An Integrated Notional Defined Contribution (NDC) Pension Scheme with Retirement and Permanent Disability

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1 TP PT An ngrad oional Dfind Conribuion (DC Pnsion Schm wih Rirmn and Prmann Disabiliy Manul Vnura-Marco and Carlos Vidal-Mliá 03/10/2014 This papr dvlops a mulisa ovrlapping gnraions modl (MOLG ha ingras oldag and prmann disabiliy ino a gnric DC framwork. n h modl, h coun balancs of paricipans who do no surviv ar disribud as inhrianc capial o h couns of h (non-disabld iv survivors on a birh cohor basis. Th modl includs ralisic dmography insofar as i aks ino coun an ag schdul of moraliy and h uncrainy concrning h iming of disabiliy, and allows for changs in h conomically iv populaion and for a larg numbr of gnraions of conribuors and pnsionrs o coxis a h momn in im. Th rsuls hivd in h numrical xampl w prsn ndors h f ha h modl rally works and show an opimal ingraion of boh coningncis ino h DC framwork. Th modl can asily b linkd o ral prics in social scuriy policis bcaus, o mnion jus a fw posiiv faurs, i could b implmnd wihou much difficuly, i would hlp o improv uarial fairnss, i would uncovr h ral cos of disabiliy and minimiz h poliical risk of disabiliy insuranc bing usd as a vo-buying mchanism. JEL: G22, H55, J26. KEYWORDS: Pay-as-you-go, Disabiliy nsuranc, Social Scuriy, Swdn Manul Vnura-Marco and Carlos Vidal-Mliá ar graful for h financial assisanc rcivd from h Spanish Minisry of h Economy and Compiivnss (Minisrio d Economía y Compiividad projc ECO Th auhors ar graful o sminar paricipans (Univrsiis of Barclona and Valncia and spcially o Francisco avarro-cabo for his hlp in calculaing h numrical xampl and Pr Hall for his English suppor. Any rrors ar nirly du o h auhors. Dparmn of Financial Economics and Acuarial Scinc, Univrsiy of Valncia, Avnida d los aranjos s.n., Valncia. (Spain. (-mail: manul.vnura@uv.s. Dparmn of Financial Economics and Acuarial Scinc, Univrsiy of Valncia, Avnida d los aranjos s.n., Valncia. (Spain. (-mail: HTUcarlos.vidal@uv.sUTH. (Corrsponding auhor. 1

2 An ngrad oional Dfind Conribuion (DC Pnsion Schm wih Rirmn and Prmann Disabiliy 1.-nroducion A noional dfind conribuion schm is a pay-as-you-go sysm (PAYG ha dlibraly mimics a financial dfind conribuion schm (FDC by paying an incom sram whos prsn valu ovr a prson s xpcd rmaining lifim quals h amoun cumulad a rirmn. hrfor has many of h faurs of an FDC schm, bu no all of hm. This yp of pnsion schm is basd on a noional coun, which is a virual coun rflcing h conribuor s individual conribuions and h ficiious rurns ha hs conribuions gnra ovr h cours of h paricipan s working lif. n principl h conribuion ra is fixd. Th DC sysm has many wll-known posiiv faurs 1, alhough hr w only highligh wo of hm ha hav o do wih h aim of his papr: has srongr immuniy agains poliical risk han mor radiional dfind bnfi (DB PAYG sysms bcaus DC incrass h financial sabiliy of h pnsion sysm by making i vry difficul for poliicians o mak promiss abou fuur rirmn bnfis. ncourags uarial fairnss and simulas conribuors inrs in h pnsion sysm as i brings o ligh any impropr or hiddn rdisribuion of bnfis o privilgd groups and rvals who rally bnfis from h lgislaion. For Holzmann & Palmr (2006 and Chłoń-Domińczak al (2012, compard o an FDC schm, h hr mos imporan diffrncs ar: Th inrnal ra of rurn (G in a gnric DC coun is a funcion of produciviy growh, labour forc growh and fors linkd o conribuion and bnfi paymn srams as opposd o h financial mark ra of rurn. Th only financial saving ha can aris undr h DC schm is in h form of a buffr fund as opposd o h fundd charr of h FDC schm. Th way h pnsion balancs of dcasd prsons ar usd (i.. inhrianc gain, also known as h survivor dividnd. n FDC schms h survivor dividnd is usually inhrid by h la conribuor s survivors. can b usd o nhanc h survivor s rirmn savings or b paid ou as a lump sum or as a phasd wihdrawal survivor bnfi. Howvr, among h counris in which DC sysms ar in pl, only Swdn applis wha is calld inhrianc gains. Ths will b chnically dfind lar. Anohr imporan diffrnc, no ofn mniond in h liraur, is h way disabiliy bnfis ar ingrad ino h schm. n mos Lain Amrican counris wih mandaory priva pnsion sysms basd on individual capializaion couns, disabiliy and survivor bnfis ar linkd o h fundd individual coun. Howvr, Wis (2006, popl who bcom disabld a a young ag migh lk sufficin capial in hir individual couns o financ an adqua disabiliy pnsion. Th sandard soluion o his shorfall problm has bn o allow disabiliy bnfi o rmain a dfind bnfi, and o adop various masurs o sich oghr h dfind bnfi (DB and dfind conribuion (DC componns of h sysm ino a cohrn whol. vrhlss, as dscribd by Rys (2010 and Krizr al (2011, big diffrncs can b found in h way h bnfis ar ingrad ino h schm. 1 S h paprs by Holzmann & Palmr (2006, Barr & Diamond (2009, Aurbh & L (2009, Aurbh & L (2011, Chłoń-Domińczak al (2012 and Holzmann al (2012 o nam jus a fw. 2

