Standard Error of Technical Cost Incorporating Parameter Uncertainty

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1 Sandard rror of echncal Cos Incorporang Parameer Uncerany Chrsopher Moron Insurance Ausrala Group Presened o he Acuares Insue General Insurance Semnar 3 ovember 0 Sydney hs paper has been prepared for Acuares Insue 0 General Insurance Semnar. he Insue Councl wshes o be undersood ha opnons pu forward heren are no necessarly hose of he Insue and he Councl s no responsble for hose opnons Chrsopher Moron he Insue wll ensure ha all reproducons of he paper acknowledge he Auhor/s as he auhor/s and nclude he above copyrgh saemen. Insue of Acuares of Ausrala AB Level 7 4 Marn Place Sydney SW Ausrala (0) f 6 (0) e acuares@acuares.asn.au w

2 Absrac he purpose of he paper s o presen a mahemacal dervaon of he varance of echncal cos ncorporang parameer uncerany arsng from he use of generalzed lnear models o esmae he componens of echncal cos. Praccal uses of he resul wll also be dscussed. Keywords: ance; echncal Cos; Parameer Uncerany; GLM; Posson; Gamma.

3 Sandard rror of echncal Cos Incorporang Parameer Uncerany. Inroducon hs paper wll lan how o calculae he sandard error of echncal Cos made up of clam frequency and clam severy models for dfferen clam ypes. he use of sascal models o esmae clam frequences and severes nroduces an addonal level of uncerany whch should be capured. echncal models for clam frequency and clam severy n General Insurance are ofen esmaed usng Generalsed Lnear Models (GLM). A common specfcaon uses a GLM wh Posson error and log-lnk for frequency and a GLM wh Gamma error and log-lnk for severy. hs parcular specfcaon wll be used as an example hroughou he paper. he eced echncal cos s a key npu no many prcng decsons and performance and monorng repors. he movaon for wrng hs paper s o derve he varance so ha can be used as an addonal npu o derve beer prcng decsons and more effecve monorng and performance repors. he paper s spl no 5 man secons: ance of Collecve Rsk usng condonal argumens wll be shown ha he varance of he collecve rsk are funcons of momens of he underlyng clam processes and he assocaed uncerany n her esmaon; GLM heory nroducon o lan some of he essenal heory for calculang momens of GLMs whch s a common mehodology used o model he underylng clam processes; Momens and her uncerany wll calculae he momens from specfc GLMs of he underlyng clams processes and presen asympoc resuls showng he uncerany n he parameer esmaes; Combnng resuls of he prevous wo secons we arrve a a more specfc formula for he varance of collecve rsk; ance of echncal Cos wll combne he prevous wo secons as well as he addve and mulplcave componens of echncal cos o derve a formula for he varance; and Praccal Applcaons wll lore some of he uses of he varance as an npu no prcng decsons and monorng and performance repors. 3

4 . ance of Collecve Rsk he fundamenal equaon of nsurance s he followng: CollecveRsk for a specfc clam ype where Random number of clams Random sze of clam We are concerned wh he varance of hs formula. he random number and sze of clams are usually esmaed by a sascal model based on hsorcal daa. In addon o he underlyng uncerany n he random processes parameer uncerany ha s are our parameer esmaes correc has also been nroduced. Usng condonal probably argumens he varance of he collecve rsk s hen: ( ) [ ( ) ] [ ( ) ] { [ ( ) ] [ ( ) ] } { [ ( ) ] } Where we have used he Law of oal ance and he Law of Ieraed xpecaons by frs condonng on he daa and parameer esmaes and hen on he random number of clams. he order of he condonng s mporan as he problem becomes dffcul o solve oherwse. ow ( ) ( ) ( ) ( ) ow as ( ) and ( ) are no funcons of we can hen ake ecaons and varances wh respec o on he above equaons: [ ( ) ] [ ( ) ] ( ) ( ) [ ( ) ] [ ( ) ] ( ) ( ) [ ( ) ] [ ( ) ] ( ) ( )

5 Sandard rror of echncal Cos Incorporang Parameer Uncerany Clearly he varance of has reduced o he momens of wh he uncerany n hese esmaes. hus and along ( ) { [ ( ) ] [ ( ) ] } { [ ( ) ] } ( ) ( ) ( ) ( ) [ ] [ ] ( ) 3. GLM heory he clam severy and clam frequency models for each of he clam ypes are modelled usng a generalsed lnear model. he sandard noaon for GLMs s: g ( ( )) g( ) ha s a GLM can be hough of as represenng each oucome of he dependen varable as beng generaed from a parcular dsrbuon n he onenal famly (he error dsrbuon). he mean ( ) of he dsrbuon depends on he ndependen varables g. hrough a lnk funcon he unknown parameers are ypcally esmaed wh maxmum lkelhood whch gves he parameers desrable properes parcularly wh respec o asympoc normaly. 4. Momens and her Uncerany Gven he commens n he GLM heory secon he underlyng processes for he clam frequency and clam severy of dfferen clam ypes wll have some underlyng dsrbuon from he onenal famly. ypcally clam frequency s assumed o be Posson dsrbued; whle clam severy s assumed o be Gamma dsrbued. A log-lnk funcon s also commonly used. ha s ~ Posson ( ) ( ) g ( ) ( ) ( ) g ( ) ( ) 5

