NDC Dynamic Equilibrium model with financial and

Transcription

1 9 July 009 NDC Dynamc Equlbrum modl wh fnancal and dmograhc rsks rr DEVOLDER, Inmaculada DOMÍNGUEZ-FABIÁN, Aurél MILLER ABSTRACT Classcal socal scury nson schms, combnng a dfnd bnf hlosohy and a ay as you go sysm, ar clarly undr hra akng no accoun h gnral dmograhc voluon of many counrs for h nx dcads An nrsng am o solv hs roblm s o manan h ay as you go mchansm bu movng o a dfnd conrbuon sysm noonal accouns or NDC schms In ordr o mlmn such schms s ncssary o dfn varous aramrs such as h noonal ra, h annuy convrson rc or h ndxaon rocdur All hs chocs ar no nural n rm of sably of h sysm Th uros of hs ar s o rsn a 3 gnraons-modl rmng o modl h nflunc of h dynamc voluon of h fnancal and h dmograhc aramrs on h qulbrum of a NDC sysm KEYWORDS Socal Scury, Noonal accoun, Dmograhc Rsk, Longvy, Acuaral Farnss Insu ds Scncs Acuarlls, Unvrsé Caholqu d Louvan, 6 ru ds Wallons, 348 Louvan la Nuv, Blgum Darmn of Fnancal Economy, Unvrsy of Exrmadura, San

2 INTRODUCTION oulaon agng and dcrasng of frly hav gnrad nd o rform socal scury sysms of nson basd on a AYG chnqu In dd h combnaon of a dfnd bnf hlosohy wh a AYG chnqu lads clarly o a major fnancal dsasr n cas of dmograhc crss A frs ossbl soluon o hs fuur challng s o swch from AYG o a fundd rgm whr acuaral qulbrum s a naural consqunc of h mchansm Bu shfng for a socal scury sysm from AYG o full fundng s no so sml and can also lad o dangrous accumulaon of fnancal rsk Anohr romsng rform sms o b h NDC schm, whch has rcnly bn adod n varous counrs as Ialy, Lava, oland and Swdn Svral rsarchs [Valds ro 999, Schrman 999 and Aurbach and L 007] hav shown howvr ha NDC mchansm, as ald n racc, dos no mly drcly qulbrum n a conx of fnancal and dmograhc uncrany Mor rcsly, rsn condons of alcaons of NDC canno guaran auomac fnancal sably wh dmograhc and longvy rsk If h auomac fnancal sably s h ably of a nson lan o adjus o fnancal shocks whou lgslav nrvnon Valdés-ro, 999, w wll show n hs work how h auomac fnancal sably s ossbl n h NDC n a modl wh dffrn gnraons and akng accoun fnancal, dmograhc and longvy rsks n a drmnsc framwork Th ar s organzd as follows In scon, w formalz h dmograhc modl wh 3 gnraons and comu n a sac nvronmn h NDC qulbrum Thn scon 3 movs o a dynamc modl whr h 3 man dmograhc and fnancal rocsss can chang on m In scon 3 w comu an qulbrad NDC schm assumng a sam amoun of nson for all gnraons of rrs Scon 4 gnralzs hs aroach by consdrng gnraon nson In arcular, w rsn varous qulbrad schms basd on dffrn chocs of h 3 man aramrs of a NDC sysm Non qulbrad chocs ar also rsnd Scon 5 concluds h ar

3 STATIC MODEL WITH 3 GENERATIONS INTRODUCTION W wll consdr hr a dynamc modl of oulaon basd on 3 gnraons: on gnraon of acv ol and wo gnraons of rrd ol Tradonally, s mor usual o work only wh gnraons n an ovrlang modl of nson [s for nsanc Vrmyln 007] Nvrhlss, n h conx of NDC and n ordr o caur h ffc of ndxaon of nson afr rrmn hs ffc bng on of h ky aramrs n a NDC schm, aars ssnal o nroduc a las wo gnraons of rrs So w suos ha a ag y ol ar nrng h oulaon as acv mmbr and rcv a salary, on whch a nson conrbuon s comud A ag y +, ol ar rrd and rcv hr frs nson amoun A ag y +, h survvors rcv a scond nson amoun, basd on an ndxaon rul afr rrmn A ag y + 3, hr s no survvor n hs gnraon and no ohr aymn occurs y y + y + y + 3 Acv rod Rrmn rod In hs oulaon, w nroduc a NDC nson lan Th uros of hs scon s o look a qulbrum condons basd on h aramrs of h nvronmn NOTATIONS AND ASSUMTIONS W nroduc h followng dmograhc lmns: L x, : numbr of ol rsn a ag x a m x y, y +, y + E L y, : numbr of ol nrng h oulaon a m W gnor moraly bwn ag y and ag y + lk n h classcal NDC aradgm whr moraly s no akn no accoun n h acv ar of h carr 3

