Clams Reservg Whe There Are Negave Values he Ruo Tragle Erque de Alba ITAM Meco ad Uversy o Waerloo Caada 7 h. Acuaral Research Coerece The Uversy o Waerloo Augus 7-0 00
. INTRODUCTION The may uceraes volved he ayme o losses maes he esmao o he requred reserves more dcul. Ye some o he esg mehods such as he oular cha-ladder are smle o aly. However has become evde ha here s a eed or beer ways o oly o esmae he reserves bu also o oba some measures o her varably. Ths has led o he develome o sochasc reservg models Taylor 000 Kass e. al. 00 Eglad ad Verrall 00 de Alba 00. The cha-ladder s used as a bechmar several o he reereces meoed above due o s geeralzed use ad ease o alcao. Ths aclaes comarso bewee mehods. However hs aer our am o o develo Bayesa mehods ha rovde resuls close o hose o he cha-ladder mehod. Raher we am a usg objecve Bayesa mehods o model boh clam esy ad severy usg some commo assumos ad o use he resulg redcve dsrbuos o esmae loss reserves allowg or egave values. I hs aer we rese a alcao o Bayesa orecasg mehods o he esmao o reserves or ousadg clams. We assume ha he me umber o erods aes or he clams o be comleely ad s ed ad ow ha aymes are made aually ad ha he develome o aral aymes ollows a sable ay-o aer. Ths s agreeme wh may esg models or clams reservg o-le geeral surace ha assume elcly or mlcly ha he rooro o clam aymes ayable he j-h develome erod s he same or all erods o org de Alba 00. The resuls are alcable o ay requecy o clam aymes years quarers ec. ad legh o ay-o' erod. We rese a Bayesa aroach o orecasg oal aggregae clams umber or amou gve daa o some develome years or several occurrece years. Esseally he daa would corresod o a ycal ru-o ragle used loss reservg. We use he erm clams reservg s mos geeral sese. I arcular we are cocered wh he suao whe here are egave values he develome ragle o he cremeal clam amous. We use sadard oao so ha Z cremeal umber or amou o eves clams he -h develome year corresodg o year o org or accde year. Thus { Z ;......s } where s mamum umber o years sub erods aes o comleely ay ou he oal umber or amou o clams corresodg o a gve eosure year. I hs aer we do o assume Z > 0 or all ad s. Mos clams reservg mehods usually assume ha s ad ha we ow he values Z or + +. The ow values are reseed he orm o a ru-o ragle Table. Negave cremeal values ca arse due o mg o resurace or salvage recoveres or remums beg cluded as egave loss amous. I could be argued ha he roblem s more wh he daa ha wh he mehods. The daa should be adjused beore alyg hese mehods o sasy regulaory requremes. I hs resec de Alba ad Bolla
00 rovde a ls o oeal adjusmes. Alhough he esmao rocedures ca be aled boh o curred ad losses ad aggregae case esmaes combed or ad clams s robably beer o use he laer sce egave values are less lely o aear. Tha s because case esmaes are se dvdually ad oe ed o be coservave resulg over-esmao he aggregae. Ths leads o egave cremeal amous he laer sages o develome. Tycally hese egave values wll be he resul o salvage recoveres aymes rom hrd ares oal or aral cacellao o ousadg clams due o al over-esmao o he loss or o ossble avorable jury decso avor o he surer rejeco by he surer or la errors. We eed revous resuls usg a ull Bayesa model. I ac wo dere models are reseed: oe o orecas he umber o ousadg clams ad oe or oal aggregae clams. The laer s eeded rom de Alba 00 o cosder egave cremeal values. The model reseed here allow he acuary o rovde o esmaes ad measures o dserso as well as he comlee dsrbuo or he reserves. The aer s srucured as ollows. Seco gves a bre descro o revous resuls releva o our aroach. Seco roduces some Bayesa coces ad her alcaos acuaral scece. Seco 4 descrbes a Bayesa model or clam amous he resece o egave values. Some eamles are gve Seco 6. All yes o model are reseed oly dscree me.. BACKGROUND For a