Fo schung am i wKöln
Band 4/2022
Collec i e De ined Con ibu ion Plans – Back es ing
Based on Ge man Capi al Ma ke Da a 1950 - 2022
Oska Goecke
Collec i e De ined Con ibu ion Plans –
Back es ing Based on Ge man Capi al Ma ke Da a 1950 - 2022
Oska Goecke
TH Köln (Uni e si y o Applied Sciences), Ins i u e o Insu ance S udies, Claudiuss asse 1,
D50678 Köln, Ge many
oska .goecke@ h-koeln.de
Abs ac
Using his o ical capi al ma ke da a o Ge many (1950-2022) we analyze and compa e
(indi idual) de ined con ibu ion (IDC-) and collec i e de ined con ibu ion (CDC) pension
plans. To his end we de ine simple asse liabili y managemen ules ha go e n a CDC
pension plan and compa e hese o IDC-plans wi h he same asse allo a ion. Ou main esul
is, ha he CDC pension plans allow o a signi ican imp o emen o he isk e u n p o ile
compa ed o indi idual pension plans. He eby we conside di e en isk measu es. This
empi ical s udy a i ms he heo e ical esul s based on s ochas ic CDC-models.
1. In oduc ion
Pension sys ems wo ldwide consis o a combina ion o Pay-as-You-Go (PAYG) sys ems and
capi al- unded sys ems. F om he mac oeconomic iewpoin a PAYG sys em is linked o he
p oduc ion ac o labo while a capi al- unded sys ems is linked o he p oduc ion ac o
capi al. Bo h sys ems ha e hei me i s and d awbacks. One a gumen in a o o capi al
unding is he ac ha in mos de eloped coun ies we obse e an aging wo king popula ion,
ha pu s p essu e on PAYG-Sys ems. Howe e , a pension und canno di ec ly in es in o he
2
04.11.2022 09:46
p oduc ion ac o capi al in he no ion o na ional income accoun ing.1 In pa icula an
in es men in o co po a e bonds only indi ec ly secu es a sha e in capi al s ock o an economy
since pa o he p o i gene a ed by he company, namely a isk p emium, goes o he
sha eholde s. Simila conside a ions apply o go e nmen bonds. Thus - a leas in heo y -
only a 100% eal in es men (s ocks and eal es a e) would insu e a ull pa icipa ion in he
p oduc ion ac o capi al. Bu a 100% equi y in es men means an unaccep able isk o mos
sa e s. So, capi al unded pension sys ems a e con on ed wi h he isk- e u n dilemma: The
capi al can be in es ed in isky asse (such as equi ies) wi h high e u ns o can be in es ed
sa ely (e.g. in go e nmen bonds) wi h signi ican lowe e u ns. The equi y isk p emium
(ERP) is a well-es ablished key igu e o gauge he a e age ex a e u n i one in es s in
equi ies a he han in bonds. Howe e , an indi idual sa e who pu s aside pa o he / his
labo income du ing wo king li e may ace a s ock ma ke c ash jus a he momen she/ he
wan s o e i e o o buy an annui y. The p oblem is ha an indi idual sa e ( he same applies
o an age coho o sa e s) canno ealize he a e age e u n on equi ies.
Collec i e de ined con ibu ion (CDC) pension sys ems y o o e come his p oblem by some
kind o collec i e ag eemen among gene a ions o sa e s aiming o edis ibu e “de ia ions”
om he a e age. 2 The idea o in e gene a ional isk ans e wi h espec o capi al ma ke
isks is old since classical wi h-p o i li e insu ance o endowmen policies can be ega ded as
an implemen a ion o some kind o in e gene a ional (capi al ma ke ) isk ans e .3 The las
yea s esea ch in o in e gene a ional isk ans e a angemen s has helped o be e
unde s and CDC pension schemes. Applying s ochas ic models (including s ochas ic
simula ion echniques) one can p o e a posi i e wel a e e ec (e.g. Go don/ Va ian 1988,
1 Only a small ac ion o he na ional capi al (machine y, eal es a e, pa en s, in ellec ual p ope y, …) a e aded
on capi al (including eal es a e) ma ke s.
2 C . Gollie (2008)
3 C . Goecke (2003)
3
04.11.2022 09:46
Gullién/ Jø gensen/ Nielsen 2006, Gollie 2008, Hoe enaa s 2008 e c.) o CDC-a angemen
o we can de i e op imal ALM-s a egies (Chen/ Kanagawa/ Zhang (2021)). Howe e , o he
bes knowledge o he au ho , he e a e only ew empi ical s udies wi h espec o CDC plans
(Wesb oom/ A ends/ Tu nock/ Ha ding 2015 wi h UK-Da a). We wan o ill his “empi ical
gap” by a back es ing based on Ge man capi al ma ke da a. E en hough we a e wo king
wi h eal wo ld capi al ma ke da a i should be clea , ha he ollowing is no a s udy o eal
CDC-pension sys ems; i is a he a s udy abou how a CDC-pension sys em would ha e
pe o med i as p ope CDC-sys em had been ins alled in he pas decades.
The au ho is well awa e o he ac ha he ollowing s udy only e e s o he Ge man ma ke
and ha his o ic ma ke da a may no be ep esen a i e o wha migh happen in u u e.
Howe e , one should keep in mind ha e en he mos elabo a ed s ochas ic model is equally
limi ed wi h espec o he p edic i e powe – he u u e is only a ealiza ion o one o
millions o possible u u e andom walks!4
The CDC-model unde lying ou back es ing is inspi ed by he con inuous ime (c. .) model
p esen ed in Goecke (2013) and we wan o check whe he he heo e ical esul s de i ed o
he c. . model emain alid o eal capi al ma ke s. To his end i s ly, we ha e o o malize a
disc e e ime (d. .) e sion and secondly, we ha e o selec p ope capi al ma ke da a.
To di e en ia e om CDC-models we will e e o (indi idual) de ined con ibu ion (IDC-)
models, whe e he sa e pa icipa es in a no mal in es men und a angemen , whe e he
indi idual pension capi al is one- o-one linked o he ma ke alue o asse s o ha pa icula
in es men und.
4 We do no discuss he a he philosophical ques ion whe he s ochas ic models build on he ma hema ical
concep o p obabili y a e in p inciple app op ia e o model economic scena ios; he undamen al p oblem is ha
he ma hema ical concep o p obabili y implici ly p esumes ha unce ain e en s (like h owing dices) can be
epea ed a bi a ily o en and ha economic scena ios a e no he esul o h owing dices bu mainly esul o
human decisions, pa ly d i en by emo ions and i a ionali ies.
