COLLECTIVE RISK MODEL IN NON-LIFE INSURANCE

Transcription

1 Ecoomic Horizos, May - Augus 203, Volume 5, Number 2, Faculy of Ecoomics, Uiversiy of Kragujevac UDC: 33 eissn www. ekfak.kg.ac.rs Review paper UDC: : ; doi: /ekohor30263D COLLECTIVE RISK MODEL IN NON-LIFE INSURANCE Zlaa Djuric* Faculy of Ecoomics, Uiversiy of Kragujevac, Kragujevac, Serbia The operaio of busiess isurace compaies, based o assumig risks of differe profiles, is accompaied by flucuaios i he busiess evirome. The complexiy of predicig a fiacial effec for claims i o-life isurace lies i he srucure of isurers liabiliies, whose amou cao be deermied a he ime of payme of he premium. By aalyzig he key isurace processes, risk heory focuses o modelig claims as he fiacial cosequeces of uforesee eves. I addiio, i provides he aswer as o how much of a premium o charge i order o avoid bakrupcy, which makes i a complex ad opical research area. The paper preses he mai resuls of he collecive risk model for he key busiess processes of olife isurace compaies: he claim umber process ad he claim amou process. I risk heory, hese are reaed as sochasic processes, which offers a wide rage of possibiliies for he modelig ad simulaio of specific busiess problems. Keywords: o-life isurace, risk heory, sochasic process, Poisso process, premium calculaio priciples JEL Classificaio: C3, C43, C46 INTRODUCTION Risk avoidace has geeraed he esablishme ad operaio of isurace compaies which provide heir clies wih opporuiies o disperse ad miimize heir losses. The isured rasfer heir risks o isurers who, by formig a large eough group of relaed risks, reduce he loss of each isured by chargig a appropriae premium. The basic source of all o-life isurers dilemmas lies i he fac ha he premiums are paid prior o he occurrece of ay adverse eves. I is herefore ecessary o assess * Correspodece o: Z. Djuric, Faculy of Ecoomics, Uiversiy of Kragujevac, Dj. Pucara 3, Kragujevac, Serbia; zdjuric@kg.ac.rs he likelihood of realizaio, as well as he moeary value of he loss ha mus be compesaed for. The heory of probabiliy ad saisics allows isurers o see uforuae eves as pheomea ha, because of cerai regulariies, ca be prediced ad modeled (Embrechs & Klüppelberg, 993). The applicaio of he risk heory i o-life isurace is a eve more powerful ool for aalyzig ad defiig very complex busiess risks. The accepace of a variey of risks has framed hree basic quesios ha o-life acuaries, above all ohers, mus focus heir aeio o i order o adequaely proec heir cusomers: How much of acceped risks ca be realized i a specific ime, or how may compesaio requess ca be expeced o he basis of he colleced isurace policies?

