ASSESSMENT OF THE RUIN PROBABILITIES

Transcription

1 Associate Professor Paul TĂNĂSESCU, PhD Departmet of Fiace Lecturer Iulia MIRCEA, PhD Departmet of Applied Mathematics The Bucharest Uiversity of Ecoomic Studies ASSESSMENT OF THE RUIN PROBABILITIES Abstract: I this paper, we aalyze the rui probability for some risk models, which is the probability that a isurer will face rui i fiite time whe the isurer starts with iitial reserve ad is subjected to idepedet ad idetically distributed claims over time. The ideal is as we are able to come up with closed form solutios for the ifiite horizo rui probability ad the fiite horizo rui probability. But, the cases where this is possible are few; therefore we must make approximatios of rui probability. I this paper, we isist o the discrete time isurace model ad o the diffusio approximatio ad so-called corrected diffusio approximatio (CDA). We aalyze the rui probability with respect to: the parameters of the idividual claim distributio ad the itesity parameter of the umber of claims process. Rui theory with debit ad credit iterest has received cosiderable attetio i recet years. I this lie, we cosider a perturbed risk model i which a curret premium rate will be adjusted i ay period (usually year) i which there are o losses ad ay surplus available at the begiig of the period is reivested. Also, we aalyze ad the iverse problem: to determie the iitial reserve whe it is give the rui probability. Keywords: Browia motio, corrected diffusio approximatio, risk process, rui probability, surplus process. JEL Classificatio: C00, G0, G30 1. Itroductio The actuarial risk model has two mai compoets: oe characterizig the frequecy of evets ad aother describig the size (or severity) of gai or loss resultig from the occurrece of a evet. The stochastic ature of both, the icidece ad severity of claims, has a essetial role for the set up of a realistic model. I

2 Paul Taasescu, Iulia Mircea examiig the ature of the risk associated with a portfolio of busiess, it is ofte of iterest to assess how the portfolio may be expected to perform over a exteded period of time. Oe approach cocers the use of rui theory. Rui theory is cocered with the excess of the icome (with respect to a portfolio of busiess) over the outgo, or claims paid. This quatity, referred to as isurer s surplus, varies i time. Specifically, rui is said to occur if the isurer s surplus reaches a specified lower boud. Oe measure of risk is the probability of a suchlike evet, clearly reflectig the volatility iheret i the busiess. This probability is called rui probability. It ca serve as a useful tool i log rage plaig for the use of isurer s fuds. The compay receives a certai amout of premium to cover its liabilities. The compay is assumed to have a certai iitial capital (risk reserve) at its disposal. Oe importat problem i risk theory is to ivestigate the rui probability, i.e. the probability that the risk busiess ever becomes egative. The ideal is as we are able to come up with closed form solutios for the ifiite horizo rui probability ad the fiite horizo rui probability. But, the cases where this is possible are few; therefore we must make approximatios of rui probability. There are various ways to model aggregate claims distributios, the time evolutio of the reserves of a isurace compay ad its claim surplus process ad to defie the probability of rui. The idea behid the diffusio approximatio is to first approximate the claim surplus process by a Browia motio with drift by matchig the two first momets. Sice Browia motio is skip-free, the idea to replace the risk process by a Browia motio igores the presece of the overshoot ad other thigs. I this paper, we isist o rui probability of discrete-time surplus process, ad o the diffusio approximatios of rui probability. I the classical risk model, the premium rate c is a fixed costat that satisfies a positive security loadig coditio, amely, c >0, ad the premium rate is irrespective of the claim experiece. However, the premium rate i practice, especially for autoisuraces, is ofte adjusted accordig to the claim experiece. The assumptio that the premium rate keeps costat is very restrictive i practice. The models from rui theory with debit ad credit iterest ad with perturbed risk have received cosiderable attetio i recet years. Also, multi-dimesioal risk theory has gaied a lot of attetio i the past few years maily due to the complexity of the problems ad the lack of closed-form results eve uder very basic model assumptios. Oe of the mai questios relatig to the operatio of a isurace compay is the calculatio of the probability of rui, ad the probability of rui before time T. The theory of martigales provides a quick way of calculatig the risk of a isurace compay. If X : 1 be a sequece of idepedet ad idetically distributed radom variables, let S S : 0 be its associated radom walk (so that S0 0

