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

1. On the (ab)use of Omega?
SYstemic Risk TOmography:
Signals, Measurements, Transmission Channels, and
Policy Interventions
M.Caporin,University ofPadova (Italy)
M. Costola,Ca'Foscari University ofVenice (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.

2. 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 measurement
with two main motivations
the capture stylized facts of financial returns such as asymmetry or
non-Gaussian density,
to evaluate managed portfolio in asset allocation.

3. Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion
Literature Review
The interest on this research field started from the seminal
contribution of Sharpe in (1966) and an increasing number of
studies 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 and
Sommersguter-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 specific
non-subjective criterion (mutual funds).

4. Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion
Literature Review
A large number of performance measures have already been
proposed and, as a consequence, related different ranks.
The identification of the most appropriate performance measure
depends on several elements, in particular the preferences of the
investor and the properties or features of the analyzed
assets/portfolios returns.
Furthermore, the choice of the “optimal” performance measure
depends on the purpose of the analysis (an investment decision, the
evaluation of manager’s abilities, the identification of management
strategies and of their impact, either in terms of deviations from the
benchmark or in terms of returns or risks).
Despite some limitations, the Sharpe (1966) ratio is still considered
as the reference performance measure (Hodges’s paradox).

5. Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion
Literature Review
In recent years, different authors widely used the performance measure
introduced 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 the
well-known Sharpe ratio, supporting their choice by the
non-Gaussianity of returns and by the inappropriateness of volatility
as 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 et
al., 2013; Hamidi et al., 2013).
As criterion function for portfolio optimization in order to introduce
downside 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).

6. 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 is
inconsistent with the Strict Second-order Stochastic Dominance
(SSSD).
2 Observe that the trade-off between return and risk, corresponding to
the Omega measure, may be essentially influenced by the mean
return.
3 Illustrate in static and dynamic frameworks that Omega optimal
portfolios can be associated with traditional optimization paradigms
depending on the chosen threshold used in the computation of
Omega.
4 Present some robustness checks on long-only asset and hedge fund
datasets.

7. 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 measures
based 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 and
left part of the support of the density of returns.
In most cases, measures belonging to this class can be re-defined as
ratios of two Power Expected Shortfalls (or Generalized Higher/Lower
Partial 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)

8. Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion
The Omega ratio
The Keating and Shadwick (2002) Omega measure, following Bernardo
and Ledoit (2000), corresponds to the case where
τ1 = 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/Lower
Partial Moments and the conditional expectation operator.
Intuitively, the Omega ratio separately considers favorable and
unfavorable potential excess returns with respect to a threshold that has
to be given (arbitrary).

9. 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 the
features of the return distribution (moments including skewness and
kurtosis).
⇒ A ranking is theoretically always possible, whatever the threshold (in
contrast 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 in
τ ∈ R,
for any couple of portfolios p = {A, B}, ΩA(·) = ΩB (·) ∀τ ∈ R, if
and only if FA(·) = FB (·), where functions Fp(·) is the CDF of the
returns on a portfolio p.
for any portfolio p (if there exists one risk-free asset p = 0 with
return r0), Ωp(·) ≤ Ω0(·) ∀τ ≤ r0, with Ω0(·) = +∞.

10. 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 may
vary according to investment objectives and individual preferences.
As mentioned by Unser (2000), we are often only interested in an
evaluation of outcomes which are “risky” thus reflecting the attitude
towards downside risk.
Usually, their values are smaller than a given target, which, for
example, is the riskless rate or the rate of a financial index
(benchmark).

11. 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 is
misleading (and not compatible) with a rational behavior.
We use a simplified frameworks, such as the Gaussian one, but also
under less stringent hypotheses on the features of the risky asset
return density.
In order to introduce the Omega “curse”, we use the definition of
consistency 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 B
then A ≤ρ B,
is inferior-consistent with the SSD criterion if and only if: A ≤ρ B
then A SSD B,
is consistent with the SSD criterion if and only if ρ is both strictly
superior- and inferior- consistent with the SD criterion.

12. 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 as
E(rA) = E(rB ).
Even though he might be confident that the Fund A is less risky
than Fund B, he would not be able to conclude that the investors
would necessarily agree with his choice, i.e. A is preferred to B.
For a measure that is strictly inferior-consistent, he would have this
certainty.

