ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY PREMIUM ADJUSTMENTS
Stéphane LOISEL Université de Lyon, Université Claude Bernard Lyon 11
Julien TRUFIN École d'Actuariat, Université Laval2
Abstract: In this paper, we consider a discrete-time ruin model where experience rating is taken into account. The main objective is to determine the behavior of the ultimate ruin probabilities for large initial capital in the case of light-tailed claim amounts. The logarithmic asymptotic behavior of the ultimate ruin probability is derived. Typical paths leading to ruin are studied. An upper bound is derived on the ultimate ruin probability in some particular cases. The influence of the number of data points taken into account is analyzed, and numerical illustrations support the theoretical findings. Finally, we investigate the heavy-tailed case. The impact of the number of data points used for the premium calculation appears to be rather different from the one in the light-tailed case. Keywords: Discrete-time ruin model, Bühlmann's model, light-tailed claims, large deviation, ultimate ruin probability, Lundberg coefficient, path to ruin, heavy-tailed claims.
1. INTRODUCTION In risk theory, one problem of interest is to compute the probability for an insurance company to be ruined. The main results of this field can be found in the books of Asmussen (2000), Dickson (2005), Grandell (1990), Kaas et al. (2008) and Rolski et al. (1999). Until recently, the increments of the surplus process of the insurance company were often assumed to be independent. This independence assumption is however not justified for a certain number of reasons, including the implementation of experience rating mechanisms by the insurance company. In this case, the financial result of a given calendar year then depends on the result(s) of one or several previous years.
1 Université de Lyon, Université Claude Bernard Lyon 1, Institut de Science Financière et d'Assurances, 50 Avenue Tony Garnier, F-69007 Lyon, France. 2 École d'Actuariat, Université Laval, Québec, Canada.
BULLETIN FRANÇAIS D’ACTUARIAT, Vol. 13, n° 25, janvier – juin 2013, pp. 73- 102 74 S. LOISEL – J. TRUFIN
In this paper, experience rating is taken into account. Allowing the premium amount to depend on past claims experience through an appropriate credibility mechanism is in line with insurance practice in most lines of business. In the ruin theory literature, only a few authors have studied ruin problem with credibility dynamics. In Asmussen (1999), the ruin model allows adapted premiums rules. More precisely, a continuous-time risk process is considered with compound Poisson claims, where the premium rate is exclusively calculated on the basis of past claims statistics. In practice, one would use a mixture between a quantity based on past claim experience and a 'a priori' premium level. The easiest way to compute credibility adjusted premiums is to use Bühlmann's credibility model. This is what Tsai and Parker (2004) have done in a discrete-time risk model. They have studied, by means of Monte-Carlo simulations, the impact of the number of past claim experience years taken into account for the calculation of the premiums on the ultimate ruin probabilities for light-tailed claims. In this paper, we deal with a framework that is similar to the one of Tsai and Parker (2004). One of the main observation made by Tsai an Parker (2004) is that for an insurance company with a large initial capital and light-tailed claims, the use of a bigger (finite) number of past claim experience years to calculate the premiums decreases the ultimate ruin probability. Among others we derive theoretical results supporting this observation until a certain number of past claim experience. These results should be of interest for risk managers and people performing an ORSA (Own Risk and Solvency Assessment) of an insurance company, as discussed in Gerber and Loisel (2012). Some of the results established in Asmussen (1999) within the continuous-time compound Poisson process are also extended to the discrete-time risk model discussed in this paper. In the particular case where the claim amounts follow a compound Poisson, we find the same results than in Asmussen (1999) for the logarithmic asymptotic behavior of the ultimate ruin probability on the one hand, and for the typical paths leading to ruin on the other hand. We also show that the story is different for heavy-tailed claim amounts. Some authors as Nyrhinen (1994) and Mikosch and Samorodnitsky (2000) have investigated asymptotic ruin results for moving average processes. Some asymptotic results derived for our particular process are similar to some obtained by these authors for this related class of processes.
ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY 75 PREMIUM ADJUSTMENTS
