Notional defined contribution pension schemes: Why does only Sweden distribute the survivor dividend?

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1 Noonal dfnd conrbuon pnson schms: Why dos only Swdn dsrbu h survvor dvdnd? Carlos Vdal-Mlá Dparmn of Fnancal Economcs and Acuaral Scnc, Unvrsy of Valnca, Avnda d los Naranjos s.n., Valnca, Span. -mal: carlos.vdal@uv.s María dl Carmn Boado-Pnas (Corrspondng auhor), Insu for Fnancal and Acuaral Mahmacs (IFAM), Dparmn of Mahmacal Scncs, Unvrsy of Lvrpool, Lvrpool L69 7ZL, UK. -mal: Carmn.Boado@lvrpool..uk Francsco Navarro-Cabo Dparmn of Bnfs-Rrmn, Towrs Wason d España S.A., Call Suro d Quñons 40-42, Madrd, Span. -mal: Francsco.Navarro@owrswason.com). Acknowldgmns María dl Carmn Boado-Pnas and Carlos Vdal-Mlá ar graful for h fnancal asssanc rcvd from h Spansh Mnsry of h Economy and Compvnss (projc ECO ). Ths projc has rcvd fundng from h Europan Unon s Svnh Framwork Programm for rsarch, chnologcal dvlopmn and dmonsraon undr gran agrmn RARE. Prlmnary vrsons of h papr wr prsnd a h Pnsons, Bnfs and Socal Scury Colloquum 2011 (PBSS 2011) n Ednburgh, a h Acuaral Thrs and Rsarchrs Confrnc (ATRC 2011, 2013) n Oxford and Kl, and a a smnar organzd by h Unvrsy of Valnca n July W ar graful o smnar parcpans and spcally o Manul Vnura-Marco and Ol Srgrn for hr valuabl commns and o Pr Hall for hs Englsh suppor. Th commns and suggsons mad by Prof Clfon and h wo anonymous rfrs wr xrmly hlpful n mprovng h papr. Any rrors ar nrly du o h auhors. 1 Elcronc copy avalabl a: hp://ssrn.com/absr=

2 Noonal dfnd conrbuon pnson schms: Why dos only Swdn dsrbu h survvor dvdnd? Absr Th am of hs papr s o analys h rol of h survvor dvdnd n noonal dfnd conrbuon (NDC) pnson schms. A prsn hs faur can only b found n h Swdsh dfnd conrbuon (DC) schm. W dvlop a modl ha ndorss h da ha h survvor dvdnd has a srong bass for nablng h NDC schm o hv fnancal qulbrum and ha no ncludng h dvdnd s a non-ransparn way of compnsang for ncrass n longvy and/or lgy coss from old pnson sysms. W also fnd ha h avrag ffc of h dvdnd rmans unchangd for any consan annual ra of populaon growh, ha conrbuors who rh rrmn ag always g a hghr rurn han h schm dos, and ha populaon growh nabls cohors wh mor yars of conrbuons o bnf o a grar xn from h dvdnd ffc. JEL: E62, H55, J26, M41. KEYWORDS: Inrnal Ra of Rurn, Fnancal Equlbrum, Longvy Rsk, Pay-As-You- Go, Publc Pnsons, Rrmn, Transparncy. 2 Elcronc copy avalabl a: hp://ssrn.com/absr=

3 Noonal dfnd conrbuon pnson schms: Why dos only Swdn dsrbu h survvor dvdnd? 1. Inroducon. Th nroducon of wha ar known as noonal (or non-fnancal) dfnd conrbuon pnson couns (NDCs) as a componn of modrn mul-pllar pnson sysms n som counrs has bn on of h man nnovaons of h las wo dcads as rgards pnson rform. Thy can b found n Ialy (1995), Kyrgyzsan (1997), Lava (1996), Poland (1999), Swdn (1999), Brazl 1 (1999) and Mongola (2000). Ohr counrs such as Grmany, Ausra, Franc, Fnland, Porugal and Norway hav ncorporad som sor of adjusmn mchansm ha can also b found n NDCs o hlp calcula or ndx h nal rrmn pnson. Accordng o Holzmann al (2012), mor counrs ncludng Egyp, Chna and Grc ar srously consdrng h nroducon of NDCs 2. A noonal dfnd conrbuon schm s a pay-as-you-go schm (PAYG) ha dlbraly mmcs a fnancal dfnd conrbuon schm (FDC) by payng an ncom sram whos prsn valu ovr a prson s xpcd rmanng lfm quals h cumulad capal a rrmn. Th prcal applcaon of NDCs frs cam abou n h arly 1990s, and snc h md- 1990s hy hav bn nroducd n a numbr of counrs. For Holzmann & Palmr (2012), NDC schms work wll from a prcal pon of vw, as shown by h xprncs of Ialy, Lava, Poland and Swdn, bu hy could b mad o work vn br. Ths papr dals spcfcally wh h so-calld survvor dvdnd (also known as nhranc gans), whch s dsrbud n h savng phas on a brh cohor bass from h coun balancs of parcpans who do no survv o rrmn n NDC pnson schms 3. Ths faur can currnly only b found n h Swdsh DC schm. Accordng o Pnsonsmyndghn (2013), h Swdsh pnson schm ams o rdsrbu rsourcs from ndvduals wh shorr-han-avrag lf spans o hos who lv longr durng h pay-ou (or dcumulaon) phas. Ths arrangmn works n xly h sam way as n DC-fundd pllars whn h bnfcary chooss h pay-ou opon of rcvng h rrmn bnf as a lf annuy. NDC schms do no allow phasd whdrawals 4, whr h rrmn savngs of dcasd bnfcars ar dsrbud o hr nhrors. Unlk wh h DC-fundd pllar, n h pay-n phas h pnson balancs of dcasd prsons ar rdsrbud h yar o survvng nsurd prsons n h sam brh cohor. Ths nlmn s complly dffrn from h nhranc opons n DC-fundd pllars, whr h cumulad capal n h ndvdual coun s dsrbud o nhrors and/or ransformd no a survvor bnf. Vry ll anon has bn pad o h subjc of h survvor dvdnd n h conomc lraur. No vn h rpor publshd by h Swdsh auhors, Pnsonsmyndghn (2013), xplans n dph s uaral foundaon and h ffc has on h schm's fnancal qulbrum. As far as w know, h papr by Boado-Pnas & Vdal-Mlá (2014) s h only on ha has shd any lgh on whhr would b jusfd o nclud h survvor dvdnd whn calculang affla pnson balancs n an NDC framwork wh no populaon changs. I concluds ha ncludng h survvor dvdnd n h calculaon of h nal pnson s by no mans rrlvan bcaus h pnson could rs by up o 21.84% dpndng on h moraly scnaro usd. Th am of hs papr s o carry ou an n-dph analyss of h rol of h survvor dvdnd n NDC schms. Wh hs am n mnd, w frs xnd h modl dvlopd by Boado- Pnas & Vdal-Mlá (2014). Scond, w show ha conrbuors who rh rrmn ag 3