4
04.11.2022 09:46
Capi al Ma ke Da a o Back es ing
The c. . model in Goecke (2013) p esumes a s ylized capi al ma ke wi h a isk- ee asse wi h
a cons an in e es a e and a isky asse d i en by a geome ic B ownian mo ion wi h cons an
d i and ola ili y. I is assumed ha he pension asse s a e a mix o he isk- ee and he isky
asse and ha he equi y a io (i.e. he ela i e pa o isky asse s) can be adjus ed
con inuously. I we wan o check he heo e ical ime con inuous model wi h eal capi al
ma ke da a, we mus ind a sui able p oxy o a isk- ee asse and a isky asse . Fo ou back-
es ing we assume ha o a pension manage an in es men in o op a ed go e nmen bonds
is ega ded as a isk- ee in es men and ha an in es men in o a well-di e si ied po olio o
equi ies is ega ded as a isky in es men wi h a posi i e ex a e u n. We ake he Ge man
go e nmen bond index REXP as a p oxy o a isk- ee bond po olio and he Ge man equi y
index DAX a p oxy o a b oadly di e si ied equi y po olio.
Clea ly, an in es men in o a REXP-Po olio is no isk- ee in he s ic sense since om
mon h o mon h we obse e unexpec ed gains o losses om ola ile ma ke in e es a es
le e aged by he du a ion o he po olio. Consis en wi h ou p agma ic de ini ion o a isk-
ee asse , we will subs i u e he cons an isk- ee in e es a e o he ime con inuous model
by he cu en yield o ou s anding go e nmen bonds is( ), which we in e p e as he expec ed
e u n o a bond in es men . The ime con inuous model implici ly p esumes a ze o-in la ion
economy. We he e o e adjus ou eal capi al ma ke da a o elimina e in la ion. To his end
we ake a sui able consume p ice index CPI( ) and calcula e p ice adjus ed index alues
REXPp and DAXp. On he basis is( ) we de ine
µ
s( ), he p ice adjus ed expec ed (log-) e u n
(based on in o ma ion up o ime ) o he mon h ollowing o a isk- ee in es men . De ails
a e explained in Appendix.
5
04.11.2022 09:46
2. A simple CDC model o back es ing
2.1. Model desc ip ion
The CDC model o ou back es ing is a disc e e ime (d. .) adop ion o he con inuous ime
(c. .) model p esen ed in Goecke (2013). The disc e e ime in e al is ∆ = 1/12 i.e. one mon h.
We conside a pension und wi h asse s and liabili ies. The asse s a ime – deno ed P( ) –
a e in es ed in a mix u e o equi ies ( ep esen ed by DAXp) and bonds ( ep esen ed by
REXPp). The liabili y side o he balance shee o ou pension und consis s o wo pa s: he
o al o all indi idual accoun s – deno ed by V( ) – and a capi al ese e – deno ed by R( ) –
which se es as a bu e . We ha e P( ) = R( ) + V( ); we assume ha always P( ) > 0 and
V( ) > 0, bu we allow o a nega i e ese e R( ) < 0 in case o an unde unding o he
pension und.
Figu e 1. Balance shee o a CDC pension unds
We assume ha a he beginning o each mon h a new gene a ion en e s wo king li e and
s a s paying con ibu ions in o he pension und. A he same ime he gene a ion o eshly
e i ed lea es he pension und aking wi h hem hei indi idual asse s, i.e. hei sha e o V( ).
We hus do no conside li e ime annui y paymen s o e i ees, ins ead we assume ha all
e i ees ecei e a one-o paymen upon e i emen . We implici ly assume ha all sa e s li e
o see hei e i emen .
6
04.11.2022 09:46
We assume ha he sum o con ibu ion o all ac i e wo ke s jus ma ches he o al o money
paid ou o he e i ees. By his assump ion we ule ou dynamic e ec s om a g owing o
sh inking popula ion. We do no ebuild he s a -up phase o a CDC sys em; ins ead we s a
wi h an ini ial balance shee (P( 0), V( 0), R( 0)) in 0 = 0 ep esen ing01.01.1950. Howe e ,
wha we do is we simula e mo e o less a o able s a ing condi ions by se ing he ini ial
ese e a io a di e en le els. As poin ed ou , we use p ice adjus ed da a. Thus, 1 Eu o
con ibu ion payed by a gene a ion o ac i e wo ke s has he same eal alue as 1 Eu o paid
ou o he same gene a ion decades la e .
No a ion and De ini ions
Ou d. . model is based on mon hly da a, i.e. ou ime uni ∆ ep esen s one mon h. The ime
index ep esen s he i s day o he mon hs wi hin he back es ing pe iod (01.01.1950 o
01.07.2022). Gene ally, we iden i y he end o a mon h wi h he beginning he ollowing
mon h. Fo example, i we conside a 480-mon h sa ing plan s a ing a he beginning o Jan.
1960 and alling due a he end o Decembe 1999, we say ha he sa ing plan s a s a =
120∆ and ends a = 600∆.
A ime , he beginning o mon h [ , +∆[, he pension manage de e mines he equi y a io
β
( ) ≥ 0. Wi hin he ime in e al [ , +∆[ we ollow a buy-and-hold s a egy, i.e. wi h espec
o he pe o mance o P( ) we assume ha
( )
() ()
()() 1 ()
() () ()
pp
pp
DAX REXP
P
P DAX REXP
ββ
+∆ +∆
+∆ = +−
o ≥ 0. (Eq 1)
Fo ≥ ∆ we de ine
()
( ): ln ()
P
P
P
µ
=
−∆
.
7
04.11.2022 09:46
Fu he mo e, a ime ≥ 0 he pension manage de e mines he decla a ion
η
( ), namely he
(log-) in e es a e o he mon h [ , +∆[, by which he indi idual accoun s a e upda ed, i.e.
( )
( ) ()exp ()V V
η
+∆ =
o
()
( ) ln ()
V
V
η
+∆
=
.
No e ha
µ
P( ) can be obse ed a ime and e e s o he in es men pe iod [ -∆, [, while
η
( ) e e s o [ , + ∆[.
β
( ) and
η
( ) a e de e mined a ime based on he in o ma ion
a ailable up o ime . Thus
β
( ),
η
( ) and
µ
P( ) can be ega ded as -adap ed p ocesses.
We de ine
() ()
( ) : ln ln 1
() ()
P R
V P
ρ
= =−−
and call i ese e a io a ime . We p e e his
log-de ini ion ins ead o R( )/P( ) because i simpli ies he no a ion. Fo example, using his
de ini ion we ge he ollowing simple ecu sion o he ese e a io:
( ) () ( ) ()
P
ρ ρµ η
+∆ = + +∆ −
. (Eq 2)
The ac ha
η
( ) is de e mined a ime while
µ
P( +∆) canno be obse ed be o e +∆ implies
ha he ALM-manage canno gua an ee a posi i e ese e a io.
The ALM-s a egy p oposed in Goecke (2013) and analyzed in Chen e.a. (2021) is
cha ac e ized by he ollowing h ee ules:
Rule 1: The pension manage ies o keep he ese e a io
ρ
( ) close o a gi en s a egic
ese e a io
ρ
s ≥ 0; we call
ˆ(): () s
ρ ρρ
= −
he ese e gap.
Rule 2: The pension manage ollows a gi en s a egic equi y a io
β
s ∈[0, 1]. Howe e ,
we allow o mon hly adjus men s o
β
( ) as a eac ion o a posi i e o nega i e
ese e gap.
