RISK-BASED CAPITAL REQUIREMENTS

Transcription

1 XXXV ASTIN Collouium 6-9 Jue 4 - Bee RISK-BASED CAPITAL REQUIREMENTS FOR PROPERTY AND LIABILITY INSURERS ACCORDING TO DIFFERENT REINSURANCE STRATEGIES AND THE EFFECT ON PROFITABILITY Ryaad Mee Daish Re Koeveje B, DK-84 Hole Tel: Fax: [email protected] Savelli Nio Caholi Uivesiy of Mila Lao Gemelli, Mila Tel: Fax: [email protected]. Absa A is heoeial simulaio model is hee applied i ode o assess he defaul is fo boh Popey ad Liabiliy muli-lie isues alo a sho-em ime hoizo. Fuhe, some Ris-Based Capial euiemes ae aalysed aodi o is measues as VaR, TVaR, UES ad Rui pobabiliy, ivesiai he impa of eihe diffee hoizo imes ad levels of ofidee. Moeove, he effe of diffee eisuae saeies o he above meioed RBC apial euiemes is deal wih, also wih efeee o he pofiabiliy of he Isue houh a Ris vs Reu ade-off aalysis. The esuls of he model show how ew RBC euiemes i o-life busiess should pay hihe aeio o he pie of eisuae oves, havi o oly a favouable so effe o he is bu also o he expeed value of he eu, ha mih iease he oal is of isue s isolvey eve i a sho ime hoizo. This may be useful i he pese debae oei he ew EU apial euiemes o be esablished i he Solvey II phase, ad fo ieal is modelli evaluai he fiaial seh of he ompay. Keywods: No-Life Isuae Solvey, Ris-Theoy simulaio models, Reisuae saeies, Ris measues, Ris ad Pofiabiliy ade-off. INTRODUCTION AND FRAMEWORK OF THE MODEL. The mai ae of he pese pape is o aalyse he is pofile of eihe a popey ad a liabiliy muli-lie isue, o boh solvey (o fiaial seh ad eu behmas, ad moeove o show also he effes of some adiioal eisuae eaies. The famewo of he model povides a is heoeial appoah whee he udewii is is maily deal wih, bei he ivesme vaiables eaded as sohasi bu wih o edi is ad wheeas he u-off is aisi fom loss eseves is o osideed as well as he oelaio bewee diffee lies. I lassial Ris-Theoy lieaue he sohasi Ris Reseve U a he ed of he yea is ive by:

2 ( U ( j U j LR TX RE RE RE [( B X E ( B X C ] DV ( j / wih oss pemiums volume (B, sohasi aeae laims amou ( X ad eeal ad auisiio expeses (E ealized i he middle of he yea, wheeas j is he sohasi aual ae of ivesme eu. Fuhemoe, owihsadi he laim esevi u-off is o osideed hee, he eu omi fom he ivesme of he iiial Loss Reseve LR - is ae io aou, wheeas he Loss Reseve a eah ime is assumed o be a osa oeffiie of he oss pemiums (LR l*b. As o eisuae, ad RE B deoes he oss pemiums volume eded o eisue wheeas RE C ae he amous of laims efuded by eisue ad he eisuae ommissios, espeively. Fuhe, boh axaio TX ad divideds DV (he lae paid o soholdes a he ed of he eleva yea ae osideed io he model. I is woh o emphasize ha fo may eeal isuae lies (e.. hid-pay liabiliy he u-off is oei he developme of he iiial esimae of laim eseve is o eliible a all ad i paie i is a addiive soue of is. Fo eah sile lie of busiess he oss pemium amou is omposed of is pemium P E( X, wih safey loadis applied as a (osa uoa λ of he is pemium λ*p ad expese loadi as a (osa oeffiie applied o he oss pemiums: RE X B P λ P B Nowihsadi he is loadi oeffiie λ is ep osa ove he whole ime hoizo; i is iiially ompued aodi o he sadad deviaio pemium piiple fo he iiial pofolio suue as follows, eadi boh explii ad implii is loadi: ( ( x [ λ E( X E( j LR] β ( X wheeas x deoes he (osa ae of axaio. I paie, he isue may as fo a oal is loadi amou (e of axaio eual o β fo eah ui of sadad deviaio of he sohasi oal amou of laims of he lie. Hee he sadad value β5% is osideed fo he ompuaio of he oal is loadi fo eah lie of busiess. Fo eah lie of busiess, he omial oss pemium volume ieases yealy by he laim iflaio ae (i ad he eal owh ae (: B ( i ( B assumed aes i ad o be osa i he osideed ime hoizo bu o eessaily he same fo all lies of busiess. pae / 6

3 Followi he olleive appoah, he aeae laims amou is ive by a ompoud poess: ( X i, i X (fo eah lie of busiess whee is he adom vaiable of he umbe of laims oued i he yea fo he eleva lie ad i, he adom laim size of he i-h laim oued a yea fo he eleva lie. Vey ofe, he umbe of laims i eeal isuae is assumed o be Poisso disibued. Havi a dyami pofolio, he Poisso paamee fo eah lie of busiess will be ieasi (o deeasi eusively yea by yea by he eal owh ae suh ha is Poisso disibued wih paamee ( depedi o he ime. O he ohe had, he simple Poisso law feuely fails o povide a saisfaoy epeseaio of he aual laim umbe disibuio. Usually he umbe of laims is affeed by ohe ypes of fluuaios ha pue adom fluuaios: eds, sho-em fluuaios ad lo-em yles. I he pese pape eds as well as lo-em yles ae diseaded ad oly sho-em fluuaios ae ae io aou. Fo his pupose a suue vaiable will be iodued o epese sho-em fluuaios i he umbe of laims. I paie he (deemiisi paamee of he simple Poisso disibuio fo he umbe of laims of yea will u o be a sohasi paamee, whee is a adom suue vaiable havi is ow pobabiliy disibuio depedi o he sho-em fluuaios i is oi o epese. Sie eds ae diseaded, he oly esiio fo he pobabiliy disibuio of is ha is expeed value has o be eual o. Fuhe, we will assume ha is Gamma disibued wih paamees (h,h. Thus, he momes of he suue vaiable ae ive by: E( ( / h γ ( / h ( Coseuely, he umbe of laims ae Neaive Biomial disibued, ieasi he sadad deviaio of as well as he sewess, hus ieasi he is of havi exessive laim umbes. Fo eah lie of busiess, he laim sizes, deoed by i,, ae assumed o be i.i.d. adom vaiables wih a oiuous disibuio - havi d.f. S( ad o be saled by oly he iflaio ae i eah yea. The momes abou he oii ae eual o: E j (, ( j j (, ( j i i E i i a j, See Bead, Peiaie ad Pesoe (984. Hee he vaiaio of he Poisso paamee fom oe ime ui o he ex is aalysed. I is woh o eall ha a suue vaiable have bee also used whe he vaiableess of he Poisso paamee fom oe is ui o he ex is o be ivesiaed (see Bűhlma (97. pae / 6

