Application of the Poisson log-bilinear projection model to the G5 mortality experience

Transcription

1 Applicaion of he Poisson log-bilinear projecion model o he G5 moraliy eperience Anoine Delwarde 1, Michel Denui 2, Monserra Guillén 3 and Anoni Vidiella-i-Anguera 4 Absrac. I is now well documened ha he human moraliy globally declined during he course of he 20h cenury. These moraliy improvemens pose a challenge for he pricing and reserving in life insurance. This paper aims o analyze he paern of moraliy decline in he G5 counries (France, Germany, Japan, UK and US). Assuming a furher coninuaion of he sable pace of moraliy decline, a Poisson log-bilinear projecion model is applied firs o aggregaed and hen o disaggregaed daa o forecas deah raes. Keywords: life insurance, age-se-specific moraliy, projeced lifeables 1 Inroducion and Moivaion As demonsraed in Benjamin & Soliman (1993) and McDonald e al. (1998), moraliy a adul and old ages reveals decreasing annual deah probabiliies, alhough no for all counries. Oeppen & Vaupel (2002) showed some hisorical misconcepions abou he limis of life epecancy. Moraliy improvemens pose a challenge for he planning of public reiremen sysems as well as for he pricing and reserving in life insurance; see, e.g., Olivieri (2001) and Coppola e al. (2000). The calculaion of epeced presen values requires an appropriae moraliy projecion o avoid underesimaion of fuure coss. Besides, moraliy improvemens pose a paricular challenge for annuiy and pensions business where paymens are made during an individual s lifeime - more so han normal life insurance. In order o proec he company from moraliy improvemens, acuaries have o resor o lifeables including a forecas of he fuure rends of moraliy (he so-called projeced ables). Differen approaches for building hese echnical bases have been developed by acuaries and demographers. Since Cramér & Wold (1935), he evoluion over ime of graduaed moraliy curves is popular for he purpose of erapolaion. One classical procedure is based on he projecion of parameers (see e.g. Benjamin & Soliman (1993)). This approach of course heavily relies on he appropriaeness of he reained parameric models (as Maeham, for insance). 1 Insiu des Sciences Acuarielles, Universié Caholique de Louvain, Rue des Wallons, 6, B-1348 Louvain-la-Neuve, Belgium 2 Insiu des Sciences Acuarielles, Universié Caholique de Louvain, Rue des Wallons, 6, B-1348 Louvain-la-Neuve, Belgium 3 Dep. Economerics RFA-IREA, Universia de Barcelonam, Diagonal, 690, Barcelona, Spain 4 Bluecap Managemen Consuling, Velzquez, 78, E Madrid, Spain Moreover, he esimaed parameers are ofen srongly dependen so ha univariae erapolaions may be misleading. An eample of he projecion of parameers approach is found in Bu & Haberman (2004). Lee & Carer (1992) proposed a simple model for describing he secular change in moraliy as a funcion of a single ime inde. This model is fi o hisorical daa. The resuling esimae of he ime-varying parameer is hen modeled and forecas as a sochasic ime series using sandard Bo-Jenins mehods. From his forecas of he general level of moraliy, he acual age-specific raes are derived using he esimaed age effecs. The main saisical ool of Lee & Carer (1992) is leas-squares esimaion via singular value decomposiion of he mari of he log age-specific observed forces of moraliy. This implicily means ha he errors are assumed o be homosedasic, which is quie unrealisic: he logarihm of he observed force of moraliy is much more variable a older ages han a younger ages because of he much smaller absolue number of deahs a older ages. Brouhns e al. (2002a,b) and Renshaw & Haberman (2003a,b) have each implemened similar alernaive approaches o moraliy forecasing based on heerosedasic Poisson error srucures. Anoher drawbac of he Lee-Carer mehodology is ha he required daa have o fill a recangular mari because of singular value decomposiion. This may pose a problem when he forma of he available daa has been modified in he pas (he acuary has o complee he daa using differen echniques which may bias he resuls). As we will see in Secion 3, he mehod used in his paper avoids hese drawbacs. However, he predicion inervals remain quie narrow. This is a common feaure of Lee-Carer predicions (see, Renshaw & Haberman, 2003b. p.135). The mehodology eamined in his paper is a mere erapolaion of pas rends. All purely erapolaive forecass assume ha he fuure will be in some sense lie he pas. Some auhors (see e.g. Guerman & Vanderhoof (1999)) severely criicized his approach because i seems o ignore underlying mechanisms. As poined ou by Wilmoh (2000), such a criique is valid only insofar as such mechanisms are undersood wih sufficien precision o offer a legiimae alernaive mehod of predicion. The undersanding of he comple ineracions of social and biological facors ha deermine moraliy levels being sill imprecise, he erapolaive approach o predicion is paricularly compelling in he case of human moraliy. The mehod described in his paper does no aemp o c BELGIAN ACTUARIAL BULLETIN, Vol. 6, No. 1, 2006

