1Department of Economics and Management“Marco Fanno”, University of Padova

massimiliano.caporin@unipd.it

Vth Conference on Computational and Financial Econometrics

- University of London, December 2011 -

“A Survey on the Four Families

of Performance Measures”

Massimiliano Caporin1 Grégory M. Jannin2

Francesco Lisi3 Bertrand B. Maillet4

2A.A.Advisors-QCG (ABN AMRO),

Variances and University of Paris-1 (PRISM)

gregory.jannin@malix.univ-paris1.fr

3Department of Statistical Sciences,

University of Padova

lisif@stat.unipd.it

The fourth author thanks the Europlace Institute of Finance for financial support. This presentation engages onlyits authors and does not necessarily reflect the opinions of their employers. The usual disclaimers apply.

4A.A.Advisors-QCG (ABN AMRO),

Variances and University of Paris-1 (CES/CNRS and EIF)

bmaillet@univ-paris1.fr

Agenda

Motivation

Literature

The Four Families of Performance Measures

Preliminary Conclusions

Extensions

A Survey on theFour Families

of PerformanceMeasures

M. CaporinG. Jannin

F. LisiB. Maillet

2 / 25

Motivation

Since the introduction of the Sharpe ratio (1966), different

measures of portfolio performance are proposed in the literature

The most complete studies so far are: Aftalion and Poncet (2003),

Bacon (2008), Cogneau and Hübner (2009a and 2009b)

The aim of this paper is to present a complete survey of portfolio

performance measures, according to four main families:

Relative performance measures;

Absolute performance measures;

Density-based performance measures;

Utility-related performance measures.

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A Survey on theFour Families

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M. CaporinG. Jannin

F. LisiB. Maillet

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Some Surveys about Performance Measurement

Bacon C., (2008), Practical Portfolio Performance, Wiley Finance Series,384 pages.

Cogneau P. and G. Hübner (2009a), “The (More Than) 100 Ways tomeasure Portfolio Performance. Part 1: Standardized Risk-adjustedMeasures”, Journal of Performance Measurement 13(4), 56-71.

Cogneau P. and G. Hübner (2009b), “The (More Than) 100 Ways tomeasure Portfolio Performance. Part 2: Special Measures andComparison”, Journal of Performance Measurement 14(1), 56-69.

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A Survey on theFour Families

of PerformanceMeasures

M. CaporinG. Jannin

F. LisiB. Maillet

Literature (1/2)

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The Four Representative Performance Measures

Sharpe W., (1966), “Mutual Fund Performance”, Journal of Business 39(1),119-138.

Jensen M., (1968), “The Performance of Mutual Funds in the Period 1945-

1964”, Journal of Finance 23(2), 389-419.

Keating C. and W. Shadwick, (2002), “A Universal Performance Measure”,Journal of Performance Measurement 6(3), 59-84.

Morningstar Inc., (2002), “Morningstar: The New Morningstar RatingTM

Methodology”, Morningstar Research Report 22/04/02 (available atdatalab.morningstar.com/Midas/PDFs/Research_StarRating.pdf), 20 pages.

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A Survey on theFour Families

of PerformanceMeasures

M. CaporinG. Jannin

F. LisiB. Maillet

Literature (2/2)

5 / 25

General Form of Relative Performance Measures(most often expressed in return per unit of risk)

where are the (rescaled) returns, P is a function that

depends upon the observed performance, and R is a

corrected risk measure of the investor’s portfolio under study,

such as R =R .

Objective: comparing the observed (rescaled) performance of

the managed portfolio per unit of risk.

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A Survey on theFour Families

of PerformanceMeasures

M. CaporinG. Jannin

F. LisiB. Maillet ,

1**

p*pp rrPM RP

The Four Families of Performance Measures (1)

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*

*

*pr

The Four Families of Performance Measures (1)

The Reward-to-variability ratio (Sharpe, 1966)

where is the expectation operator, and are

respectively the returns and the total risk of the portfolio p,

and is the risk-free asset.

Interpretation: this measure evaluates the compensation

earned by the portfolio manager, as gauged by the expected

excess return per unit of portfolio total risk.

