2. ®1999 South-Western College Publishing 1 Chapter 8 The Gain From Portfolio Diversification

3. ®1999 South-Western College Publishing 2 Optimal Portfolio Level of Risk Expected Return Determine

4. ®1999 South-Western College Publishing 3 Background Assumptions As an investor you want to maximize the returns for a given level of risk.As an investor you want to maximize the returns for a given level of risk. The relationship between the returns for assets in the portfolio is important.The relationship between the returns for assets in the portfolio is important. A good portfolio is not simply a collection of individually good investments.A good portfolio is not simply a collection of individually good investments.

5. ®1999 South-Western College Publishing 4 Risk Aversion Given a choice between two assets with equal rates of return, risk averse investors will select the asset with the lower level of risk.

6. ®1999 South-Western College Publishing 5 Evidence That Investors are Risk Averse Many investors purchase insurance for: Life, Automobile, Health, and Disability Income. The purchaser trades known costs for unknown risk of lossMany investors purchase insurance for: Life, Automobile, Health, and Disability Income. The purchaser trades known costs for unknown risk of loss Yield on bonds increases with risk classifications from AAA to AA to A….Yield on bonds increases with risk classifications from AAA to AA to A….

7. ®1999 South-Western College Publishing 6 Are all investors risk averse? Risk preference may have to do with how skewed the possible returns are that are involved - risking small amounts, but insuring large losses Do the possible outcomes push the investor into different categories of overall consumption?

8. ®1999 South-Western College Publishing 7 Definition of Risk 1. Uncertainty of future outcomes or 2. Probability of an adverse outcome We will consider several measures of risk that are used in developing portfolio theory

9. ®1999 South-Western College Publishing 8 Markowitz Portfolio Theory Formally extends the MVC from individual stocks to a portfolio contextFormally extends the MVC from individual stocks to a portfolio context Quantifies riskQuantifies risk Derives the expected rate of return for a portfolio of assets and an expected risk measureDerives the expected rate of return for a portfolio of assets and an expected risk measure Shows the variance of the rate of return is a meaningful measure of portfolio riskShows the variance of the rate of return is a meaningful measure of portfolio risk Derives the formula for computing the variance of a portfolio, showing how to effectively diversify a portfolioDerives the formula for computing the variance of a portfolio, showing how to effectively diversify a portfolio

10. ®1999 South-Western College Publishing 9 Assumptions of Markowitz Portfolio Theory 1. Investors consider each investment alternative as being presented by a probability distribution of expected returns over some holding period.

11. ®1999 South-Western College Publishing 10 Assumptions of Markowitz Portfolio Theory 2. Investors minimize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth.

12. ®1999 South-Western College Publishing 11 Assumptions of Markowitz Portfolio Theory 3. Investors estimate the risk of the portfolio on the basis of the variability of expected returns.

13. ®1999 South-Western College Publishing 12 Assumptions of Markowitz Portfolio Theory 4. Investors base decisions solely on expected return and risk, so their utility curves are a function of expected return and the expected variance (or standard deviation) of returns only.

14. ®1999 South-Western College Publishing 13 Assumptions of Markowitz Portfolio Theory 5. For a given risk level, investors prefer higher returns to lower returns. Similarly, for a given level of expected returns, investors prefer less risk to more risk. (MVC) Are these assumptions realistic? Do investors really act to “maximize their utility functions w.r.t. expected return and variance”? Do pool players pay attention to the rules and laws of physics?

15. ®1999 South-Western College Publishing 14 Assumptions of Markowitz Portfolio Theory Using these five assumptions, a single asset or portfolio of assets is considered to be efficient if no other asset or portfolio of assets offers higher expected return with the same (or lower) risk, or lower risk with the same (or higher) expected return.