2 68 Ecoomic Horizos (203) 5(2), Wha amou of moey should be provided for he payme of claims received, i.e. wha is he average expeced amou of a claim? How much of a premium o charge o he isured i order o absorb he claim ad provide icome o isurace compaies? The applicaio of he risk heory i geeral isurace is accompaied by criicisms of is limied pracical imporace i he busiess world, so ha i has log bee igored ad heoreically ad mahemaically developed maily by Scadiavia scieiss. Today, however, i is a major research challege for may mahemaicias ad acuaries due o he broad framework ad he logical coex wihi which i is possible o simulae aural flucuaios prese i real busiess processes. Solvecy II, a ew updaed se of regulaory requiremes for isurace compaies operaig i he Europea Uio, requires he complee reame ad measureme of a risk-margi based o risk, which has promoed he applicaio of he risk heory. Apar from he radiioal mehods, here is ow a eed for a ew dyamic approach based o he sochasic cocep of he realizaio of adverse eves. Isurers are geerally ieresed i oal paymes ha may follow from he isurace porfolio. If he prese value of he oal poeial payou is see as he sum of idividual paymes, we are alkig abou he idividual risk model. The secod model, which observes he aggregae amou of claims arisig from all of he colleced policies, is kow as he collecive risk model. Alhough more rece, i has sigificaly ouperformed he older, idividual, model because of is applicabiliy. This paper aalyzes he modelig of he key processes i he operaio of isurace compaies: he claim umber process ad he claim amou process. The aim of his paper is o prese he advaages ad disadvaages of he collecive risk heory i he aalysis of his problem, poiig ou he possibiliy of heir applicaio ad direcios for furher developme. Therefore, he key hypohesis cosidered i he paper is: he risk heory, alhough o very applicable i pracical work, provides a broad framework for moiorig, aalyzig ad predicig a umber of high-risk siuaios ad provides guidace for miigaig ad overcomig he problems ha may arise. Sarig from he defied objec ad purpose, he paper will firs prese he geeral model of he risk heory i geeral isurace, ad he aalyze he applicaio of Poisso process ad is modificaio i he claim umber process. The iegraio of his processes wih he process of he oal amou of paid claims makes he research area very complex, bu predomia whe seig up he basic priciples of quaifyig he premium. COLLECTIVE RISK MODEL Mahemaical models i he heory of o-life isurace aalyze claims for damages o provide a aswer as o how much premium o charge i order o avoid bakrupcy. Claims icomig o a isurace compay ca be reaed as radom variables reflecig he collecio of he adverse oucomes realizaio of isured eves (claims) o a se of real umbers (he moeary payoff amou) or as mappig X: Ω R, where Ω is he se of elemeary eves ad R is he se of real umbers. The maer ad probabiliy of he occurrece of claims or moeary paymes represes he probabiliy disribuio of hese radom variables. Every radom variable may be associaed he disribuio fucio, F X, o, which does o describe he acual oucome of he radom variable X, bu raher ells us how he possible values of X are disribued. Fucio F x : R 0,, defied wih X [ ] ( ) FX x = P ω Ω X ω x = P X x, for x R, is he disribuio fucio of radom variable X ad represes a probabiliy ha radom variable X should ake values less ha or equal o x. I he coex of isurace, if a radom variable is he amou of he claim of a isured, he disribuio fucio is a probabiliy ha he oal amou of isured damage observed will be less ha or equal o he fixed amou x. For a coiuous radom variable, he disribuio fucio is F x = P X x = g x dx, x R X where g(x) is he probabiliy desiy fucio. The mos impora umerical characerisics of radom x

