Optimal Funding of a Defined Benefit Pension Plan

Opimal Funding of a Defined Benefi Pension Plan G. Deelsra 1 D. Hainau 2 1 Deparmen of Mahemaics and ECARES Universié Libre de Bruxelles (U.L.B.), Belgium 2 Insiu des sciences acuarielles Universié Caholique de Louvain (UCL), Belgium Dynsoch+ 2007 Amserdam June 7, 2007 Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 1 / 35

Ouline 1 Inroducion 2 The financial marke cash, a rolling bond and socks 3 Liabiliies and moraliy modelling Liabiliies modelling Moraliy modelling 4 Deflaor Financial, wage and acuarial deflaor Fair value of he liabiliies 5 The opimisaion problem The objecives The opimisaion problem 6 Soluion Opimal conribuion rae and arge wealh The bes replicaing sraegy 7 Numerical example 8 Conclusions Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 2 / 35

Inroducion Pension benefis in Belgium Social securiy funded by he sae: Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 3 / 35

Inroducion Pension benefis in Belgium Social securiy funded by he sae: Acive workers finance direcly reired individuals (iner generaional solidariy) No capialisaion Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 3 / 35

Inroducion Pension benefis in Belgium Social securiy funded by he sae: Acive workers finance direcly reired individuals (iner generaional solidariy) No capialisaion Pension plans funded by he employer Capialisaion Two main pension schemes: Defined benefis and Defined conribuions Asses managed by a life insurer or by a pension fund Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 3 / 35

Inroducion Pension benefis in Belgium Social securiy funded by he sae: Acive workers finance direcly reired individuals (iner generaional solidariy) No capialisaion Pension plans funded by he employer Capialisaion Two main pension schemes: Defined benefis and Defined conribuions Asses managed by a life insurer or by a pension fund Individual savings Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 3 / 35

Inroducion Pension benefis in Belgium Social securiy funded by he sae: Acive workers finance direcly reired individuals (iner generaional solidariy) No capialisaion Pension plans funded by he employer Capialisaion Two main pension schemes: Defined benefis and Defined conribuions Asses managed by a life insurer or by a pension fund Individual savings Capialisaion Promoed by he sae via ax exempions Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 3 / 35

Inroducion Defined benefi pension plan The benefis are defined a priori, e.g. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 4 / 35

Inroducion Defined benefi pension plan The benefis are defined a priori, e.g. Life Annuiy proporional o he las salary, before reiremen. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 4 / 35

Inroducion Defined benefi pension plan The benefis are defined a priori, e.g. Life Annuiy proporional o he las salary, before reiremen. The conribuion paid in by he sponsor (he employer) has o be deermined o enable he benefis liabiliies o be me. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 4 / 35

Inroducion Defined benefi pension plan The benefis are defined a priori, e.g. Life Annuiy proporional o he las salary, before reiremen. The conribuion paid in by he sponsor (he employer) has o be deermined o enable he benefis liabiliies o be me. Normal cos (NC): esimaed conribuion for new enrans. A revised level of conribuions is assessed a he periodical acuarial valuaion when he esimaed value of asses or liabiliies changes as a resul of Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 4 / 35

Inroducion Defined benefi pension plan The benefis are defined a priori, e.g. Life Annuiy proporional o he las salary, before reiremen. The conribuion paid in by he sponsor (he employer) has o be deermined o enable he benefis liabiliies o be me. Normal cos (NC): esimaed conribuion for new enrans. A revised level of conribuions is assessed a he periodical acuarial valuaion when he esimaed value of asses or liabiliies changes as a resul of inflaion, invesmen performance, salary developmen, deah and wihdrawal of members. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 4 / 35

Inroducion Objecives of he fund manager The financial risks are beared by he employer. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 5 / 35

Inroducion Objecives of he fund manager The financial risks are beared by he employer. The longeviy risks are beared by he employer when he affiliaes choose o receive a reiremen a life annuiy. Objecives of he pension fund manager: Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 5 / 35

Inroducion Objecives of he fund manager The financial risks are beared by he employer. The longeviy risks are beared by he employer when he affiliaes choose o receive a reiremen a life annuiy. Objecives of he pension fund manager: Opimisaion of asse allocaion. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 5 / 35

