Fo schung am i wKöln
Band 7/2018
Resilience and In e gene a ional Fai ness
in Collec i e De ined Con ibu ion Pension Funds
Oska Goecke
Fo schung am i wKöln, Band 7/2018
Oska Goecke
Fo schungss elle FaRis
Resilience and In e gene a ional Fai ness in Collec i e De ined Con ibu ion
Pension Funds
Abs ac
A pension sys em is esilien i i able o abso b ex e nal ( empo al) shocks and i i is able o adap
o (long e m) shi s o he socio
-economic en i onmen . De ined bene i (DB) and de ined con ibu ion
pension plans beha e con as ingly wi h espec o capi al ma ke shocks and shi s: while DB
-plan
bene i s a e no a ec ed by ex e nal shocks hey o ally lack adap abili y wi h espec o undamen al
changes; DC
-plans au oma ically adjus o a changing en i onmen bu any ex e nal shock has a
di ec impac on he (expec ed) pensions. By adding
a collec i e componen o DC-plans one can
make hese collec i e DC (CDC)
-plans shock abso bing - a leas o a ce ain deg ee. In ou CDC
pension model we build a collec i e ese e o asse s ha se es as a bu e o capi al ma ke shocks,
e.g. s ock ma
ke c ashes. The idea is o ans e money om he collec i e ese e o he indi idual
pension accoun s whene e capi al ma ke s slump and o eed he collec i e ese e whene e
capi al ma ke a e booming. This mechanism is pa icula aluable o age
coho s ha a e close o
e i emen . I is clea ha wi hd awing asse s om o adding asse s o he collec i e ese e is
essen ially a ans e o asse s be ween he age coho s. In ou nea eali y model we in es iga e he
e ec o s ock ma ke shock
s and in e es a e (and mo ali y) shi s on a CDC- pension sys em. We
a e pa icula ly in e es ed in he ques ion, o wha ex end a CDC
-pension sys em is ac ually able o
abso b shocks and whe he he in e gene a ional ans e o asse s ia he collec i
e ese e can be
ega ded as ai .
- I -
Resilience and In e gene a ional Fai ness in
Collec i e De ined Con ibu ion
Pension Funds
Oska Goecke
- II -
Resilience and In e gene a ional Fai ness in Collec i e De ined
Con ibu ion Pension Funds
Con en
1 In oduc ion .............................................................................. 1
2 Basic Model ............................................................................. 3
2.1 Popula ion Model ............................................................................ 3
2.1.1 CDC Pension Fund ....................................................................................... 3
2.1.2 S eady S a e Popula ion and Popula ion Dynamics ...................................... 5
2.2 Liabili ies ......................................................................................... 7
2.3 Asse s ............................................................................................. 10
3 Asse Liabili y Managemen .................................................. 12
3.1 Basic Rela ions .............................................................................. 12
3.2 ALM – S a egies .......................................................................... 17
3.3 Indi idual Sa ing and Dissa ing ................................................... 24
3.4 S eady S a e Analysis .................................................................... 25
4 Resilience Tes ....................................................................... 28
4.1 S eady S a e O iginal Posi ion ....................................................... 29
4.2 Capi al Ma ke Shock .................................................................... 30
4.2.1 Capi al Ma ke Shock E ec on IDC-Plans ............................................... 30
4.2.2 Capi al Ma ke Shock E ec on CDC-Plans .............................................. 31
4.3 Capi al Ma ke Shi ...................................................................... 35
4.3.1 Capi al Ma ke Shi E ec on IDC-Plans .................................................. 35
4.3.2 Capi al Ma ke Shi E ec on CDC-Plans ................................................ 36
4.4 Mo ali y Shi ............................................................................... 40
4.4.1 Mo ali y Shi E ec on IDC-Plans ........................................................... 42
4.4.2 Mo ali y Shi E ec on CDC-Plans ......................................................... 43
5 Concluding Rema ks .............................................................. 51
Re e ences .................................................................................... 52
Figu es and Tables ....................................................................... 54
Con ac ......................................................................................... 56
- 1 -
1 In oduc ion
All o e he wo ld de ined bene i pension plans (DB-plans) a e in e ea , meaning
ha young employees en e ing wo king li e mus accep de ined con ibu ion pension
plans (DC-plans).1 The e a e se e al easons o his de elopmen , including: in-
c eased isk awa eness among employe s, in ensi ied egula ion and a low in e es
a e en i onmen .
Employees and labou unions ega d he shi om DB o DC as a massi e educ ion
o labou igh s since he in es men isk is pu on he weak shoulde s o employees.
This ac canno be denied. Howe e , one can also a gue ha he ansi ion om DB
o DC is jus p oo ha DB plans a e unsus ainable in he sense ha hey lack lexi-
bili y o adjus o a changed economic en i onmen . As a consequence, ine i able ad-
jus men s had o be made by closing old DB sys ems and in doing so pu ing he i-
nancial bu den o he obsole e DB plans on he shoulde s o he younge gene a ion.2
This gene a ion is hi wice since a he same ime he social secu i y pension sys ems
a e unde econs uc ion wi h he ob ious ou come o he young.3
Compa ed o DB-plans, pu e (indi idual) DC-plans a e “o e - eac i e” in he sense
ha pension bene i s a e di ec ly linked o he ime alue o he pension po . Equi y
ma ke shocks, shi s o he yield cu e o changing li e expec ancy ins an aneously
hi he expec ed pension o he pension in paymen .
The idea behind collec i e DC- (CDC-) plans is o in oduce a collec i e componen
o a DC-plan o bu e ex e nal shocks o shi s in o de o s abilise (expec ed) pen-
sion paymen s. The collec i e ese e in a CDC sys em can be ega ded as an unallo-
ca ed und o asse s. This und mus be ed by con ibu ions o asse e u ns. Pay-
men s in o and wi hd awals om he collec i e ese e cons i u e an in e gene a-
ional ans e o asse s.
In he ollowing we p esen a mul i gene a ion CDC-pension model including ules
o when and how he in e gene a ional ans e is o be ca ied ou . The main pu -
pose o his pape is o apply he concep o esilience o a pension sys em. Resili-
ence is he abili y o a sys em o abso b (single) ex e nal shocks and o adap o (pe -
manen ) shi s o he socio-economic en i onmen . Ou app oach allows us o explic-
i ly measu e he in e gene a ional ans e .
1 C . [OECD 2011], p. 15.
2 We ha e he same e ec i he bene i s o a DB plan emain un ouched bu he con ibu ions a e
adjus ed.
3 C . [House o Commons 2016] p. 15-16.
- 2 -
The u ili y inc easing e ec o in e gene a ional isk ans e has been p o en by
many au ho s using di e en me hods. [Go don/ Va ian 1988] use a s ylised o e lap-
ping gene a ion model o p o e ha he go e nmen should play an ac i e ole by
bo owing o sa ing in he capi al ma ke o imp o e isk alloca ion be ween gene a-
ions. [Gollie 2007] add esses he in e gene a ional isk ans e in a pension und
wi h a s able numbe o new young wo ke s eplacing he e i ees who ge a lump
sum paymen as pension bene i . Using expec ed u ili y heo y, Gollie can p o e ha
i all gene a ions sa e in o a common pension und he expec ed u ili y o e e y
gene a ion can be inc eased. [Wes e hou 2011] discusses he ques ion o how he in-
e gene a ional isk ans e in a pension sys em can be designed in such a way ha
e e y gene a ion eally akes ad an age o he sys em. [Cui e.a. 2011] a gue in he
same spi i as [Gollie 2007], howe e hei pension model is mo e ealis ic in he
sense ha hei model wo ks wi h cu en pension paymen s (ins ead o lump sum
bene i s) and hey in oduce an abso bing unding su plus, which inances he in e -
gene a ional ans e . Fu he mo e [Cui e.a. 2011] use op ion p ice echniques o
alue he in e gene a ional ans e .
Ou con ibu ion is o discuss he esilience o a CDC pension scheme wi h espec o
in e gene a ional ai ness. We say ha a pension scheme is esilien , i i is able o
abso b ex e nal (single) shocks (e.g. a c ash o ma ke alue o equi ies) and i is
able o adjus o (pe manen ) shi s (e.g. shi o in e es a es o mo ali y). I is desi -
able ha a single s ock ma ke c ash does no a ec pensions in paymen o ull ex-
end. Howe e , as in de ined con ibu ion sys em wi h no ex e nal sponso any p o-
ec ion o he g oup o pensione s is implici ly inanced by an in e gene a ional ans-
e om he young o he old. Young pa icipan s will ega d his kind o in e gene a-
ional ans e as ai because hey expec ha soone o la e he e ec s o he down
shock will be compensa ed by an up shock. Howe e , i e.g. he isk- ee in e es a e
shi s o a new lowe le el, say combined wi h a lowe in la ion a e, hen he unde -
s anding o in e gene a ional ai ness could be ha all age coho s ha e o bea he
consequences. Unde hese ci cums ances a wa ing o pension adjus men s o a cu
o pensions in paymen could be compelling om he pe spec i e o in e gene a-
ional ai ness.
The se up o his pape is as ollows. Following his in oduc ion, sec ion 2 in o-
duces ou basic pension model and sec ion 3 he asse liabili y managemen (ALM)
ules. The esilience es in sec ion 4 cons i u es he main pa o his pape . To es
he esilience o he pension sys em we ha e o de ine a s eady s a e posi ion (sec ion
4.1). Then we apply capi al ma ke shock (sec ion 4.2) and capi al ma ke shi (sec-
ion 4.3) scena ios o he sys em. Finally in sec ion 4.4 we discuss he e ec s o a
mo ali y shi .