4 Evoluon of h acv oulaon: f s h consan annual ra of ncras of h nranc funcon of h oulaon, h ol nrng h oulaon a m ar gvn by: E E0 Evoluon of h ndvdual wags: L us dfn h annual ra of ncras of h salary, h ndvdual wags a m s s0 Evoluon of h global wag mass: S + S 0 Survval robably bwn ag y and ag y + : Indvdual conrbuon: h conrbuon ra o h NDC lan s consan, so w hav: c π s W ar workng wh 3 moran aramrs: ncras of ndvdual salary, growh of h oulaon and longvy ffc hrough h survval robably Takng no accoun h saonary assumons of hs modl, w assum ha h gnraons of rrs wll rcv a m h sam amoun nson Ths assumon wll b rlaxd lar whn w wll consdr dynamc fnancal and dmograhc modls 3 EQUILIBRIUM CONDITION W wll aly on hs modl h classcal rul of ay as you go AYG: Incom of h yar Oucom of h yar sarng from h conrbuons and comung h lvl of nson gnrad by hs AYG arrangmn Thn w wll nrr h amoun of nson n rms of a noonal accoun formula 4

5 Incom for h yar : C π S0 Oucom for h yar : Toal numbr of nsonrs: L E + So h qulbrum rlaon of AYG bcoms now: L C W g for h nson amoun o ay: 0 πs0 E0 + πs0 + + πs0 + Th rlacmn ra RR s gvn by: RR π s + whch s an ncrasng funcon of h dmograhc ra, of h conrbuon ra π and a dcrasng funcon of h survval robably Ths formula of nson can b sn as a noonal accoun formula Th nson s ad a ol jus rrd who wr acv ol a m If w aly h radonal noonal accoun formula, coms: nson conrbuon x rvalorzaon c R annuy a Th conrbuon s qual o: c π s0 So a naural dnfcaon o a NDC dsgn can b dscrbd as follows: h rvalorzaon s mad usng h oal growh of salars: + R h annuy s also consan and gvn by: a + 3 Ths arcular choc nducs a scal form for h ndxaon rocss For hs w hav o comar wh knowng ha s gvn by: 5

6 π s0 + + So f w comu h rao / w oban h rul of ndxaon n qulbrum: R 4 4 OTIMAL DESIGN OF A NDC SCHEME A hs sag, n sac condons, h concluson n a drmnsc nvronmn can b summarzd as follows: f w wan an xac qulbrum n NDC, w can choos h followng aramrs: Rvalorzaon of h salars bfor rrmn o comu h frs nson: oal growh of salars ndvdual growh + growh of oulaon nson ndxaon afr rrmn: ndvdual growh of salars Comuaon of h annus: wh a dscoun ra qual o h oulaon growh and usng h survval robably Ths arcular dsgn s gnrally calld h canoncal choc Bu ohr combnaons of h 3 rocsss ar ossbl For nsanc w could ado a sam valu for h coffcn of rvalorzaon of salars and h ndxaon of nsons, corrsondng o h ncras of salars: Thn rlaon gvs h comabl valu of h annuy: R R 5 a + 6 In gnral, n hs modl h valu of h nson ndxaon rocss s fxd and gvn by formula 4 Th ohr rocsss R and a can b chosn mor arbrarly bu conssnly wh consran 4 and wll gnra fnally h sam nson If w choos a gnral rvalorzaon rocss gvn by: 6

