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A Survey on the Four Families of Performance Measures Massimiliano Caporin 1 Grgory M. Jannin 2 Francesco Lisi 3 Bertrand B. Maillet 4 1 Department of Economics and Management 2 A.A.Advisors-QCG (ABN AMRO), Variances and Marco Fanno,

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SLIDE 1

1Department of Economics and Management

“Marco Fanno”, University of Padova massimiliano.caporin@unipd.it 30th International French FInance Association Conference

  • EM Lyon, May 2013 -

“A Survey on the Four Families

  • f 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@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 only its 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 Orléans (LEO/CNRS and EIF) bertrand.maillet@univ-orleans.fr

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SLIDE 2

Agenda

Motivation Aim of the Article Literature The Four Families of Performance Measures Preliminary Conclusions Extensions

A Survey on the Four Families

  • f Performance

Measures

  • M. Caporin
  • G. Jannin
  • F. Lisi
  • B. Maillet

2 / 35

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SLIDE 3

Motivation (1/8)

Fund W is preferred => higher performance and no obvious risk

5

3 / 35 A Survey on the Four Families

  • f Performance

Measures

Source: simulations by the author.

40 60 80 100 120 140 160 12/08 03/09 06/09 09/09 12/09 03/10 06/10 09/10 12/10 03/11 06/11 09/11 12/11

Fund W (Good) Fund Z (Bad)

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SLIDE 4

Motivation (2/8)

5

4 / 35 A Survey on the Four Families

  • f Performance

Measures

Source: simulations by the author.

40 60 80 100 120 140 160 12/08 03/09 06/09 09/09 12/09 03/10 06/10 09/10 12/10 03/11 06/11 09/11 12/11

Fund X (Good) Fund Y (Bad)

Fund X is preferred => higher performance but more risky sometimes

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SLIDE 5

01/99 01/00 01/01 01/02 01/03 01/04 01/05 01/06 01/07 01/08 01/09 01/10 01/11 50 100 150 200 250 300 Fund A Fund B DJI

5 / 35

Fund A (Good) Fund B (Bad)

Source: USD Daily quotes, from 01/01/1999 to 05/13/2011. We compare in this illustration the performance of the Fund A and the Fund B – the uninformed agent – to their benchmark which is the DJI. Fund A corresponds to the informed investor who is characterized by an alpha set to .01% and a sensitivity of his portfolio to the benchmark equals to .39, while the Fund B, namely the uninformed investor, has respectively an alpha -.01% sets to and a beta equals to 1.10. We use an Ordinary Least Square method to calculate these data. Computations by the authors.

D.J. Index

A Survey on the Four Families

  • f Performance

Measures

Motivation (3/8)

Fund A is preferred => higher performance

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SLIDE 6

01/01 01/02 01/03 01/04 01/05 01/06 01/07 01/08 01/09 01/10 01/11 01/12 70 80 90 100 110 120 130 140 150 Dates Performance

6 / 35

Net Asset Values of Fund A, Fund B and the D.J. Index on the last period

Source: Illustration of Net Asset Values of Fund A (thin grey line) and Fund B (bold black line) over a one-year period. Grey areas represent market shocks. The x-axis corresponds to dates whilst the y-axis shows the performance of the two funds. Computations by the authors.

Fund A Fund B DJI 1st Shock 2nd Shock

A Survey on the Four Families

  • f Performance

Measures

Motivation (4/8)

Fund A or Fund B ? Performance and/or Risk ?

