1. INVESTMENTS | BODIE, KANE, MARCUS Chapter Seven Optimal Risky Portfolios Copyright © 2014 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

2. INVESTMENTS | BODIE, KANE, MARCUS 7-2 The investment decision: Capital allocation (risky vs. risk-free) Asset allocation (construction of the risky portfolio) Security selection Optimal risky portfolio The Markowitz portfolio optimization model Long- vs. short-term investing Chapter Overview

3. INVESTMENTS | BODIE, KANE, MARCUS 7-3 Top-down process with 3 steps: 1.Capital allocation between the risky portfolio and risk-free asset 2.Asset allocation across broad asset classes 3.Security selection of individual assets within each asset class The Investment Decision

4. INVESTMENTS | BODIE, KANE, MARCUS 7-4 Market risk Risk attributable to marketwide risk sources and remains even after extensive diversification Also call systematic or nondiversifiable Firm-specific risk Risk that can be eliminated by diversification Also called diversifiable or nonsystematic Diversification and Portfolio Risk

5. INVESTMENTS | BODIE, KANE, MARCUS 7-5 Figure 7.1 Portfolio Risk and the Number of Stocks in the Portfolio Panel A: All risk is firm specific. Panel B: Some risk is systematic or marketwide.

6. INVESTMENTS | BODIE, KANE, MARCUS 7-6 Figure 7.2 Portfolio Diversification

7. INVESTMENTS | BODIE, KANE, MARCUS 7-7 Portfolio risk (variance) 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 of two assets move together (covary) Portfolios of Two Risky Assets

8. INVESTMENTS | BODIE, KANE, MARCUS 7-8 Portfolio return: r p = w D r D + w E r E w D = Bond weight r D = Bond return w E = Equity weight r E = Equity return E(r p ) = w D E(r D ) + w E E(r E ) Portfolios of Two Risky Assets: Return

9. INVESTMENTS | BODIE, KANE, MARCUS 7-9 Portfolio variance: = Bond variance = Equity variance = Covariance of returns for bond and equity Portfolios of Two Risky Assets: Risk

10. INVESTMENTS | BODIE, KANE, MARCUS 7-10 Covariance of returns on bond and equity: Cov(r D, r E ) =  DE  D  E  D,E = Correlation coefficient of returns  D = Standard deviation of bond returns  E = Standard deviation of equity returns Portfolios of Two Risky Assets: Covariance

11. INVESTMENTS | BODIE, KANE, MARCUS 7-11 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 Portfolios of Two Risky Assets: Correlation Coefficients

12. INVESTMENTS | BODIE, KANE, MARCUS 7-12 When ρ DE = 1, there is no diversification When ρ DE = -1, a perfect hedge is possible Portfolios of Two Risky Assets: Correlation Coefficients

13. INVESTMENTS | BODIE, KANE, MARCUS 7-13 Table 7.2 Computation of Portfolio Variance From the Covariance Matrix

14. INVESTMENTS | BODIE, KANE, MARCUS 7-14 Figure 7.3 Portfolio Expected Return

15. INVESTMENTS | BODIE, KANE, MARCUS 7-15 Figure 7.4 Portfolio Standard Deviation

16. INVESTMENTS | BODIE, KANE, MARCUS 7-16 Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation

17. INVESTMENTS | BODIE, KANE, MARCUS 7-17 The minimum variance portfolio is the portfolio composed of the risky assets that has the smallest standard deviation; the portfolio with least risk The amount of possible risk reduction through diversification depends on the correlation: If  = +1.0, no risk reduction is possible If  = 0, σ P may be less than the standard deviation of either component asset If  = -1.0, a riskless hedge is possible The Minimum Variance Portfolio

18. INVESTMENTS | BODIE, KANE, MARCUS 7-18 Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs

19. INVESTMENTS | BODIE, KANE, MARCUS 7-19 Maximize the slope of the CAL for any possible portfolio, P The objective function is the slope: The slope is also the Sharpe ratio The Sharpe Ratio

20. INVESTMENTS | BODIE, KANE, MARCUS 7-20 Figure 7.7 Debt and Equity Funds with the Optimal Risky Portfolio

21. INVESTMENTS | BODIE, KANE, MARCUS 7-21 Figure 7.8 Determination of the Optimal Overall Portfolio

22. INVESTMENTS | BODIE, KANE, MARCUS 7-22 Figure 7.9 The Proportions of the Optimal Complete Portfolio

23. INVESTMENTS | BODIE, KANE, MARCUS 7-23 Security selection The first step is to determine the risk-return opportunities available All portfolios that lie on the minimum-variance frontier from the global minimum-variance portfolio and upward provide the best risk-return combinations Markowitz Portfolio Optimization Model

24. INVESTMENTS | BODIE, KANE, MARCUS 7-24 Figure 7.10 The Minimum-Variance Frontier of Risky Assets

25. INVESTMENTS | BODIE, KANE, MARCUS 7-25 Search for the CAL with the highest reward-to- variability ratio Everyone invests in P, regardless of their degree of risk aversion More risk averse investors put more in the risk- free asset Less risk averse investors put more in P Markowitz Portfolio Optimization Model

26. INVESTMENTS | BODIE, KANE, MARCUS 7-26 Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL

27. INVESTMENTS | BODIE, KANE, MARCUS 7-27 Capital Allocation and the Separation Property 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 risk-free versus the risky portfolio depends on personal preference Markowitz Portfolio Optimization Model

28. INVESTMENTS | BODIE, KANE, MARCUS 7-28 Figure 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set

29. INVESTMENTS | BODIE, KANE, MARCUS 7-29 The Power of Diversification Remember: If we define the average variance and average covariance of the securities as: Markowitz Portfolio Optimization Model

30. INVESTMENTS | BODIE, KANE, MARCUS 7-30 The Power of Diversification We can then express portfolio variance as Portfolio variance can be driven to zero if the average covariance is zero (only firm specific risk) The irreducible risk of a diversified portfolio depends on the covariance of the returns, which is a function of the systematic factors in the economy Markowitz Portfolio Optimization Model

31. INVESTMENTS | BODIE, KANE, MARCUS 7-31 Table 7.4 Risk Reduction of Equally Weighted Portfolios

32. INVESTMENTS | BODIE, KANE, MARCUS 7-32 Optimal Portfolios and Nonnormal Returns Fat-tailed distributions can result in extreme values of VaR and ES and 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 Markowitz Portfolio Optimization Model

33. INVESTMENTS | BODIE, KANE, MARCUS 7-33 Risk pooling Merging uncorrelated, risky projects as a means to reduce risk It 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 when the policies are uncorrelated Sharpe ratio increases Risk Pooling and the Insurance Principle

34. INVESTMENTS | BODIE, KANE, MARCUS 7-34 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 spreading a portfolio of fixed size across many assets, not merely adding more risky bets to an ever- growing risky portfolio Risk Sharing

35. INVESTMENTS | BODIE, KANE, MARCUS 7-35 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