Centre for Financial Risk

Transcription

1 Cnr fr Financial Risk Dubl Sh Nis Prcss and Is Applicain In Insuranc Jiwk Jang Wrking Papr 9

2 Th Cnr fr Financial Risk brings ghr rsarchrs in h Faculy f Businss & Ecnmics n uncrainy in capial marks. I has w srands. On srand invsigas h naur and managmn f financial risks facd by insiuins, including banks and insuranc cmpanis, using chniqus frm saisics and acuarial scinc. I is dircd by Asscia Prfssr Kn Siu. Th hr srand invsigas h naur and managmn f financial risks facd by hushlds and by h cnmy as a whl, using chniqus frm cnmics and cnmrics. I is dircd by Asscia Prfssr Sfan Trück. Th c dircrs prm rsarch in financial risk, and h xchang f idas and chniqus bwn acadmics and praciinrs.

3 Dubl sh nis prcss and is applicain in insuranc Jiwk Jang Dparmn f Acuarial Sudis, Faculy f Businss and Ecnmics, Macquari Univrsiy, Sydny NSW 9, Ausralia, jjang@fs.mq.du.au Absrac W cnsidr a cmpund Cx mdl f insuranc risk wih h addiinal cnmic assumpin f a psiiv inrs ra. T accmmda bh schasic claim innsiy and h im valu f claims wihin h aggrga lss, w us a dubl sh nis prcss. Using is gnrar, w driv h mmns f aggrga accumulad/discund claims whr h claim arrival prcss fllws a Cx prcss wih sh nis innsiy. Rmving h paramrs in a dubl sh nis prcss gradually, w shw ha i bcms a cmpund Cx prcss wih sh nis innsiy, a singl sh nis prcss and a cmpund Pissn prcss, rspcivly. Numrical cmparisns ar shwn bwn h mmns (i.. mans and variancs) f a cmpund Pissn mdl and hir cunrpars f a cmpund Cx mdl wih/wihu cnsidring a psiiv inrs ra. Fr ha purps, w assum ha claim sizs and primary vn sizs fllw an xpnnial disribuin rspcivly. Kywrds: Dubl sh nis prcss; a Cx prcss; schasic innsiy and im valu f claims; aggrga accumulad/discund claims.. Inrducin In insuranc mdlling, h assumpin ha rsuling claims ccur in rms f a Pissn prcss is inadqua as i has drminisic innsiy. T accmmda schasic naur f h claim arrivals frm d, windsrm, hail, bush r and arhquak in pracic, an alrnaiv pin prcss nds b usd gnra hir arrivals. Dassis and Jang (3) usd a Cx prcss wih sh nis innsiy pric caasrph rinsuranc and drivaivs. Albrchr and Asmussn (6) applid sh nis Cx prcsss n h prbabiliy f ruin. In classical risk hry, w fn implicily assum ha inrs ras ar zr wihin h claims prcss. Dlban and Hazndnck (987) xndd h classical risk hry cnsidr h c f h inrducin f inrs ra facrs, lading an xplsin f liraur in his subjc (Wilm 989; Paulsn 998; Lévillé and Garrid ; Jang 4 and Kim and Kim 7). S far hwvr n paprs accmmda bh a psiiv inrs ra claims and h schasic naur f claim frquncy wihin h claim arrival prcss. T achiv hs, w inrduc a spci c mdl ha has h fllwing srucur: P L = N X i ( Si) ; (.) i= whr L is h aggrga accumulad claim amuns up im, X i, i = ; ;, ar h claim amuns which ar assumd b indpndn and idnically disribud wih disribuin funcin H(x) (x > ), S i is h im f claim i, is h risk-fr frc f inrs ra and dal wih schasic naur f caasrphic lss arrival in pracic, w us a Cx prcss fr N. Th Cx prcss prvids xibiliy by ling h innsiy n nly dpnd n im bu als allwing i b a schasic prcss. Thrfr h Cx prcss can b viwd as a w sp

