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ECONOMIC AND BUSINESS REVIEW VOL. 13 No. 4 211 251 272 251 ADOPTION OF PROJECTED MORTALITY TABLE FOR THE SLOVENIAN MARKET USING THE POISSON LOG-BILINEAR MODEL TO TEST THE MINIMUM STANDARD FOR VALUING LIFE ANNUITIES DARKO MEDVED 1 ALEŠ AHČAN 2 JOŽE SAMBT 3 ERMANNO PITACCO 4 Absrac: For he bes esimae of life annuiy provisions, he longeviy risk of he insured populaion mus be esimaed. In his aricle, we presen an applicaion of he Poisson logbilinear model o consruc life annuiy ables for he Slovenian marke. As daa on he selecion effec of annuiy owners are no available for he Slovenian marke, we have used selecion saisics from UK daa. We hen compare hose ables wih he German annuiy ables DAV 1994 R, which are he curren minimum sandard for valuing annuiy-relaed liabiliies in Slovenia. I is shown ha curren minimum sandard underesimae longeviy risk of he insured populaion in Slovenia by 2 4%. Keywords: Solvency II, valuaion of insurance liabiliies, Lee-Carer, moraliy projecions, life epecancy JEL classificaion: G17, G23, J11 1 INTRODUCTION Solvency II has proposed major changes o he valuaion of insurance echnical provisions and has had a considerable impac on reserving processes. The Solvency II framework requires a consisen marke approach o he valuaion of insurance asses and liabiliies. In such an approach, boh asses and liabiliies should be valued a he amoun for which hey could be ransferred, or seled, beween knowledgeable and willing paries (for deails, see he Quaniaive Impac Sudy 5 [QIS5]). This is a relaively new con- 1 JMD Consuling, Kamnik, Slovenia, e-mail: darko.medved@jmd-consuling.com 2 Universiy of Ljubljana, Faculy of Economics, Ljubljana, Slovenia e-mail ales.ahcan@ef.uni-lj.si 3 Universiy of Ljubljana, Faculy of Economics, Ljubljana, Slovenia, e-mail joze.samb@ef.uni-lj.si 4 Universià di Triese, Facolà di Economia, Triese, Ialy, e-mail ermanno.piacco@econ.unis.i

252 ECONOMIC AND BUSINESS REVIEW VOL. 13 No. 4 211 cep for insurance companies, as currenly, according o eising insurance legislaion, echnical provisions are valued on a book-value basis (e.g., Medved, 2). Technical provisions represen he major par of he liabiliy side of insurance companies balance shee. Insurance companies are required o esablish echnical provisions for all fuure obligaions arising from insurance conracs. There are hree main caegories of echnical provisions in he insurance indusry: non-life-insurance obligaions, life insurance obligaions, and healh insurance obligaions. According o Solvency II, he value of echnical provisions is equal o he sum of a bes esimae and a risk margin. The bes esimae is calculaed as an epeced value of all fuure cash-ou and cash-in flows, aking ino accoun he ime value of money. Cash-flow projecions should reflec a realisic epecaion of fuure demographic, legal, medical, echnological, social, and economic developmens. The risk margin is a buffer above he bes esimae of discouned cash flows, and i proecs agains worse-han-epeced scenarios. The risk margin covers model risks, parameer risk, and srucural uncerainy. For more deails on how o calculae he bes esimae and risk margin, see he QIS5 echnical specificaion. Valuing echnical provisions of life annuiies depends mainly on projeced demographic rends. A life annuiy is a specific insurance conrac in which one pary (an insurance company), in echange for paymen of a premium, guaranees a series of paymens unil he deah of he oher pary (he insured person). The projecion of fuure moraliy improvemens has significan effecs on premium calculaion and reserving for life annuiies (see Olivieri 21). As such, annuiies are associaed wih longeviy risk, in ha decreasing moraliy raes of he insured populaion lead o an increase in he number of annuiy paymens. This aricle addresses he sochasic projecions of fuure demographic rends in Slovenia as a key parameer of bes-esimae valuaions of Slovenian annuiies. There are no official projeced moraliy ables for he Slovenian populaion. To value life annuiies, insurance companies in Slovenia mus use annuiy ables ha are based on he moraliy profile of populaions in foreign counries. The Slovenian Insurance Supervision Agency has se he German annuiy ables DAV 1994 R as he minimum sandard. This means ha insurance companies have o value heir liabiliies using DAV 1994 R annuiy ables; however, hey can use oher ables, as long as hose ables produce higher echnical provisions han he DAV 1994 R. The resul, hough, is ha he indusry sandard is o use he DAV 1994 ables for premium calculaion and reserving, and in urn, moraliy saisics from 1994 on he insured in Germany are used o value liabiliies for annuiies and pensions in Slovenia. The DAV 1994 ables were used in he German insurance indusry unil 25, when he DAV 24 R ables were inroduced (see DAV, 25). The replacemen resuled in a 1 2% increase in premiums for deferred annuiies in Germany, depending on he insured s age and se. This is a subsanial increase in premium raes, and an imporan

he cenral deah rae m () coincide, which is a direc consequence of he piecewise consan D. MEDVED, A. AHČAN, J- SAMBT, E. PITACCO ADOPTION OF PROJECTED MORTALITY FOR THE... 253 quesion for he Slovenian insurance indusry in he Solvency II framework is wheher he DAV 1994 R ables are sill sufficien or even appropriae for measuring he bes esimae of liabiliies from annuiies and pensions in Slovenia. We ry o answer ha quesion here. To achieve his goal, we implemened he Lee-Carer (LC) mehod and is eension, which is he curren sandard for acuarial modelling of fuure moraliy. The Lee-Carer mehod is a powerful approach o moraliy projecions, as i combines a demographic model wih a ime-series model. In a sochasic framework, he resuls of LC projecions consis of poin and inerval esimaes. In his respec, he LC mehod allows for uncerainy The in srucure forecass. of his aricle is as follows: In Secion 2 we presen he main feaures of he In sochasic his aricle, LC we mehodology apply boh he for LC projecing model and moraliy. he Poisson Secion log-bilinear 3 covers daa mehod, specificaion one of and he laes eensions of he basic LC mehod, o he case of Slovenia, wih moraliy daa for calibraion. he period In 1945 27. Secion 4 Using we apply a Poisson he LC log-bilinear and Poisson projecion log-bilinear on Slovenian mehods populaion daa, we build selecion annuiy moraliy ables for comparison wih DAV 1994 R o daa for and Slovenia DAV 25 and R. presen We hen use resuls, he projeced and we also moraliy eplain raes how for kappa Slovenia projecions o es are he calculaed. curren minimum sandard for valuing annuiies. Wih back-esing, we es he bes fi for he projecions. In Secion 5 we calculae he The srucure of his aricle is as follows: In Secion 2 we presen he main feaures of selecion effec on annuiy purchasers and es he minimum sandard for valuing annuiy he sochasic LC mehodology for projecing moraliy. Secion 3 covers daa specificaion liabiliies. and calibraion. Secion 6 In oulines Secion our 4 conclusions. we apply he LC and Poisson log-bilinear mehods o daa for Slovenia and presen he resuls, and we also eplain how kappa projecions are calculaed. Wih back-esing, we es he bes fi for he projecions. In Secion 5 we calculae he selecion effec on annuiy purchasers and es he minimum sandard for valuing 2 STOCHASTIC annuiy liabiliies. MORTALITY Secion 6 oulines FORECASTING: our conclusions. THEORETICAL FRAMEWORK 2.1 The Lee-Carer model 2 STOCHASTIC MORTALITY FORECASTING: THEORETICAL FRAMEWORK In 1992 Lee and Carer esablished new sandard for projecing moraliy. They proposed a 2.1 The Lee-Carer model simple model for describing he change of moraliy as a funcion of ime inde. They In 1992 Lee and Carer esablished new sandard for projecing moraliy. They proposed a simple modelled model he cenral for describing deah rae he using change he log-bilinear of moraliy model as a funcion (Lee & Carer, of ime 1992) inde. They modelled he cenral deah rae using he log-bilinear model (Lee & Carer, 1992) ln m ( ) = α + βκ + ε (1) (1), where a describes he average age paern of moraliy over ime, and b describes he deviaion where α from describes he average he average paern when age paern k varies. of moraliy In addiion, over k ime, eplains and he evoluion describes he of he level of moraliy over ime, and e, is he error erm ha reflecs he age-specific influences no capured by he model. I is assumed ha he error erm has a mean of deviaion from he average paern when κ varies. In addiion, κ eplains he evoluion of and a sandard deviaion of σ e. The basic LC assumpion is ha he force of moraliy μ () and he cenral deah rae m () coincide, which is a direc consequence of he piecewise he level consan of moraliy forces over of assumed ime, moraliy. and ε, is he error erm ha reflecs he age-specific influences no capured by he model. I is assumed ha he error erm has a mean of and a sandard deviaion of σ. The basic LC assumpion is ha he force of moraliy µ () and ε

