Estimating IBNR Claims Reserves for General. Insurance Using Archimedean Copulas

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1 Applied Mathematical Scieces Vol o HIKARI Ltd Estimatig IBNR Claims Reserves for Geeral Isurace Usig Archimedea Copulas Patrick Weke ad Carolie Ratemo School of Mathematics Uiversity of Nairobi Nairobi Keya Copyright 203 Patrick Weke ad Carolie Ratemo. This is a ope access article distributed uder the Creative Commos Attributio Licese which permits urestricted use distributio ad reproductio i ay medium provided the origial work is properly cited. Abstract Claims reservig for geeral isurace busiess has developed sigificatly over the recet past. There has always bee a slight mystery i short-term isurace cotracts of how to go about reservig for claims which have ot yet come i ad are still i some sese of figmet of the future. Isurace claims variables are o-ormally distributed ad therefore a measure that will capture the depedece amog the variables better tha the usual correlatio is employed. Oe such method is the use of copulas. The object of this paper therefore is to use the Archimedea copula Clayto copula ad Frak copula to estimate outstadig icurred but ot reported (IBNR) claim reserves. A compariso of the estimates of outstadig claim reserves obtaied from differet Archimedea copulas is also preseted. Keywords: IBNR claim reserves Archimedea copula Clayto copula Frak copula. INTRODUCTION I geeral isurace a claim is a demad for paymet of damages that may be covered uder a policy ad a reserve is a estimate of the amout of moey set aside for the evetual paymet of a claim. Paymet of a claim is what cosummates the isurace cotract. A claim is icurred whe it happes regardless of whe i the future it is paid. Reserves are classified as liabilities o the compay s balace sheet as they represet future obligatios of a isurace compay. They are importat sice they are a measure of a compay s fiacial solvecy ad improper reservig ca therefore preset a false picture of a compay s fiacial coditio. A reported claim is oe that has already bee processed to the extet that a cetral record o it is held. However claims occur

2 224 P. Weke ad C. Ratemo almost every day but are usually ot reported the same day. This may be due to the ormal time lag i reportig claims difficulties i determiig the size of the claim ad so o. The oly certaity is that such claims will come i ad that there is a duty to make provisio for them. These claims ot yet kow to the isurer but for which a liability is believed to exist at the reservig date are referred to as Icurred but ot Reported (IBNR) claims. Past claims data which should be adequate ad accurate is used to costruct estimates for the future paymets ad it cosists of a triagle of icremetal claims grouped by time of origi (whe the claim or accidet was icurred) ad developmet time (time elapsed sice the accidet). The problem is thus to complete this ru-off triagle. The oly iheret ucertaity is described by the distributio of possible outcomes ad oe eeds to arrive at the best estimate of the reserve. May classical methods of dealig with the reserve problem have bee developed all of which are based o differet coefficiet calculatios ad deal with the classical developmet triagle (for example Chai ladder method Borhuetter- Fergusso ad separatio techique). I almost ay method aalyzig the upper triagle is based o well-kow techiques from statistics. However the essetial problem to be solved is the maagemet of the risk associated with the future (the lower triagle). Most methods estimate the lower triagle cell-by-cell ad do ot pay eough attetio to the structure describig the depedecies betwee these cells. Each cell ca be cosidered as a uivariate radom variable beig part of the multivariate radom variable describig the lower triagle. Hece the IBNR reserve must be cosidered as a (uivariate) radom variable beig the sum of the depedet compoets of the radom vector describig the lower triagle. Goovaerts et. al. (200) studied various IBNR evaluatio techiques ad foud out that estimatig the correlatios from the past data ad usig them for multivariate simulatios of the lower triagle is a dagerous techique because the isurer is especially iterested i the tail of the distributio fuctio ad that a multivariate simulatio techique will oly be possible if the whole depedecy structure of the lower triagle is kow. They observed that i practice situatios where oly the distributio fuctios of each cell ca be estimated with eough accuracy but where oly limited iformatio of the depedecy structure ca be obtaied (because of iadequate data) are ecoutered. Sice the true multivariate distributio fuctio of the lower triagle could ot be determied i most cases because the mutual depedecies are ot kow or difficult to cope with they cocluded that the oly coceivable solutio is to fid upper ad lower bouds for this sum of depedet radom variables which use as much as possible of the available iformatio. Uderstadig relatioships amog multivariate outcomes is a basic problem i statistical sciece. Multivariate relatioship is limited by the basic setup that requires the aalyst to idetify oe dimesio of the outcome as the primary measure of iterest (the depedet variable) ad other dimesios as supportig

