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.

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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

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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

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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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