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STK 4500

STK 4500. Lecture January 30th, 2012 Nils F. Haavardsson. Overview. Premium (P). P. Client. Insurance company. Financial markets. Pension. Return. Overview. Premium (P). P. Client. Insurance company. Financial markets. Pension. Return. Risks( actuarial ). Mortality

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STK 4500

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  1. STK 4500 LectureJanuary 30th, 2012 Nils F. Haavardsson

  2. Overview Premium (P) P Client Insurance company Financial markets Pension Return

  3. Overview Premium (P) P Client Insurance company Financial markets Pension Return Risks(actuarial) Mortality Howlong do people live? Experience Do people live longer thanthemodelspredict? Disability Howmanypeoplebecomedisabled? Churn Willtheclientresign his account? Catastrophe Will a major catastropheoccur?

  4. Overview Premium (P) P Client Insurance company Financial markets Pension Return Risks(actuarial) Risks(financial) Mortality Howlong do people live? • Interest rate • Minimum guaranteeissues • Assetliability matching issues Experience Do people live longer thanthemodelspredict? • Stocks, bonds and real estate • Volatile returns • Assetallocationstrategies Disability Howmanypeoplebecomedisabled? Spread Are assetssufficientlydiversified? Churn Willtheclientresign his account? Currency Are currency risks hedged? Catastrophe Will a major catastropheoccur? Illiquidity Are the markets sufficientlyliquid?

  5. Overview Premium (P) P Client Insurance company Financial markets Pension Return Risks(actuarial) Risks(administrative) Risks(financial) Mortality Howlong do people live? Costs Is thebureacracymanagable? (for example, Solvency II) • Interest rate • Minimum guaranteeissues • Assetliability matching issues Experience Do people live longer thanthemodelspredict? Revisions Doesthe game change? (Life contractsarelong term) • Stocks, bonds and real estate • Volatile returns • Assetallocationstrategies Disability Howmanypeoplebecomedisabled? Spread Are assetssufficientlydiversified? Churn Willtheclientresign his account? Currency Are currency risks hedged? Catastrophe Will a major catastropheoccur? Illiquidity Are the markets sufficientlyliquid?

  6. An introductoryexample • ImagineMr (55) decides to investmoney to securehimselffinancially in his old age • He depositscapitalof S0=1 and drawstheentireamountwithearned compound interest in 15 years, on his 70th birthday • He willthenget • Supposethelikelihoodofreaching 70 given thatone’s age is 55 is 0.75. ThenMr(55)s expectedpayout is 0.75S15 • SupposeMr (55) teams up withMr (55)1 and Mr(55)2,whoare in theexact same situation as Mr (55) • The expectedamount at (55)s disposalafter 15 years is now

  7. Introduction to survivalanalysis • Survivalanalysis is in statisticslinked to eventhistoryanalysis, thatdealswith data observingindividuals over time • Outcome data consistof time ofoccurenceofevents and theeventsthatoccur • Frequently, an eventmay be considered as a transition from onestate to another, and thereforemulti-statemodelswillprovide a relevant framework for eventhistoryanalysis • Considerthefollowing simple model: Alive Dead

  8. Introduction to survivalanalysis Alive Dead • The observation for a given individualwillhere in the most simple form consistof a random variable, say T, representing time from a given origin (time 0) to theoccurenceoftheevent ’death’ • The distributionof T may be characterized by theprobabilityfunction F(t)=P(T<=t) or equivalently by thesurvivalfunctions S(t)=1-F(t)=P(T>t). • It is seenthat S(t) and F(t), respectively, correspond to being in state 0 or 1 at time t. • Ifeveryindividual is assumed to be in state 0 at time 0 then F(t) is alsothetransitionprobability from state 0 to state 1 for the time interval from 0 to t. • In continuous time thedistributionof T mayalso be characterized by thehazard rate function (1)

  9. Introduction to survivalanalysis Alive Dead • That is • Thus, is thetransitionintensity from state 0 to state 1, i.e., theinstantenousprobability per time unitofgoing from state 0 to state 1. • is oftenreferred to as thehazard rate. • In general, eventhistoryanalysisdealswithinference for transitionintensitiesand transitionprobabilities in multi-statemodels. • This includesestimation and hypothesis tests for thesequantities and analysisofregressionmodelswherethesequantitiesarerelated to explanatory variables observed for theindividuals under study (2)

  10. Exampleofsurvivalcurve (churn) S(t) for men S(t) for women Insurance customer Exit to competitor Numberofdays • Formula for expectedsurvival (usingintegration by parts): • Geometricinterpretation: Expectedsurvival is area under curveabove (3)

  11. Look for significantinteractioneffects • The churnvaries for every segment • The difference in churnbetween men and womenvaries for every segment • We say that there is an interaction effect between sex and segment related to churn Churn per professiongroup and per sex

  12. Put all importantfactorstogether in a full eventhistorymodel Variables in themodel: • Numberofproductgroups • Age ofthe policy holder • Sex ofthe policy holder • Region • …..