7 R hn h annuy convrson mus b qual o: α α a + 3 DYNAMIC MODEL WITH UNIFORM ENSION 3 ASSUMTIONS Th basc modl roosd n scon was basd on consan and sady condons for h fnancal and dmograhc aramrs Hr w consdr ha hs aramrs can flucua from on yar o anohr Mor rcsly h growh of h acv oulaon dnod by and h growh of ndvdual wags dnod by ar now funcons of m: j : growh ra of h acv oulaon for yar j j,, j : growh ra of ndvdual wags for yar j j,, Th survval robably can also chang from on gnraon o anohr: j In hs scon w wll sll assum ha h gnraons of rrs wll rcv a m a sam amoun of nson soldary bwn h rrs Ths assumon, naural n scon akng no accoun h sably of h aramrs, can now b dscussd and wll b rlaxd n scon 4 Th INCOME for h yar can now b comud as follows: Evoluon of h acv oulaon E E0x Evoluon of h ndvdual wags s s0x Evoluon of h global wags S E s S 0x + Survval robably Indvdual conrbuon for h yar c π s0x 7

8 Toal conrbuon for h yar C π E0 s0x + Th OUTCOME for h yar s gvn by: Toal numbr of nsonrs L L + L Nw gnraon of rrd ol L E0 x Gnraon rrd h yar bfor L E0x nson for h nw and old rrd gnraons 3 EQUILIBRIUM CONDITION Usng h sam qulbrum mchansm as n scon 3, w g n hs dynamc modl h followng xrsson for h amoun of nson gnralzng formula : Th rlacmn ra bcoms: x + π s0x 3 + x RR s π + Onc agan hs valu can b nrrd as a noonal accoun formula In ordr o dfn a NDC nson schms w mus choos 3 rocsss: h coffcn of rvalorzaon of h noonal accoun R h annuy a h coffcn of ndxaon of h nsons R Thn h gnrad nson s gvn n our 3 rod modl wh unform nson by h followng xrssons, h frs on basd on h noonal convrson a rrmn and h scond on lnkng succssv nson amouns wh h ndxaon rul: 8

9 c R a 3 R 33 Clarly w hav an nfn numbr of ossbls of choc for h rocsss R, a and R n ordr o rlca h valu gvn by 3 rmanng conssn wh h NDC mchansm 3 and 33 A frs naural choc cohrn wh scon canoncal choc and drcly nsrd by 3 lads o h followng dnfcaon: 3 Frs dsgn canoncal dsgn W ak as coffcn of rvalorzaon h numraor of 3 and as annuy h dnomnaor of 3: R x + a + x Th rvalorzaon of h salars s mad usng h oal growh of salars Th annuy s comud usng as dscoun ra, h oulaon growh of h as and no of h currn yar! and as survval robably, h robably as obsrvd h yar of comuaon and no a rojcd valu for h consdrd rr gnraon! Usng formula 33 w oband h conssn coffcn of ndxaon of nson x + R x x + x a x x a If h dmograhc condons ar sac growh of h oulaon and longvy, hn hs sysm rms an ndxaon of h nson n ln wh h salary ncras: R x Ths quvalnc dsaars n a dynamc dmograhc modl For nsanc, f h dmograhc growh says consan bu f h longvy s ncrasng, hn:

10 + R x + < x 3 Scond dsgn unform ndxaon bfor and afr rrmn Onc agan, xacly as n scon, h canoncal choc gnras wo dffrn rocsss of ndxaon rscvly R bfor rrmn gvn by 34 and R afr rrmn gvn by 35 W could alrnavly ask a sam valu for hs wo ndxaon rocsss sam hlosohy as formula 5 Bcaus h valu of R s fxd by h rlaons 33 and 3, w ar only fr o chang h valu of h rocss R Th choc lads so o: x + R R x x + Thn h qulbrum formula 3 gvs h valu of h annuy: a Thrd dsgn rvalorzaon n ln wh salars W could also choos o us a rvalorzaon rocss n ln wh h growh of ndvdual salars cf 5 n h sac cas: R x Th nson ndxaon s sll gvn hn by 35 and h annuy bcoms now: a + + L us rmark ha n hs modl xacly as n scon hs dffrn dsgns ar jus for dnfcaon and nrraon uross bu gnra fnally h sam nson labls gvn a m by for all gnraons [nson amoun gvn n h 3 dsgns by formula 3] Th suaon wll b qu dffrn n h nx scon 0

