NDC Dynamic Equilibrium Model with financial and ...actuaries.org/pbss/colloquia/tokyo/devolder_pierreppt.pdfin time with a focus on equilibrium between active people and retirees

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NDC Dynamic Equilibrium Model NDC Dynamic Equilibrium Model NDC Dynamic Equilibrium Model NDC Dynamic Equilibrium Model

with financial and demographic riskswith financial and demographic riskswith financial and demographic riskswith financial and demographic risks

Pierre DEVOLDER

Inmaculada DOMINGUEZ FABIAN

Aurélie MILLER

1The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009

Outline

• 1. NDC PAYG DC or DB …

• 2. DEMOGRAPHIC MODEL

• 3. STATIC EQUILIBRIUM

• 4. DYNAMIC UNIFORM PENSION MODEL

• 5. DYNAMIC GENERATION PENSION MODEL

• 6. CONCLUSION


1. NDC PAYG DC or DB…

NDC = Notional Defined Contribution Schemes

= Notional Accounts

- Attempt to reproduce the logic of individual funded pensionaccounts but now in a PAYG framework.

- Combination of : - solidarity of PAYG technique - stability of the charges in DC

( Disney (1999), Scherman (1999), Williamson(2004),…)



2 basic techniques in order to finance pension liabilities

PAY AS YOU GO FUNDING

Pensions for retirees are paid by active people

Active people financetheir own pension

Unfunded schemes Funded schemes



This fundamental choice is present as well for definedbenefit pension schemes (DB) as for defined contributions pension schemes (DC).

Pay as you go Funding

DB Classical Classical EmployeeSocial Security Benefit DB plan

DC Notional Accounts Pension saving(NDC) Accounts



By definition funded DC pension schemes are really “defined contribution” !!!

In particular, the actuarial equilibrium is always guaranteed .

NDC are much more delicate …especially in presence ofdemographic and financial risks ( between DC and DB !!!)

( Valdez Prieto ( 2000), Settergren (2001),Auerbach/Lee (2006))

Therefore the design of the NDC is crucial .


2. DEMOGRAPHIC MODEL

The Overlapping Generation Model ( OLG Model):

Stylization tool in order to capture the dynamic evolution of populationin time with a focus on equilibrium between active peopleand retirees. ( for instance Vermeylen (2007))

OLG Assumptions: model with 3 generations: - Agents have finite lives- They live in three periods :

- they are “active” , then “retired” 2 periods , then dead- when one generation becomes old, anotheryoung generation is born .



Notations and assumptions : static model :

y y+1 y+2 y+3

Entry Retirement Survival Endage age age

Survival 1 p 0probability



Notations :

π++

=

===++=

=

γ

δ

:rateoncontributi

p:2yand1yagebetweenyprobabilitsurvival

e).0(s)t(s:salarymean

e).0(E)t(E

ttimeatenteringpeopleofnumber)t,y(L)t(E

)2y,1y,yx(

ttimeatxagedpeopleofnumber)t,x(L

t

t


3. STATIC EQUILIBRIUM

tt e.e).0,y(L).0(s.

)t,y(L).t(s.)t(C:onscontributitotalγδπ=

π=

PAYG + DC scheme for this population:

Actuarial equivalence between income and outcome

Income :

Outcome :

)e.pe).(0(E).t(P

))t,2y(L)t,1y(L).(t(P:pensionstotal)2t()1t( −δ−δ +=

+++

( same pension for both generations )



annuity

tionrevalorizawithonscontributiofsum

)t(a

)t(R).1t(c)t(P =−=

δ−

δ+γ−γ

+π=

e.p1

e.e).0(s.)t(P

)1t(

Interpretation as a NDC formula :

Mean pension achieved :

δ−

δ+γ

−γ

+====

π==−

e.p1annuity)t(a

etionrevaloriza)t(R

e).0(s.oncontributi)1t(cwith )1t(



γ=−

= e)1t(P

)t(P)t(RP

Replacement rate :

Indexation process :

δ−

δ

+π==

ep1

e.

)t(s

)t(P)t(RR

Following the indexation of wages



CONCLUSION :

In order to design a NDC scheme , we must define 3 processes :

a) the indexation process ( pension indexation afterretirement): RP

b) the revalorization process of the salaries : Rc) the annuity : a



In the static model, the PAYG constraint gives the

following canonical identification :

δ−

δ+γ

γ

+==

=

e.p1)t(a

e)t(R

e)t(RP Pension Indexation : salary growth

Revalorization : salary + population growth

Canonical design of the NDC scheme

Annuity : population growth



)t(a.

