on the Ω‐ratio
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Applied Mathematics & Statistics frey@ams.sunysb.edu
OntheΩ‐Ratio
RobertJ.Frey
AppliedMathematicsandStatistics
StonyBrookUniversity
29January2009
15April2009(Rev.05)
2
OntheΩ‐Ratio
RobertJ.Frey
AppliedMathematicsandStatisticsStonyBrookUniversity
29January2009
15April2009(Rev.05)
ABSTRACT
Despite the fact that the Ωratio captures the complete shape of the underlying return
distribution, selecting a portfolio bymaximizing the value of theΩratio at a given return
threshold does not necessarily produce a portfolio that can be considered optimal over a
reasonablerangeofinvestorpreferences.Specifically,thisselectioncriteriontendstoselecta
feasible portfolio that maximizes available leverage (whether explicitly applied by the
investororrealizedinternallybythecapitalstructureoftheinvestmentitself)and,therefore,
doesnottradeoffriskandrewardinamannerthatmostinvestorswouldfindacceptable.
3
TableofContents
1. TheΩRatio.................................................................................................................................... 4
1.1 Background ............................................................................................................................................5
1.2 TheΩ LeverageBias............................................................................................................................7
1.3. ASimpleExampleoftheΩLB ..........................................................................................................8
2. ImplicationsoftheΩLB ............................................................................................................. 9
2.1. Example:Normalvs.LeptokurtoticandNegativelySkewed ............................................. 10
2.2 Example:Normalvs.Bimodal........................................................................................................ 13
2.3 TheΩLBforFactorReturnsandOtherReplicatingStrategies.......................................... 15
3. Conclusions ..................................................................................................................................17
Bibliography ........................................................................................................................................19
4
1. TheΩ ‐Ratio
In Keating and Shadwick (2002a) and (2002b) a new performancemeasure, the
Omegaratio(Ω‐ratio),isintroduced“whichcapturestheeffectsofallhighermomentsfully
and which may be used to rank and evaluate manager performance” (Keating and
Shadwick, 2002a, p. 4). Keating’s and Shadwick’s basic criticisms of many current
performance measures are difficult to argue with, and in particular, “that mean and
variance cannot capture all of the risk and reward features in a financial return
distribution”(KeatingandShadwick,2002b,p.2).
ThestandardapplicationoftheΩ‐ratioistoselectareturnthresholdandthenrank
investmentsaccording to thevalueof theirΩ‐ratios at that threshold,withhigher ratios
preferred to lower. They state, “No assumptions about risk preferences or utility are
necessary”(KeatingandShadwick,2002b,p.4).Thismethodologyisextendedtoportfolio
optimization in a straightforwardway: Select a return threshold and then construct the
portfolio with the highestΩ–ratio at that threshold. Their claim is that theΩ‐ratio is a
universalperformancemeasurebecauseitrepresentsacompletedescriptionofthereturn
distribution.
Keating and Shadwick (2002b, p. 10) note that any invertible transformation
applied to the returndistributionwill yield another valid distribution fromwhich anΩ‐
ratio can be constructed. Such transformations can be used as alternatives to utility
functions.Itisdifficulttosee,however,howsuchatransformationendsupbeingdifferent
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orsuperiortoadirectapplicationofutilitytheory.Thisideaisnotdevelopedfurthernor
doesthereappeartobeanyfurtherdevelopmentofitintheavailableliterature.
Anumberofpapershaveexploredtheapplicationof theΩ‐ratiotoselecting from
among or creating portfolios of investments whose return characteristics are not
adequately captured within the tradition mean‐variance framework. See, e.g., Barreto
(2006),Betrand(2005),Favre‐BulleandPache(2003),Gillietal. (2008)andTogherand
Barsbay(2007).
These studies, however, focus on examples that best illustrate the Ω‐ratio’s
strengths. True understanding of anymeasure demands thatwe also understand how it
breaksdown.ThenotionthattheΩ‐ratio“avoidstheneedforutilityfunctions,”asstatedin
KeatingandShadwick(2002ap.4),isfartoostrongastatement.Aswillbeshowninwhat
follows,theΩ‐ratiohasaseriousbiasthatcanleadtoexcessiverisktaking.