16. ®1999 South-Western College Publishing 15 Alternative Measures of Risk Variance or standard deviation of expected returnVariance or standard deviation of expected return –Alternatively, variation away from some benchmark (tracking error) Range of returnsRange of returns Returns below expectationsReturns below expectations –Semivariance - measure expected returns below some target –Intended to minimize the damage

17. ®1999 South-Western College Publishing 16 Expected Rates of Return Individual risky assetIndividual risky asset –Sum of probability times possible rate of return PortfolioPortfolio –Weighted average of expected rates of return for the individual investments in the portfolio

18. ®1999 South-Western College Publishing 17 Computation of Expected Return for an Individual Risky Investment

19. ®1999 South-Western College Publishing 18 Computation of the Expected Return for a Portfolio of Risky Assets

20. ®1999 South-Western College Publishing 19 Variance (Standard Deviation) of Returns for an Individual Investment Standard deviation is the square root of the variance Variance is a measure of the variation of possible rates of return R i away from the expected rate of return [E(R i )]

21. ®1999 South-Western College Publishing 20 Variance (Standard Deviation) of Returns for an Individual Investment where P i is the probability of the possible rate of return, R i

22. ®1999 South-Western College Publishing 21 Variance (Standard Deviation) of Returns for an Individual Investment Standard Deviation

23. ®1999 South-Western College Publishing 22 Variance (Standard Deviation) of Returns for an Individual Investment Variance ( 2 ) =.0050 Standard Deviation ( ) =.02236

24. ®1999 South-Western College Publishing 23 Variance (Standard Deviation) of Returns for a Portfolio Computation of Monthly Rates of Return Computation of Monthly Rates of Return

25. ®1999 South-Western College Publishing 24 Time Series Returns for Coca-Cola: 1998

26. ®1999 South-Western College Publishing 25 Times Series Returns for Exxon: 1998

27. ®1999 South-Western College Publishing 26 Times Series Returns for Coca-Cola and Exxon: 1998

28. ®1999 South-Western College Publishing 27 Variance (Standard Deviation) of Returns for a Portfolio For two assets, i and j, the covariance of rates of return is defined as: Cov ij = E{[R i - E(R i )][R j - E(R j )]}

29. ®1999 South-Western College Publishing 28 Computation of Covariance of Returns for Coca-Cola and Exxon: 1998

30. ®1999 South-Western College Publishing 29 Scatter Plot of Monthly Returns for Coca-Cola and Exxon: 1998

31. ®1999 South-Western College Publishing 30 Computation of Standard Deviation of Returns for Coca-Cola and Exxon: 1998

32. ®1999 South-Western College Publishing 31 Portfolio Standard Deviation Formula

33. ®1999 South-Western College Publishing 32 Portfolio Standard Deviation Calculation Any asset of a portfolio may be described by two characteristics:Any asset of a portfolio may be described by two characteristics: –The expected rate of return –The expected standard deviations of returns The correlation, measured by covariance, affects the portfolio standard deviationThe correlation, measured by covariance, affects the portfolio standard deviation Low correlation reduces portfolio risk while not affecting the expected returnLow correlation reduces portfolio risk while not affecting the expected return

34. ®1999 South-Western College Publishing 33 Combining Stocks with Different Returns and Risk Case Correlation Coefficient Covariance Case Correlation Coefficient Covariance a +1.00.0070 a +1.00.0070 b +0.50.0035 b +0.50.0035 c 0.00.0000 c 0.00.0000 d -0.50 -.0035 d -0.50 -.0035 e -1.00 -.0070 e -1.00 -.0070 1.10.50.0049.07 2.20.50.0100.10

35. ®1999 South-Western College Publishing 34 Combining Stocks with Different Returns and Risk Assets may differ in expected rates of return and individual standard deviationsAssets may differ in expected rates of return and individual standard deviations Negative correlation reduces portfolio riskNegative correlation reduces portfolio risk Combining two assets with -1.0 correlation reduces the portfolio standard deviation to zero only when individual standard deviations are equalCombining two assets with -1.0 correlation reduces the portfolio standard deviation to zero only when individual standard deviations are equal

36. ®1999 South-Western College Publishing 35 Constant Correlation with Changing Weights 1.10 r ij = 0.00 2.20

37. ®1999 South-Western College Publishing 36 Constant Correlation with Changing Weights

38. ®1999 South-Western College Publishing 37 Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = +1.00 1 2 With two perfectly correlated assets, it is only possible to create a two asset portfolio with risk- return along a line between either single asset

39. ®1999 South-Western College Publishing 38 Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = 0.00 R ij = +1.00 f g h i j k 1 2 With uncorrelated assets it is possible to create a two asset portfolio with lower risk than either single asset

40. ®1999 South-Western College Publishing 39 Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = 0.00 R ij = +1.00 R ij = +0.50 f g h i j k 1 2 With correlated assets it is possible to create a two asset portfolio between the first two curves

