Fo ewo d
The classical way o s ee ing a company by simple pe o mance indica o s has be-
come inc easingly subs i u ed by alue-based-managemen sys ems ha combine
isk and e u n in a easonable manne . In his con ex , he p ecise de e mina ion o
he capi al equi emen as he mos ele an ac o in p oducing insu ance co e is
ine i able. In o de o sepa a e he di e en in luences sui ably, an adequa e pe -
o mance measu emen should be ca ied ou .
In e nal models in non-li e insu ance usually ocus on he s ochas ic economic capi-
al a e one yea in o de o de e mine he capi al equi ed. In li e insu ance, he
Ma ke Consis en Embedded Value is belie ed o de ine a sui able concep o an
economic en i y alue. In non-li e insu ance, his concep also wo ks qui e well as
an al e na i e ai alue app oach.
This pape has been p oduced as a esul o a long e m p ojec ha has been ini-
ialized and execu ed by mysel oge he wi h he o he edi o s wi h u he pa icipa-
ion and con ibu ion o
S e an A ens
Vanessa Bi ne
La s Helmig
Lau a Hosse
Ch is ina Hübne
Simon Kau hold
K is ina Klein
Sonja Kohl
Sebas ian Langel
Hend ik Meye
Volha Mu askha
Anna Naumo a
Anne Neu
Jan Poll
I ana Simic
Ch is oph Wiebe
The a ge o his p ojec was o in oduce he opic “Value Based Managemen in
Non-li e Insu ance” s ep by s ep o as many people as possible enabling a p o ound
unde s anding wi hou equi ing oo much ma hema ical knowledge. The e o e, lo s
o examples ha e been de eloped, which a e no only as simple as possible, bu
also as complex as necessa y.
This p ojec would no ha e succeeded, i i had no been suppo ed by so many
in ol ed pa ies. Especially, he engagemen o he co-edi o s o his publica ion
has o be poin ed ou in his con ex .
Fu he mo e, we would like o hank M . Robe G. P ice o his ca e ul e ision o
he o iginal English documen .
Cologne, Ma ch 2013 Ma ia Heep-Al ine
Con en
1 INTRODUCTION TO VALUE-BASED MANAGEMENT ....................................1
1.1 B
USINESS
M
ODEL OF
I
NSURANCE
P
RODUCTION
...............................................2
1.1.1 Capi al E iciency due o Syne gy........................................................4
1.1.2 Risk Mi iga ion by Reinsu ance...........................................................7
1.1.3 Addi ional Syne gy due o In es men .................................................8
1.1.4 Legal F amewo k.................................................................................9
1.2 V
ALUE AND
R
ISK
-
BASED
M
ODELS
.................................................................10
1.2.1 T adi ional S ee ing Pa ame e ..........................................................10
1.2.2 Risk-based S ee ing Pa ame e ........................................................10
1.2.3 Requi ed Capi al e sus A ailable Capi al.........................................11
2 REQUIRED CAPITAL......................................................................................14
2.1 E
XTERNAL
M
ODELS
.....................................................................................15
2.2 I
NTERNAL
M
ODELS
–
B
ASIC
A
PPROACH
........................................................19
2.3 I
NTERNAL
M
ODELS
–
S
TOCHASTIC
P
ROFIT
&
L
OSS
A
CCOUNT
.........................29
2.3.1 Technical Resul – Unde w i ing Risk................................................31
2.3.2 Non- echnical Resul – Asse Risk ....................................................40
2.3.3 Non- echnical Resul – Reinsu ance De aul .....................................43
2.3.4 Non- echnical Resul – Ope a ional Risk...........................................44
2.3.5 Non- echnical Resul – Ex ao dina y Tax Dep ecia ion....................48
2.4 I
NTERNAL
M
ODELS
–
R
EQUIRED
C
APITAL
......................................................49
2.4.1 Comple e Model & Capi al Dis ibu ion..............................................50
2.4.2 Comple e Model & Capi al Dis ibu ion – Calcula ion Example .........51
2.4.3 De e mina ion o Requi ed Capi al ....................................................54
2.4.4 Capi al Alloca ion...............................................................................57
3 RISK-BASED PERFORMANCE MEASUREMENT.........................................63
3.1 U
NDERWRITING
P
ERFORMANCE
M
EASUREMENT
............................................63
3.1.1 T adi ional Pe o mance Measu emen .............................................64
3.1.2 T adi ional Pe o mance Measu emen – Calcula ion Example.........66
3.1.3 Risk-based Pe o mance Measu emen ............................................70
3.1.4 Risk-based Pe o mance Measu emen – Calcula ion Example........71
3.1.5 New Business e sus Exis ing Business – Calcula ion Example.......78
3.1.6 CoC Requi emen s and Ta ge Combined Ra ios.............................85
3.2 A
SSET
&
U
NDERWRITING
P
ERFORMANCE
M
EASUREMENT
..............................87
3.2.1 Asse Pe o mance............................................................................87
3.2.2 Asse & Unde w i ing Pe o mance ...................................................91
3.2.3 Asse & Unde w i ing Pe o mance – Sepa a e T ea men ................95
3.3 A
SSET
&
U
NDERWRITING
P
ERFORMANCE
O
PTIMIZATION
..............................100
3.3.1 P elimina y Rema ks.......................................................................101
3.3.2 Model App oach – Unco ela ed Risks............................................104
3.3.3 Model App oach – Gene al Case....................................................105
3.3.4 Calcula ion Examples......................................................................107
3.3.5 Conclusion.......................................................................................113
3.4 T
REATMENT OF
E
XTRA
D
IVIDENDS
..............................................................114
3.4.1 Cos o Capi al and Ta ge P emium...............................................114
3.4.2 Ex a Di idends Acco ding o Unde w i ing Pe o mance................115
3.4.3 Ex a Di idends Acco ding o Asse Pe o mance...........................118
3.4.4 Ex a Di idends Gi en Se e al Acciden Yea s...............................120
4 EMBEDDED VALUE AS FAIR VALUE APPROACH.....................................122
4.1 E
MBEDDED
V
ALUE IN
L
IFE
I
NSURANCE
........................................................123
4.2 H
ISTORICAL
D
EVELOPMENT
.......................................................................124
4.2.1 Applica ion o Embedded Value ......................................................127
4.2.2 Ma ke Consis en Embedded Value...............................................128
4.3 E
MBEDDED
V
ALUE IN
N
ON
-
LIFE
I
NSURANCE
................................................132
4.3.1 Di e ences be ween Li e and Non-li e Insu ance............................132
4.3.2 MCEV P inciples o Non-li e Insu ance ..........................................133
4.3.3 Gene al App oach...........................................................................139
4.4 E
MBEDDED
V
ALUE IN
N
ON
-
LIFE
I
NSURANCE
–
C
ALCULATION
E
XAMPLE
..........141
4.4.1 Example Company..........................................................................141
4.4.2 Ne Asse Value...............................................................................144
4.4.3 Value o In-Fo ce Business .............................................................147
4.4.4 Ma ke Consis en Embedded Value...............................................150
4.4.5 MCEV e sus Economic Capi al......................................................152
4.5 C
ONCLUSION
............................................................................................156
GLOSSARY..........................................................................................................158
BIBLIOGRAPHY...................................................................................................162
LIST OF FIGURES ...............................................................................................164
LIST OF ABBREVIATIONS ..................................................................................169
- 1 -
1 In oduc ion o Value-based Managemen
The concep o alue-based managemen (VBM) a ose om he ac , ha a he
end o he las cen u y companies we e becoming mo e and mo e complex. Man-
age s s a ed o conside he capi al used o in es men s as well as he capi al
cos s. Al ed Rappapo is ega ded as one o he co- ounde s o alue-based man-
agemen . His classic book “C ea ing Sha eholde Value”, which se co po a e s a -
egy in ela ion o he sha eholde alue, was published in 1986. Nowadays, his
managemen app oach is de ined as ollows:
“Value-based managemen is an app oach o managemen whe eby he com-
pany’s o e all aspi a ions, analy ical echniques, and managemen p ocesses
a e aligned o help he company maximize i s alue by ocusing managemen
decision-making on he key d i e s o sha eholde alue.”
1
The ocus o VBM is on he sha eholde alue. We ha e o bea in mind ha his is
a one-sided app oach, which does no conside he pe spec i e o o he s ake-
holde s.
Impo an o a alue-based managemen app oach is he a e o e u n, demanded
by he sha eholde s om he insu ance company. Due o se e al isks which in lu-
ence he sha eholde alue, an e ec i e alue-based managemen always includes
a compa ison o isk and he gene a ed alue.
The answe s o he ollowing ques ions p o ide he basis o a success ul alue-
based app oach:
• How can isks and alues be de ined, measu ed and compa ed?
• Which pa ame e s and echniques inc ease he sha eholde alue?
• Do he sha eholde s ge a isk adjus ed a e o e u n o hei capi al?
• Does he a e o e u n exceed he capi al cos s?
On he basic o hose aspec s he managemen o an insu ance company has o
decide, how o s ee he business acco ding o a alue-based app oach:
• Rega ding new business p emium calcula ion (p emium isk).
• Rega ding exis ing business ese e se ing ( ese e isk).
1
Sca le 2001, Value Based Managemen , p. 2.
- 2 -
Be o e ocusing on he di e en aspec s o alue-based managemen in he ollow-
ing chap e s, i is necessa y o unde s and he insu ance business model. In he
basic cons uc ion o insu ance business we ind he easons no only o he p o i -
abili y o he business model bu also o he di icul ies wi h espec o o he busi-
ness models.
1.1 Business Model o Insu ance P oduc ion
Fo a good unde s anding o he business model o insu ance he di e ences be-
ween insu ance and a ypical consume good (e.g. ca s) should be ou lined. In he
ollowing igu e he di e ences ega ding p oduc ion and sale in connec ion wi h he
alloca ion o isks a e shown.
P oduc ion Sale
Consume Good
P e Sale
T anspa ency o p oduc ion,
cos s a e almos ce ain.
ela i ely small isk
Pos P oduc ion
Volume o sales is am-
biguous.
ela i ely high isk
Insu ance Co e Pos Sale
No anspa ency o p oduc ion,
p oduc ion cos s a e ambiguous
(amoun and da e o paymen ).
ela i ely high isk
P e P oduc ion
Volume o sales is
known, possibly minimal
olume needed.
ela i ely small isk
Figu e 1: Insu ance Co e e sus a Typical Consume Good
I you conside a ypical ma e ial good (e.g. a ca ), he p oduc ion has o be inished
be o e i can be pu chased. The cos s o p oduc ion a e co e ed by he p oduce o
he good and ha e al eady been paid be o e he ca goes on sale. So he e is a isk
o sale, which implies he possibili y ha he p oduce canno sell he ca s he has
p oduced o he p ice ha co e s he p oduc ion cos s.
In con as an insu ance p oduc is an imma e ial good, which is p oduced a e he
con ac has been signed.
I du ing an ag eed pe iod o insu ance, a speci ied unce ain e en occu s, he in-
su ed pe son will be indemni ied by he insu e o he inancial loss.
2
Because
2
Ca e ; Lucas; Ralph 2000, Reinsu ance, p. 3.
- 3 -
he e is no isible p oduc ion o insu ance and he only physical i em he consume
ecei es is he policy, he achie emen s o insu ance co e e y o en seem qui e
non anspa en . Those aspec s especially imply ha nei he he policyholde no he
insu e knows
• whe he he insu ed e en will occu ,
• when i will occu and
• how (and o which ex en ) i will occu .
As a esul o his unce ain y i is di icul o calcula e he claims paymen s and o
de e mine he p emium. Due o he isk ans e be ween he insu ed pe son and
he insu ance company he insu e needs an es ima ion o he expec ed claims and
hei dis ibu ion o e ime. Because claims o en occu a e a ime lag, he insu e
has o es ablish a ese e. The accoun ing is pe o med on an acc ual basis.
The p emium in non-li e insu ance is no mally paid o a pe iod o one yea and is
cha ged di ec ly o wi hin a sho pe iod a e he insu ance con ac has been
signed o enewed.
All in all he p emiums ha e o co e he cos s o he insu ance company, which can
be di ided in o h ee ypes:
• Acquisi ion cos s (a he beginning o he con ac ),
• adminis a ion cos s (du ing he con ac ) and
• claims paymen s (a e a ime lag - some imes o se e al pe iods).
So he unde w i ing o insu ance includes he isk ha he p emium could be calcu-
la ed oo low o co e he cos s and claims paymen s o e ime. This unce ain y is
called unde w i ing isk.
To co e he unde w i ing isk, he insu e needs inancial supply: on he one hand
by he p emiums o he insu ed, on he o he hand by addi ional capi al supply. To
summa ize:
The amoun and poin in ime o u u e claims paymen s a e unce ain
and ha e o be secu ed by capi al. So capi al is he mos impo an
p oduc ion ac o o insu ance p oduc ion.
- 10 -
1.2 Value and Risk-based Models
To implemen a alue-based managemen , special models a e needed o measu e
isk and alue. As a consequence, he managemen o an insu ance company can
de i e decisions om he cu en alue and isk si ua ion.
1.2.1 T adi ional S ee ing Pa ame e
T adi ional s ee ing pa ame e s a e o en based on he balance shee in o ma ion o
an insu ance company ha is publicly a ailable. Examples o s ee ing indices a e
(ne ) p o i and i s ela ion o olume. O he s a e p emium olume, cos a io and
combined a io.
Bu all hese adi ional s ee ing pa ame e s no mally dis ega d he unde lying isk
o an insu ance unde aking. Because o he ac , ha he conside a ion o isk is
basic o a alue-based app oach, new isk based s ee ing pa ame e s mus be im-
plied.
1.2.2 Risk-based S ee ing Pa ame e
The ocus o alue-based managemen is a su icien isk analysis, as i helps o
assess he sol ency o an insu ance company and o inc ease he sha eholde
alue. In he ollowing, mode n and well known me ics o a isk-based pe o m-
ance e alua ion a e in oduced.
Re u n on Risk Adjus ed Capi al (RORAC)
As an al e na i e o he adi ional Re u n on Equi y (ROE), RORAC is a isk ad-
jus ed s ee ing pa ame e whe e he ollowing ela ion holds:
RORAC = Re u n / Requi ed Capi al.
The RORAC shows he ela ion be ween e u n and he equi ed capi al and is im-
po an o isk adjus ed pe o mance measu ing. To conside he isk, he economic
e u n is a o ed in compa ison o he e u n based on book alues (e.g. he Ge -
man GAAP e u n). I he insu e inc eases isk wi hou changing he p o i si ua ion,
he RORAC dec eases because mo e capi al is equi ed.
- 11 -
Economic Value Added (EVA
7
)
Ano he isk adjus ed s ee ing pa ame e is he Economic Value Added whe e he
ollowing ela ion holds:
EVA = Re u n – Cos o Capi al
= Re u n – Requi ed Capi al · CoC Ra io.
The EVA as an absolu e numbe shows he e u n o a business line minus cos s o
capi al. A posi i e numbe implies an added alue whe eas a nega i e EVA shows
a alue des uc ion. The cos s o capi al a e he p oduc o capi al equi ed and he
cos o capi al a io. The cos o capi al a io is he ex a di idend a io he in es o
demands and is in luenced by ex e nal and in e nal e ec s. I he isk inc eases
wi hou changing he p o i si ua ion, mo e capi al will be needed and he EVA will
dec ease. I he e u n is smalle han he equi ed capi al cos s, he EVA will be-
come nega i e and he business unp o i able. This is somewha c ucial i he CoC
a io is chosen a i icially high.
Risk Adjus ed Re u n on Capi al (RAROC)
A u he isk adjus ed pa ame e is he Risk Adjus ed Re u n on Capi al whe e he
ollowing ela ion holds:
RAROC = EVA / A ailable Capi al.
To calcula e he RAROC he equi ed capi al and a model o capi al cos s a e
needed. The RAROC is an index wi hou dimension whe eas he EVA is an abso-
lu e numbe .
1.2.3 Requi ed Capi al e sus A ailable Capi al
The ask o alue-based managemen is o compa e he a ailable and equi ed
capi al. In a su icien si ua ion he a ailable capi al is equal o he equi ed capi al
o e en highe .
7
S e n S ewa & Co. has adema ked he abb e ia ion EVA.
- 12 -
Figu e 7: Value Managemen e sus Risk Managemen
On he one hand, an e alua ion o he a ailable capi al is needed o analyze he
ac ual alue. Thus alua ion models like he secu i y p inciple (e.g. Ge man GAAP),
bes es ima e (e.g. US GAAP and pa ly IFRS ac ual s a us) o ai alue (e.g. IFRS
inal s a us) a e used o de e mine he a ailable capi al. Those a e qui e adi ional
alua ion app oaches which do no conside any isks.
On he o he hand, he equi ed capi al speci ies he amoun o capi al which is
needed o co e he isks aken by he insu ance company. Wi hin Sol ency II con-
ex , he equi ed capi al is de ined as he capi al needed o p o ec he company a
99.5% secu i y le el. Risk models o de e mine he equi ed capi al a e desc ibed in
he nex chap e .
I he amoun o a ailable capi al is lowe han equi ed capi al, he unde w i ing o
new business will no be possible in he same way as be o e. In such a case, he
insu e can unde ake he ollowing op ions o con inue business:
Reduc ion o isk
The isk o a g oss po olio can be educed by di e en echniques which a e de-
sc ibed below:
Dec ease o Volume
The isk olume can be dec eased h ough cancella ion o con ac s o p oduc s,
h ough isk exclusion o h ough limi (sum insu ed) dec ease. Bu less olume
may lead o less p o i .
Value Managemen
T adi ional App oach
A ailable Capi al
Valua ion Models
- Secu i y P inciple
- Bes Es ima e
- Fai Value
VBM
Risk Managemen
Ad anced App oach
Requi ed Capi al
Risk Models
- Ex e nal Models
- In e nal Models
- 13 -
Inc ease o P emiums
I possible, inc easing he p emium is he bes solu ion o inc easing he a ailable
capi al. Al hough he olume emains he same, he capi al and p o i si ua ion is
imp o ed. Bu ma ke compe i ion has o be aken in o accoun .
Change o Risk S uc u e
In o de o imp o e he isk s uc u e, he insu e can check i s isk po olio and
make changes o he unde w i ing. In he example men ioned abo e, he s anda d
de ia ion was 10,000 bu i i is possible o educe his o 8,000 by isk-adjus ed un-
de w i ing, he po olio size emains he same bu becomes less o a isk o he
insu e .
Pu chase o einsu ance
Reinsu ance educes he isk, bu i a ec s he p o i -si ua ion. This will be analyzed
in mo e de ail in he nex chap e .
Inc ease o capi al
The las op ion is injec ion o new capi al om he sha eholde s bu his educes he
p o i si ua ion.
The i s insu e has o de elop and imp o e me hods o measu ing isk and capi-
al, which is necessa y o secu e he isk co ec ly. The e u n should be adequa e
o a special isk s uc u e. Fo he e alua ion o a ailable capi al he balance shee
capi al (e.g. Ge man GAAP o IFRS) o he ( i ual) economic capi al (Embedded
Value in li e insu ance, sha eholde ’s ne asse alue in non-li e insu ance) can be
used.
- 14 -
2 Requi ed Capi al
The p o i and loss si ua ion o insu ance companies luc ua es om yea o yea
on he basis o isks caused by andom luc ua ions, e o s and changes. I hose
isks occu signi ican ly – conside ably highe claims paymen s han expec ed, e -
o s in he calcula ion o p emiums, changes in ex e nal in luences (e.g. judicial de-
cisions, p ice le els) – hen he p e iously collec ed p emiums a e insu icien and a
conside able loss a ises. I he claims (and adminis a ion) paymen s a e highe
han he p emiums in such yea s, he exagge a ion o loss mus be co e ed by he
insu e ’s capi al. The g ea e a company’s capi al, he lowe he dange o insol-
ency and he e o e he highe he p obabili y o a las ing gua an ee o gi en pe -
o mance p omises. This is economically desi able, because insol ency o an in-
su ance company has an impac on he whole economy. As he supply o capi al
equi es cos s, insu e s y o de e mine he (minimal) amoun o capi al which is
app op ia e acco ding o he isk. Thus – as al eady men ioned in he p e ious
chap e – he equi ed capi al is e y impo an o he alue based managemen o
an insu e .
To de e mine he equi ed capi al, he e a e se e al app oaches. These app oaches
can be di ided in o
• ex e nal models and
• in e nal models.
In some coun ies, such as Swi ze land and he Uni ed S a es, in e nal models a e
no au ho ized by he go e nmen . In he EU bo h models a e pe mi ed wi hin he
amewo k o Sol ency II egula ions.
The ollowing will explain he di e ences be ween ex e nal and in e nal models o
de e mining he equi ed capi al. The ex e nal models a e only ou lined b ie ly,
whe eas ocus will be placed upon he in e nal models as hese ha e a leas he
same equi emen s as he ex e nal models. Fu he mo e, in e nal models a e mo e
adequa e o co po a e managemen . The sec ion abou he in e nal models will be
di ided in o he h ee sub-chap e s:
• Basic app oach,
• s ochas ic p o i & loss accoun and
• equi ed capi al
- 15 -
A e explaining he basic app oach o he in e nal models, he indi idual compo-
nen s o a s ochas ic p o i & loss accoun will be cla i ied. On his basis, he me hod
o de e mining he equi ed capi al will be illus a ed.
2.1 Ex e nal Models
As men ioned abo e, ex e nal models p o ide a mo e simplis ic iew o he isk
si ua ion han in e nal models, because hey a e cha ac e ised by closed o mulas
and simpli ied bo om-up-app oaches a e usually ac o models.
Bo om-Up-App oach in his con ex implies ha he capi al equi emen mus be
de e mined sepa a ely i s o each ca ego y o isk. A e wa ds, he capi al e-
qui emen s om he indi idual isk ca ego ies a e agg ega ed o an o e all capi al
equi emen .
Based on a speci ic ime ho izon, ac o models compa e he a ailable capi al wi h
he equi ed capi al esul ing om he insu ance company’s isk posi ion. The dis-
ad an age o a ac o model is ha i does no desc ibe any quali a i e connec ions.
Thus no s a emen s can be made abou he insu ance company’s ac ual posi ion in
ela ion o isk. Ex e nal models can be di ided in o:
• sol ency models (as he Sol ency II model) and
• a ing models (as he S anda d & Poo model – S&P model)
Ra ing models o ins ance a e ela i ely simila o he sol ency models in hei cal-
cula ion, bu may no be used o de e mine he equi ed capi al. They only se e o
a ing pu poses.
