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## Performance Evaluation

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**Introduction**• Complicated subject • Theoretically correct measures are difficult to construct • Different statistics or measures are appropriate for different types of investment decisions or portfolios • Many industry and academic measures are different • The nature of active managements leads to measurement problems**Abnormal Performance**What is abnormal? Abnormal performance is measured: • Benchmark portfolio • Market adjusted • Market model / index model adjusted • Reward to risk measures such as the Sharpe Measure: E (rp-rf) / sp**Factors That Lead to Abnormal Performance**• Market timing • Superior selection • Sectors or industries • Individual companies**s**p • rp = Average return on the portfolio • rf = Average risk free rate = Standard deviation of portfolio return p Risk Adjusted Performance: Sharpe 1) Sharpe Index rp - rf s**rp = Average return on the portfolio**• rf = Average risk free rate • ßp = Weighted average b for portfolio Risk Adjusted Performance: Treynor 2) Treynor Measure rp- rf ßp**a**= Alpha for the portfolio Risk Adjusted Performance: Jensen 3) Jensen’s Measure = rp - [ rf + ßp ( rm - rf) ] a p p rp = Average return on the portfolio ßp = Weighted average Beta rf = Average risk free rate rm = Avg. return on market index port.**M2 Measure**• Developed by Modigliani and Modigliani • Equates the volatility of the managed portfolio with the market by creating a hypothetical portfolio made up of T-bills and the managed portfolio • If the risk is lower than the market, leverage is used and the hypothetical portfolio is compared to the market**M2 Measure: Example**Managed Portfolio Market T-bill Return 35% 28% 6% Stan. Dev 42% 30% 0% Hypothetical Portfolio: Same Risk as Market 30/42 = .714 in P (1-.714) or .286 in T-bills (.714) (.35) + (.286) (.06) = 26.7% Since this return is less than the market, the managed portfolio underperformed**T2 (Treynor Square) Measure**• Used to convert the Treynor Measure into percentage return basis • Makes it easier to interpret and compare • Equates the beta of the managed portfolio with the market’s beta of 1 by creating a hypothetical portfolio made up of T-bills and the managed portfolio • If the beta is lower than one, leverage is used and the hypothetical portfolio is compared to the market**T2 Example**Port. P. Market Risk Prem. (r-rf) 13% 10% Beta 0.80 1.0 Alpha 5% 0% Treynor Measure 16.25 10 Weight to match Market w = bM/bP = 1.0 / 0.8 Adjusted Return RP* = w (RP) = 16.25% T2P = RP* - RM = 16.25% - 10% = 6.25% T2P = RP/Bp – RM = 13/.8 – 10 = 6.25%**Which Measure is Appropriate?**It depends on investment assumptions 1) If the portfolio represents the entire investment for an individual, Sharpe Index compared to the Sharpe Index for the market. 2) If many alternatives are possible, use the Jensen a or the Treynor measure The Treynor measure is more complete because it adjusts for risk**Other Measures of Risk**• A very popular measure of risk is called Value at Risk or VAR. It is generally the amount you can lose at a particular confidence interval. It is only concerned with loss. • At the 95% level, E(R) – 1.65(standard dev) • At the 99% level, E(R) – 2.33(standard dev)**VAR Example**• Invest $100 at 10% with S.D. of 15%. What is the 95% Var over the year? • 10% - 1.65(15%) = -14.75% • Var = 100 * -.1475 = $14.75**Semi-variance or semi-standard deviation**This statistic only measures variations below the mean or some threshold. Semi-variance = 1/n * (summation of the average – rt)^2 where Where: n = the total number of observations below the meanrt = the observed valueaverage = the mean or target value of the data set. Standard deviation is just the square root of the above.**Semi-variance or semi-standard deviation**Example: You note the returns 3, 8, 3, 5, 11. Mean is 6. Thus 3 returns are less than the mean. The semi variance is [(6-3)^2 + (6-3)^2 + (6-5)^2]/3 = 6.33 and the semi-standard deviation is 6.33^.5 = 2.51. Compare this to the population standard deviation which is [(6-3)^2 + (6-8)^2 + (6-3)^2 + (6-5)^2+(6-11)^2]/5 = 9.6. Square root of 19.6 is 3.1%. Note that deviations above the mean are not considered, thus the large deviation(11 compared to 6) is ignored and the semi-standard deviation is actually smaller in this case.**Monte Carlo Methods**• Monte Carlo methods can be used to ask what if questions based on thousands of simulations. The range of values found during the simulations can be used to estimate possible outcomes and thus the risk associated with any position. • For example, using the mean and standard deviation of the stock market, one could simulate possible return patterns over many years to see how a portfolio may be affected.**Limitations**• Assumptions underlying measures limit their usefulness • When the portfolio is being actively managed, basic stability requirements are not met • Practitioners often use benchmark portfolio comparisons to measure performance**Performance Attribution**• Decomposing overall performance into components • Components are related to specific elements of performance • Example components • Broad Allocation • Industry • Security Choice • Up and Down Markets**Process of Attributing Performance to Components**Set up a ‘Benchmark’ or ‘Bogey’ portfolio • Use indexes for each component • Use target weight structure**Appropriate Benchmark**• Unambiguous – anyone can reproduce it • Investable • Measurable • Specified in Advance**Process of Attributing Performance to Components**• Calculate the return on the ‘Bogey’ and on the managed portfolio • Explain the difference in return based on component weights or selection • Summarize the performance differences into appropriate categories**Example, Performance Attributuion**Actual: (0.70 ´ 2.0%) + (0.20 ´ 1.0%) + (0.10 ´ 0.5%) = 1.65% Benchmark: (0.60 ´ 2.5%) + (0.30 ´ 1.2%) + (0.10 ´ 0.5%) = 1.91% Underperformance = 1.91% - 1.65% = 0.26%**Lure of Active Management**Are markets totally efficient? • Some managers outperform the market for extended periods • While the abnormal performance may not be too large, it is too large to be attributed solely to noise • Evidence of anomalies such as the turn of the year exist The evidence suggests that there is some role for active management