The Solution to the Bühlmann - Straub Model in the case of a Homogeneous Credibility Estimators

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1 5 Economy Infomatics, -4/005 The Solution to the Bühlmann - Staub Model in the case of a Homogeneous Cedibility Estimatos Lect. Viginia ATANASIU Mathematics Depatment, Academy of Economic Studies Oiginal pape, which contains a cedibility model, fo detemining the lineaized homogeneous cedibility pemiums at contact level. The fact that it is based on complicated mathematics, involving conditional expectations, needs not bothe the use moe than it does when he applies statistical tools lie SAS, GLIM. These techniques can be applied by anybody on his own field of endeavou, be it economics, medicine, o insuance. We give a athe explicit desciption of the input data fo the Bühlmann Staub model used, only to show that in pactical situations thee will always be enough data to apply cedibility theoy to a eal insuance potfolio. Keywods: homogeneous linea cedibility fomulae, Lagange multiplie, unbiased estimatos. I ntoduction In this aticle we fist give the Bühlmann Staub model see Section -, which consists of a potfolio of non-life insuance contacts. In Section we will give the assumptions of the Bühlmann Staub model and the optimal lineaized cedibility pemium is deived. In Section we give unbiased estimatos fo the stuctue paametes, such that if the stuctue paametes in the optimal lineaized cedibility pemium ae eplaced by these estimatos, a homogeneous estimato esults. In Section 3 we will show that this last estimato is in fact the optimal lineaized homogeneous cedibility estimato.. The Bühlmann Staub model Fo this model we loo upon the potfolio as epesented in Diagam. We conside a potfolio which can be subdivided in goups consisting of contacts with common is paamete, as in Diagam. Contacts Stuctue vaiables Obsevable vaiables with associated weights p e i o d t θ X (w ) X (w ) X t (w t ) Diagam Bühlmann Staub model Each contact =, is the aveage of a goup of w contacts, whee w is the weight (size) of the goup at time, with =,t. Rema: These weights aise when the contacts ae eplaced by aveages of identical contacts (with the same is paamete), and the weight then epesents the numbe of such contacts. The model consists of the stuctual vaiables θ and the obsevable vaiables X, whee =, and =, t. So, the contact consists of the set of vaiables:

2 Economy Infomatics, -4/005 (θ, X ) = θ, X, =, t, whee =, ; the contact indexed is a andom vecto consisting of a andom stuctue paamete θ and obsevations X, X,, X t, see Diagam : (θ, X ) = (θ, X,, X t ), whee =,. Of couse the vaiables X epesent the aveage of w contacts gouped togethe at time, as follows: X = w w i= X ( i), =, t and =,. The Bühlmann Staub assumptions can be fomulated as: (BS ): the contacts =, (the pais, the couples (θ, X ) with =, ) ae independent; moeove, fo evey contact =, and fo θ = θ fixed, the vaiables X,, X t ae conditionally independent. The vaiables θ,, θ ae identically distibuted. The obsevations X, =,, =, t have finite vaiance. (BS ): E(X θ ) = µ(θ ), =,, =, t (we assume that all contacts have common expectation of the claim size as a function µ( ) of the is paamete θ, whee =, ). Va(X θ ) = σ (θ )/w, =,, =, t, whee all w > 0 (apat fom the weighting facto w, we assume that the vaiance is also the same function of the is paamete),with X (i), i =,w, =,, =, t satisfying the hypotheses: (BS ) and (BS ), whee: (BS ): fo evey =, anf fo θ = θ fixed, the vaiables X (i), i =,w, =, t ae conditionally independent and identically distibuted. The vaiables θ,, θ ae identically distibuted and the obsevations X (i), i =,w, =, t, =, have finite vaiance, and: (BS ): E(X (i) θ ) = µ(θ ), i =,w, =,, =, t Va(X (i) θ ) = σ (θ ), i =, =,, =,w,t Consequence of the hypothesis (BS ): Cov(X, X q θ ) = 0, =,,,q =, t, < q. Obsevations: ) µ(θ ) is the pue net is pemium of the contact, with =,. ) the Bühlmann Staub assumptions expess the common chaacteistics of the is unde consideation. Now, we deive the optimal lineaized nonhomogeneous cedibility estimato. The optimal lineaized cedibility estimatos non-homogeneous ae given in the following theoem: Theoem : (lineaized non-homogeneous cedibility estimato in the Bühlmann Staub model) Unde the hypotheses (BS ) and (BS ) of the Bühlmann Staub model, the following optimal lineaized non-homogeneous cedibility estimato fo µ(θ ), fo some fixed, is obtained: M a = µ (θ ) = ( z )m + z M, (), t wq whee M = X w = X q denotes the individual estimato fo µ(θ ), and the esulting q= w cedibility facto fo contact is given by: z = aw / (aw + s ), with a = Va[µ(θ )], s = E[σ (θ )], m = t E[µ(θ )] as usual, whee w = w q, =,. q= To be able to use esult (), one still has to estimate the potfolio chaacteistics m, s, a. Some unbiased estimatos ae given in the following section.. Paamete estimation The estimatos obtained in the pevious section contained unnown stuctue paametes (the cedibility pemium fo this Bühlmann Staub model involves thee unnown paametes: m, s and a). So, the expessions fo these (pseudo-) estimatos ae no longe statistics. But since the contacts ae embedded in a collective of identical contacts, all po-