- 3 -
2 Basic Model
2.1 Popula ion Model
2.1.1 CDC Pension Fund
We conside a pension und o ac i e and e i ed employees. The ac i e employees
pay pe iodic con ibu ions o build up a pension capi al. A a ce ain e i emen age z
he indi idual pension capi als a e con e ed in o a li e annui y. The pension und is
exclusi ely inanced by he egula con ibu ions; he e is no ex e nal en i y ha
could s ep in i he pension und uns ou o asse s. Examples o such scena ios
would be i asse s do no pe o m as expec ed o i he e i ees li e longe han ex-
pec ed esul ing in pension bene i s ha ing o be adjus ed. In ex eme cases pension
paymen s may ha e o be cu . On he o he hand, o e pe o ming asse s o declining
li e expec ancy e en ually esul in highe pension bene i s.
In he case o a de ined con ibu ion (DC) pension und, he con ibu ions de e mine
he pension bene i s. I obse ed asse e u ns o mo ali y a es de ia e om he ex-
pec ed alues he pension bene i s ha e o be adjus ed while con ibu ions emain un-
changed. In con as , in a de inded bene i (DB) scheme, he con ibu ions would be
adjus ed bu no he p omised bene i s. The s anda d design o a DC schemes is an
indi idual DC scheme, whe e each pa icipan pays con ibu ions in o a pe sonal
pension po , a e i emen he acc ued capi al o he pension po de e mines he paid
bene i s.
To ou unde s anding he cha ac e is ic ea u e o a collec i e DC (CDC) pension
und is ha he e is a collec i e ese e, i.e. pa o he o al asse s can be used o bal-
ance unexpec ed losses on he asse side o ac ua ial losses on he liabili y side. The
ollowing FIGURE 1 shows he s ylised balance shee o he pension und. We ha e o
explain when and how he collec i e ese e is deployed and e illed.
FIGURE 1: S ylised Balance Shee
- 4 -
We assume ha employees en e he sys em a a ixed en y age x0 and ha hey e-
main in he popula ion un il dea h. I an employee dies be o e age z he balance o
he pe sonal accoun is paid ou . F om he e i emen age o z onwa ds an annui y is
paid un il he pe son dies.
He e we lis some basic no a ions wi h espec o he popula ion model:
: ime index = 0, 1, …, T
x0: ixed en y age, i no s a ed o he wise we se x0 = 20
z: ixed e i emen age, i no s a ed o he wise we se z = 65
ω
: maximal age, i no s a ed o he wise we se
115
ω
=
L( , x): numbe o pe sons o he ( , x)-coho , i.e. he numbe o pe sons who a e
x yea s old a ime . We assume ha each age coho is homogeneous,
i.e. all membe s sha e he same mo ali y isk and ha e he same pension
en i lemen s.
(, ) ( 1, 1)/ (, )
p x L x L x=++
: su i al p obabili y o he ( , x)-coho . This is
a andom a iable condi ioned o he a aible in o ma ion a ime , ob-
se able a ime +1.
ˆ(, )p x
: es ima ed su i al p obabili y o he ( , x)-coho o he ime in e al
[ , +1] based on he in o ma ion up o ime
(, )
a
p x
: ac ua ial su i al p obabili y o he ( , x)-coho . These alues a e used
o calcula ed he ac ua ial ese e o pensions due. The ac ua ial su i al
p obali ies could be bes o p uden es ima es. We do no model an ongo-
ing upda ing o pa ( , x) o ma ch he expe ienced mo ali y a e up a ce -
ain da e. Howe e , in he cou se o ou discussion we will also examine
he e ec o a mo ali y shi .
By de ini ion o
ω
we ha e
ˆ
(, ) (, ) (, ) 0
a
p p p
ωω ω
= = =
o all .
We do no model he idiosync a ic mo ali y isk, i.e. he isk ha a single pe son
dies in a ce ain ime pe iod. Ins ead, we allow o non in ege L( , x) and assume ha
( 1, 1) (, ) (, )L x L x p x+ +=
,
whe e he andom a iable
(, )p x
ep esen s he sys ema ic mo ali y isk.
- 5 -
We hink o
ˆ(, )p x
as any easonable bes es ima e o
(, )p x
. In p ac ice, he
ph ase bes es ima e does no necessa ily imply ha
( )
ˆ(, ) (, )p x p x=E
.4 We dis in-
guish be ween
(, )
a
p x
and
ˆ(, )p x
o allow o sa e y ma gins wi h espec o mo -
ali y a es.
We ega d he ini ial popula ion
( )
0
(0, ) :
L xx x
ω
≤≤
and he new en an s
( )
0
( , ): 0L x ≥
as de e minis ic.
0
1
( ): ( , )
z
A
xx
L L x
−
=
=
∑
: o al numbe o ac i e employees a ime
( ): ( , )
R
xz
L L x
ω
=
=∑
: o al numbe o e i ees a ime
(): () ()
AR
L L L = +
: o al popula ion a ime .
Fo con enience we de ine L( , x):= L( , x0) o all x < x0 and L(-1, x):= L(0, x) o all
x, assuming ha be o e ime =0 we had a s able popula ion. I no s a ed o he wise
we calib a e ou model popula ion such ha L(0, 20) = 1000.
2.1.2 S eady S a e Popula ion and Popula ion Dynamics
The bes es ima e p obabili ies
ˆ(, )
p x
a e aken om he mo ali y ables Rich a eln
2005G, 5 which a e he gene ally accep ed s anda d ables o calcula ing book e-
se es o DB- plans in Ge many. The en y age o he Rich a eln 2005G is x0 = 20
and he e minal age is
ω
= 115, i.e.
ˆ( ,115) 0p =
o all . The Rich a eln 2005G a e
de i ed om social secu i y da a o male and emale employees and comp ise ables
o all bi h coho s be ween 1891 and 2005. I indica ed we will p esen sepa a e e-
sul s o a male and a emale popula ion. Howe e , mos calcula ions a e pe o med
on he basis o a hyb id male/ emale popula ion . To his end we de ine hyb id su -
i al p obabili ies by
( )
() ( )
1
2
ˆˆ ˆ
(, ) (, ) (, )
male emale
p x p x p x= +
. One should be awa e o
he ac ha he esul ing hyb id popula ion is no he popula ion o a 50 - 50 mixed
male/ emale popula ion.
4 Fo example, in he s ochas ic CDB-model (as desc ibed in he [Cai ns e.a. 2006]) he “na u al” bes
es ima e is no necessa ily an unbiased es ima o .
5 “Re e ence ables” [Heubeck e al. 2006]
- 12 -
( )
2
1
12
( ) (1 )exp( ) exp( )
MM
PP
β µβ µ σ
+
=−++E
.
I we de ine
exp( ) 1
A
i
µ
= −
and
2
1
2
exp( ) 1
S MM
i
µσ
= +−
, we ge
1
(1 ) (1 ) (1 ) 1 ( )
A S A SA
Pi i i ii
P
ββ β
+
=− + + + =++ −
E
.
Howe e , one may con ince onesel ha
( )
1
( 1) : ln /
PP
µ
+
+=
canno be de-
composed as in (Eq. 2).
3 Asse Liabili y Managemen
3.1 Basic Rela ions
We de ine
()
(): ln ()/ () P V
ρ
=
- he log- ese e a io o simply he ese e a io.
He ha e
ρ
( ) > 0 i P( ) > V( ). In he ollowing
ρ
( ) will be he undamen al con ol
a iable o he asse liabili y managemen (ALM). Fo p ac ione s, he co e a io
P( )/V( ) a he han
ρ
( ) is aken as he indica o o he “wellbeing” o a pension
und. Clea ly, i makes no di e ence whe he we con ol
ρ
( ) o P( )/V( ). Bu , as we
will see,
ρ
( ) simpli ies no a ions. No e ha o P( )/V( ) ≈ 1 (say 0.8< P( )/V( ) <
1.2) we ha e 1 +
ρ
( ) ≈ P( )/V( ).
A ime (i.e. based on he in o ma ion up o ime ) he pension manage has o de-
dide on
σ
, he isk exposu e o he coming ime pe iod [ , +1]. I we apply a p o-
spec i e decla a ion, hen also
η
( +1) and
ε
( +1) a e de e mined a ime . I is clea
ha i we wan o gua an ee a minimum co e a io (o ese e a io) hen we mus
apply a e ospec i e decla a ion.
Fo he ollowing p oposi ions we de ine o ≥ 0:
( ) ( )
(, ) (, ) (, ) 1 (, ) (, ) (, ) 1
( , ): () () () ( )
RR
z
L xb x a x L xb x a x
w x V V B V
−−
= =
+− +
( o x ≥ z)
1(, )
( 1) : ln ( , )
(, )
xz a
p x
w x
p x
ω
π
−
=
+=−
∑
1
ˆ(, )
ˆ( 1) : ln ( , )
(, )
xz a
p x
w x
p x
ω
π
−
=
+=−
∑
- 13 -
()
( ): ()
CF
V
λ
=
( )
1 ()exp ()
() () ()
( ) : ln ln ln
() () () 1 ()
P CF P
V CF V
λρ
δλ
−−
−
= −=
−−
() () () ()
( ): () () ( )
RR
z
V V B V
V CF V
γ
+− +
= =
−+
.
Rema ks
1. Since
( )
1
( ) (, ) (, ) (, ) 1
R
xz
V L xb x a x
ω
−
=
+= −
∑
, w( ,x) is he ela i e weigh o he
( , x)-coho in VR( +). No e ha
1
(, ) 1
xz
w x
ω
−
=
=
∑
.
2.
ˆ( 1)
π
+
is he weigh ed sa e y ma gin i he ac ua ial assump ions wi h espec
o he su i al p obabili ies a e se so ha
ˆ
(, ) (, )
a
p x p x>
. I we use bes es i-
ma e su i al p obabili ies o ac ua ial alua ion we ha e
ˆ( 1) 0
π
+=
.
