Chapter 24

Irwin/McGraw-Hill

24-24-11 The McGraw-Hill Companies, Inc., 1999

INVESTMENTSFourth Edition

Bodie Kane Marcus

Portfolio PerformancePortfolio PerformanceEvaluationEvaluation

Chapter 24Chapter 24

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Bodie Kane Marcus

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

IntroductionIntroduction

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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 ReturnsDollar- and Time-Weighted Returns

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Text Example of Multiperiod ReturnsText 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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Period Cash Flow

0 -50 share purchase

1 +2 dividend -53 share purchase

2 +4 dividend + 108 shares sold

%117.7

)1(

112

)1(

5150

21

r

rr

Internal Rate of Return:

Dollar-Weighted ReturnDollar-Weighted Return

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Time-Weighted ReturnTime-Weighted Return

%66.553

25354

%1050

25053

2

1

r

r

Simple Average Return:

(10% + 5.66%) / 2 = 7.83%

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Averaging ReturnsAveraging Returns

Arithmetic Mean:

n

t

t

n

rr

1

Geometric Mean:

1)1(/1

1

nn

ttrr

Text Example Average:

(.10 + .0566) / 2 = 7.81%

[ (1.1) (1.0566) ]1/2 - 1

= 7.83%

Text Example Average:

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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

Comparison of Geometric and Comparison of Geometric and Arithmetic MeansArithmetic Means

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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

Abnormal PerformanceAbnormal Performance

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Market timing Superior selection

- Sectors or industries

- Individual companies

Factors That Lead to Abnormal Factors That Lead to Abnormal PerformancePerformance

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1) Sharpe Index

rp - rf

p

rp = Average return on the portfolio

rf = Average risk free rate

p= Standard deviation of portfolio

return

Risk Adjusted Performance: SharpeRisk Adjusted Performance: Sharpe

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MM2 2 MeasureMeasure

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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MM2 2 Measure: ExampleMeasure: 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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2) Treynor Measure rp - rf

ßp



ßp = Weighted average for portfolio

Risk Adjusted Performance: Risk Adjusted Performance: TreynorTreynor

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Risk Adjusted Performance: Risk Adjusted Performance: JensenJensen

3) Jensen’s Measure

= rp - [ rf + ßp ( rm - rf) ]p

p = Alpha for the portfolio


ßp = Weighted average Beta


rm = Avg. return on market index port.

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Appraisal RatioAppraisal Ratio

Appraisal Ratio = p / (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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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?Which Measure is Appropriate?

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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

LimitationsLimitations

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Adjusting portfolio for up and down movements in the market

Low Market Return - low ßeta High Market Return - high ßeta

Market TimingMarket Timing

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Example of Market TimingExample of Market Timing

******

**

**

**

**

**

**

****

****

******

******

****

****

rp - rf

rm - rf

Steadily Increasing the Beta

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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 AttributionPerformance Attribution

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

Process of Attributing Performance Process of Attributing Performance to Componentsto Components

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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

Process of Attributing Performance Process of Attributing Performance to Componentsto Components

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)(

&

1

11

11

BiBi

n

ipipi

n

iBiBi

n

ipipiBp

n

ipipip

n

iBiBiB

rwrw

rwrwrr

rwrrwr

Where B is the bogey portfolio and p is the managed portfolio

Formula for AttributionFormula for Attribution

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Contributions for PerformanceContributions 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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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 Complications to Measuring PerformancePerformance

Chapter 24

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