3 Assessmet of the Rui Probabilities ad S X1 X for 1), ad suppose that S with drift. It is iterestig to develop the high accuracy approximatios to the distributio of the maximum radom M max S : 0. For 0 M u u, where variable u if S u u, 1:, so that computig the tail of M is equivalet to computig a level crossig probability for the radom walk S. I isurace risk theory, P u is the probability that a isurer will face rui i fiite time whe the isurer starts with iitial reserve u ad is subjected to idepedet ad idetically distributed claims over time. Oe importat approximatio holds as 0. This asymptotic regime correspods i risk theory to the settig i which the safety loadig is small. I this case, the approximatio PM u exp u / where is valid, Var X 1. Because the right had side is the exact value of the level crossig probability for the atural Browia approximatio to the radom walk S, it is ofte called the diffusio approximatio to the distributio of M. Also, i this paper, we cosider a time depedet risk model for the surplus of a isurer, i which the curret premium will be adjusted after a year without losses ad the available amout of moey is reivested. At the same time, we also wat to derive a equatio satisfied by the survival probability ad to determie the risk reserve. The remaider of this paper is orgaized as follows. A brief literature review is give i Sectio. Sectio 3 presets a itroductio i the models of surplus process ad more detailed the assessmet of the rui probabilities. Fially, i Sectio 4, we exemplify our methods with umerical results startig our models from data of aual report o 01 of two Romaia Isurace Compay 1 ad we preset the paper cocludes with some commets.. Literature Review Recetly, several ew risk models have bee proposed i the specialized literature, i which the premium icome of a isurer is ucertai ad depeds o some radom compoets i the surplus of a isurer. For example, Dufrese ad Gerber (1991) added a diffusio to the classical compoud Poisso surplus process. The diffusio 1 ABC Asigurări Reasigurări SA (ABCAR) ad SC Geerali Româia Asigurare Reasigurare SA (GRAR)

4 Paul Taasescu, Iulia Mircea describes a ucertaity of the aggregate premium icome or a additioal ucertaity to the aggregate claims. A time-depedet premium risk model ca be foud i Asmusse (000), i which premium rates are adjusted cotiuously accordig to the curret level of a isurer's surplus. Albrecher ad Asmusse (006) ivestigated a adaptive premium that is dyamically adjusted accordig to the overall claim experiece. I additio, the depedece betwee other compoets i risk models was also studied. For istace, Albrecher ad Boxma (004) cosidered a depedet risk model i which the Poisso arrival rate of the ext claim is determied by the previous claim size. They exteded their model to a Markov-depedet risk model i which both arrival rates ad claim size distributios are determied by the state of a uderlyig cotiuous-time of Markov chai type. Furthermore, Albrecher ad Teugels (006) have studied the risk models with depedece betwee iter-claim times ad claim sizes. Sigificat results have bee achieved i models i which claims occur accordig to a Poisso process; see for e.g. Cai (007), Zhu ad Yag (008), Mitric ad Sedova (010), Asmusse ad Albrecher (010), Mitric, Bădescu ad Staford (01) ad the refereces therei. Meawhile, the reewal risk model uder such assumptios has bee studied much less frequetly i the literature durig this period. Oe otable cotributio is Kostatiides et al. (010), where the authors preset asymptotic results for the ifiite time absolute rui probability. The mai focus of the paper of Mitric, Bădescu ad Staford (01) was the aalysis of the Gerber Shiu discouted pealty fuctio (Gerber ad Shiu, 1998), startig from a geeral o-reewal risk model with costat force of iterest. They preseted a geeral methodology that leads to a tractable aalytical solutio for the Gerber Shiu fuctio (with a pealty that depeds o the deficit oly), with coefficiets that are obtaied i a recursive fashio. Moreover, they obtaied closed form solutios for the absolute rui probabilities ad the deficit at the absolute rui, extedig the results obtaied uder the classical case with expoetial claim amouts. For the diffusio approximatio, the idea to replace the risk process by a Browia motio igores the presece of the overshoot ad other thigs. Siegmud (1979) proposed a so-called corrected diffusio approximatio (CDA) that reflects iformatio i the icremet distributio beyod the mea ad variace. The objective of the corrected diffusio approximatio is to take this ad other deficits ito cosideratio. The set-up is the expoetial family of compoud risk processes with parameters. Blachet ad Gly (006) developed this method to the full asymptotic expasio iitiated by Siegmud. Fu K. A. ad Ng C.Y.A. (014) cosider a cotiuous-time reewal risk model, i which the claim sizes ad iter-arrival times form a sequece of idepedet ad idetically distributed radom pairs, with each pair obeyig a depedece structure. They suppose that the surplus is ivested i a portfolio whose retur follows a Lévy

