DOI: 10.4018/IJOCI.2018040101
International Journal of Organizational and Collective Intelligence Volume 8 • Issue 2 • April-June 2018
Copyright©2018,IGIGlobal.CopyingordistributinginprintorelectronicformswithoutwrittenpermissionofIGIGlobalisprohibited.
Financial Uncertainties
Alireza Daneshkhah, Coventry University, Coventry, UK
Amin Hosseinian-Far, The University of Northampton, Northampton, UK Omid Chatrabgoun, Malayer University, Malayer, Iran
Tabassom Sedighi, Cranfield University, Cranfield, UK Maryam Farsi, Cranfield University, Cranfield, UK
ABSTRACT Since theglobalfinancialcrash,oneofthemaintrendsinthefinancialengineeringdisciplinehas beentoenhancetheefficiencyandflexibilityoffinancialprobabilisticriskassessments.Creditors couldimmenselybenefitfromsuchimprovementsinanalysishopingtominimisepotentialmonetary losses.Analysisofrealworldfinancialscenariosrequiremodelingofmultipleuncertainquantities withaviewtopresentmoreaccurate,nearfutureprobabilisticpredictions.Suchpredictionsare essentialforaninformeddecisionmaking.Inthisarticle,theauthorsextendBayesianNetworks Pair-CopulaConstruction(BN-PCC)furtherusingtheminimuminformationvinemodelwhich resultsinamoreflexibleandefficientapproachinmodelingmultivariatedependenciesofheavy-taileddistributionandtaildependenceasobservedinthefinancialdata.Theauthorsdemonstrate thattheextendedmodelbasedonminimuminformationPair-CopulaConstruction(PCC)can approximateanynon-GaussianBNtoanydegreeofapproximation.Theproposedmethodhas beenappliedtotheportfoliodataderivedfromaBraziliancasestudy.Theresultsshowthatthe fittingofthemultivariatedistributionapproximatedusingtheproposedmodelhasbeenimproved comparedtootherpreviouslypublishedapproaches. KeywORdS
Complex Dependencies, Financial Modeling, Heavy-Tailed Densities, Non-Gaussian Bayesian Network, Vine Copula Model
INTROdUCTION
Inrecentyears,therehavebeenadvancementsintheefficiencyimprovementofcomputational algorithms for financial risk modeling(Ramachandran & Chang,2014; Zhang et al., 2018), developmentofnewmethodologiesfortacklingfinancialpredictionintheeraofBigData(Jeonet al.,2018)andtheintroductionofemergingtechnologiesandplatforms(Changetal.,2012;Chang etal.,2017)suchascloudthatcanbenefitfinancialmodelingandprediction.Therearenumerous studiesthathaveprovedtheassumptionofnormaldistributioninfinancialassetreturnswrong. Inrecentdecades,therehavebeennumerousresearchworksandcasestudiesthathaveverified
thefactthatfinancialapplicationsentailheavy-taildistribution.Thisiswheredeviationfromthe meanisfargreaterthannormaldistribution.Moreover,understandingtheimplicationsofheavy-taildistributionsiscrucial,specificallywhenassessingfinancialrisk.Thekeytofinancialrisk assessmentistomaximisetheprofitand/orreturnoninvestmentwhilstminimizingthepotential realisticrisks.Riskshaveimpact,likelihoodorprobabilitywhichwillassistusincalculatingthe riskvalue.However,catastrophesusuallyensueuponconvergenceofthemostextremeeventsand thereforecalculatingtheprobabilityofriskmanifestationisthekeytoaninformeddecisionmaking. Weshouldalsonotethatheavytailedandsuperheavytaileddistributionsdonotonlyappearin financialsettings,andtheyalsoexistinavarietyofotherapplicationsanddomains,including butnotlimitedtoenvironmentandweatherdata,electronicengineeringforinstanceheavytailed noise,hospitalpatients’staystatisticsandothers.Infinancialapplications,BayesianNetworks (BNs)andcopulasaretwocommonapproachestomodelingjointuncertaintieswithprobability distributions.Inparticular,copulashaveacclaimedmorepopularity;thisisduetofactthatwith copulawecanapproximatetheprobabilitydistributionofthedatawithheavytail,whichisinfact verycommoninfinancialapplications(Ibragimov&Prokhorov,2017). BACKGROUNd Constructionofmultivariatedistributionwouldassistusinappropriatelyexaminingdependencies betweenmultivariatedatainrealworldcomplexities.Inrecentyears,copulashavegainedpopularity inconstructingmultivariatedistributionsandsurveydependencystructures.Oneofthemain