Working Paper Number 13/15. Projections of dynamic generational tables and longevity risk in Chile

Transcription

1 Number 13/15 Projecions of dynamic generaional ables and longeviy risk in Chile Economic Analysis Madrid, April 8, 213

2 13/15 Working Paper Madrid, April 8, 213 Projecions of dynamic generaional ables and longeviy risk in Chile Auhors: Javier Alonso y David Tuesa, Diego Torres, Begoña Villamide March 7, 213 Absrac The increase in longeviy risk is leading o serious challenges for economies. Indusries such as insurance and pensions, which are mos closely relaed o he managemen of he risks of an aging populaion, have for a number of years eperienced direc effecs of his kind. To counerbalance his, hey have developed echniques for consrucing moraliy ables in order o projec he fuure rends of life epecancy a birh and hus reduce he level of uncerainy ha his marke by is naure involves. Developed counries have led echnical improvemens for consrucing hese ables, while Lain American counries have lagged behind significanly in his respec. Given ha hese counries canno ye develop ables weighed by social and medical aspecs, i is highly probable ha his siuaion will coninue. Tha is why his sudy aims o consruc a forecas for moraliy raes, based on projecion models of he ARMA (p, q) ype and non-parameric conras mehodology. The sudy is based on he case of Chile, which provides mos informaion for consrucing a model for a Lain American counry. The esimaes show ha he official moraliy ables in Chile could include significan lags by 25, which will have major negaive effecs on he pension and insurance indusry, in he hypoheical case ha hey were no updaed. In anoher eercise, using he moraliy able esimaed in his work, we found ha if pensions in Chile are no o lose heir purchasing power, he conribuion rae would have o be increased by 8 percenage poins in he case of men and 4 in he case of women. Given ha Chile is he bes developed counry in he region wih respec o moraliy ables, he negaive effecs on he res of Lain America could be even more worrisome. Key words: Pensions, insurance, longeviy risk, moraliy ables, Lain America, Chile. JEL: G23, J32, G22. Page 2

3 Madrid, April 8, Inroducion In he second half of he las cenury, he world began o show clear signs of major demographic changes ha would affec he way labor markes and, in he final analysis, he economy iself, behaved. One of hese changes is he greaer longeviy of life epecancy. As progress has been made in healh, nuriion, governmen healh policies and lifesyles, life epecancy has increased subsanially. A he sar of he 2h cenury he average life epecancy was barely over 3, and was below 5 even among he highes-income groups (Prenice, 26). As a consequence of ha, his average increased o 48 years mid-way hrough he cenury, and currenly is esimaed o be over 66 years, wih a figure of over 8 in some counries. Alhough hese average figures sill show major conrass beween differen geographical areas, paricularly beween he so-called developed and lower-income counries, i is also rue ha a very swif process of convergence is underway as he social and medical progress spreads. In fac, in he las 3 years he number of people who live o 6 years or over has doubled and i is epeced o double again by 25, when his age group will be larger han he populaion under he age of 14. The ransiion o an economic realiy wih longer-living populaions involves imporan challenges for he economy as a whole, alhough in a paricularly direc way for hose indusries ha have focused heir objecive on he coverage of risks associaed wih old age, such as pensions and insurance. The "risk" of living longer does no appear o have reached is limi ye, so i incorporaes an elemen of uncerainy wih respec o gauging he producs offered correcly. A key elemen when i comes o designing insurance producs (and hus adjusing heir respecive prices) o address he risk associaed wih inacive old age, is o have well designed moraliy ables wih projecions ha can provide he bes possible balance beween he needs of buyers and sellers. Developed counries were he firs o noe he pressures from an aging sociey, which began o make inroads firs on he public finances, and hen on he privae insurance indusries hemselves. However, developing counries are converging a an increasingly fas pace. In he 195s, he gap beween life epecancy a birh for hese wo groups of counries was around 2 years; i has now been narrowed o around 5 years. The riches counries were he firs o make significan innovaions in he design of moraliy ables, and hey were updaed consanly. The mos advanced counries such as Germany even include social and bio-medical informaion o fine-une heir esimaes. Lain America has lagged behind his progress. I could be said ha only in he las decade have serious seps been aken o improve he ables, wih he inclusion of dynamic ables. Chile has aken he lead in his respec. Neverheless, hey sill lag behind significanly because of he mehodologies used, which canno include he whole range of facors ha inerac in he projecions of moraliy ables. Wih his in mind, his paper includes an economeric analysis designed o reveal he poenial gap produced by he moraliy ables currenly used in Lain America. For his purpose we have focused on he case of Chile, because i is he counry wih mos informaion available for making he necessary calculaions. In general, he problems found in he sudy sugges ha he challenges ha could be facing oher moraliy ables in he region could be bigger, bearing in mind ha Chile is he counry ha has worked mos seriously and for he longes ime on his subjec. A difficuly for Lain American counries in general, and Chile in paricular, is ha hey do no have sufficien pas informaion on moraliy rends o consruc heir own long-erm ables. One way of dealing wih his is o compare he informaion available in hese counries' ables wih ha of oher counries for which we have more informaion. We can use sample equivalence ess o idenify hose ha offer he greaes similariies. Page 3