4 always g a hghr rurn han h schm dos, and ha populaon growh nabls cohors wh mor yars of conrbuons o bnf o a grar xn from h dvdnd ffc. Th rsuls rhd n h numrcal xampl w prsn ndors h f ha h modl rally works. Followng hs brf nroducon, n Scon 2 w prsn h modl. Is man novls ar h nroducon of changs n h growh of h v populaon and h possbly of xplorng h ffc of h survvor dvdnd on h rlaonshp bwn h ndvdual s nrnal ra of rurn (IRR) for conrbuors and h schm s IRR. In Scon 3 w prsn a complx xampl rprsnav of a gnrc NDC schm. Spcfcally, w provd a numrcal llusraon of h ffcs of h survvor dvdnd on h schm's fnancal qulbrum whn h conomcally v populaon s no consan, plus h mp of h survvor dvdnd on h ndvdual s IRR for conrbuors who rh rrmn ag. Th papr nds wh h conclusons, possbl drcons for fuur rsarch and an appndx wh som proofs of h formulas usd n Scon Th Modl. In hs scon w xnd h uaral ovrlappng gnraons modl (OLG) dvlopd by Boado-Pnas & Vdal-Mlá (2014) basd on hos frs pu forward by Srgrn & Mkula (2005), Boado-Pnas al. (2008) and Vdal-Mlá & Boado-Pnas (2013). Ths paprs wr o som xn nsprd by h counng framwork for organzng, summarzng and nrprng daa on ransfr sysms and h lf cycl dvlopd n L (1994), Wlls (1988) and Arhur & McNcoll (1978). As w wll s lar, h man xnsons o hs modl ar h nroducon of changs n h conomcally v populaon, h possbly of xplorng h ffc of h survvor dvdnd on h rlaonshp bwn h ndvdual s IRR for conrbuors who rh rrmn ag and h schm s IRR. Th modl s man faurs and assumpons ar: Th dfnd conrbuon ra crdd o h ndvdual, θ a, s fxd ovr m. Th nal pnson dpnds on h valu of h cumulad noonal coun, h xpcd moraly of h cohor n h yar h conrbuor rhs rrmn, and a fuur ndxaon ra, λ,.. pnsons n paymn ncras or dcras a an annual ra of λ. Th cumulad capal n h noonal coun rflcs h parcpan's ndvdual conrbuons and h fcous rurns hs conrbuons gnra ovr h cours of h parcpan s workng lf, plus h nhranc capal. Th coun balancs of parcpans who do no survv o rrmn ar dsrbud as nhranc capal o h couns of survvng parcpans on a brh cohor bass. Th schm dos no provd a mnmum pnson. Conrbuons and bnfs ar payabl yarly n advanc. Parcpans lvs las (w-1-x ) prods, whr (w-1) s h hghs ag o whch s possbl o survv and x s h arls ag of nry no h pnson schm. Th ag gvng nlmn o rrmn pnson, x +A, s fxd. Th ndvdual conrbuon bas grows a an annual ra of g. 4

5 Th conomcally v populaon ncrass or dcrass ovr m a an annual ra of γ, affcng all groups of conrbuors qually. Th schm's ncom from conrbuons (wag bll) also grows (dcrass) a ra G = (1+g)(1+γ)-1. Whn h pnson schm rhs a maur sa = w-1-x -A yars from ncpon, A gnraons of conrbuors and (w-(x +A)) gnraons of pnsonrs coxs a h momn n m. Th dmographc-fnancal srucur a any momn s gvn by: 1. Ag: Conrbuo rs' ags x, x 1, x 2,......x A 1, x A, x A 1, w Numbr of conrbuors by ag a m : N Pnsonrs ' ags N ( 1γ), N N ( γ),..., N N ( γ) (x, (x 1, ) (x 1, 1) 1 (x A1, ) 1 (x A1, 1 ) (x, ) ) 1 2. whr N(x k, ) N(x, ) k p 1 1, wh p x k x bng h probably ha an ndvdual agd x wll b alv a ag x +k. 3. Avrag wag by ag a m : -1-1 y(x, ) y(x, 1) ( 1 g), y(x 1, ) y(x 1, 1) ( 1 g),..., y(x A1, ) y(x A1, 1 ( 1 g) ) Ths dmographc srucur mans ha h ag-wag srucur (conrbuon bass) only undrgos proporonal changs. Th slop of h ag-wag srucur s consan. Afr h man assumpons hav bn dald, for h sak of clary hs scon wll b dvdd no hr subscons: a dscrpon of h pnson schm n h maur sa, h dfnon and calculaon of h survvor dvdnd, and h ffc of h survvor dvdnd on h schm's fnancal qulbrum Dscrpon of h pnson schm n h maur sa. Th man mplcaons of h NDC schm s bng n a maur sa ar (1) pays full bnfs o all gnraons of rrmn pnsonrs, (2) h dpndncy rao 5, dr, sablzs, and (3) h fnancal rao, fr, s consan du o h f ha h avrag pnson and h avrag conrbuon bas volv a h ra of varaon n wags. Hnc h oal conrbuon ra ( θ ) ha nsurs qualy bwn conrbuon rvnu and pnson xpndur s consan ovr m, and h schm's ncom from conrbuons s quvaln o h prsn uaral valu of h pnsons awardd n ha yar. Th nal avrag pnson n yar for hos ndvduals who rh rrmn ag x +A, P(x A, ), can b calculad as:

6 P (x A, ) T (x A, ) K N(x A,) a K (x A, ) 1 λ x A A P c1 (x A, c, ) N N (x A, ) (x A, c, ) 4. whr: N(x A, ) : Numbr of conrbuors agd x +A, whras N (x A, c,) s h numbr of conrbuors agd x +A who hav bn conrbung for h las c yars. Thrfor N A N (x A, ) (x A, c, ) c1 k wx A λ 1 1 λ a x A k px A k G : Prsn valu a ag x +A of 1 monary un of a lfm 0 1 pnson payabl n advanc and ndxd a ra, wh a chncal nrs ra qual o G. Ths s also calld h annuy for or uaral dvsor, wh k p x A bng h probably ha an ndvdual agd x +A wll b alv a ag x +A +k. K T (x A,) : Toal cumulad noonal capal a m for all ndvduals who rh ag x +A. A Thrfor T K(x A,) K(x A, c, ) N, whr K (x (x A, c, ) A, c,) rprsns h cumulad c1 noonal capal a ag x +A for on ndvdual who has bn conrbung for h las c yars. T K (x A,) K (x A,) : Avrag cumulad noonal capal a m for on ndvdual agd N(x A, ) x +A. 1 Hncforh F wll b usd o dno 1 G hroughou hs papr. Wh populaon growh of γ 0, onc h ndvdual jons h labour mark hy wll connu workng nonsop unl rrmn ag. Th only x from h labour mark n hs modl s arly dah. Thrfor hr ar A dffrn conrbuon pahways ha wll drmn A dffrn pnsons, as conrbuors mgh b workng for 1 yar, 2 yars, A- 1 yars. Conrbuons of ndvduals who d bfor rhng rrmn ag ar ncludd n h noonal capal. Consqunly, n h cas of γ 0, P (x A, c,) s h nal pnson a ordnary rrmn ag for ndvduals who nrd h labour mark a ag x +A-c,.. for on ndvdual who has bn conrbung for h las c yars, whras P(x A, ) s h avrag pnson for ndvduals who rr a h ordnary rrmn ag, hs bng a wghd avrag pnson of h A dffrn pnsons onc sld. As formula [4.] shows, n ln wh h dfnon of an NDC schm bng on ha dlbraly mmcs an FDC schm, h procss for calculang h nal amoun of h rrmn pnson s smlar o h on usd for calculang h fuur nsalmns of a dfrrd lfm annuy, a yp of annuy conr whr paymns ar no schduld o bgn unl a fuur da, n hs cas h ag of rrmn. 6

