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Understanding Optimal Risky Portfolios and Diversification

Explore the concept of diversification and portfolio risk, market and firm-specific risk, covariance and correlation, and the relationship between portfolio risk and returns. Learn how to determine the optimal risky portfolio and the importance of risk pooling and risk sharing in long-term investments.

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Understanding Optimal Risky Portfolios and Diversification

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  1. CHAPTER 7 Optimal Risky Portfolios

  2. Diversification and Portfolio Risk Market risk Systematic or nondiversifiable Firm-specific risk Diversifiable or nonsystematic

  3. Figure 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio

  4. Figure 7.2 Portfolio Diversification

  5. 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

  6. Two-Security Portfolio: Return

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

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

  9. Covariance Cov(rD,rE) = 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

  10. 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

  11. Table 7.1 Descriptive Statistics for Two Mutual Funds

  12. Three-Security Portfolio 2p = w1212 + w2212 + w3232 + 2w1w2 Cov(r1,r2) Cov(r1,r3) + 2w1w3 + 2w2w3 Cov(r2,r3)

  13. Table 7.2 Computation of Portfolio Variance From the Covariance Matrix

  14. Table 7.3 Expected Return and Standard Deviation with Various Correlation Coefficients

  15. Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions

  16. Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions

  17. 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

  18. Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation

  19. 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

  20. Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs

  21. The Sharpe Ratio Maximize the slope of the CAL for any possible portfolio, p The objective function is the slope:

  22. Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio

  23. Figure 7.8 Determination of the Optimal Overall Portfolio

  24. Figure 7.9 The Proportions of the Optimal Overall Portfolio

  25. 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

  26. Figure 7.10 The Minimum-Variance Frontier of Risky Assets

  27. Markowitz Portfolio Selection Model Continued We now search for the CAL with the highest reward-to-variability ratio

  28. Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL

  29. 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

  30. Figure 7.12 The Efficient Portfolio Set

  31. 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

  32. Figure 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set

  33. 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:

  34. Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes

  35. 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

  36. 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

  37. 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:

  38. Risk Sharing What does explain the insurance business? Risk sharing or the distribution of a fixed amount of risk among many investors

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