DUCATIO AD AMIATIO COMMITT OF TH SOCITY OF ACTUAIS COSTUCTIO AD VALUATIO OF ACTUAIAL MODLS STUDY OT TOPICS I CDIBILITY THOY by Curs Gary Dean, FCAS, MAAA Copyrgh 005 Curs Gary Dean eproduced by he Casualy Acuaral Socey and he Socey of Acuares wh persson of he auhor The ducaon and xanaon Coee provdes sudy noes o persons preparng for he exanaons of he Socey of Acuares They are nended o acquan canddaes wh soe of he heorecal and praccal consderaons nvolved n he varous subjecs Whle varyng opnons are presened where approprae, ls on he lengh of he aeral and oher consderaons soees preven he ncluson of all possble opnons These sudy noes do no, however, represen any offcal opnon, nerpreaons or endorseen of he Socey of Acuares or s ducaon and xanaon Coee The Socey s graeful o he auhors for her conrbuons n preparng he sudy noes C-4-05 Prned n USA
Preface Ths sudy noe was wren o suppleen he Credbly chaper of Foundaons of Casualy Acuaral Scence as a readng for he fourh CAS/SOA exanaon I presens poran opcs no covered n he Foundaons chaper ncludng he Bühlann-Sraub Model and nonparaerc and separaerc esaon of credbly forula paraeers The auhor would lke o hank Clve Keange, a eber of boh he CAS xanaon and Syllabus Coees, for hs suggeson o wre he sudy noe Clve provded oversgh durng he wrng and edng process, and ade helpful suggesons abou he conen and clary of exposon He also chared he coee ha revewed he sudy noe I would also lke o hank he revewers on he coee for her any valuable conrbuons ncludng Joseph Boor, ussell Greg, asser Hadd, Legh Hallwell, Suar Klugan, Waler Lowre, Marjore osenberg, and Gary Vener Gary Vener s responsble for uch of he aeral ncluded n he secon An Inuve Model for Credbly Ball Sae Unversy sudens also provded valuable coens and should be recognzed: Heaher Adas, Chuke Ounj, Doug Prle, Lor Thopson, and Melany Tower
Table of Conens Credbly Models evew of Bühlann Model Bühlann-Sraub Model saon of Credbly Forula Paraeers onparaerc Separaerc 3 Concluson Appendx A: Bühlann and Bühlann-Sraub Credbly saors Are Lnear Leas Squares saors Appendx B: Proof of Uncondonal Toal ance Forula Appendx C: onparaerc saors for he xpeced Value of he Process ance and he ance of he Hypohecal Means n he Bühlann-Sraub Model are Unbased eferences xercses Soluons o xercses 3
Credbly Models Ths sudy noe suppleens he Credbly chaper of Foundaons of Casualy Acuaral Scence as a readng for he fourh CAS/SOA exanaon Several poran opcs no covered n he Foundaons ex are presened here, ncludng he Bühlann- Sraub credbly odel and esaon of credbly forula paraeers I s assued ha he suden already has soe falary wh he aeral covered n he Credbly chaper before readng hs sudy noe The credbly odels ha wll be dscussed are ofen referred o as greaes accuracy credbly or leas squares credbly As wll be explaned laer, hese ehods aep o produce lnear esaes ha wll nze he expeced value of he square of he dfference beween he esae and he quany beng esaed Bühlann credbly wll be revewed payng parcular aenon o he splfyng assupons ha dsngush fro he ore general Bühlann-Sraub odel ha follows The second half of he sudy noe covers esaon of credbly forula paraeers when underlyng dsrbuons are unknown Before begnnng a ore rgorous sudy, an nuve dervaon of he useful Bühlann credbly odel wll be presened An Inuve Model for Credbly The acuary uses observaons of evens ha happened n he pas o forecas fuure evens or coss For exaple, daa ha was colleced over several years abou he average cos o nsure a seleced rsk, soees referred o as a polcyholder or nsured, ay be used o esae he expeced cos o nsure he sae rsk n fuure years Because nsured losses arse fro rando occurrences, however, he acual coss of payng nsurance losses n pas years ay be a poor esaor of fuure coss Consder a rsk ha s a eber of a parcular class of rsks Classes are groupngs of rsks wh slar rsk characerscs, and hough slar, each rsk s sll unque and no que he sae as oher rsks n he class In class rang, he nsurance preu charged o each rsk n a class s derved fro a rae coon o he class Class rang s ofen suppleened wh experence rang so ha he nsurance preu for an ndvdual rsk s based on boh he class rae and acual pas loss experence for he rsk The poran queson n hs case s: How uch should he class rae be odfed by experence rang? Tha s, how uch credbly should be gven o he acual experence of he ndvdual rsk? Inuon says ha wo facors appear poran n fndng he rgh balance beween class rang and ndvdual rsk experence rang: How hoogeneous are he classes? If all of he rsks n a class are dencal and have he sae expeced value for losses, hen why boher wh ndvdual 4
experence rang? Jus use he class rae On he oher hand, f here s sgnfcan varaon n he expeced oucoes for rsks n he class, hen relavely ore wegh should be gven o ndvdual rsk loss experence ach rsk n he class has s own ndvdual rsk ean called s hypohecal ean The ance of he Hypohecal Means VHM across rsks n he class s a sascal easure for he hoogeney or vce versa, heerogeney, whn he class A saller VHM ndcaes ore class hoogeney and, consequenly, argues for ore wegh gong o he class rae A larger VHM ndcaes ore class heerogeney and, consequenly, argues for less wegh gong o he class rae How uch varaon s here n an ndvdual rsk s loss experence? If here s a large aoun of varaon expeced n he acual loss experence for an ndvdual rsk, hen he acual experence observed ay be far fro s expeced value and no very useful for esang he expeced value In hs case, less wegh, e, less credbly, should be assgned o ndvdual experence The process varance, whch s he varance of he rsk s rando experence abou s expeced value, s a easure of he varably n an ndvdual rsk s loss experence The xpeced Value of he Process ance PV s he average value of he process varance over he enre class of rsks Le represen he saple ean of n observaons for a randoly seleced rsk Because here are n observaons, he varance n he saple ean s he varance n one observaon for he rsk dvded by n Gven rsk, hs varance s PV / n where PV s he process varance of one observaon Because rsk was seleced a rando fro he class of rsks, an esaor for s varance s PV / n PV / n PV / n Ths s he xpeced Value of he Process ance for rsks n he class dvded by he nuber of observaons ade abou he seleced rsk I easures he varably expeced n an ndvdual rsk s loss experence Leng µ represen he overall class ean, a rsk seleced a rando fro he class wll have an expeced value equal o he class ean µ The varance of he ndvdual rsk eans abou µ s he VHM, he ance of he Hypohecal Means There are wo esaors for he expeced value of he h rsk: he rsk s saple ean, and he class ean µ How should hese wo esaors be weghed ogeher? A lnear esae wh he weghs sung o 00 would be sae w wµ An opal ehod for weghng wo esaors s o choose weghs proporonal o he recprocals of her respecve varances Ths resuls n gvng ore wegh o he The expecaon s aken over all rsks n he class 5