approach is o assume he following: 254 κ = and 1 β = (2) ECONOMIC AND BUSINESS REVIEW VOL. 13 No. 4 211 The usual approach o esimaing he parameers is o use he leas-squares mehod. which forces α o be an average of he log of cenral deah raes over calendar years. Once Furhermore, one mus impose addiional consrains o obain a unique soluion. The usual The usual approach o esimaing he parameers is o use he ) leas-squares mehod. Furhermore, approach parameers is α o one, assume βmus and he impose κfollowing: are addiional esimaed consrains (denoed by o α obain, a unique and κ ) ), soluion. we can The forecas usual approach is o assume he following: moraliy by modelling he values of κ in he fuure wih a ime-series approach (e.g., as a κ = and (2) β = 1 (2) random walk wih a drif or an auoregressive inegraed moving average (ARIMA) model. which forces a o be an average of he log of cenral deah raes over calendar years. Once which parameers forces α a, b and k are esimaed (denoed by a, b and k ), we can forecas o be average of he log of cenral deah raes over calendar years. Once moraliy 2.2 The by Poisson modelling log-bilinear he values model of k in he fuure wih a ime-series approach (e.g., as a random walk wih a drif or an auoregressive inegraed moving ) average (ARIMA) model. parameersα, β and κ are esimaed (denoed by α, and κ ) ), we can forecas As several auhors have noed (e.g., Brouhns e al., 22; Sihole e al., 2, Lee, 2), he moraliy Lee-Carer by mehod modelling assumes he values ha random of κ errors in he are fuure homoscedasic. wih a ime-series Tha is, approach he error (e.g., erms as are a 2.2 The Poisson log-bilinear model random assumed walk o have wih finie a drif variance, or an auoregressive and wih he inegraed assumpion moving of normaliy, average (ARIMA) hey share model. he same As several auhors have noed (e.g., Brouhns e al., 22; Sihole e al., 2, Lee, 2), underlying probabiliy densiy funcion. In mos cases, his assumpion is violaed because he he Lee-Carer mehod assumes ha random errors are homoscedasic. Tha is, he error 2.2 erms The are Poisson assumed log-bilinear o have model finie variance, and wih he assumpion of normaliy, hey share he same underlying probabiliy densiy funcion. In mos cases, his logarihm of he observed moraliy rae has much greaer variabiliy a older ages han a assumpion As several is auhors violaed have younger ages. I is herefore because noed (e.g., sensible he logarihm Brouhns e o assume of al., ha he 22; he observed Sihole number moraliy e al., 2, of deahs rae Lee, follows has he much 2), he Poisson greaer variabiliy a older ages han a younger ages. I is herefore sensible o assume Lee-Carer mehod assumes ha random errors are homoscedasic. Tha is, he error erms are ha law he wih number parameer of deahs (Brouhns follows e al., 22): he Poisson law wih parameer (Brouhns e al., 22): assumed o have finie variance, and wih he assumpion of normaliy, hey share he same underlying probabiliy densiy D funcion., Poisson In ( mos ETR cases,, µ ( his )) assumpion is violaed because (3) (3) he where logarihm D, is of he he number observed of moraliy observed rae deahs has of much persons greaer aged variabiliy in year a, older ETR, ages is he han a cenral where number of persons eposed o risk, and μ () is he force of moraliy. The log of, is he number of observed deahs of persons aged in year, TR, is he he younger force of ages. moraliy I is herefore is ln μ () sensible = a + o b assume k, in ha he LC he model. number The of deahs parameers follows have he he Poisson same The meaning esimaes as of in parameers he LC model. α, β and κ are denoed cenral number of persons eposed o risk, and µ () wih α, and κ ), obained by law wih parameer (Brouhns e al., 22): is he force of moraliy. The log of he The maimising esimaes he of parameers log-likelihood a, in b model and k(3), are which denoed is given wih by a he, b following: and k, obained by maimising force of moraliy he log-likelihood is ln µ ( ) = in αmodel + βκ, which, as in is he given LC model. by he The following: parameers have he same, Poisson ( ETR, µ ( )) (3) v v v meaning as in he L( LC αβκ, model., ) = D ( α + βκ) ETR ep( α + βκ)) (4) (4),,, where, is he number of observed deahs of persons aged in year, TR, is he Because of he presence of he bilinear erm b k, i is impossible o esimae he proposed model Because wih of commercial he presence saisical of he bilinear packages ermha implemen he Poisson regression. An opion cenral for number obaining of persons he esimaes eposed is o o risk, use and κ he mehod µ, () i is impossible o esimae he proposed is he proposed force of by moraliy. Goodman The (1979), log of he who model suggesed wih commercial he ieraive saisical mehod for packages esimaing ha implemen log-linear he models Poisson wih regression. bilinear erms. An opion Wih force his of approach, moraliy is we ln define µ ( ) = he α saring + βκ, as values in he for LC parameers model. The as parameers a =, b have = he and same k for =. obaining The parameers he esimaes are hen is o esimaed use he mehod using proposed he following by Goodman ieraion: (1979), who suggesed meaning as in he LC model. he ieraive mehod for esimaing log-linear models wih bilinear erms. Wih his approach, ) ) ) we define he saring values for parameers as α =, β = and κ =. The parameers are hen esimaed using he following ieraion:

