Algorithmic portfolio tilting to harvest higher moment gains

(1)

Contents lists available atScienceDirect

Heliyon

www.elsevier.com/locate/heliyon

Research article

Algorithmic portfolio tilting to harvest higher moment gains

Kris Boudt

^a,b,c,∗

, Dries Cornilly

^d

, Frederiek Van Holle

^e

, Joeri Willems

^e

aUniversiteitGent,Sint-Pietersplein5,9000Gent,Belgium

bVrijeUniversiteitBrussel,Pleinlaan2, 1050Brussel,Belgium

cVrijeUniversiteitAmsterdam,DeBoelelaan1105, 1081Amsterdam,Netherlands

dSYZAssetManagement,RueduCommerce3, 1204Genève,Switzerland

eDegroofPetercamAssetManagement,Guimardstraat18,1040Brussel,Belgium

A R T I C L E I NF O A B S T R A C T

Keywords:

Mean-variance-skewness-kurtosis Non-normality

Portfolioallocation Tilting

Statistics Finance Banking Econometrics Operationsmanagement Business

Economics Informationscience Industry

Many financial portfolios are not mean-variance-skewness-kurtosis efficient. We recommend tilting these portfoliosin adirectionthatincreasestheirestimatedmeanandthirdcentralmomentanddecreasestheir varianceandfourthcentralmoment.Theadvantagesoftiltingcomeatthecostofdeviationfromtheinitial optimalitycriterion.Inthispaper,weshowtheusefulnessofportfoliotiltingappliedtotheequally-weighted, equal-risk-contributionandmaximumdiversificationportfoliosinaUCITS-compliantassetallocationsetting.

0. Introduction

Many of the popular risk-based portfoliosare optimized accord- ingtoanoptimalitycriterion thatdoesnottakeintoaccount higher order portfolioreturn moments(see e.g.Lee(2011)).Typical exam- ples from the literature on risk-based asset allocation include the equally-weightedandequal-risk-contributionportfolios(Maillardetal.

(2010)).Afurtherexampleisthemaximumdiversificationportfolioof ChoueifatyandCoignard(2008),whichmaximizestheratiobetween theweightedaveragestandarddeviationoftheportfoliocomponents andtheportfoliostandarddeviation.These portfoliosmaystillbeef- ficient for investors withmean-variancepreferences, asdiscussed in ArdiaandBoudt(2015).Theyare,however,likelytobeinefficientfor investorswhosepreferencesdependonthehigherordermomentsofthe portfolioreturndistribution.

*

Correspondingauthorat:UniversiteitGent,Sint-Pietersplein5,9000Gent,Belgium.

E-mailaddress:[email protected](K. Boudt).

1 Thisapproachofoptimizingnon-standardizedmomentsisconsistentwiththeresultsofScottandHorvath(1980) whichsupportpreferencesforthirdandfourth centralmomentsinsteadofthestandardizedskewnessandexcesskurtosis.Weusetiltinginthesenseofover- orunderweightingassetscomparedtobenchmark portfolios (seee.g.DarsinosandSatchell(2004)).WereferthereadertoStutzer(2000,2003),HaleyandMcGee(2006),HaleyandWhiteman(2008) andHaleyand McGee(2011) foranalternativedeﬁnitionoftiltinginaportfoliocontext,namelymodifyingtheprobabilityreturndistribution.

Severalsolutionshavebeenproposedintermsofportfolioalloca- tionmethods that take thehigher order moments intoaccount (see e.g.Jondeauetal.(2007) andBoudtetal.(2012)).Inthispaper,we studytheproblemofimprovingportfoliosthatareinefficientfroma mean-variance-skewness-kurtosis(MVSK)perspective,whereskewness andkurtosisaredefinedasthenon-standardizedthirdandfourthcen- tralportfoliomoments.Wedosobytiltingsuchinefficientportfoliosto anallocationthatismoreMVSKefficient,whilekeepingthegoodprop- ertiesofthe initialportfolioweights.¹ Theproposedmethodology is inspiredbytheshortageminimizationframeworkofBriecetal.(2007).

Weexpect MVSKportfolio tiltingtobe especially usefulin asset allocation.Wethereforeillustratethemethodologybytiltingthreecom- monrisk-basedassetallocationportfoliosinaUCITS-compliantsetting, wheretheinvestmentuniverseconsistsof12broadassetclassindices.

Weﬁnd thatthe tiltedrisk-based portfoliosachieve superiorout-of-

https://doi.org/10.1016/j.heliyon.2020.e03516

Received2December2019;Receivedinrevisedform28January2020;Accepted27February2020

2405-8440/©²⁰²⁰^TheÂuthors.^Published^byÊlsevier^Ltd.^Thisîsânôpenâccessârticleûnder^the^CC^BY^license(http://creativecommons.org/licenses/by/4.0/).

(2)

Table 1

ImpactofMVSKtiltingonportfoliocriteria.