esaor wh saller varance and less wegh o he esaor wh larger varance In any suaons hs wll resul n a nu varance esaor Please see he frs proble n he exercses a he end of he sudy noe The resulng weghs are w PV / n and PV / n VHM w VHM PV / n VHM oe ha a denonaor was chosen so ha he weghs add o one A lle algebra produces n w and PV n VHM n w PV n VHM Seng K PV / VHM, he wegh assgned o he rsk s observed ean s n w n K Ths s he falar Bühlann credbly forula wh credbly Z n / n K In hs secon, a rsk seleced fro a rang class was used o llusrae he concep of credbly In general, an ndvdual rsk or a group of rsks coes fro a larger populaon and he goal s o fnd he rgh balance beween usng he daa for he saller group and he larger populaon Many oher exaples are possble xaple An acuary calculaed ndcaed rae changes by errory for auooble nsurance The rae change ndcaon for he h errory was Cobned daa for he enre sae ndcaed ha a rae change of 0% was requred Fro hese values, credbly weghed rae change ndcaons were calculaed: Credbly weghed rae change Z x Z x 0 % ndcaon for errory The credbly weghs Z were calculaed fro he forula Z n / n K where n was he nuber of nsured vehcles n he errory durng he hree-year daa collecon perod A rgorous dervaon of he Bühlann credbly forula s provded n Appendx A 6
Prelnares and oaon The acuary uses observaons for a rsk or group of rsks o esae fuure oucoes for ha sae rsk or group In hs sudy noe, alhough he er a rsk s ofen used, he sae coens can generally be appled o a group of rsks where he group s a collecon of rsks wh soe coon characerscs The acual observaon durng e for ha parcular rsk or group wll be denoed by x, whch wll be he observaon of correspondng rando varable, where s an neger For exaple, ay represen he followng: uber of clas n perod Loss rao n year Loss per exposure n year Oucoe of he h roll of a de An ndvdual rsk s a eber of a larger populaon and he rsk has an assocaed rsk paraeer ha dsngushes he ndvdual s rsk characerscs I s assued ha he rsk paraeer s dsrbued randoly hrough he populaon and wll denoe he rando varable The dsrbuon of he rando varable depends upon he value of : f x For exaple, ay be a paraeer n he dsrbuon funcon of In he case of a Posson clas process, gh be he expeced nuber of clas Alhough he exaples n hs sudy noe wll use s ha are scalars, one can also buld odels wh as a uldensonal vecor wh each coponen of he vecor descrbng soe aspec of he ndvdual s rsk characerscs If s a connuous rando varable, he ean for gven, s he condonal expecaon, x f x dx µ, where he negraon s over he suppor of f x If s a dscree rando varable, hen a suaon should be used: x f x all x The negral noaon wll be used n general cases, bu he reader should be aware ha a suaon s called for wh dscree rando varables I wll be assued ha µ s consan hrough e for he odels consdered n hs sudy noe 3 The rsk paraeer represened by he rando varable has s own probably densy funcon pdf: f The pdf for descrbes how he rsk characerscs are 3 Ths s a ajor assupon ha s easly volaed n pracce sk characerscs can change for a varey of reasons: a young drver becoes a beer drver wh experence; a busness ay nsue rsk conrol procedures ha reduce losses; raffc denses ay ncrease n an area leadng o ncreased probables of auo accdens; and nflaon wll ncrease he coss of loss payens 7
dsrbued whn he populaon If wo rsks have he sae paraeer, hen hey are assued o have he sae rsk characerscs ncludng he sae ean µ The uncondonal expecaon of s x f, x, dx d x f x f dx d 4 x f x dx f d µ µ The condonal varance of gven s µ µ f x dx σ Ths varance s ofen called he process varance for he seleced rsk The uncondonal varance of, also referred o as he oal varance, s gven by he Toal ance forula:, or Toal ance ance of he Hypohecal Means xpeced Value of he Process ance A proof of hs forula s shown n Appendx B These conceps are bes deonsraed wh an exaple xaple The nuber of clas durng he h perod for a rsk has a Posson x e dsrbuon wh paraeer : P x The rsk was seleced a rando x! fro a populaon for whch s unforly dsrbued over he nerval 0, Ths sple dsrbuon for was chosen o ake he negraon easy I wll be assued ha s consan hrough e for each rsk Hypohecal ean for rsk wh paraeer s µ because he ean of he Posson rando varable s he paraeer Process varance for rsk wh paraeer s σ because he varance equals he paraeer for he Posson 3 ance of he Hypohecal Means VHM s 4 oe ha a subsuon for he jon densy funcon f, x, was ade usng he relaonshp f, x, f x f 8
/ d d 0 0 4 xpeced Value of he Process ance PV s d / 0 5 Uncondonal ance or oal varance s Bühlann Model VHM PV / / 7 / The Bühlann odel assues ha for any seleced rsk, he rando varables {,,,,, } are ndependenly and dencally dsrbued For he seleced rsk, each has he sae probably dsrbuon for any e perod, boh for he,,, rando varables n he experence perod, and fuure oucoes,, As Hans Bühlann descrbed, hoogeney n e s assued The characerscs ha deerne he rsk s exposure o loss are assued o be unchangng and he rsk paraeer assocaed wh he rsk s consan hrough e for he rsk The eans and varances of he rando varables for he dfferen e perods are equal and are labeled µ and σ, respecvely, as shown n he able below: Assupons of Bühlann Credbly Hypohecal Mean: µ Process ance: σ Of course he hypohecal eans and process varances wll vary aong rsks, bu hey are assued o be unchangng for any ndvdual rsk n he Bühlann odel To apply Bühlann credbly, he average values of hese quanes over he whole populaon of rsks are needed, along wh he varance of he hypohecal eans for he populaon: Populaon ean: µ µ xpeced Value of Process ance: PV σ 3 ance of Hypohecal Means: VHM µ µ µ The populaon ean µ provdes an esae for he expeced value of n he absence of any pror nforaon abou he rsk The PV ndcaes he 9