4 randmisain prcdur. A prcss is usd gnra anhr prcss N by acing is innsiy. Tha is, N is a Pissn prcss cndiinal n which islf is a schasic prcss. Lsss arising frm a caasrph dpnd n is innsiy. On f h prcsss ha can b usd masur h impac f caasrphic vns is h sh nis prcss. Prvius wrks f insuranc applicain using sh nis prcss and a Cx prcss wih sh nis innsiy can b fund in Klüpplbrg & Miksch (995), Dassis & Jang (3, 5 and 8) and Jang & Krvavych (4). Jang and Fu (8) als usd a Cx prcss wih sh nis innsiy mdl prainal risk. Th sh nis prcss is paricularly usful in lss arrival prcss as i masurs h frquncy, magniud and im prid ndd drmin h c f caasrphic vns. As im passs, h sh nis prcss dcrass as mr and mr lsss ar sld. This dcras cninus unil anhr vn ccurs which will rsul in a psiiv jump in h sh nis prcss. Thrfr h sh nis prcss can b usd as h paramr f a Cx prcss masur h numbr f caasrphic lsss, i.. w will us i as an innsiy funcin gnra a Cx prcss. W will adp h sh nis prcss usd by Cx & Isham (98): whr: = P M Y j ( Uj) ; (.) j= is h iniial valu f ha is carrid n frm caasrphic vns incurrd prviusly; fy j g j=;; is a squnc f indpndn and idnically disribud randm variabls wih disribuin funcin G (y) (y > ) and E (Y ) = (i.. magniud f cnribuin f caasrphic vn j innsiy); fu j g j=;; is h squnc rprsning h vn ims f a Pissn prcss M wih cnsan innsiy ; is h ra f xpnnial dcay. Caasrphic vns may ak lng marialis s h dcay ra may n b xpnnial. I is assumd b f his frm fr a mar f cnvninc, i.. clsd-frm xprssins f nal rsuls ar asily drivd. W als mak h addiinal assumpin ha a Pissn prcss M and h squncs fx i g i=;; and fy i g j=;; ar indpndn f ach hr. Th abv w quains can b wrin in rms f schasic di rnial quain (SDE), i.. whr and whr If w rplac wih in (.3), i bcms dl = L d dv ; (.3) P V = N X i i= d = d dc ; (.4) P C = M Y j : j=

5 d = d dv (.5) and ghr wih d = d dc ; (.6) w hav a dubl sh nis prcss. Fr dails f dubl sh nis prcss, w rfr Dassis (987). Wih h abv mdl spci cain and assuming ha L = and is sainary, in Scin w driv h mmns (i.. xpcain and varianc) f L and f L = L, whr L P = N X i S i (.7) i= is h aggrga discund claim amuns up im. T d s, w us h gnrar f h prcss ( ; ; ). In Scin 3 and Scin 4, dling h paramrs in a dubl sh nis prcss, w shw ha i bcms a cmpund Cx prcss wih sh nis innsiy, a singl sh nis prcss, a cmpund Pissn prcss and driv hir mmns rspcivly. Assuming ha claim sizs and primary vn sizs fllw an xpnnial disribuin rspcivly, w bain h xplici xprssins f hs mmns and shw hir numrical calculains. Scin 5 cnains sm cncluding rmarks.. Dubl sh nis prcss and is gnrar Th gnrar f h prcss ( ; ; ) acing n a funcin f (; ; ) blnging is dmain is givn by A f (; ; 4 f ( x; ; ) dh (x) f (; ; ) Z 4 f (; y; ) dg (y) f (; ; ) 5 ; (.) whr f : (; ) (; ) R! (; ). I is su cin ha f (; ; ) is di rniabl w.r..,, fr all, ; and ha Z f (; x; ) dh (x) f (; ; ) < and Z f (; y; ) dg (y) f (; ; ) < fr f (; ; ) blng h dmain f h gnrar A. Fr dails f nding h gnrar f h prcss ( ; ; ) applying h picwis drminisic Markv prcsss (PDMPs) hry, w rfr Dassis and Jang (8).. Expcain f h aggrga discund claim 3