4 wih ad i, muually idepede fo eah yea. The expeed laim size of a lie has bee simply deoed by m wheeas ad ae he usual is idies of he laim size disibuio. Fo popey, we disiuish bewee sile-is laims ad aasophe laims. Fuhemoe, we assume ha hee is a fixed sum isued fo eah popey poliy, ad he sile-is laims a o exeed his amou. Caasophe laims ae assumed o affe a lae umbe of poliies epesei all popey lies of busiess. Le L deoe he umbe of lies of busiess. Fo eah lie of busiess, l, L, L, le K l ( deoe he umbe of poliies i yea ad SI l ( deoe he sum isued of he eah poliy i yea. i is assumed ha he pofolio is hi by wo ypes of laims: Ris laims affei oly oe poliy ad aasophe laims affei a lae umbe of poliies fo all lies. The aeae laims amou i yea, X is ive by a ompoud poess: L l, a, (4 X l,, W l whee l, is he adom vaiable of he umbe of laims fo he l-h lie of busiess oued i he yea ad l,, he adom laim size of he -h laim fo he l-h lie oued a yea. Fuhemoe, a, is he adom vaiable of he umbe of aasophe laims oued i he yea ad W, he adom laim size of he -h aasophe laim oued a yea. Moe deails oei he aasophe losses simulaio ae epoed i Appedix III.,. A COMPARISON BETWEEN EXACT AND SIMULATION MOMENTS OF THE CAPITAL RATIO: AN EXAMPLE. Usually, he apial aio u U / B is pefeed o be aalysed isead of he oal is eseve amou. Diseadi (oly i his seio eisuae, axaio, divideds, ivesme eu fom loss eseves ad eadi he ivesme eu ae j as deemiisi ad osa ove he ime ad aual eeal expeses pefely mahed by he expeses loadi ( E B, fo a sile lie of busiess he apial aio i his paiula heoeial famewo is ive by: X (5 u u p ( λ P Ris idies of he laim size disibuio ae: a ad. ( a m ( a m a a a pae 4 / 6

5 whee ad p deoe he followi wo o-eaive joi faos: j ( i ( ( λ / p j P B ( j / The aual joi fao is depedi o he ivesme eu ae j, he laim iflaio i ad he eal owh ae ; o he ohe had fao p is depedi o he iidee of he is pemium by oss pemium (P/B, osa if expeses ad safey loadi oeffiies ( ad λ ae maiaied osa alo he ime, ieased of he eu fao fo half a yea. Afe some maipulaios, he sohasi euaio (5 of he aio u us o: h X h h (6 u u p ( λ h h Ph The heoeial expeed value, vaiae ad sewess of he apial aio U/B fo a sile lie of busiess i he above meioed seaio ae epoed i Appedix 4. Readi a sile lie isue (e.. Moo Liabiliy, havi pofolio ad eeal paamees as hose epoed i Table, he simulaio model desibed i he pevious seio has bee used ad he esuls ae fiued ou i Table, whee hey ae also ompaed wih he heoeial (exa momes, deived i he fomula epoed i Appedix I, whee also adomess of he suue vaiable is iluded. I ha simple famewo (as poved also i Appedix I, he iiial apial aio (u is affei oly he expeed values of he aio U/B i he ime hoizo bu lealy o sadad deviaio ad sewess. I addiio o ha, as o sadad deviaio of he apial aio U/B a ime, as explaied i he ex fomula: X X / / / ( u ( j ( j ( j B λ P λ ( we e i is ive by he sadad deviaio of he loss aio X/B a yea muliplied by he fao j fo half a yea. Aually, fo he Sile-lie liabiliy Isue desibed i Table, a ime he sadad deviaio of he loss aio X/B is eual o.5% ive by he sadad deviaio of he loss aio X/B (5.5% muliplied fo he aio (usually mio ha depedi o expeses ad safey loadi ( /( λ eual o.75. Fially we e he exa value of.7% oaied i Table fo he sadad deviaio a ime of he apial aio U/B. 4 A his ead see also Peiaie & Raala (98 ad Savelli ( pae 5 / 6

6 TABLE : Paamees of he Sile-lie Isue (Moo Liabiliy Paamees : STANDARD INSURER Iiial is eseve aio u,5 (* Iiial expeed umbe of laims 8. Vaiae suue vaiable. Sewess suue vaiable γ.8 Iiial expeed laim size (EUR m 6. Vaiabiliy oeffi. of 7 Safey loadi oeffi. λ. % Expese loadis oeffiie 5. % Real owh ae 5. % Claim iflaio ae i 5. % Ivesme eu ae j 4. % Loss Reseve aio l % Taxaio ae x % Divideds ae dv % (* This measue is euivale o appoximaely.5 imes he miimum EU solvey mai TABLE : Sile-Lie Isue (moo liabiliy Resuls of 4. simulaios EXACT AND SIMULATION MOMENTS OF THE CAPITAL RATIO U/B TIME EXACT MOMENTS SIMULATION MOMENTS (N4. MEAN ST.DEV. SKEW. MEAN ST.DEV. SKEW. 5. % 5. % 5.6 %.7 % %.6 % % 5.47 % % 5.46 % % 8.4 % % 8.5 % LIABILITY AND PROPERTY MULTI-LINE INSURERS: TWO CASE STUDIES IN THE GENERAL FRAMEWORK. I he ex seios he esuls of he simulaio model illusaed i he pevious seios ae epoed fo wo muli-lie Isues fo a ime hoizo of yeas (T. The fis Isue has wo diffee lies of liabiliy isuaes (moo ad ommeial liabiliy ad he seod oe has hee diffee lies of Popey isuaes (homeowes, aiulue pae 6 / 6

