March 14, Title: Change of Measures for Frequency and Severity. Farrokh Guiahi, Ph.D., FCAS, ASA

Transcription

1 March 4, 009 Tle: Chage of Measures for Frequecy ad Severy Farroh Guah, Ph.D., FCAS, ASA Assocae Professor Deare of IT/QM Zarb School of Busess Hofsra Uversy Hesead, Y 549 Eal: Farroh.Guah@hofsra.edu Phoe: ASTI 009 Hels, Flad Jue 4, 009

2 Chage of Measures for Frequecy ad Severy. Iroduco. The cooes of reu for a surace roduc are ade u Los Cos-- ure reu-- Exese, ad Rs Load. Casualy ad Proery acuares aly her owledge of raeag o calculae Loss Cos based o he relaosh Loss Cos = Frequecy Severy. I ay saces, oe s eresed o fd ou how he chages Frequecy ad/or Severy affec he chages Loss Cos. I a world whou rs, oe ay be eed o use ulvarable calculus o exla chages based o Taylor s forula LC LC dlc F, S df ds F S where LC LC F, S deoes Loss Cos, ad F ad S rerese Frequecy ad Severy resecvely. I our ucera world, oe cao rely o rules of ordary calculus,.e. equao above o assess he ac of chages o Loss Cos coge uo chages Frequecy ad Severy. Wha s eeded s a secfcao of suable robablsc odels for Frequecy ad Severy. The, a robablsc odel for Loss Cos ca be develoed based o how he sub-odel cooes for Frequecy ad Severy erac ogeher. A robablsc odel for Loss Cos would rovde for a descro of ossble Loss Cos values wh assocaed robables of aag hose values. Ieres ay le a robably dsrbuo for Loss Cos a a secfed e or he behavor of Loss Cos as evolves over e,.e., a sochasc rocess goverg he Loss Los.. Probablsc odels for Frequecy, Severy ad Loss Cos. To creae a odel for Frequecy, we beg wh he oo of a Po Process, PP. A PP s characerzed by a sequece of o-decreasg rado varables {, }. T reses he e T of occurrece of he h eve of eres, e.g., T ay be he accde dae of he h cla o a surer durg a exosure erod. Whe he er-arrval es, T T,, wh T 0 0, are deede ad decally dsrbued rado varables havg a exoeal dsrbuo wh araeer, he he PP s a hoogeeous Posso rocess wh esy. Wh a PP s assocaed a coug rocess {, 0} defed as I T, where I A deoes he dcaor rado varable whch aes value oe f he eve A occurs ad zero oherwse. Thus, cous he uber of es he eve of eres occurs he erval 0, ]. For he Posso rocess we have Pr e!, 0,, 3

3 The Frequecy a a o of e s exressed as a rae based o he forula " exosure" base deeds uo he Le of Busess cosdered. E[ ] where he ex osure ex, we cosder a odel for Severy. The cos of a surace roduc s o solely deede uo owg wheher a cla has occurred. We are also eresed owg he ac of a cla o our boo of busess. Hece, for he h cla occurrg a e T, we are eresed he sze of ha cla deoed by. A Mared Po Process, MPP, s a bvarae sequece of rado varables { T,, } where T s e of he occurrece of he h cla, ad s he corresodg aou of loss for he h cla. I s assued ha ' s are deede ad decally dsrbued rado varables dsrbued as wh a robably easure P. We use he er robably easure ad robably dsrbuo erchageably hs aer. So, he sochasc behavor of he cla sze Severy--s govered by P. I hs aer, we assue has a dscree robably dsrbuo. Fally, we wa o creae a sochasc odel for Loss Cos based o a coug rocess ad a robably easure P for he Severy. The cooud Posso rocess ay be used o rese he behavor of Loss Cos over e. A cooud Posso rocess, Y, s defed as Y 4 I 4 above, s udersood ha Y =0 wheever 0. I s useful o use he coce of a MPP, as defed above, o rovde a alerave reseao for he cooud Posso rocess. To a MPP, we assocae a coug rocess,, A, accordg o, A I T, A 5 I 5 above,, A cous he uber of es he eve of eres occurs erval 0, ] subec o he ' s beg resrced by he eve A. For a secfed ad A, 5 s a rado varable. For a secfed A,, A s a coug sochasc rocess, ad for secfed,, A s robably easure. For a cooud Posso rocess, s cusoary o assue ha he T ad are deede of each oher, ad he coug rocess relaed o {, } s a Posso rocess wh robably dsrbuo 3. We ae a furher assuo ha has a fe robably dsrbuo wh values he se x, x,, x } ad P x,. ow, we gve a alerave forula for Y as { T Y I T { x I x } I T 3