Deterioration of reference objective ℎref(𝐰|𝝁,𝚺,𝚽,𝚿,𝜻)≤ℎref(𝐰0|𝝁,𝚺,𝚽,𝚿,𝜻) +𝜅 Increase in mean return 𝐰^′𝝁≥𝐰^′₀𝝁+𝛿_𝜇

Decrease in variance 𝐰^′𝚺𝐰≤𝐰^′₀𝚺𝐰0−𝛿Σ

Increase in skewness 𝐰^′𝚽(𝐰⊗𝐰)≥𝐰^′₀𝚽(𝐰0⊗𝐰0) +𝛿Φ

Decrease in kurtosis 𝐰^′𝚿(𝐰⊗𝐰⊗𝒘)≤𝐰^′₀𝚿(𝐰0⊗𝐰0⊗𝐰0) −𝛿Ψ

Note:Theparameters𝛿_𝜇,𝛿Σ,𝛿Φ,𝛿ΨarepositivewhentheMVSKtiltedportfoliohas preferentialmomentscomparedtothebenchmark,atthecostofanincreaseof𝜅inℎref.

sample momentswhile stillshowing thecharacteristicsoftheinitial allocationcriterion.

Theremainderofthearticleis organizedasfollows.Section1in- troducesournotation.Section2presentstheproposedmethodologyfor portfoliotilting.Section3documentstheusefulnessofthemethodin aUCITS-compliantassetallocationframework.Weendthepaperwith aconclusion.Thesupplementaryappendixprovidesanexampleusing theRpackagemvskPortfolios.

1. Notation

Weconsiderthedecisionproblemofoptimizingaportfolioinvested in𝑁 assetswithweightvector𝐰≡(𝑤1,…,𝑤_𝑁)^′.Theoptimalityofthe portfolioweightsisevaluatedwithrespecttoareferenceportfolio𝐰0

andtheﬁrstfourportfolioreturnmoments.Forthoseportfolioreturn moments,accurateestimatorsforthecorrespondingcomomentmatri- ceshavebeenproposedbyMartelliniandZiemann(2010) andBoudt etal.(2020a,b),amongothers.Theinterpretationandreliableestima- tionof comomentmatricesgreater thanfouris asubject forfurther research.Itisalsoanopenresearchquestionwhethertheintegration of portfoliomoments greaterthan fourinportfolio optimizationim- provestheout-of-samplewelfareoftheinvestor(seee.g.Jondeauand Rockinger(2006)).Investorsinterestedinportfoliooptimizationusing allmomentscandirectlyoptimizetheportfolioreturndistribution,as inHaleyandMcGee(2006).Thisrequirestore-estimatetheportfolio returndistributionforeverycandidateportfoliovector𝐰.Thecompu- tationaladvantageofourMVSKtiltingportfolioisthattheoptimization doesnotrequiresuchre-estimation.

1.1. Assetreturns,theircomomentsandtheportfoliomoments

The asset returnover theinvestment horizon is denoted by 𝐫≡ (𝑟1,…,𝑟_𝑁)^′.Itsmean,covariance,coskewnessandcokurtosismatrices are

𝝁≡𝔼[𝐫], 𝚺≡𝔼[

(𝐫−𝝁)(𝐫−𝝁)^′] , 𝚽≡𝔼[

(𝐫−𝝁)(𝐫−𝝁)^′⊗(𝐫−𝝁)^′] , 𝚿≡𝔼[

(𝐫−𝝁)(𝐫−𝝁)^′⊗(𝐫−𝝁)^′⊗(𝐫−𝝁)^′] ,

(1)

where⊗standsfortheKroneckerproduct.Thesecomomentsdetermine theportfoliohigherordermoments.Deﬁne𝑚_𝑞≡𝔼[

(𝐰^′𝐫−𝐰^′𝝁)^𝑞] asthe 𝑞-thcentralportfoliomoment,implicitlydependingontheweights𝐰. Wehavethat𝑚2≡𝐰^′𝚺𝐰,𝑚3≡𝐰^′𝚽(𝐰⊗𝐰)and𝑚4≡𝐰^′𝚿(𝐰⊗𝐰⊗𝐰).

Wedenotetheportfoliotrackingerrorvolatilitybythevolatilityofthe diﬀerenceinportfoliocomposition:

TE_vol(𝐰) =√

(𝐰−𝐰0)^′𝚺(𝐰−𝐰0). (2)

1.2. Optimalityofthereferenceportfolio

Weareagnosticabouttheconstructionof thereferenceportfolio 𝐰0.Itcanbeexpert-based,algorithmicoracombinationofboth.The proposed MVSKtiltingrequires thattheinvestorelicitanoptimality criterionℎref(⋅)forwhich𝐰0istheoptimalsolution.Thiscriterionisa

functionoftheportfolioweights𝐰,theassetreturnmoments𝝁,𝚺,𝚽 and𝚿andpossiblyotherparametersstackedintothevector𝜻: 𝐰0≡argmin_𝐰_∈_ℎref(𝐰|𝝁,𝚺,𝚽,𝚿,𝜻), (3) whereisthefeasibleset.Itisalwayspossibletoﬁndacriterionfor whichthisreferenceportfolioisoptimal.Thesimplestcriterionisthe trackingerrorvolatilityin(2),whichisobviouslyminimizedwhen𝐰= 𝐰0.

