ֻ11जֻ1௹ܵ ॓ ࿐ ࿐ Б 2008ֻ2ᄅ ٳޣ൰གЌੱ֥ቋཬؽӰหवઅଆ࣮ 杨 舸,田 澎(ഈݚࢌ๙ն࿐νܵ࿐ჽ,ഈݚ200052)ᅋေ:将人寿保险产品中的退保权视作美式期权,提出了一个保单退保率分布的理论模型,用最小二乘蒙特卡罗模拟计算了分红型人寿保险在合同期内各年的退保率.研究表明:市场无风险利率a保险公司担保利率a资产波动率和保险公司盈余分配比例对退保率有影响;分红保险在到期日前会出现一个退保的高峰,在峰值过后,退保率下降并保持一段平缓的水平,直至到期日前的2~3年左右,退保率逐渐上升.ܱՍ:分红寿险;最小二乘蒙特卡罗模拟;退保率;美式期权ᇏٳোݼ: ໓ངѓ്:A ໓ᅣщݼ:1007-9807(2008)01-0095-060 ႄ ഈඍਆ۱ാིੱіࣉྛਔྩᆞ,ӮູݓຓЌག܄ඳാིੱі֥ࠎԤ.؟୍ট,൰ག൧ӆა࣮ླྀ߶Ќ(LifeInsuranceMarketingandResearchAssocia (ᇶေ൞൰གӁᇏ֥Ќ)߶ؓЌག܄ඳ֥ࣜႏӁളᇗն႕ཙ.ݓຓᄝഈൗࡀϖ୍tion,LIMRA)֩၂ུྛြࠏܒ္҂ק௹ؓྛြ֥սࣼؿളݖ؟ఏЌག܄ඳၹնਈЌႄؿӊڱາЌ౦ঃࣉྛטҰ,ѩ܄࣮҃Бۡ[8~11].ᆃུࠏБۡ๙ӈᆰّܴ႘ਔྛြଽЌੱ֥౦ঃ,҂؎,ቋᇔ֝ᇁЌག܄ඳӁ֥൙ࡱ.Ֆପൈఏ,ႵܱЌ֥໙ี၂ᆰູြࢸࠣ࿐ඌࢸ෮ܱᇿ[1].Ֆֹؓാིੱіࣉྛਔྩᆞ,ᆃུБۡಌكקଢభ֥౦ঃु,ݓ֥০ੱԩႿ၂۱໊֮ഈ.ෛሢਈٳ༅.ؓЌੱ໙ี࣮֥҂ቀ,ႄఏਔ࿐ᆀૌࣁವြ֥ष٢a০ੱ൧ӆ߄ࣉӱ֥ࡆॹ֥ྖ౿.DarބD[12]oddsaO[13]utrevilleaKuoaTsai,ೂݔໃট০ੱഈശބC[14]henᄎႨ࠹ਈࣜ࠶࿐ં,ؓЌੱնཬ,ࣼႵॖିႄؿնਈ֥Ќ,֝ᇁЌག܄ඳԛགྷӊڱາࠏമᇀӁ.ၹՎ,ႵсေໃԖ֥ٳ҃ࣉྛਔקਈٳ༅,ࡹ৫ਔЌੱაః႕ཙ᠉ၹࡗ֥ࣜဒଆ.൞, ఘࣂಯໃࡹ৫Ⴕܱ,ؓЌ໙ีᅚष࣮.ႮႿЌੱ֥նཬᆰࢤّ႘ਔЌ໙ี֥Ќੱაః႕ཙၹࡗܱ༢֥ંଆ[14]!.