FM-Formulas北美精算师SOA考试第二门FM公式汇总

ExamFM/2InterestTheoryFormulasby(/iropracyThisisacollaborationofformulasfortheinteresttheorysectionoftheSOAExamFM/CASExam2.Thisstudysheetisafreenon-copyrighteddocumentforstudentstakingExamFM/2.Theauthorofthisstudysheetisusingsomenotationthatisuniquesothatnodesignationwillrepeat.Eachdesignationhasonlyonemeaningthroughoutthesheet.FundamentalsofInterestTheoryandTimeValueofMoney()niPVFV+=1()niFVPV+=1()iid+=1vd−=1iddi=−()iv+=11dv−=1ivd=()taTheamountaninitialinvestmentof1growstobytimet≡()tATheamountaninitialinvestmentof≡()0Agrowstobytimet()()()ilntteita+⋅=+=11()()()()()ilntteAiAtA+⋅=+=1010)()tδeta⋅=(ilnδ+=1()δnnneiv−−=+=1()()()tatatδ′=()()taeduuδt=∫0()()()00tAeAduuδt=∫Effectiveinterestratewithnominalrate()miconvertiblem-thly()11−⎟⎟⎠⎞⎜⎜⎝⎛+=mmmiiEffectivediscountratewithnominalrate()pdconvertiblep-thly()pppdd⎟⎟⎠⎞⎜⎜⎝⎛−=−11NominalRateEquivalence()()ppmmδpdmidvei−⎟⎟⎠⎞⎜⎜⎝⎛−=⎟⎟⎠⎞⎜⎜⎝⎛+=−===+111111Effectiveannualrateduringthet-thyearisgivenby:ti()()()()()()1111−−−=−−−==tAtAtAtatataamountbeginningearnedamountitNotethatthet-thyearisgivenbythetimeperiod[]t,t1−Therefore,theinterestearnedduringthet-thyearisgivenby:()()()11−−=⋅−tAtAitAForequivalentmeasuresofinterestwehavethefollowingrelationship:()()()()iiiδddd2332AnnuitiesAnnuityImmediate—paymentsaremadeattheendoftheperiodAnnuityDue—paymentsaremadeatthebeginningoftheperiodAnnuityImmediate()()()iiiisnnni|n1111121−+=+++++=−−ivvvvanni|n−=+++=12i|nni|nsva⋅=()i|nni|nais1⋅+=AnnuityDue()()()()diiiisnnni|n111111−+=++++++=−dvvvanni|n−=+++=−111i|nni|nsva⋅=()i|nni|nais1⋅+=IdentitiesforAnnuityImmediateandAnnuityDue()|n|n|naiadia1+==()|n|n|nsisdis1+==|n|naa11−+=11−=+|n|nssPerpetuityivvvaai|nni|1lim32=+++==∞→∞daai|nni|1lim==∞→∞ContinuousAnnuitiesi|nni|naδiδva1=−=()i|nni|nsδiδis11=−+=∫=ntindtva0|()()∫∫=−nduuδdttpePVt00where()()0ntnδuduFVeptdt∫=∫()tp=paymentfunctionIncreasingAnnuities—Paymentsare1,2,…,n()invaIan|ni|n−=()()()()dnvaIaiIadiaIn|ni|ni|ni|n−=+==1()()()insIaiIs|ni|nni|n−=+=1()()()()dnsaIiIsdisI|ni|nni|ni|n−=+==1()()2111limiidiIaIai|nni|+===∞→∞()()21limdaIaIi|nni|==∞→∞DecreasingAnnuities—Paymentsaren,n-1,…,2,1()ianDai|ni|n−=()()()()danDaiDadiaDi|ni|ni|ni|n1−=+==()()()()isinDaiDsinninnin|||11−+=+=()()()i|nni|naDisD1+=PresentValueoftheannuitywithterms()YnXYXYXX1,,2,,−+++…⎟⎟⎠⎞⎜⎜⎝⎛−+⋅invaYaXnnin||PresentValueoftheperpetuitywithterms…,2,,YXYXX++2iYiX+AnnuitieswithTermsinGeometricProgression—()()()121,,1,1,1−+++nqqq…PresentValueis()()()()()qivqvqvqvqvVnnnn−+−=⋅+++⋅++⋅++⋅=−11111101322UsefulIdentities|kn|n|knavaa+=+()|n|