1.1 Background
TheΩ‐ratio is theratioofprobabilityweightedgainsand lossesaboutaspecifiedreturn
threshold.LettherandomvariableXrepresentthereturnofaparticularinvestment,µXits
meanandFXitsCDF.Forthesakeofexpositionalsimplicityinwhatfollows,assumethatFX
is continuous and strictly increasing over the range –∞ > x>∞. Keating and Shadwick
(2002a,p.8and2002b,p.3)definetheΩ‐ratioofXatthresholdτasshowninEquation1.
ΩX[τ ] =1− FX (x)( )dx
τ
∞
∫
FX (x)dx−∞
τ
∫
Equation 1 - The Omega Ratio
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AndagraphicalrepresentationoftheΩ‐ratiofollowsbelowinFigure1.
Figure 1 - Graphical Representation of the Omega Ratio
Forallpracticalpurposes,theΩ‐ratioisaninvertibletransformoftheCDF.Keating
andShadwick(2002a,pp.10‐12)derivesomeofitsimportantproperties.Inparticular,
that
• ΩX[µX]=1;
• ΩX[x]isamonotonedecreasingfunctionfrom(–∞,∞)to(∞,0);
• ΩX[x]isanaffineinvariantofthereturnsdistribution,i.e,ifY=a+bX,then
ΩY[a+bτ]=ΩX[τ].
AplotoftheΩ‐ratioforaStandardNormaldistributionappearsbelowinFigure2.
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Figure 2 - Omega Ratio of the Standard Normal Distribution
Severalresearchershaveinvestigateditsproperties.Kazemi,SchneeweisandGupta
(2003)showthattheΩ‐ratiocanbewrittenastheratioofacalloptiontoaputoptionwith
strikesatthethreshold.KaplanandKnowles(2004)developamoregeneralizedmeasure,
theKapparatio(Κ‐ratio),whichsubsumestheΩandSortinoratiosasspecialcases.
1.2 TheΩ LeverageBias
LetthereturnofanassetbearandomvariableXwithCDFFX,meanµXandstandard
deviation∞ > σX> 0. Assume that there exists a risk‐free asset with return rf, that an
investorcanlendorborrowatthatrateandthat0<rf,<µX.
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Denotetheleverageλastheamountinvestedintheriskyassetwith(1–λ)invested
orborrowedatrfanddesignatetherandomvariableofthereturnoftheleveragedposition
asYwithCDFFY.Note that the term leverage isused inageneralwaythatencompasses
placingsomecapital in cash,λ<1,being fully invested,λ=1,orborrowing to financea
positiongreaterthanone’scapital,λ>1.
ThemeanandstandarddeviationofYareλµX+(1–λ)rfandλσX, respectively.Of
course,bothhavethesameSharperatio.TheCDFsofXandYarerelated;viz.,FY[λx+(1–λ)
rf]=FX[x].Consequently,wecanassertthatΩY[λτ+(1–λ)rf]=ΩX[τ],and,hence,thesingle
crossingpointisΩY[rf]=ΩX[rf].
Considerthecaseforλ>1andτ>rf.ThenFX[τ]>FY[τ].Thisinturnimpliesthatthe
numerator ofΩX[τ] is less than the numerator ofΩY[τ] and the denominator ofΩX[τ] is
greater than the denominator ΩY[τ]. Thus, ΩX[τ] < ΩY[τ]. Analogous results hold for
alternaterestrictionsonλandτ.
Thus, foragiven investmentX, an investorwhouses theΩ‐ratiowitha threshold
higherthantherisk‐freerateasaselectioncriterionalwaysprefersmoreleverageonXto
less, independentofthedistributionofX.ThisiscalledtheΩ leveragebias(ΩLB) inwhat
follows.
1.3. ASimpleExampleoftheΩLB
Let the risk‐free return rf = 0.03 and consider an investment with a Normally
distributed return with mean µ = 0.10 and standard deviation σ = 0.12. Assume we
leverage3:2,i.e.,λ=1.5.AplotoftherespectiveΩ‐ratiosoftheunleveragedandleveraged
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investmentsappearsbelowinFigure3.Asexpected,thecrossoveroccursatrfafterwhich
the leveraged investment dominates the unleveraged one. Most investors would select
thresholdsgreaterthantherisk‐freerate;hence,maximizingtheΩ‐ratiowoulddrivemost
investorstomaximizetheleverageoftheirpositions.
