北美精算师 fm 教材financial mathematics习题答案 第...

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1、FinancialMathematicsSecondEditionAPracticalGuideforActuariesandotherBusinessProfessionalsByChrisRuckman,FSA&JoeFrancis,FSA,CFAPublishedbyBPPProfessionalEducationSolutionstopracticequestions–Chapter9Solution9.1Thetablebelowdescribestheprobabilitydistributionoftheaccumulatedamountattheendof10years:1,

2、000AV10Probability91,0001.051.05××=1,628.89460.2591,0001.051.07××=1,930.38220.4091,0001.051.10××=2,475.84510.35Theexpectedaccumulatedvalueis:1,628.89460.251,930.38220.402,475.84510.35×+×+×=$2,045.92Solution9.22Thetablebelowshowstheprobabilityforeachpossiblevalueof(1,000AV10).1,000(1,000)2AV10AV10Pr

3、obability1,628.89461,628.89462=2,653,297.710.251,930.38221,930.38222=3,726,375.330.402,475.84512,475.84512=6,129,808.840.35Theexpectedvalueofthesquareoftheaccumulatedvalueafter10yearsis:1FinancialMathematicsSolutionstopracticequestions–Chapter92EA[(1,000V10)]=×0.252,653,297.710.403,726,375.330.356,

4、129,808.84+×+×=4,299,307.65Thestandarddeviationoftheaccumulatedvalueis:0.50.5222Standarddeviation=−EA[(1,000VEA10)]([1,000V10])=4,299,307.652,045.92−=$336.91Solution9.3Themeanandthevarianceoftheannualyieldare:iEi==[](0.2)(0.10)(0.5)(0.12)(0.3)(0.15)t++=0.12522222s=++−(0.2)(0.10)(0.5)(0.12)(

5、0.3)(0.15)0.125=0.000325Themeanoftheaccumulatedamountattheendof6yearsis:6EA[5,000V66]=×=5,000E[AV]5,000(1.125)=$10,136.43Thevarianceis:2Var(5,000AV66)5,000=VarAV()=+5,00022[(1is)+2]62−(1+i)×6=+5,00022[(1.125)0.000325]61−(1.125)2=158,408.56Thestandarddeviationistherefore:Standarddeviation==1

6、58,408.56$398.01Solution9.4Themeanoftheaccumulatedvalueattheendofthreeyearsis:3EA[100,000V33]===100,000[EAV]100,000(1.14)$148,154.40Solution9.5StatementsIIIandIVaretheonlycorrectstatements.nI.EAV[]1n=+EinII.[EAVn]=+E(1i)nIII.EAV[]1n=+()E[]inIV.EAV[]1[n=+()Ei]StatementIisclearlynotcorrectsin

7、ceitsvaluegoestooneasnincreases(assumingthatthepossiblevaluesofiarepositivevalueslessthan1).StatementIIistheformulaforthefixedinterestratemodel.StatementIIIsimplifiestothecorrectformulaforindependentandiden