6 In his scin, assuming ha L = and is sainary, w xamin h man f h aggrga discund claim. T d s, l us bgin wih driving h xpcain f a im assuming ha is givn. If w s f (; ; ) = f () = in (.), hn w hav whr A = m ; Z m = xdh (x) <. Frm h Dynkin s frmula cndiining n, w hav E( j ; ) = Di rniaing (.) w.r.., w hav E( s j ; s )ds m s ds: (.) de( j ; ) = E( d j L ; ) m and slving his di rnial quain, w bain E( j ; ) = m s s ds: (.3) Nw ak cndiinal xpcain n in (.3) and assum ha is givn, hn w hav E( j ; ) = m s E( s j )ds: (.4) I is knwn in Jang and Krvavych (4) ha h xpcain f givn is givn by and ha if is sainary (i.. l R whr = ydg (y) <. Sing (.5) in (.4), i is givn ha E( j ; ) = m whr 6= and if is sainary (i.. l E( j ) =! ), i bcms m (.5) ; (.6)! ), i bcms ; (.7) Rplac wih in (.7) and (.8) rspcivly and fr simpliciy, l us assum ha L =. Thn w hav E(L j ) = m 4 (.8) ; (.9)

7 and if is sainary (i.. l! ), i bcms m = m s j a : (.) Muliplying bh sids in (.9) and (.) rspcivly, w hav h man f h aggrga discund claim givn ; (!) E(L j ) = m () ; (.) and if is sainary (i.. l! ), i bcms m = m a j a : (.). Varianc f h aggrga discund claim Similar Scin., assuming ha L = and is sainary, w xamin h varianc f h aggrga discund claim. T d s, l us sar wih driving h scnd mmn f a im assuming ha and ar givn. S f (; ; ) = f () = in (.), hn w can asily bain whr E( j ; ) = m s E( s s j ; )ds m s E( s j )ds;(.3) m = x dh (x) <. T bain h xprssin f (.3), rsly w nd nd h xprssin fr E( s s j ; ): sing f (; ; ) = f (; ) = in (.), w hav S E( j ; ) = () I is knwn in Jang and Krvavych (4) ha m () ()s E( s j )ds () ()s E( s j ; )ds: (.4) E( j ) = : (.5) and ha if is sainary (i.. l! ), i bcms 5

8 whr = R y dg (y) <. ; (.6) Hnc using (.5) and (.7), (.4) is givn ha E( j ; ) = () m m m m ( () m! () Assuming ha = and ha is sainary fr (.7), i bcms ()!! () ) : (.7) m m! () m! () : (.8) Rplac wih in (.8), hn w hav m m! ( ) m ( )! (.9) and muliply bh sids, h jin xpcain f L and a im is givn by m m Nw sing (.7) and (.5) in (.3), w hav m : (.) 6

9 E( j ; ) = n m n m m n 6 m 4 nm m 6 4 m 4 m n n m nm n m m m m m m m m m ! () 3 5 m : (.) W rsly assum ha = in V ar( j ; ) = E( j ; ) sainary. Thn i is givn by fe( j ; )g and ha is n m m m m! () n m m m m m n m m m m m m : (.) Rplacing wih in (.), w hav 7

10 n m m m m! ( ) n m m m m m n m m m m m m (.3) and muliply bh sids, hn h varianc f h discund aggrga claims, whr is sainary, is givn by m m m m m m m m m m m m m m m! () : (.4) I is knwn in Jang (998) ha h varianc f givn is givn by and ha if is sainary (i.. l V ar( j ) =! ), i bcms : (.5) 3. Variains frm h man f dubl sh nis prcss S = in (.5), hn w hav a cmpund Cx prcss wih sh nis innsiy as aggrga claim prcss. Puing = ihr in (.) r in (.), is man is givn by m (3.) as h im valu f claims is n lngr cnsidrd. Th sam rsul can als b fund in Dassis and Jang (3). Rplac wih in (.6) and fr simpliciy, l us assum ha =, hn (.5) bcms = s j a : (3.) 8