varably o be expeced fro observaons ade abou ndvdual rsks The VHM s a easure of he dfferences n he eans aong rsks n he populaon Because µ s unknown for he seleced rsk, he ean he esaon process I s an unbased esaor for µ, s used n µ µ The condonal varance of, assung ndependence of he gven, s σ σ The uncondonal varance of s σ µ VHM Bühlann credbly assgned o esaor s gven by he well-known forula PV Z K, where s he nuber of observaons for he rsk and K PV / VHM Mulplyng he nueraor and denonaor by VHM / gves an alernave for: VHM Z PV VHM oe ha he denonaor s jus as derved a few lnes earler Therefore Z / K can be wren as Z ance of he Hypohecal Means Toal ance of he saor µ The nueraor s a easure of how far apar he eans of he rsks n he populaon are, whle he denonaor s a easure of he oal varance of he esaor s The credbly weghed esae for µ, for,,,,, 0
µ Z Z µ The esaor µ s a lnear leas squares esaor for µ Ths eans ha { Z Z µ µ } s nzed when Z / K Appendx A proves hs 5 Bühlann-Sraub Model The requreen ha he rando varables,,,,, for a rsk be dencally dsrbued s easly volaed n he real world For exaple: The work force of a workers copensaon polcyholder ay change n sze fro one year o he nex The nuber of vehcles owned by a coercal auooble polcyholder ay change hrough e The aoun of earned preu for a rang class vares fro year o year In all of hese cases, one should no assue ha,,,,, are dencally dsrbued, alhough an assupon of ndependence ay be warraned A rsk s exposure o loss ay vary and s assued ha hs exposure can be easured Soe easures of exposure o loss are: Aoun of nsurance preu uber of eployees Payroll uber of nsured vehcles uber of clas In fac, a fundaenal prese of nsurance rang s ha exposure bases can be denfed ha are drecly relaed o he poenal for loss The Bühlann-Sraub odel assues ha he eans of he rando varables are equal for he seleced rsk, bu ha he process varances are nversely proporonal o he sze e, exposure of he rsk durng each observaon perod For exaple, when he rsk s wce as large, he process varance s halved These assupons are suarzed n he followng able: 5 The expeced squared error s nzed only f he rue values of he PV and VHM are used o calculae K If esaed values of he PV and VHM or K are used, whch s coonly done n pracce, he lnear esaor as gven above s no longer opal Ths s an advanced opc beyond he scope of hs sudy noe
Assupons of Bühlann-Sraub Credbly Perod Perod xposure Hypohecal Mean for sk per Un of xposure Process ance for sk µ σ σ The rando varables represen nuber of clas, oneary losses, or soe oher quany of neres per un of exposure, and s he easure of exposure For exaple, could be nuber of clas per house-year 6, or gh be a loss rao 7 oe ha he process varance for he rando varable decreases as he exposure ncreases xaple The annual nubers of clas for ruck drvers n a hoogeneous populaon are ndependenly and dencally dsrbued The populaon gh represen he work force of a large ruckng copany wh src hrng sandards and good safey ranng for each drver For each drver he nuber of clas per year has a ean of µ and a varance of σ The paraeer apples o every drver n he group A group of 0 drvers s seleced fro he larger populaon Wha s he expeced annual clas frequency for he group of 0 drvers? Wha s he varance of he annual clas frequency for he group? Soluon Le,,, 0 be rando varables represenng he nuber of clas n 0 year for each of he en seleced drvers Then, s he annual clas 0 frequency for he group; ha s, s he annual nuber of clas per drver The exposure s 0 and he un of exposure s one drver The expeced value and varance for he annual clas frequency for he group are 0 0 0 0 0 0 µ µ and 6 A house-year eans one house nsured for one full year I also represens wo houses each nsured for one-half year, or n houses each nsured for /n years 7 Loss rao equals losses dvded by preu In hs case preu s he easure of exposure A loss rao of 60% eans ha here are 60 n losses for each 00 of preu
3 0 0 0 0 00 0 0 σ σ In hs exaple, he exposure s he nuber of drvers n he group, whch s 0 The expeced clas frequency s he sae wheher here s one drver, 0 drvers, or 00 drvers n he group; however, he varance n he group s clas frequency s nversely proporonal o he nuber of drvers n he group How should rando varables,,, assocaed wh a seleced rsk or group of rsks be cobned o esae he hypohecal ean µ? A weghed average usng he exposures wll gve a lnear esaor for µ wh nu varance Frs defne Then, defne he weghed average ecall ha he varance of each gven s σ / For a weghed average w, he varance of wll be nzed by choosng he weghs w o be nversely proporonal o he varances of he ndvdual s; ha s, rando varables wh saller varances should be gven ore wegh So, weghs w / are called for under he curren assupons The proof s ncluded as an exercse The condonal expeced value and varance of gven rsk paraeer are µ µ, and σ σ xaple Connung he pror exaple, assue ha he nuber of drvers n he group was sx n he frs year, seven n he second year and nne n he hrd year represens he nuber of clas per drver and s he nuber of drvers n he group n years,, and 3 6 7 9
4 9 7 6 3 3 9 7 6 9 7 6 3 3 µ 4 3 3 9 7 6 9 7 6 9 9 7 7 6 6 σ σ σ σ xaple A class for workers copensaon nsurance produced he followng: Year Payroll n 00 Uns Losses Loss per xposure 00,000 300,000 300 x 0,000 00,000 8 x 3 0,000 3 30,000 67 x 3 Toal 330,000 80,000 48 x The exposure un s 00 of payroll oe ha x can be calculaed wo equvalen ways: 67 330,000 0,000 8 330,000 0,000 300 330,000 00,000 3 x x 8 x 300,000 00,000 30,000 / 330,000 The PV and VHM are defned o be PV σ and VHM µ, where he expeced value s over all rsk paraeers n he populaon eeber, he loss per un of exposure s used because he exposure can vary hrough e and fro rsk o rsk The uncondonal ean and varance of are 8 oe ha hs ehod produces 49, whch dffers fro 48 n he able because of roundng error
µ µ, and σ µ VHM PV µ s As n he spler Bühlann case, he credbly assgned o he esaor of Z ance of he Hypohecal Means Toal ance of he saor VHM PV VHM Mulplyng he nueraor and denonaor by /VHM yelds a falar lookng forula Z K The oal exposure replaces n he Bühlann forula and he paraeer K s defned as usual K PV σ VHM µ oe ha he Bühlann odel s acually a specal case of he ore general Bühlann- Sraub odel wh for all The credbly weghed esae s µ Z Z µ xaple The acuares a he Good Healh Insurance Copany calculae prospecve preus for group nsurance polces usng a Bühlann-Sraub credbly odel Analyss of Good Healh s daa led o he followng assupons for s busness: For all polces ogeher, he prospecve average annual expeced pure preu per nsured person s,400 The varance of he hypohecal ean pure preus across group plans s 500,000 The expeced value of he process varance n annual coss per nsured person s 50,000,000 5