8
04.11.2022 09:46
Rule 3: The decla a ion
η
( ) should basically ollow he expec ed a e o e u n o he
pension asse s, in pa icula a high equi y a io should esul in a highe
decla a ion. Howe e , because o Rule 1, he ac ual decla a ion is adjus ed
acco ding o he ese e gap.
These gene al ules mus be speci ied. Again, we ollow he app oach in Goecke (2013) and
de ine an asse managemen ule (AM- ule) and liabili y managemen ule (LM- ule)
depending on pa ame e s
α
∆ ≥ 0 and
θ
∆ ≥ 0 :
AM-Rule:
( )
ˆ
() () ()
s ss
β βαρ ρ βαρ
∆∆
=+ −=+
wi h side cons ain 0 ≤
β
( ) ≤ 1, 5
LM-Rule:
( )
ˆ
() () () () ()
ee
P sP
ηµθρρµθρ
∆∆
= + −= +
,
whe e
()
e
P
µ
deno es he expec ed a e o e u n o he asse s based on he in o ma ion up o
ime . We de ine
( )
22
1
2
(): () () ()
e
Ps M
ERP
µ µ β βσ
= +∆ −
, whe e
µ
s( ) deno es he
expec ed e u n o a isk- ee in es men , ERP is he equi y isk p emium and
σ
M is an
es ima e o he a e age ma ke ola ili y. We ake ERP = 5% and
σ
M = 20% - see Appendix 1
o de ails. We should keep in mind ha he calib a ion o ERP and
σ
M only has e ec on he
expec ed e u n on asse s; he ac ual pe o mance o he pension po olio P( ) is no a ec ed
by he choice o
η
( ).. Howe e , i we o e es ima e ERP, hen he decla a ion
η
( ) ends o be
oo op imis ic and he p obabili y on a nega i e ese e gap will inc ease.
Gi en ERP and
σ
M he asse liabili y managemen is ully de e mined by only ou
pa ame e s, namely
β
s,
ρ
s,
α
∆ and
θ
∆.
5 We could allow o
β
( ) > 100%, which co esponds o a deb - inanced in es men in equi ies. Howe e , in he
con ex o pension managemen a deb - inanced specula i e in es men makes li le sense apa om he ac
ha his kind o in es men migh be p ohibi ed by supe iso y au ho i ies.
15
04.11.2022 09:46
1216.64, which is 57.24% lowe ha he inal capi al o he sa ing plan ma u ing 12 mon hs
be o e (2845.21). The a e age o e all ma u ing da es is ICImean = 11.52%. Fo a
β
= 50%
equi y po olio we ge ICImax = 30.86% and ICImean = 6.06%, and o
β
= 0% we ge
ICImax = 18.24% and ICImean = 2.95%. We see ha e en o a p esumed secu e in es men in
Ge man go e nmen bonds wi hin only 12 mon hs he pension capi al o wo gene a ions o
sa e s can di e by mo e han 18%. The eason o his is, ha be ween June 2021 and June
2022 he in e es a es and in la ion a es inc eased subs an ially.
As poin ed ou , we belie e ha o assess he isk o a ce ain pension ehicle (and hence he
accep ance by consume s) i is no enough o e alua e he inal capi al bu also he whole
accumula ion pe iod. The ups and downs du ing he accumula ion pe iod de ini ely can s ess
he sa e – in o he wo ds we hink ha a sa ing plan wi h hea y ups and downs a e
pe cei ed o be iskie han a sa ing plan wi h a con inuously inc easing acc ued capi al.
To illus a e his, we look a he 40-yea s sa ing plan ha s a s on 01.04.1963 and ends on
31.03.2003 – c . Figu e 3.
Figu e 3. Acc ued capi al o a sa ing plan (1.4.1963-31.3.2003) o
β
= 0%, 50%, 100%.
16
04.11.2022 09:46
We see ha he inal pension capi al is mo e o less he same o all h ee le els o he equi y
a io.12 Howe e , in es ing in o REXPp (
β
= 0) esul s in a mo e o less con inuously
inc easing acc ued capi al, while a DAXp (
β
= 100%) in es men makes any easonable
p ojec ion o he inal pension capi al impossible. The ola ili y o he mon hly e u ns13
du ing he in es men pe iod is 3.87% (
β
= 0%), 9.23% (
β
= 50%) and 17.77% (
β
= 100%).
The isk indica o s MDD and MLD a e illus a ed in Figu e 4. Figu e 4 depic s he
accumula ion pe iod o sa ing plans ma u ing end o June 2022 o
β
= 0% and
β
= 100%.
The MDD o he
β
= 100% sa ing plan is 67.92%, since be ween end o Feb . 2000 and end
o Ma ch 2003 he acc ued capi al ell om 931.87 o 298.95 - by 67.92%. The MDD o
β
=
0% is 16.71%, espec i ely.
The MLD o
β
= 100% and 0% is ( espec i ely) 202 and 121 mon hs. The MLD o
β
= 0%
is illus a ed in Figu e 4: he acc ued capi al end o May 2012 is 728.05 and 121 mon hs la e
(end o June 2022) i is s ill lowe , namely 725.14. The ex eme high loss du a ion o a “sa e”
in es men ehicle is caused by low (p ice adjus ed) in e es a es du ing mos o he sa ing
pe iod, he inc easing in e es a es combined wi h high and in la ion in 2022. S ill ue is ha
a isk-a e se pe son paid 121 con ibu ions while a he same ime he pu chasing powe o
he o his pension capi al did no g ow.
12 The a e o e u n is 4.48% ( o
β
= 0%), 4.81% ( o
β
= 50%) and 4.18% ( o
β
=100%)
13 Annualized s anda d de ia ion o 480 mon hly (log-) e u ns
17
04.11.2022 09:46
Figu e 4. Accumula ion pe iod o a sa ing plan (1.7.1982-30.6.2022) o
β
= 0% and 100%.
3. Resul s
3.1.1. Lump sum in es men
Be o e e alua ing he sa ing plans we illus a e he e ec o he in e gene a ional isk
ans o ma ion o CDC-plans. To his end e alua e a lump-sum in es men o he o al back-
es ing pe iod (01.01.1950 o 30.06.2022) o di e en in es men ehicles:
Cons an mix po olio wi h a sha e o
β
in es ed in o DAXp and (1-
β
) in es ed in o
REXPp po olio o
β
∈ {0%, 10%, …, 100%} – abb e ia ed CM(
β
).
In es ing in o a CDC-Po olio wi h
θ
∆ = 2%,
α
∆ = 0,
ˆ(0) 0
ρ
=
and wi h an
unde lying cons an mix po olio wi h
β
∈ {0%, 10%, …, 100%} - abb e ia ed
CDC(
θ
∆= 2%,
β
)
In es ing in o a CDC-Po olio wi h
θ
∆ ∈ {0.5%, 1%, 2%, 5%, 10%, 20%, 40%, 60%
80%, 100%},
α
∆ = 0,
ˆ(0) 0
ρ
=
wi h
β
= 100% - abb e ia ed CDC(
β
=100%,
θ
∆).
