A Survey on the Four Families of Performance Measures

Transcription

1 Massimiliano Caorin 1 Grégory M. Jannin Francesco Lisi 3 Bertrand 4 1 Deartment of Economics and Management Marco Fanno, University of Padova massimiliano.caorin@unid.it A.A.Advisors-QCG (ABN AMRO), Variances and University of Paris-1 (PRISM) gregory.jannin@univ-aris1.fr 3 Deartment of Statistical Sciences, 4 A.A.Advisors-QCG (ABN AMRO), Variances University of Padova lisif@stat.unid.it and University of Orléans (LEO/CNRS, PRISM and EIF) bertrand.maillet@univ-orleans.fr Cluster Risques Financiers - University of Orléans, Aril 01 - The fourth author thanks the Eurolace Institute of Finance for financial suort. This resentation engages only its authors and does not necessarily reflect the oinions of their emloyers. The usual disclaimers aly.

2 Agenda Motivation Literature The Preliminary Conclusions Extensions / 6

3 Motivation More than 100 ways to measure ortfolio erformance 1 Definition of the four main families of measures General exressions for each family Identification and exlicit formulations for the more reresentative measures (and their close variants in Aendix) Intuition for each measure with some short criticisms Codes (MatLab, R, ) available soon on: 3 / 6

4 Literature (1/) Main and Recent Surveys about Performance Measurement Cogneau P. and G. Hübner, (009a), The (More Than) 100 Ways to measure Portfolio Performance. Part 1: Standardized Risk-adjusted, Journal Measurement 13(4), Cogneau P. and G. Hübner, (009b), The (More Than) 100 Ways to measure Portfolio Performance. Part : Secial and Comarison, Journal Measurement 14(1), Some Books about Performance Measurement Aftalion F. and P. Poncet, (003), Les techniques de mesure de erformance. Economica, 140 ages. Bacon C., (008), Practical Portfolio Performance. Wiley Finance Series, 384 ages. Cobbaut R., R. Gillet and G. Hübner, (011), La gestion de ortefeuille instruments, stratégie et erformance. De Boeck, 50 ages. 4 / 6 Knight J. and S.E. Satchell, (00), Performance Measurement in Finance. Butterworth- Heinemann, 365 ages.

5 Literature (/) The Four Reresentative Performance Share W., (1966), Mutual Fund Performance, Journal of Business 39(1), Jensen M., (1968), The Performance of Mutual Funds in the Period , Journal of Finance 3(), Keating C. and W. Shadwick, (00), A Universal Performance Measure, Journal Measurement 6(3), Morningstar, (00), Morningstar: The New Morningstar Rating TM Methodology, Morningstar Research Reort /04/0, 0 ages. 5 / 6

6 The (1) General Form of Relative Performance (most often exressed in return er unit of risk) PM P r R r r where are the (rescaled) returns, P is a function that deends uon the observed erformance, and R is a corrected risk measure of the investor s ortfolio under study, such as R R. 1, 6 Objective: exressing the observed (rescaled) erformance of the managed ortfolio er unit of risk 6 / 6

7 The (1) The Reward-to-variability ratio (Share, 1966) 1 1, S E r rf r where is the exectation oerator, and are resectively the returns and the total risk of the ortfolio, and is the risk-free asset. Interretation: this measure evaluates the comensation earned by the ortfolio manager, as gauged by the exected excess return er unit of ortfolio total risk 7 / 6