comrehesve alhough o ehausve revew o esg sochasc mehods ha ca hadle he esece o egave cremeal values see Eglad ad Verrall 00. Alhough hey rovde some Bayesa resuls mos o he mehods reseed here aroach he roblem rom he o o vew o reques or classcal sascs ad he ramewor o geeralzed lear models GLM. They rovde redcos ad redco errors or he dere mehods dscussed ad show how he redcve dsrbuos may be obaed by boosrag ad Moe Carlo mehods. From he classcal vewo hey maly cosder hree models a over-dsersed Posso a egave bomal ad a Normal aromao o he laer. They also meo he sadard log-normal model whch was roduced by Kremer 98 ad aalyzed deal Verrall 99b. They rovde a Bayesa ormulao or he Borhueer- Ferguso Techque. Eglad ad Verrall 00 emhasze ha some o he mehods reseed are beer sued or modelg ad amous or umber o clams sce curred daa whch may clude over-esmao o case esmaes leadg o egave cremeal values may cause roblems. Aer descrbg he sochasc bass or he cha-ladder mehod hey dcaes ha he Normal model has he advaage ha ca roduce esmaes or a wde rage o daa ses ad s less aeced by he resece o egaves
Table Year o Develome Year org...... - Z Z... Z Z - Z Z Z... Z Z - - Z Z... Z - : - - - Z - Z - - - Z - - The sochasc verso o he cha-ladder mehod s deed as a geeralzed lear model GLM wh a over-dsersed Posso dsrbuo Reshaw ad Verrall 998. I he over-dsersed Posso model he mea ad varace are o he same. I our revous oao mj E Zj wh a varace uco V Zj φm j ad scale arameer φ > 0 combed wh he log l uco log m j µ + α + β j. Overdserso s acheved hrough φ. Ths model reroduces he esmaes o he classcal cha-ladder mehod. Esmaes o he arameers µ ˆ αˆ βˆ j are obaed by usg a quas-lelhood aroach. Reshaw ad Verrall 998 sugges he use o Pearso resduals he GLM whe here are egave values. They o ou ha s o alcable o all ses o daa ad ca brea dow he resece o a suce umber o egave cremeal clams. The Posso assumo seems adequae or couous varables le clams amous. The egave bomal model s closely relaed o he revous oe Verrall 000. The dsrbuo he GLM s ow assumed o be a egave bomal wh mea λ j W j ad varace j λj W j φλ where W j Z ad { λ j : j... } are he cha-ladder develome acors. As beore φ s a over-dserso arameer. Ths mehod yelds esseally he same esmaes as he over-dsersed Posso. Wh a suce umber o egave cremeal clams s ossble ha some o he λ s become less ha oe ad so he varace would o es. I s he ossble ad ecessary o use a Normal aromao ad he cha-ladder resuls ca sll be reroduced. I s o recommeded o use he Normal aromao all suaos maly because real clams daa are sewed eve hough s alcao s lely o be less roublesome racce. The ormal aromao assumes he dsrbuo s ormal wh he same mea as beore ad varace φ j W j. The l uco remas he same all cases. Ths las model s see o be equvale o oe roosed by Mac j 99. I addo hese models have he dsadvaage ha hey cororae ew arameers he φ ha mus also be esmaed bu hs s he rce oe mus ay o j esmae he reserves he resece o egave values.
. BAYESIAN MODELS We do o ed o gve here a eesve revew o Bayesa mehods. Raher we wll descrbe hem very brely ad dscuss her alcaos acuaral scece seccally loss reservg. Bayesa aalyss o IBNR reserves has bee cosdered beore by Jewell 989990 Verrall 990 ad Haasru ad Arjas 996. For geeral dscusso o Bayesa heory ad mehods see Berger 985 Berardo ad Smh 994 or Zeller 97. For a dscusso o Bayesa mehods acuaral scece see Klugma 99 Maov 996 00 Scoll 00 Nzouras ad Dellaoras 00 ad de Alba 00. Here we reer oly o hose ha ca be aled o suaos where X < 0 or some. Verrall 990 aroaches he subjec o redcg ousadg clams usg herarchcal Bayesa lear models cosderg he ac ha he cha-ladder echque s based o a lear model: he wo-way aalyss o varace model ANOVA. He esseally carres ou a Bayesa aalyss o he wo-way ANOVA model o oba Bayes ad emrcal Bayes esmaes. The laer are gve