3 Z. Djuric, Collecive risk model i o-life isurace 69 variables are obaied by usig he expeced value ad variace. The expeced value or expecaio of a discree radom variable is, while + he coiuous radom variable E( X ) = xg( x) dx. The variace is ofe used as a idicaor of he homogeeiy of a populaio or a sample. For a discree radom variable, he variace is 2 2 Var X = σ = E X E X, so i is a measure of he deviaio of a radom variable from is expeced value. For a coiuous radom variable, ( ) σ = x E X g x dx. The cocep of a radom variable is idepede of ime. However, may busiess processes are o be aalyzed ad heir implemeaio moiored a radom imes, so i is ecessary ha he ime compoe should be icluded. Radom variable X, whose implemeaio is moiored i ime, is deoed by X or X(). If T R is a se of ime, he X is deermied for every a cerai family of radom variables, which defies he sochasic process. A sochasic process X = {X, R} ca be reaed as a fucio of wo variables ad defied as X: T ˣ Ω K, where K is he se of saes, a se coaiig all he values of he observed process. For a seleced ime T ad a elemeary eve ω Ω, he realizaio of he process is deoed by X (, ω). Therefore, if he ime is fixed, he he fucio ω X(,ω) is he radom variable describig he process of implemeaio i a fuure ime, whereas if he eve is fixed ω Ω, he he fucio X(,ω) describes he implemeaio of he process X over ime. This ime fucio is he realizaio or rajecory of he sochasic process. Moreover, if he se T is couable, here is a discree radom process or a series of radom variables, oherwise here is a coiuous process. For a process o model, i is ecessary ha realisic assumpios faihfully describig he basic characerisics of his problem be iroduced, as well as such ha ca mahemaically be formulaed while heir characerisics ad implicaios ca easily be proved. I he collecive risk model, oe sars from he followig hypoheses (Ramasubramaia, 2005, 2): E X = x p i i. The oal umber of claims, B, received a a ime, is a radom variable. Claims arrive a a isurace compay i imes {Ti}, valid for 0 T T 2..., ad hey are claims arrival imes; 2. Ay claim, arrived i ime Ti iduces he payme of damages X i, or a amou of he claim. Sequece {X i } is a sequece of oegaive, idepede ideically disribued radom variables (i.i.d. radom variables); 3. The claim size process {X i } ad he ime of mauriy {Ti} are idepede of each oher. The process size ad he claims umber, {X i } ad B, are also idepede. The wo mos impora processes accompayig he process of busiess isurers are he claims umber process ad he oal claim amou process. As boh processes are followed i ime, hey are sochasic processes. I addiio, he claims umber process, i.e. he umber of claims icurred, is defied: = { } B max i :T 0 i ad represes he umber of claims received a ime 0, while he process of he oal amou of claims paid is B Z = X X B = X i, 0. (2) As he deermiisic idex of he parial sum Z = X + X X is replaced wih a radom variable B(), he process Z = ( Z ( ), 0) is a radom process of parial sums, ofe referred o as he compouded or collecive process. MODELING THE NUMBER OF CLAIMS The Poisso process, iroduced i F. Ludberg (903) as a model for he claim umber process B : 0, where B() is a radom variable, has he ceral ad domia place i o-life isurace mahemaics, i he collecive risk heory i paricular. Accordig o he classical defiiio of he probabiliy heory, a ieger radom variable Y is said o have a ()

4 70 Ecoomic Horizos (203) 5(2), λ λ k Poisso disribuio if: P Y = k = e, for k! k { 0,, 2... } ad for λ > 0. The Poisso radom variable is used as a model for he umber of phoe calls per ui of ime, he umber of cars ad buses ha pass hrough a poi per ui of ime, he umber of users who accessed a web sie, he umber of radioacive paricles per ui of ime, ad so o. I has a very rare, bu very useful propery ha E(Y) = var(y) = λ. The Poisso process moiors he occurrece of a eve over ime ad he momes i which he eve occurred, so ha i has widely bee applied i he modelig of rare eves or eves for which here is o more ha oe possible realizaio i a shor period of ime. The Poisso process is a radom process, defied o a se of ime, as a family of radom variables { } B, T, where here is se T = [0, + ), if he followig is fulfilled (Ramasubramaia, 2005, 3): () B() is a o-egaive