Funded and unfunded systems: two ends of the same stick. Angrisani Massimo

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1 unded and unfunded sysems: wo ends of he same sick Angrisani Massimo Universià degli sudi di Roma "a Sapienza" acolà di Economia Diparimeno di maemaica per le Decisioni Economiche, inanziarie ed Assicuraive Via del Casro aurenziano 9, Roma hone: ax: Mail address: SUMMARY he aim of his aricle is o give evidence ha, if a ension und of he defined conribuion ype is brough back o a susainable level of liabiliy uncovered, as regard o he conribuion level, and yields yearly a revenue on he pension savings equalling he asse s ineres rae plus wage s growh rae respecively weighing as is funded and unfunded liabiliy componens on he oal pension deb, hen he und can guaranee susainabiliy hrough ime. Key words: pension sysem pay-as-you go funded sysem, social insurance

2 1. INRODUCION In 1966 Henry Aaron, following an indicaion by rof. Samuelson, proved he paradox of social securiy by which social insurance can increase he welfare of each person if he sum of he raes of growh of populaion and real wage exceeds he rae of ineres. Since hen he rus in he pay-as-you go pension sysem has largely grown, unil a fall in he demographic level ook place in almos all wesern counries, proving he unsusainabiliy, in he medium long erm, of such sysem a he curren benefi levels. he answer o he demographic problem has been he inroducion of new funded sysems hrough a second and hird pillar, permiing he firs pillar o progressively reduce is benefis which would be reinegraed by he new wo-pillar benefis. he aim of he presen aricle is o prove ha in a defined conribuion ension und, of payas-you-go ype, negaive demographic effecs can be opposed hrough he inroducion in he same sysem of a funded componen, and ha i is possible o run pension obligaions hrough a sor of mixed financial managemen capable of securing susainabiliy and more adequae benefis. In his way also he older worker generaions, who will have a good level of benefis, if compared o conribuion amouns, made possible by a favourable demographic facor, will paricipae in refunding. he new scheme will favour inergeneraional equiy in he conribuion o benefi raio, hus fulfilling he ask of solidariy on which he pay as you go pension sysem ress. he funded componen, ha we can define a differenial reserve, mus guaranee he sysem s susainabiliy wih an accepable conribuion level and, besides, providing an ineres rae, may guaranee more adequae pension saving reurns even if he populaion growh rae falls. his aricle highlighs he fac ha boh ypes, he funded and pay-as-you-go financial managemens, are no o be seen aniheical, as i commonly happens, bu working as he wo ends of he same sick, in a coninuum. I poins ou, moreover, ha he prevailing of eiher he one or he oher scheme mus follow a more favourable expeced developmen in wages or in asse ineres rae. he approach o he problem consiss in considering he sysem s oal liabiliy (composed of is laen par owards workers and is curren par owards pensioners) as he resul of wo componens, one covered by he asse, herefore funded, and he remaining par uncovered herefore unfunded. In his respec wo main hypoheses are formulaed. 1

3 he former concerns he liabiliy s unfunded componen which, respec o wages, mus be consisen wih he amoun of he conribuion level. In ha sense we speak of a susainable level of unfunded pension liabiliy. he laer concerns he revenue on he pension savings. e assume ha he revenue in quesion equals each year he convex combinaion of he asse s ineres rae and wage s growh rae which weigh as he funded and unfunded componens of oal liabiliy. e prove ha, if he sysem is brough back o a susainable level of unfunded pension liabiliy by inroducing an adequae funded componen and he revenue on he pensions savings is consisen wih he ypology of he pension liabiliy in he sense specified, hen is asse will never be zeroed and hence he sysem will be solven hrough ime. In Ialy a funded componen, subsanial bu no sufficien, is already presen in he liberal professions social securiy sysems: he arge is o make i srucural also by ransforming he sysems from defined benefis o defined conribuions. 2.SUSAINABIIY HEOREM All he funcions here conained are supposed o be coninuous and, when necessary differeniable in, + e s consider he asse s ), conrol ime inerval of he heorem. evoluion course resuling from is ineres rae a ime, from conribuions (which we consider already free from he und s managemen coss) and from pension obligaions. e indicae by: r = he asse insan ineres rae a ime, (on a yearly basis). C = Insan flow of conribuions a ime, (on a yearly basis) () = Insan flow of pensions a ime, (on a yearly basis) e assume: = () r + C () (1) 2

4 e s wrie he evoluion equaion of he ension und s oal liabiliy. e have: a = + where a = ension liabiliy a ime owards workers (in aciviy), laen deb. = ension liabiliy a ime owards pensioners, curren deb. e indicae: a = Insan flow of pension liabiliy (on a yearly basis) a ime which urns from laen ino curren. r = Insan ineres credied by he und a ime on pension liabiliy (boh curren and laen) on a yearly basis. e observe ha indicaes, as o workers, he revaluaion of oal pension savings while, as o pensioners, i indicaes he revaluaion rae of heir pensions. e s suppose, in fac, ha he iniial pension be compued in conribuion erms on he basis of acuarial equilibrium beween conribuions and pensions a a pre-paid ineres rae, echnical rae, equal o zero. he iniial pension herefore resuls from oal pension savings a reiremen ime divided by he annuiizaion divisor a zero echnical rae which herefore, in case a survivor pension is no paid ou, coincides wih his remaining average life expecancy a ha ime. r e s wrie he evoluion equaion of he pension deb componen relaive o workers: a a a = r + C In a parallel way, as o curren deb owards pensioners, i issues: a = r + () As o oal deb, we have he following evoluion equaion: () = r + C () (2) he above formula (2) provides he evoluion of he und s oal pension liabiliy, apar from accidenal deviaions, herefore un-sysemaic, in a conribuion ype pension scheme based 3