3 n Chil, Jams al (2009, disabld workrs who qualify ar guarand a DB for h balanc of hir livs: 70% of hir avrag wag (if oally disabld and 50% (if parially disabld, i.. disabiliy insuranc ops up h capial cumulad in h individual coun if h balanc is lss han h minimum rquird o financ a prmann disabiliy pnsion. Th diffrnc bwn h capial ncssary o pay h pnsion and h balanc availabl a h im of h disabiliy or dah vn is calld h addiional conribuion and is on of h main cos componns of disabiliy and survivor insuranc 2. n DCs, rirmn and prmann disabiliy ar no fully ingrad. n Swdn, for xampl, h currn rgulaions on disabiliy pnsion ar closly linkd o h old-ag pnsion sysm bu no ingrad ino i. According o Palmr (2006 and Chłoń-Domińczak al (2012, h Swdish modl for rirmn pnsion righs for prsons rciving disabiliy bnfis involvs impuing conribuions for insurd priods of disabiliy and paying hm ino h rirmn coningncy. Ths paymns, which ar mad annually from gnral ax rvnus, ar nrd as a cos for h disabiliy sysm in h counry s couns and ar par of h ransfr from sa rvnus o h DC pnsion fund. Prmann disabiliy bnfis ar convrd ino rirmn bnfis a ag n aly, disabiliy pnsions ar basd on h noional capial a h im of disabiliy, and his is ingrad aking ino coun h gap bwn h individual s ag whn h pnsion is grand and h rfrnc ag of 60 yars. Howvr, as Gronchi & isico (2006 hav poind ou, h formula usd o calcula h disabiliy pnsion provids only a wak link bwn bnfis and conribuions. Working-ag disabiliy policy oday is on of h biggs social and labour mark challngs for policy makrs and currnly occupis an imporan pl on h conomic policy agnda in many dvlopd counris. According o h OECD (2010 and Auor & Duggan (2006, disabiliy bnfi in a numbr of counris has bcom h bnfi of las rsor for popl unabl o rmain in, or nr, h labour mark. Encouragd by h conomic crisis, mos of hs counris, Burkhausr al (2013, ar now considring how bs o rform hir disabiliy pnsion schms. Many social scuriy sysms, D Jong al. (2010 and OECD (2010, f vr highr disabiliy coss. Spnding on disabiliy pnsions has bcom a significan problm for public financs in mos OECD counris. Apparn public spnding on disabiliy bnfis oals 2% of GDP on avrag ross h OECD, rising o as much as 4-5% in counris such as orway, h hrlands and Swdn 4. On avrag around 6% of h working-ag populaion rlis on disabiliy bnfis, wih his figur rhing 10-12% in som counris in h norh and as of Europ. Th US Social Scuriy Disabiliy nsuranc (USD programm, BOT (2013, is suffring srious financial problms. Sinc 2009, i has bn paying ou mor in annual bnfis han i rcivs in conribuions and inrs from is rus fund. Basd on currn growh, i is projcd o b insolvn by Burkhausr al (2013 poin ou ha h fors driving unsusainabl USD programm growh ar similar o hos ha ld o unsusainabl growh in four ohr OECD counris (Ausralia, h hrlands, Swdn and h Unid Kingdom. According o Koning & Van Vuurn (2007, mployrs and rad unions hav cooprad in h pas on h us of disabiliy bnfis as a subsiu for unmploymn and arly rirmn programms, noably in h hrlands and Swdn. n orway, Rg al (2009 find ha 2 Chilan law sipulas ha mmbrs who di or bcom disabld bfor lgal rirmn ag gnra a survivor or disabiliy pnsion rspcivly. 3 As Wis (2006 poind ou, a disabld individual s rplmn ra may incras or dcras upon rhing h rirmn ag bcaus disabiliy bnfis ar dfind by a formula, whil old ag bnfis ar a funcion of inrs ras, moraliy ras and h cumulad noional capial. 4 Th word apparn has bn usd bcaus public spnding is gnrally undrsimad du o h pric known as convrsion, Zayaz (2011, or pnsion rclassificaion, Vnura-Marco & Vidal-Mliá (

4 downsizing subsanially incrass h disabiliy nry ra of workrs in h plans affcd. Milligan & Wis (2012 poin ou ha disabiliy insuranc programms sill play a big rol in h dparur of oldr prsons from h labour forc, as many pass hrough disabiliy insuranc on hir pah from mploymn o rirmn. Bníz-Silva al (2010 find inrnaional vidnc ha h businss cycl has much o do wih xplaining boh h sock of disabiliy bnfi claimans and inflows o and ouflows from ha sock. Thy conclud ha h ris in unmploymn du o h currn global conomic crisis is xpcd o incras h numbr of disabiliy insuranc claimans. Laun & Wallnius (2013 find ha gnrous arly rirmn bnfis cra srong incnivs for arly rirmn, in larg par hrough disabiliy insuranc, in Franc, Spain, Swdn and o a lssr xn Grmany. Poliical risk also sms o play an imporan rol in disabiliy insuranc. According o Marin (2006, i nabls shor-rm poliical populariy o b hivd a h cos of long-rm susainabiliy. Easir css o arly rirmn, broadr covrag, mor gnrous rplmn incom and a mor rlaxd scrning of ligibiliy and assssmn of claims buy h immdia saisfion of inrs groups and vors. n h US, yngar & Masrobuoni (2010 provid fairly srong vidnc ha som govrnors ar using h USD as a vo-buying mchanism. Similarly in h cas of Spain, Jiménz-Marin al (2007 hav shown ha hr ar significan rgional diffrncs in h probabiliis of rciving a bnfi wihou dsrving i, which sms o suggs, alhough h auhors do no ually pu i ino words, ha prmann disabiliy bnfis hav bn usd as an lcoral ool, spcially in h lss dvlopd rgions of Spain. To sum up, in mos dvlopd counris and in a similar way o DB PAYG rirmn sysms, disabiliy insuranc (D has many complx problms ha nd o b addrssd, and as Marin (2006 poind ou, disabiliy pnsions sm o hav bcom wha migh b considrd h garbag can of h social scuriy sysm. Th aim of his papr is o dvlop a mulisa ovrlapping gnraions modl (MOLG ha ingras old-ag and prmann disabiliy ino a gnric DC framwork. n h modl, h coun balancs of paricipans who do no surviv ar disribud as inhrianc capial o h couns of surviving conribuors on a birh cohor basis. Th modl includs ralisic dmography insofar as i aks ino coun an ag schdul of moraliy and h uncrainy concrning h iming of disabiliy, and i allows for changs in h conomically iv populaion and for a larg numbr of gnraions of conribuors and pnsionrs o coxis a h momn in im. Th rsuls hivd in h numrical xampl w prsn ndors h f ha h modl rally works and show an opimal ingraion of boh coningncis ino h DC framwork. As far as w know, h modl proposd is an innovaion in his fild and w hav bn unabl o find similar modls in h conomic liraur. Th modl can asily b linkd o ral prics in social scuriy policis bcaus, o mnion jus a fw posiiv faurs, i could b implmnd wihou much difficuly, i would hlp o improv uarial fairnss, i would uncovr h ral cos of disabiliy and minimiz h poliical risk of disabiliy insuranc bing usd as a vo-buying mchanism. Th srucur of h papr is as follows. Afr his inroducion, Scion 2 prsns an uarial OLG modl ha ingras rirmn and prmann disabiliy ino a gnric DC sysm. For h sak of clariy, his scion is sparad ino hr subscions daling wih h drminaion of h yar in which h sysm rhs a maur sa, h dfiniion and drminaion of h survivor dividnd, and h ffc of h survivor dividnd on h sysm's financial quilibrium. Scion 3 shows a numrical illusraion rprsning a gnric DC pnsion sysm wih wo coningncis. This scion is dividd ino wo diffrn pars cording o h assumpions mad abou h growh of h conomically iv populaion. Scion 4 shows our conclusions and 4

