On the (ab)use of Omega? - Caporin M., Costola M., Jannin G., Maillet B. December 15, 2013.

On the (ab)use of Omega?

SYstemic Risk TOmography:Signals, Measurements, Transmission Channels, and Policy Interventions

M. Caporin, University of Padova (Italy)M. Costola, Ca' Foscari University of Venice (Italy) G. Jannin, A.A.Advisors-QCG (ABN AMRO), Variances and Univ. Paris-1 Pantheon-Sorbonne (PRISM)B, Maillet, dA.A.Advisors-QCG (ABN AMRO), Variances, Univ. La Reunion and Orleans (CEMOI, LEO/CNRS and LBI)

CFE - ERCIM 2013 – London (UK). December 15, 2013.

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

The Use of Performance Measures

The financial economics literature focuses on performance measurementwith two main motivations

the capture stylized facts of financial returns such as asymmetry ornon-Gaussian density,

to evaluate managed portfolio in asset allocation.

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Literature Review

The interest on this research field started from the seminalcontribution of Sharpe in (1966) and an increasing number ofstudies appeared in the following decades (Caporin et al., 2013).

Performance evaluation of active management; Cherny and Madan(2009), Capocci (2009), Darolles et al. (2009), Jha et al. (2009),Jiang and Zhu (2009), Zakamouline and Koekebakker (2009),Darolles and Gourieroux (2010), Glawischnig andSommersguter-Reichmannn (2010), Jones (2010), Billio et al.(2012a), Billio et al. (2012b), Cremers et al. (2012).

Performance evaluation has relevant implications:

it allows us to understand agent choices,

ranking assets or managed portfolios according to a specificnon-subjective criterion (mutual funds).

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Literature Review

A large number of performance measures have already beenproposed and, as a consequence, related different ranks.

The identification of the most appropriate performance measuredepends on several elements, in particular the preferences of theinvestor and the properties or features of the analyzedassets/portfolios returns.

Furthermore, the choice of the “optimal” performance measuredepends on the purpose of the analysis (an investment decision, theevaluation of manager’s abilities, the identification of managementstrategies and of their impact, either in terms of deviations from thebenchmark or in terms of returns or risks).

Despite some limitations, the Sharpe (1966) ratio is still consideredas the reference performance measure (Hodges’s paradox).

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Literature Review

In recent years, different authors widely used the performance measureintroduced by Keating and Shadwick (2002): The Omega.It is defined as a ratio of potential gains out of possible losses.

In the evaluation of active management strategies in contrast to thewell-known Sharpe ratio, supporting their choice by thenon-Gaussianity of returns and by the inappropriateness of volatilityas a risk measure when strategies are non-linear and active (e.g.Eling and Schuhmacher, 2007; Annaert et al., 2009; Hamidi et al.,2009; Bertrand and Prigent, 2011; Ornelas et al., 2012; Zieling etal., 2013; Hamidi et al., 2013).

As criterion function for portfolio optimization in order to introducedownside risk in the estimation of optimal portfolio weights (e.g.Mausser et al., 2006; Farinelli et al., 2008, 2009; Kane et al., 2009;Hentati and Prigent, 2010; Gilli and Schumann, 2010; Gilli et al.,2011).

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Aim of the Paper

We analyse the relevance of such approaches. In particular,

1 Show through a basic illustration that the Omega ratio isinconsistent with the Strict Second-order Stochastic Dominance(SSSD).

2 Observe that the trade-off between return and risk, corresponding tothe Omega measure, may be essentially influenced by the meanreturn.

3 Illustrate in static and dynamic frameworks that Omega optimalportfolios can be associated with traditional optimization paradigmsdepending on the chosen threshold used in the computation ofOmega.

4 Present some robustness checks on long-only asset and hedge funddatasets.

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

A General Class

The Omega measure belongs to a general class of performance measuresbased on features of the analyzed return density.Following Caporin et al. (2013),

PMP = P+(rp)×[P−(rp)

]−1, (1)

where P+(·) and P−(·) are two functions associated with the right andleft part of the support of the density of returns.In most cases, measures belonging to this class can be re-defined asratios of two Power Expected Shortfalls (or Generalized Higher/LowerPartial Moments), which reads:

PMP = H(rp, τ1, τ2, τ3, τ5, o1, o2, k1, k2)

= [−E (|τ1 − rp|o1 |τ1 − rp < τ3)](k1)−1

×

[−E (|τ2 − rp|o2 |rp − τ2 < τ4)](k2)−1.