2. THE MODEL Let us consider an insurance company with an heterogeneous group of policyholders mixing p individuals with r different risk levels. We assume that there are no observable risk characteristics enabling to distinguish the risk level of these individuals (this group of individuals can be seen as the result of a risk classification previously performed with the observable risk factors). The risk faced by the insurance company depends on the risk levels of each individual. Therefore, the number of potential portfolios for the insurer is equal to nr= p . Obviously, two portfolios with the same numbers of people within each risk level are equivalent since they present the same risk. Let us denote by ji() the risk level of the ith individual ( ip=1, , ) for the jth portfolio ( j =1, ,n ) and by qk ( kr=1, , ) the probability for a policyholder to belong to the risk level k . The probability for the insurance company to cover the portfolio j is p then given by pq= ∏ . j i=1 ji() In such situation of practical relevance, when it is not possible to rely on additional risk factors, the insurance company uses credibility premiums in order to determine the pure premium corresponding to its portfolio or equivalently to each of its policyholders. The model proposed in Bühlmann (1967) and (1969) is largely used in practice. In this paper we focus on this well-known credibility model. ()j Let us denote by Yk the annual claim amounts of the company and by Yk the independent and identically distributed annual aggregated claim amounts for portfolio j , with common distribution function Fj . The a priori mean of the Yk 's is of course given by n μ ()j =[]=YYpkj [ ] , (1) j=1 ()j where Y is a random variable with distribution function Fj , while the a priori variance breaks up in two terms: σν2 + =[]= Yak , (2) with n ()j − μ 2 aY=([]) pj (3) j=1 and
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n ν ()j =[]. Ypj (4) j=1 Parameter a measures the part of the variance coming from the heterogeneity of the portfolios, whereas parameter ν measures the part of hazard in the a priori variance. As mentioned in Tsai and Parker (2004), some casualty insurers only consider a finite number of most recent periods of claim experiences to renew the premium. Let us + denote by m ∈∪∞ {} the so-called "horizon of credibility", which corresponds to the number of periods taken into account for the calculation of the premiums. 0 So, depending on the choice of m, with the convention that =0, the premium i=1 of the company in time k −1 is given by ∑ = (1− ) + ( , ) (1+ ), (5) , , , ( , ) where min(mk ,− 1) a z = (6) km, ν +−min(mk , 1) a is Bühlmann's credibility factor and η >0 is the premium security loading. Our premium rule depends then on the parameters μ , a and ν . For simplicity, we have assumed that no historical data is available. This simplifying assumption does not influence the asymptotic results derived in the sequel. Now, if the portfolio held by the insurer is portfolio j , then the dynamics of the insurer's surplus obeys to the equation kk ()j +− ()j (7) UuCkm, =, im, Yi ii=1 =1 ()j where the Cim, 's are described by equation (5) with YYi = i , and where u is the initial capital of the company. The corresponding ultimate ruin probability may be written as ψ ()j ()=PruU()jj <0forsomek| Uu () = . (8) m km, 0,m Immediately, it appears the existence of a subset of "bad" portfolios, denoting bm , ∈ ψ ()j such that for all jbm , m ()=1u for all u . Indeed, let us define +−ημ + ()j η()j (1 )((1zzYmm ) [ ]) − m =1, (9) []Y ()j where
ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY 77 PREMIUM ADJUSTMENTS
ma z =. (10) m ν + ma ∞ η()j Obviously, for m < , m corresponds to the average security loading of the ∞ η()j η premium for km> while for m = , m = , which corresponds this time to the →∞ security loading for k . The subset bm is then defined as η()j ≤ bjmm={ =1, , n : 0}. (11) Also, in the following, the complementary subset of ("not bad") portfolios is denoted ∈ ()j ≤ μ ⊂ by bm , and we shall say that j g if and only if []Y . Clearly, we have g bm . The purpose of this paper is to investigate the behavior of the ultimate ruin ψ ()j probabilities m ()u for large u in the case of light-tailed annual claim amounts. First, the ψ ()j logarithmic asymptotic behavior of m ()u is derived. Then, typical paths leading to ruin ψ ()j are studied. Also, an upper bound on m ()u is derived in some particular cases. Afterwards, we are able to determine, for each portfolio j , the influence of "strategy" m ψ ()j on m ()u for large u . Next, whatever the portfolio j , optimal strategies, in a sense to be defined in the sequel, are deduced. As an illustration, numerical simulations are also ψ ()j performed. Finally, we also study the influence of m in first order on m ()u in the heavy-tailed case. As we shall see, the conclusion is totally different than in the light-tailed case.
