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DOI: 10.15240/ ul/001/2015-4-009
In oduc ion
The de elopmen o human socie y places
con inuously inc easing demands on i s
p oduc ion. This b ings o he socie y no only
posi i e bu also nega i e e ec s. Along wi h
he de elopmen o indus ial p oduc ion,
people ha e o ace associa ed isks. Indus ial
acciden s, which ha e occu ed ecen ly, s ill
ha e a nega i e impac on human li e, heal h,
he en i onmen and he economy.
Majo acciden s a e de i ned as e en s
esul ing om uncon olled de elopmen s
du ing an indus ial ac i i y, such as a se ious
leakage, i e o explosion, which may
immedia ely o subsequen ly lead o se ious
h ea o indi iduals in o ou side he p emises,
o o he en i onmen , in which one o mo e
haza dous subs ances a e in ol ed.
This a icle deals wi h di e en aspec s
and issues o majo acciden s wi h ocus
on he p e en ion o hei occu ence and
possible consequences. Since he isks o
majo acciden s ake ca as ophic p opo ions,
he quali y legisla ion on his issue is needed.
O e iew o legisla i e ac ions is in he sepa a e
chap e o a icle. This legisla ion imposes o
indus ial companies wi h he isk o majo
acciden he obliga ion o liabili y insu ance
o damages caused by he ealiza ion o
hese isks. The main objec i e o he a icle
is o p esen he possibili y o loss dis ibu ion
o majo acciden s and simula ion o po en ial
ex eme losses which a e necessa y o
de e mining he p emiums.
Fo he sake o illus a ion ha e been selec ed
he majo acciden s and hei consequences
since he ea ly wen ie h cen u y. The expe ience
gained om hese and o he disas e s ha e
played a signi i can ole in he de elopmen
o legisla ion in ela ion o he p e en ion and
liquida ion o consequences o majo acciden s.
Minama a, Japan (1932–1968): The
Company p oducing e ilize s eleased a o al
o 27 ons o me cu y compounds in o he
sea. The esul was a mass poisoning o local
esiden s called Minama a disease. In he wake
o his e en , 2,000 o 3,000 people died. [6]
Se eso, I aly (1976): An explosion o
a chemical eac o in he chemical plan
o Gi audan company. The company’s
managemen announced ha i had been
a common acciden and hey ailed o p o ide
in o ma ion abou he leakage o oxic
subs ances. I was as la e as se en een days
a e he acciden ha he ac o y managemen
admi ed ha abou wo kilog ams o dioxin
leaked in he ai , an amoun o poison capable
o killing 19,000 people. [5]
Bhopal, India (1984): Bhopal disas e is
conside ed o be he wo s indus ial acciden
in he wo ld. I s a ed in a plan o he U.S.
Company Union Ca bide India Limi ed,
p oducing pes icides, whe e deadly hyd ogen
cyanide gas and me hylisocyana e (MIC)
escaped. To his da e, mo e han 20,000 people
died and app oxima ely 500,000 we e inju ed
as a esul . [1]
Cuba ão, B azil (1984): An explosion in
a pe ochemical plan o Pe obas esul ed in
a i e in local slums. The amoun o demises
was due o he comple e bu ning o some si es
ne e p ecisely de e mined; i is es ima ed o be
app oxima ely 500.
San Juan Ixhua epec, Mexico (1984):
A se ies o explosions in he la ge wa ehouse
LPG Company Pe oleos Mexicanos des oyed
pa o he ci y. Abou 500 people died.
Sandoz, Swi ze land (1986): A i e in he
ag o-chemical company Sandoz wa ehouse
caused he elease o oxic ag ochemicals
in o he ai , bu also abou 30–40 ons o hese
chemicals escaped in o he Rhine Ri e . Mix u e
o subs ances in he i e con ained pes icides,
dioxins, me cu y, chlo ine compounds,
l uo escen dyes, o ganophospha es and
o he s. Ciba-Geib, chemical company ound
SIGNIFICANCE AND POSSIBILITIES
OF MAJOR ACCIDENT INSURANCE
Pa la Jind o á, Radim Jakubínský
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122 2015, XVIII, 4
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close o his si e, a emp ed o make use o his
si ua ion eleasing 400 li es o a azine in o
he i e , belie ing his will no be e ealed.