4 Madrid, April 8, 213 In our case, he sample equivalence analysis uses non-parameric ess ha compare he level of moraliy (sign es), form (runs es) and he eisence of a similar paern beween boh ables ( χ2 es). The applicaion of hese ess will allow us o deermine which counry has moraliy ables ha are saisically comparable o he Chilean, N years ahead/behind. The sudy also includes he addiional condiion ha he equivalence of he ables mus be sable and prolonged over ime, o ensure ha his equivalence is no limied o a chance episode in ime. The model projecs he moraliy raes in Chile using he ARMA (p, q) mehodology for each of he 11 ages and for each gender, wih a oal of 22 regressions. As will be seen in he sudy, here are significan differences beween he official ables and hose given by our esimaes. Finally, he work calculaes he economic impac of he use of inadequae moraliy ables. Afer his inroducion, he second secion reviews how far Lain America has progressed in he developmen of moraliy ables. The hird secion discusses differen ypes of mehodologies for projecing moraliy ables. Secion 4 presens he esimaes made o calculae he projeced moraliy ables for Chile and compare hem wih hose currenly used. Secion 5 demonsraes he economic effecs of using inadequae moraliy ables; and finally, he las secion presens he main conclusions of he sudy. 2. Moraliy ables in Lain America and longeviy risk Demography is he science ha sudies human populaions, heir characerisics and heir dynamics. The firs works o deal wih saisical demographic daa were developed by Ibn Khaldun ( ). However, he science only finally ook off in he Unied Kingdom wih he works of John Graun in he 17h cenury and Rober Malhus in he 18h. Populaion censuses are he main source of demographic informaion. Alhough censuses have been held in various culures for housands of years wih he aim of making a collecion easier, modern censuses arose a he sar of he 19h cenury in European counries such as he Unied Kingdom and Sweden. The bigges census projec in Lain America was launched when he Iner-American Saisical Insiue (IASI) launched is Census of he Americas (COTA) program, which was followed a large number of counries in he region in 195. The delay in he availabiliy of saisics, combined in many cases he need for beer echnical skills on he par of specialiss in some counries, has led o a lag in he availabiliy of informaion such as moraliy ables ha are needed o develop acuarial science. In Chile, unil 24 he RV-85 moraliy ables were used, as esablished by Circular No. 656 of he pension manager supervisor Superinendencia de Adminisradoras de Fondos de Pensiones (AFP) 1. These ables were originally designed o be applied o he US populaion, bu were adjused using CELADE daa from he 1982 census. However, saring in 2 he RV-85 ables began o show signs of being ou of dae for he calculaion of life epecancy. This deficiency led o he AFP Supervisor (Pension Supervisor) and Supervisor of Securiies and Insurance o consruc he RV-24 moraliy ables. The new ables updaed he calculaion of he programmed reiremen benefis and life annuiies using he daa of reiremen pensioners for he period , inclusive. Using adjusmen and graduaion echniques, hey deermine he probable life epecancy of fuure pensioners. The mos noable resul is ha he RV-24 ables showed he RV-85 had underesimaed life epecancy. For eample, he life epecancy of women aged 6 increased by up o 3.6 years wih he new ables. In he case of men he error was no as large, a around.51 years below he real figure. 1 For more deails, see: hp:// hp:// and he press release: hp:// Page 4

5 Madrid, April 8, 213 Colombia had been using he same moraliy able for pensioners since 1994 (RV89, based on he daa for 1989). In 2, he Colombian banking supervisor Superinendencia Bancaria de Colombia issued Circular 71/2 2 requesing informaion on he moraliy of acive conribuors, pensioners, he disabled and people wih invalidiy, wih he aim of preparing upo-dae moraliy ables. An analysis of hese daa revealed ha hese ables had o be updaed, and in 21 Resoluion 1555/21 3 resolved o replace he RV89 wih he RV8 ables. As in he case of Chile, he updaed ables in Colombia revealed ha life epecancy had been underesimaed. The RV8 calculaed a life epecancy for women aged 6 of 27 years, while he calculaions of he Naional Adminisraive Deparmen of Saisics (DANE) pu he figure a around 22, a difference of 5 years. In he case of men, he difference was no as grea, a 3.3 years. Peru had since 1993 used he Chilean RV85 ables, bu afer 13 years, in 26, Resoluion 354/26 approved he use of modified Chilean RV-24 ables. More recenly, Resoluion 17728/21 included as obligaory he use of RV-24 ables modified and adjused o he Peruvian eperience 4. The moraliy ables used o calculae life annuiies in he Meican social securiy sysem are prepared by he Naional Insurance and Finance Commission, which is he regulaor and supervisor of pension insurers. From 1997 o 28 hey used moraliy ables called EMSSA 97 (Meican Social Securiy Acive Eperience), based on a sudy projecing moraliy prepared by he Naional Populaion Council (CONAPO) for he period They were broken down by gender, bu did no ake ino accoun improvemens in moraliy. In 27 a saisical analysis of he observed moraliy of pensioners wih life annuiies was carried ou for he period This showed a greaer survival rae han epeced. As a resul, in 29 i was decided o adop he dynamic moraliy ables called EMSSA 29, which ake ino accoun improvemens in forecas life epecancy for he coming years, in accordance wih he demographic rend for increased longeviy, and using he sudy of projeced moraliy in Meico for he period carried ou by CONAPO. Char 1 sums up he differen mehodologies used o calculae moraliy ables in Chile, Peru, Colombia and Meico. Chilean and Colombian ables are clearly beer posiioned, agains an ideal benchmark of mehodologies for ables based on social and bio-medical aspecs, which would ensure a beer esimae of life epecancy and is fuure developmen. Char 1 Calculaion mehodologies of moraliy ables in Lain America a he age of 65 years old Calculaed by he naional saisical insiues Using he ables calculaed by he official populaion censuses, usually each 1 years. Significan deviaions can be made in inercensal esimaes Source: BBVA Research Case of Peru Adopion of he ables of a similar counry enering an adjusmen of more/less years Case of Chile and Colombia Weighed ables by socio-biomeric To do permanen work aspecs on eploiaion of adminisraive daa of he members of he sysem 2: For more deails see Circular 71/2: hp:// 3: hp:// 4: For more deails see he following resoluions: hp://inrane1.sbs.gob.pe/idxall/seguros/doc/resolucion/pdf/ r.pdf hp://inrane1.sbs.gob.pe/idxall/seguros/doc/resolucion/pdf/ r.pdf hp://inrane1.sbs.gob.pe/idxall/seguros/doc/resolucion/pdf/ r.pdf Page 5