3. LIMIT RESULTS FOR LIGHT-TAILED CLAIMS In this Section, for simplicity, let us assume that for all j =1, ,n , there exists ()j r()j >0 such that []erY exists for all 0 3.1 The case m < ∞ ψ ()j 3.1.1 Asymptotic behavior of m ()u Theorem 3.1 Suppose that there exists a unique positive solution to the equation −−rz(1 )μη (1 + ) rY()j (1 −+ (1 η ) z ) eemm =1, (12) ρ()j which we denote m . Furthermore, assume that for all k , the sum of the first k annual 78 S. LOISEL – J. TRUFIN k rZ losses ZYC=(()j − ) possesses a finite moment-generating function []e k for kimi=1 i , ρ()j 0 ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY 79 PREMIUM ADJUSTMENTS −−+rzY(1 (1ηη ) ) [()jj ](1 + () ) rzY (1 −+ (1 η ) ) () j eemm m=1, (15) and consequently ρ ()j ρ()j =,m (16) m −+η 1(1)zm ρ()j where m is the classical Lundberg coefficient defined as the unique positive solution of the equation −+ ()jjη () ()j eerY[](1)m []=1.rY (17) ∞ η()j Now, from classical results, it is then obvious that the inequality (>)>0m is a ρ()j necessary condition for the existence of m . Let us prove that the inequality ν (>)>0∞ η()j is equivalent to m < and j ∈b . m aη m Proposition 3.1 ν (>)>0∞⇔η()j mjb < and ∈ . (18) mmaη 1 ν Proof. () : If ∞ >>0η()j , then, obviously z < , or equivalently m < , m m 1+η aη −+μη()j −+ η and (1zYzmm ) (1 ) > [ ](1 (1 ) ) . Thus −+μη()j + η()j ((1zzYmm ) [ ])(1 ) − m =1 []Y ()j [YzzY()jj ](1−+ (1ηη ) ) + [ () ](1 + ) >1mm − []Y ()j −+η + +η − =(1 (1 )zzmm ) (1 ) 1=0. ⇐ ∈ −+μη()jj + () (): Since j bm , we have ((1zzYmm ) [ ])(1 ) > [ Y ] . Thus, ν 1 (1−+zYz )μη (1 ) > [()j ](1 −+ (1 η ) ) . Now, since m < , z < . So, η()j < ∞ mmaη m 1+η m ()j −+η and [Yz ](1 (1 )m ) > 0 . Consequently (1−+zYz )μη (1 ) [()j ](1 −+ (1 η ) ) ∞−−>=η()j mm 1> 1=0. m ()jj−+ηη () −+ [YzYz ](1 (1 )mm ) [ ](1 (1 ) ) ρ()j This result highlights the obvious fact that the existence of m implies that j ψ ()j ∈ must belong to bm , since m ()=1u for jbm . 80 S. LOISEL – J. TRUFIN ν Let us explain the reason why the horizon of credibility , denoted by m()c aη from now, constitutes a critical value for the existence of the Lundberg coefficient and ψ ()j consequently for the logarithmic asymptotic behavior of m ()u . ()c ()j From mm= , the global impact of each Yk on the surplus process for km> , ()j −+η i.e. Yzk (1 (1 )m ) , becomes negative, which is, of course, a drastic change in the ∅ ≥ ()c nature of the insured risk. Also, bm = for all mm. More fundamentally, in first order, the only effects selected of the credibility dynamics compared to the classical case m =0 are a modification of the security loading ρ()j and a decreasing of the claim amounts. Indeed, equations (15) and (16) teach us that m is identical to that of the classical ruin model with a premium security loading equals to η()j ()j −+η m and annual claim amounts given by Yzk (1 (1 )m ) . Therefore, in this associated classical ruin model, from the critical value m()c , no ruin can be observed, which explains the reason why a Lundberg coefficient does not exist in this case. ()j ()j rY ()j rY ∞ So, since []e exists for all 0 ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY 81 PREMIUM ADJUSTMENTS ()c ρ()j ρ()jj () ()jc () mm< such that m exists, we always have m < r , then we put mm= . Hence, if we replace condition (C1) by the stronger condition mm< ()j , we have now ρ()j necessary and sufficient conditions to guarantee existence and uniqueness of m such that ρ()jj () m < r , i.e. result (13). 3.1.3 Large deviations path to ruin ()j ∈ Let us assume that mm< and j bm , i.e. that result (13) is valid. The goal here is to understand how ruin occurs for large u . More particularly, we want to determine the typical shape of a path leading to ruin when the initial capital of the company is large. To this end, let us use large deviations results of Glynn and Whitt (1994). First of all, for large κρ′ ()j u , we know that ruin occurs roughly at time u /(mm ) . Next, the cumulant γ ()r Z generating function κρ′ ()j of κρ′ ()j is asymptotically changed from u/(mm ) u/(mm ) γ ()r κρ′ ()j to u/(mm ) κρ′ ()j u/(mm ) γργρημ()rrz+−()jj ()=(1) () −+ (1) − uu/(κρ′′()j )mjm /( κρ () ) im , mm mm i=1 ()jj () (rY+−+++ρη ) 1 ( zz++ z )(1 + ) mimim1, 2,min(imu+ , /κρ′ ( ( j ) ) ), m e mm u/(κρ′ ()j ) mm + ln . ()()jj i=1 ρηYzz1−+++ (++ z )(1 + ) mimim1, 2,min(imu+ , /κρ′ ( ( j ) ) ), m e mm ()j Consequently, given that ruin occurs, the claim size distribution Yk is ρ ()j exponentially tilted by the time-dependent factor m ()k , with ρκρ ()j ,for=1,,/()ku ′ ()j − m mmm ρ ()j ()=k κρ′ ()j − . m uk/(mm ) ()jj () () j 1−+zkumu (1ηρ ) , for = / κρ′′ ( )−+ 1, , / κρ ( ) m m m mm mm κρ′ ()j Hence, for large u , the drift of the path to ruin at time k (ku=1, , /mm ( ) ), δ denoted as m ()k , is given by 82 S. LOISEL – J. TRUFIN ()jj () ()j ρ ()kY Yem δμηη−+−−+ mm()=(1kz ) (1 ) (1(1 ) z m ) <0. (20) ρ()jj()kY () e m However, notice that since u →∞, the time period κρ′′()jj−+ κρ () δ (/umumm ( ) 1,/ mm ( )) where m ()k decreases with k is negligible κρ′ ()j − δ compared with the time period (1,um /mm ( ) ) where m ()k remains constant. This analysis highlights that ruin is caused by a succession of small claims, followed