Sol ency Models
Sol ency models indica e sol abili y ules o capi al adequacy o insu ance com-
panies. Besides he al eady men ioned Sol ency II model, which applies o he in-
su ance companies in he EU, he e a e o he sol ency models wo ldwide, such as
he Swiss (Swiss Sol ency Tes ) and he US-Ame ican (RBC s anda ds) sol ency
model.
To egula e capi al esou ces, Sol ency II, o example, uses a wo-s age app oach
which consis s o a Sol ency Capi al Requi emen (s age 1) and a Minimum Capi al
Requi emen (s age 2).
The Sol ency Capi al Requi emen (SCR) co esponds wi h he capi al which he
insu ance company should ha e a i s disposal in o de o ha e a high p obabili y
(a leas 99.5%) o no being echnically uined by he losses occu ing du ing he
- 16 -
ollowing pe iod o one yea . The Minimum Capi al Requi emen (MCR) e lec s he
p o ision o a minimal le el o he insu ance company's own unds and co esponds
wi h he amoun o capi al, below which he con inuance o he insu ance business
can be endange ed. A b each o he Minimum Capi al Requi emen leads o se ious
measu es, which can culmina e in a wi hd awal o he business license.
The amoun o dina ily equi ed o be main ained by he insu ance company co e-
sponds o he Sol ency Capi al Requi emen ( a ge capi al). Acco ding o Sol ency
II, his may be de e mined ei he by a uni o m so called “S anda d Fo mula“ as an
ex e nal sol ency model o by an in e nal model which will be explained in la e sec-
ions.
8
Fo a su icien capi aliza ion an insu ance company mus ha e a i s disposal a ail-
able capi al o a leas he same amoun as equi ed capi al, i.e.
A ailable Capi al / Requi ed Capi al ≥ 100 %.
Conce ning he model s uc u e, sol ency models ha e changed o e ime bu he e
a e ou main isk ca ego ies ha de e mine he gene al model amewo k.
Asse isks
De aul isks
Asse de aul
Ma ke isk
Asse de aul
Reinsu ance de aul
Cu ency isk
In e es a e isk
Unde w i ing isks Ope a ional isks
P emium isk IT ailu e
Rese e isk Managemen e o
e c.
Figu e 8: Risk Ca ego ies o Sol ency Models
9
8
Heep-Al ine a.o. (2010), p. 8-11; Heep-Al ine a.o. (2011), p. 5-9.
9
Heep-Al ine a.o. (2010), p. 12.
- 17 -
The able abo e illus a es he ou main isk ca ego ies co e ed by sol ency mod-
els (and ha mus be co e ed by all o he models a leas ) which will be explained
in he ollowing.
Asse Risks
Asse isks a e subdi ided in o special subca ego ies, e.g. asse de aul , ma ke
isk, ( o eign) cu ency isk and in e es a e isk. The asse de aul can also be
ca ego ized as de aul isk.
De aul Risks
In p inciple hose isks include he de aul o asse s as well as he de aul o ein-
su ance ( ega ded as an asse ), bu wi hin he amewo k o sol ency models he
asse de aul isk has been classi ied as an asse isk. In o de o educe he isk o
einsu ance de aul , a minimum a ing should be equi ed wi h espec o he e-
insu e chosen.
Unde w i ing Risks
Unde w i ing isks a e di ided in o p emium isk and ese e isk. The p emium isk
is limi ed exclusi ely o inco ec ly calcula ed p emiums o unusually high losses
om new business. The ese e isk is he isk ha he ese es o ou s anding
claims o he exis ing business a e oo low. An unde w i ing loss he e o e a ises, i
he calcula ed p emium o he acc ued ese es a e lowe han needed.
Ope a ional Risks
Ope a ional isks a e no o iginally insu ance-speci ic isks. They include all ope a -
ing isks which can cause losses in a business. Fo example, managemen e o s
o he ailu e o adminis a i e sys ems belong o his ca ego y.
Fo each g oup o isks conside ed, a sepa a e capi al equi emen is calcula ed.
The indi idual capi al equi emen s a e agg ega ed o ob ain he o al capi al e-
qui emen whe e di e en co ela ions a e aken in o accoun . The agg ega ion o
he indi idual isks can be dis inguished concep ually be ween wo assump ions:
1. The isks R
1, …,
R
k
wi h he capi al equi emen s C
1, …,
C
k
a e assumed
o be ully dependen on each o he as well as on he esidual isk.
2. The isks R
k+1, …,
R
n
wi h he capi al equi emen s C
k+1, …,
C
n
a e
assumed o be co ela ed wi h he co ela ions ρ
ij
.
- 18 -
In summa y he ollowing gene al agg ega ion o mula can be es ablished:
C
o al
= C
1
+ ... + C
k
+ (∑
i>k
C
i2
+ ∑
i,j>k
ρ
ij
· C
i
· C
j
)
1/2
This agg ega ion ule shall be illus a ed in he ollowing example o an insu ance
company wi h he ollowing alues:
Ope a ional Risk (OR) 100.0,
Unde w i ing Risk (UW) 400.0,
Asse Risk (A) 300.0.
The capi al equi emen s due o unde w i ing isk and asse isk a e conside ed o
be o ally independen whe e he ollowing assump ions hold wi h espec o he
capi al equi emen s due o ope a ional isk:
To ally Independen OR: C
o al
= [C
UW
² + C
A
² + C
OR
²]
1/2
= [400.0² + 300.0² + 100.0²]
1/2
= 509.9
Fully Dependen OR: C
o al
= [C
UW
² + C
A
²]
1/2
+ C
OR
= [400.0² + 300.0²]
1/2
+ 100.0
= 600.0
This example shows how much in luence he dependence s uc u e has upon he
de e mina ion o he capi al equi emen . The capi al equi emen s o he subg oups
a e iden ical in bo h a ian s bu hei dependency is di e en . Thus di e en o al
capi al equi emen s esul .
10
Ra ing Models
In he ollowing, he a ing models will be explained on he basis o he S&P model.
The S&P model (like sol ency models) is a ac o model and also esul s om a
bo om-up app oach. Conce ning he unde w i ing isk, his model ep esen s a
simple app oach consis ing o an en i y ac o , a p emium ac o and a ese e ac-
o o he de e mina ion o he capi al equi emen . I is used o a ings pu poses.
10
Heep-Al ine a.o. (2010), p. 11-14; Heep-Al ine a.o. (2011), p. 7-11.
- 19 -
The en i y ac o depends on he le el o secu i y a ge ed a he indi idual com-
pany le el. The ollowing diag am shows he en i y ac o s o he ele an a ing
ca ego ies acco ding o S&P.
11
Ra ing Class
En i y Fac o Financial Secu i y
AAA Abo e 175 % Ou s anding
AA 150 % - 174 % Excellen
A 125 % - 149 % Ve y good
BBB 100 % - 124 % Good
To ob ain a s able S&P a ing, a company should o ien owa ds he highe limi o a
ange in he calcula ion o i s capi al esou ces. Thus, possible nega i e e en s can
be abso bed wi hou being downg aded o a lowe a ing. Acco dingly, an en i y ac-
o o 125 % would indica e a s able BBB a ing a he han an A a ing.
The p emium and ese e ac o s depend on he isk s uc u e o a segmen . These
ac o s a e p o ided as ixed alues by S&P. The ollowing desc ibes he capi al
alloca ion sys em acco ding o S&P:
RC(1) = En i y Fac o · P emium Fac o · P emium,
RC(2) = En i y Fac o · Rese e Fac o · Rese e a he Begin o Pe iod 2,
…
RC( ) = En i y Fac o · Rese e Fac o · Rese e a he Begin o Pe iod .
A he beginning o he i s pe iod, he capi al equi emen is calcula ed by he mul-
iplica ion o he en i y ac o wi h he p emium ac o o each segmen and he
p emium (as olume measu e). In he ollowing pe iods, he mul iplica ion akes
place wi h he ese e ac o and he esidual ese e a he beginning o he new
pe iod (as olume measu e) ins ead o he p emium ac o and he p emium.
12
2.2 In e nal Models – Basic App oach
Wi hin he Sol ency II amewo k insu e s can also es ablish hei own in e nal
models ins ead o using ex e nal models o de e mine hei capi al equi emen s in
o de o e lec hei business isks which ha e been desc ibed in he p e ious
11
Heep-Al ine a.o. (2010), p. 56.
12
Heep-Al ine a.o. (2010), p. 55-57.
- 26 -
S ochas ic FV a = 1 Nominal Value o he Ze o Bond: 1,000.00
Du a ion o he Ze o Bond: 4
isk- ee Ra e (s ochas ic ESG) 6.0%
Risk Sp ead (s ochas ic ESG) 3.0%
FV
1
= 1,000 / (1 + 0.060 + 0.030)
4
= 708.43
Change in FV a = 1 ∆FV
1
= FV
1
– FV
0
= 708.33 – 712.99 = - 4.56
In he example ou lined, he s ochas ic ai alue o he ze o bond a e he expi a-
ion o he pe iod clea ly esul s om he simula ed isk- ee in e es a e and he
simula ed in e es a e sp ead.
16
Mon e-Ca lo Simula ions
By he means o Mon e-Ca lo simula ions, an empi ical capi al dis ibu ion a he
end o he pe iod can be gene a ed by he simula ed s ochas ic p o i and loss ac-
coun and he de e minis ic capi al a he beginning. I is necessa y ha as many
simula ions as possible will be ca ied ou in o de o show ex emely a e e en s.
Excep ional ci cums ances, such as ope a ional isks, which occu e y a ely and
whose conside a ion is ex emely impo an due o sol ency easons, can only be
shown accu a ely by a mul i ude o simula ions.
The ollowing igu e illus a es possible de elopmen s owa ds he s ochas ic capi al
a e one pe iod (based on a de e minis ic capi al o 500 a beginning) o i e simu-
la ed pa hs.
16
Heep-Al ine a.o. (2010), p. 66-68.
- 27 -
0
500
1.000
0 1
Pa h 1
Pa h 2
Pa h 3
Pa h 4
Pa h 5
Figu e 9: Capi al a e one Yea o gi en Mon e Ca lo Simula ions
17
Fo app oxima ing he dis ibu ion o he capi al a e one yea as many Mon e Ca lo
simula ions o he s ochas ic p o i and loss accoun ha e o be ca ied ou as pos-
sible. Only hen, can e y a e e en s wi h a highly nega i e in luence on he p o i
and loss accoun be conside ed.
A e a su icien la ge numbe o simula ions, he majo i y o he simula ions is dis-
ibu ed a ound he mean alue. I can be obse ed ha he dis ibu ion o he capi-
al a e one yea is limi ed a he posi i e ail, because he maximal p o i a e one
yea is limi ed. On he o he hand, ela i ely high claims paymen s may occu so
ha he dis ibu ion is ela i ely unlimi ed a he nega i e ail. As a consequence he
dis ibu ion is no mally le skewed.
F om a sol ency pe spec i e, he ocus lies on he nega i e ail o he dis ibu ion,
whe e he cases a e illus a ed in which he capi al app oaches nil and he insu -
ance company is h ea ened wi h insol ency. The scena ios in which he capi al
a e one yea is abo e he mean alue a e less di e si ied han he scena ios in
which he capi al is below he mean alue.
18
The s uc u e o a capi al dis ibu ion a he end o he pe iod is illus a ed in he ol-
lowing igu e. This dis ibu ion o capi al is ypically le skewed. This means ha he
dis ibu ion is limi ed a he posi i e ail bu uns ou a a nega i e ail.
17
Heep-Al ine a.o. (2010), p. 63.
18
Heep-Al ine a.o. (2010), p. 62-64.
- 28 -
0,00
1,25
-500 -250 0 250 500 750 1000 1250
Figu e 10: Dis ibu ion o he Capi al a e one Yea
19
The cos s o modelling a capi al dis ibu ion by an in e nal model a e ha dly jus i i-
able on he basis o he sol ency equi emen s. I is he e o e ecommendable o
use he esul s o s ee ing pu poses, as he simula ed dis ibu ion p o ides he ol-
lowing con olling in o ma ion:
• Technical Ruin de ines he SCR a = 0,
• Minimum Capi al Requi ed de ines he MCR a = 1,
• Sol ency Capi al de ines he SCR a = 1,
• Ra ing Capi al de ines he RCR a = 1.
Technical uin occu s, i he capi al a he end o he pe iod alls below ze o. Due
o he Sol ency II equi emen s he capi al a he beginning o he pe iod mus be
high enough such ha echnical uin occu s only once in 200 yea s.
The minimum capi al equi ed a he end o he pe iod de ines he nex s ee ing
le el. I he capi al a he end o he pe iod ells below his le el, his would imply
uin o he sha eholde . E en i business ac i i ies we e no p ohibi ed, he supe i-
so y au ho i ies would ake o e he managemen o he business in his case. This
would be he equi alen o an “exp op ia ion” o he owne .
The nex le el is he sol ency capi al a he end o he pe iod. I his le el is no
eached a he end o he pe iod he supe iso y au ho i ies would con ac he in-
19
Heep-Al ine a.o. (2010), p. 64.
- 29 -
su ance company and demand adequa e ac ions o sol e he p oblem by he end o
he ollowing pe iod.
Ano he impo an ocus is on secu ing he a ing capi al a he end o he pe iod.
The downg ading o a company’s a ing due o a dec ease in capi al a he end o
he yea can ha e he esul ha in he ollowing yea less business can be w i en
and ha he in es men e u ns demanded by he sha eholde s canno be no p o-
duced.
I (on he base o a simula ion model) he p obabili y o alling below an in ended
le el is oo high, he company should unde ake app op ia e managemen meas-
u es.
20
Because he minimum capi al equi ed, he sol ency le el, and he a ing le el ha e
o be e alua ed a = 1 i is necessa y o simula e he dis ibu ion also a = 2 (o o
p oceed some ype o app oxima ion) in o de o es ablish whe he he le el can be
eached in he nex pe iod.
By checking he dis ibu ion abo e, i is ob ious ha he company does no comply
wi h he sol ency equi emen s a = 0. This example will be discussed mo e in en-
si ely in he ollowing sec ions.
2.3 In e nal Models – S ochas ic P o i & Loss Accoun
21
In his sec ion, we will ocus in mo e de ail on modeling di e en isks wi hin a s o-
chas ic p o i & loss (P&L) accoun . A sho o e iew o he isk ca ego ies has al-
eady been gi en in he p e ious sec ion. Conce ning s ochas ic P&L accoun , he e
is a spli be ween
• Technical Resul (Unde w i ing Risk) and
• Non- echnical Resul wi h unde lying
o Asse Risk,
o Reinsu ance De aul Risk,
o Ope a ional Risk and
o O he Risks such as Ex ao dina y Tax Dep ecia ion.
The s ochas ic P&L is necessa y o he de e mina ion o he equi ed capi al by
s ochas ic simula ions. The mos impo an componen o he s ochas ic P & L is he
20
Heep-Al ine a.o. (2010), p. 64-66.
21
This chap e is a sho summa y o he chap e s 3 o 5 om “In e ne Modelle nach Sol ency II”,
Heep-Al ine , Kaya, K enzlin, Wel e .
- 30 -
o dina y P&L due o he yea ly business budge . An o e iew o how o model he
unde w i ing and asse isks is shown in he igu e below.
Figu e 11: Unde w i ing and Asse Risk
22
Conce ning unde w i ing and asse isks, we can di e en ia e be ween exis ing and
new business. The exis ing business is e lec ed in he exis ing ese es and he
asse s co e ing hose ese es. The ese e isk e lec s he possible ola ili y o
he exis ing business, which occu s due o he change o he ese es. The eal
claims amoun may di e signi ican ly om he es ima ed alue.
The new business is e lec ed in he incoming p emium and ou going claims. I is
impo an o calcula e he p emium isk-app op ia ely in o de o co e he claims.
The p emium isk e lec s he isk ha he p emium – e en i calcula ed app op i-
a ely – is insu icien o pay an ex ao dina y claims expe ience. Wi hin he con ex
o he s ochas ic modeling, managemen ules play an impo an ole in any case.
Those ules co e aspec s like he S a egic Asse -Alloca ion o he co e age o he
sol abili y
23
.
Apa om he unde w i ing and asse isks he e a e also o he s ochas ic in lu-
ences, which a ec he P&L esul om he non- echnical side. The mos impo an
o hem a e einsu ance de aul and ope a ional isk oge he wi h ex ao dina y ax
dep ecia ion. Those aspec s should be conside ed in an in e nal model.
22
Heep-Al ine , Ma ia 2011, In e nes Holdingmodell nach Sol ency II, p. 109
23
Nikolic, H abo szki 2012, In e p e a ion on Modelle gebnissen, p. 3
Unde w i ing &
Asse Risk
Exis ing Business New Business
Asse
Model Rese e
Model Claims
Model Po olio &
Asse Model
Technical Resul
+ Non- echnical Resul – Asse
= P & L be o e o he Risks
- 31 -
2.3.1 Technical Resul – Unde w i ing Risk
As al eady could be seen in he igu e abo e and will be illus a ed a e wa ds, he
unde w i ing isk can be spli in o
• ese e isk o he exis ing business and
• p emium isk o he new business.
Among he a ie y o models o ese e e alua ion, we can choose o ins ance a
chain ladde model. Wi h he help o a s ochas ic model, we can see possible de-
elopmen s o ou ese es. As a consequence, he chain ladde model is s ochas-
ic; hus we ob ain s ochas ic bes es ima es oge he wi h a dis ibu ion o hese
alues. In a one-pe iod-model, only he s ochas ic o he nex diagonal is ele an
so ha he ull ola ili y is no ealized.
Wi h a s ochas ic ese e model, we can measu e he ese e isk as well as he
un-o isk. The ese e isk e lec s he possible de ia ions om a gi en bes es i-
ma e and he un-o isk e lec s he possible ola ili y o he paymen pa e n.
A claims model is necessa y o de e mine he p emium isk and he un-o isk o
he new business. The p emium isk e lec s he possible insu iciency o he p e-
mium o pay he claims. The un-o isk e lec s he ola ili y o he paymen pa -
e ns o he new business.
P emium and Rese e Risk
Due o sol ency equi emen s, unde w i ing isk mus be spli in o p emium isk and
ese e isk. The p emium isk e lec s he isk o he p emium in he cu en yea
being insu icien o co e he losses. The ese e isk e lec s he isk o he e-
se e o he exis ing business a he beginning o he yea being insu icien a he
end o he yea . Consequen ly he non- echnical esul ne can be s uc u ed as ol-
lows:
Ne P emiums
– Ne Cos s
– Ne Claims Paymen s – New Business
– Alloca ion o Ne Rese es – New Business
– Ne Claims Paymen s – Exis ing Business
+ Change in Ne Rese es – Exis ing Business
_____________________________________
= Ne Non- echnical Resul
P emium Risk
Rese e Risk
- 32 -
Conce ning he ese e isk, he expec ed alue o he non- echnical esul ne
should be ze o. This means ha on a e age he paymen s should be equal o he
changes o ese e o he exis ing business. Thus, he ese e isk e lec s he
a iabili y o he exis ing business esul due o paymen s and change o ese es in
he exis ing business.
Mo eo e , he p emium and ese e isk could be dec eased signi ican ly by ein-
su ance. Fo sol ency equi emen s, i is e y impo an o measu e how he ein-
su ance dec eases he isk and hus he equi ed capi al.
The dec ease o isk by einsu ance depends on he ype o he einsu ance be-
cause he isks can be ceded p opo ionally o non-p opo ionally. In many cases, i
may no be su icien o buy only p opo ional einsu ance due o possible big claims
amoun s in he ail o a claims dis ibu ion.
New Business Model
As al eady men ioned we need a claims model o he new business o analyze he
s uc u e o he claims dis ibu ion. Mo eo e , he claims model is necessa y in o -
de o see how einsu ance a ec s he equi ed capi al. I may be qui e impo an o
model mo e han he o al claims amoun in o de o analyze he eal impac o a
einsu ance solu ion. The e should he e o e be a leas a spli in o
• Base Claims,
• Na Ca Claims and
• Majo Claims.
This spli enables us o check he e iciency o he einsu ance solu ion. Mo eo e ,
di e en einsu ance ea ies should be used o secu e hose di e en claims ypes.
Base claims ha e a high equency wi h low claims amoun . The e o e, hey need
no be einsu ed a all o only on a p opo ional basis. This ype o claims can be
es ima ed by a global dis ibu ion o agg ega e losses using, o example, he panje
ecu sion.
Na Ca claims a ise om one e en and ela e o many policy holde s. Those
e en s a e caused by na u al haza ds, which a e modeled using an e en model
wi h e en ables om ex e nal p o ide s o indi idually depending on he com-
pany’s own po olio s uc u e. The Na Ca claims a e usually einsu ed on an XL
pe occu ence basis.
- 33 -
Majo claims ha e a low equency, bu a high claims amoun . They a e so e y
ola ile ha a s ochas ic simula ion is qui e impo an .
Because o he low equency and he high amoun s, majo claims a e einsu ed on
an XL pe isk basis
24
. In o de model his ype o claims adequa ely, hey mus be
spli in o a claims numbe and a claims size model. The e o e one needs
• a equency model o he claims numbe and
• a se e i y model o he claims size.
Fo a be e unde s anding o he impac o he einsu ance on he majo claims we
will ou line hose wo model ypes in mo e de ail wi h an example.
F equency Model
To model he equency we apply a Poisson model. The Poisson dis ibu ion is one
o he simples disc e e dis ibu ions sui able o a equency model. This dis ibu ion
depends only on he Poisson pa ame e λ. I N is he numbe o claims, han
P[X = N] = (λ
N
/ N!) · e
-λ
E[X] = λ
Va [X] = λ
The Poisson pa ame e λ de ines he expec ed alue as well as he a iance o his
dis ibu ion. Being an a e age, λ does no need o be in eg al. Usually he expec ed
alue is no equal o he a iance. The e o e, i has o be checked whe he he ob-
se ed pa ame e i s in he hypo heses “expec ed alue = a iance” o no .
The p obabili ies as well as he accumula ed p obabili ies o a Poisson dis ibu ion
wi h pa ame e λ = 4.32 a e shown in he igu e below.
24
Heep-Al ine , Ma ia 2010, In e nes Modell nach Sol ency II, p. 19.
- 34 -
0%
20%
40%
60%
80%
100%
120%
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Claims Numbe
P[X = N]
P[X ≤ N]
Figu e 12: Poisson Dis ibu ion
Gi en his Poisson dis ibu ion, a andomly d awn quan ile o 62.80% esul s in an
expec ed claims numbe o i e.
