Report

CHAPTER 7 Optimal Risky Portfolios Investments, 8th edition Bodie, Kane and Marcus Slides by Susan Hine McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved. Diversification and Portfolio Risk • Market risk – Systematic or nondiversifiable • Firm-specific risk – Diversifiable or nonsystematic 7-2 Figure 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio 7-3 Figure 7.2 Portfolio Diversification 7-4 Covariance and Correlation • Portfolio risk depends on the correlation between the returns of the assets in the portfolio • Covariance and the correlation coefficient provide a measure of the way returns two assets vary 7-5 Two-Security Portfolio: Return rp rP Portfolio Return wr D D wE r E wD Bond Weight rD Bond Return wE Equity Weight rE Equity Return E (rp ) wD E (rD ) wE E (rE ) 7-6 Two-Security Portfolio: Risk w w 2wDw EE Cov(rD , rE ) 2 P 2 D 2 D 2 E 2 E 2 D = Variance of Security D 2 E = Variance of Security E Cov(rD , rE )= Covariance of returns for Security D and Security E 7-7 Two-Security Portfolio: Risk Continued • Another way to express variance of the portfolio: P2 wD wDCov(rD , rD ) wE wE Cov(rE , rE ) 2wD wE Cov(rD , rE ) 7-8 Covariance Cov(rD,rE) = DEDE D,E = Correlation coefficient of returns D = Standard deviation of returns for Security D E = Standard deviation of returns for Security E 7-9 Correlation Coefficients: Possible Values Range of values for 1,2 + 1.0 > > -1.0 If = 1.0, the securities would be perfectly positively correlated If = - 1.0, the securities would be perfectly negatively correlated 7-10 Table 7.1 Descriptive Statistics for Two Mutual Funds 7-11 Three-Security Portfolio E (rp ) w1E (r1 ) w2 E (r2 ) w3 E (r3 ) 2p = w1212 + w2212 + w3232 + 2w1w2 Cov(r1,r2) + 2w1w3 Cov(r1,r3) + 2w2w3 Cov(r2,r3) 7-12 Table 7.2 Computation of Portfolio Variance From the Covariance Matrix 7-13 Table 7.3 Expected Return and Standard Deviation with Various Correlation Coefficients 7-14 Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions 7-15 Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions 7-16 Minimum Variance Portfolio as Depicted in Figure 7.4 • Standard deviation is smaller than that of either of the individual component assets • Figure 7.3 and 7.4 combined demonstrate the relationship between portfolio risk 7-17 Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation 7-18 Correlation Effects • The relationship depends on the correlation coefficient • -1.0 < < +1.0 • The smaller the correlation, the greater the risk reduction potential • If = +1.0, no risk reduction is possible 7-19 Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs 7-20 The Sharpe Ratio • Maximize the slope of the CAL for any possible portfolio, p • The objective function is the slope: SP E (rP ) rf P 7-21 Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio 7-22 Figure 7.8 Determination of the Optimal Overall Portfolio 7-23 Figure 7.9 The Proportions of the Optimal Overall Portfolio 7-24 Markowitz Portfolio Selection Model • Security Selection – First step is to determine the risk-return opportunities available – All portfolios that lie on the minimumvariance frontier from the global minimumvariance portfolio and upward provide the best risk-return combinations 7-25 Figure 7.10 The Minimum-Variance Frontier of Risky Assets 7-26 Markowitz Portfolio Selection Model Continued • We now search for the CAL with the highest reward-to-variability ratio 7-27 Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL 7-28 Markowitz Portfolio Selection Model Continued • Now the individual chooses the appropriate mix between the optimal risky portfolio P and T-bills as in Figure 7.8 n 2 P i 1 n w w Cov(r , r ) j 1 i j i j 7-29 Figure 7.12 The Efficient Portfolio Set 7-30 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 risky portfolio is purely technical – Allocation of the complete portfolio to Tbills versus the risky portfolio depends on personal preference 7-31 Figure 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set 7-32 The Power of Diversification n • Remember: 2 P i 1 n w w Cov(r , r ) i j 1 j i j • If we define the average variance and average covariance of the securities as: 1 n 2 i n i 1 2 n 1 Cov n(n 1) j 1 j i n Cov(r , r ) i 1 i j • We can then express portfolio variance as: 1 2 n 1 2 P Cov n n 7-33 Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes 7-34 Risk Pooling, Risk Sharing and Risk in the Long Run • Consider the following: p = .001 Loss: payout = $100,000 No Loss: payout = 0 1 − p = .999 7-35 Risk Pooling and the Insurance Principle • Consider the variance of the portfolio: 1 2 n 2 P • It seems that selling more policies causes risk to fall • Flaw is similar to the idea that long-term stock investment is less risky 7-36 Risk Pooling and the Insurance Principle Continued • When we combine n uncorrelated insurance policies each with an expected profit of $ , both expected total profit and SD grow in direct proportion to n: E (n ) nE ( ) Var (n ) n Var ( ) n SD(n ) n 2 2 2 7-37 Risk Sharing • What does explain the insurance business? – Risk sharing or the distribution of a fixed amount of risk among many investors 7-38