MODELING CATASTROPHES AND THEIR IMPACT ON INSURANCE PORTFOLIOS

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1 MODELING CATASTROPHES AND THEIR IMPACT ON INSURANCE PORTFOLIOS Hélèe Cossette,* Thierry Duchese, ad Étiee Marceau ABSTRACT The authors propose a geeral idividual catastrophe risk model that allows damage ratios to be radom fuctios of the catastrophe itesity. They derive some distributioal properties of the isured risks ad of the aggregate catastrophic loss uder this model. Through the model ad rui probability calculatios, they formally illustrate the well-kow fact that the catastrophe risk caot be diversified through premium collectio aloe, as is the case with the usual day-to-day risk, eve for a arbitrary large portfolio. They also derive some risk orderigs betwee differet catastrophe portfolios ad show that the risk level of a realistic portfolio falls betwee that of a portfolio of comootoic risks ad that of a portfolio of idepedet risks. Fially, the authors illustrate their fidigs with a umerical example ispired from earthquake isurace. * Hélèe Cossette, Ph.D., is a Associate Professor, École d Actuariat, Pavillo Alexadre-Vacho, Uiversité Laval, Québec, QC G1K 7P4, Caada, hcossett@act.ulaval.ca. Thierry Duchese, A.S.A., Ph.D. is a Assistat Professor, Départemet de mathématiques et Statistique, Uiversité Laval, Pavillo Alexadre-Vacho, Québec, QC G1K 7P4, Caada, duchese@ mat.ulaval.ca. Étiee Marceau, A.S.A., Ph.D., is a Associate Professor, École d Actuariat, Pavillo Alexadre-Vacho, Uiversité Laval, Québec, QC G1K 7P4, Caada, emarceau@act.ulaval.ca. 1. INTRODUCTION Earthquakes, droughts, floods, hurricaes, witer storms, ad torado outbreaks are amog the atural catastrophes that ca produce large amouts of losses. For istace, the Natioal Climatic Data Ceter (NCDC 2001) reported that the Uited States sustaied 49 weather-related disasters from i which overall damages/costs reached or exceeded $1 billio. Of these 49 disasters, 42 occurred durig , with total damages/costs exceedig $185 billio. A importat proportio of these losses are isured losses, ad this tedecy should persist as there are more items at risk i the catastropheproe areas, a larger proportio of these risks are gettig isured, ad the value of the items isured icreases. I health isurace, the rise of health care-related costs ad of the desity of populatios i urba areas icrease the potetial for a costly epidemic. Ufortuately, the evaluatio of the probability distributio of the losses followig from such disasters ca be quite difficult, ad simple actuarial models ad methods are usually ot appropriate for such calculatios. Oe key elemet missig i the more traditioal models is the itrisic depedecy betwee the risks exposed. For example, oe hurricae will cause several correlated claims at the same time, ad risks that are geographically close to each other are likely to produce highly correlated claim amouts. I this paper, we aim to fulfill several objectives. Our first goal is to propose idividual catastrophe risk models that will be geeral ad realistic, yet tractable. We do so by geeralizig the classical idividual risk model through a mathematical formalizatio of the catastrophe computer simulatio models that ca be foud i the actuarial literature. We proceed i three steps: we (i) add catastrophic loss radom variables to the idividual risk model, (ii) add depedece betwee the catastrophic claim amouts by makig them determiistic fuctios of the catastrophe itesity, ad (iii) make the model more realistic by lettig the catastrophic loss amouts be radom fuctios of the catastrophe itesity. Our secod objective is to examie the behavior of the aggregate catastrophic loss for a portfo- 1

2 2 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 7, NUMBER 4 lio uder these three idividual catastrophe models, ad to model the impact of the presece of catastrophes o a isurace portfolio. We tackle this problem by calculatig rui probabilities ad by derivig stochastic orderigs uder the three catastrophe models. Iterestig cosequeces of these calculatios are that the usual methods of evaluatio of the distributio of the aggregate claim amouts used i risk theory still apply i the cotext of catastrophe models, ad we are able to illustrate i a formal maer the wellkow fact that the catastrophic risk caot be diversified through premium collectio aloe, as is the case with a portfolio of idepedet risks. The paper is orgaized as follows. Sectio 2 presets a review of the curret developmets i the research areas related to catastrophe isurace. The three sectios that follow Sectio 2 cotai the mai cotributios of the paper. We costruct a geeral ad realistic idividual catastrophe risk model i three steps i Sectio 3. Rui probability calculatios ad results o the odiversifiability of the catastrophe risk are obtaied i Sectio 4. Sectio 5 compares the risk portfolios from Sectio 3 through risk measures ad stochastic orderig. I Sectio 6, a umerical example based o earthquake isurace is give to illustrate the models, methods, ad results preseted i Sectios 3 to 5. We coclude with a discussio i Sectio REVIEW OF CURRENT DEVELOPMENTS IN THE AREA Sice the 1990s, various papers have cosidered depedecy betwee risks. Amog them, Dhaee ad Goovaerts (1996) ad Bäuerle ad Müller (1998) study differet orderigs betwee two portfolios of depedet risks i the idividual risk model. Based o the cocept of comootoicity (Dhaee et al. 2002a,b), Wag ad Dhaee (1998) fid the riskiest stop-loss premiums. Deuit et al. (1999) costruct stochastic bouds o sums of depedet risks. Wag (1998) (see also the discussio by Meyers 1999) suggests a set of tools for modelig ad combiig correlated risk portfolios. Most methods proposed by Wag (1998) are also applicable to the collective risk model. Aother approach to the impact of catastrophes o isurace busiess is through extreme value theory. See Beirlat ad Teugels (1992), Beirlat et al. (1996), Embrechts et al. (1997), McNeil (1997), Rootzé ad Tajvidi (1997), Reiss ad Thomas (1997), ad Resick (1997). Their approach is maily cocered with the effect of possible catastrophes o the probability distributio of the aggregate loss radom variable. Fially, a third approach cosiders modelig the loss portfolio at the idividual risk level. This icludes stochastic modelig, such as Brilliger (1993), who examied the developmet of appropriate premium rates i the case of a atural disaster, such as a earthquake, through temporal ad spatial stochastic modelig of the frequecy ad itesity of earthquakes i a give regio. Modelig is also doe by directly simulatig the effect of catastrophes o a portfolio with computer-based models. As a matter of fact, earthquake ad hurricae simulatio models have bee the tools of choice for actuaries who wated to adapt their ratemakig methods to take the risk of catastrophes ito accout. Because of the importace of losses due to atural catastrophes i the last decade (particularly hurricaes), there has already bee a good amout of work doe i the actuarial literature, ad the Forums of the Casualty Actuarial Society o ratemakig have produced some very iterestig proposals (Clark 1986, Burger et al. 1996, Walters ad Mori 1996, ad Cherick 1998). Basically, these papers propose methods to assess the effect of catastrophes o a isurace portfolio through simulatio methods based o recet meteorological or geological models. These methods represet a great improvemet over the more traditioal methods based o short-term loss data, as they are a better reflectio of the mechaism that causes the claims, ad they make better use of the recet meteorological, demographic, ad egieerig developmets ad data. Moreover, their implemetatio is relatively straightforward with the easy access to computatioal power available today. I this paper, we propose a mathematical formalizatio of the computer-based models. We propose models that are suitable for several types of catastrophe isurace (atural catastrophes, epidemics, etc.) ad that ca be iterpreted uder both a macro- ad a micro-perspective. (By macro-perspective we refer to approaches used i pricig catastrophe bods, i.e., that we have iformatio o the umber of catastrophes ad the total amout of losses by catastrophe, while the