Chemicals immedia ely caused he dea h o
aqua ic animals in he i e . 100 ons o i sh
we e killed and pollu ion o he Rhine eached
as a as he Ne he lands. [3]
Baia Ma e, Romania (2000): Many expe s
belie e ha i is Eu ope’s wo s en i onmen al
disas e since he Che nobyl explosion.
A dam holding back 100,000 cubic me e s
o con amina ed wa e bu s con amina ing
d inking-wa e supplies o mo e han 2.5 million
Hunga ians. Due o he oxici y o cyanide in
wa e , especially a ound he basin o he Tisza,
i ually all li ing o ganisms pe ished along he
i e . Fu he sou hwa ds, in he Se bian pa ,
app oxima ely 80% o aqua ic li e was killed.
Two yea s a e , he ecosys em began o e u n
o i s o iginal s a e, al hough s ill a om he
le el be o e he disas e . [5]
Enschede, The Ne he lands (2000):
Fi e a S.E. Fi ewo ks caused a subsequen
explosion o i ewo ks ha he company had
p oduced. Nume ous explosions ensued wi hin
he ollowing 30 minu es, de as a ing an a ea
o 5 km2. The i e sp ead o a neighbou ing
b ewe y. The esul ing cloud o smoke om
he wo companies was seen a a dis ance o
60 km. In his disas e , 22 people los hei li e,
including ou i e- i gh e s, and o e 940 people
we e inju ed. I des oyed app oxima ely 500
apa men s, 1,500 homes, 60 businesses. [5]
Toulouse, F ance (2001): Explosion in he
AZF chemical ac o y. I was equi alen o 20–
40 ons o TNT, i caused a emo o 3.4 on
Rich e scale and was hea d up o a dis ance
o 80 km. The explosion caused a o al o 29
dea hs, 2,500 se ious inju ies and 8,000 mino
inju ies. Damages paid ou o insu ance claims
exceeded € 1.5 billion. [5]
Wes , Texas (2013): Ammonium ni a e
exploded in a e ilize ac o y. As a esul 15
people died, nea ly 200 people we e wounded
and 150 buildings we e des oyed o damaged.
The essen ial in o ma ion, howe e , is he ac
ha he company had liabili y insu ance o
damages only in he amoun o one million
dolla s, bu he o al damage exceeded one
hund ed million U.S. dolla s. Wi h espec o he
o al damage, i may sound inadequa e ha he
company was i ned $ 118,300 agg ega ely. In
many s a es, including Texas, he e is no legal
obliga ion o he company o conclude liabili y
insu ance o he damage wi h he insu ed
sum co esponding wi h he ange o possible
damages.
1. De elopmen o he Numbe
o Acciden s
Human socie y is also exposed o he ac ion
o na u al causes, o en in he o m o na u al
disas e s. The numbe and consequences o
man-made disas e s is inc easing wi h he
de elopmen o human socie y. De elopmen o
indus ial p oduc ion also b ings abou he isk
o majo acciden s.
Figu e 1 shows he de elopmen o he
numbe o disas e s caused by human in l uence,
compa ed o na u al disas e s be ween 1970
and 2012. This da a shows a long- e m g ow h
o bo h ypes o ca as ophic e en s. The cha
shows ha om 1970 onwa ds, he yea 2010
is he i s yea whe e a highe occu ence o
na u al disas e s appea ed mo e han man-
made disas e s. This is also one o he ala ming
signals o he socie y o p o ec hemsel es
and hei en i onmen om he e ec s o man-
made disas e s.
The p ima y challenge o he company
is o ake such measu es so ha no majo
acciden s occu . Howe e , people mus be
adequa ely p epa ed o possible explosions,
i es, spills, and o he se ious aul s. Fo
his eason i is ad isable o use eme gency
scena ios. Damages a ising om indus ial
disas e s can each ex eme alues o billions
o Eu os. I a company should be able o bea
he i nancial consequences o majo acciden s,
i is necessa y o ha e insu ance ha co e s
possible damages in an adequa e amoun . A
his poin i should be no ed ha he insu ance
company i sel , howe e , canno bea he isks
in such olumes ha a e ypical o hese isks. I
is he e o e necessa y o nego ia e ensu ing o
each insu ance con ac .
2. Legisla ion a Issue P e en ion
o Majo Acciden s
The need o legisla ion o ope a o s o
indus ial acili ies and o ganisa ions, which a e
h ea ened by majo acciden s, was es ablished
immedia ely a e he i s occu ence o se ious
e en s.