3 Estimatig IBNR claims reserves 225 variables (the idepedet variables). I isurace this relatioship is ot of primary iterest as we mostly deal with joit distributio fuctios where we eed to uderstad the distributio of several variables iteractig simultaeously ad ot i isolatio of oe aother. I the IBNR problem it is ecessary to cosider the claim size ad developmet time as the two variables iteractig simultaeously. The ormal distributio has log domiated the study of multivariate distributios. More recet texts o multivariate aalysis such as (Krzaowski 988) have begu to recogize the eed for examiig alteratives to the ormal distributio setup. This is certaily true for actuarial sciece applicatios such as log tailed claims variables (Hogg ad Klugma 984) where the ormal distributio does ot provide a adequate approximatio to may datasets. There has bee a tedecy to use correlatio as if it was a all-purpose depedece measure but it is ofte misused ad applied to problems for which it is ot suitable. However empirical research i fiace ad isurace show that the distributios are seldom i the class of spherical ad elliptical distributios (Embrechts McNeil ad Strauma 999). Therefore correlatio is a rather imperfect measure of depedecy i may circumstaces ad thus the copula comes i hady as a alterative measure of depedecy. A costructio of multivariate distributio that does ot suffer from these drawbacks is based o the copula fuctio. Copulas thus are extremely helpful because they give a atural way of allowig for depedecy that is free from the drawbacks of correlatio. They are ivariat to reasoable trasformatios of radom variables ad/or their distributio fuctios. The otio of a fuctio characterizig the depedece structure betwee several radom variables comes from the work of Hoeffdig i the early forties. Other authors idepedetly itroduced related otios afterwards but it was Sklar (959) that defied copula as a fuctio that liks the multivariate distributio to fuctios of the uivariate margial distributios. Literature o the statistical properties ad applicatios of copulas i fiace ad isurace has bee developig rapidly i the recet past. Wag (997) proposes copula for modellig aggregate loss distributios of correlated isurace policies. Frees ad Valdez (998) ad Klugma ad Parsa (999) use copula to model bivariate isurace claim data. More recetly Pettere ad Kollo (2006) used the Archimedea class of copulas to model claim size of a Latvia isurace compay ad later used the bivariate Clayto copula to estimate IBNR reserves. This paper therefore focuses o how differet types of copulas ca be used to fit IBNR claims data ad the compares the estimates of outstadig claim reserves obtaied from differet Archimedea copulas for ay statistical sigificace. I sectio 2 we outlie the methodology that will be used. Discussio of results obtaied as well as estimatio ad compariso of IBNR reserves estimates will be i sectio 3. Coclusios based o the results will be i the fial sectio.