3 MODELING CATASTROPHES AND THEIR IMPACT ON INSURANCE PORTFOLIOS 3 micro-perspective refers to modelig risks at the idividual loss level.) The methods proposed i this paper ca, thus, be viewed as a compromise betwee the stochastic approach of Brilliger (1993) ad the computer-based approach from the Forums of the Casualty Actuarial Society. Whereas the latter two approaches use catastrophe models especially for premium calculatios, we further use them to make formal statemets o risk diversifiability ad risk orderig. 3. MODELING CATASTROPHE RISKS I this sectio, we cosider a group of isurace cotracts i a specific geographic area. We assume that these cotracts are exposed to oe specific catastrophe risk, such as hurricaes, earthquakes, or floods. Our approach cosists of modelig the risks idividually; this ca be related to the so-called idividual risk model, which is preseted i Pajer ad Willmot (1992), Klugma et al. (1998), or Rolski et al. (1999), for example. We defie the models over a short period of time (say, oe, three, six, or twelve moths). We first give a geeral formulatio of the portfolio of isured risks, the we cosider three differet models for the fiacial losses caused by catastrophes. While the geeral portfolio represetatio ad the first two models for the catastrophic losses are ot ew, the third model for catastrophic losses is; we preset all three models to better illustrate the fiacial impact of the risk of catastrophes later o. 3.1 Buildig a Portfolio of Risks The total losses over a give fixed period (e.g., oe year) for the i-th cotract are represeted by the r.v. X i TOT with X i TOT X i UR X i CAT, i 1, 2,...,. The r.v. X i CAT correspods to the losses due to the catastrophe risk, ad the r.v. X i UR represets all other losses due to usual risks. We defie the r.v.s X i UR ad X i CAT as X i UR M i B i,k, M i 0, i 1,..., k1 0, M i 0 ad where X i CAT M 0 C i,k, M 0 0, i 1,...,, k1 0, M 0 0 M i is the umber of usual risk losses for the cotract i over oe period. M 0 is the umber of catastrophes i the specific area over oe period. B i,k is the k-th usual risk claim amout for cotract i. C i,k is the k-th catastrophe loss amout for cotract i. We make the followig assumptios: 1. For a give cotract i, X i UR ad X i CAT are idepedet. 2. For a give cotract i, B i,1, B i,2,... are idepedet ad idetically distributed (i.i.d.) ad idepedet of M i. 3. For a give cotract i, C i,1, C i,2,...are i.i.d. ad idepedet of M X i UR ad X i UR are idepedet, i i. For the usual risk part of the portfolio, the model proposed above amouts to the traditioal idividual risk model. For the catastrophic part of the portfolio, however, we get a more flexible model. Clearly, the r.v.s X 1 CAT,...,X CAT are ot idepedet sice they are all a fuctio of M 0, the umber of occurreces of the catastrophe over oe period. Moreover, for a give catastrophe, C 1k,..., C k are ot ecessarily idepedet. It follows that X 1 TOT,...,X TOT are ot idepedet either. We defie the total (aggregate) fiacial losses for the whole portfolio of cotracts as S TOT X TOT i X UR i X CAT i S UR S CAT. I this paper, our focus will be o the aggregate amout of the catastrophic claims, S CAT, ad modelig of S UR will ot be addressed. We, thus, cosider models for S CAT X 1 CAT X 2 CAT X CAT. (1)

4 4 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 7, NUMBER 4 We ca rewrite equatio (1) as follows S CAT M 0 C CAT 1,k C CAT,k, M 0 1 k1 0, M 0 0. (2) A equivalet represetatio of S CAT is give by S CAT M 0 D CAT, M 0 1 k1 0, M 0 0, (3) where D CAT,k C CAT 1,k... C CAT,k represets the total amout of losses due to a catastrophe. The represetatio i equatios (2) ad (3) clearly shows that S CAT ca be see as a sigle risk; that is, if at least oe catastrophe occurs, the oe large fiacial loss for the compay occurs. We, therefore, have two differet perspectives i which to approach the modelig of S CAT. The first, give by equatio (1) is a micro-perspective. I the pricig of idividual isurace cotracts, we eed to use this microperspective approach, as we have to model the distributio of the idividual catastrophic loss r.v.s C CAT 1,...,C CAT, for a give catastrophe. This is the approach used whe oe simulates the effect of catastrophes o isurace portfolios usig computer catastrophe models (e.g., Walters ad Mori 1996 or Cherick 1998). The secod scale, give by equatio (3), is a macro-perspective scale. Whe this macroperspective approach is used, the distributio of the r.v. D CAT for a give catastrophe is directly modeled. Such a approach is take whe the data o total losses by catastrophe (such as Property Claim Services (PCS) idex, etc.) are available. I the pricig of catastrophe bods or other fiacial catastrophe isurace derivatives, we are iterested i directly modelig the distributio of M 0 ad D CAT (see, e.g., Schmock 1999, Harrigto ad Niehaus 1999, Christese ad Schmidli 2000, ad Cox ad Pederse 2000). Basic properties of the distributio of S CAT ca be derived. From equatio (1), we deduce ES CAT EX 1 CAT X 2 CAT X CAT EX CAT i EM 0 EC CAT i. Accordig to equatios (2) ad (3), we also have ES CAT E M0 ES CAT M 0 E M0 M 0 ED CAT EM 0 ED CAT EM 0 EC 1 CAT C CAT EM 0 EC CAT i. From equatio (1), the variace of S CAT is give by VarS CAT VarX CAT i i1 ii CovX i CAT, X i CAT. We ca also deduce the variace of S CAT from equatios (2) ad (3) VarS CAT EM 0 VarD CAT VarM 0 ED CAT 2 EM 0 VarC 1 CAT C CAT VarM 0 EC 1 CAT C CAT 2. The cumulative distributio fuctio of S CAT is obtaied from equatio (3): F S CAT x PrM0 0 PrM 0 kf CAT CAT D,1...D,k x, x 0. k1 The stop-loss premium, which is defied to be S CAT d ES CAT d ], where (u) max(u, 0), is the pure premium for a stop-loss reisurace cotract with a give retetio level d 0. Fially, from equatio (3), the momet-geeratig fuctio (m.g.f.) of S CAT is give by S CAT t M0 l D CAT t. (4) 3.2 Models for the Idividual Catastrophic Claims We will cosider three differet models for the fiacial losses due a catastrophe. As we shall see,