7 Th balancd conrbuon ra, θ, maks spndng on pnsons qual o h rvnu from conrbuons, and hnc: Aggrga conrbuons A1 θ y k 0 (x+k, ) N (x+k, ) λ N (x a A,) x A -1 P(x A, ) N (x Ak, ) F k0 Expndur on pnsons Thrfor onc h maur sa s rhd, h mro conrbuon ra, θ, s consan from an uaral pon of vw and can b xprssd as h produc of h dpndncy rao and h fnancal rao: k 5. θ dr fr P (x A, ) A1 Expndur on pnsons w-x A1 k0 N y(x +k, ) N (x+k, ) k 0 Conrbuon bas (x Ak, ) F k θ 1... θ Dfnon and drmnaon of h survvor dvdnd whn h conomcally v populaon ncrass. As n h Swdsh NDC schm, w follow h prncpl ha h monary un conrbud s pad ou n h form of a rrmn or old-ag bnf, bu no ncssarly o h ndvdual who mad h conrbuons. An cumulad survvor dvdnd s ncludd for any ndvdual who rhs rrmn ag. Th coun balancs of parcpans who do no survv o rrmn ar dsrbud as nhranc capal o h couns of survvors on a brh cohor bass 6. In hs modl, h amoun of h survvor dvdnd can manly b quanfd usng formulas [7.] and [8.] Formula [7.] shows h mahmacal xprsson of h cumulad survvor dvdnd a rrmn ag a m for an ndvdual who blongs o h nal group (ha nrd h schm a ag x ) and has hrfor conrbud snc nrng h pnson schm: D A1 A-1 Ak Ak (x A, A, ) K (x A, A, ) θ a y(x k, A k ) ( G) D(x k, A k ) ( G) 1 1 k 0 k 1 K(x A, A,) D whr k, Ak ) s h dvdnd dsrbud a m -A+k for ndvduals agd x +k. (x 7. As mnond abov, h cumulad survvor dvdnd a a spcfc ag s h poron of h crdd coun balancs of parcpans rsulng from h dsrbuon, on a brh cohor bass, of h coun balancs of parcpans who do no survv o rrmn. In ohr words s h dffrnc bwn crdd capal K(x A, A, ), whch ncluds conrbuons and ndxaon on conrbuons for mmbrs from h sam cohor who dd, and ndvdual crdd noonal capal K (x A, A, ). Whn h conomcally v populaon grows ovr m (s Scon 2.1.), hr ar A dffrn conrbuon pahways ha wll drmn A dffrn conrbuon profls. 7

8 Formula [7.] hrfor nds o b modfd o coun for h dffrn profls. Formula [8.] quanfs h avrag cumulad dvdnd a ag x +A, akng no coun h dffrn A conrbuon profls: N 1 (x A, ) θ D A1 a k0 N (x A, ) A D c1 (x k, Ak) y (x A, c, ) N (x A, ) N (x k, Ak) (x A, c, ) ( 1 G) K Ak (x A, ) A K K N c1 (x A, c, ) (x A, ) (x A,c, ) In shor, whn h v populaon ncrass, h cumulad dvdnd for h ndvdual who rhs rrmn ag dpnds on h moraly ras by ag, h ra of populaon growh and h numbr of yars conrbud Th ffc of h survvor dvdnd on h schm's fnancal qulbrum. Th survvor dvdnd plays a crucal rol n h schm's fnancal qulbrum and - as shown n Appndx 1 - s asy o prov h quvalnc bwn h mro (balancd) conrbuon ra, θ, and h crdd ndvdual conrbuon ra, θ, n h modl. a If h amoun of h pnson s drmnd from h ndvdual noonal capal whou consdrng h survvor dvdnd, hn h nw balancd conrbuon ra, θ *, and h crdd ndvdual conrbuon ra, θ a, ar dffrn bcaus h rrmn bnfs ar srcly lowr han hy should b (as h survvor dvdnd s no dsrbud among h survvors). Th rlaon bwn boh conrbuon ras can b xprssd usng h so-calld dvdnd ffc, as shown n formula [9.]: θ a θ * K K (x (x A,) A,) θ * Dvdnd 0 ffc D (x A,) K (x A,) P * (x θ P A, ) (x A, ) whr K (x A,) s h avrag ndvdual cumulad noonal capal a m for ndvduals agd x +A, whou akng no coun h survvor dvdnd, and P s h avrag (x A, ) pnson of an ndvdual who rrs a h ordnary ag a, agan whou consdrng h dvdnd. Th dvdnd ffc masurs hr h ncras n h nal rrmn pnson afr ncluson of h survvor dvdnd or h dcras n h balancd conrbuon ra f h dvdnd s no ncludd n h pnson calculaon. Thrfor * θ a θ bcaus h schm savd mony by no ncludng h survvor dvdnd. If θ θa wr conrbud nsad of θ *, h pnson schm would connuously cumula fnancal rsrvs bcaus gnorng h survvor dvdnd producs savngs whn longvy s consan ovr m. In prc hs rsrvs could fnanc h ncras n spndng on pnsons rsulng from ncrass n longvy and/or lgy coss from old pnson sysms. Indd, n Poland and Lava hs rvnus provd funds for ohr socal nsuranc commmns ha hav no spcfc sourc of fundng. Boh counrs dcdd o nroduc fundd componns and, as a rsul, h rvnu for h pay-as-you-go pllars was 8

9 rducd du o conrbuons bng ransfrrd o fundd couns. Hnc h nhranc gans hlp o covr h doubl paymn burdn. In yar, assumng consan longvy, h amoun of h schm s surplus, S, s asy o quanfy bcaus s conncd o h survvor dvdnd, spcfcally o h avrag cumulad dvdnd: s A * ( θ a θ ) 1 y k0 (x +k, ) N (x +k, ) N (x A, ) D (x A,) ( K(x A, ) K(x A, ) ) 10. Th cos of longvy masurd as h ncras n lf xpcancy a rrmn ag, Δx A, ha could b fnancd f h dvdnd wr no dsrbud, n h smpls cas whr λ G, can b quanfd followng h rasonng prsnd n Appndx Numrcal xampl. Ths scon prsns a numrcal xampl o llusra h mporan rol playd by h survvor dvdnd n h NDC framwork. To do hs w bascally us h closd-form xprssons dvlopd n Scon 2. Indvduals ar assumd o jon h labour mark a ag 16 (x, ) onwards and conrbu 16% ( θ a ) of hr conrbuon bas unl hy rh 65 (x +A). Th ndvdual conrbuon bass grow a an annual cumulav ra (g) of 1.6%, whl h rrmn pnson, onc sld, s consan n ral rms ( λ 0 ). For h purposs of comparson, h moraly abls 7 usd ar hos for Poland n 2009 (PL), Lava n 2010 (LT) and Swdn (SW) n Basln cas: zro populaon growh In h cas of zro populaon growh, all ndvduals nr h labour mark a ag 16 and work connuously unl rrng a ag 65. Th amoun of pnson payabl o ndvduals agd 65 s drmnd, afr 49 conrbuon yars, cordng o h formulas shown n h prvous scon. Undr hs scnaro h balancd conrbuon ra ( θ ) s 16% and concds wh h crdd conrbuon ra ( θ a ). Th frs vrcal axs of Fgur 1 (s afr Tabl 1 blow) shows h numbrs of conrbuors and pnsonrs by ag dpndng on h moraly scnaro (SW, PL or LT) n a maur sa from a cross-sconal pon of vw. Th scond vrcal axs rprsns h conrbuon bas and pnson srucur undr h hr dffrn moraly scnaros (SW-w&p, PL-w&p and LT-w&p). Th man valus makng up h schm's fnancal qulbrum undr h hr moraly abls ar shown n Tabl 1. [Insr Tabl 1] [Insr Fgur 1] I can b sn ha h moraly parn has a sgnfcan ffc on h amoun of h rrmn pnsons provdd by h schm. Undr h moraly scnaro wh h hghs lf xpcancy (Fgur 1, SW, n blu), h uarally far pnsons ar much lowr han hos 9