advantagesofthecopulafunctionistoseparatedependencystructurefrommarginaldistributions. Moreover, by using copula function, some quantities such as tail dependency which is the dependencybetweenextremevaluesofthevariables,canbeobtained.Amoreflexiblemultivariate copulaknownasthevinecopulamodelhasbeenrecentlydeveloped(Bedfordetal.,2016)for modelingmultivariatedependency.Thishierarchicalgraphicalmodelwasfirstlyintroducedin (Harry,1997)andlaterwasformulatedin(Bedford&Cooke,2001;Bedford&Cooke,2002).Its structureisbasedondecompositionofamultivariatedensityintoacascadeofbivariatecopula. Paircopulaconstructionsolvesthelimitationinconstructionofmultivariatecopula,andfurthermore considersthedependencybetweenpairofvariables.Bedfordetal.(2016)enhancedtheflexibility ofthevinemodelbyproposinganon-parametricbivariatecopula.Theyproposedanalternative methodwhichwasbasedonusingminimuminformationcopulas.Theirproposedapproximation couldofferanylevelofprecisionbasedonthemodelconstraints.Incontrast,ourmethodoffers greaterflexibilityinspecifyingcopulas.Itisstraightforwardtoimplement,withtheonlytechnical assumptionofconsideringthecopuladensityascontinuousandnon-zero.Anappropriateapproach tobuildanentropycopulaorspecifyingdependencyconstraintsisbymoments(Daneshkhahet al.,2012).Momentscanbespecifiedeitheronthecopulaorontheunderlyingbivariatedensity. Thesemomentconstraintsareconsideredasreal-valuedfunctions∅1,...,∅k thatarerequiredto acceptexpectedvaluese1,…,ek,respectively.Theexpectedvaluesarethencomputedbasedon theprovidedconstraintsi.e.provideddataorexperts’beliefs.Inthispaper,weimprovethefitted multivariatedensityapproximationproposedin(Baueretal.,2012;Bauer&Czado,2012)using anewlydevelopedapproximationmethodbasedontheentropymethod.Theconditionalandjoint probabilitiesofthisBNcanbederivedbyconstructinganentropycopulabetweenthenodesof interestsgiventheirparents’sets.AnentropycopulacanberepresentedintermsofPolynomial Series (PS), and more flexible ones including Orthonormal Polynomial Series (OPS) and OrthonormalFourierSeries(OFS).Wedemonstratethattheapproximationaccuracywillbenotably increasedusingentropycopula.Weverifyourclaimbycomparingourapproximationwiththe resultsillustratedin(Baueretal.,2012)tomodeltheglobalportfoliodatafromtheperspectiveof anemergingmarketinvestorlocatedinBrazil.
THe PROPOSed APPROXIMATION MeTHOd Thestandardmultivariatecopulaisnotcapableofmodelingmultivariatefinancialdata.An alternativetothestandardcopulamodelisthePCCmodelwhichismoreflexibleandefficientfor multivariatedata.Nevertheless,PCCmodeloptimizationbecomesacomputationallychallenging, whenthenumberofvariablesincreases.Baueretal.(2012)attemptedtoresolvethisbycapturing conditionalindependenceswithindata,andproposedanewmodelcalledBayesianNetworkPCC (BN-PCC).ThisnewmodelisstructurallymoreflexiblethanPCCbecauseofcapturingconditional independences within a data structure. Additionally, the challenge of computing conditional distributionsingraphicalmodelsfornon-Gaussiandistributionscanbeeasedusingbivariatecopulas. Ourproposedmethodextendsthisapproachfurtherthroughusingminimuminformationvinemodel, withaviewtocomputingmultivariatedependenciesofheavy-taileddistributionandtaildependence (asobservedinthefinancialdata)inamoreefficientandflexibleapproach.Ourproposedmodel basedonminimuminformationPCCcanapproximateanygivennon-GaussianBNtoanyrequired degreeofapproximation.UnlikethemethoddevelopedbyBaueretal.(2012),theproposedmodel isnotrestrictedtotheuseofparametricpair-copulamodels.Inourapproachpair-copulascanbe approximatedusingthemaximumentropyconceptgiventhelimitedobserveddatabytruncating thecorrespondingpolynomials/basesafterkterms,inordertomeettherestrictionsimposedby thedataandproblemunderstudy.Inthecasestudyapplication,weexaminethreedifferentbases: ordinarypolynomial,orthonormalandFourierseriesandproposethebestfittingmodelbasedona goodness-of-fitcriterion.