2 incorporae assumpions abou advances in medical science or specific environmenal changes: no informaion oher han previous hisory is aen ino accoun. This means ha his approach is unable o forecas sudden improvemens in moraliy due o he discovery of new medical reamens or revoluionary cures including anibioics. Similarly, fuure deerioraions caused by epidemics, he appariion of new diseases or he aggravaion of polluion canno ener he model. The acuary has o eep his in mind when he ses his reinsurance program. Our resuls allow o compare he moraliy eperience observed in he G5 counries in he same model and o produce forecass. An esimaed average deah rae and a common inde of moraliy decline can be obained from he analysis, which is essenial for economiss. Mos financial and insurance decisions are aen on he basis of a worldwide view, more han on a regional or paricular locaion. From our analysis, one can obain baseline moraliy forecass from he pooled populaion of five counries, bu a he same ime, one can see he influence of each gender, age, ime rend and counry on he moraliy forecas. The observed pas behaviour of he G5 is summarized in a single model and he comparison of each counry specific effec becomes much easier. The reason for choosing he G5 counries was simpliciy, bu a he same ime, he resuls are ineresing for mulinaional decision maers ha require a general moraliy forecas. When planning long-erm producs, economic organizaions need o forecas he fuure longeviy of a wide economic area of influence. Mulinaional companies wih pension schemes and members in differen counries where moraliy is changing o a differen eend are also ineresed in he opic. 1.1 Noaion We analyze he changes in moraliy as a funcion of boh age and ime. Henceforh, T () is he remaining lifeime of an -aged individual in calendar year ; his individual will die a age + T () in year + T (). q () is he probabiliy ha an -aged individual dies in calendar year, i.e. q () = Pr[T () 1]. p () = 1 q () is he probabiliy ha an -aged individual in calendar year reaches age + 1, i.e. p () = Pr[T () > 1]. µ () is he moraliy force a age during calendar year. e () is he epeced remaining lifeime of an individual aged in year. a () is he pure premium of a life annuiy sold o an - year-old individual in year. E is he eposure-o-ris a age during year, i.e. he oal ime lived by people aged in year. D is he number of deahs recorded a age during year, from an eposure-o-ris E. L is he number of individuals aged on January 1 of year. 1.2 Daa The unique source of daa is he Human Moraliy Daabase, available on The daa used in his paper have been downloaded from his websie on December 12, Specifically, we obained hree daa series for each counry: deahs, Eposure-o-Ris and moraliy raes by year of deah. The daa consis of gender-specific abulaions of deahs and eposures-o-ris from he G5 counries: France, Germany, Japan, UK (England & Wales) and USA. The age runs from 0 o 110+ for all counries by group of 5, i.e. 0 4, 5 9, 10 14,..., , The counries are hen grouped o consiue he so-called G5 by simply adding he number of deahs and eposures-o-ris of he 5 differen counries. Daa are available from min = 1960 o ma = 1997 for all he G5 members. 1.3 Agenda Secion 2 describes original Lee-Carer model and is Poisson specificaion. A maimum lielihood parameers esimaion is inroduced and applied o he whole G5 populaion daa. The ime componen erapolaion is also performed using an ARIMA ime series model. In Secion 3 we swich o he model inroduced by Wilmoh & Valonen (2002). Each G5 counry is now viewed as a value of a covariae. Weighed sums of squares are analysed and he wo gender specific ime series are erapolaed in order o creae projeced lifeables. As epeced, age is he mos imporan facor deermining moraliy rae. The ime effec is more relevan han he counry effec if weighs are aen ino accoun, which is a sign of convergence. In oher words, he ime horizon is more imporan han he counry, bu since he counry effec is no negligible, he differences beween counry-specific deah raes increase wih ime. 2 Poisson log-bilinear mehodology 2.1 Lee-Carer classical mehodology Before describing he Poisson model, we firs recall he basic feaures of he classical Lee-Carer approach. Lee & Carer (1992) proposed a simple model for describing he secular change in moraliy as a funcion of a single ime inde. The mehod describes he log of a ime series of age-specific deah raes as he sum of an age-specific componen ha is independen of ime and anoher componen ha is he produc of a ime-varying parameer reflecing he general level of moraliy, and an age-specific componen ha represens how rapidly or slowly moraliy a each age varies when he general level of moraliy changes. The mehod is in essence a relaional model ln µ () = α + β κ + ǫ () (1) where µ () denoes he observed force of moraliy a age during year, he ǫ () s are homosedasic cenered er- 55