Inroducion Objecives of he fund manager The financial risks are beared by he employer. The longeviy risks are beared by he employer when he affiliaes choose o receive a reiremen a life annuiy. Objecives of he pension fund manager: Opimisaion of asse allocaion. Opimisaion of conribuion adjusmens in order o minimize he oal unanicipaed cos for he sponsor. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 5 / 35

The financial marke The financial marke cash, a rolling bond and socks The financial marke is made up of hree asses: Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 6 / 35

The financial marke The financial marke cash, a rolling bond and socks The financial marke is made up of hree asses: cash, a rolling bond and socks. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 6 / 35

The financial marke The financial marke cash, a rolling bond and socks The financial marke is made up of hree asses: cash, a rolling bond and socks. The reurn of he cash accoun is he risk free rae r and is modelled by a Vasicek model. Consider a 2-dimensional ) sandard Brownian moion W Pf = (W r,pf, W S,Pf defined on a complee probabiliy space (Ω f, F f, P f ). Under Q f, he risk free rae is he soluion of he following SDE: dr = a.(b σ r. λ ) r r ).d + σ r. (dw r,pf + λ r.d, }{{ a}}{{} b Q where W r,qf is a Wiener process under Q f, and wih a, b, σ r and λ r consans. dw r,qf Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 6 / 35

The financial marke cash, a rolling bond and socks Consider a rolling bond of mauriy K whose price is denoed R K. This bond is a zero coupon bond coninuously rebalanced in order o keep a consan mauriy and is price obeys o he dynamics: dr K R K = r.d σ r.n(k). (dw r,pf = r.d σ r.n(k).dw r,qf where n(k) is a funcion of he mauriy K : n(k) = 1 a. (1 e a.k ). ) + λ r.d A sock wih price process S is modelled by a geomeric Brownian moion and is correlaed wih he ineres raes flucuaions: ds ) ) = r.d + σ Sr. (dw r,pf + λ r.d + σ S. (dw S,Pf + λ S.d S = r.d + σ Sr.dW r,qf and wih σ Sr, σ S and λ S consans. + σ S.dW S,Qf Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 7 / 35

Liabiliies modelling Liabiliies and moraliy modelling Liabiliies modelling We consider one cohor of members enered ino he fund a he same age x and earning he same salary, denoed (A ). The evoluion of he individual salary is correlaed o he financial marke: da A = µ A ().d + σ Ar dw r,pf + σ AS dw S,Pf + σ A.dW A,Pa where µ A () is he average growh of he salary and W A,Pa is a Wiener process defined on a probabiliy space (Ω a, F a, P a ), ha represens he inrinsic randomness of he salary and is independen of W r,pf and W S,Pf. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 8 / 35

Liabiliies modelling Liabiliies and moraliy modelling Liabiliies modelling We consider one cohor of members enered ino he fund a he same age x and earning he same salary, denoed (A ). The evoluion of he individual salary is correlaed o he financial marke: da A = µ A ().d + σ Ar dw r,pf + σ AS dw S,Pf + σ A.dW A,Pa where µ A () is he average growh of he salary and W A,Pa is a Wiener process defined on a probabiliy space (Ω a, F a, P a ), ha represens he inrinsic randomness of he salary and is independen of W r,pf and W S,Pf. All members reire a he age x + T and in case of deah, no benefis are paid. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 8 / 35

Liabiliies modelling Liabiliies and moraliy modelling Liabiliies modelling We consider one cohor of members enered ino he fund a he same age x and earning he same salary, denoed (A ). The evoluion of he individual salary is correlaed o he financial marke: da A = µ A ().d + σ Ar dw r,pf + σ AS dw S,Pf + σ A.dW A,Pa where µ A () is he average growh of he salary and W A,Pa is a Wiener process defined on a probabiliy space (Ω a, F a, P a ), ha represens he inrinsic randomness of he salary and is independen of W r,pf and W S,Pf. All members reire a he age x + T and in case of deah, no benefis are paid. Each pensioner will receive a coninuous annuiy whose rae B is a fracion, α, of he las wage: B = A T.α. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 8 / 35