5 Assessmet of the Rui Probabilities process. Whe the claim-size distributio is domiatedly-varyig tailed, they obtaied asymptotic estimates for the fiite- ad ifiite horizo rui probabilities. 3. Surplus Process ad Assessmet of the Rui Probabilities We are iterested i the surplus process U t, t 0 i cotiuous-time (or its discrete-time versio,ut, t 0,1, 0 ), which measures the surplus of the portfolio at time t. We begi at time zero with U u (or i discrete case U0 u ), the iitial surplus (iitial reserve, risk reserve). The surplus at time t is U t u Pt At S t (or i discrete-time versio Ut u Pt At St ), where P t, t 0 (i discrete time Pt, 0,1, measures all premiums collected up to time t, St, t 0,1, At, t 0 (i discrete time At, 0,1, t ) is the premium process which S t, t 0 (i discrete time ) is the loss process, which measures all losses paid up to time t, ad t ) is the earig process, which measures all earigs of ivestmet icome up to time t. We make the followig assumptio: Pt (or P ) may deped o S r (or S ) for r t (for example, t divideds based o favorable past loss experiece may reduce the curret premium). Defiitio 3.1 The cotiuous-time, fiite-horizo survival probability is give by, 0 for all 0 0 u P U t t U u, the cotiuous-time, ifiitehorizo survival probability is give by 0 for all 0 0 r u P U t t U u, the discrete-time, fiite-horizo survival probability is give by u, P U 0 for all t 0,1,, U u, the discrete-time, ifiite-horizo t 0 survival probability is give by u PUt 0 for all t 0,1, U0 u cotiuous-time, fiite-horizo rui probability is give by u, 1 u, cotiuous-time, ifiite-horizo rui probability is give by u 1 u, the, the, the

6 Paul Taasescu, Iulia Mircea u, 1 u,, ad discrete-time, fiite-horizo rui probability is give by the discrete-time, ifiite-horizo rui probability is give by u 1 u. A) The geeral mathematical model of a isurace risk i cotiuous-time is composed of the followig objects: X of idepedet ad idetically distributed radom a) A sequece i i 1,,3, variables, havig the commo distributio fuctio F ad a fiite mea m. The radom variable (r.v.) X is the cost of the i th idividual claim. i b) The stochastic process N N t; t 0, the compay i the time iterval X are idepedet objects. i The total amout of claims i 0,t is The risk process Y Y t ; t 0 N t is the umber of claims paid by 0,t. The coutig process N ad the sequece N() t S() t X. (1) i1 is defied by Y t c t S t i, () where c 0 is the costat premium rate per uit time. U t u Y t, (3) Thus, the isurer s surplus at time t is where u U0 is the iitial capital. Also ; 0 U t t is the risk process. Whe the premium rate is ot a costat, we obtai a geeralized model. Thus, if the premium at the momet t is fuctio t ct, the Y t cxdx S t (4) k X We deote mk E X i, k 1,,3,., ad M X E e the momet geeratig fuctio (mgf) of the radom variable X. Note that m m1. We cosider that there exists a costat (average amout of claim per uit 1 Nt as.. time) such that Xi. (5) t t i1 We defie the safety loadig as the relative amout by which the premium c rate c exceeds, thus. (6) I the classical risk model, the process N is a homogeeous Poisso process with itesity (arrival rate), so that the surplus process of a isurer is described by 0