One of Good Healh s clens had he followng experence durng a one-year perod wh coss adjused o reflec prospecve coss Group Polcy Insured Persons Cos per Insured Person 40 3,000 Calculae a credbly weghed pure preu for group polcy Soluon K 50,000,000/500,000 500, µ,400, 40 40 3,000, Z 343 40 500 saed Pure Preu 3433,000 343,400,59458 saon of Credbly Forula Paraeers Boh he Bühlann and Bühlann-Sraub odels requre a deernaon of he paraeer K In pracce here are several ways hs s done: Judgenally selec K A larger K gves less credbly o he ndvdual saple ean and ore credbly o he populaon ean A saller K gves ore credbly o he ndvdual saple ean, bu he saple ean ay change sgnfcanly fro one easureen perod o anoher producng a flucuang esae For exaple, he laes hree years of daa ay be used o calculae As an old year rolls off o be replaced by a ore recen year of daa, he value of can drascally change Selec a K value ha would have worked bes n pror applcaons of he odel If one s ryng o esae,, fro Z Z µ, one approach would be o nze x Z x Z µ where he acual oucoes for year are copared wh he credbly weghed forecass for all rsks n he populaon One could also use soe oher funcon of he dfference beween oucoes and forecass such as he su of absolue errors or he su of absolue values of he percenage errors 3 Aep o deerne he PV and VHM coponens of K For he reander of hs sudy noe, 3 wll be dscussed 6
To calculae he xpeced Value of he Process ance PV and he ance of he Hypohecal Means VHM he followng are needed: Process varances σ for each rsk n he populaon Hypohecal eans µ for each rsk n he populaon 3 Dsrbuon funcon for o calculae he PV σ and VHM µ her all of he above us be esaed fro he daa, or else splfyng assupons us be ade Suppose ha here are ndependen rsks o be observed for separae e perods as represened by he rando varables n he able below: Te Perod sk sk Paraeer : : : : : oe ha anoher subscrp has been added o he rando varables because he dscusson now concerns ulple rsks The s are rando varables represenng he losses or nuber of clas for rsk durng e perod and, for now, each rsk has ndependen oucoes Assocaed wh each s a ean µ such ha µ The expeced values of each of he oucoes for any seleced rsk are assued o be equal The followng addonal assupons are also ade: For any seleced rsk, he s are ndependen gven The oucoes for any rsk are ndependen of any oher rsk 3 The rando varables,,, are ndependen and dencally dsrbued fro a coon dsrbuon f For a rando varable seleced a rando fro he able he expeced value s µ µ 7
To calculae he uncondonal varance of a randoly seleced, he oal varance forula us be used: onparaerc saon In he nonparaerc case, no assupons are ade abou he for or paraeers of he dsrbuons of, nor are any assupons ade abou he dsrbuon of he rsk paraeers Of course, for he Bühlann odel s assued ha for any gven rsk, he rando varables {,,, }, represenng he oucoes for dfferen observaons, are ndependenly and dencally dsrbued wh dencal eans and varances The oucoes for dfferen rsks are also ndependen In he Bühlann- Sraub odel, he oucoes for rsk have he sae eans bu he process varances are nversely relaed o he exposure oe ha he nuber of observaons has a subscrp ndcang ha he nuber of observaons can vary by rsk n he Bühlann- Sraub odel To apply he odels n Secons and, soe nforaon abou he probably dsrbuons f x and f s requred Alhough he exac dsrbuon funcons are no needed, he PV and VHM us be obaned n order o calculae K and hen he credbly Z In pracce he PV and VHM are ofen unknown The PV and VHM can be esaed fro he daa for he Bühlann odel or he ore coplcaed Bühlann-Sraub odel The esaon procedures are soees referred o as eprcal Bayesan procedures or, equvalenly, eprcal Bayes esaon Bühlann Model saes of he PV and VHM can be ade fro eprcal observaons abou a saple fro he populaon of rsks Assue ha here are rsks n he saple and separae observaons wll be ade for each rsk The rando varables n he lefhand secon of he followng able represen he oucoes: Te Perod sk sk s Saple Mean σ sk s Saple Process ance σ : : : : : : σ 8
The s are unbased esaors for each rsk s ean µ, and he σ s are unbased esaors for each rsk s process varance σ oe ha he dvsor s raher han for he saple varances because saple eans raher han rue eans µ are used o copue he xpeced Value of he Process ance PV The ndvdual saple process varances σ can be cobned o produce an unbased esae for he expeced value of he process varance of he populaon To esae hs, he average of he ndvdual saple process varances s copued: PV σ ance of he Hypohecal Means VHM In he pror able he s are esaors for he unknown hypohecal eans µ and hese values can be used o esae he varance of he hypohecal eans Because he rsks are all ndependen, he s are ndependen rando varables An unbased esaor for he varance of s where However, hs s OT he esae for he VHM The oal varance forula gves 9 The frs er on he rgh can be splfed by nong ha µ The followng relaonshp can splfy he second er on he rgh: Subsung no he forula σ / µ σ /
The frs er on he rgh s jus he VHM and he second er s he PV / earrangng ers yelds VHM PV / Subsung n he unbased esaors for he quanes on he rgh produces an unbased esaor for he VHM, VHM / To esae VHM µ, he quany needs o be adjused downward by PV / because he process varance ncreases he varably of he esaes for µ The larger he average process varance, he larger he necessary correcon o o esae VHM µ Because of hs subracon, s also possble ha he esaor VH M ay be negave Wha does ean for a VHM o be negave? Because varances us be nonnegave, one usually concludes ha zero s a reasonable esae for he VHM and ha he eans of he ndvdual rsks are all he sae There s no eprcal evdence ha he rsk eans are dfferen fro one anoher xaple Two rsks were seleced a rando fro a populaon sk had 0 clas n year one, 3 clas n year wo, and 0 clas n year hree: 0,3,0 The clas by year for sk were,, In hs case, and 3 x 0 3 0 / 3, x / 3 5 / 3, and x 5 / 3/ 4 / 3 σ 0 3 0 /3 3 σ 5/ 3 5/ 3 5/ 3 /3 / 3 PV σ σ / 3 / 3/ 5 / 3 { 4 / 3 5 / 3 4 / 3 } 5 / 3 / 3 / 3 V HM The saple eans for he wo rsks are close relave o he szes of he saple process varances The calculaed VH M s negave, so a value of zero wll be assued The hypohecal eans are ndsngushable Ths ples a credbly facor Z 0 Credbly weghed esaors for he rsk eans can be derved usng he forulas 0