4 226 P. Weke ad C. Ratemo Suppose that a distributio fuctio F 2. METHODOLOGY dimesioal radom vector X = ( X X 2... X ) has ( x x... x ) = Pr( X x X x... X x ) oe ca decompose F ito uivariate margials of distributio fuctio called copula. X i i = 2... ad aother Defiitio: Copula Iformally a copula C is a joit distributio fuctio defied o the uit square with uiform margials (or margis). Formally defie I as the uit iterval I = [ 0] F as ay oe-dimesioal distributio fuctio ad C a distributio fuctio of the uiform distributio 0 the [ ] [ F( x) ] R F ( x) = C x. This ca be carried over to two ad higher dimesioal distributio fuctios F (with F F2... F deotig the oe-dimesioal margial distributio fuctios). That is where ( ) x ( x x... x ) C[ F ( x ) F ( x )... F ( x )] F = (2.) x... 2 x R Here C deotes the distributio fuctio o [ 0 ] with uiform margis. That is C ( u u... u ) = Pr( U u U u... U u ) where U is a (ex-ate) uiform radom variable whereas u is the correspodig (ex-post) realizatio. Thus equatio (2.) breaks up a multivariate distributio ito i. The oe-dimesioal margial distributio fuctios F F2... F ii. The depedece structure (Copula)..

5 Estimatig IBNR claims reserves 227 Propositio: Properties of copulas A copula is ay fuctio : [ 0] [ 0]. ( u u2... u ) i [ ] C ( u u... 2 ) 0. u = C satisfyig the properties 0 if atleast oe compoet u i is zero the 2. For u [ 0 ] C(... u... ) = u i (2... ) i i i 3. [ u u ] [ u u ]... [ u u ] dimesioal rectagles i [ 0 ] 2... ( ) i= 2 i i+ i i ( u u... u ) 0 C. 2 i i 2 i Theorem: Sklar (959) Let F x ) F ( x )... F ( x ) be a give cotiuous margial distributio fuctios. ( 2 2 x x 2 x R The for every ( ). If F ( x x... ) F 2 x is a joit distributio with fuctio with margis x ) F ( x )... F ( x ) the there exists a uique copula C such that ( 2 2 ( x x... x ) C( F ( x ) F ( x )... F ( x )) F = (2.2) 2. Coversely if C is ay copula the fuctio F defied i equatio (2.2) is a joit distributio fuctio with margis F x ) F ( x )... F ( x ). ( 2 2 For o-cotiuous F x ) F ( x )... F ( x ) C is uiquely defied o ( 2 2 RageF x ) RageF ( x )... RageF ( x ). ( 2 2 If we cosider the radom variables X ad Y such that F ( x y) = Pr( X x Y y) F ( x) = Pr( X x) F ( x) = Pr( Y y) 2 the we call C the copula of X ad Y. Sklar s theorem through the statemet ( x y) C( F ( x) F ( )) F = 2 y

6 228 P. Weke ad C. Ratemo splits the joit probability distributio ito the margials ad a copula so that the latter oly represets the depedece betwee X ad Y. From this modellig separatio it follows that also i the estimatio or calibratio phase oe ca idetify the margials ad at a secod stage specify the copula fuctio. 2.. Bivariate Archimedea copula cocepts If we cosider the radom variables X ad Y such that F the ( x y) ( x y) = Pr( X x Y y) F ( x) = Pr( X x) F ( x) = Pr( Y y) 2 F ca be writte i terms of a copula ad it s margial distributios as F ( x y) = C( F ( x) F ( y) ) = C( u v) = Pr( U u V v) 2 (2.3) the we call C the copula of X ad Y. If we represet the copula i the followig form C where : [ 0] [ 0 ] ( u v) ψ ( ψ ( u) ψ ( v) ) = (2.4) ψ is a cotiuous strictly decreasig ad covex fuctio such that ψ ( ) = 0 ad ψ (0) = ad is such that the fuctio ψ has a iverse [ 0 ] [ 0 ] ψ : with the same properties like ψ except that ψ (0) = ad ψ ( ) = 0 the the copula is referred to as Archimedea. A list of Archimedea copulas ad their properties ca be foud i Nelse (2007). I this paper we have examied three copulas Clayto Frak ad Gumbel copulas ad their properties are show i table 2. below. Table 2.: Archimedea copulas ad associated properties Clayto Frak Gumbel Geerator t e t ( l t) ψ (t) l e Iverse ( + s ) l [ + e ( ) ] s e exp( s ) geerator ψ (s) Bivariate ( + ) v u v ( e )( e ) copula + u + v ( u v) e C ψ u ( )

