Advaces Mathematcal ad Computatoal Methods Credblty remum Calculato Motor Thrd-arty Lablty Isurace BOHA LIA, JAA KUBAOVÁ epartmet of Mathematcs ad Quattatve Methods Uversty of ardubce Studetská 95, 53 ardubce CZCH RUBLIC bohda.lda@upce.cz http://www.upce.cz Abstract: - Improvg the qualty of premum calculato methods s a effectve factor reducg the surace techcal rsk of surace compay. Wth the developmet of the compettve surace market motor thrd-party lablty surace former socalst coutres the Bühlma-Straub credblty model has a wde rage of possbltes to be used. I ths artcle we deal wth ts applcato based o real data of fve Slovak surace compaes. Keyword: - Credblty premum, Motor thrd-party lablty surace, mprcal Bayes credblty model II, Credblty factor, Credblty pure premum Itroducto The orgal oly oe state surace compay the Czech ad the Slovak Republcs had a moopoly posto motor thrd-party lablty surace utl 5. Wth the emergece of a compettve market ths mportat type of surace a typcal stuato for the use of emprcal Bayes credblty models was created. mprcal Bayes credblty theory s the collectve ame for the vast lterature whch has developed sce Bühlma ad Straub`s (97. Although ths model s a bass for other more specfc models such as herarchcal, multdmesoal or regresso credblty models, ths artcle we deal wth oedmesoal Bühlma-Straub credblty model ad ts applcato. The problem for the surace compay s to determe the pure premum for the comg year ad there are two extreme postos:. the surace compay do ot has ay ow data or could decde to gore the past data from the polcy tself ad base the pure premum equal to µ o the almost certaly larger amout of data from smlar polces;. the surace compay could decde to gore the data from smlar polces ad charge a pure premum equal to x based solely o past data from the polcy tself. A credblty premum represets a compromse betwee the two extreme stuatos gve above. The credblty premum formula s: Z x Z ( c ( µ where Z s the credblty factor. Ths factor s the weght put o the data from the rsk tself ad depeds o the amout of past data avalable from the polcy tself. The value of Z s betwee zero ad oe ad should crease from year to year as more data are obtaed. The mprcal Bayes Credblty Model II (BCT II Our problem s to estmate a pure premum for a rsk, gve some data. Let Y, Y,..., Y,..., Y deote the aggregate clams successve year from ths rsk ad be a correspodg sequece of kow costats,,...,,...,. Costat,,,..., s terpreted as a measure of the amout of busess the -th year, for example the premum come or the umber of polces ssued year. The ISB: 978--84-7-3 59
Advaces Mathematcal ad Computatoal Methods Y,,,..., ( are stadardsed varables by removg the effect of dfferet busess levels. We assume that the dstrbuto of each depeds o a fxed, but ukow parameter θ. I accordace wth Bayesa approach to estmato we cosder θ as radom varable. Assumptos of the BCT II model:. Varables / θ, / θ,..., / θ are depedet but ot ecessarly detcally dstrbuted, θ θ does ot deped. ( ad ( o. Wth these two assumptos we ca defe: s ( θ ( θ m (3 ( θ ( θ (4. ervato of the Credblty remum We defe a pure premum year,,,...,, as ( Y θ ( θ m( θ (5 Because of a lear combato wth costats Q s kow, we eed estmate ( θ m as a a a... a ( a...,, a, a, a