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Learn about optimal risky portfolios and the importance of diversification in reducing portfolio risk. Explore concepts such as market risk, systematic risk, firm-specific risk, covariance, correlation, and the efficient frontier. Discover how to determine the optimal overall portfolio and the capital allocation lines.
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CHAPTER 7 Optimal Risky Portfolios
Diversification and Portfolio Risk Market risk Systematic or nondiversifiable Firm-specific risk Diversifiable or nonsystematic
Figure 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio
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
Two-Security Portfolio: Risk = Variance of Security D = Variance of Security E = Covariance of returns for Security D and Security E
Two-Security Portfolio: Risk Continued Another way to express variance of the portfolio:
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
Correlation Coefficients: Possible Values Range of values for 1,2 + 1.0 >r> -1.0 If r = 1.0, the securities would be perfectly positively correlated If r = - 1.0, the securities would be perfectly negatively correlated
Three-Security Portfolio 2p = w1212 + w2212 + w3232 + 2w1w2 Cov(r1,r2) Cov(r1,r3) + 2w1w3 + 2w2w3 Cov(r2,r3)
Table 7.2 Computation of Portfolio Variance From the Covariance Matrix
Table 7.3 Expected Return and Standard Deviation with Various Correlation Coefficients
Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions
Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions
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
Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation
The relationship depends on the correlation coefficient -1.0 << +1.0 The smaller the correlation, the greater the risk reduction potential If r = +1.0, no risk reduction is possible Correlation Effects
Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs
The Sharpe Ratio Maximize the slope of the CAL for any possible portfolio, p The objective function is the slope:
Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio
Markowitz Portfolio Selection Model Security Selection 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 Selection Model Continued We now search for the CAL with the highest reward-to-variability ratio
Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL
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
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 T-bills versus the risky portfolio depends on personal preference
Figure 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set
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:
Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes
Risk Pooling, Risk Sharing and Risk in the Long Run Consider the following: Loss: payout = $100,000 p = .001 No Loss: payout = 0 1 − p = .999
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
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:
Risk Sharing What does explain the insurance business? Risk sharing or the distribution of a fixed amount of risk among many investors