1,p p f pS E r r

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A Survey on theFour Families

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M. CaporinG. Jannin

F. LisiB. Maillet

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The Main Sharpe-based Performance Measures

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A Survey on theFour Families

of PerformanceMeasures

M. CaporinG. Jannin

F. LisiB. Maillet

Author Name RescaledPerformance

RiskMeasure

CorrectionCoefficient

Morey and Vinod(2001)

DoubleSharpe Ratio

ܧ ݎ െ ݎStandardDeviation

ௌߪଵ

Zakamouline andKoekebakker(2009)

Adjusted forSkewness SharpeRatio

ܧ ݎ െ ݎStandardDeviation

൛ͳ ଷǡ × ݏ ݎ × 3ଵ

× ܧ ݎ െ ݎ ൈ ߪଵ ൟ

ଵଶ

Israëlsen(2005)

Reward-to-absolute-excess return Ratio

ܧ ݎ െ ݎStandardDeviation

ݏ^1 ܧ ݎ െ ݎ

Sharpe(1994)

InformationRatio

ܧ ݎ െ ݎTracking

Error-

Treynor(1965)

Reward-to-volatilityRatio

ܧ ݎ െ ݎ Beta -

The Four Families of Performance Measures (1)

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Performance Measures based on Other Risk Measures

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A Survey on theFour Families

of PerformanceMeasures

M. CaporinG. Jannin

F. LisiB. Maillet

Author Name RescaledPerformance

CorrectedRisk Measure

Yitzhaki(1982)

Gini Ratio ܧ ݎ െ ݎ Gini coefficient

Darolles et al.(2009)

L-performanceTL-moment of

order 1TL-moment of order 2

Konno and Yamazaki(1991)

Mean AbsoluteDeviation Ratio

ܧ ݎ െ ݎ Mean Absolute Deviation

Caporin and Lisi(2011)

Reward-to-range Ratio ܧ ݎ െ ݎ Range

Young(1998)

Minimax Ratio ܧ ݎ െ ݎ Minimax Portfolio

Dowd(2000)

Reward-to Value-at-Risk Ratio

ܧ ݎ െ ݎ Value−at−Risk

Sortino and Satchell(2001)

Reward-to k-LowerPartial Moments Ratio

ܧ ݎ െ ݎLower Partial Moment

of order k

Martin and McCann(1989)

Ulcer PerformanceIndex

ܧ ݎ − ݎAverage Squared

Weekly Drawdowns

The Four Families of Performance Measures (1)

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The Main Limits of Relative Performance Measures

Rankings are consistent only if portfolio returns are ellipticallydistributed and/or the representative agent has a quadraticutility function (except for Zakamouline and Koekebakker,2009)

Use of derivative instruments (fat-tailed and skewed returndistributions) or/and strategies with (time-varying) leverageeffects yield to misleading conclusions (see Kao, 2002; Aminand Kat, 2003a and 2003b; Gregoriou and Gueyie, 2003)

A Survey on theFour Families

of PerformanceMeasures

M. CaporinG. Jannin

F. LisiB. Maillet

The Four Families of Performance Measures (1)

10 / 25

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General Form of Absolute Performance Measures(expressed in return)

where is a transformation function (generally linear),

are the (rescaled) returns, is a function that depends upon

the observed performance and is a function that is related

to the theoretical performance of the investor’s portfolio under

study, conditionally to a set of information denoted .

Objective: comparing the observed (rescaled) performance of the

managed portfolio to its theoretical performance, considering a

model.

,, ** pth

pp rrPM PPΓ

A Survey on theFour Families

of PerformanceMeasures

M. CaporinG. Jannin

F. LisiB. Maillet

The Four Families of Performance Measures (2)

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P

thP

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The Alpha (Jensen, 1968)

where is the expectation operator, are the returns of

the portfolio p, are the returns of the market portfolio m,

is the risk-free rate, is the sensitivity of investor’s

portfolio returns with respect to market portfolio returns.

Interpretation: this measure assesses the extra-performance,

realized by the portfolio manager, given its sensitivity to

systematic risk.