U gen alks on a new EU di ec i e on
whole egula o y amewo k in ensu ing he
sa e y o haza dous ins alla ions s a ed a e
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he explosion o cyclohexane in he ac o y
NYPRO L d. in Flixbo ough (UK, 1974). O e
he nex wo yea s in he Eu opean Communi y
amewo k occu ed h ee addi ional se ious
chemical acciden s: Beek (Ne he lands, 1975),
Man edonia (I aly, 1976) and Se eso (I aly,
1976). [5]
In he ligh o hese ca as ophic acciden s,
i was clea ha new legisla ion o imp o e
he sa e y o indus ial si es, planning o
eme gencies o -si e acciden s and dealing
wi h he aspec s o b oade egional and c oss-
bo de indus ial sa e y is needed. Di ec i e
Se eso, p epa ed in Feb ua y 1977, which
was adop ed by he Council o Minis e s o he
Eu opean Communi y on 24 6 h, 1982, is he
esul o hose e o s. The ollowing measu es
ha e o be adop ed by indi idual Membe
S a es no la e han Janua y 8 h, 1984. [18]
The Di ec i e applies o he p e en ion
o majo acciden s which may be caused by
ce ain indus ial ac i i ies, and o limi hei
consequences o a man and he en i onmen .
I ocuses on he con e gence o he measu es
aken by he Membe S a es in his a ea. A icle
1 de i nes e ms such as indus ial ac i i y,
ope a o , majo acciden s and haza dous
subs ances.
The Di ec i e was modi i ed wice, in 1987
Di ec i e 87/216/EEC o 19 Ma ch 1987
(O i cial Jou nal No L 85 o 28 Ma ch 1987) and
he 1988 Di ec i e 88/610/EEC o 24 No embe
1988 (OJ L 336 o 7 Decembe 1988). Bo h
amendmen s aimed o ex end he scope o his
Di ec i e, la gely in o de o include he s o age
o haza dous subs ances. [18]
Changes occu ed in esponse o a majo
acciden in he Union Ca bide ac o y in Bhopal,
India in 1984 and acciden s in he Sandoz
wa ehouse in Basel, Swi ze land in 1986.
Se eso I does no apply o nuclea acili ies
and plan s p ocessing adioac i e subs ances
and ma e ials, mili a y equipmen , p oduc ion
and sepa a e s o age o explosi es, gunpowde
and ammuni ion, mining and o he mining
ope a ions, equipmen used o he disposal o
oxic and haza dous was e, which a e subjec
o Communi y law, i hei aim is o p e en
majo acciden s.
As he Council o Eu ope and
ep esen a i es o he Go e nmen s o he
Membe S a es si ing in he Council s essed
he need o a mo e e ec i e implemen a ion o
Di ec i e 82/501/EEC and called o a e iew
o he di ec i e which, i necessa y, included
a possible ex ension o he p o ince scope o
Di ec i e and a g ea e exchange o in o ma ion
in his i eld be ween Membe S a es. Also
he need o imp o ed managemen o isks
and acciden s was s essed. In addi ion, he
acciden in Bhopal and Mexico highligh ed he
dange posed by he p oximi y o esiden ial
Fig. 1: De elopmen o he numbe o disas e s 1970–2012
Sou ce: own
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124 2015, XVIII, 4
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buildings and dange ous a eas. Las bu no
leas , om he impo ance and bene i s o
in o ming he indi idual Membe S a es was on
Decembe 9 h, 1996, a new Di ec i e 96/82/EC
accep ed, known as he Se eso II. [6]
Due o se e e indus ial acciden s in
Toulouse, F ance, in Baia Ma e, Romania and
Enschede, in he Ne he lands and conclusions
o s udies on ca cinogens and subs ances
dange ous o he en i onmen was Se eso II
ex ended by Di ec i e 2003/105/EC. The new
Di ec i e equi es Membe S a es o ensu e
a e y de ailed app aisal o isks by using
possible acciden scena ios ha co e isks
a ising om s o age and p ocessing ac i i ies in
mining, s o age o py o echnics and explosi es
s o age o ammonium ni a e based e ilize s [5]
The main eason o eplacing he
Se eso II Di ec i e is a change in he sys em
o classi i ca ion o dange ous subs ances
es ablished by he Eu opean Di ec i e No.
