Portfolio optimization with warren and bill

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Portfolio optimization with warren and bill

  1. 1. Stodder, Efficient Frontier<br />Portfolio Optimization MethodologyHow to Find that Efficient Frontier!<br />Theoretical & “Real Life” Examples<br />
  2. 2. Concept of Beta<br />
  3. 3.
  4. 4. Security Market Line Equation<br />Required Return=Risk Free + Risk Premium<br />on Stock iRate on Stock i<br />Required Return=Risk Free + βi(Market Risk)<br />on Stock iRate Premium<br />Ri= Rrf+ βi(Rm - Rrf)<br />
  5. 5. Beta of the Market = 1<br />βi= (Ri– Rrf)/(Rm - Rrf)<br /> So if Ri = Rm, βi = βm then<br />βm = (Rm– Rrf)/(Rm - Rrf) = 1<br />
  6. 6. The Efficient Frontier<br />Non-Diversifiable<br />Risk<br />
  7. 7. How do We Find the Efficient Frontier?<br />Basic Strategy:<br />Find the Standard Deviation(σi) and Mean Return(μi) of every stock Stock i.<br />For any given rate of return, find the minimal standard deviation portfolio that can achieve that return. <br />
  8. 8. At what point do we have least risk?<br />
  9. 9. WHY DIVERSIFY?<br />
  10. 10. Use More Than One Basket for Your Eggs<br /> The Axiom<br /> The Concept of Risk Aversion Revisited<br /> Preliminary Steps in Forming a Portfolio<br /> The Reduced Security Universe<br /> Security Statistics<br /> Interpreting the Statistics<br /> The Role of Uncorrelated Securities<br /> The Variance of a Linear Combination<br /> Diversification and Utility<br /> The Concept of Dominance<br />
  11. 11. The Efficient Frontier<br /> Optimum Diversification of Risky Assets<br /> The Minimum Variance Portfolio<br /> The Effect of a Risk free Rate<br /> The Efficient Frontier with Borrowing<br /> Different Borrowing and Lending Rates<br /> Naive Diversification<br /> The Single Index Model<br />
  12. 12. <ul><li> Don’t put all your eggs in one basket.
  13. 13. Failure to diversify may violate the terms of a fiduciary trust.
  14. 14. Risk aversion seems to be an instinctive traitin human beings.</li></li></ul><li>Preliminary Steps in Forming a Portfolio<br />Identify a collection of eligible investments known as the security universe.<br />Compute statistics for the chosen securities. <br />e.g. mean of return<br /> variance / standard deviation of return<br /> matrix of correlation coefficients<br />
  15. 15. Preliminary Steps in Forming a Portfolio<br />Insert Figure 16-1 here.<br />
  16. 16. Preliminary Steps in Forming a Portfolio<br />Insert Figure 16-2 here.<br />
  17. 17. Preliminary Steps in Forming a Portfolio<br /> Interpret the statistics. <br />1. Do the values seem reasonable?<br />2. Is any unusual price behavior expected to recur?<br />3. Are any of the results unsustainable?<br />4. Low correlations: Fact or fantasy?<br />
  18. 18. wherexi = the proportion invested in security i<br />The Role of Uncorrelated Securities<br /><ul><li>The expected return of a portfolio is a weighted average of the component expected returns.</li></li></ul><li>The Role of Uncorrelated Securities<br />Insert Table 16-5 here.<br />
  19. 19. two-security<br />portfolio risk<br />= riskA + riskB+ interactive risk<br />The Role of Uncorrelated Securities<br /><ul><li>The total risk of a portfolio comes from the variance of the components and from the relationships among the components.</li></li></ul><li> better<br />performance<br />expected return<br />risk<br />The Role of Uncorrelated Securities<br /><ul><li>Investors get added utility from greater return. They get disutility from greater risk.
  20. 20. The point of diversification is to achieve a given level of expected return while bearing the least possible risk.</li></li></ul><li>The Role of Uncorrelated Securities<br /><ul><li>A portfolio dominates all others if no other equally risky portfolio has a higher expected return, or if no portfolio with the same expected return has less risk.</li></li></ul><li>Efficient frontier<br />impossible<br />portfolios<br />expected return<br />dominated<br />portfolios<br />risk <br />(standard deviation of returns)<br />The Efficient Frontier : <br />Optimum Diversification of Risky Assets<br /><ul><li> The efficient frontier contains portfolios that </li></ul>are not dominated.<br />
  21. 21. single security<br />with the highest<br />expected return<br />expected return<br />minimum variance<br />portfolio<br />risk (standard deviation of returns)<br />The Efficient Frontier : <br />The Minimum Variance Portfolio<br /><ul><li>The right extreme of the efficient frontier is a single security; the left extreme is the minimum variance portfolio.</li></li></ul><li>The Efficient Frontier : <br />The Minimum Variance Portfolio<br />Insert Figure 16-6 here.<br />