5 discusss som issus ha would hav o b akn ino coun whn puing h modl ino pric. Th papr nds wih h bibliographical rfrncs. 2. Th Modl This scion dvlops a mulisa ovrlapping gnraions modl (MOLG ha ingras rirmn and prmann disabiliy ino a gnric DC sysm aking ino coun h survivor dividnd. To a gra xn h modl includs ralisic dmography, Bommir & L (2003, insofar as i aks ino coun an ag schdul of moraliy and h uncrainy concrning h iming of disabiliy and allows for changs in populaion. W build on h modls dvlopd by Boado-Pnas & Vidal-Mliá (2014 and Vnura-Marco & Vidal-Mlia (2014, basd on hos firs pu forward by Srgrn & Mikula (2005, Boado-Pnas al. (2008 and Vidal-Mliá & Boado-Pnas (2013. Boado-Pnas & Vidal-Mliá (2014 dvlop a modl o show whhr i would b jusifid o includ h survivor dividnd whn calculaing affilia pnsion balancs in a gnric DC framwork. Thy conclud ha h survivor dividnd has a srong financial basis which nabls h mro conribuion ra applid o b h sam as h individual ra crdid. Th modl by Vnura-Marco & Vidal-Mlia (2014 prsns a horical bas for applying a Swdish yp uarial balanc sh (ABS o boh rirmn and disabiliy coningncis in a DB PAYG sysm, hrby aking a sp owards filling h larg gap in h liraur in his ara. Thy indica ha hir modl has many ohr prical implicaions which could b of inrs no only o DB sysms bu also o DCs. Ths paprs wr o som xn inspird by h couning framwork for organizing, summarizing and inrpring daa on ransfr sysms and h lif cycl dvlopd in L (1994, Willis (1988 and Arhur & Mcicoll (1978. Th main saring assumpions ar: Th affilias conribu for rirmn and disabiliy coningncis. Thr is a dfind conribuion ra (fixd ovr im o cop wih boh coningncis. Th iniial disabiliy and rirmn pnsions dpnd on h valu of h cumulad noional coun, h xpcd moraliy of h cohor in h yar h conribuor bcoms disabld and a noional impud fuur indxaion ra λ, i.. pnsions in paymn incras or dcras a an annual ra of λ. Th capial cumulad in h noional coun rflcs h paricipan's individual conribuions and h ficiious rurns hs conribuions gnra ovr h cours of h paricipan s working lif, plus h inhrianc capial. Th coun balancs of paricipans who do no surviv o rirmn ar disribud as inhrianc capial o h couns of h (non-disabld iv survivors on a birh cohor basis. Th cumulad noional capial is no spli ino coningncis bcaus prmann disabiliy is considrd a yp of compulsory arly rirmn for halh rasons. Th sysm dos no provid a minimum pnsion. is assumd ha conribuions and bnfis ar payabl yarly in advanc. Paricipans livs las (w-1-x priods, whr (w-1 is h highs ag o which i is possibl o surviv and x is h arlis ag of nry ino h sysm. 5

6 Th ag giving nilmn o rirmn pnsion, x +A, is fixd. This assumpion dos no imply loss of gnraliy bcaus, as w will s lar for h disabiliy coningncy, h ags ha giv nilmn could b dfind as an inrval. As rgards disabiliy pnsion, i is supposd ha iniially h ags ha giv nilmn ar o b found in ag inrval [x +1, x +A] 5. Th ag inrval is lar widnd o [x +A+1, w- 1]. Th only rason for a disabld workr s bnfi o rmina is hrough h dah of h pnsionr. W do no ak convrsions or rcovris ino coun, i.. convrsion and rcovry ras ar null in our modl. Th conribuion bas (coinciding wih arnings grows a an annual ra of g. Th conomically iv populaion incrass or dcrass ovr im a an annual cumulaiv ra of γ, affcing all groups of conribuors qually. Th sysm's incom from conribuions (wag bill growh also grows (dcrass a ra G = (1+g(1+γ-1. Whn h sysm rhs h maur sa = w-x -1 yars from incpion, A gnraions of conribuors and (w-(x +1 gnraions of pnsionrs coxis a h momn in im. Onc h main assumpions hav bn daild, for h sak of clariy his scion will b dividd ino hr subscions daling wih h drminaion of h yar in which h sysm rhs a maur sa, h dfiniion and drminaion of h survivor dividnd, and h ffc of h survivor dividnd on h sysm's financial quilibrium Dscripion of h sysm and drminaion of h yar in which i rhs a maur sa. Diagram 1 shows h rlaionships (ransiions bwn h various collcivs (sas ha will b sparad in h modl. Th diffrnc bwn his modl and h on found in Vnura-Marco & Vidal-Mlia (2014 is ha h pnsion sysm is DC insad of DB PAYG and h survivor dividnd is xplicily akn ino coun. Wih rgard o h modl dvlopd by Vidal-Mliá al (2013, a nw sa - disabiliy - is inroducd, along wih h nw rlaionships shown by brokn lins in h diagram. W work wih a simplifid yp of mulipl sa ransiion modl, Habrman & Pico (1999, which is a probabiliy modl ha dscribs a subjc's movmns among various sas: iv (a, disabld (i, rird (r and dad (d 5 ndd a prson of x yars may bcom disabld afr having paid conribuions and hrfor sars o rciv disabiliy pnsion a ag x +1 yars. Similarly, a prson of x +A-1 yars may bcom disabld a ha ag afr conribuing and will hrfor rciv bnfi for bing disabld a ag x +A yars. 6

7 Rcovris ar no possibl Diagram 1 DC schm wih prmann disabiliy Conribuor = Aciv (a x + k x + A-1 x + A-1 Disabld (i Rird (r x + k x + A-1 Convrsions ar no allowd x + k x + 1 x + k x + A Dad (d 1.-Transiion probabiliis: Th discr modl could b xprssd as a four-sa non-homognous Markov chain wih h following ransiion probabiliis, in which no mor han on ransiion wihin a yar is assumd: aa p x k, h probabiliy ha an iv prson agd x +k will rh ag x +k+1 bing iv, ai p x k, h probabiliy ha an iv prson agd x +k will bcom disabld during h yar, ar p x k, h probabiliy ha an iv prson agd x +k will b rird on yar lar, ad p x k, h probabiliy ha an iv prson agd x +k will di during h yar, i i px k, h probabiliy ha a disabld prson agd x +k will rh ag x +k+1 in h sam sa, i r p x k, h probabiliy ha a disabld prson agd x +k will b rird on yar lar, id p x k, h probabiliy ha a disabld prson agd x +k will di during h yar, rr p x k, h probabiliy ha a rird prson agd x +k will rh ag x +k+1 in h sam sa, rd p x k, h probabiliy ha a rird prson agd x +k will di during h yar. 7

8 2.-Ag: Conribuors' ags x, x 1, x 2,..., x A 1, x A, x A 1,... w-1 Pnsionrs' ags (rirmn x 1, x 2,..., x A 1,x A, x A 1,... w-1 Pnsionrs' ags (disabiliy W assum ha h affilia canno conribu and rciv pnsion in h sam yar. f an individual bcoms disabld a conribuion ag x +k [x, x +A-1], h corrsponding disabiliy pnsion payabl will b a ag x +k+1 [x +1, x +A]. 3.-umbr of conribuors by ag a im : (x, 0 (x,, ( 1γ, (x 1, 0 ( 1γ,..., (x A1, 0 (x 1,,..., (x A1, ( 1γ aa aa whr (x k, (x, kpx and k p x is h sabl-in-im raio bwn h numbrs of individuals agd x and x +k yars. Sabl raios or probabiliis includ h dcrmns du o dah and disabiliy associad wih h ag, wih h possibiliy of a rurn o iv lif no bing considrd. is a diffrn mar whn i coms o considring dcrmns or nw nris du o migraory movmns; hs ar includd in paramr γ Avrag wag (avrag conribuion bas by ag a im : y (x, 0 y (x,, y (x 1,, , ( 1 g, y(x 1, 0 ( 1 g, , y(x A1, 0 y (x A1, ( 1 g 3. Th dmographic framwork abov implis ha h ag-wag srucur only undrgos proporional changs. Th slop of h ag-wag srucur is consan. 5.-umbr of disabld popl: n ag inrval [x +1, x +A] a = 1 whr: (x ai aa ai p (x, k- px p k, 1 (x k-1, 0 xk xk 1 4. ai p x k- 1 is h probabiliy ha an iv prson agd x +k-1 will bcom disabld during h yar. aa k- px 1 is h probabiliy ha an iv prson agd x will rh ag x +k-1 bing iv. (x k, 1 is h numbr of popl who bcom disabld in yar of ag x +k, bcoming disabld as far as h sysm is concrnd bcaus hir disabiliy rally bgan in h prvious priod [0, 1. For 2 and ag inrval [x +1, x +A] w nd o considr wo yps of disabld popl: hos agd x +k yars who bcam disabld in h currn yar, k,, and hos whos disabiliy bgan x 8