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SLIDE 7

7 / 35

Statistics of Fund A and Fund B on the last period

Fund A Fund B Ranking

MPPM ( = 2) 12.69% 13.31% B MPPM ( = 3) 11.50% 11.58% B

  • Ann. Performance

15.20% 16.47% B

  • Ann. Volatility

15.79% 18.09% A Skewness 2.12

  • 2.44

A Kurtosis 9.86 16.06 A Max Drawdown

  • 10.00%
  • 20.00%

A Value-at-Risk 95%

  • 1.06%
  • 1.68%

A Value-at-Risk 99%

  • 1.54%
  • 4.09%

A Sharpe 1.04 1.01 A Omega 5.99 5.78 A Sortino 3.35 1.92 A Kappa 3 2.46 1.13 A Calmar 1.63 0.92 A Burke .44 .32 A Sharpe-Omega .13 .12 A Treynor .34 .23 A Treynor-Black .26 .15 A Graham-Harvey .09 .08 A Sterling .83 .62 A Cornell 11.94% 10.94 A Israëlsen 1.04 1.05 A Jensen-Alpha 11.23% 10.30% A RAP 20.18% 19.57% A MRAP 34.09% 23.20% A SRAP 9.55% 8.94% A Ulcer 2.01 1.65 A Ziemba 1.70 .61 A

Fund B is however preferred by some performance measures (MPPM)

A Survey on the Four Families

  • f Performance

Measures

Motivation (5/8)

According to the main criteria, Fund A is preferred

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SLIDE 8

1 5 6

8 / 35

Iso-MPPM Frontiers displaying the Quantity of Over-volatility required for a Given Over-performance for reversing the MPPM Ranking

.00 .50 1.00 1.50 2.00 .00 1.00 2.00 Over-performance Over-volatility

ρ=2 Sharpe Ratio Fund B ρ=10 ρ=9 ρ=8 ρ=7 ρ=6 ρ=5 ρ=4 ρ=3

Source: Illustration of over-performance (the y-axis) and over-volatility (the x-axis), both expressed in %, of returns in Fund B (symbolized by the blue star), compared to Fund A, and ranking frontiers (solid lines) computed from the Manipulation-Proof Performance Measure (see, Ingersoll et al., 2007) when varying the risk aversion coefficient from 2 to 10 (is equal to 3 in the original paper). Computations by the authors.

.50 1.00 A Survey on the Four Families

  • f Performance

Measures

Motivation (6/8)

risk performance

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SLIDE 9

Motivation (7/8)

Performance depends on the investor’s preferences and risk profiles Strong link between performance and inflows/outflows in funds Performance measures published in the academic literature are based on different dimensions (total risk, systematic risk sensitivity, drawdowns, return density, VaR, CAPM, APT, utility function, etc.) More than 100 ways to measure portfolio performance…

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  • f Performance

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Motivation (8/8)

5

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  • f Performance

Measures

1st Family: Relative Performance Measures

3 1966 4 2005 4 2001 5 2009 7 2009 5 2006 3 1994 4 2005 3 1965 4 1965 2008 1994 N 2005 3 2000 2003 3 2002 3 2003 3 1998 3 1982 2 2009 3 1991 3 2011 3 2001 3 1991 2006 3 2004 4 2005 3 1989 3 1994 3 1991 1996

2nd Family: Absolute Performance Measures

4 1968 5 1972 4 1999 5 2000 1998 2007 2009 4 1972 8 1970 6 1966 6 1981 1986 6 1973 1973 6 1979 8 1993 10 1997 1993 7 1998 4 1999 5 1991 N 1996 5 1987 5 2001 6 2002 3 1989 1979 4 1997 2005 3 1997 3 1997 5 1999 2 1981

3rd Family: Density-based Performance Measures

2 2000 2 2002 3 2006 3 2006 4 1999 4 2008 3 2004 6 2004 2010 2010

4th Family: Utility-based Performance Measures

2 2002 3 2000 4 2005 4 2007 2 2010

Legend: See Caporin M., G. Jannin, F. Lisi and B. Maillet, (2012), "A Survey on the Four Families of Performance Measures", 29 pages.

