Portfolio Performance Evaluation Chapter 24

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

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

Text Example of Multiperiod Returns Period Action 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

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:

Time-Weighted Return Simple Average Return: (10% + 5.66%) / 2 = 7.83%

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%

Geometric & Arithmetic Means Compared Past Performance - generally the geometric mean is preferable to arithmetic Predicting Future Returns from historical returns Use a weighted average of arithmetic and geometric averages of historical returns if the forecast period is less than the estimation period Use geometric if the forecast and estimation period are equal

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 Factors of abnormal performance Market timing Superior selection Sectors or industries Individual companies

Risk Adjusted Performance: Sharpe 1) Sharpe Index rp - rf  p rp = Average return on the portfolio rf = Average risk free rate p = Standard deviation of portfolio return 

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: 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

Risk Adjusted Performance: Treynor 2) Treynor Measure rp - rf ßp rp = Average return on the portfolio rf = Average risk free rate ßp = Weighted average for portfolio

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.

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

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

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

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

Example of Market Timing rp - rf * * * * * * * * * * * * * * * * * * * * * rm - rf * * Steadily Increasing the Beta

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

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

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

Contributions for Performance Contribution for asset allocation (wpi - wBi) rBi + Contribution for security selection wpi (rpi - rBi) = Total Contribution from asset class wpirpi -wBirBi

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

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