9 arlir or survivors agd x +k yars who coninu from prvious yars, x k,. Th srucur for h numbr of popl who bcam disabld during h yar is always givn by: 1 ai -1 ai x k, x k-1, -1 p x k 1 x k-1, 0 ( 1γ p x k 5. Afr ag x +A+1 yars, all h disabld in h sysm ar by dfiniion considrd survivor disabld bcaus, onc h sa of iviy disappars, nobody can bcom disabld for h purposs of h sysm. Thrfor, and always for 2, as far as h coninuing disabld ar concrnd a disincion has o b mad bwn wo ag inrvals, [x +2, x +A] 6 and from ag x +A+1 yars onwards. Th srucur of h survivor disabld in [x +2, x +A], whos voluion will dpnd on survival ii probabiliis px k-1, which ar diffrn from hos for h iv populaion 7 aa, px k-1, incorporas all hos who bcam disabld in succssiv arlir priods and hav survivd. n gnral, whn all h disabld popl who bgan in = 1 hav did, his mans ha w-x, and hrfor from hr on for all his disabiliy band w g k <, S k- ii 1 k-spx s 6. s1 x k, x s, -ks s whr sa. ii k-s x s p is h probabiliy ha a disabld prson agd x +s will rh ag x +k in h sam Th oal numbr of disabld for h ag in can b calculad by: S k -1-k s ii x k, x k, x k, x s, 1 1 γ k-s px s x s, -k s k-s px s 7. smáx1,k- 1 s1 From ag x +A+1 yars onwards, no mor nw disabld popl ar akn ino coun, and so for ag inrval [x +A+1, w-1], i.. k {1, w-1- (x +A}, w g: A S ii x A, -k x A, -k k px A x s, -ks-a (x Ak, A-s ii x s k ii x A s1 According o h saring assumpions, h amoun of iniial pnsion for disabiliy paid a ag x +k wih k {1,, A} and c {1,, k} is: K 1 (x k, c, P Kx k1,c1,1 θa yx k1,1 1 G λ a whr x k1,c1, : Accumulad noional capial a im -1 for on individual agd x +k-1 who has bn conribuing for h las c -1 yars, whr for k = 1 x k k p p ii n k = 1 h disabld ar always nwly disabld as hy com from ag x in -1, and hrfor (x +1, = (x +1,. 7 According o Pio (2012, h moraliy of disabld popl conains an xra-moraliy rm and can b rprsnd ihr as a spcific moraliy (via h appropria numrical abls or paramric moraliy laws or via adjusmns o h sandard ag parn of moraliy. Plamondon al (2002 show ha considring spcific moraliy for prmannly disabld popl is sandard pric in social scuriy. 9

10 and for k = A (x P A, c, Whras for h rirmn coningncy: (x 1, 1, -k1 P θa y x, k λ a x 1 K x A1, c 1, 1 θ a yx A1, 1 λ a ( 1G 1 G x A R (x A, c, P K x R a A,c, λ x A p Kx A,c, θa y x A,c, G p R λ a x A ad x A 1 aa x A1 12. λ a R λ a wih x k and x A rspcivly bing annuiy fors (Af D, Af R, i.. h prsn valu of a lifim annuiy for h disabld or rird of 1 monary uni pr yar payabl in advanc and growing a ral ra, valud a ag x +k yars and ag x +A yars, wih a chnical inrs ra qual o d=g. n formula [12] h rm in dail in Scion 2.2. p p ad x A 1 aa x A1 1 is h annual inhrianc gain for ha will b dvlopd For c=k, hs pnsions would b for popl wih a full conribuion hisory, i.. hos who nr h labour mark a h arlis ag x and xi agd x + k-1 and x + A yars for disabiliy and rirmn rspcivly. (x k, R (x A, P is h avrag pnsion for disabld individuals a ag x +k, wih k {1, 2,, A}, whil P is h avrag pnsion for individuals who rir a h ordinary rirmn ag. Th formr is a wighd pnsion of h k diffrn disabiliy pnsions onc sld, whil h lar is a wighd pnsion of h k diffrn rirmn pnsions onc sld. Thrfor, for k {1,, A} and c {1,, k} and (x k, P k c (x k, c, P (x k, (x k, c, (x k 1, K θ a y a (x k 1, 1 λ x k ( 1 G R (x A, P A c R (x A, c, P (x A, c, K (x A, (x A, R λ a x A Th oal cumulad noional capial in yar for h gnraion agd x +k, K T (x k, includs conribuions mad by all conribuors in 1 plus h crdid coun balancs of conribuors in yar 1 corrsponding o hos dying in h priod [ 1, and iv conribuors in yar, boh capializd for a priod. Howvr, w hav o rmov h crdid coun balancs of iv paricipans in yar -1 who bcom disabld during h yar bcaus hy rciv h disabiliy pnsion in yar, i.. conribuions allocad o h disabld hav o b dducd from oal conribuions:

11 k-1 k 1 T aa ad K (x k, θ a yx s, x s, pxh pxh 15. s0 hs f w includ conribuions mad in yar for h gnraion agd x +k yars and ak ino coun T [ ] T formula [15], w g K (x k, K (x k, θa y(x k, (x k,. Hnc, wih h oal cumulad noional capial in yar for h gnraion agd x +k, wih conribuions for im bing includd a ag x +k {x +1,, x +A-1} for all conribuors who rh ha ag, h spnding on disabiliy pnsions in yar w x 1 for bnficiaris agd x +k yars is: T [ ] K (x k 1, 1 ( 1 G (x k 1,1 [ ] K (x k1, 1 ( 1G P(x k, 1 Af (x k, (x k, 1 ( 1 γ 1 P (x k, 1 ( 1 g Th spnding on nw rirmn pnsions awardd in yar and h amoun of h annual avrag pnsion paid a rirmn ag ar: K T (x A, 1 K (x A, R (x A, R (x A, P(x A, (x A, (x A, Af Af 17. whr K T (x A, is h oal cumulad noional capial a ag x +A of all h conribuors who rh ha ag: K T (x A, T [ ] K(x A1, 1 K θa T ai (x A1, 1 θ y 1 G1 p A1 s0 y a x A1, 1 x A1, 1 A1 aa ad x s, x s, pxh px h hs x A1 n h financially susainabl DC framwork, h spnding on pnsions has o b qual o h aggrga incom from conribuions cording o balancd ra θ, and hrfor: 18. θ 1 A1 k0 A y k1 s1 (x +k, (x s, P (x +k, (x s, F A k1 s1 k Ak-s Aks (x s, P p ii xs (x s, P F R (x A, k-s ks -1 k0 p ii xs (x Ak, F k whr F 1 G paymn and G. This lads o: is an indxaion for which dpnds on (indxaion of pnsions in A1 A K (x θ y G k 1, 1 a (x k, (x k, K 1 1 T θ y 1 (x A, 20. k 0 (x +k, (x +k, k 1 11