ᇗӱ؇Ч໓ؓՎࣉྛંྟӇ൫,ؓЌੱն,෮ၛᆌؓЌੱ֥࣮ࣼཁ֤Ⴍູᇗေ.ಖط,ՖၘႵ֥໓ངটु,ؓЌ໙ี֥ཌྷܱ࣮ཬ֥ٳ҃ࣉྛ࣮.ᄝ҂ॉ੮ඵວࠣࢌၞٮႨᇶေࠢᇏᄝؓЌ֥קࡎٚ૫[2~5],ؓЌੱ֥֥౦ঃ༯,ᄎႨ௹ಃં,ิԛٳޣ൰གЌ࣮ࢠഒ.ႵႿՎੱٳ֥҃ંଆ,ѩႨቋཬؽӰหवઅٚ,Ч໓ؓٳޣ൰གᄝކ௹ଽ۲୍֥Ќੱնཬࣉྛ࣮مଆٳ҃ࢲݔ..Ԯ֥ടࠣЌੱ֥࣮ᇶေ൞۴ऌၛສ֥ࣜဒඔऌটщᇅാི(Ќ)ੱі,ѩ๙ݖՎіؓ1 ࠎЧଆЌག܄ඳ֥Ќੱࣉྛყҩ,ఃٚمোරႿщᇅളଁі.ᆃᇕാིੱіቋᄪ֥ႵLintonാིੱіࡌקࣁವ൧ӆ൞ປಆଉ҈֥.ࠧ҂թᄝࠣM[7]oorheadാིੱі[6].1975୍,Brzezinskiؓྐ༏҂ؓӫaඥ൬aࢌၞӮЧބછॢ֥ཋᇅ֩౦ ൬۠ರ௹:2003-12-19;ྩרರ௹:2007-07-03.ቔᆀࡥࢺ:ဗ Ჲ(1972∀),ଳ,ᇜದ,Ѱൖު,Emai:lgyang428@;
∀96∀ܵ ॓ ࿐ ࿐ Б2008୍2ᄅ.ྙՎຓ,Ч໓ߎࣉ၂҄ࡌקЌದ൞ྟ֥,Pg(t)=Pg(t-1)(1+rg)(2)ЌದބЌག܄ඳ൞ڄགᇏྟ֥.ٳޣЌག֥၂۱หׄ൞ЌದॖၛٳཚЌག ໙ี૭ඍ܄ඳ֥႔০.ೂޅಒק܄ඳ႔০,۲۱Ќག܄ඳቓॉ੮Ԛൈख़,Ќದ๙ݖᴗ࢚ЌٮཟЌم҂࣐ཌྷ,ط၂Ϯ္҂ؓຓ܄ष.Ч໓࿖ӻག܄ඳܓઙ၂ٺ௹ཋູT֥ٳޣ൰ག,ࠆ֤ࡎ࿃໗ࡲࣜႏ֥ჰᄵ,ࡼЌག܄ඳ֥႔০S(t)ࡥ߄ᆴູP0֥ሧӁ.๙ݖӻႵᆃٺሧӁ,Ќದࡼູሧ൬ၭࡨ୍֥֒քЌᆦڱPg(t),ѩַҀഈ߶ᄝކ֞௹ರTࠆ֤Ќག܄ඳ֥ᆦڱP(t).၂୍؇܄ඳၭު֥ഺჅ%,ࠧՎຓ,ೂݔᄍྸЌದᄝ֞௹ರభิభᇏᆸކS(t)=A(t)-Pg(t)+B(t-1)(1+r),ᄵЌದॖࠆ֤P(t)֥ᆦڱ(t=1,2,#,(3)T-1).ఃᇏ,P(t)ູЌག܄ඳؓЌದ֥ڵᅏ.ՖЌದ֥࢘؇ഈु,္ॖၛФ൪ູൔᇏ:B(t-1)(1+r)ູഈ၂୍؇Ќག܄ඳ֥ၭ.Ќದ֥ሧӁՖط,Ќག܄ඳ֥ቋᇔڵᅏ(ؓЌದط,ఃնཬ౼थႿЌག܄ඳ֥ࣜႏሑঃ.