PV K, Z and µ Z Z VHM K Alhough he esaors P V and VH M are unbased esaors for he xpeced Value of he Process ance and he ance of he Hypohecal Means, he esaed value Ẑ for he credbly Z s no unbased In pracce, he above Ẑ s generally acceped as a reasonable esae for he credbly wegh xaple Two rsks were seleced a rando fro a populaon Over a four-year perod, sk had he followng clas by year: 0,0,,0 The clas by year for sk were:,,0, Calculae credbly weghed esaes for he expeced nuber of clas per year for each rsk Soluon x 0 0 0 / 4 / 4, x 0 / 4 5 / 4, and x / 4 5/ 4/ 3/ 4 σ 0 / 4 0 / 4 / 4 0 / 4 /4 / 4 σ 5/ 4 5/ 4 0 5/ 4 5 / 4 /4 / PV σ σ / / 4 // 7 / {/ 4 3/ 4 5 / 4 3/ 4 } 7 / / 4 7 / 48 V HM PV 7 / K 8 /7 VHM 7 / 48 4 Z 7 / 4 K 4 8/7 µ Z Z 7 / 4 / 4 7 / 4 3/ 4 9 / 48 3958 µ Z Z 7 / 4 5 / 4 7 / 4 3/ 4 53/ 48 04 Bühlann-Sraub Model The Bühlann-Sraub Model s ore coplcaed because a rsk s exposure o loss can vary fro year o year, and he nuber of years of observaons can change fro rsk o rsk The reason ha Bühlann-Sraub can handle varyng nubers of years s because he nuber of years of daa for a rsk s refleced n he oal exposure for he rsk
The able below shows represenng cla frequency, loss rao, or average cos per exposure n he op half of he cell, and he correspondng nuber of exposures for he rsk durng he sae e perod n he boo half: sk Perods of xperence : : : : The nuber of perods of experence can vary by rsk; for exaple, for sk here are perods of experence copared wh perods for sk The experence perods do no have o sar a he sae e eher For exaple, he frs experence perod for sk gh be n Year Y whereas he frs experence perod for sk ay be Year Y xaple ABC Insurance, Inc sells denal nsurance plans o copanes wh fewer han one hundred eployees An acuary s analyzng he nuber of clas per eployee Lookng a he frs copany n her fle, she sees ha he copany has hree full years of plan coverage In he frs year here were 40 eployee-years wh 84 clas, n he second year here were 44 eployee-years wh 88 clas, and n he hrd year here were 4 eployee-years wh 05 clas Desgnang hs seleced copany as sk, hen: 84 clas / 40 eployee-years clas/eployee-year 88 clas / 44 eployee-years 0 clas/eployee-year 3 05 clas / 4 eployee-years 5 clas/eployee-year The exposures are 40 eployee-years, 44 eployee-years, and 3 4 eployee-years The nex able shows esaors for rsk eans and varances:
3 sk Perods Toal xposure Saple Mean Saple sae for σ σ σ : : : : : σ Toal / PV / σ In he Bühlann-Sraub odel, he ean s assued o be consan hrough e for each rsk : µ s an unbased esaor for he ean of rsk : µ µ ecall fro Secon ha he process varance of s nversely proporonal o he exposure: σ / Ths eans ha for rsk, σ, for o To provde soe ovaon for he process varance esaes n he las colun of he able above, s helpful o wre ou he defnon of varance as µ σ, for o Sung boh sdes over and dvdng by he nuber of ers yelds µ σ, or
σ µ The quany µ s unknown, so s used nsead n he esaon process Ths reduces he degrees of freedo by one so s replaced by n he denonaor: σ Thus, an unbased 9 esaor for σ s σ If and each rsk has he sae nuber of years of daa, hen he esaors σ ach hose fro he Bühlann odel xpeced Value of he Process ance and The PV can be esaed by cobnng process varance esaes σ of he rsks If hey are cobned wh weghs w /, hen w σ PV Ths s an unbased esaor for he PV as shown n Appendx C A way o guess he denonaor s o observe ha here are ndcaon of degrees of freedo However, here are esaors ers added ogeher n he nueraor, an for he ndvdual 9 A proof s no very dffcul and can be odeled afer he sandard proof ha he saple varance wh dvsor s an unbased esaor for he varance 4
eans, and ha reduces he degrees of freedo by So, he dvsor s ance of he Hypohecal Means The hypohecal ean for rsk s µ The varance of he hypohecal eans can be wren as VHM µ µ where µ µ Because oucoes of he rando varables and are esaors for µ and µ, respecvely, a good sarng pon for developng an esaor for he varance of he hypohecal eans s: oe ha each er s weghed by s oal exposure over he experence perod However, hs s no unbased An unbased esaor s VHM PV / Showng ha hs esaor s unbased akes several seps and he deals are ncluded n Appendx C Wh he Bühlann-Sraub odel, he easure o use n he credbly forula s he oal exposure for rsk over he whole experence perod oe ha he npus n he P V and VH M forulas are easured per un of exposure The forulas o copue credbly weghed esaes are K PV, Z and Z Z VHM K µ xaple Carpenry conracors A and B had nsurance polces coverng pckup rucks Over a four-year perod he followng was observed: Year Insured _Y Y Y Y3_ A uber of Clas 3 0 Insured Vehcles B uber of Clas 0 Insured Vehcles 4 3 5
sae he expeced annual cla frequency for each nsured usng he Bühlann-Sraub odel Soluon The rando varables represenng cla frequency and he correspondng exposures are as follows: Year Insured _Y Y Y Y3_ A Clas per xposure 3/ 3 4 0 xposure uber of Vehcles 3 4 B Clas per xposure / /3 3 0 xposure uber of Vehcles 4 3 3 The cla frequency s uber of Clas/Insured Vehcles The frs able shows nuber of clas ha are he values for A 7, B 4 3 9, and 7 9 6 x A 3 0 / 7, x B 0 / 9 / 3, and x 7 9/ 3/6 5 / 8 σ A 3/ 0 /4 / σ B 4/ / 3 3/ 3 / 3 0 / 3 /3 / 6 4 / 3 / 6 PV / 30 3667 4 3 oe: You can su he seven ers nsde he brackes n he expressons for σ A and σ and hen dvde he oal by 43 Ths saves he sep of dvdng by he ndvdual degrees of freedo and hen undong when calculang he PV esae B 7 5/8 V HM 9/ 3 5/ 8 / 30 6 7 9 6 757 PV K 3667 / 757 087 VHM 6
7 939 77035/8 7703 7703 087 7 7 A A A A µ K Z 388 885/8 3 88/ 88 087 9 9 B B B B µ K Z Balancng he esaors I s desrable n any cases for he esaors Z Z µ, when weghed ogeher, o equal he overall saple ean An exaple s an experence rang plan The aoun of preu charged o pay for losses per un of exposure for he h rsk s µ The average loss per exposure for all rsks s To ake he experence rang plan balanced, e, for he su of he peces o add up o he oal, he goal s Z Z In general hs gh no happen Pung µ n for he copleen of credbly yelds µ Z Z Z Z µ Z Z µ µ Z Z The lef-hand sde can be splfed by nong ha Z K K K K Z The weghed average of he credbly esaes for he rsks wll equal f
