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CHAPTER 6

CHAPTER 6. OPTIMAL RISKY PORTFOLIOS. Top-down process Capital allocation between risky portfolio and risk-free assets Asset allocation across broad asset classes (bond/ stock) Security selection of individual assets within each asset class Risky portfolio (E/D )

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CHAPTER 6

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  1. CHAPTER 6 OPTIMAL RISKY PORTFOLIOS

  2. Top-down process • Capital allocation between risky portfolio and risk-free assets • Asset allocation across broad asset classes (bond/ stock) • Security selection of individual assets within each asset class • Risky portfolio (E/D ) • Risky portfolio and risk-free asset (E/D/Rf)

  3. 6.1 Diversification and Portfolio risk

  4. Diversification and Portfolio Risk • Consider a portfolio composed of only one stock, what would be the sources of risk? What if add one stock? • Insurance principle • Market risk • Risk that remains even after extensive diversification, attributable to market wide risk sources • Systematic or non-diversifiable • Firm-specific risk • Risk that can be eliminated by diversification • Diversifiable or nonsystematic

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

  6. Figure 7.2 Portfolio Diversification

  7. 6.2 Portfolio of Two Risk Assets

  8. Portfolios of Two Risky Assets • Study the efficient diversification • Construct risky portfolios to provide the lowest possible risk for any given level of expected return • Consider a portfolio comprised of two mutual funds, a bond portfolio D, and a stock fund E

  9. Portfolios of Two Risky Assets

  10. Portfolios of Two Risky Assets • Rate of return on the portfolio: • Expected rate of return on the portfolio: • Variance of the rate of return on the portfolio:

  11. Two-Security Portfolio: Return

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

  13. Portfolios of Two Risky Assets • Covariance: • Another way to express variance of the portfolio:

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

  15. Portfolios of Two Risky Assets • Variance is reduced • Correlation coefficient • Portfolio variance

  16. Portfolios of Two Risky Assets Range of values for rD,E -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

  17. Portfolios of Two Risky Assets • When Perfect positive correlation • Portfolio variance: • Portfolio SD is no bigger than the weighted average of the individual security SD

  18. Portfolios of Two Risky Assets • When perfect positive correlation • When perfect negative correlation

  19. Portfolios of Two Risky Assets • Hedge asset: negative correlation with the other assets in the portfolio • Expected return is unaffected by correlation, standard deviation is less than the weighted average of the component standard deviation • portfolio of less than perfectly correlated assets always offer better risk-return opportunities than the individual component securities on their own • The lower the correlation, the greater the gain in efficiency.

  20. Portfolios of Two Risky Assets • Perfect hedged position • When To solve

  21. Portfolios of Two Risky Assets Experiment with different proportions

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

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

  24. Figure 7.4 Portfolio Standard Deviation as a Function of Investment Proportions Minimum variance portfolio

  25. Portfolios of Two Risky Assets • The Minimum-Variance Portfolio • In the Example ,what is the minimum level • Diversification effect: Minimum-variance portfolio has a standard deviation smaller than that of either of the individual component assets • Pass through the two undiversified portfolios of WD=1,WE=1

  26. 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

  27. Figure 7.5 Portfolio Expected Return as a Function of Standard Deviation PORTFOLIO OPPORTUNITY SET: For any pair of wD and wE

  28. Portfolios of Two Risky Assets

  29. Portfolios of Two Risky Assets

  30. Portfolios of Two Risky Assets • Portfolio Opportunity set: • All combination of portfolio expected return and std deviation that can be constructed from the two available assets • The best portfolio will depend on risk aversion

  31. 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 When r = -1.0 , perfect hedging opportunity and maximum advantage from diversification Correlation Effects

  32. 6.3 Asset Allocation with Stocks, Bonds and Bills

  33. Extending to Include Riskless Asset • Asset allocation across the three key asset classes: • Stocks, bonds, risk-free assets • Risk-free T-bills yielding 5% • Two possible CAL, comparing their reward-to-variability ratio • CAL A • CAL B

  34. The Optimal Risky Portfolio B: Wd=0.7,wE=0.3 A: minimum variance portfolio wD=0.82,wE=0.18

  35. Figure 7.7 The Opportunity Set of the Debt and Equity Funds with the Optimal CAL and the Optimal Risky Portfolio Tangency portfolio, yield the CAL with highest feasible reward-to-volatility ratio, the optimal risky portfolio to mix with T-bills

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

  37. The Optimal Risky Portfolio • Maximize the slope of the CAL • The tangency portfolio is the optimal portfolio to mix with T-bills

  38. The Optimal Risky Portfolio • The solution for the optimal risky portfolio (see xls : optimal risky portfolio)

  39. The Optimal Complete Portfolio • The optimal complete portfolio • Specify the return characteristics of all securities (expected return, variance, covariance) • The opportunity set of risky assets • The optimal risky portfolio: The CAL tangent with the opportunity set • Use the investor’s degree of risk aversion

  40. The Optimal Complete Portfolio • Investor with risk aversion A=4 • Percentage in bonds=0.7439*0.4=0.2976 • Percentage in stocks=0.7439*0.6=0.4463

  41. The Complete Portfolio Indifference Curve Opportunity Set of Risky Assets

  42. Figure 6.8 The Complete Portfolio – Solution to the Asset Allocation Problem

  43. Extending Concepts to All Securities • The optimal combinations result in lowest level of risk for a given return • The optimal trade-off is described as the efficient frontier • These portfolios are dominant

  44. 6.4 The Markowitz Portfolio Selection Model

  45. Security Selection • Generalize the portfolio construction problem to many risky assets and a risk-free asset • To determine the risk-return opportunities available (Minimum-variance frontier) from the set of risky assets • Involve the risk-free asset (CAL tangent to the efficient frontier) and get the optimal risky portfolio P • Choose the appropriate mix between P and T-bill

  46. Security Selection • Minimum-variance frontier • A graph of the lowest possible variance that can be attained for a given portfolio expected return • Efficient frontier of risky assets • The part of the frontier that lies above

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

  48. Figure 7.11 The Efficient Frontier of Risky Assets with the Optimal CAL search for the CAL with the highest reward-to-variability ratio

  49. Figure 7.12 The Efficient Portfolio Set

  50. Markowitz Portfolio Selection Model Now the individual chooses the appropriate mix between the optimal risky portfolio P and T-bills as in Figure 7.8

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