5 MODELING CATASTROPHES AND THEIR IMPACT ON INSURANCE PORTFOLIOS 5 the third approach is the more realistic oe; it is a mathematical formalizatio of some of the computer simulatio models used for catastrophe isurace pricig (e.g., Walters ad Mori 1996 or Cherick 1998). The first two approaches ca be cosidered as opposite extremes with respect to the level of depedece betwee the catastrophic claim amouts: The first approach assumes idepedece betwee these amouts, while the secod approach assumes complete depedece. I order to simplify the presetatio, we assume that oly oe catastrophe ca occur i a specific area over a year. The models ca be easily adapted to the case where more tha oe catastrophe ca occur, but i most practical applicatios, the probability of more tha oe catastrophe i a regio i a year is egligible. I each model j (j 1, 2, 3), the r.v. X i CAT(j) represets the costs related to the catastrophe protectio, which is defied by X i CAT j C i CAT j, M 0 1 0, M 0 0, where M 0 is a Beroulli r.v. with mea q, ad C i CAT(j) represets the fiacial losses for the cotract i if a catastrophe occurs. We assume that the fiacial loss C i CAT(j) is expressed as a proportio of the property value; that is, C i CAT j U i CAT j b i, where b i is the value of the property i ad U i CAT(j) [0, 1] is called the loss proportio or the damage ratio. (I subsectios to 3.2.3, we give a differet formulatio of U i CAT(j) for each model j (j 1, 2, 3).) We ca, thus, write X i CAT(j) as X i CAT(j) b i Y i CAT(j), with CAT Y j i U CAT i j, M 0 1 0, M 0 0. The expectatio ad the variace of X CAT(j) i are EX i CAT j EM 0 EC i CAT j b i EM 0 EU i CAT j, ad VarX i CAT j EM 0 VarC i CAT j VarM 0 EC i CAT j 2 b i 2 EM 0 VarU i CAT j VarM 0 EU i CAT j 2 }, for i 1,2,...,. The m.g.f. of X i CAT(j) is give by Xi CAT j t M0 l Ci CAT j t, where M0 deotes the m.g.f. of M 0. We also have that CovX i CAT j, X i CAT j EX i CAT j X i CAT j for i i {1,2,...,}. First, with ad EX i CAT j EX i CAT j, EX i CAT j X i CAT j E M0 EX i CAT j X i CAT j M 0, EX i CAT j X i CAT j M 0 1 EC i CAT j C i CAT j The, b i b i EU i CAT j U i CAT j EX i CAT j X i CAT j M EX i CAT j X i CAT j E M0 M 0 EC i CAT j C i CAT j It follows that b i b i EM 0 EU i CAT j U i CAT j. CovX i CAT j, X i CAT j b i b i EM 0 EU i CAT j U i CAT j EM 0 2 EU i CAT j EU i CAT j, for i i {1,2,...,}. Note that, uder this specificatio of X i CAT(j), the stop-loss premium for S CAT(j) becomes S CAT j d qbtot E V CAT j d b TOT, where b TOT b 1... b correspods to the total exposure i isured property value of a isurace compay i a give area ad V CAT j U 1 CAT j b 1 b TOT U CAT j b b TOT ca be see as a aggregate measure of the damage ratio for the whole portfolio.

6 6 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 7, NUMBER Model with Idepedet Damage Ratios I the first model (j 1), the r.v.s U CAT(1) i, i 1,..., that represet the damage ratios are assumed idepedet. This implies that CAT( Cov(U j) CAT( i, U j) i ) 0 ad CovC i CAT j, C i CAT j b i b i CovU i CAT j, U i CAT j 0 for i i {1,2,...,}. It follows that ad EX i CAT j X i CAT j b i b i EM 0 EU i CAT j EU i CAT j CovX i CAT j, X i CAT j b i b i EM 0 EU i CAT j U i CAT j EM 0 2 EU i CAT j EU i CAT j } b i b i EM 0 EU i CAT j EU i CAT j EM 0 2 EU i CAT j EU i CAT j } b i b i EU i CAT j EU i CAT j EM 0 EM 0 2, for i i {1,2,...,}. Also, D CAT(1) C 1 CAT(1)... C CAT(1) correspods to a sum of idepedet r.v.s. Therefore, the m.g.f. of S CAT(1) i equatio (4) becomes S CAT1 t M0 l D CAT1 t M0 l Ci CAT1 t M0 l CAT1 Ci t. Obviously, this model is ot completely realistic, sice by assumig that the damage ratios are idepedet, they are ot iflueced by the itesity of a catastrophe. I the followig subsectio, a first attempt is made to take the itesity of the catastrophe ito accout Damage Ratios as Determiistic Fuctios of Catastrophe Itesity I this secod model (j 2), the damage ratio U CAT(2) i is a determiistic fuctio of a r.v. I that represets the itesity of the catastrophe felt i the specific geographic area of the isured risks of the portfolio; that is, U CAT(2) i i (I), where i : 3 [0, 1], with beig the rage of I. We suppose that the r.v. I has c.d.f. F I ad is idepedet of M 0. I reality, the itesity may be a fuctio of several radom factors relatig to a catastrophe. The defiitio of i depeds o the characteristics of the covered property i (e.g., type of buildig structure), but usually i is a positive, icreasig fuctio of the catastrophe itesity. This implies that, if two properties i ad i have the same characteristics, the i i, ad thus, P[U i CAT(2) (x) U i CAT(2) (x)i x] 1. We have that E[U i CAT(2) U i CAT(2) ] E I [ i (I) i (I)] for i i {1,2,...,}. It follows that CovC i CAT2, C i CAT2 b i b i EM 0 E I i I i I E I i IE I i I. The c.d.f. of S CAT(2) is F S CAT2 x PrM0 0 PrM 0 1F D CAT2 x 1 q qf I x, where (x) {y : b i i (y) x}. It follows that the m.g.f. of S CAT(2) is S CAT2 t M0 l D CAT2 t M0 l E I e t b i ii 1 q q e t b i i df I. This model is still ot quite realistic, as two properties with the same characteristics are ulikely to icur the exact same damage ratio upo occurrece of a catastrophe Damage Ratios as Radom Fuctios of Catastrophe Itesity For the third approach (j 3), the proportio U CAT(3) i is ot a determiistic fuctio of the catastrophe itesity aymore, but rather a r.v. whose distributio is coditioal o the itesity of the catastrophe. Of course, this coditioal distributio will also deped o the characteristics of the risk isured. For example, the ATC-13 report (Applied Techology Coucil 1985) gives probability mass fuctios for the damage ratios as a fuctio of buildig type ad earthquake itesity o the modified Mercali scale. Mathematically, this ca be writte as