KZ KZ Z Z µ Balance can be acheved by usng a credbly weghed µ credbly as he copleen of xaple The pror exaple produced: Weghs Credbles Saple Means B A 7 6 9 6 Credbly saes Z A 7703 A µ 939 B 88 Z A B B µ 388 3 5 Weghed 650 68 8 Average The overall saple ean s 650 bu he weghed average of he credbly esaes s 68, whch s % below he overall saple ean The credbly weghed average s Z 7703 88/ 3 µ 6579 7703 88 Z ecalculang he credbly esaes: A µ 7703 77036579 94 and B µ 88/ 3 886579 3944 The weghed average s 7/694 9/63944 650, whch equals Separaerc saon Assung ha he rando varables have a parcular dsrbuonal for can splfy he calculaons For exaple, he probably dsrbuon for gh be Posson or bnoal If s he nuber of clas per exposure and s he nuber of exposures for rsk, hen he produc s he nuber of clas for rsk n e perod 8
Furherore, assue ha he nuber of clas s Posson dsrbued If he rsk paraeer s he ean nuber of clas per exposure, hen s boh he ean and varance of he nuber of clas for exposure, Dvdng hrough by yelds By defnon, µ and σ The resul s ha wh he Posson assupon, follows ha µ σ Because he ean and process varance are equal for each rsk, he sae s rue for he expeced values wh µ σ PV The esaor for he ean µ µ s µ, and he pror forula eans ha hs sae esaor can be used for he PV under he Posson assupon xaple The nforaon n he pror exaple wll be used along wh he addonal assupon ha he nuber of clas for each rsk s Posson dsrbued As calculaed prevously, x 7 9/ 3/6 5/ 8 Ths s he esae for he overall ean, so under he Posson assupon follows ha PV x 5 / 8 65 Whou he Posson assupon, he PV esae was 3667 Connung on wh he calculaons, V HM 7 5/8 9/ 3 5/8 6 7 9 6 5/8 49 PV 5/8 K 43737 VHM 49 7 Z A A 655 µ 655 6555/8 8558 A K 7 43737 A 9 Z B B 6730 µ 6730/ 3 67305/8 487 B K 9 43737 B 9
Of course, one could use he balancng procedure f balanced esaes were approprae In he exaple above, one can copue he P V eher fro he saple process varances of he daa or wh a Posson assupon Unlke he pror exaple, he nex exaple requres an assupon abou he clas process o esae he P V xaple Durng a hree-year perod, a group of,000 auo polces generaed he followng clas profle: Toal uber of Clas n Three Years uber of Polces 0 533 30 05 3 4 5 8 ach polcy was n he group for he enre hree years and nsured exacly one auooble The expeced nuber of clas per year for each nsured s assued o be consan fro year o year and he acual nuber of clas per year follows a Posson dsrbuon Deerne a credbly esaor for he expeced annual clas frequency for a polcy ha had no clas over he enre hree-year perod Do he sae for a polcy ha had fve clas Soluon Whou he Posson assupon, addonal nforaon would be needed o solve hs proble To esae he PV, would be necessary o know he nuber of clas durng each of he hree years for each of he,000 polces Ths would allow he calculaon of a saple process varance for each nsured, and hen an overall average process varance could be copued If each nsured has a Posson dsrbuon, hen as saed prevously µ σ PV, and µ can be used as an esaor for he PV whch produces 5330 30 05 3 4 85 PV 80,0003 The 3 n he denonaor s requred because s he average annual clas frequency 30
The spler Bühlann odel can be used because all of he exposures are dencally one so ha, and VHM PV / VHM PV / oe ha,000 and ha here are 533 ers n he suaon wh 0, 30 ers wh / 3, 05 ers wh / 3, ec The hrees n he denonaors are requred because he annual frequency s he nuber of clas over he hree-year perod dvded by hree 5330 8 30/ 3 8 05/ 3 8 3/ 3 8 4/3 8 85/3 8 VHM 8/ 3 000 PV 80 3 V HM 099, K 46, Z 075 VHM 099 K 3 46 clas 5 clas µ 0 0750 07580 807 µ 0755/ 3 07580 565 3 Concluson The Credbly chaper of Foundaons of Casualy Acuaral Scence along wh hs sudy noe provdes a basc educaon n credbly heory As wh os acadec presenaons, any of he exaples are dealzed and do no address soe essy reales ha ake dffcul o esae credbly odel paraeers In pracce, usng he os precse credbly paraeer esae, versus a reasonable esae, should no affec fnal resuls ha uch Mahler dscusses hs n 5 Credbly heory produces a lnear leas-squares esaor and hs can soees be a ajor source of error 0 For exaple, suppose ha for soe lne of nsurance, class raes vary fro a low of 00 per exposure o as uch as 0000 per exposure An error of 00 n he class rae would no be so bad for he 0000 rae bu would be a huge error for a class ha deserved a 00 rae In he Bühlann and Bühlann-Sraub odels, s he sze of he error ha aers so ha a 00 error n he 00 rae ges he sae wegh as a 00 error n he 0000 rae aher han nzng he squared errors as Bühlann and Bühlann-Sraub credbly do, ay be preferable o nze he squared relave errors, e percenage errors One way o accoplsh hs s o ake logarhs of values and hen nze squared errors rrors n logs are relave errors n he orgnal scale Anoher alernave s o use class loss raos nsead of pure preus 0 Gary Vener provded hs exaple 3
I can be dffcul o denfy a good copleen of credbly, he quany ha ges ulpled by Z In he basc Bühlann and Bühlann-Sraub odels, hs quany s he populaon or class ean o whch he rsk belongs Wha are he consequences f hs populaon or class ean s also hghly varable and ndcaed nsurance raes flucuae sgnfcanly? The selecon of a good copleen of credbly s soees par ar and par scence Boor denfes crera o consder when choosng a copleen of credbly n 6 The odels covered n hs sudy noe assue ha a rando varable, represenng an poran quany for a rsk, has a consan ean µ hrough e and ha he varance σ s also consan In realy, a rsk s characerscs ay shf hrough e Ths suaon s addressed by Mahler n 7, 8, and 9 sk heerogeney s anoher praccal ssue ha us be consdered A bg rsk s no necessarly he su of saller ndependen rsks For exaple, a rsk wh 00,000 n annual preu ay no behave lke he su of en ndependen 0,000 preu rsks, even hough he rsks coe fro he sae rang classfcaon Suppose he acuary s lookng a loss raos Ofen, he varance n he loss rao for he 00,000 rsk wll be greaer han he varance n he loss rao for he su of en ndependen 0,000 rsks A larger rsk can have dfferen rsk characerscs The larger rsk should receve less credbly han pled by he sple / K forula Mahler addresses hs n 9 Also called copleen for credbly 3