3.
( 1)
π
+
and
ˆ( 1)
π
+
only depend on he su i al p obabili ies o he coho o
e i ees.
( 1)
π
+
measu es o wha ex en he expe ienced and he ac ua ially
p esupposed mo ali y a es di e ge. I he ac ua ial assump ions include sa e y
ma gins hen
( 1)
π
+
is expec ed o be posi i e.
We ega d
ˆ( 1)
π
+
as he bes es ima e o
( 1)
π
+
based on in o ma ion up o
ime . As p ac i ione s we do use he ph ase “bes es ima e” a he gene ously.
In pa icula we do no s ipula e ha
( )
ˆ
( 1) ( 1)
ππ
+= +E
. One should no e
ha
( )
ˆ(, ) (, )p x p x=E
o all x and does no imply ha
( )
ˆ
( 1) ( 1)
ππ
+= +E
.
4. I
ˆ
(, ) (, )
a
p x p x=
hen
ˆ( 1) 0
π
+=
and
1
(, )
( 1) ln ( , )
ˆ(, )
xz
p x
w x
p x
ω
π
−
=
+=−
∑
.
( 1)
π
+
can be in e p e ed as he weigh ed mo ali y e ec .
5.
λ
( ) can be in e p e ed as he liquidi y a io, he a io o ou going money o he
o al liabili ies. No e ha
λ
( ) < 1 since CF( ) < V( ).
6. Since V( ) > CF( ) (by de ini ion)
δ
( ) is well de ined p o ided P( ) > CF( ).
7. No e ha
ρ
( ) = 0 implies
δ
( ) = 0. Fo
ρ
( ) > 0
δ
( ) is posi i e and inc easing
in CF( ) and o
ρ
( ) < 0
δ
( ) is nega i e and dec easing in CF( ).
δ
( )
- 14 -
measu es he e ec o he cash low CF( ) on he ese e a io
ρ
( ). CF( ) has
no e ec on he absolu e alue o he ese e P( ) – V( ), bu CF( ) ≠ 0 e ec s
he ese e a io. I CF( ) < 0, which is ypical o a young popula ion, he e-
se e a io will dec ease. This e ec is simila o he s ock dilu ion e ec when
addi ional common sha es a e issued. A sha e buy-back p og am has an oppo-
si e e ec . So we call
δ
( ) he s ock e ec . As we will see below, he s ock e -
ec will be posi i e i he pension sys em is in a s eady s a e. I is also posi i e
i he pension sys em is unwinding.
8.
γ
( ) can be in e p e ed as he weigh ed age bu den.
γ
( ) = 0 means ha he e
a e no pension liabili ies, and
γ
( ) = 1 implies ha he e a e no liabili ies o
ac i e wo ke s.
P oposi ion 1
I
η
( +1) is he p o i pa icipa ion o he indi idual pension accoun s and i he
pensions a e adjus ed by
ε
( +1), hen we ha e he ollowing ecu sions o he liabil-
i ies:
( )
( 1) exp ( 1) ( )
AA
V V
η
+= + +
(Eq. 4)
( )
( 1) exp ( 1) ( 1) ( )
RR
a
V V
ε µπ
+= ++ − + +
(Eq. 5)
I
( 1) ( 1) ( 1)
a
ε η µπ
+= +− + +
, hen
()( ) ( )
( 1) exp ( 1) ( ) ( ) exp ( 1) ( )V V CF V
ηη
+= + − = + +
(Eq. 6)
( 1) ( ) ( 1) ( 1) ( )
ρ ρµ η δ
+− = +− ++
. (Eq. 7)
P oo
To p o e (Eq. 4) we use de ini ion (Eq. 1) and he ac ha ( , x0) = 0:
01
( 1) ( , 1) ( 1, )
z
A
xx
V L x x
= +
+= − +
∑
( ) ( )
01
exp ( 1) ( , 1) ( , 1)
z
xx
L x x c
η
= +
= + − −+
∑
( )
0
1
exp ( 1) () (, ) (, )
z
xx
C L x x
η
−
=
=++
∑
( )
( )
exp ( 1) () () () ()
A
z
V D V C
η
= + −−+
.
- 15 -
To e i y (Eq. 5) we ake (Eq. 2) and use he de ini ion o w( , x) and
( 1)
π
+
and he
ecu sion o ä ( , x):
( )
( )
( )
( )
( )
1
1
1
( 1) ( 1, 1) ( 1, 1) ( 1, 1)
(, ) 1
exp ( 1) (, ) (, ) (, ) (, )
(, )
exp ( 1) () () () (, )
(, )
exp ( 1) ( 1) () () () .
R
xz
a
xz a
R
az
xz a
R
az
V L xb xa x
a x
p xL xb x p x
p x
V V B w x
p x
V V B
ω
ω
ω
µε
µε
µε π
−
=
−
=
−
=
+= ++ ++ ++
−
= ++
= ++ + −
= + +− + + −
∑
∑
∑
(Eq. 6) and (Eq. 7) ollow di ec ly om (Eq. 4) and (Eq. 5) and he de ini ion o
δ
( ).
♦
Rema k
I we de e mine
η
( +1) and
ε
( +1) e ospec i ely, i.e. on he basis o in o ma ion
up o ime +1, hen acco ding o (Eq. 7)
η
( +1) and
ε
( +1) can be de ined such ha
any p ede e minded ese e le el
ρ
( +1) can be eached. Fo example, i we de ine
( 1) ( 1) ( )
η µδ
+= ++
and
( 1) ( 1) ( 1)
a
ε η µπ
+= +− + +
, hen
( 1) ( )
ρρ
+=
. I
his was ou ALM-s a egy, we wouldn’ need a collec i e ese e! Howe e , in his
se ing capi al ma ke isks and he mo ali y isk would di ec ly a ec he indi idual
accoun s o pensions. The main bene i o a collec i e sys em, namely he in e gene -
a ional isk sha ing, would hen no be enabled.
Fixing
η
( +1) and
ε
( +1) a ime (and no a ime +1) e lec s he idea o de ined
ambi ion.6 This is a ac i e o sa e s and e i ees because hey know in ad ance,
how hei con ibu ions a e acc ued and how he pensions a e adjus ed.
In P oposi ion 1 we ha e se
( 1) ( 1) ( 1)
a
ε η µπ
+= +− + +
, which can only be de-
e mined e ospec i ely. Thus o a p ospec i e decla a ion we ha e o eplace
( 1)
π
+
by
ˆ( 1)
π
+
.
P oposi ion 2
I in he si ua ion o P oposi ion 1 we de ine
ˆ
( 1) ( 1) ( 1)
a
ε η µπ
+= +− + +
hen
( )( )
1
( 1) exp ( 1) ( ) ( )
V Y V CF
η
+
+= ++ −
(Eq. 8)
6 c. . [Day e al. 2014]
- 16 -
11
( 1) ( ) ( ) ( ) ( 1)
XY
ρ ρ σ µσ δ η
++
+− = − + + − +
, (Eq. 9)
whe e
()
()
( )
1
ˆ
: ln 1 ( ) exp ( 1) ( 1) 1
Y
γ ππ
+
= + +− + −
.
P oo
The de ini ion o
ε
( +1) oge he wi h P oposi ion 1 shows ha
() ( )
( ) ( )
( ) ( )
( )
( )
1
( 1) ( 1) ( 1)
ˆ
exp ( 1) ( ) exp ( 1) ( 1) ( 1) ( )
() () ˆ
exp ( 1) ( ) exp ( 1) ( 1)
() ()
ˆ
exp ( 1) ( ) 1 ( ) exp ( 1) ( 1) 1
exp ( 1) ( ) .
AR
AR
AR
V V V
V V
V V
V
V V
V
Y V
η ηππ
η ππ
η γ ππ
η
+
+= ++ +
= + ++ ++ +− + +
++
= + + + +− +
++
= + + + +− + −
= ++ +
(Eq. 9) is a di ec consequence o (Eq. 8) and he de ini ion on
δ
( ).
♦
Rema k
1. (Eq. 9) will be he basis o he ALM-s a egies which a e p esen ed in he nex
sec ion. The change o he ese e a io,
ρ
( +1)-
ρ
( ), can be b oken down in o
he s ochas ic capi al ma ke e ec
1
( 1) ( )
X
µ µσ σ
+
+= +
he s ochas ic longe i y e ec Y +1
he s uc u al s ock e ec
δ
( )
he p o i pa icipa ion
η
( ).
2. Admi edly, he de ini ion o Y +1 is a li le bi clumsy, bu i se es pe ec ly o
isola e he longe i y isk. Y +1 depends on he weigh ed age bu den
γ
( ) and he
di e ence be ween he es ima ed and he obse ed longe i y e ec
ˆ( 1) ( 1)
ππ
+− +
.
I
γ
( ) = 0 hen Y +1 = 0, and i
γ
( ) = 1 hen
1ˆ( 1) ( 1)
Y
ππ
+= +− +
.
Using he 2nd o de Taylo app oxima ion o he unc ion
( )
( )
ln 1 ( ) exp( ) 1
γ
∆ + ∆−
, we ge he ollowing app oxima ion:
( )
( )
1
12
()1 1 ()
Y
γγ
+≈∆ + ∆ −
wi h
ˆ
: ( 1) ( 1)
ππ
∆= + − +
. (Eq. 10)
3. I
ˆ( 1) 0
π
+=
, and especially i
ˆ
(, ) (, )
a
p x p x=
o all x, hen
1
1
(, )
ln 1 ( ) 1 ( , )
(, )
xz a
p x
Y w x
p x
ω
γ
−
+
=
=+−
∑
.
