Chapter 18

Portfolio Performance EvaluationCopyright 2010 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin 18-2

18.1 Risk-Adjusted Returns

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Types of management revisitedPassive management1. Capital allocation between cash and the risky portfolio2. Asset allocation within the risky portfolio.How passive the management actually is varies

Change allocations in 1 or 2 according to perceptions of risk to keep current with portfolio goals.Active managementForecasting future rates of return on either/both asset classes and individual securities

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Performance evaluationPurpose:Are the returns worth the risk and the fees?MeasuresAverage return by itself is an insufficient measure. Why?Average return may not = expected returnRisk return relationshipMajor component of returns is the market performanceNeed a measure of abnormal performance

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Abnormal PerformanceHow is abnormal performance measured?

Comparisons to peer groupsRank fund performance within a given categoryBenchmark portfolio such as an indexReally need measures that consider risk:Calculate reward to risk measures such as the alpha, Sharpe Measure (or variations)

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Factors That Lead to Abnormal PerformanceSuccessful across asset allocationsSuperior allocation within each asset class

Sectors or industriesOverweight better performing sectors, underweight poorer performers

Individual security selectionPick the right stocks, those with performance better than expected

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Risk Adjusted Performance

ptPMtPPt eRRTo measure abnormal performance, must measure normal performance:The single index model can be used:

PMtPP )E(R)E(RFrom this the expected return [E(RP)] can be found:

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Risk Adjusted Performance: JensenMPPP R-R

market the on return excess AverageRbeta Portfolio

portfolio the on return excess verageARalpha Portfolio

MPPP

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Risk Adjusted Performance: Jensen (cont.)Measure of abnormal returnMust establish statistical significance via regressionPossible conceptual problem:Greater abnormal return may be due to greater riskUnless you can hedge out the risk with short sales one should rank portfolio performance by the adjusted alpha which is p / p

MPPP R-R

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Measuring Risk Adjusted Performance: Sharpe Ratios1) Historical Sharpe Ratios

pfp rrRatios Sharpe

deviation standard Portfoliorate free risk Averager

portfolio the on return Averagerpfp

Use:When choosing among competing portfolios that will not be mixed.In practice:Used when one manager handles the (entire) portfolio.

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Risk Adjusted Performance: Treynor2) Historical Treynor Ratios

pfp rrRatios Treynor

beta Portfoliorate free risk Averager

portfolio the on return Averagerpfp

Use:Evaluate a portfolio when portfolio is a piece of a larger portfolio that has different managers.

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M2 MeasureVariation of the Sharpe Ratio that is easier to interpret.Concept of the Sharpe is easy to interpret but the Sharpe number is not.Developed by Modigliani and Modigliani; hence M2Use M2 to compare performance of a managed portfolio (MP) with a market index.The M2 measure creates a hypothetical complete portfolio that is composed of T-bills and the managed portfolio that has the same standard deviation as the market index.

pfp rrRatios Sharpe

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M2 Measure: ExamplePurpose: Create a complete portfolio w/ same risk as the market

Market = WMP = & WT-bills = E(rPComplete) =Since this return is more than the market the managed portfolio outperformed the market on a risk adjusted basis by 4.25%.

25% / 40% = 62.5% 37.5%

Managed Portfolio Market TManaged Portfolio Market T--billbillReturnReturn 32%32% 18%18% 6%6%Stan. DevStan. Dev 40%40% 25%25% 0%0%

(0.625)(0.32) + (0.375)(0.06) = 22.25%

WMP MP + (1 WMP) T-bill25% = WMP40% + (1 WMP)0;

M2 = 22.25% - 18% = 4.25%18-14

Information RatioWhen evaluating a portfolio to be mixed with a position in the passive benchmark portfolio we must draw on insights of the Treynor-Black Model (See Chapter 6)Choose the portfolio, which when combined with the passive benchmark, generates an efficient frontier with the best return per unit of risk as measured by the standard deviation. This is found by combining the Sharpe ratio of the benchmark M with the information ratio of portfolio P: 2

pP2MOptimal SsqrtS

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Summary of measures and usagePerformance Measure Definition Application

Sharpe pR / as the When choosing among portfolios competingoptimal risky portfolio

Treynor Rp / When ranking many portfolios that will be mixed to form the optimal risky portfolio

Information ratio p / eWhen evaluating a portfolio to be mixed with a position in the passive benchmarkportfolio

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More on AlphaAlpha and the Sharpe measure

Positive alpha does not guarantee a higher Sharpe than the benchmark because SM( -1) < 0.Necessary but not a sufficient condition for net performance improvementThe alpha must be large enough to offset increase in residual risk from moving away from the diversified optimum.