Se e i y Model
To model he claims amoun we apply a Pa e o model. I is a con inuous dis ibu-
ion, which depends on he pa ame e s K ( h eshold) and α (Pa e o pa ame e ). The
Pa e o pa ame e α mus be posi i e; i de e mines how as he dis ibu ion unc ion
ends o 100%.
The a iance and he expec ed alue depend on he pa ame e s α and K. Mo e-
o e , his dis ibu ion is e y special because i α is less han 1 we do no ha e an
expec ed alue; i α is less han 2 we do no ha e a a iance e c. Especially he
ollowing ela ions hold:
P[X < x] = 1 – (x / K)
- α
E[X] = (α · K) / (α – 1)
Va
[
X
] = (α · K
2
) / [(α – 1)
2
· (α – 2)]
The Pa e o dis ibu ion is sui able o modelling he majo claims amoun , because i
s a s a a h eshold K (co esponding o an excess poin o an XL ea y). Mo e-
- 35 -
o e , he Pa e o dis ibu ion has a ela i ely hea y ail, which can be seen in he
igu e below:
0%
50%
100%
150%
0 5 10 15 20 25 30
Claims Size
Densi y Func ion
Dis ibu ion Func ion
Figu e 13: Pa e o Dis ibu ion
The densi y unc ion and he dis ibu ion unc ion o he Pa e o dis ibu ion shown in
he igu e abo e a e based on he ollowing pa ame e :
Pa e o Pa ame e α 3.57
Th eshold K 5.00
E[X] = (3.57 · 5.00) / (3.57 – 1)
= 6.94.
Va [X] = (3.57 · 5.00
2
) / [(3.57 – 1.00)
2
· (3.57 – 2.00)]
= 8.57.
So a , we ha e modelled he expec ed claims numbe and he expec ed claims
size. Assuming independency be ween claims numbe and claims size we ob ain
E[S] = E[N] · E[X]
E[S] = 4.32 · 6.94 = 29.98
- 42 -
In he ollowing, we would like o demons a e how he asse isk can in luence he
FV. We he e o e ocus on he in e es a e isk and he sp ead isk.
Calcula ion Example – In e es Ra e Risk
In o de o demons a e an example o he in e es a e isk we conside a isk- ee
ze o bond gi en he ollowing pa ame e :
De e minis ic FV a = 0 Nominal Value o he Ze o Bond: 1,000.00
Du a ion o he Ze o Bond: 5
Risk- ee Ra e (de e minis ic) 4.0%
FV
0
= 1,000 / (1 + 0.040)
5
= 821.93
S ochas ic FV a = 1 Nominal Value o he Ze o Bond: 1,000.00
Du a ion o he Ze o Bond: 4
Risk- ee Ra e (s ochas ic ESG) 6.0%
FV
1
= 1,000 / (1 + 0.060)
4
= 792.09
Change in FV a = 1 ∆FV
1
= FV
1
– FV
0
= 792.09 – 821.93 = - 29.83
The inc ease o he isk- ee in e es a e om 4.0% o 6.0% p oduces a loss o
29.83.
Calcula ion Example – Sp ead Risk
In he second example we would like o show he impac o a change o he isk
sp ead gi en he ollowing si ua ion:
- 43 -
De e minis ic FV a = 0 Nominal Value o he Ze o Bond: 1,000.00
Du a ion o he Ze o Bond: 5
Risk- ee Ra e (de e minis ic) 4.0%
Risk Sp ead (de e minis ic) 3.0%
FV
0
= 1,000 / (1 + 0.040 + 0.030)
5
= 712.99
S ochas ic FV a = 1 Nominal Value o he Ze o Bond: 1,000.00
Du a ion o he Ze o Bond: 4
Risk- ee Ra e (s ochas ic ESG) 4.0%
Risk Sp ead (s ochas ic ESG) 5.0%
FV
1
= 1,000 / (1 + 0.040 + 0.050)
4
= 708.43
Change in FV a = 1 ∆FV
1
= FV
1
– FV
0
= 708.33 – 712.99 = - 4.56
The inc ease o he isk sp ead om 3.0% o 5.0% p oduces a loss o 4.56.
2.3.3 Non- echnical Resul – Reinsu ance De aul
Any loss a ising om einsu ance de aul basically depends on he p obabili y and
he size o such de aul as well as he olume o einsu ance w i en. The p obabili y
o RI de aul can be de e mined by he c edi wo hiness o einsu e s in ol ed in
business ela ionship wi h p ima y insu e .
In i s u n, he c edi wo hiness o a pa icula einsu e can be classi ied acco ding
o his c edi a ing. I is possible ha he discha ge o CoC gi en an e icien ein-
su ance solu ion can dec ease signi ican ly, i he selec ed einsu e has a poo
c edi wo hiness.
The ceded pa o capi al cos s ep esen s a discha ge in capi al cos s. The isk
ma gin, on he con a y, is a bu den on capi al cos s; i depends on he einsu e ’s
c edi wo hiness o a ing. In o he wo ds, he isk ma gin is a p ice o he po en ial
RI de aul .
The ollowing able illus a es an example o a possible insu e 's RI s uc u e ac-
co ding o he a ings o he einsu e s acco ding o he de aul p obabili ies (e.g.
p o ided by a ing agencies):
- 44 -
Figu e 20: S uc u e o a Reinsu ance Po olio
As i can be seen, mos einsu ance con ac s in his example a e concluded wi h
einsu e s ha ing a ings om AAA o BBB. The a e age de aul p obabili y is
1.02%, which e lec s an a e age BB einsu ance s uc u e. This, howe e , may be
c ucial o an indus ial insu e .
The o e all de aul p obabili y as well as he RI exposu e change a e a pe iod o
one yea and should be modeled s ochas ically, e.g. in he ollowing way:
De e min. RI de aul a = 0 RI Exposu e: 1,000.0
A e age De aul P obabili y: 1.02%
Expec ed De aul : 10.2
S och. RI De aul a = 1 RI Exposu e: 1,050.0
A e age De aul P obabili y: 5.00%
Expec ed De aul : 57.5%
The expec ed de aul has changed signi ican ly due o a e y high s ochas ic eali-
za ion o he a e age RI de aul p obabili y.
2.3.4 Non- echnical Resul – Ope a ional Risk
Ope a ional isks addi ionally a ec he P&L esul . They a ise om business isks
and a e no insu ance speci ic (e.g. IT-de aul s, managemen mis akes and w ong
p ocess o ganiza ion). Any ope a ional isk a ec s he balance shee nega i ely
ei he as cash low in he cu en pe iod o in o m o a bad deb ese e a he end
o he pe iod.
By law, Ge man insu e s o example ha e o p o ide in o ma ion abou hei ope a-
ional isks in he appendices o he annual epo s – usually in o m o a so called
- 45 -
“ isk map”. In o de o c ea e such a map he insu e s ha e o iden i y, e alua e,
and con ol hei own isks.
Howe e , due o he lack o s a is ical da a his can be done based solely on he
sys ema ic sel -assessmen . An example o a quan i a i e isk map is shown in he
igu e below:
Risk
No. Amoun P obab. Exp. Value STD wi h
Co . o 0%
1
10
0.1%
0.01
0.32
2 50 0.1% 0.05 1.58
3 100 0.1% 0.10 3.16
4 500 0.1% 0.50 15.80
5 10 1.0% 0.10 0.99
6 50 1.0% 0.50 4.97
7 100 1.0% 1.00 9.95
8 500 1.0% 5.00 49.75
9 10 10.0% 1.00 3.00
10
50
10.0%
5.00
15.00
13.26 55.64
To al
Figu e 21: Quan i a i e Risk Map
The insu e es ima es, acco ding o a sel -assessmen , he amoun A and p obabil-
i y P o a isk occu ence. The expec ed alue EV and he s anda d de ia ion STD
o a single isk can be calcula ed as ollows:
EV = A · P
STD = A · (P · (1 - P))
1/2
The expec ed alue is linea ; he e o e he o al alue can be calcula ed by jus add-
ing he indi idual alues. The s anda d de ia ion does no beha e in a linea ash-
ion. Howe e , aking in o accoun an assumed a e age co ela ion o 0% he o al
alue can be calcula ed as ollows:
STD(X
1
+ … + X
n
) = (VAR(X
1
) + … + VAR(X
n
))
1/2
In he gi en example he o al expec ed alue o ope a ional isk equals 13.26 and
he o al s anda d de ia ion equals o 55.64. I is e y much e iden ha ope a ional
isks ha e a e y high coe icien o a ia ion (de ined as CV = STD / EV), which
- 46 -
equals 419.9% in his case. Fo no mal P&L isks, he CV is ypically below 100%.
Because o his, he densi y unc ion is highly igh skewed. In he igu e below an
app oxima ion o he dis ibu ion by a logno mal dis ibu ion is illus a ed.
0.0%
20.0%
40.0%
60.0%
80.0%
100.0%
120.0%
0.0 50.0 100.0
Figu e 22: Dis ibu ion Func ion o Ope a ional Risks
The dis ibu ion unc ion con e ges slowly agains 100% because o he high coe i-
cien o a ia ion. This dis ibu ion e lec s he ac ha he expec ed losses due o
ope a ional isks a e qui e low. On he o he hand, he e a e e y high ealiza ions
ha ing a big impac on he isk si ua ion o an en i y. In he ollowing cha he isk
map acco ding o he gi en example is illus a ed.
Amoun
high
medium
low
P obabili y low medium high
1
2
3
4
5
6
7
8
9
10
Figu e 23: Quali a i e Risk Map
- 47 -
The amoun o possible ope a ional isks is shown on he e ical axis and he
p obabili y o hei occu ence on he ho izon al axis. Bo h alues a e di ided in o
h ee classes: low, medium and high. In o al he e is a classi ica ion in h ee di e -
en isk a eas:
• high isks ed a ea,
• medium isks yellow a ea,
• low isks g een a ea.
The ed a ea ep esen s he highes isks. Any isk loca ed in his a ea occu s wi h
a medium o high p obabili y and causes a middle o high loss. Insu e s should ake
app op ia e measu es in o de o educe o elimina e he numbe o such isks o o
educe he amoun o loss.
The yellow a ea desc ibes medium isks. These isks ha e ei he high p obabili y
o occu ence combined wi h small amoun o loss o low p obabili y o occu ence
wi h high le el o damage. The insu e s should cons an ly moni o hese isks and
p e en any mo emen om he medium isk a ea in o he high isk a ea.
The g een a ea ep esen s a low dange a ea, whe e only isks wi h low p obabili y
o occu ence and small amoun o expec ed losses a e loca ed. The isks wi hin
his a ea do no eally imply a high dange , bu hey should no mo e in o o he a -
eas.
In o de o show he impac o ope a ional isks mo e accu a ely we will model he
equi y o an insu e wi h and wi hou inclusion o ope a ional isks.
1,000.00 350.00 Equi y
650.00
Liabili ies
To al 1,000.00 1,000.00 To al
Asse s
Asse s
1,000.00
336.74
Equi y
650.00 Liabili ies
13.26 Bad Deb Rese e
To al
1,000.00 1,000.00 To al
Asse s
Asse s Liabili ies
Liabili ies
Figu e 24: Balance Shee Excluding & Including Ope a ional Risks
- 48 -
A i s glance i can be ecognized ha inclusion o ope a ional isks as a bad deb
ese e immedia ely leads o a lowe ac ual capi al o 336.74 compa ed o 350.
All balance shee posi ions will be simula ed on he assump ion o logno mal dis i-
bu ion wi h he ollowing pa ame e :
Expec ed
Value Coe . o
Va ia ion S anda d
De ia ion
Asse s 1,000.00
10.0%
100.00
Liabili ies 650.00
12.5%
81.25
Ope a ional Risks 13.26
419.9%
55.64
The capi al excluding ope a ional isks esul s as di e ence be ween asse s and
liabili ies, while he capi al including ope a ional isks is addi ionally educed by he
simula ed isks. The esul s on he basis o 5,000 simula ions a e shown in he ol-
lowing able:
excl. OR incl. OR in %
Expec ed Values 351.97 339.24 96.38%
Ruin P obabii y 0.67% 1.29% 192.54%
Requi ed Capi al 369.22 423.22 114.63%
Capi al Dis ibu ion
Figu e 25: Simula ed Capi al & Ruin P obabili y
The equi ed capi al unde a VaR app oach co esponds o he expec ed alue mi-
nus he 0.5%-quan ile. The inclusion o ope a ional isks in ou example is e lec ed
in he inc ease o capi al equi ed - by 14.6% om 369.22 o 423.22 while he a ail-
able capi al dec eases only by 3.6%. Thus, he inclusion o ope a ional isks in-
c eases he uin p obabili y and he capi al equi ed disp opo ionally; ope a ional
isks ha e a conside able impac .
2.3.5 Non- echnical Resul – Ex ao dina y Tax Dep ecia ion
The ex ao dina y ax dep ecia ion occu s only in ex eme si ua ions and has a e y
nega i e impac on he P&L esul .
I a company obse es a loss, he e is usually a "nega i e" ax bu den in o m o a
“loss ca ied o wa d”. This loss can be balanced agains u u e p o i s. In a ma ke
alue model his can be ea ed as a de e ed ax asse on he economic balance
shee . I he e is no u he u u e p o i expec ed, hen his de e ed ax asse has o
be w i en o ex ao dina ily.
- 49 -
Any in e nal model should include sui able managemen ules o ea such ex ao -
dina y ax dep ecia ion. The e is a “minimal ule” o w i e o i he capi al is only
co e ed by de e ed ax asse s. Compa e he igu e below.
300.00
Equi y
50.00 Liabili ies
De e ed Tax 350.00
To al 350.00 350.00 To al
Asse s Liabili ies
Figu e 26: Ex ao dina y Tax Dep ecia ion
In his scena io, he company owns “ ax asse s” o 350 co e ing an equi y o 300.
Gi en such a si ua ion, he company is mo e o less insol en so ha de e ed axes
o 350 ha e o be w i en o . The equi y a e dep ecia ion equals -50; he company
is insol en .
This example e lec s he ac ha ax e ec s do no p e en a uin. In his case, he
company won’ be sa ed om insol ency by he ax au ho i y. Tax e ec s can only
smoo h he P & L esul s, bu no hing mo e.
The ex ao dina y ax dep ecia ion may p oduce ex eme non-linea e ec s. Thus, i
is by no means clea how much capi al a company has o injec (in case o a de i-
ciency) o can ex ac (in case o a edundancy) acco ding o sol ency equi e-
men s.
2.4 In e nal Models – Requi ed Capi al
In he p e ious chap e s, he mos ele an ma hema ical and economic basic p in-
ciples o an in e nal isk model ha e been de eloped. I has been explained how o
model he indi idual componen s o a s ochas ic p o i and loss accoun by Mon e
Ca lo simula ions. In his sec ion, all in o ma ion will be combined o an o e all
model. To ob ain he equi ed capi al we ha e o pe o m he ollowing s eps:
• Me ging he indi idual model componen s o an o e all model by using
managemen ules.
• Pe o ming a simula ion un o de e mine he empi ical o e all dis ibu ion.
• E alua ion o he empi ical o e all dis ibu ion o de e mine he equi ed
capi al by he Value a Risk (VaR) o he Tail Value a Risk (TVaR) P inciple.
• Alloca ion o he equi ed capi al o he isk in luences ( op-down app oach).
- 50 -
As p e iously explained, he de e minis ic capi al a he beginning o he pe iod and
he s ochas ic p o i and loss accoun simula ed by Mon e Ca lo simula ions a e
used o calcula e he capi al a he end o he pe iod whe e he choice o inpu pa-
ame e is undamen al in his con ex .
2.4.1 Comple e Model & Capi al Dis ibu ion
The s ochas ic p o i and loss (P&L) due o he basic equa ion discussed be o e
consis s mainly o s ochas ic p o i & loss con ibu ions and he espec i e inpu pa-
ame e bu i is also de e mined by managemen ules.
Managemen Rules
Managemen ules a e non-s ochas ic elemen s o an o e all model ha a ec he
income s a emen . They se e as a u he basis o business decisions. In he
modeling p ocess he co po a e s a egy should be designed wi hou unnecessa y
complexi y. The ollowing managemen ules we e applied in all ou calcula ions:
• All asse s like s ocks a e conside ed as accumula ed wi hou any liquid
di idend ou go.
• All liquid acc uals a e in es ed in sho e m isk- ee pape s un il he end o
he yea .
• Sho loans o co e nega i e liquidi y can also be pe o med on a isk- ee
base.
• Di idends om subsidia ies o o pa en companies a e no aken in o
accoun .
29
I should be poin ed ou ha he impac o managemen ules is no e y s ong in a
sho e m calcula ion. Bu in conside a ion o se e al pe iods, managemen ules
can ep esen signi ican ac o s which in luence he esul s.
Inpu Pa ame e
Wi h ega d o he o e all model, he ollowing ypes o pa ame e ha e o be con-
side ed:
• Ma ke pa ame e (e.g. ma ke in e es a e).
• Co po a e pa ame e (e.g. ax a e).
29
Heep-Al ine , E olgso ien ie e Un e nehmenss eue ung, Vo lesungsssk ip , 2012.
- 51 -
• P o i & Loss speci ic pa ame e (e.g. asse s, p emium-income, claims
ese e, einsu ance s uc u e).
• Co ela ion pa ame e (e.g. be ween isk- ee a e and sp ead o ixed-income
bonds)
A e all inpu pa ame e and managemen ules ha e been es ablished, Mon e
Ca lo simula ions can be pe o med based on he calcula ion scheme.
2.4.2 Comple e Model & Capi al Dis ibu ion – Calcula ion Example
In his sec ion a simpli ied s ochas ic p o i & loss accoun model will be es ablished
and used so ha a capi al alloca ion a he end o he pe iod can be de e mined.
Mon e Ca lo simula ions a e based on andom expe imen s, which a e ca ied ou
by using sui able andom numbe s. I should be no ed ha a su icien numbe o
simula ions ha e o be gene a ed, in o de o p oduce s able esul s. Mon e Ca lo
simula ions es ablish an empi ical dis ibu ion which se es as an app oxima ion o
he heo e ical dis ibu ion. The quali y o he app oxima ion depends on he numbe
o simula ions.
Based on he dis ibu ion, he capi al needs o he company a e de e mined. Finally
he capi al is alloca ed by using a op-down app oach o indi idual model compo-
nen s.
The ollowing igu e shows he inpu pa ame e s o he example, which will be ana-
lyzed u he in mo e de ail.
Pa ame e A e age Coe . o
Va ia ion
Ma ke In e es Ra e 4%
10%
Capi al a Begin 500
Tax Ra e 35%
Op. Risk (in % o P emium) 5%
250%
P emium 1,000
2%
Cos Ra io 20%
10%
Loss Ra io 70%
35%
Figu e 27: Inpu Pa ame e o he Calcula ion Example
In he example desc ibed in his sec ion he ollowing componen s o a P&L accoun
a e modelled in a simpli ied way:
- 58 -
o capi al han low- isk segmen s. In consequence, due o hei highe capi al ol-
ume, high- isk segmen s ha e o gene a e mo e p o i (in absolu e alues).
In he ollowing sec ions, we would like o ou line di e en ma hema ical me hods o
alloca e he capi al in an insu ance company. Howe e , be o e any me hod can be
applied, we ha e o de e mine he equi ed capi al whe e wo di e en app oaches
can be used acco ding o he Value a Risk o Tail Value a Risk p inciple. The
Value a Risk a a 99.5% secu i y le el is used in Sol ency II. The Tail Value a Risk
is used e y o en in in e nal models.
P opo ional Alloca ion
The P opo ional Alloca ion is he simples app oach o alloca e he capi al in a non-
li e insu ance company wi hou la ge calcula ion e o , because he syne gy e ec
is alloca ed p opo ionally. S ochas ic p ope ies a e no conside ed in his ap-
p oach. The capi al is calcula ed by using he ollowing ma hema ical o mula:
RC
i,mod
= RC
i
· RC
ges
/ ∑ RC
i
whe e
RC
i,mod
Capi al Alloca ion pe Single Risk
RC
i
Requi ed Capi al pe Single Risk (wi hou syne gy e ec s)
RC
ges
Requi ed Capi al a Company Le el
∑ RC
i
Sum o all Single Requi ed Capi al (wi hou syne gy e ec s)
The disad an age o his app oach is ha no isk s uc u e and no dependence be-
ween single isks is conside ed.
Adjus men o Risk-Le el
The basic assump ion o his app oach is he educ ion o he secu i y-le el o sin-
gle con ibu ions so ha he sum adds o he o al capi al equi emen . Wi h his
app oach in mind, he ollowing o mula applies:
RC
i,mod
= RC
i,β
wi h ∑ RC
i,β
= RC
ges,α
RC
i,mod
Capi al Alloca ion pe Single Risk
RC
i,β
Requi ed Capi al a a Secu i y Le el β pe Single Risk
∑RC
i,β
Sum o Requ. Capi al a a Secu i y Le el β o all Single Risks
RC
ges,α
Capi al Requi emen o he company a a Secu i y Le el α
- 59 -
This app oach o Risk-Le el Adjus men akes s ochas ic p ope ies in o accoun . In
compa ison o a p opo ional alloca ion, he isk si ua ion in he ail a ea is modeled
mo e adequa ely. A disad an age o his app oach is ha i is no linea .
Co a iance Algo i hm
Fo he Co a iance Algo i hm, he co a iance con ibu ions o he indi idual compo-
nen s o he o e all a iance a e calcula ed wi h he aid o a co ela ion ma ix. The
capi al is alloca ed acco ding o he co a iance con ibu ions, see he ollowing ig-
u e wi h an alloca ion algo i hm on he base o he RC acco ding o TVaR p inciple.
Techn. Non- ope a . Capi al
Resul Techn. Risk a End
Resul a e Tax
Requi ed Capi al 664.4 13.4 87.7 765.4
in %
86.8%
1.8%
11.5%
100.0%
Figu e 33: Co a iance Algo i hm
All in all, he Co a iance Algo i hm ep esen s a ela i ely simple and easily appli-
cable me hod o he alloca ion o capi al, which conside s he isk in an adequa e
manne . A disad an age is ha his algo i hm pu s a disp opo iona e amoun o
weigh on high isks.