The Main Performance Measure Other Performance Measures directly based on the Main One Alternative Performance Measures

GH2

Cornell

THE TAXONOMY TABLE OF PERFORMANCE MEASURES

Henriksson-Merton Modigliani- Modigliani

RAP SRAP

Graham-Harvey

HM GH1 MRAP

Lobosco Graham-Harvey

RVAR

Dowd Watanabe Kaplan-Knowles

κ

RLPM SM RCVAR YO

Young Scholtz-Wilkens

U-P GL

αZB

Black
  • DR
Biglova-Ortobelli- Rachev-Stoyanov Biglova-Ortobelli- Rachev-Stoyanov Ortobelli-Rachev- Biglova-Stoyanov

RMDD

Ortobelli-Rachev- Biglova-Stoyanov Gemmill-Hwang- Salmon

LAPH MRAR

Fama-French

RE

Cantaluppi-Hug

CM

Chauveau-Maillet Martin-Rachev- Siboulet

αC

Carhart

αMcD

McDonald Pogue-Solnik- Rousselin Ang-Chua

αPSR αAC αFF αEGDH αHS αLB

  • αFS

K SHARAD

Ferson-Schadt Elton-Gruber-Das- Hlavka Hwang-Satchell

αCK

Connor-Korajczyk Sortino-Stachell Leland Aftalion-Poncet Keating-Shadwick Morningstar Ingersoll-Spiegel- Goetzmann-Welch Brown-Kang-Lee Kaplan
  • Sortino-Meer-
Plantinga Farinelli-Tibiletti

αAP

SE

Srivastava- Essayyad

GT UIP

Hübner Stutzer Muralidhar Gemmill-Hwang- Salmon

BA

Bacon

IS2

Israëlsen

ERR IR

Sortino-Meer

PI

Bernardo-Ledoit Caporin-Lisi

Author(s)

MCV

Moses-Cheyney-Veit Muralidhar

CAP OE

Briec-Kerstens- Jokung

GRR RADD

αHM

PPW

Grinblatt-Titman

LAPS FT RR

Henriksson-Merton

C IS1 L ASSR

Israëlsen

YI T S DS

Briec-Kerstens

BK

Brennan

αB

Treynor-Mazuy

αTM

Morey-Vinod

T2 ASKSR

Zakamouline- Koekebakker Zakamouline- Koekebakker Sharpe Treynor Treynor

Parameter Number Publication Date

W1 C B

Darolles-Gouriéroux- Jasiak Watanabe

SHORT NAME

W2 RE / RC

Morey-Morey Sharpe Ziemba

DRS

Yitzhaki

αJ

NS

Jensen Fama Kestner

AB

Alexander-Baptista

RMVAR

Favre-Galeano

KY

Konno-Yamazaki Young Burke Martin-McCann
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SLIDE 11

Aim of the Article

Definition of the four main families of measures General expressions for each family Identification and explicit formulations for the more representative measures (and their close variants in Appendix) Intuition for each measure with some short criticisms Codes (MatLab, R,…) available soon on: www.performance-metrics.eu

1

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  • F. Lisi
  • B. Maillet

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SLIDE 12

Main and Recent Surveys about Performance Measurement

Cogneau P. and G. Hübner, (2009a), “The (More Than) 100 Ways to measure Portfolio

  • Performance. Part 1: Standardized Risk-adjusted Measures”, Journal of Performance

Measurement 13(4), 56-71. Cogneau P. and G. Hübner, (2009b), “The (More Than) 100 Ways to measure Portfolio

  • Performance. Part 2: Special Measures and Comparison”, Journal of Performance

Measurement 14(1), 56-69.

Some Books about Performance Measurement

Aftalion F. and P. Poncet, (2003), Les techniques de mesure de performance. Economica, 140 pages. Bacon C., (2008), Practical Portfolio Performance. Wiley Finance Series, 384 pages. Cobbaut R., R. Gillet and G. Hübner, (2011), La gestion de portefeuille – instruments, stratégie et performance. De Boeck, 520 pages. Knight J. and S.E. Satchell, (2002), Performance Measurement in Finance. Butterworth- Heinemann, 365 pages.

A Survey on the Four Families

  • f Performance

Measures

  • M. Caporin
  • G. Jannin
  • F. Lisi
  • B. Maillet

Literature (1/2)

12 / 35

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SLIDE 13

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, (2002), “Morningstar: The New Morningstar RatingTM Methodology”, Morningstar Research Report 22/04/02, 20 pages.