1272/2008 o Decembe 16 h 2008, abou
classi i ca ion, labelling and packaging o
subs ances and mix u es. I was also necessa y
o cla i y and upda e ce ain pa s o he
di ec i e, o imp o e he implemen a ion and
en o cemen o he Di ec i e. In o al, in he
Se eso III (2012/18/EU) is lis ed 32 easons.
This Di ec i e was adop ed on July 4 h, 2012,
published July 24 h, 2012, coming in o o ce
on Augus 13 h 2012, o be implemen ed
by May 31s , 2015 wi h he excep ion o he
implemen a ion o he A icle 30, wi h he la es
da e Feb ua y 14 h, 2014. This pa applies o
hea y uel oils.
Se eso III Di ec i e is ex ended o onsho e
unde g ound gas s o age acili ies. The
Se eso II di ec i e de i nes 8 concep s: plan ,
equipmen , ope a o , haza dous subs ances,
majo acciden haza ds, isks and wa ehouse.
The new Se eso III Di ec i e de i nes 19
concep s, including 7 om he p e ious di ec i e
(excluding concep s o e).
The Se eso I, II and III, which a e
g adually eleased by he Eu opean Economic
Communi y, he Eu opean Communi y and he
Eu opean Union, a e inco po a ed by indi idual
Membe S a es in o hei na ional legisla ion.
In he Czech Republic, he law No. 59/2006
Coll is being add essed. The legisla ion also
includes access o isk assessmen , eme gency
scena ios and plans, he need o insu ance,
public access o in o ma ion and many o he
se ious measu es.
Acco ding o Law No. 59/2006 Coll., abou
he p e en ion o majo acciden s, a se e e
acciden is de i ned as an abno mal, pa ially o
o ally uncon ollable, spa ially and empo ally
bounded e en , such as a majo leakage,
i e o explosion, which occu ed o he o igin
is imminen ly h ea en in he con ex o wi h
he use o he building o acili y in which
he haza dous subs ance is manu ac u ed,
p ocessed, used, anspo ed o s o ed, and
leading o se ious dange o se ious impac on
he li es and heal h o people, li es ock and he
en i onmen o ha m o p ope y. [17]
This law pe ained o app oxima ely 150
indus ial companies in he Czech Republic
and es ablished basic obliga ions o ope a o s
o hese objec s. I can be said ha his law
ep esen ed a signi i can con ibu ion o he
p e en ion om majo acciden s in he Czech
Republic. Howe e , i is clea ha mos
companies we e no su i cien ly p epa ed o ul i l
obliga ions s emming om his law; he e o e
he sa e y documen a ion was in la ge numbe s
epea edly e u ned o ep ocessing [1].
Law No. 59/2006 Coll. was pa ially
amended se e al imes and by he 1s o Ma ch
in 2010 came in o e ec he law No. 488/2009
Coll. ha amends he law No. 59/2006 Coll.,
abou he p e en ion o majo acciden s
caused by dange ous chemicals o chemical
p epa a ions and abou amending he law No.
258/2000 Coll., abou he p o ec ion o public
heal h and amendmen o some ela ed laws.
[17]
Legisla ion o liabili y o damages incu ed
as a esul o majo acciden is newly ound in
§ 12. The ope a o is obliged o conclude new
insu ance wi hin 100 days om he en y in o
o ce o he decision on he app o al o he
secu i y so wa e o secu i y epo s. The limi
o indemni y mus e l ec he ange o possible
impac s o se e e acciden s, which a e cu en ly
lis ed in he app o ed secu i y p og am o in an
app o ed sa e y epo . [17]
In he new legisla ion a e speci i ed
indi idual adminis a i e o enses in a sepa a e
§ 36, which is used in § 37, which deals wi h
i nes, while in he p e ious law, he indi idual
adminis a i e o enses a e lis ed igh a he
indi idual le el o i nes.
The legisla ion o majo acciden s in he
Czech Republic uses a o al o i e ins umen s
o p o ec ion, namely:
 Inclusion o an objec o de ice.
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The obliga ion o he ope a o o p epa e
a lis o haza dous subs ances; p opose
ca ego iza ion o g oup A o B, o handle
he p o ocol on non-inclusion.
 Risk analysis, secu i y p og am and epo .
The obliga ion o he ope a o o
pe o m analysis and e alua ion o isk
o a majo acciden and on i s basis o
p ocess sa e y p og am o p e en ion
o majo acciden s o he g oup A,
g oup B, hen a sa e y epo .