whch mmze: ( [ m ( θ a a a a ] By solvg equatos a..., a,... a ad usg the relatos (proof Waters, 994 ( m( θ ( m( θ [ ( m( θ ] k ( m( k ( θ [ ( m( θ ] ( ( s ( θ m( θ we get a ad ( [ ( m( ] θ a,,,..., the forms ( m( θ a φ (7 φ k k where a k,,,..., (8 φ k ( s ( θ ( m( θ φ. uttg (7 ad (8 to (, the estmate of the pure premum per ut of volume of rsk s ( m( θ ( s ( θ ( m( θ ( s ( θ ( m( θ ( m( θ Y (9 whch ca be rewrtte as credblty premum where ( m( θ Z ( Z ( m( θ / ( Z, ( s ( θ ( m( θ. arameter s stmato To calculate the pure premum for -th rsk we eed to estmate ( m( θ, ( m( θ, ( s ( θ from sutable set of data. For the purposes of these parameters estmato we regard the partcular -th rsk as oe of a set of rsks. We assume that for each of the rsks we have observed values Y of the aggregate clams past years,,...,. Furthermore, we kow the values of weghts,,,...,,,,...,. We assume that for each,,..., the dstrbutos of Y depeds o the ukow parameter θ, whch s fxed for each,,...,. Varables θ, θ,..., θ for each,,..., are depedet, but ot ecessarly detcally dstrbuted. arameters of rsks θ, θ,..., θ are depedet ad detcally dstrbuted. So fuctos. ISB: 978--84-7-3
Advaces Mathematcal ad Computatoal Methods ( θ m( θ ( θ s ( θ are the same for all ad ( m( θ, ( m( ( s ( θ them smply as ( m( θ ( m( θ s ( θ. θ, are depedet of so we shall to deote We wll use the followg otato:,, ( * The proposed estmators are show below: est ( s ( θ ( m( ( est θ ( ( ( m( θ ( (3 est (4 ( * ( The credblty pure premum for -th rsk s ( m( / Z ( Z ( θ (5 where credblty factors Z are dfferet for each rsk,,,...,, calculated by Z ( ( s ( θ m θ ( ( 3 Applcato of the BCT II Model Motor Thrd-arty Lablty Isurace The data below (Table show the aggregate clams for motor thrd-party lablty surace sx Slovak surace compay years -. Table Total clams Y ( thousads of Isurace Years comp. 7 8 9 Allaz 45,8 43,75 5,48 5, 47,4 ČSOB 4,88 3,8,,3,9 Geeral,44 3,8, 3, 7,9 KOO 43,4 55, 7,74 7,9 5,9 Uqa,8 3,, 7,99 9, Wuste 4,88 4,4 4,88,4 3,98 Source: Aual Reports -, Slovak Isurace Assoc. Table cotas the umbers of polces for ths type of surace busess for each rsk,,,..., ad each year,,,..., 5. These costats we wll use as the weghts to calculate the emprcal Bayes premums for all the compaes the comg year base o the data Table. Table umber of uderwrtg polces ( thousads Isurace Years comp. 7 8 9 Allaz 73, 73,4 79, 745, 7,7 ČSOB,9 7,8 5, 5, 55,8 Geeral 59,5 57, 9, 8, 85,7 KOO 44,5 58, 3, 595,5 8,8 Uqa 35, 59, 9,8,9 4,3 Wuste 5,9 8, 78, 4,8,4 Source: Aual Reports -, Slovak Isurace Assocato Supportg calculatos based o the data Table ad Table 4 of the stadardzed varables by relato ( cotas Table 3. Table 3 Supportg calculatos Isurace compay Allaz 383394,534 ČSOB 33,47 Geeral 753,5 KOO 78979,994 Uqa 4847,8773 Wuste 34748,847 Source: Ow calculatos Totals of colums Table 3 we get characterstcs 885 ad,973. Table 4 Stadardzed values of Isurace Years comp. 7 8 9 Allaz,3,59,7,7,8 ČSOB,48,5,47,45,39 Geeral,4,54,,7,93 KOO,98,,,,8 Uqa,,,5,75,8 Wuste,9,4,,58,37 Source: Ow calculatos ISB: 978--84-7-3