6

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  • f Performance

Measures

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  • G. Jannin
  • F. Lisi
  • B. Maillet

Literature (2/2)

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SLIDE 14

General Form

  • f

Relative Performance Measures (most often expressed in return per unit of risk) where are the returns, P is a function that depends upon the observed performance, R is a risk measure of the investor’s portfolio, and are two specific threshold returns, and is a correction term. Objective: expressing the observed (rescaled) performance of the managed portfolio per unit of corrected risk

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A Survey on the Four Families

  • f Performance

Measures

  • M. Caporin
  • G. Jannin
  • F. Lisi
  • B. Maillet

The Four Families of Performance Measures (1)

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( )

⋅

( )

( )

1 1 2

,

p p p p

PM r r c τ τ

−

  = − × − ×   P R

p

r

( )

⋅

1

τ

2

τ

p

c

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SLIDE 15

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 p f r

S E r r σ

−

  = − ×  

2 1

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  • f Performance

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The Four Families of Performance Measures (1)

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SLIDE 16

The Main Sharpe-based Performance Measures**

2 1 3 6 4

A Survey on the Four Families

  • f Performance

Measures

  • M. Caporin
  • G. Jannin
  • F. Lisi
  • B. Maillet

Authors Name Rescaled Performance Risk Measure Correction Coefficient

Morey and Vinod (2001) Double Sharpe Ratio

  • Standard

Deviation

  • Zakamouline and

Koekebakker (2009) Adjusted for Skewness Sharpe Ratio

  • Standard

Deviation 1 1 ,

3

  • Israëlsen

(2005) Reward-to- absolute-excess return Ratio

  • Standard

Deviation 1^

! 1

Sharpe (1994) Information Ratio

  • Tracking

Error

  • Treynor

(1965) Reward-to- volatility Ratio

  • Beta
  • 16 / 35

**where corresponds to the investor’s relative preferences for skewness, is the skewness of the underlying return distribution and is the sign function.

,

  • ∙

( )

1 p

r τ − P

( )

2 p

r τ − R

p

c

The Four Families of Performance Measures (1)

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SLIDE 17

Performance Measures based on Other Risk Measures

2 1 3 6 4

A Survey on the Four Families

  • f Performance

Measures

  • M. Caporin
  • G. Jannin
  • F. Lisi
  • B. Maillet

Authors Name Rescaled Performance Corrected Risk Measure Yitzhaki (1982) Gini Ratio

  • Gini coefficient

Darolles et al. (2009) L-performance TL-moment of

  • rder 1

TL-moment of order 2 Konno and Yamazaki (1991) Mean Absolute Deviation Ratio

  • Mean Absolute Deviation

Caporin and Lisi (2011) Reward-to-range Ratio

  • Range

Young (1998) Minimax Ratio

  • Minimax Portfolio

Dowd (2000)* Reward-to Value-at- Risk Ratio

  • Value−at−Risk

Sortino and Satchell (2001) Reward-to o-Lower Partial Moments Ratio

  • Lower Partial Moment
  • f order o

Martin and McCann (1989)** Ulcer Performance Index

  • Average Squared

Weekly Drawdowns

17 / 35

*See also Favre and Galeano (2002) who use the modified Value-at-Risk and Martin et al. (2003) who refers to the Conditional Value-at-Risk. **See also Burke (1994), Young (1991) and Kestner (1996) which respectively based their measures on the Total Squared Monthly Drawdowns, the Maximum Drawdown and the Average Yearly Maximum Drawdowns.

The Four Families of Performance Measures (1)

( )

1 p

r τ − P

( )

2 p p

r c τ − × R

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SLIDE 18

Some Main Limits of Relative Performance Measures

Rankings are consistent only if portfolio returns are elliptically distributed and/or the representative agent has a quadratic utility function Use of derivative instruments (fat-tailed and skewed return distributions) or/and strategies with (time-varying) leverage effects yield to misleading conclusions (see Kao, 2002; Amin and Kat, 2003a and 2003b; Gregoriou and Gueyie, 2003)

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  • f Performance

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The Four Families of Performance Measures (1)

*

*except for Zakamouline and Koekebakker (2009).