Moraliy modelling Liabiliies and moraliy modelling Moraliy modelling A ime = 0, he size of he cohor is denoed n x. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 9 / 35

Liabiliies and moraliy modelling Moraliy modelling Moraliy modelling A ime = 0, he size of he cohor is denoed n x. The moraliy process (N ) is defined as in Møller (1998) on a probabiliy space (Ω m, F m, P m ) and is assumed o be independen from he filraion generaed by W r,pf, W S,Pf, W S,Pa. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 9 / 35

Moraliy modelling Liabiliies and moraliy modelling Moraliy modelling A ime = 0, he size of he cohor is denoed n x. The moraliy process (N ) is defined as in Møller (1998) on a probabiliy space (Ω m, F m, P m ) and is assumed o be independen from he filraion generaed by W r,pf, W S,Pf, W S,Pa. N poins ou he oal number of deahs observed ill ime and is given by n x N = I (T i ) i=1 where I is an indicaor funcion, T 1, T 2,..., T nx are exponenially disribued random variables modelling he remaining lifeimes of he affiliaes and where he moraliy rae of his jump process is denoed by µ x+. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 9 / 35

Liabiliies and moraliy modelling Expeced number of survivors Moraliy modelling The expeced number of survivors under P m is equal o he curren number of survivors imes a survival probabiliy: ( s ) E ((n x N s ) F m ) = (n x N ). exp µ(x + u).du. }{{} s p x+ Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 10 / 35

Liabiliies and moraliy modelling Expeced number of survivors Moraliy modelling The expeced number of survivors under P m is equal o he curren number of survivors imes a survival probabiliy: ( s ) E ((n x N s ) F m ) = (n x N ). exp µ(x + u).du. }{{} s p x+ s p x+ is he acuarial noaion for he probabiliy ha an individual of age x + survives ill age x + s. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 10 / 35

Deflaor Deflaor Financial, wage and acuarial deflaor Le (Ω, F, P) be he probabiliy space resuling from he produc of he financial, wage and moraliy probabiliy spaces: Ω = Ω f Ω a Ω m F = F f F a F m N P = P f P a P m where he sigma algebra N is generaed by all subses of null ses from F f F a F m. The marke of pension fund liabiliies is incomplee owing o he presence of wo unhedgeable risks: he salary risk and he moraliy risk. The pricing of pension fund liabiliies is hence done under a probabiliy measure Q which is equal o he produc of Q f, Q a and Q m. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 11 / 35

Deflaor Deflaor Financial, wage and acuarial deflaor Le (Ω, F, P) be he probabiliy space resuling from he produc of he financial, wage and moraliy probabiliy spaces: Ω = Ω f Ω a Ω m F = F f F a F m N P = P f P a P m where he sigma algebra N is generaed by all subses of null ses from F f F a F m. The marke of pension fund liabiliies is incomplee owing o he presence of wo unhedgeable risks: he salary risk and he moraliy risk. The pricing of pension fund liabiliies is hence done under a probabiliy measure Q which is equal o he produc of Q f, Q a and Q m. An insurer s deflaor is here composed of hree elemens called abusively he financial, wage and acuarial deflaors, and is an exension of he deflaors used in Hainau and Devolder (2006b). Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 11 / 35

Deflaor and bond price Deflaor Financial, wage and acuarial deflaor The deflaor used o price liabiliies, wrien H(, s) is in our seings he produc of he financial, wage and acuarial deflaors: H(, s) = exp ( s 0 r u.du ) ( ( ( dq f dq a,λa dq m,h ( exp dp )s ).( f dp )s. a dp (.( )s m. 0 r dq f dq a,λa dq m,h u.du dp f ) dp a ) dp m ) Remark ha he expecaion of he deflaor H(, s) is equal o he price of a zero coupon bond, denoed B(, s): B(, s) = ( E (H(, s) F ) = E Q e R ) s ru.du F = ( ) exp β.(s ) + n(s ).(β r ) σ2 r.n(s )2 4.a where β = b Q σ2 r 2.a 2 = b σ r. λ r a σ2 r 2.a 2 Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 12 / 35