7 Estimatig IBNR claims reserves 229 Table 2.: Archimedea copulas ad associated properties (cotiued) Parameter \ 0 Kedall s τ + 2 Spearma s Complicated form ρ [ ) { } [ ) \ { 0} [ ) 4 2 D ( ) D ( ( D ( )) ( )) 2 No closed form x k k t where Dk ( x) = k dt is the Debye fuctio. t x e 0 We will use these Archimedea copulas for the estimatio of IBNR reserves i sectio IMPLEMENTATION AND RESULTS Appropriate distributio models will first be fitted to the radom variables of iterest ad the best fit determied by the Kolmogorov-Smirov test. A appropriate Archimedea copula model for the data will the be foud usig the method due to Geest ad Rivest (200) ad goodess of fit of the model determied both graphically ad aalytically usig QQ-plots ad Kolmogorov- Smirov test statistic respectively. Average claim size i each developmet time uit will the be foud by simulatio usig the obtaied copulas. The umber of claims occurrig i each time uit will also be foud ad a distributio fitted. This distributio will the be used to estimate the average umber of claims i each time uit. Fially reserves will be obtaied by multiplyig average claim size i each developmet time uit to the average umber of claims reported i each time uit ad to the umber of time uits. Compariso of the estimated reserves will the be doe usig the Ma-Whitey U test statistic. I this paper we have used published data from Taylor ad Ashe (983) which was also used by Verrall (99 998) Mack (993) ad Reshaw ( ). The data is i icremetal form. The developmet time has legth oe year ad used claims are for a period of te years. A claim will be characterized by two radom variables developmet time ad claim size. The aalysis ad fittig of copulas will be doe usig S-plus Fimetrics module ad calculatios doe usig excel.

8 230 P. Weke ad C. Ratemo 3.. Fittig Distributios to Claim Size ad Developmet Time To have a idea about the desirable shape of the families of distributios that ca be used we calculated descriptive statistics for both radom variables ad are show as i table 3. below. Table 3.: Descriptive Statistics for Developmet Time ad Claim Size Statistic Developmet time Claim size Mea Media Mode Std. Deviatio Sample Variace Skewess Kurtosis Rage Miimum Maximum Sample Size From the descriptive statistics the media ad the mode values for both variables are ot close ad the rage is also big. As a result the variables will ot be exactly ormally distributed. They exhibit a slightly loger tail to the right tha the ormal distributio as ca be see from the skewess values ad also slightly more peaked as idicated by their respective kurtosis values. The best fittig distributios for the radom variables see i the graphs below was obtaied by comparig the cumulative distributio fuctios (CDFs) of the radom variables with the CDFs of a hypothesized distributio. For developmet time the best fittig distributio was obtaied by a logormal distributio with μ =. 6 ad σ = 0.7 with a Kolmogorov test statistic value of 0.3 correspodig to a p value of which allowed us ot to reject the hypothesized distributio. Claim size was also best fit by a logormal distributio with μ = 3. 6 ad σ = 0.66 with a Kolmogorov test statistic value of 0. with a correspodig p value of

Estimatig IBNR claims reserves 23 Figure 3.

Size S Relatioship betwee the two variables was

9 Estimatig IBNR claims reserves 23 Figure 3.: Empirical ad Hypothesized CDFs for Developmet Time Figure 3.2: Empirical ad Hypothesized CDFs for Claim Size S Relatioship betwee the two variables was also determied usig Kedall s tau t ad liear correlatio coefficiet. Kedall s tau value betwee the radom variables was compared to a liear correlatio coefficiet of This implies a slight egative relatioship. Figure 3.3 beloww shows this relatioship betwee the variables.