4 x I T x, x A, A 6 where A s he sgleo eve x },, A s defed by 5, ad we defe as, A. { oe ha, A rereses he uber of clas whe he corresodg cla aous are exacly he value x. The followg rears are releva o Y as gve by 6: Rear. The uber of ers he suao 6 s fxed, beg equal o. Rear. Sce he eves A, A,, A are uually exclusve, he he rado varables, A,, A,,, A are deede for ay gve, see Las ad Brad 995. Ths fac s useful whe oe res o deere he dsrbuo of Y. Rear 3. The dsrbuo of, A s a Posso wh esy P A where s he esy of he uresrced Posso rocess, ad P A s he robably of he eve A based o he robably easure P rereseg he dsrbuo of, see Las ad Brad. s he uber of clas whe he corresodg cla aous are x. ay also be vewed as a resrced verso of he coug rocess. Ths o of vew s referred o as hg of a Posso rocess. Rear 4. Y, as gve by 6, s a weghed su of Posso rocesses wh dfferg eses. Afer hs relary dscusso of a cooud Posso rocess, le us cosder he oo of a chage of easure as relaes o Frequecy, Severy ad Loss Cos. I hs aer, here are hree sochasc elees of eres whch we are eres o ow how hey chage subec o chages her uderlyg robably easures. These rado elees are, a Posso rocess relaed o Frequecy; rereseg he Severy; ad fally Y, a cooud Posso rocess, reseg he Loss Cos. 3. Chage of easures for Frequecy, Severy ad Loss Cos. We beg by cosderg he chage of easure for Severy frs. Recall ha he rado varable, rereseg he Severy ay be reseed by alerave robably easures. So, f, ally has a robably easure P, ad laer o a alerave robably easure Q, he we exress sybolcally hs chage of easure as P Q. A chage of easure reles uo a fudaeal heore easure heory ow as Rado- ody Theore. Before, we ca sae he Rado-ody Theore, we eed o defe he coce of he absolue couy of wo easures. We say ha he easure Q s absoluely 4

5 couous wh resec o he easure P, ad wre hs as Q P, f Q A 0 wheever P A 0 for he eve A. Thus, P ad Q agree o eves whch have robably of zero. ow, we sae a verso of Rado-ody Theore suable for robably easures, see Jacod ad Proer 00. Rado-ody Theore: Le P ad Q be wo robably easures defed o he sae easure sace. If Q P, he here exss a oegave rado varable Z such ha for eve A. Moreover, Z s uque a.s. Q A Z dp E { Z I A} A P ad Z s referred o as Rado-ody dervave. P We wre dq Z, dp The followg wo coes are releva wh resec o he Rado-ody Theore. Frs, f we sar wh a robably easure P ad a oegave easurable fuco, Z, he s o hard o show ha Q A as defe by Q A Z dp s a se fuco ad s deed a easure. The sgfcace of Rado-ody heore s ha coverse resul s also vald. Tha s, gve wo easures Q ad P, wh Q P, he here exss a easurable fuco Z such ha Q ad P are relaed accordg o Q A Z dp. The Rado-ody saes he exsece of Z whou rovdg a exlc exresso for Z. Secod, whe P s a dscree robably dsrbuo easure, corresodg o ag values a deuerable se, he s easy o gve a exlc exresso for Z, he Rado- ody dervave. For havg a dscree dsrbuo, we have PrQ x Q x Z,,, 7 Pr x P x The above P Z ' s ay be vewed as lelhood raos. I hs aer, he chage of easure for Severy,, as gve by 7 above ay be saed by q Z,,,, 8 where Pr P x, subec o A A ad q s slarly defed as q PrQ x. ex, we cosder he chage of easure for Frequecy. Sce, Frequecy s aly deered by a Posso rocess, we cosder ex he chage of easure for a Posso rado varable. The Posso rado varable or rocess s characerzed by esy araeer, so we dcae sybolcally a chage of easure for he Posso rocess by. Posso rado varable s a dscree rado varable ad we ca aly forula 7 o derve a chage of easure for a Posso rocess by usg 3 5