18
04.11.2022 09:46
Fo each po olio we e alua e he a e o e u n and he annualized ola ili y. The annualized
ola ili y (Vola) is calcula ed as
()
()
12 ln :0
P
P
S De T
+∆
⋅ ≤ ≤ −∆
o cons an mix
po olio in es men and as
()
()
12 ln :0
V
V
S De T
+∆
⋅ ≤ ≤ −∆
o in es men in o V( ).
To s a wi h, Figu e 5 jus illus a es he in e play o
µ
P( ),
η
( ), and he ese e gap
ˆ() ()
s
ρρρ
= −
. In his example he unde lying po olio is a DAXp in es men (
β
=100%)
and he pa ame e s o he CDC-model a e
( )
ˆ
, , , (0) (100%, 2%, 0, 0)
s
βθα ρ
∆∆
=
.
The smoo hing e ec o he in e gene a ional isk ans e is ob ious; he annualized ola ili y
o he
µ
P( )-pa hs is 17.83% and 2.05% o he
η
( )-pa hs – a educ ion o mo e han 88%!
This e u n smoo hing is only possible i we allow o a ola ile ese e gap. Since o
β
s = 100% and
α
∆ =0
ˆˆ
() () () () 0.03 ()
e
Ps
η µ θρ µ θρ
∆∆
= + = + ∆+
we ge as a ough
es ima ion
( ) ( )
ˆ
() ()S dDe S dDe
ηθ ρ
∆
≈
, i we neglec he impac o he
µ
s( )- e m.
Ac ually, in his example we ha e
( )
()S dDe
η
= 0.59% and
( )
ˆ()S dDe
θρ
∆
= 0.545%.
The mean e e ing cha ac e o he ALM- ules insu e ha he ese e a io always e u ns o
he s a egic le el. This e ec is e en mo e p onounced in p ac ice since equi y bea ma ke s
ha e always be ollowed by bull ma ke s and ice e sa.
19
04.11.2022 09:46
Figu e 5. (
µ
P( ),
η
( ),
ρ
( ) -
ρ
s) o 0 ≤ ≤ T
Fo he h ee ypes o in es men ehicles explained abo e (CM(
β
), CDC(
θ
∆ = 2%,
β
),
CDC(
β
= 100%,
θ
∆)) we calcula e isk- e u n p o iles, whe e isk is measu ed as he
ola ili y and he e u n as he (annualized) a e o e u n o e he back- es ing pe iod. The
ola ili y- e u n p o ile o CM(
β
) con i ms he isk- e u n dilemma, i.e. ha an inc easing
(expec ed) e u n comes along wi h inc easing isk (he e: ola ili y).
As poin ed ou he in es men in a pu e REXPp po olio is no isk ee wi h espec o
ola ili y. Howe e , ola ili y – as de ined he e - migh o e s a e he in es men isk, i we
wan o e alua e long- e m in es men s. DAXp and REXPp a e no ully co ela ed, so he e is
a di e si ica ion e ec be ween bo h.
The isk- e u n p o ile o CDC(
θ
∆= 2%,
β
) shows he isk-mi iga ing e ec o bu e ing
capi al ma ke a ia ions. The e is also a isk-mi iga ing e ec o a pu e REXPp -po olio (
β
= 0) since he collec i e ese e bu e s sho e m in e es a e luc ua ions.
20
04.11.2022 09:46
The a es o e u n o CDC(
θ
∆= 2%,
β
) a e sligh ly highe han he a es o e u n o CM(
β
)
due o he ac ha he inal ese e gaps a e sligh ly nega i e.
The isk- e u n p o ile o CDC(
β
=100%,
θ
∆) shows ha
θ
∆ is he c ucial ALM-pa ame e o
manage he smoo hing e ec . We do no include he case
θ
∆ = 0 because in his case he
ese e gap is jus a “ andom” walk and no mean e e ing.
Figu e 6. Re u n- ola ili y-p o ile o di e en IDC-/ CDC-in es men ehicles
Cons -Mix-Po olio
CDC(
θ
∆ = 2%,
β
)
CDC(
β
= 100%,
θ
∆)
β
a e o
e u n
Vola
β
a e o
e u n
Vola
θ
∆
a e o
e u n
Vola
0%
3.07%
3.84%
0%
3.18%
0.75%
0.5%
7.03%
1.01%
10%
3.61%
3.91%
10%
3.73%
0.78%
1%
7.10%
1.39%
20%
4.13%
4.73%
20%
4.25%
0.85%
2%
7.19%
2.05%
30%
4.61%
5.99%
30%
4.73%
0.95%
5%
7.28%
3.43%
40%
5.07%
7.48%
40%
5.18%
1.08%
10%
7.32%
4.98%
50%
5.49%
9.08%
50%
5.60%
1.22%
20%
7.32%
7.00%
60%
5.87%
10.76%
60%
5.99%
1.37%
40%
7.27%
9.84%
70%
6.23%
12.49%
70%
6.34%
1.53%
60%
7.25%
12.37%
80%
6.54%
14.24%
80%
6.66%
1.70%
80%
7.24%
14.94%
90%
6.83%
16.03%
90%
6.94%
1.87%
100%
7.25%
17.77%
100%
7.07%
17.83%
100%
7.19%
2.05%
Table 1: Da a unde lying Figu e 6
21
04.11.2022 09:46
Figu e 7 and Table 2 analyze he beha io o he ese e gap. Fo CDC(
θ
∆ = 2%,
β
) he mean
o he ese e gap is close o ze o asse ing he mean- e e sion cha ac e o he ese e
p ocess. Figu e 7 also shows ha he dis ibu ion o he ese e a io is oughly symme ic.
This is no he case i
θ
∆ alls below 2%, because hen
θ
∆ is o low o o ce he ese e gap
back o ze o.
Figu e 7. E alua ion o
{ }
ˆ( ), 0,..., T
ρ
=
o di e en CDC- a ian s.
