Portfolio Performance Evaluation

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Chapter 24

• Portfolio PerformanceEvaluation

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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 management leads to
  measurement problems

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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

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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

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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
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Time-Weighted Return
Simple Average Return (10 5.66) / 2 7.83
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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
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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

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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

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Factors That Lead to Abnormal Performance

• Market timing
• Superior selection
• Sectors or industries
• Individual companies

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Risk Adjusted Performance Sharpe

• 1) Sharpe Index

rp - rf
?
p
?
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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

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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
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Risk Adjusted Performance Treynor

• 2) Treynor Measure

rp - rf ßp
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Risk Adjusted Performance Jensen
3) Jensens 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.

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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
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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

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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

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Market Timing

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

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Example of Market Timing
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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

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Process of Attributing Performance to Components

• Set up a Benchmark or Bogey portfolio
• Use indexes for each component
• Use target weight structure

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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

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Formula for Attribution
Where B is the bogey portfolio and p is the
managed portfolio
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Contributions for Performance
Contribution for asset allocation (wpi -
wBi) rBi Contribution for security selection
wpi (rpi - rBi) Total Contribution
from asset class wpirpi -wBirBi
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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