10 undr h ohr moraly scnaros (PL, n blk, and LT, n gry). Consqunly undr h λ SW moraly scnaro, h annuy dvsor, a x A, usd o calcula h amoun of h nal rrmn pnson s hghr (17.23) han hos undr h ohr moraly scnaros (15.05 and for PL and LT rspcvly). Gvn ha h nal assumpons ar analogous for all hr scnaros, moraly ras play a crucal rol n hvng h schm's fnancal qulbrum. As Tabl 1 shows for h SW scnaro, h schm s dpndncy ra ( dr ) s hghr (38.86%) han for h ohr wo scnaros (30.20% and 27.57% for PL and LT rspcvly), whras h schm s fnancal ra ( fr ) s lowr (41.18% as agans 52.98% and 58.04% for PL and LT rspcvly). Ths rsuls com naurally from formula [6.]. If hr s a dfnd conrbuon ra, θ, fxd ovr m, and f h dpndncy rao s drmnd by h moraly scnaro, hn h fnancal rao has o b adapd cordngly. For h SW scnaro, h amoun of h pnson onc sld a rrmn ag, β(x A, ), amouns o 68.77% of h avrag salary. Th mp of h dvdnd ffc, D, on h nal pnson s no vry sgnfcan as would only rs by 7.39% afr h balancs of parcpans who do no survv o rrmn ar dsrbud. For h sam scnaro, f h survvor dvdnd wr no ncludd n h calculaon of h nal rrmn pnson, a dscrpancy would ars bwn h crdd conrbuon ra qual o 16%, θ, and h ra ncssary o fnanc h pnson, θ *, n hs cas 14.90%. As can b sn n formula [9.], h drc lnk bwn boh ras s h dvdnd ffc ( θ D *)1. θ Th mp of h dvdnd ffc, D, on h nal pnson s no nsgnfcan for h ohr wo scnaros, wh h pnson rsng by 18.32% usng h Polsh moraly abls (PL) and 23.12% usng h Lavan abls (LT). As a rsul, h rplmn ras rhd ar also hghr du o ncluson of h dvdnd. Thr s a drc rlaon bwn h dvdnd ffc and h rplmn ras. Indd, wh h varabls provdd n Tabl 1, s asy o s β ha (x ( A, ) D ) 1. β (x A, ) As sn n Tabl 1, f h survvor dvdnd wr no ncludd n h calculaon of h nal rrmn pnson, h pnson schm could handl an unxpcd ncras n lf xpcancy a rrmn ag ( Δx A) of bwn 1.48 and 3.76 yars, dpndng on h scnaro, bfor xhausng h cumulad surplus. Agan, h dvdnd ffc can b usd o xplan h ncras n lf xpcancy a rrmn ndd o nuralz h lk of Δ dvdnd. Gvn hs, s no dffcul o chck n Tabl 1 ha x A D 3.2. Populaon changs Th growh n populaon mans ha h rrs gnraon can b spl no A dffrn cohors, whos common for s h numbr of yars conrbud snc jonng h labour mark. Ths scon xplors wo addonal assumpons abou h ra of populaon growh: (1) ha h numbr of conrbuors of all ags grows a an annual ra of γ =0.01 ovr m (hncforh PL+), and (2) ha h numbr of conrbuors of all ags dcrass by an annual ra of γ =-0.01 (hncforh PL-). Th Polsh moraly scnaro (rfrrd o n h prvous scon as PL) s akn as a rfrnc whn analysng h ffc of populaon changs, whhr ncrass or dcrass, bcaus s survvor dvdnd has an avrag ffc on h rsuls, fallng bwn h ohr wo scnaros. x A 10

11 Th rsuls prsnd n h prvous sub-scon ar rcalculad akng no coun h ffc of populaon changs undr h wo addonal assumpons dscrbd abov. Fgur 2 s ncludd lar n hs sub-scon for a br undrsandng of wha happns whn h conomcally v populaon grows. In addon, h rlaonshp bwn h IRR for conrbuors by yars of conrbuons and h IRR for h schm slf s sudd undr h populaon growh scnaro. Th man rsuls ar shown n Fgurs 3 and 4. Th man valus makng up h schm's fnancal qulbrum undr h hr populaon scnaros ar shown n Tabl 2. [Insr Tabl 2] Wha rally draws h anon s ha h rao bwn h numbrs of conrbuors and pnsonrs, h rao bwn h avrag salary and pnson, and h ffc of h survvor dvdnd rman unchangd whn h conomcally v populaon s no consan ovr m. Howvr, h xplanaon s obvous. Gvn ha h conrbuon ra s h sam undr h hr populaon scnaros, h rao bwn h numbr of pnsonrs and h numbr of conrbuors (dr ) mus also b h sam bcaus dpnds on h moraly scnaro and, cordng o quaon [11.] n Appndx 1, h rao sablzs bcaus boh groups volv (ncras or dcras) a xly h sam ra as populaon growh (γ). Thrfor, cordng o quaon [6.], h schm's avrag rplmn ra (fr ) has o b h sam for all hr populaon scnaros. Th ra of populaon growh has a drc ffc on h susanabl schm s ra of rurn (G), whch n h NDC framwork largly drmns h amoun of h nal rrmn bnf provdd by h schms. Consqunly, h hghr h populaon growh, h hghr h amoun of cumulad noonal capal, whch mpls a hghr amoun of nal pnson. [Insr Fgur 2] Th schm's susanably drvs from an adjusmn o h avrag nal pnson, drcly λ lnkd o h annuy for, a x A, and h cumulad noonal capal rhd a rrmn ag. Wh an annual ncras n populaon of γ =0.01 and for a gvn amoun of crdd noonal capal a rrmn ag, h nal amoun of h pnson awardd wll b hghr han wh a dcras n populaon of 1% ( γ =-0.01) or consan populaon growh of 0% ( γ =0). Ths s bcaus h annuy dvsor usd s lowr (13.73 as agans for h dcras n populaon and for consan populaon rspcvly). Th hghr h valu of G, h lowr h valu of h annuy dvsor. In our xampl, gnraon mmbrs who rr a ag 65 could com from any of 49 dffrn cohors dpndng on h numbr of yars conrbud. Ths drmns 49 (A) dffrn amouns of pnson ha s h gnraon s avrag nal pnson, lnkd o h avrag numbr of yars conrbud by hos who rh rrmn ag. Thrfor growh n h conomcally v populaon brngs abou changs n h avrag yars conrbud (AYC), as shown n Tabl 2. Undr h assumpon ha h numbr of conrbuors of all ags grows a an annual ra of γ =0.01 ovr m, h AYC s as opposd o 49 for γ =0 or γ =-0.01, whr all h conrbuors who rh rrmn ag sard workng a h 11

12 nry ag of 16,.. A yars ago. I can b sad ha h valu assgnd o γ has an nvrs nflunc on h AYC for h gnraon ha rrs a m. In sp of h growh n populaon ha brngs abou a rducon n h AYC (s Tabl 2), h avrag rplmn ra, β(x A, ), rhd for PL + s vn hghr (95.07%) han n h ohr wo cass (78.68% and 86.75% for PL - and PL rspcvly). Th daa provdd n Tabl 2 clarly show us ha h amoun of h pnson wh an qual numbr of yars of conrbuons for h cas of posv populaon growh s much hghr han for h cas of sabl populaon and hghr han for a dcras n populaon. Ths s o b xpcd gvn ha h schm s susanabl rurn wh populaon growh s hghr. Anohr undrlyng ssu s whhr or no h varaon n populaon has an nflunc on h dvdnd ffc. Accordng o h daa shown n Tabl 2, h dvdnd ffc rmans consan for any valu of γ, bu w nd o sudy wha happns whn h populaon ncrass. In h cas of populaon growh hr s a vcor of pnsons - A dffrn pnsons - so s mporan o fnd ou whhr h mp of h dvdnd rmans consan for cohors blongng o h sam gnraon of rrs whn hr ar changs n h ra of populaon growh. Th answr can b sn n Fgur 2, whch shows h dvdnd ffc for h of h A cohors ha mak up h rrs gnraon undr h assumpon ha h populaon grows annually a a consan ra of 0.01, 0.02 or For h valus of γ assumd, h AYCs ar 38.97, and rspcvly, gvn ha γ has an nvrs nflunc on h AYC for h gnraon ha rrs a m. Fgur 2 shows ha, for xampl, undr h assumpon ha h populaon grows annually a a consan ra of 0.01 (rprsnd by h sold blk ln), hos conrbuors who rh rrmn ag havng sard workng a h arls ag possbl,.. ag 16, bnf from a dvdnd ffc of 19.07%. Ths s hghr han h dvdnd ffc wh zro populaon growh (18.32%) bu lowr han for h scnaros n whch h populaon grows annually a a consan ra of 2% or 4%, rprsnd by h sold gry and blu lns rspcvly. Undr h scnaro wh h hghs populaon growh, h dvdnd ffc for conrbuors who sard workng a h arls ag rhs 20.99%, whras for h nrmda assumpon s 19.77%. And s no only conrbuors wh 49 conrbud yars ha oban a hghr han avrag dvdnd ffc; many ohr conrbuor cohors wh fwr yars of conrbuons also oban a hghr survvor dvdnd. For xampl, as shown n Fgur 2 undr h scnaro wh h hghs populaon growh, h dvdnd ffc for hos cohors wh mor han 33 yars of conrbuons s hghr han h avrag, whras for hos cohors wh fwr han 33 conrbud yars s lowr. Undr h ohr wo populaon scnaros, h numbr of yars conrbud ar 38 and 42 for γ=0.02 and γ=0.01 rspcvly. Populaon growh hrfor nabls cohors wh mor yars of conrbuons o bnf o a grar xn from h dvdnd ffc. Th mor h numbr of conrbuors grows, h largr h pnson for cohors wh mor yars of conrbuons compard o wha would hav bn whou ncludng h survvor dvdnd. Nvrhlss, mus b srssd ha h avrag ffc of h dvdnd rmans consan for any valu of γ. -Th conrbuors IRR and s rlaon o h schm s IRR (G). To provd a numrcal llusraon, w wll look a h rlaonshp bwn h conrbuors IRR and ha for h schm slf undr h populaon growh scnaro. 12