- 17 -
4. The pension adjus men
ˆ
( 1) ( 1) ( 1)
a
ε η µπ
+= +− + +
can be ega ded as
ai , since he e is no sys ema ic ans e o capi al be ween he young and he
old. I he ac ua ial su i ing p obabili ies pa( , x) a e calcula ed wi h sa e y
ma gins, hen he ini ial pensions b( , z) a e lowe compa ed o a bes es ima e
pension. Then
ˆ( 1)
π
+
ensu es ha he sa e y ma gins a e (on a e age) e-
unded o he coho o e i ees. Howe e , wi hin he coho o e i ees high
sa e y ma gins wi h espec o pa( , x) do ha e a edis ibu ional e ec , since
highe pension adjus men s a e unila e ally a ou able o long li ing e i ees.
3.2 ALM – S a egies
We now come o he ques ion o how o con ol he CDC-pension und desc ibed
abo e. Capi al ma ke oppo uni ies and isks, mo ali y a es and he numbe o new
en an s a e exogenous a iables, o which only he capi al ma ke isk can be con-
olled o a ce ain ex en . Ou CDC-pension und is sel inancing in he sense ha
he e is no ou side ins i u ion ha can s ep in i capi al ma ke s pe o m ex emely
badly o people li e much longe han expec ed. On he o he hand, he pension
membe can be su e ha e e y con ibu ion paid in o he sys em is exclusi ely used
o dea h o pension bene i s.
Since he pension und i sel does no gua an ee any bene i s, he e mus be some
good a gumen s o employees o en us hei con ibu ion o such a sys em. Ac u-
ally, he only good eason o en e such a collec i e sys em is ha he employees
ha e a good chance o ge a be e isk- e u n p o ile han in an indi idual sa ing and
dissa ing a angemen .
Be o e p esen ing ALM- ules o he CDC pension und, le us s a e some p inciples
ha he ALM has o comply wi h:
P inciple 1: The bene i s a pe son ecei es a e calcula ed on he basis o hei pe -
sonal pension capi al a e i emen age. Especially all pension membe s wi hin
an age coho a e ea ed equally.
The idea behind his p inciple is ha he sole pu pose o he collec i e elemen
in he CDC plan is o enable an in e empo al isk ans e . Thus, in he absence
o isk a CDC plan should be no hing bu a simple DC plan wi h a one- o-one
co espondence be ween con ibu ions and bene i s on he indi idual le el.
Ou CDC model complies wi h P inciple 1 since he pensions a e calcula ed on
he basis o accumula ed con ibu ion and u he mo e,
η
( ) and
ε
( ) apply
equally o ac i e wo ke s and e i ees espec i ely.
- 18 -
P inciple 2: I mus be ensu ed ha P( ) ≥ V( ), i.e.
ρ
( ) ≥ 0. We hink o a capi al
unded sys em, which in gene al means ha pension bene i s a e p e unded by
egula con ibu ions. In con as , in a pay-as-you-go p o ision sys em he cu -
en ly paid bene i s a e co e ed by cu en ly incoming con ibu ions. Ins ead o
P( ) ≥ V( ) o all , we could equi e ha a any ime all pension liabili ies can
be se led e en i he e a e no u he con ibu ions. Howe e , in a sys em wi h
no gua an ees he exp ession “all pension liabili ies” is a he ague o has o
be made p ecise. In ou model he unde s anding o VR( ) is ha his is he ac u-
a ial ese e unde he assump ion ha he cu en ly paid pensions a e kep
cons an in u u e. No e ha VR( ) is no he ma ke consis en alue o he pen-
sion liabili ies since we do no p ice he pension und’s implici op ion o in-
c ease o educe u u e paymen s i ci cums ances equi e.
In ou model CDC pension sys em we can ensu e P( ) ≥ V( ) only i we allow o a
e ospec i e decla a ion. In case o a p ospec i e decla a ion P( ) ≥ V( ) can only be
ensu ed wi h a ce ain deg ee o p obabili y. Thus, in he case o a p ospec i e decla-
a ion we ha e o ake a weakened e sion o P inciple 2:
P inciple 2’: I mus be ensu ed ha P( ) ≥ V( ), bu o a ansi ional pe iod P( ) <
V( ) is accep ed p o ided measu es a e aken o es o e ull unding.
P inciple 3: No age coho is sys ema ically p e e ed o pu a a disad an age
compa ed o o he s (in e gene a ional equi y). The equi emen o gene a ional
equi y is undamen al o any pension sys em - capi al unded o pay-as-you-
go. This issue is widely discussed in li e a u e.7
Admi edly, P inciples 2, 2’ and 3 a e pu in a he ague e ms. They con ey he
idea o a “ ai ” pension sys em, bu ai ness is no an ac ua ial concep . A his poin ,
i is wo h o men ion he undamen al concep , which John Rawls (1921-2002)
wo ked ou in his seminal book “A Theo y o Jus ice”. He add esses he p oblem o
jus ice be ween gene a ions om an abs ac pe spec i e so ha his ules a e no di-
ec ly applicable o a unded pension scheme.8 Howe e , his idea o a social con ac
ag eed upon behind he “ eil o igno ance” (“in he o iginal posi ion”) can be ap-
plied o he ques ion o a ai pension scheme. Behind he eil o igno ance people
do no know in ad ance whe he hei gene a ion will be lucky o unlucky wi h e-
spec o he indi idual li e span and o he u u e de elopmen o capi al ma ke s.
7 C . [Eu opean Union 2016] IORP II Di ec i e, A icle 7.
8 Rawls explici ly add essed he issue o in e gene a ional ai ness – c . [Rawls 1971], Chap e 44, pp.
251-258.
- 19 -
P inciple 2 equi es ha he ALM has o con ol he co e a io P( ) / V( ) o - which
is equi alen - he ese e a io
ρ
( ) . I
ρ
( ) h ea ens o all below ze o o some
h eshold
ρ
min measu es ha e o be aken , e.g. pension cu s and/ o he educ ion o
he isk exposu e on he asse side. In e gene a ional equi y (P inciple 3) equi es
ha
ρ
( ) is also capped abo e, since an un easonable la ge ese e a io indica es ha
he e is a sys ema ic ans e om he old o he young.
As poin ed ou , gene a ional equi y equi es ha
ˆ
( 1) ( 1) ( 1)
a
ε η µπ
+= +− + +
, o h-
e wise he e would be a sys ema ic income ans e be ween old and young.
In ou se ing a easible e ospec i e ALM ule is a ule which a ime (on he in o -
ma ion up o ime ) de e mines
η
( ) and
σ
, such ha P inciples 1, 2 and 3 a e sa is-
ied. A easible p ospec i e ALM- ule is a ule which a ime (on he in o ma ion up
o ime ) de e mines
η
( +1) and
σ
, such ha P inciples 1, 2’ and 3 a e sa is ied.
We can hink o a wide ange o ALM-s a egies ha comply wi h he abo e p inci-
ples. The ALM ules we use he e a e aken om [Goecke 2013]. The model p e-
sen ed he e is ime con inuous and es ic ed o he accumula ion phase. Bu he
basic ea u es can be ans e ed he disc e e case. In pa icula we adop he idea o a
s a egic ese e a io
ˆ
ρ
and a s a egic isk exposu e
ˆ
σ
. The pai
ˆˆ
(, )
ρσ
ep esen s
a s a e o equilib ium in he sense ha i we obse e a ese e a io
ˆ
()
ρρ
=
hen we
choose
ˆ
σσ
=
as he isk exposu e.
η
( ) is chosen such ha he ese e a io emains
unchanged p o ided capi al ma ke e u ns and mo ali y a es a e jus as expec ed.
Ano he ea u e aken om [Goecke 2013] is ha whene e
ˆ
()
ρρ
≠
we adjus
σ
and
( 1)
η
+
in dependence o he ese e gap
ˆ
()
ρρ
−
.
We now s a e ou basic ALM-s a egy in he p ospec i e e sion. Fo eal numbe s
max
ˆˆ
(, ,, , )a
ρσ θσ
we de ine
(ALM 1)
( )
max
ˆ
ˆ( ) and 0
a
σσ ρ ρ σσ
=+ − ≤≤
(ALM 2)
( )
ˆ
( 1) ( ) () ()
η µσ δ θ ρ ρ
+= + + −
(ALM 3)
ˆ
( 1) ( 1) ( 1) .
a
ε η µπ
+= +− + +
Rema ks:
1. The ese e gap
ˆ
()
ρρ
−
a he han he ese e a io
ρ
( ) is he decisi e con-
ol a iable o he CDC-pension sys em. Howe e , due o he s ock e ec
δ
( )
he absolu e le el o
ρ
( ) does ha e in luence on he p ocess.
- 20 -
2. (ALM 1) is mo i a ed by he ollowing conside a ions. Suppose a ime we
de e mine he isk exposu e
σ
unde he side cons ain , ha wi h p obabili y
1-
α
he ese e a io does no all below
ρ
min, i.e.
( )
min
( 1)
ρ ρα
+≤ ≤
P
.
Fo
( )
ˆ
( 1) ( ) () ()
η µσ δ θ ρ ρ
+= + + −
- c. . (ALM 2) - his is equi alen o
()
()
1 1 min
ˆ
() ()
XY
σ ρ ρ θρ ρ α
++
−≤ − + −≤
P
.
Le VaR
α
> 0 deno e he
α
- alue a isk o X +1, i.e.
( )
1
X VaR
αα
+≤− =P
, hen
o Y +1 = 0 (i.e. neglec ing he mo ali y isk) we ge
()
min
ˆˆ
(1 ) ( )
VaR
α
ρρ θρ ρ
σ
− +− −
≤
.
Thus, i we seek maximal isk exposu e unde he cons ain
( )
min
( 1)
ρ ρα
+≤ ≤P
, hen we ha e o de ine
( )
ˆ
ˆ()
a
σσ ρ ρ
=+−
wi h
min
ˆ1
ˆ,a
VaR VaR
αα
ρρ θ
σ
−−
= =
.