7 ad ommeial popey. I he ex hese Isues will be deoed by espeively Liabiliy Muli-lie Isue (LMI ad Popey Muli-lie Isue (PMI. The feaues of hese wo muli-lie Isues (epoed i Table. ae o daw by a speifi eal daa se bu ae assumed o he base of paial mae aalyses ad paial modelli made by woi paies 5. As eads he mai Isue s paamees i is woh o emphasize ha: - laims umbe paamees: fo eah lie aodi a Poisso disibuio wih a suue vaiable disibued as a Gamma disibuio wih ideial paamees (i paie he laim ou has a Neaive Biomial disibuio. The disibuio is diffee fo eah lie, aodi diffee values of he expeed umbe of laims ad he vaiae of he suue vaiable (fom % fo moo liabiliy o 4% fo popey lies; - laim size paamees: fo liabiliy lies he laim size is disibued as a Loomal vaiable, whee he measue of vaiabiliy oeffiie has a ea elevae o explai he vaiabiliy ad sewess of he apial aio disibuio U/B. Readi he popey lies, he sile is laims ae assumed o be Lo-omal disibued he oly diffeee is ha he laim amous ae uaed sie hey a o exeed he sum isued of he affeed poliy. The aasophe laims ae assumed o be Paeo disibued (oe-paamee ad ae affei all lies of busiess i he same eve. Sie we will lie o sudy popoioal eaies wih he eded shae of eah poliy o deped o he size of he poliy (Suplus eaies, i is impoa ha we ae able o spli eah aasophe laim o he lass of busiess. Please efe o Appedix III o see how his poblem is hadled; - ivesme eu: he aual ae is eaded as a sohasi vaiable, by a auoeessive model (desibed i Appedix II, wih a expeed ae of 4%; - eal owh: is assumed osa ad eual o 5% fo all lies of busiess of boh mulilie Isues; - laims iflaio: he aual ae is assumed o be osa ad eual o 5% fo he liabiliy isuaes ad smalle (% fo popey isuaes; - loss eseves: ae assumed o be a osa aio of he oss pemiums of he yea, % fo he liabiliy Isue ad % fo he Popey Isue; - safey loadi: havi assumed β5%, he safey loadi oeffiie is ompued aodi he fomula ( i seio. The oeffiies fo he popey lies (7.5% ae io aou he edued elevae of he loss eseve (% of pemiums, isead of % fo he Liabiliy isue. The oeffiie fo he Commeial Liabiliy (4.7% is siifialy lae ha i ase of Moo Liabiliy (.% beause of he diffee (6 isead of 7. I is woh poii ou ha he oeffiie λ ae ep osa fo he eie ime hoizo (T. Ayway, i is woh o poi ou ha a peiodial aual adjusme of he safey loadi oeffiie would have o affeed siifialy he esuls beause of oly a slih eduio of he aio (X/E(X as he expeed umbe of laims ae ieasi yea by yea (as a effe of he posiive eal owh of he pemium volume, emidi ha he pooli effe is less eleva whe a o eliible suue vaiable (o divesifiable is is pese as i his ase. - expeses loadi: 5% fo boh liabiliy ad popey lies; - axaio ae: fla ae of 5% is assumed 5 A his ead a speial efeee has bee he model paamees used by he IAA Isue Solvey Woi Pay i A Global Famewo fo Isue Solvey Assessme, daf May pae 7 / 6

8 - divideds ae: a he ed of eah yea is assumed o be paid o soholdes % of he (posiive aual esul, e of axaio, Noe ha o efeee o he apial aio of he yea is osideed, beause he measue of divideds is o affeed by eihe a healhy o a daeous sae of he Isue. TABLE : PARAMETERS OF THE LIABILITY MULTI-LINE INSURER Paamees : LOB (MOTOR LIABILITY LOB (COMMERCIAL LIABILITY TOTAL Iiial Capial aio u. % Iiial expeed umbe of laims 8... Vaiae suue vaiable,. Sewess suue vaiable γ.8.5 Iiial expeed laim size (EUR m Vaiabiliy oeffi. of laim size 7 6 Loss Reseve Raio l %. % Expese loadis oeffiie 5 % 5. % Bea oeffiie β 5 % 5. % Safey loadi oeffi. λ. % 4.7 % Real owh ae 5. % 5. % Claim iflaio ae i 5. % 5. % Ivesme eu ae (expe. value j 4, % 4, % Taxaio ae x 5 % Divideds ae dv % Iiial Ris Pemium (mill Eu P 8,, 4, Iiial Goss Pemiums (mill Eu B 47, 48,9 96, Readi he eisuae saeies, he Liabiliy Muli-lie Isue will have he hoie bewee Quoa Shae eay wih ommissio eual o he expese loadi ( RE 5% ad a ulimied Exess of Loss ove wih a pemium ae eual he is pemium plus a fao (% of he sadad deviaio of he losses o he ove ad iludi a fuhe loadi a % fo pofi, oss ad boeae. pae 8 / 6

