1. 0 Portfolio Management 3-228-07 Albert Lee Chun Evaluation of Portfolio Performance Lecture 11 2 Dec 2008

2. Albert Lee Chun Portfolio Management 1 Introduction As portfolio managers, how can we evaluate the performance of our portfolio? As portfolio managers, how can we evaluate the performance of our portfolio? We know that there are 2 major requirements of a portfolio manager’s performance: We know that there are 2 major requirements of a portfolio manager’s performance: 1. The ability to derive above-average returns conditioned on risk taken, either through superior market timing or superior security selection. 2. The ability to diversify the portfolio and eliminate non-systematic risk, relative to a benchmark portfolio.

3. Albert Lee Chun Portfolio Management 2 Today Performance Measurement Risk Adjusted Performance Measures Measures of Sharpe, Treynor and Jensen Measures of Skill and Timing Performance Measurement Risk Adjusted Performance Measures Measures of Sharpe, Treynor and Jensen Measures of Skill and Timing Attribution de performance Attribution de performance Concept de mesures ajustées pour le risque Mesures de Sharpe, Treynor et Jensen Mesure des habilités de timing

4. Albert Lee Chun Portfolio Management 3 Averaging Returns Arithmetic Mean: Geometric Mean: Example: (.10 +.0566) / 2 = 7.83% [ (1.1) (1.0566) ] 1/2 - 1 = 7.808% Example: 17-3

5. Albert Lee Chun Portfolio Management 4 The arithmetic average provides unbiased estimates of the expected return of the stock. Use this to forecast returns in the next period. The fixed rate of return over the sample period that would yield the terminal value is know as the geometric average. The geometric average is less than the arithmetic average and this difference increases with the volatility of returns. The geometric average is also called the time-weighted average (as opposed to the dollar weighted average), because it puts equal weights on each return. Geometric Average 17-4

6. Albert Lee Chun Portfolio Management 5 Dollar-weighted returns Internal rate of return. Internal rate of return. Returns are weighted by the amount invested in each stock. Returns are weighted by the amount invested in each stock. Time-weighted returns Not weighted by investment amount. Not weighted by investment amount. Equal weighting Equal weighting Geometric average Geometric average Dollar- and Time-Weighted Returns 17-5

7. Albert Lee Chun Portfolio Management 6 Example: Multiperiod Returns PeriodAction 0Purchase 1 share of Eggbert’s Egg Co. at $50 0Purchase 1 share of Eggbert’s Egg Co. at $50 1Purchase 1 share of Eggbert’s Egg Co. at $53 1Purchase 1 share of Eggbert’s Egg Co. at $53 Eggbert pays a dividend of $2 per share 2Eggbert pays a dividend of $2 per share 2Eggbert pays a dividend of $2 per share Sell both shares for $108 Sell both shares for $108 17-6

8. Albert Lee Chun Portfolio Management 7 PeriodCash Flow 0-50 share purchase 1+2 dividend -53 share purchase 2+4 dividend + 108 shares sold Internal Rate of Return: Dollar-Weighted Return Dollar Weighted: The stocks performance in the second year, when we own 2 shares, has a greater influence on the overall return.

9. Albert Lee Chun Portfolio Management 8 Time-Weighted Return [ (1.1) (1.0566) ] 1/2 - 1 = 7.808% 17-8 Time Weighted: Each return has equal weight in the geometric average. Geometric Mean:

10. Albert Lee Chun Portfolio Management 9 Performance Measurement

11. Albert Lee Chun Portfolio Management 10 Early Performance Measure Techniques Portfolio evaluation before 1960 Portfolio evaluation before 1960 Once upon a time, investors evaluated a portfolio’s performance based purely on the basis of the rate of return. Research in the 1960’s showed investors how to quantify and measure risk. Grouped portfolios into similar risk classes and compared rates of return within risk classes.