3 ror erms and where he parameers are subjeced o he consrains κ = 0 and β = 1 (2) ensuring model idenificaion. The model (1) is fied o a mari of age-specific observed forces of moraliy using singular value decomposiion (SVD). Specifically, he α s, β s and κ s are such ha hey minimize ( ) 2. ln µ () α β κ (3), I is worh menioning ha model (1) is no a simple regression model, since here are no observed covariaes in he righ-hand side. The minimizaion of (3) consiss in aing for α he row average of he ln µ () s, and o ge he β s and κ s from he firs erm of a SVD of he mari ln µ () α. This yields a single ime-varying inde of moraliy κ. When he model (1) is fi by minimizing (3), inerpreaion of he parameers is quie simple: α : he fied values of α eacly equals he average of ln µ () over ime so ha ep α is he general shape of he moraliy schedule; β : represens he age-specific paerns of moraliy change. I indicaes he sensiiviy of he logarihm of he force of moraliy a age o variaions in he ime inde κ. In principle, β could be negaive a some ages, indicaing ha moraliy a hose ages ends o rise when falling a oher ages. κ : represens he ime rend. The acual forces of moraliy change according o an overall moraliy inde κ modulaed by an age response β. The shape of he β profile ells which raes decline rapidly and which slowly over ime in response of change in κ. The error erm ǫ (), wih mean 0 and variance σǫ 2 reflecs paricular age-specific hisorical influence no capured in he model. The resuling esimae of he ime-varying parameer is hen modeled and projeced as a sochasic ime series using sandard Bo-Jenins mehods. From his forecas of he general level of moraliy, he acual age-specific raes are derived using he esimaed age effecs. Before proceeding direcly o modeling he parameer κ as a ime series process, he κ s are adjused (aing α and β esimaes as given) o reproduce he observed number of deahs D, ha is he κ s solve D = E ep( α + β κ ). (4) So, he κ s are reesimaed so ha he resuling deah raes (wih he previously esimaed α and β ), applied o he acual ris eposure, produce he oal number of deahs acually observed in he daa for he year in quesion. There are several advanages o maing his second sage esimae of he parameers κ. In paricular, i avoids sizable discrepancies beween prediced and acual deahs (occurring because he firs sep is based on logarihms of deah raes). Oher advanages are discussed by Lee (2000). The ime facor κ is inrinsically viewed as a sochasic process and Bo-Jenins echniques are hen used o esimae and forecas κ wihin an ARIMA imes series model. 2.2 Poisson log-bilinear model According o Alho (2000), he model described in equaion (1) is no well suied o he siuaion of ineres. As already menioned, he main drawbac of he OLS esimaion via SVD is ha he errors are assumed o be homosedasic. This is relaed o he fac ha for inference we are acually assuming ha he errors are normally disribued, which is quie unrealisic. The logarihm of he observed force of moraliy is much more variable a older ages han a younger ages because of he much smaller absolue number of deahs a older ages. The approach of Brouhns e al. (2002a) consiss in subsiuing Poisson random variaion for he number of deahs for an addiive error erm on he logarihm of moraliy raes. I is worh o menion ha he Poisson disribuion is well-suied o moraliy analyses; see e.g. Brillinger (1986) and Mc Donald (1996a,b,c) for more deails. I has been successfully applied by Renshaw & Haberman (2003b) and Sihole e al. (2000) o he forecasing of moraliy rends. We now consider ha ( ) D Poisson E µ () wih µ () = ep (α + β κ ) (5) where he parameers are sill subjeced o he consrains (2). The force of moraliy is hus assumed o have he same logbilinear form lnµ () = α + β κ as in he Lee-Carer model. The meaning of he α, β, and κ parameers is essenially he same as in he classical Lee-Carer model. Only he random par of he model is modified. 2.3 Maimum lielihood esimaion Insead of resoring o SVD for esimaing α, β and κ, we now deermine hese parameers by maimizing he loglielihood L(α,β,κ) based on model (5). Le us denoe as D = E[D ] = E ep(α + β κ ) he epeced number of deahs a age during year. Then, L(α,β,κ) { ( DD = ln ep( D ) D! )} = { D ln D D } ln {D!} = { } D (α + β κ ) E ep(α + β κ ) + consan. Because of he presence of he bilinear erm β κ, i is no possible o esimae he proposed model wih commercial saisical pacages ha implemen Poisson regression. Here, we 56