by a succession of moderately large claims. ψ ()j 3.1.4 Upper bound on m ()u In this subsection only, we take into account a possible claim experience of h years ()j ()jj () say, i.e. (YY−+hh12,,, −+ Y0 ), because, this time, as we see in the following, it will impact the result. Hence, equation (5) becomes k−1 Yi −+μ ikmh=max(−− ,1 ) +η Czzkm,,,=(1 km ) km (1), (21) min(mh ,+− k 1) with min(mh ,+− k 1) a z =. (22) km, ν ++−min(mh , k 1) a Let bmh=min( , ), and let us denote ψ ()j (uyy , , , ) = Pr U()jjjj < 0forsomek | UuYyYy () = , () = , , () = . (23) mb01−−km,10,m 0 0− b 1 b ψ ()j We are now in position to derive an upper bound on mb(,uy01 , , y− ) based on the same approach than in Gerber (1982). ()j ∈ Theorem 3.3 Assume that mm< and j bm . Then −ρ()j m uˆ ψ ()j ≤ e mb(,uy01 , , y− ) , (24) −ρ()j Uˆ eTmT|<∞ where ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY 83 PREMIUM ADJUSTMENTS ( ) ( ) ( ) = , +(1+ )[ ( , ̅ +⋯+ , ̅ )+ ( , ̅ +⋯+ , ̅ )+⋯+ ( ) ˆ ()j ( , )( , ̅ +⋯+ ( , ), ̅ )], uUˆ = 0 , and TkU=inf{ |km, <0}, zkm, with z = . km, min(mh ,+− k 1) Proof. First, we note that ˆ ()jj++η () ++ + () j ++ UUkkmkmmkmkmm=(1)[(km,1 Yz k++1, z , )( Yz k− ++− 1, z 1, ) ++()j ++ Yzzmax(km−+ 1,1 − h ) ()]km++−+−1, m max( km 1,1 hm ), ()jj+− () = UCYkm−1, km, k ++η ()jj ++ + () ++ (1 )[Yzkk (km++1, z kmm , ) Yz−1 ( km ++− 1, z kmm 1, ) ++()j ++ Yzzmax(km−+ 1,1 − h ) ()]km++−+−1, m max( km 1,1 hm ), ( ) ∑ = ( ) +(1− ) (1 + ) + (1 + ) ( , ) − ( ) , , , ( , ) ++η ()jj ++ + () ++ (1 )[Yzkk (km++1, z kmm , ) Yz−1 ( km ++− 1, z kmm 1, ) ++()j ++ Yzzmax(km−+ 1,1 − h ) ()]km++−+−1, m max( km 1,1 hm ), ()jj++η () ++ + () j ++ =(1)[(UYzzYzzkm−−1, k 1km,1,,2, k+− m m )( k − 2 km k +− m m ) ++()j ++ Yzzmax(km−− ,1 h ) ()]km,max(,1), m+−− k m h m +−μη + −()j −+ η ++ (1zYzzkm,1,, ) (1 )k (1 (1 )( k++ m k mm )) ˆ +−μη + −()j −+ η ++ =Uzkkm−++1, (1 ) (1 ) Yk (1 (1 )( z kmkmm 1,, z )). Φ ()j Hence, if we denote by k the sigma-algebra generated by {,=1,2,,}Yii k, we have −−ρρ()jjˆˆ () −−ρμηρη()jjj(1zYzz ) (1 + ) ()() (1 −+++ (1 )( )) mkUUΦ mk−1 mkm,1,, m kmkmm++ eeee|=k−1 . Now, let us define −−μη +()j −+ η ++ rz(1km,1,, ) (1 ) rY (1 (1 )( z k++ m z k mm )) hrekm, ()= e . ≥−+ ≤ Obviously, for kmh1, hrhrkm, ()= m (). Since zzkm−1, km , and ≥ ≤≤ zzkm−1, km , , we clearly have hrhrhrkm−1,() km , () m (). Consequently, by (12), ρρ()jj≤ () hhkm, () m m ()=1 m . Thus, we get 84 S. LOISEL – J. TRUFIN −−ρρ()jjˆˆ () − ρ () j ˆ mkUUΦ≤ mk−−11ρ()j mk U eehe|=kkm−1 () . −ρ()j ˆ So, the process {}e mkU is a super-martingale. Let w be a positive integer. By the Optional Stopping Theorem (considering the stopping time Tw∧ =min{, Tw }),we get −−ρρ()jjˆˆ () − ρ () j ˆ − ρ () j ˆ − ρ () j ˆ mUU0 ≥+≥ mTw∧ mT U mw U mT U e e=. eITw≤≤ eI Tw> eI Tw Hence letting w →∞, we have −−ρρ()jjˆˆ () − ρ () j ˆ mmTUU0 ≥∞∞∞ mT Uψ ()j eeTTeTuyy|< Pr[<]= | ≥ ()j ∈ 3.1.5 The case mm and j bm ≥ ()j ∈ Now, let us consider the case mm and j bm . For the next result, let us ψ ()j ψ ()j ρ()j ρ()j ρ()j ρ()j η()j η ()j η()j denote m ()u by m,η ()u , m by m,η , m by m,η , m by m,η and m by η ()j η η ()j m,η to reveal explicitly the dependence in . Obviously, notice that m,η increases with η ρ()j η . Hence, we know by classical results that m,η also increases with and consequently ρ()j m,η by (16). Furthermore, let us define ν ηηηη(,1) ≤(j) =min :m,η >0andm< . (25) aη and ν ηηηη(,2) ≤(j) =max :m,η >0andm< . (26) aη Proposition 3.2 Assume that ρρ()jj< r()j ≤ () . Then, we have mm,,ηη(,1) (,2) 1 ψ ()j ≤− ()j lim sup lnm,η (ur ) . (27) u→∞ u ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY 85 PREMIUM ADJUSTMENTS ηηη(,1)≤ (,2) ρ()j ()j Proof. Obviously, there exists < such that m,η = r . For all η(,1) ≤ηη < , we have 11ψψρ()j ≤−()j ()j limsupln()ln()=.m,η uu lim mm,,ηη uu→∞uu →∞ Letting η ↑η , we obtain (27). ()j ()j λ rY As a particular, let us assume that YExp ()j . In this case, []e exists all λ j for r < λ and is equal to . Clearly, r()j = λ . Now, since j λ − j j r 1 ()j λ lim ln Pr[U1,m < 0] = j (28) u→∞ u ψ ()j ≥ ()j and that m ()uU Pr[1,m <0], result (27) becomes 11ψψ()jj () −λ liminfln()=supln()=.mmjuu lim (29) uu→∞uu →∞ 3.1.6 Conclusion ∈ ψ ()j In conclusion, on the one hand, if jbm then m ()=1u for all u . On the other ∈ ()j ψ ()j hand, if j bm , then for mm< , the logarithmic asymptotic behavior of m ()u is given by equation (13) while for ()>∞≥mm()j , the logarithmic asymptotic behavior of ψ ()j m ()u is this time characterized by equation (27). ∞ 3.2 The case m = ∞ Clearly, with zm =1, i.e. m = , the results of the previous section have no meaning. This is the reason why this important case has to be treated separately. ()j 3.2.1 Asymptotic behavior of ψ ∞ ()u Let us state an analogue of Theorem 3.1. Theorem 3.4 Suppose that there exists a unique positive solution to the equation 1 rY()j (1++ (1η )ln x ) dxln e = 0, (30) 0 ρ()j rZk which we denote ∞ . If we assume that for all k , []e exists for 0 86 S. LOISEL – J. TRUFIN 1 ()j ()j lim lnψρ∞∞ (u ) =− . (31) u→∞ u 1 rZ Proof. Again, it suffices to show that κ∞ (re ) :=lim →∞ ln [k ] exists for k k ρ()j κρ()j 0 ( ) ( ) ∑ ∑ ( ) , ln [ ] = − (1 + ) (1 − , )+ln . Now we have ∑ ( ) ( ) −(1+ ) = ( ) 1 − (1 + ) , . , −1 Hence we get ( ) , ∑ ( ) ∑ ln [ ]=− (1+ ) (1 − , )+ln . Thus, we have k 11rZk −+ημ − limln [er ] = (1 ) lim (1 zi,∞ ) →∞kk →∞ kki=1 kk−1 z +∞ ()j −+η l 1, rYi 1(1 ) l 1 ili=1 = + lim ln e . k→∞ k It is obvious that k 1 − lim (1zi,∞ ) = 0. →∞ k k i=1 Furthermore, note that kk−−11 z +∞ a l 1, =. lal+ν li== li Since we have ll+1 aa a dt≤≤ dt, llal++νν al−1 al + ν we find that ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY 87 PREMIUM ADJUSTMENTS k−1 ak+−+ν zl+∞1, a(1) k ν ln≤≤ ln . ai+−+ν l a(1) i ν li= Consequently, letting k →∞, we get k−1 zl+∞1, ai − =ln . lak+ν li= So, we obtain ( ) 1 1 ∑ ( ) lim ln [ ]= lim ln . → → Hence we have ai k rY ()j 1(1)ln++η +ν +ν 11rZk ak a ak limln [ee ] = lim ln →∞kkaak →∞ +ν kki=1 ()j ai k rY 1(1)ln++η a ak+ν =ln[]lim e →∞ ak +ν k i=1 1 rY()j ()1(1)ln++η x =lndx e . 0 ()j It appears that result (31) requires only existence and uniqueness of ρ∞ (since the ()jj () condition ρ∞ < r is always fulfilled). ()j Equation (30) and so ρ∞ does not depend on parameters μ , a and ν . It means →∞ ψ ()j that for u , the logarithmic asymptotic behavior of m ()u is insensitive to the initial ()jj++ () YY1 k −1 premium and to the speed with which premiums C ∞ become (1+η ) . k, k −1 N As a particular case, let us consider YU()j , where N is Poisson i=1 i β distributed with parameter and the Ui 's are i.i.d. random variables. We assume that N and the Ui 's are independent. We have ( ) [ ]= ∑ = ∑ | = ∑ Pr[ = ] = ∑ [ ] Pr[ = ] = ∑ Pr[ = ] = ∑ Pr[ = ] = [ ] = ( ) (32) 88 S. LOISEL – J. TRUFIN β r − since eerN= (1) e . So, the equation (30) becomes 11()j rY(1++ (1ηη )ln x )ββ rU (1 ++ (1 )ln x ) − dxln e = dx e = 0. (33) 00 We recognize the equation (2.2) in Asmussen (1999). In fact, this is not surprizing ()j since ρ∞ is insensitive to parameters μ , a and ν . ()j 3.2.2 The Lundberg coefficient ρ∞ ∞ ηη()j ∅ Since m = , we know that m =>0 for all j , i.e. bm = . ()j Proposition 3.3 Assume that κ∞′′ ()>0r . Then, ρ∞ exists and is unique. κ ()j ++η Proof. Obviously, we have ∞ (0) = 0 . Denoting Yx()1(1)ln by Zx , we have rZ rZ rZ Zexx Ze I + Ze x I ≤ 11xxZxZ xx>0 0 κ∞′ ()=rdx = dx rZ rZ 00eexx rZ Zex I+ ZI ≤ 1 xZxx>0 xZ 0 ≥ dx . rZ 0 e x Hence we get 1 κηη′ ()jj++ − () ∞ (0) = [YdxxY ] 1 (1 ) ln = [ ] < 0 0 since η >0, and κ∞′ ()r becomes positive for large r . Now, since κ∞′′ ()>0r by assumption, the proof is over. 3.2.3 Large deviations path to ruin As previously, we aim at knowing how ruin occurs for the case m = ∞ . Of course, ()j ruin occurs roughly at time u /(κρ∞∞′ ). Now, the cumulant generating function γ γ ()j ()r of Z ()j is asymptotically changed from ()j ()r to u/(κρ∞∞′ ) u/(κρ∞∞′ ) u/(κρ∞∞′ ) ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY 89 PREMIUM ADJUSTMENTS ()j u/(κρ∞∞′ ) γργρημ+−()jj () −+ − ()jj()rrz∞∞ () ()=(1) (1) i , ∞ uu/(κρ∞∞′′ ) /( κρ ∞∞ ) i=1 ( ) ( ) ( ) / ( ) , ( ) ( ) ∑ ( ) / ( ) + ∑ ln ( ) . (34) ( ) ( ) / ( ) , ( ) ∑ ()j Consequently, given that ruin occurs, the claim size distribution Yk is this time ()j exponentially tilted by the factor ρ∞ ()k , where ()j u/(κρ∞∞′ )1− ()j zl+∞1, ()j ρη∞∞()=1k −+ (1 ) ρ . (35) l lk= Since u →∞, we get ()jjk () ρη∞∞()=1k ++ (1 )ln ρ . (36) κρ′ ()j u /(∞∞ ) − 1 ()j ()j 1+η The factor ρ∞ ()k is negative as long as ku Figure 1: Typical path to ruin for the case m = ∞ 90 S. LOISEL – J. TRUFIN ψ ()j 3.3 Impact of the insurer's strategy m on m ()u for large u ψ ()j ψ ()j ∞ 3.3.1 Comparison of m ()u and m+1()u for large u (with m < ) Let us assume that the insurer's strategy is to take an horizon of credibility m < ∞ . The objective is to determine for large u if it is beneficial to take one year more to renew ψ ()j ψ ()j the premium. In other words, the objective is to compare m ()u and m+1()u for large u . ∈ For jbm , we have the following Proposition: Proposition 3.4 ψψ()j ≤()j m+1()uum ()=1forallu. (37) ⊆ ∈ η()j ≤ Proof. It suffices to show that bbmm+1 . Let jbm+1 . By definition, m+1 0 . −+μ ()j ′ ()j − μ η()j ≤ Let hx()=(1 x ) x [ Y ]. We have hx()=[ Y ] >0 since m+1 0 means ()jj≥+ημ − + () η [YzzY ] (1 )((1mm++11 ) [ ]) and >0. Thus, since zzmm+1 > , we have ηη()j +−+−≤hz()mm η hz (+1 ) η()j ∈ m =(1 ) 1<(1 ) 1=m+1 0, or equivalently, jbm . []YY()jj [] () ∈ This means that if jbm , it is beneficial to pass from an horizon of credibility m to an horizon m +1, whatever the initial capital u . ∈ ψ ()j Now, for j bm , it could seem a priori that the comparison of m ()u and ψ ()j m+1()u for large u differs according to the involved portfolio j . Indeed, two opposite a priori behaviours seem to take shape. If j ∈ g , then hx′()≤ 0 and consequently ηη()j ≥ ()j ψψ()j ≤ ()j m m+1 , which seems to be in favor of m ()uum+1 () whatever u and in ∈ ′ particular for large u . But, on the other hand, if j bgm \ , then hx()>0 and ηη()j ()j consequently m < m+1 , which seems this time to be in favor of the inequality ψψ()j ≤ ()j m+1()uum () whatever u . For mm< ()j , we have the next Proposition: ≥ Proposition 3.5 There exists a positive constant u0 such that for all uu0 , we ψψ()j ≤ ()j have m+1()uum (). (38) ∈ ()j ⊆ ∈ Proof. We know that j bm and mm< . Since bbmm+1 , jbm+1 . Hence, we have to consider two cases. Firstly, let us assume that mm+1< ()j . By Property 3.2, our result is proved. Secondly, let us assume that mm+≥1 ()j . Hence, we know that ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY 91 PREMIUM ADJUSTMENTS 1 ψρ()j − ()j lim lnmm (u ) = u→∞ u and 1 ψ ()j ≤− ()j lim sup lnm+1 (ur ) , u→∞ u ρ()jj () and since m < r , this completes the proof. ∈ ηη()j ≥ ()j Thus, even if j g and hence m m+1 , (38) holds for large u . In fact, this is not surprizing. With light-tailed claims, the event ruin occurs after many atypically large claims. At most m is large, at most the premium reacts favorably to this large claims, i.e. η()j η()j m+1 becomes larger than m in such circumstances. These facts are confirmed by our analysis of the path leading to ruin. Indeed, it teaches us that ruin is caused by a succession ρ()j of many atypically large claims more dangerous as m is large since m increases with m. For mm≥ ()j , we cannot conclude anything like that, since the only result we have is Equation (27). ∈ In conclusion, assuming that the strategy of the company is m, if jbm , then it is always beneficial for large u to increase by 1 the horizon of credibility. Furthermore, if ∈ ()j ≥ ()j j bm , for mm< , the same conclusion comes while for mm , we cannot certify that this is also the case. 3.3.2 Sub-optimal strategies m for large u Let us investigate sub-optimal strategies m for large u . Definition 3.5 A strategy m1 is said to be sub-optimal compared to a strategy m2 if for all portfolio j , we have ψψ()jj()uu≤ () () for u →∞. mm21 Definition 3.6 A strategy m1 is said to be sub-optimal if there exists a strategy m2 such that m1 is sub-optimal compared to m2 . We have the next Proposition: ()j ∞ Proposition 3.6 mm< min{:jm()j >0} like m = are sub-optimal. ()j Proof. In view of Propositions 3.4 and 3.5, it is clear that mm< min{:jm()j >0} is ()jj () sub-optimal. For m = ∞ , it is also obvious since ρ∞ < r . 92 S. LOISEL – J. TRUFIN As we have seen, for a given portfolio j , increasing m decreases the number of "bad portfolios" and provides better premium adaptation to scenarios leading to ruin, at least as long as mm< ()j . An extension of this reasoning to all m would allow to think that all m < ∞ is sub-optimal compared to m = ∞ . However, Proposition 3.6 shows that this is not the case. In fact, as u →∞, all m < ∞ becomes negligible compared to ruin time T . Hence, the premiums will be able to react "quickly" (relatively to T ) and "significantly" (at least for m "large" but finite) to a succession of large claim amounts, whatever the time where these claims occur and whatever the claim amounts observed before. But for m = ∞ , obviously, this is not the case anymore. Indeed, as time goes by, the premium reaction will become less and less effective to finish by not be able anymore to thwart several important successive claim amounts. This phenomenon is even truer if the claim amounts observed initially are small. This is well highlighted by the analysis of the path leading to ruin. 3.3.3 Numerical illustration Let us consider a case where n =3. Let us assume that Y ()j ( j =1,2,3) are λ λ 2 Exponentially distributed, with mean 1/ j and variance 1/ j . We then have μ λλ++ λ λμ−+−+−22 λ μ λμ 2 = (1/11 )ppp (1/ 2 ) 2 (1/ 33 ) , ap= (1/112233 ) (1/ ) p (1/ ) p νλ22++ λ λ 2 and = (1/11 )pp (1/ 22 ) (1/ 33 ) p. λ λ λ Let us assume that 1/1 = 3/4, 1/2 =1, 1/3 = 5/4, pp12==1/3 and η ∈ ∈ =0.1. We have j =1,2 g and jb=3 m as long as m <30. The critical horizon of credibility m()c =250. For our numerical purpose, let us compare strategies ∞ ψ (3) m = 0, 2,10, 250,1000 and . Of course, for all u , m ()=1u for m =0,2 and 10 . Carrying out 100000 simulations on an horizon of 10000 time periods, we obtain the results below, also shown on Figures 2 to 3. ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY 93 PREMIUM ADJUSTMENTS ψ (1) m ()u u 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 m =0 43.758% 29.996% 20.566% 14.086% 9.844% 6.754% 4.604% 3.218% 2.268% 1.566% 1.120% m =2 44.438% 30.316% 20.572% 13.850% 9.474% 6.352% 4.232% 2.866% 1.952% 1.326% 0.890% m =10 45.892% 31.502% 21.348% 14.278% 9.568% 6.308% 4.072% 2.696% 1.772% 1.124% 0.724% m = 250 46.936% 32.404% 22.034% 14.822% 9.920% 6.582% 4.252% 2.810% 1.828% 1.156% 0.758% m =1000 46.942% 32.408% 22.036% 14.827% 9.922% 6.583% 4.252% 2.810% 1.828% 1.156% 0.758% m = ∞ 46.948% 32.412% 22.042% 14.832% 9.924% 6.584% 4.252% 2.810% 1.828% 1.156% 0.758% u 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 m =0 0.764% 0.540% 0.386% 0.260% 0.200% 0.148% 0.086% 0.060% 0.046% 0.036% m =2 0.608% 