Present Values and Accumulations NGUS S - PDF document

PresentValuesandAccumulationsS.MVolume3,pp.1331–1336EncyclopediaOfActuarialScience(ISBN0-470-84676-3)EditedbyJozefL.TeugelsandBjørnSundtJohnWiley&Sons,Ltd,Chichester,2004 PresentValuesandEffectiveInterestMoneyhasatimevalue;ifweinvest$1today,weexpecttogetbackmorethan$1atsomefuturetimeasarewardforlendingourmoneytosomeoneelsewhowilluseitproductively.Supposethatweinvest$1,andayearlaterwegetback$.Theamountinvestediscalledthe,andwesaythateffectiverateofinterestperyear.Evidently,thisdenitiondependsonthetimeunitwechoosetouse.Inarisklessworld,whichmaybewellapproximatedbythemarketforgoodqualitygovernmentbonds,willbecertain,butiftheinvestmentisrisky,uncertain,andourexpectationattheoutsettoreceivecanonlybeintheprobabilisticsense.Wecanregardtheaccumulationofinvestedmoneyineitheraretrospectiveorprospectiveway.Wemaytakeagivenamount,$say,tobeinvestednowandask,asabove,towhatamountwillitaccumulateyears?Or,wemaytakeagivenamount,$say,requiredinyears’time(tomeetsomeliabilityperhaps)andask,howmuchweshouldinvestnow,sothattheaccumulationinyears’timewillequal?Thelatterquantityiscalledthepresentvalueyears’time.Forexample,iftheeffectiveannualrateofinterestisperyear,thenweneedtoinvest$1now,inordertoreceive$1attheendofoneyear.Instandardactuarialnotation,isdenoted,andiscalledthediscountfactor.Itisimmediatelyclearthatinadeterministicsetting,accumulatingandtakingpresentvaluesareinverseoperations.Althoughatimeunitmustbeintroducedinthedenitionof,moneymaybeinvestedoverlongerorshorterperiods.First,consideranamountof$1tobeinvestedforcompleteyears,atarateperyeareffective.simpleinterest,onlytheamountoriginallyinvestedattractsinterestpaymentseachyear,andyearstheaccumulationis$ni)compoundinterest,interestisearnedeachyearontheamountoriginallyinvestedestalreadyearned,andafteryearstheaccumu-lationis$ni)0),anastuteinvestorwillturnsimpleinterestintocompoundinterestjustbywithdrawinghismoneyeachyearandinvestingitafresh,ifheisabletodoso;thereforetheuseofsimpleinterestisunusual,andunlessotherwisestated,interestisalwayscompound.Giveneffectiveinterestofperyear,itiseasilyseenthat$1investedforanylengthoftimewillaccumulateto$.Thisgivesustheruleforchangingthetimeunit;forexample,ifitwasmoreconvenienttousethemonthastimeunit,interestofperyeareffectivewouldbeequivalenttointerest1permontheffective,becauseChangingInterestRatesandtheForceofInterestTherateofinterestneednotbeconstant.Todealwithvariableinterestratesinthegreatestgener-ality,wedenetheaccumulationfactorA(t,s)betheamounttowhich$1investedattimeaccumulatebytime�st.Thecorrespondingdis-countfactorisV(t,s),theamountthatmustbeinvestedattimetoproduce$1attime,andclearlyV(t,s)/A(t,s).Thefactthatinterestiscom-poundisexpressedbytherelationA(t,s)A(t,r)A(r,s)trs.(forceofinterestattime,denoted
(t),isdened
(t) ,t),t) t=d ,t).(Therstequalitygivesanordinarydifferentialequa-tionfor,t),whichwithboundarycondition1hasthefollowingsolution:,t)
(s),t)
(s)Thespecialcaseofconstantinterestratesisnowgivenbysetting
(t),aconstant,fromwhichweobtainthefollowingbasicrelationships:i).( PresentValuesandAccumulations ThetheoryofcashowsandtheiraccumulationsandpresentvalueshasbeenputinaverygeneralframeworkbyNorberg[10].NominalInterestInsomecases,interestmaybeexpressedasanannualamountpayableinequalinstalmentsduringtheyear;thentheannualrateofinterestiscalledForexample,underanominalrateofinterestof8%peryear,payablequarterly,interestpaymentsof2%oftheprincipalwouldbemadeattheendofeachquarter-year.AnominalrateofperyeartimesduringtheyearisdenotedThisisequivalenttoaneffectiverateofinterestofper1year,andbytheruleforchangingtimeunit,thisisequivalenttoeffectiveinterestof/m)1peryear.RatesofDiscountInsteadofsupposingthatinterestisalwayspaidattheendoftheyear(orothertimeunit),wecansup-posethatitispaidinadvance,atthestartoftheyear.Althoughthisisrarelyencounteredinprac-tice,forobviousreasons,itisimportantinactuarialmathematics.Theeffectiverateofdiscountperyear,,isdenedbyi/(,andreceiv-ingthisinadvanceisclearlyequivalenttoreceivinginarrears.WehavethesimplerelationNominalratesofdiscountmayalsobedened,exactlyasforinterest.AnnuitiesCertainWeoftenhavetodealwithmorethanonepayment,forexample,wemaybeinterestedintheaccumula-tionofregularpaymentsmadeintoabankaccount.Thisissimplydone;bothpresentvaluesandaccu-mulationsofmultiplepaymentscanbefoundbysummingthepresentvaluesoraccumulationsofeachindividualpayment.isaseriesofpaymentstobemadeatdenedtimesinthefuture.Thesimplestarelevelannuities,forexample,ofamount$1perannum.Thepaymentsmaybecontingentontheoccurrenceornonoccurrenceofafutureevent–forexample,aisanannuitythatispaidaslongastherecipientsurvives–butiftheyareguaranteedregard-lessofevents,theannuityiscalledanannuitycertainActuarialnotationextendstoannuitiescertainasfollows:Atemporaryannuitycertainisonepayableforalimitedterm.Thesimplestexampleisalevelannuityof$1peryear,payableattheendofeachofthenextyears.Itsaccumulationattheendyearsisdenoted n ,anditspresentvalueattheoutsetisdenoted n .Wehave n =nŠ1r=0(1+=(1+Š1 i)a n =nr=1vr=1Švn Therearesimplerecursiverelationshipsbetweenaccumulationsandpresentvaluesofannuitiescertainofsuccessiveterms,suchas n+1 i)s n+1 =v+ ,whichhaveveryintuitiveinterpretationsandcaneasilybeveriedAperpetuityisanannuitywithoutalimitedterm.Thepresentvalueofaperpetuityof$1peryear,payableinarrear,isdenoted  ,andbytakingthelimitinequation(5)wehave  .Theaccumulationofaperpetuityisundened.Anannuitymaybepayableinadvanceinsteadofinarrears,inwhichcaseitiscalledan.Theactuarialsymbolsforaccumulationsandpresentvaluesaremodiedbyplacingapairofdotsoverthe.Forexample,atemporaryannuity-dueof$1peryear,payableyearlyforyearswouldhaveaccumulation n yearsorpresentvalue n atoutset;aperpetuityof$1peryearpayableinadvancewouldhavepresentvalue  ;andsoon.Wehave n =nr=1(1+=(1+Š1 d)¨a n =nŠ1r=0vr=1Švn d)¨a  =1 d) PresentValuesandAccumulations Annuitiesarecommonlypayablemorefrequentlythanannually,saytimesperyear.Alevelannu-ityof$1peryear,payableinarrearstimesayearforyearshasaccumulationdenoted yearsandpresentvaluedenoted outset;thesymbolsforannuities-due,perpetu-ities,andsoonaremodiedsimilarly.Wehave =(1+Š1 ia =1Švn i¨s =(1+Š1 d¨a =1Švn Comparing,forexample,equations(5)and(10),wendconvenientrelationshipssuchas =i i n Inprecomputerdays,whenallcalculationsinvolvingaccumulationsandpresentvaluesofannuitieshadtobeperformedusingtablesandlogarithms,theserelationshipswereuseful.Itwasonlynecessarytotabulate n n ,andtheratiosi/ii/d,ateachannualrateofinterestneeded,andallvaluesof couldbefound.Inmoderntimesthistrickissuperuous,since,forexample, canbefoundfromrstprinciplesastheaccumulationofanannuityof,payableinarrearsfortimeunitsataneffectiverateofinterestof1pertimeunit.Accordingly,the isincreasinglyofhistoricalinterestonly.Afewspecialcasesofnonlevelannuitiesariseoftenenoughsothattheiraccumulationsandpresentvaluesareincludedintheactuarialnotation,namely,arithmeticallyinc-reasingannuities.Anannuitypayableannuallyyears,ofamount$inthethyear,hasaccumulationdenoted(Is) andpresentvalue(Ia) ifpayableinarrears,or ifpayableinadvance.(Is) =¨s n Šn (Ia) =¨a n Š i =¨s n Šn d =¨a n Š (Is) (andsoon)isavalidnotationforincreas-ingannuitiespayabletimesayear,butnotethatthepaymentsareofamount$1duringtherstyear,$2duringthesecondyearandsoon,thearithmeticallyincreasingsequence$1,$3/m,...atintervalsof1year.Thenotationforthelatteris (andsoon).Intheory,annuitiesorothercashowsmaybepayablecontinuouslyratherthandiscretely.Inpractice,thisisrarelyencounteredbutitmaybeanadequateapproximationtopaymentsmadedailyorweekly.Intheinternationalactuarialnotation,continuouspaymentisindicatedbyabarovertheannuitysymbol.Forexample,anannuityof$1peryearpayablecontinuouslyforhasaccumulation n andpresentvalue n .We n
