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A meta-measure of performance related to both investors and investments characteristics

A Correction to this article was published on 26 August 2021

This article has been updated

Abstract

We introduce hereafter a new flexible meta-measurement of portfolio performance, called the Generalized Utility-based N-moment measure, relying both on a characterization of the whole return distribution and on the set of preferences of the investor, which is adapted to analyze the performance of hedge funds. It could also serve as the basis of a Fraudulent Behavior Index aiming to detect fraudulent funds.

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Change history

Notes

  1. See also Markowitz (2014) for a complete survey.

  2. See Kolm et al. (2014) for the main trends on applications in operational research for portfolio optimizations.

  3. See also, for instance, Cherny and Madan (2009), Capocci (2009), Darolles et al. (2009), Jha et al. (2009), Jiang and Zhu (2009), Stavetski (2009), Zakamouline and Koekebakker (2009), Darolles and Gouriéroux (2010), Glawischnig and Sommersguter-Reichmannn (2010), Billio et al. (2012, 2013, 2015), Joenväärä et al. (2013), Cremers et al. (2013), Smetters and Zhang (2014), Kadan and Liu (2014), Ferson and Lin (2014) and Brown et al. (2010). To our best knowledge, the most comprehensive works dedicated to performance assessment, published in the two last decades, are those of Knight and Satchell (2002), Amenc and Lesourd (2003), Aftalion and Poncet (2003), Le Sourd (2007), Bacon (2008a, b), Cogneau and Hübner (2009a, b), Fischer and Wermers (2012), and Caporin et al. (2014).

  4. See, for instance, the controversy about the consistency of the Sharpe and Omega ratio and the second-order stochastic dominance criterion (Cf. Hodges 1998; Fong 2016; Balder and Schweizer 2017; Klar and Müeller 2018; Caporin et al. 2018; Bi et al. 2019).

  5. See Kimball (1990, 1992, 1993) and Scott and Horvath (1980).

  6. And when we also know that the Keating and Shadwick (2002) Omega measure, in some sense, disregards risk—see Caporin et al. (2018).

  7. Called Gram–Charlier type A, Edgeworth, Mac Laurin or Taylor expansion, depending on the field, context, ways of regrouping terms, and reference function or distribution.

  8. See, e.g., Jarrow and Rudd (1982), Corrado and Su (1996), Bakshi et al. (1997), Bakshi and Madan (2000), Jondeau and Rockinger (2001), Jurczenko et al. (2004), Martin et al. (2005), Lim et al. (2006), Corrado (2007), León et al. (2009), Andreou et al. (2010), Jha and Kalimipalli (2010), Tanaka et al. (2010), Del Brio and Perote (2012), Schlögl (2013), Chateau (2014) and Lin et al. (2015).

  9. See, e.g., Arrow (1964), Feldstein (1969), Samuelson (1970), Jean (1971, 1973), Arditti and Levy (1972), Rubinstein (1973), Borch (1974), Ingersoll (1975), Loistl (1976), Simkowitz and Beedles (1978), Levy and Markowitz (1979), Scott and Horvath (1980), Pulley (1983), Kroll et al. (1984), Dittmar (2002), Briec et al. (2007) and Martellini and Ziemann (2010). See also Jurczenko and Maillet (2006) for further references.

  10. We refere here to the survey by Jurczenko and Maillet (2006) on the theoretical foundations of a mean–variance–skewness–kurtosis decision criterion, discussing the conditions of convergence and the potential drawbacks of expansions.

  11. We refere here to the survey by Jurczenko and Maillet (2006) on the theoretical foundations of a mean–variance–skewness–kurtosis decision criterion, discussing the conditions of convergence and the potential drawbacks of expansions.

  12. Please see our Web Appendix C (available on demand to the authors) for a list of the main utility functions expressed with the first four moments of returns and a table of elements’ decomposition. Note here that \( U_{i}^{\left( 0 \right)} \left( . \right) = U_{i}^{{}} \left( . \right) \) per convention, and that the term in \( U_{i}^{\left( 1 \right)} \left( . \right) \) disappears since E [rp − m1, p (rp)] = 0 per definition. Pre-multiplying the first term (in \( U_{i}^{{}} \left( . \right) \)) by m1,p(rp)1(for a non null mean), leads to the artificial but compact form of the following Eq. (5) with (1) a sum of terms with n = 1 to N, whilst (2) the sum of terms goes from n = 0 to N in Eqs. (1) and (3) the Jensen inequality residual term in Eq. (10) below such as:

    \( \tilde{\xi }_{p} = \mathop \sum \limits_{n = 2}^{N} \left( {n!} \right)^{ - 1} U_{i}^{\left( n \right)} \left[ {E\left( {r_{p} } \right)} \right]\left[ {r_{p} - E\left( {r_{p} } \right)} \right]^{n} + \tilde{\varepsilon }_{N + 1} \left( {r_{p} } \right), \)

    with a sum of terms going from n = 2 to N.

  13. In our Web Appendix K (available on demand to the authors), we have checked the accuracy of this expression with the main utility functions, using the first four moments of returns and a coefficient of risk aversion (a) equal to 3.

  14. See our Web Appendix C (available on demand to the authors) for a decomposition of special cases of the HARA class, such as utility functions belonging to the CARA and CRRA classes.