6 Madrid, April 8, Projecion mehodologies for moraliy ables The firs works o projec moraliy can be raced back o 1875, when he Swiss asronomer H. Gylden fied a sraigh line over a sequence of general moraliy raes of he Swiss populaion for he period and hen erapolaed i as a projecion (Piacco e al., 29). Following he classificaion of models used by Camarda (28) we can mainly disinguish parameric and non-parameric models. 3.1 Models for projecing moraliy ables We will now briefly discuss each of he models used o projec moraliy ables. However, we firs have o define some general measuremens of moraliy ha will be used laer: Le us define X as a random non-negaive and coninuous variable ha describes he fuure life span of individuals. Three funcions characerize and describe he disribuion of X : he funcion of probabilisic densiy, he funcion of survival and he risk raes (or moraliy risk raes). The survival funcion is defined as he probabiliy ha an individual will survive more han years, using he formula: Equaion 1 S ( ) Pr( X ) f ( ) d Equaion 1 is in urn he complemen of he cumulaive disribuion funcion, i.e.: Equaion 2 S( ) 1 F( ) Using Equaion 1 and Equaion 2 we ge Equaion 1 f ( ) S ( ) Anoher fundamenal funcion is he "risk funcion", which measures he probabiliy ha a cerain even may occur o an individual over ime; in our case i is he probabiliy ha an individual dies. The probabiliy ha an individual over he age of 35 will no live longer han 75 years is given by he following condiional probabiliy: Pr(35 X 75 X 35) F(75) F(35) S(35) In general erms, he oucome probabiliy (risk funcion) is he limi of he condiional probabiliy when he inerval becomes infiniely small, i.e. Page 6

7 Madrid, April 8, 213 Equaion 4 h( ) Pr( lim X X ) lim F( ) S( ) F( ) f ( ) S( ) d ln S( ) d The las equivalence is given using Equaion 3 and he derivaion rule for logarihmic funcions. Using Equaion 4 we can derive an ideniy ha relaes he survival funcion S () o he risk funcion h () Equaion 5 h( ) d ln S( ) d h( u) du ln S( ) h( u) du ep d ln S( u) du du h( u) du S( ) S()= ep[ -H() ] Equaion H ( ) h( u) du is called he cumulaive risk funcion. Ne we will describe each of he models. 3.2 Parameric models: Age-based models Gomperz (1825): Gomperz was one of he firs o model moraliy as a level of risk. He observed ha in he age range of 3 o 8, moraliy risk increased eponenially as age increased. He herefore suggesed ha he level of risk should be modeled as follows: h ( ) a b e Where is he age of he individual wih relaion o a baseline age, a is he moraliy rae a age "" (generally 3 years), and b is he rae of increase in moraliy per era year of life. Using Equaion3 and Equaion 5 we can derive he densiy funcion for he Gomperz disribuion. Equaion 6 f ( ) a e b b u b 1) ep a e du a e ep b a( e b Equaion 7 f ( ) a e a(1 e b b b ) ep Page 7

8 Madrid, April 8, 213 Makeham (186): Makeham eended Gomperz's equaion by including a consan ha absorbed he moraliy risk ha is independen of age: h ( ) c a b e In his case as well, following he same procedure as above, we can derive he probabiliy densiy funcion: Equaion 8 f ( ) a e c a(1 e b b b ) ep This model, as in he case of Gomperz, only models moraliy for adul ages. The models presened below aim o model he special characerisics of all ages, i.e. he high rae of infan moraliy, which falls drasically in he iniial years of life; a booming-ou during youh and adulhood and an upurn in moraliy afer 8 years. The models of Perks (1932), Thacher (1999): logisic models have been among he main models proposed for he behavior of moraliy raes a ages of 8 and over. Perks was he firs o propose a logisic modificaion o he Gomperz-Makeham models. The modified equaion is: Equaion 9 h( ) c a 1 e e b b As we can see, his funcion covers ha of Makeham (when ) and ha of Gomperz ). The model proposed by Thacher is similar o ha of Perks. (when and c Models ha describe he behavior of all age groups Heligman and Pollard (198) have consruced a model ha aims o describe he behavior of he moraliy rae in all age groups: Equaion 1 h( ) A C ( B) De 2 E(ln ln F ) GH 1 GH Where A, B,..., H are he parameers of he model. This paramerizaion makes he calculaion difficul, and i is also difficul o give a meaning o he esimaed parameers. Siler (1983) proposed a model ha inegraes 3 models of moraliy o describe he behavior of all he age groups wih 5 parameers. Anson (1988) proposed a fifh-degree polynomial o represen he moraliy risk raes in humans Age and ime-based models The above models only capured he fac ha he moraliy rae will change as age increases. In his secion, we will commen he models ha have been developed o include he ime dynamic of moraliy raes Relaional models These models relae a reference (sandard) moraliy rae, for eample ha obained using he Gomperz model, o he moraliy rae of he populaion a ime "" and a specific age. Page 8

9 Madrid, April 8, 213 Himes e al. (1994) proposed a relaional model of he ype: Y ( ) I J Where Y j () is he logisic ransformaion of he moraliy raes for each age, is he componen of moraliy independen of age and ime, I is a variable dummy for age, J is a variable dummy for he year analyzed, is he age-specific moraliy rae ha is independen of he passage of ime, due o progress in quesions of healh) The Age-Period-Cohor Model (APC) is he moraliy rae corresponding o ime (such as These models are developed wih he aim of separaing ou changes in he moraliy rae due o hree demographic coordinaes: age, period and cohor. Mahemaically his model suggess ha he naural logarihm of he moraliy raes may be represened as follows: Equaion 11 ln( m) Logarimo naural Tasa de Moralidad c Efeco edad Efeco endencia emporal Efeco Cohore Efeco edad y iempo 1,...,m 1,...,T c 1,..., m T 1 The problem wih his model is ha of he oal 2m+2n-1 parameers o be calculaed (in oher words, m parameers for age, plus T parameers for ime and m+t-1 parameers for he cohors), only 2m+2n-4 are idenifiable (if we esimae m-1 parameers of age, he m can be obained as a linear combinaion of he ohers, as is he case wih ime and he cohor, so we have 3 parameers ha are linear combinaions of he previous ones) The Lee-Carer model Lee and Carer (1992) reduced he compleiy of he ATC model presened above and ransformed i ino a bilinear one (i.e. adding a muliplicaive ineracion beween he demographic dimensions ha affec moraliy). Equaion 12 ln(m ) 1,...,m 1,...,T Page 9