 Plan o he physical p o ec ion o he
building o acili y.
 In e nal and ex e nal eme gency plan.
 Liabili y insu ance. [8]
3. Majo Acciden Insu ance
Fo insu ance o majo acciden s in he
Czech Republic i is compulso y o conclude
con ac ual insu ance, wi h espec o he Law
No. 59/2006 Coll. Fo se ious indus ial acciden
insu ance, § 12 o Law No 59/2006 Coll. mus
be abided by, whe e he limi s o insu ance a e
se . The le el o limi o insu ance bene i mus
e l ec he ange o possible impac s o se e e
acciden s, which a e cu en ly lis ed in he
app o ed secu i y p og am o in an app o ed
sa e y epo . The le el o limi o insu ance
bene i ag eed by he ope a o o he es ing
o he ope a ion s age should e l ec he ange
o possible impac s o majo acciden based
on he esul s o isk analysis and assessmen ,
submi ed o he Regional O i ce. [17]
To al insu ance p emium is de e mined by
he ela ion:
CP = ZP * KSP * KRM + DN, (1)
whe e:
CP is he o al insu ance p emium,
ZP is he basic insu ance, including pa icipa ion,
KSP is he coe i cien o pa icipa ion,
KRM is he isk ac o ,
DN a e addi ional cos s.
Fo calcula ion o he isk ac o , a pa ial
isk analysis need be ca ied ou , he esul o
he e alua ion is h, whe ein:
1 ≤ h ≤ 5, (2)
while 1 is he bes a ing and 5 he wo s a ing;
co esponding wi h he highes possible isk.
Subsequen ly, he o e all coe i cien o he
isk ac o KRM is se , whe e:
0.3 ≤ KRM ≤ 6. (3)
The o al insu ance p emium inc eases
when inc easing he sum and inc easing ela i e
pa icipa ion dec eases he o al insu ance
p emium. The basic p oblem in de e mining he
p emium acco ding o (1) is co ec iden i i ca ion
o he basic p emium. Fo his aim, he insu ance
company needs o know he ex en o damage
p obabili y models including in o ma ion abou
possible ex eme losses. In he nex sec ion,
he issue o modeling and simula ion o ex eme
damages is deal wi h.
3.1 Modelling and Simula ion
o he Ex eme Losses
Se ious indus ial acciden s a e o en classi i ed
as ex eme, ca as ophic damages wi h he
insu ance claims amoun ing o billions o eu os.
Ex eme alues heo y is used o assess
he isks o highly imp obable e en s. To such
e en s belong se ious indus ial acciden s
and also a ious na u al disas e s (hu icanes,
l oods, ea hquakes, i es, e c.) and man-made
disas e s, including nuclea acciden s and
e o ism.
As al eady shown in i gu e 1, he amoun o
ca as ophic e en s has a endency o inc ease.
The o al insu ance bene i o insu ance
companies is made up o 80% paymen o
damages om such e en s, while hei sha e
in he o al numbe o insu ance bene i is
app oxima ely 20%. Insu ance companies a e
o ced o inno a e and con inually e alua e hei
app oaches o isk assessmen and indi idual
insu abili y o isks associa ed wi h he design
o insu ance p oduc s. Bo h hese ac s a e o
cou se signi i can ly e l ec ed in he p ices o
insu ance.
To be able o model and simula e he ex eme
losses o majo acciden s, he ollowing sec ions
p o ide a heo e ical p ocedu e o p ac ical
use o quan ile unc ion and o de s a is ics o
he simula ion o ex eme losses. Mo e de ails
abou ca as ophic isk managemen and abou
modeling ca as ophic loses you can i ne in [14]
o [16].
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126 2015, XVIII, 4
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3.2 Quan ile Model
In he po olios o se ious indus ial acciden s
insu ance is he p obabili y o occu ence o
he ex eme losses highe in compa ison wi h
con en ional insu ance po olio. These claims
ha e a la ge i nancial impac on insu ance
companies, so i is impo an o he insu e o know
he p obabili y model ha adequa ely desc ibes he
insu ance losses also in he igh -hand ail.
Fo he ex eme losses modelling he long
o hea y ailed p obabili y dis ibu ions a e
used. The Pa e o dis ibu ion is o en used as
a model o claim amoun s needed o ob ain
well- i ed ails [12]. Random a iable X has
a dis ibu ion wi h he hea y ail, i applicable:
limx→∞eλx P(X > x) = limx→∞ eλx F  (x) = ∞,
o λ > 0. (4)
Quan ile unc ions applica ion is one way o
modelling he claim amoun s and subsequen
simula ion o ex eme losses.
