CHAPTER 7 Optimal Risky Portfolios Investments 8 th

CHAPTER 7 Optimal Risky Portfolios Investments, 8 th edition Bodie, Kane and Marcus Slides by Susan Hine Mc. Graw-Hill/Irwin Copyright © 2009 by The Mc. Graw-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 7 -6

Two-Security Portfolio: Risk = Variance of Security D = Variance of Security E = Covariance of returns for Security D and Security E 7 -7

Two-Security Portfolio: Risk Continued • Another way to express variance of the portfolio: 7 -8

Covariance Cov(r. D, r. E) = DE D E 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

Three-Security Portfolio 2 p = w 12 12 + w 22 12 + w 32 32 + 2 w 1 w 3 Cov(r 1, r 2) Cov(r 1, r 3) + 2 w 2 w 3 Cov(r 2, r 3) 7 -11

Table 7. 2 Computation of Portfolio Variance From the Covariance Matrix 7 -12

Table 7. 1 Descriptive Statistics for Two Mutual Funds 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: 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 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 • Remember: • If we define the average variance and average covariance of the securities as: • We can then express portfolio variance as: 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: • 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: 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