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Mas e P og am in S a is ics and In o ma ion Managemen
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Ins i u o Supe io de Es a ís ica e Ges ão da In o mação
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Es ima ion o Technical Rese es
using S ochas ic Me hods
hAnalysis and Risk Managemen i
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Mas e o Science in Mas e in hS a is ics and In o ma ion
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Es ima ion o Technical Rese es
using S ochas ic Me hods
hAnalysis and Risk Managemen i
B una Raquel Sebas ião Gonçal es
Disse a ion submi ed in pa ial ul illmen
o he equi emen s o he deg ee o
Mas e o Science in Mas e in hS a is ics and In o ma ion
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Es ima ion o Technical Rese es using S ochas ic Me hods
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Abs ac
Insu ance companies need o ha e p ope echnical ese es in o de o s ay sol en ,
due o he liabili ies ha hey ake esponsibili y o . The no ion o wha a e p ope
ese es is a subjec s udied by insu e s and academics.
Fo he non-li e insu ance he claim ese es, amongs o he ypes o ese es, ha e
g ea signi icance. Hence he exis ence o many kinds o ese ing me hods, ei he
s ochas ic o de e minis ic.
This wo k aims o compa e he es ima es o claim ese es using he Thomas Mack
and he Boo s ap me hods o he wo ke s compensa ion insu ance.
Keywo ds: Claims ese ing; Chain Ladde ; Thomas Mack; Boo s ap.
ii
Resumo
Companhias de segu os p ecisam de e p o isões écnicas ap op iadas de o ma a se
man e em sol en es, dados os iscos que es as omam esponsabilidade. A noção de o
que é uma p o isão adequada é um ema es udado pelas segu ado as.
Pa a o amo Não-Vida a p o isões de sinis os, de en e ou os ipos de p o isões,
êm g ande signi icância. Consequen emen e, exis em á ios mé odos pa a a sua de e -
minação, que es ocás icos, que de e minís icos.
O p opósi o des e abalho é a compa ação das es ima i as das p o isões de si-
nis os no amo de aciden es de abalho usando os mé odos de Thomas Mack e de
Boo s ap.
Pala as-cha e: P o isões pa a Sinis os; Chain Ladde ; Thomas Mack; Boo s ap.
ix

1
In oduc ion
Insu ance companies ha e o deal wi h isk managemen and one o he key decisions is
o build and main ain p ope echnical p o isions so ha hey can assu e he ul ilmen
o u u e esponsibili ies owa ds policyholde s and hi d pa ies.
In Non-Li e insu ance, he mos impo an echnical p o isions a e he ones asso-
cia ed wi h claim amoun s. This is due o he cos o some claims which linge o ge
se led and o he s ha we e no immedia ely epo ed.
The con inuous e o o imp o e he accu acy o he es ima ion o claims is a
wo hy ask when i comes o he p o i abili y o he insu ance indus y, and a needed
one in o de o ensu e i s sol ency, S aub and G ubbs, 1998.
The p oblem ha insu ance companies ace is how o es ima e he p ope claim e-
se es so ha whene e hese a e no enough i can jeopa dize he companies’ sol ency
and migh a ise inancial p oblems. I he ese es a e subs an ial, he p o i abili y
migh be in luenced in some sensi i e way, so, i is impo an o ha e he igh amoun
o p o isions.
Thus, he main ques ions ha come ac oss insu ance companies a e he ollowing:
• Wha migh be he adequa e amoun in he claim ese es?
• How a e hese ese es es ima ed?
To ge hese answe s, se e al s ochas ic me hods ha e been cons uc ed wi h s ong
heo e ical backg ound making possible o no only es ima e ese es bu also o ob ain
e o measu emen s and build con idence in e als a ound hese es ima ions. The e-
o e, insu ance companies may use di e en me hods in o de o be e es ima e claim
ese es.
Fo ou wo k, we will be using he wo di e en s ochas ic me hods ha a e as
ollows:
•
Chain Ladde /Thomas Mack me hod which is he mos simple and amous
dis ibu ion- ee me hod;
1
CHAPTER 1. INTRODUCTION
•
Boo s ap Me hod ha is a simula ion algo i hm based me hod o c ea e pseudo-
da a h ough e-sampling wi h eplacemen ;
The da ase used o his analysis was ex ac ed om he ASF s a is ical a chi e
and i will be used he Rso wa e o cons uc each model whils se ing ou he
assump ions.
