1、INVESTMENTS | BODIE, KANE, MARCUS Copyright 2011 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin CHAPTER 7 Optimal Risky Portfolios INVESTMENTS | BODIE, KANE, MARCUS The Investment Decision Top-down process with 3 steps: 1.Capital allocation between the risky portfolio and r
2、isk-free asset 2.Asset allocation across broad asset classes 3.Security selection of individual assets within each asset class INVESTMENTS | BODIE, KANE, MARCUS Diversification and Portfolio Risk Market risk Systematic or nondiversifiable Firm-specific risk Diversifiable or nonsystematic INVESTMENTS
3、 | BODIE, KANE, MARCUS Figure 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio INVESTMENTS | BODIE, KANE, MARCUS Figure 7.2 Portfolio Diversification INVESTMENTS | BODIE, KANE, MARCUS Covariance and Correlation Portfolio risk depends on the correlation between the returns of
4、 the assets in the portfolio Covariance and the correlation coefficient provide a measure of the way returns of two assets vary INVESTMENTS | BODIE, KANE, MARCUS Two-Security Portfolio: Return Portfolio Return Bond Weight Bond Return Equity Weight Equity Return p DEDE P D D E E r r w r w r wwrr ()()
5、() pDDEE E rw E rw E r INVESTMENTS | BODIE, KANE, MARCUS = Variance of Security D = Variance of Security E = Covariance of returns for Security D and Security E Two-Security Portfolio: Risk EDEDEEDD rrCovwwww,2 22222 p 2 E 2 D ED rrCov, INVESTMENTS | BODIE, KANE, MARCUS Two-Security Portfolio: Risk
6、Another way to express variance of the portfolio: 2 (,)(,)2(,) PDDDDEEEEDEDE w w Cov rrw w Cov r rw w Cov rr INVESTMENTS | BODIE, KANE, MARCUS D,E = Correlation coefficient of returns Cov(rD,rE) = DE D E D = Standard deviation of returns for Security D E = Standard deviation of returns for Security
7、E Covariance INVESTMENTS | BODIE, KANE, MARCUS Range of values for 1,2 + 1.0 -1.0 If = 1.0, the securities are perfectly positively correlated If = - 1.0, the securities are perfectly negatively correlated Correlation Coefficients: Possible Values INVESTMENTS | BODIE, KANE, MARCUS Correlation Coeffi
8、cients When DE = 1, there is no diversification When DE = -1, a perfect hedge is possible DDEEP ww D ED D E ww 1 INVESTMENTS | BODIE, KANE, MARCUS Table 7.2 Computation of Portfolio Variance From the Covariance Matrix INVESTMENTS | BODIE, KANE, MARCUS Three-Asset Portfolio 112233 ()( )( )( ) p E rw
9、E rw E rw E r 2 3 2 3 2 2 2 2 2 1 2 1 2 www p 3 , 2323 , 1312, 121 222wwwwww INVESTMENTS | BODIE, KANE, MARCUS Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions INVESTMENTS | BODIE, KANE, MARCUS Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportion
10、s INVESTMENTS | BODIE, KANE, MARCUS The Minimum Variance Portfolio The minimum variance portfolio is the portfolio composed of the risky assets that has the smallest standard deviation, the portfolio with least risk. When correlation is less than +1, the portfolio standard deviation may be smaller t
11、han that of either of the individual component assets. When correlation is -1, the standard deviation of the minimum variance portfolio is zero. INVESTMENTS | BODIE, KANE, MARCUS Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation INVESTMENTS | BODIE, KANE, MARCUS The amount of
12、possible risk reduction through diversification depends on the correlation. The risk reduction potential increases as the correlation approaches -1. If = +1.0, no risk reduction is possible. If = 0, P may be less than the standard deviation of either component asset. If = -1.0, a riskless hedge is p
13、ossible. Correlation Effects INVESTMENTS | BODIE, KANE, MARCUS Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs INVESTMENTS | BODIE, KANE, MARCUS The Sharpe Ratio Maximize the slope of the CAL for any possible portfolio, P. The objective function is the slope: The sl
14、ope is also the Sharpe ratio. () Pf P P E rr S INVESTMENTS | BODIE, KANE, MARCUS Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio INVESTMENTS | BODIE, KANE, MARCUS Figure 7.8 Determination of the Optimal Overall Portfolio INVESTMENTS |
15、BODIE, KANE, MARCUS Figure 7.9 The Proportions of the Optimal Overall Portfolio INVESTMENTS | BODIE, KANE, MARCUS Markowitz Portfolio Selection Model Security Selection The first step is to determine the risk- return opportunities available. All portfolios that lie on the minimum- variance frontier
16、from the global minimum-variance portfolio and upward provide the best risk-return combinations INVESTMENTS | BODIE, KANE, MARCUS Figure 7.10 The Minimum-Variance Frontier of Risky Assets INVESTMENTS | BODIE, KANE, MARCUS Markowitz Portfolio Selection Model We now search for the CAL with the highest
17、 reward-to-variability ratio INVESTMENTS | BODIE, KANE, MARCUS Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL INVESTMENTS | BODIE, KANE, MARCUS Markowitz Portfolio Selection Model Everyone invests in P, regardless of their degree of risk aversion. More risk averse investors
18、put more in the risk-free asset. Less risk averse investors put more in P. INVESTMENTS | BODIE, KANE, MARCUS Capital Allocation and the Separation Property The separation property tells us that the portfolio choice problem may be separated into two independent tasks Determination of the optimal risk
19、y portfolio is purely technical. Allocation of the complete portfolio to T- bills versus the risky portfolio depends on personal preference. INVESTMENTS | BODIE, KANE, MARCUS Figure 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set INVESTMENTS | BODIE, KANE, MARCUS The Pow
20、er of Diversification Remember: If we define the average variance and average covariance of the securities as: 2 11 ( ,) nn Pijij ij ww Cov r r 2 2 1 11 1 1 ( ,) (1) n i i nn ij ji j i n CovCov r r n n INVESTMENTS | BODIE, KANE, MARCUS The Power of Diversification We can then express portfolio varia
21、nce as: 22 11 P n Cov nn INVESTMENTS | BODIE, KANE, MARCUS Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes INVESTMENTS | BODIE, KANE, MARCUS Optimal Portfolios and Nonnormal Returns Fat-tailed distributions can result in extreme values of VaR and ES a
22、nd encourage smaller allocations to the risky portfolio. If other portfolios provide sufficiently better VaR and ES values than the mean-variance efficient portfolio, we may prefer these when faced with fat-tailed distributions. INVESTMENTS | BODIE, KANE, MARCUS Risk Pooling and the Insurance Princi
23、ple Risk pooling: merging uncorrelated, risky projects as a means to reduce risk. increases the scale of the risky investment by adding additional uncorrelated assets. The insurance principle: risk increases less than proportionally to the number of policies insured when the policies are uncorrelate
24、d Sharpe ratio increases INVESTMENTS | BODIE, KANE, MARCUS Risk Sharing As risky assets are added to the portfolio, a portion of the pool is sold to maintain a risky portfolio of fixed size. Risk sharing combined with risk pooling is the key to the insurance industry. True diversification means spre
25、ading a portfolio of fixed size across many assets, not merely adding more risky bets to an ever-growing risky portfolio. INVESTMENTS | BODIE, KANE, MARCUS Investment for the Long Run Long Term Strategy Invest in the risky portfolio for 2 years. Long-term strategy is riskier. Risk can be reduced by selling some of the risky assets in year 2. “Time diversification” is not true diversification. Short Term Strategy Invest in the risky portfolio for 1 year and in the risk-free asset for the second year.