Deflaor Financial and wage deflaor Financial, wage and acuarial deflaor The financial deflaor H f (, s) a ime for a cash flow paid a ime s is equal o he produc of he discoun facor and of he change of measure: H f (, s) = exp ( s 0 r u.du ). ( exp ) 0 r u.du. ( dq f dp ( )s f dq f dp f ) The wage deflaor a insan, for a paymen occurring a ime s : H a (, s) = ( dq a,λa dp a )s ( dq a,λa dp a ) = exp ( 12 s s ). λ a,u 2.du λ a,u.dwu A,Pa Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 13 / 35

Deflaor Financial and wage deflaor Financial, wage and acuarial deflaor The financial deflaor H f (, s) a ime for a cash flow paid a ime s is equal o he produc of he discoun facor and of he change of measure: H f (, s) = exp ( s 0 r u.du ). ( exp ) 0 r u.du. ( dq f dp ( )s f dq f dp f ) The wage deflaor a insan, for a paymen occurring a ime s : H a (, s) = ( dq a,λa dp a )s ( dq a,λa dp a ) = exp ( 12 s s ). λ a,u 2.du λ a,u.dwu A,Pa By he incompleeness caused by he salary risk, λ a,u will be chosen in he sequel o be some (arbirary) consan. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 13 / 35

Acuarial Deflaor Deflaor Financial, wage and acuarial deflaor The second source of incompleeness is he moraliy risk. For any F m -predicable process h s, such ha h s > 1, an equivalen acuarial measure Q m,h is defined by he random variable soluion of he SDE: ( ) dq m,h d dp m ( ) ( dq m,h = dp m.h.d N 0 ( ) dq m,h = dp m.h.dm ) (n x N u ) µ(x + u)du Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 14 / 35

Acuarial Deflaor Deflaor Financial, wage and acuarial deflaor The second source of incompleeness is he moraliy risk. For any F m -predicable process h s, such ha h s > 1, an equivalen acuarial measure Q m,h is defined by he random variable soluion of he SDE: ( ) dq m,h d dp m ( ) ( dq m,h = dp m.h.d N 0 ( ) dq m,h = dp m.h.dm ) (n x N u ) µ(x + u)du We adop he noaion λ N,u = (n x N u ).µ(x + u) for he inensiy of jumps. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 14 / 35

Acuarial Deflaor Deflaor Financial, wage and acuarial deflaor The acuarial deflaor a insan, for a paymen occurring a ime s, is defined by: H m (, s) = ( dq m,h dp m )s ( dq m,h dp m ) ( s s ) = exp ln (1 + h u ).dn u h u.λ N,u.du Under Q m,h, he expeced number of survivors a ime s is equal o he number of survivors a ime muliplied by a modified probabiliy of survival s px+ h : E Qm,h ((n x N s ) F m ) = (n x N ). exp } ( s ) µ(x + u).(1 + h u ).du {{ s p h x+ Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 15 / 35

Acuarial Deflaor Deflaor Financial, wage and acuarial deflaor The acuarial deflaor a insan, for a paymen occurring a ime s, is defined by: H m (, s) = ( dq m,h dp m )s ( dq m,h dp m ) ( s s ) = exp ln (1 + h u ).dn u h u.λ N,u.du Under Q m,h, he expeced number of survivors a ime s is equal o he number of survivors a ime muliplied by a modified probabiliy of survival s px+ h : E Qm,h ((n x N s ) F m ) = (n x N ). exp } ( s ) µ(x + u).(1 + h u ).du {{ s p h x+ In he sequel of his work, we resric our field of research o a consan process h u = h. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 15 / 35

Fair value of he liabiliies Deflaor Fair value of he liabiliies The fair value a ime of he liabiliies a he dae of reiremen, denoed L, is defined as he expecaion of he deflaed value of fuure conribuions and benefis. If T m is he maximum ime horizon of he insurer s commimens, L is equal o: L = E ( T T m ) H(, s).c s.ds + H(, s). (n x N s ).B.ds F. T Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 16 / 35

Fair value of he liabiliies Deflaor Fair value of he liabiliies The fair value a ime of he liabiliies a he dae of reiremen, denoed L, is defined as he expecaion of he deflaed value of fuure conribuions and benefis. If T m is he maximum ime horizon of he insurer s commimens, L is equal o: L = E ( T T m ) H(, s).c s.ds + H(, s). (n x N s ).B.ds F. T The fair value a he reiremen dae T of he liabiliies is given by: ( Tm ) L T = E H(T, s). (n x N s ).B.ds F T T = (n x N T ).α.a T. Tm T s T px+t h.b(t, s).ds. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 16 / 35