4 have implemened an unidimensional Newon mehod proposed by Goodman (1979) for esimaing log-linear models wih bilinear erms. In ieraion sep ν + 1, a single se of parameers is updaed fiing he oher parameers a heir curren esimaes using he following updaing scheme θ (ν+1) = θ (ν) L(ν) / θ 2 L (ν) / θ 2 where L (ν) = L( θ (ν) ). In our applicaion, here are hree ses of parameers; ha is, he α, he β, and he κ erms. The updaing scheme is as follows, saring wih α (0) (0) = 0, β = 1, and κ (0) = 0 (random values can also be used) α (ν+1) = α (ν) κ (ν+1) = κ (ν) β (ν+1) = β (ν) ( D ( D ( D ) (ν,ν,ν) D D (ν,ν,ν) ) (ν+1,ν,ν) D β(ν) ( β(ν) ) 2 D (ν+1,ν,ν) D (ν+1,ν,ν+1) D (ν+1,ν,ν+1) ( ) κ (ν+1) κ (ν+1) ) 2 where D (να,ν β,ν κ) = E ep( α (να) + β (ν β) κ (νκ) ) is he esimaed number of deahs afer ν α updaes of parameer α, ν β updaes of parameer β and ν κ updaes of parameer κ. The crierion used o sop he procedure is a very small increase of he log-lielihood funcion. Le us menion ha i is also possible o opimize he Poisson lielihood by monioring he associaed deviance, as described in Renshaw & Haberman (2003b). The ML esimaions of he parameers have o be adaped in order o fulfill he Lee-Carer consrains: specifically, we swich from α, β, κ o he new parameers by ransformaion α α + β κ κ ( κ κ) β β β β where κ is he mean of iniial κ s and β is he sum of iniial β s. The new esimaes fulfill he consrains and provide he same D since he value of α + β κ is no alered by ransformaion. Conrarily o he classical Lee-Carer approach (where SVD is applied o ransformed moraliy raes), he error applies direcly on he number of deahs in he Poisson regression approach. There is hus no need of a second-sage esimaion lie (4). Indeed, differeniaing he loglielihood wih respec o α gives he equaion D = D = E ep( α + β κ ). (6) So, he esimaed κ s are such ha he resuling deah raes applied o he acual ris eposure produce he oal number of deahs acually observed in he daa for each age. Sizable discrepancies beween prediced and acual deahs are hus avoided. We apply he Poisson modelling o he whole G5 populaion daa. The Poisson esimaes α, β and κ involved in (5) are depiced in Figures e 04 1 e 03 1 e 02 1 e 01 1 e+00 1 e 04 1 e 03 1 e 02 1 e 01 1 e+00 Figure 1. Average deah raes ep( α ) for he G5 counries (op: males, boom: females). 57

5 Figure 2. Raes of moraliy decline β for he G5 counries (op: males, boom: females). Figure 3. Inde of moraliy decline κ for he G5 counries (op: males, boom: females). 58