22
04.11.2022 09:46
min max
10%-
quan ile
90%-
quan ile
median mean S dDe
β
CDC(
θ
∆ = 2%,
β
)
0%
-16.30%
13.22%
-7.08%
8.44%
0.96%
1.11%
6.10%
10%
-17.56%
15.57%
-7.89%
9.19%
1.16%
2.06%
6.61%
20%
-20.85%
19.45%
-9.59%
11.38%
1.40%
1.77%
8.00%
30%
-24.51%
28.98%
-11.42%
14.60%
1.67%
2.02%
9.91%
40%
-28.26%
39.34%
-13.47%
18.21%
1.96%
2.13%
12.10%
50%
-33.92%
49.72%
-15.71%
22.00%
2.29%
2.30%
14.45%
60%
-42.75%
60.13%
-17.76%
25.82%
2.64%
2.20%
16.90%
70%
-51.79%
70.57%
-20.06%
29.21%
3.02%
2.44%
19.42%
80%
-61.04%
81.04%
-21.98%
33.21%
3.43%
2.34%
21.99%
90%
-70.52%
91.54%
-24.08%
37.82%
3.86%
2.50%
24.60%
100%
-80.24%
102.08%
-25.97%
41.99%
4.31%
2.57%
27.25%
θ
∆
CDC(
β
= 100%,
θ
∆)
0.5%
-68.06%
148.90%
-25.83%
64.45%
14.81%
9.96%
37.03%
1%
-74.87%
128.83%
-31.54%
46.95%
7.94%
4.34%
33.14%
2%
-80.24% 102.08% -25.97% 41.99% 4.31% 2.57% 27.25%
5%
-74.13%
66.37%
-20.31%
26.60%
1.86%
1.04%
19.41%
10%
-63.83%
49.03%
-15.91%
18.37%
0.95%
1.10%
14.33%
20%
-50.69% 35.23% -10.78% 12.24% 0.46% 0.88% 10.12%
40%
-39.39%
23.85%
-7.89%
8.36%
0.22%
0.43%
7.12%
60%
-33.52% 18.79% -6.29% 6.83% 0.14% 0.61% 5.97%
80%
-30.65%
17.65%
-5.86%
6.13%
0.10%
0.51%
5.41%
100%
-29.84%
19.15%
-5.40%
5.81%
0.08%
0.29%
5.14%
Table 2: Da a unde lying Figu e 7
3.1.2. Analyzing 40-yea s sa ing plans
Figu e 8 depic s he inal capi al o 391 (o e lapping) 480-mon hs IDC-sa ing plans wi h
cons an equi y a io
β
= 0%, 50% and 100% - shown as solid lines. The co esponding inal
capi als o CDC-pension plans wi h
θ
∆ = 2% and
β
= 0%, 50% and 100% a e shown as
b oken lines. The di e ence be ween he solid and he b oken line indica es he deg ee o
in e gene a ional ans e o asse s.
23
04.11.2022 09:46
Figu e 8. Final capi al o 40-yea sa ing plans o di e en IDC-/ CDC- a ian s; he x-
axis shows he ma u i y da e
The hea y ups and downs o IDC-sa ing plans wi h
β
= 100% indica e ha e en o a long-
e m sa ing plan i is nea ly impossible o p edic he ou come. As poin ed ou we wan o
e alua e long- e m sa ing p ocesses unde he aspec o in e gene a ional ai ness, ha ing in
mind, ha a capi al unded sys em should enable a ai pa icipa ion in he economic capi al.
To his end we in oduced he isk measu e in e gene a ional imbalance (IGI) – in ela ion o
he a e o e u n, – c . he isk- e u n p o iles in Figu e 9. The e is some esemblance o he
isk- e u n p o iles in Figu e 6. No e ha he ange o a es o e u n o IDC-sa ing plans in
Table 3 ( om 3.97% o 6.66%) smalle han he ange in Table 1 ( om 3.07% o 7.07%).
This is due o he ac ha in Table 3 we analyze he a es o e u n o 480-mon hs sa ing
pe iods and ha he a e o e u n o each o hese is al eady a weigh ed a e age. The
in e gene a ional isk ans e is pa icula ly s ong wi h espec o he maximal IGI. He e we
obse e ha inc easing
β
( om 0% o 60%) comes along wi h a dec easing o he maximal
IGI. Wi h espec o he CDC(
β
= 100%,
θ
∆) sa ing plans we no ice o
θ
∆ = 0.5% an
24
04.11.2022 09:46
abno mal isk- e u n p o ile. Again, his is jus he consequence ha he mean e e ing e ec
is o low.
Figu e 9. A e age ( op) and maximal (bo om) in e gene a ional imbalance (IGI) o
IDC- and CDC-sa ing plans.
31
04.11.2022 09:46
a io. No e ha o
θ
∆ = 1 we ha e
ˆ
() () ()
e
P
ηµ ρ
= +
and hus he ola ili y o
( )
ˆ()
ρ
is
e y close o he ola ili y o
( )
()
η
.
4.1.2. Asse adjus men pa ame e
α
∆
The ollowing P oposi ion is he mo i a ion o ou LM-Rule
( )
(): ()
ss
β βαρ ρ
∆
=+−
and i
helps o calib a e he pa ame e s
β
s and
α
∆ . Suppose a ime we wan o con ol he “ uin
p obabili y” ha he ese e
ρ
( +∆) alls behind a gi en minimum le el
ρ
min , hen he
ollowing holds:
P oposi ion 15
I we assume ha
( ) () () ( )
dis
e
P P M
WW
µ µ βσ
+∆
+∆ − −
=
,16 we ge
( )
min
min
ˆ
(1 ) ( )
ˆˆ
( ) () ()
s
s
M
ρ ρ θρ
ρ ρ ρρ βσ
∆
− +−
+∆ ≤ − =Φ −
∆
and
( )
( )
min
( ) () () ()
ss
ρ ρ ρ ε β βαρ ρ
∆
+∆ ≤ ≤ ⇔ ≤ + −
, (Eq 4)
wi h
min
1
:
s
s
M
u
ε
ρρ
βσ
−
−
=
∆
,
1
1
:
Mu
ε
θ
ασ
∆
∆
−
−
=∆
and
1
1
: (1 )u
ε
ε
−
−
=Φ−
, he 1-
ε
quan ile o he
s anda d no mal dis ibu ion. He e (W ) deno es a (disc e e) s anda d Wiene p ocess.
◊
Rema k: The assump ion
( )
( ) () ()
dis
e
P P M
WW
µ µ βσ
+∆
+∆ − −
=
is da ing o se e al easons.
Fi s ly, e en o
β
( ) = 0
( ) ()
e
PP
µµ
+∆ −
bea s an in e es a e isk, secondly, he mon hly
15 The p oo is s aigh o wa d and le o he eade .
16 “
dis
=
” s ands o „has he same dis ibu ion as”
32
04.11.2022 09:46
e u ns o a pu e equi y po olio a e only app oxima ely no mal-dis ibu ed. Thi dly, gi en
ρ
( )
he dis ibu ion o
( ) ()
e
PP
µµ
+∆ −
is no independen o he -his o y – in o he wo ds, eal
wo ld capi al ma ke s a e no e icien .17
Example: Fo
σ
M = 20%,
ρ
s −
ρ
min = 20%,
ε
= 1% and
θ
∆ = 2% we ge
u1-
ε
= 2.3263 ,
min
1
148.91%
s
s
Mu
ε
ρρ
βσ
−
−
= =
∆
,
1
17.2965
M
u
ε
θ
ασ
∆
∆
−
−
= =
∆
.
This means ha – a leas in heo y - e en a deb inanced equi y in es men (
β
> 100%)
would be allowed, i we equi e ha in only 1 o 100 mon hs he ese e gap alls below
- 20%.
As can be seen om Figu e 15 he “ uin p obabili ies” (he e: he p opo ion o case whe e he
ese e gap alls below – 20%) a e much highe han he P oposi ion abo e sugges s. The
eason is ha log- e u ns o equi y ma ke s a e no no mally dis ibu ed.18
17 Fo example, “i a ional exube ances” - analyzed by Nobel lau ea e Robe Shille (Shille 2014) - s ongly
indica e ha equi y ma ke s a e no e icien .
18 I we ake
σ
M = 40% ins ead o
σ
M = 20% in he example abo e he uin p obabili ies acco ding Eq 4
app oxima ely co espond o he back es ing esul s in Figu e 15.