7 Assessmet of the Rui Probabilities Nt () U() t u c t X (7) i1 i We will use the mea value priciple i order to compute the et premiums, thus c (1 ) m. (8) I the cotiuous-time case, we have the followig: Defiitio 3. The rui momet T is T if t 0 U t 0. (9) Defiitio 3.3 The rui probability with respect to iitial reserve u ad the safety loadig is, if 0 0, u P U t U u g c. (10) t0 The rui probability as a fuctio of iitial reserve is if 0 0 t0 u P U t U u (11) I this case we have the followig propositios. Propositio 3.1. Assume that (5) holds. i) If 0 sup S t c t, the a.s. ad hece u 1 ii) If 0 large u. 0t for all u., the sup S t ct a.s. ad hece u 1 0t ad g 1 rx Let hr e 1 df x 0 for all sufficietly m. The adjustmet coefficiet (or Ludberg expoet) R is the smallest positive solutio of the equatio: hr cr 0. (1) Propositio 3.. If the adjustmet coefficiet R exists, the: Ru RS a) the rui probability is u, e E e 1, (13) where S( ) ( C( ) )) represets the severity of the loss at the momet of rui;

8 Paul Taasescu, Iulia Mircea b) (Cramer s asymptotic rui formula) u u, ~ C e u, (14) m where C. M ' R m 1 X Corollary: If the idividual claim follows a expoetial distributio with parameter, the u, e g u g We will focus o a approach of estimatig a risk process ad the rui probability usig the Browia motio. I some particular cases, it is obtaied the diffusio approximatio, which ca be derived from approximatig the risk process with a Wieer process (Browia motio) with drift. It regards the way the surplus process Y t based o the compoud Poisso process is related to the Wieer t process. Take a limit of the process t. Y t as the expected umber of dowward jumps becomes large ad simultaeously the size of the jumps becomes small (i.e. ad 0, where the jump size is X V, so that V has fixed mea ad variace). Because the Wieer process with drift is characterized by the ifiitesimal mea ad ifiitesimal variace, we impose the mea ad variace fuctios to be the same for both processes. Thus, c m ad m is M exp t c M 1. Therefore X Yt 0. The mgf of Yt lim MYt exp t t, which is the mgf of the ormal distributio N t, t. Let fuctio time iterval t ad variace fuctio 0,. Let T t u W t W t, t 0 deote the Wieer process with mea t. We cosider the probability of rui i a if 0. The probability of rui before is t, if u P T P W t u. 0 t 0

9 Assessmet of the Rui Probabilities Propositio 3.3. The probability of rui before is u u u u, e. (15) Lettig, the ultimate rui probability is diffusio approximatio). u m u m u e e (i.e. the Corollary: The probability desity fuctio of the time legth util rui is give by f T 3 u u e, 0. Hece to obtai the expected time util rui, give that it occurs, the idea behid the diffusio approximatio is to first approximate the claim surplus process by a Browia motio with drift by matchig the two first momets. Cosider that the claim sizes are idepedet ad idetically distributed o-egative radom variables with cumulative distributio fuctio F ad fiite mea m ad fiite variace. Thus the stadard diffusio approximatio is m u DA u, expu m (16) For light-tailed radom walk problems Siegmud (1979) derived a correctio which was adapted to rui probabilities by Asmusse ad Biswager (1997). A alterative coverig also certai heavy-tailed cases was give of Hoga (1986). The result will be a approximatio of the type 4 u m1 m3 m1 m 3 m1 u CDA u, 1 exp u 3 (17) 3 m 3 m m whe m5, where m i is the i-th momet of F (evidet m1 m). It is so-called corrected diffusio approximatio of the rui probability. Whe F is the Uiform(0,b) u distributio fuctio we obtai: DA u, exp3 (18) b