Intheapplication,weconsiderthreeexamplesofℎref-functions.The ﬁrstequalstheHerﬁndahlindexoftheportfolioweights:

ℎEW(𝐰|𝝁,𝚺,𝚽,𝚿,𝜻) =

∑𝑁 𝑖=1

𝑤²_𝑖, (4)

whichisminimizedbytheequally-weightedportfolio.DeMigueletal.

(2009) refertothisportfolioasthe1∕𝑁 allocation.Theportfoliouses anaiveapproach todiversiﬁcationasitignores theheterogeneityin the componentreturndistributionswhenoptimizingtheweights.The secondobjectivefunctionis thesumofsquared deviationsof the𝑁 percentageriskcontributionswithrespectto1∕𝑁:

ℎERC(𝐰|𝝁,𝚺,𝚽,𝚿,𝜻) =

∑𝑁 𝑖=1

(

%𝑅𝐶_𝑖− 1 𝑁

)2

, (5)

with%𝑅𝐶𝑖≡^𝑤_𝐰^𝑖^[′^𝚺𝐰𝚺𝐰^]^𝑖 thepercentagevolatilitycontributionofasset𝑖. Thesolutiontothisoptimizationistheequal-riskcontributionportfolio (Maillardet al. (2010)). Thethird objectivethat we consider is the negativeofthediversiﬁcationratio(ChoueifatyandCoignard(2008)):

ℎDR(𝐰|𝝁,𝚺,𝚽,𝚿,𝜻) = −𝐰^′√ diag(𝚺)

√𝐰^′𝚺𝐰 , (6)

wherediag(𝚺)= (𝚺11,…,𝚺_𝑝𝑝)^′isthe𝑁× 1vectorcontainingthediago- nalelementsof𝚺.

Notethattheframeworkhasalarge scope andalsoincludesthe mean-varianceefficientportfoliosof Markowitz(1952) andthemini- mummodifiedValue-at-RiskandExpectedShortfallportfoliosofBoudt etal.(2008,2012),amongothers.Moreover,thefunctionℎref isnot necessarilytheoneusedtodetermine𝐰0.Forexample,itispossible toconstructthemaximumdiversificationportfoliowithweights𝐰0by minimizingℎDR,butuseℎref=TE_volwhentiltingtheportfolio.

2. MVSKportfoliotilting

Weassumethattheinvestoriswillingtosacriﬁceamargin𝜅ofthe valueofthereferenceobjectivefunctioninordertoimprovetheportfo- lioperformanceintermsofanincreaseinthemeanandskewnessand adecreaseinthevarianceandthekurtosis.Weimposetheseimprove- mentsthroughinequalityconstraints,ascanbeseeninTable1,where theminimumlevelsofimprovementsinthemean,variance,skewness andkurtosisare𝛿_𝜇,𝛿Σ,𝛿Φ,and𝛿Ψ,respectively.

Ourapproachtosimultaneouslylookforimprovementsinexpected return,variance,skewnessandkurtosisiscompatiblewiththeaccepted directionofpreferencesfortheﬁrstfourmoments(seee.g.Scottand Horvath(1980))andiscloselyrelatedtotheutility-basedframework (JondeauandRockinger(2006),Harveyetal.(2010)). Intheutility- based approach to non-Gaussian preferences,the expected utility is

(3)

oftenapproximatedbyaTaylorexpansion,ormoregenerally,afunc- tion

_𝜶(𝒘) =𝛼1𝒘^′𝝁−𝛼2𝒘^′𝚺𝒘+𝛼3𝒘^′𝚽(𝒘⊗𝒘) −𝛼4𝒘^′𝚿(𝒘⊗𝒘⊗𝒘). (7) Thevector𝜶containspositivevaluesandgovernstherelativeimpor- tance of thehigher moments withrespect tothe expectedutility of theinvestor.Suchanobjectivefunctionisgenerallynon-concaveand henceitisimpossibletoguaranteeaglobalmaximum.Inthisregard, Briecetal.(2007) proveaprimal-dualrelationshipbetweentheprimal, shortage-basedapproachfollowedinthispaperandthedualapproach ofdirectlymaximizinganon-concaveexpectedutilityapproximation.

2.1. Choiceof𝛿parameters

Inthecurrent formulationwe haveat leastthree unknowns: the tiltedportfolioweights𝐰, theminimumimprovementlevels𝛿(⋅) and thesacriﬁceparameter𝜅.Inprinciple,wecanassumethattheinvestor willelicitthese.Inpractice,however,thisisnotrealisticandadata- drivencalibrationisrequired.Toachievethis,wesimplifytheproblem byassumingthatthevalueof𝜅isﬁxedandthatthe𝛿(⋅)’sarefunctions ofasingle𝛿.