he ieraive mehod for esimaing log-linear models wih bilinear erms. Wih his approach, ) we define he saring values for parameers as α =, D. MEDVED, A. AHČAN, J- SAMBT, E. PITACCO ADOPTION OF PROJECTED MORTALITY FOR THE... 255 are hen esimaed using he following ieraion: ) ) β = and κ =. The parameers ) ( D D ) ) ) ) ) α α β β κ κ n,, ) ) n+ 1 n n+ 1 n n+ 1 n = + ), =, n = D, ) ) ( D D ) β ) ) ) ) κ κ β β α α n+ 1 n+ 1,, ) ) n+ 2 n+ 1 n+ 2 n+ 1 n+ 2 n+ 1 = + ) ),, n 1 2 n = = + ( β ) D, ) ) ( D D ) κ ) ) ) β β, κ n+ 2 n+ 2,, n+ 3 n+ 2 = + ) ) n+ 2 2 n+ 2 ( κ ) D, ) ) ) = κ, α = α n + 3 n + 2 n + 3 n + 2 (5) (5) 2.3 Projecing fuure d moraliy d d 2.3 Projecing fuure moraliy κ= ϕ1 κ+... + ϕp κ+ ξ+ ψ1ξ 1 + ψqξ q (6) To To obain obain esimaes esimaes of of fuure fuure moraliy, moraliy, one one mus mus esimae esimae he he dynamics dynamics of of kappa kappa for for boh 2 boh where men ϕand p women, ψ q (Lee-Carer,, and ξ is a Gaussian 1992;). As whie-noise several auhors process, have such noed ha (e.g., σ ξ > Car- er, 1996; Lee, 2), k. In mos men and women (Lee-Carer, 1992;). As several auhors have noed (e.g., Carer, 1996; Lee, can be regarded as a sochasic process, modelled by fiing an ARIMA(p,d,q) insances, model. The dynamics of k can hus be described as follows: 2), κ he appropriae ime-series model akes a simpler form, such as can be regarded as a sochasic process, modelled by fiing an ARIMA(p,d,q) d d d κ = κ 1 + c + ξ + ψξ κ 1= derived ϕ1 κfrom +... + model ϕp κarima(,1,1). + ξ+ ψ1ξ 1 + ψthe qξ qconsan erm c (6) indicaes (6) model. The dynamics of κ can hus be described as follows: 2 where he average ϕ p O, ψannual q, and change ξ is a of Gaussian κ whie-noise process, such ha σ ξ >. In mos and presens he forecass of he long-erm change in insances, he appropriae ime-series model akes a simpler form, as k = 2 where ϕ k 1 + c + p, ψ q, and ξ is a Gaussian whie-noise process, such ha σ ξ >. In mos ξ + moraliy. ψξ 1 derived On he from basis model of he ARIMA(,1,1). resuls of he The ime-series consan erm model, c we indicaes can obain he average forecass of annual insances, change he of k appropriae and presens ime-series he forecass model of he long-erm akes a change simpler in moraliy. form, such On as he fuure basis of moraliy he resuls and of is he momens ime-series from he model, following: we can obain forecass of fuure moraliy κand = κis momens + c + ξ + ψξ from he derived following: from model ARIMA(,1,1). The consan erm c indicaes 1 1 he average annual change µ of ( + κ n) and = ep( presens α)ep( he βκ forecass + n) of he long-erm (7) change (7) in moraliy. On he basis of he resuls of he ime-series model, we can obain forecass of 3 DATA 3 DATA fuure moraliy and is momens from he following: Populaion moraliy daa by age and se for he period 1971 28 were provided by he Populaion moraliy daa by age and se for he period 1971 28 were provided by he Saisical Office of he Republic of Slovenia. For he 1971 198 period, here are some minor Saisical discrepancies Office of beween he Republic µ he daa used and he official cumulaive daa for he same ( + n) of = ep( Slovenia. α)ep( For βκ he + n) 1971 198 period, here (7) are some years, especially for 1972 and 1973. The discrepancies are in he range of 1 persons, which minor is negligible. discrepancies beween he daa used and he official cumulaive daa for he same years, Again, 3 especially DATA we denoe for 1972 by ETR and, he 1973. eposure The discrepancies o risk a age are on in he las range birhday of 1 during persons, year which. is The eposure o risk refers o he oal number of person-years in a given populaion over a calendar Populaion negligible. year, moraliy and i is daa esimaed by age by and he se number for he of he period populaion 1971 28 aged were in provided he middle by he of he calendar year (1 July of each year); ha is, hose who reached age beween 1 July Saisical Office of he Republic of Slovenia. For he 1971 198 period, here are some of he previous year and 3 June of he observing year. By D, we denoe he number of minor Again, discrepancies we denoe by beween TR he, he daa eposure used and o risk he a official age cumulaive on he las daa birhday for he during same year years,. especially for 1972 and 1973. The discrepancies are in he range of 1 persons, which is The eposure o risk refers o he oal number of person-years in a given populaion over a negligible. calendar year, and i is esimaed by he number of he populaion aged in he middle of he

256 ECONOMIC AND BUSINESS REVIEW VOL. 13 No. 4 211 deahs recorded a age las birhday during calendar year. Then, he maimum likelihood esimaor for μ () (force of moraliy) equals ) D µ () = ETR,, (8) (8) Assuming a consan force of moraliy for nonineger years, we have μ () = m () (see Piacco e al., 29). Wih his assumpion, we can consruc Slovenian moraliy ) ) Assuming a consan force of moraliy for nonineger years, we have µ daa () = m() (see for he period 1971 28. Piacco e al., 29). Wih his assumpion, we can consruc Slovenian moraliy daa for he For our analysis, we also needed moraliy daa for he Slovenian populaion before 1971. The period Human 1971 28. Moraliy Daabase (HMD) conains average cenral moraliy raes for 193 1933, 1948 1952, 1952 1954, and 196 1962. We used his informaion o inerpolae m () for he years 1945 197. We calculaed a log regression line beween 1932 and 1985 and made an inerpolaion beween 1945 and 197 using a 95% confidence inerval. We For chose our 1945 analysis, as a we saring also needed year for moraliy our analysis daa for because he Slovenian he generaion populaion born before 1971. his The year is highly likely o have already reired. Human Moraliy Daabase (HMD) conains average cenral moraliy raes for 193 1933, Slovenian moraliy daa have some irregulariies ha had o be adjused before we could We use 1948 1952, apply he hem for forecasing. 1952 1954, mehod ha For and Denui eample, 196 1962. and Goderniau a very We old used (25) ages, his he informaion proposed o daa have o erapolae very inerpolae deah ) low risk m eposures, which leads o large sampling errors and highly volaile crude deah raes. For ( ) raes for a eample, he very years old risk 1945 197. ages. Following eposures We for calculaed his approach, men varied a beween log regression he deah raes 567 in line 1971 beween for very and 13 1932 old ages in and 27. 1985 were For and esimaed he made 1971 198 according period, no daa are available for age groups older han 85. Therefore, we needed a mehod o erapolae a survival funcion a very old ages, wihou requiring accu- an inerpolaion o he beween logisic formula 1945 and proposed 197 using here. a 95% Parameers confidence were inerval. chosen We in such chose a way 1945 as as o a rae We maimise moraliy apply saring year he daa for fi. mehod for ha par Denui of he and populaion. Goderniau Recen (25) moraliy proposed o sudies erapolae sugges deah ha raes our analysis because he generaion born before his year is highly likely o he force of moraliy is slowly increasing a very old ages and approaching a relaively fla a shape very (e.g., old ages. Piacco Following e al., 29). his approach, In oher words, he deah he raes eponenial for very rae old of ages he were moraliy esimaed have already reired. increase a very old ages is no consan bu declines. according The log-quadraic o he logisic regression formula model proposed is defined here. as follows: Parameers were chosen in such a way as o We apply he mehod ha Denui and Goderniau (25) proposed o erapolae deah maimise he fi. raes Slovenian a very moraliy old ages. daa Following have some his irregulariies approach, he ha deah had o raes be adjused for very before old ages we were could use 2 esimaed according o he logisic ln qˆ ( formula ) = a + b proposed + c + εhere., Parameers were chosen (9) in such hem a way for as forecasing. o maimise For he eample, fi. a very old ages, he daa have very low risk eposures, The log-quadraic regression model is defined as follows: which leads o large sampling errors and highly volaile crude deah raes. For eample, risk The where log-quadraic he one-year regression deah model probabiliy is defined a ime as follows: wih ε, is independen and normally eposures for men varied beween 567 in 1971 and 13 in 27. For he 1971 198 period, 2 disribued, wih a mean of ln qˆ and ( ) = variance a + b of + c σ +. εif, ω is an age limi, hen (9) we (9) have a no daa are available for age groups older han 85. Therefore, we needed a mehod o where consrain: he one-year q deah probabiliy a ime wih e,1 is independen and normally ω () = 1. To ensure he concave behaviour of ln qˆ disribued, erapolae wih a survival a mean funcion of and a variance very old of ages, σ 2 ( ), we implemened a second where he one-year deah probabiliy a ime. If wihou w is an requiring ε age limi, accurae hen we moraliy have a consrain: consrain: q w daa for, is independen and normally () = 1. To ensure he concave behaviour of ln qˆ (), we implemened a second consrain: ha par of he populaion. Recen moraliy sudies 2 sugges ha he force of moraliy is disribued, wih a mean of and variance of σ. If ω is an age limi, hen we have a slowly increasing a very old ages and approaching a relaively fla shape (e.g., Piacco e al., consrain: qω () = 1. To ensure he concave q() behaviour =ω = of ln qˆ ( ), we implemened (1) a (1) second 29). In oher words, he eponenial rae of he moraliy increase a very old ages is no consrain: consan bu declines. We obained he opimal fi (highes 2 ) wih ω = 13 and a saring smoohing age = 75. However, we used only he resuls ha q we () obained =ω = from age 85 onward. (1)