The objecives The opimisaion problem The objecives We recall ha: The benefis are financed during he accumulaion phase: The employer agrees o pay coninuously a oal conribuion rae c in order o finance he benefis. The normal cos, NC, is he arge level of conribuion and is calculaed a = 0. One of he pension fund manager s goal is o mainain c as close as possible o NC. The second objecive pursued by he pension plan manager is o obain a value of he asses as close as possible o L T, he marke value of he liabiliies a he ime of reiremen. The arge oal asse value is denoed X T. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 17 / 35

The opimisaion problem The opimisaion problem The opimisaion problem Opimisaion problem and value funcion V (, x, n, a) = [ T min E c, X T A (x) where u 1 and u 2 are consan weighs and wih ] u 1. (c s NC) 2.ds + u 2.( X T L T ) 2 F, X = x Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 18 / 35

The opimisaion problem The opimisaion problem The opimisaion problem Opimisaion problem and value funcion V (, x, n, a) = [ T min E c, X T A (x) where u 1 and u 2 are consan weighs and wih Budge consrain A (x) = ] u 1. (c s NC) 2.ds + u 2.( X T L T ) 2 F, X = x ) {((c s ) s [,T ], X T such ha ( T ) } E H(, s).c s.ds + H(, T ). X T F x (1). Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 18 / 35

Lieraure The opimisaion problem The opimisaion problem Acuarial lieraure: Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 19 / 35

Lieraure The opimisaion problem The opimisaion problem Acuarial lieraure: Haberman and Sung (1994, 2005), Boulier e al. (1995), Josa Fombellida and Rincon-Zapaero (2004, 2006), Cairns (1995, 2000), Huang and Cairns (2006). Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 19 / 35

Lieraure The opimisaion problem The opimisaion problem Acuarial lieraure: Haberman and Sung (1994, 2005), Boulier e al. (1995), Josa Fombellida and Rincon-Zapaero (2004, 2006), Cairns (1995, 2000), Huang and Cairns (2006). As he marke is incomplee, he fac ha X T belongs o A (x) doesn guaranee ha his process is replicable by an adaped invesmen policy. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 19 / 35

Lieraure The opimisaion problem The opimisaion problem Acuarial lieraure: Haberman and Sung (1994, 2005), Boulier e al. (1995), Josa Fombellida and Rincon-Zapaero (2004, 2006), Cairns (1995, 2000), Huang and Cairns (2006). As he marke is incomplee, he fac ha X T belongs o A (x) doesn guaranee ha his process is replicable by an adaped invesmen policy. We inspire us upon he approach of Brennan and Xia (2002), see also Hainau and Devolder (2006a, b). Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 19 / 35

Soluion Maringale mehod of Cox-Huang Opimal conribuion rae and arge wealh Following Brennan and Xia (2002), we will use he Maringale mehod and minimize firs wih respec o he conribuions and he associaed erminal arge wealh. Le y R + be he Lagrange muliplier associaed o he budge consrain a insan and define he Lagrangian by: ) L (, x, n, a, (c s ) s, X T, y = (2) ( T ) E u 1. (c s NC) 2.ds + u 2.( X T L T ) 2 ) F y. ( ( T )) x E H(, s).c s.ds + H(, T ). X T F. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 20 / 35

Soluion Opimal conribuion rae and arge wealh Opimal conribuion rae and arge wealh Under echnical condiions, he opimal conribuion rae and arge wealh are: cs = y 1.H(, s). + NC 2.u 1 X T = y 1.H(, T ). + L T. 2.u 2 The opimal Lagrange muliplier, y, is such ha he budge consrain (1) is binding: y = E (H(, T ).L T F ) x NC. T E (H(, s) F ) ds 1 2.u 1. T E (H(, s) 2 F ) ds + 1 2.u 2.E (H(, T ) 2 F ). (3) Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 21 / 35