The investorwishes tomaximize this 𝛿, thereby achieving more preferableportfoliomomentsatamaximumcostof𝜅 inthereference objectivefunction.ThegeneralMVSKportfoliotiltingproblemisthen:

maximize

𝛿∈ℝ,𝐰∈ 𝛿

subject to ℎref(𝐰|𝝁,𝚺,𝚽,𝚿,𝜻)≤ℎref(𝐰0|𝝁,𝚺,𝚽,𝚿,𝜻) +𝜅, 𝐰^′𝝁≥𝐰^′₀𝝁+𝛿_𝜇,

𝐰^′𝚺𝐰≤𝐰^′₀𝚺𝐰0−𝛿Σ,

𝐰^′𝚽(𝐰⊗𝐰)≥𝐰^′₀𝚽(𝐰0⊗𝐰0) +𝛿Φ,

𝐰^′𝚿(𝐰⊗𝐰⊗𝐰)≤𝐰^′₀𝚿(𝐰0⊗𝐰0⊗𝐰0) −𝛿Ψ,

(8)

where𝛿𝜇,𝛿Σ,𝛿Φ,and𝛿Ψareallincreasingfunctionsof𝛿.Wedenote thisas(𝛿_𝜇,𝛿Σ,𝛿Φ,𝛿Ψ)^′=𝑔(𝛿).Thethreshold𝜅determinesthemaximum deteriorationinthereferencefunction.Intheextremecaseofsetting 𝜅= ∞,wehavetheMVSKoptimizationsetupofBriecetal.(2007).

2.2. Choiceof𝜅parameter

Theoptimizationin (8) yields theoptimalvalueof 𝛿for agiven valueof𝜅,andincreasing𝜅 alwaysincreasesthevalueof𝛿.Togauge thistrade-oﬀbetweenlowvaluesof𝜅beingmoreinlinewiththeinitial portfolioobjectiveandhighervaluesof𝛿yieldingpreferablemoments, we recommendvisualizing 𝛿on agrid of𝜅 values thatranges from zerotoapre-setupperbound𝜅.Basedonthis,aninvestororportfolio managercanselectavalueof𝜅thatlimitsthedecreaseinthereference objectiveforasigniﬁcantgaininmoments.Thischoicewilldependon thelevel,slopeandcurvatureofthecurve.

Inthecontextofportfoliotilting, itisalsonaturaltoset𝜅 atthe highest possiblevaluethatrespectsanupperbound ontheportfolio trackingerrorvolatilityTE_vol.Thismeansthatwelookforthehighest improvementintheMVSKobjectives(asmeasuredby𝛿)forwhichthe tiltedportfolioissuﬃcientlyclosetotheinitialportfolio. TheMVSK portfoliotilingproblemisthen:

maximize

𝛿∈ℝ,𝐰∈ 𝛿

subject to TE_vol(𝐰)≤𝜏, 𝐰^′𝝁≥𝐰^′₀𝝁+𝛿_𝜇, 𝐰^′𝚺𝐰≤𝐰^′₀𝚺𝐰0−𝛿Σ,

𝐰^′𝚽(𝐰⊗𝐰)≥𝐰^′₀𝚽(𝐰0⊗𝐰0) +𝛿Φ,

𝐰^′𝚿(𝐰⊗𝐰⊗𝐰)≤𝐰^′₀𝚿(𝐰0⊗𝐰0⊗𝐰0) −𝛿Ψ,

(9)

where(𝛿_𝜇,𝛿Σ,𝛿Φ,𝛿Ψ)^′=𝑔(𝛿)and𝜏 determines themaximumtracking errorvolatilityofthetiltedportfolioascomparedtothereferenceport- folio.

2.3. Shrinkageestimationofcomoments

Inordertoimplementtheproposedframework,reliableestimates ofthecomoments 𝝁,𝚺, 𝚽and𝚿arerequired.One possibilityis to useparametricconditionslinkingthemeanestimatedreturntothees- timatedcomoments,asinArdiaandBoudt(2015).Anotherpossibility istocombinethesample estimatoranda parametricstructuredma- trixusingtheapproachof shrinkageestimation,asrecommendedby LedoitandWolf(2003),MartelliniandZiemann(2010) andBoudtet al.(2020a).Ingeneral,theshrinkageestimatorforthecentralmoments 𝜽isgivenby

̂𝜽= (1 −̂𝛼^⋆)̂𝜽sample+̂𝛼^⋆̂𝜽structured, (10) witĥ𝜽samplethesamplemomentsand̂𝜽structuredthemomentsundera structuredestimationapproach.Severalchoicesoftargetsarepossible.

Inthispaper,weusethestructuredmomentsundertheassumptionthat themarginalsareindependent,whichisapopularchoiceinpractice.

Thevalueof̂𝛼^⋆isdeterminedinadata-drivenwaybyminimizing themeansquaredestimationerror:

𝛼^⋆= arg min

𝛼∈[0,1]𝔼

[‖‖‖(1 −𝛼)̂𝜽sample+𝛼̂𝜽structured−𝜽‖‖‖

2]

, (11)

forwhichaconsistentestimatorisderivedinLedoitandWolf(2003) forthecovariancematrixandin Boudtetal.(2020a) forthehigher moments.