30
Co-Measu e Algo i hm
The Co-Measu e Algo i hm is based on he linea i y o he condi ional expec ed
alue so ha he Algo i hm is sui able when he capi al equi emen is de e mined
by he Tail Value a Risk p inciple. The Co-Measu e Algo i hm is de ined by he ol-
lowing o mula:
C
1
= C
0
+ ∑ PL
i
E [C
1
] = C
0
+ ∑ E[PL
i
]
TVaR
α
[C
1
] = C
0
+ ∑ E[PL
i
| PL ≤ VaR
α
[PL]]
RC
α
= ∑ (E[PL
i
] - E[PL
i
| PL ≤ VaR
α
[PL]])= ∑ RC
i,α
C
1
Capi al a e one yea
E [C
1
] Expec ed alue o capi al a e one yea
TVaR
α
[PL] Tail Value a Risk wi h isk le el α
RC
α
Capi al equi emen wi h isk le el α a e one yea
30
Nguyen 2008, Handbuch de we - und isikoo ien ie en S eue ung on Ve siche ungsun e neh-
men, p. 218.
- 60 -
The Co-Measu e Algo i hm is a mode n s a is ical app oach wi h good ma hema i-
cal p ope ies. A disad an age is ha i may alloca e ex emely high capi al e-
qui emen s o highe isks. The e o e al e na i e app oaches should be conside ed
i necessa y, o example he Shapley Algo i hm.
31
Shapley Algo i hm
The Shapley Algo i hm is a game heo e ical me hod which de e mines he capi al
need o a isk h oughou he accession o an al eady exis ing collec i e. This
me hod is a combina ional p ocedu e whe e all possible N! combina ions o N isks
a e aken in o accoun . In a po olio wi h a wide numbe o isks his me hod
causes eno mous calcula ion e o .
Fo cla i ica ion, he Shapley Algo i hm will be explained wi h he ollowing example
gi en h ee isks X, Y and Z. In case o no mally dis ibu ed isks he equi ed capi-
al is p opo ional o he s anda d de ia ion (STD) in such a way ha we ocus on
his isk measu e in he ollowing. We ha e he ollowing ma ginal con ibu ions:
1. I X is conside ed as he i s isk:
M
x
= STD (X)
2. I X is conside ed as he second isk a e he isk Y:
M
X|Y
= STD (X+Y) – STD (Y)
3. I X is he las isk:
M
X|Y+Z
= STD (X+Y+Z) – STD (Y+Z)
4. Combina ion o all isk con ibu ions
R
X
= ⅓ · M
X
+ ⅓ · (½ · M
X|Y
+ ½ · M
X|Z
) + ⅓ · M
X|Y+Z
5. The o e all isk is desc ibed as ollowed:
R
X
+ R
Y
+ R
Z
= R
X+Y+Z
= STD(X+Y+Z)
O e all, he Shapley Algo i hm ecei es a wide ange o accep ance. Because o
he eno mous calcula ing e o due o he la ge numbe o isks, he p ac ical appli-
ca ion o his me hod is ques ioned. I we use he a iance ins ead o he s anda d
31
Heep-Al ine ; Hake ; Lazic; Wes e mann e al. 2011, In e nes Holdingmodell nach Sol ency II-
Sch i ü Sch i zu einem in e nen Holdingmodell, p.26-27.
- 61 -
de ia ion as a isk measu e, hen he Shapley Algo i hm deli e s he Co a iance
Algo i hm.
Compa ison o Alloca ion Me hods
32
The ollowing able p esen s an o e iew o he main a ibu es o di e en alloca-
ion me hods as well as hei ad an ages and disad an ages.
Alloca ion
Me hod Ad an ages Disad an ages
P opo ional Al-
loca ion • Simple handling • No conside a ion o
s ochas ic p ope ies
Adjus men o
Risk Le el
• Conside a ion o
s ochas ic p ope ies
• Rela i ely complex
• No Linea i y
• Big isks demand high capi al
Co a iance
Algo i hm
• Conside a ion o
s ochas ic p ope ies
• Genuine accep ance
• Applica ion in many
s anda d models
• Linea app oach
• Big isks demand high capi al
• Does no i o he VaR o
TVaR p inciple
Co-Measu e Al-
go i hm • Conside a ion o
s ochas ic p ope ies
• Linea i y
• Cohe ence
• Big isks demand high capi al
• Low accep ance o esul s
• Elimina ion o small isks
• Fi s only o he TVaR p inciple
Shapley
Algo i hm
• In ui i e alloca ion
algo i hm
• Widely accep ed
• Equali y p inciple
• Highly complex calcula ion
• Calcula ion ime
The Co a iance Algo i hm is a e y manageable app oach, because i p esen s a
ela i ely simple and easily execu able me hod o capi al alloca ion ha also con-
32
Heep-Al ine ; Kaya; K enzlin; Wel e e al. 2010, In e ne Modelle nach Sol ency II - Sch i ü
Sch i zum in e nen Modell in de Schaden e siche ung, 2010, p. 222.
- 62 -
side s he isk in an adequa e way. Howe e , he me hod only ep esen s a linea
dependency be ween he isks ha is no adequa e in e e y case.
The use o he P opo ional Alloca ion is e y easy, bu he dependencies be ween
he isks and he isk si ua ion in he ail a ea a e no conside ed.
Wi h espec o he Adjus men o Risk-Le el, s ochas ic p ope ies a e also consid-
e ed and he isk si ua ion in he ail a ea is indica ed mo e accu a ely. Howe e ,
his me hod is no linea .
O he me hods like he Co-Measu e Algo i hm o he Shapley Algo i hm seem o be
a ac i e app oaches, bu hey a e no always applicable, because business seg-
men s ca ying big isks demand high capi al (Co-Measu e Algo i hm) o because
he me hod demands a g ea compu ing ime in case o a high numbe o isks
(Shapley Algo i hm).
Cos o Capi al
The equi ed capi al is he cen al inpu ac o o he business model o insu ance.
In his sec ion he de e mina ion o he equi ed capi al has been explained in mo e
de ail. Cos s o Capi al de ine he p ice o p o iding his inpu ac o . In he ollow-
ing igu e he mechanism o calcula e he Cos o Capi al is illus a ed:
Pe iod
Requi ed Capi al
Ex a Di idend
Cos o Capi al
= 0 = 1 = 2
ED(1)
ED(2)
RC(1) RC(2) RC(3) RC(n)
ED(n)
…
…
…
= n-1 = n
…
…
…
ED(n-1)
Figu e 34: Cos o Capi al (CoC)
As he igu e illus a es Cos o Capi al can be de ined as he p esen alue o ex a
di idends (in he sense o a isk sp ead) on he Requi ed Capi al ha is needed o
secu e he isk co e age. In he ollowing sec ions he CoC will be desc ibed in
mo e de ail.
- 63 -
3 Risk-Based Pe o mance Measu emen
In he p e ious sec ion we saw how an insu ance company can de e mine i s e-
qui ed capi al and how his capi al can be alloca ed o se e al isk in luences. This
chap e p esen s i s ly he managemen o unde w i ing. Subsequen ly, i illus-
a es how insu ance companies can con ol hei o al po olio including he capi al
in es men . The las sec ion desc ibes he pe o mance op imiza ion. To sum up,
he ollowing opics a e ea ed:
• Unde w i ing Pe o mance,
• Asse Pe o mance.
Fu he mo e, he sec ion dealing wi h unde w i ing pe o mance is sepa a ed in o
he ollowing wo di e en app oaches:
• T adi ional Pe o mance Measu emen ,
• Risk-based Pe o mance Measu emen .
In o de o unde s and he di e ence be ween hose wo app oaches, de ailed ex-
amples a e discussed.
3.1 Unde w i ing Pe o mance Measu emen
One pa o unde w i ing pe o mance measu emen consis s in he de ini ion o
guidelines o subsc ibe he isk. P o i abili y analyses a e used o e i y he hose
guidelines. These analyses ake place be o e he unde w i ing (new business) o
a e wa ds du ing he execu ion (exis ing business). The e a e wo pe spec i es: “A
p io i” in o de o a i a new business o “a pos e io i” o con ol an exis ing busi-
ness. The igu e below illus a es he ime ho izon o a p o i abili y analysis.
Figu e 35: New Business e sus Exis ing Business
=0 =1 =n
New Busi-
ness
A p io i
Exis ing Bu-
siness
A pos e io i
…
- 64 -
The ollowing sec ion ocuses on he "a p io i" unde w i ing analysis wi h espec o
new business. The e is a conside a ion o he a ge alues a he beginning o he
unde w i ing pe iod. The ollowing alues ha e o be es ima ed:
• The claims amoun ,
• adminis a ion and o he cos s,
• cos s o capi al,
• isk- ee in e es a e and
• equi ed capi al.
Wi h his inpu da a we can de e mine he p emium and check whe he he segmen
is p o i able o no .
3.1.1 T adi ional Pe o mance Measu emen
This sec ion s a s wi h he adi ional app oach o p emium calcula ion. I is only
based on he esul s o he unde w i ing p ocess and does no include he expec ed
in es men income. This will be e alua ed sepa a ely and does no in luence di-
ec ly he p emium calcula ion. In p ac ice, he p emium calcula ion is in luenced by
mo e ac o s e.g. he impac s o compe i ion policy.
New Business
Acco ding o he adi ional app oach he p emium has o co e he adminis a ion
cos s, he ul ima e claims amoun and an addi ional p o i ma gin. In non-li e insu -
ance i is assumed ha he e is usually a p o i ma gin be ween wo and h ee pe -
cen .
33
The ollowing ela ion holds:
Adminis a ion Cos s
+ Ul ima e Claims Amoun
+ P o i Ma gin
= P emium
This p emium is he basis o assessing p o i abili y. In his assessmen , usually he
echnical esul o he combined a io is calcula ed. These e ms a e explained la e .
33
Heep-Al ine (2010), p. 45
- 65 -
Exis ing Business:
In o de o assess p o i abili y he unde w i ing esul is de e mined. Addi ionally i
migh be conside ed ha he expec ed p o i ma gin could be ul illed as calcula ed
in he p emium. The “a pos e io i” unde w i ing esul is de ined as ollows:
P emium
- Adminis a ion Cos s
- Claims Amoun
= Unde w i ing Resul
Ano he me hod o p o i abili y assessmen is he conside a ion o he combined
a io as a combina ion o loss a io and cos a io. Bo h, he cos a io and he loss
a io a e al eady used as an indica o o a po olio assessmen . The loss a io is
he ela ionship o claims paymen s o ecei ed p emiums. The cos a io ep esen s
he ela ion o adminis a i e cos s e sus ecei ed p emiums. The combined a io
is calcula ed as ollows:
Combined Ra io = (Adminis a ion Cos s + Claims Amoun ) / P emium.
In an unde w i ing pe spec i e, he combined a io should be less han 100% o
deli e ing a e u n. In p ac ice, he combined a io a ies widely be ween di e en
b anches.
Bo h key indica o s o he adi ional app oach a e easy o de e mine and easy o
unde s and. Bu jus he unde w i ing is conside ed and no he capi al in es men .
A conside a ion o he cash lows is usually no pe o med. Bu o he insu ance
business, i is cha ac e is ic ha he paymen s ha e o be paid wi h a ime delay o
he p emium income. Because o ha he inancial esou ces a e no needed in o al
and can be in es ed in he capi al ma ke bea ing in e es . This can compensa e a
nega i e unde w i ing esul . Bu he adi ional app oach does no conside his as-
pec adequa ely.
The e o e he adi ional pe o mance measu emen may no assess whe he an
achie ed p o i abili y is su icien .
The examples desc ibed in he ollowing assume a e age claims and cos pay-
men s. These a e only s a is ical pa ame e s which may no ealize in p ac ice. I
hese a ia ions canno be compensa ed by he collec i e, he insu ance company
has o compensa e an un a ou able claim expe ience by he p o ision o capi al.
The adi ional app oach does no show which le el o isk should be secu ed by
- 66 -
capi al and how much excess e u n he insu ance company has o gene a e in o -
de o use his capi al. Thus, he adi ional pe o mance measu emen does no
conside all impo an aspec s.
3.1.2 T adi ional Pe o mance Measu emen – Calcula ion Example
In his sec ion an example o a liabili y segmen is discussed wi h espec o he a-
di ional pe o mance measu emen wi h he ollowing inpu pa ame e :
P emium 1,000.0
Cos Ra io 25.0%
Loss Ra io 80.0%
Du a ion 3
The p emium income o 1,000 is ecei ed a he beginning o he i s pe iod. Addi-
ional p emium paymen s do no occu . I is assumed ha he e a e cos s o 25% o
he p emium and a loss a io o 80%. Due o a secu i y p inciple he claim ese e
is ini ially cons i u ed wi h 900 (o e ese a ion). The du a ion (e.g. he a e age
paymen du a ion) is 3 yea s. When he cos a io and he loss a io a e summed
up, i esul s a combined a io o 105%. I will be checked i his business can be a
all p o i able o i he insu ance company su e s a loss.
A e an example wi h espec o a single acciden yea , we conside a egula
p emium income esul ing om an inc easing o a dec easing po olio o e se e al
acciden yea s. Finally, we conside he impac o in e es s.
P o i & Loss E ec – Single Acciden Yea
This example assumes a cons an po olio on he base o a single acciden yea .
The ollowing able shows he de elopmen o he liabili y segmen o he inancial
yea s 1 o 4 whe e he p emiums a e eco ded as an income in he i s yea . An
amoun o 25% o he p emium is sub ac ed immedia ely as cos s. Also, a claims
ese e o 900 is es ablished.
- 67 -
1 2 3 4
Inc. Exp. Inc. Exp. Inc. Exp. Inc. Exp. Inc. Exp.
P emiums 1,000 0 0 0 1,000 0
Cos s 250 0 0 0 0 250
Claim Paymen s 0 0 0 800 0800
Claim Rese es 900 0 0 -900 0 0
Sum 1,000 1,150 00000-100 1,000 1,050
Combined Ra io
Financial Yea
115.0% 105.0%
To al
Figu e 36: Income & Expenses – Single Acciden Yea
The esul in he i s yea co e s an income o 1,000 and expenses o 1,150 and
esul s a combined a io o 115% o his inancial yea . In he nex wo yea s he e
a e no cash lows, so ha he claims ese es emain unchanged un il he ou h
yea . Because o he dissolu ion o he o e ese ed claim ese e in his yea , he
insu ance company ge s an income o 100. The example ends in he ou h yea ,
because he e a e no addi ional incomes / expenses. In o al, he insu ance com-
pany ecei es an income o 1,000 and expenses o 1,050. The e is a combined
a io o 105% in yea 4.
Wi hou he conside a ion o in es men income, a segmen wi h a combined a io
abo e 100% can ne e p oduce a posi i e esul .
The example should be modi ied, because a cons an po olio o only one acciden
yea is no ypical o he insu ance business.
P o i & Loss E ec – Se e al Acciden Yea s
We conside now a egula p emium income o e se e al acciden yea s which e-
sul s in an inc easing o a dec easing po olio. Assuming an annual g ow h o 10%,
we ob ain he ollowing able:
G ow h 10% Inc. Exp. Inc. Exp. Inc. Exp. Inc. Exp. Inc. Exp.
Accid. Yea 1 1,000 1,150 00000-100 1,000 1,050
Accid. Yea 2 1,100 1,265 00001,100 1,155
Accid. Yea 3 1,210 1,392 0 0 1,210 1,271
Accid. Yea 4 1,331 1,531 1,331 1,398
Accid. Yea 5 1,464 1,537
Sum 1,000 1,150 1,100 1,265 1,210 1,392 1,331 1,431 6,105 6,410
41 2 3
Combined Ra io 115.0% 115.0%
Financial Yea
115.0% 107.5% 105.0%
To al
Figu e 37: Income & Expenses – Se e al Acciden Yea s, 10% Inc ease
The combined a io in he i s inancial yea is again 115%. Because o he con-
s an inc easing cos s and p emium income, he combined a io does no change in
he nex wo inancial yea s. F om he ou h yea , when he i s claims a e se led,
he combined a io o he inancial yea s dec eases o 107.5% whe e he combined
- 74 -
In his example, a S&P-capi al-alloca ion-model wi h he ollowing inpu da a is
used:
S&P Company Le el 125.0%
S&P P emium Ra e 27.0%
S&P Rese ing Ra e 10.0%
Ex a Di idend Ra e 6.0%
The alloca ion o capi al demons a es he isk- ela ed capi al demand o di e en
lines o business. The amoun o alloca ed capi al depends on he conside ed seg-
men and on he company’s a ge a ing. The chosen mul iplie o 125% is used o
companies wi h a s ong BBB- a ing as a ge a ing. Cos o Capi al o 6% is e-
qui ed o compensa e he isk bea ing, analog o he Swiss sol ency model.
Pe iod P emium Claims Rese e P em. Fac . Res. Fac . To al En . Fac .
27.0% 10.0% 125.0%
1 1,000.0 0.0 270.0 0.0 270.0 337.5
20.0 0.0 800.0 0.0 80.0 80.0 100.0
30.0 0.0 800.0 0.0 80.0 80.0 100.0
40.0 800.0 800.0 0.0 80.0 80.0 100.0
50.0 0.0 0.0 0.0 0.0 0.0 0.0
To al 800.0
Base o Capi al Alloca ion Capi al Alloca ion due o S&P
Figu e 45: S&P Alloca ion o Capi al o Gene al Liabili y wi h CR = 105.0%
The nex s ep is o calcula e he cos o capi al. Cos s o capi al on he equi ed
capi al p o ided a e equi ed a he end o a pe iod. The CoC is ob ained by mul i-
plying he equi ed capi al a he beginning o a pe iod wi h 6%. The discoun ed ex-
a di idends add up o he Capi al Cos s in o al.
Pe iod Accumul. Discoun Requi ed Amoun
Capi al CoC Ra e 6.0% o Co e .
4.00% Nominal Discoun ed
1 100.00% 337.5 735.4
2 96.15% 100.0 20.3 19.5 0.0
3 92.46% 100.0 6.0 5.5 0.0
4 88.90% 100.0 6.0 5.3 -697.4
5 85.48% 0.0 6.0 5.1 0.0
6 82.19% 0.0 0.0
To al 35.5 38.0
Cos o Capi al wi h
Begin o he Pe iod
Figu e 46: Requi ed Cos o Capi al o Gene al Liabili y wi h CR = 105.0%
- 75 -
The p esen alue o amoun o co e age and he sum o he discoun ed cos o
capi al a e now known. They need o be compa ed in o de o disco e whe he he
analysed segmen is p o i able enough. Ob iously, he p esen alue o amoun o
co e age is highe han he equi ed cos o capi al. Thus, he segmen is su i-
cien ly p o i able.
In he nex example he segmen mo o insu ance – i e and he is conside ed o
illus a e he impac o a di e en cash low s uc u e.
Pa ially Comp ehensi e
In con as o he liabili y segmen , he main cha ac e is ic o he pa ially comp e-
hensi e segmen is he low p obabili y o la e claims and he quick claim se lemen .
The e o e, he e is jus a sho du a ion in he conside ed segmen . To analyse his
segmen , he same inpu da a as be o e is conside ed:
P emium 1,000.0
Expense Ra e 25.0%
Combined Ra io 105.0%
Ma ke In e es Ra e 4.0%
The nominal iew on his segmen esul s in he same nega i e echnical esul o -
50 as be o e. The di e en cash low s uc u e does no play any ole a his s age.
Pe iod
CF in %
A e age
Du a ion
P emium
Cos s
Resul
Incu ed Fu u e
1 80.0% 0.5 1,000.0 250.0 640.0 110.0
2 20.0% 1.5 0.0 0.0 160.0 -160.0
30.0% 2.5 0.0 0.0 0.0 0.0
4 3.5 0.0 0.0 0.0 0.0
50.0% 4.5 0.0 0.0 0.0 0.0
To al 100.0% 0.7 1,000.0 250.0 0.0 800.0 -50.0
Nominal Values
Claims
Figu e 47: Nominal Cash Flow o Pa ially Comp ehensi e wi h CR = 105.0%
In con as o he liabili y segmen , 80% o he claims paymen s a e paid in he i s
yea and 20% in he second yea . The nex able shows he impac on he amoun
o co e age by conside ing he p esen alues.
- 76 -
Pe iod
Accumul.
Discoun
P emium Cos s Resul
4.00% Incu ed Fu u e
1 98.06% 980.6 245.1 627.6 107.9
2 94.29% 0.0 0.0 150.9 -150.9
3 90.66% 0.0 0.0 0.0 0.0
4 87.17% 0.0 0.0 0.0 0.0
5 83.82% 0.0 0.0 0.0 0.0
To al 980.6 245.1 0.0 778.4 -43.0
Discoun ed Values
Middle o he Pe iod
Claims
Figu e 48: Discoun ed Cash Flow o Pa ially Comp ehensi e wi h CR = 105.0%
The conside a ion o he p esen alues has a posi i e e ec on he liquid esul .
Howe e , in con as o he gene al liabili y segmen , he e ec is no posi i e
enough so he esul is s ill nega i e. Consequen ly his segmen is no p o i able
e en unde he modi ied pe cep ion.
The calcula ion o he p esen alues cla i ies ha he modi ied pe cep ion only has
a low impac on segmen s wi h a sho du a ion. The cash low s uc u e de e mines
he p o i abili y o a business.
The nex ques ion is how he combined a io should be changed o ensu e su icien
p o i abili y. The a ge combined a io depends on he capi al alloca ion policy o an
en i y and he e o e i is di e en o di e en ypes o insu e s. The ollowing pa-
ame e s in pa icula de e mine he a ge combined a io:
• Segmen cha ac e is ics (like ola ili y o du a ion),
• company’s secu i y le el (de e mining he S&P mul iplie ),
• equi ed ex a di idend and
• ma ke in e es a e.
The able below indica es ha a posi i e liquid esul de i es om a combined a io
o 99.6% in he pa ially comp ehensi e segmen .
Pe iod
CF in %
A e age
Du a ion
P emium
Cos s
Resul
Incu ed Fu u e
1 80.0% 0.5 1,000.0 250.0 596.5 153.5
2 20.0% 1.5 0.0 0.0 149.1 -149.1
30.0% 2.5 0.0 0.0 0.0 0.0
4 3.5 0.0 0.0 0.0 0.0
50.0% 4.5 0.0 0.0 0.0 0.0
To al 100.0% 0.7 1,000.0 250.0 0.0 745.6 4.4
Nominal Values
Claims
Figu e 49: Nominal Cash Flow o Pa ially Comp ehensi e wi h CR = 99.6%
- 77 -
The smalle combined a io a ises om a educ ion o he expec ed claims pay-
men s (due o a ela i e inc ease in p emium a es). Because o he lowe combined
a io he echnical esul is now posi i e. The nex able illus a es he e ec o dis-
coun ing.