13 Gnrally spakng, h conrbuors IRR - wh or whou h survvor dvdnd - dpnds on ag of nry o h labour mark and h m h calculaon s don (a ag of nry, a rrmn ag, s yars afr nrng h labour mark, c.). As w wll s lar, hs s bcaus h ag rhd by h conrbuor s vry mporan whn compung h IRR. Th mp of hs wo condonng fors undrlyng h rsuls of h conrbuors IRR ar xplord n Fgurs 3 and 4. For h hr moraly scnaros (SW, sold blu ln, PL, sold blk ln, and LT, sold gry ln), Fgur 3 prsns h xpcd IRR (x A, A-K, ) for conrbuors who rh rrmn ag akng no coun h survvor dvdnd and dsngushng by ag of nry o h labour mark,.. cordng o h numbr of yars conrbud. Th fgur also compars h cohor IRRs wh h schm s IRR (G, brokn blk ln), whch s h sam as h rurn ha conrbuors would g f h survvor dvdnd wr no ncludd whn calculang h nal rrmn pnson. As xpcd, h IRR dcrass as h ag of nry o h labour mark ncrass bcaus h conrbuors bnf lss from h dsrbuon of dcasd prsons conrbuons. I can also b obsrvd ha, dpndng on h moraly scnaro, h IRR vars bwn 3.41% for SW (h scnaro wh h lows moraly ras) and 5.41% for LT (h scnaro wh h hghs moraly ras) a a labour mark nry ag of 16. Ths dffrnc bwn moraly scnaros nds o dcras as h ag of nry o h labour mark ncrass. If h ndvdual jons h labour mark a ag 56, h IRR vars bwn 2.97% for SW and 3.74% for LT. [Insr Fgur 3] Hnc h ndvduals IRR vars sgnfcanly dpndng on h numbr of conrbuon yars for h cohor whn h sam gnraon, and conrbuors who rh rrmn ag always g a hghr rurn han h schm bcaus hr noonal capal also ncluds conrbuons from afflas who d bfor rhng rrmn ag. Fgur 4 shows h rsuls of h sudy s for cohors of conrbuors who jond h labour mark a h arls ag possbl,.. 16 n our xampl, dffrnang by ag aand. [Insr Fgur 4] Th oucoms whn h survvor dvdnd s akn no coun, IRR(sd), ar (x s, ) rprsnd n Fgur 4 as SW (sold blu ln), PL (sold blk ln) and LT (sold gry ln). I can b sn ha h IRR ncrass wh ag, du o h f ha h survvors cumulad noonal capal ncrass yarly as a rsul of h dsrbuon of h dvdnd. Th rsuls whn h survvor dvdnd s no akn no coun ar rprsnd n Fgur 4 as SW* (brokn blu ln), PL* (brokn blk ln) and LT* (brokn gry ln). A maxmum rurn qual o G s only hvd by survvors who rh rrmn ag, and hrfor h schm prmannly cumulas rsrvs f h dvdnd s no ncludd whn calculang h pnson. Th rsuls shown n Fgur 4 do no xly mach on of h sad proprs for NDCs cordng o Palmr (2006): Propry 1. A any m h prsn valu of an ndvdual s lfm bnf quals h ndvdual s coun balanc. For h parcpan and a all ms, h amoun n h coun, K, s h prsn or xpcd valu of hs or hr bnf. Th valu of h coun s drmnd by h ndvdual s 13

14 own conrbuons and h sysm s nrnal ra of rurn; and Damond (2006): An NDC s supposd o provd bnfs for dffrn cohors ha hav a prsn dscound valu ha quals h valu of h coun, usng h nrnal ra of rurn (IRR) (of h sysm) for a dscoun ra. Whn hy say h sysm s nrnal ra of rurn or h nrnal ra of rurn (IRR) (of h sysm), boh auhors sm o b rfrrng o wha n hs papr has bn dfnd as G. Hnc h propry sad by Palmr (2006), IRR=G, s only fulflld n wo spcfc cass for h conrbuon cohor ha jons h labour mark a h arls possbl ag. Cas 1: Whn h IRR s compud a ag of nry no h labour mark undr h assumpon ha h survvor dvdnd s akn no coun o calcula h rrmn bnf. Cas 2: Whn h IRR s valud a rrmn ag undr h assumpon ha h survvor dvdnd s no akn no coun o calcula h rrmn bnf. Boh cass ar dnfd n Fgur 4 for conrbuors who nrd h labour mark a h arls possbl ag. For all ohr cass, dpndng on whhr or no h schm aks h survvor dvdnd no coun whn calculang h rrmn pnson, h xpcd IRR for dffrn conrbuors s (vry) dffrn from G, and hs dffrnc dpnds on h numbr of yars xpcd o b conrbud and h survval probabls arbud o afflas. 4. Concludng commns and drcons for fuur rsarch. Among hos counrs n whch NDC schms hav bn nroducd, only Swdn appls wha s known as h survvor dvdnd. Surprsngly ll anon has bn gvn o hs subjc n h conomc lraur and no vn h rpor publshd by h Swdsh auhors xplans n any dph why hs survvor dvdnd s appld. As far as w ar awar, h papr by Boado-Pnas & Vdal-Mlá (2014) s h only on ha has shd any lgh on whhr would b jusfd o nclud h survvor dvdnd whn calculang affla pnson balancs n an NDC framwork. In h prsn papr w hav xndd hr modl o coun for changs n h conomcally v populaon, and h ffc of h survvor dvdnd on h rlaonshp bwn h ndvdual s IRR for conrbuors who rh rrmn ag and h schm s IRR. W fnd ha whn h v populaon changs, h modl ndorss h da ha h survvor dvdnd has a sound bass whch nabls h NDC schm o hv fnancal qulbrum. To pu anohr way, h papr dmonsras ha h survvor dvdnd nabls h balancd conrbuon ra appld o b h sam as h ndvdual crdd ra. A smlar oucom was rhd by Palmr (2012) rgardng h quvalnc bwn hs wo conrbuon ras n an NDC framwork, bu whou xplcly consdrng h ffc of h survvor dvdnd. W also fnd ha h avrag ffc of h survvor dvdnd rmans unchangd for any consan annual ra of populaon growh, ha conrbuors who rh rrmn ag always g a hghr rurn han h schm dos, and ha populaon growh nabls cohors wh mor yars of conrbuons o bnf o a grar xn from h dvdnd ffc. Th hghr h numbr of conrbuors, h hghr h pnson for hos cohors wh mor yars of conrbuons, compard o wha hy would hav rcvd whou ncluson of h survvor dvdnd. On h prcal sd, can b sad ha h numrcal xampl dvlopd n h papr s clos o raly. I confrms ha our modl rally works bcaus h rsuls mak sns and provd us wh som usful valus rgardng h magnud of h dvdnd ffc, h unxpcd ncras n lf xpcancy a rrmn ag ha h NDC schm could handl 14