3.Illustrationinassetallocation

Inthisapplication,wefocus on variance-skewness-kurtosis(VSK) eﬃcientportfolios,aimingtodecreasethevarianceandfourthcentral momentwhileincreasingthethirdcentralmoment.Wedonotincorpo- ratetheestimatedexpectedreturnssincetheirestimationprecisionis typicallylowcomparedtotheprecisioninestimatingtheportfoliovari- ance,skewnessandkurtosis.Wefurtherusethefollowingparameters fortiltingtheportfolioinordertoimprovethevariance,skewnessand kurtosis:

VSK∶ (𝛿𝜇, 𝛿Σ, 𝛿Φ, 𝛿Ψ)

=𝛿× (−∞,𝐰^′₀𝚺𝐰0,|𝐰^′₀𝚽(𝐰0⊗𝐰0)|,𝐰^′₀𝚿(𝐰0⊗𝐰0⊗𝐰0)). (12) TheVSKtiltingaboveisequivalenttoMVSKtiltingincaseofhomoge- neousexpectedassetreturns(i.e.,𝝁=𝜇𝜄𝑁,with𝜄𝑁 the𝑁-dimensional vectorofonesand𝜇thecommonexpectedreturn).Notethatin(12) we letthefunction𝑔(𝛿)dependinaspecificwayonthebenchmarkportfo- lio𝐰0.Alternativespecificationsmaybemoredesirabledependingon theapplication.Inparticular,whentheminimumvariance,maximum skewnessorminimumkurtosisportfolioisusedasbenchmarkportfo- lio,thecalibrationneedstobemodifiedsuchthatadeteriorationofthe second,thirdorfourthmomentportfolioperformanceisallowed.This ispossiblebysetting 𝛿Σ,𝛿Φ,or𝛿Ψ to−∞,whilesetting ℎref(⋅)tothe objectivefunctioncorrespondingtotheminimumvariance,maximum skewnessorminimumkurtosisportfolio.

3.1. Data

Thedataconsistofweeklyreturnsof12broadscaleinvestmentin- dicesovertheperiodJanuary26,2001untilMarch15,2019.Allseries areexpressedinEUR.Table2providesanoverviewoftheindicesalong withtheirmean,volatility,skewnessandkurtosisoverthefullperiod.

Theannualvolatilitiesliebetween3%foriBoxxEURCorporateand21%

forMSCI EmergingMarkets.Thestandardizedskewness is generally

(4)

Table 2

Summarystatisticsoftheweeklyreturnsoftheassetclassindices.

geom. mean volatility skewness ex. kurt.

1 MSCI Europe 2.82 0.19 -0.71 7.46

2 MSCI USA 4.55 0.18 -0.34 3.05

3 MSCI Japan 1.63 0.19 -0.11 1.31

4 MSCI Emerging Markets 7.35 0.21 -0.15 4.97

5 EPRA Eurozone 5.16 0.19 -1.29 9.01

6 JPM EMU 4.64 0.04 -0.14 2.16

7 iBoxx EUR Corporate 4.50 0.03 -0.94 5.41

8 Bloomberg Barclays Global Aggregate Treasuries 3.17 0.06 0.57 4.01

9 JPM EMBI Global Diversiﬁed Composite 7.29 0.11 -0.12 4.87

10 Bloomberg Barclays Global High Yield 6.96 0.10 -0.21 4.70

11 Bloomberg Barclays US Corporates 4.37 0.10 0.29 1.46

12 Bloomberg Barclays World Govt Inﬂation Linked 4.34 0.07 0.11 1.39 Note:Wereporttheannualizedgeometricmean(geom.mean,%),theannualizedstandarddeviation (volatility),thestandardizedskewnessandexcesskurtosis(ex.kurt.)overtheperiodJanuary26,2001 untilMarch15,2019.

negative,exceptforBarclaysGlobalAggregateTreasuries,USCorpo- ratesandWorldGovtInﬂationLinked,forwhichitisslightlypositive.

Allassetsshowheavy-tailedbehavior andhaveapositiveexcesskurto- sis.

3.2. Illustrationofportfoliotilting

Figs.1and2illustratetheconceptofVSKportfoliotiltingforthe lastdayofoursample,namelyMarch15,2019.Thereferenceportfolio is themaximumdiversiﬁcationportfolio(DR), andthemomentsare estimatedasoutlinedinbelow.

Thedotsin Fig.1representdiﬀerentrandomlygeneratedfeasible portfolios andprovide anidea of theset in which we are optimiz- ing.WehighlighttheDRportfolioandassociatedtiltedportfolioswith 𝜅= −(0.01,0.025,0.05)ℎDR(𝐰0).Whenthemargin𝜅increases,thetilted portfoliosattainahigherskewness whiledecreasing invarianceand kurtosis.

Fig.2ashowsthejointimprovementsinthemomentsmeasuredby theobjectivevalue𝛿in(8).Itisremarkablehowmuchimprovement is possiblewhileonlymarginallydecreasingthediversiﬁcationratio.

