McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 24 Portfolio Performance Evaluation

24-2 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 Introduction

24-3 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 Dollar- and Time-Weighted Returns

24-4 Text Example of Multiperiod Returns PeriodAction 0Purchase 1 share at $50 1Purchase 1 share at $53 Stock pays a dividend of $2 per share 2Stock pays a dividend of $2 per share Stock is sold at $54 per share

24-5 PeriodCash Flow 0-50 share purchase 1+2 dividend -53 share purchase 2+4 dividend shares sold Internal Rate of Return: Dollar-Weighted Return

24-6 Time-Weighted Return Simple Average Return: (10% %) / 2 = 7.83%

24-7 Averaging Returns Arithmetic Mean: Geometric Mean: Text Example Average: ( ) / 2 = 7.83% [ (1.1) (1.0566) ] 1/2 - 1 = 7.81% Text Example Average:

24-8 Past Performance - generally the geometric mean is preferable to arithmetic mean because it represents a constant rate of return we would have needed to earn in each year to match actual performance over past period. Predicting Future Returns from historical returns arithmetic is preferable to geometric mean, because it is an unbiased estimate of the portfolio’s expected return. In contrast while the geometric mean is always less than the arithmetic mean, it constitutes a downward- biased estimator of the stock’s expected return. Geometric & Arithmetic Means Compared

24-9 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 (r p -r f ) /  p Abnormal Performance

24-10 Market timing Superior selection Sectors or industries Individual companies Factors Leading to Abnormal Performance

) Sharpe Index r p - r f p r p = Average return on the portfolio r f = Average risk free rate p = Standard deviation of portfolio return   Risk Adjusted Performance: Sharpe

24-12 M 2 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

24-13 M 2 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

) Treynor Measure r p - r f ß p r p = Average return on the portfolio r f = Average risk free rate ß p = Weighted average  for portfolio Risk Adjusted Performance: Treynor

24-15 = r p - [ r f + ß p ( r m - r f ) ] Risk Adjusted Performance: Jensen 3) Jensen’s Measure p p = Alpha for the portfolio r p = Average return on the portfolio ß p = Weighted average Beta r f = Average risk free rate r m = Avg. return on market index port.  

24-16 Appraisal Ratio (Information Ratio) Appraisal Ratio =  p /  (e p ) Appraisal Ratio divides the alpha of the portfolio by the nonsystematic risk Nonsystematic risk could, in theory, be eliminated by diversification

24-17 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 Which Measure is Appropriate?

24-18 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. Limitations

24-19 Adjusting portfolio for up and down movements in the market Low Market Return - low ßeta High Market Return - high ßeta Market Timing

24-20 Example of Market Timing * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * r p - r f r m - r f Steadily Increasing the Beta

24-21 Decomposing overall performance into components Components are related to specific elements of performance Example components Broad Allocation Industry Security Choice Up and Down Markets Performance Attribution

24-22 Set up a ‘Benchmark’ or ‘Bogey’ portfolio Use indexes for each component Use target weight structure Attributing Performance to Components

24-23 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. Attributing Performance to Components

24-24 Where B is the bogey portfolio and p is the managed portfolio Formula for Attribution

24-25 Contributions for Performance Contribution for asset allocation (w pi - w Bi ) r Bi + Contribution for security selection w pi (r pi - r Bi ) = Total Contribution from asset class w pi r pi -w Bi r Bi

24-26 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 Complications to Measuring Performance

24-27 Style Analysis Based on regression analysis Examines asset allocation for broad groups of stocks More precise than comparing to the broad market Sharpe’s analysis: 97.3% of returns attributed to style