Pe iod
Accumul.
Discoun
P emium Cos s Resul
4.00% Incu ed Fu u e
1 98.06% 980.6 245.1 584.9 150.5
2 94.29% 0.0 0.0 140.6 -140.6
3 90.66% 0.0 0.0 0.0 0.0
4 87.17% 0.0 0.0 0.0 0.0
5 83.82% 0.0 0.0 0.0 0.0
To al 980.6 245.1 0.0 725.5 9.9
Discoun ed Values
Middle o he Pe iod
Claims
Figu e 50: Discoun ed Cash Flow o Pa ially Comp ehensi e wi h CR = 99.6%
The e is now an amoun o co e age o 9.9. I should be checked whe he his is
enough o co e he equi ed cos s o capi al. As be o e, his can be con olled by
using he S&P- capi al alloca ion model. The e o e, in he i s ins ance, he emain-
ing ese e a he beginning o he second pe iod needs o be calcula ed.
Pe iod Accumul. Discoun P emium Rese e
4.00% Single Accum.
1 98.06% 1,000.0 596.5 596.5 0.0
2 94.29% 0.0 149.1 745.6 149.1
3 90.66% 0.0 0.0 745.6 0.0
4 87.17% 0.0 0.0 745.6 0.0
5 83.82% 0.0 0.0 745.6 0.0
To al 745.6
Base o Capi al Alloca ion
Middle o he Pe iod Claims
Figu e 51: Base o Capi al Alloca ion o Pa ially Comp ehensi e wi h CR = 99.6%
The S&P and he CoC model can be used wi h he same da a as be o e wi h he
excep ion ha he e a e o he S&P ac o s o pa ially comp ehensi e.
S&P Company Le el 125.0%
S&P P emium Ra e 12.0%
S&P Rese ing Ra e 12.0%
Ex a Di idend Ra e 6.0%
The nex able illus a es he calcula ion o he alloca ed capi al based on he p e-
mium and ese e ac o and he en i y ac o de ining he company’s secu i y le el.
- 78 -
Pe iod P emium Claims Rese e
P em. Fac .
Res. Fac .
To al
En . Fac .
12.0% 12.0% 125.0%
1 1,000.0 596.5 120.0 0.0 120.0 150.0
20.0 149.1 149.1 0.0 17.9 17.9 22.4
30.0 0.0 0.0 0.0 0.0 0.0 0.0
40.0 0.0 0.0 0.0 0.0 0.0 0.0
50.0 0.0 0.0 0.0 0.0 0.0 0.0
To al 745.6
Base o Capi al Alloca ion Capi al Alloca ion due o S&P
Figu e 52: S&P Alloca ion o Capi al o Pa ially Comp ehensi e wi h CR = 99.6%
Re e ing o he calcula ion o he p esen alue o amoun o co e age i has o be
checked whe he he equi ed cos o capi al is highe o lowe han he amoun o
co e age, see he ollowing able.
Pe iod Accumul. Discoun Requi ed Amoun
Capi al CoC Ra e 6.0% o Co e .
4.00% Nominal Discoun ed
1 100.00% 150.0 150.5
2 96.15% 22.4 9.0 8.7 -140.6
3 92.46% 0.0 1.3 1.2 0.0
4 88.90% 0.0 0.0 0.0 0.0
5 85.48% 0.0 0.0 0.0 0.0
6 82.19% 0.0 0.0 0.0
To al 9.9 9.9
Cos o Capi al wi h
Begin o he Pe iod
Figu e 53: Requi ed Cos o Capi al o Pa ially Comp ehensi e wi h CR = 99.6%
I can be seen ha he p esen alues o amoun o co e age and o cos s o capi al
a e equal wi h espec o he chosen combined a io. Consequen ly he business is
p o i able and 99.6% is he a ge combined a io.
3.1.5 New Business e sus Exis ing Business – Calcula ion Example
In o de o ind ou possible miscalcula ions in a segmen and o s a sui able coun-
e measu es a e wa ds, i is impo an o compa e he ac ual alues wi h he a ge
alues.
Gene al Liabili y – New Business
“A pos e io i” i should be checked, i he pa ame e es ima ed “a p io i” i wi h he
ealized alues up o he poin in ime . The inpu alues – such as a e age claims
his o y o isk- ee in e es a es – may ha e e ol ed di e en ly han p edic ed be-
o e. I he e a e nega i e de ia ions om he expec ed esul , he insu ance com-
pany should pe o m a de ailed analysis, in o de o p e en o ecas e o s in u-
u e.
- 79 -
In case o nega i e de ia ions i should dis inguished whe he unde w i ing isk o
capi al in es men isk is a ibu able. Misjudgemen s o he unde w i e ega ding
damage and loss expe ience is pa o he unde w i ing isk and all o he esponsi-
bili y o he unde w i e , bu o ecas e o s in e ms o in es men should no be
a ibu ed o he unde w i e .
The igu e below illus a es he “a p io i” conside a ion whe e all u u e liabili ies a e
only es ima ed and discoun ed o he s a ing poin = 0.
Figu e 54: New Business a = 0
34
The ollowing calcula ion example illus a es an “a p io i“ conside a ion whe e only
es ima ed alues a e used. A ime =0, he ollowing da a inpu a e gi en:
Ma ke In e es Ra e 4.0%
P emium 1,000.0
Cos Ra io 25.0%
Combined Ra io 97.5%
S&P Company Le el 150.0%
S&P P emium Ra e 27.0%
S&P Rese e Ra io 10.0%
Ex a Di idend 12.0%
34
Heep-Al ine , Ma ia: Ausgewähl e Aspek e de we o ien ie en Un e nehmenss eue ung in de
Schaden e siche ung; p. 63.
0
+1 T
Assessmen o Fu u e Liabili ies
Discoun ing o Fu u e Cash Flow
- 80 -
The equi ed ex a di idend o 12% and he capi al alloca ion o 150% e e o he
a ge alues o an indus ial insu e , because he e a e inc eased equi emen s
wi h espec o he e u n in con as o a mu ual insu ance company. The ollowing
able illus a es he es ima ed cash low si ua ion a he beginning o he conside a-
ion ime pe iod.
Pe iod CF in % A e age
Du a ion P emium Cos s Resul
Incu ed Fu u e
1 30.0% 0.5 1,000.0 250.0 217.5 532.5
2 25.0% 1.5 181.3 -181.3
3 20.0% 2.5 145.0 -145.0
4 15.0% 3.5 108.8 -108.8
5 10.0% 4.5 72.5 -72.5
To al 100.0% 2.0 1,000.0 250.0 0.0 725.0 25.0
Nominal Values
Claims
Figu e 55: Nominal Cash Flow a =0
In he i s s ep he nominal cash low is calcula ed. Independen on he du a ion o
he liabili ies he o al cash balance is 25.0.
Pe iod Accumul. Discoun P emium Cos s Resul
4.00% Incu ed Fu u e
1 98.06% 980.6 245.1 213.3 522.2
2 94.29% 170.9 -170.9
3 90.66% 131.5 -131.5
4 87.17% 94.8 -94.8
5 83.82% 60.8 -60.8
To al 980.6 245.1 0.0 671.2 64.2
Discoun end Values
Middle o he Pe iod Claims
Figu e 56: Discoun ed Cash Flow a =0
In he second s ep he discoun ed cash low is calcula ed. In his case (depending
on he du a ion o he liabili ies) he cash balance will inc ease up o 64.2. I mus be
checked whe he he calcula ed cash balance is su icien enough wi h espec o
he ex a di idend equi emen s. In he nex able he capi al alloca ion acco ding o
S anda d & Poo s model is calcula ed a ime =0.
Pe iod P emium Claims Rese e P em. Fac . Res. Fac . To al En . Fac .
27.0% 10.0% 150.0%
1 1,000.0 217.5 270.0 270.0 405.0
2 181.3 507.5 50.8 50.8 76.1
3 145.0 326.3 32.6 32.6 48.9
4 108.8 181.3 18.1 18.1 27.2
5 72.5 72.5 7.3 7.3 10.9
To al 725.0
Base o Capi al Alloca ion Capi al Alloca ion due o S&P
Figu e 57: Capi al Alloca ion a =0
- 81 -
The equi ed cos s o capi al a e calcula ed in he ollowing able acco ding o he
equi ed cos o capi al a io o 12% in o de o check he a ge ul illmen .
Pe iod Accumul. Discoun Requi ed Amoun Ta ge
Capi al CoC Ra e 12.0% o Co e . Ful illmen
4.00% Nominal Discoun ed in %
1 100.00% 405.0 522.2
2 96.15% 76.1 48.6 46.7 -170.9
3 92.46% 48.9 9.1 8.4 -131.5
4 88.90% 27.2 5.9 5.2 -94.8
5 85.48% 10.9 3.3 2.8 -60.8
6 82.19% 1.3 1.1
To al 64.3 64.2 100.0%
Cos o Capi al wi h
Begin o he Pe iod
Figu e 58: Ta ge Ful illmen a =0
The able shows he a ge ul illmen in he “a p io i” conside a ion a abou 100.0%.
Gene al Liabili y – Exis ing Business
“A pos e io i” he cash lows CF
1,
CF
2,…,
CF
o pas liabili ies
ha e ealized and he
cash lows CF
+1,
CF
+2,…,
CF
T
o u u e liabili ies ha e o be es ima ed acco ding o
a modi ied o ecas . The pas cash lows mus be accumula ed up o whe e he
u u e cash lows ha e o be discoun ed back o , see he igu e below.
Figu e 59: Exis ing Business a
35
In he case ha he ealized and u u e cash lows a e unknown o di icul o es i-
ma e, he ollowing app oxima ion scheme can be used:
35
Heep-Al ine , Ma ia: Ausgewähl e Aspek e de o ien ie en Un e nehmenss eue ung in de Scha-
den e siche ung; p. 63.
-1
+1
0
Assessmen o Fu u e Liabili ies
Valua ion o Pas Liabili ies
Accumula ion o Pas Cash Flow Discoun ing o Fu u e Cash Flow
- 82 -
• The “a p io i” es ima ed cash low pa e n un il ime can be calib a ed o
100% and used as cash low pa e n o he incu ed liabili ies.
• The “a p io i” es ima ed cash low pa e n s a ing om ime + 1 can be
calib a ed o 100% and used as cash low pa e n o he u u e liabili ies.
The ealized cash lows un il ime mus be accumula ed wi h he ealized isk- ee
in e es a es
1,
2,…,
, and he es ima ed u u e cash lows om ime mus be dis-
coun ed wi h he es ima ed isk- ee in e es a es
+1,
+2,…,
T.
In his con ex , i can be wo ked app oxima ely wi h a ixed a e age in e es a e o
he pas and a ixed a e age in e es a e o he u u e. As an app oxima ion, i is
also possible o “ ix” he capi al alloca ion o he “a p io i” alloca ion.
In he example conside ed be o e, he claims and in e es a e expe ience and es-
ima ion ha e de eloped a ime = 2 in he ollowing way:
Realized Ma ke In e es Ra e 3.75%
Es ima ed Fu u e In e es Ra e 3.50%
Realized Incu ed Claims Paymen 400.0
Es ima ed Fu u e Claims Paymen 350.0
A =2 he ealized ma ke in e es a e o 3.75% is di e en o he ini ially es ima ed
isk- ee ma ke in e es a e o 4%. The p ospec i e ma ke in e es a e is es i-
ma ed wi h 3.5%.The expec ed loss a io o e he o al un-o pe iod is es ima ed
as 75%. Those de elopmen s will ha e a nega i e impac on p o i abili y.
In his example, he cash low pa e n will be app oxima ed in he way p e iously
desc ibed. The change in claims expe ience leads o nominal liquid balance o ze o.
Pe iod P emium Cos s Resul
Incu ed Fu u e Incu ed Fu u e
1 30.0% 1,000.0 250.0 218.2 531.8
2 25.0% 181.8 -181.8
3 20.0% 155.6 -155.6
4 15.0% 116.7 -116.7
5 10.0% 77.8 -77.8
To al 55.0% 45.0% 1,000.0 250.0 400.0 350.0 0.0
Cash Flow in % Nominal Values
Claims
Figu e 60: Nominal Cash Flow a =2
In a second s ep he incu ed alues will be accumula ed un il =2 wi h he ealized
ma ke in e es a e o 3.75% as ollowing:
- 83 -
• Fo pe iod 1: (1+0.0375)
1.5
= 1.0568,
• Fo pe iod 2: (1+0.0375)
0.5
= 1.0186.
The es ima ed u u e cash lows will be discoun ed o = 2 wi h he es ima ed u u e
isk- ee in e es a e o 3.50% as ollows:
• Fo pe iod 3: (1 + 0.035)
-1/2
= 0.9892,
• Fo pe iod 4: (1 + 0.035)
-3/2
= 0.9497,
• Fo pe iod 5: (1 + 0.035)
-5/2
= 0.9176.
Accumula ion o incu ed pas cash lows and discoun o es ima ed u u e cash
lows esul s o an o e all discoun ed cash balance o 41.7.
Pe iod Accumul. Discoun P emium Cos s Resul
3.75% 3.50% Incu ed Fu u e
1 105.68% 1,056.8 264.2 230.6 562.0
2 101.86% 185.2 -185.2
3 98.29% 152.9 -152.9
4 94.97% 110.8 -110.8
5 91.76% 71.4 -71.4
To al 1,056.8 264.2 415.8 335.1 41.7
Discoun end Values
Middle o he Pe iod Claims
Figu e 61: Accumula ed / Discoun ed Cash Flow a =2
Al hough he liquid balance a e accumula ion and discoun ing is posi i e, i mus
be checked howe e , whe he he liquid balance is su icien wi h ega ds o he
equi ed capi al cos s. The ollowing able illus a es he ese e a he beginning o
he pe iod, which is needed as a base o he capi al alloca ion.
Pe iod P emium Claims Rese e P em. Fac . Res. Fac . To al En . Fac .
27.0% 10.0% 150.0%
1 1,000.0 218.2 270.0 270.0 405.0
2 181.8 531.8 53.2 53.2 79.8
3 155.6 350.0 35.0 35.0 52.5
4 116.7 194.4 19.4 19.4 29.2
5 77.8 77.8 7.8 7.8 11.7
To al 750.0
Capi al Alloca ion due o S&PBase o Capi al Alloca ion
Figu e 62: Capi al Alloca ion a =2
- 90 -
Ob iously he combina ion wi h 90% o Asse 1 gene a es a minimal isk (ex-
p essed in e ms o he s anda d de ia ion). The igu e abo e shows a ypical
Ma kowi z alloca ion wi h he e icien boa de and one ine icien combina ion. Only
he alloca ions on he bo de a e e icien .
Calcula ion Example - P e e ence-Sys ems
The e a e a lo o di e en e icien po olios. Howe e , which alloca ion should a
company choose? The chosen po olio should i wi h he indi idual company p e -
e ences. A p ope alloca ion can be de e mined by using p e e ence unc ions. Two
classic p e e ence unc ions (in he con ex o alue based managemen ), which
we e in oduced in he i s chap e , will be de e mined a e one yea and will be
used o ind ou he bes solu ion:
RORAC = Expec ed Value / Requi ed Capi al
≈ E / (
α
· STD),
EVA = Expec ed Value – Capi al Cos s
≈ E – k ·
α
·STD
wi h α he isk le el and E he expec ed alue. In he ollowing, a p i a e lines in-
su e will be examined by using hese wo p e e ence sys ems. This insu e in es s
in he wo asse classes as be o e and concen a es only on isk li e insu ance. In
his case, he e is a ( ela i ely) sa e ou low o liabili ies wi h he amoun o 1,100
and a s anda d de ia ion nea o ze o. We also assume a cos o capi al a io o
7.5% and a BBB- a ing (con o ming o he Sol ency II – Secu i y Le el o 99.5%).
Sha e STD
Asse 1
Asse s
Liabili ies
Resul
100% 1,100 1,100 0 100
90% 1,110 1,100 10 98
80% 1,120 1,100 20 105
70% 1,130 1,100 30 119
60% 1,140 1,100 40 139
50% 1,150 1,100 50 163
40% 1,160 1,100 60 188
30% 1,170 1,100 70 215
20% 1,180 1,100 80 243
10% 1,190 1,100 90 271
0% 1,200 1,100 100 300
Expec ed Value
Figu e 70: Risk / Re u n Analysis gi en wo Asse s (2)
- 91 -
In his case, he isk e u n p o ile is qui e simila o he isk e u n p o ile o he in-
es men company only in es ing in wo asse s wi h he di e ence ha all expec ed
cumula ed alues a e educed by 1,100, whe e, wi h espec o he combina ion o
0% Asse 1 and 100% Asse 2, we ob ain he ollowing esul s:
RC = 2.58 · 300 = 773
RORAC = 100 / 773 = 12.9%,
EVA = 100 – 7.5% · 773 = 42
In he ollowing able, all RORAC and EVA combina ions a e lis ed in such a way
ha an op imal alue can be de i ed.
Sha e STD RC RORAC EVA
Asse 1
99.50%
7.50%
100% 100 258 0.0% -19
90% 98 252 4.0% -9
80% 105 270 7.4% 0
70% 119 308 9.8% 7
60% 139 359 11.1% 13
50% 163 419 11.9% 19
40% 188 485 12.4% 24
30% 215 554 12.6% 28
20% 243 625 12.8% 33
10% 271 699 12.9% 38
0% 300 773 12.9% 42
Figu e 71: Capi al Alloca ions – P i a e Line Insu e
Al hough di e en me hods we e used, he same op imal esul s can be obse ed in
his calcula ion example. Acco ding o RORAC, he alloca ion wi h 0% o Asse 1
and 100% o Asse 2 is he mos p o i able wi h 12.9% ex a di idend. The maxi-
mum EVA is also obse ed gi en 100% Asse 2.
RORAC as well as EVA p o ide p e e ence sys ems in o de o ind an op imal de-
cision among all e icien po olios.
3.2.2 Asse & Unde w i ing Pe o mance
Looking a a p i a e line insu e wi h wo Asse s and wo lines o business (LoB) in
his sec ion, he ollowing inpu si ua ion can be assumed:
- 92 -
Co el.
10% expec ed STD
Asse 1 1,100 100
Asse 2 1,200 300
a e one yea
Co el.
10% expec ed STD
LoB 1 1,075 100
LoB 2 975 300
a e one yea
Figu e 72: Inpu Da a gi en wo Asse s and wo LoB
A co ela ion o 10% is gi en be ween he asse s as well as be ween he liabili ies,
excluding a co ela ion be ween asse s and liabili ies.
Because o ou a iables, he e a e many di e en possible combina ions. Due o
his ac , we will only ake in o accoun alloca ions wi h 50% s eps. In he ollowing
able he expec ed alues and s anda d de ia ions a e lis ed bu only o 0% 50%
and 100% combina ions:
Exp.
Asse 1 LoB 1 Asse s Liab. To al Resul
100% 100% 100 100 141 25
50% 100% 163 100 191 75
0% 100% 300 100 316 125
100% 50% 100 163 191 75
50% 50% 163 163 230 125
0% 50% 300 163 341 175
100% 0% 100 300 316 125
50% 0% 163 300 341 175
0% 0% 300 300 424 225
Sha e STD
Figu e 73: Risk / Re u n Analysis gi en wo Asse s and wo LoB
Gi en a y% sha e o asse 1 and a x% sha e o liabili y 1 we ob ain he ollowing
o mulas:
E = (y · 1,100 + (1 – y) · 1,200) – (x · 1,075 + (1 – x) · 975),
VAR(A) = (y · 100)
2
+ ((1 – y) · 300)
2
+ 2 · 10% · y · 100 · (1 – y) · 300
VAR(L) = (x · 100)
2
+ ((1 – x) · 300)
2
+ 2 · 10% · x · 100 · (1 – x) · 300
VAR = VAR(A) + VAR(L)
Gi en a combina ion wi h 50% o Asse 1 and 50% o LoB 1 we ob ain he ollowing
esul s:
- 93 -
E = (50% · 1,100 + 50% · 1,200) – (50% · 1,075 + 50% · 975)
=125
STD = (163
2
+ 163
2
)
0.5
= 230
Please no ice ha he s anda d de ia ion o 163 gi en a combina ion o 50% Asse
1 has been calcula ed be o e. (The same calcula ion applies o he s anda d de ia-
ion gi en a combina ion o 50% LoB 1.)
Calcula ion Example – P i a e Line Insu e
In he ollowing sec ion, di e en business models will be checked wi h espec o
he gi en inpu da a – a p i a e line insu e and a einsu e . The esul s o he p i-
a e line insu e a e lis ed below wi h espec o all combina ions o 0%, 50 % and
100 % o Asse 1 o LoB 1.
Exp. RC RORAC EVA
Asse 1 LoB 1 Resul 99.50% 7.50%
100% 100% 25 364 6.9% -2
50% 100% 75 492 15.2% 38
0% 100% 125 815 15.3% 64
100% 50% 75 492 15.2% 38
50% 50% 125 593 21.1% 81
0% 50% 175 879 19.9% 109
100% 0% 125 815 15.3% 64
50% 0% 175 879 19.9% 109
0% 0% 225 1,093 20.6% 143
Sha e
Figu e 74: RoRAC and EVA Op imum – P i a e Line Insu e
The RoRAC op imum is achie ed in “ he middle” gi en a combina ion o 50% Asse
1 and 50% LoB 1 whe e he EVA op imum is achie ed a “ he bounda y” gi en 0%
Asse 1 and 0% LoB 1. This is a qui e ex eme combina ion wi h a high capi al e-
qui emen and i is no clea , i such a high capi al amoun is a ailable.
In o al, he si ua ion is much mo e complex and less anspa en han in he case o
wo asse s. One he one hand i is s ill possible o exclude ine icien alloca ions bu
on he o he hand an e iciency cu e canno be easily iden i ied.
Looking a he isk e u n p o iles o he combina ions p e iously analyzed, “ isu-
ally” e icien combina ions can be iden i ied, bu he e a e s ill a lo o ine icien
combina ions. In he igu e below all combina ions o 0%, 25%, 50%, 75% and
100% Asse 1 o LoB 1 a e lis ed.
