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Optimal Risky Portfolios- Asset Allocations

Optimal Risky Portfolios- Asset Allocations. BKM Ch 7. Asset Allocation. Idea from bank account to diversified portfolio principles are the same for any number of stocks Discussion A. bonds and stocks B. bills, bonds and stocks C. any number of risky assets.

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Optimal Risky Portfolios- Asset Allocations

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  1. Optimal Risky Portfolios- Asset Allocations BKM Ch 7

  2. Asset Allocation • Idea • from bank account to diversified portfolio • principles are the same for any number of stocks • Discussion • A. bonds and stocks • B. bills, bonds and stocks • C. any number of risky assets Bahattin Buyuksahin, JHU , Investment

  3. Diversification and Portfolio Risk • Market risk • Systematic or nondiversifiable • Firm-specific risk • Diversifiable or nonsystematic Bahattin Buyuksahin, JHU , Investment

  4. Figure 7.1 Portfolio Risk as a Function of the Number of Stocks in the Portfolio Bahattin Buyuksahin, JHU , Investment

  5. Figure 7.2 Portfolio Diversification Bahattin Buyuksahin, JHU , Investment

  6. 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 Bahattin Buyuksahin, JHU , Investment

  7. Two-Security Portfolio: Return Bahattin Buyuksahin, JHU , Investment

  8. = Variance of Security D = Variance of Security E = Covariance of returns for Security D and Security E Two-Security Portfolio: Risk Bahattin Buyuksahin, JHU , Investment

  9. Two-Security Portfolio: Risk Continued • Another way to express variance of the portfolio: Bahattin Buyuksahin, JHU , Investment

  10. 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 Bahattin Buyuksahin, JHU , Investment

  11. 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 Bahattin Buyuksahin, JHU , Investment

  12. Table 7.1 Descriptive Statistics for Two Mutual Funds Bahattin Buyuksahin, JHU , Investment

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

  14. Asset Allocation • Portfolio of 2 risky assets (cont’d) • examples • BKM7 Tables 7.1 & 7.3 • BKM7 Figs. 7.3 (return), 7.4 (risk) & 7.5 (trade-off) • portfolio opportunity set (BKM7 Fig. 7.5) • minimum variance portfolio • choose wD such that portfolio variance is lowest • optimization problem • minimum variance portfolio has less risk • than either component (i.e., asset) Bahattin Buyuksahin, JHU , Investment

  15. Table 7.2 Computation of Portfolio Variance From the Covariance Matrix Bahattin Buyuksahin, JHU , Investment

  16. Table 7.3 Expected Return and Standard Deviation with Various Correlation Coefficients Bahattin Buyuksahin, JHU , Investment

  17. Figure 7.3 Portfolio Expected Return as a Function of Investment Proportions Bahattin Buyuksahin, JHU , Investment

  18. Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions Bahattin Buyuksahin, JHU , Investment

  19. Minimum Variance Portfolio as Depicted in Figure 7.4 Bahattin Buyuksahin, JHU , Investment 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

  20. Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation Bahattin Buyuksahin, JHU , Investment

  21. 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 Bahattin Buyuksahin, JHU , Investment

  22. Figure 7.6 The Opportunity Set of the Debt and Equity Funds and Two Feasible CALs Bahattin Buyuksahin, JHU , Investment

  23. The Sharpe Ratio Bahattin Buyuksahin, JHU , Investment Maximize the slope of the CAL for any possible portfolio, p The objective function is the slope:

  24. Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio Bahattin Buyuksahin, JHU , Investment

  25. Figure 7.8 Determination of the Optimal Overall Portfolio Bahattin Buyuksahin, JHU , Investment

  26. Asset Allocation • Finding the optimal risky portfolio: II. Formally • Intuitively • BKM7 Figs. 7.6 and 7.7 • improve the reward-to-variability ratio • optimal risky portfolio  tangency point (Fig. 7.8) • Formally: Bahattin Buyuksahin, JHU , Investment

