 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-1 Irwin/McGraw-Hill Efficient Portfolio Frontier

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-2 Irwin/McGraw-Hill Risk Reduction with Diversification Number of Securities St. Deviation Market Risk Unique Risk

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-3 Irwin/McGraw-Hill 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

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-4 Irwin/McGraw-Hill  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 Two-Security Portfolio: Risk

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-5 Irwin/McGraw-Hill  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 Covariance

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-6 Irwin/McGraw-Hill Range of values for  1, >  1,2 > -1.0 If  = 1.0, the securities would be perfectly positively correlated If  = - 1.0, the securities would be perfectly negatively correlated Correlation Coefficients: Possible Values

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-7 Irwin/McGraw-Hill E(r p ) = W 1 r 1 + W 2 r 2  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 Two-Security Portfolio

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-8 Irwin/McGraw-Hill Two-Security Portfolios with Different Correlations  = 1 13% %8%8 E(r) St. Dev 12%20%  =.3  = -1

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-9 Irwin/McGraw-Hill 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 Portfolio Risk/Return Two Securities: Correlation Effects

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-10 Irwin/McGraw-Hill Cov(r 1 r 2 ) W1W1 = + - 2Cov(r 1 r 2 ) 2 W2W2 = (1 - W 1 )   2 E(r 2 ) =.14=.20Sec 2 12 =.2 E(r 1 ) =.10=.15Sec 1     Minimum-Variance Combination 2 2 2

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-11 Irwin/McGraw-Hill W1W1 = (.2) 2 - (.2)(.15)(.2) (.15) 2 + (.2) 2 - 2(.2)(.15)(.2) W1W1 =.6733 W2W2 = ( ) =.3267 Minimum-Variance Combination:  =.2

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-12 Irwin/McGraw-Hill 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

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-13 Irwin/McGraw-Hill W1W1 = (.2) 2 - (.2)(.15)(.2) (.15) 2 + (.2) 2 - 2(.2)(.15)(-.3) W1W1 =.6087 W2W2 = ( ) =.3913 Minimum -Variance Combination:  = -.3

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-14 Irwin/McGraw-Hill 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

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-15 Irwin/McGraw-Hill  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

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-16 Irwin/McGraw-Hill r p = Weighted average of the n securities  p 2 = (Consider all pairwise covariance measures) In General, For an n-Security Portfolio:

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-17 Irwin/McGraw-Hill 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 Extending Concepts to All Securities

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-18 Irwin/McGraw-Hill The Minimum-Variance Frontier of Risky Assets E(r) Efficient frontier Global minimum variance portfolio Minimum variance frontier Individual assets St. Dev.

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-19 Irwin/McGraw-Hill Portfolio Selection & Risk Aversion E(r) Efficient frontier of risky assets More risk-averse investor U’’’ U’’ U’ Q P S St. Dev Less risk-averse investor

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-20 Irwin/McGraw-Hill Estimating the expected future returns Estimating the variance-covariance matrix - Sample period: how long is too short? - Relevance: is the future the same as the past? Estimating the Frontier in Practice

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-21 Irwin/McGraw-Hill The optimal combination becomes linear A single combination of risky and riskless assets will dominate Extending to Include Riskless Asset

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-22 Irwin/McGraw-Hill ObjectivesConstraintsPolicies Return RequirementsLiquidityAsset Allocation Risk ToleranceHorizonDiversification RegulationsRisk Positioning TaxesTax Positioning Unique NeedsIncome Generation Portfolio Policies in Practice

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-23 Irwin/McGraw-Hill Type of InvestorReturn RequirementRisk Tolerance Individual and Personal Trusts Life CycleLife Cycle Mutual FundsVariableVariable Pension FundsAssumed actuarial rateDepends on payouts Endowment FundsDetermined by income Generally needs and asset growth to conservative maintain real value Matrix of Objectives

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-24 Irwin/McGraw-Hill Type of InvestorReturn RequirementRisk Tolerance Life InsuranceSpread over cost of Conservative funds and actuarial rates Nonlife Ins. Co. No minimum Conservative BanksInterest Spread Variable Matrix of Objectives (cont’d)

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus 8-25 Irwin/McGraw-Hill Liquidity - Ease (speed) with which an asset can be sold and created into cash Investment horizon - planned liquidation date of the investment Regulations - Prudent man law Tax considerations Unique needs Constraints on Investment Policies