Conditional Probability and the Collective Risk Model

Transcription

1 Codoal Probably ad he Collecve Rs odel Legh J. Hallwell, FCAS, AAA Absrac. Oe of he mos powerful ad profoud ools of casualy acuaral scece s he collecve rs model S K. I s wdely used by casualy acuares, especally by hose he feld of resurace. early oe hudred pages of oe sadard exboo (Klugma, [998, Chaper ) hardly suffce o survey he geuy wh whch acuares ad scholars have aalyzed. uch of her aalyss proceeds from he applcao of codoal probably o he so-called dvdual rs model S K. Ths paper peeraes deeper o boh codoal probably ad he collecve rs model, dervg ew sghs o hgher momes ad her geerag fucos. Parcular aeo s devoed o he fourh mome of he collecve rs model, for whch o formula seems prevously o have bee publshed. A appedx exeds codoal probably o a ovel echque of loss developme. Keywords: codoal probably, momes, cumulas, collecve rs model.. ITRODUCTIO Ths paper apples codoal probably o he momes of he collecve rs model. I he ex seco we wll se forh defos of codoal momes ad co-momes, ad hrd seco wll derve formulas whch ucodoal momes are expressed erms of codoal oes. ex, he fourh seco, afer explag why momes hgher ha he hrd are o addve, we wll roduce a addvy-resorg adusme ow as a cumula. I he ffh seco we wll apply codoal cumula formulas o he collecve rs model o seze he prze of a maageable formula for s fourh cumula. Fally, he sxh seco we wll expla he cumula geerag fuco, ad show s usefuless relag cumulas o momes ad dervg cumulas of he collecve rs model. Casualy Acuaral Socey -Forum, Sprg 0

2 Codoal Probably ad he Collecve Rs odel. DFIITIOS Le be a radom varable wh fe mea [ [ he h mome of as [ ( ). For posve eger, defe. Here we have defed wha probably heory calls he ceral momes of. Of course, he frs ceral mome s zero. The secod mome s he varace, ad he hrd s he sewess. We shall call he fourh mome he uross. ow le deoe a eve whch codos he probably dsrbuo of. We ca he spea of he codoal h mome [ [ ( ), where [. Furhermore, as a mulvarae exeso, we ca defe he h co-mome as [ ( ) C,K,. Sce he order of he co-mome s he umber of s argumes, s superfluous o subscrp he defo as C. The frs co-mome s zero; he secod s he covarace. We shall call he hrd ad he fourh co-momes he co-sewess ad he co-uross. A co-mome of radom varables s zero, f ay of hem s cosa, sce ha case oe of he facors he Π operaor wll always be zero. The same radom varable may appear he argume ls more ha oce; as a specal The reader should o cofuse [ for he h mome wh (), he mome geerag fuco of. Some defe uross as he fourh cumula, ( ) [ [ ( ), also ow as excess uross because he uross of he ormal dsrbuo s hree mes he square of s varace. Somemes (e.g., Day [99, ) sewess ad uross are defed as wha we would call coeffces of sewess ad uross,.e., he momes or cumulas srpped of dmeso by dvdg hem by he hrd ad fourh powers of he sadard devao. I geeral, ( [ ) [( ) [ [. Codog a oe level of expecao should by defaul cascade o he ex or esed level, ad so o. The edecy o dsregard hs equaly may dcae a defec he acceped oao. I helps (a leas helps hs auhor) o regard ucodoal expecao as codoal upo he uversal eve V: [ [ V. Oe mus be wary of such msaes as equag C [,, ad C [, co-mome wh a secod., whch cofuses a hrd Casualy Acuaral Socey -Forum, Sprg 0

3 Codoal Probably ad he Collecve Rs odel case, C,, ( ) ( ) mes 678 K [ [. The codoal co- mome s C [, ( [ ),K.. UCODITIOAL OTS I TRS OF CODITIOAL Our purpose here s o derve formulas ha express ucodoal momes erms of momes ad co-momes codoal upo. Ths begs wh he ey sequece: [ ( ) [ ( ) [ [ [ [ {( ) ( )} ex we expad hs accordg o he bomal heorem: [ [ [ {( ) ( )} ( ) ( ) [ [( ) ( ) [ [( ) ( ) [ [ ( ) The fourh le follows from he hrd because ( ) behaves as a cosa wh he esed expecao, ad so ca be ae ousde. Of hese erms, he ( ) h s zero, sce ( )[ [ 0 o-vashg erms, amely:. Hece, hs bomal form, he h mome has Casualy Acuaral Socey -Forum, Sprg 0