33
04.11.2022 09:46
Figu e 15: P obabili y ha
ˆ()
ρ
alls below -20% o di e en
α
∆-le els wi h unde lying
ALM pa ame e s
θ
∆ = 2% and
β
s = 100%/ 75%/ 50% ( ed/ o ange/ blue
column).
Back es ing esul s in Figu e 15 show ha he LM-pa ame e
α
∆ is a decisi e ins umen o
manage he ese e gap. Figu e 15 also indica es ha high
α
∆-le els could be
coun e p oduc i e a leas o
β
s = 75% o
β
s = 50%.
5. Conclusion and ou look
Ou analysis shows ha collec i e de ined con ibu ion (CDC) schemes a e mo e han jus a
nice idea. Based on he heo e ical model p esen ed in (Goecke, 2013) we could p o e ha he
isk e u n p o ile o CDC plans is much be e han ha o DC plan. E en in a wo s case
scena io CDC plans pe o m be e han indi idual DC plans. Clea ly, he e is no gua an ee
ha an excellen pe o mance obse ed in he pas will ecu in u u e. Bu i should be
s essed ha ou analysis is based on obse a ions o 721/2 yea s. Wi hin his ime span, we
ha e obse ed ex eme si ua ions such as he oil p ice c isis 1973, s ock ma ke c ashes and
34
04.11.2022 09:46
bubbles, high in la ion a es, and ex eme sho and long e m in e es a es.19 The s eng h o
he CDC plan is ha i is sel adjus ing due o esilience ac o s in he ALM ules. Wi h
espec o he undamen al objec i e o ensu e a ai pa icipa ion in p oduc ion ac o capi al,
a CDC plan is supe io o a pension plan wi h an in e es a e gua an ee. Looking back in o
he economic his o y, we should ealize ha long e m in e es a e gua an ees ei he a e
wo hless o a e unbea able o he wa an o . Pension sys ems mus be adjus able o he
economic eali y!
I has been c i icized ha a CDC plan is jus a ze o-sum game: he ad an age o one
gene a ion o sa e s is he disad an age o he o he . This iew o ally igno es he p inciple
o isk sha ing and insu ance. A CDC scheme is an “insu ance” con ac whe e he sa e pays
a p emium (in o m o con ibu ions in o he collec i e ese e) and ecei es bene i s (in o m
o paymen s ou o he collec i e ese e). In A Theo y o Jus ice John Rawls in oduced he
concep o ai ness unde he eil o igno ance.20 Following Rawls’ heo y a ansgene a ional
con ac be ween sa e s will be ega ded as ai , i he sa e s we e o ag ee upon his con ac
p o ided hey did no know in ad ance which gene a ion hey belong o – i.e. unde he eil o
igno ance. F om his pe spec i e a CDC plan is ai . Howe e , i is also clea ha a sa e
migh eel un ai ly ea ed i she o he was obliged o en e a hea ily unde unded CDC
scheme.
CDC schemes a e no an all-pu pose answe o he pension challenge in an aging socie y.
The e a e appa en limi a ions. Fo example a CDC plan can only wo k i he pa icipan s
canno wi hd aw money a will. Con ac ual compliance is essen ial o in e gene a ional isk
ans e . Tha does no mean ha CDC plans a e Ponzi plans. Whene e he low o new
19 The 1-mon h money ma ke a es anged be ween 13.33% (Sep . 1973) and – 0.6% (Dec. 2021).
20 C . (Rawls 1991), in pa icula Sec ion 24 (pp. 118 .) and Sec ion 44 (pp. 251 .)
35
04.11.2022 09:46
en an s s ops, a CDC plan can be con e ed in o a simple DC plan by se ing he a ge
ese e a io o ze o.
Ou back es ing analysis conside ed only he accumula ion phase. We assumed ha he
weal h a e i emen is paid ou and ein es ed ou side he und. An appa en ex ension o he
model is o include a decumula ion phase and hen analyze he pension isk, e.g. he ola ili y
o pensions in paymen . I would be equally in e es ing o analyze popula ion dynamics
(g owing o sh inking gene a ions, winding o ) in a CDC scheme.
We hope a leas , ha ou analysis has added one mo e a gumen in a ou o CDC plans as a
good al e na i e o DC and DB plans.
36
04.11.2022 09:46
Appendix: Da a Desc ip ion and Calib a ion o ERP and
σ
M
We ake REXP as a p oxy o a po olio o Ge man go e nmen bonds and DAX as a p oxy
o a well-di e si ied po olio o Ge man equi ies. REXP and DAX a e bo h pe o mance
indices calib a ed such ha REXP(end o yea 1987) = 100 and DAX(end o yea 1987) =
1000.
The index REXP is calcula ed on he basis o a po olio 30 di e en ic ious Ge man
go e nmen bonds wi h coupon a es 6%, 7.5% and 9% and ma u i ies be ween 1 and 10
yea s.21 The a e age Macauly du a ion is 5.02, 4,81 and 4.62 o in e es a es 0%, 3% and
6%, espec i ely. Clea ly, an in es men in o a REXP-Po olio (o in Ge man go e nmen
bonds) is no isk- ee in he s ic sense since om mon h o mon h a pension asse manage
will expe ience unexpec ed gains o losses om ola ile ma ke in e es a es le e aged by he
du a ion o he po olio. Consis en o ou p agma ic de ini ion o a isk- ee asse , we will
subs i u e he cons an isk- ee in e es a e o he c. . model by he cu en yield o
ou s anding go e nmen bonds (desc ibed in de ail below), which we in e p e as he expec ed
e u n o a REXP-in es men .
DAX is a weigh ed pe o mance index comp ising he 40 bigges Ge man join s ock
companies.22
End o mon h index alues o REXP and DAX a e p o ided by Deu sche Bundesbank om
end o Jan. 1967 (REXP) and end o Dec. 1987 (DAX) onwa ds.23 The end o mon h index
alues (based on he las ixing on he las ading da e o he pa icula mon h) is conside ed
21 Deu sche Bundesbank (2021); S a is ische Fach eihe Kapi alma k kennzahlen, Janua 2022, p. 17 .
22 C . Deu sche Bö se AG (2022).
23 Time se ies BBK01.WU3141 and BBK01.WU046A
37
04.11.2022 09:46
o be iden ical wi h he index alue o he i s ading da e o he ollowing mon h jus be o e
he i s ixing.
Rema k on no a ion: We use he ime index ep esen ing any poin in ime be ween 0 = 0
( ep esen ing he beginning o he i s ading day o Jan. 1950) and T = 870∆ ( ep esen ing
he beginning o he i s ading day o July 2022). Since we ha e a disc e e ime (d. .) model
based on mon hly da a, we some imes w i e = 01.mm.yyyy 24 ins ead o = k ∆ 25 jus o
make anspa en wha da e is e e ing o.