The eupon, we will ha e as Chap e 2 he in oduc ion o some concep s, such as
some gene al aspec s o claim ese es, how do claims p ocess h ough ime, how he
da a is p esen ed and some no a ions ha will be used.
In Chap e 3, he e will be a heo e ical explana ion o he Chain Ladde / Thomas
Mack me hod and i s assump ions ollowed by a p ac ical applica ion and e i ica ion
o he assump ions o ou da ase .
In Chap e 4, he Boo s ap me hod will be explained and hen applied o he
da ase .
2
2
Basic Concep s
2.1 The Claims P ocess
A Non-Li e insu ance policy is de ined as a con ac be ween wo pa ies, he insu e
and he insu ed whe e, on he occu ence o special e en s he insu e , no mally an
insu ance company, has o pay a gi en amoun o money, once o se e al imes, o he
insu ed. These amoun s depend on he peculia ci cums ances o hose e en s, Taylo ,
2012.
The ollowing igu e ep esen s he claims p ocess since i s occu ence o i s clo-
su e.
Figu e 2.1: Claims P ocess h ough ime
A ime
1
he e is an e en ha gene a es a claim and i is called he da e o
occu ence. La e in ime,
2
, he insu ed no i ies he insu e o he claim, Taylo ,
2012.
I is no expec ed o he insu e o pay he insu ed immedia ely. The paymen s in
imes
3
,
4
and
5
a e made a e some necessa y p oceedings, namely he alida ion
and assessmen o he claim. When he insu e conside he end o claim paymen s,
he ac ion on he claim is comple e and he ile is closed a ime 6.
E en ually, he need o open he ile may a ise,
7
. This can happen due o ap-
pea ance o new ele an in o ma ions ela ed o he claim, and, i i is necessa y, he
3
CHAPTER 2. BASIC CONCEPTS
insu e migh ha e o pay again o he insu ed,
8
, in o de o ec i y he case, and hen
he ile is closed again,
9
. No e ha hese
k,k
= 7
,
8
,
9, may occu du ing se e al yea s.
The insu e has he obliga ion o build p o isions, a he end o each p ac ice,
ha aim o cope wi h he paymen s o claims ha al eady happened and we en’ ye
epo ed o he insu e , and he e migh be some claim iles ha a e no ye closed and
he paymen s a e due in he nex accoun an s yea .
In he in e al be ween
1
and
2
, he claim is said o be incu ed bu no epo ed
(IBNR), since he e en ha led o a claim al eady happened bu he insu e does no
know bu he la e s ill akes esponsibili y o he liabili y, Taylo , 2012.
When he insu e is al eady no i ied ha he e has been a claim bu does no know
he eal amoun ha is going o pay o he insu ed, [
2
,
6
], he p o ision o his sum
is called incu ed bu no enough ese ed (IBNER).
2.2 Da a
The ASF (Au o idade de Supe isão de Segu os e Fundos de Pensões) is he Po uguese
na ional au ho i y esponsible o he conse a i e and beha iou al supe ision o
he insu ance, einsu ance and pension unds ac i i ies and i s managing en i ies,
“Au o idade de Supe isão de Segu os e Fundos de Pensão”, n.d. In ega ds o his
wo k, we ha e e ie ed he paid amoun s o claims o acciden s a wo k insu ances,
in housand o Eu os om 2007 un il 2016.
To es ima e he p o ision amoun s, insu ance companies should own his o ical
da a ega ding claim cos s ha , h ough s a is ical p ocedu es, could deduce i s de el-
opmen o he u u e.
This da a is o ganized and o ms a wo-dimensional a ay whose shape is, no mally,
iangula as we can see in Figu e 2.2, Taylo , 2012.
Yea o acciden De elopmen Yea
0 1 2 ··· j··· n−1n∞
0X0,0X0,1X0,2··· X0,j ··· X0,n−1X0,n X0,∞
1X1,0X1,1X1,2··· X1,j ··· X1,n−1
2X2,0X2,1X2,2··· X2,j ···
··· ··· ··· ··· ··· ···
i Xi,0Xi,1Xi,2···
··· ··· ··· ···
n−1Xn−1,0Xn−1,1
n Xn,0
Figu e 2.2: Run-O T iangle
Each ow o he ma ix ep esen s he yea o acciden , o when ha he insu e
was no i ied. The columns o he ma ix ep esen s he de elop yea o he claims, i.e.
he numbe o yea s ha ake place since he occu ence o he claims un il i s se led
4
2.3. ESTIMATION METHODS
by he insu e . The las column,
∞
, is called ul ima e and holds he in o ma ion o all
occu ed claims, bu whose se lemen will only be done a e nde elopmen yea s.
E en hough he mos common pe iod o ime is yea s, he da a can be ep esen ed
by semes e , imes e o e en mon hs.
The quan i ies obse ed in he de elopmen pe iod
j
wi h he ime o acciden
i
a e deno ed by Xi,j and he ones ha we a e going o s udy a e:
•Xi,j
=
Ci,j
consis s in he paymen s made in he de elopmen pe iod
j
ega ding
he claims a isen in ime i, in o he wo ds he inc emen al paymen s;
•Xi,j
=
Ai,j
consis s in he paymen s made in he de elopmen pe iod
j
, included,
ega ding he claims a isen in ime i, i.e. he cumula i e paymen s;
The ac o gi en by Ai,j is easily ob ained by
Ai,j =
j
X
k=0
Ci,k.(2.1)
As we said be o e, he main goal is o es ima e he compensa ion ha he insu e
has o pay o he insu ed. So ha , wi h he gi en da a, i is possible o p o ide he
cu en liabili y ela i e o he claims ha occu ed in yea
i
, and i will be deno ed by
Ri, gi en by
Ri=Ai,∞−Ai,n−i,0≤i≤n,
whe e
Ai,n−i
is he o al paid amoun om yea
i
un il he mos ecen yea in
conside a ion o he claims.
Consequen ly he o al amoun o he p o ision will be gi en by
R=
n
X
i=0
Ri.
2.3 Es ima ion Me hods
Th ough he na u e o claims and he his o ical da a o insu ance companies, he
es ima es o claim ese es can be ob ained using a case by case app oach o s a is ical
me hods.
The case by case app oach is used o es ima e ese es indi idually o each claim
p ocess and demands o know he claim in hand. This is mo e sui able o si ua ions
whe e he e is a lack o p ope da a o said da a doesn’ ha e app op ia e ea u es o
use s a is ical me hods.
Whene e he numbe o claims in an insu e po olio is big enough, s a is ical
me hods a e used o es ima e he claim ese es. The choice o he me hod is made
5

CHAPTER 2. BASIC CONCEPTS
h ough he assessmen o he impac ha se e al ex e nal and in e nal elemen s migh
ha e in he es ima ion, such as, da a homogenei y and s a is ical na u e o es ima o s.
In o de o o esee he o al amoun o claims ha happened in each yea and, hus,
he p o ision amoun o hold oday we need o es ima e he lowe iangle o he Run-
O T iangle in Figu e 2.2. In o de o do so, we a e going o use some usual me hods
such as he Chain Ladde and Boo s ap me hods.
2.4 Con idence In e als Es ima ion
When es ima ing he amoun o ese es we can use bo h de e minis ic and s ochas ic
models, bu wi h he la e we can also quan i y he deg ee o unce ain y o hese
es ima ions and wi h hese e o measu emen s we can assess he quali y o he models’
adjus men , hus scale he eliabili y o he a ained esul s. Th ough his, we can build
con idence in e als, in o de o see he a iance o he ese es es ima es, and i will
allow us o do as a s ic analysis as we wan . Usually, we use as an e o measu emen
he Mean Squa ed E o (MSE) which will be de ined la e .
In o de o build a con idence in e al o an es ima ion o he o al ese e,
ˆ
R
,
we need o know he dis ibu ion o he andom a iable
R
, which usually is a e y
di icul , and some imes, an impossible ask. So, in p ac ice, h ough he Cen al Limi
Theo em, i is assumed ha
R
ollows an asymp o ically No mal Dis ibu ion wi h
mean µand a iance σ2,
R∼Nµ,σ2,
whe e
µ
=
ˆ
R
and
σ2
=
[
MSE ˆ
R
. He e,
ˆ
R
is he es ima ion o he o al ese e and
[
MSE ˆ
Ri s mean squa ed e o gi en by he usage o he s ochas ic model.
In hese condi ions, he con idence in e al o he es ima ion o
ˆ
R
o a con idence
le el o 1 −αis gi en by:
ˆ
R−Φ1−α
2·q[
MSE ˆ
R;ˆ
R+Φ1−α
2·q[
MSE ˆ
R(2.2)
whe e,
Φ1−α
2
is he p obabili y quan ile o 1
−α
2
om he No mal Dis ibu ion (0,1).
Some imes, when he in e io limi o he con idence in e al is nega i e o he
S anda d E o (SE) is signi ican ly high, ha is
c
SE ˆ
R=q[
MSE ˆ
R>0.5·ˆ
R,
we ha e o conside ha he andom a iable
R
ollows a Logno mal Dis ibu ion, wi h
µ
and
σ2
as pa ame e s, whose i s wo momen s a e he same as he es ima ed alues
by he model,
R∼Logno mal µ,σ2
6
2.4. CONFIDENCE INTERVALS ESTIMATION
whe e,
E(R)
=
eµ+σ2
2
and
V(R)
=
e2µ+σ2·eσ2−1
. This means ha he pa ame e s
o he dis ibu ion a e gi en by









E(R)=ˆ
R
V(R)=MSE ˆ
R⇔








µ+σ2
2=lnˆ
R
e2µ·eσ2·eσ2−1=
[
MSE ˆ
R
⇔








µ=lnˆ
R−σ2
2
e2ln(ˆ
R)−σ2
2·eσ2·eσ2−1=
[
MSE ˆ
R
⇔










µ=lnˆ
R−σ2
2
σ2=ln [
MSE(ˆ
R)
ˆ
R2+ 1!
In his case he con idence in e al will be gi en by
eµ−Φ1−α
2·√σ2;eµ+Φ1−α
2·√σ2
We can, in a simila way, de ine he con idence in e als o he es ima ions o
ˆ
Ri,0≤i≤n, as well o he unknown amoun s Ci,j o Ai,j, (i,j) : n−i+ 1 ≤j≤n+ 1.
7
3
Chain Ladde Me hod/ Thomas
Mack
The Chain Ladde is conside ed o be he mos amous me hod o es ima e claim e-
se es. This is because i is a simple app oach and i is dis ibu ion- ee, meaning
ha a e li le assump ions o conside , in his case he e a e h ee assump ions. Com-
plemen a y o his, is i also known ha ese es es ima ed using he Chain Ladde
me hod o he mos ecen acciden yea s a e e y sensi i e o a ia ions in he ob-
se ed da a, Mack, 1993.
Rega ding he assump ions o he Chain Ladde me hod, he i s one s a es ha
he e a e de elopmen ac o s ha ep esen how claims om a ce ain yea o acciden
e ol e in each de elopmen yea . We aim o es ima e hese de elopmen ac o s he
bes we can o each acciden yea and he ones ega ding he mos ecen yea in
ou da a allows us o es ablish a o al paymen amoun , a e all possible yea s o
de elopmen un il ha ecen yea .
The second assump ion comes due o he ac ha his algo i hm does no ake
in o accoun any dependence be ween acciden yea s, we can addi ionally assume ha
he cumula i e paymen s o di e en acciden yea s a e independen . Howe e , in
p ac ice, he independence o he acciden yea s can be skewed by ce ain calenda
yea e ec s like majo changes in claims handling o in case ese ing.
The las , and hi d, assump ion unde lying he Chain Ladde me hod is associa ed
wi h he a iance o he es ima o s o
Ci,j
. Those ha should be conside ed a e he
ones wi h he lowes a iance.
When s udying his me hod, Thomas Mack de eloped a s ochas ic model, wi h
simila esul s o he adi ional and de e minis ic Chain Ladde me hod, which has, as
an ad an age, he means o es ima e he mean squa ed oo ha explains he a ia ion
o esul s.
Since we a e wo king wi h claims om insu ance o acciden s a wo k we ha e o
ake in o accoun a ail ac o , i.e. we ha e o conside ha he amoun
Ai,n
is no he
las claim and he las de elopmen ac o is highe ha 1.
9
CHAPTER 3. CHAIN LADDER METHOD/ THOMAS MACK
H0: The occu ence yea s a e independen
s
H1: The occu ence yea s a e no independen
wi h a signi icance le el o 5%, e e y ime he ollowing does no happen
Z∈E(Z)−Φ−1
1−α
2·pV(Z);E(Z)+Φ−1
1−α
2·pV(Z)(3.9)
whe e
Z
is he andom a iable gi en by
Z
=
Z1
+
Z2
+
···
+
Zk−1
, and unde he null
hypo hesis we ha e ha
E(Z)=X
k
E(Zk)and V(Z)=X
k
V(Zk)
wi h
E(Zk)=a
2− a−1
!a
2a
and
V(Zk)=a(a−1)
4− a−1
!+E(Zk)−(E(Zk))2,
whe e is he bigges in ege less o equal o (a−1)/2.
Summa izing he a o emen ioned, i
Z
does no all in he con idence in e al
abo e we canno p oceed wi h he implemen a ion o he model.
3.3.3 Thi d Assump ion
To e i y he hi d assump ion gi en by
(3.4)
we need o ake in o conside a ion ha
he condi ional a iance o
Ai,j+1
is di ec ly p opo ional o
Ai,j
wi h
σ2
j
as a cons an
o p opo ionali y. Wi h he known sec ion o he Run-o T iangle we can es ima e
VAi,j+1|Ai,0,...,Ai,j h ough
EAi,j+1 −EAi,j+1|Ai,0,...,Ai,j2(3.1)
=
(3.1)
=Ai,j+1 −Ai,j · j2≈
≈Ai,j+1 −Ai,j ·ˆ
j2,
0≤i≤n−1,0≤j≤n−1
The es ima ed alues o he condi ional a iance should come close o
Ai,j ·σ2
j
, i
(3.4)
is e i ied, i.e., he ob ained esiduals will also be p opo ional, o come close, o
Ai,j.
I (3.4) e i ies we will ha e
16

3.4. PRACTICAL APPLICATION
Ai,j+1 −Ai,j ·ˆ
j2≈Ai,j ·σ2
j
o
Ai,j+1 −Ai,j ·ˆ
k
pAi,j ≈σj
When ob ained h ough his, he alues will no longe be linked o
Ai,j
, so i will
be enough o ep esen he o de ed pai

Ai,j+1 −Ai,j ·ˆ
j
pAi,j
,Ai,j, o a ixed j(3.10)
in a plo o see i he se o ob ained poin s do no p esen any ype o end. I his
happens, hen we can accep he assump ion gi en by (3.4) as alid.
Howe e , i he e is a end he assump ion will no be e i ied and consequen ly we
mus ge o he jes ima o s, o , i necessa y ejec he implemen a ion o his model.
3.4 P ac ical Applica ion
Le ’s conside ed he ollowing Run-o T iangle as ou de elopmen iangle wi h inc e-
men al paid claims om acciden s a wo k, since 2007 un il 2016. All alues o ou
da a a e in housands o Eu os and all he esul s ha we will see below we e ob ained
using he e sion 0.2.14 o he ChainLadde package o RC an, Gesmann e al., 2021.
Figu e 3.1: Inc emen al paid claims
To apply he Thomas Mack Model o ou da a we need o change he Run-O
T iangle in o de o ha e cumula i e paid claims.
17
CHAPTER 3. CHAIN LADDER METHOD/ THOMAS MACK
Figu e 3.2: Cumula i e paid claims
The i s hing we need o do is o e i y i ou da a ollows all he assump ions
implied in he Model and o ha we need o calcula e he de elopmen ac o s,
j
,
gi en by (3.2) and a e he ollowing
Figu e 3.3: Es ima ion o De elopmen Fac o s
Using he linea eg ession explained a he end o 3.1.1, we’ e ob ained a ail ac o
o 1
.
001286 and since his is close o 1 we will no be using i , he e o e he Run-o
T iangle in Figu e 3.1 will be conside ed as closed.
As said in Subsec ion 3.3.1, we will s a by analysing he ollowing plo s o see he
linea ela ion unde lying he assump ion gi en by (3.1).
Figu e 3.4: Da a adjus men o j
As we can see in Figu e 3.4, o each column
j
, he line wi h slope o
j
seems
o be well adjus ed o he da a and o ha , we can assume ha ou da a ollows
he assump ion in
(3.1)
, which says ha he e is p opo ionali y be ween adjacen
de elopmen yea s, .
18
3.4. PRACTICAL APPLICATION
The o he cha ac e is ic o he i s assump ion ha we need o p o e is ha he in-
di idual de elopmen ac o s
Ai,j/Ai,j−1
and
Ai,j+1/Ai,j
a e no co ela ed by applying
a Spea man es o hese indi idual de elopmen ac o s. By compu ing
Ai,j+1/Ai,j
o
each en y o he Run-o T iangle and we ge he ollowing ou pu .
Figu e 3.5: Indi idual de elopmen ac o s
To do he Spea man es we mus , as explained abo e, ank he indi idual de elop-
men ac o s in wo di e en ways in o de o build i,j and si,j.
(a) i,j (b) si,j
Figu e 3.6: Indi idual de elopmen ac o s anked
Using
(3.5)
we ge , ha o each
j
, he es ima ions o he Spea man’s co ela ion
coe icien is
Figu e 3.7: Indi idual de elopmen ac o s
And he inal es ima ion o Tis 0.07108844.
So, as we said be o e, when he e is no co ela ion i is expec ed o
E(T)= 0 and V(T)=1
28 = 0.03571429
he e o e, he con idence in e al o 50% o he es ima ion o Twill be
[−0.1274666;0.1274666]
Since ou es ima i e belongs o his in e al we will no ejec he null hypo hesis
wi h a le el o s a is ical signi icance, meaning ha we do no ejec he hypo hesis o
he non-co ela ion o he de elopmen ac o s.
19
CHAPTER 3. CHAIN LADDER METHOD/ THOMAS MACK
Nex , we will p o e i ou de elopmen ac o s a e independen o all o he de el-
opmen yea s as s a ed by
(3.3)
, and o his we will implemen wha was desc ibed in
Subsec ion 3.3.2.
We al eady ha e he Run-O T iangle wi h he indi idual de elopmen ac o s,
Figu e 3.5, so we only need o ank each indi idual de elopmen ac o by column
om he smalles o he la ges .
Figu e 3.8: Indi idual de elopmen ac o s anked om smalles o la ges
As a o emen ioned, he anked indi idual de elopmen ac o s a e assigned o he
L
o
S
se s, i hey a e abo e o below he median, espec i ely, and a e labelled wi h
a "
L
"o "
S
". When an elemen is equal o he median i is labelled wi h an "
∗
"and is
igno ed, meaning i does belong o nei he o he se s.
Figu e 3.9: Elemen s o he Land Sse s
Le
Sj
and
Lj
be he numbe o elemen s belonging o he
S
and
L
se s, espec i ely,
om a jdiagonal.
20
3.4. PRACTICAL APPLICATION
j SjLjZj a EZjEZj
0 0 1 0 1 0 0 0
1 1 1 1 2 0 0.5 0.25
2 0 2 0 2 0 0.5 0.25
3 3 1 1 4 1 1.25 0.44
4 3 1 1 4 1 1.25 0.44
5 3 3 3 6 2 2.06 0.62
6 3 4 3 7 3 2.41 0.55
7 5 3 3 8 3 2.91 0.80
8 2 4 2 6 2 2.06 0.62
To al 14 12.94 3.97
Table 3.1: Resul s o Zjand Z a iables
F om he esul s on Table 3.1 we can calcula e he lowe and highe limi o ou
con idence in e al gi en by (3.9),
[10.80648,18.60758]
Since, ou
Z
is in he con idence in e al we don’ ejec he null hypo hesis, mean-
ing we don’ ha e any s a is ical eason o s a e ha he occu ence yea s a e no
independen .
Fo he hi d assump ion, we will look in o a se ies o plo s ha ep esen he
o de ed pai in (3.10) o each j.
Thus, o ou da ase he plo s o he poin s gi en by (3.10) a e he ollowing,
Figu e 3.10: Weigh ed Residuals
As we can see, i appea s ha he e is no ype o end, meaning ha he assump ion
(3.4) is alid.
Now ha all he assump ions unde lying he Thomas Mack Model ha e been
p o ed we can, he e o e, begin o apply he Model o ou da a.
21

CHAPTER 3. CHAIN LADDER METHOD/ THOMAS MACK
As a o emen ioned, he Model es ima es he u u e amoun s
Ai,j
, hus he p o i-
sions es ima es
Ri
. The es ima ions o he de elopmen ac o s,
ˆ
j
, calcula ed eso ing
o (3.2) a e ep esen ed in Figu e 3.3.
Thus, he lowe iangle can now be comple ed using (3.1), which gi es us,
Figu e 3.11: Full T iangle wi h es ima ed u u e claim amoun s
The ea e , he es ima ed ese e o each acciden yea is
Figu e 3.12: Es ima ed ese e by Acciden Yea
Consequen ly he o al ese e is 195285.8.
Now o measu e he a iabili y o
ˆ
Ri
using (3.5), we need o es ima e
σ2
j
using (3.6),
so
Figu e 3.13: Es ima ion o σ2
j, 1 ≤j≤8
Since 0
.
7753622
>
0
.
05117787, we a e in condi ions o use (3.7) o ex apola e he
las es ima o , ˆ
σn−1, which alue is 0.003378157.
Finally, he a iabili y o he es ima o s o Rigi en by (3.5) is
Figu e 3.14: Es ima ion o
MSE(ˆ
Ri)
F om Sec ion 2.4 and om he es ima ions abo e, we ge he es ima ion o he
con idence in e als gi en by (2.2) o each occu ence yea .
Figu e 3.15: Es ima ion o con idence in e al o Ri
22
4
Boo s ap Me hod
The Boo s ap me hod was in oduced by B andlen E on, in 1979, Taylo , 2012, and
i s essence is o ake a sample o a iangle o paid claims om an unknown dis i-
bu ion in o de o ob ain in o ma ion abou u u e claim paymen s, o ese e, by
c ea ing la ge amoun o se s o pseudo-da a h ough e-sampling wi h eplacemen
om he obse ed da a in he iangle, England and Ve all, 2002. These se s ha e he
same unde lying dis ibu ion so we can a ain, o each se , s a is ics o in e es and
he dis ibu ion o hose s a is ics, such as, he mean o each se o pseudo-da a, he
s anda d de ia ion o he se o means and es ima e he s anda d e o o he mean.
Ini ially, Boo s apping was only used o ge es ima ion e o s o ese e es ima es
om he Chain Ladde Model. La e , some adjus men s we e made o he Model in
o de o simula e he p ocess e o and allowing o ge a p edic i e dis ibu ion.
When we apply he boo s ap, i is expec ed o he da a o be independen and
iden ically dis ibu ed, in o de o he e-sampling wi h eplacemen o happen om
he o iginal da a. Since we ha e a eg ession ype p oblem, ou da a is independen
(as p o ed in he subsec ion 3.3.2) bu i is no iden ically dis ibu ed. To su pass his
p oblem i is a s anda d p ocedu e o boo s ap he esiduals om he inc emen al
amoun s,
Ci,j
ins ead, because hese a e app oxima ely independen and iden ically
dis ibu ed, wi h a p ope de ini ion o he esiduals dismissing he s uc u e o he
Run-O T iangle.
23
CHAPTER 4. BOOTSTRAP METHOD
The ea e , we illus a e he main s eps o he Boo s apping
Figu e 4.1: Boo s ap Me hod
The Da a T iangle is ou da a and he g ey a ea in Model and Rese e is he p ojec ion
o u u e paymen s and he es ima ed ese es using he Boo s ap me hod and he
iangle deno ed by (2) is a i ed model o he pas da a. The di e ence be ween
iangles (1) and (2) gi es one se o Residuals, iangle (3), meaning ha ou da a and
he i ed da a may di e , Lowe, 1994.
I he da a is a implemen a ion o a andom p ocess, ano he execu ion o he
p ocess migh lead o ano he se o da a ha di e s om he model, hence ano he
se o esiduals.
When p oducing a ious se s o da a wi h esiduals e-sampling and adding hem
o he i ed model we c ea e many se s o possible da a iangles, deno ed as Pseudo-
Da a. Fo each iangle he ese ing me hod is implemen ed, p oducing a se ies o
ese e es ima ed Pseudo-Rese es. A e se e al i e a ions we ge a la ge collec ion o
Pseudo-Rese es, which will ha e a speci ic dis ibu ion wi h i s mean and a iance.
The la e gi es us a measu e o a iabili y o he ese e es ima e,Lowe, 1994.
4.1 Boo s ap Me hod
Acco ding o, England and Ve all, 2002, we need o p epa e ou da a be o e boo s ap-
ping. Since he da a (Da a T iangle (1) in Figu e 4.1) does no ollow any dis ibu ion
ha we know o , we will be using he Chain Ladde / Thomas Mack (Chap e 3) o ge
he de elopmen ac o s om he cumula i e paymen s.
24
4.1. BOOTSTRAP METHOD
Then, i is equi ed o build a new Run-o T iangle (Model and Rese e (2) in Figu e
4.1) wi h cumula i e i ed alues o he obse ed da a, using a backwa ds ecu sion
s a ing om he la es obse ed cumula i e paymen in he main diagonal di ided by
he calcula ed de elopmen ac o s.
Meaning, he new i ed alues, Di,j,0≤i≤n,0≤j≤n−i, a e gi en by
Di,n−i=Ai,n−iand Di,j−1=Di,j
ˆ
j
(4.1)
Nex , in o de o assess he quali y o he i ed alues we will calcula e he unscaled
esiduals ha will be used du ing he e-sampling using he Pea son esiduals
P
i,j
gi en
by
P
i,j =Ci,j −ˆ
mi,j
pˆ
mi,j
(4.2)
whe e,
Ci,j
a e he inc emen al paymen s om he obse ed da a and
ˆ
mi,j
a e he
inc emen al i ed paymen s (4.1).
We now need o adjus he Pe son esiduals (Residuals (3) in 4.1), o eplica e he
bias co ec ion using an analy ic app oach, conside ing he deg ees o eedom , i.e.
he numbe o obse a ions minus he numbe o pa ame e s,
P adj
i,j = n
1
2n(n+ 1) −2n+ 1 · P
i,j (4.3)
Also using he deg ees o eedom we can calcula e he Pea son scale pa ame e
φ
,
φ=Pn
iPn−i
j P
i,j
1
2n(n+ 1) −2n+ 1 (4.4)
Now, we will begin he algo i hm po ion o he Boo s ap Me hod wi h an i e a i e
loop whe e we c ea e a new pas Run-O T iangle by e-sampling he adjus ed Pe son
esiduals wi h eplacemen , hus ob aining a se o pseudo-da a.
Fo his i s s ep we need o compu e (4.2) o Ci,j, i.e.,
Ci,j = P adj
i,j ·pmi,j +mi,j
Consecu i ely, we p oduce he cumula i e o m o he pseudo-da a and use he
Chain Ladde me hod upon i , gene a ing he es ima e o he u u e cumula i e pay-
men s.
Then, we go back o he beginning o he i e a i e loop and c ea e a new pseudo-
da a wi h e-sampled adjus ed Pe son esiduals wi h eplacemen .
Fo each comple ed pseudo-da a we compu e he espec i e ese e o each occu -
ence yea and hus a o al ese e, as we did in he in he Thomas Mack me hod and
s o e all he in o ma ion, his will o m a p edic i e dis ibu ion. We can also calcula e
a iabili y measu es om he s o ed in o ma ion.
25
CHAPTER 5. CONCLUSION
i is 7,305 hund eds o eu os which ep esen s 3% and 4%, espec i ely. The highe
alue o he Boo s ap migh be due o he andomness o he esiduals and he a ious
se s pseudo-da a.
Conside ing he 95% o con idence le el in Table 5.1, we can assume om he
con idence in e al ha a cau ious alue o he ese e would be 204,336 hund eds o
eu os. Howe e , his alue can no be aken ligh ly because he e a e o he ac o s ha
migh in luence hese es ima ions, minding ha he ac ua y has o add ess he eali y
o he company as well.
The es ima ion o ese es is e y impo an o he ac ua ial equilib ium o he
non-li e sec ion o insu ance companies. In o de o be p uden , i is use ul o apply
s ochas ic me hods, ins ead o de e minis ic me hods, due o hei ad an ages, pa icu-
la ly, being able o de ine con idence in e als so we can choose he bes ese e alue
wi h high con idence.
F om he Thomas Mack me hod we can see ha hese ype o me hods a e also
build wi h igo ous s a is ical ounda ions, which he da a has o o e come in o de o
use es s o e alua e he adjus men s made o i .
Neglec ing he use o he bene i s o s ochas ic me hods migh lead o es ima e
a ese e ha could be good enough bu wi h low ce ain y, pu ing he insu ance
company in isk o sol ency.
Some imes o be e es ima e ou ese es he es ima ion me hods mus e lec he
aspec s o di e en insu ance ypes. Fo his we can use a ail ac o , ha should be
selec ed h ough he cu es o adjus men o he de elopmen coe icien s. In ou case,
his adjus men was no made due o he insigni icance o he es ima ed ail ac o .
Besides he ad an ages men ioned, hese me hods also ha e a low need o compu-
a ional powe which makes hem mo e p ac ical.
Finally, i is impo an o poin ou ha ex e nal ac o s, such as legal, economical
and social ha e o be aken in o accoun since hese a e no co e ed by he es ima ions
me hod and migh al e he u u e ese es.
32

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33
h2021iEs ima ion o Technical Rese es using S ochas ic Me hods B una Gonçal es