A𝜅 of1%resultsinarelativeimprovementofatleast15%,whereas increasing𝜅to5%yieldsarelativeimprovementofaround23%.This isalsoseeninthebottomrowofFig.1,wherethelargestdiﬀerence betweensubsequentportfoliosoccurswhen𝜅= 1%.

Forthesameportfolios,theeﬀectontheweightsisillustratedin Fig.2b.Thetiltedportfolioslowertheequityallocation(assets1,2, 3,and4)andincreasethebondcomponentthroughGlobalHighYield (asset 10).Asmentioned bya referee,bootstrappingcan be usedto evaluatetheestimationuncertaintyoftheoptimizedweightsortoro- bustifytheweightswithrespecttoestimationuncertainty.Wereferthe interestedreadertoScherer(2002) forareview.

3.3. Out-of-sampleresults

We now turn toanalyzing the out-of-sample effects of portfolio tilting on portfolio performance whenthe reference portfolio is ob- tainedbyminimizingℎEW,ℎERCandℎDR.ForℎEWandℎERC,wecon- siderdeviationsof𝜅= 0.01,0.05,0.10.ForℎDR,wetake𝜅 proportional tothediversification ratioof themaximumdiversification portfolio;

𝜅= −(0.01,0.025,0.05)ℎDR(𝐰0). Inaddition tothese fixed 𝜅, we auto- matically selectthe tilted portfolioswith maximum annualtracking error volatilities of 0.5% and 1%. We adoptthe industry-practice of rollingfive-yearestimationwindowsandestimatethemomentsusing the shrinkageapproach toward thetarget thatassumes independent marginals.Beforeestimation,wewinsorizetheobservationsfollowing theprocedureinBoudtetal.(2008).Theportfoliosarelong-onlyand fullyinvestedwithamaximumweightof20%inasingleasset,comply- ingwithUCITSregulationsinafund-of-fundssetting.Inpractice,the indicescanbereplacedbyfundscoveringdifferentassetclasses.

Fig.3aillustratesthatthecovarianceandcokurtosisshrinkagecoef- ﬁcientsarestableovertimeandputmostoftheweightonthesample covarianceandcokurtosisestimators.Thecoskewnessshrinkagecoef- ﬁcientismorevolatileandoccasionallyputsallweightonthetarget.

ThisisinlinewiththefindingsofMartelliniandZiemann(2010) and Boudtetal.(2020a).Thecourseofthediversificationratioovertimeis showninFig.3b.Notethelargedecreaseofthediversificationratioin theperiod2014to2018duetoanincreaseintheaveragecorrelation betweentheindicesfromaround24%to50%.

Table3reportssummarystatisticsforthebenchmarkandtiltedport- folios.Inallthreesettings,weobserveadecreaseinvolatilityandfourth centralmoment(𝜓)andanincreaseinthirdcentralmoment(𝜙),which isexactlywhatVSKportfoliotiltingaimstoachieve.Theeﬀectismore pronouncedwhen𝜅 increases, butitcomesat thecostof aslightly loweraverage return.As ameasure forthe trade-oﬀ,we followthe recommendationofZakamoulineandKoekebakker(2009) andusethe skew-adjustedSharperatio,

ASSR=SR

√

1 +skewness

3 SR, (13)

wheretheSharperatio(SR)andskewnessarecomputedontheannual out-of-sample returns.Thelast column of Table3shows thattilting improvestheASSRinallcases.

ItisfurtherimportanttonoticeinTable3thatportfoliotiltingalso improvesthedownsiderisk,sincethemaximumdrawdowndecreases comparedtothebenchmarkportfolio.InthecaseoftheDRportfolio, themaximumdrawdownevengoesfrom-15.6%to-11.1%whenallow- inga 5%lowerdiversiﬁcationratio. Theeﬀect onturnoverishighly dependentonthebenchmarkportfolio. DuetotheVSKportfoliobe- ingslightlymoreconcentrated,theturnovergenerallyincreaseswith𝜅. However,turnovercanbedealtwithinpracticebyimposinganaddi- tionalconstraintintheportfoliooptimizationstep.

Fig.4aillustratestheminimumimprovementinthevariance,skew- nessandkurtosisasmeasuredby𝛿in(8) whentiltingtheportfolios.

Itisclearthatmostoftheimprovementhappensforlowvaluesof𝜅, suchas𝜅= 1%.Increasing𝜅 isstillbeneficial,butitseffectisdamp- ened.For𝜅= 1%,Fig.4bshowsthedifferenceindrawdownovertime betweenthemaximumdiversificationportfolioandthetiltedportfolio.