၂Ϯط,Ќག܄ඳ߶ഡק၂۱քЌ০ੱᄵ൞ఃሧӁ)ॖၛіൕູP(t)=Pg(t)+max[0, S(t)](4)rg,P(t)୍֥ᄹӉੱ҂֮ႿھքЌ০ੱ.๙ӈ,Ќག܄ඳ߶ࡼٳޣ൰གᅬ෮ؓႋ֥ሧӁൔᇏ: ູЌག܄ඳ֥႔০ٳб২(ႻӫٳሧႿࣁವ൧ӆ.ᄝ୍؇ଌ,֒Ќག܄ඳ֥ޣб২),ડቀ &[0,1].ھᆴᄀն,ؓЌದᄀሧӁള႔০ൈ,Ќག܄ඳ߶ࡼ႔০ο၂ק֥бႵ০.οᅶݓ∋۱ದٳޣЌགࣚෘܿק(), ֥২ْߎ۳Ќದ,ՖطЌᆣਔЌದॖၛٳཚ౼ᆴ҂֤ཬႿ.Ќག܄ඳၹູሧطջট֥൬ၭ(ӫೂݔᄍྸЌದิభᇏᆸކ(ࠧᄍྸЌ ০ҵၭ!∃).ᆃဢ,Ќག܄ඳ֥ሧӁᄝᆦڱЌದᆭದЌ),ପહؓЌದطᄝࡗ[0,T]ഈ,ࡼު֥ഺჅ҆ٳࠧܒӮЌག܄ඳ֥ၭ,࠺ູ߶࿊ᄴቋႪ֥ᇏᆸൈࡗၛఃሧӁࡎᆴቋն߄.࠺Bᆃ၂Ќൈࡗູ!*,ᄵրࢳ֥ቋႪ߄໙ีॖၛ݂(t).Ќག܄ඳ֥ሧӁaၭބؓЌದ֥ڵᅏᆀᆭࡗ֥ܱ༢ॖၛႨі1টࡥֆіൕࢲູ.і1Ќག܄ඳࡥၞሧӁڵᅏі!*=inf{t|P(t)Tab-rtle1SimpleBalancesheet ∗supEQ[eP(t)|Pt]}(5)t&[0,T]ሧӁڵᅏൔᇏ:{Pt}ູtൈख़ܱႿP(t)֥෮Ⴕྐ༏ࠢ,P(t)A(t)EQ[+]ູڄགᇏྟ༯֥௹ຬ.B(t)൙ൌഈ,!*൞၂۱ቋႪ๔ൈ,൞ෛࠏэਈ. =A(t) =A(t)ᄝᆃ၂ቋႪ๔ൈ֥໙ีᇏ,ႅݣਔᄝ[0,T]ࡗഈЌੱնཬ֥ٳ҃౦ঃ.ၹՎ,๙ݖؓ!*֥࠹ ଆ֥ࡹ৫ෘ,ॖၛࠆ֤ЌֆЌੱնཬ֥ٳ.҃ऎุ֥ࢳॉ੮Ќག܄ඳࡼሧӁሧႿປಆٳ߄֥ݖӱ,ࡼᄝ༯၂҆ٳࣉྛབྷ༥ંඍ.ሧቆކ֥౦ྙ.ഡሧӁ֥э߄ڛՖࠫޅ҃ᄎ,ࠧ2 ଆ֥ࢳdA(t)=rA(t)dt+ A(t)dWQ(t)(1)ఃᇏ:rູڄག০ੱa ູሧӁ֥ѯ,ੱrބ नູӈඔ,WQູѓሙົବݖӱ. ࢳٚم֥࿊ᄴᄝtൈख़Ќག܄ඳؓЌದ֥ڵᅏ౼थႿ֒൰གӁॖၛᄝކ֞௹ರభᆳྛ,ၹطॖ୍֥႔০,҂ି֮ႿఃքЌᆴ,࠺ູPg(t),ఃᆴູၛ൪ቔૅൔ௹ಃ.