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Portfolio Management 3-228-07 Albert Lee Chun

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

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Portfolio Management 3-228-07 Albert Lee Chun

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  1. Portfolio Management3-228-07Albert Lee Chun Evaluation of Portfolio Performance Lecture 11 2 Dec 2008

  2. Introduction • 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: 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. Today • Performance Measurement Risk Adjusted Performance MeasuresMeasures of Sharpe, Treynor and JensenMeasures of Skill and Timing • Attribution de performance Concept de mesures ajustées pour le risque Mesures de Sharpe, Treynor et Jensen Mesure des habilités de timing

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

  5. Geometric Average 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. 17-4

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

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

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

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

  10. Performance Measurement

  11. Early Performance Measure Techniques • 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. Peer Group Comparisons • 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. • 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.

  13. Treynor Portfolio Performance Measure

  14. Treynor (1965) • 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. • 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. • As the portfolio becomes diversified, unique risk diminishes.

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

  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.

  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. • Hence, it implicitly assumes a completely diversified portfolio. • 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. • A portfolio negative Beta will have a negative Treynor measure. • Also known as the Treynor Ratio.

  18. T-Lines Q has higher alpha, but P has steeper T-line. P is the better portfolio. 17-18

  19. Sharpe Portfolio Performance Measure

  20. Sharpe Measure • 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 larger the measure the better, as the portfolio earned a higher excess return per unit of total risk.

  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 startedcalling it the Sharpe Ratio.

  22. SML vs. CML • 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. • 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. • Thus, the Sharpe measure evaluates the portfolio manager in terms of both return performance and diversification.

  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). • 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).

  24. Jensen Portfolio Performance Measure

  25. Jensen’s Alpha • Alpha is a risk-adjusted measure of superior performance • 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. • 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.

  26. Multifactor Jensen’s Measure Measure can be extended to a multi-factor setting, for example:

  27. Information Ratio

  28. Information Ratio 1 • 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.

  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.

  30. Portfolio Tracking Error Excess Return relative to benchmark portfolio b Average Excess Return Variance in Excess Difference Tracking Error

  31. Information Ratio • 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. Information to noise ratio. • Annualized IR

  32. Information Ratios

  33. M2 Measure

  34. M2 Measure Developed by Leah and her grandfather Franco Modigliani. M2 = rp*- rm rp* is return of the adjusted portfolio that matches the volatility of the market index rm. 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.

  35. M2 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% Since the return of the portfolio is less than the market, M2 is negative, and the managed portfolio underperformed the market. 17-35

  36. M2 of Portfolio P 17-36 17-36

  37. Excess Returns for Portfolios P and Q and the Benchmark M

  38. Performance Statistics 17-38

  39. Which Portfolio is Best? • It depends. • If P or Q represent the entire portfolio, Q would be preferable based on having higher sharp ratio and a better M2. • 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).

  40. Style Analysis

  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

  42. Sharpe’s Style Portfolios for the Magellan Fund 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. 17-42

  43. Fidelity Magellan Fund Returns vs Benchmarks Fund vs Style and Fund vs SML 19.19% Impact of positive alpha on abnormal returns. 17-43

  44. Average Tracking Error for 636 Mutual Funds Bell shaped 17-44

  45. Market Timing

  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

  47. Returns from 1990 - 1999 17-47

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

  49. Performance of Bills, Equities and Timers Beginning with $1 dollar in 1926, and ending in 2005....

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