7 MODELING CATASTROPHES AND THEIR IMPACT ON INSURANCE PORTFOLIOS 7 PU CAT3 i ui x p iux, u u 1,...,u k, x, p iux 1. u We let p iux deped o i to make explicit the fact that these probabilities will deped o the characteristics (e.g., buildig type, age) of the i-th risk. Note that the coditioal distributio of CAT(3) U i eed ot be discrete; Cossette et al. (2002) give a similar model with a cotiuous coditioal beta distributio for the damage ratios caused by wid. Obviously, two properties with the same characteristics will ot ecessarily icur the same damage ratio, i.e., P[U CAT(3) i (x) U CAT(3) i (x)i x] is ot ecessarily equal to 1 for two properties i ad i with idetical characteristics, as was the case with the secod model. We make the assumptio that coditioal o I, U CAT(3) 1,..., U CAT(3) are idepedet. We the have EU i CAT3 U i CAT3 E I EU i CAT3 U i CAT3 I E I EU i CAT3 IEU i CAT3 I, for i i {1,2,...,}. It follows that CovC i CAT3, C i CAT3 b i b i EM 0 E I EU i CAT3 IEU i CAT3 I b i b i EM 0 2 EU i CAT3 EU i CAT3, for i i {1,2,...,}. The c.d.f. of S CAT(3) is F S CAT3 x PrM 0 0 PrM 0 1F D CAT3 x 1 q qf C1 CAT3 C CAT3 x 1 q q F C1 CAT3 C CAT3 I xdf I. It follows that the m.g.f. of S CAT(3) is S CAT3 t M0 l D CAT3 t M0 I l E Ci It CAT3 q 1 q l Ci It I CAT3 df. The third model is the most realistic of the three models itroduced i this sectio. Uder this model, the damage ratio is iflueced by the itesity of the catastrophe (which was ot the case with the first model), but is still allowed to fluctuate from property to property eve if these properties share the same characteristics (which was ot the case with the secod model). Now that we ca model the catastrophe risk, we ca evaluate its impact o the fiacial risk posed by a isurace portfolio. 4. PORTFOLIO RISK MANAGEMENT We cosider the global risk of a portfolio of a isurace compay. A approach based o risk measures will be preseted i Sectio 5. I this sectio, we first argue that lookig at idividual premiums will ot detect the risk iduced by catastrophes i a portfolio. We the examie the behavior of the aggregate fiacial losses as the umber of cotracts withi the portfolio icreases, ad illustrate how the catastrophe risk caot be diversified by icreasig the size of the portfolio. 4.1 Idividual Premiums ad Rui Probability Let i CAT(j) (X i CAT(j) ) deote the loaded premium associated to catastrophe coverages of cotract i uder model j (j 1, 2, 3). We exclude expeses ad profit compoets from the premium calculatios. We assume that the i CAT(j) s are computed uder separate premium priciples. (For a survey of the premium calculatio priciples, see, e.g., Gerber 1979, Dayki et al. 1994, or Rolski et al ) Geerally, premium calculatios are preseted i the cotext of coverages excludig catastrophe risks (see Casualty Actuarial Society 1996 for a survey o the computatio of such premiums). I this sectio, we apply these priciples to the computatio of i CAT(j). The simplest priciple is the pure premium priciple where i CAT(j) E[X i CAT(j) ] for i 1, 2,...,. More geerally, the loaded premium is greater tha the pure premium ad the differece is called the safety margi (or safety loadig): i CAT(j) i CAT(j) E[X i CAT(j) ] where i CAT(j) are assumed positive. The relative safety margi i CAT(j) is defied by

8 8 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 7, NUMBER 4 CAT j i CAT j i CAT EX j i. Amog the premium calculatio priciples, we fid The expected value priciple: CAT(j) i (1 CAT(j) i )E[X CAT(j) i ], with CAT(j) i 0; The variace priciple: CAT(j) i E[X CAT(j) i ] CAT(j) i Var[X CAT(j) i ], with CAT(j) i 0; CAT(j) The stadard deviatio priciple: i E[X CAT(j) CAT(j) CAT(j) i ] i Var[X i ], with c i 0. It is worth poitig out that ay of these premium priciples will yield idetical premiums for each of the three risk models cosidered i Sectio 3. This is due to the fact that the differece betwee the risk models is ot i the margial distributios of each isured risk, but i their joit distributio. Hece, the above premium priciples fail to detect the differece betwee the three models. We must, therefore, cosider the stochastic behavior of the aggregate fiacial losses. We do so through rui probability calculatios, which formally illustrates the well-kow fact that idepedet risks are diversifiable, whereas catastrophic risks are ot. I this paper, by rui probability we mea the probability that the isurace compay does ot meet its fiacial commitmets over a fixed period of time (e.g., the ext year). Note that we do ot take ito accout ay risk reserve or special allocatio from the surplus (or capital) i our defiitio of the rui probability. Oe questio of iterest is the behavior of the rui probability as the umber of isured cotracts withi the portfolio of the isurace compay icreases. We assume that the first two momets of ay claim amout radom variables X i are fiite ad strictly positive. These assumptios are reasoable i practice, sice amouts isured o idividual cotracts are fiite. To simplify the presetatio, we also assume that the relative safety margis i the premiums are equal for all cotracts of the portfolio. 4.2 Diversificatio uder Idepedet Risks We show i Propositio 1 that, if the relative safety loadig is positive, the probability of ot havig eough moey to cover the total claims teds to 0 as the umber of risks teds to ifiity, whereas Propositio 2 shows that this same probability coverges to 1 whe the safety loadig is egative. I both propositios, we cosider a sequece of positive idepedet idividual cotract loss amout radom variables Y 1,..., Y. We assume that the first two momets are fiite ad strictly positive; that is, there exist real umbers a 1, a 2, b 1, b 2 such that 0 a 1 E[Y i ] a 2 ad 0 b 1 Var(Y i ) b 2. Let T be the aggregate fiacial loss radom variable, defied as T Y 1... Y, with ET EY i (5) VarT VarY i. (6) Defie the rui probability for the portfolio of cotracts as Pr T i, where E[(1 )Y i ] is the premium for the i-th cotract. Propositio 1 If the safety margi 0, the the rui probability goes to 0 whe 3. Propositio 2 If the safety margi 0, the the rui probability goes to 1 whe 3. The previous two propositios cofirm the commo kowledge that idepedet risks are diversifiable, as log as the umber of risks is large ad the premium charged for each risk is superior to its expected value. The proofs of all the propositios of Sectio 4 are relegated to the Appedix. 4.3 Diversificatio uder Catastrophic Risks I the ext three propositios, we cosider a sequece of positive catastrophic idividual cotract loss amout radom variables X 1 CAT(j),..., X CAT(j), j 1, 2, 3. We assume that the first two