აᆺିᄝ֞௹ರᆳྛ֥ൔ௹∃൙ൌഈ,Ќದ҂ࣇॖၛࠆ֤০ҵၭ,طߎॖၛࠆ֤ၹൌ࠽ඵວੱ֮Ⴟყק֥ඵວੱջট֥ ඵҵၭ!,ၛࠣၹൌ࠽֥ႏြٮႨ֮Ⴟყק֥ႏြٮջট֥ ٮҵၭ!.ູ໙ีࡥ߄,Ч໓ᇏ҂ॉ੮ඵວੱބٮႨ֥႕ཙ,ၹط,Ќֆ֥ٳޣࣇႮሧ൬ၭӁള.%۴ऌ∋Ќག܄ඳҍༀᇅ؇(֥ܿק,Ќག܄ඳؿള୍֥؇,ॖၛႨ༯၂୍؇֥০ᄝ෮֤ඥభҀ.)ᇏݓЌགࡓܵჴ߶໓ࡱ(Ќࡓؿ[2003]67ݼ).
ֻ1௹ဗ Ჲ֩:ٳޣ൰གЌੱ֥ቋཬؽӰหवઅଆ࣮∀97∀ಃཌྷб,ૅൔ௹ಃ֥קࡎбࢠ.ն؟ඔ֥ૅൔॖၛଆЌག܄ඳሧӁ֥ѯ.ᄜ۴ऌൔ(2)∀(4),௹ಃીႵࢳ༅ࢳ,ສສေࢹᇹඔᆴٳ༅֥ٚمࠆ࠹ෘ֤֞K่Ќֆࡎᆴэ֥ਫ਼ࣥPki(i=0,1,2,.֤෮ၛ,Ч໓Ⴈඔᆴٚمࣉྛଆ֥ࢳ.#,N,k=1,2,#,K),ఃᇏPi=P(i+,t).หवઅଆ൞ඔᆴଆ࠹ෘᇏႋႨࢠູܼؓႿൔ௹ಃ,ᆺିᄝ֞௹ರᆳྛ,ၹطᆺླ֥ٗ၂ᇕٚم.൞,ႮႿఃऎႵభཟଆ֥หॉ੮֞௹ರTൈख़֥౦.ྙಖط,ૅൔ௹ಃсྶॉ,ׄ҂ିޓݺֹಒקऎႵުཟםսෆ෬หᆘ֥ૅ੮ิభᆳ.ྛࠧᄝ၂ൈख़,ӻႵದࡼбࢠ৫ࠧᆳൔ௹ಃקࡎ,ၹطӉ௹ၛট၂ᆰФಪູᆺൡႨႿྛ௹ಃ֥ࡎᆴ(ࠧଽᄝࡎᆴ)ა࿃ӻႵھ௹ಃൔ௹ಃ],ط҂ൡႿૅൔ௹ಃ[15][16.ಖط,Tilley֥ࡎᆴ(ࠧ௹ຬࡎᆴ).֒௹ಃ֥ଽᄝࡎᆴնႿ௹০Ⴈหवઅଆؓ၂۱ࠎႿॖᆦڱޣ০֥ૅൔຬࡎᆴൈ,Ќದࡼ࿊ᄴิభᆳྛކ,ڎᄵࡼुי௹ಃࣉྛܙᆴ,ਔӉ௹ၛটหवઅٚ࿃ӻႵ.ၹط,ᄝ၂ൈख़,Ќֆ֥௹ຬࡎᆴӮູم҂ିႨႿૅൔ௹ಃקࡎٳ༅֥Ԯܴ.TथҦ֥ܱ.illey֥ٚمູหवઅଆႋႨႿૅൔ௹ಃקࡎࡹ৫ႮႿหवઅଆऎႵభཟ,ྟၹՎ,ᄝiൈఏ၂۱ࠎԤख़Ќֆ֥௹ຬࡎᆴޓࠆ.ູ֤ࢳथᆃ၂,,ႮႿ֥࠹ෘݖӱླေ֥թԥॢࡗބᄎෘ۽ቔਈ٤ӈն,ՖطھٚمൌགྷఏLongstaffބ[19]SchwartzิԛႨ؟ཛൔކھൈটٳख़֥௹ຬࡎᆴ,ࠧ,Ⴕൈമᇀ൞҂ॖି֥.