- 94 -
Exp. RC RORAC EVA
Asse 1 LoB 1 Resul 99.90% 15.00%
100% 100% 25 437 5.7% -41
50% 100% 75 590 12.7% -14
0% 100% 125 977 12.8% -22
100% 50% 75 590 12.7% -14
50% 50% 125 711 17.6% 18
0% 50% 175 1,055 16.6% 17
100% 0% 125 977 12.8% -22
50% 0% 175 1,055 16.6% 17
0% 0% 225 1,311 17.2% 28
Sha e
0
50
100
150
200
250
0 100 200 300 400 500
Risk
Re u n
Figu e 75: Risk / Re u n P o ile gi en wo Asse s and wo LoB
In he nex calcula ion example he RoRAC and EVA op ima o a einsu e wi h
di e en CoC pa ame e is analyzed.
Calcula ion Example - Reinsu e
The esul s o he einsu e a e lis ed in he able below. As p e iously s a ed, only
combina ions wi h 0%, 50% and 100% o Asse 1 o LoB 1a e shown in he igu e.
The RORAC op imum is achie ed o a combina ion wi h 50% Asse 1 and a 50%
LoB 1. The EVA op imum is achie ed o he “ex eme” combina ion wi h 0% Asse
1 and 0% LoB 1. The op imal combina ions a e he same as be o e whe eas he
op imal alues a e di e en .
Figu e 76: RoRAC and EVA Op imum – Reinsu e
- 95 -
A po olio wi h wo isky b anches and wo isky asse classes c ea es a di e en
e u n s uc u e han a po olio wi h wo isky asse s and one mo e o less isk- ee
liabili y. The si ua ion in he second case e lec s he classical Ma kowi z app oach
whe eas he (mo e complex) si ua ion in he i s case is mo e ealis ic.
Calcula ion Example - Conclusion
The capi al cos a e is p ede ined by he managemen . Bu he e is an unce ain y
abou he achie abili y o his goal. The decisi e ac o is he ma ke , which can be
ha dly in luenced by he insu e . In o de o achie e a speci ied a ge , he insu e
would ha e o inc ease he p emium o he capi al ma ke e u ns. Bu he insu e
has o ocus on he ma ke p ices and he compe i o s in o de o be compe i i e.
Ne e heless, a ela i ely low capi al cos a e would be una ac i e o po en ial
in es o s. As a consequence, companies a e o ced o se almos unachie able
goals.
Le ’s ha e a look a he p e iously examined einsu e wi h a secu i y le el o 99.9%
and a capi al cos a e o 15%. I is ques ionable whe he his capi al cos a e is
app op ia e. A his secu i y le el he insu e expec s one de aul wi hin 1,000 yea s.
I you compa e ha ac wi h he ela i i y high capi al cos a e he p opo ionali y
be ween isk and e u n is doub ul.
The a e o e u n de ined by he managemen is o en ou side he e iciency cu e.
The e a e wo possible measu es o “p oduce” achie able combina ions: Fi s ly, he
insu e can educe he aimed ex a di idend. Howe e , he insu e is in a compe i-
i e si ua ion and in es o s could be dissa is ied wi h he e u n on in es men .
The e o e, i is no so easy o educe he a e o e u n. Secondly, hey could ake
mo e isks, bu hen he a ge ed secu i y le el would no be eached.
By using he EVA me hod some ques ions occu . Is i easonable o de ine a nega-
i e EVA alue as a des uc ion o capi al? In o de o speci y his issue one could
look a he igu e abo e. An ex a di idend o 12.8% (addi ionally o he isk- ee
e u n) wi h nega i e EVA is obse ed. The RORAC p oduced is qui e high, so he
in e p e a ion as capi al des uc ion seems o be doub ul.
3.2.3 Asse & Unde w i ing Pe o mance – Sepa a e T ea men
In he p e ious sec ion i was demons a ed how he simul aneous op imiza ion o
unde w i ing and capi al in es men can be managed in an insu ance company.
Due o he ac ha insu ance is a co-p oduc , you can i ually spli an insu ance
company in o wo pa s: Unde w i ing and asse managemen . Rega ding he asse
managemen , he ollowing asse s a e assumed in he ollowing:
- 96 -
Re u n Accum. CV Absolu e
Asse 1 4.0% 1,040 0.0% 0
Asse 2 10.0% 1,100 30.0% 330
Expec ed S d. De ia ion
Figu e 77: Inpu Da a – A ailable Asse Po olio
I is ob ious ha Asse 1 ep esen s a isk- ee asse . The e o e no capi al is
needed o secu e his asse . In con as o his, Asse 2 is a isky asse which de-
mands capi al o secu e he asse . Wi h his in mind, wo s a egies will be checked:
In he i s case, he insu e only in es s in he isk- ee asse (S a egy 1). Al e na-
i ely, he insu ance company in es s also in he isky asse (S a egy 2). Compa -
ing hose wo s a egies enables a p ope s ee ing o he po olio acco ding o un-
de w i ing and in es men impac on he isk. These s a egies will be discussed on
he basis o he ollowing ques ions:
• Wha is he equi ed capi al o bo h s a egies a a de aul le el o 0.2 %?
• Which s a egy p o ides a highe e u n on Risk Adjus ed Capi al?
• Which s a egy is he bes ?
Taking in o accoun a isk- ee in es men o he equi ed capi al a he beginning o
he pe iod as well as he expec ed esul due o in es men and unde w i ing, we
ob ain he ollowing ela ionships o bo h s a egies:
RC = ( · STD – (E(A) – E(L)) / (1 + )
STD he o e all s anda d de ia ion,
E(A) he expec ed alue o he asse s a he end o yea ,
E(L) he expec ed alue o he liabili ies a he end o he yea and
he isk- ee in e es a e.
These ela ionships a e de i ed om he dis ibu ion o he capi al a he end o he
pe iod unde an assump ion o no mally dis ibu ed asse s and liabili ies.
Calcula ion Example – S a egy 1 wi h isk- ee Asse s
As p e iously s a ed, he insu ance company in es s solely in a isk- ee asse , o
example a go e nmen bond wi h a de aul isk o almos ze o. We assume ha he
o al p emium income is a he beginning o he yea . Fu he mo e all cos expendi-
u e is a he end o he yea . As a esul o his assump ion, he p emium income
can be ully in es ed o e a pe iod o one yea . (This assump ion can be achie ed
in any case by conside a ion o sui able p esen alues.)
- 97 -
CV Absolu e
Asse 1 100.0% 4.0% 1,040 0.0% 0
Asse 2 0.0% 10.0% 0 30.0% 0
LoB 1 100.0% 98.5% 985 15.0% 148
To al 55 148
Expec ed S d. De ia ion
Figu e 78: Inpu Da a – S a egy 1
Fo Asse 2 and he line o business, a no mal dis ibu ion is assumed. Mo eo e
he e is no co ela ion be ween asse s and liabili ies. The equi ed capi al is calcu-
la ed acco ding o he p e iously speci ied o mula whe e o a secu i y le el o
99.8 % he ac o equals o 2.88 so ha
RC = (2.88 · 148 – 55) / 1.04 = 357
RORAC = (E(A) – E(L)) / RC = 55 / 357 = 15.4%
To al Re u n = + RORAC = 4.0% + 15.4% = 19.4%
The insu ance company needs a equi ed capi al o 357 based on a de aul isk o
0.2 %. An a e age o al RoRAC o 19.4% is achie ed.
Calcula ion Example – S a egy 2 wi h isky Asse s
In his case he insu ance company in es s 65% in he isk- ee asse and 35% is
in es ed in he isky asse class. The da a needed o he ollowing calcula ions a e
lis ed in he able below.
CV Absolu e
Asse 1 65.0% 4.0% 676 0.0% 0
Asse 2 35.0% 10.0% 385 30.0% 116
LoB 1 100.0% 99.0% 990 15.0% 149
To al 71 188
Expec ed S d. De ia ion
Figu e 79: Inpu Da a – S a egy 2
Fo comple eness i should be no ed ha in s a egy 2 he e is also no co ela ion
be ween capi al in es men and he line o business. Due o he isky asse , he
company has o p o ide mo e capi al. Fu he mo e, he addi ional cos s o he
complex asse managemen a e e lec ed in he combined a io, which is 0.5 pe -
cen age poin s highe han o s a egy 1.
- 98 -
Based on hose esul s i is possible o calcula e he equi ed capi al and he e u n
on in es men . Conce ning he p e iously men ioned o mula, we can calcula e wi h
espec o he 99.8% secu i y-le el as ollows:
RC = (2.88 · 188 – 71) / 1.04 = 452
RORAC = 71 / 452 = 15.7 %
To al Re u n = 15.7 % + 4.0 % = 19.7
I he esul s o he di e en s a egies a e compa ed, he ollowing di e ences can
be obse ed. Following s a egy 1, he company has o p o ide capi al o 357 and
achie es a o al RoRAC o 19.4%. Unde s a egy 2, a highe capi al o 452 mus
be p o ided because o he isky asse s. This is ewa ded wi h a ma ginally be e
e u n on in es men o 19.7%. Does his esul imply ha s a egy 2 is be e han
s a egy 1? In he ollowing sec ion i will be e i ied whe he he esul s s ay alid
when he pa ame e s a e changed.
Calcula ion Example – Impac o Pa ame e Change
To analyze he impac o pa ame e change, he p incipal scena ios o bo h s a e-
gies a e main ained, bu he ollowing pa ame e s a e changed:
• Reduc ion o he isk- ee in e es a e.
• Inc ease o he Combined Ra io.
• Reduc ion o he expec ed e u n o asse 2.
• Inc ease o he s anda d de ia ion o asse 2
• Inc ease o he s anda d de ia ion o he LoB.
Due o hose changes, bo h s a egies mus be analyzed wi h espec o o al Ro-
RAC and he de aul isk. The ollowing able shows he impac o he pa ame e
changes (wi h unchanged capi al a s a ):
- 99 -
Pa ame e RoRAC Ruin
Old New P ob.
Risk F ee In e es Ra e 4.0% 3.0% 15.6% 0.26%
Combinded Ra io 98.5% 99.5% 16.6% 0.26%
Expec ed Re u n Asse 2 1,100 1,075 19.4% 0.20%
S d. De ia ion o Asse 2 330 550 19.4% 0.20%
S d. De ia ion o B anch 15.0% 20.0% 19.4% 1.52%
Value
Figu e 80: Impac o Pa ame e Changes (1)
Due o he ac ha s a egy 1 in es s only in isk- ee asse s, he e is no impac on
he RoRAC and he de aul isk when he expec ed e u n dec eases and he s an-
da d de ia ion inc eases wi h espec o asse 2. Rela i e o his scena io, only he
inc ease o he LoB ola ili y p oduces c ucial esul s.
In he ollowing able he impac s o he pa ame e changes a e illus a ed o he
second s a egy:
Pa ame e RoRac Ruin
Old New P ob.
Risk F ee In e es Ra e 4.0% 3.0% 17.3% 0.24%
Combinded Ra io 99.0% 100.0% 17.5% 0.25%
Expec ed Re u n Asse 2 1,100 1,075 17.8% 0.30%
S d. De ia ion o Asse 2 330 550 19.7% 1.30%
S d. De ia ion o B anch 15.0% 20.0% 19.7% 0.91%
Value
Figu e 81: Impac o Pa ame e Changes (2)
S a egy 2 is highly a ec ed by he isky asse , in such a way ha any inc ease o
he s anda d de ia ion p oduces ano he c ucial impac in his case.
Be o e he conside a ion o pa ame e changes, s a egy 2 seemed o p o ide a
(sligh ly) be e pe o mance han s a egy 1. A e he conside a ion o pa ame e
changes s a egy 2 seems o be mo e ola ile and o p oduce mo e c ucial si ua-
ions wi h espec o he sol ency equi emen s.
Asse & Unde w i ing Pe o mance – Sepa a e T ea men o Pe o mance
In es men exclusi ely in isk- ee asse s is no in any case sa is ac o y o insu -
ance companies and hei ambi ious e u n a ge s. Due o his ac , a company
mus some imes in es in isky asse o inc ease he p o i abili y due o syne gy e -
ec s. In such a case, i is impo an o di e en ia e be ween he pe o mance o he
unde w i ing and o he asse managemen . In he ollowing i will be discussed how
unde w i ing and asse managemen con ibu e o an o e all pe o mance, see he
ollowing able.
- 106 -
ρ
00
… ρ
0n
φ
00
… φ
0m
… ρ
ik
… … φ
il
…
ρ
n0
… ρ
nn
φ
n0
… φ
nm
φ
00
…φ
0n
ψ
00
…ψ
0m
… φ
jk
… … ψ
jl
…
φ
m0
… φ
mn
ψ
m0
… ψ
mm
Figu e 85: Co ela ion Ma ix
37
We can ace he gene al case o he al eady conside ed cases by decomposing
he symme ic co ela ion ma ix C o C = M
T
· D · M (Cholesky decomposi ion) wi h
M an uppe iangula ma ix and D a diagonal ma ix.
0 0 0 0 0
0 0 0 0 0
= x
0 0 0 0 0
x
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
Figu e 86: Cholesky Decomposi ion o he Co ela ion Ma ix
38
All a iables will be unco ela ed when we ans o m all asse s and liabili ies acco d-
ing o he Cholesky decomposi ion. The ans o med combined a ios can be in e -
p e ed as combined a ios o a linea combina ion o he liabili ies, bu he ans-
o med e u ns a e mixed e ms now.
Conce ning he gene al case, we will concen a e on he solu ions wi h espec o
he RORAC, because he solu ions wi h espec o he EVA a e qui e complex. In
many cases such a solu ion does no exis .
Rega ding he RORAC op imiza ion in he gene al case, he ollowing o mulas a e
ob ained conce ning he pa ial de i a i es o S by x
i
o y
j
:
∂S/∂x
i
= (∑(ρ
ik
· σ
i
– ρ
0k
· σ) · x
k
· σ
k
+
(∑(φ
il
· σ
i
– φ
0l
· σ) · y
l
· τ
l
) / S =: X
i
*/ S
∂S/∂y
j
= (∑(φ
jk
· τ
j
– φ
0k
· τ) · x
k
· σ
k
+
(∑(ψ
jl
· τ
j
– ψ
0l
· τ) · y
l
· τ
l
) / S =: Y
j
*/ S
37
Heep-Al ine , 2011, Pe o manceop imie ung des (B u o) Neugeschä s in de Schaden e siche-
ung, p. 9.
38
Heep-Al ine , 2011, Pe o manceop imie ung des (B u o) Neugeschä s in de Schaden e siche-
ung, p 10.
- 107 -
The o mulas in he unco ela ed case a e simpli ied e sions o he gene al o mu-
las. In he gene al case, we ob ain o all i, j, k, l > 0
X
i
*/∆
i
= X
k
*/∆
k
= –Y
j
*/∆c
j
= –Y
l
*/∆c
l
E · X
1
* = ∆
1
· S
2
whe e he i s equa ions a e linea ones. Subs i u ing S
2
in a sui able way, ano he
linea equa ion is ob ained
( – c) · X
1
* = ∆
1
· (σ · (∑ ρ
0k
· x
k
· σ
k
+ ∑ φ
0l
· y
l
· τ
l
) +
τ · (∑ φ
0k
· x
k
· σ
k
+ ∑ ψ
0l
· y
l
· τ
l
)).
In o al, oge he wi h he wo no maliza ion equa ions ∑ x
i
= 1 = ∑ y
j
we now ob ain
(m + n + 2) linea equa ions wi h (m + n + 2) a iables x, x
1
, …, x
n
and y, y
1
, …, y
m
,
which a e sol able o he gene al case (i.e. wi h he excep ion o singula pa ame-
e cons ella ions). Only in special cases can he solu ion be desc ibed explici ly. A
solu ion is no au oma ically easible because some coe icien s may no be be-
ween ze o and one.
In some special cases, simpli ied solu ions will be ob ained ha can be ea ed in
EXCEL. In he nex sec ion, some esul s conce ning special cases a e discussed.
3.3.4 Calcula ion Examples
This sec ion discusses some examples. In his con ex , he e m “ easible” op imum
implies ha a local maximum exis s in which all coe icien s a e be ween ze o and
one. Maximum alues a he bounda ies o allowed combina ions a e no consid-
e ed a his poin .
Calcula ion Examples - Unco ela ed Asse s and Liabili ies
Fo he i s example, he assump ion is o in es he new business p emium in o
wo isky asse classes, which a e no co ela ed wi h each o he . The ollowing in-
pu da a is assumed:
S anda d
De ia ion
Cumula ed
Re u n
Asse 1 10.0%
107.5%
Asse 2 20.0%
112.5%
Figu e 87: Cumula ed Re u ns
- 108 -
Fo simplici y, only he unde w i ing o a single isk- ee line o business wi h a
combined a io o 100.0% is conside ed (unde w i ing jus a ela i ely low isk ca e-
go y in a e y la ge collec i e.) Unde hese assump ions, he model p o ides as
ollows, he classical Ma kowi z app oach o asse s:
S anda d
De ia ion
Re u n
Asse 1 10.0%
7.5%
Asse 2 20.0%
12.5%
Figu e 88: Uncumula ed Re u ns
Conce ning he uncumula ed e u ns, he ollowing able p o ides he RORAC and
he EVA op imum gi en a capi al cos pa ame e o c = 30.0% wi h espec o EVA.
S anda d
De ia ion
Re u n RORAC
Op imum
EVA Op .
c = 30.0%
Asse 1 10.0%
7.5%
70.6%
35.3%
Asse 2 20.0%
12.5%
29.4%
64.7%
Figu e 89: RORAC and EVA Op imum gi en Risk- ee Liabili ies
Wi h espec o he EVA op imum he e is signi ican ly mo e in es men in he iskie
asse class han wi h espec o he RORAC op imum. Fu he mo e, only he abso-
lu e alue is op imized wi hou conside a ion o he capi al equi emen s. In he spe-
ci ic case, he capi al equi emen gi en he EVA op imum is signi ican ly highe
han he capi al equi emen in case o he he RORAC op imum. Possibly, he
company is no able o deposi he equi ed capi al o he EVA op imum.
The RORAC op imum can be illus a ed as a angen poin on he isk / e u n cu e
wi h a line h ough he o igin as ollows:
- 109 -
0%
5%
10%
15%
0% 5% 10% 15% 20%
Risk
Re u n
Figu e 90: RORAC Op imum
The EVA op imum is de ined as a angen poin on he isk / e u n cu e wi h a line
whose slope equals he cos o capi al pa ame e c. Because he slope o he isk /
e u n cu e does no all below an asymp o ic alue, such a angen poin does no
exis o small pa ame e c, see he ollowing igu e.
0%
5%
10%
15%
0% 5% 10% 15% 20%
Risk
Re u n
0%
5%
10%
15%
0% 5% 10% 15% 20%
Risk
Re u n
Figu e 91: EVA Op imum o c = 30.0%, bu no EVA Op imum o c = 10.0%
I he cos o capi al pa ame e c is oo small hen only a maximum alue exis s a
he bounda y wi h 100% in es men in he isky asse – ega dless o he amoun o
capi al equi ed.
- 110 -
Now he example will be ex ended by unde w i ing wo lines o business (unco e-
la ed wi h each o he and wi h he asse classes) wi h he ollowing isk / e u n p o-
ile in he op imiza ion app oach:
Combined
Ra io S anda d
De ia ion
LoB 1
97.0%
20.0%
LoB 2
98.5%
10.0%
Figu e 92: Combined Ra ios
The ollowing able illus a es he asse and liabili y sha es wi h espec o he RO-
RAC op imum:
STD Cum. Re-
u n Sha e
STD Comb.
Ra io Sha e
Asse 1 10.0%
107.5%
64.5%
LoB 1 20.0%
97.0%
24.7%
Asse 2 20.0%
112.5%
35.5%
LoB 2 10.0%
98.5%
75.3%
Figu e 93: RORAC Op imum gi en Risky Asse s and Lines o Business
The sha es o he asse classes ha e changed wi h he inclusion o he lines o
business in he op imiza ion, because he op imiza ion app oach (e en wi h unco -
ela ed asse s and liabili ies) is no independen o he asse and liabili y isk / e u n
p o iles. In his pa icula case, he e is a maximum because he e u ns a e posi-
i e. Wi h he inclusion o isky lines o business, he e is a highe weigh wi h e-
spec o he isky asse due o a highe deg ee o di e si ica ion.
An e en s onge shi wi h espec o he weigh o he isky asse and liabili y
classes a e obse ed a e EVA op imiza ion – ega dless o he amoun o capi al
equi ed.
STD Cum. Re-
u n Sha e
STD Comb.
Ra io Sha e
Asse 1 10.0%
107.5%
12.9%
LoB 1
20.0%
97.0%
40.1%
Asse 2 20.0%
112.5%
87.1%
LoB 2
10.0%
98.5%
59.9%
Figu e 94: EVA Op imum gi en Risky Asse s and Liabili ies o c = 30.0%
- 111 -
The ables show ha he e a e signi ican di e ences be ween he EVA and RO-
RAC op imum. Fo he EVA op imiza ion app oach he e a e highe weigh s o he
iskie posi ions.
I has al eady been men ioned ha in he p esen model app oach he conside a ion
o isk- ee asse s is no p oblem because he e is usually isk in he liabili y posi-
ions. In ac , i is an independen business decision, o in es he cash lows esul -
ing om he echnical esul in isky o iskless in es men s. In his case a isky in-
es men is no necessa ily “be e ” han a isk- ee in es men .
Thus, he example is ex ended by including a isk- ee asse wi h an in e es a e o
4.0% in o de o check how he weigh s o asse and liabili y classes will change:
STD Cum. Re-
u n Sha e
STD Comb.
Ra io Sha e
Risk F ee 0.0%
104.0%
22.4%
LoB 1
20.0%
97.0%
40.1%
Asse 1 10.0%
107.5%
12.9%
LoB 2
10.0%
98.5%
59.9%
Asse 2 20.0%
112.5%
87.1%
Figu e 95: RORAC Op imum gi en a Risk- ee Asse
The isk- ee asse only ecei es a ela i ely small weigh because o he high di-
e si ica ion.
Wi h espec o EVA, he esul s a e qui e complex. The e is no solu ion o he
(qui e ealis ic) capi al cos pa ame e c = 30.0%. To ob ain a solu ion in he classi-
cal sense, he pa ame e has o be signi ican ly inc eased, e.g. c = 60.0%.
STD Cum. Re-
u n Sha e
STD Comb.