13. 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 return
distribution of Fund B has twice the volatility of that of Fund A.

14. Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion
SISSD
Proposition: Consider returns on two assets A and B, with
non-degenerated densities, such as the asset A Second-order
Stochastically 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 Stochastic
Dominance Inconsistent (SISSDI) when conditions are met.
Proof: see the paper.

15. 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 of
the portfolio A are strictly lower that its relative losses in terms of
cumulative densities.
In the case 2, Fund B is Omega-preferred because relative gains of
the portfolio B are strictly higher that its relative losses with regard
to cumulative densities.
In the case 3, Fund A is chosen by Omega since relative gains of
Fund A are strictly higher than its losses in terms of cumulative
densities.
This means that the choice of funds according to Omega is directly
dependent on the threshold.

16. 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 a
density of the same (elliptical) family, with identical means
E(rA) = E(rB ) = µ, but different volatilities such as σ(rA) < σ(rB ), the
Strict Second-order Stochastic Dominance implies that the ranking in
terms 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 Stochastic
Dominance Inconsistent (SISSDI in cases 2 and 3).
Proof: see the paper.

17. 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 it
determines the ranking of the funds and reflects her investment
choice,
In particular, when studying the case of two symmetric densities
(simple illustration of Gaussian laws) such as E(rA) = E(rB ) = µ, we
may face to an irrational ordering when the chosen threshold is high.
We can also show that even in a more complex setting based on two
asymmetric and leptokurtic (lognormal) densities (with some
uncertainty), the results remains strictly identically.
The Omega ratios of two (or several) funds, which are characterized
by similar mean returns but different volatilities, will be equal.
This latter fact leads us to a more general study on the trade-off between
expected return and risk when Omega is driving allocation and
investment choices.

18. 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 levels
for 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 extra
over-performance for the same amount of over-volatility to reverse
the fund rankings obtained with the Omega measure,
for a threshold equal to 10.00%, we only require 100 basis points of
extra over-performance for 400 basis points of over-volatility (greedy
agent),
lower the threshold and the closer decisions using the Sharpe ratio
and the Omega criterion.

19. 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(σ2
m + v2
p )]∆t + (σm˜+ υp ˜η
√
∆t)
where µm is the market portfolio return, αp is the extra-performance
generated by the manager, σm is the market portfolio total risk, υp
corresponds to the residual portfolio specific risk, ∆t is the data
frequency, ˜ 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 them
according their mean,
we compute the associated Sharpe ratios and Omega measures for
each threshold and determine how often these measures conclude as
the ordering given by their mean. mean returns.

20. 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 for
Fund A than for Fund B (second column) and the frequency to which the
Omega measure concludes the opposite (third to fifth columns) according
to several thresholds: 10.00%, 5.00% and .00%.

21. 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 the
Omega criterion with other classical paradigms, in a static and dynamic
way.
empirical illustrations of properties based on, first, realistic
simulations
secondly, on three different market databases used in the literature
on portfolio optimization (namely Hentati and Prigent, 2010;
Darolles et al., 2009; DeMiguel et al., 2009).

22. Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion
An Illustration of a Misleading Choice of Ω

23. Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion
Performance of the 5 indexes - Hentati and Prigent (2010)

24. Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion
Performance of 5 Ptfs Optimized (Static Analysis)

25. Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion
Performance of 5 Ptfs Optimized (Static Analysis)

26. Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion
Performance of 5 Ptfs Optimized (Dynamic Analysis)

27. Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion
Ω Ptfs Optimized with different τ (Static Analysis)

28. Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion
Ω Ptfs Optimized with different τ (Dynamic Analysis)

29. Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion
Robustness Check

30. Introduction The Ω ratio Inconsistency of Ω Ω as an Optimization Criterion? Conclusion
Conclusion
The Omega ratio appears strongly sensitive to the threshold and
thus may yield to obvious contradictions.
The Omega criterion can lead to obvious under-optimization
solutions in some realistic cases.
On some databases, investment strategies based on Omega do not
add real values compared to other classical paradigms
On some other datasets, the Omega-based optimal portfolio is
similar to:
a Maximum mean return-based optimal portfolio for a high threshold;
a Minimum volatility-based optimal portfolio for a low threshold.

31. This project has received funding from the European Union’s
Seventh Framework Programme for research, technological
development and demonstration under grant agreement n° 320270
www.syrtoproject.eu