(n
a n =n0(1+tdt=n0eŠt=1Švn
a  =0(1+tdt=0eŠt=1 Increasingcontinuousannuitiesmayhavearateofpaymentthatincreasescontinuously,sothatattherateofpaymentis$peryear,orthatincreasesatdiscretetimepoints,forexample,arateofpaymentthatislevelat$peryearduringthyear.Theformerisindicatedbyabarthatextendsoverthe,thelatterbyabarthatdoesnot.Wehave =nŠ1r=01)r+1r(1+Štdt=¨s n Šn
PresentValuesandAccumulations =nŠ1r=01)r+1r(1+tdt=¨a n Š
( I =n0+Štdt= s n Šn
( I =n0+tdt= a n Š Muchoftheaboveactuarialnotationservedtosimplifycalculationsbeforewidespreadcomputingpowerbecameavailable,anditisclearthatitisnowatrivialtasktocalculateanyofthesepresentvaluesandaccumulations(exceptpossiblycontinuouscashows)withasimplespreadsheet;indeedrestrictionssuchasconstantinterestratesandregularpaymentsarenolongerimportant.Onlyunderveryparticularassumptionscananyoftheaboveactuarialformulaebeadaptedtononconstantinterestrates[16].Forfulltreatmentsofthemathematicsofinterestrates,see[8,9].AccumulationsandPresentValuesUnderTheremaybeuncertaintyaboutthetimingandamountoffuturecashows,and/ortherateofinterestatwhichtheymaybeaccumulatedordis-counted.Probabilisticmodelshavebeendevelopedthatattempttomodeleachoftheseseparatelyorincombination.Manyofthesemodelsaredescribedindetailinotherarticles;herewejustindicatesomeofthemajorlinesofdevelopment.Notethatwhenweadmituncertainty,presentval-uesandaccumulationsarenolongerequivalent,astheywereinthedeterministicmodel.Forexample,ifapaymentof$1nowwillaccumulatetoaran-domamount$inayear,Jensen’sinequality()showsthatE[1[1X].Infact,theonlywaytorestoreequalityistoconditionon,inotherwords,toremovealltheuncer-tainty.Financialinstitutionsareusuallyconcernedwithmanagingfutureuncertainty,sobothactuarialandnancialmathematicstendtostresspresentval-uesmuchmorethanaccumulations.Lifeinsurancecontractsdenepaymentsthatarecontingentuponthedeathorsurvivalofoneormoreindividuals.Thesimplestinsurancecon-tractssuchaswholelifeinsuranceguaranteetopayaxedamountondeath,whilethesim-plestannuitiesguaranteealevelamountthrough-outlife.Forsimplicity,wewillsupposethatcashowsarecontinuous,anddeathbenetsarepayableatthemomentofdeath.Wecan(a)rep-resentthefuturelifetimeofapersonnowagebytherandomvariable;and(b)assumeaxedrateofinterestofperyeareffective;andthenthepresentvalueof$1paidupondeathistherandomvariable,andthepresentvalueofanannuityof$1perannum,payablecontinu-ouslywhiletheylive,istherandomvariable Tx principleofequivalencestatesthattwoseriesofcontingentpaymentsthathaveequalexpectedpresentvaluescanbeequatedinvalue;thisisjustthelawoflargenumbers(Probability)appliedtorandompresentvalues.Forexample,inordertondtherateofpremium thatshouldbepaidthroughoutlifebythepersonnowage,weshouldsolve Px Tx Infact,theseexpectedvaluesareidenticaltothepresentvaluesofcontingentpaymentsobtainedbyregardingthelifetableasadeterministicmodelofmortality,andmanyofthemarerepre-sentedintheinternationalactuarialnotation.Forexample,E[ andE[ Tx ]= .Calcu-lationoftheseexpectedpresentvaluesrequiresasuitablelifetable(LifeTableLifeInsurance).Inthismodel,expectedpresentvaluesmaybethebasisofpricingandreserv-inginlifeinsuranceandpensions,butthehighermomentsanddistributionsofthepresentvaluesareofinterestforriskmanagement(see[15]foranearlyexample,whichisaninterestingreminderofjusthowradicallythescopeofactuarialsciencehasexpandedsincetheadventofcomputers).Formoreonthisapproachtolifeinsurancemath-ematics,see[1,2].Formorecomplicatedcontractsthanlifeinsurance,suchasdisabilityinsuranceorincomeprotectioninsurance,multiplestatemodelsweredevelopedandexpectedpresentvaluesofextre-melygeneralcontingentpaymentswereobtainedassolutionsofThiele’sdifferentialequationsLifeInsuranceMathematics)[4,5].Thisdevelopmentreacheditslogicalconclusionwhen