APPLICATION OF THe CASe STUdy
OP,OPSandOFSbasisfamiliesareusedtoapproximatethemultivariatedistributionassociated withtheselectedPCCDAGstructurecorrespondingtocasestudyportfoliodata.Inthissection,we attempttodemonstratethesuperiorflexibilityofourapproachcomparedtothemethodproposed byBaueretal.(2012). Example:Inthisexample,wehavedeployedthesamedatasetusedinMendesetal.(2010)and inChatrabgounetal.(2018)withaviewtoclearlyoutlineapproximationmethodandforcomparison purposes.Thedataconsistsofconcurrentdailylog-returns.ThedimensionsareBraziliancomposite hedgefundindex(ACI),along-terminflation-indexedBraziliantreasurybondsindex(IMA-C), Brazilianstockindexwiththe100largestcapitalizationcompanies(IBRX),indexoflargeworld stockscomputedbyMSCI(WLDLg),indexofsmallcapitalizationworldcompaniescomputedby MSCI(WLDSm),andindexoftotalreturnsonUStreasurybondscomputedbyLehmanBrothers Barra(LBTBond).ThesedataaregatheredforthewindowsbetweenJanuarythe2nd,2002andOctober the20th,2008inwhich1629dataitemsarecollected. Inthefirststep,theserialcorrelationwithinthesix-timeseriesareclearedaway,topursuethe practiceofindependenceofvariableobservationsovertime.Henceforth,theserialcorrelationin theconditionalmeanandtheconditionalvariancearemodelledbyanAR(1)andaGARCH(1,1) model(Aasetal.,2009)respectively. TogenerateaPCC-DAGapproximationfittedtothisdatasetusingentropydistributionsand basedonthedifferentbasis,thereistherealchallengeofconnectingDAGmodelstovines.Tospecify DAGstructureinourdata,weapplystructurelearningalgorithmssuchasthePCalgorithmtothe data∅−1(.),where∅denotesthestandardnormalcdf.Suchatransformationisrequired,asthe
assessmentofconditionalindependenceat%5significancelevelbythePCalgorithmassumes normality.Asanalternativeapproach,expertknowledgeisfrequentlyexploitedtodefineDAG (Kurowicka&Cooke,2006).Therearealsostructureselectionalgorithmsfornon-GaussianDAG’s availableinBaueretal.(2012)whicharesimilarlybasedonthePCalgorithm.WeadopttheDAG structurepresentedinFigure1byapplyingthePCalgorithm.Moreover,thepresentedstructurefor
non-GaussianavailableDAGin(Baueretal.,2012)producesthesameresults.ConsideringtheDAG, wedecomposethemultivariatedensityofthedatabyapplyingTheorem1inordertoderivePCC-DAGstructurei.e.giventhepresentedDAG,Baueretal.’stheoremBaueretal.(2012)prescribes whichpaircopulasneedtobespecifiedinthedefinitionofourmodel.Notethatvariable1(ACI)has threeparents(2(IMA),3(IBrX),and5(WldSm))astheorderoftheparentsarebasedontheheuristic ruleofmodellingstrongbivariatedependencespriortoweakdependences.Ourdecisionwasbased onestimatesτˆofKendall’sτvariable1,5(τ=0.209),variable1,3(τ=0.197),andvariable1,2 (τ=0.127),respectively.Similarrulecanbeappliedforvariables3(IBrX)anditsparents(2(IMA) and4(WLdLg)basedonτˆasτˆ32=0.0858,andτˆ34=0.424.Also,variable5havetwoparents (3(IBrX)and4(WIdIg))whichareτˆ53=0.402andτˆ54=0.75.Basedontheseordering,according tothemeasureofdependenciesKendall’s,theresultingmultivariatedensitydecompositionis: f x x f x c F x F x c F x i i i 1 6 1 6 1 6 15 1 1 5 5 45 4 4 , ,… , , − …