D. MEDVED, A. AHČAN, J- SAMBT, E. PITACCO ADOPTION OF PROJECTED MORTALITY FOR THE... 257 We obained he opimal fi (highes R 2 ) wih w = 13 and a saring smoohing age = 13. However, we used only he resuls ha we obained from age 85 onward. Furhermore, some deahs raes were equal o (i.e., here were no deahs in he observed period), which happens quie ofen a younger ages because of he small populaion. Because we used logarihms of deah raes for forecasing, we implemened adjusmen echniques o obain posiive values. In paricular, we used inerpolaion echniques wih neighbour cenral deah raes o obain he bes esimae for such cases. 4 FORECASTING MORTALITY USING POPULATION MORTALITY STATISTICS FOR SLOVENIA In his secion, we presen he resuls using he sochasic mehods inroduced in Secion 2. On he basis of back-esing, we decided on wo mehods for projecing fuure moraliy: he Lee-Carer model and he Poisson log-bilinear model based on he daa from 1971 o 28 described in Secion 3. The code for his secion was programmed using Malab sofware. 4.1 The original Lee Carer model vs. he Poisson log-bilinear model Firs, we presen he resuls of he model inroduced in Brouhns e al. (22) using he Poisson log bilinear regression approach. The resuls of ha model are compared wih he resuls of he original Lee-Carer (1992) model. In he analysis we used daa from he Slovenian Saisical Office, adjused as described in Secion 3. The resuls obained from he Poisson log bilinear model are represened in Figure 1 by a green line, and he resuls obained from he original Lee-Carer model are represened by a blue line. As Figure 1 shows, he beas from boh mehods ehibi highly erraic behaviour, regardless of he mehod used. This is mainly a consequence of he small populaion and relaively low number of boh eposures and deahs in he Slovenian populaion as compared wih larger counries. Figure 1 shows ha over he pas 4 years, he bigges improvemens in moraliy were in he age group of minors, especially newborns and children beween he ages of 1 and 14 years. In his age group, he discrepancy beween he LC and Poisson log bilinear models is also greaes. In he case of he Poisson log bilinear model, bea is somewha larger for his age group han in he LC mehod, and i is slighly lower for mos of he oher age groups. The beas indicae rends similar o hose in oher counries, wih moraliy improvemens being greaes in he lower age groups.

258 ECONOMIC AND BUSINESS REVIEW VOL. 13 No. 4 211 Figure 1: Bea() as a funcion of age (male): he Poisson vs. he LC model Figure 1: Bea() as a funcion of age (male): he Poisson vs. he LC model.35 male.3.25 bea().2.15.1.5 2 4 6 8 1 12 age Figure 2: Alpha as a funcion of age (male): he Poisson vs. he LC model Figure 2: Alpha as a funcion of age (male): he Poisson vs. he LC model As Figure 1 shows, he beas from boh mehods ehibi highly erraic behaviour, reg of he mehod used. This is mainly a consequence of he small populaion and relaive -2 number of boh eposures and deahs in he Slovenian populaion as compared wih counries. Figure 1 shows ha over he pas 4 years, he bigges improvemens in mo alpha() -1-3 -4 were in he age group of minors, especially newborns and children beween he ages of -5-6 14 years. In his age group, he discrepancy beween he LC and Poisson log-bilinear m -7 is also greaes. In he case of he Poisson log-bilinear model, bea is somewha larger f -8 male age group -9 han in he LC mehod, and i is slighly lower for mos of he oher age group 2 4 6 8 1 12 age beas indicae rends similar o hose in oher counries, wih moraliy improvemens The alphas in boh models indicae ha here are hardly any differences in he calculaed values. The only difference is a small discrepancy in alphas for children beween he The greaes alphas in he in boh lower models age groups. indicae ha here are hardly any differences in he calcu ages of 2 and 1 years. Overall, he resuls of boh models are similar o hose for oher values. The only difference is a small discrepancy in alphas for children beween he age and 1 years. Overall, he resuls of boh models are similar o hose for oher coun

D. MEDVED, A. AHČAN, J- SAMBT, E. PITACCO ADOPTION OF PROJECTED MORTALITY FOR THE... 259 counries. Moraliy is relaively high for newborns and drops considerably for minors. For male eenagers and young aduls, moraliy increases wih a noiceable hump (i.e., he esoserone hump) around age 2. Afer ha age, moraliy is consan or slighly decreases unil age 3, when i sars o increase again almos linearly ino he oldes ages, as observed by Cairns e al. (26). Figure 3: Kappa as a funcion of year (male): he Poisson vs. he LC model Figure 3: Kappa as a funcion of year (male): he Poisson vs. he LC model 4 male 3 2 1 kappa() -1-2 -3-4 -5-6 197 1975 198 1985 199 1995 2 25 21 year Figure 3 shows ha wih boh mehods, kappa decreases subsanially over he observed period (1971 28). During ha ime, here is a coninuous improvemen in moraliy Figure 3 shows ha wih boh mehods, kappa decreases subsanially over h for all age groups (see also Figure 1). Given he average value of bea of.1, moraliy has, on average, more han halved in he observed period. Of course, some age groups (e.g., minors) period eperienced (1971 28). a much During greaer ha moraliy ime, here decline is han a coninuous he average, improvemen whereas in mor older age groups had improvemens below or well below he average. Looking a he rend age for kappa, groups we can (see see also ha Figure wih he 1). LC Given mehod, he kappa average decreases value almos of linearly, bea of.1, mora whereas for he Poisson log-bilinear model, i seems o increase a an even higher rae and ehibis a mild curvaure. average, more han halved in he observed period. Of course, some age groups (e Turning o he resuls for females, he rend in kappa is similar o ha for males. Kappa decreases eperienced almos a linearly much greaer over ime moraliy under boh decline mehods. han Again, he average, boh mehods whereas older age yield similar resuls, wih only sligh differences in calculaed values for some years. Overall, improvemens we can conclude below ha he or resuls well below of he wo he mehods average. are Looking relaively a robus he rend for for kappa, he values of kappa. The resuls for females alpha, bea, and kappa are presened in Appendi ha 1. wih he LC mehod, kappa decreases almos linearly, whereas for he P bilinear model, i seems o increase a an even higher rae and ehibis a mild curva Turning o he resuls for females, he rend in kappa is similar o ha for ma

26 ECONOMIC AND BUSINESS REVIEW VOL. 13 No. 4 211 4.2 Projecing kappa using he Poisson log-bilinear model In his secion, we presen he resuls of projecing kappa using he Poisson log-bilinear model (Brouhns e al., 22). In his case, he value of c is equal o 2.43 (for a definiion of c, see Secion 2.3). To check he validiy of he model, we eamined he saisical properies of he residuals. Table 5 (in Appendi 2) shows ha we canno rejec he hypohesis of normally disribued residuals. Boh he Jarque-Berra es and he values of kurosis and skewness indicae ha he normal disribuion is a good approimaion for he disribuion of he residuals. Furhermore, in looking a Q-saisics for auocorrelaion, we observed ha here is no saisically significan auocorrelaion. The previous analysis suggesed ha for Slovenian moraliy saisics, a random walk wih drif model is suiable for modelling he esimaed k. This is no surprising, as a similar observaion can be found in many oher counries (see, e.g., Tuljapurkar e al., 2). Figure 4: Kappa() as a funcion of ime (males): he Poisson model Figure 4: Kappa() as a funcion of ime (males): he Poisson model 5 scenarios male -5-1 kappa() -15-2 -25-3 -35 196 198 2 22 24 26 28 21 year Figure 4 reveals a fairly srong downward rend in moraliy. For he low-moraliy scenario, he rend is somewha higher as a resul of he greaer-han-epeced decrease in kappa. Likewise, for he high-moraliy scenario, he decrease in kappa is somewha lower. We can see ha in boh cases he rend is negaive, which means ha he reducion in kappa Figure is highly 4 reveals saisically a fairly significan. srong downward rend in moraliy. For he low-moraliy scena Le us he now rend consider is somewha he resul for higher females. as Again, a resul he ARIMA(,1,1) of he greaer-han-epeced proves adequae for decrease in kap modelling he dynamics of kappa. Likewise, for he high-moraliy scenario, he decrease in kappa is somewha lower. We see ha in boh cases he rend is negaive, which means ha he reducion in kappa is hig