12. Albert Lee Chun Portfolio Management 11 Peer Group Comparisons This is the most common manner of evaluating portfolio managers. This is the most common manner of evaluating portfolio managers. Collects returns of a representative universe of investors over a period of time and displays them in a box plot format. Collects returns of a representative universe of investors over a period of time and displays them in a box plot format. Example: “US Equity with Cash” relative to peer universe of US domestic equity managers. Example: “US Equity with Cash” relative to peer universe of US domestic equity managers. Issue: There is no explicit adjustment for risk. Risk is only considered implicitly. Issue: There is no explicit adjustment for risk. Risk is only considered implicitly.

13. Albert Lee Chun Portfolio Management 12

14. Albert Lee Chun Portfolio Management 13 Treynor Portfolio Performance Measure

15. Albert Lee Chun Portfolio Management 14 Treynor (1965) Treynor (1965) developed the first composite measure of portfolio performance that included risk. Treynor (1965) developed the first composite measure of portfolio performance that included risk. He introduced the portfolio characteristic line, which defines a relation between the rate of return on a specific portfolio and the rate of return on the market portfolio. He introduced the portfolio characteristic line, which defines a relation between the rate of return on a specific portfolio and the rate of return on the market portfolio. The beta is the slope that measures the volatility of the portfolio’s returns relative to the market. The beta is the slope that measures the volatility of the portfolio’s returns relative to the market. Alpha represents unique returns for the portfolio. Alpha represents unique returns for the portfolio. As the portfolio becomes diversified, unique risk diminishes. As the portfolio becomes diversified, unique risk diminishes.

16. Albert Lee Chun Portfolio Management 15 A risk-adjusted measure of return that divides a portfolio's excess return by its beta. The Treynor Measure is given by Treynor Measure The Treynor Measure is defined using the average rate of return for portfolio p and the risk-free asset.

17. Albert Lee Chun Portfolio Management 16 Treynor Measure A larger Tp is better for all investors, regardless of their risk preferences. Because it adjusts returns based on systematic risk, it is the relevant performance measure when evaluating diversified portfolios held in separately or in combination with other portfolios.

18. Albert Lee Chun Portfolio Management 17 Treynor Measure Beta measures systematic risk, yet if the portfolio is not fully diversified then this measure is not a complete characterization of the portfolio risk. Beta measures systematic risk, yet if the portfolio is not fully diversified then this measure is not a complete characterization of the portfolio risk. Hence, it implicitly assumes a completely diversified portfolio. Hence, it implicitly assumes a completely diversified portfolio. Portfolios with identical systematic risk, but different total risk, will have the same Treynor ratio! Portfolios with identical systematic risk, but different total risk, will have the same Treynor ratio! Higher idiosyncratic risk should not matter in a diversified portfolio and hence is not reflected in the Treynor measure. Higher idiosyncratic risk should not matter in a diversified portfolio and hence is not reflected in the Treynor measure. A portfolio negative Beta will have a negative Treynor measure. A portfolio negative Beta will have a negative Treynor measure. Also known as the Treynor Ratio. Also known as the Treynor Ratio.

19. Albert Lee Chun Portfolio Management 18 T-Lines 17-18 Q has higher alpha, but P has steeper T-line. P is the better portfolio.

20. Albert Lee Chun Portfolio Management 19 Sharpe Portfolio Performance Measure

21. Albert Lee Chun Portfolio Management 20 Similar to the Treynor measure, but uses the total risk of the portfolio, not just the systematic risk. Similar to the Treynor measure, but uses the total risk of the portfolio, not just the systematic risk. The Sharpe Ratio is given by The Sharpe Ratio is given by The larger the measure the better, as the portfolio earned a higher excess return per unit of total risk. The larger the measure the better, as the portfolio earned a higher excess return per unit of total risk. Sharpe Measure

22. Albert Lee Chun Portfolio Management 21 Sharpe Measure It adjusts returns for total portfolio risk, as opposed to only systematic risk as in the Treynor Measure. Thus, an implicit assumption of the Sharpe ratio is that the portfolio is not fully diversified, nor will it be combined with other diversified portfolios. It is relevant for performance evaluation when comparing mutually exclusive portfolios. Sharpe originally called it the "reward-to-variability" ratio, before others started calling it the Sharpe Ratio.