Unfunded liabiliies Soluion Opimal conribuion rae and arge wealh The numeraor of (3) represens precisely he unfunded liabiliies, denoed by T UL = E (H(, T ).L T F ) x NC. E (H(, s) F ) ds, (4) } {{ } ā,t namely he par of he benefis ha are no ye financed: he expeced fair value of reserves less he curren asse value and less he normal cos imes a financial annuiy ā,t of mauriy T. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 22 / 35

Soluion Opimal conribuion rae and arge wealh Opimal conribuion rae and arge wealh The opimal conribuion process c and he erminal arge wealh X T depend on he unfunded liabiliies UL : Opimal conribuion and arge wealh where cs F (, s) = UL +NC 2u }{{ 1 } amorisaion rae X T = UL F (, T ) + L T 2u 2 Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 23 / 35

Soluion Opimal conribuion rae and arge wealh Opimal conribuion rae and arge wealh The opimal conribuion process c and he erminal arge wealh X T depend on he unfunded liabiliies UL : Opimal conribuion and arge wealh where cs F (, s) = UL +NC 2u }{{ 1 } amorisaion rae X T = UL F (, T ) + L T 2u 2 H(, s) F (, s) = 1 2.u 1. T E (H(, v) 2 F ) dv + 1 2.u 2.E (H(, T ) 2 F ) > 0 Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 23 / 35

Value funcion Soluion Opimal conribuion rae and arge wealh The value funcion depends on he square of unfunded liabiliies: V (, x, n, a) = UL 2 1 u 1. T E (H(, s) 2 F ) ds + 1 u 2.E (H(, T ) 2 F ) (5) Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 24 / 35

Soluion The bes replicaing sraegy The opimal arge wealh is no hedgeable As L T is a funcion boh of he moraliy and of he salary which are no replicable, i is easily seen ha X T is no hedgeable. An approach o obain a replicable wealh approximaing he opimal arge wealh X T is o projec i on he space of replicable processes, and herefore o use a Kunia-Waanabe decomposiion, see also Hainau and Devolder (2006a). Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 25 / 35

Soluion The bes replicaing sraegy The opimal arge wealh is no hedgeable As L T is a funcion boh of he moraliy and of he salary which are no replicable, i is easily seen ha X T is no hedgeable. An approach o obain a replicable wealh approximaing he opimal arge wealh X T is o projec i on he space of replicable processes, and herefore o use a Kunia-Waanabe decomposiion, see also Hainau and Devolder (2006a). Our reasoning in his paper is based on dynamic programming (see e.g. Fleming and Rishel 1975 for deails) and is also applied in Hainau and Devolder (2006b). Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 25 / 35

Soluion The se of replicable processes The bes replicaing sraegy Le (π S, π R ) denoe respecively he fracion of he wealh invesed in socks and rolling bonds and define Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 26 / 35

Soluion The se of replicable processes The bes replicaing sraegy Le (π S, π R ) denoe respecively he fracion of he wealh invesed in socks and rolling bonds and define The se of replicable processes A π (x) = {((c s ) s [,T ], X T ) e R T T r s.ds.x T = x + T + e R s ru.du.πs S.X s.ds s + (π S ) (π R ) F adaped : e R s ru.du.c s.ds T e R s ru.du.π R s.x s.dr K s }. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 26 / 35

Soluion The se of replicable processes The bes replicaing sraegy Le (π S, π R ) denoe respecively he fracion of he wealh invesed in socks and rolling bonds and define The se of replicable processes A π (x) = {((c s ) s [,T ], X T ) e R T T r s.ds.x T = x + T + e R s ru.du.πs S.X s.ds s + (π S ) (π R ) F adaped : e R s ru.du.c s.ds T e R s ru.du.π R s.x s.dr K s }. By definiion, he se A π (x) is included in A (x) and ) ) dx = ((r + π S.ν S + π R.ν R.X + c.d + π S.σ S.X.dW S,Pf ( ) + π S.σ Sr π R.σ r.n(k).x.dw r,pf. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 26 / 35