(2)

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

The Omega ratio

The Keating and Shadwick (2002) Omega measure, following Bernardoand Ledoit (2000), corresponds to the case whereτ1 = τ2 = τ3 = τ4 = τ and o1 = o2 = k1 = k2 = 1.Therefore,

Ωp(τ) = E (|rp − τ ||rp > τ)× [E (|rp − τ ||rp > τ)]−1

= (GHPMrp,τ,τ,1)× (GLPMrp,τ,τ,1)−1

= H(rp, τ, τ, τ, τ, 1, 1, 1, 1),

(3)

where GHPMrp,τ,τ,1 and GLPMrp,τ,τ,1 are, respectively, the Higher/LowerPartial Moments and the conditional expectation operator.

Intuitively, the Omega ratio separately considers favorable andunfavorable potential excess returns with respect to a threshold that hasto be given (arbitrary).

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

The Omega ratio

The main advantage of the Omega measure is that it incorporates all thefeatures of the return distribution (moments including skewness andkurtosis).

⇒ A ranking is theoretically always possible, whatever the threshold (incontrast to the Sharpe Ratio).

Furthermore, it displays some properties such as (see Kazemi et al., 2004;Bertrand and Prigent, 2011):

for any portfolio p (with a symmetric return distribution),Ωp(τ) = 1 when τ = E (rp),

for any portfolio p, Ωp(·) is a monotone decreasing function inτ ∈ R,

for any couple of portfolios p = A,B, ΩA(·) = ΩB(·) ∀τ ∈ R, ifand only if FA(·) = FB(·), where functions Fp(·) is the CDF of thereturns on a portfolio p.

for any portfolio p (if there exists one risk-free asset p = 0 withreturn r0), Ωp(·) ≤ Ω0(·) ∀τ ≤ r0, with Ω0(·) = +∞.

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

The Omega ratio

In this setting, the threshold τ must be exogenously specified as it mayvary according to investment objectives and individual preferences.

As mentioned by Unser (2000), we are often only interested in anevaluation of outcomes which are “risky” thus reflecting the attitudetowards downside risk.

Usually, their values are smaller than a given target, which, forexample, is the riskless rate or the rate of a financial index(benchmark).

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Inconsistency of the Omega when Ranking Funds

The use of Omega that leads investors to create risky asset rankings ismisleading (and not compatible) with a rational behavior.

We use a simplified frameworks, such as the Gaussian one, but alsounder less stringent hypotheses on the features of the risky assetreturn density.

In order to introduce the Omega “curse”, we use the definition ofconsistency in the sense of the Strict Stochastic Dominance (SSD,in short).

SSD, Danielsson et al. (2008)

A risk measure denoted ρ,

is superior-consistent with the SSD criterion if and only if A SSD Bthen A ≤ρ B,

is inferior-consistent with the SSD criterion if and only if: A ≤ρ Bthen A SSD B,

is consistent with the SSD criterion if and only if ρ is both strictlysuperior- and inferior- consistent with the SD criterion.

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Inconsistency

Let us suppose that a risk manager relies on a risk measure denoted ρ,which is not SSD inferior consistent.

We assume that he has the choice between a Fund A and a Fund B,which are characterized by identical mean returns such asE (rA) = E (rB).

Even though he might be confident that the Fund A is less riskythan Fund B, he would not be able to conclude that the investorswould necessarily agree with his choice, i.e. A is preferred to B.

For a measure that is strictly inferior-consistent, he would have thiscertainty.

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Simulation of two Gaussian Density Functions

Both Funds have exactly the same average daily return, but the returndistribution of Fund B has twice the volatility of that of Fund A.

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

SISSD

Proposition: Consider returns on two assets A and B, withnon-degenerated densities, such as the asset A Second-orderStochastically Dominates the asset B (noted A SSD B).The ranking in terms of Omega of the two assets satisfies the following:

1 If∫ +∞−∞ [FB(x)− FA(x)]dx = 0, a necessary and sufficient condition

for having A ≺Ω B is that∫ +∞τ

[1− FA(x)]dx −∫ +∞τ

FA(x)dx < 0;

2 If∫ +∞−∞ [FB(x)− FA(x)]dx > 0, a sufficient condition for having

A ≺Ω B is that∫ +∞τ

FB(x)dx −∫ +∞τ

[1− FB(x)]dx < 0;

3 If∫ +∞−∞ [FB(x)− FA(x)]dx = 0, a sufficient condition for having

A Ω B is that∫ +∞τ

[1− FA(x)]dx −∫ +∞τ

FA(x)dx > 0;

and thus the Omega criterion is Strict Inferior Second-order StochasticDominance Inconsistent (SISSDI) when conditions are met.Proof: see the paper.

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Ω and SISSDI

We notice that the Omega measure is SISSDI in the first two cases,

In the case 1, Fund B is Omega-preferred because relative gains ofthe portfolio A are strictly lower that its relative losses in terms ofcumulative densities.

In the case 2, Fund B is Omega-preferred because relative gains ofthe portfolio B are strictly higher that its relative losses with regardto cumulative densities.