6 2.4 Modelling he inde of moraliy As in he Lee-Carer mehodology he ime facor κ is inrinsically viewed as a sochasic process. Bo-Jenins echniques are herefore used o esimae and forecas κ wihin an ARIMA(p, d, q) imes series model, which aes he general form d κ = ρ + Θ q(b)ǫ Φ p (B) where B is he delay operaor, B(κ ) = κ 1, B 2 (κ ) = κ 2,... ; = 1 B is he difference operaor, κ = κ κ 1, 2 κ = κ 2κ 1 + κ 2,... ; Θ q (B) is he Moving Average polynomial, wih coefficiens θ = (θ 1,θ 2...θ q ); Φ p (B) is he Auoregressive polynomial, wih coefficiens φ = (φ 1,φ 2...φ p ); ǫ is whie noise wih variance σ 2 ǫ. The parameers of he models are ρ, θ, φ and σ ǫ. The order of he model is seleced according o he AIC crierion. ARIMA(1,1,1) is hus chosen for males and ARIMA(0,1,1) for females. The mehod we use o obain esimaes for he ARIMA parameers is condiional leas squares. Males Variable Coefficien Sd. Error ρ -0,2709 0, φ 1-0, , θ 1 0, , Females Variable Coefficien Sd. Error ρ -0,3645 0, θ 1-0, ,15576 Forecased values of ime parameers κ (here, is beyond he observaion period) and 95% predicion inervals can be seen on Figure 4. As is discussed in he ne secions, he parameer esimaes of he Poisson model and he κ forecass can be used o obain projeced age-specific moraliy raes. Projeced deahs raes are displayed in Figure An analysis based on disaggregaed daa: he Wilmoh-Valonen Model 3.1 Model Le us now see each counry as a par of he whole G5 populaion. We consider he model inroduced by Wilmoh & Valonen (2002); he five G5 counries are now referred o by a value of a counry facor. For each se separaely he model becomes D Poisson ( E µ () ) where µ () = ep ( α + α + α + α + (β + β )κ ) Figure κ wih predicions and confidence inervals (op: males, boom: females). 59

7 and refers o he counry. Since we eep only one ime paern by se, moraliies of every counry are supposed o decline in he same way. This model allows o analyse he level and age paern of moraliy by counry, he general ime paern of moraliy change, and he speed and age paern of moraliy change by counry. As for he Lee-Carer model, he erapolaion of esimaes κ gives fuure moraliy raes for given gender, age, ime and counry. The main ineres of mehod consiss in he esimaion of an unique ime series (or wo if each gender is reaed separaely) which gives moraliy raes for all counries and age-ime caegories. This ime we impose he following consrains: α = α = 0 for all (7) α = α = β = 0 for all (8) κ = 0 (9) κ ma κ min = 1 (10) ma min where = 1979, o ensure he idenifiabiliy of he model. Noe ha his model yields a huge number of parameers (P) and consrains (C): α α α α β β κ Toal P C Thus we have o esimae 320 parameers fulfilling 55 consrains, bu he number of daa poins is equal 8740, ha is he number of counries imes he number of age groups imes wo, because of he wo genders, imes he number of available years Figure 5. Projeced deah raes (op: males, boom: females). 3.2 Parameers esimaion The parameers α, α, α, α, β, β and κ are esimaed wih he maimum lielihood mehod. Esimaes for G5 populaion are depiced in Figures Figure 7 shows very differen counry-specific deah raes. Evoluion of hese raes loos also very differen from one counry o anoher as we can see in Figure 9. Greaer β indicaes ha Japanese moraliy declines faser han in he oher counries. Figures 8 and 9 show some ereme values a he highes ages, ha could be smoohed or adjused. This conclusion is quie similar from he one in Tuljapurar, Li, & Boe (2000) who applied Lee-Carer mehodology o each G7 counry separaely. They obained very close ime paerns for si of he seven counries; only Japan looed o have faser declining moraliy. Counry specific β were also of he same order despie several dispariies. Now we can analyse conribuions of facors age, ime and counry in he model. I is useful o compue he sum of squares in order o quanify he relaive imporance of age, ime and counry in he model. We refer he reader o he Appendi for deails abou he derivaion of he formulas for he unweighed and weighed sums of squares. 60