10 Madrid, April 8, 213 The variance of is assumed consan for all ages and periods (his assumpion is relaed in some varians of his model). This model requires addiional resricions for is calculaion. Commonly 1 T 1 n ln( m ) is imposed, so we focus he model on he average moraliy raes for he period under analysis. Using his resricion, we can inerpre ha represen he fied effec of age or deviaions wih respec o he mean observed over ime. is a moraliy rae ha varies over ime. This is one of he mos commonly used models for a variey of demographic purposes and i is considered as he sandard model for modeling and predicing moraliy raes. Wha is aracive abou he model is ha he moraliy rae obained rend. The procedure used for he projecion is as follows: 1) Calculae,, y 2) Model as an ARIMA process sums up he "global" 3) Wih he esimaed ARIMA model, projec he values of 4) Use he esimaed values of y ogeher wih he projecions of projeced moraliy raes. o obain he As we will see in he following secion our model could be classified wihin his las group (parameric age-period models). 4. Descripion of he mehodology applied in his sudy For he projecion eercises, he moraliy raes were esimaed using ARMA(p, q) models for each of he 11 ages and for each gender, wih a oal of 22 ARMA(p,q)regressions: Equaion 13 (1-Φ 1 B Φ 2 B 2 - Φ p B p )y = (1+θ 1 B + θ 2 B 2 + θ p B p )u + α + β + λ 2 + δ 3 Where y represens he Naperian logarihm of he moraliy rae for he populaion of a cerain age and se, B is he lag operaor (By = Y -1 ), represens he ime rend and u is he par no capured by he ARMAX model. We assume ha u is duly saionary, in oher words he average independen of he ime and equal o zero and he auocovariance of order s are only affeced by he lapse of ime beween he periods and do no depend on he ime. In simple erms, we assume ha here is a dynamic ha can be esimaed, and ha consiss of wo pars: 1) a deerminisic rend ha may be approimaed by a mahemaical equaion; and 2) a sochasic rend whose rajecory is influenced by he pas values of he moraliy rae and presen and pas innovaions. The firs par of his dynamic (he deerminisic rend) is considered in he ARMAX model as an eogenous eplanaory variable (X), while he second "sochasic rend" is modeled wih an auoregressive moving-average model. The above wo paragraphs can be shown in algebraic erms. Page 1

11 Madrid, April 8, 213 Solving Equaion 11 and carrying ou basic algebraic manipulaions we obain he variaion of he moraliy rae ( y ): y = ( 1 y y p y -p ) +(1 + 1 B + 2 B p B p ) u +α + β y -1 = ( 1 y y p y -p-1 ) +(1 + 1 B + 2 B p B p ) u -1 +α + β(-1) + (-1) 2 + (-1) 3 y -1 = ( 1 y y -2 + p y -p )+( u + 1 u u p u -p-1 )+(β+ (2-1)+ ( )) Variación Tendencia esocásica Tendencia deerminisa asa de moralidad We do no impose a specific funcional formula o esimae he deerminisic rend; raher, we selec a formula ha adaps bes o he daa (we es wihin all he linear, quadraic and cubic equaions). This selecion is carried ou using he Schwarz informaion crierion, which is consisen, i.e. for large samples i will end o selec he correc model if he assumpions are correc. In algebraic erms he Schwarz crierion is: Equaion 14 SBIC = ln ( 2 ) + (k/t) ln T However, for he purposes of he calculaion we will use an equivalen formula. Equaion 15 SBIC ln T 1 u k T ˆ ln T Equaion 15 is obained using he fac ha E u ), which implies ( T uˆ 2 1 ˆ ; in T addiion, as for all he models analyzed T is a consan, he only variaion being k, minimizing equaion 15 is equivalen o minimizing equaion Non-parameric ess. Sample equivalence es. The Chilean case As we commened above, when a deermined counry does no have reliable informaion available on moraliy raes, i can adop hose of anoher counry as is own, assuming ha he wo populaions are similar. The differences in life epecancy may be adjused by advancing/delaying he years wih which he curren moraliy of he counry in quesion is compared. This mehodology would no be appropriae for he case of Chile, as he counry already has is own moraliy ables (RV24 and RV29) ha have been prepared following a major echnical effor. However, RV4 and RV9, which are updaed every five years, do no have a sufficien saisical hisory o projec he relevan improvemen facors in he long erm. The goal is herefore o projec he Chilean moraliy ables o measure he longeviy risk ha may occur in he fuure. To do so, we will compare he curren moraliy ables of Chile wih hose counries ha have ables of sufficien qualiy and regulariy of informaion so ha by Page 11