3 Economy 3 Infomatics, -4/005 viding independent infomation on the stuctue distibution, it is possible to give unbiased estimatos of these quantities, so we can eplace the unnown stuctue paametes by estimates. In this section, we conside diffeent contacts, each with the same stuctue paametes: m, s and a, so we can estimate these quantities using the statistics of the diffeent contacts. Some unbiased estimatos z m = M 0 = X zw = W w (whee: z. = z z. s = ( t ) = w s, s = fo the stuctue paametes: m, s and a, ae given in the following theoem. So, we will povide some useful estimatos fo the stuctue paametes: m, s and a in the following theoem: Theoem : (paamete estimation in the Bühlmann Staub model) The estimatos: ) () (X s X w ) (3) a = w.. w ( X w X ww) ( ) s /( w.. w ) (4) t w (whee: w.. = w = w q, X ww = X w ) ae unbiased estimatos of the coesponding stuctue paametes, i.e. = = q= = w.. E( m ) = m, E( s ) = s, E( a ) = a. Rema: In case m in () is estimated by M 0, we obtain a homogeneous linea combination of all obsevable vaiables, giving an unbiased estimate of m. The following section shows that this happens to give the optimal unbiased homogeneous lineaized cedibility esult. 3. Bühlmann Staub model fo homogeneous cedibility estimatos Replacing the stuctue paamete m by an unbiased estimate esults in a homogeneous cedibility estimato. In Section 3, we will show that this last estimato is in fact the optimal lineaized homogeneous cedibility estimato. Now, we deive the optimal lineaized homogeneous cedibility estimato. The optimal lineaized cedibility estimatos, homogeneous ae given in the following theoem: Theoem 3: (homogeneous cedibility estimatos in the Bühlmann Staub model) Min E + µ ( θ ) m c i ( X i m) α c, α, The solution of the following minimization poblem: t Min E µ ( θ ) ci X i, (5) c i= = (c =(c i ) ), such that E[µ(θ )]=c E( i X i (6), is M a = ( z )M 0 + z M (7), whit z as in Theoem. Poof: Let be fixed. The unbiasedness estiction (6) can be witten as c i = (8), because E(X i ) = E[µ(θ )] = m. We inset it in the expectation in (5), and add it to the function to be optimized with a Lagange multiplie α / m. The following poblem esults: c i (9) i )