15 f h survvor dvdnd wr no ncludd n h calculaon of h nal rrmn pnson, and h conrbuors IRR compud cordng o a s of dffrn scnaros. In shor, hs opc s parcularly mporan for h dsgn of pnson rforms and hrfor hs rsarch could hav a consdrabl mp for hos counrs ha ar currnly rhnkng h srucur of hr publc pnson sysms. Takng h survvor dvdnd xplcly no coun ncrass h polcal arvnss of h rform by provdng hghr nal rrmn bnfs. Th ssu of ransparncy s also mporan, bcaus no ncludng h dvdnd mans ha sysms nd o cumula fnancal rsourcs (as n counrs such as Poland, Ialy and Lava) as a non-ransparn way o proc hr sysms agans h longvy rsk and/or o fnanc lgy coss from formr pnson arrangmns. Fnally, basd on h modl prsnd n hs papr, a numbr of mporan drcons for fuur rsarch can b dnfd. Frs, h covrag of any unxpcd ncras n longvy n cass whr h SD s no dsrbud could b xplord furhr n ordr o valua whhr h SD s a ponal soluon for covrng longvy rsk n NDCs. Incrass n longvy can b rflcd no only by an ncras n lf xpcancy a rrmn ag bu also by a dcras n moraly ras or an ncras n survval probabls. Probabls, manwhl, can b valuad as bng hr consan ovr m or ag-spcfc dpndng on h moraly modl. Th scond drcon would conss of valuang h mp of nroducng a mnmum pnson on h schm's fnancal qulbrum. Accordng o Holzmann & Palmr (2006), NDC schms should b supplmnd wh a mnmum ncom (pnson) guaran. For Barr & Damond (2009), h purpos of pnsons s o provd an adqua ncom sram whn h ndvdual s unabl o work du o dsably or rrmn, so would b a good da o nroduc a mnmum pnson n ordr o manan a mnmum sandard of lvng. Thrd, would b nrsng o dsgn a fully ngrad NDC modl wh rrmn and prmann dsably. An NDC schm s wdly dfnd as a PAYG schm ha dlbraly mmcs an FDC schm. In mos counrs wh mandaory ndvdual capalzaon couns (Rys (2010)), dsably nsuranc s fully ngrad no h FDC schm. A h sam m, cordng o Auor & Duggan (2006), OECD (2010) and Burkhausr al. (2013), dsably nsuranc s a bg challng for polcy makrs oday. Hnc, gvn ha NDC schms hav posv faurs ha could hlp o mprov h ffcncy of dsably nsuranc, would b usful o dvlop a horcal modl ha fully ngrad h dsably conngncy no an NDC framwork. Th mhodology dvlopd by Vnura-Marco & Vdal-Mlá (2014) could b a rfrnc for dsgnng hs ngrad modl. Fnally, nsuranc nnovaon could b ncorporad no h modl, as proposd by Muraugh al (2001) and Brown & Warshawsky (2013) for fundd sysms, by ngrang rrmn and long-rm car (LTC) annus. Th NDC framwork could b usful for hs purpos. Ths suggson sms from h f ha LTC as a conrbuory conngncy has bn provdd n h Grman conrbuory pnson sysm (Rohgang (2010)) snc h md-1990s. Barr (2010) also gvs sound rasons for xndng socal scury o provd mandaory covr for LTC. 15

16 5. Rfrncs Arhur, W.B. & McNcoll, G. (1978). Samulson, Populaon and Inrgnraonal Transfrs. Inrnaonal Economc Rvw 19(1), Aurbh, A. J. & R. D. L (2011). Wlfar and Gnraonal Equy n Susanabl Unfundd Pnson Sysms. Journal of Publc Economcs, 95 (1-2), Aurbh, A. J. & R. D. L (2009). Noonal Dfnd Conrbuon Pnson Sysms n a Sochasc Conx: Dsgn and Sably. In J. Brown, J. Lbman & Ws, D. (Eds), Socal Scury Polcy n a Changng Envronmn (chapr 2). Unvrsy of Chcago Prss: Chcago. Auor, D & M. C. Duggan (2006). Th Growh n h Socal Scury Dsably Rolls: A Fscal Crss Unfoldng. Journal of Economc Prspcvs, 20 (3), Barr, N. (2010). Long-rm Car: A Suabl Cas for Socal Insuranc. Socal Polcy and Admnsraon, 44 (4), Barr, N & Damond, P. (2009). Pnson Rform: A Shor Gud. Oxford Unvrsy Prss. Nw York. Boado-Pnas, M. C. & C. Vdal-Mlá (2014). Nonfnancal dfnd conrbuon pnson schms: s a survvor dvdnd ncssary o mak h sysm balancd? Appld Economcs Lrs, 21(4), Boado-Pnas, M.C. & Vdal-Mlá, C. (2012). Th Acuaral Balanc of h Pay-As-You-Go Pnson Sysm: h Swdsh NDC modl vrsus h US DB modl. In Holzmann, R., E. Palmr & Robalno, D. (Eds.), NDC Pnson Schms n a Changng Pnson World, Volum 2 (chapr 23, pp ). Washngon D.C.: World Bank. Boado-Pnas, M.C, Valdés-Pro, S. & Vdal-Mlá, C. (2008). An Acuaral Balanc Sh for Pay-As-You-Go Fnanc: Solvncy Indcaors for Span and Swdn. Fscal Suds, 29, Börsch-Supan, A. H. (2006). Wha Ar NDC Sysms? Wha Do Thy Brng o Rform Srags? In Holzmann, R. & Palmr, E. (Eds), Pnson Rform: Issus and Prospcs for Noonal Dfnd Conrbuon (NDC) Schms (chapr 3, pp ). Washngon, DC: World Bank. Brown, J. & M. Warshawsky (2013). Th lf car annuy: a nw mprcal xamnaon of an nsuranc nnovaon ha addrsss problms n h marks for lf annus and longrm car nsuranc. Th Journal of Rsk and Insuranc, 80, (3), Burkhausr, R., M. C. Daly, D. McVcar & R. Wlkns (2013). Dsably Bnf Growh and Dsably Rform n h U.S.: Lssons from Ohr OECD Naons, Fdral Rsrv Bank of San Francsco. Chłoń-Domńczak, A., Franco, D. & Palmr, E. (2012). Th Frs Wav of NDC Takng Sock Tn Yars plus Down h Road. In Holzmann, R., Palmr, E. & Robalno, D. (Eds)., NDC Pnson Schms n a Changng Pnson World, Volum 1 (chapr 2, pp ) Washngon, DC: World Bank. Damond, A. (2006). Dscusson of Concpualzaon of Non-Fnancal Dfnd Conrbuon Sysms. In Holzmann, R. & Palmr, E. (Eds), Pnson Rform: Issus and Prospcs for Noonal Dfnd Conrbuon (NDC) Schms (chapr 5, pp ). Washngon, DC: World Bank. 16