12 Madrid, April 8, 213 projecing he ables of he counry in quesion, we can obain ones for Chile, under cerain crieria of sample equivalence. Forfar e al. (1988) have proposed a classic mehod of comparing moraliy ables associaed wih differen eperiences. We can use his o esablish he crierion ha would allow us o sae ha wo moraliy ables of wo differen counries are saisically equivalen. The auhors propose non-parameric ess ha compare he level of moraliy (signs es), he form (runs es) and he eisence of a similar paern beween boh ables (χ2 es). The applicaion of hese ess will allow us o check wha counry presens moraliy ables ha are saisically comparable o he Chilean, N years ahead/behind. We also apply an addiional condiion: ha he equivalence of he ables mus be sable and prolonged over ime, so ha his equivalence is no limied o a casual episode in ime. Based on he eperiences of moraliy in Chile (TCH) and in he counry used for comparison (TPC), for which here are survivor series TCH R and TPC R o age, where he oal number of deahs a age is A A TCH A TPC, he probabiliy of deah would herefore be TCH A TCH TPC q and q TPC respecively. R TCH A TPC R For he purpose of applying non-parameric ess, we will begin by defining he null hypohesis and he alernaive hypohesis. The null hypohesis is ha here is no difference beween he moraliy of he group (TCH) and he moraliy of he comparaive counry (TCP) (in oher words, any difference beween he moraliy of he wo populaions is due o he sample or chance). The alernaive hypohesis is ha here is a difference beween he wo ses of daa. Thus if i is rejeced, i means ha he daa do no come from he same populaion. In his work, we have chosen a deermined level of significance of.5, wih one or wo ails, as indicaed. The specific saisical form of es is seleced according o each es. Mos of he ess used are based on he hypohesis ha he number of deahs a each age A X, can approimae a normal disribuion, for A X 5, which may no occur for ereme ages. If his does no occur, he ages mus be grouped ogeher unil he hypohesis is verified. Tha is why he daa have been grouped ogeher for ages X 84. Once he es saisic has been deermined, we formulae he decision rule. This involves deermining a number ha separaes he region where H is no rejeced from he region of rejecion. This number is called he criical value and is deermined by using he probabiliy disribuion associaed wih he es saisic, as well as he level of significance. The decision rule is ha i is no rejeced if he calculaed value of he es saisic is lower han he criical value. Essenially, his rule indicaes ha if here are major differences beween he wo moraliy eperiences, he null hypohesis H should be rejeced. Oherwise, he hypohesis is no rejeced Signs es In his es, he difference beween he probabiliy of deah a age, for each of he populaions TCH and TPC. q q is calculaed TCH TPC X The null hypohesis H is ha here is no difference beween he moraliy of group TCH (he moraliy of he Chilean populaion) and he moraliy of he group in he comparaive counry Page 12

13 Madrid, April 8, 213 (TPC). In oher words, any difference in erms of he level of moraliy of he comparison populaions is due o he sampling. The alernaive hypohesis is ha here is a difference beween he wo ses of daa. If H is rejeced, means ha he daa do no come from he same populaion. For he purpose of deermining wheher o rejec H, a level of significance is chosen. In our case a level of significance of.5 wih wo ails is seleced. The number of posiive signs (NP) is chosen as he es saisic. On he hypohesis ha i is rue, here will be an equal probabiliy (p=5%) of he differences being posiive or negaive. NP herefore follows a binomial disribuion of N parameers, where in our case N is he number of age classes and p=5%: N! 1 P(NP = r) =, r =, 1, 2... N r! (N - r)! 2 N E[NP] = N 2 y V[NP] = N 4 For he purposes of deermining wheher o rejec H,a level of significance of 5% has been chosen, and H is rejeced if P(NP = r),25 or if P(NP = r,25.. In oher words, he hypohesis ha he moraliy of each populaion is similar for very low values or very high for posiive signs Runs es In his es, he null hypohesis H esablishes ha here is no difference in he form of he moraliy able for he group TCH and he moraliy of group TCP (in oher words, any difference beween he moraliy of he wo populaions is due o he sampling). The alernaive hypohesis H 1 is ha here is a difference beween he wo ses of daa. Thus if i is rejeced, i means ha he daa do no come from he same populaion. TCH TPC This es also calculaes he difference beween he probabiliies of deah a age q q, for each of he populaions TCH and TPC and i is verified wheher he signs of he difference are posiive or negaive. Le n 1 be he number of posiive signs and n 2 he number of negaive signs, wih n1 n2 N. The es saisic chosen is he number of groups wih one or more consecuive deviaions of he same sign (NR), which can be epressed as: X If r=2k (if i is even): P(NR = r) = 2(n 1-1)! (k - 1)! (n 1 - k)! (n 2-1)! (k - 1)! (n 2 - k)! n 1! n 2! N! Page 13

14 Madrid, April 8, 213 If r=2k+1 (if i is odd): (n P(NR = r) = 1-1)! (n 2-1)! n 1!n 2! + (k - 1)! (n 1 - K)! k! (n K)! N! (n 1-1)! k! (n 1 - K - 1)! (n 2-1)! (k - 1)! (n 2 - K)! n 1!n 2! N! The average number of runs is deermined by: E[NR] = 2n 1 n 2 N + 1 And is variance is: Var[NR] = 2n 1 n 2 (2n 1 n 2 - N) N 2 (N - 1) For he purposes of deermining wheher o rejec H, a level of significance of 5% has been chosen and H is rejeced if P(NP = r),25 or if P(NP = r, The chi2 es 2 In his es, he null hypohesis H esablishes ha here is no difference beween he moraliy of he group TCH and he moraliy of group TCP (in oher words, any difference beween he moraliy of he wo populaions is due o he sampling). The alernaive hypohesis H 1 is ha here is a difference beween he wo ses of daa. If H is rejeced, i means ha he daa do no come from he same populaion. The following raios are calculaed for he purpose of applying he chi-squared es: Z TCH X A R TCH X TCH R TCH X p q q and Z TPC X A R TPC X TPC R TPC X p q q The es saisic is given by he following epression: χ2 = [(Z X TCH ) 2 + (Z X TPC ) 2 ] If H is rue, he es saisic 2 follows a chi-squared disribuion wih N degrees of freedom. For he purposes of deermining wheher o rejec H, a level of significance of 5% has been 2 chosen and herefore H is rejeced if The daa The mos eensive informaion available on moraliy ables is included in he Human Moraliy Daabase (HMD). The HMD5 represens a join effor beween he Deparmen of Demography a he Universiy of California, Berkeley and he Ma Plank Insiue for Demographic Research o prepare and compile deailed daa on populaion and moraliy and made available for he whole scienific communiy. Currenly here is informaion for 37 counries, including Chile. 5 (hp:// Page 14