I we use he Black Scholes amewo k o he capi al ma ke (c . Rema k 1 o
sec ion 2.3), hen X is no mally dis ibu ed wi h a iance 1 and expec a ion 0.
On he basis o he Sol ency 2 secu i y le el o 1-
α
= 99.5% we ge VaR
α
=
2.5758. Suppose ha he egula o allows a empo a y unde unding o 90%
and a “no mal” unding a io o 115%, hen
ρ
min = ln(0.9) = -10.54%,
ˆ
ρ
=
ln(1.15) = 13.98%, and
min
ˆ1
ˆ0.0952 and a
VaR VaR
αα
ρρ θ
σ
−−
= = =
.
Assuming ha a b oadly di e si ied po olio o s ocks has a ola ili y o abou
19%,
ˆ0.095
σ
=
co esponds o an equi y a io o abou 50%.
The ques ion o how o calib a e
θ
, we will answe in iew o P op. 3, below.
3. Pa ame e a in (ALM 1) de e mines he adjus men speed wi h espec o he
isk exposu e. Fo a = 0 we ha e a cons an mix s a egy h oughou he ime
ho izon. I a > 0 hen he isk appe i e o he asse alloca ion changes in line
wi h he posi i e o nega i e ese e gap
ˆ
()
ρρ
−
. The case a < 0 co esponds
o a massi e an i cyclic in es men s a egy, because we hen inc ease he isk
exposu e a e bad expe ience wi h he pension asse s. Howe e , his s a egy
massi ely inc eases he isk o encoun e ing nega i e ese e a ios.
- 21 -
4. The side cons ain 0 ≤
σ
≤
σ
max allows us o keep he isk exposu e wi hin
easonable limi s. In ou se ing
σ
= 0 implies a isk ee in es men . We
should be awa e ha e en a po olio o AAA- go e nmen bonds is no isk
ee, since bond p ices a e d i en by ma ke in e es a es. So in p ac ice we
mus choose a
σ
no below some
σ
min > 0. We could skip he uppe bound
σ
max i we allowed o le e age ins umen s. Howe e usually hese ins u-
men s a e p ohibi ed o pension unds.
5. Acco ding o (ALM 1) and (ALM 2) he isk exposu e and he p o i pa icipa-
ion a e linea unc ion o he ese e gap. Since P( )/V( ) ≈ 1+
ρ
( ) o 0.8 ≤
P( )/V( ) ≤ 1.2, we can say ha isk exposu e and p o i pa icipa ion a e ap-
p oxima ely linea ly dependen o he ese e gap. Since a low co e a io
P( )/V( ) << 1 is gene ally ega ded as mo e c i ical han a high co e a io, he
ansi ion om P( )/V( ) o ln(P( )/V( )) is a leas plausible.
6. In (ALM 2)
η
( +1) has h ee componen s:
µ
(σ ) ensu es a ai pa icipa ion in he po olio e u ns. All pension
membe s di ec ly sha e he expec ed asse e u ns.
δ
( ) ensu es ha he capi al e u ns om he collec i e ese e a e e enly
edis ibu ed o he pension membe s.
The e m
( )
ˆ
()
θρ ρ
−
ep esen s an in e gene a ion isk ans e . I he
obse ed ese e a io alls behind he a ge a io, hen all membe s ha e
o pu ex a money aside o ill he gap. I he e is a posi i e ese e gap
hen all membe s ge an equal sha e. I is ob ious ha he gene a ion o
young employees would p e e a s ong ese e because his allows a
highe isk exposu e and, in he long un, a highe e u n on in es men .
The pensione s would be a he eluc an o s eng hen he collec i e e-
se e. In [Goecke 2013] his e m ( o
θ
> 0) ensu es he mean e e ing
p ope y o he s ochas ic p ocess
ρ
( ). Economically,
θ
< 0 makes no
sense; i is also clea ha wi h
θ
= 0 we had no con ol o e he ese e.
The case
θ
> 1 implies an o e eac ion – c . P op. 3 below.
7. As poin ed ou , he p ospec i e decla a ion in (ALM 2) canno ensu e ha P( )
≥ V( ). In o de o sa egua d a minimum ese e a io
ρ
min, we can de ine a e -
ospec i e a ian o (ALM 2) by
( )
( )
()
1 1 min
ˆ
( 1) ( ) () () , ()
e o
Min X Y
η µσδ θρρσ ρρ
++
+= + + − − + −
.
- 28 -
The compa ison o CDC- and IDC-pension a angemen s mus ake in o accoun ha
membe s o a CDC- plan ecei e addi ional e u ns om he collec i e ese e,
namely he s ock e ec
δ
which is posi i e p o ided
ρ
>0.
To measu e he e ec we calcula e TVIDC(x) and TVCDC(x), he ime alue o u u e
(dea h and pension) bene i s minus u u e con ibu ions o membe s he x-coho in
he IDC and CDC-case. Then TVCDC(x) - TVIDC(x) measu es he e ec o he ex a e-
u n o
ln(e e e )
ρ µ ρµ
δ
+
=− +−
om he collec i e ese e.
Figu e 5 illus a es his o
µ
= 0.025,
µ
a = 0.01,
ρ
=0.15 and
δ
≈ 0.0041. Fo exam-
ple, an employee, aged x0 = 20, en e ing he CDC plan will ecei e mo e bene i s
wi h a ime alue o abou 4.39 con ibu ion a es. This is exac ly he ime alue o
he addi ional e u n o
δ
.
FIGURE 5: Value added pe head (TVCDC(x) - TVIDC(x)) in a CDC-pension scheme
in s eady s a e wi h a cons an capi al ma ke e u n o
µ
=0.025, a ese e a io o
ρ
= 0.15 and a con ibu ion a e o c = 1.
4 Resilience Tes
A pension sys em is esilien , i i is able o abso b ex e nal (single) shocks and
adap o a (las ing) shi o he economic en i onmen . Ou esilience es wo ks as
ollows: We s a om a s eady s a e si ua ion and hen apply a shock o a shi sce-
na io and analyze he e ec s on he pension bene i s. In a DC pension sys em all dis-
u bances om ou side mus be compensa ed by adjus ing he pension bene i s. In he
IDC- e sion we do no allow o isk ans e be ween gene a ions, so he IDC- e -
sion will se e as a e e ence model o e alua e di e en ALM-s a egies o he CDC-
model.
- 29 -
4.1 S eady S a e O iginal Posi ion
We assume ha ou pension sys em s a s om a s eady s a e posi ion11 wi h ollow-
ing pa ame e s:
annual con ibu ions c = 1 payable om age x0= 20 un il age z-1 = 64
cons an capi al ma ke e u ns
( ) 0.025
µµ
= =
a s a iona y popula ion wi h ime independen su i al p obabili ies
( 1)
ˆ
(, ) (, ) (, ) ( ) ()
a
Lx
p x p x p x px Lx
+
= = = =
,
whe e p(x) a e he male/ emale hyb id su i al p obabili ies as desc ibed in
sec ion 2.1.2. We hen ha e
ˆ
() () 0
ππ
= =
o all .
cons an numbe o new en an s L(x0) = L(20) = 1000
ixed ac ua ial in e es a e o
µ
a = 0.01 and annui y ac o s o x ≥ z
0
()
( ) : exp( )
()
x
a
k
Lx k
ax k
Lx
ωµ
−
=
+
= −
∑
; ä(z) = 17.9249
pensions in paymen a e adjus ed a he a e o
ε
=
µ
-
µ
a = 0.015.
Fo he IDC-model he acc ued pension capi al a he age o x: x0 ≤ x ≤ z = 65 is hen
( )
0
exp ( ) 1
( ): 1 exp( )
xx
x
µ
µ
−−
=−−
,
and he ini ial s eady s a e pension is
( ) 84.2531
( ) 4.7003
17.9249()
z
bz az
= = =
.
To make IDC- and CDC-plans compa able we assume ha in he CDC-case we s a
wi h a s eady s a e ese e e a io
ˆ0
ρ
=
. Then in he s eady s a e si ua ion pensions
and dea h bene i s a e iden ical o IDC- and CDC-plans. We de ine
0
() () o
:() () () o
x
Lx x x x z
PLx xax z x
ω
≤<
=≤≤
and
( )
0
( 1) () () o
:0 o
x
Lx Lx x x x z
Dzx
ω
−− <≤
=<≤
o be he s eady s a e pension capi al o he x-coho and he dea h bene i o hose
who die be ween age x-1 and x. We deno e by P he o al s eady s a e pension capi al
and V he o al s eady s a e pension liabili ies. Unde he assump ion ha
ˆ0
ρ
=
we
11 c . sec ion 3.4
- 30 -
calcula e
00
11
z
xx
xx xx
PV P D
ω
=+=+
= = +
∑∑
= 2 530 615 + 5 851 = 2 536 466. No e ha by
ou con en ion P and V comp ise he dea h bene i s o he deceden s o he o ego-
ing yea .
4.2 Capi al Ma ke Shock
A capi al ma ke shock is associa ed wi h an equi y ma ke c ash o boom. S a ing
om a s eady s a e si ua ion wi h a cons an in es men e u n o
µ
= 0.025 we as-
sume ha a ime T0 (i.e. a he end o [T0-1, T0]) we obse e a e u n o
µ
+
µ
∆ wi h
µ
∆ = +0.2 (“up-scena io”) o
µ
∆ = -0.2 (“down-scena io”). In he ollowing ou wo d-
ing always e e s o he down-scena io, howe e he de i ed o mula s apply o ei-
he cases.
