efficient portfolio frontier
DESCRIPTION
Efficient Portfolio Frontier . St. Deviation. Unique Risk. Market Risk. Number of Securities. Risk Reduction with Diversification. Two-Security Portfolio: Return. 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 - PowerPoint PPT PresentationTRANSCRIPT
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-11
Efficient Portfolio Efficient Portfolio Frontier Frontier
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-22
Risk Reduction with DiversificationRisk Reduction with Diversification
Number of Securities
St. Deviation
Market Risk
Unique Risk
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-33
rp = W1r1 + W2r2
W1 = Proportion of funds in Security 1W2 = Proportion of funds in Security 2r1 = Expected return on Security 1r2 = Expected return on Security 2
1
n
1iiw
Two-Security Portfolio: ReturnTwo-Security Portfolio: Return
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-44
p2
= w121
2 + w222
2 + 2W1W2 Cov(r1r2)
12 = Variance of Security 1
22 = Variance of Security 2
Cov(r1r2) = Covariance of returns for Security 1 and Security 2
Two-Security Portfolio: RiskTwo-Security Portfolio: Risk
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-55
1,2 = Correlation coefficient of returns
Cov(r1r2) = 12
1 = Standard deviation of returns for Security 12 = Standard deviation of returns for Security 2
CovarianceCovariance
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-66
Range of values for 1,2
+ 1.0 > 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: Correlation Coefficients: Possible ValuesPossible Values
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-77
E(rp) = W1r1 + W2r2
p2
= w121
2 + w222
2 + 2W1W2 Cov(r1r2)
p = [w1
212 + w2
222 + 2W1W2 Cov(r1r2)]1/2
Two-Security PortfolioTwo-Security Portfolio
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-88
Two-Security Portfolios withTwo-Security Portfolios withDifferent CorrelationsDifferent Correlations
= 1
13%
%8E(r)
St. Dev12% 20%
= .3
= -1
= -1
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-99
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: Portfolio Risk/Return Two Securities: Correlation EffectsCorrelation Effects
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-1010
1
1 2
- Cov(r1r2)
W1=
+ - 2Cov(r1r2)
2
W2 = (1 - W1)
2E(r2) = .14 = .20Sec 2 12 = .2E(r1) = .10 = .15Sec 1
Minimum-Variance CombinationMinimum-Variance Combination
2 2
2
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-1111
W1=
(.2)2 - (.2)(.15)(.2)
(.15)2 + (.2)2 - 2(.2)(.15)(.2)
W1 = .6733
W2 = (1 - .6733) = .3267
Minimum-Variance Combination: Minimum-Variance Combination: = .2 = .2
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-1212
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 Minimum -Variance: Return and Risk with with = .2 = .2
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-1313
W1=
(.2)2 - (.2)(.15)(.2)
(.15)2 + (.2)2 - 2(.2)(.15)(-.3)
W1 = .6087
W2 = (1 - .6087) = .3913
Minimum -Variance Combination: Minimum -Variance Combination: = -.3 = -.3
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-1414
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 Minimum -Variance: Return and Risk with with = -.3 = -.3
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-1515
2p = W1
212 + W2
2
+ 2W1W2
rp = W1r1 + W2r2 + W3r3
Cov(r1r2)
+ W323
2
Cov(r1r3)+ 2W1W3
Cov(r2r3)+ 2W2W3
Three-Security PortfolioThree-Security Portfolio
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-1616
rp = Weighted average of the n securitiesp
2 = (Consider all pairwise covariance measures)
In General, For an In General, For an n-Security Portfolio:n-Security Portfolio:
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-1717
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 SecuritiesExtending Concepts to All Securities
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-1818
The Minimum-Variance FrontierThe Minimum-Variance Frontierof Risky Assetsof Risky Assets
E(r)
Efficientfrontier
Globalminimumvarianceportfolio Minimum
variancefrontier
Individualassets
St. Dev.
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-1919
Portfolio Selection & Risk AversionPortfolio Selection & Risk AversionE(r)
Efficientfrontier ofrisky assets
Morerisk-averseinvestor
U’’’ U’’ U’
Q
PS
St. Dev
Lessrisk-averseinvestor
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-2020
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 PracticeEstimating the Frontier in Practice
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-2121
The optimal combination becomes linear
A single combination of risky and riskless assets will dominate
Extending to Include Riskless AssetExtending to Include Riskless Asset
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-2222
Objectives Constraints Policies
Return Requirements Liquidity Asset Allocation
Risk Tolerance Horizon Diversification
Regulations Risk Positioning
Taxes Tax Positioning
Unique Needs Income Generation
Portfolio Policies in PracticePortfolio Policies in Practice
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-2323
Type of Investor Return Requirement Risk Tolerance
Individual and Personal Trusts Life Cycle Life Cycle
Mutual Funds Variable Variable
Pension Funds Assumed actuarial rate Depends on payouts
Endowment Funds Determined by income Generallyneeds and asset growth to conservativemaintain real value
Matrix of ObjectivesMatrix of Objectives
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-2424
Type of Investor Return Requirement Risk Tolerance
Life Insurance Spread over cost of Conservativefunds and actuarial rates
Nonlife Ins. Co. No minimum Conservative
Banks Interest Spread Variable
Matrix of Objectives (cont’d)Matrix of Objectives (cont’d)
Irwin/McGraw-Hill
The McGraw-Hill Companies, Inc., 1999
INVESTMENTSFourth Edition
Bodie Kane Marcus
8-8-2525
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 PoliciesConstraints on Investment Policies