ThecauseofthedifferenceisfoundintheweightsgiveninFig.4cand 4d.Theclearestdifferenceisthatthetiltedportfolioreducedtheover- allallocationtoequities(assets1,2,3and4)whileincreasingthebond component.Moreover,thetiltedportfoliore-allocatespartofthebond exposure.Themaindifferences here aretheweightsof GlobalHigh Yield(asset10)andUSCorporates(asset11).TheDRportfoliodoes notinvestinGlobalHighYield,whilethetiltedportfolioisinvestedin theperiods2006-2009and2016-2019.Sincebotharehighlycorrelated andhaveaboutthesamevolatility,thisisduetoadifferenceinhigher ordercomomentsinrelationtotheothercomponentsoftheportfolio.

(5)

Fig. 1.Eﬀect of VSK tilting when the reference portfolio is the maximum diversiﬁcation portfolio.

Weveriﬁedthatthechoiceofwindowlength,shrinkagetargetand pre-processingdoesnotaltertheconclusionpresentedinthissection.

Robustnesschecksareavailablefromtheauthorsuponrequest.

4. Conclusion

This paper presents a mean-variance-skewness kurtosis (MVSK) portfoliotiltingframeworkdesignedtomodifyportfolioweightsin a directionthatimprovestheﬁrstfourmomentsoftheportfolioreturn distribution. UndertheMVSKportfolio tiltingframework,theinitial

portfolio is assumed tobe optimal withrespect toan initial objectivefunction.However,itispossibletojointlyimprovethemoments atthecostofonlyaslightdeteriorationinthisinitialobjectivefunc- tion.Hence,portfoliotiltingisatrade-oﬀbetweenthedeteriorationin portfolioconstructionobjectiveontheonehand,andtheimprovements intheportfolioreturnmomentsontheother.Theempiricalapplication showstheimprovementsinout-of-sampleperformancewhentiltingthe equally-weighted,equal-risk-contributionandmaximumdiversiﬁcation portfoliosinaUCITS-compliantassetallocationsetting.Theseperfor- mancegainsarelow-hanging-fruitforallinvestors.Apracticalpolicy

(6)

Fig. 2.Eﬀect of portfolio tilting on portfolio moments and weights.

Fig. 3.Portfolio statistics over time.

Table 3

Portfoliosummary:variance-skewness-kurtosistilting.

geom. mean volatility 𝜙 𝜓 MD Turn. ASSR

DR 4.506 0.052 -0.222 0.228 -15.605 86.713 0.671

𝜅= 0.01 4.318 0.049 -0.197 0.190 -13.603 85.650 0.694

𝜅= 0.025 4.326 0.048 -0.171 0.166 -12.392 92.869 0.718

𝜅= 0.05 4.473 0.047 -0.147 0.145 -11.053 99.652 0.761

𝜏= 0.005 4.462 0.049 -0.200 0.195 -13.834 81.661 0.694

𝜏= 0.01 4.547 0.047 -0.157 0.153 -11.627 90.682 0.755

ERC 4.838 0.053 -0.248 0.264 -14.106 42.527 0.691

𝜅= 0.01 4.666 0.050 -0.161 0.176 -11.846 63.846 0.738

𝜅= 0.025 4.611 0.049 -0.130 0.148 -10.795 89.080 0.755

𝜅= 0.05 4.563 0.048 -0.119 0.134 -10.045 125.653 0.762

𝜏= 0.005 4.706 0.050 -0.186 0.194 -12.401 82.228 0.711

𝜏= 0.01 4.688 0.048 -0.144 0.153 -10.768 119.563 0.749

EW 5.447 0.084 -1.305 2.007 -26.562 55.653 0.509

𝜅= 0.01 5.175 0.066 -0.559 0.713 -19.264 50.015 0.590

𝜅= 0.025 4.824 0.059 -0.333 0.402 -16.088 95.241 0.617

𝜅= 0.05 4.670 0.050 -0.197 0.211 -12.570 73.487 0.705

𝜏= 0.005 5.335 0.079 -1.092 1.566 -25.251 59.247 0.520

𝜏= 0.01 5.226 0.074 -0.902 1.203 -23.890 68.179 0.532

Note:Wereporttheannualizedgeometricmean(geom.mean,%),theannualizedstandarddeviation(volatility),theskewness(𝜙, 10⁻⁶)andkurtosis (𝜓,10⁻⁷),maximumdrawdown(%), annualturnover(Turn.)andskew-adjustedSharperatio(ASSR)basedonannualreturns.The DR,ERCandEWportfoliosarereported,alongwiththeperformanceoftheirtiltedversionsfor 𝜅= 0.01,0.05,0.10andtiltingwithamaximumannualtrackingerrorvolatility𝜏of0.5%and1%.

(7)

Fig. 4.Portfolio statistics over time.

recommendationisthereforethatregulatorsinthefieldofinvestorpro- tectionshouldpromoteknowledgeofhowhigherordermomentsaffect financialportfoliooutcomesandhowtheycanbeoptimized.Inorderto facilitatethisprocess,wehavereleasedanopensourceimplementation intheRpackagemvskPortfolios.

Declarations

Authorcontributionstatement

K.Boudt,D.Cornilly,F.VanHolleandJ.Willems:Conceivedand designedtheexperiments;Performed theexperiments;Analyzedand interpretedthedata;Contributedreagents,materials,analysistoolsor data;Wrotethepaper.