9 TABLE 4: PARAMETERS OF THE PROPERTY MULTI-LINE INSURER Paamees : LOB (HOMEOWNERS PROPERTY LOB (AGRICULTURE PROPERTY LOB (COMMERCIAL PROPERTY TOTAL Iiial Capial aio u. % Iiial Numbe of poliies Sum Isued pe poliy (EUR Expeed loss feuey pe is losses 8. % 6. % 8. % Iiial expeed umbe of laims Vaiae suue vaiable,4.4.4 Sewess suue vaiable γ Iiial expeed laim size (EUR 6 m Vaiabiliy oeffi. of laim size 4 8 Caaophe PML (mill EUR 8, 8, 48, 56, Exp.. of aasophe laims CAT Caasophe laims Paeo paam α. Cumulaio fao.5 Loss Reseve Raio l. %. %. % Expese loadis oeffiie 5. % 5. % 5. % Bea oeffiie β 5. % 5. % 5. % Safey loadi oeffi. λ 7.5 % 7.5 % 7.5 % Real owh ae 5. % 5. % 5. % Claim iflaio ae i. %. %. % Ivesme eu ae (expe. value j 4, % 4, % 4, % Taxaio ae x 5 % Divideds ae dv % Iiial Ris Pemium (mill EUR P 49, 7,5 4,, - heeof sile is losses 44,7,6 4,4 5,7 - heeof a losses 4,6 7,,7 4, Iiial Goss Pemiums (mill EUR B 77, 4, 67,6 88, Readi he eisuae saeies, i is assumed he Popey Muli-lie Isue will have he hoie o he mae amo Quoa Shae eay wih ommissio 7%, a Suplus eay wih ommissio 6% ad a Exess of Loss pe is ove wih a pemium ae eual he is pemium plus a fao (% of he sadad deviaio of he losses o he ove ad iludi a fuhe loadi a % fo pofi, oss ad boeae. I all ases he Popey Muli-lie 6 Befoe uaio i he sum isued. pae 9 / 6

10 Isue s eeio will be poeed by a Caasophe Exess of Cove wih a ae o lie se usi a sadad exposue ai mehod. 4. THE RESULTS OF THE SIMULATION MODEL REGARDING THE RBC REQUIRED ACCORDING SOME DIFFERENT RISK MEASURES. A well ow oe-sided appoah o is evaluaio is he Value-a-Ris (VaR widely used whe he is elies o he ouee of ufavouable eves suh as isolvey ae o be esimaed. Tha id of appoah has a soud baoud i auaial lieaue, whee Capiala-Ris (CaR ad pobabiliy of ui have usually bee he mai pillas i solvey aalyses. The VaR measue is widely used i fiae, bu wih a ime spa muh moe edued (e.. days ompaed wih isuae is aalyses, i whih a ime spa of eihe o yeas is usually adoped. I isuae solvey, VaR a be summaized as he maximum loss fo a isue ove a ae hoizo wihi a ive ofidee level (e.. 99%; i ohe wods i deoes a moeay amou fo he is of maai a isuae ompay. Assumi o have o iiial apial (U, he VaR fo he hoizo ime (, wih a ofidee level eual o -ε (e.. 99.% is he ive by: VaR(, U ( ε whee U ε ( is he ε-h peeile of he Ris Reseve amou a ime (wih a ofidee level euied i he aalysis usually ahe lae, a leas 95%. I ase his is measue is used fo RBC euiemes a ime, he ivesme eu omi fom he iiial apial euied should be also iseed io he fomula. I his famewo, whee he ivesme eu is eaded as sohasi, he expeed value of he ivesme eu has bee adoped. Fuhemoe, RBC euiemes should be ompaed wih he iiial oss pemiums volume (B of he Isue, i ode o be able o ompae he oupu amo diffee isues ad seaios. Coseuely, expessed as a aio of he iiial oss pemiums, ha is measue a be wie as: ε (7 b VaR ε (, RBC B VaR ε (, U ε ( B [ E( j ] uε ( whee he peeile aio u ε ( ad he joi fao ae espeively: U ε ( E( j uε ( ( ( i B I ase of a muli-lie isue wih diffee owh aes ad laim iflaio i, he fao will esul as a weihed aveae aodi he elevae of eah diffee lie of busiess. pae / 6

11 I ase is pefeed o use he TVaR as a is measue i ode o ead moe popely he lef ail of he is eseve disibuio (o euivalely he ih ail of he loss disibuio -U, he aio of he RBC euieme a be expessed as follows: (8 b whee we have: TVaR ε (, RBC B TVaR ε (, U TVaR ε ( B [ E( j ] u TVaR ε ( U ( TVaR TVaR ε ( E U ( / U ( < U ε( E u / u < uε ( B uε ( ( B A hid is measue hee eaded is he UES (Uodiioal Expeed Shofall, ai io aou eihe he pobabiliy of he eve oui ad he maiude of he esuli shofall. The UES a yea is he ive by: UES( E [ max(, U ] P( U < EU ( u < E u / u < B / U < P As meioed befoe, we a expess also his is measue i a elaive way, ompaed o oss pemiums of he yea: UES( B (9 ues E[ max(, u ] ( I he ee auaial lieaue is emphasized how his is measue a be eaded as he is pemium of a isuae oa whih would ove he shofall of he ompay i ase i ous, bei he sum of he expeed shofalls (ui defiis weihed by hei pobabiliies. I piiple, his is he udisoued pemium o pay by he soholdes fo a uaaee of solvey a ime wihou mae available ay iiial is apial. Fuhemoe, he UES is a oe-sided is measue, lie he semi-vaiae, i whih defiis ae iluded bu supluses ae ioed. As o saey paamees (ime hoizo ad level of ofidee o uaify he Ris-Based Capial by VaR o TVaR is measues, he ex simulaio esuls have osideed a ime hoizo of, o yeas, wih a level of ofidee of eihe 99.%, 99.5% o 99.9%. The esuls ae fiued ou also e of eisuae aodi he fomula (. A his sae, oly Quoa Shae aaemes ae osideed: - fo he Liabiliy Muli-lie Isue a % Quoa Shae wih eisuae ommissios eual o 5% (oiide wih he die expeses ae of eded pemiums has bee osideed; - fo he Popey Muli-lie Isue a 4% Quoa Shae ove wih ommissio 7% has bee osideed. The eeio is poeed by a Caasophe Exess of Loss ove i exess of he yealy expeed amou of aasophe losses up o.5 imes he pobable maximum loss (PML. Howeve, i is assumed ha i is possible o e losses up o imes he PML, suh hee is a (small is of havi isuffiie oveae. pae / 6