4 Codoal Probably ad he Collecve Rs odel Casualy Acuaral Socey -Forum, Sprg 0 [ [ ( ) [ [ [ [ ( ) [ ( ) [ 0 However, a hs po we have o expressed he ucodoal mome erms of codoal momes ad co-momes; we mus express he expecao wh he Σ operaor as a co-mome. Leg [ [ ζ, ad rememberg ha frs ceral momes are zero, we derve: [ [ [ [ ( ) [ ( ) [ [ [ [ { }( ) [ ( ) [ [ [ [ { }( ) [ ( ) [ ( ) [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [.,,,,,, mes mes C C K 678 K ζ ζ ζ ζ ζ Hece, for, o express he h ucodoal mome he desred codoal form requres ( ) 0, max o-vashg erms. The frs Σ operaor does o come

5 Codoal Probably ad he Collecve Rs odel o play ul, ad he secod ul. A leas he complexy does o crease afer he fourh mome. The secod, hrd, ad fourh momes follow readly from he geeral formula: [ [ [ [ [ [ [ [ [ [ cf. Klugma [998,9 Sew [ [ [ [ C [ [, [ [ [ [ Sew[ Cov[ [, [ Sew[ [ Kur [ [ [ [ C [ [, [ 6C [ [, [, [ 6 [ [ [ [ [ [ [ Kur[ Cov[ Sew[, [ 6Cosew[ [, [, [ 6 [ [ [ [ Kur[ [ Casualy Acuaral Socey -Forum, Sprg 0 5

6 Codoal Probably ad he Collecve Rs odel Casualy Acuaral Socey -Forum, Sprg 0 6. OTS VRSUS CUULATS Tha he codoal expresso of he h mome requres ( ) 0, max erms dcaes a bed he road bewee ad. I s hardly cocdeal ha momes beyod he hrd are o addve. If ad are depede radom varables wh meas ad ν, he h mome of her sum s: [ ( ) ( ) { } [ ( ) ( ) { } [ ( ) ( ) [ ( ) [ ( ) [ [ [ [ [ [ [ [ [ [ [ ν ν ν ν The Σ operaor dsrups he addvy whe s rage s o-empy,.e., whe, or. So, wh depedece, he frs hree momes are addve; he fourh mome he erm [ [ [ [ [ [ 6 6 dsrups he addvy, a erm aalogous o [ [ [ [ 6 he uross formula. everheless, adusmes o momes hgher ha he hrd ca obvae he dsrupo ad resore addvy. Such adused momes are ow as cumulas. The fourh-order adusme s:

7 Codoal Probably ad he Collecve Rs odel [ [ [ [ 6 [ [ [ [ [ 6 [ [ ( [ [ ) [ [ [ [.. Thus, he fourh cumula, [ [ [ Kur [ [ κ s addve. 5 Sce he collecve rs model volves sums of depede radom varables, he fourh cumula, whch we shall call he excess uross, wll prove more useful ha he fourh mome. Is codoal expresso s: skur [ Kur[ [ [ Kur[ Cov[ Sew[, [ 6Cosew [ [, [, [ 6 Kur 6Cosew ( ) [ [ [ [ Kur[ [ [ [ [ [ [ Kur[ Cov[ Sew[, [ 6Cosew [ [, [, [ [ [ [ [ [ [ [ Kur[ [ [ Cov Sew[, [ [ [ [, [, [ Kur[ [ [ [ [ [ [ [ [ skur[ Cov[ Sew[, [ 6Cosew [ [, [, [ skur[ [ [ [ 5 The formulas for hgher-order cumulas become creasgly more complcaed. The reader ca verfy he addvy of he ex wo cumulas accordg o he defos (cf. Seco 6): 0 Sew κ κ 5 6 [ [ [ [ 5 [ [ 5 [ κ [ 0Sew[ 5 [. 6 Casualy Acuaral Socey -Forum, Sprg 0 7