Fo REXP- and DAX-index alues no published by Deu sche Bundesbank, we use backwa d
p ojec ions om di e en sou ces:
Backwa d P ojec ion o DAX( ) ( 0 ≤ ≤ 01.01.1988)
We use he backwa d p ojec ion o Gielen (1994),26 adjus ed such ha he index alue
ma ches DAX(01.01.1988) = 1000.
Backwa d P ojec ion o REXP( ) ( 0 ≤ ≤ 01.01.1967)
Fo 01.01.1950 ≤ ≤ 01.01.1967 we use he ollowing o mula o backwa d p ojec ion
( )
1
12 1( )
() ( ) 1 () 1 ()
D
s
s
s
i
REXP REXP i i
−+ +∆
= +∆ ⋅ + ⋅
+
, (Eq A1)
wi h REXP(01.02.1967) = 21.24,27 is( ) = isk ee in e es a e obse ed a ime and D =
4.80.28
We in e p e is( ) o be he yield o an in es men in o Ge man go e nmen bonds held o
ma u i y in es ed a ime . Howe e , he ac ual e u n on in es men , calcula ed a ime
24 The i s o a mon h is iden i ied wi h he beginning o he i s ading da e o his mon h.
25 k coun s he numbe o mon hs a e 01.01.1950, i.e. k = 12·(yyyy-1950) + mm-1
26 Gielen, G ego (1994): Können Ak ienku se noch s eigen? Lang is ige T endanalyse des deu schen
Ak ienma k es, Gable -Ve lag, Wiesbaden 1994
27 REXP index alue o end o Jan. 1967 acco ding o Deu sche Bundesbank da a.
28 4.8 equals Macauly-Du a ion o REXP-po olio o an in e es a e o 3.19%.
38
04.11.2022 09:46
+ ∆, depends also on he in e es a e is( +∆) and he du a ion o he bond po olio. We
s ipula e ha he ma ke alue a ime +1 o a Ge man go e nmen bond in es men o
100€ a ime is
( )
1/12 1 ()
100€ 1 ( ) 1( )
D
s
s
s
i
i i
+
⋅+ ⋅
++∆
. This is he mo i a ion o (Eq A1).
Risk ee In e es Ra es: is( ) ( 0 ≤ ≤ T):
01/1950 ≤ ≤ 01/1953: Mean alue o Cen al Bank lomba d and discoun a e29
02/1953 ≤ ≤ 01/1954: Cu en yield o ou s anding public bonds (5.0% coupon)30
02/1954 ≤ ≤ 02/1956: Cu en yield o ou s anding public bonds (5.5% coupon)31
03/1956 ≤ ≤ 03/1960: Cu en yield o ou s anding public co e ed bonds32
04/1960 ≤ ≤ 07/2022: Cu en yield o ou s anding Ge man go e nmen bonds.33
No e ha e.g. is( ) is he a e age o obse ed yields o he o egoing mon h; is( ) can be
obse ed ime .
Money-Ma ke Index MMI( )
We calcula e a Money Ma ke index based on he ollowing money ma ke a es iMM( )
published by Deu sche Bundesbank:
01/1950 ≤ ≤ 01/1960: day- o-day money (a e age o minimum and maximum)34
02/1960 ≤ ≤ 01/1999: 1-mon h-money (mon hly a e age)35
02/1999 ≤ ≤ T: 1-mon h-EURIBOR (mon hly a e age).36
We se MMI( = 01/1950):= 100 and de ine ecu si ely
( )
1
12
( ) () 1 ()
MM
MMI MMI i +∆ = ⋅ +
.
29 Deu sche Bundesbank, Time Se ies BBK01.SU112 and BBK01.SU113
30 Calcula ed om published ma ke alues o co e ed bonds; Wi scha und S a is ik, Mon sbe ich e 06/1953
(p. 256*), 12/1953 (p. 676*) and 12/1954 (p. 652*)
h ps://www.s a is ischebiblio hek.de/mi / ecei e/DESe ie_mods_00000012.
31 E alua ion o Deu sche Bundesbank Mon hly Repo s Ap il – Dec. 1956
32 Deu sche Bundesbank, Time Se ies BBSIS.M.I.UMR.RD.EUR.MFISX.B.A150.A.R.A.A._Z._Z.A
33 Deu sche Bundesbank, Time Se ies BBSIS.M.I.UMR.RD.EUR.S1311.B.A604.A.R.A.A._Z._Z.A
34 Deu sche Bundesbank, Time Se ies BBK01.SU0102 and BBK01.SU0103
35 Deu sche Bundesbank, Time Se ies BBK01.SU0104
36 Deu sche Bundesbank, Time Se ies BBK01.SU0310
39
04.11.2022 09:46
Implici ly we he eby assume ha a ime we can in es in a 1-mon h money ma ke pape
wi h in e es a e iMM( ).
Consume P ice Index CPI( ) o = 01.01.1949 o = 01.07.2022:
CPI( ) is he conca ena ion o he ollowing wo ime se ies:
01.01.1949 ≤ ≤ 01.01.1991: Consume P ice Index o Wes Ge many 37
01.02.1991 ≤ ≤ 01.07.2022: Consume P ice Index38.
Figu e A1: P ice Index, Pe o mance o a DAX-, REXP- and a Money Ma ke -
in es men , no malized a 100 o 0 =1.1.1950, log-scaled
Figu e A1 illus a es he pe o mance o a DAX-, REXP- and MMI-in es men s a ing wi h
an ini ial capi al o 100 on 01.01.1950. To illus a e he e e se p ojec ion o REXP in Figu e
A1 we ha e added a o wa d p ojec ion o REXP ( om 01/1967 onwa ds) using o mula (Eq
37 S a is isches Bundesam (2021), P eise, Ve b auche p eise ü Deu schland, Lange Reihen ab 1948, „F ühe es
Bundesgebie , P eisindex ü die Lebenshal ung, 4-Pe sonen-Haushal e on A bei e n und Anges ell en mi
mi le em Einkommen, index basis = 100 (a e age o yea 1995). This ime se ies has been escaled such ha
he index alue o 01/1991 equals 86.9; he escaled index alue a e ounded o wo decimal poin s.
38 S a is isches Bundesam , Fachse ie 17, Reihe 7 ( ime se ies 61111-0002), index basis = 100 (a e age o yea
2015)
40
04.11.2022 09:46
A1) – see do ed line. Ob iously REXP can well be app oxima ed by using only one in e es
a e (namely is( ) as desc ibed abo e) and a sui able es ima ion o he a e age (Macauly-)
du a ion.
P ice adjus ed indices REXPp( ), DAXp( ) and MMIp( ) o = 01.01.1950 o 01.07.2022:
0
()
() () ()
p
CPI
REXP REXP CPI
=
,
0
()
() () ()
p
CPI
DAX DAX CPI
=
,
0
()
() () ()
p
CPI
MMI MMI CPI
=
wi h 0 = 01.01.1950 ≤ ≤ 01.07.2022
P ice adjus ed in e es a es
µ
s( ) and
µ
s( ) o = 01.01.1950 o 01.07.2022:
( )
( 12 )
1
() ln 1 ()
12 ( )
ss
CPI
i CPI
µ
−∆
= +
We use he log-in e es a e on mon hly basis o simpli y he no a ion. We in e p e
µ
s( ) as
he expec ed eal e u n o a isk- ee in es men a ime o he ollowing mon h [ , + ∆] on
he basis o he obse ed yield is( ) and he expe ienced dep ecia ion o he o egoing yea .39
µ
s( ) can be calcula ed on he basis o in o ma ion up o ime .