12 TABLE 5. Liabiliy ad Popey Muli-lie Isues Resuls of 4..simulaios RBC AND UES RATIOS (% ACCORDING DIFFERENT LEVELS OF CONFIDENCE AND TIME HORIONS NO REINSURANCE RISK MEASURES LIABILITY MULTI-LINE INSURER PROPERTY MULTI-LINE INSURER TIME HORION TIME HORION T T T T T T b(var i % 99. % % % b(tvar i % 99. % % % ues i % WITH REINSURANCE Liabiliy: % QS fo eah lie ad RE 5% Popey: 4% QS fo eah lie ad RE 7%. Reeio poeed by Ca XL EUR mill xs EUR 8.4 mill a ROL 5.85% RISK MEASURES LIABILITY MULTI-LINE INSURER PROPERTY MULTI-LINE INSURER TIME HORION TIME HORION T T T T T T b(var i % 99. % % % b(tvar i % 99. % % % ues i % Noe: also i ase of Reisuae he aios ae ive as eihe RBC o UES amou divided by iiial oss pemiums. 5. EFFECTS OF DIFFERENT REINSURANCE STRATEGIES ON THE REQUIRED RBC. Fo Liabiliy lies diffee Quoa Shaes eaies ad ulimied Exess of loss oves have bee osideed. The Quoa Shae Teaies ae all havi a eisuae ommissio of 5% (eual o he expese aio. The pemium fo he exess of loss oves ae alulaed as he is pemium fo he laye plus % of he sadad deviaio, ad fially loaded wih % fo expeses, boeae ad pofi. pae / 6

13 Fo Popey lies diffee Quoa Shaes eaies, Suplus eaies ad pe is Exess of loss oves ovei up o he sum isued have bee osideed. The Quoa Shae Teaies ae all havi a eisuae ommissio of 7%, ad he Suplus eaies ae all havi a eisuae ommissio a 6%. Give he eeio R of he suplus eay he eaied shae fo eah lie of busiess is alulaed as ( R / α mi, fo l, L, L l SI l by assumi fo eah lie of busiess, all poliies ae havi he same sum isued. The pemium fo he pe is exess of loss oves ae alulaed as he is pemium (alulaed exaly usi he paamees i he d.f. S( fo he laye plus % of he sadad deviaio, ad fially loaded wih % fo expeses, boeae ad pofi. I addiio o ha, he eeio is always poeed by a aasophe exess of loss ove i exess of he expeed yealy expeed a losses e of popoioal eisuae (i is assumed ha pe is exess of loss oves will have so lae exess pois ha hey will have o majo impa o he aasophe losses ad up o.5 imes he oal PML fo all lies. Sie we ae assumi ha i is possible o have losses up o imes he PML e of popoioal eisuae, hee is a mio is ha he ompay i ase of a vey seious aasophe will o have suffiie oveae i he op. I ohe wods, he a XL ove will hae as he popoioal poamme haes. The pemium fo he aasophe exess of loss oves ae alulaed as he is pemium (alulaed usi a exposue appoah fo he laye plus % of he sadad deviaio, ad fially loaded wih 5% fo expeses, boeae ad pofi. Fiue shows he oespodee bewee he RBC aio (hose houh he TVaR 99% is measue wih T ad he eded pemium fo diffee eeios o he eisuae poammes. I is woh o meio ha wihou ay eisuae ove he RBC aio would be.5% ad 6.5% of oss pemiums fo Liabiliy ad Popey muli-lie Isue espeively. Fo he Popey Isue, e of oly CaT XL ove he aio is siifialy edui o 4.4%. Fo he Liabiliy muli-lie Isue, he aphs fo he QS eaies ae liea beause of eisuae ommissio (5% ideial o he expese aio. Theefoe he aual esuls e of he QS eaies ae popoioal o he aual esuls i ase of o eisuae. Clealy a liea behaviou would o be i foe i ase of eihe moe favouable o moe ufavouable eisuae ommissios, I ase of exess of loss, he RBC aio ieases fo deeasi exess pois (ad heeby ieasi eded pemiums. Oe easo is ha he XL pie is uie expesive ad oveie (o edue he is dow oly uil a eai eeio, ad afe ha he os is so hih ompaed wih he eded is ha he expeed esul fo he isue is eaive. pae / 6

14 FIGURE : Coespodee bewee he RBC aio (houh TVaR 99% is measue ad he eded pemium fo diffee eeios o he eisuae poammes fo T (esuls of 4. simulaios. Liabiliy RBC 5% 4% % % % % % % 4% 6% 8% % Ceded pemiums QS XL Popey RBC 5% 4% % % % % % % 4% 6% 8% % Ceded pemiums QS & Ca XL Suplus & Ca XL Ris ad Ca XL Fo Popey muli-lie Isue, all he aphs sas a eded pemiums eual o.5% whih is he pie fo he aasophe exess of loss i ase of o popoioal eisuae. The aph fo he QS eaies is o loe sily liea: he eisuae ommissio is o loe ideial o he expese aio, ad he QS eaies ae ombied wih a Ca XL ovei he eeio. The aph showi he Suplus eaies beas a 5% eded pemiums his is beause his poi oespods o a eeio a EUR 4. whih is he sum isued fo he seod lie pae 4 / 6