When espec ing speci i c basic ules,
ad an ageous ea u es o quan ile unc ions
allow o combine and edi unc ions in
such a way ha he esul ing shape is non-
dec easing, quan ile unc ion again, wi h
s a is ically in e p e ed pa ame e s [7], [15].
Quan ile unc ion Q (p) is de i ned o each
eal p, 0 ≤ p ≤ 1 by ela ion:
Q (p) = xp , o which F(xp) = p. (5)
Quan ile unc ion Q (p) is hen de i ned as
he in e se o he dis ibu ion unc ion F (x). The
alue xp is called p-quan ile. By di e en ia ing
he unc ion Q (p) by p we ob ain he quan ile
densi y unc ion:
q(p) = dQ (p)
dp ,0 ≤ p ≤ 1. (6)
O de s a is ics play a key ole in modelling
using quan ile unc ions. A mo e de ailed
explana ion quan ile models can be ound o
example in [4], [13] and [15].
3.3 Simula ion o he Ex eme Values
When simula ing ex eme alues, i is p ima ily
necessa y o use he p og am o gene a ing
pseudo andom numbe s. Basic pseudo andom
numbe s a e wi hin he in e al <0, 1>
and ep esen a andom obse a ion om
a con inuous uni o m dis ibu ion on his in e al.
Quan ile unc ion o a uni o m dis ibu ion on
his in e al <0, 1> is exp essed as:
S(p) = p o 0 ≤ p ≤1. (7)
I using a andom numbe gene a o , we can
gene a e independen alues om a uni o m
dis ibu ion on he in e al <0, 1>. In his way,
he gene a ed pseudo- andom numbe s and
hei use in he p obabilis ic model o any ype
is called simula ion [7].
The basis o such simula ions is
Q- ans o ma ion ule. I z = T(x) is a non-
dec easing unc ion o x and Q (p) is he
quan ile unc ion, hen also T(Q(p)) is a quan ile
unc ion [7]. I is a non-dec easing unc ion o
T(x) a quan ile unc ion Q (p) o any dis ibu ion,
he applica ions o Q- ans o ma ion ule o he
case o uni o m dis ibu ion quan ile unc ion
S (p) = p we can simula e he alue o x om
dis ibu ion wi h quan ile unc ion Q (p) as:
xi = Q(ui) o i = 1, 2, …, n, (8)
whe e u1, u2,…, un a e simula ed alues om
a uni o m p obabili y dis ibu ion on he in e al
<0, 1>. Subs i u ing ui o quan ile unc ion xi =
Q(ui), we ob ain he a anged alues x(i), ha
gua an ee a non-dec easing shape o he
unc ion Q(ui).
The g ea ad an age o simula ion using he
quan ile unc ion is ha i also allows o simula e
only he highes alues in uppe ail wi hou
necessi y o simula ion he cen al alues o
andom a iable.
We assume he igh -hand ail o he
p obabili y dis ibu ion. By [7] and [13] i is
possible o simula e he highes alue as:
x(n) = Q(u(n)),whe e u(n) = n
1
n, (9)
while n is a andom numbe om he in e al
<0, 1>. I he sequence o he ans o med
a iables is de i ned in he o m:
u(n) = n
1
n
u(n–1) = ( n – 1 ) 1
n –1 *
u(n)
u(n–2) = ( n – 2 ) 1
n –2 *
u(n – 1)
.
.
.
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127
4, XVIII, 2015
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whe e i, o i = n, n-1, n-2,, ..., a e a plu ali y
o alues gene a ed as a andom selec ion om
a uni o m dis ibu ion, hen by de i ni ion alues
ui, o i = n, n-1, n-2..., we ge an inc easing
sequence:
u(i–1) < ui . (11)
Values u(i) o m an o de ed sequence o
alues om a uni o m dis ibu ion. I we ge one
alue u(n), he ela ion o he simula ion has he
o m:
u(m) = ( m ) 1
m
* u(m+1) , o m = n–1,n–2,…
(12)
The la ges obse a ions o a iable X a e
hen simula ed as:
x(n) =Q(u(n) ) ,
x(n–1) =Q(u(n–1) ) ,
x(n–2) =Q(u(n–2) ) ,
(13)
⁞
3 .4 Simula ion o Ex eme Losses
o Majo Acciden s
In his pa , heo e ical knowledge abou
quan ile unc ion is applied on he da a acqui ed
om in o ma ion abou majo acciden s
om he p e ious chap e and applied o
simula e ex eme losses. In o al, 27 damages
calcula ed in eu os ha e been selec ed om
he in o ma ion sys ems o majo acciden s
(EMARS, ZEMA, ARIA and PZHP). All o hem
ook place o e he yea s 2008–2010 and hey
a e o de ed in Table 1.
