Bayesian Credibility for Excess of Loss Reinsurance Rating. By Mark Cockroft 1 Lane Clark & Peacock LLP

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1 By Cly o c o Lo Rc Rg By M Coco L Cl & Pcoc LLP GIRO coc 4 Ac Th pp c how o v cly wgh w po- pc-v o c o lo c. Th po co o Poo-Po ol ch wh po G o. Kywo c o lo c g By cly Poo Po G po Acowlg cl I wol l o h ho who hv wg h pp wh co clco ppop. o y poly. Th vw coclo p h pp y ow.. Ioco Rg c o lo ol c c po y cog h c cl o h pc-g o y copg h pol g ch po-g. Rly o h wo ho g o. So h goo o why o ho o o pply gv ccc o pl gc chg h pol cvly vlg h pc-g ho. B h g c h o vl o why ho h l cly h oh ll cly. Th pp l o wo plh pp y Bhl 967 P & Mhz 99 whch l cly pply o o c cl. Th wo h pp o ggg cl o ol ly. Aggg lo ch ggg cl ggg l p po coo o co h pp. Th pp l wh pc-g po-g o ol c. Tlpho:

2 Th o h pp v co. F ol h pol oc h oo. Th h wo o l cly c poly cl g By G/Poo cly ol. I Sco 4 h pp co l By cly ol o cl vy. I Sco 5 h pp cl cl vy co ol o By cly. Flly pccl co co o wh pc- po-g ho hv ly ogh.. Th pol Rc g o h o lo o ol c ly wh cl pp l.. ly -. Th o lo o h go-p FG gov y p o p. I g p h pol lly w hll p o h -p c. Th po po h h p h gv o gvg h cl o vc σ cool o. Th lo ov o cl N whch o gv poo N. Th o p clly. Th clc o h N ] whch copo o h pc l vo. Th o ovo N l o vl whch w p o h cl z o hv ow po poo o. Th pp co h cly l c lo o h By poo h h pol o h c ly. A l o o h oα β. 3. L cly By c ly cl co. Th ollowg ho poo o Bhl 967: Tho 3. Th l o c lo o h o wh h pol o ov h cly : wh σ ρ. ρ V Poo: To h o g l o c lo h o h w yg o α β whch c w α β β. Th α β. So h ol o β V β.

3 o wh pc o β g h vv l o zo gv β V V. Th p clly o whch ollow h: σ. Pg β gv h cly h o. P & Mhz 99 pply Bhl ho o h o cl p y o lo IBNR lv po ollow. I h N h o cl h Poo o wh p whch h po G o wh p. I lo h h y o cl hv ov o cl Σ. Ily w c h h po-g o o cl N ] ] / h pc-g N /. L 3. Th By poo o lo G wh p. Poo: N ~ Po o N ~ o wh z z z I N h By poo o o h N L Th h o o G o wh p.

4 Th ollowg l o Tho 3.3 o Coolly 3.8 clv o P & Mhz 99. Tho 3.3 h ccc c ov h l o N h cly o. wh h lo h o h By poo o. Poo: Th gl o o h cly ollow cly o Bhl ho Tho 3. ov og h po-g pc-g ov. Th o o ollow o og h h o ov h o y h o Poo o wh p σ h h G po o wh p h / vc /. To pov h h h o h By poo o N ] w o o L 3. h h poo o G wh p o: N ] whch h o. Tho 3.4 Th cool o o N Ngv Bol o wh p /. Poo: I N p h cool o o N h N whch h o o Ngv Bol wh p /. To pply h cly l o c ly o N o h o cl h c o h poly h gv FG cl c. I h pcl c wh ow o o h ollowg l ho pplcl.

5 L 3.5 N lo Poo wh p. Poo: N P N N P N P. Wh h o h poly h o cly h o o h Ngv Bol o wh p -. Hc: N P whch h o o h Poo.. L 3.6 h G o wh p /. Poo: o P P o /. /.. Hc: o o P / /. Rg o h ov cl o h y o cl o h o cl h c ch y Σ. A o w hv h ollowg o N : h po-g N ] // h pc-g N /.