a credbly erreao. Two alerave ormulaos are cosdered oe wh o ror ormao ad aoher where he uses a secc ror dsrbuo or he arameers. More recely Bayesa resuls are rovded Eglad ad Verrall 00 oably or he Borhueer-Ferguso B-F echque. The Borhueer-Ferguso echque s useul whe here s sably he rooro o ulmae clams ad he early develome years so ha he cha-ladder echque yelds usasacory resuls. The dea s o use eeral ormao o oba a al esmae o ulmae clams. I he radoal B-F mehod use s made elcly o erec ror eer owledge o row arameers ulmae clams. Ths s he used wh he develome acors o he cha-ladder echque o esmae ousadg clams. Ths s clearly well sued or he alcao o Bayesa mehods whe owledge s o erec Eglad ad Verrall 00. I may brea dow he resece o egave values Verrall 00. Mac 000 rovdes a summary o he echque. Nzouras ad Dellaoras 00 cosder varous comeg models usg Bayesa heory ad Marov cha Moe Carlo mehods. Clam cous are used order o add a urher herarchcal sage he model wh log-ormally dsrbued clam amous. I a rece aer de Alba 00 reses a model or aggregae clams by searag umber o clams ad average clams whch are also assumed log-ormally dsrbued. I hs aer we ollow esseally he aroach o he laer. A sadard measure o varably s redco error deed as he sadard devao o he dsrbuo o ossble reserves. I he Bayesa coe he usual measure o varably s he sadard devao o he redcve dsrbuo o he reserves. Ths s a aural way o dog aalyss he Bayesa. I hs aer our am s o oba o oly hs sadard devao bu also show he comlee redcve dsrbuo. 4
4. A BAYESIAN MODEL FOR AGGREGATE CLAIMS I hs seco we rese a model or he uobserved aggregae clam amous ad hece he ecessary reserves or ousadg clams. Le he radom varable Z rerese he value o aggregae clams he -h develome year o accde year... The Z are ow or + + ad we assume Y Z log X M + δ log +δ where we ca use alerave seccaos or X. The arameer δ correcs he values so as o mae ossble o ae logarhms. The rs oe o hese seccaos we cosder s o le X be he umber o closed clams he -h develome year corresodg o year o org. I hs case M Z / X s he corresodg average clam. Ths s he srucure used Taylor ad Ashe 98. The secod seccao we cosder hs aer s o le X X or all....e. X s some measure o eosures each year o org e.g he sze o orolo year. I s used as a sadardzg measure o busess volume. Ths s he ormulao used mos o he reereces e.g. Verrall 990. A ossble hrd seccao would be o use o model aggregae clam amous whou cludg ay ormao o umber o clams. Tha s use M Z or equvalely le X.... Ths seccao s equvale o he model o Doray 996. Sce he rs ormulao o he hree meoed above s more geeral X deeds o we shall cosder more deal. I eher case we assume addo ha Y * j j Log M + δ µ + α + β + ε ε ~ N 0...... ad + + so ha M ollows a hree arameer log-ormal dsrbuo.e. M ~ LN µ + α + β δ ad y * µ α β e[ Log M + δ µ α β ]. I s well ow ANOVA ha cera resrcos mus be mosed o he arameers order o aa esmably. We use he alerave assumo ha α β 0. Also we dee T +/ umber o cells wh ow clam ormao he uer ragle; ad uow. U T -/ umber o cells he lower ragle whose clams are L I s well ow ha esmao he hree arameer log-ormal dsrbuo ca be 5
very usable Crow ad Shmzu 988. Hece we wll use he role lelhood wh δ relaced by s ML esmaor as gve hs reerece o age say δˆ ad dee y Log M + δˆ. We he carry ou he res o he aalyss wh hs value relaced. Usg mar oao he model ca be wre as ollows: y W + ε ε ~ N 0 I where y { y ;... + + } s a T U -dmeso vecor ha coas all he observed values o Y µα... α β... β ' s he vecor o arameers ε s he T U vecor o errors ad W s he T U desg mar o he model. Now y W.. δˆ T U e[ y W ' y W ] 4 where he vecors.. '.. ' coa he ow daa he ragle. We use hs seccao he ollowg secos. I he ollowg sub-secos he reader s reerred o de Alba 00 or deals sce may o he resuls are esseally he same. Also wha ollows we use drec Moe Carlo smulao as descrbed Aed B o he aoremeoed reerece. 