ູक़ڛTilleyٚم֥ᇅჿྟ[17],BroadieބGlassermanؓTE(Vi)=EQ-r,t[eVi+1|Pi]illey֥ٚمࣉྛਔڿࣉ,ಯಖթᄝሢࢠնྐ༏թԥ−2ai1+ai2Pi+ai3(Pi)(6)֥҂ቀൔᇏ:E(Vi)ູᄝiൈख़Ќֆ֥௹ຬࡎᆴ,ai1,ai2,,Ⴍః൞ᄝଆݖӱᇏ,ෛሢླေଆ֥ॖิభᆳྛರ௹ඔਈ֥ᄹࡆ,෮ླ֥թԥਈބᄎෘai3ູiൈख़Ќֆ௹ຬࡎᆴ֥݂߭༢ඔ,ఃᆴॖၛ۽ቔਈࡼӯགྷԛᆷඔᄹӉ,෮ၛಯಖ҂൞၂๙ݖ༯૫֥ቋཬؽӰ݂߭ࠆ.֤۱ٳམ֥Ϸم.ູՎ,GrantaVؓၩ၂่ٳޣ൰གڵᅏਫ਼ࣥPki,ᄝၩൈoraބW[18]eeksࡼऎႵުཟෆ෬หᆘ֥ܿ߃නམख़i,࿃ӻႵھކ֥௹ຬࡎᆴEk(Vi)ॖіವೆหवઅٚمॿࡏᇏ,֤ં൞ᄝྐ༏թൕູԥॢࡗބᄎෘ۽ቔਈ֥ေഈ,ߎ൞ᄝଆ֥ࣚEk2(Vi)=ai1+ai2Pki+ai3(Pki)+∀ki(7)ಒӱ؇ഈႵޓն֥ڿࣉ.ᄝՎࠎԤഈൔᇏ:∀kiູҗҵ.๙ݖ෮Ⴕڵᅏਫ਼֥ࣥҗҵ,LongstaffބSchw[19]artzิԛቋཬؽӰหवઅଆ(leastٚބቋཬ,ࠧ๙ݖ.k2(∀i)֥ᆴቋཬ,ॖ֤֞KsquaresMonteCarlosimulation)࠹ෘૅൔ௹ಃ.ൔ(7)ᇏ؟ཛൔҕඔ֥ܙ࠹ᆴa^i1a^i2ބa^i3.Վൈ,ਫ਼ࣥھٚمᄝ࠹ෘਈބթԥਈഈႵ٤ӈն,ၹط֤֞ࢠܼ֥ٗႋႨkᄝiൈख़Ќֆ֥௹ຬࡎᆴܙ࠹ᆴ^kE(Vi)іൕູ.ԢਔหवઅଆEk^(Vi)=a^i1+a^i2Pki+a^i3(Pk2i)(8),ӈႨ֥ඔᆴ࠹ෘ֥ٚمߎႵ۬ׄمބႵཋҵٳم֩.൞,აᆃਆ۱ٚمᄝࠆ֤Ќֆ௹ຬࡎᆴ֥ܙ࠹ᆭު,ؓЌֆཌྷбЌੱٳ֥҃ܙ࠹ᄵбࢠಸၞ,༯૫ࣼটؓՎࣉྛ,หवઅଆऎႵਆ۱Ⴊ൝:၂൞бࢠਲࠃ,ၞႿൌགྷބڿࣉ;ؽ൞ଆܙ࠹֥ѓሙ༂ҵࠣඪૼ.൬৻؇ა໙ี֥ົඔऎႵࢠ఼֥৫ؓႿਫ਼ࣥk,ᄝ֞௹ರభ֥၂ൈख़,бࢠ,ྟՖطିܔ۷ݺֹႨႿ؟ൈࡗࢨ؍a؟ѓ֥эਈ֥ۚົEk^(Vi)აPki֥նཬ.֒Pki/Ek^(Vi)ൈ,ᄵᄝھൈစളᆣಊ֥קࡎ໙.