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Fina2802: Investments and Portfolio Analysis Spring, 2010 Dragon Tang

Lecture 9 Capital Allocation February 9, 2010 Readings: Chapter 6 Practice Problem Sets: 1-5,12-14,26-28. Fina2802: Investments and Portfolio Analysis Spring, 2010 Dragon Tang. Asset Allocation. Objectives :

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Fina2802: Investments and Portfolio Analysis Spring, 2010 Dragon Tang

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  1. Lecture 9 Capital Allocation February 9, 2010 Readings: Chapter 6 Practice Problem Sets: 1-5,12-14,26-28 Fina2802: Investments and Portfolio AnalysisSpring, 2010Dragon Tang Chapter 6: Asset Allocation

  2. Asset Allocation • Objectives: • Characterize the risk and return of portfolios containing risky and risk-free assets. • Evaluate the performance of a passive strategy. Chapter 6: Asset Allocation

  3. Asset allocation Security selection Portfolio Selection Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  4. John Bogle: “Asset allocation accounts for 94% of the differences in pension fund performance” Identify investment opportunities (risk-return combinations) Choose the optimal combination according to investor’s risk attitude Asset Allocation Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  5. Portfolios: Basic Asset Allocation • The complete portfolio is composed of: • The risk-free asset: Risk can be reduced by allocating more to the risk-free asset • The risky portfolio: Composition of risky portfolio does not change (market portfolio) • This is called Two-Fund Separation Theorem. • The proportions depend on your risk aversion. Chapter 6: Asset Allocation

  6. Risk-free Investment Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  7. Example: Risky investment with ten possible rates of return Stock Returns Are Uncertain Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  8. Risk and Risk Premium • Risk-free rate: determined by demand/supply and intermediaries (such as Fed) • Risk premium (=Risky return –Risk-free return) • Example • The expected return on the S&P500 is 9% • The return on a 1-month T-bill is 3% • The risk premium is 6% (9%-3%) • Risk aversion E(rP) – rf = ½ A σp2 Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  9. Complete Portfolio Expected Return Example: Let the expected return on the risky portfolio, E(rP), be 15%, the return on the risk-free asset, rf, be 7%. What is the return on the complete portfolio if all of the funds are invested in the risk-free asset? What is the risk premium? 7% 0 What is the return on the portfolio if all of the funds are invested in the risky portfolio? 15% 8% Chapter 6: Asset Allocation

  10. Complete Portfolio Expected Return Example: Let the expected return on the risky portfolio, E(rP), be 15%, the return on the risk-free asset, rf, be 7%. What is the return on the complete portfolio if 50% of the funds are invested in the risky portfolio and 50% in the risk-free asset? What is the risk premium? 0.5*15%+0.5*7%=11% 4% Chapter 6: Asset Allocation

  11. Complete Portfolio Risk Premium In general: Equal to 0 Chapter 6: Asset Allocation

  12. Portfolio Standard Deviation where sc- standard deviation of the complete portfolio sP - standard deviation of the risky portfolio srf - standard deviation of the risk-free rate y - weight of the complete portfolio invested in the risky asset Chapter 6: Asset Allocation

  13. Portfolio Standard Deviation Example: Let the standard deviation on the risky portfolio, sP, be 22%. What is the standard deviation of the complete portfolio if 50% of the funds are invested in the risky portfolio and 50% in the risk-free asset? 22%*0.5=11% Chapter 6: Asset Allocation

  14. We know that given a risky asset (p) and a risk-free asset, the expected return and standard deviation of any complete portfolio (c) satisfy the following relationship: Capital Allocation Line Where y is the fraction of the portfolio invested in the risky asset Chapter 6: Asset Allocation

  15. Capital Allocation Line • Risk Tolerance and Asset Allocation: • More risk averse - closer to point F • Less risk averse - closer to P Chapter 6: Asset Allocation

  16. Slope of the CAL S is the increase in expected return per unit of additional standard deviation S is the reward-to-variability ratio or Sharpe Ratio Chapter 6: Asset Allocation

  17. Slope of the CAL Example: Let the expected return on the risky portfolio, E(rP), be 15%, the return on the risk-free asset, rf, be 7% and the standard deviation on the risky portfolio, sP, be 22%. What is the slope of the CAL for the complete portfolio? S = (15%-7%)/22% = 8/22 Chapter 6: Asset Allocation