))t(R.).(1t(c)t(P

αα−=

δ−

δ+γ

γ

+==

=

e.p1)t(a

e)t(R

e)t(RP

Remark :Even if, in this static model, this canonical identification seems natural, a lot of other identifications are possible ( giving off course finally the same pension amount !!). In fact, the processes R(t) and a(t) can be multiplied by any positive constant ( …presentation purpose…) :

For instance :

δ−δ−

γ

γ

+===

=

2e.pe)t(a

e)t(RP)t(R

e)t(RP

Same indexation


4. DYNAMIC UNIFORM PENSION

yprobabilitsurvival:p

wagesofrategrowth:

populationofrategrowth:

γ

δ

Static model Dynamic model

t

t

t

p

,...)2,1t;tyear(

γ

=δ

2 cases for the pension amount : - uniform pension at time t- 2 generations of retires

receiving different pensions



δ−

δ+γ−γ

+π=

e.p1

e.e).0(s.)t(P

)1t(

Uniform pension – equilibrium condition :

1t

tt

e.p1

e.)exp().0(s.)t(P

t

1t

1ii

−δ−

δ+γ−

=

+

γπ=

∑

static

dynamic



)t(a

)t(R).1t(c)t(P

−=)1t(P

)t(P)t(RP

−=

1t

tt

e.p1

e.)exp().0(s.)t(P

t

1t

1ii

−δ−

δ+γ−

=

+

γπ=

∑

NDC canonical interpretation :

?

1t

1t

t

t

e.p1)t(a

e

e.

)t(a

)1t(a.e)t(RP)exp()t(R

t

tt

−

−

δ−

δ

δγ

+=

−=δ+γ=



NDC canonical interpretation : conclusions

1°) R(t): revalorization of salaries:as expected : salary + population growth

2°) a(t): conversion annuity :as expected : population + probability

( … but time related !!!) 3°) RP(t ) : indexation of the pension :

not as expected !!! No more simply equal to salary growth



1t

t

t

e

e.

)t(a

)1t(a.e)t(RP

−δ

δγ −=

New indexation rule :

SalaryGrowth Demographic

and longevity risk

The expected indexation coefficient is corrected by 2 terms : - a ratio of successive annuities- a ratio of successive demographic growths

Demographicrisk



)t(Re

e.

)t(a

)1t(a.e)t(RP

1t

t

t =−=−δ

δγ

)(1t

2t1t1t e.pe)t(a −−− δ+δ−−

δ− +=

te)t(R γ=

Other possible NDC identifications :

EXAMPLE 1 : uniform indexation before and after retirement :

EXAMPLE 2 : revalorization in line with salary growth :

)(t

1ttt e.pe)t(a −δ+δ−δ− +=


5. DYNAMIC GENERATION PENSION

pensionuniform)t(P =

generationretiredoldtheforpension)t(P

generationretirednewtheforpension)t(P

2

1

==

Equilibrium condition :

)t(P).t,2y(L)t(P).t,1y(L)t(s.).t,y(L 21 +++=π

….Infinite number of different equilibrated schemes !!!



Additional constraint : NDC definition :

The first pension is computed using at retirement a NDC formula . The second pension is then a consequence of the equilibrium relation.

Different NDC identifications will now lead to different pension amounts( balance between the 2 generations) !!!

General NDC approach :

The pension for the new generation is given by :

)t(a

)t(R).1t(c)t(P1

−=

The processes R(t) and a(t) must be chosen.



)t,2y(L

)t,1y(L

)t(P

1.

)t,2y(L

)t(C

)t(P

)t(P

11

2

++−

+=

Then the pension for the old generation is given by the equilibrium condition :

The indexation process is given by the relation :

)1t(P

)t(P)t(RP

1

2

−=



te.p1)t(a

)exp()t(R

1t

tt

δ−++=

δ+γ=

Canonical choice : ( model 1 ):

)t(a

)1t(a.

p

p.e)t(RP

t

1tt−= +γ

Natural

Indexation

Longevitycorrection

Annuitycorrection

Naturalrevalorization

Annuity withProspectiveLife table



1t

tt

p1)t(a

)exp()t(R

++=δ+γ=

Other possible choices : ( model 2 ):

)t(a

)1t(a.

p

p.e)t(RP

t

1ttt−= +γ+δ

Indexation =

revalorizationLongevitycorrection

Annuitycorrection


Annuity Life Expectancy



tep1)t(a

)exp()t(R

t

tt

δ−+=

δ+γ=

Other possible choices : ( model 4 ):

)t(a

)1t(a.e)t(RP t

−= γ

Natural

Indexation Annuitycorrection


Annuity with presentLife table


6. CONCLUSION

In a dynamic model the indexation process after retirement can not be chosen so naturally.

Different choices can be done with respect to the annuity conversion and the revalorization of salaries.

Then in an equilibrium model, the indexation afterretirement becomes a constraint and is generally not equalto a natural indexation . Correction terms mainly based on an annuity ratio appear systematically.


FURTHER RESEARCH

- Extension in a more sophisticated demographic model with “many” ages;

- Stochastic approach for the financial and demographic background;

- Introduction of an Equilibrium Fund in this stochasticframework.

THANK YOU


Prof. Pierre DEVOLDERACTUUniversité Catholique de Louvain ( UCL)6 Rue des Wallons1348 LOUVAIN la NEUVEBELGIUM

Mail : [email protected]

URL :http://www.actu.ucl.ac.be/staff/devolder/pdevolder.html

Corresponding Author

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NDC Dynamic Equilibrium Model with financial and ...actuaries.org/pbss/colloquia/tokyo/devolder_pierreppt.pdfin time with a focus on equilibrium between active people and retirees