PresentValuesandAccumulations lifehistorieswereformulatedasprocesses,inwhichsettingthefamiliarexpectedpresentvaluescouldagainbederived[6]aswellascomputationallytractableequationsforthehighermoments[13],anddistributions[3]ofpresentvalues.Allofclassicallifeinsurancemathematicsisgeneralizedveryelegantlyusingcountingprocesses[11,12],aninterestingexampleofJewell’sadvocacythatactuarialsciencewouldprogresswhenmodelswereformulatedintermsofthebasicrandomeventsinsteadoffocusingonexpectedvalues[7].Alternatively,orinaddition,wemayregardtheinterestratesasrandom(Interest-rateModeling),anddevelopaccumulationsandpresentvaluesfromthatpointofview.Undersuitabledistributionalassumptions,itmaybepossibletocalculateorapproximatemomentsanddistributionsofpresentvaluesofsimplecontingentpayments;forexample,[14]assumedthattheforceofinterestfollowedasecond-orderautoregressiveprocess,while[17]assumedthattherateofinterestwaslog-normal.TheapplicationofsuchstochasticassetmodelsAsset–LiabilityModeling)toactuarialproblemshassincebecomeextremelyimportant,butthederivationofexplicitexpressionsformomentsordistributionsofexpectedvaluesandaccumulationsisnotcommon.Complexassetmodelsmaybeappliedtocomplexmodelsoftheentireinsurancecompany,anditwouldbesurprisingifanalyticalresultscouldbefound;asaruleitishardlyworthwhiletolookforthem,instead,numericalmethodssuchasMonteCarlosimulationareused(StochasticSimulationReferences[1]Bowers,N.L.,Gerber,H.U.,Hickman,J.C.,Jones,D.A.&Nesbitt,C.J.(1986).ActuarialMathematics,TheSocietyofActuaries,Itasca,IL.[2]Gerber,H.U.(1990).LifeInsuranceMathematicsSpringer-Verlag,Berlin.[3]Hesselager,O.&Norberg,R.(1996).Onprobabilitydis-tributionsofpresentvaluesinlifeinsurance,Mathematics&Economics,35–42.[4]Hoem,J.M.(1969).MarkovchainmodelsinlifeBl¨atterderDeutschenGesellschaftf¨urVer-,91–107.[5]Hoem,J.M.(1988).TheversatilityoftheMarkovchainasatoolinthemathematicsoflifeinsurance,inTransactionsofthe23rdInternationalCongressofActuaries,Helsinki,,pp.171–202.[6]Hoem,J.M.&Aalen,O.O.(1978).Actuarialvaluesofpaymentstreams,ScandinavianActuarialJournal38–47.[7]Jewell,W.S.(1980).Generalizedmodelsoftheinsur-ancebusiness(lifeand/ornon-lifeinsurance),inTrans-actionsofthe21stInternationalCongressofActuaries,ZurichandLausanne,,pp.87–141.[8]Kellison,S.G.(1991).TheTheoryofInterest,2ndEdition,Irwin,BurrRidge,IL.[9]McCutcheon,J.J.&Scott,W.F.(1986).AnIntroductiontotheMathematicsofFinance,Heinemann,London.[10]Norberg,R.(1990).Paymentmeasures,interest,anddiscounting.Anaxiomaticapproachwithapplicationstoinsurance,ScandinavianActuarialJournal14–33.[11]Norberg,R.(1991).Reservesinlifeandpensioninsur-ScandinavianActuarialJournal3–24.[12]Norberg,R.(1992).Hattendorff’stheoremandThiele’sdifferentialequationgeneralized,ScandinavianActuar-ialJournal2–14.[13]Norberg,R.(1995).Differentialequationsformomentsofpresentvaluesinlifeinsurance,Insurance:Mathe-matics&Economics,171–180.[14]Pollard,J.H.(1971).Onuctuatinginterestrates,Bul-letindeL’AssociationRoyaledesActuairesBelges68–97.[15]Pollard,A.H.&Pollard,J.H.(1969).Astochasticapproachtoactuarialfunctions,JournaloftheInstituteofActuaries,79–113.[16]Stoodley,C.L.(1934).Theeffectofafallinginterestrateonthevaluesofcertainactuarialfunctions,TransactionsoftheFacultyofActuaries,137–175.[17]Waters,H.R.(1978).Themomentsanddistributionsofactuarialfunctions,JournaloftheInstituteofActuaries,61–75.SeealsoInterest-rateModelingInsuranceMathematicsS.M