Chapter 6 Efficient Diversification

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. r p = W 1 r 1 + W 2 r 2 W 1 = Proportion of funds in Security 1 W 2 = Proportion of funds in Security 2 r 1 = Expected return on Security 1 r 2 = Expected return on Security 2 Two-Security Portfolio: Return W i  i=1n = 1

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved.  p 2 = w 1 2  w 2 2  W 1 W 2 Cov(r 1 r 2 )  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

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Covariance  1,2 = Correlation coefficient of returns  1,2 = Correlation coefficient of returns Cov(r 1 r 2 ) =    1  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

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Correlation Coefficients: Possible Values If  = 1.0, the securities would be perfectly positively correlated If  = - 1.0, the securities would be perfectly negatively correlated Range of values for  1, <  < 1.0

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved.  2 p = W 1 2  W 2 2    + 2W 1 W 2 r p = W 1 r 1 + W 2 r 2 + W 3 r 3 Cov(r 1 r 2 ) + W 3 2  3 2 Cov(r 1 r 3 ) + 2W 1 W 3 Cov(r 2 r 3 ) + 2W 2 W 3 Three-Security Portfolio

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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:

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. E(r p ) = W 1 r 1 + W 2 r 2 Two-Security Portfolio  p 2 = w 1 2  w 2 2  W 1 W 2 Cov(r 1 r 2 )  p = [w 1 2  w 2 2  W 1 W 2 Cov(r 1 r 2 )] 1/2

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved.  = 0 E(r)  = 1  = -1  =.3 13% 8% 12%20% St. Dev TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Portfolio Risk/Return Two Securities: Correlation Effects Relationship depends on correlation coefficient -1.0 <  < +1.0 The smaller the correlation, the greater the risk reduction potential If  = +1.0, no risk reduction is possible

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved Cov(r 1 r 2 ) W1W1 W1W1 = = Cov(r 1 r 2 ) 2 2 W2W2 W2W2 = (1 - W 1 ) Minimum Variance Combination  E(r 2 ) =.14 =.20 Sec 2 12 =.2 E(r 1 ) =.10 =.15 Sec 1        2

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. W1W1W1W1 = (.2) 2 - (.2)(.15)(.2) (.15) 2 + (.2) 2 - 2(.2)(.15)(.2) W1W1W1W1 =.6733 W2W2W2W2 = ( ) =.3267 Minimum Variance Combination:  =.2

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. r p =.6733(.10) (.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  =.2    

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. W1W1W1W1 = (.2) 2 - (.2)(.15)(-.3) (.15) 2 + (.2) 2 - 2(.2)(.15)(-.3) W1W1W1W1 =.6087 W2W2W2W2 = ( ) =.3913 Minimum Variance Combination:  = -.3

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. r p =.6087(.10) (.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  = -.3    

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. E(r) The minimum-variance frontier of risky assets Efficientfrontier Globalminimumvarianceportfolio Minimumvariancefrontier Individualassets St. Dev.

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Extending to Include Riskless Asset The optimal combination becomes linear A single combination of risky and riskless assets will dominate

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. E(r) CAL (Global minimum variance) CAL (A) CAL (P) M P A F PP&FA&F M A G P M  ALTERNATIVE CALS

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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) /   E(R P ) - R f ) /  P   E(R A ) - R f ) /    Regardless of risk preferences combinations of P & F dominate

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Single Factor Model R i = E(R i ) + ß i F + e 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

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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   e rrrr i fm i i fi   

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Let: R i = (r i - r f ) R m = (r m - r f ) R m = (r m - r f ) Risk premium format R i =  i + ß i (R m ) + e i Risk Premium Format

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Estimating the Index Model Excess Returns (i) SecurityCharacteristicLine Excess returns on market index R i =  i + ß i R m + e i

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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 = Systematic + Unsystematic

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Measuring Components of Risk  i 2 =  i 2  m 2 +  2 (e i ) where;  i 2 = total variance  i 2  m 2 = systematic variance  2 (e i ) = unsystematic variance

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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

McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Advantages of the Single Index Model Reduces the number of inputs for diversification Easier for security analysts to specialize