ieger radom variable which is rue B ( 0) 0, 0, which meas ha here is o claim a ime = 0; { } (2) B is a o-decreasig process, i.e. if B() is he ieger ( ): 0 0 s <, he B() B(s), where B() - B(s) deoes ( ( ) 0 <, he ) B() ( ( ) B(s), ) ( λ ) k λ P B + s B s = k = P B = k = e k! he umber of claims received i he ime ierval B() - B(s), for s <, is he umber of claims (s, ]; received i he ierval (s, ]. (3) B : 0 has idepede icremes such ha for 0 < < 2 <... < < umber of claims received i he disjoi iervals B( ), B( 2 ) - B( ),..., B( ) - B( -), for =, 2,... are idepede radom variables; (4) a probabiliy of he arrival of a cerai umber of claims a a ime ierval depeds oly o he legh of he ierval, so ha he claim umber process has saioary icremes, i.e. for 0 s < ad h > 0, idepede radom variables B() - B(s) ad B(+h) - B(s+h) have he same disribuio; (5) a probabiliy of he arrival of wo or more claims i a paricular ime ierval is egligibly low, i.e. P(B(h) 2) = o(h), ie. P(B(+h) B() 2) = o(h), where o(h) is a ifiiely small size o( h) wih he propery lim = 0 ; h 0 h (6) i a very shor ime, a probabiliy of he arrival of a reques is approximaely proporioal o he legh of he ierval, so here is a λ > 0 such ha P(B(h)=) = λh + o(h), whe 0. Number λ is he claim arrival rae. Alhough he Poisso process is o he mos realisic process for he claim umber process due o is may aracive ad applicable properies developed ad deeced over several decades, i is a referece poi i modelig. The limiaio of he sadard Poisso process ca be reduced ad he models expaded by various modificaios of he sadard Poisso process, aalyzed i deail by J. F. C. Kigma (993). Thus, for modelig he claims umber process, here are wo, much broader ad more realisic, oher processes ha appear: he reewal process ad he mixed Poisso process. For he formulaio ad mahemaical modelig of he claims umber process, we sar from he followig assumpios, which are boh aural ad ecessary (Mikowa, 200, 3): B() 0 The defiiio of he Poisso process implies ha, for each sochasic process, which icludes he claim umber process, { B( ): 0} for s 0, k = 0,, 2,... is valid: ( ) P B + s B s = k = λ k ( ) λ = P B = k = e k! or he claim umber process is a homogeeous Poisso process, wih he rae of he arrival of claim λ, where λ is a cosa. The proof of his resul ad differe approaches performig i ca be foud i he works of may auhors, such as N. L. Bowers e al (997), C. T. Dayki e al (994), S. Klugma e al (998). For he claim umber process, from he aspec of isurace, he ime bewee he arrivals of wo cosecuive claims is also impora. If he arrival ime (3)

5 Z. Djuric, Collecive risk model i o-life isurace 7 of he -h claim or he waiig ime of he -h claim is defied by:, =,2,.., T 0 = 0, (4) i ca be assiged a series of ime bewee wo successive claims A i, defied by A i = T i T i-. Aalogously o hese defiiios, i follows where { 0 } T = if : B = { } { } s : T > s = B s = 0 λ P A > s = P B s = = e 0 Iducively, i ca be see ha, for he Poisso process { B( ): 0} wih a growh rae λ, he radom variables A i are idepede radom variables, expoeially disribued, wih parameer λ, ie A i : Ɛ( λ), so ha E( A i ) =, i, λ > 0. λ As T = A + A A is he sum of radom variables wih expoeial disribuio, i meas ha he arrival ime of he -h claim T has a gamma disribuio, T : Г(,λ) (Rolski e al, 999). Oe of he key characerisics of Poisso processes { B( ): 0} is ha he ime bewee he arrivals of wo successive claims is a radom variable wih a expoeial disribuio wih rae λ. Aoher impora feaure of he process B : 0 is ha hese imes are idepede. These wo feaures provide us aoher way of geeralizig he Poisso process. Specifically, we ca assume ha oegaive, idepede radom variables A i wih he same disribuio ca have whaever, eiher a discree or a absoluely coiuous disribuio. This assumpio leads us o he reewal process (Asmusse, 2000), which provides greaer flexibiliy i choosig a ime schedule for A i. Ulike he Poisso process, where B() has a Poisso disribuio for each, i he reewal process, his propery does o apply, so he