Figu e A2: Pe o mance o a p ice-adjus ed DAX-, REXP- and Money Ma ke -
in es men o 100 (log-scaled).
39 We p e e a p ice adjus men on a yea ly basis o elimina e seasonal e ec o he p ice index.
Publika ions eihe „Fo schung am i wKöln“
Die Ve ö en lichungen de Online-Publika ions eihe "Fo schung am i wKöln" (ISSN: 2192-8479)
we den übliche weise übe Cologne Open Science (Publika ionsse e de TH Köln) e ö en lich . Die
Publika ionen we den hie du ch übe na ionale und in e na ionale Biblio hekska aloge,
Suchmaschinen sowie ande e Nachweisins umen e e schlossen.
Alle Publika ionen sind auch kos enlos ab u ba un e www.i w-koeln.de.
2022
3/2022
Knobloch, Miebs: Ak uelle He aus o de ungen an das ak ua ielle und inanzielle Risikomanagemen
du ch COVID-19 und die anhal ende Nied igzinsphase. P oceedings zum 16. FaRis & DAV-
Symposium am 10. Dezembe 2021
2/2022
Knobloch: Ein Po olio on inhomogenen Ma ko -Ke en mi Abhängigkei ss uk u
1/2022
Ins i u ü Ve siche ungswesen: Fo schungsbe ich ü das Jah 2021
2021
4/2021
Ins i u ü Ve siche ungswesen: Risiko im Wandel als He aus o de ung ü die
Ve siche ungswi scha
3/2021
Völle , Mülle -Pe e s: Insu Tech Ka e i wKöln 2021 - Bei äge zu Insu Techs
und Inno a ion am i wKöln
2/2021
Knobloch: Die quan i a i e Risikobewe ung bei einem Po olio on dicho omen Risiken mi hil e des
zen alen G enzwe sa zes
1/2021
Ins i u ü Ve siche ungswesen: Fo schungsbe ich ü das Jah 2020
2020
7/2020
Mülle -Pe e s, Schmid , Völle : Re olu ionie en Big Da a und KI die Ve siche ungswi scha ? 24.
Kölne Ve siche ungssymposium am 14. No embe 2019
6/2020
Schmid : Küns liche In elligenz im Risikomanagemen . P oceedings zum 15. FaRis & DAV Symposium
am 6. Dezembe 2019 in Köln
5/2020
Mülle -Pe e s: Die Wah nehmung on Risiken im Rahmen de Co ona-K ise
4/2020
Knobloch: Modellie ung eine Can elli-Zusage mi hil e eine bewe e en inhomogenen Ma ko -Ke e
3/2020
Mülle -Pe e s, Ga ze : Todsiche : Die Wah nehmung und Fehlwah nehmung on All ags isiken in de
Ö en lichkei
2/2020
Völle , Mülle -Pe e s: Insu Tech Ka e i wKöln 2020 - Bei äge zu Insu Techs
und Inno a ion am i wKöln
1/2020
Ins i u ü Ve siche ungswesen: Fo schungsbe ich ü das Jah 2019
2019
5/2019
Mude s: Risiko und Resilienz kollek i e Spa p ozesse – Back es ing au Basis deu sche und US-
ame ikanische Kapi alma k da en 1957-2017
4/2019
Heep-Al ine , Be g: Mik oökonomisches P oduk ionsmodell ü Ve siche ungen. Teil 2:
Rendi emaximie ung und Ve gleich mi klassischen Op imie ungsansä zen.
3/2019
Völle , Mülle -Pe e s: Insu Tech Ka e i wKöln 2019 - Bei äge zu Insu Techs und Inno a ion am
i wKöln
2/2019
Rohl s, Pü z, Mo awe z: Risiken des au oma isie en Fah ens. He aus o de ungen und
Lösungsansä ze ü die K z-Ve siche ung. P oceedings zum 14. FaRis & DAV-Symposium am
7.12.2018 in Köln.
1/2019
Ins i u ü Ve siche ungswesen: Fo schungsbe ich ü das Jah 2018
2018
7/2018
Goecke: Resilience and In e gene a ional Fai ness in Collec i e De ined Con ibu ion Pension Funds
6/2018
Miebs: Kapi alanlages a egien ü die bAV – He aus o de ungen ü das Asse Managemen du ch
das Be iebs en ens ä kungsgese z. P oceedings zum 13. FaRis & DAV Symposium am 8. Dezembe
2017 in Köln
5/2018
Goecke, Heep-Al ine , Knobloch, Schiegl, Schmid (H sg.): FaRis a ICA 2018 – Con ibu ions o he
In e na ional Cong ess o Ac ua ies 2018 in Be lin. Bei äge on FaRis Mi gliede n zum Wel kong ess
de Ak ua e om 4. bis zum 8. Juni 2018 in Be lin
4/2018
Knobloch: Die P ade eine bewe e en inhomogenen Ma ko -Ke e - Fallbeispiele aus de
be ieblichen Al e s e so gung
3/2018
Völle , Mülle -Pe e s: Insu Tech Ka e i wKöln 1/2018 - Bei äge zu Insu Techs und Inno a ion am
i wKöln
2/2018
Schmid , Schulz: Insu Tech. P oceedings zum 12. FaRis & DAV Symposium am 9. Juni 2017 in Köln
1/2018
Ins i u ü Ve siche ungswesen: Fo schungsbe ich ü das Jah 2017
2017
8/2017
Ma e ne, Pü z: Al e na i e Capi al und Basis isiko in de S anda d o mel (non-li e) on Sol ency II
7/2017
Knobloch: Kons uk ion eine un e jäh lichen Ma ko -Ke e aus eine jäh lichen Ma ko -Ke e - Eine
Ve allgemeine ung des linea en Ansa zes
6/2017
Goecke, Oska (H sg.): Risiko und Resilienz. P oceedings zum 11. FaRis & DAV Symposium am 9.
Dezembe 2016 in Köln
5/2017
G undhö e , D euw, Quin , S egemann: Bewe ungspo ale - eine neue Quali ä de Konsumen en-
in o ma ion?
4/2017
Heep-Al ine , Meh ing, Rohl s: Bewe ung des e ügba en Kapi als am Beispiel des Da enmodells
de „IVW P i a AG“
3/2017
Mülle -Pe e s, Völle : Insu Tech Ka e i wKöln 1/2017 - Bei äge zu Insu Techs und Inno a ion am
i wKöln
2/2017
Heep-Al ine , Mülle -Pe e s, Schimikowski, Schnu (H sg.): Big Da a ü Ve siche ungen. P oceedings
zum 21. Kölne Ve siche ungssymposium am 3. 11. 2016 in Köln
1/2017
Ins i u ü Ve siche ungswesen: Fo schungsbe ich ü das Jah 2016