D. MEDVED, A. AHČAN, J- SAMBT, E. PITACCO ADOPTION OF PROJECTED MORTALITY FOR THE... 261 Figure 5: Kappa() as a funcion of ime (females): he Poisson model Figure 5: Kappa() as a funcion of ime (females): he Poisson model 5 scenarios female -5 kappa() -1-15 -2-25 -3 196 198 2 22 24 26 28 21 year As one can see from he corresponding able in Appendi 2, he residuals obained using he ARIMA(,1,1) model are uncorrelaed and approimaely normally disribued. Figure As 5 shows one can ha see he difference from he beween corresponding scenarios increases able in in Appendi ime, bu he 2, esimaes he residuals obaine remain fairly close. ARIMA(,1,1) model are uncorrelaed and approimaely normally disribued 4.3 Projecing kappa using he LC model shows ha he difference beween scenarios increases in ime, bu he esimaes re We now consider he case of modelling kappa under he original Lee-Carer model. We can see close. ha he simple ARIMA(,1,) process seems o be appropriae for he esimaed k for males. The resuls, presened in Table 5 (in Appendi 2), indicae ha he residuals are no auocorrelaed, whereas he skewness, kurosis, and Jarque-Berra ess indicae ha 4.3 he Projecing hypohesis of kappa normally using disribued he LC residuals model canno be rejeced. Anoher imporan consequence of he residual es is ha, in he LC model, he sandard deviaion of residuals is greaer han in he Poisson log-bilinear model. The difference beween We now consider he case of modelling kappa under he original Lee-Carer mod he forecass is hus much smaller in he case of he Poisson log-bilinear model. The resuls see ha reveal he ha simple he rend ARIMA(,1,) is somewha lower process han for seems he Poisson o be model; appropriae in conras, hough, he sandard deviaion of he residuals is greaer han in he Poisson mod- for he esima el. The values for kappa under he high-moraliy scenario are hus greaer in he original males. The resuls, presened in Table 5 (in Appendi 2), indicae ha he residu LC model han in he Poisson model. The same goes for he cenral endency scenario. The values for he low-moraliy scenario in he LC model are comparable wih he values in auocorrelaed, he Poisson model. whereas he skewness, kurosis, and Jarque-Berra ess indica hypohesis of normally disribued residuals canno be rejeced. Anoher consequence of he residual es is ha, in he LC model, he sandard deviaion of

262 ECONOMIC AND BUSINESS REVIEW VOL. 13 No. 4 211 In conras o males, he dynamics for females canno be modelled as an ARIMA(,1,) model bu can be fied bes by an ARMA(2,2) model. In his case, as presened in Table 6 (in Appendi 2), he residuals are no auocorrelaed (a leas up o he relevan number of lags) and can be assumed o be normally disribued. Once again, we observe ha he sandard deviaion of kappa esimaes is significanly higher han in he Poisson log-bilinear model. This could be an argumen in favour of using Brouhns e al. o model fuure moraliy on Slovenian moraliy saisics. Namely, wih he Poisson log-bilinear model, he confidence inerval of esimaes is much narrower han in he LC model. 4.4 Back-esing For fuure projecions, we needed a quaniaive assessmen of boh models before deciding which model would beer capure he moraliy rend. Of course, we could no do his graphically, so we used he back-esing echnique o deermine which model would be mos appropriae for esimaing fuure moraliy. More precisely, we esed he models agains real daa (in our case, number of deahs) for he period 21 28. In he firs sep, we fi he model parameers o he daa for he period 1971 2. In he second sep, we used he values of he parameers obained in sep 1 o predic he number of deahs D, in he period 21 28. We hen used several sandard indicaors of fi o compare he mehods. Table 1: Comparison of mehods using back-esing for he 21 28 period (males) LC Poisson log-bilinear MSE 21 2.8 MPE 11.7 11.6 R 2.955.96 Source: SORS and own work Firs, we eamine he resuls for males for he period 21 28. A comparison of he mehods using back-esing for ha period (males) in Table 1 reveals he supremacy of he Poisson log-bilinear mehod, which is even more convincing for females. Of all variaion in he number of deahs for he period 21 28, 99% can be eplained by his mehod (see Table 2). Table 2: Comparison of mehods using back-esing for he 21 28 period (females) LC Poisson log-bilinear MSE 18.5 11.5 MPE 11.2 7 R 2.975.99 Source: SORS and own work

D. MEDVED, A. AHČAN, J- SAMBT, E. PITACCO ADOPTION OF PROJECTED MORTALITY FOR THE... 263 On he basis of he back-esing analysis, we concluded ha he Poisson log-bilinear model fi he acual cenral deah raes beer han he original LC model, and herefore we Thus, used ha when model projecing o forecas he Slovenian values of populaion kappa under moraliy. differen scenarios for obaining he Thus, esimaes when of projecing fuure moraliy, he values we can of kappa use he under following differen relaionship: scenarios for obaining he esimaes of fuure moraliy, we can use he following relaionship: m (28 + ) = m (28) ep( β ( κ κ )) (11) (11) 28+ 28 where m (28 + ) is he cenral deah rae for year (28 + ) and age. Formula is essenially where man (28 erapolaion + ) is he cenral of he classical deah rae Lee-Carer for year (28 model + using ) and he age projecions. Formula of (11) is kappa obained from ARIMA models. In deermining fuure moraliy, we mus ake ino essenially accoun he an erapolaion uncerainy of of our he esimaes. classical Lee-Carer We herefore model consruced using he hree projecions scenarios of kappa ha differ wih respec o kappa values used when making projecions. Under he besesimae obained scenario, from ARIMA we deermined models. In fuure deermining values of fuure kappa moraliy, by aking we kappa mus ake o be ino equal accoun o he he epeced value derived from he ARIMA model. Using equaion, we yielded fuure values uncerainy of kappa of hrough our esimaes. he following We herefore relaionship: consruced k hree 28+ = kscenarios 28 + c. ha differ wih respec o kappa values used when making projecions. Under he bes-esimae scenario, we In he high-moraliy scenario, we obained fuure values of kappa by using he following relaionship: k 28+ = k 28 + c + 2s e. In he low-moraliy scenario, we obained deermined fuure values of kappa by aking kappa o be equal o he epeced value derived fuure values of kappa by assuming lower-han-epeced values of kappa. In his case, we obained from he he ARIMA fuure values model. of Using kappa equaion wih k 28+ (11), = we k 28 yielded + c fuure 2s e. values of kappa hrough he following relaionship: κ = 28 κ + 28 c. + 5 TESTING THE BEST-ESTIMATE VALUATION OF A LIFE ANNUITY 5.1 Cohor vs. period life ables In he high-moraliy scenario, we obained fuure values of kappa by using he following To calculae he presen value of a fuure obligaion arising from life annuiy paymens, acuaries relaionship: birhday mus (denoed develop κ as a life able (also called a able or acuarial able). A life 28+ = q κ). Life 28 + cables + 2σ ε are. derived In he from low-moraliy observed scenario, and projeced we obained moraliy fuure raes, able shows for each age he probabiliy ha a person of ha age will die before his or her values which ne birhday can of kappa be presened (denoed by assuming he qfollowing lower-han-epeced ). Life ables mari: are derived values from of kappa. observed In his and case, projeced we obained moraliy raes, which can be presened in he following mari: he fuure values of kappa wih κ = 28 κ + 28 c 2σ. + ε q ( min ) L q ( min n) L q ( min ma ) M (12) (12) 5 TESTING THE BEST-ESTIMATE M VALUATION OF A LIFE ANNUITY q ( ma ) q ( ma n) q ( ma ma ) 5.1 Cohor vs. period life ables where {q where ()}, (,..., n ) represens observed moraliy raes, and {q ()}, ( n + 1,..., ma ) To calculae he presen value of a fuure obligaion arising from life annuiy paymens, represens projeced moraliy raes. By n we denoe he base year from which projecions acuaries are made. mus The develop sequence a life able q (), (also q +1 ( called + 1),... a moraliy is a cohor able able. acuarial The sequence able). A q life (), able q +1 (), q(), q } +2 ()... ( is, K a period, n) represens able. This leads observed o he consrucion moraliy of wo ypes of raes, life ables. and shows for each age he probabiliy ha a person of ha age will die before his or her ne q( ), ( n + 1, K, ma ) represens projeced moraliy raes. By n we denoe he base year from which projecions are made. The sequence q( ), q+ 1( + 1), K is a cohor able. The

