An Extended Lee-Carter Model for Mortality Differential by Long-Term Care Status

Transcription

1 Proposal submied o he 2016 ARIA Annual Meeing An Eended Lee-Carer Model for Moraliy Differenial by Long-Term Care Saus Asuyuki Kogure a, Shinichi Kamiya b, Takahiro Fushimi a a Faculy of Policy Managemen, Keio Universiy, Japan b Nanyang Business School, Nanyang Technological Universiy, Singapore Absrac This paper aims o propose a new mehodology o forecas moraliy raes of he longerm care (LTC) populaion wih longeviy risk A major obsacle o devising such a mehod is lack of daa on he number of deahs in LTC populaions, which prevens us from using he convenional moraliy model such as he Lee-Carer model To overcome his difficuly, we propose an eended Lee-Carer model wih a erm represening moraliy differenial due o LTC saus which does no require he daa on he number of deahs in LTC populaions We apply he proposed model o he daa from he Japanese long-erm care insurance sysem Our preliminary resuls shows ha he proposed mehod capures he heerogeneiy in he moraliy rae beween he LTC sauses properly We plan o deliver a more deailed invesigaion a he conference 1 Inroducion Increased human lifeime is accompanied by a greaer chance of becoming disabled, which could require significan addiional coss for long-erm care (LTC henceforh) Therefore, needs for LTC have been increased and financing is cos has been an imporan opic for developed economies Impaired annuiies, enhanced annuiies, and life annuiies ha pay ou a higher rae once an insured becomes LTC disabled may be considered he soluions, which share he characerisic ha pays more o insured wih lower life epecancy which is evaluaed based on insured s paricular healh problems Thus, he key elemen of he soluions is how he moraliy is relaed o healh saes However, sudy on he complicaed moraliy dynamics is limied due o lack of daa Using Ialian naional daa on healh and moraliy, Levanesi and Menziei (2012) show ha he premium raes variabiliy is higher for LTC insurance han for enhanced annuiy wih increased annuiy rae for LTC disabled This demonsraes naural hedging beween life annuiies and LTC insurance Gourierou and Lu (2013) propose an approach o model he enry ino LTC as a unobserved jump in he moraliy inensiy and o derive LTC hazard and moraliy raes from he underlying moraliy daa wih or wihou reference o LTC daa This paper aims o propose a new mehod o forecas moraliy raes of he long-erm care (LTC) populaion wih longeviy risk A major obsacle o devising such a mehod is lack of daa on he number of deahs in LTC populaion, which prevens us from using he Corresponding auhor kogure@sfckeioacjp

2 convenional moraliy model such as he Lee-Carer model To overcome his difficuly, we propose an eended Lee-Carer model wih a erm represening moraliy differenial due o LTC saus which does no require he daa on he number of deahs in LTC populaions We apply he proposed model o he daa from he Japanese long-erm care insurance sysem Our preliminary resuls shows ha he proposed mehod capures he heerogeneiy in he moraliy rae beween he LTC sauses properly The paper is organized as follows In Secion 2, we model moraliy differenials for LTC populaions in he Lee-Carer model The esimaion approach is proposed in Secion 3 In Secion 4 we implemen he model for predicion purpose using he Japanese moraliy and LTC daa Secion 5 concludes 2 Lee-Carer Modeling for differenial by LTC saus Le E j and D j denoe he populaion size and he number of deahs for age ( = min,, ma ) wih LTC saus j in year ( = min,, ma ) Here j ranges from 1(=leas severe saus) up o J(=mos severe saus) Then he oal populaion size and he oal number of deahs for age in year is given by J J D = D j, E = E j D E j (j = 1,, J) are observed, bu D j are no available Suppose ha he force of moraliy of an individual aged a ime in LTC saus j follows he Lee-Carer law µ j = ep{α j + β j κ j } (1) Here we assume ha α j, which represens he age effec of LTC saus j, is wrien as α j = γ + η j We epec ha he moraliy rae becomes higher as he LTC saus ges worse on average Thus we impose a monooniciy consrain: η 1 η 2 η J, (2) which makes sure ha α j is monoone increasing wih j for each κ j may be inerpreed as he common period effec across age due o facors such as medical skills or public healh Thus we assume ha i does no depend on j β j represens he sensiiviy of he moraliy o changes in κ β j may depend on boh and j in general Here for simpliciy we assume ha i depends only on Thus (1) becomes µ j = ep {γ + η j + β κ }, (3) Then we assume ha he moraliy of he oal populaion of age in year is given as he geomeric average of (3): J µ = (µ j ) w j = ep γ + w j η j + β κ, j where w j E j /E If all he LTC saus parameers η j are equal: η 1 = η 2 = = η J = η, 2