15 of busiess (Aiulual. Fo eeios a EUR 4. o hihe (i.e. eded pemiums less ha 5%, homeowes as well as aiulual is fully eaied by he isue, ad oly he ommeial lie of busiess is affeed by he suplus eay ad hee is o majo iease i he RBC aio by ieasi he eeios o hihe ha EUR 4,. 6. THE IMPACT OF THE RBC ON THE PROFITABILITY OF PROPERTY AND LIABILITY INSURERS. Hee he expeed Reu o Euiy (RoE will be used as a measue fo he Isue s pefomae ad le R (, T deoe he expeed RoE all ove he full ime hoizo (,T. I he famewo hee assumed, whee boh divideds ad axaios ae pese (as sohasi vaiables beause of sohasi eoomi aual esuls, i is ive by: T T U T DVh U U T DVh ( (, h h R T E E U U I is woh o emphasize ha his measue is o affeed a all by he divided poliy adoped by he maaeme i he ime hoizo, beause if o divideds ae disibued he is eseve amou U is ieasi of he same amou. I his seio we ae osidei he followi eisuae saeies: Popey Isue: - No eisuae - Quoa shae 4% wih eisuae ommissio 7% ombied wih aasophe exess of loss EUR mill xs EUR 8.4 mill a ROL 5.85%, eisaeme a % - Suplus eay eeio EUR. wih eisuae ommissio 6% (a suplus wih his eeio oespods appoximaely o ede he same amou of pemiums as he 4% QS above ombied wih aasophe exess of loss EUR 7.7 mill xs EUR. mill a ROL 6.7%, eisaeme a % addiioal pemium - Pe is exess of loss EUR 84. xs EUR 6. a pemium ae 7.5% ombied wih aasophe exess of loss EUR 69.9 mill xs EUR 4. mill a ROL 5.85%, eisaeme a % addiio pemium - Caasophe exess of loss EUR 69.9 mill xs EUR 4. mill a ROL 5.85%, eisaeme a % addiio pemium Liabiliy Isue: - No eisuae - Quoa shae % wih eisuae ommissio 5% - Quoa shae % wih eisuae ommissio 5% - Ulimied exess of loss i exess of EUR 7. a pemium ae 7.57% pae 5 / 6

16 Fo eah eisuae saey applied, he miimum euied b as desibed above is applied as iiial apial aio u. The esuls of hese eisuae saeies applied o he 4. simulaios ae show i Table 6 ad 7. TABLE 6. Liabiliy Muli-lie Isue Resuls of 4..simulaios RATE ON EQUITY (ROE FOR DIFFERENT REINSURANCE STRATEGIES WITH INITIAL RISK RESERVE RATIO EQUAL TO THE REQUIRED RBC REINSURANCE STRATEGY INITIAL RISK RESERVE U RBC(T VAR 99.5% INITIAL RISK RESERVE U RBC(T TVAR 99% TIME HORION TIME HORION T T T T T T NO REINSURANCE b (% ues ( Fiie ui pob (% Expeed fiie-ime RoE (% NET OF QS % b (% ues ( Fiie ui pob (% Expeed fiie-ime RoE (% NET OF QS % b (% ues ( Fiie ui pob (% Expeed fiie-ime RoE (% NET OF RISK XL b (% ues ( Fie ui pob (% Expeed fiie-ime RoE (% pae 6 / 6

17 TABLE 7. Popey Muli-lie Isue Resuls of 4..simulaios RATE ON EQUITY (ROE FOR DIFFERENT REINSURANCE STRATEGIES WITH INITIAL RISK RESERVE RATIO EQUAL TO THE REQUIRED RBC REINSURANCE STRATEGY INITIAL RISK RESERVE U RBC VAR 99.5% INITIAL RISK RESERVE U RBC TVAR 99% TIME HORION TIME HORION T T T T T T NO REINSURANCE b (% ues ( Fiie ui pob (% Expeed fiie-ime RoE (% NET OF QS 4% b (% ues ( Fiie ui pob (% Expeed fiie-ime RoE (% NET OF SURPLUS b (% ues ( Fiie ui pob (% Expeed fiie-ime RoE (% NET OF RISK XL b (% ues ( Fie ui pob (% Expeed fiie-ime RoE (% NET OF CAT XL b (% ues ( Fiie ui pob (% Expeed fiie-ime RoE (% I Fiue, he ade-off Ris vs Reu of he wo muli-lie isues is illusaed fo all he yeas of he ime hoizo, whee he measue of is has bee deoed by RBC of he yea wih iiial eseve aio eual o he alulaed TVaR 99% fom Table. Fo Liabiliy Isue he esuls show how he QS eaies have o impa o he fiie ime expeed RoE R (, T beause he ommissio ae is ideial o he expese aio. The oly (aual impa is ha he RBC edues aodi o lowe eeios of he QS. The seleed XL oa is slihly moe effeive i edui he RBC, ad he RoE is less beause a loadi is iluded i he eisuae pemium. Fo Popey i is see ha he 4% QS eay is moe effeive i edui he is ha he Suplus eay wih eeio EUR. (he eded pemiums fo he wo eaies ae pae 7 / 6

18 appoximaely he same. Oe of he easos is ha he suplus eay does o poe aais ay vaiabiliy i he yealy losses fo homeowes lie of busiess sie his lie is eaied %. Howeve, he expeed RoE is bee fo he QS eay maily beause ommissio ae fo he QS eay is hihe ha fo he Suplus eay (he eisue will usually pay less ommissio fo a moe u-balaed eay. The aleaive eplai he popoioal eaies by a pe is exess of loss poeio is see o be oo expesive a aleaive he expeed RoE is muh less ad he RBC is hihe. FIGURE Ris vs Reu Tade-off: fiie-ime Expeed RoE vs RBC aio (TVaR 99%, aodi o eded pemiums fo diffee eeios o he eisuae poammes fo T (esuls of 4. simulaios wih u b fo eah sile eisuae saey. Liabiliy 8% Fiie ime RoE 6% 4% % % % % % % 4% 5% b (TVaR 99% No eisuae Ne of QS % Ne of QS % Ne of XL Popey Fiie ime RoE 8% 6% 4% % % % % % b (TVaR 99% No eisuae Ne 4% QS Ne of suplus Ne of pe is XL Ne of a xl oly A simila ade-off is show i Fiue. Hee he Uodiioal Expeed Shofall (UES of he yea is used as is measue ad he pofiabiliy defied by he fiie ime expeed RoE R (, T. Simila olusios a be made fo his aph. pae 8 / 6