232 P. Weke ad C. Ratemo Figure 3.3: Locatio of the t Claim Sizes i Developmet Time T We fitted the copulas to our data usig a o-parametrii the thesis by de Matteis (200).

10 232 P. Weke ad C. Ratemo Figure 3.3: Locatio of the t Claim Sizes i Developmet Time T We fitted the copulas to our data usig a o-parametrii the thesis by de Matteis (200). The copula parameter estimates thatt were obtaied usig Kedall s tau t obtaied above were ad.870 for the Clayto (Family ) Gumbel (Family 4) ad Frak (Family 5) copulas respectively of which oly the Clayto ss ad Frak s parameter estimates were valid. To see the fit of these copulas graphically the t fittig procedure due to Geest ad Rivest (993) outlied coditioal distributio of Y give X agaist stadard uiform quatiles are plotted.

Estimatig IBNR claims reserves 233 Figure 3.

From the QQ-plots above the best fittig copula for

The choice of the two is justified by the p value of

poits were simulated 0 times from the Clayto ad Frakk

11 Estimatig IBNR claims reserves 233 Figure 3.4: Graphs for the Approximatio of the Three Copulas From the QQ-plots above the best fittig copula for the data iss the Clayto followed by the Frak copula. The choice of the two is justified by the p value of the Kolmogorov test statistic s which was for the Clayto C copula ad for the Frak copula Estimatig IBNR 200 poits were simulated 0 times from the Clayto ad Frakk copulas ad average claim sizes forr each developmet time obtaied. These averages are show i table 3.2.

12 234 P. Weke ad C. Ratemo Table 3.2: Simulated Average Claim Sizes for each Developmet time Developmet Time Clayto Frak The umbers of claims happeig i each year of origi were also examied ad a distributio fitted as show below. Table 3.3: Statistics of Number of claims happeig i a year Mea Std. Error of Mea Media Mode Std. Deviatio Sample Variace Skewess Kurtosis Rage Miimum Maximum Sample Size 0.00 The best fittig distributio for the umber of claims happeig i a year was obtaied by the logormal distributio with μ = ad σ =. 0 with a Kolmogorov test statistic value of with a correspodig p value of Average umber of claims happeig i each year of origi is the foud by multiplyig the value of the logormal desity fuctio for the umber of claims happeig i each year of origi by the average umber of claims happeig i a year ad also the average plus 2 ad 3 stadard deviatios which will be represeted by Dev Dev Dev 2 ad Dev 3 respectively ad by the legth of the iterval (i our case year). Results are show i table 3.4 below.

13 Estimatig IBNR claims reserves 235 Table 3.4: Average umber of claims for the Four Variatios Year of Logormal PDF*Dev PDF*Dev PDF*Dev 2 PDF*Dev 3 origi PDF IBNR reserves are the calculated by multiplyig the simulated average claim size i each developmet time by the average umber of claims i each developmet time uit ad to the umber of time uits for all the four variatios. The results are show i table 3.5 below. Table 3.5: Estimated IBNR Reserves usig the Clayto Copula Year of PDF*Dev PDF*Dev PDF*Dev 2 PDF*Dev 3 origi Total Fially we compare the four total estimated IBNR reserves from both copulas usig the Ma-Whitey U statistic. The test statistic value was correspodig to a p value of implyig that the estimates obtaied from the two copulas are ot sigificatly differet.