8 1 e 04 1 e 03 1 e 02 1 e 01 1 e+00 1 e 04 1 e 03 1 e 02 1 e 01 1 e+00 Figure 6. ep( α + α ): deah raes by se (op: males, boom: females). 1 e 04 1 e 03 1 e 02 1 e 01 1 e+00 1 e 04 1 e 03 1 e 02 1 e 01 1 e+00 SS WSS Males Females Males Females Age 98.01% 98.00% 95.85% 96.74% Time 0.72% 0.89% 2.37% 2.24% Counry 1.27% 1.12% 1.78% 1.01% -Counry main 0.15% 0.07% 1.02% 0.44% -Counry age 0.52% 0.71% 0.44% 0.25% -Counry ime 0.60% 0.34% 0.32% 0.32% Figure 7. ep( α + α + α + α ): deah raes by se and counry. Solid: France, dashed: Germany, doed: Japan, dodash: UK, long dash: USA (op: males, boom: females). Table 1. Sum of squares (SS) and weighed sums of squares (WSS). 61

9 Figure 8. β: rae of moraliy decline by se (op: males, boom: females). Figure 9. β + β : rae of moraliy decline by se and counry. Solid: France, dashed: Germany, doed: Japan, dodash: UK, long dash: USA (op: males, righ: females). 62

10 Table 1 decomposes he oal variabiliy in differen sources. We can see ha age accouns for he mos par of he oal variabiliy, for males and females. Time and counry effecs are much less imporan. 3.3 Moraliy predicions Figure κ : inde of moraliy decline by se (op: males, boom: females). The ARIMA model used o erapolae he ime series in he fuure is chosen by he AIC crierion. We seleced an ARIMA(2,1,1) process for males and ARIMA(0,1,1) for females. Resuls of he esimaion are displayed ne: Males Variable Coefficien Sd. Error ρ -1, , φ 1 0, , φ 2 0, , θ 1-0, ,0433 Females Variable Coefficien Sd. Error ρ -1, , θ 1-0, , Figure 11 depics he projeced κ s, ogeher wih he 95% confidence inervals, separaely for men and women. Wih he projeced κ s, we can now generae fuure moraliy raes, separaely for men and women and for each of he G5 counries. This is eemplified in Figures 12 and 13 for years 2000 and 2020 (he calendar year 1980 has been added o have an idea of he evaluaion wih ime). We see from Figures 12 and 13 ha he differences beween counry-specific deah raes increase wih ime. Japan seems o have paricularly low deah raes. 4 Discussion Thans o he model inroduced by Wilmoh & Valonen (2002) we obained counry specific projeced deah raes assuming a general se specific ime paern. Conrarily o previous inernaional sudies abou moraliy improvemens, an inegraed Poisson regression model is used o forecas he fuure deah raes applying o G5 counries. The forecas is driven by a single ime inde. This inde is projeced o he fuure using ARIMA echniques. The main advanage of he counry specific model in secion 3 is ha one can analyze he paern for each counry individual rend and compare i o he mean rend ha is obained wih he Lee-Carer resuls presened in secion 2. So, i is easier o spo if he rend for one paricular counry our group of counries diverges from he res. In order o have confidence inervals on parameers α, α, α, α, β, β and κ (and hen on moraliy saisics as life epecancy e ()), i could be useful o adap he boosrapping mehod inroduced by Brouhns, Denui & Vermun (2002b) and Brounhs, Denui & Vaneilegom (2004). Resampling each counry able of deah numbers should deermine if 63

11 e 05 1 e 03 1 e 01 1 e 05 1 e 03 1 e 01 1 e 05 1 e 03 1 e 01 Figure 11. κ wih predicions and confidence inervals (op: males, boom: females). Figure 12. Projeced moraliy for males = 1980, 2000, 2020 (from op o boom). Solid: France, dashed: Germany, doed: Japan, dodash: UK, long dash: USA. 64