PPMMP )1(SSS = Correlation between RP.RM

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More on AlphaAlpha and the Treynor measure

Positive alpha does not guarantee a higher Treynor ranking because you have to know the Beta as well.

PPMP TT

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More on AlphaAlpha and the Information Ratio:

Positive alpha does not guarantee a higher square of the information ratio because higher alpha may come with higher residual risk.

PePRatio nInformatio

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Alpha Capture & TransportIf an analyst finds an undervalued security and invests in it, market moves may wipe out any gains.

Can hedge out market risk via shorting stock index or stock index futures to establish a market neutral position.Process is called alpha capture or alpha transport.When short positions and leverage are allowed a significant non-zero alpha is a sufficient condition for an improvement in the Sharpe and information ratio.18-20

Performance measures for P, Q, M & cash

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Evaluation with a multi-index modelEvidence indicates we should use a multi-index model such as the Fama-French model to establish the expected return:

This allows an estimation of alpha:ptPHMLtHMLSMBtSMBMtPPt errRR

PHMLtHMLSMBtSMBMtPPt rrRR

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T2 (Treynor Square) MeasureUsed to convert the Treynor Measure into percentage return basisMakes it easier to interpret and compareEquates the beta of the managed portfolio with portfolio made up of T-bills and the managed portfolioIf the beta is lower than one, leverage is used and the hypothetical portfolio is compared to the market

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T2 ExamplePort. P. Market

Risk Prem. (r-rf) 13% 10%Beta 0.80 1.0Alpha 5% 0%Treynor Measure 16.25 10Weight to match Market w = M/ P = 1.0 / 0.8Adjusted Return RP* = w (RP) = 16.25%T2P = RP* - RM = 16.25% - 10% = 6.25%

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18.2 Style Analysis

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Style AnalysisComplex method of performance evaluation introduced by William SharpeRecent studies of mutual fund performance show that > 90% of variation allocations to bills, bonds and stocks.

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Style AnalysisExplaining percentage returns by allocation to styleStyle Analysis has become popular with the industry

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the Magellan Fundvariability is explained by asset allocation among these factors, & actually only _____ variables explain The remaining _____ is due to security selection.

97.5%7

2.5%three

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Risk-Adjusted Ratings

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RatingCompanies are put into peer groups based on Morningstar style definitions Risk adjusted fund performance is ranked and then Stars are assigned according to the following table (5 stars is the highest rating)

Star ratings are highly correlated to Sharpe measure rankings

Percentile Stars0-10 1

10-32.5 232.5-67.5 367.5-90 490-100 5

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Category RARs and Excess Return Sharpe Ratios

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18.4 Risk Adjustment With Changing Portfolio Composition

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Problems with performance measuresMeasures assume a fund maintains a constant level of riskParticularly problematic for funds that engage in active asset allocationSharpe M = 0.4First 4 qtrs Sharpe = 0.52nd 4 qtrs Sharpe = 0.5Overall Sharpe = 0.37What happened?

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Problems with performance measuresIn a large universe of funds, some funds will have abnormal performance in every period just by chance.

Requires statistical workVolatility is quite high and creates large errors in estimation

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18.5 Performance Attribution Procedures

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Performance AttributionDecomposing overall performance into componentsPerformance is determined by specific portfolio choices, Major performance determinants:

Broad asset allocation among types of securities,Industry weighting in equity portfolio,Security choice,Timing of purchases and sales.18-36

Process of Attributing Performance to ComponentsUse indexes for each component in the bogeyUse target weight structure in the bogey

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Process of Attributing Performance to Componentsthe managed portfolioExplain the difference in return based on component weights or selectionSummarize the performance differences into appropriate categories

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5.34%Total0.4857%23%Cash

1.89%7%Bonds7.28%70%Equity

Monthly ReturnWeightComponent

Performance Attribution ExampleManaged portfolio with monthly return of 5.34%.