8 The (1) The Main Share-based Performance ** Authors 1 Name Rescaled Performance Morey and Vinod (001) 3 Zakamouline and Koekebakker (009) 4 Double Share Ratio Adjusted for Skewness Share Ratio P r Risk Measure R ݎ ݎ ܧ Standard Deviation ݎ ݎ ܧ Standard Deviation r Correction Coefficient c ߪ r ଵ 1 ቄͳ ଷǡ ଷ ݎ 3 ଵ ଵ ଵ ݎ ܧ ݎ ଵ ଶቋ ߪ Israëlsen (005) Reward-toabsolute-excess return Ratio ݎ ݎ ܧ Standard Deviation 1 ] ݎ ሻ ݎሺܧሾ ݏ 1^ Share (1994) 6 Information Ratio ݎ ݎ ܧ Tracking Error - Treynor (1965) Reward-tovolatility Ratio - Beta ݎ ݎ ܧ 8 / 6 **where ଷǡ corresonds to the investor s relative references for skewness, ଷ ݎ is the skewness of the underlying return distribution and ݏ is the sign function.

9 9 / 6 The (1) (198) Performance based on Other Risk 1 Yitzhaki 3 6 Authors Name Rescaled Performance Darolles et al. (009) Konno and Yamazaki (1991) Caorin 4 and Lisi (011) Young (1998) Dowd (000)* Sortino and Satchell (001) Martin and McCann (1989)** Corrected Risk Measure Gini Ratio ܧ ݎ ݎ Gini coefficient L-erformance Mean Absolute Deviation Ratio TL-moment of order 1 TL-moment of order ܧ ݎ ݎ Mean Absolute Deviation Reward-to-range Ratio ܧ ݎ ݎ Range Minimax Ratio ܧ ݎ ݎ Minimax Portfolio Reward-to Value-at- Risk Ratio Reward-to o-lower Partial Moments Ratio Ulcer Performance Index ܧ ݎ ݎ Value at Risk ܧ ݎ ݎ Lower Partial Moment of order o ܧ ݎ ݎ Average Squared Weekly Drawdowns *See also Favre and Galeano (00) who use the modified Value-at-Risk and Martin et al. (003) who refers to the Conditional Value-at-Risk. **See also Burke (1994), Young (1991) and Kestner (1996) which resectively based their measures on the Total Squared Monthly Drawdowns, the Maximum Drawdown and the Average Yearly Maximum Drawdowns. P r R r

10 The (1) Some Main Limits of Relative Performance Rankings are consistent only if ortfolio returns are ellitically distributed and/or the reresentative 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, 00; Amin and Kat, 003a and 003b; Gregoriou and Gueyie, 003) 10 / 6 *excet for Zakamouline and Koekebakker (009).

11 The () General Form of Absolute Performance 1(exressed in return) th PM, Γ P r P r, 3 where 6 is a transformation function (generally linear), 4are the (rescaled) returns, P the observed erformance and th P is a function that deends uon is a function that is related 5 to the theoretical erformance of a model reference ortfolio, conditionally to a set of information denoted. r 11 / 6 Objective: comaring the observed (rescaled) erformance of the managed ortfolio to its theoretical erformance, considering a model

12 The () 1 The Alha (Jensen, 1968) J E r r Er r, f m f,m where is the exectation oerator, are the returns of the ortfolio, are the returns of the market ortfolio m, is the risk-free rate, is the sensitivity of investor s ortfolio returns with resect to market ortfolio returns. 6 Interretation: this measure assesses the extra erformance, realized by the ortfolio manager, given its sensitivity to systematic risk 1 / 6

13 The () Fama (197) The Main Jensen-tye Performance * Authors 1 Name Rescaled Performance P r Theoretical Performance Net Selectivity Index ݎሺܧ ሻ ݎ ݎ ܧ ݎ ߪ ଵ ߪ P th r Black (197) 3 Treynor 4and Mazuy (1966) Henriksson and Merton (1981) Zero-beta CAPM Market Timing Model Parametric Market Timing Model ǡ௭ ߚ ௭ ݎ ܧ ݎ ܧ ௭ ݎ ܧ ݎ ܧ ݎ ݎ ܧ ଵǡǡ ߚ ݎ ݎ ܧ ݎ ݎ ܧ + ଶ ଶǡǡߚ ݎ ܧ ଵǡǡ ߚ ݎ ݎ ݎ ܧ ݎ ݔ ݎ ܧ ǡͳ ߚ ଶǡǡ 13 / 6 Connor and Korajczyk (1986) 6 Ferson and Schadt (1996) Multi-factor Model Multi-factor Conditional Model ǡ ߚ ܨ ܧ ݎ ݎ ܧ ଵ ݐ ǡ ߚ ܨ ܧ ݎ ݎ ܧ *where ݎ ௭ are the returns of the zero-beta ortfolio, ߚ ଵǡǡ and ߚ ଶǡǡ are the selectivity and the market timing coefficients, ܨ is the k-th loading factor and ߚ ǡ ݐ is the sensitivity of the ortfolio to k-th the loading factor at time t. ଵ