6 Where s usually fxed a bu n he presence of over- or under-dsperson hen he dsperson parameer needs o be ncorporaed n he varance. And g g Gamma ~ d. Where g comes from he GLM heory secon and s he dsperson parameer as esmaed durng he model fng process. 4. Parameer Uncerany o ncorporae uncerany n he parameer esmaes we frs noe some dsrbuonal properes specfcally relevan o log-lnk GLMs esmaed hrough maxmum-lkelhood: ~ ~ ~ ~ Lognormal Lognormal g W W W Where he frs lne follows from he fac ha maxmum-lkelhood esmaors are asympocally normal and he second and hrd follow from sandard resuls. ow usng sandard resuls for a Lognormal dsrbuon we have:

7 Sandard rror of echncal Cos Incorporang Parameer Uncerany 7 5. Combnng Resuls Usng all he nformaon prevously and applyng he approprae subscrps for clam frequency and for clam severy and nong ha { } represens he daa and parameer esmaes we arrve a: Followng he logc n Brockman & Wrgh (99) we noe ha and are populaon means as esmaed by maxmum-lkelhood of a generalsed lnear model. Gven hs hey are asympocally normal and are ndependen due o he parameers no overlappng. herefore acklng he frs erm: ow he second erm: Where we have used he ndependence of and and have gnored parameer uncerany n he dsperson parameers and.

8 Furher subsuon yelds Combnng all our resuls ogeher we have: { } 6. ance of echncal Cos Up unl hs pon we have gnored he fac ha he rsk premum s made up of he collecve rsks of a number of dfferen clam ypes. We can and our ermnology o nclude he subscrp represenng clam ype. hus RP Independence beween he dfferen clam ypes follows because each clam and s sze belongs o only one clam ype. Models for he clam frequency and severy for each of he dfferen clam ypes are also based on muually dsjon daa ses each conssng of a sample of ndependen clams. hus we can calculae he varance of he rsk premum as: { } RP

9 Sandard rror of echncal Cos Incorporang Parameer Uncerany he rsk premum (workng loss) s jus one componen of overall echncal cos. here are oher mulplcave and addve componens. For example we mgh have he followng dollar amouns: CH Fxed Re ns ClamsHandlng xpense($per Clam) FxedPolcy xpens e($per Polcy) Rensurance xpense($ per Polcy) And he followng varable amouns: Comm Inv Pr of Commsson(%of echncal Cos) Invesmen Income(%of echncal Cos) Requred Rae of Reurn(%of echncal Cos) hus he echncal cos can be represened by he followng formula: C C Comm CH Comm Pr of Pr of Inv Fxed Inv Re ns he varance of he echncal cos s hen: CH ( ) Fxed Comm ( C) CH Comm Pr of Inv We have already calculaed above whch leaves: ( ) { [ ( ) ] [ ( ) ]} { [ ] [ ]} { ( ) ( ) ( )} Subsung he resuls above no ( C) yelds our desred resul. Re ns Pr of Inv 9

10 7. Concludng Remarks Componens of he above formulae namely fed values and varance-covarance of he parameer esmaes can easly be exraced from mos commercal packages ncludng SAS and MBLM. hus he calculaon of he sandard error can be added as an addonal sep durng he sandard predcon process for calculang he eced echncal cos for each polcy. 8. Praccal Applcaons he eced echncal cos s a key npu no many prcng decsons and performance and monorng repors. he varance of he esmae however s a dmenson no ofen consdered. Smplscally wo polces wh he same eced echncal cos may have dfferen varance. By gnorng he varance we would consder he wo polces equal from a echncal perspecve whch would lead o a poorer resul han f he polcy wh lower varance was argeed. By applyng he formula derved n hs paper o he busness s possble o arrve a beer prcng decsons and performance and monorng repors. Some of he oher applcaons nclude: An npu n FA capal models allowng beer quanfcaon of uncerany for each of he polces; Confdence nervals can be added o exsng sascs used n monorng porfolo performance whch enables clearer oson and consderaon of uncerany; Volaly n prof allows he denfcaon of segmens whch are conrbung o volaly n prof resuls; and Rsk based KPIs allows KPIs o be derved whch gve proper consderaon o he varably of he wren busness and hus allows farer comparson beween porfolos and lnes of busness.

11 Sandard rror of echncal Cos Incorporang Parameer Uncerany References Barne G Odell & Zehnwrh B 008 Meanngful Inervals Casualy Acuaral Socey Forum 008 hp:// Brockman M J & Wrgh S 99 Sascal moor rang: makng effecve use of your daa Journal of he Insue of Acuares [JIA] (99) 9: hp:// Wackerly Mendenhall III W & Scheaffer R L 008 Mahemacal Sascs wh Applcaons Sevenh edon uxbury homson Brooks/Cole. Wood G R Confdence and Predcon Inervals for generalsed lnear accden models eparmen of Sascs Macquare Unversy hp:// predcon_nervals.pdf