12 1 e 05 1 e 03 1 e 01 1 e 05 1 e 03 1 e 01 counry specific parameers are significanly differen one o anoher. From he projeced moraliy in 2020, one can see ha Japan has low deah raes for adul males and young females compared o he oher counries. On he oher hand, France has lower moraliy raes for females above 40 years, a phenomenon ha has been observed in he pas which produces large life epecancies for women. We do no discuss he behaviour of cenennials. There is very lile informaion in ha age group. The resuls show ha for some age and gender groups, he projeced behaviour indicaes very similar moraliy raes across he G5 counries. For male over 60 years old, he differences beween counries are small. On he oher hand, in he age inerval beween 20 and 40 years, which poins a he young acive populaion group, one can see some male over moraliy in France and he UK. An ineresing oupu of he resuls is he predicion of he inde of moraliy decline obained for male and female separaely. If correcly implemened his may be used as a reference indicaor of moraliy improvemens of a wide geographical and economic area. ACKNOWLEDGEMENTS Anoine Delwarde was suppored by a research gran from he Fonds de la Recherche pour l Indusrie e l Agriculure - FRIA. This financial suppor is graefully acnowledged. Michel Denui hans he suppor of Belgian Governmen under he Proje d Acion de Recherches Concerées 04/ FEDER SEC and BBVA are also acnowledged. REFERENCES 1 e 05 1 e 03 1 e 01 Figure 13. Projeced moraliy for females = 1980, 2000, 2020 (from op o boom). Solid: France, dashed: Germany, doed: Japan, dodash: UK, long dash: USA. [1] Alho, J.M.(2000). Discussion of Lee (2000). Norh American Acuarial Journal 4, [2] Benjamin, B. & Soliman, A.S.(1993). Moraliy on he Move. Insiue of Acuaries, Oford. [3] Brillinger, D.R. (1986). The naural variabiliy of vial raes and associaed saisics. Biomerics 42, [4] Brouhns, N., Denui, M. & Van Keilegom, I. (2004). Boosrapping he Poisson log-bilinear model for moraliy projecion. Scandinavian Acuarial Journal, in press. [5] Brouhns, N., Denui, M. & Vermun, J. (2002a). A Poisson log-bilinear regression approach o he consrucion of projeced lifeables. Insurance: Mahemaics & Economics 31, [6] Brouhns, N., Denui, M. & Vermun, J.K. (2002b). Measuring he longeviy ris in moraliy projecions. Bullein of he Swiss Associaion of Acuaries, [7] Bu, Z. & Haberman, S. (2004). Applicaion of Fraily-Based Moraliy Models Using Generalized Linear Models. As in Bullein, 34, 1, [8] Coppola, M., Di Lorenzo, E. & Sibillo, M. (2000). Ris sources in a life annuiy porfolio: decomposiion and measuremen ools. Journal of Acuarial Pracice 8, [9] Cramér, H. & Wold, H. (1935). Moraliy variaions in Sweden: A sudy in graduaion and forecasing. Scandinavian Acuarial Journal 18, [10] Goodman, L.A. (1979). Simple models for he analysis of associaion in cross-classificaions having ordered caegories. Journal of he American Saisical Associaion 74,