(
)
=∏
( )
×(
( ) ( )
)
×( )
,, , , ( F x c F x F x c F x F x c 5 5 46 4 4 6 6 34 3 3 4 4 1( )
(
)
×(
( ) ( )
)
×(
( ) ( )
)
× 33 5 1 5 1 5 3 5 3 5 12 35 1 35 1 3 5 2 | | | | | | | , ( | )) ( | , , F x x F x x c F x x x F(
)
×(
)
335( | , ))x2 x x3 5 Wenowderivetheentropycopulaeassociatedwithsomeofthemomentconstraintsbetween copulavariables1,2,3,4,5,6inthedensitydecompositionabove.Weinitiallyconstructmaximum entropycopulasforunconditionalcopulac c c c15, , ,46 34 45.Subsequently,itisimperativetoestablish thebasesthatshouldbeselectedandthenumberofdiscretizationpointsineachcase.Wehave outlinedtheapproachfortheunconditionalcopulac15.Theapproachforunconditionalcopulas c c c46, ,34 45issimilar.BasisfunctionscouldbechosenbasedonthemethoddescribedinDaneshkhah etal.(2012)i.e.startingwithsimplebases,andmovingtomorecomplexones,andincludingthem untilwearesatisfiedwithourapproximation.OurOPbasisfunctionsareasfollows:Figure 1. The DAG structure for Brazilian market investor’s daily log returns; the structure has six dimensions and the log returns are occurring in the same period of time
ψ1
() () () () () () () () () ()
. ψ1 . ,ψ1 . ψ2 . ,ψ2 . ψ1 . ,ψ1 . ψ3 . ,ψ3 . ψ1 . , . . , . . , . . , . . , . . ψ2() () () () () () () () () ())
ψ2 ψ2 ψ3 ψ3 ψ2 ψ1 ψ4 ψ4 ψ1 , . . , . . , . . , . . , . . ψ1() () () () () () () () () ())
ψ5 ψ5 ψ1 ψ2 ψ4 ψ4 ψ2 ψ3 ψ3 ,… OPSbasisfunctionconstructedusingGram-Schmidtprocess: ϕ1() () () () () () () () () ()
. ϕ1 . ,ϕ1 . ϕ2 . ,ϕ2 . ϕ1 . ,ϕ1 . ϕ3 . ,ϕ3 . ϕ1 . ,, . . , . . , . . , . . , . . ϕ2() () () () () () () () () ())
ϕ2 ϕ2 ϕ3 ϕ3 ϕ2 ϕ1 ϕ4 ϕ4 ϕ1 , . . , . . , . . , . . , . . ϕ1() () () () () () () () () ())
ϕ5 ϕ5 ϕ1 ϕ2 ϕ4 ϕ4 ϕ2 ϕ3 ϕ3 ,… Here,wedefineϕi()
. over[0;1]asfollows(Daneshkhahetal.,2012): ϕ0( )
u =1 ϕ ϕ ϕ ϕ 0 0 1 0 1 0 1 2 0 u u u u du u du u u n j n n j j j n j n( )
= −( )
( )
( )
− = − =∑
∫
∫
. −∑
∫
∫
( )
( )
( )
≥ 1 0 1 0 1 2 1 u u du u du u n n j j j ϕ ϕ ϕ TheOFSbasisfunctionsaresimilarlypresentedas: ∅1()
. ∅1()
. ,∅1()
. ∅2()
. ,∅2()
. ∅1()
. ,∅1()
. ∅3()
. ,∅3()
. ∅1()
. ,, . . , . . , . . , . . , . . ∅2()
∅2()
∅2()
∅3()
∅3()
∅2()
∅1()
∅4()
∅4()
∅1(
))
, . . , . . , . . , . . , . . ∅1()
∅5()
∅5()
∅1()
∅2()
∅4()
∅4()
∅2()
∅3()
∅3(
))
,… wherethegeneralformof∅1()
. isgivenby: ∅0( )
u =1,∅1( )
u = cos2( )
2πu ,∅2( )
u = sin2( )
2πu , ∅3( )
u = cos2( )
4πu ,∅4( )
u = sin2( )
4πu , ∅5( )
u = cos2( )
6πu ,∅6( )
u = sin2( )
