Upper and lower bounds for sums of random variable

Insurance:MathematicsandEconomics27(2000)151–168UpperandlowerboundsforsumsofrandomvariablesqRobKaasa;b;,JanDhaenea;b,MarcJ.Goovaertsa;baInstituteofActuarialScience,UniversityofAmsterdam,Roetersstraat11,1018WBAmsterdam,NetherlandsbCRIR,KULeuven,BelgiumReceivedMarch2000;receivedinrevisedformJuly2000AbstractInthiscontribution,theupperboundsforsumsofdependentrandomvariablesX1CX2C


CXnderivedbyusingcomonotonicityaresharpenedforthecasewhenthereexistsarandomvariableZsuchthatthedistributionfunctionsoftheXi,givenZDz;areknown.Byasimilartechnique,lowerboundsarederived.Anumericalapplicationforthecaseoflognormalrandomvariablesisgiven.©2000ElsevierScienceB.V.Allrightsreserved.Keywords:Dependentrisks;Comonotonicity;Convexorder;Cash-flows;Presentvalues;Stochasticannuities1.IntroductionInsomerecentarticles,Goovaerts,Denuit,Dhaene,MüllerandseveralothershaveappliedtheoryoriginallystudiedbyFréchetinthepreviouscenturytoderiveupperboundsforsumsSDX1CX2C


CXnofrandomvariablesX1;X2;:::;Xnofwhichthemarginaldistributionisknown,butthejointdistributionoftherandomvectorX1;X2;:::;Xniseitherunspecifiedortoocumbersometoworkwith.Theseupperboundsareactuallysupremainthesenseofconvexorder.Theconceptofconvexorderiscloselyrelatedtothenotionofstop-lossorderwhichismorefamiliarinactuarialcircles.Bothexpresswhichoftworisksisthemoreriskyone.AssumingthatonlythemarginaldistributionsoftheXiaregiven(orused),theriskiestinstanceSuofSoccurswhentherisksX1;X2;:::;Xnarecomonotonous.Thismeansthattheyareallnon-decreasingfunctionsofoneuniform(0,1)randomvariableU,andsincethemarginaldistributionmustbePr[Xix]DFi.x/,thecomonotonousdistributionisthatofthevectorF−11.U/;F−12.U/;:::;F−1n.U/.InthiscontributionweassumethatthemarginaldistributionofeachrandomvariableX1;X2;:::;Xnisknown.Inaddition,weassumethatthereexistssomerandomvariableZ,withaknowndistributionfunction,suchthatforanyiandforanyzinthesupportofZ,theconditionaldistributionfunctionofXi,givenZDz,isknown.WewillderiveupperandlowerboundsinconvexorderforSDX1CX2C


CXn,basedontheseconditionaldistributionfunctions.Twoextremesituationsarepossiblehere.OneisthatZDS,orsomeone-to-onefunctionofit.ThentheconvexlowerboundforS,whichequalsE[SjZ],willjustbeSitself.TheotheristhatZisindependentofallX1;X2;:::;Xn.InthiscaseweactuallydonothaveanyextrainformationatallandtheupperboundforSisjustthesamecomonotonousboundasbefore,whilethelowerboundreducestothetrivialboundE[S].ButinsomeqPaperpresentedattheFourthConferenceonInsurance:MathematicsandEconomics,Barcelona,24–26July2000.Correspondingauthor.E-mailaddress:[e-mail](R.Kaas).0167-6687/00/$–seefrontmatter©2000ElsevierScienceB.V.Allrightsreserved.PII:S0167-6687(00)00060-3152R.Kaasetal./Insurance:MathematicsandEconomics27(2000)151–168cases,andthelognormaldiscountprocessofSection5isagoodexample,arandomvariableZcanbefoundwiththepropertythatbyconditioningonitwecanactuallycomputeanon-triviallowerboundandasharperupperboundthanSuforS.InSection2,wewillpresentashortexpositionofthetheoryweneed.Section3givesupperbounds,Section4improvedupperbounds,aswellaslowerbounds,bothappliedtothecaseoflognormaldistributionsinSection5.Section6givesnumericalexamplesoftheperformanceofthesebounds,andSection7concludes.2.SometheoryoncomonotonousrandomvariablesLetF1;F2;:::;Fnbeunivariatecumulativedistributionfunctions(cdfsinshort).Fréchetstudiedtheclassofalln-dimensionalcdfsFXofrandomvectorsX.X1;X2;:::;Xn/withgivenmarginalcdfsF1;F2;:::;Fn,whereforanyrealnumberxwehavePr[Xix]DFi.x/;iD1;2;:::;n.Inthispaper,wewillconsidertheproblemofdeterminingstochasticlowerandupperboundsforthecdfoftherandomvariableX1CX2C


CXn,withoutrestrictingtoindependencebetweenthetermsXi.WewillalwaysassumethatthemarginalscdfsoftheXiaregiven,andthatallcdfsinvolvedhaveafinitemean.Thestochasticboundsforrandomvariableswillbeintermsof“convexorder”,whichisdefinedasfollows:Definition1.ConsidertworandomvariablesXandY.ThenXissaidtoprecedeYintheconvexordersense,notationXcxY,ifandonlyifforallconvexrealfunctionsvsuchthattheexpectationsexist,wehaveE[v.X/]E[v.Y/]:Itcanbeproven,seee.g.ShakedandShanthikumar(1994),thattheconditioninthisdefinitionisequivalentwiththefollowingcondition:E[X]DE[Y];E[X−d]CE[Y−d]Cforalld;whereE[Z]CisanotationforE[maxfZ;0g].Usinganintegrationbyparts,theorderingconditionbetweenthestop-losspremiumsE[X−d]CandE[Y−d]CcanalsobeexpressedasZ1d.1−FX.x//dxZ1d.1−FY.x//dxforalld:IncaseXcxY,extremevaluesaremorelikelyforYthanforX.Intermsofutilitytheory,XcxYentailsthatlossXispreferredtolossYbyallriskaversedecisionmakers,i.e.E[u.−X/]E[u.−Y/]forallconcavenon-decreasingutilityfunctionsu.Thismeansthatreplacingthe(unknown)distributionfunctionofalossXbythedistributionfunctionofalossYcanbeconsideredasanactuariallyprudentstrategy,e.g.whendeterminingreserves.Fromtheaboverelation,weseeimmediatelythatddxfE[X−x]C−E[Y−x]CgDFX.x/−FY.x/:Thus,tworandomvariablesXandYwithequalmeanareconvexorderediftheircdfscrossonce.Thislastconditioncanbeobservedtoholdinmostconceivableexamples,butitiseasytoconstructinstanceswithXcxYwherethecdfscrossmorethanonce.Itfollowsim