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## Chapter 24

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**Chapter 24**Portfolio PerformanceEvaluation 24-1**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 management leads to measurement problems 24-2**Dollar- and Time-Weighted Returns**Dollar-weighted returns • Internal rate of return considering the cash flow from or to investment • Returns are weighted by the amount invested in each stock Time-weighted returns • Not weighted by investment amount • Equal weighting 24-3**Text Example of Multiperiod Returns**PeriodAction 0 Purchase 1 share at $50 1 Purchase 1 share at $53 Stock pays a dividend of $2 per share 2 Stock pays a dividend of $2 per share Stock is sold at $108 per share 24-4**Dollar-Weighted Return**Period Cash Flow 0 -50 share purchase 1 +2 dividend -53 share purchase 2 +4 dividend + 108 shares sold Internal Rate of Return: 24-5**Time-Weighted Return**Simple Average Return: (10% + 5.66%) / 2 = 7.83% 24-6**Averaging Returns**Arithmetic Mean: Text Example Average: (.10 + .0566) / 2 = 7.81% Geometric Mean: Text Example Average: [ (1.1) (1.0566) ]1/2 - 1 = 7.83% 24-7**Comparison of Geometric and Arithmetic Means**• Past Performance - generally the geometric mean is preferable to arithmetic • Predicting Future Returns- generally the arithmetic average is preferable to geometric • Geometric has downward bias 24-8**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) / p 24-9**Factors That Lead to Abnormal Performance**• Market timing • Superior selection • Sectors or industries • Individual companies 24-10**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 p 24-11**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 24-12**M2 Measure: Example**Managed Portfolio: return = 35% standard deviation = 42% Market Portfolio: return = 28% standard deviation = 30% T-bill return = 6% Hypothetical Portfolio: 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 24-13**rp = Average return on the portfolio**• rf = Average risk free rate • ßp = Weighted average for portfolio Risk Adjusted Performance: Treynor rp - rf ßp 2) Treynor Measure 24-14**Risk Adjusted Performance: Jensen**3) Jensen’s Measure = rp - [ rf + ßp ( rm - rf) ] p = Alpha for the portfolio p rp= Average return on the portfolio ßp = Weighted average Beta rf = Average risk free rate rm = Avg. return on market index port. 24-15**Appraisal Ratio**Appraisal Ratio = ap / s(ep) Appraisal Ratio divides the alpha of the portfolio by the nonsystematic risk Nonsystematic risk could, in theory, be eliminated by diversification 24-16**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 or the Treynor measure The Treynor measure is more complete because it adjusts for risk 24-17**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 24-18**Market Timing**Adjusting portfolio for up and down movements in the market • Low Market Return - low ßeta • High Market Return - high ßeta 24-19**rp - rf*** * * * * * * * * * * * * * * * * * * * rm - rf * * * Steadily Increasing the Beta Example of Market Timing 24-20**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 24-21**Process of Attributing Performance to Components**Set up a ‘Benchmark’ or ‘Bogey’ portfolio • Use indexes for each component • Use target weight structure 24-22**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 24-23**Formula for Attribution**Where B is the bogey portfolio and p is the managed portfolio 24-24**Contributions for Performance**Contribution for asset allocation (wpi - wBi) rBi + Contribution for security selection wpi (rpi - rBi) = Total Contribution from asset class wpirpi -wBirBi 24-25**Complications to Measuring Performance**• Two major problems • Need many observations even when portfolio mean and variance are constant • Active management leads to shifts in parameters making measurement more difficult • To measure well • You need a lot of short intervals • For each period you need to specify the makeup of the portfolio 24-26