5 on an acuarial conribuion - pension equilibrium, provided ha demographic evaluaions are correc. e poin ou ha, in our assumpions, he whole conribuion amoun urns ino und pension liabiliy, ha is ino pension benefis for subscribers, including, for example, also he conribuions of hose who died in he course of heir working years. e indicae by ν he quoien a ime beween oal pension liabiliy and curren debi, i.c. ν = hence ( ) = ; i issues ν ( ) 1 ν If we furhermore suppose, in order o simplify he maer wihou losing he overall view, ha pension benefis are no exended o survivors 1, we ake: γ = weighed average residual life expecancy a ime relaive o he pensioners populaion. he weigh relaive o each pensioner equals ha of his pension benefi on ha of curren pensions whole number. I issues: 1 = γ and herefore 1 [( γ ) ( ) ] ( ) = ν e define he unds s unfunded pension liabiliy is uncovered par, ha is: AYG = ( ) ( ( ) ( )) If we suppose ha he populaion of he insured is made up of workers wih a wages of heir own, le s pu: β () () AYG = () where = Insan flow of wages a ime (yearly basis), ( ) 0, for e s furhermore define, as o conribuions α = conribuions rae a ime ( α ( ) > 0) >. 4

6 aking ino accoun he hypohesis ( ) > 0, for, we prove he following: heorem e s suppose ha a ime resuls ( ) > 0 and ha he unfunded liabiliy proves susainable, as regard o he conribuion level, in oher erms i saisfies he condiion: a) γ β ( ) ν ( ) α for : he following condiion holds: b) he und pays a pension liabiliy wih an ineres rae equalling: r = r() + for hen, if saring from ime zeroed, i.e. > 0, > Evidence: a conribuion rae ( ) α ( ) α = is applied, he asse can be e s firs verify ha, if saring from ime he insan ineres rae on pension liabiliy is provided by (b), hen i follows: r paid by he und β ( ) β = where > Since AYG β =, e verify ha, for β = e have AYG() = 0 5

7 AYG = ()= (as a resul of (2) and (1)) r + C ( ) ( ) r( ) C( ) + ( ) = = = r ()( r ) e observe ha, from (b), i also issues: for r () r = ( ) As a resul of wha indicaed above, i follows: AYG AYG = 0 and herefore AYG = 0, for for β = for >. and hence β ( ) e reformulae (1) expressing conribuions by means of wage and conribuion rae, ha is C ()= α and, besides, wriing pension benefis saring from he oal pension liabiliy, we have: = () r + C () 1 = () r + α γ ()= = = [ ]= 1 () r + α ( γ ν ( ) ) ( ) [ ] ( ) ( ) 1 1 () r + α ( γ ν ) ( ( ) ( ) ) ( γ ν ) ( ) = [ ] γ ( ) ( ) () r + α ( ) ( γ ( ) ν ( ) ) ( ) = ν Replacing AYG wih β (), we have: [ ] + α() = r ( γ ν ) ( ) ( ) 1 AYG 1 β ( ) () () ν 1 (1 bis) γ 6

8 Considering ha from (b), follows ( ) β ( ) he conribuion rae ( ) α( ) = r() β = for >, and ha, saring from ime, α = is applied, i issues ha: β ( ) () () ν 1 + α( ) for γ ν γ ih iniial condiion, for =, we obain: 1 τ 1 ( ) ( ) r τ dτ r s ds γ( τ) ν( τ) γ( s) ν( s) β( ) ( ) = e ( ) + e ( τ) α( ) dτ γτντ ()() aking ino accoun hypohesis a), formula (3) implies ha he asse ( ), acive in, can be zeroed. e can say he asse ( ) is a differenial reserve, respec o he conribuion level ( ) for he und a ime if hypohesis a) is verified. OBSERVAION In heorem, we can apply a non consan conribuion rae α () ha verifies his condiion γ β ( ) ν α for. In his way, we can eliminae he hypohesis a) and however he heorem hesis works. for (3) α, 3. CONCUSIONS In his paper we prove ha i is possible o ensure he equilibrium of defined conribuion pension sysems wih a componen of pension deb unfunded if, wih respec o wages, he unfunded componen is susainable as regard o he conribuion level and he sysems provide an ineres rae on pensions savings coheren wih he funded and unfunded deb componens. 7

9 γ equals he average weighed remaining life expecancy of all pensions in course a ime, aking ino accoun, as regards primary pensions, also he expeced duraion of he correlaed survivor benefi and is possible inferior amoun wih respec o he primary pension benefi. he weigh relaive o each pension equals ha of his pension benefi on ha of curren pensions whole number. 1 hen also a survivor pension is paid ou in addiion o a primary pension, ( ) 8

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