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SLIDE 19

2 1 3 5 6 4

General Form

  • f

Absolute Performance Measures (expressed in return)

where Π ∙,∙ is a transformation function (generally linear), are the returns, $ ∙

is a function that depends upon the

  • bserved performance and $%& ∙

is a function that is related to the theoretical performance of a model reference portfolio, conditionally to a set of information denoted Ω,

and are two specific threshold returns. Objective: comparing the observed (rescaled) performance of the managed portfolio to its theoretical performance, considering a

model

( ) ( )

1 2

, ,

th p p p

PM P r P r τ τ   = Π − − Ω   Γ

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( )

⋅ P

( )

⋅

th

P

The Four Families of Performance Measures (2)

p

r

1

τ

2

τ

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SLIDE 20

2 1 3 5 6 4

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 the Four Families

  • f Performance

Measures

  • M. Caporin
  • G. Jannin
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  • B. Maillet

( )

[ ]

( )

[ ]

,

,m p f m f p J p

r r E r r E β α × − − − =

20 / 35

The Four Families of Performance Measures (2)

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SLIDE 21

The Main Jensen-type Performance Measures*

2 1 3 6 4

A Survey on the Four Families

  • f Performance

Measures

  • M. Caporin
  • G. Jannin
  • F. Lisi
  • B. Maillet

Authors Name Rescaled Performance Theoretical Performance Fama (1972) Net Selectivity Index

  • (

*

Black (1972) Zero-beta CAPM

+ ( +

),+ Treynor and Mazuy (1966) Market Timing Model

  • ̅(,-

)

.,,( ̅(,-

  • )

.,,( Henriksson and Merton (1981) Parametric Market Timing Model

  • ̅(,-

)

.,,( /0

̅(,-, 0 )

.,,( Connor and Korajczyk (1986) Multi-factor Model

  • 2 3

4 ),56 7 48

Ferson and Schadt (1996) Multi-factor Conditional Model

  • 2 3

4 ),56 9 7 48

21 / 35

*where

+ are the returns of the zero-beta portfolio, )

.,,( and ) .,,( are the estimated selectivity and the market timing coefficients, 3

4 is the k-th

loading factor and ),56 9 is the sensitivity of the portfolio p to k-th the loading factor at time t.

The Four Families of Performance Measures (2)

( )

1 p

r τ − P

( )

2 th p

r τ − Ω P

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SLIDE 22

Other Miscellaneous Absolute Performance Measures*

2 1 3 6 4

A Survey on the Four Families

  • f Performance

Measures

  • M. Caporin
  • G. Jannin
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  • B. Maillet

Authors Name Performance Measure Henriksson and Merton (1981) Non-parametric Market Timing Model :

9 : 9 1

Moses et al. (1987) Diversification-adjusted Alpha Measure ;

< =,(

  • Grinblatt and Titman

(1989) Positive Period Weighting Measure 2 >-

,-

  • ?
  • 8

Modigliani and Modigliani (1997) Risk-Adjusted Performance Measure

1 @,( @,(

Cantaluppi and Hug (2000) Efficiency Ratio

A

  • Muralidhar

(2001) Correlation-Adjusted Performance

B >B A >A

+

1 >B >A

Muralidhar (2002) Skill, History And Risk- Adjusted Measure C

B >B A >A

+

1 >B >A

D E

The Four Families of Performance Measures (2)

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( )

( )

1 2

,

th p p

r r τ τ   Π − − Ω   P P

*where and are the probabilities (resp. up and down) of correct forecasts of the portfolio manager about market variations, evaluates the diversification premium earned by the manager, corresponds to the diversification level of the investor’s portfolio compared to that of the market portfolio, depends on the correlation coefficient between the investor’s portfolio and his benchmark and is linked to the length of observations.

:

9

: 9 =,( @,( F G >B

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SLIDE 23

The Main Limits of Absolute Performance Measures

Performance computed with these measures are strongly influenced by the reference portfolio (market portfolio or proxy) Almost all of them assumes stability of the systematic risk sensitivities of the investor’s portfolio over time Disregards skewness and kurtosis of the studied portfolio returns which may alter rankings when using investment strategies based on derivative instruments

A Survey on the Four Families

  • f Performance

Measures

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The Four Families of Performance Measures (2)

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* **

* except for Ferson and Schadt (1996). ** except for Hwang and Satchell (1998).