Figure 3 - Effects of Leverage on a Normally Distributed Investment
2. ImplicationsoftheΩLB
One may well counter that the whole point of the Ω‐ratio is to select “better” return
distributions. In theexample above the leveragedandunleveraged investments areboth
Normallydistributed.Inaportfoliooptimizationinwhichleverageisnotafactormightnot
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theΩ‐ratiostillselectaportfoliowhosereturndistributionisthebesttrade‐offofup‐and
downsideperformancevisàvistheinvestor’sreturnthreshold?
Theanswer isno,orat leastnotnecessarily.Givenmultiple investmentstrategies,
thosewithotherwise lessdesirabledistributions can improve theirΩ‐ratioby takingon
moreleverage.Unfortunately,thisincreaseinleveragecanalsomeangreaterexposureto
thoseaspectsoftheinvestment’sreturnthatmadeitlessdesirable.
Leveragemayoccurinternallywithinafundorcompany;therefore,aninvestorwho
isnotexplicitlyallowingleverageattheportfolio levelmayneverthelessfindit implicitly
represented in the candidates available. Thus, the ΩLB can drive portfolio selection
towardschoicesthathavehighlevelsofleverage,whetherinternalorexternal.
Finally, if we consider a factormodel as a suitable representation of the returns
across a set of investments, then higher factor exposures, which closely resemble the
effectsofleverage,alsotendtoresultinhigherΩ‐ratiosbecauseoftheΩLB.Theseissues
willbeexploredinthefollowingsections.
2.1. Example:Normalvs.LeptokurtoticandNegativelySkewed
WewillfollowtheapproachfollowedinseveralintroductorypapersontheΩ‐ratio
anduseittocomparedifferentreturndistributions.Considerthreepotentialinvestments:
A,BandC.Aplotoftheirreturns’PDFsisbelowinFigure4.
The returns of A are Normally distributed with mean µA = 0.1175 and standard
deviationσA =0.1047.The returnsofB are representedby a finitenormalmixturewith
component weights wB = (0.95, 0.05)T, component means µB = (0.13, –0.12)T, and
componentstandarddeviationsσB=(0.085,0.15)T.ThemeanandstandarddeviationofB
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arethesameasthatofA.InvestmentCisaleveragedversionofBwithλ=1.5,achievedby
borrowingattherisk‐freeraterf=0.03.Thissituationcanariseif,e.g.,BandCrepresent
different share classes of the same underlying hedge fund, or if B and C are separate
investmentsthatfollowsimilarstrategieswithCrunningatahigherrisk‐level.
Figure 4 - PDFs of Investments A, B and C
ThemeanandstandarddeviationofBarethesameasthatofA.InvestmentsBandC
arenegativelyskewedandleptokurtotic.AllthreeinvestmentshavethesameSharperatio.
A plot of the lower tails of the CDFs below in Figure 5 clearly illustrates dramatic
differencesindownsiderisk.Theprobabilityofalossthatisgreaterthan40%isabout1%
forCbutitisnegligibleforAandB.
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Figure 5 - CDFs of Loss Tails of Investments A, B and C
Forthresholdsgreaterthanroughly0.05,acomparisonoftheΩ‐ratiosofAandB,
shownbelowinFigure6,doesnotindicateamarkedpreferenceforAorB.Forthresholdsτ
≥0.037,CdominatesbothA andB.Now it is certainlypossible thatsome investorswith
modestreturnthresholdsmayprefertoleverageupB toC,despitethefactthatitslower
tailismateriallyfatterthanthatofA’s;however,itisnottenabletoassertthatallwould.In
fact,manywouldbeterrifiedofdoingso.
InvestmentswithlessdesirabledistributionscanimprovetheirΩ‐ratiobytakingon
more leverage. Thus, the ΩLB can drive portfolio selection towards choices that have
undesirable distributions but high levels—whether external or internal—of leverage.
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Figure 6 - Omega Ratios of Investments A, B and C
2.2 Example:Normalvs.Bimodal
A second, more extreme example explores this further. Consider three potential
investments:D,EandF.ThereturnsofDareNormallydistributedwithmeanµD=0.1and
standarddeviationσD=0.155.ThereturnsofEcanbemodeledasafinitenormalmixture
with component weights wE = (0.5, 0.5)T, component means µE = (0.25, –0.05)T, and
componentstandarddeviationsσE=(0.04,0.04)T.InvestmentFisaleveragedversionofE
withλ =1.5, achievedbyborrowingat the risk‐free rate rf =0.03.All three investments
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have the same Sharpe ratio. A plot comparing the distributions ofD, E and F is below.