Fundingstatement

The authors gratefully acknowledge support from the Research Foundation - Flanders (FWOresearch grant𝐺023815𝑁 andPhD fel- lowship1114117𝑁).

Competingintereststatement

Theauthorsdeclarenoconﬂictofinterest.

Additionalinformation

Theanalysiscanbereproducedusingthecodeavailableathttps://

github.com/cdries/mvskPortfolios.

Supplementarycontentrelatedtothisarticlehasbeenpublishedon- lineathttps://doi.org/10.1016/j.heliyon.2020.e03516.

References

Ardia,D.,Boudt,K.,2015.Impliedexpectedreturnsandthechoiceofamean-variance eﬃcientportfolioproxy.J.Portf.Manag. 41(4),68–81.

Boudt,K.,Carl,P.,Peterson,B.G.,2012.Assetallocationwithconditionalvalue-at-risk budgets.J.Risk 15(3),39–68.

Boudt,K.,Cornilly,D., Verdonck,T.,2020a. A coskewnessshrinkage approachfor estimatingthe skewnessoflinear combinationsof randomvariables.J.Financ.

Econom. 18(1),1–23.

Boudt,K.,Cornilly,D.,Verdonck,T.,2020b.Nearestcomomentestimationwithun- observedfactors.J.Econom.https://doi.org/10.1016/j.jeconom.2019.12.009.forth- coming.

Boudt,K.,Peterson,B.G.,Croux,C.,2008.Estimationanddecompositionofdownside riskforportfolioswithnon-normalreturns.J.Risk 11(2),79–103.

Briec,W.,Kerstens,K.,Jokung,O.,2007.Mean-variance-skewnessportfolioperformance gauging:ageneralshortagefunctionanddualapproach.Manag.Sci. 53(1),135–149.

Choueifaty,Y.,Coignard,Y.,2008.Towardmaximumdiversiﬁcation.J.Portf.Manag. 35 (1),40–51.

Darsinos,T.,Satchell,S.,2004.Measuringstyletiltinganddecomposingstylerisk.J.

AssetManag. 5(1),64–71.

DeMiguel,V.,Garlappi,L.,Uppal,R.,2009.Optimalversusnaivediversiﬁcation:how ineﬃcientisthe1/nportfoliostrategy?Rev.Financ.Stud. 22(5),1915–1953.

Haley,M.R.,McGee,M.K.,2006.TiltingsafetyﬁrstandtheSharpeportfolio.FinanceRes.

Lett. 3(3),173–180.

Haley,M.R.,McGee,M.K.,2011.“KLICing”thereandbackagain:portfolioselectionusing theempiricallikelihooddivergenceandHellingerdistance.J.Empir.Finance 18(2), 341–352.

Haley,M.R.,Whiteman,C.H.,2008.Generalizedsafetyﬁrstandanewtwistonportfolio performance.Econom.Rev. 27(4–6),457–483.

Harvey,C.R.,Liechty,J.C.,Liechty,M.W.,Müller,P.,2010.Portfolioselectionwithhigher moments.Quant.Finance 10(5),469–485.

(8)

Jondeau,E.,Poon,S.-H.,Rockinger,M.,2007.FinancialModelingUnderNon-Gaussian Distributions.SpringerScience&BusinessMedia.

Jondeau,E.,Rockinger,M.,2006.Optimalportfolioallocationunderhighermoments.

Eur.Financ.Manag. 12(1),29–55.

Ledoit,O.,Wolf,M.,2003.Improvedestimationofthecovariancematrixofstockreturns withanapplicationtoportfolioselection.J.Empir.Finance 10(5),603–621.

Lee,W.,2011.Risk-basedassetallocation:anewanswertoanoldquestion?J.Portf.

Manag. 37(4),11–28.

Maillard,S.,Roncalli,T.,Teïletche,J.,2010.Thepropertiesofequallyweightedrisk contributionportfolios.J.Portf.Manag. 36(4),60–70.

Markowitz,H.,1952.Portfolioselection.J.Finance 7(1),77–91.

Martellini,L.,Ziemann,V.,2010.Improvedestimatesofhigher-ordercomomentsand implicationsforportfolioselection.Rev.Financ.Stud. 23(4),1467–1502.

Scherer,B.,2002.Portfolioresampling:reviewandcritique.Financ.Anal.J. 58(6), 98–109.

Scott,R.C.,Horvath,P.A.,1980.Onthedirectionofpreferenceformomentsofhigher orderthanthevariance.J.Finance 35(4),915–919.

Stutzer,M.,2000.Aportfolioperformanceindex.Financ.Anal.J. 56(3),52–61.

Stutzer,M.,2003.Portfoliochoicewithendogenousutility:alargedeviationsapproach.

J.Econom. 116,365–386.

Zakamouline,V.,Koekebakker,S.,2009.Portfolioperformanceevaluationwithgen- eralizedSharperatios:beyondthemeanandvariance.J.Bank.Finance 33(7), 1242–1254.