3 hen we have µ = ep {γ + η + β κ }, which is he sandard Lee-Carer model by wriing α for γ + η Thus he proposed model is an eension of he sandard Lee-Carer model so as o incorporae moraliy differenial by LTC saus I is ineresing o noe ha he proposed model corresponds o he common facor model of Li and Lee (2005) and he sraified Lee-Carer model in Bu and Haberman (2009) Our model differs wih hese models in wo respecs One is ha our model does no require he deah number daa D j The oher is ha we impose he monooniciy condiion (2) 3 Esimaion 31 Normal model Le m D E denoe he moraliy rae To esimae {γ, η j, β, κ } we assume ha he logarihm of he moraliy y log m follows a normal model iid y = log µ + ε, ε N(0, σ 2 ) J = γ + w j η j + β κ + ε = γ + w η + β κ + ε (4) where ε s are independen errors assumed o be idenically disribued as he normal disribuion wih mean 0 and he variance σ 2 and w In a mari form (4) is wrien as w 1 w 2 w J η 1, η η 2 η J y = γ 1 NT + W η + β κ + ε, (5) where y = y min y ma 1, 1 NT =, W = 1 w min w ma, κ = κ min κ ma, ε = ε min ε ma and N T ma min Maimum likelihood under he equaliy and inequaliy consrains As in he sandard Lee-Carer model we make he assumpions β = 1, κ = 0 (6) 3

4 for he parameer idenificaion Then he maimum likelihood esimaion is equivalen o minimizing ( y γ w ) 2 η β κ f({γ }, η, {β }, κ) = = (y γ 1 NT W η β κ) (y γ 1 NT W η β κ) (7) subjec o he consrains (2) and (6 ) To solve his consrained minimizaion problem we consider a Lagrangian funcion ( ) L({γ }, η, {β }, κ; λ 1, λ 2 ) = 1 2 f({γ ( }, η, {β }, κ)+λ 1 β 1 +λ 2 1 NT κ ) J 1 + µ j (η j η j+1 ), where λ 1, λ 2, µ 1, µ 2,, µ J 1 are Lagrangian muliples Then he Karush-Kuhn-Tucker (KTT) condiions are: L = 0 γ L η = 0 L = 0 β β = 1 κ = 0 µ 1 0, µ 2 0,, µ J 1 0 µ 1 (η 1 η 2 ) = 0, µ 2 (η 2 η 3 ) = 0,, µ j 1 (η J 1 η J ) = 0, When J is small, we can solve he consrained minimizaion problem direcly from he KTT condiions For eample, suppose J = 4 and le η1, η 2, η 3, η 4 denoe he opimal values of η 1, η 2, η 3, η 4 Then we need o consider he following eigh cases: Case 1 η 1 < η 2 < η 3 < η 4 Then i holds ha µ 1 = µ 2 = µ 3 = 0 and he minimizaion problem will be wihou he monooniciy condiion Case 2 η1 < η 2 < η 3 = η 4 Then i holds ha µ 1 = µ 2 = 0 and he inequaliy consrain η 3 η 4 becomes he equaliy consrain η 3 = η 4 Thus we merge he LTC sauses 3 and and 4 ino one saus, 3 = {3, 4}, say, and solve he minimizaion problem for he hree LTC sauses 1, 2 and 3 wihou he monooniciy resrain Case 3 η 1 < η 2 = η 3 < η 4 Then i holds ha µ 1 = µ 3 = 0 and he inequaliy consrain η 2 η 3 becomes he equaliy consrain η 2 = η 3 Thus we merge LTC sauses 2 and and 3 ino one saus, 2 = {2, 3}, say, and solve he minimizaion problem for he hree LTC sauses 1, 2 and 3 wihou he monooniciy resrain 4