disribuio for B() is geerally o kow, ad he deermiaio of he probabiliy of he eve B()= is reduced o he deermiaio of he expecaio of he radom variable B() (Pajer & Willmo, 992). Also, as for he arrival ime of he -h claims T = Ai, i holds ha: s (5) (6) I geeral, i is difficul o deermie he disribuio for T bu we kow ha, if A i : Ɛ(λ) he T : Г(,λ) ad if A i : Poi(λ) he T : Poi(,λ). Sudies by may scieiss i he field of he reewal process (Klig & Goovaers, 993) have led o a powerful mahemaical heory he reewal heory, which allows you o very precisely deermie he expeced umber of requess E(B()) for a large. Accordig o he sric law of large umbers, if he expecaio of he ime of he arrivals of wo successive claims E(A i ) = λ - fially, he Also, accordig o he elemeary reewal heorem, he followig applies: The mos accurae iformaio abou he pedig arrival imes of claims is give i Blackwell s reewal heorem, accordig o which (7) (8) (9) (0) So, he expeced umber of reewals per ierval (, +h], for a sufficiely large, is proporioal o he legh of he ierval ad idepede of. The basic premise ha he average rae of claim occurrece is cosa is o realisic sice he claim arrival is ofe depede o he weaher. Lookig a he parameer λ as a fucio of ime, he model of a homogeeous Poisso process ca be exeded o a o-homogeeous Poisso process. I also sars wih a zero, has idepede icremes for which i is rue ha for 0 s <, icreme B() - B(s) is Poisso disribued wih he parameer re, ucio μ T s B = lim B = lim λ ( ) = λ E B ( ( ]) λ E B, + h h, λ = 0 λ y dy. Furhermo- y dy is a fucio of he mea value of a o-homogeeous Poisso process, for some o-egaive measurable fucio λ. If he fucio of he mea value is liear, i.e. µ() = λ, i is a homogeeous Poisso process; oherwise i is a s

6 72 Ecoomic Horizos (203) 5(2), o-homogeeous oe. By iroducig he iesiy fucio λ(), he arrival process ca be moiored ad modeled accordig o seasoal reds as well. If claims come from heerogeeous isured groups, he arrival claim rae varies from oe policy o aoher, so ha λ() ca be viewed as a radom variable Λ(), > 0. The se Λ, 0 is a sochasic process ad herefore, he process { B( ): 0} is a double sochasic Poisso process. Treaig he λ as a radom variable idepede of ime, his sochasic process B : 0 is a mixed Poisso process, which is a eve more powerful geeralizaio of geeral Poisso processes. The mixed Poisso process loses some properies of he Poisso process (icremes are muually depede, he disribuio for B() i geeral is o Poisso s), bu i provides may more choices ha he rajecories of he Poisso process ad he reewal process (Gradell, 997). MODELING THE TOTAL CLAIM AMOUNT PROCESS Aalyzig he claims process is exeded whe he cosideraio icludes o jus he umber of he claims received bu he size of he claims demads iduce, oo. The sum of idividual claims or he aggregae amou of he claims is a key problem, boh i pracice ad i heoreical discussio. I fac, as he umber of claims ad he amou of he claim are sochasic variables, here is a double sochasic model of he aggregae amou of claims. Depedig o he selecio of he claim umber process B, here are differe models for he oal claim amou process uil he mome of ime : B Z = X X N = X i, 0 () Oe of he mos popular ad useful models i o-life isurace mahemaics is Cramer-Ludberg s model (Cramer, 955), which combies he claim amou ad he arrival ime, wih he followig assumpios (Mikosch, 2009, 8): { } The claim umber process B = max i :T 0 i is a homogeeous Poisso process wih rae λ > 0, i which claims are realized i arrival imes 0 T T 2... ; The claim received a he ime T i iduces he payme of damage X i. Sequece {T i } is a sequece of oegaive, idepede radom variables wih he same disribuio fucio; Sequeces {X i } ad {T i } are muually idepede. If we cosider ha he discoued sum i.e. he prese value of he cumulaive amou of claims i he ime ierval [0, ]: B Vi Z = e X, 0 (2) 0 i where r > 0 is he ieres rae, i Cramer-Ludberg s model, he expeced amou ecessary for selig he claims received i he observed ime ierval is B( ) rt i r E e Xi = λ e E X (3) r