Figure 7 - PDFs of Investments D, E and F
Theplotbelowcompares theΩ‐ratiosof these threealternatives. It shows thatD
and E interweave with one another, although for thresholds τ<0.1 D is the clear
preference. As expected, the leveraged F dominates its unleveraged counterpart E for
thresholdsτ>rf.However,italsodominatestheNormallydistributedDforthresholdsτ>
0.054.
TheeffectofleveragingupEtoFtakesadistributionwithmodesat–0.05and0.25
and produces onewithmodes –0.09 and 0.36. Using theΩ‐ratio as a selection criterion
wouldleadinvestorswithmodestthresholdstoselectahighlyleveragedinvestmentwith
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considerabledownsiderisk.FormanyinvestorsapreferenceforFoverDandEmaymake
sense,but, as is the case for theexample in theprevious section, it isnot likely tomake
senseforallofthem.Thesingleparameteravailabletoinvestors,thethreshold,isn’tmuch
useinarticulatingdifferentinvestors’preferencesinthisregard.
Figure 8 - Omega Ratios of Investments D, E and F
2.3 TheΩLBforFactorReturnsandOtherReplicatingStrategies
Investment returns can typically be replicated by their exposures to a limited
number of factors plus and uncorrelated idiosyncratic component. In cases in which
traditional factormodels have not proven adequate, researchers have found forms of so
called alternative or exotic beta—such as hedge fund or commodity indices (Litterman
16
2005)—ordynamictradingstrategies—whichusetraditionalinvestmentstoreplicatethe
propertiesofalternativeinvestments(KatandPalaro,2005,2006aand2006b).Thus,even
whennon‐linearitiesormultipleregimesareinvolved,aswithalternativeinvestmentsor
direct investment inderivatives, the returns canoftenbemodeledbya smallnumberof
basis functions, perhaps including indices or trading strategies, qua factor returns.
HeightenedfactorexposurescanhaveaneffectontheΩ‐ratiosimilartothatofincreased
leverage.
Consider,forexample,asinglefactormodelssuchastheCapitalAssetPricingModel
(CAPM).UndertheCAPM,investmentreturnscanbemodeledbytheirexposuretoasingle
market factor.Letxi(t)be thereturnof investment i at timet,βi theexposureof i to the
systematicmarketreturnm(t)andεi(t)anidiosyncraticrandomerrorthatisuncorrelated
withm(t)andotherinvestments’errors(i.e.,Cov[m(t),εj(t)]=0andCov[εi(t),εj(t)]=0for
j≠i). The CAPM asserts that xi (t) − rf = βi (m(t) − rf ) + εi (t) . Rearranging terms yields
xi (t) = βim(t) + (1− βi )rf + εi (t) . Thus, except for the idiosyncratic component, the
relationshipbetweenxiandmisthesameasthatofthereturnofmleveragedbyafactorλ
=βi.
Once a moderate level of diversification has been achieved the impact of the
idiosyncraticrisksisdiversifiedaway.Undertheseconditionsandforthresholdsτ >rf,one
would expect that the ΩLB would drive portfolio construction relentlessly towards a
portfoliowithhighβ exposure. Inmorecomplexmulti‐factormodels theeffectwouldbe
thesameandwouldoccuracrossthemultiplefactorspresent.
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3. Conclusions
The work of Kaplan and Knolwes (2003)mentioned earliermay help to provide
additionalinsightintothoseconditionsinwhichtheΩ‐ratiomaybreakdown.Theyshow
that theΩ‐ratio is related toΚ1‐ratio (the ratio of excess expected return to threshold
dividedbythefirstlowerpartialmomentwithregardtothreshold)asshowninEquation2.