19 FIGURE. Ris vs Reu Tade-off: fiie-ime Expeed RoE vs UES aio, aodi o eded pemiums fo diffee eeios o he eisuae poammes fo T (esuls of 4. simulaios wih u b fo eah sile eisuae saey. Liabiliy 8% Fiie ime RoE 6% 4% % %.%.%.% ues No eisuae Ne of QS % Ne of QS % Ne of XL Popey Fiie ime RoE 8% 6% 4% % %.%.%.4%.6% ues No eisuae Ne 4% QS Ne of suplus Ne of pe is XL Ne of a xl oly pae 9 / 6

20 7. THE IMPACT OF THE RBC ON THE PROFITABILITY OF PROPERTY AND LIABILITY INSURERS. Clealy, he Isues hee eaded ae o epeseaive as sadad (liabiliy ad popey isues i he mae ad fuhemoe he is heoeial model hee applied implies a eai deee of simplifiaios, oei e.. he laim esevi u-off is, he dyami pemium ai, he oelaio deee bewee he iss, he ivesme model, he asse alloaio ad he edi is. Howeve, he esuls of he model oei RBC euiemes ad pofiabiliy i he pevious seios show ha he EU miimum solvey mai should be siifialy ieased by he EU leislaio i he fuue, also payi majo aeio o pii odiios iluded i he eisuae eaies, affei o oly he vaiae of he poess bu he expeed esul of he yea oo, wih ufavouable effes o isue s solvey i some ases. Fo isae, he esuls show ha a Popey Isue should o udewie ay is wihou a Caasophe XL ove ohewise a exemely hih is apial is euied. The pe is XL ove seems oly o be suiable fo uie lae eeios i ode o be oo expesive, ad i does o offe suffiie oveae of hih vaiabiliy i he umbe of laims. Fuhemoe, i ase of a Quoa Shae eay, he miimum apial euieme should o be always deeased aodily he edi uoa o eisuae; i ase of ufavouable eisuae ommissios he isue s is is o deeasi i he same measue of he eded uoa ad fo lae ime hoizos ( o 5 yeas i mih also be (o supisily lae ha he is measue wihou ha ove. Fially, he ew Euopea RBC euiemes should o edue siifialy he pofiabiliy of he ompay ohewise a mio iees o he isuae mae would ou by fiaial ivesos. pae / 6

21 REFERENCES Auaial Advisoy Commiee o he NAIC Popey & Casualy Ris-Based Capial Woi Goup Hama e al. [99]: Popey-asualy is-based apial euiemes a oepual famewo, Casualy Auaial Soiey Foum; Aze P., Delbae F., Ebe J.M., Heah D. [999]: Cohee Measues of Ris, Mahemaial Fiae 9 (July, -8; Biish Geeal Isuae Solvey Goup [987]: Assessi he solvey ad fiaial seh of a eeal isuae ompay, The Joual of he Isiue of Auaies, vol. 4 p. II, Lodo; Bűhlma H. [97]: Mahemaial Mehods i Ris Theoy, Spie-Vela, New Yo; Coo, R.D. ad M.E.Johso (98: A Family of Disibuios fo Modelli No-ellipially Symei Mulivaiae Daa. Joual of he Royal Saisial Soiey, B4, 98, -8. Cous S.M., Thomas T. [997]: Modelli he impa of eisuae o fiaial seh, Biish Auaial Joual, vol., pa. III, Lodo; Dayi C.D., Hey G.B. [99]: Maai ueaiy i a eeal isuae ompay, The Joual of he Isiue of Auaies, vol. 7 p. II, Lodo; Dayi C.D., Peiäie T., Pesoe M. [994]: Paial Ris Theoy fo Auaies, Chapma & Hall, Lodo; IAA Isue Solvey Woi Pay (: A Global Famewo fo Isue Solvey Assessme, daf May Joio P. []: Value a Ris, d ediio MGaw Hill, New Yo; Kluma S, Paje H., Willmo G. [998]: Loss Models Fom Daa o Deisios, Joh Wiley & Sos, New Yo; Meyes G., Klie F., Lalode D.: The Aeaio ad Coelaio of Isuae Exposue, Casualy Auaial Soiey Foum, Summe Műlle Woi Pay - Repo of he EU Isuae Supevisoy Auhoiies [997]: Solvey of isuae udeais; Peiäie T, Raala J. [98]: Solvey of isues ad eualizaio eseves, Isuae Publishi Compay Ld, Helsii; Peiäie T., Bosdoff H., Pesoe M., Raala J., Ruohoe M. [989]: Isuae solvey ad fiaial seh, Fiish Isuae Taii ad Publishi Compay, Helsii; Savelli N. []: A Ris Theoeial Model fo assessi he Solvey pofile of a Geeal Isue, XXXIV ASTIN Collouium, Auus, Beli. Wa, Shau S. (998: Aeaio of Coelaed Ris Pofolios: Model ad Aloihms. Poeedis of he Casualy Auaial Soiey, LXV, Wie soms Euope (II (: Aalysis of 999 Losses ad Loss Poeial. Muih Re. pae / 6

22 pae / 6 APPENDIX Exa momes of he apial aio U/B i a simplified famewo Expeed value ( if p u u E if p u λ λ Vaiae Remidi we have deoed by p he aio / ( j p λ, he vaiae of he solvey aio B U u / is ive by: ( ( ( z p P X p u ( wih ( ( ( ( i j s I he usual ase, s we have ( ( ( ( ( ( ( ( s s p s p p u z z