  15. Some interesting related works, however, also show that the ratio \( \left[ {U_{i}^{\left( 3 \right)} \left( . \right)/U_{i}^{\left( 1 \right)} \left( . \right)} \right] \) is also linked to a risk aversion characteristic of a rational agent, who makes an arbitrage between the first and the third moments (Cf. Crainich and Eeckhoudt 2008).

  16. Lajeri-Chaherli (2004) proposes an expansion to the order five, mentioning the fifth-order risk as being the “edginess”, whilst Caballé and Pomansky (1996) refine even further the expansion to the N-th order, referring to the “risk aversion of order N”, as an analogue to the traditional classical Absolute Risk Aversion (see also Eeckhoudt and Schlesinger 2006).

  17. Please refer to our Web Appendix C (available on demand to the authors) for a decomposition to the fourth moment of the most common utility functions.

  18. Because the measure is merely applied both to homogeneous investments and for similar individuals.

  19. There also exists a link between the new Generalized Utility-based N-moment measure of performance and the Cumulative Prospect Theory when considering the investor’s sensitivities as modified (subjective) probabilities associated with the distribution of non-distorted returns.

  20. Our measure, following Billio et al. (2013), shares some similarities with the (Nth order) General Ranking Measure (see Smetters and Zhang 2014, p. 9).

  21. See Eq. (8) for the precise expressions of the λn,i in the case of a HARA-type of utility function (and to our Web Appendix C—available on demand to the authors, for a decomposition to the fourth moment of the most common utility functions.

  22. Note here that these measures belong to one of the four main families of performance measures identified in the survey of Caporin et al. (2014).

  23. See our Web Appendix B (available on demand to the authors) for a full sketch of the algorithm.

  24. Results for an illustration of Iso-MPPM curves when considering over-skewness and over-performance are available upon request.

  25. Table 1 is an exact replication of Table 5 (on page 1534) in Ingersoll et al. (2007), using their simulation scheme. See also our Web Appendix E for more details (available on demand to the authors).

  26. We first start by defining a “neutral” agent for whom the scalar products, respectively, between the first four sensitivities and the first four average moments of the studied sample are strictly identical. Secondly, we specify four different categories of investors characterized by a high sensitivity to only one of the four moments (ceteris paribus). More precisely, we have a greedy investor, denoted GUN4,G,p, who is focused on the mean, a risk-averse agent, named GUN4,RA,p, with a high sensitivity to the variance, a prudent one, called GUN4,P,p, characterized by a significant preference to the third moment and a very temperate investor, alias GUN4,T,p, who severely dislikes the fourth moment (Cf. Caporin et al. 2014, for the definition of the different performance measures used in the following tables).

  27. Using the same hypothesis defined in Ingersoll et al. (2007), we compute the Jensen (1968) alpha assuming a systematic risk sensitivity of informed and uninformed managers’ portfolios equal to 1.

  28. Table 2 is also an exact replication of Table 6 (on page 1535) in Ingersoll et al. (2007), using their simulation scheme. See also our Web Appendix F (available on demand to the authors) for more details.

  29. The 30 ranked portfolios correspond, as a mere illustration, to a random sample of 15 informed and 15 uninformed managers whose portfolio returns respect the simulation scheme defined in Table 1 (Cf. Ingersoll et al. 2007). For each measure of performance in this table, the value and rank of funds are reported in the various columns of Panel A. Funds are sorted according to their Sharpe ratios (first two columns). Then, for the other measures (following columns), related relative ranks are presented. For instance, the highest Sharpe ratio fund is also the best, according to the Jensen, MPPM2 and MPPM3 measures, whilst it is the third fund in terms of Sharpe, which the best according to the MPPM4 measure, and the 20th fund for the MPPM5. In this sample also, the 30th fund is the worst fund, whatever the measure.

  30. The 30 ranked portfolios correspond to 15 informed and 15 uninformed managers whose portfolio returns respect the simulation scheme defined in Table 1 (Cf. Ingersoll et al. 2007).

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Acknowledgements

We are grateful to Philippe Bertrand, Stephen Brown, Massimiliano Caporin, Benjamin Hamidi, Georges Hübner, Christophe Hurlin, Leónid Kagan, Robert Kosowski, Patrick Kouontchou, Jan Pieter Krahnen, Patrice Poncet, Jean-Luc Prigent, Thierry Roncalli, Olivier Scaillet, Christian Schlag, Kent Smetters and Spyridon Vrontos for suggestions, help and encouragements when preparing this article. We are also grateful to the participants to the Vth Conference on Computational and Financial Econometrics, the XIth Conference on Advances in Financial Econometrics, the Quantitative Finance Seminar at the University of Padua (Padova, April 2013), the VIIth IRMC, and the Finance Seminar at the SAFE Goethe University of Frankfurt (Frankfurt, October 2014) for valuable comments. We also greatly thank here Gregory Jannin for some previous intensive collaborations on this topic, and Zhining Yuan for research assistance. Pelizzon also thanks the Leibniz Institute for Financial Research SAFE for financially sponsoring this research. A previous preliminary version circulated under the title “Portfolio Performance Measure…” (Cf. Billio et al. 2013). Resources linked to this article are available on: www.performance-metrics.eu. The general disclaimer applies.

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Billio, M., Maillet, B. & Pelizzon, L. A meta-measure of performance related to both investors and investments characteristics. Ann Oper Res 313, 1405–1447 (2022). https://doi.org/10.1007/s10479-020-03771-w

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Keywords

  • Performance measurement
  • Hedge funds
  • Higher-moments
  • Statistical expansion