Appendx A: Bühlann and Bühlann-Sraub Credbly saors are Lnear Leas Squares saors A rsk s seleced a rando fro a populaon and observaons are o be ade These oucoes are represened by he rando varables {,,, } ach oucoe has he sae ean µ where s he rsk paraeer assocaed wh he seleced rsk In he Bühlann odel he rando varables are ndependenly and dencally dsrbued and he saple ean s gven by In he Bühlann-Sraub odel, he rando varables all have he sae ean µ, bu he condonal varances condonal on rsk paraeer are nversely proporonal o he exposure, whch can vary fro observaon o observaon In hs case he saple ean s defned o be, where I follows ha µ for boh odels If he expecaon s calculaed over he enre populaon wh possbly dfferen rsk paraeers, hen µ µ, where µ s he populaon ean The oucoe of he rando varable wll be used o esae µ In parcular, he goal s o fnd a lnear esaor a b wh he wo consans seleced o Mnze a b µ The expecaon s over and If were he lnear leas squares esaor for µ, hen a 0 and b would nze he above expeced value Mnzng he expecaon wll sar wh a rearrangeen of ers: a b µ a b bµ bµ µ µ b µ a b 33
b µ b µ a b µ a b µ b µ b{ µ }{ a b µ } { a b µ } The ddle er s zero because b{ µ }{ a b µ } b{ a b µ }{ µ } b{ a b µ } { µ } 0 Alhough he above looks ugly, he pon s ha { µ } 0 because he expeced value of condonal on s µ So far he resul s a b µ b { µ } { a b µ } Only he second er on he rgh nvolves a Wha value of a wll nze? { a b µ } a a b µ b µ a a b µ b µ Takng he paral dervave wh respec o a of he rgh-hand sde and seng equal o zero, and replacng µ by he populaon ean µ, yelds Subsung hs expresson n for a : a bµ a b µ b { µ } b { µ µ } The frs er followng b s PV / for he Bühlann odel or PV / for he Bühlann-Sraub odel The followng relaonshps show hs for he Bühlann odel: { µ } { µ } / The er followng b s he VHM, so for he Bühlann odel follows ha 34
a b µ b PV / b VHM Mnzng he expeced value on he lef s equvalen o nzng he rgh-hand sde The dervave of he rgh-hand sde wh respec o b s whch yelds b PV / b VHM 0, VHM b PV VHM Ths can be rewren as PV b, K PV K VHM VHM If he expressons calculaed above for he wo consans are subsued no he esaor a b, hen a b µ K K Ths s Bühlann s for For he Bühlann-Sraub odel, he s replaced by he oal exposure 35
Appendx B: Proof of Uncondonal Toal ance Forula Toal ance Forula: Le W represen ; hen he forula leads edaely o W W W { } { } Slarly, { } yelds { } oe ha n equaons and here s a coon er on he rgh-hand sdes, hough of oppose sgns, { } Addng and ogeher yelds { } The expecaons on he rgh-hand sde can be rewren as { } { } provng ha, 36
Appendx C: onparaerc saors for he xpeced Value of he Process ance and he ance of he Hypohecal Means n he Bühlann-Sraub Model are Unbased Assupons: ach of ndependen rsks has an assocaed rsk paraeer For he h rsk here are observaon perods and he rando varable represens he observaon for rsk n e perod 3 The nuber of exposures for he h rsk n e perod s and he su for perods s The average of he over observaon perods s defned o be The ean for all rsks s wh 4 The expeced value of s consan hrough e for a gven rsk wh rsk paraeer : µ for o 5 The varance of for a gven rsk wh rsk paraeer s nversely proporonal o he aoun of exposure: xpeced Value of he Process ance σ / for o The expeced value of he esaor for he xpeced Value of he Process ance s PV The nneros su n he nueraor can be rewren as See Secon 37
38 µ µ µ µ µ µ The ddle and las ers can be cobned usng and, so µ µ The condonal expecaon of he frs er on he rgh s σ σ µ Because and µ, follows ha µ σ σ σ Cobnng he resuls gves σ σ σ Subsung hs back no he equaon a he begnnng of he secon: PV σ
39 PV σ PV Ths las equaon shows ha he esaor for he PV s unbased ance of he Hypohecal Means The expeced value of he esaor for he ance of he Hypohecal Means s PV VHM / The suaon n he nueraor on he rgh-hand sde can be rewren as µ µ µ µ µ µ The ddle and las ers can be cobned usng and, so µ µ Takng he expecaon of boh sdes yelds µ µ The frs varance on he rgh can be expanded as
40 Because µ, he frs er s he ance of he Hypohecal Means, µ Wrng ou he varance nsde he second er yelds σ σ The resul s σ µ VHM PV / Ths resul s also useful n calculang as follows: PV VHM PV VHM Pung he peces ogeher yelds PV VHM PV VHM Splfyng and cobnng ers gves PV VHM Pung hs resul no he equaon for he expeced value of he esaor for he ance of he Hypohecal Means yelds PV PV VHM VHM / The frs secon of hs appendx showed PV PV, leavng VHM VHM
FCS Only a few seleced references are lsed here The suden new o credbly heory should fnd o 4 parcularly helpful, 3 and 4 nclude exensve references Bühlann, Hans, xperence ang and Credbly, ASTI Bullen, Vol 4, o 3, 967, pp 99-07 A classc paper ha nroduced he Bühlann odel Herzog, Thoas, Inroducon o Credbly Theory, ACT Publcaons A well organzed survey of credbly heory 3 Klugan, Suar A, Harry H Panjer and Gordon Wllo, Loss Models: Fro Daa o Decsons, John Wley & Sons Provdes a ore rgorous reaen of credbly heory and ncludes os he aeral covered n hs sudy noe 4 Mahler, Howard C and Curs Gary Dean, Credbly, Foundaons of Casualy Acuaral Scence, Fourh don, Casualy Acuaral Socey, 00, Chaper 8 An nroducon o credbly heory wh any exaples and probles 5 Mahler, Howard C, An Acuaral oe on Credbly Paraeers, Proceedngs of he Casualy Acuaral Socey, Vol LIII, 986, pp-6 xplans ha reasonable, as opposed o precse, credbly forula paraeers are generally suffcen 6 Boor, Joseph A, The Copleen of Credbly, Proceedngs of he Casualy Acuaral Socey, Vol LIII, 996, pp -40 Dscusses crera o consder when choosng a copleen of credbly 7 Mahler, Howard C, An xaple of Credbly and Shfng sk Paraeers, Proceedngs of he Casualy Acuaral Socey, Vol LVII, 990, pp 5-308 Ths readng and he nex wo consder suaons where a rsk s characerscs shf hrough e 8 Mahler, Howard C, A Markov Chan Model of Shfng sk Paraeers, Proceedngs of he Casualy Acuaral Socey, Vol LIV, 997, pp 58-659 9 Mahler, Howard C, Credbly wh Shfng sk Paraeers, sk Heerogeney, and Paraeer Uncerany, Proceedngs of he Casualy Acuaral Socey, Vol LV, 998, pp 455-653 As s le ndcaes, hs paper exends credbly heory o several suaons where spler credbly odels are no approprae 4