8 Codoal Probably ad he Collecve Rs odel 5. OTS OF TH COLLCTIV RISK ODL The collecve rs model, whch casualy acuares mus sudy for her examaos, s soc--rade, especally he feld of resurace. I cosders aggregae loss S as he sum of a radom umber of depede, decally dsrbued clams: S K. Here we wll apply our codoal formulas 6 o derve he frs four cumulas of S erms of hose of ad. Because he are depede ad decally dsrbued, as well as due o he addvy of cumulas, [ S [, [ S [ Sew [ S Sew[, ad [ S skur[ [, Sew [, ad [, skur. Because [, skur are cosas, we may remove hem from momes codoal upo, beg careful o rase hem o he power of he codoal momes. The frs cumula, he mea, s rval: [ S [ [ S [ For he secod, he varace: [ S [ [ S [ [ S [ [ [ [ [ [ [ [. [ [ [. very acuary a some me leared hs formula; o may remas famlar. However, he hrd mome s o suded, ad hece, o commoly ow: Sew S [ [ Sew[ S Cov[ [ S, [ S Sew[ [ S [ Sew[ Cov[ [, [ Sew[ [ [ Sew[ Cov[, [ [ Sew[ [ [ Sew[ [ [ [ Sew[ [ 6 We wll chage he omeclaure of hese formulas so as o agree wh ha of he collecve rs model,.e., S wll appear sead of, ad sead of. I hs seco wll represe he severy of a clam. Casualy Acuaral Socey -Forum, Sprg 0 8

9 Codoal Probably ad he Collecve Rs odel oeheless, hs formula appears Par [996, 77 ad Klugma [998, 98. Las, we derve he excess uross, whose formula we have o see pr before: skur [ S [ skur[ S Cov[ Sew[ S, [ S 6Cosew [ [ S, [ S, [ S skur[ [ S [ [ S [ skur[ Cov[ Sew[, [ 6Cosew [ [, [, [ skur[ [ [ [ [ skur[ Cov[, Sew[ [ 6Cosew [,, [ [ skur[ [ [ [ [ skur[ [ Sew[ [ 6Sew [ [ [ skur[ [ [ [. 6. TH CUULAT GRATIG FUCTIO The mome geerag fuco of a sum of depede radom varables equals he produc of her mome geerag fucos. Sce logarhms cover mulplcao o addo, s aural o cosder he logarhm of he mome geerag fuco, whch has come o be ow as he cumula geerag fuco ψ (c.g.f.),.e., ( ) l [ e Is dervaves a zero are called cumulas: 7 [ κ [ ( 0) oe aoher: ψ ψ ψ.. If he are depede of ( ) l e l e l [ e l [ e ψ ( ). 7 I Seco we roduced cumulas as momes adused o resore addvy. Ths hardly suffces for a defo, ad we have o prove he exsece ad he uqueess of he adusme. The dervaves of he c.g.f. a zero cosue a proper defo of cumula, ad he Taylor-seres argume of hs seco ca be made o a rgorous proof of he uqueess of he adusme. Casualy Acuaral Socey -Forum, Sprg 0 9

10 Codoal Probably ad he Collecve Rs odel Sce dffereao s a lear operaor, he cumula of a sum of depede radom varables equals he sum of he cumulas of he radom varables. The frs hree cumulas equal he mea, he varace, ad he sewess. Bu equaly ceases wh he fourh cumula: [ [ [ ( ) [ [ κ (Day [99, ad Hallwell [00, 65). Here we wll show he relevace of he c.g.f., o () he expresso of cumulas erms of momes ad () he momes of he collecve rs model. Frs, he Taylor-seres expaso of he c.g.f. embeds he cumulas: ψ [ ( ) ( ) ψ 0 ψ ( 0)! 0 κ [!. [ The ceral momes of, [ ( ), are smlar coeffces he Taylorseres expaso of he mome geerag fuco of : ( ) [ [! 0 [!. e We ca combe hese wo equaos o relae he cumulas ad he momes: κ ( ) ( ) [! ψ ( ) l [ l [ l [ e e e l [!. Bu he logarhm has s ow Taylor-seres expaso for < x, vz.: l ( x) x x x x K ( ) x. So he relaoshp ca be expressed as wo polyomals : Casualy Acuaral Socey -Forum, Sprg 0 0