15 Madrid, April 8, 213 Wilmoh e al. (27) offer a broad and deailed descripion of he mehodology used in he daabase. According o Canudas-Romo (28) he hisorical moraliy series for Chile included in he HMD covers he period from 1992 o 25. The auhor saes ha he main reasons for resricing he HMD series o he period since 1992 is ha only in he las wo censuses of 1992 and 22 is he level of incorrec age informaion under 3%, and a leas 9% of he deahs during his period were cerified by a docor. The resuls Comparing he Chilean ables from 1992 o 25 wih hose of 22 counries6 in he daabase on which here is sufficien hisorical informaion o carry ou economeric projecions, we have obained resuls ha allow us o make a long-erm projecion for he case of Chile. Table 1 shows he counries in which he hypohesis of equivalence for he hree ypes of ess proposed (signs, runs and chi) are no rejeced. However, hese counries would fail an addiional condiion ha we have imposed on ourselves: ha he equivalence relaion in he ables should be sable over ime and ha i should cover a sufficien number of years. The only counries ha comply wih his condiion would be New Zealand in he case of men, and Ausria in he case of women. The dynamic generaional ables for Chile are equivalen o Ausria -4 years for men and o New Zealand -6 years for women. Table 1 Non-parameric ess o idenify similariies a he age of 65 years old Men /Ausria Signs Runs es Chi Chile Women / New Zeland Signs Runses Chi Chile =The null hypohesis of sample equivalence is acceped. =The null hypohesis of sample equivalence is rejeced. Source: BBVA Research 6 The counries for which we have carried ou he sample equivalen ess are: Spain, Denmark, Slovenia, Porugal, Finland, Ireland, he Unied Kingdom, Germany, Belgium, New Zealand, Holland, Ausria, Norway, Canada, Israel, Sweden, Ausralian France, Iceland, Ialy, Swizerland and Japan. Page 15

16 Madrid, April 8, 213 The long-erm relaionship esed using he available daa shows ha for he period for men and in he case of women, in mos cases (a green "1" in Table 1 above), he hree ess are significan and do no rejec he sample equivalence hypohesis. Only in a few cases (a whie "") is he equivalence hypohesis rejeced in one of he ess. A es of he robusness is ha he mos saisically poen es (chi) only fails in he case of women in The generaional ables for Ausria and New Zealand have been available since 1948, so we can make he appropriae economeric analyses o projec he ables of hese counries in he long-erm, and hus hose of Chile, wih heir corresponding gender lags. The projecions based on he ARMAX model by specific ages offer a life epecancy of 9.91 years in 25, while he Chilean Naional Insiue of Saisics (INE) projecs a life epecancy a birh of years (see Char 2). If we ake he official esimaes of he INE (82.14) and compare i wih he esimaes of Europop for he case of Ausria (86.5) he life epecancies in Chile and Ausria can be seen o diverge by around 4 years, unlike wha one would epec given he hisorical rend of hese wo counries. In fac, one can see in Char 3 ha saring in 1985, he difference beween he projecions of life epecancy in Ausria and Chile have remained relaively consan a around 1.6 years on average o 28. Char 2 Life epecancy projecions a he age of BBVA-Model Projecion Real Official Value-Chile Source: BBVA Research Projecion Official Projecion-Chile Eurosa Projecion-Ausria Page 16

17 Working Paper Madrid, April 8, 213 Char 3 Convergence of he life epecancies Ausria Chile Source: To sum up, he INE projecions for Chile show a divergence in he life epecancy of Chile and Ausria in 25, which is conrary o he empirical evidence observed for he las 5 years, boh in he projecion carried ou by Eurosa and in ha carried ou for his paper. These discrepancies and he een of he deviaion may generae uncerainy: for he insurance indusry when calculaing life annuiies; and in erms of a possible insufficiency of he conribuion raes needed o achieve an adequae replacemen rae for fuure pensioners due o an underesimaed life epecancy. 5. The economic impac of he use of inadequae moraliy ables A life annuiy is an insurance by which, in echange for a premium, an insurance company guaranees he paymen of a regular income unil he righs epire. This income o be paid by he insurance company includes paymen of ineres a wha is called a echnical ineres rae. When he ime for reiremen comes, a pension saver ransfers he accumulaed capial in his or her individual capializaion accoun o an insurance company and he company assumes he financial and longeviy risks associaed wih he produc. The fundamenal variables of he life annuiy business are, firs, he financial variables, ineres raes; and second, changes in he moraliy raes and, in paricular, longeviy risk. The insurance company guaranees an ineres rae hroughou he lifeime of he insurance. From he financial poin of view, he insurance company herefore has o be able o inves he funds in financial asses ha can guaranee is commimen o he insured pary. The ime horizon of he operaion is also uncerain, depending basically on he life epecancy of he insured pary. If he life epecancy of a populaion increases wih respec o he life epecancy considered a he ime he raes for he produc were se, his would involve a loss for he company, as i would have o pay more monhly paymens han i had iniially calculaed. Therefore, he life annuiy business mus ake changes in he underlying variables ino accoun if i is o be viable, as an unfavorable change in any of hem will resul in he amoun charged being insufficien a paricular momen of ime. The produc's fundamenal variables have o be deermined when seing he raes for life annuiies. This involves defining he biomeric bases; in oher words, how he moraliy of he populaion will progress over ime and wha ineres rae will be guaraneed. Below we analyze in deail how a life annuiy is calculaed, and how he changes in he underlying variables, above all changes in moraliy, affec his. Page 17