11 Muliply in (3.) hn i bcms = a j a : (3.3) Cnsidring drminisic claim innsiy, h man f claim sizs and h risk-fr frc f inrs ra, hn (3.) and (3.3) ar h mans f aggrga accumulad/discund claims rspcivly whr h claim arrival prcss fllws a Pissn prcss. Th sam rsul can als b fund in Jang (4). S = ihr in (3.) r in (3.3), hn i is givn by (3.4) as h im valu f claims is n lngr cnsidrd. This is h man f aggrga claims whr h claim arrival prcss fllws a Pissn prcsss. Fr h numrical cmparisn, l us dn =, = m and = in (3.)-(3.4). Using an xpnnial disribuin f G(y) and H(x) fr vn jump and claim siz rspcivly, i.. (.), (3.), (3.3) and (3.4) ar givn by g(y) = y and h(x) = x wih >, > ; a ; (Sh nis Cx wih psiiv inrs) (3.5) j a ; (Sh nis Cx wih n inrs) (3.6) a ; (Pissn wih psiiv inrs) (3.7) j a : (Pissn wih n inrs) (3.8) If w cmpar bwn (3.5) and (3.7) (r bwn (3.6) and (3.8)), w can asily nic a di rnc ha ariss du schasic claim innsiy (i.. h sainary man f sh nis innsiy f a Cx prcss, ) and drminisic claim innsiy (i.. h man f a Pissn prcss, ). Obviusly, if = ; (3.5) and (3.7) (r (3.6) and (3.8)) hav h sam valus. Nw l us illusra h calculains f h mans f a cmpund Pissn mdl and hir cunrpar f a cmpund Cx mdl wih/wihu cnsidring a psiiv inrs ra. Exampl 3. Th paramr valus usd calcula h mans ar and = :; = 5; = :5; = = :5; = 4, = :: Th calculains f h mans f a cmpund Pissn mdl and hir cunrpar f a cmpund Cx mdl wih/wihu cnsidring a psiiv inrs ra ar shwn in Tabl 3.. 9

12 Tabl 3.. Man Sh nis Cx wih psiiv inrs 783:3 Sh nis Cx wih n inrs 8: Pissn wih psiiv inrs 4877: Pissn wih n inrs 5: Th nx xampl shws h c n h mans f Cx wih psiiv inrs cas causd by changs in h valu f, and rspcivly. Exampl 3. Using h sam paramr valus in Exampl 3., h calculains f h mans f Cx wih psiiv inrs cas causd by changs in h valus f, and ar shwn in Tabl Tabl 3.. Man = : 39; 6 = : 39; 6 = :5 7; 83:3 = 3; 9:6 = ; 95:8 Tabl 3.3. Man = 3; 9:6 = 4 7; 83:3 = 9; 58 = 39; 6 = 5 97; 54 Tabl 3.4. Man = : 78; 33 = : 7; 83:3 = :5 ; 56:7 = 78:33 = 39:6 Tabl 3. shws ha h highr h magniud f cnribuin f primary vn j innsiy (i.. h lwr valu), h highr h man. Tabl 3.3 indicas ha h highr h primary vn arrival ra (i.. h highr valu), h highr h man. Tabl 3.4 shws ha h lwr h im prid ndd drmin vns cs (i.. h lwr valu), h highr h man. Exampl 3.3 Using h sam paramr valus in Exampl 3., h calculains f h mans f Cx wih psiiv inrs cas a ach valu f h insananus ra ar shwn in Tabl 3.5. Exampl 3.4 L! in (3.5) and (3.7), hn w hav Tabl 3.5. Man = : 7; 96: = :3 7; 88: = :5 7; 83:3 = :7 7; 76:4 = :9 7; 65:9 and ; (Sh nis Cx wih psiiv inrs wih in ni im hrizn) (3.9) ; (Pissn wih psiiv inrs wih in ni im hrizn) : (3.) Using h sam paramr valus in Exampl 3., Tabl 3.6 shws h acuarial n prmiums f h discund aggrga claims whn h im hrizn gs fr bh sh nis Cx and Pissn cas

13 Tabl 3.6. Sh nis Cx wih psiiv inrs wih in ni im hrizn Pissn wih psiiv inrs wih in ni im hrizn Man 6; ; 4. Variains frm h varianc f dubl sh nis prcss Similar Scin 3, if w s = in (.4), i is givn by m as h im valu f claims is n lngr cnsidrd. Th sam rsul can als b fund in Jang and Fu (9). Rplac wih in (.6) and fr simpliciy, l us assum ha =, hn (.5) bcms = s j a : (4.) Muliply in (4.), hn i bcms (4.) = a j a : (4.3) Cnsidring drminisic claim innsiy, h man f claim sizs and h risk-fr frc f inrs ra, hn (4.) and (4.3) ar h variancs f aggrga accumulad/discund claims rspcivly whr h claim arrival prcss fllws a Pissn prcss. Ths rsuls can als b fund in Jang (4). S = ihr in (4.) r in (4.3), hn i is givn by (4.4) as h im valu f claims is n lngr cnsidrd. I is h varianc f aggrga claims whr claim arrival prcss fllws a Pissn prcss. Fr h numrical cmparisn, l us dn =, = m and = in (4.)-(4.4). Using an xpnnial disribuin f G(y) and H(x) fr vn jump and claim siz rspcivly, i.. (.4), (4.), (4.3) and (4.4) ar givn by g(y) = y and h(x) = x wih >, > ; a j a ( ) a j a! () a, (Sh nis Cx wih psiiv inrs) (4.5) j a