ีՎຓ,Ⴈหवઅଆॖၛख़ᇏᆸކ(Ќ);ّᆭ,ᄵ࿃ӻႵކ.Ֆᇔ֤֞ૄ่ଆਫ਼ࣥഈ௹ಃᆳྛ֥ൈࡗ,ູෛࠏݖׄष,ᇯ҄߭םսᇀԚൈख़,࿊౼ቋཬ֥ᆳӱ֥ቋႪ๔ൈ໙ีิ܂ਔඔᆴࢳ.෮ၛ,Ч໓ࡼҐྛൈࡗູھਫ਼ࣥഈ௹ಃ֥ቋႪᆳྛൈࡗ,ࠧЌႨLongstaffބSchwartzิԛ֥ቋཬؽӰหवઅൈࡗ!*k.م࣮ٳޣ൰ག֥Ќ.ੱႨဢ֥ٚم,࠹ෘ෮Ⴕਫ਼ࣥഈ֥ቋႪᆳྛ ࢳݖӱ൮༵ࡼൈࡗൈࡗ.ࣉ၂҄,ؓਫ਼ࣥk,jik=1,֒i=!*k,ॖ[0,T]ࣉྛ߄,౼ൈࡗࡗۯjik=0,֒i0!*kູ,t,T=N+,t.۴ऌൔ(1),ᄎႨหवઅٚم֤֞ᄝֻiൈख़ھ௹ಃФᆳྛۀੱ֥ܙ࠹ᆴູ
∀98∀ܵ ॓ ࿐ ࿐ Б2008୍2ᄅKЌႵࠒ֥ࠞቔႨ.prob(!=i)=.jikK(i=1,2,#,N)k=1/(9)ᆃ۱ۀੱູࠧھൈख़༯Ќੱ֥ܙ࠹ᆴ.3 ࠹ෘࢲݔࠣٳ༅ູਔࣉ၂҄ඪૼЧ໓֥ଆ,ᄝՎ۳ԛ၂۱ൌ࠽֥ෘ২.Ч໓࿊ᄴაG[3]rosenބJorgensenཌྷ֥ҕඔ,ࠧP0=100,B0=0,T=20,r=,rg=0 045, =0 15, =0 75.ٳљॉ੮ڄག০ੱraЌག܄ඳքЌ০ੱrgaሧӁѯੱ ࠣ႔1ڄག০ੱؓЌੱ֥႕ཙჅٳб২ ؓЌੱ֥႕ཙ.֒ॉ੮ଖ၂ၹ֥႕ཙൈ,ڿэھၹѩܥקఃҕඔ҂э.ࣉྛ(ᇏҕඔູP0=100,B0=0,T=20,หवઅଆൈ,ଆ100000ՑሧӁѯ֥ਫ਼rg=, =, =)ࣥ,ಖުૄ୍֥Ќ.ੱᄝՎࠎԤഈᇗگࣉྛ1000Ց,౼नᆴ,֤֞ቋᇔࢲݔ.ೂ1∀4෮ൕ.Ֆଆ࠹ෘ֥ࢲݔഈु,ٳޣ൰གᄝ֞௹ರభ֥ଖ၂ൈख़߶ԛགྷ၂۱Ќ֥ۚڂ,ᄝڂᆴݖ,ުЌੱ༯ࢆѩЌӻᄝ၂۱ࢠ֥֮ඣ,ᆰᇀ֞௹ರభ֥2~3୍ቐႷ,ЌੱႻᇯࡶഈശ.Ќۚڂԛགྷ֥ൈࡗބڂᆴնཬ൳ڄག০ੱraքЌ০ੱrgaሧӁ֥ѯੱ ބ႔Ⴥٳб২ ֩ၹ֥႕ཙ.൧ӆڄག০ੱrؓЌੱ֥႕ཙೂ1෮ൕ.Ֆᇏॖၛुԛ,֒ڄག০ੱ֮ႿքЌ০ੱ2քЌ০ੱؓЌੱ֥႕ཙൈ,ЌದࡼӻႵЌֆᆰᇀ֞௹ರ(ႮႿᆃ่౷ཌᆺᄝ֞௹ರᆳྛ,ၹطაxᇠᇗކ,ᇏᆺ(ᇏҕඔູP0=100,B0=0,T=20,่౷ཌ).