Ra io Sha e
Risk ee 0.0%
104.0%
-119.8%
LoB 1
20.0%
97.0%
31.7%
Asse 1 10.0%
107.5%
136.8%
LoB 2
10.0%
98.5%
68.3%
Asse 2 20.0%
112.5%
83.1%
Figu e 96: EVA Op imum gi en a Risk F ee Asse o c = 60.0%
As can be seen in he able abo e, he e is a solu ion o he EVA app oach bu i is
no easible in he sense ha each sha e is posi i e and less han one. The e is
such a d ama ic es uc u ing o asse s ha in p inciple he ecommenda ion is no
o buy he isk- ee asse s, bu a he o bo ow i , in o de o ake e en mo e o he
- 112 -
iskie asse classes o he po olio. This e y simple example shows ha he EVA
op imiza ion may be much mo e c ucial han he RORAC op imiza ion.
Calcula ion Examples – Co ela ed Asse s and Liabili ies
In his sec ion, he e ec s o co ela ions will be analyzed . Only he RORAC op imi-
za ion will be analyzed because o he high complexi y o he EVA op imiza ion in
connec ion wi h e y c ucial solu ions.
Fo his pu pose, he case o he wo p e iously in oduced isky asse s and lines o
business is conside ed, whe e i is now assumed ha he wo asse s and liabili ies
a e co ela ed wi h 50% o each o he . This example al eady con ains essen ial ea-
u es o he gene al case:
STD Cum. Re-
u n Sha e
STD Comb.
Ra io Sha e
Asse 1 10.0%
107.5%
63.0%
LoB 1
20.0%
97.0%
11.1%
Asse 2 20.0%
112.5%
37.0%
LoB 2
10.0%
98.5%
88.9%
Figu e 97: RORAC Op imum gi en Risky Asse s & Liabili ies wi h 50.0% Co .
The use o co ela ions changes he weigh s, such ha he lowe - isk posi ions ob-
ain a highe weigh . I he co ela ions a e oo high, i is no longe a easible solu-
ion.
In a inal s ep, his example is ex ended e en mo e by including he isk- ee asse
wi h an in e es a e o 4.0%. This example will con ain he whole complexi y o he
gene al case excep he co ela ions be ween asse s and liabili ies.
STD Cum. Re-
u n Sha e
STD Comb.
Ra io Sha e
Risk F ee 0.0%
104.0%
36.4%
LoB 1
20.0%
97.0%
9.1%
Asse 1 10.0%
107.5%
33.3%
LoB 2
10.0%
98.5%
90.9%
Asse 2 20.0%
112.5%
30.3%
Figu e 98: RORAC Op imum wi h a Risk F ee Asse wi h 50.0% Co ela ion.
I a isk- ee asse in he RORAC op imiza ion is in ol ed (assumed he asse and
liabili ies a e co ela ed o each o he ) his asse ob ains a high weigh . I is also
ema kable o see ha he inclusion o a isk- ee asse in luences no only he
weigh o he asse s, bu also he weigh o he liabili ies.
- 113 -
3.3.5 Conclusion
Unde some simpli ied model assump ions, he e is always an op imum o isky
in es men s wi h espec o he RORAC. In he special case o wo isky in es -
men s, one can ep esen his as angen ial poin on he isk / e u n cu e wi h a
line h ough he ze o poin .
I he EVA as a p e e ence unc ion is used, hen in he simple case o wo isky in-
es men s he e will no always be a easible solu ion. In such a case a solu ion is
ob ained as a angen ial poin on he isk / e u n cu e wi h a line Yield = cons an
+ c · STD. This angen poin does no exis i c is oo small. Thus, he op imum is
he maximum a he bounda y – ega dless o he capi al equi emen . Highe cos
pa ame e c p oduce solu ions, bu may be un ealis ic.
This esul indica es some doub s conce ning he con enience o he EVA as a use-
ul KPI, because e en in he simples case a solu ion depends on he choice o he
capi al cos pa ame e . The ac ha only e y high capi al cos pa ame e s p oduce
an op imum, should be c i ically e alua ed.
A (simpli ied) RORAC op imiza ion will gene a e solu ions, which a e accep able bu
no necessa ily in he sense ha all pa ame e s a e be ween ze o and one. I he e
a e jus co ela ions be ween asse s and liabili ies, we will immedia ely ecognize
ha op imal combina ions o asse s will be in luenced by he cha ac e is ics o he
asse s hemsel es and he cha ac e is ics o he liabili ies (and ice e sa). I is
possible o summa ize g oups o segmen s o ha e a smalle dimension in he op-
imiza ion app oach in o de o ob ain easible solu ions.
In gene al, he e is a solu ion o he RORAC op imiza ion p oblem. I is, howe e ,
e y di icul o in e p e .
As has al eady been men ioned, he conside a ions ou lined he e a e no necessa -
ily sui able o he op imiza ion o complex einsu ance s uc u es. The app oach
p esen ed he e can only be used o deli e a simpli ied model o ha e be e s a -
ing alues o al e na i e calcula ions in in e nal models. Ne e heless, some impli-
ca ions o alue-based managemen can be iden i ied.
- 114 -
3.4 T ea men o Ex a Di idends
In his chap e , he ea men o ex a di idends will be discussed. The e o e, i is
good o keep in mind he ollowing equi alence equa ion, which e lec s all ele an
aspec s o a i a ing be o e a con ac is w i en:
P esen Value o P emiums = P esen Value o Claims
+ P esen Value o Cos s
+ P esen Value o Ex a Di idends.
This chap e concen a es on he co ec dis ibu ion o di idends o he sha eholde
a e ha ing unde w i en he con ac . Cos s o capi al included in he p emium
should only be dis ibu ed o he sha eholde when hey ha e been ea ned. This will
be analyzed by means o some calcula ion examples in he ollowing sec ions.
3.4.1 Cos o Capi al and Ta ge P emium
Ex a di idends a e de ined as he e u n on equi ed capi al abo e he Risk- ee
in e es a e. The Cos o Capi al as p esen alue o all ex a di idend depends on
he capi al alloca ion a he beginning o a pe iod as well as on he equi ed ex a
di idend a he end o a pe iod.
We ha e analyzed in a p e ious sec ion how a ge combined a ios and hus a ge
combined p emiums depend on di e en le els o isk- ee in e es a es wi h e-
spec o di e en CoC models e lec ing di e en insu ance ma ke s. We did no
conside any change in he CoC equi emen s gi en di e en le els o isk- ee in-
e es a es, compa e he able below.
Ma ke In e es Ra e 2.0% 4.0% 6.0%
Ex a Di idend 13.0% 11.0% 9.0%
To al Yield 15.0% 15.0% 15.0%
P esen Value Claims 88.7 79.6 72.0
P esen Value Cos s 29.1 28.3 27.6
P esen Value Ex a Di idends 31.7 24.2 18.0
P esen Value P emium 149.5 132.1 117.6
P emium 151.0 134.7 121.1
Figu e 99: Ta ge P emiums gi en a Fixed To al Yield
- 115 -
I he o al yield is ixed han he e is s ong dependency on he di e en le els o
isk- ee in e es a es wi h espec o he a ge p emiums.
Ma ke In e es Ra e 2.0% 4.0% 6.0%
Ex a Di idend 8.0% 11.0% 14.0%
To al Yield 10.0% 15.0% 20.0%
P esen Value Claims 88.7 79.6 72.0
P esen Value Cos s 29.1 28.3 27.6
P esen Value Ex a Di idends 19.5 24.2 28.0
P esen Value P emium 137.3 132.1 127.6
P emium 138.7 134.7 131.4
Figu e 100: Ta ge P emiums gi en a Va iable To al Yield
The model wi h a iable o al yields esul s less- ola ile a ge p emiums in he case
o a change in he isk- ee in e es a e. This concep could be used o s abilize he
p emium calcula ion.
3.4.2 Ex a Di idends Acco ding o Unde w i ing Pe o mance
The ea men o ex a di idends will be explained by a calcula ion example. Fi s ly,
he unde w i ing pe o mance will be aken in o accoun ; a e wa ds he impac o
isky in es men s will be analyzed. Bo h sides should be conside ed sepa a ely,
because unde w i ing and asse managemen ac independen ly om each o he in
an insu ance company.
Fo his example an indus ial insu e was chosen. No mally, an indus ial insu e
has highe yield expec a ions han an insu e in a pe sonal lines business – o ex-
ample o ge a good “A” a ing. The model calcula ion is based on he ollowing in-
pu pa ame e :
Ma ke In e es Ra e
P emium
Cos Ra io
Claims Ra io
En i y Fac o
P emium Fac o
Rese e Fac o
CoC Ra e
4.0%
1,000.0
25.0%
73.3%
150.0%
22.0%
12.0%
10.0%
The nex igu e illus a es he a-p io i unde w i ing cash lows wi h a a ge ul ill-
men o 100.0%. The capi al alloca ion o 330 a he beginning is ob ained by mul i-
plying he p emium o 1,000 wi h he en i y ac o o 150% and he p emium ac o
- 122 -
4 Embedded Value as Fai Value App oach
The on-going change o a alue based managemen equi es app op ia e key ig-
u es and s ee ing sys ems. In e nal models in non-li e insu ance a e usually based
on he economic capi al a e one yea as s ochas ic a ge unc ion acco ding o he
immedia e ealiza ion o asse s and liabili ies a ma ke alues. Tha does no al-
ways cons i u e a ealis ic hypo hesis. The di ec liquida ion o all asse s and liabili-
ies would esul in high discoun s on he asse s espec i ely in high su cha ges on
he liabili ies. Especially he exis ence o ma ke alues o loss ese es seems
illusiona y. In addi ion, he ensions in he inancial ma ke s lead o dis o ions in
ma ke p ices (e.g. in o m as a liquidi y p emium). These ci cums ances do no e-
lec adequa ely he medium o long e m alue si ua ion o an insu ance com-
pany.
39
Gi en he Embedded Value EV (as an al e na i e app oach o e alua ing co po a e
economic capi al) he ai alues o asse s and liabili ies will be ealized only o e
ime acco ding o a i ual un-o . Thus, “modelling” o a i ual ex e nal in es o o
he insu ance po olio is no eques ed. This app oach leads o he ollowing ad an-
ages and disad an ages:
• As a consequence o ic ional and o he cos s, he EV is lowe han he
di ec ly a ibu able economic alue.
• The EV is mo e ealis ic because he e would be discoun s in he case o
selling he po olio.
• The EV eac s less sensi i ely o ma ke p ice luc ua ions.
The e o e, he EV-app oach is well es ablished in he ypical long e m business o
li e insu ance. In non-li e insu ance, howe e , his concep is ac ually no well es ab-
lished, al hough he e a e i s applica ions wi hin he in eg a ed s ee ing o he
whole business. Thus, i is consequen o conside also he EV wi hin he alue
based managemen in non-li e insu ance a a middle- e m pe spec i e. This will
esul in a cohe en iew on isk s ee ing a g oup le el. Especially wi hin he
amewo k o Sol ency II he insu ance g oups a e in e es ed in a consis en com-
pany s ee ing sys em. The e o e a ha moniza ion o modelling app oaches be ween
he li e and non-li e segmen s is equi ed.
40
This chap e desc ibes in wha way and o wha ex en he EV concep could apply
o non-li e insu ance. Fi s , o a be e unde s anding, he me hodology and de el-
39
Heep-Al ine , K ause (2012), p. 2.
40
Heep-Al ine , Be g (2012).
- 123 -
opmen o he EV in li e insu ance will be explained. A e ha , an app oach o
ans e he idea o non-li e insu ance is p esen ed. The ollowing example o he
ic ional p ope y/casual y insu e named “Felda inge B andkasse” will illus a e an
EV calcula ion. Finally, based on he esul s a conclusion and ou look is gi en. The
explana ions and desc ip ions in his chap e a e mainly based on he esea ch e-
sul s o he coope a ion be ween he wo king g oup “Embedded Value Non-li e” o
he Ge man Associa ion o Ac ua ies (Deu sche Ak ua e einigung) and he mas e
s uden s a he Ins i u e o Insu ance S udies in Cologne (Ins i u ü Ve si-
che ungswesen de FH Köln).
41
4.1 Embedded Value in Li e Insu ance
The e u n p o ile in li e insu ance dis inguishes om o he lines o business be-
cause o i s long- e m cha ac e . Typically he high acquisi ion cos s a he begin-
ning o a con ac will be amo ized o e ime by he p o i s in u u e yea s, see he
igu e below.
Figu e 110: Annual P o i o a Li e Insu ance Con ac
42
The e o e, li e insu e s in a pe iod o g ow h show an ope a ing loss due o i s high
a e o new business. I concludes ha he annual epo ed gain om income
s a emen s does no e lec he adequa e alue o he li e insu ance po olio. Thus,
u u e cash lows ha e o be aken in o accoun o he alua ion o con ac s. The
EV conside s he p esen alue o all u u e p o i s o he insu ance po olio and
ega ds he long- e m na u e o he business.
To summa ize his, he EV is an indica o o he p ospec i e ea nings po en ial o a
li e insu ance company and is he key pe o mance igu e o he sha eholde s and
po en ial in es o s.
41
Heep-Al ine (2012).
42
Gü le (2012), p. 7.
- 124 -
4.2 His o ical De elopmen
James Ande son published he basic concep ual idea in he yea 1959.
43
Based on
his isola ed p ojec ion o u u e cash lows he EV app oach e ol es cons an ly o e
ime. Today he EV is a gene ally accep ed indica o in li e insu ance and his is
why mos companies publish an addi ional EV epo beside he legal epo ing e-
qui emen s.
44
The EV es ima es he alue o he company, based on i s cu en ne
wo h plus he p esen alue o u u e p o i s minus cos s. The es ima ion o u u e
cash lows equi es an ex ensi e se o assump ions. Fo example, he u u e in e -
es a es, in la ion, policyholde beha iou and mo ali y ha e o be conside ed. A -
emp s o ha monize and imp o e he ini ial concep o he adi ional Embedded
Value (TEV) led o he concep o he Eu opean Embedded Value (EEV) and inally
o he Ma ke Consis en Embedded Value (MCEV).
45
T adi ional Embedded Value (TEV)
The TEV co esponds o he alue o he adjus ed equi y (ne asse alue) plus he
P esen Value o Fu u e P o i s (PVFP) o he co e ed business minus he Cos o
Capi al (CoC).
Figu e 111: T adi ional Embedded Value
46
The sepa a e componen s o he adi ional Embedded Value a e explained in mo e
de ail below.
Ne Asse Value (NAV)
The NAV is he book alue acco ding o gene ally accep ed accoun ing p inciples
(e.g. Ge man GAAP) o he equi y adjus ed wi h alua ion ese es (di e ence be-
ween he ma ke alues and he accoun ing alues) and he di idends o he
sha eholde s which a e included in he balance shee p o i . The NAV is di ided in o
he Requi ed Capi al (RC) and he F ee Su plus (FS). The RC is demanded o ex-
ample as a sol ency capi al by he insu ance supe ision o by he a ing agencies.
43
Ande son (1959).
44
PWC, p. 1.
45
Heep-Al ine ; K ause (2012), p. 7.
46
Gü le (2012), p. 8-10.
- 125 -
Cos o Capi al (CoC)
The CoC co esponds o an adequa e in e es on he RC. Fo he pu pose o p o-
iding capi al o he insu ance company he sha eholde s demand an app op ia e
e u n on he in es ed capi al (Risk Discoun Ra e, RDR). The ac ual in es men
income on he RC is usually lowe han he expec ed isk discoun a e (RDR). Fu -
he mo e, he pa icipa ion o he policyholde s as well as he axes on he in es -
men income on he RC should be conside ed.
P esen Value o Fu u e P o i s (PVFP)
An essen ial elemen o he TEV is he de e minis ic PVFP. Fo he calcula ion, he
ollowing assump ions a e used:
• The insu ance po olio is in un-o .
• The p o i and loss accoun and he balance shee will be p ojec ed o e he
p ede ined p ojec ion pe iod.
• The u u e new business will no be conside ed.
• The in es men income on equi y is no aken in o accoun .
As a esul , he u u e p o i s a e de e mined. The ollowing discoun ing calcula ion
uses he RDR and he PVFP is hen iden i ied and quan i ied.
Eu opean Embedded Value (EEV)
An ea lie lack o clea guidelines o he de e mina ion o he TEV made compa a-
bili y be ween he di e en companies complica ed o in es o s and sha eholde s.
In he yea 2004 he so called CFO o um, comp ising he 21 chie inancial o ice s
o he mos impo an Eu opean insu ance g oups, es ablished he Eu opean Em-
bedded Value P inciples (EEVP). The EEVP se down 12 gene al binding ules.
47
Fo ins ance, beside he h ee componen s o he TEV, he EEV conside s he Time
Value o Op ions and Gua an ees (TVOG) as an addi ional ac o , see he igu e
below.
Figu e 112: Eu opean Embedded Value
48
47
CFO Fo um, Eu opean Embedded Value P inciples.
48
Heep-Al ine (2012), p. 18.
- 126 -
The gua an ees mainly e e o ixed p omised inancial gua an ees. An example o
op ions is he igh o cancella ion o policyholde s o he lump sum op ion in annu-
i y insu ances. The e o e, he de e minis ic pe spec i e o capi al ma ke scena ios
is insu icien o assessing he TVOG app op ia ely. Especially he e alua ion o
inancial gua an ees needs a s ochas ic asse / liabili y p ojec ion model o e lec
he ola ili y o he inancial ma ke s.
49
I is necessa y o de elop managemen
ules, e.g. o de e mining he pa icipa ion o he policyholde s on in es men in-
comes and o an assump ion o u u e policyholde s’ beha iou . Fu he mo e, he
EEVP equi es a consis en calcula ion o he RDR and homogeneous publica ion
s anda ds.
50
Ma ke Consis en Embedded Value (MCEV)
The componen s o he MCEV align wi h he EEV. Fu he mo e, cos s o non-
hedgeable isks mus be aken explici ly in o accoun . Because he EEVP did no
sol e he p oblem o an app op ia e and objec i e RDR su icien ly, he CFO Fo um
published he Ma ke Consis en Embedded Value P inciples (MCEVP) in June
2008 in o de o b ing g ea e consis ency and imp o ed disclosu e o he EEV. The
MCEVP include 17 “Key p inciples”, 145 “A eas o guidance”
51
and a “Commen a y
on P inciples & Guidance (Basis o Conclusions).”
52
The MCEV is cu en ly he
mos sophis ica ed and ha monized EV concep . I alues asse s and liabili ies on a
ma ke -consis en basis. Asse s a e alued a he amoun o which hey can be
sold a he ime o alua ion. The liabili ies, which a e no aded and illiquid, a e
alued by a eplica ing po olio o o he adequa e ma hema ical echniques. The
MCEVP also equi e a consis en alua ion o he TVOG simila o he p icing o
op ions and o he de i a i es on capi al ma ke s (Black & Scholes). Fu he mo e,
cos s o non-hedgeable isks mus be aken explici ly in o accoun .
Bu he discussion abou he igh me hodology and assump ions s ill con inues. In
Oc obe 2009, he CFO Fo um published an amendmen o he MCEV P inciples o
allow o he inclusion o an illiquidi y p emium. Fu he mo e, in Decembe 2011 a
p ess elease was issued by he CFO Fo um o ake accoun o he cu en so e -
eign deb ma ke condi ions in EV epo ing as an ini ial s ep owa ds he expec ed
con e gence o MCEV wi h he de eloping Sol ency II egula o y amewo k.
53
Howe e , he discussion is s ill on-going as o how and o which p oduc s such illiq-
49
CFO Fo um, Basis o Conclusions Eu opean Embedded Value P inciples, p. 15.
50
Gü le (2012), p. 10-11.
51
CFO Fo um, MCEV P inciples & Guidance.
52
CFO Fo um, MCEV Basis o Conclusions.
53
www.c o o um.nl /embedded_ alue.h ml.
- 127 -
uidi y p emiums should be applied as well as how so e eign deb ma ke condi ions
should be aken in o accoun unde Sol ency II.
54
4.2.1 Applica ion o Embedded Value
The Embedded Value applies in he ollowing a eas o he insu ance business:
55
E alua ion o a Company:
The EV is an al e na i e app oach o he e alua ion o a li e insu ance company
wi h a mo e signi ican exp essi eness han he classical igu es. The e o e, i is he
main componen in he nego ia ion p ocess o me ge s and acquisi ion ansac ions.
Company s ee ing:
As he ma e ial pa o in e nal li e insu ance models, he EV de e mines he e-
qui ed isk capi al and is he e o e a main pa o he isk managemen , especially
conside ed in he amewo k o Sol ency II. The equi ed capi al is calcula ed based
on sensi i i ies, s ess scena ios and he equi ed secu i y le el.
Mo emen Analysis:
Wi h a mo emen analysis as a ool o a alue-added analysis, he sepa a e im-
pac s o a change in he EV could be examined ex pos . The Mo emen Analysis is
an impo an ool o he pe o mance measu emen and he e alua ion o he man-
agemen o a li e insu ance company. The e o e, he MCEV a he end o he yea
(EoY) will be compa ed a pos e io i wi h he MCEV a he beginning o a yea
(BoY). The easons o he change o he alue will be analysed indi idually. The
ollowing igu e shows a schema ic example o a Mo emen Analysis.
54
Munich Re, Ma ke Consis en Embedded Value Repo 2011, p. 3.
55
DAV, Embedded Value in de Schaden e siche ung, p. 6.
- 128 -
Non- inancial
Expe ience
a iances
MCEV
BoY
Opening
adjus men s
Non- inancial
assump ion
changes
Misma ching
p o i
Re u n on asse s
no backing
Liabili ies: “unwind”
New
Business
Unexplained
MCEV
EoY
Di idends o
sha eholde s
ma ke -consis en compensa ion o aking ALM-
isk: elimina e o measu ing e u n
Figu e 113: Mo emen Analysis
56
Di e en ac o s lead o an inc easing / dec easing MCEV. Fo example, he change
o he non- inancial assump ions wi h ega d o he u u e has educed he alue.
On he o he hand, ac o s like he de ia ion be ween he ealized and he es ima ed
non- inancial assump ions, he o e pe o mance o in es men ea ning and he
alue added by new business, lead o a highe MCEV a he end o he yea .
4.2.2 Ma ke Consis en Embedded Value
The MCEV is he p esen alue o sha eholde s’ in e es s in he ea nings dis ibu -
able om asse s alloca ed o he co e ed business a e making su icien allow-
ance o he agg ega e isks in ol ed. When calcula ing he MCEV he ollowing
p inciples ha e o be conside ed:
57
Closed Fund P ojec ion:
In opposi e o he App aisal Value
58
(AV) he EV does
no conside u u e new business. The EV and he exis -
ing insu ance po olio will be p ojec ed in un-o .
Bes Es ima e: The calcula ion is based on ealis ic assump ions.