9 MODELING CATASTROPHES AND THEIR IMPACT ON INSURANCE PORTFOLIOS 9 momets are fiite ad strictly positive, i.e., there exist real umbers a 1, a 2, b 1, b 2 such that 0 a 1 E[X CAT(j) i ] a 2 ad 0 b 1 Var(X CAT(j) i ) b 2. Let S CAT(j) be the aggregate fiacial loss radom variable, defied as S X 1 CAT(j) CAT(j)... S CAT(j) with CAT ES j CAT EX j i. Defie the rui probability CAT(j) for the portfolio of cotracts as CAT j Pr S CAT j CAT i j, where i CAT(j) (1 )E[X i CAT(j) ] is the premium for the catastrophe coverage for the i-th cotract. I the remaider of Sectio 4.3, we examie the behavior of CAT(j) for the catastrophe models with idepedet damage ratios (j 1), with damage ratios as determiistic fuctios of catastrophe itesity (j 2), ad with damage ratios as radom fuctios of catastrophe itesity (j 3). This will provide a formal argumet to support the fact that catastrophe risk caot be diversified i the same fashio as the risk of usual dayto-day busiess. Ideed, we will prove that a isurace or reisurace compay caot diversify the fiacial risk caused by catastrophes through premium icome aloe, eve with a arbitrary large portfolio of isured risks Idepedet Damage Ratios I the followig propositio, we demostrate that, uder the model with idepedet damage ratios, the rui probability teds to q, the probability that a catastrophe occurs: Propositio 3 If 0 EC i CAT1 EX CAT1 i 1 1 1, q the the rui probability CAT(1) teds, whe 3, to the probability that at least oe catastrophe occurs; that is, lim CAT1 1 PrM 0 0 q. 3 Notice that the upper boud E C i CAT1 EX CAT1 i 1 o the relative safety margi correspods to the case where the premium i CAT(1) is equal to E[C i CAT(1) ], the expected idividual catastrophic loss give a catastrophe happes. This is a reasoable boud as it is hard to imagie a isurace compay chargig a premium greater tha E[C i CAT(1) ]. Withi the cotext of this simple model, we have show that the catastrophe risk caot be fully diversified like the ocatastrophe risks ca Damage Ratios as Determiistic Fuctios of Catastrophe Itesity Recall that the damage ratio r.v.s U CAT(2) 1,..., U CAT(2) are positive icreasig fuctios i of a r.v. I, the itesity of the catastrophe. We ow show i this case that the rui probability is greater tha a strictly positive fractio of the probability that a catastrophe occurs. Propositio 4 If 0 EC i CAT2 EX CAT2 i 1 1 q 1, there exists a strictly positive real umber c with 0 c 1, such that lim CAT2 q c 0. 3 To illustrate the result of the previous propositio, cosider a portfolio of property isurace where b i b ad the damage ratio r.v.s U i CAT(2) are idetically distributed U i CAT2 U CAT2, which meas that i (I) (I) for i 1,2,...,. The, for ay, CAT2 Pr S CAT2 i CAT2 q PrbI 1 q bei q PrI 1 q EI.

10 10 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 7, NUMBER 4 For the positive ad icreasig fuctio (u), defie 1 ( ) as its iverse. It leads to CAT2 q PrI 1 q EI, q PrI 1 1 q EI,, where (1 ) q E[(I)] E[(I)] by assumptio ad Pr(I 1 ((1 ) q E[(I)])) is strictly positive. The, we obtai the rui probability CAT2 q PrI 1 1 q EI, qc,, where c Pr(I 1 ((1 ) q E[(I)])) ca be computed give the fuctio Damage Ratios as Radom Fuctios of Catastrophe Itesity Recall that the damage ratio r.v.s U CAT(3) 1,..., U CAT(3) are positive radom fuctios i of a r.v. I, the itesity of the catastrophe. We ow show i this case that the rui probability teds to a strictly positive fractio of the probability that a catastrophe occurs. Propositio 5 If 0 EC i CAT3 EX CAT3 i 1 1 1, q there exists a strictly positive real umber c with 0 c 1, such that CAT3 lim Illustratio q c 0. The results of Propositios 4 ad 5 may feel couterituitive at first, sice the rui probability uder Model 2 is lower tha the probability of rui uder Model 3. The followig example helps to uderstad the meaig of these propositios better. For simplicity, suppose that the catastrophe itesity is a discrete radom variable that may take o oe of seve distict values. The damage ratios are assumed discrete o {1/5, 2/5,...,5/5} (respective probabilities 0.26, 0.33, 0.24, 0.13, 0.04). Usig the results from Appedix A, we ca compute the c.d.f.s of S CAT(j), j 1, 2, 3. These c.d.f.s are plotted for a portfolio of 150 risks, each of isured value 10, with a probability of catastrophe i the year equal to 0.1 i Figure 1. The model is such that E[S CAT(j) ] 70.6 ad E[D CAT(j) ] 706, j 1, 2, 3. The most obvious characteristic of the distributio of S CAT(j) is that it has a probability mass of 0.9 at 0, sice the probability of o catastrophe is 0.9. I the idepedet damage ratio case, we see that the distributio of S CAT(1), give a catastrophe occurs, is cocetrated aroud E[S CAT(1) ], as the law of large umbers would suggest. I the case of determiistic damage ratios, whe a catastrophe occurs, all damage ratios take o the same value ad we get a c.d.f. for S CAT(2) with a staircase appearace. For istace, if the damage ratios all take o the value 1/5, the expected value of S CAT(2) is 300, the poit of the first jump i the staircase c.d.f. Note that, give a catastrophe occurs, the distributio of S CAT(2) is a lot more spread out aroud E[D CAT ] 706 tha is the case with idepedet damage ratios. For radom damage ratios, we see that the c.d.f. of S CAT(3), give a catastrophe occurs, is cocetrated aroud seve poits, each poit correspodig to a possible value of the catastrophe itesity. This makes sese as, give a catastrophe of a specific itesity, S CAT(3) amouts to a sum of idepedet radom Figure 1 Cumulative Distributio Fuctio of the Aggregate Catastrophic Loss for the Three Models Note: Solid lie: idepedet damage ratios ( j 1), dotted lie: damage ratios determiistic ( j 2), dashed lie: damage ratios radom ( j 3).