41. ®1999 South-Western College Publishing 40 Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = 0.00 R ij = +1.00 R ij = -0.50 R ij = +0.50 f g h i j k 1 2 With negatively correlated assets it is possible to create a two asset portfolio with much lower risk than either single asset

42. ®1999 South-Western College Publishing 41 Portfolio Risk-Return Plots for Different Weights Standard Deviation of Return E(R) R ij = 0.00 R ij = +1.00 R ij = -1.00 R ij = +0.50 f g h i j k 1 2 With perfectly negatively correlated assets it is possible to create a two asset portfolio with almost no risk R ij = -0.50

43. ®1999 South-Western College Publishing 42 The Efficient Frontier The efficient frontier represents that set of portfolios with the maximum rate of return for every given level of risk, or the minimum risk for every level of returnThe efficient frontier represents that set of portfolios with the maximum rate of return for every given level of risk, or the minimum risk for every level of return Frontier will be portfolios of investments rather than individual securitiesFrontier will be portfolios of investments rather than individual securities –Exception: the asset with the single highest return

44. ®1999 South-Western College Publishing 43 Efficient Frontier for Alternative Portfolios Efficient Frontier A B C E(R) Standard Deviation of Return

45. ®1999 South-Western College Publishing 44 The Efficient Frontier and Investor Utility An individual investor’s utility curve specifies the trade-offs he is willing to make between expected return and riskAn individual investor’s utility curve specifies the trade-offs he is willing to make between expected return and risk The slope of the efficient frontier curve decreases steadily as you move upwardThe slope of the efficient frontier curve decreases steadily as you move upward These two interactions will determine the particular portfolio selected by an individual investorThese two interactions will determine the particular portfolio selected by an individual investor

46. ®1999 South-Western College Publishing 45 The Efficient Frontier and Investor Utility The optimal portfolio has the highest utility for a given investorThe optimal portfolio has the highest utility for a given investor It lies at the point of tangency between the efficient frontier and the utility curve with the highest possible utilityIt lies at the point of tangency between the efficient frontier and the utility curve with the highest possible utility

47. ®1999 South-Western College Publishing 46 Selecting an Optimal Risky Portfolio X Y U3U3 U2U2 U1U1 U 3’ U 2’ U 1’ Figure 8.10

48. ®1999 South-Western College Publishing 47 Gains From Diversification Lower CorrelationLower Correlation Smaller Portfolio VarianceSmaller Portfolio Variance Larger Gains From DiversificationLarger Gains From Diversification For a Given Set of WeightsFor a Given Set of Weights

49. ®1999 South-Western College Publishing 48 How Many Stocks Are Required For Adequate Diversification? Benefits achieved even with “naïve” or evenly weighted diversificationBenefits achieved even with “naïve” or evenly weighted diversification The More Stocks, the BetterThe More Stocks, the Better Increasing Transaction CostsIncreasing Transaction Costs 10-15 Stocks Sufficient10-15 Stocks Sufficient 90% of Maximum Benefit with 12-1890% of Maximum Benefit with 12-18 Most Benefits Achieved with 10Most Benefits Achieved with 10 Diminishing Benefits with Additional StocksDiminishing Benefits with Additional Stocks

50. ®1999 South-Western College Publishing 49 “A Little Diversification Goes A Long Way” As the Number of Assets Increases the Incremental Contribution to Variance Reduction Becomes Smaller and Smaller As the Number of Assets Increases the Incremental Contribution to Variance Reduction Becomes Smaller and Smaller

51. ®1999 South-Western College Publishing 50 Mutual Funds Diminishing Benefits of More Stocks are Still PositiveDiminishing Benefits of More Stocks are Still Positive –There is some gain Low Cost of DataLow Cost of Data Large Funds Have to Invest in Many StocksLarge Funds Have to Invest in Many Stocks –Avoid buying and selling affects –Regulation M Own  5% of any company’s stockOwn  5% of any company’s stock

52. ®1999 South-Western College Publishing 51 Unrelated Industries Significant Risk ReductionSignificant Risk Reduction –Achieved by diversifying across different industries Stocks From Same IndustryStocks From Same Industry –Highly positively correlated Stocks From Different IndustriesStocks From Different Industries –Negative correlation