  22. 22. Efficient frontier:<br />Rf to M to C<br />impossible<br />portfolios<br />C<br />expected return<br />M<br />dominated<br />portfolios<br />Rf<br />risk (standard deviation of returns)<br />The Efficient Frontier : <br />The Effect of a Riskfree Rate<br /><ul><li>When a riskfree investment complements the set of risky securities, the shape of the efficient frontier changes markedly.</li></ul>E<br />D<br />
  23. 23. The Efficient Frontier : <br />The Effect of a Riskfree Rate<br />In capital market theory, point M is called the market portfolio.<br />The straight portion of the line is tangent to the risky securities efficient frontier at point M and is called the capital market line.<br />Since buying a Treasury bill amounts to lending money to the U.S. Treasury, a portfolio partially invested in the riskfree rate is often called a lending portfolio.<br />
  24. 24. Efficient frontier:<br />the ray from Rf through M<br />impossible<br />portfolios<br />borrowing<br />expected return<br />M<br />lending<br />dominated<br />portfolios<br />Rf<br />risk (standard deviation of returns)<br />The Efficient Frontier with Borrowing<br /><ul><li>Buying on margin involves financial leverage, thereby magnifying the risk and expected return characteristics of the portfolio. Such a portfolio is called a borrowing portfolio.</li></li></ul><li>Efficient frontier : RL to M, the curve to N, then the ray from N<br />impossible<br />portfolios<br />expected return<br />N<br />M<br />RB<br />dominated<br />portfolios<br />RL<br />risk (standard deviation of returns)<br />The Efficient Frontier : <br />Different Borrowing and Lending Rates<br /><ul><li>Most of us cannot borrow and lend at the same interest rate.</li></li></ul><li><ul><li> As portfolio size increases,</li></ul>total portfolio risk, on average, declines. After a certain point, however, the marginal reduction in risk from the addition of another security is modest.<br />total risk<br />Nondiversifiable risk<br />number of securities<br />The Efficient Frontier : Naive Diversification<br /><ul><li>Naive diversification is the random selection of portfolio components without conducting any serious security analysis.</li></ul>20<br />40<br />
  25. 25. The Efficient Frontier : Naive Diversification<br />The remaining risk, when no further diversification occurs, is pure market risk. <br />Market risk is also called systematic risk and is measured by beta.<br />A security with average market risk has a beta equal to 1.0. Riskier securities have a beta greater than one, and vice versa.<br />
  26. 26. The Efficient Frontier : The Single Index Model<br />A pairwise comparison of the thousands of stocks in existence would be an unwieldy task. To get around this problem, the single index model compares all securities to a benchmark measure.<br />The single index model relates security returns to their betas, thereby measuring how each security varies with the overall market.<br />
  27. 27. The Efficient Frontier : The Single Index Model<br /><ul><li>Beta is the statistic relating an individual security’s returns to those of the market index.</li></li></ul><li>The Efficient Frontier : The Single Index Model<br /><ul><li>The relationship between beta and expected return is the essence of the capital asset pricing model (CAPM), which states that a security’s expected return is a linear function of its beta.</li></li></ul><li>The Efficient Frontier : The Single Index Model<br />Insert Figure 16-11 here.<br />
  28. 28. The Efficient Frontier : The Single Index Model<br />Insert Figure 16-12 here.<br />
  29. 29. <ul><li> Use More Than One Basket for Your Eggs
  30. 30. The Axiom
  31. 31. The Concept of Risk Aversion Revisited
  32. 32. Preliminary Steps in Forming a Portfolio
  33. 33. The Reduced Security Universe
  34. 34. Security Statistics
  35. 35. Interpreting the Statistics
  36. 36. The Role of Uncorrelated Securities
  37. 37. The Variance of a Linear Combination
  38. 38. Diversification and Utility
  39. 39. The Concept of Dominance</li></li></ul><li>What did we learn?<br /><ul><li> The Efficient Frontier
  40. 40. Optimum Diversification of Risky Assets
  41. 41. The Minimum Variance Portfolio
  42. 42. The Effect of a Riskfree Rate
  43. 43. The Efficient Frontier with Borrowing
  44. 44. Different Borrowing and Lending Rates
  45. 45. Naive Diversification
  46. 46. The Single Index Model</li></li></ul><li>Appendix: Arbitrage Pricing Theory<br /><ul><li> Theory presumes that market return is determined by a number of distinct, unidentifiable macroeconomic factors
  47. 47. Four factors that make the market move:
  48. 48. The economy
  49. 49. Fed policy
  50. 50. Valuation
  51. 51. Investor sentiment</li></li></ul><li>Appendix: Arbitrage Pricing Theory<br />
  52. 52. Appendix: Arbitrage Pricing Theory<br />

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