13 [11] Guerman, S. & Vanderhoof, I.T. (1999). Forecasing changes in moraliy: a search for a law of causes and effecs. Norh American Acuarial Journal 2, [12] Lee, R.D. (2000). The Lee-Carer mehod of forecasing moraliy, wih various eensions and applicaions. Norh American Acuarial Journal 4, [13] Lee, R.D. & Carer, L. (1992). Modelling and forecasing he ime series of US moraliy. Journal of he American Saisical Associaion 87, [14] McDonald, A.S., Cairns, A.J.C., Gwil, P.L. & Miller, K.A. (1998). An inernaional comparison of recen rends in moraliy. Briish Acuarial Journal 4, [15] Oeppen, J. & Vaupel, J.W. (2002). Broen limis o life epecancy. Science 296, 5570, [16] Olivieri, A. (2001). Uncerainy in moraliy projecions: an acuarial perspecive. Insurance: Mahemaics & Economics 29, [17] Renshaw, A.E. & Haberman, S. (2003a). Lee-Carer moraliy forecasing wih age specific enhancemen. Insurance: Mahemaics & Economics 33, [18] Renshaw, A. & Haberman, S. (2003b). Lee-Carer moraliy forecasing: a parallel generalized linear modelling approach for England and Wales moraliy projecions. Applied Saisics 52, [19] Sihole, T.Z., Haberman, S. & Verrall, R.J. (2000). An invesigaion ino parameric models for moraliy projecions, wih applicaions o immediae annuians and life office pensioners daa. Insurance: Mahemaics & Economics 27, [20] Tuljapurar, S., Li, N. & Boe, C. (2000). A universal paern of moraliy decline in he G7 counries. Naure 405, [21] Wilmoh, J.R. (2000). Demography of longeviy: pas, presen and fuure rends. Eperimenal Geronology 35, [22] Wilmoh, J.R. & Valonen, T. (2002). A parameric represenaion of moraliy differenials over age and ime. EAPS seminar, Trends in differenials in morbidiy and moraliy: Analysis and eplanaion, Ponignano, Ialy, April 2001 APPENDIX Formulas for he sums of squares In his appendi, we derive all he formulas used o ge he values displayed in Table 1. In order o derive he relaive imporance of each regressor, we need o decompose he oal sum of squares as he sum of he conribuion o he sum of squares by each variable. Given ha here are ineracions, he decomposiion is no sraighforward. The compuaions are easier by modifying consrains (9)-(10) ino κ = 0 (11) κ 2 = 1. (12) The new esimaes can easily be obained by he ransformaion where a = α α a b β α α a b β β 1 b β β 1 b β κ a + bκ ) (38 2 κ2 1 ( κ ) b = a 38 κ = ( κ 2 ( κ ) 2 38 Le us define he following sums of squares: SSAge = 38 5 SSTime = 5 β 2 α 2 ) 1 2. SSCounry = ( α ) ( α ) 2 SSToal =,, + ( β ) 2,, ( α + α + α + ( β + β ) κ ) 2 The oal sum of squares SSToal can easily be decomposed in hree componens relaive o age (SSAge), ime (SST ime) and counry (SSCounry). One drawbac of his formulaion is o give equal weigh o each daa cell (,,), regardless of he size of he populaion involved. I could be beer o analyze he weighed sums of squares defined as follows: WSSAge = WSSTime = WSSCounry = +, w α 2 w β2 w ( α ) 2 +, w, ( β ) 2 w, ( α ) 2 WSSToal =,, w,, ( α + α + α where he weighs are w = w = +( β + β ) κ ) 2, E,, E, E,, E, w = E,, E w, = w w w,, = w w w. The weighs values are displayed in Tables 2, 3 and 4. One can prove W SST oalage = WSSAge + WSSTime + WSSCounry 66

14 if he parameers saisfy new consrains w α = w α = 0 (13) w α = w α = w β = 0 (14) w κ = 0 (15) w κ 2 = 1. (16) Noe ha he sum of weighs is 1. Compue c a, c b and γ as ( w 2 ) 1/2 c a = 1 + ( w ) 2 c b = c a / w γ = w α + w α + w w α c a w w (β + β c ). b Then, he parameer ransformaions ha saisfy he new consrains are c a + c b β 1 c b (β + β 1 c b (β w β ) w β ) α α + w α + w α c a w (β + β c ) γ b α α + w α + w α c a w (β + β c ) γ b α α c a c b (β + β ) w (α c a c b (β + β ))) + w (α c a (β + β c ))) b w w (α c a (β + β c )). b Thus, he (weighed) sums of squares pariion he oal variabiliy (aen o 100%) ino componens aribuable o age, ime and counry. Table 2. w Males Females % 2.233% % 2.258% % 2.287% % 2.315% % 2.341% % 2.365% % 2.388% % 2.410% % 2.432% % 2.453% % 2.475% % 2.499% % 2.523% % 2.546% % 2.567% % 2.586% % 2.604% % 2.622% % 2.640% % 2.659% % 2.678% % 2.696% % 2.712% % 2.728% % 2.743% % 2.758% % 2.773% % 2.789% % 2.805% % 2.824% % 2.843% % 2.863% % 2.883% % 2.903% % 2.921% % 2.937% % 2.954% % 2.970% Se-specific weighs associaed wih each calendar year. 67

15 w Males Females % 7.078% % 7.232% % 7.425% % 7.510% % 7.494% % 7.337% % 7.121% % 6.816% % 6.440% % 5.995% % 5.599% % 5.250% % 4.853% % 4.357% % 3.646% % 2.759% % 1.808% % 0.902% % 0.306% % 0.063% % 0.008% % 0.001% % 0.000% Table 4. w Males Females UK 9.596% 9.570% US % % France % % Japan % % Germany % % Se-specific weighs associaed wih each counry. Table 3. Se-specific weighs associaed wih each age. 68