Isurers are geerally ieresed i he order of he magiude of Z(), ad cosequely i he disribuio fucios for Z(). Sice deermiig he disribuio for Z() is a very complicaed problem, he soluio lies i a simulaio model ad i obaiig rough esimaes of he mea ad he variace of Z(). The expecaio of he oal amou of paid claims idicaes is average size. Assumig idepedece bewee X i ad B, if E(B()) ad E(X ) are fial, i ca easily be obaied ha: B E( Z ) = E E Xi B = = E B EX = E B E X ( ) (4) As i Cramer-Ludberg s model he process B() is a homogeeous Poisso process, he E(B()) = λ, where λ is he iesiy rae of a homogeeous Poisso process, so ha from (4) we obai: E(Z()) = λe(x ) (5) To have more complee iformaio abou he disribuio of Z(), we should combie iformaio abou he expecaio wih he variace Var(Z()), for which he followig is valid (Mikosch, 2004):

7 Z. Djuric, Collecive risk model i o-life isurace 73 Var(Z()) = E(B())Var(X ) + + Var(B())(E(X )) 2 (6) As i Cramer-Ludberg s model i holds ha E(B()) = Var(B()) = λ, we obai: Var(Z()) = λe(x 2 ) (7) Ye aoher impora model for he process {Z() : 0} was iroduced by Sparre-Aderse (Aderse, 957) ad is implicaios have bee sudied by may auhors (Sharif & Pajer, 995; Gees e al, 2003), for whom he process B : 0 is a reewal process. I he reewal model, however, he deermiaio of he esimaes of expecaios ad variaces is difficul ad does o give such cocree resuls. We have see ha, accordig o he sric law of large umbers, if he expecaio of he arrival imes of wo cosecuive claims E(Ai) = λ - E <, he ( B ) λ, whe. This meas ha: E(Z()) = λe(x )(+o()), (8) ad Var(Z()) = λ[var(x ) + Var(A ) λ 2 (E(X )) 2 ](+o()) (9) Based o hese resuls, we fid ha he expecaio ad variace asympoically grow almos liearly as a fucio of ime. This iformaio ca be very useful i he pracical deermiaio of premiums sufficie for he seleme of losses, he size of Z(). PREMIUM CALCULATION PRINCIPLES The amou of he moey he isured pays he isurer, as a compesaio for risk, is a premium. Risks ad a premium are closely relaed o each oher sice a premium amou is deermied by a average size of a risk, whose every chage mus be refleced i he amou of such a premium. Lookig a a premium as a moeary payme from he isurer, i he coex of he above processes, i is obvious ha a isurace compay will be operaig a a loss if a premium is less ha he expeced amou of paymes i.e. if p() < Z(). As we have obaied i previous argumes ha E(Z()) = λe(x )(+o()),, i is logical o deermie a premium so ha: p() = λe(x )(+ρ) (20) where ρ is a posiive cosa ad represes safey loadig or a charge for securiy. I he risk heory, here are priciples which all premiums p() should saisfy, kow as he premium priciples. To deermie a premium, as he mappig of ucerai fuure losses oo heir fiacial equivale, acuaries have developed a umber of mehods for he deermiaio of he premium priciples (Albers, 999; Dickso, 99; Ladsma e al, 200), he followig oes beig basic: The priciple of e premium is a basic priciple, accordig o which p() = E(Z()). I does o iclude safey loadig, because acuaries ofe assume ha here is pracically o risk if he isurer sells eough of ideically disribued ad idepede policies; The priciple of he expeced value is based o he previous oe, bu icorporaes proporioal safey loadig. Accordig o his priciple, p() = (+ρ) E(Z()) for some ρ > 0. This priciple is geerally used i life isurace. The applicaio of his priciple i o-life isurace is limied due o a high heerogeeiy of acceped risks; The priciple of variace is based o he assumpio ha p() = E(Z()) + αvar(z()), for some α > 0, i which a safey margi is proporioal o he variace of expeced losses; The priciple of sadard deviaio, also based o he e premium priciple, is ofe used i o-life isurace. Accordig o his priciple, he expeced value of a loss mus be covered by a premium icludig safey loadig, which is proporioal o he sadard deviaio of he expeced damage, i.e. ( ) p = E Z + α Var Z, for α > 0. Due o is lieariy whe i comes o proporioal chages i claims, his priciple is mosly used i propery ad casualy isurace.