  13. Process for eventhistorymodel Dataanalysisofeachfactorthatmayaffectchurn Analysisofpossibleinteractioneffectsbetweenthefactors Developmentof full eventhistorymodel Numberofdays

  14. Survivalmodelling: actuarialnotation Actuarialevaluationon a time increment T countslifelength as Y=LT where T may be a year, quarter or month and L an integer. The probabilitydistributionof L is known as a lifetable and is usuallyspecifiedthroughtheconditionalprobabilities On theleftthesurvivalprobability is thelikelihoodof living k periods longer whereasthemortalityonthe right is theprobabilitythattheindividual dies during the last period. Bothquantitiesdependoncurrent age l. Note thefollowingrecursions: And for themortalities (4) survivalprobabilities mortalities (5) (6)

  15. Survivalmodelling: actuarialnotation (5) and (6) can be obtained by notingthatifb>a>=l+kthen (7) is used to get (5) and (6). First (5) is obtained letting (7) (8)

  16. Survivalmodelling: actuarialnotation (6) is obtainedinserting b=l+k+1 and a=l+k (9)

  17. Mortalityintensitymodelling The occurenceofdeath and othereventsmay be modelledusingeventhistoryanalysistechniques (introducedearlierthissession) Thesetechniques have beengreatlysophisticatedthe last 40 years The knowledgeobtainedcan be veryuseful, sinceknowledgeregardingtheeventintensitiesmay be refinedusingregressiontechniques Where is a vectorofregression variables Unfortunately, thistechniquerequires a lot of data (iesurvival data onindividuallevel) So thismay be an importantreasonwhythistechnique has not beenwidely used in thelifeinsuranceindustry. Instead, thelifeinsuranceindustry has sincelongbeenusingtheGomperz-Makehammodel, specified as ie a parametriccurve. The parameters in (10) areestimatedusing ML (maximumlikelihood) techniques. Note thatthismethoddoes not require data onindividuallevel (10)

  18. A numericalalgorithm for expectedsurvival The formula for expectedsurvival is see (3) for details. An algorithm for expectedsurvivalcan be derived, usingtheapproximation Algorithm 12.1., Expectedremainingsurvival • 0 Input {pl}, h, l=y/h, le • 1 P <-1 and E <- ½ • 2 For k=l,…,le do • P<-Ppk and E<-E+P • 3 Return E <-Eh (11) (12)

  19. Applyingthealgorithm for expectedsurvival Algorithm 12.1 implemented in Excel (R is recommended). Resultdisplayedbelow Age

  20. Single life arrangements Two basic cash flowsarethosethatpersists as long as the policy holder is alive and thosethatareone-timeupondeath. and Survivalprobability at time k, given alive at time 0 (at age l0) Agreedpayment at time k Paymentswhilealive Death probability at time k, given alive at time 0 (at age l0) Agreedpayment at time k Payment (one-time) upondeath

  21. Howmortality risk affectsvalue The equivalenceprinciple at time k: Ifthe present valueof all payments is 0 thenthe present valueofpastpaymentsequalsthe present valueoffuturepayments time k Pastpayments (retrospective) Paymentsahead (prospective) The accountsarecredited becausesomebody dies early Retrospective reserve Prospective reserve Insured dies Insuredalive

  22. Life insurancenotation Supposeonemoneyunit is receivedeachof K time periods, butceaseswhentheindividual dies. The money is received at the end ofeachperiod Ifthemoney is received in advance, theexpressionbecomes The expressionsarerelated In arrearswhilealive In avancewhilealive

  23. Computingmortality-adjustedannuities Algorithm 12.2 The coefficients 0 input l0, K, d=1/(1+r), {ql } 1 a<-1, lb<-1, l<-l0-1 2 for k=0,…,K-1 repeat a<-a+b and l<-l+1 b<-b(1-ql)d 3 Return a and a..<-a+b-1. (a and a.. arenow )

  24. An example: Mr(55) revisited Consider MR (55), receivingoneunitofmoney for K periods. Assumethefollowingvalues for thedifferent parameters

  25. Result from Mr(55) revisited The net present value for payments in advancepresented to theleft. The net present value for payments in arrearspresented to the right. In both cases the cash flowsarecompared to cash flowswithoutthemortalityassumption The NPV withoutmortality is 4% greaterthanthe NPV in advance. The NPV withoutmortality is 15% greaterthanthe NPV in arrears. Weseethatthedifferencebetweenthetwo cash flows is greaterwhenpaymentsaredone in arrears. Weseetheeffectofmortalityaccumulate more severly in the case ofarrearsthan in advance, sincethemoney is received at the end oftheperiod and not in advance In advance In arrears 4% 15% K K

  26. Commoninsurance arrangements Life annuitiesSuppose an individual has saving at retirementlr and purchases an annuitythat lasts K periodswithinterruption by death. The mortalityadjustedvalueof a unitannuity is then and thepension s receivedeachperiod is determined from

  27. Commoninsurance arrangements DefinedbenefitsSavingsarebuilt up to support a given pension. Supposetheindividualentersthe arrangement at age l0 and retires at lr to receive a fixedpension s broken off by death or after K periods. With all payments in advancethe present valueofthescheme at the time it is drawn up is

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