Chapter 27 Evaluation of Portfolio Performance

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Chapter 27Evaluation of Portfolio Performance
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What is Required of a Portfolio Manager?

• Above-average returns within a given risk class.
  
  
  
  
  
  
• Portfolio diversification to eliminate
  unsystematic risk.

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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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Composite Portfolio Performance Measures

• Treynor Measure
  
  
  
  
  
• Sharpe Measure
  
  
  
  
  
• Jensen Measure

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Treynor Portfolio Performance Measure

• First composite measure of portfolio performance
  that included risk.
  
  
  
  
  
  
• Utilized Characteristic Line.

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Treynors Measure (T)

• Where
• Ri Average rate of return for portfolio i
  during a specified time period.
• RFR Average rate of return on a risk-free
  investment during the same time period i.
• Bi Slope of the funds characteristic line
  during time period i.

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Demonstration of Comparative Treynor Measures

• Assume that over the past 10 years
  
  
  
  
  
  
  
• Rm 0.12, (SP 500 Return)
• RFR 0.04, (90 day T-Bill Rate)
  
  
  
  
  
• Therefore, Tm (0.12 - 0.04) / 1 0.08

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Treynor Example

• Now assume that over the same period, portfolio
  managers A, B, and C had the following
  performance.

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Computing T Values Yields

• TW (0.09 - 0.04) 0.90 0.055
• TX (0.14 - 0.04) 1.05 0.095
• TY (0.16 - 0.04) 1.20 0.100
• TM (0.12 - 0.04) 1.00 0.080

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Performance Plotted on SML
TC
TB
TA
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Negative T Values

• Two Causes
  
  
  
  
  
  
  
  
  
• - Returns less than RFR and a positive beta
• - Returns above RFR and a negative beta
  
  
  
  
  
• Plot on SML

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Sharpe Portfolio Performance Measure

• Where
• Si Sharpe portfolio performance measure for
  portfolio i.
• RFR Average rate of return on risk-free assets
  during the same time period.
• Ri Average rate of return for portfolio i
  during a specified time period.
• ?i Standard deviation of the rate of return for
  portfolio i during the time period.

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Demonstration of Comparative Sharpe Measures

• Assume that over the past 10 years

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Sharpe Example

• Now assume that over the same period, portfolio
  managers A, B, and C had the following
  performance.

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Computing S Value Yields

• SA (0.09 - 0.04) 0. 09 0.55
• SB (0.14 - 0.04) 0.11 0.909
• SC (0.16 - 0.04) 0.12 1.00
• SM (0.12 - 0.04) 0.10 0.80

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Performance Plotted on CML
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Treynor versus Sharpe Measures

• Beta vs. Standard Deviation
• Treynor - Beta
• Sharpe - Standard Deviation
  
  
  
  
  
• Ranking differences from different
  diversification levels.
  
  
  
  
  
  
• SML vs. CML

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Jensen Portfolio Performance Measure

• Based on CAPM
• E(Rj) RF ?j E(Rm) - RF
  
  
  
  
  
• Where
• E(Rj) The expected return on security j
• RF The one-period risk-free interest rate.
• ?j Systematic risk (beta) for security j.
• E(Rm) Expected return on the market portfolio.

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Jensen Portfolio Performance Measure

• Rjt RFt ?j Rmt - RFt Ujt gt Rjt - RFt
  ?j Rmt - RFt Ujt
• (E(Ujt) 0 if CAPM hold.)
• If we run a regression
• Rjt - RFt ?j ?j Rmt - RFt ?jt
• What is E(?j )?

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Applying the Jensen Measure

• Requires use of different RF, Rm, and Rj, for
  each period.
  
  
  
  
  
• Assumes portfolio is well diversified and only
  considers systematic risk.
  
  
  
  
  
• Regression of (Rj- RF) on (Rm - RF) to find
  alpha.
• R2 can be useful as a measure of 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, use 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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Performance Attribution Analysis (PAA)

• Overall Performance
• f (Asset Allocation, Security Selection).
  
  
  
  
  
• Performance Attribution Analysis
• - Allocation Effect
• - Selection Effect

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Value Added Performance

• Allocation Effect Si wai - wpi x Rpi - Rp
• Selection Effect Si wai x Rai - Rpi
  
  
  
  
  
• Where
• wai ,wpi Investment proporations given to the
  i-th market segment in the managers actual
  portfolio and the benchmark portfolio,
  respectively.
• Rai ,Rpi Investment return to the i-th market
  segment in the managers actual portfolio and
  the benchmark, receptively.
• Rp Total return to the benchmark portfolio.

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Example
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Example

• Overall Actual Return (0.59.7)(0.389.1)(0.
  125.6)8.98
• Overall Benchmark Return (0.68.6)(0.309.2)
  (0.105.4)8.46
• Total value added8.98-8.460.52

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Example

• Allocation Effect
• (-0.1)(8.6-8.46) (0.08)(9.2-8.46)
  (0.02)(5.4-8.46) -0.02
• Selection Effect
• (0.5)(9.7-8.6) (0.38)(9.1-9.2)
  (0.12)(5.6-5.4)0.54
• Total value addedAllocation Effect Selection
  Effect(-0.02)(0.54)0.52

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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