8 74 Ecoomic Horizos (203) 5(2), CONCLUSION Isurace compaies are isiuios absorbig udesirable effecs of heir users risks. Due o he ifluece of poliical ad legal as well as social ad climaic facors, rapid chages i he busiess ad ecoomic evirome require a comprehesive ad dyamic risk reame, especially i o-life isurace. Therefore, H. Cramer said ha he goal of risk heory is o provide a mahemaical aalysis of he flucuaios i he isurace busiess ad o sugges various meas of proecio agais heir adverse effecs (Cramer, 930, 7). The oldes approach o his problem is he idividual-risk heory. I observes idividual isurace policies, wih differe characerisics ad risk profiles, so ha he overall risk of doig busiess is obaied via he summig of all he claims arisig from he eire porfolio of isurace policies. However, he claims arise radomly, so he risk process is a sochasic process. Thus, he collecive risk model, based o he applicaio of sochasic processes i isurace, has a very impora role i he developme of academic acuarial sciece. I his model, claims are reaed aggregaely, a he level of he porfolio as a whole. Alhough he risk process is cosidered as oe of he simpler forms of sochasic processes, here is sill much o do o have i applied. The mahemaical foudaio has applied some ecessary, however urealisic, assumpios i he model cosrucio ad developme for boh he claim umber process ad he oal amou of claims paid process. Despie heir broad sigificace, he mai disadvaages ad limiaios of heoreical cosideraios perai o he deermiaio of he disribuio fucio which realisically reflecs he saisics of isurers. The execued simulaios of he proposed model use some of he kow disribuio fucios, which ca almos ever represe isurers porfolio adequaely. Today, a large umber of papers focus o he deermiaio of he geeral disribuio fucios, which will icrease he correspodece of he obaied resuls wih a realiy (Cossee e al, 2002; Embrechs e al, 997; Kaas e al, 200). Moreover, a lo of work is focused o he cosrucio of a model which will iclude iflaio i deermiig he oal amou of he compesaio paid. I order o have i pracically applied, which is he direcio i which his heory is o furher develop, i is ecessary ha he fac ha claims are o paid a he same ime or immediaely afer he arrival of a reques o a isurace compay should be ake io accou. Also, special aeio should be paid o he coss accompayig he reame ad seleme of claims. The mai resuls of he collecive risk heory, which are preseed i he paper, are idicaive of a wide rage of he modificaios, modelig ad simulaios of eves ha may occur. The mai disadvaage of heoreical cosideraios, icludig his paper, is heir currely limied applicabiliy i he pracical busiess evirome. However, as he rage of he risk of o-life isurers i a icreasigly urbule busiess evirome is i a cosa icrease, real cosequeces ca o loger be prediced by usig oly busiess saisics. I is idispuable ha he collecive risk model represes a broad scieific field, egagig umerous scieiss producig growigly cocree resuls whe he covergece of heory ad cocree busiess problems are cocered. Hece, he combiaio of visualizaio ad sochasic acuarial experiece is a srog mechaism o solve a icreasigly complex isurers risk. REFERENCES Albers, W. (999). Sop-loss premiums uder depedece. Isurace: Mahemaics ad Ecoomics 24, Aderse, E. S. (957). O he collecive heory of risk i case of coagio bewee claims. Bullei of he Mahemaics ad is Applicaio, 2, Asmusse, S. (2000). Rui Probabiliies. Sigapore: World Scieific. Bowers, N. L., Gerber, H. U., Hickma, J. C., Joes, D. A., & Nesbi, C. J. (997). Acuarial Mahemaics. Schaumburg, Illiois: Sociey of Acuaries. Cossee, H., Gaillardez, P., Marceau, E., & Rihoux, J. (2002). O wo depede idividual risk models. Isurace: Mahemaics ad Ecoomics, 30, Cramer, H. (930). O he mahemaical heory of risk. Sockholm, Skadia Jubilee Volume.