xercses Credbly Models ando varables and Y are ndependen and boh have he sae ean value: Y µ The varances of and Y are, respecvely, σ and Y σy Defne he rando varable Z o be a lnear cobnaon of and Y wh Z w w Y Gven ha Z µ, prove ha Z wll be nzed f / σ / σ weghs are seleced such ha w and w Y ; / σ / σ Y / σ / σy ha s, choose weghs nversely proporonal o he varances of each rando varable Two urns conan a large nuber of balls wh each ball arked wh one nuber fro he se {0,,4} The proporon of each ype of ball n each urn s dsplayed n he able below: Urn uber on Ball denoed by _0 _4_ A 60% 30% 0% B 0% 30% 60% An urn s randoly seleced and hen a ball s drawn a rando fro he urn The nuber on he ball s represened by he rando varable a Calculae he hypohecal eans or condonal eans A and B b Calculae he varance of he hypohecal eans: c Calculae he process varances or condonal varances A and B d Calculae he expeced value of he process varance: e Calculae he oal varance or uncondonal varance and show ha equals he su of he quanes calculaed n b and d Bühlann Model Many exercses are ncluded n 4, he Mahler and Dean Credbly chaper of Foundaons of Casualy Acuaral Scence 3 You are gven: Two rsks have he followng severy dsrbuons: 4
Aoun of Cla Probably of Cla Aoun for sk Probably of Cla Aoun for sk 50 05 07,500 03 0 60,000 0 0 sk s wce as lkely o be observed as sk A cla of 50 s observed Deerne he Bühlann credbly esae of he second cla aoun fro he sae rsk Course 4 Fall 003 #3 Bühlann-Sraub Model Use he followng nforaon for exercses 4-6 Two urns conan a large nuber of balls wh each ball arked wh one nuber fro he se {0,,4} Balls are drawn fro he urns wh replaceen The proporon of each ype of ball n each urn s dsplayed n he able below: Urn uber on Ball _0 _4_ A 60% 30% 0% B 0% 30% 60% 4 Suppose ha urn A s seleced and n balls are drawn fro he urn a Wha s he expeced average value of he n balls? b Wha s he varance of he average value of he balls? 5 An urn s seleced a rando, wo balls are drawn fro he urn and he average value of he wo balls s recorded a Wha s he expeced value of he process varance PV of? b Wha s he varance of he hypohecal eans VHM of? 6 An urn s seleced a rando Durng he frs round, wo balls were drawn and he average value of he wo balls was 0 Durng he second round, four balls were drawn fro he sae urn as n he frs round and he average of he four balls was 5 Anoher ball wll be drawn fro he sae seleced urn Usng he Bühlann-Sraub credbly odel, wha s he esaed value of he ball? 43
7 Bogus Adversng, Inc has a sall flee of copany cars ha vares n sze fro year o year All of he copany drvers receve he sae ranng and s assued ha each car n he flee has he sae expeced annual accden frequency ha reans consan hrough e Ths expeced accden frequency per car s unknown bu has a unfor dsrbuon on he nerval 0, The nuber of clas for a car s Posson dsrbued Durng he las hree years Bogus had he followng clas experence: Year Cars n Flee Toal uber of Clas Y 4 Y 5 Y 0 If Bogus expecs o have hree cars n he flee nex year, use Bühlann-Sraub credbly o esae he oal nuber of clas nex year 8 You are gven four classes of nsureds, each of who ay have zero or one cla, wh he followng probables: Class uber of Clas _0 I 09 0 II 08 0 III 05 05 IV 0 09 A class s seleced a rando wh probably ¼, and four nsureds are seleced a rando fro he class The oal nuber of clas s wo If fve nsureds are seleced a rando fro he sae class, esae he oal nuber of clas usng Bühlann-Sraub credbly Course 4 Fall 00 #3 9 You are gven he followng nforaon on large busness polcyholders: Losses for each eployee of a gven polcyholder are ndependen and have a coon ean and varance The overall average loss per eployee for all polcyholders s 0 The varance of he hypohecal eans s 40 v The expeced value of he process varance s 8,000 v The followng experence s observed for a randoly seleced polcyholder: 44
Year Average Loss per uber of ployee ployees 5 800 0 600 3 5 400 Deerne he Bühlann-Sraub credbly preu per eployee for hs polcyholder Course 4 Fall 00 #6 0 For each of he n ndependen rando varables,,, n he followng are rue: µ and σ / The weghed ean s defned o be w n wh n w a Prove ha µ b Prove ha s nzed by choosng weghs w / where n You are gven: The nuber of clas ncurred n a onh by any nsured has a Posson dsrbuon wh ean λ The cla frequences of dfferen nsureds are ndependen The pror dsrbuon s gaa wh probably densy funcon: v 6 00λ 00λ e f λ 0λ Monh uber of Insureds uber of Clas 00 6 50 8 3 00 4 300? Deerne he Bühlann-Sraub credbly esae of he nuber of clas n Monh 4 Course 4 Fall 003 #7 onparaerc saon Two vehcles were seleced a rando fro a populaon and he followng cla couns were observed: 45
Vehcle uber of Clas durng Year Year Year Year 3 Year 4 0 0 3 3 You are neresed n he annual clas frequency of each vehcle Use eprcal Bayesan esaon procedures o do he followng: a sae he expeced value of he process varance P V for he nuber of clas n one year b sae he varance of he hypohecal eans V H M c Calculae he credbly weghed esae of he annual clas frequency for each vehcle 3 Two edu-szed nsurance polces produced he followng losses over a hree-year perod: Insured Annual Losses Year Year Year 3 5 4 3 5 6 7 You are ryng o esae he expeced annual losses for each nsured Assung ha he oal exposures for each polcy are equal and rean consan hrough e, use eprcal Bayesan esaon procedures o do he followng: a sae he expeced value of he process varance P V for one year of losses b sae he varance of he hypohecal eans VH M c Calculae he credbly weghed esae of he annual losses for each nsured 4 An nsurer has daa on losses for four polcyholders for 7 years The loss fro he h polcyholder for year j s j You are gven ha 4 7 j j 3360 4 330 Usng nonparaerc eprcal Bayes esaon, calculae he Bühlann credbly facor for an ndvdual polcyholder Course 4 Sprng 000 #5 and Fall 00 # 46
5 YZ Insurance Copany offers a janoral servces polcy ha s raed on a per eployee bass The wo nsureds shown n he able below were randoly seleced fro YZ s polcyholder daabase Over a four-year perod he followng was observed: Year Insured _Y Y Y Y3_ A uber of Clas 3 3 o of ployees B uber of Clas 0 o of ployees 4 4 4 sae he expeced annual cla frequency per eployee for each nsured usng he eprcal Bayes Bühlann-Sraub esaon odel 6 You are gven he followng nforaon on owng losses for wo classes of nsureds - aduls and youhs: xposures Year Adul Youh Toal Y,000 450,450 Y,000 50,50 Y,000 75,75 Y3,000 5,5 Toal 5,000,000 6,000 Pure Preu Year Adul Youh Toal Y 0 5 755 Y 5 4400 Y 6 5 7340 Y3 4 3667 Toal 3 0 467 You are also gven ha he esaed varance of he hypohecal eans s 75 a Deerne he nonparaerc eprcal Bayes credbly pure preu for he youh class, usng µ as he copleen of credbly b Deerne he nonparaerc eprcal Bayes credbly pure preu for he youh class, usng he ehod ha preserves oal losses Course 4 Fall 000 #7 7 You are gven he followng experence for wo nsured groups: 47