12 can b said ha h sysm's incom from conribuions is quivaln o h prsn uarial valu of h pnsions awardd in ha yar (commimns h sysm aks on wih pnsionrs who hav jus rird or bcom disabld, i.. h cumulad noional capial blonging o nw bnficiaris in yar. From yar w-x -1= couning from h sysm s incpion, h raio bwn h numbr of pnsionrs ( D, R and h numbr of conribuors (C ( dr sabilizs and h avrag pnsionavrag conribuion bas quoin (, ( fr R P P is alrady consan du o h f ha h W W numraor and dnominaor volv qually (a h ra of variaion in wags. Thrfor h conribuion ra, also calld h mro conribuion ra, is h produc of h dmographic dpndncy raio and h financial raio (h sysm s avrag rplmn ra: θ S fr dr R R R dr R P D P R P D P R fr, fr θ θ 21. R dr W C W C W C 2.2.-Dfiniion and drminaion of h survivor dividnd. Lik wih h Swdish DC modl, w follow h principl ha h monary uni conribud is paid ou in h form of rirmn bnfi bu no ncssarily o h individual who mad h conribuions. Th main diffrnc bwn h Swdish DC modl and ours is ha w considr wo ingrad coningncis. Thrfor, for h individual who bcoms disabld or rhs rirmn ag, hr is an cumulad survivor dividnd. Th coun balancs of paricipans who do no surviv o rirmn ar disribud as inhrianc capial on a birh cohor basis o h couns of surviving conribuors. For populaion growh, γ 0, for ag x +k hr ar k diffrn conribuion rajcoris as conribuors migh b working for 1 yar, 2 yars, k yars. Th only xis considrd ar dah and disabiliy. Thrfor: k (x k, (x k, c, c For a givn crdid conribuion ra, θ a, h cumulad survivor dividnd (or cumulad inhrianc gain a ag x + k in for on conribuor who blongs o h iniial group and has conribud sinc nring h sysm, D(x k, k,, is h diffrnc bwn h cumulad noional capial, K(x k, k,, including conribuions and indxaion on conribuions from mmbrs of h sam cohor who did whil iv (no disabld, and h individual cumulad noional capial, K(x i k, k,. Th cumulad survivor dividnd, a a spcific ag, is h porion of h crdid coun balancs of paricipans rsuling from h disribuion, on a birh cohor basis, of h coun balancs of paricipans who do no surviv o rirmn whil iv. n his cas for k {1,, A-1}: D k s1 (x k, k, D K (x s, k s (x k, k, ( 1 G k 1 k s θ a y(x ( 1 G s, k s s0 k s θ a k 1 s0 i K(x k, k, y (x s, k s ( 1 G k s f (x k, k,

13 Bcaus h cumulad noional capial, K (x k, k,, can b xprssd as: K (x k, k, f (x k, k, k1 k1 ad ks p xh θ a y(x s,ks ( 1 G 1 aa s 0 hs px h 24. k whr 1 ad p 1 x h f (x k, k, is h cumulaiv inhrianc gain for, and h rsul is a aa hs p xh formula ha is vry similar in srucur o h formula usd by h Swdish auhoriis for h DC sysm, which only includs h rirmn coningncy 8. Similarly for k = A: D (x A, A, K θ K (x A, A, A1 a s0 y A-1 θ a y(x ( 1 G s, A s s 0 (x s, As (x A, A, K K i (x A, A, ( 1 G A1 As (x A-1, A-1, -1 f θ As (x k, A, θ a y(x s, A s ( G 1 s0 a y A s1 1 (x A-1, -1 As D (x s, As ( 1 G D f (x k, A, ( 1 G (x A, and bcaus a ag x +A yars hr ar no mor conribuions, h cumulad noional capial in yar is drmind from h cumulad noional capial in yar -1, capializd for on priod using noional ra G, plus h inhrianc gains gnrad ovr h priod [-1, Th survivor dividnd whn h conomically iv populaion incrass. Assuming ha h populaion changs a ra γ 0, i.. conribuors ar joining h sysm a all ags, if on conribuor nrs h sysm a ag x +s, hy will conribu for (k-s yars, k{s+1,, A-1}, unil h ag hy bcom disabld whn hir noional capial will b: K k 1 kh (x k, ks, θa y(x s,k s ( 1 G f (x k, k-s, hs k whr 1 ad p 1 x r f (x k, k-s, is h cumulaiv inhrianc gain for, or unil rirmn aa r h p x r ag, A, whn hir noional capial will b: As S Pnsionsmyndighn (2013, Appndix A. nhrianc gain fors for h nkomspnsion. 13

14 K(x A-, A-s-, 1-1 θ y ( G D 1 K (x A, A-s, a (x A-1, -1 (x A, Finally, for k {1,, A}, h avrag cumulad dividnd can b xprssd cording o h inhrianc gain for: givn ha and k 1 ks y(x s, ks ( 1G i (x k, K (x k, (x k, θa f K ks (x k, k, s0 ( 1 γ D T k 1 k s k 1 K (x k, y(x s,k s ( 1 G ai K (x k, θ a 1 p aa k s xh 30. (x k, s0 k s p x s 1 γ hs Ti k 1 ks K (x k, y(x s,ks ( 1 G i K θ 31. (x k, a ks (x k, s 0 1γ 2.3.-Th ffc of h survivor dividnd on h sysm's financial quilibrium. Th rlaionship bwn h crdid conribuion ra and h balancd ra cording o quaion [18.] is: A1 A G K T [ ] ai T θ y K p (x A, k0 (x +k, (x +k, k1 (x k1, -1 xk Equaion [32.] xprsss ha in h maur sa rhd, h sysm's incom from conribuions (rirmn and disabiliy a ar quivaln o h prsn uarial valu of h pnsions awardd in ha yar (commimns ha h sysm aks on wih pnsionrs who hav jus rird and wih hos who bcom disabld, i.. h cumulad noional capial blonging o h nw bnficiaris in yar for boh coningncis. This mans ha liabiliis o pnsionrs and conribuors rmain consan ovr im. can b dmonsrad ha h scond mmbr of quaion [32.], h amoun of crud noional capial blonging o nw bnficiaris in yar, machs h firs mmbr, h sysm's incom from conribuions. Consqunly, θ A1 k 0 y (x +k, (x +k, θ A T ai K px k G k, - 1 k (x A1 θ θ a a s0 y T K (x x s, x s, So wha dos his rsul imply for h sysm's financial quilibrium? f h amoun of h pnsion is drmind using h individual noional capial wihou considring h survivor dividnd, hn h balancd conribuion ra and crdid ra ar diffrn sinc h bnfis ar sricly lowr han hy could b (as h survivor dividnd is no disribud among h survivors. Th amoun of h pnsion ignoring h survivor dividnd is calculad for disabiliy and rirmn rspcivly as follows: A,

15 and P i_ (x k, P i_r (x A, k c1 P A c i_ (x k, c, (x k, (x A, (x k, c, Thrfor h spnding on pnsions is: K i (x k 1, 1 i (x A, R λ a x A θ a y a (x k 1, 1 λ x k ( 1 G (x A1, 1 λ x A 34. i_r P(x i A, c, (x A, c, K K(x A, θa y ( 1G R a A P k1 c1 a P A i K (x k, θa y(x k, G (x k, K k1 k i_ (x +k, c, (x +k, c, λ x k A c1 i_r (x +A, c, R (x +A, c, a Ti (x A, and his has o b qual o h aggrga incom from conribuions cording o h nw balancd ra θ : * θ * A1 k0 y (x +k, (x +k, Disabiliy λ x k 36. A Rirmn Ti ai Ti K p 1 G K 37. k1 (x k 1, -1 xk1 Th prvious xprssion, afr som algbra, can b rwrin as: Dividnd for (x A, A T ai T D(x k 1, -1 pxk 1 1 G D (x A, * k θ 1 1 A 38. Ti ai Ti K (x px k G K k, (x A, k 1 Dividnd ffc D Consqunly Df 1 rsuls from no including h survivor dividnd in h calculaion of h * conribuion ra. Thrfor a du o h sysm s savings afr h non-inclusion of h θ θ survivor dividnd. D is a wighd avrag of h wo dividnd ffcs for boh coningncis, h wighing bing h balancd conribuion ras by coningncy as par of h oal balancd conribuion ra for h sysm, θ *. f θ θa wr conribud insad of θ *, h sysm would coninuously cumula financial rsrvs bcaus ignoring h survivor dividnd producs savings. n pric hs rsrvs could financ h incras in spnding on pnsions rsuling from incrass in longviy. Thy could vn b usd as a sourc of financ for ohr social scuriy commimns wih no spcifid sourc,.g. lgy coss from old pnsion sysms. Df 15