The portfolio was comprised of

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Performance of the Managed Portfolio

component indices with the given weights.The Bogey return represents the return on an unmanaged portfolio

The Bogey weights are representative of a standard portfolio for the typical risk tolerance of the given type of client or for the typical fund in this category

Bogey Performance and Extra Return Component Benchmark Weight Index Return during Month Bonds (Shearson-Lehman Index) 0.3 1.45% Equity (S&P500 Index) 0.6 5.81% Cash (Money Market) 0.1 0.48% Return of the Managed Portfolio 5.34% - Return of the Bogey Portfolio 3.97% Extra Return of the Managed Portfolio 1.37%

3.97%8%)(0.10)(0.41%)(0.60)(5.85%)(0.30)(1.4ReturnBogey

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Performance of the Managed Portfolio

indices with the given weights.The Bogey return represents the return on an unmanaged portfolio.

The Bogey weights are representative of a standard portfolio for the typical risk tolerance of the given type of client or for the typical fund in this category.

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Performance Attribution Example

Contribution to performance from broad asset allocation decision Note this uses the index return, not the actual managed portfolio return

Superior performance is generated by overweighting investments in classes that perform better than the bogey.

=== 0.3099%Contribution of asset allocation:

-0.4537%0.48-3.97 = -3.49%0.130.100.23Cash0.1840%5.81-3.97 = 1.84%0.100.600.70Stock0.5796%1.45-3.97 = -2.52%-0.230.300.07Bonds

ContributionIndex ReturnMinus BogeyExcess WeightBenchmark WeightActual WeightMarketxxx

A. Contribution of Asset Allocation to Performance

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Performance Attribution

Contribution to performance from broad asset allocation decisionNote that Column 4 is the INDEX return, not the actual managed portfolio return.

Superior performance is generated by overweighting investments in classes that perform better than the bogey.

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Performance Attribution1.0611%Contribution of Selection

0.000013%0.230.0000570.00480.004857Cash1.0290%0.700.01470.05810.0728Stock0.0308%0.070.00440.01450.0189Bonds

Contribution toPerformancePortfolioWeightExtraReturnIndexReturnPortfolioReturnMarketB. Contribution of Selection to Total Performance

Material superior performance in both Bond and Stock SectorsTable B delineates contribution to performance of both sector and security selection

xxx

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Performance Attribution

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Performance Attribution

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Sector Selection within the Equity MarketWeights

1.290%Contribution of Sector Allocation-0.027%0.30%-0.08950.1090.0195Technology-0.017%2.60%-0.00670.1420.1353Energy0.111%5.00%0.02210.2180.2401Credit Sensitive 1.997%10.00%0.19970.2040.4037Consumer Noncyclicals-0.355%8.80%-0.04030.1250.0847Consumer Cyclicals-0.243%4.10%-0.05930.0780.0187Capital Goods0.262%7.00%0.03740.0410.0784Business Services-0.437%6.90%-0.06340.0830.0196Basic Materials

Contribution ofSector AllocationSectorReturnExcessWeightS&P 500PortfolioSector

C. Contribution of Equity Sector Allocation to Total Performance

xxxxxxxx

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Sector Selection within the Equity Market

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Portfolio Attribution: Summary

1.3710%Total Extra Return on the portfolio0.0013%23.00%0.0057%c. Cash Extra Return

0.031%7.00%0.4400%b. Bonds Extra Return1.029%70.00%1.470%

Weights0.180%ii. Security Selection1.290%i. Sector Allocation

a. Equity Extra Return2. Selection0.3099%1. Asset Allocation

D. Portfolio Attribution Summary

The Security Selection component is inferred as follows: _________________________________Sector allocation accounted for ______ of the total so Security Selection must have resulted in _____________________.