6πu , Followingtheexplanationstoselectbasisfunctioninanoptimalmanner,weaddthebasis functionsbyusingstepwisemethodoutlinedinDaneshkhahetal.(2012).Inthismethod,ateach phase,log-likelihoodofaddingeachadditionalbasisfunctionisassessed.Subsequently,functionthatdevelopsthelargestincreaseinthelog-likelihoodisincluded.Additionally,toobtainoptimal results,firstfourbasesshouldbeconsidered(Daneshkhahetal.,2012). WecannowcomposetheentropycopuladensityC15withrespecttotheuniformdistributions giventhecorrespondingOP,OPSandOFSconstraintsabove.Wewouldinitiallyrequireidentifying thenumberofdiscretizationpoints(gridsize).Alargergridsizeprovidesanimprovedapproximation tocontinuouscopulawiththeexpenseofmorecomputationtime.Similarly,inourscenario,the approximationbecomesmoreprecise,ifweruntheD AD1 2algorithmwithfurtheriterations, nevertheless,thiswouldconsumemorecomputationalresourcesincludingtime.Therefore,itcould beconcludedthatthenumberofiterationswilldependonthegridsize.Weconsidertheapproximation errorsintherange1 10× −1to1 10× −24.Thus,thelargerthenumberofgridpointsused,thelarger
thenumberofiterationsrequiredforconvergence;thisisthecaseoveranyerrorlevel.Anygridsize followsthesamepatternwhereinitiallyalargeincreaseinthenumberofiterationsisrequiredto enhanceaccuracyandasmallerincreaseisneededwhentheerrorissmaller.Wehaveselectedagrid sizeof200 200× throughoutthisexample.Consideringgridsize,numberofiterationsanderrorsize rules(outlinedabove),wecanderivetheentropycopulac15associatedwiththechosenconstraints. Expectationsαoftheselectedbasis,Lagrangemultipliesvalues(parametervalues)λ andLog-LikelihoodaresummarizedinTable1.Log-Likelihood(L)forPS,OPS,andOFSbasisare93.49, 98.59,and38.76,respectively.ThecorrespondingcopulasintermsoftheOP,OPSandOFSbases areplottedinPanels(a),(b),and(c)inFigure2respectively. OneofthemainadvantagesofusingOPSandOFSbasesovertheordinarypolynomialseries (Bedfordetal.,2016)isthattheD AD1 2algorithmconvergesinaswiftermanner.Thisisdueto propertyoforthogonalbases(OPS&OFS)whereaddingnewbasesdoesnotalterthealready usedLagrangecoefficientsinthekernel.Ontheotherhand,thisisnotthecasewhenPSbasesis applied[6]tocalculatetheentropycopula.Therefore,whenOPisused,thereisaneedtorun D AD1 2algorithmeachtimeanewbaseisaddedtothealreadychosenbases.Thisisduetochanges inparametervalueseachtimenewtermsareadded.Therefore,moreiterationsarerequiredforthe D AD1 2algorithmtoconverge.TheoptimisationtimerequiredfortheD AD1 2algorithmusing theOPSandOFSbasesare9.83&8.89secondsrespectively,whereasthistimeforthePSbases is29.87seconds.
Theotherunconditionalcopulainthedecomposition(7)i.e.c46,c34andc45couldbesimilarly calculated.Followingastepwisemethod,PS,OPSandOFSbasesalongwiththeircorresponding constraints,resultingLagrangemultipliers,andLog-Likelihood(L)aregiveninTable2.
Theapproximatedmaximumentropycopulafortheseunconditionalcopula,intermsofthePS, OPSandOFSbases,areillustratedinPanelsofFigure3.