23. Albert Lee Chun Portfolio Management 22 SML vs. CML Treynor’s measure uses Beta and hence examines portfolio return performance in relation to the SML. Treynor’s measure uses Beta and hence examines portfolio return performance in relation to the SML. Sharpe’s measure uses total risk and hence examines portfolio return performance in relation to the CML. Sharpe’s measure uses total risk and hence examines portfolio return performance in relation to the CML. For a totally diversified portfolio, both measures give equal rankings. For a totally diversified portfolio, both measures give equal rankings. If it is not a diversified portfolio, the Sharpe measure could give lower rankings than the Treynor measure. If it is not a diversified portfolio, the Sharpe measure could give lower rankings than the Treynor measure. Thus, the Sharpe measure evaluates the portfolio manager in terms of both return performance and diversification. Thus, the Sharpe measure evaluates the portfolio manager in terms of both return performance and diversification.

24. Albert Lee Chun Portfolio Management 23 Price of Risk Both the Treynor and Sharp measures, indicate the risk premium per unit of risk, either systematic risk (Treynor) or total risk (Sharpe). Both the Treynor and Sharp measures, indicate the risk premium per unit of risk, either systematic risk (Treynor) or total risk (Sharpe). They measure the price of risk in units of excess returns per each unit of risk (measured either by beta or the standard deviation of the portfolio). They measure the price of risk in units of excess returns per each unit of risk (measured either by beta or the standard deviation of the portfolio).

25. Albert Lee Chun Portfolio Management 24 Jensen Portfolio Performance Measure

26. Albert Lee Chun Portfolio Management 25 Alpha is a risk-adjusted measure of superior performance Alpha is a risk-adjusted measure of superior performance This measure adjusts for the systematic risk of the portfolio. This measure adjusts for the systematic risk of the portfolio. Positive alpha signals superior risk-adjusted returns, and that the manager is good at selecting stocks or predicting market turning points. Positive alpha signals superior risk-adjusted returns, and that the manager is good at selecting stocks or predicting market turning points. Unlike the Sharpe Ratio, Jensen’s method does not consider the ability of the manager to diversify, as it is only accounts for systematic risk. Unlike the Sharpe Ratio, Jensen’s method does not consider the ability of the manager to diversify, as it is only accounts for systematic risk. Jensen’s Alpha

27. Albert Lee Chun Portfolio Management 26 Multifactor Jensen’s Measure Measure can be extended to a multi-factor setting, for example:

28. Albert Lee Chun Portfolio Management 27 Information Ratio

29. Albert Lee Chun Portfolio Management 28 Information Ratio 1 Using a historical regression, the IR takes on the form Using a historical regression, the IR takes on the form where the numerator is Jensen’s alpha and the denominator is the standard error of the regression. Recalling that Note that the risk here is nonsystematic risk, that could, in theory, be eliminated by diversification.

30. Albert Lee Chun Portfolio Management 29 Information Ratio 2 Measures excess returns relative to a benchmark portfolio. Sharpe Ratio is the special case where the benchmark equals the risk-free asset. Risk is measured as the standard deviation of the excess return (Recall that this is the Tracking Error) For an actively managed portfolio, we may want to maximize the excess return per unit of nonsystematic risk we are bearing.

31. Albert Lee Chun Portfolio Management 30 Portfolio Tracking Error Excess Return relative to benchmark portfolio b Average Excess Return Variance in Excess Difference Tracking Error

32. Albert Lee Chun Portfolio Management 31 Information Ratio Excess return represents manager’s ability to use information and talent to generate excess returns. Excess return represents manager’s ability to use information and talent to generate excess returns. Fluctuations in excess returns represent random noise that is interpreted as unsystematic risk. Fluctuations in excess returns represent random noise that is interpreted as unsystematic risk. Information to noise ratio. Annualized IR Annualized IR

33. Albert Lee Chun Portfolio Management 32 Information Ratios

34. Albert Lee Chun Portfolio Management 33 M 2 Measure

35. Albert Lee Chun Portfolio Management 34 M 2 Measure Developed by Leah and her grandfather Franco Modigliani. M 2 = r p* - r m M 2 = r p* - r m r p* is return of the adjusted portfolio that matches the volatility of the market index r m. It is mixed with a position in T-bills. If the risk of the portfolio is lower than that of the market, one has to increase the volatility by using leverage. Because the market index and the adjusted portfolio have the same standard deviation, we may compare their performances by comparing returns.