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General Form of Density-based Performance Measures (expressed in scalar terms – no unit) where are the returns, $ ∙ and relate to a specific (respectively right and 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 the Four Families

  • f Performance

Measures

  • M. Caporin
  • G. Jannin
  • F. Lisi
  • B. Maillet

( ) ( )

1

,

p p p

PM r r

− + −

  = ×   P P The Four Families of Performance Measures (3)

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( )

⋅

+

P

( )

⋅

−

P

p

r

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SLIDE 25

Most of the Density-based Performance Measures can be expressed such as:

where corresponds to a threshold for computing gains and for calculating losses, is another threshold specifying the right part of the support of the return density (i.e. gains) and is another threshold associated with the left part (i.e. losses), and are intensification constants, and are normalizing constants.

A Survey on the Four Families

  • f Performance

Measures

  • M. Caporin
  • G. Jannin
  • F. Lisi
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( )

1 2 3 4 1 2 1 2

, , , , , , , ,

p p

PM r

  • o k k

τ τ τ τ =H

The Four Families of Performance Measures (3)

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( ) ( ) ( ) ( )

3 4 1 2 1 2

1 1 1 2 k k

  • p

p p p p p

r f r d r r f r d r

τ τ

τ τ

− −∞ −∞

    = − − − × −            

∫ ∫

1

τ

3

τ

4

τ

1

  • 2
  • 1

k

2

k

( )

1 2 2 4 2 1 3 1

1 1 , , , , , ,

p p

k k r

  • r
  • PES

PES

τ τ τ τ − −

    = − × −      

1 2 1 3 1 2 4 2

1 1 , , , , , ,

,

p p

k k r

  • r
  • GHPM

GLPM

τ τ τ τ −

    = ×    

2

τ

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SLIDE 26

The Omega measure (Keating and Shadwick, 2002) where

corresponds to the return on a portfolio p and

  • is the risk-free rate.

Interpretation: it compares the potential gains

  • f

the managed portfolio over its potential losses, both defined according to a threshold

( )

1 , , ,1 , , ,1

, , , , ,1,1,1,1 ,

p p

p r r p

O GHPM GLPM r

τ τ τ τ

τ τ τ τ

−

    = ×     = H

A Survey on the Four Families

  • f Performance

Measures

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  • G. Jannin
  • F. Lisi
  • B. Maillet

The Four Families of Performance Measures (3)

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SLIDE 27

The Main Density-based Performance Measures*

2 1 3 6 4

A Survey on the Four Families

  • f Performance

Measures

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Authors Name Rescaled Performance Downside Performance

Bernardo and Ledoit (2000) Gain-Loss ratio Farinelli and Tibiletti (2008) One-sided Risk Measure Biglova et al. (2004) Rachev ratio Biglova et al. (2004) Generalized Rachev ratio

The Four Families of Performance Measures (3)

27 / 35

*where

is the risk-free rate and HB with I J 1,2 corresponds to intensification constants.

( )

p

r

+

P

( )

p

r

−

P

, , ,1

p f f

r r r

GHPM

, , ,1

p f f

r r r

GLPM

1 1

1 , , ,

p

  • r

r r o

GHPM    

2 2

1 , , ,

p

  • r

r r o

GHPM    

( )

3

, ,

p f

r r

ES

τ −

−

4

, ,

p f

r r

ES

τ

−

( )

3 1

, , ,

p f

r r

  • PES

τ −

−

4 2

, , ,

p f

r r

  • PES

τ

−

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SLIDE 28

The Main Limits

  • f

Density-based Performance Measures

Performance measures are highly related to the threshold (risk free rate, Minimum Acceptable Return or Value-at-Risk) The link with the utility function of the studied agent is not straightforward Density are subject to model risk

A Survey on the Four Families

  • f Performance

Measures

  • M. Caporin
  • G. Jannin
  • F. Lisi
  • B. Maillet

The Four Families of Performance Measures (3)

28 / 35

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SLIDE 29

General Form of Utility-based Performance Measures (expressed in return per unit of util) where are the returns, is the expectation operator,

L ∙

is a value (or utility) function, M ∙ is a specific function that depends upon the performance of the investor’s portfolio and is a specific threshold return. Objective: evaluating the portfolio performance from explicit representative value (or utility) functions.