Going Conce n: All assump ions base on a con inued business ope a-
ion.
Fu he Conside a ion Regula o y and legal amewo ks and con inuous man-
agemen ules ha e o be aken in o accoun .
56
DAV-A bei sg uppe EV Sach: Embedded Value in de Schaden e siche ung. Be ich an
den Ausschuss Schaden e siche ung DAV. S and 16. Sep embe 2010.
57
Gü le (2012), p. 7.
58
The AV can be in e p e ed as EV plus Goodwill.
- 129 -
The MCEV componen s co espond essen ially wi h he EEV bu use a close clas-
si ica ion le el o he single componen s. The MCEVP dis inguish be ween he ol-
lowing componen s o EV.
59
Figu e 114: Ma ke Consis en Embedded Value
60
The di e en componen s o he Ma ke Consis en Embedded Value a e desc ibed
in mo e de ail in he ollowing:
Ne Asse Value
The Ne Asse Value is di ided in o he componen s Requi ed Capi al ( ha has o
be kep wi hin he company) and F ee Su plus ( ha can be paid ou ).
Requi ed Capi al
RC is he ma ke alue o capi al alloca ed o he co e ed business. I equals a
leas he egula o y sol ency capi al, bu may be highe o mee in e nal isk capi al
models o a ing a ge s. RC is angible and may be dis ibu ed o e ime as liabili-
ies un-o .
F ee Su plus
FS is he ma ke alue o capi al alloca ed o he co e ed business bu no equi ed
o suppo he in- o ce co e ed business a he alua ion da e. FS is angible and
may be dis ibu ed immedia ely. The FS is a esidual amoun . To calcula e he FS,
an analysis o he whole equi y is needed.
59
CFO Fo um, MCEV P inciples & Guidance and Munich Re (2011), p. 19 .
60
Heep-Al ine , Ju zi (2012), p. 23.
- 130 -
Equi y o an insu ance company is de ined as he di e ence be ween all asse s and
liabili ies. FS equals he sub ac ion be ween equi y and RC (unde conside a ion o
ax and sha eholde di idends).
Fo a Ge man insu ance company, he componen s FS and RC can be concluded
om he Ge man GAAP balance shee . Bu an adjus men o he balance shee po-
si ions is necessa y o ge o he equi ed ma ke alue iew.
Value o In-Fo ce
The Value o In-Fo ce co e ed business (VIF) consis s o he PVFP, TVOG, FCRC
and CRNHR.
P esen Value o Fu u e P o i s
The PVFP is he p esen alue o u u e local s a u o y (e.g. Ge man GAAP) sha e-
holde a e - ax p o i s eme ging om he business co e ed on he condi ion ha all
economic and non-economic assump ions a e me . The e o e, he ollowing ac o s
a e essen ial o de e mine he u u e insu ance po olio de elopmen :
• In es men Income,
• de elopmen o cos and claims,
• cancella ion beha iou o policyholde s,
• dynamics on inancial ma ke s,
• eimbu semen om einsu ance and
• isk discoun a e.
The assump ions based on he Bes Es ima e p inciple ha e o be made o each
line o business and p oduc indi idually. Fu he mo e, he calcula ion o u u e p o -
i s conside s he going conce n assump ion. The assump ions made a e assumed
o be adequa e o he u u e as well on an in la ion-adjus ed basis. Based on he
assump ions, he calcula ion p ocedu e o he PVFP ollows hese s eps:
1. De e mina ion o he ne p o i be o e ax, based on he unde w i ing and
in es men esul s.
2. De e mina ion o he ne p o i a e ax o each pe iod unde e iew.
3. Discoun ing wi h he RDR o he beginning o he p ojec ion.
- 131 -
Time Value o Financial Op ions and Gua an ees
Pa icipa ing li e business is gene ally cha ac e ized by op ions and gua an ees,
which a e s ongly dependen on he inancial ma ke s (e.g. a minimum in e es a e
o a minimum le el o bonus is gua an eed o he policyholde ). The pa icipa ing
ea u es a e usually a combina ion o con ac ual o legal cons ain s and manage-
men disc e ion ha has o ake compe i i e p essu e o ma ke p ac ice in o ac-
coun . The calcula ion o TVOG should be based on a s ochas ic a ia ion o u u e
economic condi ions using me hods and assump ions consis en wi h he unde lying
EV.
F ic ional Cos s o Requi ed Capi al
FCRC e lec he axa ion cos s o isk- ee in es men on he asse s backing e-
qui ed capi al as well as he cos s o he in es men managemen o hose asse s.
Cos o Residual Non-Hedgeable Risks
CRNHR a e Cos o Capi al o all ( esidual) isks ha ha e no been conside ed in
he ma ke alue o a isk componen ye . The ollowing igu e illus a es he ange
o he MCEV componen s.
RC FS PVFP CoC FC TVOG MCEV
Figu e 115: Componen s o he MCEV
61
61
Own igu e based on Munich Re (2011), p. 4.
- 138 -
Figu e 119: In-Fo ce Business and Renewals
67
In he i s case, he con ac is comple ed be o e he balance shee da e and hus
be o e he da e o he MCEV app oach. The e o e, his insu ance con ac is a ib-
u ed o he In-Fo ce po olio un il he end o he epo ing yea .
As can be seen in he second imeline, he e is a enewal on he balance shee
da e. The con ac om he p e ious yea is con inued au oma ically, which implies
ha his con ac is assigned o he enewals and in luences he alue o he con-
inuing business.
Looking a he hi d example, he signing o he con ac ook place be o e he bal-
ance shee da e. The con ac is he e o e a ibu ed o he In-Fo ce business, be-
ginning a he balance shee da e and ending a he ma u i y da e. A e ha he
con ac will be alloca ed o he enewals.
This dis inc ion o he exis ing con ac s is o eno mous impo ance o he po olio
de elopmen in non-li e insu ance. Fo example, he po olio alue is in luenced by
he app op ia e assump ions ega ding he lapse a e and claims cos . These as-
sump ions mus be made indi idually o e e y line o business, which also means a
high eliance on unce ain planning assump ions and hus a dependency on he
business policy o a company.
In mo o insu ance he loss a io may inc ease due o he loss o good isks. Poli i-
cal ac o s could also lead o w ong assump ions, as hey a e de i ed om he pas
67
Heep-Al ine (2012), p. 48.
- 139 -
and he e o e ha e no alidi y o he u u e. The sc apping p emium o 2009 led o
a signi ican ly highe numbe o ca sales. This would mean a highe po olio loss
as i was calcula ed o he Embedded Value.
Model Shocks
Model shocks a e isola ed changes in indi idual inpu pa ame e s. They se e o
he comp ehension (sensi i i y) and es ing (plausibili y) o he calcula ed MCEV.
Especially sensi i i ies o he MCEV gi e a good i s imp ession o i s alue d i e s
and c i ical success ac o s. Fo pa ame e s ha a e s ongly in luenced subjec-
i ely, such as he Cos o Capi al his is i al. Especially, sensi i i ies ha e an
added signi icance as hey a e published in he IFRS consolida ed inancial s a e-
men s
68
.
In li e insu ance, p ede ined model shocks ha e o be applied, indica ing a change
in single calcula ion pa ame e s. Fo non-li e insu ance, such model shocks a e
also necessa y o achie e a be e unde s anding o he dependency o he Embed-
ded Value o he a ious inpu pa ame e s.
Possible model shocks (among o he s), ha could ha e a signi ican impac on he
MCEV, a e lis ed in he ollowing:
• Inc ease o Cos s o Capi al,
• inc ease o Tax a e,
• Inc ease o Cos Ra ios,
• Change in he Risk Discoun Ra e,
• Inc ease o Adminis a i e Cos s,
• P emium educ ions as well as
• Change in Claims Rese es.
The gene al app oach o calcula e an Embedded Value in non-li e insu ance is ex-
plained in he nex sec ion.
4.3.3 Gene al App oach
In he igu e below i is illus a ed how he MCEV can be de i ed as a balance shee
p ojec ion o e he o al p ojec ion ho izon. The illus a ion is based on Ge man
GAAP, bu i is applicable o o he gene ally accep ed accoun ing p inciples, oo.
68
See IFRS 4, 39A.
- 140 -
Figu e 120: Ge man GAAP Balance P ojec ion o he MCEV Calcula ion
69
In a i s modelling s ep, s a ing wi h he balance a = 0, a F ee Su plus o in a
wo s case scena io a F ee De ici , is ealized as an immedia e ex ao dina y pay-
ou . Thus, he company keeps he Requi ed Capi al a Ma ke Value a e wa ds.
The ex ao dina y payou akes place h ough a ealiza ion o hidden ese es o
liabili ies a ec ing he ne income as well as a wi hd awal o injec ion o capi al wi h
espec o he di e ence be ween balance equi y and Requi ed Capi al wi hou a -
ec ing he ne income.
A e he ex ao dina y payou , he emaining Requi ed Capi al is in es ed isk- ee
wi h he esul ha no hidden ese es will exis in he ollowing pe iods. In ansi ion
o e e y u he pe iod, he balance will be adjus ed due o changes in he P o i and
Loss Accoun and he wi hd awal o ee Requi ed Capi al.
A he s a ing poin o he p ojec ion no ex ao dina y payou happens o po en ial
hidden ese es on liabili ies as hese a e disclosed o e ime. Due o he p ojec-
ions he Requi ed Capi al as well as he liabili ies a e educed o e ime.
69
Heep-Al ine (2012), p.54.
- 141 -
To ge o he Embedded Value, he p esen alue o all p o i s and losses and all
capi al wi hd awals will be calcula ed on he basis o he in e es a e cu e adjus ed
by he CRNHR.
4.4 Embedded Value in Non-li e Insu ance – Calcula ion Example
The aim o his sec ion will be a p esen a ion o he me hodical app oach o he de-
e mina ion o he MCEV using a ic i ious insu ance company - he so-called
“Felda inge B andkasse” (FBK).
Ini ially, he ic i ious model-company will be in oduced including i s balance and all
ele an inpu pa ame e s, which a e needed o de e mine he MCEV. Followed by
a ew calcula ion examples, he ansi ion o he MCEV will be ou lined using he
calcula ed key a ios. A compa ison be ween he MCEV and he economic capi al
will sum up his sec ion.
4.4.1 Example Company
S a ing poin o he ic i ious insu ance company is he ollowing Ge man GAAP
balance o he FBK.
Asse s Liabili ies
Book Values Asse s 236,139 48,236 Ge man GAAP Equi y
Asse s backing SHE 48,236
Asse s backing Liab. 187,903 187,903 Book Values Rese es
153,952 Claims Rese es
33,951 Equaliza ion Rese e
Tax Recei ables 0 0 Tax Rese e
236,139 236,139
Figu e 121: Ge man GAAP Balance a = 0
70
All in es men s a e spli i ually in o Asse s Backing Liabili ies (ABL) and Asse s
Backing Sha eholde s Equi y (ABSE). The book alues o ABL wi h an amoun o
187,903 co e he echnical p o isions. The ABSE amoun s o 48,236 and co e
he Ge man GAAP equi y. Bo h, ABSE and ABL a e assumed o be in es ed in isk-
ee ze o bonds wi h edundancies / de iciencies acco ding o he selec ed yield
cu e. Mo eo e , he FBK is subjec o a ax a e o 32%. The Ge man GAAP bal-
ance shee o he FBK se es as he s a ing balance o he p ojec ion o su pluses
in he p ojec ion model.
The nex igu e lis s all ele an inpu da a, such as ese es and p emiums, which
a e c ucial o he MCEV de e mina ion. I also displays he Ge man GAAP balance
70
Heep-Al ine (2012), p.57.
- 142 -
wi h book alues a = 0. Besides his, he ic i ious company has only wo lines o
business, hi d-pa y mo o ehicle insu ance and homeowne s insu ance.
Posi ion Thi d Home To al
Pa y Owne s
Ea ned P emiums 92,218 37,485 129,703
Book Value Claims Rese e 142,839 11,113 153,952
Bes Es ima e Claims Rese e 88,331 7,043 95,374
in % o Booked Claims Rese es 61.8% 63.4% 62.0%
Book Value Equaliza ion Rese e 26,863 7,088 33,951
in % o Booked Claims Rese es 18.8% 63.8% 22.1%
Book Value Technical Rese e 169,702 18,201 187,903
Book Value Asse s 236,139
Redundancy / De iciency in % 2.0%
Ge man GAAP Equi y 48,236
Figu e 122: Inpu Da a – Example Company
71
Conce ning he Bes Es ima e Rese es o he exis ing business (e alua ed by sui -
able ma hema ical algo i hms) and he claims expe ience o he new business he
ollowing cash low assump ions apply:
123456
Old Rese e 26.76% 20.02% 14.56% 10.59% 7.70% 5.60%
New Business 68.54% 10.62% 7.04% 4.66% 3.09% 2.05%
Cash-Flow in % a e … Yea s
Figu e 123: Cash Flow Pa e n a = 0
72
Addi ionally, he e is a u he need o inpu pa ame e o ca y ou a MCEV p ojec-
ion, especially
• global pa ame e s,
• p ojec ion in o ma ion and
• Requi ed Capi al in o ma ion including Cos s o Capi al in o ma ion.
71
Heep-Al ine (2011), p.124.
72
Heep-Al ine (2012), p.59.
- 143 -
The global pa ame e s comp ise a isk- ee yield cu e whe e he implici o wa d
a es can be deduc ed om he spo a es as illus a ed in he ollowing able.
0 1 2 3
Spo a es 3.92% 4.70% 4.53% 4.51%
Fo wa d a es 4.70% 4.36% 4.48%
Du a ion
Figu e 124: Yield Cu e a = 0
73
Conce ning he asse s uc u e a = 0, i is assumed ha he FBK has only in-
es ed in isk- ee ze o bonds wi h he ollowing cha ac e is ics:
a e age du a ion o ixed income secu i ies 4,57
a e age in e es a e o ixed income secu i y 5.00%
hidden ese es in he book alues a =0 2.00%
in es men cos s in % o he ma ke alues 0.20%
The hidden ese es o 2.00% esul wi h espec o he chosen yield cu e. The
pe cen age o hidden ese es would change, i i we e based on a di e en yield
cu e wi h di e en in e es a e s uc u es.
Fu he inpu pa ame e s a e needed o de e mine he Requi ed Capi al, which is
needed o gene a e he MCEV o he ic i ious company. The ollowing assump ions
a e made:
• Pa ame e wi h espec o he SCR calcula ion,
• 175% co e age due o Ra ing Requi emen s,
• CoC Ra io o 6% wi h espec o he Sol ency Capi al.
The de e mina ion o he SCR is based on he QIS 5 s udy
74
. P emium isk, ese e
isk and he co ela ion be ween bo h isks depend on he in e nal model o he
FBK.
In he ollowing, he MCEV p ojec ions a e ca ied ou only o he exis ing business
in o de o be consis en wi h he usual de ini ion o economic capi al in non-li e in-
73
Heep-Al ine (2012), p.61
74
Fo mo e in o ma ion see h ps://eiopa.eu opa.eu/consul a ions/qis/insu ance/quan i a i e-impac -
s udy-5/index.h ml
- 144 -
su ance. Fu he mo e, we conside a “ i ual” un-o (e.g. wi hin o he business op-
e a ions) such ha only claims egula ion cos s (co e ed in he Bes Es ima e Re-
se es) and in es men cos s occu , bu no ope a ional cos s o new business. Op-
e a ional cos s a e no included in he p ojec ions.
The p ojec ed de elopmen o he Claims Rese es (Ge man GAAP as well as Bes
Es ima e) and he Equaliza ion Rese es is illus a ed in he ollowing igu e.
Posi ion 0 1 2 3
To al Paymen s 25,518 19,095 13,887
BE-Rese e 95,374 69,855 19,095 36,873
Ge man GAAP Rese e 153,951 112,760 50,761 59,520
Ope a ional Expenses 0 0 0 0
Equaliza ion Rese e 33,951 24,867 18,070 13,126
Value
Figu e 125: P ojec ion o Rese es wi hou Renewals
75
The p ojec ed Ge man GAAP Rese es esul om he p ojec ed Bes Es ima e
Rese e (acco ding o i s cash low pa e n) a e applica ion o he ini ial o e -
ese ing pe cen age (acco ding o he de ined managemen ules), especially
Ge man GAAP Rese e ( ) = BE-Rese e ( ) · O e - ese ing-Ra e ( )
The p ojec ed Equaliza ion Rese e esul s om he p ojec ed Ge man GAAP Re-
se e a e applica ion o he Equaliza ion Ra e, especially:
Equaliza ion Rese e ( ) = Ge man GAAP Rese e ( ) · Equaliza ion Ra e ( )
A e hose ese e p ojec ions he p ojec ion o he RC and he CRNHR can be
pe o med.
4.4.2 Ne Asse Value
In his sec ion, he Ne Asse Value (as a sum o he Requi ed Capi al and he F ee
Su plus) will be calcula ed on he base o he p e iously speci ied inpu pa ame e
o he FBK.
75
Heep-Al ine (2012), p.73.
- 145 -
Requi ed Capi al
Acco ding o he managemen ules o he FBK, he Requi ed Capi al is he maxi-
mum o he
• Requi ed Capi al o co e he SCR & MCR acco ding o Sol ency II and he
• Requi ed Capi al o co e he sol ency ma gin acco ding o Sol ency I wi h a
equi ed 175% o e load.
The igu e below shows he p ojec ion o he Requi ed Capi al o se e al yea s:
Posi ion 012
(1) Ma ke Value ABL 191,666 139,997 101,092
(2) Discoun ed BE Rese e 83,454 61,263 44,426
(3) P ojec ion o Risk Ma gin 3,454 2,514 1,815
(4) SCR incl. 175% O e load 32,130 23,586 17,104
(5) Requi ed Capi al = MAX[(2)+(3)+(4)-(1);0] 0 0 0
Pe iod
Figu e 126: Requi ed Capi al wi hou Renewals (1)
76
Requi ed Capi al o co e he SCR is only needed i he hidden ese es a e insu -
icien o co e he Sol ency II equi emen s. As shown abo e he ABL ma ke al-
ues a e su icien o mee Sol ency II equi emen s such ha he e is no capi al e-
qui ed due o his aspec .
Fu he mo e, Requi ed Capi al o co e he MCR should be calcula ed. The MCR
is se as 50% o he SCR based on Sol ency II: See he nex igu e.
Posi ion 012
(1) SCR 18,360 13,478 9,774
(2) MCR = (1) · 50% 9,180 6,739 4,887
(3) Requi ed Capi al = (2) 9,180 6,739 4,887
Pe iod
Figu e 127: Requi ed Capi al wi hou Renewals (2)
77
As a nex s ep, he Requi ed Capi al acco ding o Sol ency I in combina ion wi h a
equi ed co e age o 175% is calcula ed as
76
Heep-Al ine (2012), p.77.
77
Heep-Al ine (2012), p. 78.
- 146 -
Requi ed Capi al = 175% · MAX [P emium-Index; Claims-Index; 2,200].
The p ojec ions o he Requi ed Capi al due o Sol ency I wi h a co e age o 175%
a e lis ed in he igu e below.
Posi ion 012
(1) P emium Index 21,814 0 0
(2) Claims Index 24,236 15,327 6,337
(3) Sol ency Ma gin = MAX [(1);(2);2,200] 24,236 15,327 6,337
(4) Requi ed Capi al = 175% · (3) 42,412 26,823 11,090
Pe iod
Figu e 128: Requi ed Capi al wi hou Renewals (3)
78
Finally, all he h ee s eps ha e o be combined in o de o de e mine he Requi ed
Capi al in o al o ul il all sol ency and a ing equi emen s o he company: See he
igu e below.
Posi ion 012
(1) Requi ed Capi al SCR 0 0 0
(2) Requi ed Capi al MCR 9,180 6,739 4,887
(3) Requi ed Capi al Sil ency I 42,412 26,823 11,090
(4) Requi ed Capi al = MAX[(1);(2);(3)] 42,412 26,823 11,090
Pe iod
Figu e 129: Requi ed Capi al wi hou Renewals (4)
79
The alue o he o al Requi ed Capi al dec eases quickly wi h espec o he gi en
un-o Scena io. A = 3 he alue o he Requi ed Capi al al eady amoun s o he
minimum. As a esul , he ic i ious insu ance company needs a Requi ed Capi al o
42,412 o all unde w i en isks in = 0.
F ee Su plus
The F ee Su plus is he second componen de e mining he Ne Asse Value. In he
balance a = 0, an ini ial Ge man GAAP equi y o 48,236 is gi en. F om his s a -
ing poin , he F ee Su plus can be calcula ed by he ollowing app oach:
78
Heep-Al ine (2012), p. 78.
79
Heep-Al ine (2012), p. 79.
- 147 -
• The ini ial Ge man GAAP equi y includes 2% o hidden asse ese es (=
996). By ealizing hose ese es a ax o 32% has o be paid, which esul s
in an a e - ax alue o 657.
• The di e ence be ween he ini ial Ge man GAAP equi y and he Requi ed
Capi al can be ea ed as a ax- ee capi al wi hd awal. The di e ence
be ween bo h alues is 5,824 = 48,236 – 42,412.
I we combine all calcula ions ca ied ou in his sec ion we ob ain he ollowing Ne
Asse Value o he FBK:
F ee Su plus = 5,824 + 657
= 6,481
Requi ed Capi al = 42,412
Ne Asse Value = 48,893
Nex , he Value o In-Fo ce Business o he FBK will be calcula ed by ca ying ou
he Ge man GAAP balance shee p ojec ions o e he p ojec ion pe iod.
4.4.3 Value o In-Fo ce Business
Calcula ing he Value o In-Fo ce Business depends on di e en economic pa ame-
e s. One way o calcula e he VIF is o calcula e he P esen Value o Fu u e P o i s
and sub ac he sum o Cos s o Residual Non-Hedgeable Risks, he Time Value o
Op ions and Gua an ees and he F ic ional Cos s. Ano he way is shown in he ol-
lowing igu e illus a ing he p ojec ion esul s a = 1.
Posi ion Value
(1) To al Resul a e Capi al Remo al 40,156
(2) Cos o Capi al 1,102
(3) F ee Su plus 0
(4) Rep oduc ion o Requi ed Capi al 17,581
(5) VIF = (1) - (2) - (3) - (4) 21,474
Figu e 130: VIF Resul a = 1 wi hou Renewals
80
The p ojec ion o all discoun ed VIF esul s o e he p ojec ion pe iod esul s in a
o al VIF o 67,527.
80
Heep-Al ine (2012), p.105.