4. LOSS RESERVING USING CLAIMS PER PERIOD We wa o esmae or oba he dsrbuo o aggregae clams or accde year gve ormao o a leas oe year ha has ully develoed ad erhas o m revous * comleely ow accde years. Le Z or. Hece he ru-o Z j ragle seu we are really eresed esmag j * Z... gve * Z X ad Z...... wh + +. Codog o X s arcularly mora sce laer we combe he resuls obaed here wh he margal oseror dsrbuo o he * * X o esmae ousadg aggregae clams. Now le R Z Z or *... wh a -+ so ha Z a s he accumulao o Z u o he laes develome erod ad R he oal o he aggregae clams rocess or he develome years or whch s uow boh corresodg o busess year. Hece usg 4 ad he same assumos abou he dsrbuo o he umber o clams ad oao as Seco 4 de Alba 00 as well as rom deedece o he umber o clams ad he average clam er cell he jo d s a 6
7 + ˆ W y ˆ W...... y δ δ where + s where deoes a -+-dmesoal mulomal. We assume he arameers are deede a-ror ad secy o-ormave rors de Alba 00. The jo oseror dsrbuo s he see o be + ] W y ' W y e[ ˆ D... T U δ * * M. 5 where D rereses all he ow ormao cluded he oseror dsrbuo.e. } ˆ y W..... { D δ. We ca rewre 5 as... D D D D where D D D. Sce '...... β β α µα ad rom he rs acor 5 we ca wre ˆ] ' ˆ' e[ W W D ad ˆ] ˆ' e[ W y W y D T U +
ˆ wh W' W W' y. Ths s he square-roo vered-gamma dsrbuo Berardo ad Smh 994 age 9. Furhermore recallg a -+ ad rom Aed A de Alba 00 D a a D To comue he reserves or he ousadg aggregae clams we eed o esmae he lower oro o he ragle. We do hs by obag he mea ad varace o he redcve dsrbuo. Hece or each cell we have: E Z D E M D E M D E D because o he deedece o M ad... where M { M ;... + + }. The he Bayes esmae o ousadg clams or year o busess s E > + Z D. The Bayes esmaor o he varace he redcve varace or ha same year s oo cumbersome o derve. Hece we use drec smulao rom he oseror dsrbuos o geerae a se o N radomly geeraed values or he umber o clams each cell o he uobserved lower rgh ragle j.. > + or j N. The also or j N we rs geerae radom values o j j j he y ad rom hem or he average ayme M e{ y } δˆ or each oe o hose same cells ad ally or he corresodg edg aggregae loss ayme j j j Z M.. ad > +. These values clude boh arameer varably ad rocess varably. Thus we ca comue a radom value o he oal j j requred reserves R Z. The mea ad varace ca be comued as N j N R R j R ad R R. N j N N j. 8
The sadard devao R hus obaed s a esmae or he redco error o he umber o clams o be ad. The smulao rocess has he added advaage ha s o ecessary o oba elcly he covaraces ha may es bewee arameers. They are deal wh mlcly. 4. LOSS RESERVING USING A MEASURE OF EXPOSURE PER ACCIDENT YEAR I hs sub-seco we rese he secod seccao or equao. Now le X X some eosure acor or he ow eosures each year o org Verrall 990 Reshaw ad Verrall 994 Eglad ad Verrall 999. Sce X are o loger radom varables we oly eed o model M as beore. The eressos or Var Z D ad Cov Zs Z D wll smly somewha bu hey wll sll be cumbersome o comue. Hece we oba he redcve dsrbuo by j j smulao bu we oly eed o geerae he samles or he M e{ y } δˆ ad he requred reserves er cell wll ow be Z X M all.. > +. Usg hs rocedure we ca oba he redcve dsrbuo o he reserves ha wll be comarable o hose gve he reereces bu wh he advaage o havg he comlee redcve dsrbuo. Noce ha sce he global model s arrved a usg all he ormao o ad hs vecor does o aear whe usg X X eosures he here s o eed o model X whe s deed hs way. Furher smlcao ca be aaed here s o ormao o clams er cell or eosure. I ha case we would have X or all ad. j j 5. APPLICATION I hs seco we rese wo ses o daa ha coaa egave values Table ad Table. These have bee eesvely used o esmae he reserves by dere mehods. Table comes rom Mac 994 ad has oly oe egave value. Table s ae rom Verrall 99b ad cludes hree egave values. Tables 4 ad 5 rese he resuls o alyg dere mehods o hese ses. I Table 4 we comare he resuls o alyg he cha-ladder mehod he over-dsersed Posso ad our Bayesa smulao. As eeced he requred reserves are eacly he same or he rs wo. We also clude he Boosra esmaor o he sadard devao as gve Eglad ad Verrall 00. Ths las reerece also rovdes he reserves esmaed wh he B-F mehod whch are 5000; lower ha he ohers. All he sadard devsos are well he same rage wh he Bayesa oe slghly larger. The more srg derece s he reserves. The 9