11 Codoal Probably ad he Collecve Rs odel Casualy Acuaral Socey -Forum, Sprg 0 [ [ ( ) [ [ ( ) [.!!!!! l! κ achg coeffces of decal polyomals mus be equal. I s he las expresso o he rgh sde of he fal equao ha complcaes he machg; however, s quarc ad hgher. Hece, he frs hree cumulas mus be he mea, he varace, ad he sewess. Ad he formula for hgher cumulas begs as [ [ K κ. As a example of hgher-order machg, we wll derve he uross formula. A fourh power of arses he las expresso oly from powers of wo, or as : [ [ ( ) [ ( ) [ ( ) { } [ [ [ [ [. 8!!!!! κ κ The ffh cumula s a lle more complcaed, sll volvg, bu obaed wce as ad. The sxh cumula volves as,, ad, as well as as. Ths c.g.f. echque s arguably he eases way o derve he formulas of foooe 5. Secod, we wll derve he c.g.f. of he collecve rs model S K, beg mdful of he chage omeclaure (cf. foooe 6): ( ) ( ) ( ) [ ( ) [ ( ) ( ) ( )( ). l l l e e e S S S ψ ψ ψ ψ ψ ψ ψ ψ o So he c.g.f. of he aggregae loss s he composo of he cumula geerag fucos of frequecy ad severy (Day [99, 59). Ths s he mos elega way o derve he

12 Codoal Probably ad he Collecve Rs odel aggregae cumulas, ad s more effce ha he codoal-mome echque of Seco 5. To show hs, we wll derve he frs wo momes. ψ S ( ) ψ ( ψ ( ) ) ψ ( ) [ S ψ ( 0) ψ ( ψ ( 0) ) ψ ( 0) ψ ( 0) ψ ( 0) [ [ ψ S ( ) ψ ( ψ ( ) ) ψ ( ) ψ ( ψ ( ) ) ψ ( ) [ S ψ ( 0) ψ ( 0) ψ ( 0) ψ ( 0) ψ ( 0) [ [ [ [. S S Curous ad ambous readers, performg he hrd ad fourh dervaves, ca verfy he formulas Seco 5 for he aggregae sewess ad excess uross. 7. COCLUSIO We have show how ucodoal momes ca be expressed erms of codoal momes ad co-momes. Adusg momes o cumulas allowed us o form farly smple formulas for he sewess ad he excess uross of he collecve rs model. These formulas ca also be derved drecly from he cumula-geerag fuco. Acuares who have bee reluca o apply he mehod of momes o us he frs wo momes of he collecve rs model ca ow wh hese formulas f more versale dsrbuos o more ha wo momes. Oe ough o be more comforable wh exrapolaos o he rgh al of a aggregae loss dsrbuo afer havg cosdered s sewess ad uross. Asde from he collecve rs model, a codog paro ca chage he momes of a sum of depede radom varables whou chagg her ucodoal momes. The appedx shows how hs ca be doe loss reservg. oreover, f some amou of capal or rs marg were allocaed o a mome, codog would allow a sub-allocao o he paros. Casualy Acuaral Socey -Forum, Sprg 0

13 Codoal Probably ad he Collecve Rs odel Acowledgme The auhor graefully acowledges he help of revewers Jaso Coo, FCAS, ad va Glsso, FCAS, from whose commes hs paper has much mproved. RFRCS [. Day, C. D., T. Peäe, ad. Pesoe, Praccal Rs Theory for Acuares, Lodo: Chapma & Hall, 99. [. Hallwell, Legh J., The Valuao of Sochasc Cash Flows, CAS Forum (Resurace Call Papers, Sprg 00), pp. -68, [. Klugma, Suar A., H. Paer, ad G. Wllmo, Loss odels: From Daa o Decsos, ew or: Joh Wley & Sos, 998. [. Par, Gary S., Resurace, Foudaos of Casualy Acuaral Scece, Thrd do, Casualy Acuaral Socey, 996, pp Casualy Acuaral Socey -Forum, Sprg 0