6 Pr e Pr! [/ e! = e So he Rado-ody dervave rocess s hs case s ] Z e 9 Fally, we wa so derve a chage of easure for a cooud Posso rocess, Y, by cosderg chage easures for Frequecy ad Severy cobed. We wre sybolcally he chage of easure for Y by, P, Q. A cooud Posso Process s a exale of a sochasc rocess wh us. Udersadg he oo of chage of easure for sochasc rocesses wh us requres he owledge of sochasc calculus ad he alcao of he Grsaov s Theore. The eresed reader ay cosul Shreve 004 for furher forao o hs subec. I hs exoso, we dscuss he oo of a chage of easure for a cooud Posso rocess a foral ad uve fasho. We use he forula 6 for Y. The values of Pr Y y wll be o-zero oly f y for y x x x, ad s of he Gve, Y has a uloal robably dsrbuo accordg o... wh 0, Based o Rear 3, s a Posso rocess wh esy. 0 Usg Rear,,,, are deede rado varables. Alyg 0, ad 9, we have he Rado-ody dervave Z for q Z ex q Furherore, due o deedece resul gve by, we have as 6

7 Pr, Q,,,, Q Pr 3.a Z Pr, 3.b P e q Pr, P q { e } Pr, P,,, 3.c 3.d I 3.a, we used he fac ha ' s are deede based o he ew easure Pr, Q. A chage of easure s used for he Posso varable 3.b. I 3.c, we used he fac ha q o slfy our exresso. Ad, fally 3.d we used he oo of deedece for ' s uder he orgal easure Pr, P q We eed o ae a fal aduse o he roduc er,, aearg rgh-hadsde of 3.d. I would be srucve o llusrae hs o wh referece o a exale. Suose 3 ad 4 wh,, 3. The, aog,, 3, ad 4, we have oe sace of beg x wh robably ; wo saces of beg x wh robably ; ad oe sace beg x3 wh robably 3. oe, we have assged robables accordg o P. A slar assge of robables accordg o he easure Q s also vald. Le us defe P o be whe he realzed value of s x for,,3, 4. Slarly, we defe Q. Aga, we oe ha,,,3, 4 are decally dsrbued uder eher easure P or easure Q. The, for our exale, we ca wre. q q q 3 q Q P 7

8 Wh hs refee, he q q 4 Cobg 3.d wh 4, gves he Rado-ody dervave Z for he cooud Posso rocess as Z e q 5 To suarze, he chage of easure, as gve by Rado-ody dervave s 9 for he Frequecy oly; s 8 for he Severy oly; ad s 5 for Frequecy ad Severy cobed. Refereces Jacod, Jea ad Proer, Phll 003. Probably Esseals. Secod Edo. Srger, ew Yor. Las, G. ad Brad, A Mared Po Processes o he Real Le. The Dyac Aroach. Srger, ew Yor. Shreve, Seve E Sochasc Calculus for Face II Couous-Te Models. Srger, ew Yor. 8