14 ; (Sh nis Cx wih n inrs) (4.6) a ; (Pissn wih psiiv inrs) (4.7) j a : (Pissn wih n inrs) (4.8) In Scin 3, w fund ha if = ; (3.5) and (3.7) (r (3.6) and (3.8)) hav h sam valus, i.. E Pissn L = E Cx L : Hwvr, i ds n hld in rms f h variancs cmparing bwn (4.5) and (4.7) (r bwn (4.6) and (4.8)). This implis ha h disribuin f h aggrga discun claims wih rspc a Cx prcss has havir ail han is cunrpar wih rspc a Pissn prcss, i.. V ar Pissn n L () < V ar Cx n L () Nw l us illusra h calculains f h variancs f a cmpund Pissn mdl and hir cunrpar f a cmpund Cx mdl wih/wihu cnsidring a psiiv inrs ra. Exampl 4. Th paramr valus usd calcula h variancs ar and : = :5; = 4, = :; = :; = :5; = = = 8 ha prvids us wih h sam mans f aggrga discun claims rgardlss f h lss arrival prcss N. Th calculains f h variancs f a cmpund Pissn mdl and hir cunrpar f a cmpund Cx mdl wih/wihu cnsidring a psiiv inrs ra ar shwn in Tabl 4.. Tabl 4.. Varianc Sh nis Cx wih psiiv inrs ; 995; 4 Sh nis Cx wih n inrs 3; 48; Pissn wih psiiv inrs ; 56; 7 Pissn wih n inrs ; 6; Tabl 4. shws ha vn if h mans f aggrga discun claims ar h sam rgardlss f h lss arrival prcss N, h varianc f aggrga discun claims frm a cmpund Cx mdl is alms wic highr han is cunrpar. Thrfr if insuranc cmpanis mply man-varianc principl fr hir prmium calculains, a cmpund Cx mdl rs highr prmium han is cunrpar. Tabl 4. jusi s ha insuranc cmpany can cnsidr using a cmpund Cx mdl accmmda schasic naur f h claim arrivals frm d, windsrm, hail, bush r and arhquak in pracic. Exampl 4. Using h sam paramr valus in Exampl 4., h calculains f variancs f Cx wih psiiv inrs cas causd by changs in h valus f, and ar shwn in Tabl

15 Tabl 4.. Varianc = : 3; 758; ; = : 44; 433; = :5 ; 995; 4 = ; 9; 5 = 47; 7 Tabl 4.3. Varianc = ; 497; 7 = 4 ; 995; 4 = 7; 488; 5 = 4; 977; = 5 37; 44; Tabl 4.4. Varianc = : 3; 398; = : ; 995; 4 = :5 564; = 64; 7 = 9; 34 Tabl 4. shws ha h highr h magniud f cnribuin f primary vn j innsiy (i.. h lwr valu), h highr h varianc. Tabl 4.3 indicas ha h highr h primary vn arrival ra (i.. h highr valu), h highr h varianc. Tabl 4.4 shws ha h lwr h im prid ndd drmin vns cs (i.. h lwr valu), h highr h varianc. Exampl 4.3 Using h sam paramr valus in Exampl 4., h calculains f h variancs f Cx wih psiiv inrs cas a ach valu f h insananus ra ar shwn in Tabl 4.5. Exampl 4.4 L! in (4.5) and (4.7), hn w hav Tabl 4.5. Varianc = : 3; 6; 7 = :3 3; 55; 3 = :5 ; 995; 4 = :7 ; 937; = :9 ; 88; and ( ), (Sh nis Cx wih psiiv inrs wih in ni im hrizn) (4.9) ; (Pissn wih psiiv inrs wih in ni im hrizn) : (4.) Using h sam paramr valus in Exampl 4., Tabl 4.6 shws h variancs f h discund aggrga claims whn h im hrizn gs fr bh sh nis Cx and Pissn cas. Tabl 4.6. Sh nis Cx wih psiiv inrs wih in ni im hrizn Pissn wih psiiv inrs wih in ni im hrizn Varianc 9; 33; 3; ; 3