16 3.-umrical illusraion. This scion shows h rsuls obaind for a numrical xampl rprsnaiv of h modl dvlopd in h prvious scion. Mor spcifically, for h hr gnric DC schms analysd, w prsn h main valus ha mak up h sysm's quilibrium including h conribuion ras assignd o h coningncy, h dpndncy raio, h financial raio and h dividnd ffc. W pay spcial anion o h assumpions mad abou h moraliy ra for disabld popl and h disabiliy incidnc ra, which largly drmin h conribuion ra assignd o disabiliy. Th ffcs of populaion changs on h dividnd ffc by cohor is also analysd in dail. This scion is dividd ino wo diffrn pars cording o populaion growh. Par a assums ha h iv populaion will rmain consan. This is in lin wih h assumpion mad by h Swdish auhoriis whn valuing h sysm s asss and liabiliis, Pnsionsmyndighn (2013. Manwhil Par b incorporas populaion changs ovr im. a Baslin cas: h iv populaion will rmain consan Our saring poin is h numrical xampl dvlopd by Boado-Pnas & Vidal-Mliá (2014 for an DCr (rirmn schm afr h inclusion of h survivor dividnd. is assumd ha individuals can join h labour mark from ag 16 upwards, ha h crdid conribuion ra is consan and qual o 16% and ha h fixd rirmn ag for all individuals is 65, i.. h highs ag ha individuals can join h labour mark is 64. This iniial sysm, DCr, is xndd from h sar by adding a disabiliy coningncy, so h rsuling schm is now calld DCdr. Howvr, wih h aim of mphasizing h imporan rol of h assumpions mad abou h moraliy ra for disabld popl and h disabiliy incidnc ra, in h numrical xampl w work wih wo ingrad schms: DCdr1 and DCdr2. Wih rgard o h disabiliy coningncy, a conribuor who bcoms disabld in yar -1 rcivs an iniial disabiliy pnsion basd on formulas [10.] and [11.], i.. h cumulad noional capial a im -1 dividd by h disabiliy annuiy for corrsponding o h ag of h disabld prson. is imporan o rmmbr ha a conribuor who bcoms disabld a ag 64, h las ag a which i is possibl o conribu, dspi having mad xly h sam conribuions, would rciv an iniial pnsion ha was diffrn from (highr han h iniial rirmn pnsion bcaus h annuiy divisor is no h sam for boh coningncis. n h cas of zro populaion growh, i is supposd ha all individuals nr h labour mark a ag 16 and work coninuously unil riring a ag 65 unlss hy di or bcom disabld. Th moraliy abl 9 usd for h iv populaion (conribuors and rirmn bnficiaris is h sam in all hr schms (DCr, DCdr1 and DCdr2. Figur 1 shows h moraliy ras (in blk, firs vrical axis for iv conribuors and rirmn pnsionrs by ag. 9 Obsrvd moraliy ras for Poland in 2009, obaind from h Human Moraliy Daabas (hp:// 16

17 Moraliy and disabiliy ras by ag Moraliy ra Dr basd on EVK. Tabls Dr basd on Spanish Social Scuriy daa Th diffrnc in longviy nds o dcras noably wih h incras in h ag of individuals Disabiliy incidnc ra (Dr Ag Mr ivs Mr Ds1 Mr Ds2 Dr1 Dr2 Figur 1: Moraliy and disabiliy incidnc ras by groupd ag srucur. DCdr1 (blu in Figur 1 and DCdr2 (rd in Figur 1 show h diffrncs in moraliy ras for disabld popl (Ds (MR Ds1 and MR Ds2 in Figur 1, firs vrical axis and disabiliy incidnc ras (Dr by ag (Dr1 and Dr2 in Figur 1, scond vrical axis. Th disabiliy incidnc ra can b dfind as h raio bwn h nw bnficiaris awardd bnfis h yar and h disabiliy-xposd populaion 10. Th disabiliy incidnc ras ar basd on Spanish Social Scuriy xprinc (Dr1 and EVK abls (Dr2 which rly on h Swiss fdral govrnmn plan (no longr in xisnc. As mniond in Scion 2 and shown in Figur 1, disabld popl hav a lowr lif xpcancy han iv popl, bu h diffrnc in longviy nds o dcras noably wih h incras in h ag of h individuals 11. Th moraliy ra for disabld popl has bn drivd from ha for h iv populaion by adding an xra-moraliy ra which dcrass wih h ag of h individuals 12. Howvr, as Pico (2012 poins ou, h picur is much mor complx givn ha h moraliy of disabld popl basically dpnds on h caus and svriy of hir disabiliy. Th voluion of h pnsionr and conribuor collcivs is shown in Figur 2 as a prcnag of h iniial group (conribuors agd Th disabiliy incidnc ra should no b confusd wih h disabiliy prvalnc ra, h lar bing h raio bwn h numbr of disabld pnsionrs in currn-paymn saus h yar and h insurd-workr populaion (conribuors. 11 Th RP-2000 Moraliy Tabls graduad by h US Sociy of Acuaris, SOA (2000, show h sam moraliy ra for halhir annuians and disabld mal pnsionrs from ag 90 onwards. 12 According o OSF (2011, h moraliy ras for mal and fmal Canadian disabiliy bnficiaris agd 55 o 59 ar on avrag fiv o six ims highr han h moraliy ras for h gnral populaion for ha ag group and for h sx. Similar rlaionships ar obsrvd for ohr ag groups. 17