  27. Asset Allocation 18 • formally (continued) Bahattin Buyuksahin, JHU , Investment

  28. Asset Allocation 19 • Example (BKM7 Fig. 7.8) • 1. plot D, E, riskless • 2. compute optimal risky portfolio weights • wD = Num/Den = 0.4; wE = 1- wD = 0.6 • 3. given investor risk aversion (A=4), compute w* • bottom line: 25.61% in bills; 29.76% in bonds (0.7439x0.4); rest in stocks Bahattin Buyuksahin, JHU , Investment

  29. Figure 7.9 The Proportions of the Optimal Overall Portfolio Bahattin Buyuksahin, JHU , Investment

  30. Markowitz Portfolio Selection Model Bahattin Buyuksahin, JHU , Investment • 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

  31. Markowitz Portfolio Selection Model • Combining many risky assets & T-Bills • basic idea remains unchanged • 1. specify risk-return characteristics of securities • find the efficient frontier (Markowitz) • 2. find the optimal risk portfolio • maximize reward-to-variability ratio • 3. combine optimal risk portfolio & riskless asset • capital allocation Bahattin Buyuksahin, JHU , Investment

  32. Markowitz Portfolio Selection Model • finding the efficient frontier • definition • set of portfolios with highest return for given risk • minimum-variance frontier • take as given the risk-return characteristics of securities • estimated from historical data or forecasts • n securities ->n return + n(n-1) var. & cov. • use an optimization program • to compute the efficient frontier (Markowitz) • subject to same constraints Bahattin Buyuksahin, JHU , Investment

  33. Markowitz Portfolio Selection Model • Finding the efficient frontier (cont’d) • optimization constraints • portfolio weights sum up to 1 • no short sales, dividend yield, asset restrictions, … • Individual assets vs. frontier portfolios • BKM7 Fig. 7.10 • short sales -> not on the efficient frontier • no short sales -> may be on the frontier • example: highest return asset Bahattin Buyuksahin, JHU , Investment

  34. Figure 7.10 The Minimum-Variance Frontier of Risky Assets Bahattin Buyuksahin, JHU , Investment

  35. Markowitz Portfolio Selection Model Continued Bahattin Buyuksahin, JHU , Investment We now search for the CAL with the highest reward-to-variability ratio

  36. Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL Bahattin Buyuksahin, JHU , Investment

  37. Markowitz Portfolio Selection Model Continued Bahattin Buyuksahin, JHU , Investment Now the individual chooses the appropriate mix between the optimal risky portfolio P and T-bills as in Figure 7.8

  38. Figure 7.12 The Efficient Portfolio Set Bahattin Buyuksahin, JHU , Investment

  39. Capital Allocation and the Separation Property Bahattin Buyuksahin, JHU , Investment • 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

  40. Figure 7.13 Capital Allocation Lines with Various Portfolios from the Efficient Set Bahattin Buyuksahin, JHU , Investment

  41. The Power of Diversification Bahattin Buyuksahin, JHU , Investment • Remember: • If we define the average variance and average covariance of the securities as: • We can then express portfolio variance as:

  42. Table 7.4 Risk Reduction of Equally Weighted Portfolios in Correlated and Uncorrelated Universes Bahattin Buyuksahin, JHU , Investment

  43. Risk Pooling, Risk Sharing and Risk in the Long Run Loss: payout = $100,000 p = .001 No Loss: payout = 0 1 − p = .999 Bahattin Buyuksahin, JHU , Investment Consider the following:

  44. Risk Pooling and the Insurance Principle Bahattin Buyuksahin, JHU , Investment • 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

  45. Risk Pooling and the Insurance Principle Continued Bahattin Buyuksahin, JHU , Investment 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:

  46. Risk Sharing Bahattin Buyuksahin, JHU , Investment • What does explain the insurance business? • Risk sharing or the distribution of a fixed amount of risk among many investors

  47. An Asset Allocation Problem Bahattin Buyuksahin, JHU , Investment

  48. An Asset Allocation Problem 2 • Perfect hedges (portfolio of 2 risky assets) • perfectly positively correlated risky assets • requires short sales • perfectly negatively correlated risky assets Bahattin Buyuksahin, JHU , Investment

  49. An Asset Allocation Problem 3 Bahattin Buyuksahin, JHU , Investment

  50. CHAPTER 8 Index Models

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