14 Codoal Probably ad he Collecve Rs odel APPDI A Codoal Probably ad Clam Developme A rece assgme spurred our eres he subec of hs paper. We had a ls of he case reserves of abou 00 clams, ad were sasfed ha he oal IBR 8 for hem was zero,.e., [ IBR 0. Bu addo, we waed some measure of he varace. The clams semmed from a uusual exposure, ad we deemed o oher daa sources approprae. Sce we assumed he average developme o be zero, he clam ls self could serve as a emprcal dsrbuo f wh momes ± σ. Regardg he clams as depede, we mgh decde he momes of he oal upad loss (.e., case plus IBR) o be ± σ, or oal IBR o be 0 ± σ. Bu hs gores he lelhood of ra correlao,.e., ha afer developme large clams ed o say large, ad small clams ed o say small. Hece, σ s a maxmal value. Therefore, we decded o order he clams by her case reserves ad o srafy hem o 0 groups of approxmaely 0 clams. Belogg o a sraum s he eve ha codos a clam s probably desy as f. Sce srafcao provdes o ew [ formao, f ( x) f ( x). The we assumed ha each clam would develop as follows: wh probably p s dsrbuo would rema ha of s sraum ad wh probably q p would mgrae radomly. Cosequely, he dsrbuo of a developed clam s a mxure of dsrbuos; wh probably p he developed clam s dsrbued as f ad wh probably q as f. Le be he developed amou of clam. xg s easy wh mome geerag fucos. 8 IBR meas Icurred Bu o ough Repored (or Reserved). Cf. Par [996, 50. Casualy Acuaral Socey -Forum, Sprg 0

15 Codoal Probably ad he Collecve Rs odel The mome geerag fuco of codoal upo he sraum of s ( ) p ( ) q ( ). The overall, or ucodoal, mome geerag fuco of a developed clam s: [ [ p ( ) q ( ) [ ( ) q ( ) ( ) ( ) p p ( ) q ( ) ( ). Sce equaly of mome geerag fucos mples decal dsrbuos, s dsrbued as. Sce we have provded o ew formao, hs coservao of dsrbuo s fg. Bu he reader may ow be woderg how he varace of oal IBR ca chage despe he coservao of he overall dsrbuo. The paradox s resolved wh a dsco: varace peras o he sum of clams, whereas coservao peras o her mxure, more accuraely, o he mxure of her dsrbuos. The overall or ucodoal varace [ [ [ s coserved, bu s apporome bewee [ [ ad depeds o. A he oe exreme, a blu or o-dscrmag srafcao ells ohg abou : [ [. I hs case: [ [ [ [ [ 0. [ [ [ [ [ Casualy Acuaral Socey -Forum, Sprg 0 5

16 Codoal Probably ad he Collecve Rs odel Coversely, f he varace of he codoal mea s zero, [ [ [ mus be cosa, or. So he codoal dsrbuos of a blu srafcao ed o be dsgushable as o her frs wo momes. I hs case he varace of he sum eds oward he maxmal σ. A he oppose exreme, s so fe or dscrmag ha [ 0. The: [ [ [ [ [ [ [ 0 [. Ths meas ha he all he varace s bewee he sraa, o varace s wh a sraum. I hs case he varace of he sum eds oward he mmal value of zero. To borrow ad mx oos from opcs ad credbly, he blu srafcao passes he whe lgh of zero credbly; he fe srafcao le a prsm refracs lgh o he specrum of full credbly. Sce we wll be codog o mgrao Μ, we wll drop ad spea of he h sraum. Le here be s sraa, ad le π > 0 be he probably for a clam o be he h sraum, as deermed by he acual poro of clams ha sraum. Though he sraa eed o o be balaced, or of equal populao, π. We may model developed clam of he h sraum as follows. Radomly draw oe udeveloped clam from each sraum s dsrbuo; hese,, s are depede. The form a usrafed or average clam as he choce of wh probably π. Fally, flp a Beroull co wh probably p of heads. If he co lads heads, le equal ; oherwse, le equal. s Casualy Acuaral Socey -Forum, Sprg 0 6

