ndc dynamic equilibrium model with financial and...
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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
2The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
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),…)
3The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
1. NDC PAYG DC or DB…
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
4The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
1. NDC PAYG DC or DB…
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
5The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
1. NDC PAYG DC or DB…
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 .
6The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
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 .
7The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
2. DEMOGRAPHIC MODEL
Notations and assumptions : static model :
y y+1 y+2 y+3
Entry Retirement Survival Endage age age
Survival 1 p 0probability
8The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
2. DEMOGRAPHIC MODEL
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
9The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
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 )
10The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
3. STATIC EQUILIBRIUM
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(
11The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
3. STATIC EQUILIBRIUM
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
12The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
3. STATIC EQUILIBRIUM
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
13The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
3. STATIC EQUILIBRIUM
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
14The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
3. STATIC EQUILIBRIUM
)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
15The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
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
16The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
4. DYNAMIC UNIFORM PENSION
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
17The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
4. DYNAMIC UNIFORM PENSION
)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=
18The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
4. DYNAMIC UNIFORM PENSION
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
19The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
4. DYNAMIC UNIFORM PENSION
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
20The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
4. DYNAMIC UNIFORM PENSION
)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− +=
21The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
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 !!!
22The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
5. DYNAMIC GENERATION PENSION
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.
23The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
5. DYNAMIC GENERATION PENSION
)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
−=
24The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
5. DYNAMIC GENERATION PENSION
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
25The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
5. DYNAMIC GENERATION PENSION
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
Naturalrevalorization
Annuity Life Expectancy
26The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
5. DYNAMIC GENERATION PENSION
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
Naturalrevalorization
Annuity with presentLife table
27The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
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.
28The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
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
29The 4The 4thth PBSS ColloquiumPBSS Colloquium TOSHI CENTER Hotel, Tokyo, Japan TOSHI CENTER Hotel, Tokyo, Japan –– 44--6 October 20096 October 2009
Prof. Pierre DEVOLDERACTUUniversité Catholique de Louvain ( UCL)6 Rue des Wallons1348 LOUVAIN la NEUVEBELGIUM
Mail : Pierre.Devolder@uclouvain.be
URL :http://www.actu.ucl.ac.be/staff/devolder/pdevolder.html
Corresponding Author
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