18 Madrid, April 8, Seing pension levels for life annuiies Le z be he probabiliy of dying a age ; he probabiliy of dying in he following years is q+1, q+2,..., q+h. These probabiliies are published in ables derived from moraliy sudies covering a specific populaion. Le p+h be he probabiliy of being alive a ime + h, condiioned on he individual being alive a ime. This probabiliy is defined as follows: p (1 q h i h Le r be he echnical rae, i.e. he ineres rae guaraneed in he life annuiy. The discoun facor for ime h is defined as: i ) fd h (1 1 r) h The echnical capial required, CTN, o buy a life annuiy is defined as he curren value discouned a he echnical rae of he pension weighed by he probabiliy of being alive. In general, for a person of age his is epressed as follows: CTN i N pension* p i * fd i where N is he maimum age included in cerain biomeric bases. As can be seen in he above calculaions, once he biomeric bases, he echnical rae and he amoun of he pension have been se, he echnical capial needed o buy a life annuiy is fully deermined. Equally, once he biomeric bases, he echnical rae and a cerain amoun of capial have been se, he pension payable can be deduced, as shown below: pensión N i CTN P i X * fd i where N is he maimum age included in cerain biomeric bases. 5.2 Sensiiviy analysis Once he pension or echnical capial needed o buy a life annuiy has been deermined, he amoun ha he insurance company will pay o he insured pary is esablished. In oher words, he process of seing pension raes includes he esablishmen of he cos of he operaion under cerain biomeric and marke hypoheses. The erm of he operaion is uncerain and will depend on he biomerics. In addiion, he operaion of life annuiies is generally very long-erm, which involves a grea deal of uncerainy associaed wih marke variables. During he lifeime of he operaion, he underlying variables change, and as a resul he rae se iniially may no be sufficien o cover he associaed coss of he operaion. Page 18

19 CTN Annual profis and losses Annual pension Annual profis and losses Working Paper Madrid, April 8, 213 The following eercise analyzes how changes in he biomerics affec he pension. The baseline scenario is se as he pension ha a man reaching reiremen age a 65 will receive, aking ino accoun he biomeric bases of he moraliy able RV29, a echnical rae of 3.5% and a capial saved of 1, moneary unis. The able is hen modified, muliplying he probabiliies of moraliy by a percenage and revaluing he pension while mainaining he oher variables consan. Char 4 shows he resuls obained as follows: Char 4 Losses and gains in he indusry and pension benefis in he even of variaions in moraliy (in USD) Source: BBVA Research Profis and losses Moraliy variaion Annual pension I can be seen ha if we consider able RV29 as he baseline scenario of moraliy, he reducion in moraliy (lef) involves an insufficien rae and increased moraliy (righ) involves a sufficien rae. Char 5 shows how he echnical capial required varies as he moraliy varies for a pre-se pension. Char 5 Technical Capial in he indusry and pension benefis in he even of variaions in moraliy (in USD) 1,15 1,1 1,5 1, ,95 99, , Moraliy variaion Profis and losses Source: BBVA Research CTN The above char shows ha mainaining pensions consan and being able o guaranee a cerain sandard of living as moraliy changes and life epecancy increases means he accumulaed capial mus necessarily increase. Once he raes and sensiiviies associaed wih his process have been clarified, he objec of his sudy is now o quanify he pension rae defici for he Chilean pension sysem over ime. The reiremen age in Chile is 65 for men and 6 for women, so hese model poins were used for all he calculaions. For he biomeric bases, we used moraliy ables projeced from 1954 o 25 o deermine he pension rae over ime. The ineres rae used is 3.5%. Char Page 19

20 Working Paper Madrid, April 8, shows how life epecancy will change for a man aged 65 and a woman aged 6 over ime using he projecions calculaed in his work. Char 6 Life epecancy Man 65 Woman 6 Source: BBVA Research I can be seen ha alhough life epecancy for a man aged 65 in 1954 was 76.5, in 25 i will have risen o 86. Similarly, a woman aged 6 in 1954 had a life epecancy of 78, and in 25 i will be 9.5. The increased life epecancy, or he equivalen reducion in moraliy, means ha for a specific amoun of capial, he monhly income o be received by a person who reires a a cerain poin in ime will be greaer han he income o be received by anoher person who reires a a laer ime. Given he above, Char 7 shows he differences in he pensions o be received by a man aged 65 and a woman aged 6 who reire beween 1954 and 25. Char 7 Annual income for a man aged 65 (USD) 22, 2, 18, 16, 14, 12, 1, Source: BBVA Research A man aged 65 who reired in 1954 wih 19, dollars saved would receive a pension of an annual 2, dollars. Currenly, he same man aged 65 would receive a pension of 16, dollars, and his amoun would fall o 12,7 dollars in 25. In oher words, he fac ha moraliy falls over ime involves a reducion of 2% in pensions over he ne 4 years. Analogously o he above case, a woman aged 6 who reired in 1954 wih savings of 167, dollars would receive an annual income of 12,6 dollars. Currenly his pension would fall o 1,3 dollars and in 25 i will be 8,7 dollars. This represens a reducion of 15% over he ne fory years (see Char 8). Page 2

21 Working Paper Madrid, April 8, 213 Char 8 The annual income of a woman aged 6 years (in USD) 14, 13, 12, 11, 1, 9, 8, Source: BBVA Research In he previous secion we saw ha here are discrepancies in he predicions made by differen insiuions; in he specific case of Ausria and Chile, i is noable ha according o some insiuions, he convergence of life epecancies in he wo counries will be reversed over he ne 4 years. This is perhaps due o differen calculaion mehodologies. The quesion o ask, herefore, is Who is righ? The risk of an incorrec answer o his quesion is very relevan for he pension indusry. A simulaion eercise shows us ha a 1% deviaion in he moraliy ables used for calculaing life annuiies would resul in losses for he indusry ha could amoun o 6 million dollars in 217 (see Char 9). These losses would be mainly due o wo facors: he effec of greaer longeviy of pension savers and he effec of he use of inadequae pension raes. Char 9 Sysemic risk simulaion in he use of moraliy ables wih a 1% error in he moraliy ables of Chile (in million dollars) Source: BBVA Research In addiion, increased life epecancy could mean ha pension savers in many Lain American counries may no be making sufficien conribuions o heir pension funds. An error in esimaing his life epecancy could imply ha pension savers would have o disribue heir accumulaed balances in a life annuiy over a longer period of ime, so heir available income would reduce heir replacemen rae and hus heir sandard of living. If era conribuions were no made, fuure generaions could see heir reiremen pensions reduced by nearly 5% due o increased life epecancy (see Char 1). Page 21