Bayesa resul s much hgher. I s ear he oe ha oe would oba by alyg a sraghorward logormal model wh some adjusmes or he egave value Mac 994. Clearly he cha-ladder does o seem o be aeced by he egave value. Fgure shows he redcve dsrbuo o he reserves or Year o Org Row ad or he oal. They are clearly sewed secally he rs oe. Aalyss or he oher Rows show smlar resuls bu hey are o cluded or he sae o brevy. Ths may be oe cause or hs derece. Table 5 rovdes he resuls o alyg he he same hree mehods as above bu ow we clude also hose o he sraghorward hree arameer log-ormal model whou usg he varace correco whe esmag he clam amous er cell. Ths s he colum labeled log-normal. We also clude he resul o gorg he cells wh he egave values.e. reag hem as mssg values. I hs case here o bg dereces he larges beg he laer. I s eresg o oe ha some o he mehods yeld egave reserves or some o he accde years bu he oal s osve. Fgure shows he redcve dsrbuo or accde year o ael accde year 6 mddle ael ad or he oal boom ael. They are all much less sewed ha he revous eamle. Ths may ela why he dereces bewee he resuls o he dere mehods are smaller. The Bayesa mehod reseed here cosues a aealg alerave o clams reservg mehods he resece o egave values cremeal clams or some cells o he develome ragle. Furher aalyss s eeded o clary some o he dereces whch may be warraed whe he daa s very sewed as he rs eamle. O he oher had hs mehod wll o brea dow eve he resece o a cosderable umber o egave values. Aoher o or research wll be o avod he use o he role lelhood ad carry ou a ully Bayesa aalyss o he roblem. Table 4 5 6 7 8 9 0 50 57 68 898 74 64 88 599 54 7 06 479 570 6 87-0 67 55 40 558 488 68 594 479 649 60 4 5655 5900 4 5500 59 658 984 5 09 847 67 6 786 5 6 5 49 557 97 7 557 46 696 68 8 5 5596 665 9 6 0 06 Source: Mac 994 0
Table 4 5 6 7 8 9 0 90089 66666 464 4687 6475 6996 59 540684 0757 5896 5087 5645 40574 6480 67897 656985 4584 700 54 79066 9855 7700-478 540 774 87086-4998 8004 74657 09788 48 05-88 4 4894 08 6448 48680 0 57 06749 697658 6507 5 678 78404 88068 684 480 56 05508 646 6 0684 0987 89684 5849 6977 9 07976 7 76 9044 587769 66087 6866 445 8 59 8554 48580 454 47587 9 97658 495 69009 45097 0 584 9089 50706 4570 4 09 Source: Verrall 99b Table 4 Comarso o Resuls. Mac Daa Cha-L Over-dsersed Posso Bayesa Row Reserves Reserves Sd. Dev. Boosra Reserves Sd. Dev. 54 54 556 695 89 759 67 67 0 4 55 548 4 66 66 775 99 55 8 5 747 747 77 577 96 6 649 649 440 56 48 8 7 545 545 4 09 5065 5 8 0907 0907 50 55 654 604 9 0650 0650 6075 608 46 7790 0 69 69 987 644 464 5458 TOTAL 55 55 89 967 69459 788 Table 5 Comarso o Resuls. Verrall Daa Row Bayesa Cha-ladder Log-Normal ODP Mssg 790 8470 996 8470 859 5860 405 06 405 0697 4 650 8700 57 8700 4754 5 8600 864 57646 864 40880 6 64900 8846 946 8846 9098 7 4900 05768 7008 05768 486 8 8000 0797 998 0797 990 9 500 065 0468 065 8498 0 79900 47809 067784 47809 7070 9500 6644 68769 6644 40788 4900 9887 7500 9887 557668 TOTAL 94000 946747 99708 946747 5685
Fgure 000 Frequecy 500 0 0 5000 0000 5000 Row 600 500 400 Frequecy 00 00 00 0 0 00000 Toal Reserves 00000
Fgure 00 Frequecy 00 00 0-000 0 000 000 Row 00 00 Frequecy 00 0-000 -000 0 000 000 000 Row 6 400 00 Frequecy 00 00 0-0000 0 0000 0000 0000 40000 Toal Reserves
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