36. Albert Lee Chun Portfolio Management 35 M 2 Measure: Example Managed Portfolio: return = 35%st dev = 42% Market Portfolio: return = 28%st dev = 30% T-bill return = 6% Hypothetical Portfolio: 30/42 =.714 in P (1-.714) or.286 in T-bills Return = (.714) (.35) + (.286) (.06) = 26.7% M 2 Since the return of the portfolio is less than the market, M 2 is negative, and the managed portfolio underperformed the market. 17-35

37. Albert Lee Chun Portfolio Management 36 M 2 of Portfolio P 17-36

38. Albert Lee Chun Portfolio Management 37 Excess Returns for Portfolios P and Q and the Benchmark M

39. Albert Lee Chun Portfolio Management 38 Performance Statistics 17-38

40. Albert Lee Chun Portfolio Management 39 Which Portfolio is Best? It depends. It depends. If P or Q represent the entire portfolio, Q would be preferable based on having higher sharp ratio and a better M 2. If P or Q represent the entire portfolio, Q would be preferable based on having higher sharp ratio and a better M 2. If P or Q represents a sub-portfolio, the Q would be preferable because it has a higher Treynor ratio. If P or Q represents a sub-portfolio, the Q would be preferable because it has a higher Treynor ratio. For an actively managed portfolio, P may be preferred because it’s information ratio is larger (that is it maximizes return relative to nonsystematic risk, or the tracking error). For an actively managed portfolio, P may be preferred because it’s information ratio is larger (that is it maximizes return relative to nonsystematic risk, or the tracking error).

41. Albert Lee Chun Portfolio Management 40 Style Analysis

42. Albert Lee Chun Portfolio Management 41 Style Analysis Introduced by William Sharpe 1992 study of mutual fund performance 91.5% of variation in return could be explained by the funds’ allocations to bills, bonds and stocks Later studies show that 97% of the variation in return could be explained by the funds’ allocation to set of different asset classes. 17-41

43. Albert Lee Chun Portfolio Management 42 Sharpe’s Style Portfolios for the Magellan Fund 17-42 Monthly returns on Magellan Fund over five year period. Regression coefficient only positive for 3. They explain 97.5% of Magellan’s returns. 2.5 percent attributed to security selection within asset classes.

44. Albert Lee Chun Portfolio Management 43 Fidelity Magellan Fund Returns vs Benchmarks 17-43 Fund vs Style and Fund vs SML Impact of positive alpha on abnormal returns. 19.19%

45. Albert Lee Chun Portfolio Management 44 Average Tracking Error for 636 Mutual Funds 17-44 Bell shaped

46. Albert Lee Chun Portfolio Management 45 Market Timing

47. Albert Lee Chun Portfolio Management 46 Perfect Market Timing A manager with perfect market timing, that shifts assets efficiently across stocks, bonds and cash would have a return equal to A manager with perfect market timing, that shifts assets efficiently across stocks, bonds and cash would have a return equal to

48. Albert Lee Chun Portfolio Management 47 Returns from 1990 - 1999 Year Lg Stocks T-Bills 1990-3.207.86 199130.665.65 19927.713.54 19939.872.97 19941.293.91 199537.715.58 199623.075.58 199828.585.11 199921.044.80 17-47

49. Albert Lee Chun Portfolio Management 48 Switch to T-Bills in 90 and 94 Switch to T-Bills in 90 and 94 Mean = 18.94%, Standard Deviation = 12.04% Invested in large stocks for the entire period: Invested in large stocks for the entire period: Mean = 17.41% Standard Deviation = 14.11 With Perfect Forecasting Ability 17-48

50. Albert Lee Chun Portfolio Management 49 Performance of Bills, Equities and Timers Beginning with $1 dollar in 1926, and ending in 2005....

