 Measuring Components of Risk 

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 27 Total Risk = Systematic Risk + Unsystematic Risk Systematic Risk/Total Risk =  2 ß i 2  m 2 /  2 =  2  i 2  m 2 /  i 2  m 2 +  2 (e i ) =  2 Examining Percentage of Variance

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 28 Advantages of the Single Index Model Reduces the number of inputs for diversification Easier for security analysts to specialize