Using he s a is ical package
STATGRAPHICS Cen u ion XV by he
Kolmogo o -Smi no es we ha e ound ou
long ailed p obabili y dis ibu ions well i ed
o da a in Table 1. Resul s o his es show
Table 2.
Acco ding o p- alues, he losses a e
bes i ed by Pa e o dis ibu ion model in he
Eu opean o m [12]:
p = F(x) = 1 – ( a
x
)b (14)
By he STATGRAPHICS Cen u ion ou pu ,
pa ame e s o his p obabili y model a e
de e mined as a = 2,000,000; b = 0.774826.
The aim is o simula e he highes i e losses,
conside ing wen y majo acciden s ha ha e
happened. Fo his, he simula ion o ex eme
alues h ough quan ile unc ion will be used.
Quan ile unc ion o Pa e o dis ibu ion
can be de i ned as a unc ion in e ed o he
dis ibu ion unc ion (14) in he o m:
xp = Q(p) = a
(1–p)1
b
(15)
2,000,000 2,000,000 2,120,000 2,500,000 2,650,000
3,000,000 3,000,000 3,400,000 3,500,000 4,000,000
4,000,000 4,500,000 5,000,000 5,100,000 6,000,000
7,090,000 8,400,000 10,000,000 12,000,000 12,000,000
14,000,000 14,500,000 15,000,000 21,050,000 32,000,000
36,000,000 435,000,000
Sou ce: he in o ma ion sys ems o majo acciden s
Tab. 1: Indi idual i nancial ange o se ious indus ial acciden s (in eu o)
Loglogis ic
(3-Pa ame e ) Logno mal Logno mal
(3-Pa ame e )
Pa e o
(2-Pa ame e )
DN s a is ics 0.203818 0.137538 0.110012 0.084416
P-Value 0.212352 0.686701 0.899466 0.990618
Sou ce: own calcula ions
Tab. 2: Resul s o he Kolmogo o -Smi no es
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128 2015, XVIII, 4
Finance
The s a is ical so wa e STATGRAPHICS
Cen u ion andomly gene a ed i e numbe s
wi hin he in e al <0, 1> om an e en
dis ibu ion. The ollowing calcula ions we e
done in MS Excels, using he o mulae (10),
(12) and (13). The esul o simula ion o he
highes i e losses ou o wen y indemni ies
gi en, including he p ocedu e, is p o ided in
Tables 3 and 4.
Figu e 2 shows he simula ed damage
x=Q(u), u he jus o each o de s a is ics
X(20), X(19), …, X(16) and also hei median alues
x0.5 and quan iles x0.005 and x0.995. In [7] and [13]
is de i ed he calcula ion o he s a ed quan iles
including be a in e sion unc ion, which is why
acqui ed ou comes a e p esen ed only.
Table 4 makes i clea ha he highes
amoun o claim is om he in e al
<13,127,418.93 €; 88,829,890,943.26 €>
wi h he p obabili y α=0.99, while he median
o he highes damage eaches he alue o
156,773,540.72 €.
These esul s a e use ul o he pu poses o
insu ance and einsu ance.
3.5 Reinsu ance o he La ges Claims
Ex eme isks, simula ed in he p e ious
chap e , need o be ensu ed h ough
a combina ion o nume ous einsu ance ypes.
One o he mos used combina ions includes
a einsu ance when he applica ion o he
insu ance company’s own e en ion is ollowed
by quo a einsu ance. A highe ie is ensu ed
h ough non-p opo ional WXL/R einsu ance,
and also al e na i e o ms o einsu ance can
be use.
As shows Figu e 1, man-made disas e s
ha e been inc easingly ex ensi e la ely. Thei
modelling and simula ion is bene i cial o isk-
managemen policies o insu ance companies
and ackling c ucial issues o hei insu ance
and einsu ance.