53. ®1999 South-Western College Publishing 52 Investment Weights Affect the Portfolio’s VarianceAffect the Portfolio’s Variance Minimize Portfolio’s RiskMinimize Portfolio’s Risk Determine the Investor’s Risk ExposureDetermine the Investor’s Risk Exposure If  = -1, 1 Strategy of Weights Can Yield a Portfolio with No RiskIf  = -1, 1 Strategy of Weights Can Yield a Portfolio with No Risk

54. ®1999 South-Western College Publishing 53 Weights For The 2 Asset Portfolio W A + W B = 1 W A + W B = 1 W B = 1 - W A W B = 1 - W A W A + (1 - W A ) = 1 or 100% W A + (1 - W A ) = 1 or 100% only need to solve for one variable only need to solve for one variable also have equivalent restriction for more-than-2-asset case also have equivalent restriction for more-than-2-asset case

55. ®1999 South-Western College Publishing 54 Minimize Portfolio’s Risk Selecting Assets With Low CorrelationSelecting Assets With Low Correlation Balancing the Investment WeightsBalancing the Investment Weights Degree of CorrelationDegree of Correlation –Influences the portfolio’s level of risk –Lower the correlation, the larger the gain from diversification

56. ®1999 South-Western College Publishing 55 Calculating the Efficient Frontier Constrained optimization problemConstrained optimization problem –Objective function minimized subject to constraints (see handout) Portfolio optimizationPortfolio optimization –Minimize  p s.t. E(R) and weight constraints –Can also max. “alpha” from benchmark s.t. “tracking error” and weight constraints Index and “Enhanced Index” FundsIndex and “Enhanced Index” Funds Weight constraints can get very complexWeight constraints can get very complex –Esp. used by “quant” firms

57. ®1999 South-Western College Publishing 56 MVP Minimum Variance PortfolioMinimum Variance Portfolio Portfolio With the Smallest Variance From the Mean-Variance FrontierPortfolio With the Smallest Variance From the Mean-Variance Frontier Mean-Variance FrontierMean-Variance Frontier –Efficient frontier Dominate portfolioDominate portfolio –Inefficient frontier

58. ®1999 South-Western College Publishing 57 What Is The Implication Of Diversification For Portfolio Management? Not All Diversification Strategies are DesirableNot All Diversification Strategies are Desirable Some Strategies are InefficientSome Strategies are Inefficient No Investor Would Select Portfolios From the Segment Below Point MVPNo Investor Would Select Portfolios From the Segment Below Point MVP see next slide see next slide

59. ®1999 South-Western College Publishing 58 MVP Inefficient Frontier Efficient Frontier Standard Deviation E(R)

60. ®1999 South-Western College Publishing 59 Two Assets With Different Correlations The Higher the Correlation, the Smaller the Gain From DiversificationThe Higher the Correlation, the Smaller the Gain From Diversification As Correlation Declines, Risk Reduction IncreasesAs Correlation Declines, Risk Reduction Increases –Implies larger risk reduction because of diversification

61. ®1999 South-Western College Publishing 60 Complicated Portfolio Choices More Than Two Assets to Choose FromMore Than Two Assets to Choose From Assets do not Have the Same Mean or VarianceAssets do not Have the Same Mean or Variance Nonzero correlations prevailNonzero correlations prevail Must Find the Mean-Variance FrontierMust Find the Mean-Variance Frontier Identify the Efficient and Inefficient SetsIdentify the Efficient and Inefficient Sets

62. ®1999 South-Western College Publishing 61 Estimation Issues Results of portfolio allocation depend on accurate statistical inputsResults of portfolio allocation depend on accurate statistical inputs Estimates ofEstimates of –Expected returns –Standard deviation –Correlation coefficient Among entire set of assetsAmong entire set of assets With 100 assets, 4,950 correlation estimatesWith 100 assets, 4,950 correlation estimates Estimation risk refers to potential errorsEstimation risk refers to potential errors

63. ®1999 South-Western College Publishing 62 Estimation Issues With assumption that stock returns can be described by a single market model, the number of correlations required reduces to the number of assetsWith assumption that stock returns can be described by a single market model, the number of correlations required reduces to the number of assets Single index market model:Single index market model: b i = the slope coefficient that relates the returns for security i to the returns for the aggregate stock market R m = the returns for the aggregate stock market

64. ®1999 South-Western College Publishing 63 Estimation Issues If all the securities are similarly related to the market and a b i derived for each one, it can be shown that the correlation coefficient between two securities i and j is given as: