Estimation of Technical Reserves using Stochastic Methods

MEGI Mes ado em Es a ís ica e Ges ão da In o mação Mas e P og am in S a is ics and In o ma ion Managemen NOVA In o ma ion Managemen School Ins i u o Supe io de Es a ís ica e Ges ão da In o mação Uni e sidade No a de Lisboa 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 Managemen i 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 Managemen i Es ima ion o Technical Rese es using S ochas ic Me hods Copy igh © B una Raquel Sebas ião Gonçal es, In o ma ion Managemen School, NOVA Uni e si y Lisbon. The In o ma ion Managemen School and he NOVA Uni e si y Lisbon ha e he igh , pe pe ual and wi hou geog aphical bounda ies, o ile and publish his disse a ion h ough p in ed copies ep oduced on pape o on digi al o m, o by any o he means known o ha may be in en ed, and o dissemina e h ough scien i ic eposi o ies and admi i s copying and dis ibu ion o non-comme cial, educa ional o esea ch pu poses, as long as c edi is gi en o he au ho and edi o . 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 Bibliog aphy Au o idade de Supe isão de Segu os e Fundos de Pensão. (n.d.). h ps://www.as .com.p . Accessed: 2019-06-25. (Ci . on p. 4). England, P. D., & Ve all, R. J. (2002). S ochas ic claims ese ing in gene al insu ance. B i ish Ac ua ial Jou nal,8(3), 443–518. (Ci . on pp. 23, 24). Gesmann, M., Mu phy, D., Zhang, Y. (, Ca a o, A., Wu h ich, M., Concina, F., & Dal Mo o, E. (2021). Chainladde : S a is ical me hods and models o claims ese ing in gene al insu ance. R package e sion 0.2.14. Re ie ed om h ps://CRAN.R- p ojec .o g/package=ChainLadde . (Ci . on pp. 17, 26) Lowe, J. (1994). A P ac ical Guide To Measu ing Rese e Va iabili y Using : Boo s ap- ping , Ope a ional Time And A Dis ibu ion-F ee App oach. Gene al Insu ance Con en ion. (Ci . on p. 24). Mack, T. (1993). Dis ibu ion- ee Calcula ion o he S anda d E o o Chain Ladde Rese e Es ima es. ASTIN Bulle in,23(2), 213–225. doi:10.2143/as .23.2.200509 2. (Ci . on pp. 9, 11, 12) S aub, E., & G ubbs, D. (1998). The acul y and ins i u e o ac ua ies claims ese ing manual. olume 1 and 2. ASTIN Bulle in,28(2). doi:10.1017/S05150361000124 72. (Ci . on p. 1) Taylo , G. (2012). Loss ese ing: An ac ua ial pe spec i e. Sp inge Science & Business Media. (Ci . on pp. 3, 4, 23). Thisdocumen wasc ea edusing he(pd /Xe/Lua)L A T EXp ocesso ,basedon heNOVA hesis empla e,de elopeda heDep.In o má icao FCT-NOVAbyJoãoM.Lou enço.Lou enço,2021 Lou enço,J.M.(2021).TheNOVA hesisTempla eUse ’sManual.NOVAUni e si yLisbon.Re ie ed omh ps://gi hub.com/joaomlou enco/no a hesis/ aw/mas e / empla e.pd .(Ci .onp.33) 33 h2021iEs ima ion o Technical Rese es using S ochas ic Me hods B una Gonçal es