See Table B

The total equity extra return = 1.47%1.29%1.47% - 1.29% or 0.18%

Table B

x =x =x =

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Portfolio Attribution: Summary

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18.6 Market Timing

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Market TimingAdjust the asset allocation for movements in the market

Shift between stocks and money market instruments or bondsWith perfect ability to forecast behaves like an optionLittle evidence of market timing ability

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Figure 18.8 Rate of Return of a Perfect Market Timer

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Value of Market TimingInvest $1 on December 1, 1926

*Perfect Timing: Every month 100% of the funds are placed in either stocks or cash based on which would have the higher return.

Strategy Value in 2008 Geom Avg. ReturnMoney Markets $20 3.71%Stocks $1,626 9.44%Perfect Timing* $36,699,302,473 34.54%

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With Imperfect Ability to ForecastTakes a long time horizon to judge the abilityJudge proportions of correct calls

Bull markets and bear market calls

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Market Timing & Performance MeasurementTimer adjusts portfolio for up and down movements in the market

Low Market Return - low etaHigh Market Return - high eta

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Characteristic Lines

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Selected Problems

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Problem 1

The alphas for the two portfolios are:A = B = Might add A if you believe the alpha is significant, might short B if you believe its alpha is significant, should check Treynor.

E(r) Portfolio A 11% 10% 0.8 Portfolio B 14% 31% 1.5 Market index 12% 20% 1.0 Risk-free asset 6% 0% 0.0 11% [6% + 0.8(12% 6%)] = 0.2%14% [6% + 1.5(12% 6%)] = 1.0%

a.

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Problem 1

If you hold only one of the two portfolios, then the Sharpe measure is the appropriate criterion:SA = SB = Therefore, using the Sharpe criterion, Portfolio A is preferred.

E(r) Portfolio A 11% 10% 0.8 Portfolio B 14% 31% 1.5 Market index 12% 20% 1.0 Risk-free asset 6% 0% 0.0

0.510%6%11% 0.2631%6%14%b.

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Problem 1M2 Measure for A:

Market = WA = & WT-bills = E(rPComplete) =M2 measure for A =

E(r) Portfolio A 11% 10% 0.8 Portfolio B 14% 31% 1.5 Market index 12% 20% 1.0 Risk-free asset 6% 0% 0.0

20% / 10% = 2 -1(2)(0.11) + (-1)(0.06) = 16%

WA A + (1 WA) T-bill20% = WA10% + (1 WA)0;

16% - 12% = + 4%

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Problem 1M2 Measure for B:

Market = WB = & WT-bills = E(rPComplete) =M2 measure for B =

20% / 31% = 0.645 0.355(0.645)(0.14) + (0.355)(0.06) = 11.16%

WB B + (1 WB) T-bill20% = WB31% + (1 WB)0;

11.16% - 12% = -0.84%

E(r) Portfolio A 11% 10% 0.8 Portfolio B 14% 31% 1.5 Market index 12% 20% 1.0 Risk-free asset 6% 0% 0.0

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Problem 2

(0.70)(2.0%) + (0.20)(1.0%) + (0.10)(0.5%) = 1.65%(0.60)(2.5%) + (0.30)(1.2%) + (0.10)(0.5%) = 1.91%1.65% 1.91% = -0.26%

(Underperformance)

a. Actual: Bogey: Relative performance =

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Problem 2

Security Selection:

-0.39%Contribution of security selection:0.00%0.100.0%0.5%0.5%Cash-0.04%0.20-0.2%1.2%1.0%Bonds-0.35%0.70-0.5%2.5%2.0%EquityContributionPortfolio WeightExcess PerformanceIndexPerformancePortfolioPerformanceMarketb.

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Problem 2

0.130%Contribution of asset allocation:0.000%-1.41%0.000.100.10Cash0.071%-0.71%-0.100.300.20Bonds0.059%0.59%0.100.600.70Equity

ContributionIndex ReturnMinus BogeyExcess WeightBenchmark WeightActual WeightMarket

-0.26%Excess performance0.13%Asset allocation-0.39%Security selection

Summary

c. Asset Allocation