17 Holzmann, R. & Palmr, E. (2012). NDC n h Tns: Lssons and Issus. In Holzmann, R., Palmr, E. & Robalno, D. (Eds), NDC Pnson Schms n a Changng Pnson World, Volum 1, par I, Takng Sock of Lssons and Issus (chapr 1, pp. 3-30). Washngon, DC: World Bank. Holzmann, R., Palmr, E. & Robalno, D. (2012). Non-fnancal Dfnd Conrbuon Pnson Schms n a Changng Pnson World: Vol. 1, Progrss, Lssons, and Implmnaon. Washngon, DC: World Bank. Holzmann, R. & Palmr, E. (2006). Pnson Rform: Issus and Prospcs for Noonal Dfnd Conrbuon (NDC) Schms. Washngon, DC: World Bank. Krzr, B. A., S. J. Kay, & T. Snha (2011). Nx Gnraon of Indvdual Accoun Pnson Rforms n Lan Amrca. Socal Scury Bulln, 71(1), 20 (3), L, R. (1994). Th formal dmography of populaon agng, ransfrs, and h conomc lf cycl. In Marn, L.G. & Prson, S.H. (Eds), Dmography of Agng (pp. 8-49). Washngon, DC: Naonal Acadmy Prss. Muraugh, C., B. Spllman, & M. Warshawsky (2001). In Scknss and n Halh: An Annuy Approh o Fnancng Long-Trm Car and Rrmn Incom. Journal of Rsk and Insuranc, 68 (2), OECD (2010). Scknss, dsably and work: Brakng h barrrs. A synhss of fndngs ross OECD counrs. Pars: OECD. Palmr, E. (2006). Wha s NDC? In Holzmann, R. & Palmr, E. (Eds.), Pnson Rform: Issus and Prospcs for Noonal Dfnd Conrbuon (NDC) Schms (chapr 2, pp.17-33) Washngon, DC: World Bank. Palmr, E. (2012). Gnrc NDC: Equlbrum, Valuaon and Rsk Sharng. In Holzmann, R., Palmr, E. and Robalno, D. (Eds), NDC Pnson Schms n a Changng Pnson World Volum 2, par III: Solvncy, Lqudy, and Sably of NDC schms (chapr 19, pp ). Washngon D.C.: World Bank. Rys, G. (2010). Mark dsgn for h provson of socal nsuranc: h cas of dsably and survvors nsuranc n Chl. Journal of Pnson Economcs and Fnanc, 9 (3), Ross, J. (2004). Undrsandng h Dmographc Dvdnd. USAID. POLICY Projc, Fuurs Group. Rohgang, H. (2010). Socal nsuranc for long-rm car: an valuaon of h Grman modl. Socal Polcy and Admnsraon, 44(4), Srgrn, O. & Mkula, B.D. (2005). Th Ra of Rurn of Pay-As-You-Go Pnson Sysms: A Mor Ex Consumpon-Loan Modl of Inrs. Th Journal of Pnsons Economcs and Fnanc, 4 (2), Th Swdsh Pnson Sysm. Orang Annual Rpor (2013). Ed. Gudrun Ehnsson, Swdsh Pnsons Agncy (Pnsonsmyndghn), Sockholm. Vnura-Marco, M. & C. Vdal-Mlá (2014). An Acuaral Balanc Sh Modl for Dfnd Bnf Pay-As-You-Go Pnson Sysms wh Dsably and Rrmn Conngncs. ASTIN Bulln 44 (2), Vdal-Mlá, M.C. & Boado-Pnas, M.C. (2013). Complng h Acuaral Balanc for Pay- As-You-Go Pnson Sysms. Is Br o us h Hddn Ass or h Conrbuon Ass? Appld Economcs, 45:10,

18 Vdal-Mlá, C., Boado-Pnas, M.C. & Srgrn, O. (2009). Auomac Balanc Mchansms n Pay-As-You-Go Pnson Sysms. Th Gnva Paprs on Rsk and Insuranc: Issus and Prc. 33 (4), Vdal-Mlá, C., Boado-Pnas, M.C. & Srgrn, O. (2010). Insrumns for Improvng h Equy, Transparncy and Solvncy of Pay-As-You-Go Pnson Sysms: NDCs, ABs and ABMs. In Mcocc, M., Grgorou, G.N. & Masala, G.B. (Eds), Pnson Fund Rsk Managmn. Fnancal and Acuaral Modllng (chapr 18, pp ). Chapman & Hall/CRC Fnanc Srs. Vdal-Mlá, C., Domínguz-Fabán, I. & Dvsa-Carpo, J.E. (2006). Subjcv Economc Rsk o Bnfcars n Noonal Dfnd Conrbuon Accouns (NDCs). Th Journal of Rsk and Insuranc 73, Whhous, E.R. (2010). Dcomposng Noonal Dfnd-Conrbuon Pnsons: Exprnc of OECD Counrs Rforms. OECD Socal, Employmn and Mgraon Workng Paprs, No. 109, OECD Publshng. Wlls, R. (1988). Lf Cycls, Insuons and Populaon Growh: A Thory of h Equlbrum Inrs Ra n an Ovrlappng-Gnraons Modl. In L, R.D., Arhur, W.B. & Rodgrs, G. (Eds), Economcs of Changng Ag Dsrbuons n Dvlopd Counrs (pp ). Oxford: Clarndon Prss. Wllamson, J.B. (2004). Assssng h Pnson Rform Ponal of a Noonal Dfnd Conrbuon Pllar. Inrnaonal Socal Scury Rvw, 57 (1),

19 Appndx 1: Proofs of som formulas/quaons ncludd n Scon Fnancal and dpndncy raos n h maur sa. Onc h maur sa s rhd, h rao bwn h numbr of pnsonrs and h numbr of conrbuors - (dr ) - sablzs bcaus boh groups volv (ncras or dcras) xly qual o ra γ: dr PnsonrsR wx A1 k0 N(x Ak, ) ( 1γ) 1 ( 1 γ) N(x +k, 1) k0 Conrbuors C wx A1 N(x Ak, ) ( γ) -1k 1 1 k0 A1 A1 k0 N (x +k, ) k dr 1... dr Also, h sysm's avrag rplmn ra, xprssd by h fnancal rao, s alrady consan du o h f ha h numraor and dnomnaor volv qually (a h ra of varaon n wags): R C 11. fr Expndur on pnsons P (x A, ) w1 x A k N (x Ak, ) ( 1γ) k 0 Pnsonrs R Aggrga conrbuon bas A1 k 0 y w1 x A (x +k, ) A1 k 0 N N N(x +k, ) k 0 C Conrbuors (x Ak, ) (x +k, ) F k P W P W fr Rlaonshp bwn conrbuon ras. Th rlaonshp bwn h crdd conrbuon ra and h balancd ra cordng o formulas [4] and [6] s: A1 y(x+k, ) k0 N (x+k, ) A1 Aggrga conrbuons Ak θa N y ( G) (x k, Ak) (x k, Ak) 1 A1 k0 λ N(x A, ) ax A θ y(x +k, ) N λ (x+k, ) N(x A,) a x A k0 Expndur on pnsons 13. Thrfor s asy o s ha θ θ. a Th amoun of h pnson gnorng h survvor dvdnd s calculad as follows: 19

20 P (x A, ) K (x A, ) θ a A1 k 0 y (x k, Ak ) a λ x A ( 1 G) Ak A c1 P (x A, c, ) N N (x A, ) (x A, c, ) 14. whr P s h avrag pnson, whou akng no coun h survvor dvdnd, of (x A, ) an ndvdual who rrs a h ordnary ag a. I s a wghd avrag pnson dpndng on h A dffrn pnsons ha can b awardd. Thn, P Aggrga conrbu ons w1x-a A1 * (x y(x+k, ) N(x+k, ) k 0 k N A, ) (x Ak, ) F θ k 0 Expndur on pnsons and subsung h xprsson for P (x A, ), w g: 15. θ a θ * K K (x A,) (x A,) θ * Dvdnd ffc D (x A,) K (x A,) Th ncras n lf xpcancy a rrmn ag, Δx A f h dvdnd wr no dsrbud. If h survvor dvdnd s ncludd: P (x A, ) K (x A, ) D(x A,) λ a x A Howvr, gvn h assumpon ha λ x A K a (x A, ) λ x A P (x A, ) λ G, can b shown ha: wx A1 wx A1 k F k px A k px A 1 k 0 k 0, ha could b fnancd K a (x A, ) λ x A x A 17. a 18. whr x A s h cura xpcaon of lf for an ndvdual agd x +A,.. h xpcd numbr of compl yars rmanng for an ndvdual agd x +A o lv. Hnc formula [19] should ncssarly b fulflld o nuralz h survvor dvdnd: whr: Thrfor: P (x A, ) P K(x A, ) (x A, ) f andonly f P(x A, ) ' x A ' x 20. A x A ' Δ x A x A x A 20