51. Albert Lee Chun Portfolio Management 50 Value of Imperfect Forecasting Suppose you are forecasting rain in Seattle. If you predict rain, you would be correct most of the time. Suppose you are forecasting rain in Seattle. If you predict rain, you would be correct most of the time. Does this make you a good forecaster? Certainly not. Does this make you a good forecaster? Certainly not. We need to examine the proportion of correct forecasts for rain (P1) and the proportion of correct forecasts for sun (P2). We need to examine the proportion of correct forecasts for rain (P1) and the proportion of correct forecasts for sun (P2). The correct measure of timing ability is The correct measure of timing ability is P = P1 + P2 – 1 An forecaster who always guesses correctly will show P1 = P2 = P =1, whereas on who always predicts rain will have P1 = 1, P2 = P = 0. An forecaster who always guesses correctly will show P1 = P2 = P =1, whereas on who always predicts rain will have P1 = 1, P2 = P = 0.

52. Albert Lee Chun Portfolio Management 51 If an investor holds only the market and the risk free security, and the weights remained constant, the portfolio characteristic line would be a straight line. Adjusting portfolio weights for up and down movements in market returns, we would have: Low Market Return - low weight on the market - low ßeta Low Market Return - low weight on the market - low ßeta High Market Return – high weight on the market - high ßeta High Market Return – high weight on the market - high ßeta Identifying Market Timing 17-51 Henriksson (1984) showed little evidence of market timing. Evidence of market efficiency.

53. Albert Lee Chun Portfolio Management 52 Characteristic Lines: Market Timing 17-52 No Market Timing Beta Increases with Return Two Values of Beta

54. Albert Lee Chun Portfolio Management 53 Testing Market Timing The following regression equation, controls for the movements in bond and stock markets, and captures the superior market timing of managers The following regression equation, controls for the movements in bond and stock markets, and captures the superior market timing of managers Gamma was found to be equal to.3 and statistically significant, suggesting that TAA managers were able to time the markets. Gamma was found to be equal to.3 and statistically significant, suggesting that TAA managers were able to time the markets. However, the study also found a negative alpha of -.5. However, the study also found a negative alpha of -.5.

55. Albert Lee Chun Portfolio Management 54 Performance Attribution Analysis

56. Albert Lee Chun Portfolio Management 55 Selectivity The basic premise of the Fama method is that overall performance of a portfolio can be decomposed into a portfolio risk premium component and a selectivity component. The basic premise of the Fama method is that overall performance of a portfolio can be decomposed into a portfolio risk premium component and a selectivity component. Selectivity is the portion of excess returns that exceeds that which can be attained by an unmanaged benchmark portfolio. Selectivity is the portion of excess returns that exceeds that which can be attained by an unmanaged benchmark portfolio. Overall performance = Portfolio Risk Premium + Selectivity Overall performance = Portfolio Risk Premium + Selectivity

57. Albert Lee Chun Portfolio Management 56 Overall performance = Portfolio Risk Premium+ Selectivity Overall Performance Portfolio Risk Premium Selectivity Selectivity measures the distance between the return on portfolio p and the return on a benchmark portfolio with beta equal to the beta of portfolio p.

58. Albert Lee Chun Portfolio Management 57 Attribution Analysis Portfolio managers add value to their investors by 1) selecting superior securities 2) demonstrating superior market timing skills by allocating funds to different asset classes or market segments. Attribution analysis attempts to distinguish is the source of the portfolio’s overall performance. Total value added performance is the sum of selection and allocation effects.

59. Albert Lee Chun Portfolio Management 58 Where B is the bogey portfolio and p is the managed portfolio. Formula for Attribution 17-58 Set up a ‘Benchmark’ or ‘Bogey’ portfolio

60. Albert Lee Chun Portfolio Management 59 Allocation Effect Asset Allocation Effect Asset Allocation Effect Captures the manager’s decision to over or underweight a particular market segment i. Captures the manager’s decision to over or underweight a particular market segment i. Overweighting a segment i when the benchmark yield is high is rewarded. Overweighting a segment i when the benchmark yield is high is rewarded.