6 Tho 3.7 h pcg coo h l o N h cly o. wh / / h lo h o h By poo o. Poo: By pplyg L o Tho 3.3 plcg h vl wh p wh /. Coolly 3.8 I cly oooc cg < h. Poo: Sc / lg o lg vl o. Tho 3.9 Th cool o o o cl cg Ngv Bol wh p /. Poo: By pplyg L o Tho 3.4 plcg h vl wh p wh /. I w l h h ow o.. w l co co o o p η h L o log pply. P & Mhz 99 o vg h c hogh y g h η p whch ol ollow. L 3. I η p h cool pco o h vc o oh l o η. Poo: Fo h cool pco o h : η η η η. Fo h cool pco o h vc σ η w o h h coo o L 3.5 pply c w cog cool o η o gv p h poly o cl cg ow. Hc η Poo h vc o η h h. Th cool vc howv.

7 L 3. I Y p o vl h: VY.] Y].V]].VY]VY].V]. Poo: By o o VY.]: V Y. ] Y. ] Y. ] Y Y ] ]. Y ]. ] V Y ]. ] ] Y ]. ] Y ]. V ] ]. V Y ] V Y ]. V ] Y ]. ] V ] Y ]. ] L 3. Th cool vc o h pc vl o c w V η η η V. Poo: g L 3.: V η V η. V V V V η η η η η V η η V V η Tho 3.3 Th l o N h cly o η. η η wh η η η η. CV η. No: o o vl Y CVY] h coc o vo VY] ½ /Y]. Poo: A o h gl o o h cly ollow cly o Bhl ho Tho 3.. Th pc-g o h po-g ollow o h coo o h cool L 3.. No h σ η η o L 3. y L 3.:

8 V η η V η η CV η Fo Tho 3. h cly co c w η wh η. σ η η V η η η CV η η. CV η Hc h cly co c h pc vl o η c.. wh cg c. Howv h coc o vo ow o y c wh h h pc vl c v y. Th p h l o wh c oly g o cl yhg l lg po gg h o choc o cl vy.. h η co. Th plo h P & Mhz By cl vy o c ly. Poly h o coo vy cv c po-g h Po cv. Th y vo volvg g o p lghly g o o h poly y co. W hll coc o o o low. Fh h h o o whch p o oh hol h c o po o wh o h o hll. I h c o h Po o po-g c w coc wh h popoo o h ll o c ly. Hc h hp p o py coc. W ho poo o o po cocg h pol ol Sco o o ov cl. I co 5 w hll gl h o o c c o lo ly wh o o ov cl.

9 Th cl whch p clly wh Po o p. Th Po h h o gv y h o co: F. h po o o G vl wh p. L co h poo o o h cool o o. Th ollowg l o FG cl lo o o Hlg 993 lhogh h pcc o c ly oly: Tho 4. Th By poo o lo G wh p Σ l -l. Poo: ~ P o. ~ o y Ψ. Ψ h By poo o o h h o: l l l l L Ψ Ψ Th h o o G o wh p Σ l -l. Th cool o h lghly o o h ogl o.

10 Tho 4. Th cool o o Y - wh Y Po o wh p. Poo: I p h cool o o h l l l l l y l/] o y - y y. Hc y y y y y y whch h Po o y. Th l o h cool o hly v. I howv vy pccl: w ow hv clo o o h o o h o cl h vy o cl gv G po o. L co ow cl o c ly. L h o vl y h c o ov y: Th ollowg l wll-ow vy l: g Po o h hp o h o o h c cl h hp h o o h FG cl. Fh oc w ow h poo p o h FG cl h poo p o h c cl h p wh h o o h c po.

11 Tho 4.4 h Po o wh p. Poo: Fo h o o : P P P < P. Th l h whv cly w c v g FG cl vy w c o c ly. Howv w co pcc pply h cly poc Sco 3 gh o cl vy. Th ov cl o o z lo h o ovo. 5. By cly o c ly cl Fo l c ly w lly o hv FG cl ov o hhol o co po T. W col o ol ll FG cl h ohg o g o g c o lo o cly wghg og o. Tho wh w l o FG cl w lg o oly h o FG cl h c T wh o lo o g. W ow poo o h po o h ll pol o ggg cl o o c o lo c: Th o FG cl y N ccog o Poo o p. h po G o wh p. Th cl v p o ch oh o N clly o Po wh p. h po G o wh p. Th y o cl wh ov o cl Σ ov cl z y.