5 Case 4 η 1 = η 2 < η 3 < η 4 Then i holds ha µ 2 = µ 3 = 0 and he inequaliy consrain η 1 η 2 becomes he equaliy consrain η 1 = η 2 Thus we merge LTC sauses 1 and and 2 ino one saus, 1 = {1, 2}, say, and solve he minimizaion problem for he hree LTC sauses 1, 2 and 3 wihou he monooniciy resrain Case 5 η1 < η 2 = η 3 = η 4 Then i holds ha µ 1 = 0 and he inequaliy consrain η 2 η 3 η 4 becomes he equaliy consrain η 2 = η 3 = η 4 Thus we merge LTC sauses 2, 3 and 4 ino one saus, 2 = {2, 3, 4}, say, and solve he minimizaion problem for he hree LTC sauses and 1 and 2 wihou he monooniciy resrain Case 6 η 1 = η 2 < η 3 = η 4 Then i holds ha µ 2 = 0 and he inequaliy consrains η 1 η 2 and η 3 η 4 become he equaliy consrains η 1 = η 2 and η 3 = η 4 Thus we merge LTC sauses 1 and 2 ino one saus 1 = {1, 2}, say, and LTC sauses 3 and 4 ino one saus 3 = {3, 4}, say, and solve he minimizaion problem for he hree LTC sauses and 1, 2 and 3 wihou he monooniciy resrain Case 7 η 1 = η 2 = η 3 < η 4 Then i holds ha µ 3 = 0 and he inequaliy consrain η 1 η 2 η 3 becomes η 1 = η 2 = η 3 Thus we merge LTC sauses 1, 2 and 3 ino one, 1 = {1, 2, 3}, say, and and solve he minimizaion problem for he hree LTC sauses and 1, 2 and 3 wihou he monooniciy resrain Case 8 η 1 = η 2 = η 3 = η 4 Then i holds ha η 1 = η 2 = η 3 = η 4 Thus we merge all he saus ino one and solve he minimizaion problem, which is equivalen o he sandard Lee-Carer model Le f k denoe he minimized value of (7) for case k and le D denoe he subclass of he eigh cases for which he monooniciy condiion (2) holds Then we choose he one ha minimizes f k among he cases for which he monooniciy condiion (2) saisfies The minimizaion problem for each case is solved by considering a Lagrangian funcion of he form ) L({γ }, η, {β }, κ; λ 1, λ 2 ) = 1 2 f({γ }, η, {β }, κ) + λ 1 ( β 1 + λ 2 ( 1 NT κ ) (8) The soluion o his minimizaion is given by he following algorihm: 5

6 Algorihm for he esimaion 1 Le he iniial values of γ, β, κ be γ (0), β (0) κ (0) We may use he esimaes of he sandard Lee-Carer model as he iniial values 2 For each ν = 1, 2, (a) Updaing η ( ) 1 η (ν) = W W W ( y γ (ν 1) 1 NT β (ν 1) κ (ν 1)) (b) Updaing γ (c) Updaing β γ (ν) = 1 y + 1 N T N T w η (ν) β = κ(ν 1) (y γ (ν) W η (ν) ) κ (ν 1) κ (ν 1) β (ν) = β 1 β + 1 N X N X (d) Updaing κ κ = β (y γ (ν) W η (ν) ) (β(ν) ) 2 κ (ν) = κ 1 κ N T 3 We repea he above unil each parameer values converges Anoher way o implemen he monooniciy consrain (2) is o add he penaly erm for he monooniciy o (8) as discussed in Tibshirani, Hoefling and Tibshirani (2011) Tha is, we consider o minimize ( ) 1 2 f({γ ( }, η, {β }, κ) + λ 1 β 1 + λ 2 1 N κ ) J 1 + λ 3 (η j+1 η j ) + wih + indicaing he posiive par, + 1( > 0) When leing µ 3 o, we obain he soluion o o he original consrained minimizaion problem 4 Applicaion 41 Daa We use he daa from he public long-erm care insurance sysem in Japan which sared in April 2000 o deal wih he increase in demand for he nursery care associaed wih he rapid ageing Under he sysem hose in need of nursery care are caegorized ino seven sages: required suppor 1 and 2 and required care 1 o 5 For he presen analysis we divide he oal populaion ino four LTC saus LTC saus 1 corresponds o people no in need of nursery care LTC 6