4.2.1 Capi al Ma ke Shock E ec on IDC-Plans
Ins an ly upon obse a ion o he capi al ma ke shock he indi idual pension ac-
coun s and he annui ies a e adjus ed. Conside he (T0, x)-coho , i.e. he gene a ion
o pe sons aged x a ime T0. Fo x0 ≤ x ≤ z he pe sonal pension capi al a T0 will be
’(x):= exp(
µ
∆) (x) ins ead o (x). A e T0 he annual e u n is again
µ
, he e o e
he esul ing annui y (z-x yea s la e ) will be cu by ac o
( )( )
()
1 exp ( ) 1 exp( )
()
x zx
z
µµ
∆
− −−
- c . FIGURE 6.
Fo x > z he due pension will be b’(x) = exp(
µ
∆) b(x) ins ead o b(x). F om ime T0+1
onwa ds pensions will again be adjus ed by
ε
=
µ
-
µ
a .
The capi al ma ke shock has he s onges e ec on pe sons aged z o olde . Thei
bene i s would be cu by abou 18% compa ed o he p e-shock le el.
- 31 -
FIGURE 6: Down-shock scena io (
µ
∆= -0.2) o IDC-plans: E ec on he expec ed
pension le el, depending on he age x a ime T0.
4.2.2 Capi al Ma ke Shock E ec on CDC-Plans
In he s eady s a e scena io we ha e a cons an expec ed e u n
ˆ
()
µσ µ
=
and no ex-
e nal dis u bances, i.e. X = Y = 0 and
ˆ0
ρ
=
. Then
ρ
( ) =
δ
( ) = 0 and
η
( +1) =
µ
o all < T0. In he s eady s a e o iginal posi ion o all ages x he coho pension
capi al Px and he indi idual pension capi al Px /L(x) coincide wi h he ime alue o
u u e bene i s minus con ibu ions.
Now conside a single in e es a e shock a ime T0 (i.e.
0
ˆT
X
σµ
∆
=
). Applying ule
(ALM 2) we ha e
( 1) () ()
η µ δ θρ
+=+ +
o all and
η
(T0) =
µ
. The e o e a
ime T0 nei he he indi idual accoun s (x) no he due pensions (x) a e a ec ed.
Howe e , he o al pension capi al a ime T0 alls o exp(
µ
∆) P and
ρ
(T0) =
µ
∆ and
0
( ) ln( )T ee e
µ µµ
µ
δ
∆∆
−−
= +−
. By P oposi ion 3 we know ha
0
( ) (1 )k
Tk
ρ θµ
∆
+=−
,
so o 0<
θ
< 2
ρ
( ) con e ges o
ˆ0
ρ
=
. In he special case
θ
=1 we ge
0
( 1) 0T
ρ
+=
and
η
(T0+1) =
µ
+
µ
∆ +
δ
(T0) =
( )
ln 1 ( 1)ee
µ
µ
µ
∆
++ −
. No e ha
ε
( ) =
η
( ) -
µ
a .
Due o he non- i ial s ock e ec
δ
( ) he e is no simple o mula o
η
( ). The e o e
we jus illus a e
η
( ) o he down-scena io (
µ
∆ = -0.2) o di e en le els o
θ
- c .
FIGURE 7.
- 32 -
FIGURE 7: E ec o a capi al ma ke down-shock (
µ
∆= - 0.2) a ime T0 on he e-
se e a io
ρ
( ) ( op cha ) and he p o i pa icipa ion
η
( ) (bo om cha ) o al-
e na i e le els o
θ
.
Fo
θ
= 0 he ese e a io will emain a he le el o
ρ
= -0.2 o e e . This means
ha all u u e gene a ions ha e o pay he bill: Due o he nega i e s ock e ec we
ha e
η
=
µ
-
δ
=
µ
-
ln( )ee e
µ µµ
µ
∆∆
−−
+−
= 2.04% ins ead o
µ
= 2.5%.
Fo 0 <
θ
< 2 he ese e a io e u ns o he s eady s a e le el. I we wan ed o a oid
a nega i e p o i pa icipa ion, we would ha e o choose
θ
≤ 0.1 wi h he conse-
quence ha i akes abou 7 yea s o hal e he a e -shock ese e gap o 20%.
Ou goal is o measu e he in e gene a ional e ec s o a CDC-plan compa ed o an
IDC-plan in a shock scena io. To his end o each (T0, x)-coho we calcula e
TVCDC (T0, x), he ime alue o u u e bene i s minus u u e con ibu ions. No e ha
o IDC-plans he ime alue equals he coho ’s pension capi al i.e.
0
( ) ( ) o
( , ) exp( ) exp( ) ()()() o
IDC x
Lx x x z
TV T x P Lx xax x z
µµ
∆∆
<
= = ≥
.
- 33 -
We ake
00
(): ( ,) ( ,)
CDC IDC
TV x TV T x TV T x∆= −
as a measu e o he in e gene a ional
asse ans e o he (T0, x)-coho , and ∆TV(x)/L(x) as he indi idual e ec .
Fig. 8 illus a es he in e gene a ional edis ibu ion in he down-shock scena io. Le
us conside he CDC-plan wi h
θ
= 0.2. Then he s eady s a e pension capi al is maxi-
mal o he (T0, z)-coho – we ge Pz = 75798. In he IDC-case he pension capi al
alls o exp(
µ
∆) Pz = 62058. In he CDC-case he pension capi al o he (T0, z)-coho
emains unchanged a e he shock, bu he ime alue o u u e pensions educes o
64206. This means ha he CDC-plan causes an in e gene a ional edis ibu ion o
∆TV(z) = 64206 - 62059 = 2147 in a ou o he (T0, z)-coho .
FIGURE 8: ∆TV(x) o age coho s 0 ≤ x ≤ 115 o di e en le els o
θ
o a down-
shock scena io (
µ
∆= - 0.2).
The e is an addi ional (small) edis ibu ion e ec in a ou o he dea h bene i s pay-
able a ime T0 a e he shock. While in he CDC-case he o al dea h bene i is no
a ec ed a T0 , in he IDC-case he dea h bene i is educed by ac o exp(
µ
∆). I is
clea ha he o al e ec o e all gene a ions (including u u e gene a ions o new en-
an s) mus be ze o.
I we look a he e ec s pe capi a we see ha he posi i e o nega i e edis ibu ion
e ec s amoun s o a mul iple o he egula con ibu ion (which is 1 in ou calcula-
ions) – c . Fig. 9. Fo example, in he case
θ
= 0.2 and
µ
∆ = -0.2 each single membe
o he (T0, z)-coho ecei es a ans e o ∆TV(z)/ L(65) = 2.39. In he ex eme case
θ
= 0 he ese e will emain a he a e shock le el o -0.2 o e e so ha all u u e
gene a ions will be cha ged. This ex eme case again shows ha a CDC-sys em could
be misused by he gene a ion 50+, who may ha e s ong in luence on ALM-decisions
and who a e p one o pos pone unpleasan decisions.
- 34 -
FIGURE 9: Redis ibu ion e ec pe head (
()/ ()TV x L x∆
) o di e en le els o
θ
o a down-shock scena io (
µ
∆= - 0.2).
The ollowing able shows he in e gene a ional edis ibu ion o a single capi al
ma ke down and up shock in ela ion o he p e-shock o al pension capi al. No e
ha o
θ
= 0 he ese e a io will emain a he le el a ime di ec ly a e he shock.
i.e
ρ
( ) =
µ
∆= - 0.2 o all ≥ T0.
Capi al Ma ke Down Shock
(
µ
∆ = -0.2)
Capi al Ma ke Up Shock
(
µ
∆ = +0.2)
θ
Redis ibu ion o he
olde gene a ion
Bene icia y
age coho s
Redis ibu ion o he
younge gene a ion
Bene icia y
age coho s
0 9.11% ≥46
11.86%
≤58
0.1 3.52% ≥59
4.67%
≤61
0.2 2.17% ≥62
2.86%
≤63
0.3 1.56% ≥64
2.06%
≤63
0.4 1.22% ≥64
1.62%
≤64
0.5 1.00% ≥65
1.33%
≤64
0.6 0.86% ≥65
1.13%
≤64
0.7 0.75% ≥65
0.99%
≤64
0.8 0.66% ≥65
0.87%
≤64
0.9 0.60% ≥65
0.79%
≤64
1.0 0.54% ≥65
0.71%
≤64
TABLE 1: O e all edis ibu ion e ec in % o o al p e-shock pension capi al in a-
ou o he olde gene a ion (down-scena io) o younge gene a ion (up-scena io)
o di e en le els o
θ
.
- 35 -
4.3 Capi al Ma ke Shi
We now wan o analyse he e ec o an in e es a e shi on a pu e bond po olio.
We analyse a sudden bu pe manen in e es a e shi om
µ
o
µ
':=
µ
+
µ
shi om
some ime T0 onwa ds. We associa e his s ylised si ua ion wi h a non-expec ed deci-
sion o he cen al bank o adjus in e es a es.12 This in e es a e shi has hen wo
e ec s: Fi s ly, new ixed income in es men s bea an in e es a e o
µ
' ins ead o
µ
,
and secondly, he e is a p ice e ec on exis ing bond in es men s. I he in e es a e
shi occu s a he beginning o he ime pe iod [T0, T0 +1] hen he ma ke alue o a
bond po olio wi h an a e age du a ion o D will chance by ac o ≈ exp(-D
µ
shi )
ins an ly a e he shi . F om ime T0 onwa ds all asse s including new in es men s
will ha e a e u n o
µ
'. I D > 0 hen he in e es a e up/down shi esul s in a single
down/up shock ollowed by a pe manen up/down shi .