lifeime. Therefore, o calculae he bes esimae of an insurance annuiy, we had o adjus A life annuiy purchaser is, mos likely, a healhy TR ˆ mperson () wih paricularly low moraliy moraliy A in life he annuiy firs projecions years purchaser of o he include is, life mos annuiy SMR his likely, = effec paymen a healhy in he and, sprojecions. an d in general, a longer-han-average TR ˆ Piacco e al. (29) sugges m person () wih paricularly low moraliy (15) in he epeced lifeime. Therefore, o calculae he bes esimae of an insurance annuiy, we had following firs o adjus years model moraliy of he for life age projecions annuiy 6 and older: paymen o include and, his in effec general, in he a projecions. longer-han-average Piacco e epeced al. (29) sugges he following model for age 6 and older: A lifeime. annuiy Therefore, purchaser o calculae is, mos likely, he bes a healhy esimae person of an wih insurance paricularly annuiy, low we moraliy had o adjus in he ) LIM ) HMD ln m ( ) = f( ) + ln m ( ) + ε (16) (16) firs moraliy years projecions of he life o include annuiy his paymen effec in and, he in projecions. general, a Piacco longer-han-average e al. (29) sugges epeced he HMD LIM Here we denoe by m () he populaion cenral deah raes and by m () he annuiy lifeime. following Here purchaser Therefore, model we denoe cenral by deah o age ) HMD calculae 6 and older: m () raes. This he bes leads esimae us o produce of an insurance SMR in annuiy, he form ) LIMwe e fˆ() had, which o adjus he populaion cenral deah raes and by m () he annuiy can be used o adop moraliy projecions for he insurance marke. Slovenia moraliy moraliy projecions o include his effec in he projecions. Piacco e al. (29) sugges he eperience for annuiy purchasers ) LIM is no direcly ) available, HMD and because pension ˆ reform purchaser cenral deah raes. ln mthis leads us o produce SMR in he form e ( ) ( ) = f( ) + ln m ( ) + ε, which (16) can be sared following only model 1 years for ago, age here 6 and are older: no adequae saisical daa for he conclusion regard- used o adop moraliy projecions for he insurance marke. Slovenia moraliy eperience ) HMD ) LIM Here we denoe by m () ) LIM he populaion cenral ) HMD deah raes and by m () he annuiy for annuiy purchasers is ln no mdirecly ( ) = favailable, ( ) + ln mand ( because ) + ε pension reform sared (16) only 1 life able includes differen generaions in a single able. In a cohor life able, a life annuiy is calculaed on he basis of observed birh cohor, which means ha for each generaion, we can 264 ECONOMIC AND BUSINESS REVIEW VOL. 13 No. 4 211 consruc one life able. Cohor life ables beer eplain improvemens in moraliy for each generaion In a period separaely, life able, so he for presen bes-esimae value of a calculaions, life annuiy akes cohor ino life accoun ables age-specific are used (see Piacco moraliy raes a age and older ages observed in a given calendar year. So a period life e al., 29). able We includes can calculae differen an generaions annuiy of size in a 1 single ha is able. payable In a cohor yearly life a he able, beginning a life annuiy of each is year calculaed on he basis of observed birh cohor, which means ha for each generaion, we while can consruc an insured one is alive life able. from Cohor he following: life ables beer eplain improvemens in moraliy for each he generaion age cohor separaely, life able, so for we bes-esimae firs chose calculaions, he base cohor life birh ables year are τ. We hen To calculae used (see Piacco e al., 29). 1, k = calculaed We he can life calculae able an by annuiy ω aking of diagonal size 1 ha probabiliies is payable yearly k 1 from a birh he beginning year τ of as each follows: year k To calculae he age a&& cohor ( τ ) = life able, we firs chose he base cohor (1 ibirh ) year τ. We hen k (1 q j( τ j), k + (14) = + + + > calculaed while an he insured life able is alive by from aking diagonal probabiliies from he j= following: birh year τ as follows: We can calculae an annuiy l+ 1 ( τof ) size = (11 ha q( τis + payable )) lyearly ( τ) a he beginning of each (13) year (13) 5.2 Selecion effec 1, k = ω We while can calculae an insured an is annuiy alive from of he size k 1 following: 1 ha is payable yearly a he beginning k a&& of each year ( τ ) = (1 i) while The an sandardised insured alive moraliy from k he raio following: (1 q (SMR) j( τ j), k + (14) = + + + > is used as an inde for comparing moraliy j= eperiences beween wo groups: 1, acual deahs in a paricular k = ω populaion (e.g., life annuiy k 1 k a&& ( τ ) = (1 i) (14) owners) wih epeced deahs, k if (1 sandard q j( τ age-specific j), k moraliy + (14) 5.2 Selecion effec = + + + > raes were o be applied. j= The SMR sandardised defined moraliy as follows: raio (SMR) is used as an inde for comparing moraliy 5.2 5.2 eperiences Selecion Selecion effec beween effec wo groups: acual deahs in a paricular populaion (e.g., life annuiy TR ˆ m () The sandardised moraliy raio (SMR) The owners) sandardised wih epeced moraliy deahs, raio is used as an inde for comparing moraliy SMR if sandard = (SMR) is age-specific used as an moraliy inde for raes comparing were o (15) be moraliy applied. eperiences beween wo groups: acual deahs in san ˆ a d TR paricular () populaion (e.g., life annuiy eperiences The owners) SMR is wih defined beween epeced as wo follows: groups: deahs, acual if sandard deahs in age-specific a paricular moraliy populaion raes (e.g., were life o annuiy be m applied. The SMR is defined as follows: owners) wih epeced deahs, if sandard age-specific moraliy raes were o be applied. A life annuiy purchaser is, mos likely, a healhy person wih paricularly low moraliy in he ETR ˆ m () The SMR is defined as follows: SMR = (15) (15) san d firs years of he life annuiy paymen ETR and, mˆ in general, () a longer-han-average epeced purchaser cenral deah raes. This leads us o produce SMR in he form ˆ e ( ), which can be