12 W yg o pc l c o lo ly o pp l cl.. -. W c h o o h By poo o o o Ψ N ollow g h pc o N : l l l l N L L Ψ Ψ Hc h By poo o o h poc o h By poo o o h cl h cl v h w w pvo co. Fo plcy w c w y l / o h h cog p o po poo p Po Poo Σ y Th w c w ow h po o o cl o h c ly o L 3.6 / wh P /. L h cl vy o h ly o < I o o c h l o h o h ggg ly cl o ccog o Tho 3. w o clcl pco o h wo cl o o h po oo.

13 L o h p p wh / w h h cl o o h o o ggg cl gv. Th w o o. Th ollowg l o Hlg 993. L 5. Th cool o o h og h pol c clcl cvly ] ] o wh ] h h cl o o gl cl cool o h vl o. Poo S Goov l 984 p. Hc w c h wo cl o o ]. L 5. Poo. ] N ] ] N p. - ] ]- Fo L 5. ] ] ]. Tho. L 5.3 Poo // //. Follow o h po o L 5. g h pc o / o. L 5.3 pl w o coc o h cl o o h vl cl gv h po o. Th ollowg l L 5.4 L 5.5 Tho 5.6 o Hlg 993. No h h pcl c

14 o g g vl go h y Hlg c po o P l o y l vl o h pcl c y ly go wh cog o o pco o. L 5.4 y p Poo I w coo o h g < h ] Bgg ogh h o gv:

15 Th o h po o hc l zo. L 5.5 y h w g l wh g. Poo W c p / - pol: l l l Sg h o h o o L 5.4 gv: l l l Th l o o h po h o o g -. No h o g < g. Tho w c chg h g o o -. Tho 5.6 o gv y: g l l Poo Fo L 5.4: g l Sg h o h po o o o L 5.3 gv:

16 g g l l G wh p o: v v Tho: g l l. I o o g h l pco w ] w w vo o L L 5.7 c w : Poo Fo L 5.3 // Sg h l o L 5.4: whch wh gv h l o. Fo gv co : I v h ollowg hol o y : v v v

17 So w c l po o - wh g vly ollow: wh o h g clv o co: z z z o g : Th ov collp g l h o l o wh h o wll volv h gl: I h c g z l l h p o Gl o whch c vl ho h o hol h h. L o Ω o - vly ov o g. So w c : v v v Ω. Tho 5.8 ] c w : ] ] ] ] ] ] ] ] ] ] Ω Ω l l l l l l l l l l l l l ]

18 wh / /. Poo Fo L 5.7: ] Fo w o co h po p o o h co. To o h w o co h c < ply. I < : :. Coog o h vl o : < < ] W c p / - pol: l l l

19 Sg h po gv: < < l l l l ] pg o - : < < l l ] G wh p o: v v v < pov h v.

20 So g Ω o wh / /: { ] ] ] ] ] ] ] ] ] Ω Ω l l l l l l l l l l l l l l ] whch c g. Now w hv ll h l o cly wgh. Rcll o Tho 3. h h l c lo h By cly wh σ ρ ρ V. I o h wo cl o o h vl o wh ] ρ ρ. o Tho 5.6 ] o Tho Pccl coo W hv c olo o clcl h By cly o po pc o c o lo ly. I o pl o lg wh v p o o ] lo c v cop cly. Howv o llow gc pco g co clclg o y. I pcc h co c opp o o ch o wh o gc lo o pco. Howv h l wol h y c wh po pc ho o c ly lg h cly wghg clg: Th po cv h cl o vl hoc cl whch y cl cl o h coc g.

21 O wy o g hgh ly y pplyg ILF co o low ly h h g pc ho o volvg coo o pc po ho. Th pc ho y l o h lo o o po ho. Th po ho y h o llow o o h poly o vg c. Ayo g h cly wghg co o h wll o p h pccl coo. Rc Bhl H pc Rg Cly ASTIN Bll Vol Hlg Ol A Cl o Cog Po wh Applco o c-o-lo Rc ASTIN Bll Vol Goov M.J. Vyl F. Hzoc J. Ic P Noh- Holl A-Nw Yo 984 P Gy Mhz Ic Cly o Ty Rc c Pcg CAS co Pp 99

Convergence tests for the cluster DFT calculations

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