7 saus 2 consiss of required suppor 1, 2 and required care 1 LTC saus 3 consiss of required care 2 and 3 LTC saus 4 consiss of required care 4 and 5 We regard he populaion size in each LTC saus as he number of recipiens in he same LTC saus For years from 2001 o 2014 he numbers of recipiens in each sage are available for age class of 40-65, 65-o 69, 70-74, 75-79, 80-84, and 90+ We linearly inerpolae he numbers of recipiens for age class o obain he numbers of recipiens a each age 42 Resuls The values of he objecive funcion (7) are repored in Tables 1 and 2 The corresponding values of η j s are lised in Tables 3 and 4 We see ha he objecive funcion is minimized for case 5 among he cases for which he monooniciy condiion (2) saisfies boh in he male and he female populaions Case 5 consiss of wo LTC sauses, one corresponding o people no in need of nursery care and he oher corresponding o people in need of nursery care The esimaed parameers for case 5 are depiced in Figures 1 and 2 We see a clear moraliy differenial beween he wo LTC sauses I is ineresing o observe ha he sandard Lee- Carer model and he eended model give almos idenical values for β and κ case f rank case f rank Table 1: Objecive funcion (male) Table 2: Objecive funcion (female) Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 η η η η Table 3: Esimaed η j for each case (male) Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 η η η η Table 4: Esimaed η j for each case (female) 7

8 α, γ, γ + η 1, and γ + η 2 β α (Sandard Model) γ (Eended Model) γ + η 1 (Sae1, Eended Model, η 1 = ) γ + η 2 (Sae2, Eended Model, η 2 = ) β (Sandard Model) β (Eended Model) age age κ κ (Sandard Model) κ (Eended Model) year Figure 1: Case 5 (male) 8

9 α, γ, γ + η 1, and γ + η 2 β α (Sandard Model) γ (Eended Model) γ + η 1 (Sae1, Eended Model, η 1 = ) γ + η 2 (Sae2, Eended Model, η 2 = ) β (Sandard Model) β (Eended Model) age age κ κ (Sandard Model) κ (Eended Model) year Figure 2: Case 5 (female) 9

10 43 Forecasing moraliy raes We plan o forecas moraliy raes based on ime series predicion of he calendar parameers κ This can be wrien as follows: µ,+k,j = ep{ˆγ + ˆη j + ˆβ κ +k }, k = 1, 2, where κ +kj represens he forecas period effecs The mos common choice for ime series erapolaion mehods applied in he Lee-Carer framework are he ARIMA(p, d, q) processes, which is ( ) ( ) p q 1 ϕ i L i (1 L) d κ = δ θ i L i ε i=1 where L represens he shif operaor and {ε } are error erms In he majoriy of applicaions ARIMA(0, 1, 0), he random walk wih drif, is he sandard choice, which can be epressed as: κ = κ 1 + δ + ε Esimaed κ for he presen analysis given in Figures 1 and 2 do no look like a random walk We are currenly invesigaing which model is suiable for our purpose 5 Concluding remarks In his paper we have proposed an eended Lee-Carer model wih a erm represening moraliy differenial by LTC saus We apply he proposed model o he daa from he Japanese longerm care insurance sysem Our preliminary resuls have demonsraed he heerogeneiy in he moraliy rae beween he LTC sauses We plan o deliver a more deailed invesigaion a he conference One problem wih he presen approach is ha he parameer uncerainy is no aken ino consideraion To deal wih his problem, we may formulae he whole implemenaion by a Bayesian framework as discussed in Kogure and Kurachi (2010) References [1] Bu, Z and S Haberman (2009) llc: a collecion of R funcions for fiing a class of Lee- Carer moraliy models using ieraive fiing algorihms (Repor No Acuarial Research Paper No 190) London, UK: Faculy of Acuarial Science and Insurance, Ciy Universiy London [2] Gourerou, C and Liu Y (2013) Long-Term Care and Longeviy, DISCUSSION PAPER PI-1306, The Pensions Insiue Cass Business School Ciy Universiy London [3] Hoermann, G and J Rus (2008), Enhanced annuiies and he impac of individual underwriing on an insurer s profi siuaion, Insurance: Mahemaics and Economics 43 (1), [4] Kogure, A and Y Kurachi (2010), A Bayesian approach o pricing longeviy risk: wih applicaions o Japanese moraliy daa, Insurance: Mahemaics and Economics, 46 (1), [5] Levanesi, S, and M Menziei (2012), Managing longeviy and disabiliy risks in life annuiies wih long erm care, Insurance: Mahemaics and Economics, 50 (3), i=1 10

11 [6] Li, N and R Lee (2005), Coheren moraliy forecass for a group of populaions: An eension of he lee-carer mehod, Demography 42 (3), [7] Tibshirani, R, H Hoefling, R Tibshirani (2011), Nearly-Isoonic Regression, Technomerics 53 (1),