14 The () Other Miscellaneous Absolute Performance * 1 Grinblatt 3 and Titman Authors Name Performance Measure Henriksson and Merton (1981) Moses et al. (1987) (1989) Non-arametric Market Timing Model Diversification-adjusted Alha Measure Positive Period Weighting Measure th r, P P r ଵ ݐ ଶ ݐ ͳ ߙ ߟ ǡ ଵ ݓ ௧ ݎ ǡ௧ ݎ ௧ ଵ Modigliani 4 and Modigliani (1997) Risk-Adjusted Performance Measure ܧ ݎ ͳ ߣ ǡ ݎ ߣ ǡ Cantalui and Hug (000) Efficiency Ratio ܧ ݎ ݎ ܧ ݎ ݎ ଵ Muralidhar (001) 6 Correlation-Adjusted Performance ܧ ݎ ݓ ܧ ݎ + ݎ ͳ ݓ ݓ ݓ Muralidhar (00) Skill, History And Risk- Adjusted Measure { ܧ ݎ ݓ ܧ ݎ ݓ + ݎ ͳ ݓ ݓ 14 / 6 *where ଵ ݐ and ଶ ݐ are the robabilities (res. u and down) of correct forecasts of the ortfolio manager about market variations, ߟ ǡ evaluates the diversification remium earned by the manager, ߣ ǡ corresonds to the diversification level of the investor s ortfolio comared to that of the market ortfolio, ܥ deends on the correlation coefficient between the investor s ortfolio and his benchmark and ݓ is linked to the length of observations.

15 The () The Main Limits of Absolute Performance Performance comuted with these measures are strongly influenced by the reference ortfolio (market ortfolio or roxy) Almost all of them assumes stability of the systematic risk * sensitivities of the investor s ortfolio over time Disregards skewness and kurtosis of the studied ortfolio returns which may alter rankings when using investment ** strategies based on derivative instruments 15 / 6 * excet for Ferson and Schadt (1996). ** excet for Hwang and Satchell (1998).

16 The (3) General Form of Density-based Performance (exressed in scalar terms no unit) PM P r P r 1, r where are the (rescaled) returns, P is a function that deends uon the observed erformance and P is a measure focusing on a secific (left) art of the suort of the density of returns. Objective: comaring the observed (rescaled) erformance of the managed ortfolio to an exression deending on its losses 16 / 6

17 The (3) PM Most of the Density-based Performance can be exressed such as: H,,,, o, o, k, k k o1 o 1 ES 1, * o 1 ES * o 3, xr, 1 4, x r, 1 1 k / 6 x * r, i r i where corresonds to the ortfolio order statistics in excess of a threshold for comuting gains (for i = 1) and losses or risk (for i =), 3 is a threshold secifying the right art of the suort of the return density (i.e. gains) and 4 is another threshold associated with the left art (i.e. losses), o1 ando are intensification constants, k1 andk are normalizing constants.