23 pae / 6 Sewess ( P X P X P X P X P X P X u γ µ γ γ ( ( ( ( (, ( ( ( s v w γ wih ( ( ( ( i j v We will always have a eaive sewess ( u γ uless γ is eaive, i ha ase he si of ( u γ is depedi of he paamees. I eeal, we will have γ. I he usual ase whe,,, w v s, he sewess of he solvey aio a be wie as (, ( ( ( s s v v w w u γ γ If is Gamma(h,h ad is loomal disibued (wih wo paamees, oseuely is Neaive Biomial disibued ad we have γ. The ( ( ( m a a a m a ad he he aio: ( I his ase we have (i he usual ase wih,,, w v s :

24 pae 4 / 6 ( ( 4 ( ( ( s s v v w w u γ I he speial ase wih o owh (i.e., he fomula beomes moe simple (sie w v s, ad we ae able o mae fuhe aalyses. ( ( 4 u γ APPENDIX II Simulai he sohasi eu o ivesme As eads he sohasi ivesme eu, we assume he aual ae j of he eleva yea (, follows a AR poess depedi also o he iflaio: ( ( j i b i b j j b j j ε whee i deoes he iflaio ae, havi a AR( poess as: ( i i i a i i ε ad ε is he oise em of he poess fo yea (ideial fo boh poesses i ad j, havi assumed ha all oise ems ae i.i.d. ad ε disibued as a Gamma(4;. Aodi he assumpios made, he oise em of eah yea has expeed value, sadad deviaio ad sewess. The values used i he simulaios fo he above meioed paamees ae as follows: 4% j % i b. mea-evei divi fao fo eu a.65 mea-evei divi fao fo iflaio b b.5 j.5 oise em oeffiie i he eu poess i.8 oise em oeffiie i he iflaio poess. Fuhemoe, a miimum osai has bee fixed fo eihe iflaio (-% ad eu ae (.5%.

25 APPENDIX III Simulai he aasophe losses The umbe of aasophe laims i yea The aasophe laim amous, deoed by disibuio fuio ( W a, W, is assumed o be Poisso disibued wih mea a,., ae assumed o be i.i.d. adom vaiables wih a oiuous F a ad o be saled by he laims iflaio i as well as he owh ae i eah yea. The s h momes abou he oii ae eual o: E W i E W s s s j s s (, ( ( (, ( ( a, s, wih a, ad W, bei muually idepede fo eah yea. I ode o be able o sudy he effe of vaious popoioal oas, heeude suplus eaies, i is impoa o ow he spli of eah aasophe laim o eah busiess lie (,, W The maial disibuio of F W l, o suvivo fuio W l,, S W l, (AIII- ( w whee l,, wl FW l, l, W,, L,, i a L L wih W, Wl is assumed o be he Paeo disibuio wih umulaive disibuio fuio α, ad S ( w F ( w l, wl, Wl l,, Wl, l,. Fuhemoe, i is assumed ha o aasophe laim a exeed imes he yeas PML (Pobable Maximum Loss. Theefoe, he Paeo disibuio is uaed i imes PML fo eab lie of busiess: l w, (AIII- ( l, W wl,, ad S ( ( α W wl FW wl F l, l, PMLl α, l, l,,,. α l,,. Fo suh aasophe eves, he L lies of busiess ae o idepede, ad we will heefoe use a mehod desibed i Wa (998 o fid he joi umulaive disibuio fuio. Le ( U, L,U L be a L-dimesioal uifom disibuio wih suppo o he hypeube (, L he joi umulaive disibuio fuio ad havi (AIII- F ( β ( u,, u L β U,, U L ul L L L L l β whee u j (,, l, L, L, ad > β. As show by Coo ad Johso (98, ( β ( u, L, u mi[ u, u ] lim U,, U L L L β F L, L, pae 5 / 6

26 Thus, he oelaio appoahes o is maximum (i.e. o-moooiiy whe β deeases o zeo; he oelaio appoahes o zeo whe β ieases o ifiiy. Coo ad Johso (98 also ave he followi simple simulaio aloihm fo he mulivaiae disibuio ive by (AIII-: Sep. Le Y,,Y L be idepede a eah has a Expoeial disibuio wih mea. Sep. Le have a Gamma (, Sep. The he vaiables β disibuio. β l l, (AIII-4 U [ Y ], l, L L have a joi umulaive disibuio fuio., Fo a se of abiay maial disibuios wih suvivo fuios fuio by S,, W S WL L, we a defie a joi suvivo L / β W,, W L W l L L L L l l (AIII-5 S ( w,, w S ( w Coside he as of aeai L lies of busiess (, L, W. If we assume ha (, L, W have a mulivaiae disibuio ive by (AIII-5, a simulaio of by Sep 4. Ive he (, o U,U L i (AIII-4 usi ( W,, S W l β W,, L,, W,,,, WL,, S L by alulai, W L,. ( ( U ( ( PML α α l W l U l l l α l l W,, L,, L a easily be implemeed if he Paeo disibuio is o uaed i imes he PML. I he mulivaiae uifom disibuio ive by (AIII-, all oelaios ae posiive. Euopea Widsom Losses The Geo Riss Reseah depame i Muih Re has made some aalysis of he wie soms i Euope i 99 ad 999. The esuls ae published i Muih Re (. Fo a umbe of ouies i Euope fo house owe iss, fo aiulual iss ad fo ommeial iss, hey have esimaed he elaio bewee loss aio, he loss feuey ad he aveae loss/effeed poliy, espeively, ad he wid speed. They have also esimaed maps showi he wid speed i vaious pas of eah ouy fo widsom seaios wih eu peiod of yeas. Whe ombii hese umbes wih he maps, he followi able of yeas widsom PML s ae foud. Based o hese fiues, he paamees fo simulai he aasophe laims ae seleed i ode o obai - Caasophe PML eual o EUR 8 mill, EUR 8 mill ad EUR 48 mill fo households, aiulual ad ommeial lies of busiess, espeively. - A aasophe fao (expeed a losses i p of he oss pemiums a 6%, 6% ad 4% fo households, aiulual ad ommeial lies of busiess, espeively. pae 6 / 6