2 1 3 5 4

( )

{ },

p p

PM E U r τ   = −  

  • G

A Survey on the Four Families

  • f Performance

Measures

  • M. Caporin
  • G. Jannin
  • F. Lisi
  • B. Maillet

The Four Families of Performance Measures (4)

29 / 35

( )

⋅ G

( )

U ⋅ ( )

⋅

p

r τ

slide-30
SLIDE 30

The Morningstar Risk-Adjusted Return (Morningstar, 2002) where ∙ is the expectation operator,

are the returns of

a portfolio p, N is the risk aversion coefficient of the investor (with N O 0). Interpretation: incorporating the behavior

  • f

the agent, through a Power Utility Function, for assessing the portfolio performance, given a risk aversion coefficient

2 1 5

( )

12

1 1,

A A p p

MRAR E r

− −

  = + −    

A Survey on the Four Families

  • f Performance

Measures

  • M. Caporin
  • G. Jannin
  • F. Lisi
  • B. Maillet

The Four Families of Performance Measures (4)

30 / 35

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SLIDE 31

The Main Utility-based Performance Measures*

2 1 3 6 4

A Survey on the Four Families

  • f Performance

Measures

  • M. Caporin
  • G. Jannin
  • F. Lisi
  • B. Maillet

Authors Name Transformed Performance Value (or utility) Function Stutzer (2000) Performance Index

argmax

UVWX

∗

ZH ∙ [0\ ]

A

Kaplan (2005) Lambda measure

argmax

UVWX

∗

∙ ]

A

^

Ingersoll et al. (2007) Manipulation- Proof Performance Measure

1 N ∆9 Z ∙ 1

1

  • `

The Four Families of Performance Measures (4)

31 / 35

( ) ⋅ G

( )

p

U r τ −

*where is a constant, is a penalty function, is the frequency of observations and is the risk aversion coefficient.

] ^ ∆9 N

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SLIDE 32

The Main Limits

  • f

Utility-based Performance Measures

Strongly dependent on the utility function (exponential, power

  • r logarithmic) characterizing the behavior of the studied

investor Highly sensitive to the investor’s attitude towards risk through the risk aversion coefficient (which can be time-varying) Rankings obtained with these measures are directly related to the benchmark (risk free rate or proxy)

A Survey on the Four Families

  • f Performance

Measures

  • M. Caporin
  • G. Jannin
  • F. Lisi
  • B. Maillet

The Four Families of Performance Measures (4)

32 / 35

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SLIDE 33

2 1 3 5 6 4

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

  • f performance measures, in which most of the performance

measures fits

A Survey on the Four Families

  • f Performance

Measures

  • M. Caporin
  • G. Jannin
  • F. Lisi
  • B. Maillet

33 / 35

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SLIDE 34

2 1 3 5 6 4

Extensions

On our on-going research agenda, we have planned to add new performance measures (30 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 the Four Families

  • f Performance

Measures

  • M. Caporin
  • G. Jannin
  • F. Lisi
  • B. Maillet

34 / 35

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SLIDE 35

2 1 3 5 6 4

A Website dedicated to Performance Measurement…

A Survey on the Four Families

  • f Performance

Measures

  • M. Caporin
  • G. Jannin
  • F. Lisi
  • B. Maillet

35 / 35

See more on www.performance-metrics.eu

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SLIDE 36

We are grateful to Christophe Boucher, Georges Hübner, Patrick Kouontchou, Constantin Mellios, Patrice Poncet and Hélène Raymond for help and encouragement in preparing this work.

Bertrand B. Maillet

Thank you for your attention... “A Survey on the Four Families

  • f Performance Measures”

30th International French FInance Association Conference

  • EM Lyon, May 2013 -