11 4 DYNAMIC MODEL WITH GENERATION ENSION 4 ASSUMTION Insad of assumng as n scon 3 a unform amoun of nson for all h rrs a a sam momn, w wll now mov o a gnraon nson modl In a dynamc modl, h gnraons hav a dffrn hsory and also a dffrn fuur hrough changng survval robabls; so s now surrsng o gv hm dffrn amouns of nson W wll us h followng noaons: nson for h nw rrd gnraons a m : nson for h old rrd gnraons a m : 4 EQUILIBRIUM CONDITION Havng wo dffrn lvls of nson bu only on qulbrum rlaon h classcal AYG budg consran, w mus mos an addonal condon n ordr o dfn h schm For hs, w wll sar from a naural NDC formulaon for h nw nsons 4 Calculaon of h nson for h nw rrs Th nson of h nw rrs s comud alyng h ur noonal accoun calculus: c R a or: R s0x π 4 a whr h rvalorzaon rocss R and h annuy a mus b dfnd by h sysm 4 Calculaon of h nson for h old gnraon, h yar bfor Smlarly hs nson as bn comud h yar bfor followng h sam logc: 0x R π s 4 a 43 Calculaon of h nson for h old gnraon n In ordr o comu h nson for h old gnraon n, w us h AYG qulbrum condon:

12 C L L + So h amoun s gvn xlcly by: L L C and L L L C R a E E R s a E s E x x x x 0x x 0x π π Fnally w oban + x x R a 43 Ths rao comars h lvl of nson for h gnraons of rrs a a sam momn In scons and 3 hs rao was by dfnon qual o 44 Calculaon of h ndxaon rocss W can also comu h ndxaon rocss for h old gnraon; afr comuaon, coms: + + x x R a a a R R R ARTICULAR CHOICES OF EQUILIBRATED NDC ARAMETERS In ordr o dfn on arcular qulbrad schm w mus choos valus for h 3 basc rocsss R, R and a such as o fulfll rlaon 44 Thr ar an nfn numbr of such ossbl qulbrad NDC sysms Hr ar som ossbl cohrn schms Ohr ossbls can b off cours consdrd! L us rmark ha h dffrn chocs of aramrs ar hr mor han jus an dnfcaon xrcs as n scon 3; hy gnra now dffrn valus of nson and scally a dffrn rao of nson bwn h wo gnraons of rrs W wll rsn sx modls whch can b summarzd n h followng abl:

13 roscv robably rsn robably Basc cas Modl Modl 4 Lf xcancy Modl Modl 5 Salary Indxaon Modl 3 Modl 6 43 Modls wh roscv robabls W frs consdr 3 modls whr h annuy s comud usng roscv robabls 43 Modl canoncal choc h rvalorzaon a h oal growh of salars: R x + h annuy s gvn by: a + x 3 h ndxaon of nson s soluon of h qulbrum rlaon 44 Thn h rlacmn ra for h nw gnraon of rrs s gvn by: + RR π s + Th nson for h old rrs bcoms: + x + + x + + x + 45 x For h old gnraon, h ndxaon rocss R s gvn by : R + + x + a x x 46 + x a Th naural ndxaon by h growh of salary s corrcd by h rao of wo succssv annus and by h rao of wo succssv survval robabls 43 Modl lf xcancy h rvalorzaon a h oal growh of salars R x + h annuy s gvn by h lf xcancy: a + + 3

14 Th nson for h old gnraon s now: x + + x + + x + 47 Th ndxaon rocss bcoms: R + a x + 48 a 433 Modl 3 rvalorzaon wh salary growh h rvalorzaon a h ndvdual growh of salars: R x h annuy s gvn by: a + Th nson for h old gnraon s now: + x x Th ndxaon rocss has h followng mor comlcad form: R a + + x + a Modls wh rsn robabls Th 3 modls rsnd jus bfor can b dvlod wh annus comud wh rsn survval robabls nsad of roscv robabls; h 3 cass consdrd jus bfor bcoms hn: 43 Modl 4 canoncal choc h rvalorzaon a h oal growh of salars: R x + h annuy s gvn by: a + x 3 h ndxaon of nson s soluon of h qulbrum rlaon 44 Th nson for h old rrs bcoms: x x + + x + x Th ndxaon rocss R s gvn by : 4 4

15 R + x a x x 4 + x a Th naural ndxaon by h growh of salary s corrcd by h rao of wo succssv annus Rmark: fourh dsgn bs Ths fourh dsgn lads o dffrn nsons for h gnraons a a sam momn [cf 45] If w had chosn for h annuy h followng valu: + x a hn w would hav oband xacly h cas of unform nson dvlod n scon 3 [cf choc 3 4] 43 Modl 5 lf xcancy h rvalorzaon a h oal growh of salars R x + h annuy s gvn by h lf xcancy: a + Th nson for h old gnraon s now: x x + x Th ndxaon rocss bcoms: a R x + 44 a 433 Modl 6 rvalorzaon wh salary growh h rvalorzaon a h ndvdual growh of salars: R x h annuy s gvn by: a + Th nson for h old gnraon s now: x x + 45 Th ndxaon rocss has h followng mor comlcad form: R a + x + a 46 5