18 The (3) The Omega measure (Keating and Shadwick, 00) 1, O E r r E r r where E is the conditional exectation oerator, corresonds to the return on a ortfolio and threshold... is a Interretation: it comares the otential gains of the managed ortfolio over its otential losses, both defined according to a threshold 18 / 6

19 The (3) The Main Density-based Performance * 1 Authors Name Rescaled Performance Bernardo and Ledoit 3 (000) Gain-Loss ratio P r Downside Performance P, 0 ݎ ݎ ݔ ܧ ǡͳ ݎ ݎ ݔ ܧ r Farinelli 4and Tibiletti (008) One-sided Risk Measure ݎ ݎ భ ܯ ܪ ଵ ݎ ݎ మ ܯ ܮ భ 1 మ ଵ మ Biglova et al. (004) Rachev ratio ݎ ܧ ݎ ݎห ݎ భ ǡ ݎ ܧ ݎ ݎห ݎ మ ǡ 6 Biglova et al. (004) Generalized Rachev ratio 1 భ ݎ ݎ ܧ భ ݎห ݎ భ ǡ 1 మ ݎ ݎ ܧ మ ݎห ݎ మ ǡ 19 / 6 *where ݎ is the risk-free rate, ܣ ܯ is the Minimum Accetable Return, with 1, corresonds to intensification constants.

20 The (3) The Main Limits of Density-based Performance Performance measures are highly related to the threshold (risk free rate, Minimum Accetable Return or Value-at-Risk) The link with the utility function of the studied agent is not straightforward Density are subject to model risk 0 / 6

21 The (4) 1 3 General Form of Utility-based Performance (exressed in return er unit of util) where are the (rescaled) returns, is the exectation oerator, V is a value (or utility) function and is a 4 5 r secific function that deends uon the erformance of the investor s ortfolio. PM G E V r, G Objective: evaluating the ortfolio erformance from exlicit reresentative value (or utility) functions. 1 / 6

22 The (4) 1 The Morningstar Risk-Adjusted Return (Morningstar, 00) 1 A 1 A MRAR E r, 5 where is the exectation oerator, are the returns of a ortfolio, (with ) is the risk aversion coefficient of the studied investor. Interretation: incororating the behavior of the agent, through a Power Utility Function, for assessing the ortfolio erformance, given a risk aversion coefficient / 6

23 The (4) The Main Utility-based Performance * 1 Authors Name Transformed Performance 3 Stutzer (000) G Value (or utility) Function Performance Index ఏఢԹ ష V r Kalan (005) 4 Lambda measure ఏఢԹ ష Gemmill et al. (006) Loss-Averse Performance ଵ ଶ ଵ ଵ ଶ భ మ Ingersoll 6 et al. (007) Maniulation- Proof Performance Measure ଵ ଵ ଵ *where ߠ is a constant, ߜ is a enalty function, ଵ and ଶ are constants, ݐο is the frequency of observations and ܣ is the risk aversion coefficient. 3 / 6

24 The (4) The Main Limits of Utility-based Performance Strongly deendent on the utility function (exonential, ower or 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 roxy) 4 / 6

25 Preliminary Conclusions 1 We roose a Survey of erformance measures in a uniform and comrehensive framework through four main families clearly identified, namely relative, absolute, density-based and utilityrelated erformance measures 3 We define these main families according to two essential criteria: 4 the unit in which it is exressed, and the way the measure is built 5 We formulate a general exression for each of the four categories of erformance measures, in which most of the erformance 6 measures fits 5 / 6

26 Extensions 1 On our on-going research agenda, we have lanned to add new erformance measures (30 extra or so), in order to comlete our actual Survey 3 Next ste will be to erform some emirical alications in order to comare the roerties of collected measures (rank correlation 4 tests, ersistence, lucky versus star funds studies, etc.) 5 We would like also to comlement our intuitions about a new erformance measure (called Generalized Performance Measure ), 6which can be seen as a generalization of the four main categories resented in this aer. 6 / 6

27 Grégory M. Jannin We are grateful to Christohe Boucher and Patrick Kouontchou for hel and encouragement in rearing this work. Thank you for your attention... Cluster Risques Financiers - University of Orléans, Aril 01 -