10 Paul Taasescu, Iulia Mircea 9 u 3 u CDA u, 1 exp3 (19) 4b 4 b I this formula we ca ormalize b. Aother risk model is the Sparre Aderse model. This satisfies the followig hypotheses: i) The claim sizes 1,,... form a sequece of i.i.d. radom variables with commo distributio fuctio B that has a fiite mea 0 ; ii) Occurrece times T1, T,... are idepedet of, 1, hece iteroccurrece times 1 T 1, T T 1,, are i.i.d. radom variables idepedet of, 1. We assume that U0 u is a iitial risk reserve, ad that the isurace compay receives a sum that equals c per uit time determiistically (i.e. the itesity of the gross risk premium c 0 ). Let U be the level of the risk process just after the th payoff. Therefore, we have U U1 c, 1. Let Y c, 1. By G we deote a distributio fuctio (d.f.) of a radom variable 1 where F is the d.f. of a radom variable c 1. Let Y, the Gu Bu ydf y 0 Ec, 1 1 be a relative E k k the rui momet u if 0: S u. The,, probability of rui before the th payoff, ad u P u safety loadig. We assume 0. We deote S0 0, S 1 Y, 1, ad 1 u P u is the is the probability of rui i ifiite time. Let M sup 0 S. I this case M is fiite almost surely. Thus, we have u PM u ad u, Pmax S k u. We deote by G I the x itegrated tail distributio of G, i.e. GI x G ydy G y dy, x 0, k where G 1 G. If G I belogs to the sub expoetial class S, the the approximatio for the probability of rui i ifiite time is give by 1 u~ G y dy as u. u

11 Assessmet of the Rui Probabilities B) We cosider a discrete-time isurace model. Let the icremet i the surplus process i period (usually year) t be defied as Wt Pt At St, where: P t is the premium collected i the tth period, S t is the losses paid i the tth period, A t is ay cash flow other tha the premium ad the paymet of losses, the most sigificat cash flow is the earig of ivestmet icome o the surplus available at the begiig of the period. The surplus at the ed of the tth period U t t U t P t A t S t u P j A j S j. (0) is the 1 j1 Let the assumptio that, giveut 1, the radom variable W t depeds oly upo Ut 1 ad ot upo ay other previous experiece. For the discrete-time isurace model, we evaluate the rui probability usig the method of covolutios. The calculatio of rui probability u, t P Ut 0U0 u is recursively, usig distributio of U t. Suppose that we obtaied the discrete probability fuctio (pf) of oegative surplus Ut 1: t1 f j PUt1 u j, j 1,,, where uj 0, j u, t 1 PUt1 0 U0 u. Let g j, k PWt wj, k Ut1 u j u t u t g f obtai,, 1 P U a g f ad We shall use t1 t j, k j j1 wj, k u j a. S t X i i1 claim of cotract i. For. The the rui probability is. The to t1 j, k j (1) j1 wj, k u j 0, where is the umber of isurace cotracts ad X i is S t we shall use the bouds sm mi X ad 1i 0 S SM max X i. I particular case, whe X :, we have 1i 1 p p 0 S 0 S sm : ad 1 p p SM :. 1 p 1 1 p i