We wan o check how IDC- and CDC-plans adap o his pe manen change o he
capi al ma ke . In ou wo ding we concen a e on a down shi scena io (
µ
shi < 0). I
is quie ob ious ha , cum g ano salis, in an up-shi scena io he same happens in he
o he di ec ion. To keep he a ian s o ou calcula ions in limi s we do no adjus he
ac ua ial in e es a e
µ
a , so ha he annui isa ion ac o s ä(x) emain unchanged.
I no s a ed o he wise ou nume ical examples a e calcula ed on he basis o
µ
= 2.5%,
µ
shi = -1.0%,
µ
'=
µ
+
µ
shi = 1.5%,
µ
a = 1%,
ε
' =
µ
' -
µ
a = 0.5%.
Fu he mo e, we conside he p ice e ec due o he in e es a e shi by assuming
ha he ime alue o asse s change by ac o D := exp(-D⋅
µ
shi ) o D = 0, D = 5 and
D = 10.
4.3.1 Capi al Ma ke Shi E ec on IDC-Plans
F om ime T0 onwa ds he indi idual pension capi al bea s in e es a he lowe a e
µ
' = 1.5%, pensions in paymen a e adjus ed by
ε
' = 0.5%.
We illus a e he e ec o he g oup o ac i e membe s – c . FIGURE 10 below. Fo
example a pe son aged x = 20 o younge a ime T0 will be a ec ed mos , because
hey expe ience he lowe in e es a es o he whole accumula ion phase. Thei pen-
sion capi al a age z = 65 will be 64.75 ins ead o 84.25, ha is abou 77% o he p e-
shi le el. This is independen o he du a ion o he unde lying asse s. Fo olde
12 Ac ually cen al banks can only de e mine he sho e m in e es a es, long e m in e es a es can
only be in luenced indi ec ly.
- 36 -
membe s he posi i e du a ion e ec o D > 0 can o e compensa e he educed u-
u e e u ns. Howe e , a e e i emen he pensions a e only adjus ed by
ε
' = 0.5%
ins ead o
ε
= 1.5%.
Pension in paymen will expe ience a single inc ease by ac o D ollowed by e-
duced pension adjus men s. Fo olde pensione s he du a ion e ec a ime T0 migh
o e compensa e he educed adjus men a e.
FIGURE 10: Down-shi scena io (
µ
shi = -1%) o IDC-plans: E ec on he ex-
pec ed pension le el a age z, depending on he age x a ime T0 and he du a ion.
Le TVIDC (x, T0) deno e he ime alue o u u e bene i s (pension and dea h bene i )
minus con ibu ions a ime T0 immedia ely a e he shi . I we wan o calcula e
TVIDC (x, T0) ma ke consis en ly i mus be calcula ed on he basis o he shi ed dis-
coun a e
µ
+
µ
shi . Then clea ly TVIDC (T0, x) = D Px , whe e Px deno es he p e-shi
pension-capi al o he x-coho .
4.3.2 Capi al Ma ke Shi E ec on CDC-Plans
We assume ha he pension managemen ins an ly ecognises he in e es a e shi as
pe manen . Acco ding o (ALM 2) o ≥ T0
( )
ˆ
( 1) () ()
η µ δ θρ ρ
′
+= + + −
.
Due o he in e es a e shi , a ime T0 he asse s ha e o be e alued. As abo e, we
assume ha P':= D P is he alue o asse s immedia ely a e e alua ion. Acco d-
ingly, a e e alua ion we ha e
0
( ) ln
shi
P
TD
V
ρµ
′
= = −
and
- 37 -
( )
0
( ) ln ln ln 1 exp( )(1 )
D shi
P CF P
T D
V CF V
δ µµ
′′
−
= − =− −+
−
.
He e we used he ac ha in he s eady s a e si ua ion CF = (1-exp(-
µ
)) P. Following
(ALM 2) we ge
()()
0 00
ˆ
( 1) ( ) ( ) ln 1 exp( )(1 ) (1 )
D shi
T TT D
η µ δ θρ ρ µ µ θ µ
′′
+ = + + − = + − − +−
and
00 0
0
00 0
( )exp( )
( 1) () ln ()
( )exp( ( 1))
( 1) () ().
P CF
TT T
V CF T
TT T
µ
ρρ ρ
η
µ η δ θρ
′′
−
+− = −
−+
′
= − ++ =−
We could ha e de i ed his di ec ly om (Eq. 11) o P op. 3. Mo e gene ally we ge
0
( ) (1 )
k
shi
Tk D
ρ θµ
+ =−−
.
Fo D = 0 he ese e a io is no a ec ed a all. Due o he non- i ial s ock e ec o
D > 0 he e is no simple o mula o
η
( ) o > T0+1. So we jus p esen nume ical
esul s – c . FIGURE 11. Fo D > 0 we obse e an inc ease o he ese e a io a ime
T0. A e T0 he ese e is d awn down depending on he speed pa ame e
θ
.
- 44 -
1
( 1) (1 ) ( )
Y
ρ θρ
+
+=− +−
,
( 1) () ()
η µ δ θρ
+=+ +
and
( 1) ( 1)
a
εηµ
+= +−
. (Eq. 19)
In pa icula
0
0
()
T
TY
ρ
= −
and
0
()T
ηµ
=
.
Di e en om he IDC-case, he e all age coho s a e ea ed equally and he coho
o ac i e wo ke s becomes in ol ed.
The e ec on he p o i pa icipa ion
η
( ) and he ese e a io
ρ
( ) is mo e complex
han in he case o a capi al ma ke shi , since he age p o ile o he popula ion
changes and i akes
ω
– x0 = 95 yea s un il a new s eady s a e popula ion is eached -
see FIGURE 15.
FIGURE 15: E ec o a mo ali y down-shi (∆ =+0.5) on he ese e a io
ρ
( )
( op cha ) and p o i pa icipa ion
η
( ) (bo om cha ) in a CDC-pension sys em
o S a egy 1 o al e na i e le els o
θ
.
- 45 -
Since om ime T0 onwa ds, he age bu den
γ
( ) and he weigh s w( , x) de ia e om
he s eady s a e alues, he Y -p ocess is no i ial. Clea ly, o
θ
= 0
ρ
( ) does no
con e ge, nei he does Y . One may check ha
0
0
0
( ) (1 )
k
j
T kj
j
Tk Y
ρθ
+−
=
+=− −
∑
. So we
can deduce ha o 0 <
θ
< 2,
ρ
( ) con e ges p o ided Y con e ges. The ollowing
able shows he new s eady s a e alues o
ρ
and
η
. E.g. o
θ
= 0.2 and ∆ = +0.5 he
CDC-sys em will con e ge o a new s eady s a e wi h a pe manen nega i e ese e
o
ρ
= -3.17%. The s eady s a e p o i pa icipa ion (
η
= 1.79%) alls behind he capi-
al ma ke e u n (
µ
= 2.50%) because he mo ali y shi has o be inanced yea by
yea and u he mo e due o he nega i e ese e we ha e a nega i e s ock e ec (in
his case
δ
= -0.0788%).
Δ = + 0.5
Δ = -0.5
θ
η
ρ
TV (x0=20)
η
ρ
TV (x0=20)
10%
1.72%
-6.25%
-3.8082
3.47%
7.66%
5.7557
20%
1.79%
-3.17%
-3.2344
3.35%
3.76%
4.3964
30%
1.81%
-2.12%
-3.0320
3.31%
2.49%
3.9811
40%
1.82%
-1.59%
-2.9286
3.29%
1.86%
3.7799
50%
1.83%
-1.28%
-2.8658
3.28%
1.49%
3.6611
60%
1.83%
-1.06%
-2.8237
3.27%
1.24%
3.5828
70%
1.84%
-0.93%
-2.7934
3.27%
1.08%
3.5272
80%
1.84%
-0.80%
-2.7706
3.26%
0.93%
3.4857
90%
1.84%
-0.71%
-2.7529
3.26%
0.82%
3.4535
100%
1.84%
-0.64%
-2.7386
3.26%
0.74%
3.4279
TABLE 4: S a egy 1: P o i pa icipa ion (
η
), ese e a io (
ρ
) and ime alue o
u u e bene i s minus con ibu ions (TW) o new en an s (x0=20) in he
adjus ed s eady s a e a e a mo ali y shi o ∆ = +/- 0.5.
We now u n o he ques ion o o wha ex en a mo ali y shi induces a ans e o
weal h be ween he age coho s. To his end we i s calcula e he ime alue
TVCDC (T0, x) o u u e bene i s (including dea h bene i s) minus u u e con ibu ions
o each (T0, x)-coho immedia ely a e he shi occu ed. Then he di e ence
TVCDC (T0, x) - TVIDC (T0, x) is a sui able igu e o measu e he in e gene a ional
weal h ans e . We also calcula e
00
0
( ,) ( ,)
( ,)
CDC IDC
TV T x TV T x
LT x
−
, he indi idual con i-
bu ion (posi i e o nega i e) o he in e gene a ional ans e .
Conside o example he (T0, z)-coho . A ime T0 we obse e mo e su i o s han
expec ed, namely
( 1)
() ()
( 1)
pz
L z Lz
pz
∆
∆
−
=−
ins ead o L(z). The o al pension capi al
- 46 -
o his coho is L∆(x) (x) = 76059.39 o bo h, he IDC and he CDC-case. In he
IDC-case all u u e pensions a e paid om his capi al s ock. In he CDC-case he
pensions a e adjus ed acco ding o (Eq.19). Fo
θ
= 0.2 he ime alue o all pensions
paid o he (T0, x)-coho amoun s o 83171.67. The di e ence 6706.07 is he in e -
gene a ional weal h ans e in a ou o he (T0, x)-coho , which comes o an indi-
idual ans e o 7.88. In o he wo ds, each single membe ecei es a subsidy o
abou eigh con ibu ion a es.