11 MODELING CATASTROPHES AND THEIR IMPACT ON INSURANCE PORTFOLIOS 11 variables. For example, for the smallest of the seve values of the catastrophe itesity, the coditioal value of S CAT(3) is 480, the ceter poit of the first climb of the c.d.f. of S CAT(3). Notice that the spread aroud the distributio of S CAT(3) about E[D CAT ], give a catastrophe, is itermediate betwee that of the distributios of S CAT(1) ad S CAT(2). From the c.d.f.s, we ca calculate rui probabilities for various values of the risk loadig. Whe 200%, all three rui probabilities CAT(j) CAT(2) 0.10, j 1, 2, 3. Whe 400%, 0.074, while CAT(j) 0.10, j 1, 3. Whe CAT(3) CAT(2) 600%, 0.076, 0.074, ad CAT(1) Hece, despite very large safety margis, the rui probabilities remai positive. The results derived i Sectio 4.3 strogly suggest that compaies sellig protectio agaist the risk of catastrophes seek protectio themselves, either through reisurace or isurace derivatives (see, e.g., Schmock 1999, Harrigto ad Niehaus 1999, Christese ad Schmidli 2000, Cox ad Pederse 2000, ad Cox, Fairchild, ad Pederse 2000). However, rui probabilities aloe caot be used to assess the dagerousess of a portfolio. Ideed, the portfolio with determiistic damage ratios seems to be more risky, give the spread of its c.d.f. i Figure 1, but its rui probability is the lowest. I Sectio 5, we quatify the effect of the spread i the c.d.f.s of S CAT(j) i terms of measures of the risk level of the portfolios. 5. COMPARISON BETWEEN THE CATASTROPHE MODELS We ow wat to assess the riskiess of a portfolio whe catastrophes are possible. We will do so by comparig the risk level of the realistic catastrophe model (third model), proposed i Sectio 3, to the risk level of portfolios based o the other two more extreme models (first model ad secod model) of Sectio 3. We will rak the risk levels of these portfolios usig risk measures, to be reviewed i Sectio 5.1 for completeess. To make sure that the portfolios are comparable, we will costruct the portfolios so that the risks isured agaist catastrophes have the same margial distributios i all three portfolios i Sectio 5.2. We use existig theory o risk orderig to derive theoretical orderigs betwee the three portfolios i Sectio 5.3. We illustrate these orderigs with umerical examples i Sectio Risk Measures I this sectio, we look at meas of assessig the effect of potetial catastrophes o the riskiess of a portfolio. A risk measure is a system that allows us to quatify or compare risks (Wirch 1999). To compare the risk levels of our catastrophe models, we will cosider three risk measures used i risk maagemet: the stop-loss premium, the value at risk (VAR) ad the coditioal value at risk (CVAR) (also called the coditioal tail expected loss) Coheret Risk Measures Several differet measures of riskiess have bee proposed i the literature. Sice differet measures may lead to differet risk orderigs, it is preferable to restrict our class of potetial risk measures to a set of risk measures that satisfies miimal requiremets. Artzer et al. (1998, 1999) ad Wirch (1999) give five properties that are desirable for risk measures. They call risk measures that satisfy these properties coheret risk measures. DEFINITION 1 A coheret risk measure has the followig properties: Property 1: The risk measure must be limited above by the maximum possible et loss. Property 2: The risk measure must be subadditive. Property 3: The risk measure must be multiplicative by a scalar. Property 4: The risk measure must be idepedet of the size of possible gais. Property 5: The risk measure must be scalar additive. We examie separately three risk measures. While the CVAR is coheret, the stop-loss premium ad the VAR are ot; we cosider them, oetheless, as they are widely used risk measures i practice Stop-Loss Premium The stop-loss premium, defied as S (d) E[(S d) ], is the pure premium for a stop-loss reisurace cotract with a give retetio level d 0.

12 12 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 7, NUMBER Value at Risk The VAR is a popular risk measure i risk maagemet ad actuarial sciece. I the actuarial literature, it is also referred to as the maximal probable loss. DEFINITION 2 The VAR with a cofidece level associated to the r.v. S is defied by VAR S ifx R : F S x, where 0 1. The VAR is a popular risk measure, eve though it is ot a coheret risk measure. The properties 2 ad 3 (Wirch 1999) stated above are ot satisfied by this risk measure. For further iformatio o the VAR see, for example, Embrechts et al. (2002) ad Hürlima (2003) Coditioal Value at Risk The CVAR, also called the coditioal tail expectatio, is a coheret risk measure proposed by Artzer et al. (1998, 1999) as a alterative to the VAR (see, e.g., Wirch 1999 ad Hürlima 2001). DEFINITION 3 Let 0 1. The CVAR (S), with a cofidece level associated to the r.v. S, isdefied by CVAR S ESS VAR S. As metioed, for example, i Hürlima (2001), we have CVAR S ESS VAR S VAR S sdf S s PrS VAR S 1 1 F S sds VAR S. VARS A popular measure of the risk of death of a idividual i demography ad life actuarial sciece is the residual life expectacy e S x ES xs x Sx 1 F S x (see, e.g., Klugma et al. (1998) or Bowers et al. 1997). The CVAR ca be expressed as a fuctio of the residual life expectacy ad VAR CVAR S e Sm VAR S VAR S, or, equivaletly, CVAR S SVAR S 1 F S VAR S VAR S SVAR S VAR S. 5.2 Costructio of Three Comparable Portfolios We ow costruct three portfolios of risks, oe portfolio comig from each of the three models preseted i Sectio 3. To make the comparisos meaigful, we have to costruct the portfolios so that the radom variables represetig the damage ratios, U 1 CAT,..., U CAT, have the same margial distributios i each portfolio. The portfolios have the same structure, where we model the catastrophic claim amout for the i-th risk, X i CAT,by X i CAT b iu i CAT, M 0 1 0, M 0 0, where M 0 is a Beroulli(q) radom variable that takes o value 1 if, ad oly if, there is a catastrophe affectig the portfolio i the period of iterest. We will costruct the portfolios so that the joit distributio of U 1 CAT(j),...,U CAT(j) will be differet for each portfolio. We preset the portfolios i reverse order (j 3, 2, 1), as it is simpler to costruct portfolios with idetical margial distributios for the U i CAT(j) this way Realistic Portfolio We start by costructig a portfolio that is a relatively realistic represetatio of a group of risks subject to potetial catastrophes. This costructio is based o the third model (j 3) from Sectio We first suppose that there is at most oe catastrophe affectig the portfolio i the year with probability q. The idicator of such a catastrophe is a radom variable M 0 havig a Beroulli distributio with mea q. Give a catastrophe, its itesity, I, has a distributio with c.d.f. F I o the positive real lie. Coditioal o a observed itesity I x, we suppose that the damage ratios U i CAT(3), i 1,..., are idepedet radom variables, with U i CAT(3) havig coditioal c.d.f. at u give by F Ui CAT3 Iux. Thus, give M 0, U 1 CAT(3),..., U CAT(j) are depedet radom vari-