61. Albert Lee Chun Portfolio Management 60 Selection Effect Security Selection Effect Security Selection Effect Captures the stock picking ability of the manager, and rewards the ability to form specific market segment portfolios. Rewards the manger for placing larger weights on those segments where his portfolio outperforms the benchmark portfolio in that particular segment. Captures the stock picking ability of the manager, and rewards the ability to form specific market segment portfolios. Rewards the manger for placing larger weights on those segments where his portfolio outperforms the benchmark portfolio in that particular segment.

62. Albert Lee Chun Portfolio Management 61 Performance of the Managed Portfolio 17-61

63. Albert Lee Chun Portfolio Management 62 Performance Attribution 17-62

64. Albert Lee Chun Portfolio Management 63 Sector Selection within the Equity Market 17-63

65. Albert Lee Chun Portfolio Management 64 Portfolio Attribution: Summary 17-64

66. Albert Lee Chun Portfolio Management 65 Global Benchmark Problem (Optional)

67. Albert Lee Chun Portfolio Management 66 Benchmark Error Market portfolio is difficult to approximate Market portfolio is difficult to approximate Benchmark error Benchmark error can effect slope of SML can effect calculation of Beta greater concern with global investing problem is one of measurement Note: Sharpe measure not as dependent on market portfolio as the Treynor measure and others relying on Beta. Note: Sharpe measure not as dependent on market portfolio as the Treynor measure and others relying on Beta.

68. Albert Lee Chun Portfolio Management 67 Differences in Betas Two major differences in the various beta statistics: Two major differences in the various beta statistics: For any particular stock, the beta estimates change a great deal over time. There are substantial differences in betas estimated for the same stock over the same time period when two different definitions of the benchmark portfolio are employed.

69. Albert Lee Chun Portfolio Management 68

70. Albert Lee Chun Portfolio Management 69 Global Benchmark Problem - SML

71. Albert Lee Chun Portfolio Management 70 Global Benchmark Problem - SML

72. Albert Lee Chun Portfolio Management 71 Global Benchmark Problem - SML

73. Albert Lee Chun Portfolio Management 72 Bond Portfolio Performance Measures (Optional)

74. Albert Lee Chun Portfolio Management 73 Bond Portfolio Measures Returns-Based Bond Performance Measurement Returns-Based Bond Performance Measurement Early attempts to analyze fixed-income performance involved peer group comparisons Peer group comparisons are potentially flawed because they do not account for investment risk directly. How did the performance levels of portfolio managers compare to the overall bond market? How did the performance levels of portfolio managers compare to the overall bond market? What factors lead to superior or inferior bond-portfolio performance? What factors lead to superior or inferior bond-portfolio performance?

75. Albert Lee Chun Portfolio Management 74 Fama-French Measure Fama and French extended their 3-factor equity pricing model with 2 additional factors to account for the return characteristics of bonds Fama and French extended their 3-factor equity pricing model with 2 additional factors to account for the return characteristics of bonds TERM – captures the term premium in the slope of the yield curve. TERM – captures the term premium in the slope of the yield curve. DEF – captures the default premium in the credit spread between corporate bonds and treasuries. DEF – captures the default premium in the credit spread between corporate bonds and treasuries. These two bond factors are the dominate drivers of bond portfolio returns. These two bond factors are the dominate drivers of bond portfolio returns.

76. Albert Lee Chun Portfolio Management 75 Seven Bond Portfolios

77. Albert Lee Chun Portfolio Management 76 Bond Performance Attribution A Bond Market Line A Bond Market Line Need a measure of risk such as beta coefficient for equities Difficult to achieve due to bond maturity and coupon effect on volatility of prices Composite risk measure is the bond’s duration Duration replaces beta as risk measure in a bond market line

78. Albert Lee Chun Portfolio Management 77 Bond Market Line

79. Albert Lee Chun Portfolio Management 78

80. Albert Lee Chun Portfolio Management 79 That’s all for today!