14 236 P. Weke ad C. Ratemo 4. CONCLUSION I this paper we have cosidered two variables developmet time ad claim size which are both logormally distributed. A critical aalysis of their bivariate distributio revealed that oly two copula the Clayto ad Frak out of the three Archimedea copulas examied could be used to study the bivariate data. It is observed that estimates of outstadig IBNR claim reserves from the Clayto copula were slightly lower tha the estimates from the Frak copula ad this could be attributed to the fact that the Clayto copula is a better fit compared to the Frak copula. However the Ma-Whitey U statistic test showed that the estimates obtaied from the two copulas are ot sigificatly differet. Therefore from these results we coclude that if oly oe copula which statistically fit a give set of multivariate data is required the ay of the two copulas ca be picked for estimatio sice each oe will statistically give the same results. REFERENCES [] R. L. Borhuetter ad R. E. Ferguso The Actuary ad IBNR Proceedigs of the Casualty Actuarial Society [2] U. Cherubii E. Luciao ad W. Vecchiato Copula Methods i Fiace Wiley Chichester [3] R. De Matteis Fittig Copulas to Data Diploma Thesis Istitute of Mathematics Uiversity of Zurich 200. [4] P. Embrechts A. McNeil ad D. Strauma Correlatio ad Depedece i Risk Maagemet Properties ad Pitfalls Cambridge Uiversity Press 200. [5] P. Eglad ad R. Verrall Stadard Errors of Predictio i Claims Reservig: A Compariso of Methods Geeral Isurace Covetio ad ASTIN Colloquium Glasgow Scotlad 998. [6] E. W. Frees ad E. A. Valdez Uderstadig Relatioships Usig Copulas North America Actuarial Joural 2 (988) 25. [7] C. Geest ad L. Rivest (993). Statistical Iferece Procedures for Bivariate Archimedea Copulas. Joural of the America Statistical Associatio 88 (993) [8] M. J. Goovaerts J. L. M. Dhaee V. E. Borre ad R. Redat Some remarks o IBNR evaluatio techiques Belgia Actuarial Bulleti [9] C. Hipp Depedece Cocepts i Fiace ad Isurace: Copulas Karlsruhe Uiversity [0] S. A. Klugma H. Pajer G. Veter ad G. Willmot Loss Models: From Data to Decisios Upublished Moograph 984.

15 Estimatig IBNR claims reserves 237 [] S. A. Klugma ad R. Parsa Fittig bivariate loss distributios with Copulas. Isurace Mathematics ad Ecoomics 24 (999) [2] W. J. Krzaowski Priciples of Multivariate Aalysis: A User s Perspective Oxford Uiversity Press Oxford 988. [3] T. Mack Distributio free calculatio of the stadard error of chai ladder reserve estimates ASTIN Bulleti 23 (993) [4] R. B. Nelse Depedece Modelig with Archimedea Copulas Departmet of Mathematical Scieces Lewis ad Clark College [5] R. B. Nelse A Itroductio to Copulas. Spriger Series i Statistics [6] G. Pettere ad T. Kollo Modellig Claim Size i Time via Copulas. Proceedigs of the 28th Iteratioal Cogress of Actuaries ICA [7] G. Pettere Modellig Icurred But Not Reported Claim Reserve Usig Copula Proceedigs of the 28th Iteratioal Cogress of Actuaries ICA [8] A. E. Reshaw Chai Ladder ad Iteractive Modellig (Claims Reservig ad GLIM) Joural of the Istitute of Actuaries 6 (989) [9] A. E. Reshaw O the secod momet properties ad the implemetatio of certai GLIM based stochastic claims reservig models Actuarial Research Paper No. 65 Departmet of Actuarial Sciece ad Statistics City Uiversity Lodo 994. [20] G. C. Taylor ad F. R. Ashe Secod Momets of Estimates of Outstadig Claims Joural of Ecoometrics 23 (983) [2] R. J. Verrall O the Estimatio of Reserves from Logliear Models Isurace Mathematics ad Ecoomics 0 (99a) [22] R. J. Verrall (99b). Chai Ladder ad Maximum Likelihood Joural of the Istitute of Actuaries 8 (99b) Received: November 202

Statistics for Economics & Business

Statistics for Economics & Business Statistics for Ecoomics & Busiess Cofidece Iterval Estimatio Learig Objectives I this chapter, you lear: To costruct ad iterpret cofidece iterval estimates for the mea ad the proportio How to determie