12 Paul Taasescu, Iulia Mircea Here, we aalyze the iverse problem: to determie the iitial reserve whe it is give the rui probability. The followig otatios will be used: u - the risk reserve, mi u - the miimum reserve risk, α - the accepted probability of rui, - the safety loadig factor of the risk premium, X i, i=1,,... - the idepedet ad idetical distributed radom variables describig the claims or losses, with expectatio E(X)=m ad variace Var(X)=σ, - the umber of isurace cotracts, S X is the aggregate demad or claim. Calculatig the risk premium o the basis of the mea PRIM 1 m. The value priciple, the premium icome or reveue PRIM is coditio the accepted rui probability should fulfill is: PS PRIM u As S m m u S m m u. (), usig Chebyshev s iequality, we get: PS m m u (3) m u u From () ad (3), it follows that: mi m ad ucheb σ m. (4) From the o-egativity coditio of the reserve, we obtai a upper boud of the load mi factor. As well, we have: max ucheb, m 4m (5) Let us deote S m Z, the, usig (), we obtai PZ m u 1 ad from the Cetral Limit Theorem (CLT), we get: i1 i u z m ad 1 mi CLT u z m (6) 1 where z is the cuatile of order 1 of the stadard ormal distributio, 1 z1 1. Here, we ca use the mai result of Schulte (01) that the volume of the Poisso-Vorooi approximatio behaves asymptotically like a Gaussia radom

13 Assessmet of the Rui Probabilities variable if the itesity of the Poisso poit process goes to ifiity. A alterative approach would be to apply the uderlyig geeral cetral limit theorem directly to the ifiite Wieer Itô chaos expasio, which gives a sum of a ifiite umber of expected values of products of multiple Wieer Itô itegrals as a upper boud. z Similarly as above, we obtaied a upper boud of the load factor: 1. m mi z1 CLT We also have: max u, (7) 4 m 4. Numerical illustratio ad coclusios From 01 Aual Report of Romaia Isurace Compay (GRAR) we observe that exist isurace policies which produce the rui: gross writte premiums for RCA are 66,991,57 lei, but the gross idemity paymets for RCA are 13,099,067 lei, however gross writte premiums for fire policies are 96,961,35 lei ad gross idemity paymets for them are 13,909,967 lei. We give a first sceario for the discrete-time model. Suppose that aual losses ca assume the values 0,, 4, 8, ad 10 moetary uits (m.u.), with probabilities 0.3, 0.3, 0., 0.1, ad 0.1, respectively ad losses are paid at the ed of the year. Further suppose that the iitial surplus is 5 m.u., ad a premium of 3 m.u. is collected at the begiig of each year. Iterest is eared at 4% o ay surplus available at the begiig of the year. I additio, a rebate of 0.5 u.m. is give i ay year i which there are o losses. We wat to determie the rui probability at the ed of each of the first three years. We usig formula (1), the obtaied results are gived i Table 1. Table 1. The surplus process ad the rui probabilities Ut 1 (m.u.) t 1 w j,1, gj,1 j f 0;0.3 w, g j, j, ;0.3 w, g j,3 j,3 4;0. w, g j,4 j,4 8;0.1 w, g j,5 j,5 10; (7.8;0.3) (6.3;0.3) (4.3;0.) (0,3;0.1) (<0;0.1) 5,1 0.1

14 Paul Taasescu, Iulia Mircea (10.758; 0.09) (9.58; 0.09) (7.58; 0.06) (3.58; 0.03) (1.58; 0.03) (9.198; 0.09) (7.118; 0.06) (.958; 0.03) ( ; 0.07) (1.491; 0.07) ( ; (6.0091; (3.991; ( ; 0.07) ( ; 0.07) ( ; ( ; ( ; (7.698; 0.09) (5.618; 0.06) (1.458; 0.03) 5, 0.1 (1.3091; 0.07) ( ; 0.07) (8.6691; (4.5091; (.491; ( ; 0.07) (9.1051; 0.07) ( ; (.88051; ( ; (5.698; 0.06) (3.618; 0.04) (1.698; 0.03) (<0;0.03) (<0;0.0) (<0;0.0) (<0;0.0) (<0;0.01) (<0;0.01) ( ; (8.7491; (6.6691; 0.01) (.5091; (0.491; ( ; (7.1051; ( ; 0.01) ( ; ( ; 0.01) (6.3091; (4.7491; (.6691; (4.3091; (.7491; (0.6691; (<0;0.003) (<0;0.003) (<0;0.003) (<0;0.003) ( ; (3.1051; ( ; (.68051; (1.1051; (<0; (<0;0.003) (<0;0.003) (.51731; ( ;