16 5 Cnclusin T accmmda schasic naur f h claim arrivals frm caasrphic vns, such as d, windsrm, hail, bush r and arhquak, w cnsidrd a Cx prcss wih sh nis innsiy. Fr h addiinal cnmic assumpin f a psiiv inrs ra claims, h dualiy rsul bwn h aggrga accumulad claims and singl sh nis prcss was usd. Ths w spci cains mad a dubl sh nis prcss and using is gnrar, w drivd h mmns f aggrga discund claims. Rmving h paramrs in a dubl sh nis prcss, w shwd ha i bcam a cmpund Cx prcss wih sh nis innsiy, a singl sh nis prcss and a cmpund Pissn prcss, rspcivly. Numrical cmparisns wr shwn bwn h mmns f a cmpund Pissn mdl and hir cunrpars f a cmpund Cx mdl wih/wihu cnsidring a psiiv inrs ra. Fr ha purps, w assumd ha claim sizs and primary vn sizs fllw an xpnnial disribuin rspcivly. Rfrncs Albrchr H. & Asmussn S. (6): Ruin prbabiliis and aggrga claims disribuins fr sh nis Cx prcsss, Scandinavian Acuarial Jurnal, 86-. Cx, D. R. and Isham, V. (98) : Pin Prcsss, Chapman & Hall, Lndn. Dassis, A. (987): Insuranc, Srag and Pin Prcss: An Apprach via Picwis Drminisic Markv Prcsss, Ph.D. Thsis, Imprial Cllg, Lndn. Dassis, A. and Jang, J. (3): Pricing f caasrph rinsuranc & drivaivs using h Cx prcss wih sh nis innsiy, Financ & Schasics, 7/, Dassis, A. and Jang, J. (5) : Kalman-Bucy lring fr linar sysm drivn by h Cx prcss wih sh nis innsiy and is applicain h pricing f rinsuranc cnracs, Jurnal f Applid Prbabiliy, 4/, Dassis, A. and Jang, J. (8): Th disribuin f h inrval bwn vns f a Cx prcss wih sh nis innsiy, Jurnal f Applid Mahmaics and Schasic Analysis, Aricl ID Dlban, F., Hazndnck, J. (987): Classical risk hry in an cnmic nvirnmn, Insuranc: Mahmaics and Ecnmics 6, Jang, J. (998): Dubly schasic pin prcsss in rinsuranc and h pricing f caasrph insuranc drivaivs. Ph.D Thsis. Lndn Schl f Ecnmics and Pliical Scinc. Jang, J. (4): Maringal apprach fr mmns f discund aggrga claims. Jurnal f Risk and Insuranc 7 (),. Jang, J. and Krvavych, Y. (4): Arbirag-fr prmium calculain fr xrm lsss using h sh nis prcss and h Esschr ransfrm, Insuranc: Mahmaics & Ecnmics, 35/, 97-. Jang, J. and Fu, G. (8) : Transfrm apprach fr prainal risk managmn: VaR and TCE, Jurnal f Oprainal Risk, 3(), Jang, J. and Fu, G. (): Masuring ail dpndnc fr aggrga cllaral lsss using bivaria cmpund Cx prcss wih sh nis innsiy, submid a Annals f Acuarial Scinc. Kim, B. and Kim H.-W. (7): Mmns f Claims in a Markvian Envirnmn. Insuranc: Mahmaics and Ecnmics, 4(3), Klüpplbrg, C. and Miksch, T. (995) : Explsiv Pissn sh nis prcsss wih applicains risk rsrvs, Brnulli,, Lévillé, G., Garrid, J. (): Mmns f cmpund rnwal sums wih discund claims, Insuranc: Mahmaics and Ecnmics 8, 7 3. Paulsn, J. (998): Ruin hry wih cmpunding asss: a survy, Insuranc: Mahmaics and Ecnmics,

17 Willm, G.E. (989): Th al claims disribuin undr in ainary cndiins, Scandinavian Acuarial Jurnal,. 5