18 Consan populaion, conribuors and pnsionrs by ag 100% As a % of h iniial group 90% 80% 70% 60% 50% 40% 30% 20% Pak of disabld pnsionrs, group agd 65 Diffrncs in individuals 10% 0% Ag Cr DCr Cr DCrd1 Pr DCr Prd DCdr1 T DCdr1 Pr DCdr1 Cr DCrd2 Prd DCdr2 Pr DCrd2 T DCdr2 Figur 2: Evoluion of h collcivs by groupd ag srucur. Figur 2 shows h voluion of conribuors and pnsionrs for h hr schms: DCr (conribuors (Cr DCr and rirmn pnsionrs (Pr DCr in blk; DCdr1 (conribuors (Cr DCrd1, disabiliy pnsionrs (Prd DCrd1, rirmn pnsionrs (Pr DCrd1 and oal (T DCrd1 in blu; and, DCdr2 (conribuors (Cr DCrd2, disabiliy pnsionrs (Prd DCrd2, rirmn pnsionrs (Pr DCrd2 and oal (T DCrd2 in rd. can b sn ha in h nw modl (DCrd1 and DCrd2 hr ar wo yps of bnficiary, disabiliy pnsionrs and rirmn pnsionrs, and ha h collcivs as a whol ar smallr han h bas sysm bcaus h disabld hav a lowr lif xpcancy. Diffrncs by ag ar shown in h graph by llipss and rh hir maximum a ag 65, afr which hy ar dcrasing. Th hr collcivs would only coincid undr h addiional assumpion of qual longviy for boh disabld and non-disabld (iv or rird. f populaion growh had a posiiv valu, hn givn h way in which disabiliy is drmind, h growh ra for h disabld would b lowr han ha for h conribuing populaion. Th diffrncs bwn h DCrd1 and DCrd2 collcivs basically aris du o h diffrncs in h moraliy ras for disabld popl, as shown in Figur 1. Figur 3 shows h voluion of avrag pnsions and iniial pnsions by ag and avrag pnsions by coningncy for h hr schms. Th avrag disabiliy pnsion (APd1 and APd2 by ag is growing givn ha a highr pnsion is awardd whn mor conribuions hav bn mad. Th maximum valu is rhd a ag 64, from which im no mor disabiliy pnsions can b awardd, and hrfor for h rirmn ags h amoun is dcrasing bcaus onc h pnsion is awardd i rmains consan in ral rms. Th iniial disabiliy pnsion (Pd1 and Pd2 by ag is also growing, and h diffrncs bwn boh can b xplaind by h annuiy divisors usd o calcula hm ha ak ino coun diffrn longviy for h disabld. So wha abou rirmn pnsions? As can b sn in Figur 3, h avrag rirmn pnsion (Pr rprsns his valu for h hr schms bcaus, dspi h vry diffrn assumpions for h disabiliy coningncy and alhough DCr dos no covr disabiliy, h avrag rirmn pnsion rmains virually h sam in all hr schms. Th avrag oal pnsion by rirmn (APTr also rmains virually h sam for all hr schms. This can b considrd a sound rsul for our modl and indicas an opimal ingraion of boh coningncis ino h DC framwork. 18

19 As a % of h sysm`s avrag wag 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Pnsions by ag Diffrncs in h iniial disabiliy pnsions ar du o h annuiy divisors usd in h schm Th modl provids virually h sam avrag rirmn pnsion in h hr schms Ag APd1 APd2 Pd1 Pd2 APTd1 APTd2 Pr APTr Figur 3: Avrag and iniial pnsions by groupd ag srucur. Th main valus making up h sysm's quilibrium undr h hr gnric DC schms (DCr, DCdr1 and DCdr2 ar shown in Tabl 1. W adop h assumpion ha h conribuion ra is h sam for all schms, bu whn disabiliy is ingrad ino hm (DCdr1 and DCdr2, h conribuion ra is assignd o h coningncy as a proporion of h spnding on pnsions pr coningncy as par of oal spnding. Th conribuion ra assignd o disabiliy (3.97% in DCdr1 vrsus 5.08% in DCdr2 largly dpnds on h disabiliy incidnc ras and moraliy ras of disabld popl by ag, which drmin h sabl prvalnc disabiliy ra of h schm (7.88% in DCdr1 vrsus 10.50% in DCdr2, hough h avrag disabiliy pnsion also mars. Tabl 1: DCdr sysm wih wo coningncis: som slcd valus. Comparison wih DCr ms DCr DCdr1 DCdr2 Disabiliy Rirmn Sysm Disabiliy Rirmn Sysm θ a θ dr fr D θ * TD ( yars A r (yars A c ( yars x ( yars p ( yars c p ( yars r Bas scnario wih G=(1.016(1.00-1=0.016 Th hr schms ar in financial quilibrium bcaus h conribuion ra (s formula [23.] is h produc of h financial raio ( fr and h dpndncy raio ( dr, and hs raios prsn sligh 19

20 variaions ross h schms. n h ingrad schms h coningncy akn individually is also in financial quilibrium. As shown by Boado-Pnas & Vidal-Mlia (2014, h ffc of including h survivor dividnd ( D on h iniial pnsion is by no mans irrlvan, and h pnsion riss by 18.32% in h DCr schm. Th ingraion of disabiliy ino h DC framwork (DC kps h dividnd ffc high. Howvr, as h nw coningncy dcrass h wighd avrag ag a which conribuions o h sysm cas yars in h DCr plan agains yars in h DCdr1 schm - h dividnd ffc is smallr. Th dividnd ffc for h ingrad sysm, DCdr, is a wighd avrag of h dividnd ffcs for boh coningncis, h wighing bing h spnding on pnsions by coningncy (wihou including h survivor dividnd as par of oal spnding. As shown in h prvious scion, h dividnd ffc can b calculad as ha par of h oal cumulad noional capial originaing from conribuions mad by dcasd conribuors which blongs o nw bnficiaris in h sam yar, dividd by h yarly oal spnding on pnsions (wihou including h survivor dividnd. Alhough w do includ h survivor dividnd whn calculaing h amoun of h rirmn and disabiliy pnsions, if i wr no includd, hn a discrpancy would aris bwn h crdid conribuion ra qual o 16% and h ra ncssary o financ h bnfis, θ *, 13.78% and 13.75% for h DCdr1 and DCdr2 schms rspcivly. Thrfor, as alrady shown in h prvious scion, his numrical xampl illusras h quivalnc bwn h mro balancd conribuion ra and h crdid individual conribuion ra in h nw DCdr framwork inroducd in his papr, and h fundamnal rol playd by h survivor dividnd in hiving h sysm's financial quilibrium. Tabl 1 also shows h valus for h urnovr duraion (TD, a wll-known concp usd for compiling h ABS of DC sysms 14. Vnura-Marco & Vidal-Mliá (2014 dvlopd h sysm s xpcd avrag TD for a DB PAYG schm wih rirmn and disabiliy bnfis. s applicaion o DCdr schms is almos immdia. Th sysm s TD is inrprd as h numbr of yars xpcd o laps bfor h commid liabiliis wih conribuors and pnsionrs for rirmn and disabiliy ar complly rnwd a h currn conribuion lvl. Eh monary uni nrs h sysm as if i wr paid by a conribuor of A c yars and rmains wihin h conribuion liabiliy unil rirmn ag is rhd (pay-in. is hn rcivd by h pnsionr of A r yars afr rmaining wihin h liabiliy o pnsionrs during h pay-ou. A sysm's TD can b calculad ihr as a wighd avrag of h TDs for boh coningncis, h wighing bing h spnding on pnsions by coningncy as par of oal spnding, or as h D D diffrnc bwn h wighd avrag of h avrag ags of disabiliy (Ar Ac and R R rirmn (A r A c, h wighings hr bing spnding on pnsions pr coningncy as a par of oal spnding and h avrag ag of h conribuors. 13 For dails on how o calcula h wighd avrag ag a which conribuions cas, x, inrsd radrs can consul Vnura-Marco & Vidal-Mliá (2014, formula [50.]. 14 Th lgal dfiniion and spcific formulas usd in h Swdish DC sysm can b found in Pnsionsmyndighn (2013. This concp iniially appars in conncion wih h conribuion ass (CA for DCs, h gnral oulin of which can b found in Srgrn (2001 and (2003, whil in Srgrn & Mikula (2005 boh concps ar modlld in coninuous im, giving horical suppor. Th sarch for valid xprssions o apply o DB PAYG sysms bgan wih Boado-Pnas al (2008, coninuing wih Vidal-Mliá al (2009, which in addiion links o h concp of auomaic balanc mchanisms (ABMs. Vidal-Mliá & Boado-Pnas (2013 obain h analyical propris of h CA and confirm is soundnss as a masur of a PAYG schm s asss. 20