In the case 3, Fund A is chosen by Omega since relative gains ofFund A are strictly higher than its losses in terms of cumulativedensities.

This means that the choice of funds according to Omega is directlydependent on the threshold.

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Inconsistency under Symmetry and Equality of Means

If the two assets mentioned in the Proposition have returns with adensity of the same (elliptical) family, with identical meansE (rA) = E (rB) = µ, but different volatilities such as σ(rA) < σ(rB), theStrict Second-order Stochastic Dominance implies that the ranking interms of Omega of the two assets will be:

1 A Ω B, if τ < µ,

2 A ≈Ω B, if τ = µ,

3 A ≺Ω B, if τ > µ,

and thus the Omega criterion is Strict Inferior Second-order StochasticDominance Inconsistent (SISSDI in cases 2 and 3).Proof: see the paper.

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Preliminary Conclusion

The Omega ratio is inconsistent in the sense of the SSSD.

the threshold should be linked to the agent’s preferences since itdetermines the ranking of the funds and reflects her investmentchoice,

In particular, when studying the case of two symmetric densities(simple illustration of Gaussian laws) such as E(rA) = E(rB) = µ, wemay face to an irrational ordering when the chosen threshold is high.We can also show that even in a more complex setting based on twoasymmetric and leptokurtic (lognormal) densities (with someuncertainty), the results remains strictly identically.

The Omega ratios of two (or several) funds, which are characterizedby similar mean returns but different volatilities, will be equal.

This latter fact leads us to a more general study on the trade-off betweenexpected return and risk when Omega is driving allocation andinvestment choices.

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Iso-Omega curves for various thresholds τ

For a given τ , an Iso-Omega curve corresponds to identical Omega levelsfor various portfolios characterized by different µ and σ2.

when the threshold is equal to .00%, the trade-off is very close to 1(as for the Sharpe ratio), it requires 100 basis points of extraover-performance for the same amount of over-volatility to reversethe fund rankings obtained with the Omega measure,for a threshold equal to 10.00%, we only require 100 basis points ofextra over-performance for 400 basis points of over-volatility (greedyagent),lower the threshold and the closer decisions using the Sharpe ratioand the Omega criterion.

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Contradiction in choice when E (rB) ≤ E (rA)

Following Goetzmann et al.(2007) we simulate,

rp = exp

[µm + αp − .5(σ2m + v2

p )]∆t + (σm ε+ υp η√

∆t)

where µm is the market portfolio return, αp is the extra-performancegenerated by the manager, σm is the market portfolio total risk, υpcorresponds to the residual portfolio specific risk, ∆t is the datafrequency, ε and η are Gaussian random variables.We define four different profiles of investors,

αp = .00% and υp = .20%.

αp = 1.00% and υp = .20%, 2.00%, 20.00%.Then,

we randomly choose two portfolios among all these and order themaccording their mean,

we compute the associated Sharpe ratios and Omega measures foreach threshold and determine how often these measures conclude asthe ordering given by their mean. mean returns.

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Contradiction in choice when E (rB) ≤ E (rA)

This table displays the frequency to which the Sharpe ratio is higher forFund A than for Fund B (second column) and the frequency to which theOmega measure concludes the opposite (third to fifth columns) accordingto several thresholds: 10.00%, 5.00% and .00%.

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Omega as an Optimization Criterion?

We compare the performance of portfolios optimized according to theOmega criterion with other classical paradigms, in a static and dynamicway.

empirical illustrations of properties based on, first, realisticsimulations

secondly, on three different market databases used in the literatureon portfolio optimization (namely Hentati and Prigent, 2010;Darolles et al., 2009; DeMiguel et al., 2009).

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

An Illustration of a Misleading Choice of Ω

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Performance of the 5 indexes - Hentati and Prigent (2010)

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Performance of 5 Ptfs Optimized (Static Analysis)

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Performance of 5 Ptfs Optimized (Static Analysis)

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Performance of 5 Ptfs Optimized (Dynamic Analysis)

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Ω Ptfs Optimized with different τ (Static Analysis)

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Ω Ptfs Optimized with different τ (Dynamic Analysis)

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Robustness Check

Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion

Conclusion

The Omega ratio appears strongly sensitive to the threshold andthus may yield to obvious contradictions.

The Omega criterion can lead to obvious under-optimizationsolutions in some realistic cases.

On some databases, investment strategies based on Omega do notadd real values compared to other classical paradigms

On some other datasets, the Omega-based optimal portfolio issimilar to:

a Maximum mean return-based optimal portfolio for a high threshold;a Minimum volatility-based optimal portfolio for a low threshold.

This project has received funding from the European Union’s Seventh Framework Programme for research, technological

development and demonstration under grant agreement n° 320270

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