ΩX[τ ] = Κ1[τ ]+1 =µX − τ
(x − τ )dF[x]−∞
τ
∫+1
Equation 2 - Relationship Between the Omega and Kappa Ratios
Atagiventhresholdτ,theΩ‐ratioobscuresmuchofthedownsidetailstructure.Itis
difficult toseehowpickingτestablishesan investor’srisk‐rewardpreferencescompared
with,forexample,definingautilityfunction.Whileisittruethatbyvaryingτonecangain
moreinsight,onehastowonderifamorenaturalandintuitiveinsightcouldnotbegained
byadirectcomparisonofthereturndistributionsthemselves.
TheΩ‐ratio is not a “universal performancemeasure” as claimed in Keating and
Shadwick (2002a). To be sure, other measures, such as the Sharpe ratio, have their
deficiencies. The Sharpe ratio, however, is neutral towards leverage; it does not actively
seekitout.
Unfortunately, formodestreturnthresholds, theΩLBcandriveportfoliostowards
investments with more highly leveraged capital structures even when those choices
involvesignificantdownsiderisk.Thismaybewhatsomeinvestorssometimeswant,butit
isdifficulttoarguethatitiswhatallinvestorsalwayswant.
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AMathematica®(Version7)notebookcontainingallofthecode,computationsand
graphicsproducedinthispapercanbefoundat http://www.ams.sunysb.edu/~frey/Research/.
Included are some useful tools for computing andworkingwithΩ‐ratios. The notebook
alsocontainsadditionalexamplesnotusedinthispaper.
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Bibliography
Barreto, Susan L., 2006, “From Omega to Alpha” A New Ratio Arises,” Inside Edge, Hedgeworld.
http://www.s4capital.com/Hedgeworld%20Article%20011706%20-%20Omega.pdf
Betrand, Jean-Charles, and Damien Berlemont, 2005, “In the land of the hedge fund, the Sharpe ratio is no longer king,” Sinopia, HSBC Global Asset Management.
http://www.sinopia-group.com/internet/Sinop2007/docs/NewRiskIndicator.pdf
Favre-Bulle, Alexandre, and Sébastien Pache, 2003, “The Omega Measure: Hedge Fund Portfolio Optimization,” MBF Master’s Thesis, University of Lausanne—Ecole des HEC.
http://www.performance-measurement.org/FBPa03.pdf
Gilli, Manfred, Enrico Schumann, Giacomo di Tollo and Gerda Calej, (2008), “Constructing Long/Short Portfolios with the Omega ratio,” Research Paper Series No. 08-34, Swiss Finance Institute,
Kaplan, Paul D., and James A. Knowles, 2003, “Kappa: A Generalized Downside Risk-Adjusted Performance Measure,” Performance-Measurement.Org.
www.performance-measurement.org/KaKn04.pdf.
Kat, Harry M., and Helder P. Palaro, 2005, “Who Needs Hedge Funds? A Copula-Based Approach to Hedge Fund Return Replication,” Alternative Investment Research Centre Working Paper Series, Cass Business School, City University, London.
Kat, Harry M., and Helder P. Palaro, 2006a, “Replication and Evaluation of Fund of Hedge Fund Returns,” Alternative Investment Research Centre Working Paper Series, Cass Business School, City University, London.
Kat, Harry M., and Helder P. Palaro, 2006b, “Replication and Evaluation Hedge Fund Returns Using Futures,” Alternative Investment Research Centre Working Paper Series, Cass Business School, City University, London.
Kazemi, Hossein, Thomas Schneeweis and Raj Gupta, 2003, “Omega as a Performance Measure,” CISDM, Isenberg School of Management, Amherst, MA.
http://cisdm.som.umass.edu/research/pdffiles/omega.pdf
Keating, Con, and William F. Shadwick, 2002a, “A Universal Performance Measure,” The Finance Development Centre, London.
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http://www.isda.org/c_and_a/pdf/GammaPub.pdf
Keating, Con, and William F. Shadwick, 2002b, “An Introduction to Omega,” The Finance Development Centre, London.
Litterman, Bob, 2005, “Active Alpha Investing,” Open Letter to Investors, Goldman Sachs Asset Management, NY.
http://www2.goldmansachs.com/gsam/pdfs/USI/education/aa_beyond_alpha.pdf
Togher, Steve, and Tambe Barsbay, 2007, “Fund of Hedge Funds Portfolio Optimization Using the Omega Ratio,” The Monitor, Investment Management Consultants’ Association.
http://www.fortigent.com/uploads/published_articles/07Monitor%20JulyAug%20Togher-Barsbay.pdf
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