A Survey on theFour Families

of PerformanceMeasures

M. CaporinG. Jannin

F. LisiB. Maillet

,,mpfmfpJp rrErrE

The Four Families of Performance Measures (2)

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The Main Jensen-type Performance Measures

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A Survey on theFour Families

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M. CaporinG. Jannin

F. LisiB. Maillet

Author Name RescaledPerformance

TheoreticalPerformance

Fama(1972)

1. Selectivity Measure2. Risk Compensation Measure

ܧ ݎݎ ǡߚ

ݎ ǡߚݎ

Black(1972)

Zero-betaCAPM

ܧ ݎ െ ܧ ௭ݎ ܧ ݎ െ ܧ ௭ݎ ൈ ǡ௭ߚ

Treynor andMazuy (1966)

Market TimingModel

ܧ ݎ െ ݎܧ ݎ െ ݎ ൈ ଵǡǡߚ

+ ܧ ݎ െ ݎଶൈ ଶǡǡߚ

Henriksson andMerton (1981)

ParametricMarket Timing Model

ܧ ݎ െ ݎܧ ݎ െ ݎ ൈ ଵǡǡߚ

� ݔ ݎ െ ܧ ݎ ǡͲ ൈ ଶǡǡߚ

Connor andKorajczyk (1986)

Multi-factorCAPM

ܧ ݎ െ ݎ ܧ ܨ ൈ ǡிೖߚ

ୀଵ

Ferson andSchadt (1996)

Multi-factor ConditionalCAPM

ܧ ݎ െ ݎ ܧ ܨ ൈ ǡிೖߚ ݐ

ୀଵ

The Four Families of Performance Measures (2)

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Other Miscellaneous Absolute Performance Measures

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A Survey on theFour Families

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M. CaporinG. Jannin

F. LisiB. Maillet

Author Name Performance Measure

Henriksson and Merton(1981)

Non-parametricMarket Timing Model ଵ ݐ ଶ ݐ െ ͳ

Moses et al.(1987)

Diversification-adjusted AlphaMeasure

ߙ× ߣ

ଵ

Grinblatt and Titman(1989)

Positive Period WeightingMeasure

×௧ݓ ǡ௧െݎ ݎ

௧ୀଵ

Modigliani and Modigliani(1997)

Risk-Adjusted PerformanceMeasure

ܧ ݎ × ͳ ǡߣ െ ݎ ൈ ǡߣ

Cantaluppi and Hug(2000)

Efficiency Ratio ܧ ݎ െ ݎ × ܧ ݎ െ ݎଵ

Muralidhar(2001)

Correlation-AdjustedPerformance

ܧ ݎ ൈ ݓ × ܧ ݎ ൈ ݓ+ ݎ × ͳെ െݓ ݓ

Muralidhar(2002)

Skill, History And Risk-Adjusted Measure

{ ܧ ݎ ൈ ݓ × ܧ ݎ ൈ ݓ+ ݎ × 1 − −ݓ ݓ ൟ ×

The Four Families of Performance Measures (2)

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The Main Limits of Absolute Performance Measures

Performance computed with these measures are stronglyinfluenced by the reference portfolio (market portfolio orproxy)

Almost all of them assumes stability of the systematic risksensitivities of the investor’s portfolio over time (except forFerson and Schadt, 1996)

Disregards skewness and kurtosis of the studied portfolioreturns which may alter rankings when using investmentstrategies based on derivative instruments (except for Hwangand Satchell, 1998)

A Survey on theFour Families

of PerformanceMeasures

M. CaporinG. Jannin

F. LisiB. Maillet

The Four Families of Performance Measures (2)

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General Form of Density-based Performance Measures(expressed in scalar terms – no unit)

where are the (rescaled) returns, is a function that

depends upon the observed performance and is a

measure focusing on a specific (left) part of the support of the

density of returns.

Objective: comparing the observed (rescaled) performance of

the managed portfolio to an expression depending on its

losses.

A Survey on theFour Families

of PerformanceMeasures

M. CaporinG. Jannin

F. LisiB. Maillet

,1

**

ppp rrPM PP

The Four Families of Performance Measures (3)

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P

P

The Omega measure (Keating and Shadwick, 2002)

where is the cumulative distribution function, are

the returns of the portfolio p and is a threshold.

Interpretation: it compares the potential gains of the

managed portfolio over its potential losses, both defined

according to a threshold.