13 MODELING CATASTROPHES AND THEIR IMPACT ON INSURANCE PORTFOLIOS 13 ables with margial c.d.f.s F U1 CAT3,...,FU CAT3, where F Ui CAT3 u 0 F Ui CAT3 IuxdF I x, i 1,..., Portfolio Based o Comootoic Damage Ratios We ow costruct a portfolio based o the secod model (j 2) of Sectio We first defie the property of comootoicity (see, e.g., Wag ad Dhaee 1998, Wag 1998, Bäurle ad Müller 1998, ad Deuit et al. 2002). DEFINITION 4 A vector of r.v.s, deoted Z cm (Z 1 cm,...,z cm ), with margial c.d.f. F Zi, is said to be comootoic if oe of the three followig coditios is fulfilled: 1. The c.d.f. of Z cm is give by F Z cmx mif Z1 x 1,...,F Z x, x R ; 2. We have Z cm F 1 Z1 U,...,F 1 Z U, where U Uif(0, 1); 3. There exists a r.v. V ad icreasig fuctios f 1,..., f m such that Z i cm f i (V) for i 1, 2,...,. We make our costructio so that the damage ratios have the same margial distributios as i the realistic portfolio, but are comootoic; that is, U CAT(2) CAT(2) 1,..., U ca be writte as 1 (I),..., (I) for some r.v. I ad icreasig fuctios 1,...,. To do so, we simply let U i CAT2 F 1 U i CAT3 FI x, i 1,...,, (7) 1 where F CAT3 Ui u ify : F CAT3 Ui y u is the iverse of the c.d.f. of U CAT(3) i ad F I is the c.d.f. of I, the catastrophe itesity. It is the easy to verify that uder these assumptios, U CAT(3) 1,..., U CAT(3) ad U CAT(2) 1,..., U CAT(2) are vectors of radom variables with idetical margial distributios but with a differet joit distributio. It is also obvious that the damage ratios as defied i equatio (7) satisfy the defiitio of comootoic radom variables. This portfolio with comootoic proportios will be show to be the riskiest of our portfolios. It is also a portfolio that leads to relatively simple calculatios. It will, therefore, be a coveiet upper boud for stop-loss premiums ad other quatities of iterest for realistic portfolios. Moreover, we will see i the umerical example of Sectio 6 that this upper boud is surprisigly tight Portfolio with Idepedet Damage Ratios We ow costruct a portfolio based o the first model (j 1) of Sectio I this costructio, the damage ratios agai have the same margial distributios as i the realistic portfolio, but this time we let these damage ratios be idepedet. This is simply doe by lettig V 1,...,V be idepedet uiform radom variables o [0, 1], the by settig U CAT1 1 i F CAT3 Ui V i, i 1,...,. Agai, oe easily sees that we have idepedet radom variables U 1 CAT(1),..., U CAT(1) with margial c.d.f.s give by F U1 CAT3,...,FU CAT3, respectively. 5.3 Orderig of Risks We compare the three portfolios o the basis of stochastic orders ad depedece orders. Orderig of risks is ofte applied, for example, i the establishmet of stochastic bouds, ad ca also be used to rak portfolios accordig to their dagerousess Basic Results We first preset two defiitios of stochastic orders betwee uivariate r.v.s, which are ofte used i actuarial sciece (for details, see, e.g., Bäurle ad Müller 1998, Rolski et al. 1999, ad Kaas et al. 2001). DEFINITION 5 Let X ad X be two r.v.s such that E[X] ad E[X]. The, X precedes X uder stochastic domiace order, deoted X sd X, iff X (x) F X (x), for all x R. DEFINITION 6 Let X ad X be two r.v.s such that E[X] ad E[X]. The, X precedes X uder stop-loss order, deoted X sl X, ife[(x d) ] E[(X d) ], for all d R. Cosider ow two vectors of r.v.s X (X 1,..., X ) ad X (X 1,...,X ) where, for each i, X i ad X i have the same margial distributio (i.e., X i X i for i 1,2,...,). Defie also S X 1... X ad S X 1... X. Because of

14 14 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 7, NUMBER 4 the assumptios o X ad X, we have E[S] E[S]. Whe we wat to compare X ad X, weuse depedece orders (see, e.g., Shaked ad Shatikumar 1994 ad Joe 1997 for details o depedece orders). The, based o a give relatio betwee X ad X, we ca compare S ad S.Ifwe wat to establish that S precedes S uder stoploss order, we first eed to show that X precedes X uder the so-called supermodular order. The supermodular order is a depedece order which was itroduced i the actuarial cotext by Müller (1997) ad Bäurle ad Müller (1998), the examied by Deuit et al. (2002), for example. It allows the compariso of radom vectors with the same margials. We first eed to defie a supermodular fuctio. DEFINITION 7 A fuctio g: R m 3 R is said supermodular if g x 1,...,x i ε,...,x j,...,x m g x 1,...,x i ε,...,x j,...,x m g x 1,...,x i,...,x j,...,x m g x 1,...,x i,...,x j,...,x m is true for all x (x 1,...,x m ) R m,1i j m ad all ε, 0. This defiitio is a extesio to the otio of covexity for a fuctio d : R 3 R. Propositio 6 If g is twice differetiable, the g is supermodular, if ad oly if, 2 x i x j gx 0, for all x R m ad 1 i j m. PROOF See, for example, Marshall ad Olki (1998) or Bäurle ad Müller (1998). e m For example, the fuctios g(x 1,...,x m ) x i, g(x 1,...,x m ) m (x i d) 2 ad g(x 1,..., x m ) ( m x i d) are supermodular. Let us defie the supermodular order as preseted i Bäurle ad Müller (1998): DEFINITION 8 Let X ad X be two vectors of r.v.s X (X 1,..., X ) ad X (X 1,...,X ) where, for each i, X i ad X i have the same margial distributios (i.e., X i X i for i 1,2,...,). The, X precedes X uder the supermodular order, deoted X sm X, if E[g(X )] E[g(X )] for all supermodular fuctios g, provided their expectatios exist. We poit out that the supermodular order ca be applied whe we compare two vectors of r.v.s with the same margials. The supermodular orderig is used whe we wat to compare r.v.s with differet degrees of depedece. The relatioship betwee supermodular orderig ad comootoicity is explored i Bäurle ad Müller (1998) ad Goovaerts ad Dhaee (1999). We ow state the followig importat result: Propositio 7 If X sm X, the S sl S. PROOF See, for example, Bäurle ad Müller (1998). e The followig propositio correspods to Loretz s iequality. Lemma 8 Let X ad X cm be vectors of r.v.s give by X (X 1,...,X ) ad X cm (X 1 cm,...,x cm ) where, for each i, X i ad X i cm have the same margial distributios (X i X i cm, for i 1,2,...,), ad where X 1 cm,..., X cm are comootoic. The, X precedes X cm uder the supermodular order, deoted X sm X cm. PROOF See, for example, Bäurle ad Müller (1998). e Accordig to Lemma 8, the comootoicity correspods to the strogest depedece relatio uder supermodular orderig. Lemma 9 Let X ad X IND be vectors of r.v.s give by X (X 1,...,X ) ad X IND (X IND 1,...,X IND ) where, for each i, X i ad X IND i have the same margial distributios (X i X IND i, for i 1,2,...,) ad the compoets of X IND are idepedet. We assume that the compoets of X are positively correlated. The, X IND precedes X uder the supermodular order, deoted X IND sm X.