  18. Risk Real Sharp Asset Class Prem.(%) Retn(%) Ratio Sm Stk 13.9 14.6 0.35 Lg Stk 9.3 8.9 0.45 LT Gov 1.9 2.6 0.23 T-Bills --- 0.7 --- Historical Risk-Return Trade-off Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  19. Measuring Risk-Return Trade-off • Mean-Variance Plot Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  20. Borrowing y > 1 where rB is the borrowing rate Usually borrowing rate> lending rate Chapter 6: Asset Allocation

  21. Borrowing Example Example: Let the expected return on the risky portfolio, E(rP), be 15%, the return on the risk-free asset, rf, be 7%, the borrowing rate, rB, be 9% and the standard deviation on the risky portfolio, sP, be 22%. What is the slope of the CAL for the complete portfolio for points where y > 1? S=(15%-9%)/22%=6/22 Note: For y £ 1, the slope is as indicated above if the lending rate is rf. Chapter 6: Asset Allocation

  22. Investment Opportunity Set with Differential Borrowing and Lending Rates Chapter 6: Asset Allocation

  23. Passive Strategies • Assumes securities are fairly priced • Avoids cost of security analysis • Indexing - value-weighted portfolio • Assume that the search for mispriced securities • (performed by active strategies) keeps prices fair Chapter 6: Asset Allocation

  24. Capital Market Line (CML) SPECIAL CASE OF CAL (I.e., P=MKT) The line provided by one-month T-bills and a broad index of common stocks (e.g. S&P500) Consequence of a passive investment strategy based on stocks and T-bills Chapter 6: Asset Allocation

  25. Which Portfolio to Choose? E(r) P3? E(Rm) = 12% M P2? P1? S=0.45 rf = 3% F  0 20% Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  26. Key Determinant of Asset Allocation:Attitude towards Risk • Risk Preference • Risk averse • Require compensation for taking risk • Risk neutral • No requirement of risk premium • Risk loving • Pay to take risk • Utility Values: A is risk aversion parameter Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  27. U = E ( r ) – 1/2A s2 Where U = utility E ( r ) = expected return on the asset or portfolio A = coefficient of risk aversion s2 = variance of returns Utility Function Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  28. Utility Scores of Alternative Portfolios for Investors with Varying Risk Aversion Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  29. The Trade-off Between Risk and Returns of a Potential Investment Portfolio Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  30. Utility Values of Possible Portfolios for an Investor with Risk Aversion, A = 4 Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  31. Figure 6.2 The Indifference Curve Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  32. Utility Indifference Curves A=5 A=2 U1 Increasing utility U2 Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  33. Greater levels of risk aversion lead to larger proportions of the risk free rate. Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets. Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations. Risk Aversion and Asset Allocation Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  34. Investor’s Willingness to Pay for Catastrophe Insurance Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  35. Spread Between 3-Month CD and T-bill Rates Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  36. Find the Optimal Allocation • Solve the maximization problem: • Two approaches: • Try different w1 • Use calculus: • Solution: Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  37. When rm-rf increases, w1 increases When A increases, w1 decreases When m increases, w1 decreases W1 is constant when all three are fixed Asset Allocation Rules: Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  38. Illustration of Solution If A=4 then w1=0.56 E(r) Utility indifference curves (A=4) P3 E(Rm) = 12% M P2 8% P1! S=0.45 rf = 3% F  11.2% 0 20% Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  39. Which Portfolio to Choose? • For Jack, risk aversion A = 2, the optimal choice is 112.5% (of total capital) in the market, financed by selling short 12.5% (of total capital) in T-bills • For Jill, risk aversion A = 5, the optimal choice is 45% in the market and 55% in T-bills. • The weight in the market: Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  40. Utility Comparison 45% 113% Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  41. Utility Levels for Positions in Risky Assets for an Investor with Risk Aversion A = 4 Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  42. Utility as a Function of Allocation to the Risky Asset, y Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  43. Finding the Optimal Complete Portfolio Using Indifference Curves Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  44. Expected Returns on Four Indifference Curves and the CAL Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  45. Average Annual Return on Stocks and 1-Month T-bills; S. Dev. and Reward to Variability of Stocks Over Time Chapter 6: Asset Allocation Chapter 6: Asset Allocation Fin 2802, Spring 08 - Tang

  46. Summary • Capital allocation line (CAL) • All combinations of the risky and risk-free asset • Slope is the reward-to-variability ratio • Capital market line (CML) • Passive strategy • Market index portfolio as the risky asset • Risk aversion determines position on the capital allocation line • Next: Market Efficiency Chapter 6: Asset Allocation

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