Ex eme- alue simula ion is gene ally used
in non-p opo ional la ges claims einsu ance
LCR(p) o ECOMOR(p) einsu ance [13].
Conclusion
This a icle is de o ed o he analysis o
majo indus y acciden s. The isks o majo
acciden s and hei consequences each
ca as ophic dimensions. Tha is a eason
o equi e a quali y legisla i e modi i ca ion
ocused on he p e en ion and liquida ion o
he consequences. The SEVESO di ec i es
No. I, II and III we e sequen ially eleased
n 1/n 1/n u(n) Q(u(n))
0.6489213 20 0.050000 0.978610 0.978610 285,810,153.50 €
0.2682335 19 0.052632 0.933086 0.913127 46,829,591.89 €
0.7592468 18 0.055556 0.984815 0.899261 38,682,768.63 €
0.3568497 17 0.058824 0.941186 0.846373 22,437,953.57 €
0.4219863 16 0.062500 0.947504 0.801942 16,165,793.88 €
Sou ce: own calcula ions
Tab. 3: Simula ion o op i e damages in wen y insu ance indemni ies
Q(BETAINV(0.5; ;n- +1)) = x0.5 Q(BETAINV(0.005; ;n- +1)) = x0.005 Q(BETAINV(0.995; ;n- +1)) = x0.995
156,773,540.72 € 13,127,418.93 € 88,829,890,943.26 €
50,049,792.19 € 8,804,780.17 € 1,732,291,027.22 €
27,432,632.79 € 6,806,971.44 € 366,468,591.03 €
18,216,901.58 € 5,613,897.44 € 147,262,811.20 €
13,353,730.93 € 4,810,419.62 € 78,297,378.84 €
Sou ce: own calcula ions
Tab. 4: Quan iles o o de ing s a is ics
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by he Eu opean Economic Communi y, he
Eu opean Communi y and he Eu opean Union
a e implemen ed by indi idual membe s a es
in o hei na ional legisla ions. In he Czech
Republic hey a e ensh ined in he Ac No.
59/2009 Coll. a his junc u e. Fo he membe
s a es o he EU, OECD and UNECE he e is
an in o ma ion sys em called EMARS. This
sys em collec s da a abou ope a o s subjec
o he ele an laws and abou majo acciden s.
EMARS p o ides basic in o ma ion abou pas
acciden s also o he gene al public.
Fo he companies in he EU a ea he e is
a legal obliga ion o ake ou liabili y insu ance
wi h insu ed sum ha has o co espond o he
ex en o he possible damage. In he Czech
Republic his obliga ion is es ablished by Ac
No. 59/2006 Coll. I is necessa y o secu e
e e y insu ance con ac , because he possible
damage can each ca as ophic dimensions.
Since he e a e big isks in such cases no only
he classical p o ec ion bu also a combina ion
o he al e na i e isk ans e me hods need
o be used. Among he ART me hods ha can
be used o hedge he isk o majo acciden s
belongs he secu i iza ion o insu ance isks.
In he a icle is used da a ela ing o 27
majo acciden s o he applica ion o he
ex eme damage simula ion and modelling
me hods. Wi h he s a is ic p og amming
sys em STATGRAPHICS Cen u ion XV by
using Kolmogo o -Smi no es he Pa e o
dis ibu ion in Eu opean o m was es ablished
as he bes model o goodness o i and also
he pa ame e alues o his dis ibu ion we e
es ima ed. Knowing he p obabili y dis ibu ion
o he amoun o damages will enable he
insu ance company o de e mine he basic
p emium and o decide o he op imal ensu ing.
Owing o ca as ophic damages o majo
acciden s, he concluding pa deals wi h
modelling and simula ion o possible ex eme
damage. In his pa was used quan ile unc ion
o Pa e o dis ibu ion in he Eu opean o m.
Simula ion using quan ile unc ion enables
es ima ion o in e als o he highes damages.
This can be used in deciding abou app op ia e
ypes o disp opo ional einsu ance and
o insu ance company’s ca as ophic isks
managemen .
This pape was suppo ed in e ms o he
p ojec SGS FES 2014 SGSFES_2014003,
en i led “Vědecko- ýzkumné ak i i y
Sys émo ém inžený s í a in o ma ice”.
Fig. 2: G aphic o m o he ex eme losses simula ion
Sou ce: Own calcula ions
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