֒൧ӆڄག০ੱۚႿքЌ০ੱൈ,r=, =, =)Ќದॖିᄝ֞௹ರభЌ.ෛሢڄག০ੱ֥ശ,ۚЌۚڂԛགྷ֥ൈࡗิభ,ڂᆴ֥նཬ༯ࢆ.ॖၛुԛ,൧ӆڄག০ੱؓЌྛູ֥႕ཙ൞චᇗ֥.၂ٚ૫,֒ڄག০ੱശۚൈ,Ќֆ֥ࡎᆴ༯ࢆ,Ќದ჻ၩࣉྛᄜವሧ,ՖطႄఏЌྛູ֥ᄹࡆ.ਸ਼၂ٚ૫,֒ڄག০ੱശۚൈ,Ќག܄ඳ֥ሧ൬ၭ္߶ཌྷႋᄹࡆ.ႮႿٳޣ൰གॖၛٳཚЌག܄ඳ֥ሧ൬ၭ,Ќֆ֥ࡎᆴ߶ၹՎᄹࡆ,ՖطЌྛູࡨഒ.ሹ֥টु,ڄག০ੱശۚ,ЌႵᄹࡆ.ᆃ۱ࢲݔіૼ,൧ӆ০ੱؓٳޣ൰གЌੱ֥႕ཙა௴๙֥൰གӁ҂3ሧӁѯੱؓЌੱ֥႕ཙ.၂Ϯط,֒൧ӆ০ੱശۚൈ,௴๙൰གӁႮႿЌֆࡎᆴ༯ࢆ,Ќੱࡼෛ০ੱശۚطᄹࡆ(ᇏҕඔູP0=100,B0=0,T=20,.ၹՎr=,rg=, =),ٳޣӁؓߏࢳၹ൧ӆ০ੱഈശطջট֥
ֻ1௹ဗ Ჲ֩:ٳޣ൰གЌੱ֥ቋཬؽӰหवઅଆ࣮∀99∀Ќག܄ඳ႔Ⴥٳб২ ؓЌੱ֥႕ཙೂ4෮ൕ.ෛሢ႔Ⴥٳб২֥༯ࢆ,ЌੱႵ෮ᄹࡆ.Ֆ࠹ෘࢲݔഈु,႔Ⴥٳб২,ؓЌྛູ֥႕ཙҵљѩ҂ٳཁᇷ.ଢభ,ݓ֥၂ུЌག܄ඳ൫၇ौิۚٳޣб২টၝᇅЌྛ.ູЧ໓֥࣮іૼ,ᆃ၂ቓمི֥ݔᄝંഈѩ҂ཁᇷ.4 ࢲ ંЧ໓ᄝࣁವ൧ӆ൞ປСଉ҈֥,ࠧ҂թ4႔Ⴥٳб২ؓЌੱ֥႕ཙFᄝྐ༏҂ؓӫaඥ൬aࢌၞӮЧބછॢ֥ཋᇅ֩౦(ᇏҕඔູP0=100,B0=0ঃ֥ࡌק༯,࣮ٳޣ൰གᇏྟЌದ֥Ќ,T=20,r=,rg=, =)ྛ.ູॉҳਔڄག০ੱaЌག܄ඳքЌ০ੱaሧӁ൬ၭѯੱࠣ႔Ⴥٳб২ؓЌੱ֥႕ཙ.Ќག܄ඳքЌ০ੱrgؓЌੱ֥႕ཙೂ2࣮ࢲݔіૼ:෮ൕ.ෛሢքЌ০ੱ֥ശ,ۚЌۚڂԛགྷ֥ൈࡗ1)ٳޣЌགᄝ֞௹ರభ߶ԛགྷ၂۱Ќ֥ཟު၍.֒քЌ০ੱrg=ൈ,Ќۚڂᄝֻۚڂ,ෛުЌੱႵ෮༯ࢆѩЌӻࢠ֮ඣ,ᆰᇀ9୍ԛགྷ,ڂᆴնཬູ%ቐႷ;֒քЌ০ੱrg֞௹ರభ֥2~3୍ቐႷ,ЌੱႻᇯࡶഈശ;ഈശ֞ൈ,ЌۚڂӾᇀֻ11୍,ڂᆴն2)ڄག০ੱؓЌ֥႕ཙ൞චᇗ֥,࠻Ⴕཬູ6%ቐႷ;քЌ০ੱ࿃ഈശ֞ൈ,ЌࡆЌ֥၂૫,ႻႵࡨഒЌ֥၂૫;ა௴๙൰ۚڂᄝֻ13୍ԛགྷ,ڂᆴູ9%ቐႷ;֒քЌ০ੱགӁཌྷб,ٳޣӁ۷Ⴕ০ႿߏࢳЌ؛;ഈശᇀڄག০ੱൈ,Ќྛູؿള.