Le ’s now look a he (T0, x0)-coho . A ime T0 he pension capi al is ze o. In he
IDC-case all membe s o his coho know ha e e y Eu o hey pay in o he sys em
bea s an in e es a e o
µ
= 0.025 and will be paid back – a leas on a e age. In he
CDC- egime (
θ
= 0.2) we ge TVCDC (T0, x0) = -3 528.47 and an indi idual ans e o
-3.53. This means ha a new en an has o ealise ha mo e han 3 o he u u e con-
ibu ion a es a e ans e ed o he old gene a ion.
The ollowing FIGURE 16 illus a es he in e gene a ional ans e on coho -le el in-
cluding coho s o unbo n. I is clea ha he o al sum aken o e all exis ing and u-
u e gene a ions mus add up o ze o.
FIGURE 16: In e gene a ional edis ibu ion a e a mo ali y down-shi (∆ = +0.5)
in a CDC-pension sys em o S a egy 1 o age coho s x ≥ -80 and o
θ
= 0/ 0.1/
0.2/ 0.4/ 1.
FIGURE 17 shows he ans e on indi idual le el o x ≥ 0. Since he old age coho s
ha e ewe membe s he indi idual e ec is mo e signi ican .
- 47 -
FIGURE 17: In e gene a ional edis ibu ion pe head a e a mo ali y down-shi
(∆ = +0.5) in a CDC-pension sys em o S a egy 1 o ages x ≥ 0 and o
θ
=0.1/
0.2/ 0.6/ 0.8.
S a egy 2 (Ins an Recogni ion)
Ins an ecogni ion means ha a ime T0 he liabili ies in he balance shee a e ad-
jus ed o comply wi h he new su i al p obabili ies. Bu he bene i s payable a T0
(pensions and dea h bene i s) emain unchanged i.e.
η
(T0) =
µ
and
ε
(T0) =
µ
-
µ
a .
Fu he mo e, we assume ha he new pensions o he (T0, z)-coho a e calcula ed on
he basis o ä(z). Howe e om T0+1 onwa ds we apply äΔ(z).
Le P( ) esp. V( ) deno e he o al o asse s esp. liabili ies a ime ≥ T0 . By ou
con en ion P( ) and V( ) include he dea h bene i payable in o ac i e wo ke s
who die in [ -1, ]. Thus we ha e P(T0) = P, he s eady s a e alue o asse s. Le us
deno e by L´(x) he numbe su i o s o he (T0-1, x-1)-coho a e he mo ali y
shi .
P oposi ion 5
( )
0
( 1) ( )
( ) () () () 1 () ()()
( 1) ( )
xz
px ax
VT V L z Lz z Lxax x
px ax
ω
∆∆
=
−
′
=+− + −
−
∑
(Eq. 20)
P oo
( )
0
1
0
( ) ( 1)() () ( 1) () () ()()
z
xx xz
VT Lx x L z Lz z L xa x x
ω
−
∆
= =
′′
= − + −− +
∑∑
( )
( )
0
1
( 1)() () ( 1) () ()()()
( 1)
() () () () () ()()
( 1)
z
xx xz
xz
Lx x Lz Lz z Lxax x
px
L z Lz z a x ax Lx x
px
ω
ω
−
= =
∆∆
=
= − + −− +
−
′
+− + −
−
∑∑
∑
- 48 -
Since
( )
0
1
( 1)() () ( 1) () ()()()
z
xx xz
V Lx x Lz Lz z Lxax x
ω
−
= =
= − + −− +
∑∑
we ge (Eq. 20).
♦
No e ha o ∆ > 0
( )
() () () 0
L z Lz z
′−<
and
( 1) ( ) 10
( 1) ( )
px ax
px ax
∆∆
−−>
−
.
I he mo ali y shi is ecognised ins an ly, he e ec on he ese e a io and he
p o i pa icipa ion s ongly esembles he si ua ion a e a capi al ma ke down
shock. We illus a e he e ec s in FIGURE 18 below. As in Figu e 7 we see ha o
θ
> 0 he ese e a io will g adually e u n o he s eady s a e le el
ˆ0
ρ
=
.
FIGURE 18: E ec o a mo ali y down-shi (∆ =+0.5) on he ese e a io
ρ
( )
( op cha ) and p o i pa icipa ion
η
( ) (bo om cha ) in a CDC-pension sys em
o S a egy 2 (ins an ecogni ion) o al e na i e le els o
θ
.
As o S a egy 1 we measu e he in e gene a ional weal h ans e by compa ing he
ime alue o u u e bene i s minus con ibu ions o he (T0, x)-coho s. The ins an
ecogni ion o he mo ali y shi (∆ = +0.5) has a mild e ec on he (T0, x)-coho s
- 49 -
o x ≥ z since hei pensions a e only indi ec ly a ec ed ia educed
ε
( ). Howe e
hose who en e e i emen a T0+1 o la e ha e o endu e a double impac : i s ly he
p o i pa icipa ion and u u e pension inc eases will go down o e ill he ese e and
secondly, hei ini ial pensions a e calcula ed on he basis o he shi ed mo ali y.
This is illus a ed in FIGURE 19 and 20 below. We no ice a sha p cu a age x = 65
which is a esul o he ac ha due o he ins an ecogni ion o he mo ali y shi ,
om ime T0 onwa ds all new pensions a e calcula ed on he basis o he shi ed mo -
ali y a es. We see ha he edis ibu ional e ec o a mo ali y shi di e s clea ly
om ha o a capi al ma ke shock- compa e Figu e 19/ 20 and Figu e 8/ 9.
FIGURE 19: In e gene a ional edis ibu ion a e a mo ali y down-shi (∆ = +0.5)
in a CDC-pension sys em o S a egy 2 o age coho s x ≥ 0 and o
θ
=0.1/ 0.2/
0.6/ 0.8.
FIGURE 20: In e gene a ional edis ibu ion pe head a e a mo ali y down-shi
(∆ = +0.5) in a CDC-pension sys em o S a egy 2 o ages x ≥ 0 and o
θ
=0.1/
0.2/ 0.6/ 0.8.
- 50 -
Compa ison o S a egy 1/ 2 (delayed/ ins an ecogni ion)
The a oidance o cu ing pensions in paymen seems o be a ouchs one o a pension
plan. Acco dingly, he manage s o a pension plan will be e y eluc an o ac ually
cu pensions. I we look a he e ec o S a egy 1 o 2 on he pensions in paymen
(c . FIGURE 21) hen i is clea ha he “p oc as ina ion policy” (S a egy 1) is e y
a ac i e. We know om he analysis abo e ha S a egy 1 shi s he bu den o
longe li e expec ance o u u e gene a ions, who inhe i an e e nal loan om he old.
FIGURE 21: Pension le el o pensions in paymen a e a mo ali y down-shi
(∆ = +0.5) o S a egy 1 and 2 o di e en le els o
θ
. 100% ma ks he p e-shi
pension le el.
Bo h s a egies imply a massi e weal h ans e be ween he gene a ions. TABLE 5
below shows he weal h ans e in a ou o he olde gene a ions as a p opo ion o
he s eady s a e o al pension capi al (= 2543840). S a egy 1 u ns ou o p oduce a
s onge ans e han S a egy 2. I we compa e he age coho s ha p o i om he
ans e we see ha S a egy 1 is a ac i e o ac i e employees aged 54 and o e .
One may guess ha o many pension plans hese age coho s a e dominan in he
ep esen a i e bodies, so one migh expec ha in eal li e he e will be a s ong en-
dency o pos pone he upda ing o he mo ali y ables.
- 51 -
Mo ali y Shi (∆ = +0.5)
S a egy 1
(delay ecogni ion)
S a egy 2
(ins an ecogni ion)
θ
Redis ibu ion in % o
o al pension asse s
bu dened
age coho s
Redis ibu ion in % o
o al pension asse s
bu dened
age coho s
0
13.78%
x≤40
8.88%
x≤64
0.1
10.12%
x≤50
6.97%
x≤64
0.2
9.22%
x≤52
6.31%
x≤64
0.3
8.81%
x≤53
6.00%
x≤64
0.4
8.58%
x≤53
5.82%
x≤64
0.5
8.44%
x≤54
5.71%
x≤64
0.6
8.34%
x≤54
5.63%
x≤64
0.7
8.27%
x≤54
5.57%
x≤64
0.8
8.21%
x≤54
5.53%
x≤64
0.9
8.17%
x≤54
5.49%
x≤64
1.0
8.13%
x≤54
5.46%
x≤64
TABLE 5: O e all edis ibu ion e ec om young o old o a mo ali y shi (∆ =
+0.5) o S a egy 1 and S a egy 2 in % o o al p e-shi pension capi al.
Nei he S a egy 1 no 2 should be he choice in p ac ice! The e a e good a gumen s
o apply a mixed s a egy by adjus ing mo ali y a es s ep by s ep.
5 Concluding Rema ks
The p ima y pu pose o collec i e DC-plans is smoo h away he ups and downs o
capi al ma ke e u ns, which a e pa icula ly ola ile o s ock ma ke s. I he e is no
ex e nal ins i u ion o s ep in i equi ies slump, he smoo hing can only be done by
some kind o in e gene a ional isk ans e . In e gene a ional isk ans e is going
on since decades bu in gene al unila e ally a he cos o he younge gene a ion. The
shi om DB- o DC-plans is only one example. So he challenge is o ind ules ha
allow o a ai isk ans e be ween age coho s. Ou p oposal o such ules is
guided by he concep o esilience. We apply hese ules o se e al capi al ma ke
shock and shi scena ios and o mo ali y shi scena ios. We measu e he in e gene -
a ional e ec s; so we ha e ins umen o measu e in e gene a ional equi y. Ou con-
clusion is ha collec i e DC-plans a e he be e al e na i e compa ed o pu e DC-
plans.
- 52 -
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