Risk Managemen of a DB Underpin Pension Plan Kai Chen upervisor: Mary Hardy Acknowledge he UW Insiue for Quaniaive Finance and Insurance CKER ARC Travel Gran for heir uppor
Ouline Inroducion and Background Risk Managemen Hedging raegies Numerical Resuls Commens and Fuure Work Augus, 006 @ 4s ARC
Inroducion of Curren Pension ysems Two basic kinds of pension plans Defined Benefi (DB) and Defined Conribuion (DC) Pension Reforms Three-dimensional classificaion(lindbeck, 003): Acuarial vs Non-Acuarial Funded vs Unfunded Defined Benefi vs Defined Conribuion Augus, 006 @ 4s ARC 3
Inroducion and Background Greaer of benefi a DB and DC hybrid plan DC plus DB guaraneed minimum The payoff of he guaranee is same as an exchange opion payoff. Eg. McGill Universiy, WLU offer a hybrid pension plan wih minimum DB guaranee. Augus, 006 @ 4s ARC 4
Descripion of Benefis (DB) The DB benefis depend on he salary a exi age xr: = () D xe, xr ( ) α xr n xr xe axe+ D xe, xr ( ) α n ) xr ( xr xe a xe + :The acuarial value a age xe+ :The accrual rae :The salary a age xr :The years of service :The annuiy facor Augus, 006 @ 4s ARC 5
Descripion of Benefis (DC) The DC benefis depend on he monhly conribuion: A xe, xr ( + h) = Axe, xr ( ) ( + hf ) + c h xe+ Where c h :The conribuion rae :The ime inerval Augus, 006 @ 4s ARC 6
Risk Managemen Our problem Valuaion and risk managemen of Max(D-A,0) a exi. Assume no exis Given reiremen ime T=xr-xe, he value of guaranee a enry age xe is, e r( xr xe) Max( Dxe, xr ( T ) Axe, xr ( T ),0) Augus, 006 @ 4s ARC 7
Augus, 006 @ 4s ARC 8 Exchange Opion Raes of reurn of wo asses A European-ype opion wih mauriy ime T Payoff a mauriy ime T Price a any ime, ], [ = + = i dz d d i i i i i σ α,0) ( ),, ( Max T w = ) ( / ) / ln( ) ( ) ( ),, ( σ ρ σ σ σ σ σ σ σ + = = + = = T d d T T d d N d N w
Mone Carlo Hedging A any ime, change he value of DB and DC accouns he projeced value of guaranee calculae dela rebalance a ime +h Augus, 006 @ 4s ARC 9
raegy EAN/Aggregae Cos Mehod Final alary Fuure ervice Fuure Conribuions Too Volaile! Augus, 006 @ 4s ARC 0
raegy Projeced Uni Credi(PUC) Cos Mehod Final alary Pas ervice Pas Conribuions Eg. Enry age 35, Curren age, Normal Reiremen Age 65 65 () s( u Projeced value of DB: α 5 5 a e Projeced value of DC: 65 f ( u) du Ae Projeced value of guaranee: ) du Max (0, α 5 5 a ( ) e 65 s ( u ) du A e 65 f ( u ) du ) Augus, 006 @ 4s ARC
raegy 3 Tradiional Uni Credi(TUC) Cos Mehod Curren alary Pas ervice Pas Conribuions Eg. Enry age 35, Curren age, Normal Reiremen Age 65 () Projeced value of DB: α 5 5 a Projeced value of DC: 65 f ( u) du Ae Projeced value of guaranee: Max(0, α 5 5 a () A e 65 f ( u) du ) Augus, 006 @ 4s ARC
Hedging Coss A ime, assume dela hedging shares are, and The value of hedging fund is Δ,, + Δ,, A ime (+) -, he value of hedging fund is Δ,, + + Δ,, + A ime (+), he value of hedging fund is Δ, +, + + Δ, +, + A ime (+), he hedging cash flow CF + is Δ Δ, ( Δ, + Δ, ), + + ( Δ, + Δ, ), + Augus, 006 @ 4s ARC 3
Two ways o calculae hedging coss Discoun all hedging cash flows and amorize by he salary-relaed annuiy j j e rj e rj CF j j A ime, calculae he proporion of he hedging cash flows of he salary a ime, hen calculae he average n j CF j / j Augus, 006 @ 4s ARC 4
Parameers Base Projecions Mean of alary Growh Rae: 0.04 d of alary Growh Rae : 0.0 Mean of Crediing Rae : 0.0 d of Crediing Rae : 0.0 Conribuion Rae : 0.5 Discoun Rae : 0.05 Augus, 006 @ 4s ARC 5
Resuls Comparison (% of alary) Augus, 006 @ 4s ARC 6
Monhly Hedging Cos Augus, 006 @ 4s ARC 7
Quanile Augus, 006 @ 4s ARC 8
cenario Tess (raegy, Hedging Cos ) Augus, 006 @ 4s ARC 9
Conclusions Hedging coss depend on many facors, bu are no very sensiive o he enry age or he salary growh rae. andard errors depend on he enry age and increase when he enry age increases. Augus, 006 @ 4s ARC 0
Fuure Work Use inflaion-linked bonds hedge salary and explore basis risk. Inroduce sochasic ineres and annuiy. Augus, 006 @ 4s ARC
Thank You! Augus, 006 @ 4s ARC