Risk Management of a DB Underpin Pension Plan

Transcription

1 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

2 Ouline Inroducion and Background Risk Managemen Hedging raegies Numerical Resuls Commens and Fuure Work Augus, 4s ARC

3 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, 4s ARC 3

4 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, 4s ARC 4

5 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, 4s ARC 5

6 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, 4s ARC 6

7 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, 4s ARC 7

8 Augus, 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

9 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, 4s ARC 9

10 raegy EAN/Aggregae Cos Mehod Final alary Fuure ervice Fuure Conribuions Too Volaile! Augus, 4s ARC 0

11 raegy Projeced Uni Credi(PUC) Cos Mehod Final alary Pas ervice Pas Conribuions Eg. Enry age 35, Curren age, Normal Reiremen Age () 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, 4s ARC

12 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, 4s ARC

13 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, 4s ARC 3

14 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, 4s ARC 4

15 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, 4s ARC 5

16 Resuls Comparison (% of alary) Augus, 4s ARC 6

17 Monhly Hedging Cos Augus, 4s ARC 7

18 Quanile Augus, 4s ARC 8

19 cenario Tess (raegy, Hedging Cos ) Augus, 4s ARC 9

20 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, 4s ARC 0

21 Fuure Work Use inflaion-linked bonds hedge salary and explore basis risk. Inroduce sochasic ineres and annuiy. Augus, 4s ARC

22 Thank You! Augus, 4s ARC