17 Codoal Probably ad he Collecve Rs odel The mea of he developed clam s [ [ [ Μ p[ q[ o he formula of Seco, he varace s: [ [ [ Μ [ [ Μ Μ p p p p p [ q[ Μ ( [ [ ) q( [ [ ) [ q[ ( q[ q[ ) q( p[ p[ ) [ q[ pq( [ [ ). Μ. Accordg Sce cosderaos of ra correlao drew us o hs model, we should also deerme he covarace of wh. Tag he ex formula whou proof, 9 we have: Cov [, [ Cov[, Μ Cov [ Μ, [ Μ Μ Μ [. The secod erm o he rgh sde of he equao s zero. For he mgrao Μ does o affec he expecao of, ad he covarace of somehg wh a cosa s zero. Hece, [ [ Cov[ Μ Cov,,. I he followg reduco, we mus cosder ha he Μ radom mgrao ca reur (Ρ) wh probably π o he h sraum. Aga, a covarace erm becomes zero due o he mmuy of o Ρ: 9 The proof hges o [ [ [ {( ) ( )}{ ( ν ) ( ν ν) } Cov,. Casualy Acuaral Socey -Forum, Sprg 0 7

18 Codoal Probably ad he Collecve Rs odel Cov [, [ Cov[, Μ Μ pcov pcov pcov pcov pcov [, qcov[, { } [, q [ Cov[, Ρ Cov[ [ Ρ, [ Ρ [, q [ Cov[, Ρ s [, qπ Cov[, [, qπ Cov[, ( p qπ ) [. Ρ Ρ The covarace of he developed clam amou wh he udeveloped s posve, bu he correlao coeffce s more formave: Ρ Corr [, [, [ [ Cov ( p qπ ) [ [ [ ( p qπ ) [ [. The correlao creases wh respec o p, he probably ha he dsrbuo of a developed clam remas ha of s sraum, from a mmum of [ [ π for p 0 o a maxmum of for p. I seems ha Corr[ p,. However, hs s Pearso correlao, whereas we are cocered wh ra, or Spearma, correlao. Because a aalyc aswer eluded us, we resored o smulao. Keepg wh he assgme ha spurred our eres, we smulaed,000 eraos of he developme of he egers from o 00 s 0 groups of 0 cosecuve egers over a rage of omgrao probables p from 0% o 00% seps of 5%. We radomly permued he egers wh each group hs aloe would suffce f p ad er-group mgraos were mpossble. Bu he we flpped he Beroull co for each eger, mared whch places were he mgrag als, ad radomly permued amog hose places her egers. Casualy Acuaral Socey -Forum, Sprg 0 8

19 Codoal Probably ad he Collecve Rs odel The we calculaed he ra correlao for ha erao, ad averaged over all he eraos. The able below coas he resul: #Ier #Goups #IGrp p RaCorr, % 0.000, % 0.06, % 0.00, % 0.6, % 0.0, % 0.50, % 0.0, % 0.9, % 0.96, % 0.6, % 0.97, % 0.550, % 0.60, % 0.68, % 0.696, % 0.77, % 0.797, % 0.8, % 0.895, % 0.9, % Ideed, seems ha he ra correlao approxmaes p. oeheless, cao exacly equal p. For a a ear 00% probably of o mgrag, permuao wh each group sll dsrups a perfec correlao. Therefore, we suspec RaCorr(p) o sar ou a zero wh a slope of uy, bu o be slghly cocave (.e., o have a egave secod dervave) so ha loses groud o p as p creases o oe. I sum, as p, he probably of o mgrag (.e., he probably for he dsrbuo of a clam o rema ha of s sraum) approaches zero, srafcao becomes rreleva. Regardless of how he clams are srafed, hey wll all develop accordg o he overall dsrbuo. Ths wll produce a aggregae sadard devao approachg he maxmal σ. Ad f here were oly oe sraum, mgrao would be from overall o overall, ad Casualy Acuaral Socey -Forum, Sprg 0 9

20 Codoal Probably ad he Collecve Rs odel he aggregae sadard devao aga would be σ. Bu as p creases, ad as he sraa become arrower, he aggregae sadard devao decreases. I he exreme, wh oe clam per sraum (beer, wh zero varace wh each sraum) ad p, he aggregae sadard devao s zero. Poderg hese relaos wh wo momes led us o he dea of addg hgher momes o he codoal dsrbuos, ad hece o reag he hgher momes of he collecve rs model. Alhough we do o ed for hs o be a paper o a ew developme mehod, he reader ca see how hs clam-by-clam mehod ca be employed o apporo momes of loss ha mesh wh ay desred aggregae momes, as well as o oba useful suboals, e.g., by accde year. Casualy Acuaral Socey -Forum, Sprg 0 0