22 Acuarial income (1954=1) Life epecancy a 65 years old Working Paper Madrid, April 8, 213 Char 1 Evoluion in life epecancy and he pensions of he men Source: BBVA Research Acuarial Income Life epecancy In anoher simulaion eercise, we have calculaed wha he conribuion rae would have o be in Chile o mainain he curren replacemen rae (see Char 11). The conribuion rae would have o increase by an average of 8 percenage poins in he case of men and 4 percenage poins for women on curren levels. The difference by gender can be eplained by he greaer relaive increase in men's life epecancy compared wih women (a convergence can be observed beween he genders), and because men have higher salaries and hus have o accumulae a higher capial balance o mainain he replacemen rae. Char 11 Simulaion of required conribuion raes o mainain curren rae of subsiuion Men Source: BBVA Research Women All hese resuls poin o he need o calculae good long-erm dynamic moraliy ables o miigae he longeviy risks for boh he indusry and pension savers. If Chile, which currenly has he bes ables in he region, is facing hese risks, he need for oher counries is even more pressing. 6. Conclusions Increased life epecancy a birh poses major challenges for sociey, given he imporan ransformaions ha will influence he ways in which supply and demand will inerac in he fuure. Indusries ha are mos closely linked o longeviy risks, such as insurance and pensions, have been incorporaing major changes in heir business processes o allow hem o inerac as well as possible. One of he key elemens of hese markes is he appropriae projecions of moraliy raes, based on a se of variables ha anicipae he rends in fuure life epecancy and hus o ensure a correc esimae of he risks. The "ar" of consrucing Page 22

23 Madrid, April 8, 213 moraliy ables has been developed over ime. I requires a deailed and ongoing analysis of all hose aspecs ha affec he possibiliy of living longer, as well as anicipaing fuure rends based on curren evens. A number of developed counries have developed a working hisory based on a long radiion of collecing hisorical moraliy daa, which has allowed hem o narrow he gaps wih he real siuaion. However, his progress has a very limied hisory in Lain America. As a resul, he region is faced wih major challenges in he fuure o incorporae echnical progress and he informaion needed for he consrucion of improved ables. This work has used he case of Chile o reveal how he moraliy ables in Lain America may be lagging behind realiy, and he consequen economic effecs on he insurance and pension indusry. The calculaions have only been made for Chile, as i is he counry wih mos available informaion for applying he calculaion mehodology used in he projec. The srucure of he moraliy ables in Chile has been compared wih ha of he 22 counries where here is sufficien hisorical informaion o carry ou economeric projecions. This analysis uses non-parameric ess ha compare he level of moraliy (signs es), he form (runs es) and he eisence of a similar paern beween boh ables (χ2 es). The counries ha are saisically mos similar o he Chilean case are New Zealand in he case of men and Ausria in he case of women. The dynamic generaional ables for Chile are equivalen o Ausria -4 years for men and New Zealand -6 years for women. The long-erm relaionship esed using he available daa shows ha for he period for men and in he case of women, in mos cases he hree ess are significan and do no rejec he sample equivalence hypohesis. Only in a few cases is he equivalence hypohesis rejeced in one of he ess. A es of robusness is ha he saisically mos poen es (chi) fails only in he case of women for he year The generaional ables for Ausria and New Zealand are available from 1948, which has enabled he appropriae economeric analyses o projec he ables of hese counries in he long-erm; because of he equivalence his mean he same could be done for Chile, wih he corresponding lag by gender. For he projecion eercises, he moraliy raes were calculaed using ARMA(p, q) models for each of he 11 ages and for each gender, wih a oal of 22 ARMA(p,q) regressions. The resuls offer an average life epecancy (men and women) of 9.91 years in 25, while he INE in Chile projecs a life epecancy a birh of years. If we ake he official esimaes of he INE (82.14) and compare hem wih he esimaes of Europop for he case of Ausria (86.5) i is noiceable ha he life epecancy of Chile and Ausria diverge by around 4 years, unlike wha one would epec given he hisorical rend of hese wo counries. In fac, one can see ha saring in 1985, he difference beween he projeced life epecancies for Ausria and Chile have remained relaively consan a around 1.6 years on average unil 28. Thus he INE projecions for Chile would show a divergence beween he life epecancy of Chile and Ausria in 25 ha is conrary o he observed empirical evidence over he las 5 years, in boh he Eurosa projecion and in his paper. These discrepancies and he een of he deviaion may generae uncerainy: for he insurance indusry when calculaing life annuiies; and in erms of a possible insufficiency of he conribuion raes needed o achieve an adequae replacemen rae for fuure pensioners due o an underesimaed life epecancy. A simulaion eercise shows us ha a deviaion of 1% in he moraliy ables used for calculaing life annuiies would resul in losses for he indusry ha could amoun o 6 million dollars in 217. These losses would mainly be due o wo facors: he effec of greaer longeviy of he pension savers and he use of inadequae pension raes. In addiion, he increased life epecancy could imply ha pension savers in many Lain American counries are no making sufficien conribuions o heir pension funds. An error in esimaing his life epecancy would mean ha pension savers would have o disribue heir accumulaed balances in a life annuiy over a longer period of ime, so heir available income would reduce heir replacemen rae and hus heir sandard of living. If era conribuions were no made, Page 23