Advaces Mathematcal ad Computatoal Methods I Table 5 there are values ( for each combato of,,...,,,,..., 5. Last colum of ths table cotas the sums A of these values for each row. Table 5 Values ( ad A 7 8 9 A 5,4 5,74,,7 3,44 53,3,4 5,43,,9 3, 9,47 3 34, 7,7,7 8,77 4,8 9, 4 54,7 4,8 5,57 83,99,5 49, 5,57,89,5 3,75 5,93 5, 4,4,7,,9 8,73,4 Source: Ow calculatos As the sum of the values A last colum we get 5 ( A 544,58 Table Supportg values Isurace comp. Allaz 38 ČSOB 385 Geeral 84 KOO 83587 Uqa 388 Wuste 33787 Source: Ow calculatos Table cotas the supportg values to calculate * by (. Usg the sum of the values last colums of table, that s 55844, we get * 55844 848, 9 Table 7 Values ( ad B 7 8 9 B 3,3 78,3,, 3,3 4, 45,9,5 7,7 3, 5, 7, 3 48,8 4,5 3,3,9 97, 4,4 4 355,8 94, 9,7 57, 94, 45, 5, 3,7,3, 3,5 4, 7,,9 4,,, 5,3 Source: Ow calculatos of values table 7 wll be 5 ( 548,7 ow we ca estmate parameters of BCT II model by relatos (, (3, (4: ( m(, 973 est θ est est ( s ( θ ( m( θ * ( 5 848,,859 * 544,58,9 9 548,7 4 544,58 Table 8 cotas for each surace compay values of factor credblty ad value of credblty pure premum per ute of rsk, whch we have calculated by relatos ( ad (5. Table 8 Values of credblty pure premums Isurace compay Credblty factor Z Credblty pure premum ( Allaz,9988 5,3 ČSOB,933 48,8 Geeral,94 5,8 Koop,9955 8,8 UIQA,93958 8,83 Wüsterot,9935 3,33 Source: Ow calculatos Coclusos Formula (5 s a smple matter to calculate the emprcal Credblty premum for each surer f we have estmated ( m( θ, ( m( θ, ( s ( θ. The credblty factor s a measure of how much relace we are prepared to place o data from the rsk tself. It s a creasg fucto of the umber of years, for whch data are avalable ad s asymptotcally equal to. ( stmated s ( θ varace of the data from each rsk ad ( m( θ s a measure of the mea of the s a measure of varace betwee rsks. The larger are these values, the less relable are data from rsk tself or from other rsks. I Table 7 there are values ( ad the sums B of these values for each row. Total sum ISB: 978--84-7-3
Advaces Mathematcal ad Computatoal Methods Refereces: [] Bühlma, H., Straub,, Glaubwürdgket für Schadesätze. Mtteluge der Veregug Schwezerscher Verscherugs-mathematker 7, 97, pp. 33. [] Bühlma,H., Gsler, A., Course Credblty Theory ad ts Applcatos, Berl: Sprger, 5. [3] Kaas, R., Goovaerts, M., haee, J., eut, M., Moder Actuaral Rsk Theory, Bosto: Kluwer Academc ublshers,. [4] Gerber, H. U., A Itroducto to Mathematcal Rsk Theory, S.S. Hueber Foudato for Isurace ducato, Moograph umber 8, Rchard. Irw Ic. Homewood, Illos 979. [5] Herzog, T.., Itroducto to credblty theory. 3rd. edto, Wsted, Coectcut: ACT ublcatos, 999. [] acáková, V., Aplkovaá postá štatstka (Appled Isurace Statstcs, Bratslava: Iura dto, 4. [7] acáková, V., Šoltés,., Šoltésová, T.,Kredblý odhad škodove frekvece (Credblty stmato of Clam Frequecy. koome a Maagemet M, II, /9, pp. -. [8] Straub,., o-lfe Isurace Mathematcs, Zőrch: Sprger-Verlag, 988. [9] Šoltés,, acáková, V., Šoltésová, T., Vybraé kredblé regresé modely v havarom posteí (Selected Credblty Regresso Models Accdet Isurace, koomcký časops, Vol. 54, o.,, pp. 8-8. [] Tse Y. K., olfe Actuaral Models, Theory, Methods ad valuato, Cambrdge: Cambrdge Uversty ress, 9. [] Waters, H. R., Credblty Theory, dburgh: Herot-Watt Uversty, 993. [] Waters, H. R., A Itroducto to Credblty Theory, Lodo ad dburgh: Isttute of Actuares ad Faculty of Actuares, 994. ISB: 978--84-7-3 3