9 Z. Djuric, Collecive risk model i o-life isurace 75 Cramer, H. (955). Collecive risk heory: a survey of he heory from he poi of view of he heory of sochasic process. 7h Jubilee Volume of Skadia Isurace Compay. Sockholm, 5 92, Dayki, C. D, Peikäie, T., & Pesoe, M. (994). Pracical Risk Theory for Acuaries. Lodo, UK: Chapma & Hall. Dickso, D. C. M. (99). The probabiliy of ulimae rui wih a variable premium loadig a special case. Scadiavia Acuarial Joural, Embrechs, P., & Klüppelberg, C. (993). Some Aspecs of Isurace Mahemaics Theory of Probabiliy ad is Applicaio. Theory ov Probabiliy ad Is Applicaio, 38, Embrechs, P., Kluppelberg, C., & Mikosch, T. (997). Modellig Exremal Eves for Isurace ad Fiace. New York, NY: Spriger. Gees, C., Marceau, E., & Mesfioui, M. (2003). Compoud Poisso approximaio for idividual models wih depede risks. Isurace: Mahemaics ad Ecoomics, 32, Gradell, J. (997). Mixed Poisso Processes. Lodo, UK: Chapma & Hall. Kaas, R., Goovaers, M., Dhaee, J., & Deui, M. (200). Moder Acuarial Risk Theory. Boso, USA: Kluwer Academic Publishers. Kigma, J. F. C. (993). Poisso Processes. Oxford: Claredo Press. Klig, B. M., & Goovaers, M. (993). A oe o compoud geeralized disribuios. Scadiavia Acuarial Joural,, Klugma, S., Pajer, H. H., & Willmo, G. E. (998). Loss Models: from Daa o Decisios. New York, NY: Joh Wiley. Ladsma, Z., & Sherris, M. (200). Risk measures ad isurace premium priciples. Isurace: Mahemaics ad Ecoomics, 29(), Ludberg, F. (932). Some supplemeary research o he collecive risk heory. Skadiavisk Akuarieidskrif, 5, Mikosch, T. (2004). No-Life Isurace Mahemaics: A Iroducio wih Sochasic Processes. Berli, Germay: Spriger. Mikowa, L. (200). Isurace Risk Theory. Lecure Noes. from Pajer, H. H., & Willmo, G. E. (992). Isurace Risk Models. Schaumburg, Illiois: Sociey of Acuaries. Ramasubramaia, S. (2005). Poisso process ad isurace: a iroducio. Prepared for a series of lecures give a a Refresher course i Applied Sochasic Processes, held a he Idia Saisical Isiue, New Delhi, from hp://www. mah.iisc.ere.i Rolski, T., Schmidli, H., Schmid, V., & Teugels, J. (999). Sochasic Processes for Isurace ad Fiace. New York, NY: Wiley ad Sos. Shari, A. H., & Pajer, H. H. (995). A improved recursio for he compoud geeralize Poisso disribuio. Mieiluge der Vereiigug Schweizerischer Versicherugsmahemaiker,, Received o 5 h July 203, afer revisio, acceped for publicaio o 26 h Augus 203 Zlaa Đurić works as a eachig assisa i he disciplies of Mahemaics i Ecoomics ad Fiacial ad Acuarial Mahemaics a he Faculy of Ecoomics of he Uiversiy of Kragujevac. Her key research ieres is he applicaio of he mahemaical apparaus o ecoomic problems, paricularly fiacial mahemaics ad isurace models.