15 MODELING CATASTROPHES AND THEIR IMPACT ON INSURANCE PORTFOLIOS 15 PROOF See, for example, Müller (1997) ad Dhaee ad Goovaerts (1996). e Applicatio to Our Cotext We ow apply the above results o the stochastic orders ad the depedece orders to compare the catastrophe models. Propositio 10 We have U CAT(1) sm U CAT(3) sm U CAT(2). PROOF Clearly, the compoets of U CAT2 U 1 CAT2, U 2 CAT2,...,U CAT2 are comootoic, sice they are determiistic fuctios of the itesity of the catastrophe. The compoets of U CAT(3) are defied by commo mixture which implies that they are positively correlated. The, we have U CAT(1) sm U CAT(2) ad U CAT(3) sm U CAT(2) from lemmas 8 ad 9. e Propositio 11 We have S CAT1 sl S CAT3 sl S CAT2. from Propositio 10. From Kaas et al. (2001, Chap. 10), equatio (8) leads to S CAT(1) sl S CAT(3) sl S CAT(2). e I the last propositio, we show that the third model for catastrophe risks, the more realistic oe, is bouded below by the first model ad bouded above by the secod model. This implies that, for ay retetio level d 0, we have S CAT1d S CAT3d S CAT2d. (9) Hürlima (2003) shows that this further implies that CVAR S CAT1 CVAR S CAT3 CVAR S CAT2, (10) for 0 1. The relatios (9) ad (10) are illustrated i the umerical example of Sectio 6. These relatios are useful whe oe wats to establish stochastic bouds for the distributio of the third model. Ufortuately, we caot coclude that F S CAT(1)(x) F S CAT(3)(x) F S CAT(2)(x) for all x 0, which would have implied that VAR (S CAT(1) ) VAR (S CAT(3) ) VAR (S CAT(2) ) for 0 1. Stochastic bouds o VAR (S CAT(2) ) ca still be obtaied, as is explaied i Deuit et al. (1999). PROOF First, we have B CAT(1) sm B CAT(3) sm B CAT(2), sice B i CAT(j) b i U i CAT(j) ad because the supermodular order is preserved uder scalar multiplicatio (see Bäurle ad Müller 1998). The, it follows that D CAT1 sl D CAT3 sl D CAT2 (8) 6. NUMERICAL ILLUSTRATION OF THE THREE RISK MODELS We keep the costructio of the three portfolios as described i Sectio 5. We cosider a portfolio of 300 isured risks divided ito three classes of 100 risks each, with all the risks i a give class havig the same characteristics (say, same build- Table 1 Coditioal Probabilities of Damage Ratios for Buildig Type u p 1,1 (u) p 1,2 (u) p 1,3 (u) p 1,4 (u) p 1,5 (u)

16 16 NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 7, NUMBER 4 Table 2 Coditioal Probabilities of Damage Ratios for Buildig Type u p 2,1 (u) p 2,2 (u) p 2,3 (u) p 2,4 (u) p 2,5 (u) ig type). Each class cotais 25 risks of isured value 1, 25 risks of isured value 2, 25 risks of isured value 3, ad 25 risks of isured value 4 (here oe uit could represet, say, $100,000). Thus, the total isured value of this portfolio is b TOT 750. The catastrophe occurrece idicator M 0 is a Beroulli radom variable with mea q 0.2 (this value seems somewhat excessive, but it is coveiet for illustrative purposes). Withi the framework of the third model, we assume that, give a catastrophe occurs, the r.v. I, represetig the catastrophe itesity, takes o a value i { 1, 2, 3, 4, 5 } with a mass probability fuctio give by Pr(I 1 ) 0.2, Pr(I 2 ) 0.4, Pr(I 3 ) 0.2, Pr(I 4 ) 0.15 ad Pr(I 5 ) The coditioal mass probability of the damage ratio r.v.s give I i are provided i Tables 1, 2, ad 3. These coditioal mass probabilities are ispired from the ATC-13 report (Applied Techology Coucil 1985), which uses earthquake data from Califoria to model the distributio of the damage ratios as a fuctio of earthquake itesity. If the property covered by the isurace cotract i belogs to buildig type l, welet PrU i CAT3 ui k p l,k u, u 0.1, 0.2,..., 1.0, k 1, 2,..., 5. We derive the mass probability fuctio (see Table 4) for U i CAT(3) ad, cosequetly, of U i CAT(1) ad U i CAT(2) for the three types of buildigs. I Table 5, we provide the expectatio ad the stadard deviatio of the damage ratios U CAT(j) (j 1, 2, 3) for the three types of buildig. I the cotext of the three models, the expected aggregate fiacial losses for the whole portfolio is 80.05; that is, E[S CAT(1) ] E[S CAT(2) ] E[S CAT(3) ] ad, give that a catastrophe occurs, the expected aggregate fiacial losses for the whole portfolio is ; that is, Table 3 Coditioal Probabilities of Damage Ratios for Buildig Type u p 3,1 (u) p 3,2 (u) p 3,3 (u) p 3,4 (u) p 3,5 (u)