1

1 ,p p p p pO F r dr F r dr

A Survey on theFour Families

of PerformanceMeasures

M. CaporinG. Jannin

F. LisiB. Maillet

The Four Families of Performance Measures (3)

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The Main Density-based Performance Measures

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A Survey on theFour Families

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M. CaporinG. Jannin

F. LisiB. Maillet

Author Name RescaledPerformance

DownsidePerformance

Farinelli andTibiletti(2008)

One-sidedRisk Measure

ܧ ݔ ݎ െ ǡͲݎ ܧ ݔ ݎ െ ,ݎ 0

Sortino et al.(1999)

Upside-potential ratio

భܯܪ െݎ ݎ ܮ మܯ െݎ ݎ

Biglova et al.(2004)

Rachev ratio ܧ ݎหݎ െ భǡ ܧ ݎหݎ െ మǡ

Biglova et al.(2004)

GeneralizedRachev ratio

ܧ ݔ െݎ, 0భหݎ െ భǡ ܧ ݔ െݎ, 0

మหݎ െ మǡ

Kazemi et al.(2004)

Sharpe-Omega ratio

ܧ ݎ െ ݎ ܧ ݔ െݎ ,ݎ 0

The Four Families of Performance Measures (3)

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The Main Limits of Density-based PerformanceMeasures

Performance measures are highly related to the threshold (riskfree rate, Minimum Acceptable Return or Value-at-Risk)

The link with the utility function of the studied agent is notstraightforward

Density are subject to model risk

A Survey on theFour Families

of PerformanceMeasures

M. CaporinG. Jannin

F. LisiB. Maillet

The Four Families of Performance Measures (3)

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General Form of Utility-based Performance Measures

(expressed in return per unit of util)

where are the (rescaled) returns, is the expectation

operator, is a value (or utility) function and is a

specific function that depends upon the performance of the

investor’s portfolio.

Objective: incorporating investor's preferences and risk

profiles, through value (or utility) functions, when assessing

the portfolio performance.

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,*pp rEPM VG

A Survey on theFour Families

of PerformanceMeasures

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F. LisiB. Maillet

The Four Families of Performance Measures (4)

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G V

The Morningstar Risk-Adjusted Return (Morningstar

Inc., 2002)

where is the expectation operator, are the returns of

a portfolio p, (with ) is the risk aversion coefficient of

the studied investor.

Interpretation: incorporating the behavior of the agent,

through a Power Utility Function, for assessing the portfolio

performance, given a risk aversion coefficient.

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1 ,p pMRAR E r

A Survey on theFour Families

of PerformanceMeasures

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F. LisiB. Maillet

The Four Families of Performance Measures (4)

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The Main Utility-based Performance Measures

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A Survey on theFour Families

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Author Name TransformedPerformance

Value (or utility)Function

Stutzer (2000)PerformanceIndex ఏఢԹష

∗

Kaplan(2005)

Lambdameasure ఏఢԹష

∗

Gemmill et al.(2006)

Loss-AversePerformanceMeasures

ଵ

ଶଵ

ଵ భ

ଶ మ

Ingersoll et al.(2007)

Manipulation-ProofPerformanceMeasure

ଵ

ଵ ଵఒ

The Four Families of Performance Measures (4)

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The Main Limits of Utility-based PerformanceMeasures

Strongly dependent on the utility function (exponential, poweror logarithmic) characterizing the behavior of the studiedinvestor

Highly sensitive to the investor’s attitude towards risk throughthe risk aversion coefficient (which can be time-varying)

Rankings obtained with these measures are directly related tothe benchmark (risk free rate or proxy)

A Survey on theFour Families

of PerformanceMeasures

M. CaporinG. Jannin

F. LisiB. Maillet

The Four Families of Performance Measures (4)

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Preliminary Conclusions

We propose a Survey of performance measures in a uniform and

comprehensive framework through four main families clearly

identified, namely relative, absolute, density-based and utility-

related performance measures

We define these main families according to two essential criteria:

the unit in which it is expressed, and the way the measure is built

We formulate a general expression for each of the four categories

of performance measures, in which most of the performance

measures fits

A Survey on theFour Families

of PerformanceMeasures

M. CaporinG. Jannin

F. LisiB. Maillet

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Extensions

On our on-going research agenda, we have planned to add new

performance measures (20 extra or so), in order to complete our

actual Survey

Next step will be to perform some empirical applications in order to

compare the properties of collected measures (rank correlation

tests, persistence, lucky versus star funds studies, etc.)

We would like also to complement our intuitions about a new

performance measure (called “Generalized Performance Measure”),

which can be seen as a generalization of the four main categories

presented in this paper.

A Survey on theFour Families

of PerformanceMeasures

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F. LisiB. Maillet

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We are grateful to Christophe Boucher and Patrick Kouontchou for help and

encouragement in preparing this work.

Grégory M. Jannin

Thank you for your attention...

Vth on Computational and Financial Econometrics

- University of London, December 2011 -

“A Survey on the Four Families

of Performance Measures”