21 S S A sysm's TD is also h sum of h wighd pay-in, p c, and pay-ou, p r, duraions of on monary uni in h sysm for h yar s conribuions and is basd on populaion daa obaind from a cross-scion, no from an xplici projcion. Th TD for rirmn in h ingrad schms (34.03 and yars for DCdr1 and DCTdr2 rspcivly is slighly diffrn o h bas sysm's TD (33.08 yars. This coms abou du o h sligh chang in h avrag ag of h conribuors afr considring dcrmns hrough disabiliy. Th sysms' TDs do chang mor noicably (31.9 and 32.3 yars for DCdr1 and DCTdr2 rspcivly du o h inroducion of disabiliy, which maks h wighd avrag ag a which h las conribuion is mad bwn 5 and 8 yars arlir han for h rirmn coningncy. To nd hs commns rgarding Tabl 1, i is worh mnioning ha our xampl is qui clos o raliy, no only bcaus h OLG modl dvlopd works simulanously wih 49 and 85 gnraions of conribuors and pnsionrs rspcivly, bu also bcaus h rsuling valus for h urnovr duraion - bwn 31.9 and 32.4 yars for h ingrad sysm - diffr vry lil from hos calculad by Sgrn & Mikula (2007 for a larg group of counris (32.7 yars. b Populaion changs: h iv populaion will no rmain consan Th DCdr1 schm is akn as a rfrnc whn analysing h ffc of iv populaion changs, whhr incrass or dcrass. Two addiional assumpions ar xplord in his scion: 1 h numbr of conribuors of all ags grows a an annual ra of γ =0.01 ovr im (hncforh DCdr1+, and 2 h numbr of conribuors of all ags dcrass by an annual ra of γ =-0.01 ovr im (hncforh DCdr1-. Tabl 2 shows ha alhough h raio bwn h numbrs of conribuors and pnsionrs ( dr and h raio bwn h avrag salary and pnsion ( fr chang du o variaions in h iv populaion, h ffc of h survivor dividnd ( D rmains unchangd for boh coningncis. Th sysm's susainabl rurn (G drivs from an adjusmn o h avrag iniial pnsions ha D λ ar awardd in h cas, dircly linkd o h annuiy fors a 15 R λ x k and a x A and h cumulad noional capial rhd a rirmn or disabiliy ag. Dspi h growh in populaion, h avrag iniial pnsions (rirmn and disabiliy for DCdr1+, xprssd in Tabl 3 hrough h avrag rplmn ra for h coningncy ( D 16 R β(x k, and β(x A,, ar highr han in h ohr wo cass. Th growh of h conomically iv populaion modifis h avrag yars of conribuion (AYC. As can b sn in Tabl 2, h avrag conribuor, awardd a pnsion in yar, has bn conribuing for yars as opposd o yars for DCdr1 and DCdr1-. All conribuors who rh rirmn ag ar considrd o hav sard working a h nry ag of 16, i.. A yars ago. Likwis all conribuors who bcom disabld a ag x +k yars sard working k yars ago. f h populaion grows ovr im, h rirs gnraion and h gnraion of disabld popl can b spli ino A and k diffrn cohors rspcivly, whos common for is h numbr of yars conribud sinc joining h labour mark. 15 This is a wighd avrag calculad from h disabiliy pnsions awardd in yar. 16 Lik D λ ax k, h avrag rplmn ra for h disabiliy coningncy is a wighd avrag calculad from h disabiliy pnsions awardd in yar. 21

22 Tabl 2: DCdr sysm wih survivor dividnd whn h iv populaion will no rmain consan: som slcd valus. ms DCdr1 DCdr1+ DCdr1- D R S D R S D R S θ a θ dr fr D θ * n.a n.a n.a n.a n.a n.a D λ a x k R λ a x A AYC D β(x k, n.a n.a n.a R β n.a n.a n.a (x A, G n our xampl, gnraion mmbrs who rir a ag 65 could com from 49 diffrn cohors dpnding on h numbr of yars conribud. This drmins 49 (A diffrn amouns of pnsion ha s h avrag iniial pnsion of h gnraion, linkd o h avrag numbr of yars conribud by hos who rh rirmn ag. Similarly, gnraion mmbrs who bcom disabld a ag 46 could com from 30 diffrn cohors dpnding on h numbr of yars conribud. Figur 4 shows iniial rirmn and disabiliy pnsions awardd a h ordinary rirmn ag brokn down by yars conribud as a prcnag of h sysm's avrag wag. Th pnsion amoun for rirmn (R DCdr1+ and disabiliy (D DCdr1+ wih an qual numbr of conribuion yars whn h conomically iv populaion grows ovr im is much highr han for DCdr1 (ARP DCdr1 and ADP DCdr1, in blu, and highr for DCdr1 han for DCdr1- (ARP DCdr1- and ADP DCdr1-, in rd. This is only o b xpcd givn ha h susainabl rurn of h sysm wih populaion growh (dclin is highr (lowr and, if h schm is wll dsignd, auomaically incrass (dcrass h amoun of bnfis awardd o rirmn and disabiliy pnsionrs. 22

23 niial rirmn and disabiliy pnsions awardd a ag 65 As a % of h sysm`s avrag wag Disabiliy Rirmn f h conomically iv populaion grows ovr im, h gnraions can b spli ino cohors cording o Yars of conribuions D DCdr1+ R DCdr1+ ADP DCdr1+ ARP DCdr1+ ADP DCdr1 ADR DCdr1 ADP DCdr1- ADR DCdr1- Figur 4: Brakdown of iniial rirmn and disabiliy pnsions by yars conribud. n lin wih h papr by Vidal-Mliá al (2013, anohr ssnial concrn is whhr or no h variaion in populaion has an influnc on h dividnd ffc. For h DCr modl hy find ha, dspi h f ha h dividnd ffc rmains consan for any valu of γ, growh in h conomically iv populaion nabls cohors wih mor yars of conribuions o bnfi o a grar xn from h dividnd ffc, i.. h mor conribuors hr ar, h largr h rirmn pnsion for hos cohors wih mor yars of conribuions compard o wha i would hav bn wihou including h survivor dividnd. Can h sam cohor ffc b sn in h ingrad DC modl wih disabiliy? Th answr can b found in Figurs 5 and 6, which show h ffc of h survivor dividnd for h of h cohors ha mak up h pnsionr gnraion undr h assumpion ha h conomically iv populaion grows a a consan ra of 1%, 2% or 4% pr yar. Th valu assignd o γ has an invrs influnc on h avrag numbr of yars conribud for h pnsionr gnraion ha rirs a im. For a valu of γ = 0.01, as Tabl 3 shows, h avrag numbr of yars conribud (AYC is (disabiliy, (rirmn and (sysm. Th AYCs for γ = 0.02 and γ = 0.04 ar (26.90, 31.67, and (19.89, 22.20, rspcivly. Figur 5 shows ha h growh of h conomically iv populaion nabls cohors wih mor yars of conribuions o bnfi o a grar xn from h dividnd ffc. ndd som cohors g a highr dividnd ffc han h avrag dividnd hivd by h sysm. Also, rirmn pnsionrs bnfi from a highr ffc han disabiliy pnsionrs bcaus, alhough boh yps of pnsion ar awardd in h sam yar, h conribuors who bcom disabld ha yar do no bnfi from a disribuion of h survivor dividnd ha yar. 23