ሹ֥ඪ3)ෛሢЌག܄ඳքЌ০ੱ֥ശۚ,ሹ֥Ќট,քЌ০ੱശ,ۚЌੱ༯ࢆ.ੱ༯ࢆ,ЌۚڂӾ,ڂᆴഈശ;ሧӁѯੱ ؓЌੱ֥႕ཙೂ3෮ൕ.ॖ4)ෛሢሧӁѯੱ֥ิۚ,Ќۚڂ֥ԛགྷၛुԛ,ෛሢѯੱᄹࡆ,Ќۚڂิభ,ڂᆴնൈࡗิభ,ڂᆴնཬഈശ;ႮႿݓሧЧ൧ӆѯཬᄹࡆ,Ќྛູࡆख.֒ѯੱޓཬ,ೂ =ੱࢠն,ၹطݓЌག܄ඳႋؓЌ໙ีჍၛ0 05ൈ,ѩໃԛགྷޓૼཁ֥Ќۚڂ;֒ѯੱܱᇿ;࿃ഈശൈ,Ќۚڂԛགྷ֥ൈࡗᇯࡶิభ,ڂᆴն5)Ќག܄ඳ႔Ⴥٳб২༯ࢆ,Ќྛູᄹཬ္ᄹࡆ.ᆃ၂ࢲݔࡼؓݓ൰ག൧ӆႵ၂ק֥ࡆ,ః႕ཙѩ҂ٳཁᇷ.ၹՎ,Ќག܄ඳ൫ᆷ֝ၩၬ.აݓຓӮඃ֥ሧЧ൧ӆཌྷб,ݓ֥ሧ๙ݖิۚٳޣб২টၝᇅЌྛ,ູఃིݔᄝЧ൧ӆѯੱࢠն,൰གӁ֥Ќ໙ีၹط߶ંഈѩ҂ཁᇷ.۷ᇗུ.ෛሢЌག܄ඳሧ౻֥֡ຉॺ,ໃটЌЧ໓֥࣮ࢲݔ൞ᄝໃॉ੮ඵວੱބࢌၞٮགሧࣁᆰࢤࣉೆܢௐؽࠩ൧ӆӮູॖ,ିၹՎ,ЌႨ֥౦ྙ༯֤֞,֥ษંؽᆀؓЌੱ֥႕ཙ,ࡼག܄ඳᄝሧࣁ ೆ൧!భ۷ႋھܱᇿЌ໙.ี൞၂۱ᆴ֤ࣉ၂࣮֥҄ൌ࠽໙.ีҕॉ໓ང:[1]BriysE,:Debunkingsomecommonpitfalls[J].JournalofRiskandIn surance,1997,64:673∀694.[2]BriysE,:Pricingandregulatoryimplications[J].GenevaPa personRiskandInsuranceTheory,1994,19:53∀72.[3]GrosenA,:Theimpactofinterestguarantees,surrenderoptions,
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