effec. As a resul, we chose an alernaive soluion inroduced by he Associazione Nazionale fra la Imprese Assicurarici (25), which has been used o build moraliy saisics for he Ialian insurance annuiy indusry. D. MEDVED, A. AHČAN, J- SAMBT, E. PITACCO ADOPTION OF PROJECTED MORTALITY FOR THE... 265 The idea is o use SMR from anoher populaion wih similar characerisics as he populaion ing he selecion effec. As a resul, we chose an alernaive soluion inroduced by he Associazione for which we Nazionale wan o fra inroduce la Imprese he selecion Assicurarici effec. (25), As we which see from has been (16), used in general, o build SMR moraliy saisics for he Ialian insurance annuiy indusry. RC depends on age. Le us denoe by SMR () he reference counry s sandardised moraliy The idea is o use SMR from anoher populaion wih similar characerisics as he populaion raio for beween which we he wan insured o inroduce and general he populaion selecion effec. for he As paricular we see from year, in. We general, can hen RC SMR depends on age. Le us denoe by SMR () he reference counry s sandardised moraliy calculae raio life beween insurance he marke insured cenral and deah general raes populaion as follows: for he paricular year. We can hen calculae life insurance marke cenral deah raes as follows: ) LIM RC ) HMD m () = SMR () m () (17) (17) 5.3 5.3 Applying Applying he selecion he selecion effec effec o o Slovenian moraliy projecions RC To obain SMR (), RC To obain SMR we included an elemen of selecion ha emerged from daa from he (), we included an elemen of selecion ha emerged from daa from he Unied Kingdom, where annuiy and pension marke income is well developed. We used UK Unied daa for Kingdom, he period where 1999 22 annuiy and colleced pension by marke he Coninuous income is Moraliy well developed. Invesigaion We used UK Bureau and published in number 23 of he Coninuous Moraliy Invesigaion Repors (29), daa for which he perains period 1999 22 o he eperience colleced of by porfolios he Coninuous of immediae Moraliy and Invesigaion deferred life Bureau annuiies. The 1999 22 moraliy invesigaion presens he so-called series base moraliy and published ables in adoped number by 23 UK of he acuaries. Coninuous The Moraliy saisical Invesigaion base is eensive: Repors i (29), involves which more han 2 million lives eposed o risk. perains o he eperience of porfolios of immediae and deferred life annuiies. The 1999 In paricular, we used he moraliy invesigaion of life office pensioners (insured o deferred 22 annuiies) moraliy - invesigaion PNM ables presens for men he and so-called PNF ables series for base women, moraliy which ables show adoped he by moraliy raes for each age from 2 o 12 years, disinguished beween lives (i.e., heads UK acuaries. The saisical base is eensive: i involves more han 2 million lives eposed insured) and amouns (i.e., weighed by he benefi). o risk. By comparing he moraliy of UK insured lives wih hose of he oal UK populaion (aken from English Life Table No. 16, 2 22), we were able o quanify he increased survival of he insured populaion. To ake ino accoun he impac of economic wealh of he In paricular, insured on we selecion, used he moraliy we weighed invesigaion moraliy of raes life office by he pensioners size of annuiy, (insured as o has deferred been saisically proved in oher markes (e.g., Germany; see DAV, 25). annuiies) - PNM ables for men and PNF ables for women, which show he moraliy Such a selecion facor is srucured o represen he moraliy of he insured s deferred annuiy. In he case of an immediae annuiy, furher seleciviy should be added. As a resul, we added an era selecion facor, calculaed as he raio beween he moraliy of deferred annuiy owners and immediae annuians in he Unied Kingdom. Figure 6 shows he combined selecion facors for immediae annuians used for he populaion moraliy ables for Slovenia. Saring from he paricular generaion cohor life able derived from hisorical daa and sochasic projecions described in he previous secion, and hen in applying he selecion facors, we obained he projeced and seleced moraliy able for his paricular

266 ECONOMIC AND BUSINESS REVIEW VOL. 13 No. 4 211 generaion. Such derived life ables are considered he bes esimae of an annuiy purchaser in Slovenia. These raes can be compared wih he curren minimum sandard: DAV 1994 R. Figure 6: Selecion facor for Slovenia annuiy owners Figure 6: Selecion facor for Slovenia annuiy owners Selecion facors selecion in %.5.6.7.8.9 1. males females 2 4 6 8 1 12 Age Source: own work Source: own work 5.4 Tesing he minimum sandard 5.4 Tesing he minimum sandard In his secion, we compare he Slovenian annuiy life able wih DAV 24 and DAV 1994. Looking a he resuls presened in Table 3, he ne single premium for an annuiy based on he In Poisson his secion, model we is up compare o 12% less he han Slovenian he DAV annuiy 24 single life able premium wih annuiies. This gap is a consequence of he 15% margin incorporaed in he new German ables DAV 24 and DAV 1994. (DAV, 25) Looking and he a fac he ha resuls moraliy presened raes derived in Table from 3, he Poisson ne single model premium represen for an annuiy based on he bes esimae of fuure annuian moraliy. Those ables canno be direcly compared in his respec. he Poisson Under he model bes-esimae is up o 12% scenario, less we han deermined he DAV fuure 24 single kappa premium values by annuiies. This gap is aking kappa o be equal o he epeced value. By comparing he low-moraliy scenario wih DAV a 24, consequence we observed of only he 15% minor margin differences incorporaed raes (a in 1 7% he higher new German single premium in he case of DAV 24). ables (DAV, 25) and he fac ha moraliy raes derived from he Poisson model represen he bes esimae of The comparisons also show ha for curren generaions who have no ye reired, he DAV 1994 fuure ables underesimae annuian moraliy. he bes-esimae Those ables annuiy canno in be Slovenia direcly by compared 2%. For males in his respec. Under he who will reire in he fuure (deferred annuians), he difference is almos 4%. As a consequence, we bes-esimae believe ha scenario, he DAV 1994 we deermined ables should fuure no be kappa used values for he bes-esimae by aking kappa o be equal o he valuaion of annuiy liabiliies in he Solvency II framework. epeced value. By comparing he low-moraliy scenario wih DAV 24, we observed only minor differences in raes (a 1 7% higher single premium in he case of DAV 24).

D. MEDVED, A. AHČAN, J- SAMBT, E. PITACCO ADOPTION OF PROJECTED MORTALITY FOR THE... 267 Table 3: Immediae annuiy: Age a issue 6/birh year 195 (annuiy sars in 21) Ne single premium R94 R4 Poisson model, cenral raes Poisson, low-moraliy scenario R94/Poisson Male 18.1488 2.36943 18.5437 18.9897.978 Female 2.49378 22.24 2.9845 21.7616.977 Table 4: Deferred annuiy: Age a issue 6/birh year 198 (annuiy sars in 24) Ne single premium R94 R4 Poisson model, cenral raes Poisson, low-moraliy scenario R94/Poisson Male 19.7256 22.9455 2.46949 21.28829.963 Female 22.81581 24.42694 22.87369 23.58179.997 Noes: R94 DAV 1994 annuiy life able; R4 DAV 24 annuiy life able; Poisson model Slovenian annuiy life able based on he Poisson log-bilinear model. 6 CONCLUSION In his aricle, we have presened an applicaion of he Lee-Carer mehodology o calculae he bes-esimae value of an insurance annuiy in Slovenia. In paricular, we focused on forecasing life epecancies on a ime-series basis. We esed wo differen sochasic mehods for forecasing moraliy (basic Lee Carer and Poisson log-bilinear). On he basis of back esing analysis, we concluded ha he Poisson log-bilinear model provides a beer fi han he original Lee-Carer model for pas observed cenral deah raes for Slovenia. We herefore used a Poisson log-bilinear model o forecas moraliy. Given ha a life annuiy purchaser is, mos likely, a healhy person wih longer-han-average life epecancy, we also incorporaed he selecion effec ino he resuls. Because Slovenian moraliy saisics for annuiy purchasers are no direcly available, we chose selecion saisics from he UK eperience and compared hem wih German saisics. By muliplying he selecion facor, which depends on age, wih cohor populaion moraliy raes, we derived he bes esimae of seleced moraliy raes for an annuiy purchaser. We hen compared hose raes wih he curren minimum sandard in Slovenia: he DAV 1994 R moraliy raes. The ne single premium based on he Poisson model is 2 4% higher han ha calculaed by he curren minimum sandard in Slovenia. Therefore, he DAV 1994 R annuiy ables are inappropriae for he bes-esimae valuaion of annuiy liabiliies in he Solvency II framework. In oher words, echnical provisions for annuiies based on he DAV 1994 R ables are underesimaed by 2 4%, which is no insignifican. Afer 21 December 212, he use of only unise ables will be allowed for premium calculaion. This is also he case for life annuiies. In his respec, furher research is needed o ake ino accoun male and female selecion in he Poisson framework.