0.408% 0.276% 0.214% 0.150% 0.088% 0.054% 0.042% 0.018% 0.010% m =10 0.450% 0.308% 0.220% 0.148% 0.094% 0.046% 0.020% 0.010% 0.006% 0.000% m = 250 0.478% 0.318% 0.236% 0.160% 0.094% 0.050% 0.024% 0.008% 0.005% 0.000% m =1000 0.478% 0.318% 0.236% 0.160% 0.094% 0.050% 0.024% 0.008% 0.004% 0.000% m = ∞ 0.478% 0.318% 0.236% 0.160% 0.094% 0.050% 0.024% 0.008% 0.004% 0.000% ψ (2) m ()u u 0 1 2 3 4 5 6 7 8 9 10 m =0 82.416% 69.122% 57.854% 48.526% 40.738% 34.052% 28.682% 24.060% 20.122% 16.894% 14.206% m =2 82.208% 67.124% 55.578% 45.370% 37.024% 30.182% 24.640% 19.976% 16.290% 13.186% 10.594% m =10 82.932% 68.260% 52.710% 40.310% 30.182% 22.138% 16.162% 11.560% 8.202% 5.890% 4.152% m =250 91.820% 79.506% 64.450% 49.480% 36.514% 25.882% 18.026% 12.112% 7.926% 5.256% 3.402% m =1000 92.023% 79.808% 64.905% 49.754% 36.901% 26.228% 18.273% 12.300% 8.109% 5.422% 3.529% m = ∞ 92.232% 80.108% 65.160% 50.184% 37.088% 26.424% 18.486% 12.528% 8.278% 5.542% 3.636% u 11 12 13 14 15 16 17 18 19 20 m =0 11.792% 9.918% 8.234% 6.904% 5.742% 4.812% 4.070% 3.374% 2.834% 2.372% m =2 8.688% 7.016% 5.686% 4.618% 3.746% 3.018% 2.462% 1.970% 1.598% 1.328% m =10 2.866% 2.016% 1.452% 1.004% 0.738% 0.538% 0.374% 0.280% 0.188% 0.134% m =250 2.190% 1.360% 0.860% 0.554% 0.380% 0.224% 0.140% 0.088% 0.060% 0.038% m =1000 2.170% 1.304% 0.823% 0.528% 0.367% 0.203% 0.128% 0.084% 0.058% 0.037% m = ∞ 2.286% 1.395% 0.865% 0.530% 0.365% 0.202% 0.127% 0.084% 0.058% 0.037% 94 S. LOISEL – J. TRUFIN ψ (3) m ()u u 0 2 4 6 8 10 12 14 16 18 20 m =0 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% m =2 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% m =10 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% m =250 99.992% 99.648% 96.894% 88.370% 73.964% 58.720% 42.814% 29.850% 20.238% 13.378% 8.816% m =1000 99.994% 99.790% 97.490% 89.582% 75.532% 57.010% 41.046% 28.186% 18.682% 12.006% 7.636% m = ∞ 99.996% 99.810% 96.894% 88.370% 75.124% 57.010% 41.046% 28.186% 18.682% 12.006% 7.636% u 22 24 26 28 30 32 34 36 38 40 m =0 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% m =2 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% m =10 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% 100.000% m =250 5.714% 3.660% 2.440% 1.566% 1.034% 0.704% 0.486% 0.330% 0.240% 0.142% m =1000 4.708% 2.862% 1.784% 1.048% 0.644% 0.412% 0.250% 0.146% 0.098% 0.038% m = ∞ 4.708% 2.862% 1.784% 1.048% 0.644% 0.412% 0.250% 0.146% 0.098% 0.038% On the one hand, asymptotically, whatever the portfolio j , we see that the larger m ψ ()j ≥≤()jc () is, the smaller m ()u is, even for mm() m . This confirms our theoretical ≥ ()j findings and intuition, namely that (also for mmmin{:jm()j >0} ) m is sub-optimal compared to m +1. On the other hand, for small initial capital u , a bigger value of m may this time increase in certain cases the ultimate ruin probability. Obviously, this observation is not unexpected. Remark Clearly, the numerical illustrations can not highlight the fact that m = ∞ is sub- optimal. 4. LIMIT RESULTS FOR HEAVY-TAILED CLAIMS We have seen that in the case where the Lundberg coefficient exists, adjusting premium with credibility may increase this one. As typical sample paths for which ruin occurs exhibit a sequence of moderately large claims, there is quite often enough time to compensate early losses with credibility-adjusted future premiums. On the opposite, in the ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY 95 PREMIUM ADJUSTMENTS heavy-tailed case, it is often said that ruin is likely to be caused by one large claim (see for example Theorems 1.1 and 1.2 in Asmussen and Klüppelberg (1996) in the classical risk model). It is logical to think that credibility plays a less important role for heavy-tailed claim amount distributions. In this Section, we focus on the regular variation case. Definition 4.1 The cumulative distribution function F with support (0,∞ ) belongs to the regular variation class if for some α >0 Fxy() −α lim =,yfory >0. (39) x→∞ Fx() ∈ ∈∞ We note F −α . The convergence is uniform on each subset yy[,)0 ∞ ( 0 96 S. LOISEL – J. TRUFIN λ +∞ ψ ()uFxdx (1− ()), − λμ W c W u where c is the (continuous-time) premium income rate, λ is the intensity of the Poisson process, FW is the cumulative distribution function of the individual claim amounts and μ α()j W is the expected individual claim amount. In the -regularly varying case with α()j >2 (this means that ()j −→∞−α 1()FxW x asx , this corresponds to λ 1 −+α()j ψ ()uu 1 . cont − λμ ()j c W α −1 This result may be adapted to the discrete-time model, with constant 1 -period premium income and independent, identically distributed (aggregate) claim amounts on each period, with α()j -regularly varying distribution: if the safety loading is positive, for α()j ψ >1, there exists C >0 such that the (discrete-time) probability of ruin disc ()u ()j ψ −+α 1 →∞ satisfies disc ()uCu as u . To show this, one might for example use arguments developed in Rullière and Loisel (2004) and Lefèvre and Loisel (2008). ψ To show that there exist C1 >0 and a function 1 such that ()j ψψ()j ≥→∞ ψ −+α 1 m ()u111 () u and () u C u as u , let us consider a modified model, in which future earned premiums due to each claim are ()j anticipated and earned at the claim instant: if claim amount Yk occurs at time k , the sum ()j of future credibility premiums associated to Yk corresponds to km+ ()j +η Yzk (1 ) im, . ik=1+ ()j In our modified model, each claim amount Yk is replaced with km+ ()j ()j −+η YYkk=1(1) zim, , ik=1+ and premium income ( ) ∑ = (1− ) + ( , ) (1+ ) , , , ( , ) is replaced with −+μ η Czkm, = (1km, ) (1 ). ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY 97 PREMIUM ADJUSTMENTS For each sample path, if ruin occurs in the modified model, then it occurs as well in the initial model as credibility adjusted premiums are the same in both models but are km+ received later in the modified model. Since from km=1+ , zz= , we have ik=1+ im, m ψ from Embrechts and Veraverbeke (1982) that the ruin probability 1()u in the modified ()j ψ −+α 1 →∞ model satisfies 11()uCu for some C1 >0 as u as long as the safety loading is positive in the modified model, which is guaranteed by condition (42). ψ To show that there exist C2 >0 and a function 2 such that ()j ψψ()j ≤→∞ ψ −+α 1 m ()uuanduCuasu222 () () , ψ ()j let us rewrite m ()u as ψ ()j ()=PruimUorimU(∃≤ 1 ≤ ,()jj <0) ( ∃ > , () <0) . m im,, im This means that ψ ()j (uimUimU )≤∃≤≤ Pr 1 ,()jj < 0 +∃ Pr > , () < 0 . m im,, im −α()j −+α()j The first term is Ou() and so ou()1 . The second term Pr∃imU > ,()j < 0 im, satisfies Pr∃imU > ,()j < 0 im, iim− ≤∃()jj + + +ημη () ++ − Prim > , Ykl > u / 2 u / 2 zmm (1 ) Y (1 )(1 zi ) kl=1 =1 iim− ∃+−++++−()jjηημ () =Pr>,im Ylk (1(1))>/2/2(1)(1) zmm Y u u zi lim=1−+ k =1 iim− = Pr[∃+−+im > , Y()j (1 z (1η )) Y ()j lkm lim=1−+ k =1 δδ ++ ++ημ −− >/2uiu /2(1)(1) zm i ], 22 where δημ+−−()j −+ η = (1 ) (1zYmm ) [ ](1 z (1 )) > 0 from Condition (42). 98 S. LOISEL – J. TRUFIN Hence, since for any sequences of nonnegative r.v.'s ()Xii≥1 and ()Zii≥1 , and for any sequences of nonnegative numbers ()aii≥1 and ()bii≥1 , we have ∃≥+ +≤∃≥ +∃≥ Pr[iXZab 1,iiii > ] Pr[ iXa 1, ii > ] Pr[ jZb 1, j > j ], we get that Pr∃imU > ,()j < 0 im, i δ ≤∃Prim > , Y()j > u / 2 + i l 2 lim=1−+ im− δ +∃Prim > ,(1 − z (1 +ηημ )) Y()j > u / 2 ++ (1 ) (1 − z ) − i . mm k 2 k=1 ()j −+α 1 →∞ The second term is equivalent to Cu3 as u for some constant C3 >0, because it corresponds to a ruin probability with claim size distribution in RV ()α ()j and with a positive safety loading (from (42)). The first term i δ Pr∃+im > , Y()j > u / 2 i l 2 lim=1−+ satisfies i δδi ∃+≤∃+()jj () PrimYu > ,ll > / 2 i Pr immYu > ,max > / 2 i , 22−+ lim=1−+ lim=1 which leads to i δδ Pr∃+≤∃+−imYu > ,()jj > / 2 i Pr immYuim > , () > / 2 ( ) l 22 i lim=1−+ δ ∃+()j =Pr imYu >0,i > /2 i . 2 This may be rewritten as i δδ u Pr∃+≤∃+im > , Y()jj > u / 2 i Pr i > 0, Y () > i . l 222 i mm lim=1−+ Obviously, we have ∞ uuδδ Pr∃+≤+iY > 0,()jj > i Pr Y () > i . ii22mm 22 mm i=1 ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY 99 PREMIUM ADJUSTMENTS Now, by Lemma 5.2 in Foss et al. (2009), we obtain ∞ umuδ 2 PrYiY()j >+→∞ Pr()j > asu , i 22mmδ 2 m i=1 which completes the proof. ∈ Let us conclude in saying that, on the one hand, if jbm then ψψ()j ≤ ()j m+1()uum ()=1 by Proposition 3.4 (clearly also valid for the heavy-tailed case), but ∈ on the other hand, this time, if j bm , then an increasing of the horizon of credibility has ψ ()j ()c asymptotically no impact on m ()u in first order, at least for mm< . ACKNOWLEDGMENTS This work is partially funded by the Research Chair Actuariat Durable sponsored by Milliman, the research chair Management de la modélisation sponsored by BNP Paribas Assurance and the ANR (reference of the French ANR project : ANR-08-BLAN-0314-01). Julien Trufin also thanks the financial support of Risk dynamic. REFERENCES S. ASMUSSEN. On the ruin problem for some adapted premium rules. MaPhySto Research Report No.5. University of Aarhus, Denmark, 1999. S. ASMUSSEN. Ruin Probabilities. World Scientific, 2000. S. ASMUSSEN and C. KLÜPPELBERG. 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Rough descriptions of ruin for a general class of surplus processes. Advances in Applied Probability, 30(4):1008–1026, 1998. T. ROLSKI, H. SCHMIDLI, V. SCHMIDT and J. TEUGELS. Stochastic Processes for Insurance and Finance. John Wiley and Sons, 1999. D. RULLIÈRE and S. LOISEL. Another look at the Picard-Lefèvre formula for finite- time ruin probabilities. Insurance: Mathematics and Economics, 35(2): 187–203, 2004. C. TSAI and G. PARKER. Ruin probabilities: classical versus credibility. NTU International Conference on Finance, 2004. ULTIMATE RUIN PROBABILITY IN DISCRETE TIME WITH BÜHLMANN CREDIBILITY 101 PREMIUM ADJUSTMENTS FIGURES ψ (1) Figure 2: Numerical results for m ()u ψ (2) Figure 3: Numerical results for m ()u 102 S. LOISEL – J. TRUFIN ψ (3) Figure 4: Numerical results for m ()u