Table 1. The minimally informative copula given moment constraints for OP, OPS, and OFS bases between 1 and 5
Method Base
(
α α α α1, , ,2 3 4)
(
λ λ λ λ1, , ,2 3 4)
LPS
(
ψ ψ ψ ψ ψ ψ ψ ψ1 1, 2 1, 5 5, 1 2) (0.27,0.18,0.04,0.19) (14.2,-7.9,3.5,-4.1) 93.49 OPS (ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ1 1, 2 2, 4 2, 2 4) (0.29,0.13,0.08,0.07) (0.31,0.09,0.08,0.04) 95.59Figure 2. The minimally informative copula given moment constraints between variable 1 and 5; Panel (a): PS basis, Panel (b): OPS basis, and Panel (c): OFS basis
Table 2. The minimally informative copula given moment constraints for C46, C34, and C45
Method Base
(
α α α α1, , ,2 3 4)
(
λ λ λ λ1, , ,2 3 4)
L PS PS:(ψ ψ ψ ψ ψ ψ ψ ψ1 1, 5 5, 5 1, 1 4) OPS:(ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ1 1, 2 2, 4 2, 5 5) OFS:(∅ ∅2 2,∅ ∅ ∅ ∅ ∅ ∅1 1, 2 4, 4 2) (0.23,0.02,0.06,0.08) (-0.18,0.13,-0.06, 0.06) (-0.11,0.1,-0.08,-0.07) (1.4,6.5,-4.7,-4.6) (-0.18,0.12,-0.06, 0.06) (-0.11,0.1,-0.08,0.02) 44.19 51.03 30.37 OPS PS:(ψ ψ ψ ψ ψ ψ ψ ψ1 1, 1 2, 2 5, 2 1) OPS:(ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ1 1, 2 2, 5 3, 1 2) OFS:(∅ ∅2 2,∅ ∅ ∅ ∅ ∅ ∅1 1, 4 2, 2 4) (0.29,0.21,0.08,0.21) (0.57,0.35,0.1,-0.07) (0.35,0.3,0.19,0.01) (36,27.5,10.4,-5.3) (0.73,0.23,0.09,0.01) (0.4,0.3,0.2,-0.003) 379.02 392.4 245.49 OFS PS:(ψ ψ ψ ψ ψ ψ ψ ψ1 1, 5 5, 1 2, 1 4) OPS:(ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ1 1, 2 2, 3 3, 3 1) OFS:(∅ ∅2 2,∅ ∅ ∅ ∅ ∅ ∅1 1, 2 4, 3 1) (0.32,0.07,0.23,0.15) (0.88,0.78,0.67,-0.01) (0.8,0.7,0.1,0.09) (144,-18.4,-96.3, 42.3) (2.8,0.73,0.67,-0.01) (1.6,1.2,0.52,-0.001) 1479.6 1506.3 1366.1Now,theconditionalcopulasc13 5| ,c23 4| andc35 4| cansimilarlybeapproximatedusingthe entropyapproach.Weonlyillustrateconstructionoftheconditionalmaximumentropycopula betweenc13 5| ,c23 4| andc35 4| canbesimilarlyapproximatedinasimilarway.Tocalculatethis
copula,wedividethesupportof5intosomearbitrarysubintervalsorbinsandthenconstruct theconditionalcopulawithineachbin.Todothis,weselectbasesinthesamewayasforthe unconditionalcopulas;wethenfitthecopulatothecalculatedmeanvaluesorconstraints.We haveusedfourbinssothatthefirstcopulaisfor13|5∈(0,0.25).Theotherbinsare13|5∈ (0.25,0.5),13|5∈(0.5,0.75),and13|5∈(0.75,1).Thisprocesscanbesimilarlyfollowedfor theremainingbins. Thelog-likelihoodoftheoverallNon-GaussianPCCDAGmodelusingthePS,OPSand OFSbases,derivedbyaddingthelog-likelihoodsofthecopulasconstructedabove,are2390.44, 2669.69and2093.75,respectively.Now,sincethecomparisonbasedoncomparingthelog-likelihoodofpresentednonparametricmodelinthispaperandtheparametricmodelofBaueret al.(2012)isnotsufficient,andthemodelcomplexitymeasuredbythenumberofparametersis leftwithoutconsideration.Therefore,wecomparethesemethodsbasedontheAkaikeinformation criteria(AIC)whichincludesthemodelcomplexity.TheAICoftheoverallNon-GaussianPCC-DAGmodelusingthePS,OPSandOFSbasesare-4780.88,-5339.38and-4187.24,respectively. ThesevaluesareconsiderablylessthantheAICofthefittedNon-GaussianPCC-DAGmodels tothedatausingBaueretal.(2012)method,(withAICequalsto-3078.62).Thecorresponding resultsareoutlinedinTable3.