268 ECONOMIC AND BUSINESS REVIEW VOL. 13 No. 4 211 REFERENCES Associazione Nazionale fra la Imprese Assicurarici (25). IPS55 Base demografica per le assicurazioni di rendia, Luglio. Brouhns N., Denui M. & Vermun J. K. (22). A Poisson log-bilinear regression approach o he consrucion of projeced lifeables. Insurance: Mahemaics and Economics 31 (22) 373 393. Cairns, A. J. G., Blake, D. & Dowd, K. (26b). A wo-facor model for sochasic moraliy wih parameer uncerainy: heory and calibraion. The Journal of Risk and Insurance, 73 (4), 687 718. Carer, L. R. (1996). Forecasing U.S. moraliy: a comparison of Bo Jenkins ARIMA and srucural ime series models. The Sociological Quarerly, 37 (1), 127 144. Denui, M. & Goderniau, A.-C. (25). Closing and projecing life ables using log-linear models. Bullein of he Swiss Associaion of Acuaries, (1), 29 48. European commission, Inernal marke and services DG (21). QIS5 Technical Specificaions. hps://eiopa. europa.eu/fileadmin/_dam/files/consulaions/qis/qis5/qis5-echnical_specificaions_2176.pdf (accesed May 2, 211). Goodman, L. A. (1979). Simple models for he analysis of associaion in cross-classificaions having ordered caegories. Journal of he American Saisical Associaion, 74, 537 552. Insiue of Acuaries and he Faculy of Acuaries (29). Coninuous Moraliy Invesigaion Repors (29), No. 23. Insurance Supervision Agency. Decision on deailed rules and minimum sandards relaing o calculaion of echnical provisions, Uradni lis RS, š. 3/1, 69/1, 85/5, 66/28. hp://www.a-zn.si/documens/acs/en/ azn22en.pdf (accessed May 2, 211). DAV-Unerarbeisgruppe Rennerserblichkei (25). Herleiung der DAV-Serbeafel 24 R für Renenversicherungen, Bläer der DGVFM, XXVII, 199-313. Lee, R. D. & Carer, L. R. (1992). Modelling and forecasing U.S. moraliy. Journal of he American Saisical Associaion, 87/419, 659-671. Lee, R. (2). The Lee-Carer Mehod for foresing moraliy, wih various eensions and applicaions. Norh American Acuarial Journal, 4, 8-93. Medved, D. (2). Technical provisions and accouning soluions in insurance companies. Revizor, 11 (9), 11-38. Olivieri, A. (21). Uncerainy in moraliy projecions: an acuarial perspecive. Insurance: Mahemaics and Economics, 29, 231 245. Piacco. E. e al. (29). Modelling Longeviy Dynamics for Pension and Annuiy Business. Oford Universiy Press. Sihole, T. Z., Haberman, S. & Verrall, R. J. (2). An invesigaion ino parameric models for moraliy projecions, wih applicaions o immediae annuians and life office pensioners daa. Insurance: Mahemaics and Economics, 27, 285 312. Tuljapurkar, S., Li, N., & Boe, C. (2). A universal paern of moraliy decline in he G7 counries. Naure, 45, 789 792.

Appendi 1: Comparing he resuls of boh mehods for females alphas and beas Figure 7: Bea() as a funcion of age (females): he Poisson vs. he LC model Appendi 1: Comparing he resuls of boh mehods for females alphas and beas D. MEDVED, A. AHČAN, J- SAMBT, E. PITACCO ADOPTION OF PROJECTED MORTALITY FOR THE... 269 Appendi Figure 1: 7: Comparing Bea() he as a resuls funcion of boh female of mehods age (females): for females he alphas Poisson and beas vs. he LC model.3 Figure 7: Bea() as a funcion of age (females): he Poisson vs. he LC model.25.3 female.2.25 bea().15.2 bea().1.15.5.1.5 2 4 6 8 1 12 age 2 4 6 8 1 12 Figure 8: Alpha as a funcion of age (females): he Poisson vs. he LC model Figure 8: Alpha as a funcion of age (females): he Poisson vs. he LC model Figure 8: Alpha as a funcion female of age (females): he Poisson vs. he LC model -1-2 female -3-1 -4-2 alpha() -5-3 -6-4 alpha() -7-5 -8-6 -9-7 -1-8 2 4 6 8 1 12-9 age -1 2 4 6 8 1 12 Figure 9: Kappa as a funcion of year age (females): he Poisson vs. he LC model

Kurosis 2.642194 2.22411 3.41657 2.718712 27 ECONOMIC AND BUSINESS REVIEW VOL. 13 No. 4 211 Figure 9: Kappa as a funcion of year (females): he Poisson vs. he LC model 4 female 3 2 1 kappa() -1-2 -3-4 -5-6 197 1975 198 1985 199 1995 2 25 21 year Appendi 2: Residuals Table 5: Summary saisics for he ARIMA model Poisson model LC model males female males female Mean 2.88E-16-3.24E-16-5.96E-16 -.52288 Median -.37543.199426-3.33E-15.18313 Maimum 6.586123 6.81238 1.12461 7.569289 Minimum -7.55693-6.42445-11.78582-11.8728 Sd. Dev. 3.349367 3.93559 4.88982 4.813961 Skewness -.49737 -.28232 -.49455 -.559959

D. MEDVED, A. AHČAN, J- SAMBT, E. PITACCO ADOPTION OF PROJECTED MORTALITY FOR THE... 271 Appendi 2: Residuals Table 5: Summary saisics for he ARIMA model Poisson model LC model males female males female Mean 2.88E-16-3.24E-16-5.96E-16 -.52288 Median -.375432.199426-3.33E-15.18313 Maimum 6.586123 6.81238 1.12461 7.569289 Minimum -7.55693-6.42445-11.78582-11.8728 Sd. Dev. 3.349367 3.93559 4.88982 4.813961 Skewness -.49737 -.28232 -.49455 -.559959 Kurosis 2.642194 2.22411 3.41657 2.718712 Jarque-Bera.212627.9336 1.36536 2.7 Probabiliy.899143.627192.595551.367878 Sum 1.95E-14-7.99E-15-2.13E-14-1.882381 Sum Sq. Dev. 43.8574 344.5238 86.7658 811.976 No. of observaions 37 37 37 36 Source: SORS and own work Table 6: LC model residuals for kappa males females AC PAC Q-Sa Prob AC PAC Q-Sa Prob 1 -.21 -.21 1.7712.183.15.15.434.51 2 -.89 -.139 2.984.35.63.53.5952.743 3.52.1 2.2112.53 -.331 -.348 5.1487.161 4 -.75 -.8 2.4579.652 -.93 -.27 5.5211.238 5.34.295 6.6169.251.191.295 7.136.211 6 -.122 -.9 7.386.293.98 -.73 7.572.271 7.15.71 7.3194.396.18 -.6 8.1237.322 8 -.213 -.284 9.5842.295 -.359 -.272 14.42.71 9.34 -.25 9.6439.38 -.43.94 14.513.15 1.21.54 11.87.298.18.266 15.128.127 11 -.214 -.94 14.338.215.48.219 24.232.12 12.179.159 16.178.183.55 -.234 24.47.18 13 -.193 -.73 18.426.142 -.38.97 24.493.27 14.1 -.16 18.426.188 -.227.35 27.71.16 15 -.1 -.211 18.433.241 -.134 -.122 28.874.17 16.7.151 18.77.281

272 ECONOMIC AND BUSINESS REVIEW VOL. 13 No. 4 211 Table 7: Brouhns e al. residuals for modelling kappa males females AC PAC Q-Sa Prob AC PAC Q-Sa Prob 1 -.255 -.255 2.6156.16 -.72 -.72.29.648 2 -.118 -.196 3.1852.23 -.161 -.167 1.2718.529 3 -.89 -.195 3.5248.318 -.5 -.79 1.3799.71 4 -.29 -.161 3.5618.469 -.13 -.53 1.387.846 5.261.181 6.6234.25.2 -.8 1.455.924 6 -.225 -.153 8.9865.174 -.94 -.113 1.8171.936 7 -.3 -.6 8.9868.254.13.114 2.634.917 8 -.177 -.257 1.54.229.35.24 2.6912.952 9.16 -.3 11.866.221 -.154 -.127 3.9196.917 1.48 -.76 11.99.286.128.134 4.7928.95 11 -.23 -.238 14.929.186.155.16 6.1273.865 12.31.213 2.473.59.152.214 7.4576.826 13 -.156 -.42 21.929.56 -.251 -.155 11.245.59 14.46 -.84 22.63.77 -.87 -.47 11.717.629 15 -.25 -.28 22.14.15.133.75 12.877.612 16 -.38 -.16 22.21.137 -.64 -.38 13.157.661