Soluion Dynamic programming principle The bes replicaing sraegy For a small sep of ime, he dynamic programming principle saes ha: V (, x, n, a) = E [ + u 1. (cs NC) 2.ds+ +V ( +, X ) +, N +, A + F ]. Given ha ( X ) is he process minimizing he value funcion, any oher process (X ) A π (x) A (x) verifies he inequaliy: [ + V (, x, n, a) E u 1. (cs NC) 2.ds+ +V ( +, X +, N +, A + ) F ]. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 27 / 35

Iô s lemma and generaor Soluion The bes replicaing sraegy Using Io s lemma for jump processes: E (V ( +, X +, N +, A + ) F ) = V (, x, n, a) + E ( + G π (s, X s, N s, A s ).ds F ) + ( + ) E (V (s, X s, N s, A s ) V (s, X s, N s, A s )) dn s F where G π (s, X s, N s, A s ) is he generaor of he value funcion. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 28 / 35

Soluion The bes replicaing sraegy Deriving G π (, X, N, A ) wih respec o π S and π R leads o: The bes replicaing sraegy π S = π R = ( νr.σ Sr σs 2.σ r.n(k) + ν ) S σs 2. UL X }{{} consan ( νs.σ Sr ν R + σ AS σ S. E (H(, T ).L T F ) X (6) ( )) σs 2.σ r.n(k) + σr 2.n(K) 2. 1 + σ2 Sr σs 2. UL X }{{} consan ( σar σ r.n(k) σ ) AS.σ Sr. E (H(, T ).L T F ) σ S.σ r.n(k) X }{{} consan + 1 n(k). V Xr. 1. (7) V XX X }{{ } correcion erm Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 29 / 35

Parameer values Numerical example We consider a male populaion, age 50 of n 50 = 10.000 affiliaes. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 30 / 35

Parameer values Numerical example We consider a male populaion, age 50 of n 50 = 10.000 affiliaes. Iniial wage: A =0 = 2500 Euro. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 30 / 35

Parameer values Numerical example We consider a male populaion, age 50 of n 50 = 10.000 affiliaes. Iniial wage: A =0 = 2500 Euro. Age of reiremen: 65 years. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 30 / 35

Parameer values Numerical example We consider a male populaion, age 50 of n 50 = 10.000 affiliaes. Iniial wage: A =0 = 2500 Euro. Age of reiremen: 65 years. Annuiy received a reiremen: 20% of A T. Normal cos 2.676.300 The moraliy raes are given by: µ(x) = a µ + b µ.c x µ a µ = ln(s µ ) b µ = ln(g µ ). ln(c µ ) where he parameers s µ, g µ, c µ ake he values showed in he able: s µ : 0.999441703848 g µ : 0.999733441115 c µ : 1.116792453830 Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 30 / 35

Numerical example Oher parameers: a 12.72% σ SR -0.10% b 3.88% ν S 5.35% σ r 1.75% µ A 2.00% λ r -2.36% σ Ar 2.00% r =0 2.00% σ AS 2.00% K 8 years µ Q A 2.00% ν R 2.77% σ A 5.00% λ S 34.94% λ a -4.54% σ S 15.24% h 0.0 Table: Parameers. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 31 / 35

Conribuion raes Numerical example Mone Carlo simulaion: 5000 scenarios generaed. Figure: Conribuion raes. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 32 / 35

Invesmen proporions Numerical example Figure: Asse mix for u 1 = 1 and u 2 = 10. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 33 / 35

Conclusions Conclusions The opimal conribuion rae depends on he unfunded liabiliies. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 34 / 35

Conclusions Conclusions The opimal conribuion rae depends on he unfunded liabiliies. The amorizaion of hose unfunded liabiliies is a funcion of he weighs u 1 and u 2 defining he employer s preferences. Posiions in risky asses decrease when we approach o he mauriy. Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 34 / 35

Conclusions Conclusions The opimal conribuion rae depends on he unfunded liabiliies. The amorizaion of hose unfunded liabiliies is a funcion of he weighs u 1 and u 2 defining he employer s preferences. Posiions in risky asses decrease when we approach o he mauriy. A quadraic uiliy penalizes wihou disincion posiive and negaive spreads Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 34 / 35

Conclusions Thank you for your aenion! Griselda Deelsra and Donaien Hainau () Def. Benefi Pension Plan: opimisaion June 7, 2007 35 / 35