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5 4 Economy Infomatics, -4/005 Since (9) is the minimum of a positive definite quadatic fom, it suffices to find a solution with all patial deivatives equal to zeo. Taing the deivative with espect to c i gives fo i =,, =, t : α + Cov[µ(θ ), X i ] = α + δ i a = c i c i Cov(X i, X i ) (a + δ s / w i ), i =,, =, t Using the covaiances (see [], the chapte 8, o []), one obtains the following system of equations: (Cov[µ(θ ), X i ] = δ i a, Cov(X i, X i ) = a + δ s /w i, Cov(X i, X i ) = 0 if i i ). These equations can be simplified as follows: α + δ i a = ac i. + s c i / w i (0), whee c i. = c i. Multiplying each equation with w i and summing these equations ove the index, gives fo each i : (α + δ i a) w i. = c i. aw i. + s c i. So, c i. = (α + δ i a) w i. / (s + aw i, ) (). Inseting () into (0) gives an expession fo c i : c i = (α + δ i a) ( aw /(aw i i. + s )) w i /s. = (α + δ i a) ( z i )w i /s Fom this the estimato (7) fo µ(θ ) becomes: ( α + δ a)( z ) µ ( θ ) = w i /s X i, whee still α has to be detemined in such a way that (6) holds, too. Summing all the c i. of (), one gets: c = α z a + z = αz./a + z = i. / and the esulting value fo α = a( z ) / z., inseted in (), gives afte some algebaic manipulations the following optimal estimato fo µ(θ ): M a = µ (θ ) = ( z ) X zw + z X w () (fo moe details, see [], fom Refeences). So, the theoem is poven. Rema: Finally we want to ema that in case one uses the fomula: M a = ( - z ) M 0 + z M we have E(M a ) m in case the estimatos fom Theoem ae used, because then z is dependent of M and M 0. Of couse the attactive popety of unbiasedness is lost this way, but we can still expect the esulting estimatos to be good. Fo instance when an estimato is a maximum lielihood estimato fo a paamete, so ae functions of it fo these functions of the paamete. Conclusions One liely choice in the minimization poblem: Min E µ θ ) g( X,..., X ), giving eas- g ( ) {[ ] } ( t ily computable pemiums, is: g(x,, X t ) = c 0 + t i= = c i X i, leading to so-called lineaized cedibility esults. Anothe possibility is to limit oneself to unbiased homogeneous linea estimatos, by equiing additionally c 0 = 0 and: E[µ(θ )] = c i E(X i ) Poceeding this way one gets homogeneous linea cedibility fomulae. By the equiement of unbiasedness the sum of the cedibility pemiums equals the global pemium on the top-level. In this pape we demonstated that the estimatos obtained fo the pue net is pemium on contact level ae the best lineaized cedibility estimatos, homogeneous fo the Bühlmann Staub model, using the geatest accuacy theoy.

6 Economy Infomatics, -4/005 5 So, the aticle povides the means to calculate the cedibility pemiums at contact level, which epesents the most ecent developments in Bayesian cedibility theoy. They cetainly pesent the only solution whee insuance industy faces iss with basic is chaacteistics that cannot be assigned to any established collective o with coveage unde cicumstances not ealie met. Refeences [] V. Atansiu, Contibutions to the cedibility theoy; thesis of doctoate, Univesity of Buchaest Faculty of Mathematics, 000. [] V. Atanasiu, Estimatoii liniai şi neomogeni de cedibilitate din modelul Bühlmann Staub, Studii şi Cecetăi de Calcul Economic şi Cibenetică Economică, /00, XXXV. [3] V. Atanasiu, Estimatoii de cedibilitate din modelul clasic al lui Bühlmann, Infomatica Economică, volumul III, n. 0, timestul II / 999. [4] V. Atanasiu, Estimaea paametilo stuctuali în modelul de cedibilitate linia şi neomogen al lui Bühlmann, Infomatica Economică, volumul IV, n. (4) / 000. [5] V. Atanasiu, Estimatoii de cedibilitate omogeni din modelul clasic al lui Bühlmann, Infomatica Economică, volumul V, n. (8) / 00. [6] M.J. Goovaets, R. Kaas, A.E. Van Heewaaden, T. Bauwelincs, Insuance Seies, volume 3, Effective actuaial methods, Univesity of Amstedam, The Nethelands, 99. [7] T. Pentiäinen, C.D. Dayin, M. Pesonen, Pactical Ris Theoy fo Actuaies, Univesité Pieé et Maie Cuie, 990. [8] B. Sundt, An Intoduction to Non Life Insuance Mathematics, Veöffentlichungen des Instituts fü Vesicheungswissenschaft de Univesität Mannheim Band 8 (VVW Kalsuhe), 990.