mmnaaivv−=−()()()|n|n|nanIaDa1+=+|nnaiv1+=nnnnnnnvvvaaaa+=−−==1112||2||2()()()1111112||2||2++=−+−+==nnnnnnniiissssIftheinterestratevaries:()()()naaaa|n12111+++=()()()()()()nanaanaanas|n+++=21Ifthecompoundingfrequencyoftheinterestexceedsthepaymentfrequencyofkyears—Useanequivalentinterestrateoverkyears:()11−+=kijIfthepaymentfrequencyexceedsthecompoundingfrequencyoftheinterest—(1)Useanm-thlyannuity()()|nmm|naiia=()()|nmm|nsiis=()()|nmm|nadda=()()||nmmnsdds=(2)Useanequivalentinterestrateeffectiveoverthepaymentperiod:()111−+=mij()j|nmi|naa=()j|nmi|nss=()j|nmi|naa=()j|nmi|nss=Ifthepaymentsaremn,,m,m21…,thenthepresentvalueis()()()mni|nmi|ninvaIa−=Ifthepaymentsare22221mn,,m,m…,thenthepresentvalueis()()()()()mnmi|nmi|nminvaaI−=LoanRepayment—AmortizationAmortizationMethod—whenapaymentismade,itmustbefirstappliedtopayinterestdueandthenanyremainingpartofthepaymentisappliedtopayprincipleNotationLamountoftheloan≡nnumberofpaymentperiods≡APamountoflevelpaymentattheendoftheperiod(amortizedpayment)≡()kPloanpaymentattimek≡ieffectiveinterestrateperpaymentperiod≡kBbalanceattimek,balanceafterk-thpayment.Notethat≡LB=0kPprinciplepaidinpayment≡()kPkIinterestpaidinpayment≡()kPUsefulEquationsforLevelPaymentsinAaPL|⋅=inAaLP|=ProspectiveMethodiknAkaPB|−⋅=RetrospectiveMethod()ikAkksPiLB|1⋅−+=(tktkiBB+=+1)and()tktkiPP+=+1kkAIPP+=()111+−−−=⋅=knAkkvPBiI1+−⋅=−=knAkAkvPIPPUsefulEquationsforNon-LevelPayments()()()nnvPvPvPL+++=221()()()()()kkknnkkkPiBvPvPvPB−+=+++=−−++112211−⋅=kkBiI()kkkkkBBIPP−=−=−1LoanRepayment—SinkingFundSinkingFundLoan(SFL)—accumulatemoneyinaseparatefundbymakingapayment,inadditiontotheregularinterestpayment,everyperiod.NotationLamountoftheloan≡nnumberofpaymentperiods≡ieffectiveinterestrateperpaymentperiodbytheborrowertothelender≡jeffectiveinterestrateearnedbytheborrowerinthesinkingfund≡SDperiodicsinkingfunddeposit(SFD),assumedtobelevel≡SPperiodicoutlaybytheborrower=interestpaymenttolender+SFD≡kSsinkingfundbalanceafterk-thdeposit≡kLnetloanbalanceattimek≡UsefulEquationsj|nSsDL⋅=j|nSsLD=j|nSSsLLiDLiP+=+=j|nj|kj|kSkssLsDS=⋅=j|kSksDLL⋅−=NetPrincipalPaid()1111−−−+=⋅−⋅=−kSj|kSj|kSkkjDsDsDSSNetInterestPaidj|kSksjDLijSLi11−−⋅−=−NotesonLoansAmortizedLoan—overtimeinterestpaiddecreasesandprincipalpaidincreasesSFL—foreachoutlayinterestpaidtolenderisconstantInstallmentLoan—overtimeinterestpaiddecreaseswhiletheprincipalpaidisconstantBondsBonds—interestbearingsecurities;basicallyloansfromlendersperspectiveCallableBond—abondthatcanbepaidoff(called)beforematurityNotationFparvalue≡rcouponrate(interestra