Figure 3. The minimally informative copula given moment constraints, Panel (a):C46 for PS basis, Panel (b):C46 for OPS basis, Panel (c):C46 for OFS basis, Panel (d):C34 for PS basis, Panel (e):C34 for OPS basis, Panel (f):C34 for OFS basis, Panel (g):C45 for PS basis, Panel (h):C45 for OPS basis, and Panel (i):C45 for OFS basis
CONCLUSION Gaussiandistributionsaregenerallyusedformodelingandcomputingfinancialassetreturns,risk assessmentofcapitalallocationbybanks,andestimatingrisksassociatedwithfinancialportfolios inactuarialscience.However,theexistinginternalGaussianmodelsarelimitedwhenitcomesto inferencefromtails.AsopposedtonormalGaussiandistributions,copulasareknowntobeasuitable andpowerfulmeansforovercomingtheflawsintheexistingtechniques.Anexamplefortheapplication ofcopulasintheabovementionedareas,wouldbetheclaimallocationsandfees’assignmentsfor investigators,experts,etc.aspartofAllocatedLossAdjustmentExpense(ALAE)processes.An additionalcasefortheapplicationofcopulas,wouldberiskassessmentsconductedbybanksand creditinstitutionsforcreditandmarketevaluationsandjudgements;anexistingflawwithmanyof theexistingtechniques,knowntobeinternalbottom-upapproaches,forsuchrisksassessments,is thatthosetechniquesarenotcapableofmodelingjointdistributionofnon-identicalrisks. Therearenon-identicalapproachestoinferenceinmultivariatedistributions.BN’sandcopulas aregenerallyverysuitableformodellingsuchprobabilitydistributions.Intheapplicationswheretail propertiesareimportantforpredictiveprobabilisticmodeling,manyoftheexistingtechniquesare limitedandinadequate.Oneofthewell-knowntechniquesthatcannicelyinferfromtailpropertiesis themultivariateGaussiancopula.Asstatedabove,manyofthecurrenttechniquesusedforfinancial applicationmodelling,assumeanormalGaussiandistributionofeventsforsimplifyingthecomplex natureofthefinancialscenarios(asdiscussedin(Bedfordetal.,2016;Chang,2014;Changetal., 2015)).Theproposedmethodologyforutilisingvinestructureforapproximation,wouldenablethe modellertosimplyestablishnon-constantconditionalcorrelations,andminimisethechanceofrisk underestimation.Inthispaper,weextendedthenovelmethodoriginallypresentedbyBaueretal. (2012)toapproximateamultivariatedistributionbyanyNon-GaussianPCC-DAGstructure.The noveltyofourapproachlieswithintheuseofentropycopulasforapproximationforanylevelof precision.Furthermore,thisapproximationoffersflexibilityandmanageability.Theflexibilityofthe approachiswhereanyfunctioncouldbechosenforentropycopulaconstructions.Nevertheless,two conditionsshouldbemetwithintheDAG;themultivariatedensityiscontinuousandisnon-zero.In ourmethod,forapproximatingamultivariatedistribution,aDAGstructure,abasisfamily,andthe expectedvaluesforcertainfunctionsarerequiredtobeidentified.Finally,wherethereisaneedto performacomputerizedanalysisofthebasisfunctions,ourmethodofferstheabilityforapproximating andmodelingofrelativelymorecomplexscenarios. ACKNOwLedGMeNT ThisresearchwassupportedbyfundingfromtheUKEngineering&PhysicalSciencesResearch Council(StrategicPackage:CentreforPredictiveModellinginScienceandEngineering-GrantNo. EP/L027682/1)isacknowledged.
Table 3. Comparison between different models
Type of copula AIC
Baueretal.(2012)method -3078.62
entropycopulabasedonOFSbasis -4187.24
entropycopulabasedonPSbasis -4780.88
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Amin Hosseinian-Far holds the position of Senior Lecturer in Business Systems & Operations at the University of Northampton, United Kingdom. In his previous teaching experience, Amin was a Staff Tutor at the Open University. Prior to working at the Open University, he was an academic member of staff working at Leeds Beckett University where he was a Senior Lecturer and Course Leader in the Faculty of Arts, Environment and Technology. He received his BSc (Hons) in Business Information Systems from the University of East London, an MSc degree in Satellite Communications and Space Systems from the University of Sussex, a Postgraduate Certificate in Research and a PhD degree titled ’A Systemic Approach to an Enhanced Model for Sustainability’ which he acquired from the University of East London. Dr. Hosseinian-Far holds Membership of the Institution of Engineering and Technology (IET), Senior Fellowship of the Higher Education Academy (HEA), and Fellowship of the Royal Society of Arts (RSA). Dr. Amin Hosseinian-Far is also the founding editor and the editor-in-chief for the International Journal of Strategic Engineering.