16 44 NON EQUILIBRATED SCHEMES AND STABILIZATION FUND Ohr dsgns could b consdrd whou h qulbrum condon Thn h sysm wll gnra ach yar a dfc or a gan whch can b ncororad no a sablzaon fund W dno by F h valu of hs fund a m F 0 wll rrsn h nal amoun avalabl n h fund Thn h rcursv voluon of h fund s gvn by: whr s h rurn of h fund F + F + C L + 47 W could for nsanc hnk a h followng dsgn: h rvalorzaon rocss bfor rrmn s qual o h growh of ndvdual salary: R x h annuy s gvn by s naural valu : a + 3º h ndxaon afr rrmn s h sam as n modl I s asy o show ha hs arcular choc s no mor soluon of h qulbrum rlaon CONCLUSION AND FURTHER RESEARCH In hs ar w hav dvlod an ovrlang modl of nson wh 3 gnraons A NDC nson sysm s manly basd on 3 rocsss: rvalorzaon of salars, valu of h annuy and ndxaon of h nsons Basd on h classcal quaon of AYG, w hav oband an qulbrum quaon for hs hr aramrs A lo of cohrn and dffrn schms hav bn rsnd For nsanc, a basc sysm wha w hav calld h canoncal choc consss of h followng choc: h rvalorzaon of salars follows h gnral voluon of h oal salars; h annuy s comud usng roscv robably and obsrvd dmograhc growh Thn w hav showd ha a naural ndxaon of nsons on ndvdual growh of salars was no qulbrad and has o b corrcd by wo dmograhc raos rflcng h chang n longvy Ohr combnaons of aramrs hav also bn consdrd Th mhodology usd n hs ar s basd on drmnsc scnaros Takng no accoun h fuur uncrany of h hr basc aramrs, movng o a sochasc modlzaon wll b an moran oc o b addrssd n h nar fuur 6

17 REFERENCES AUERBACH AJ and LEE RD 006: Noonal Dfnd Conrbuon nson Sysm n a Sochasc Conx, NBER Workng ar BOADO-ENAS M, DOMINGUEZ-FABIAN I and VIDAL-MELIA C 007: Noonal Dfnd Conrbuon NDC: Solvncy and Rsk n San, Inrnaonal Socal Scury Rvw, 60, 4 DISNEY R 999: Noonal Accoun as a nson sragy: an valuaon, Th World Bank HOLZMANN R, ORENSTEIN M and RUTKOWSKI 003: nson Rform n Euro: rocss and rogrss, h World Bank HOLZMANN R and ALMER, E 006: nson Rform: Issus and roscs for Non-fnancal Dfnd Conrbuon NDC Schms World Bank ublcaons ALMER E 999: Ex from h Labor Forc for oldr workrs: can h NDC nson Sysm hl?, Th Gnva ars on Rsk and Insuranc,, 4 SCHERMAN KG 999: Th Swdsh nson Rform, Inrnaonal Labour Offc, Gnva SETTERGREN O 00: Th Auomac Balanc Mchansm of h Swdsh nson Sysm, a non chncal nroducon, Naonal Socal Insuranc Board, Swdn TURNER J 004: Indvdual Accouns: Lssons from Swdn, Inrnaonal Socal Scury Rvw, 57 VALDES-RIETO S 999: Th Economcs of nsons, onfca Unvrsdad Caólca d Chl VALDES-RIETO S 000: Th Fnancal Sably of a Noonal Accoun nson, Scandnavan Journal of Economcs, 0 VERMEYLEN K 007: Th Ovrlang Gnraons Modl and h nson Sysm, wwwbookbooncom VIDAL-MELIÁ C; DOMÍNGUEZ-FABIÁN I and DEVESA-CARIO JE 006: Subjcv Economc Rsk o Bnfcars n Noonal Dfnd Conrbuon Accouns Journal of Rsk and Insuranc, 73, 3 WILLIAMSON JB 004: Assssng h nson rform onal of a noonal dfnd conrbuon llar, Inrnaonal Socal Scury Rvw, 57 7