15 Assessmet of the Rui Probabilities ( ; ( ; ( ; 0.01) ( ; (<0; ( ; 0.01) ( ; 0.01) (.87731; 0.008) (<0;0.004) (<0;0.004) ( ; ( ; (.19091; (<0;0.003) (<0;0.003) ( ; (.63091; 5, ( ; (<0;0.003) (<0;0.003) For secod sceario, we suppose te policies with aual loss ca assume the value 0. m.u. with probability 0.6, ad loss is paid at the ed of the year. Further suppose that the iitial surplus is 1 m.u., ad a premium of 0.5 m.u. is collected at the begiig of each year. Iterest is eared at 4% o ay surplus available at the begiig of the year. I additio, a rebate of 0.1 m.u. is give i ay year i which there is o loss. We determie the dow border of rui probability ( db ) at the ed of each of the first four years. Usig sm we obtaied the results i Table. Table. The dow border of the rui probabilities U t1 t 1 f j wj,1, g j,1 wj,, g j, (m.u.) (0; ) (; ) 1 1 (1.56; ) (<0; ) db 1, (.0384; ) ( ; ) db 1, ( ; ) ( ; ) ( ; ) (<0; ) db 1, ( ; ( ; ) )

16 Paul Taasescu, Iulia Mircea ( ; (<0; ) ) ( ; ) (<0; ) db 1, Secodly, we suppose that i cotiuous-time risk model F is Uiform(0,1) distributio fuctio. Usig formulae (18)-(19) we obtaied values by a process of approximatio for rui probabilities. We give these values i Table 3, Table 4 ad Figure 1. Table 3. The diffusio approximatios u, u\ DA Table 4. The corrected diffusio approximatios u, CDA u\ Iitial reserve u i moetary uits

17 Assessmet of the Rui Probabilities Approximatios of rui probabilities 1,00 0,90 0,80 0,70 0,60 0,50 0,40 0,30 0,0 0,10 0,00 u=1 u= u=5 u=10 u=15 u=0 DA θ=0.05 DA θ=0.0 CDA θ=0.05 CDA θ=0. Figure 1. Diffusio approximatios u, ad u, Thirdly, we determie the iitial reserve for give rui probability. For this, we use formulae (4) ad (6). Let us cosider the idividual loss described by the 0 S discrete radom variable Xi :, where p We suppose that the umber q p of isurace cotracts is The values i Table 5 show that for probability of rui withi a accepted domai, the amout of reserves obtaied by Chebyshev s iequality are much higher tha those obtaied by the CLT, tes of times higher. DA CDA

18 Paul Taasescu, Iulia Mircea Table 5. The amout of the miimum reserves S (m.u.) mi u CLT mi u Ceb (m.u.) (m.u.) , We remark that: i) I the case there is o charge of safety loadig ( =0), the reserve determied either by Chebyshev s iequality or by CLT is ubouded relative to the umber claims. ii) The ratio betwee the maximum of miimum reserves equals the ratio betwee the maximum umber of claims ad depeds oly o the rui probability: mi max uche 1 ratio. mi max uclt z1 There are applicatios for which the diffusio approximatio delivers poor results, therefore proposed a corrected diffusio approximatio that reflects iformatio i the icremet distributio beyod the mea ad variace. The first problem is to fid the expected value of the maximum of a radom walk with small, egative drift, ad the secod problem is to fid the distributio of the same quatity. Sice Browia motio is skip-free, the idea to replace the risk process by a Browia motio igores the presece of the overshoot ad other thigs. The objective of the corrected diffusio approximatio is to take this ad other deficits ito cosideratio. Aother icoveiet of the diffusio approximatio is that is close to zero, ad we wat to cosider the give risk process with more safety loadig. Ackowledgemet This research was supported by CNCS-UEFISCDI, Project umber IDEI 303, code PN-II-ID-PCE

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