21 wh Δ bng h ncras n lf xpcancy a h ordnary rrmn ag, masurd n x A yars, ha would nuralz h ffc of h survvor dvdnd. 21

22 Tabl 1: NDC schm wh survvor dvdnd: som slcd valus. Dscrpon varabl Noaon PL LT SW Crdd ra=balancd ra %( θ a θ ) Balancd ra whou SD % θ * Dvdnd ffc % D Dmographc rao % dr Fnancal rao % fr Annuy dvsor λ Lf xpcancy Chang n lf xpcancy x a A x (yars) x A Δ (yars) A Rplmn ra wh SD % β (x A, ) Rplmn ra whou SD % β (x A, ) Rrmn ag x +A (yars) 16+49= Bas scnaro wh G=(1.016)(1.00)-1=

23 Tabl 2: NDC sysm wh survvor dvdnd and populaon changs: som slcd valus. Dscrpon varabls Noaon PL + PL - PL Balancd ra whou SD %( θ a θ ) Dvdnd ffc % θ * Crdd ra=balancd ra % D Dmographc rao dr % Fnancal rao Annuy dvsor Rplmn ra wh SD fr % λ a A x % β(x A, ) Rplmn ra whou SD % β (x A, ) Avrag yars conrbud AYC Susanabl schm s ra of rurn % G

24 Zro populaon growh Conrbuors and pnsonrs 100,000 90,000 80,000 70,000 60,000 50,000 40,000 30,000 20,000 10,000 Sam wag srucur by ag Conrbuors Ag Pnsonrs Dffrn pnson srucur by ag 1,600 1,400 1,200 1, Wags and pnsons SW PL LT SW w&p PL w&p LT w&p Fgur 1: Srucur of conrbuors, pnsonrs, wags and pnsons undr dffrn moraly abls. Zro populaon growh Conrbuors and pnsonrs 100,000 90,000 80,000 70,000 60,000 50,000 40,000 30,000 20,000 10,000 Sam wag srucur by ag Conrbuors Ag Pnsonrs Dffrn pnson srucur by ag 1,600 1,400 1,200 1, Wags and pnsons SW PL LT SW w&p PL w&p LT w&p Fgur 1: Srucur of conrbuors, pnsonrs, wags and pnsons undr dffrn moraly abls. 24

25 21% 18.32%, dvdnd ffc wh zro populaon growh PL wh populaon growh From 33 yars of conrbuons onwards h dvdnd ffc s hghr % 19.77% Dvdnd ffc 18% 15% 38 yars of conrbuons. 42 yars of conrbuons % 12% Yars of conrbuons 0.01, yars 0.02, yars 0.04, yars PL Fgur 2: Dvdnd ffc for PL wh populaon growh by yars of conrbuons. 21% 18.32%, dvdnd ffc wh zro populaon growh PL wh populaon growh From 33 yars of conrbuons onwards h dvdnd ffc s hghr % 19.77% Dvdnd ffc 18% 15% 38 yars of conrbuons. 42 yars of conrbuons % 12% Yars of conrbuons 0.01, yars 0.02, yars 0.04, yars PL Fgur 2: Dvdnd ffc for PL wh populaon growh by yars of conrbuons. 25

26 IRR for conrbuors who rh rrmn ag 5.50% 5.41% 5.00% 3.74% 4.50% 4.76% IRR 4.00% 3.50% 3.41% 3.49% 2.97% 3.00% 2.50% 2.00% 1.50% Susanabl rurn of h sysm = G =2.62% Workng nry ag SW PL LT G Fgur 3: Expcd IRR (x A, A-K, ) akng no coun h survvor dvdnd, wh γ=0.01 and by ag of nry o h labour mark. IRR for conrbuors who rh rrmn ag 5.50% 5.41% 5.00% 3.74% 4.50% 4.76% IRR 4.00% 3.50% 3.41% 3.49% 2.97% 3.00% 2.50% 2.00% 1.50% Susanabl rurn of h sysm = G =2.62% Workng nry ag SW PL LT G Fgur 3: Expcd IRR (x A, A-K, ) akng no coun h survvor dvdnd, wh γ=0.01 and by ag of nry o h labour mark. 26

27 5.50% Evoluon of IRR for conrbuors wh "A" yars conrbud 5.00% 4.50% CASE 2 IRR 4.00% 3.50% CASE % 2.50% 2.00% 1.50% Ag aand SW PL LT G SW* PL* LT* Fgur 4: Evoluon of IRR(x s, ) by ag aand, wh γ=0.01, for conrbuors who jond h labour mark a ag % Evoluon of IRR for conrbuors wh "A" yars conrbud 5.00% 4.50% CASE 2 IRR 4.00% 3.50% 3.00% 2.50% 2.00% 1.50% CASE Ag aand SW PL LT G SW* PL* LT* Fgur 4: Evoluon of IRR(x s, ) by ag aand, wh γ=0.01, for conrbuors who jond h labour mark a ag In Brazl, h NDC rform was adopd only for h prva scor. 2 S, for xampl, Lndbck & Prsson (2003), Wllamson (2004), Börsch-Supan (2006), Holzmann & Palmr (2006), Vdal-Mlá al (2006), Aurbh & L (2009), Vdal-Mlá al (2010), Whhous (2010), Aurbh & L (2011), Chłoń-Domńczak al (2012) and Holzmann al (2012). 27

28 3 I s mporan no o confus h concp of h survvor dvdnd (whch manly dpnds on moraly ras a spcfc ags) wh h so-calld dmographc dvdnd, Ross (2004), whch s lnkd o frly dynamcs. 4 S Krzr al (2011). On possbl way of dsgnng programmd or phasd whdrawal s whn h affla rcvs a pnson chargd on h balanc of hr ndvdual capalzaon coun whch, by rmanng undr h rsponsbly and managmn of h admnsraor, allows h rr o bnf from h rurn on h fund. Th pnson s fxd for prods of on yar and h amoun s calculad by akng no coun h balanc of h ndvdual coun, h chncal ra of nrs dfnd by law and h lf xpcancy of h workr and hr famly cordng o lgal moraly abls. 5 In hs modl h rms dpndncy rao, old-ag rao and dmographc rao ar usd as synonyms gvn ha vrybody parcpas n h labour mark. 6 A conrbuor can rh rrmn ag as v or dsabld. In h cas of Swdn, h currn rgulaons on dsably pnson ar closly lnkd o h old-ag pnson sysm. Accordng o Chłoń-Domńczak al (2012), h Swdsh modl for rrmn pnson rghs for prsons rcvng dsably bnfs s o mpu and pay conrbuons for nsurd prods of dsably no h rrmn conngncy. Ths paymns, mad annually from gnral ax rvnus, ar nrd on h counry s couns as a cos for h dsably sysm and ar par of h ransfr from sa rvnus o h NDC pnson fund. Dsably bnfs ar convrd no rrmn bnfs a ag Only obsrvd moraly ras ar usd and no h populaon srucur by ags. 8 Yars wh h las nformaon avalabl cordng o h Human Moraly Daabas (hp:// Da cssd