Securitization of Motor Insurance Loss Rate Risks

Transcription

1 criizaion of Moor Insranc Loss Ra Risks aan Ba and Cangki Kim * Absrac W ry o ransfr loss ra risks in moor insranc o capial mark. W s ranc cniq o dg moor insranc risks. As an xampl, w focs on AXA and ir scriizaion of Frnc moor insranc in 005. og is applicaion is nw, is ransacion is basd on a concp similar o CDO. rancs of bonds ar consrcd on basis of xpcd loss raio from moor insranc policy oldrs grops. As a consqnc w dvlop moor loss ra bonds sing srcr of synic CDOs. copon paymns of ac ranc dpnd on lvl of loss ras of ndrlying moor insranc pool. W sow an ingral formla for a risk adsd pric of loss ranc conrac wr loss disribion is modlld wi discond compond oisson procss. sscr ransform is cosn for a risk adsd masr and Forir invrsion mod is sd o calcla pric of moor claim ra scriis. pricing mods of rancs ar illsrad, and possibl sggsions o improv pricing mod or dsign of s nw scriis follow. Ky words: criizaion, Risk ransfr, Moor Insranc Loss Ras, CDOs. Dparmn of aisics and Acarial cinc, Univrsiy of Wsrn Onario, London, Onario, Canada, N6A 5B7. mail : ba@wo.ca * Dr. Cangki Kim is Lcrr a Acarial dis, Facly of Bsinss, Univrsiy of Nw o Wals, ydny NW 05 Asralia. l: , mail: c.kim@nsw.d.a

2 Inrodcion - For moor insranc providrs, fr claims canno b complly prdicd. is risk, mismac of acal claims from os anicipad, is a significan on and ms b managd. - In 005, AXA pionrd is sragy, slling UR 00 million of bonds, as scriizaion of ir moor insranc porfolio. inc iss of innovaiv moor insranc scriis from AXA, moor scriizaion as bn rciving anion considrably - criizaion ransfrs risks o capial marks, wr r is grar capaciy o absorb s risks compard o rinsranc mark. - Moor insranc scriizaion also cras nw invsmn opporniis, providing grar divrsificaion o radiional asss normally offrd. Invsors ar givn frdom o coos among rancs of bonds wi diffrn risk raings. - W considr srcr of a synic collaralizd db obligaion CDO for scriizaion of moor insranc loss ras. - W driv pricing formlas for scriizaion of moor insranc loss ra risks sing CDO ranc pricing mods ndr a risk adsd masr followd by nmrical xampls and discssions.

3 Caracrisics of AXA Moor Insranc criizaion - rior o discssion of gnral moor insranc scriizaion w firs considr AXA moor scriizaion o illsraiv ssnial caracrisics of is rvolionary sysm. Figr : implifid rcr of Ovrall ransacion AXA rmim Bnfi C Rinsrr Copon pls rincipal ric AAA/AAA A/A BBB/BBB- rmim pls ric Copon C pls rincipal NR Risk-fr Ass - AXA aks rinsranc of is individal moor insranc porfolio rog a pcial rpos Company C. AXA pays a prmim o C for rinsranc srvic. C isss svral bonds in r rancs rad by AAA/AAA, A/A, BBB/BBB- and non-rad NR ranc. pric is ransfrrd o invsors as copon paymns pls principal. 3

4 - no oldrs ar rqird o pay a pric o C in xcang for olding bonds. C invss prmim incom a i rcivs from AXA and procds a i rcivs from no oldrs ino a risk-fr ass. From risk-fr ass, C rcivs a copon paymn pls principal. r is n som bnfi a is rcivd by AXA from C. is coningn paymn is masrd by diffrnc bwn ral loss xprincd and xpcd loss wr no bnfi is rcivd by AXA in cas is figr is ngaiv. abl. Fars of nos issd in AXA Moor criizaion qiy ranc C Nos B Nos A Nos Amon ro 33.7 ro 7.0 million ro 67.3 million ro 05.7 million million Raing &/Fic NR BBB/BBB- A/A AAA/AAA Risk ransfr rsolds* Loss raio riggr 3.5% Loss raio riggr Loss raio riggr.8% Loss raio riggr 9.8% ranc iz* 3.5 poins of loss raio -3.5%; 0%.8 poins of loss raio 0%;.8% 7.0 poins of loss raio.8%; 9.8%.0 poins of loss raio 9.8%;0.8% *005 rsolds Loss raio riggr Fars ar from AXA Financial rocion

5 Figr. No rancing by Loss Raio AXA robabiliy Dnsiy xpcd loss raio 50% Loss Raio % Mdian Man Loss raio riggr, X% robabiliy Dnsiy Unrad Class C Class B Class A 05.7m 67.3m 7.0m 33.7m Loss Raio % X - 3.5% X% X.8% X 9.8% X 0.8% *Amons ar in ros. 5

6 3 Bnfis of Moor Insranc criizaion A. moor insranc company - criizaion allows moor insranc company o dg ir ndrlying loss risk, ransfrring i o invsors. - I is an addiional sorc of fnds for pary scriizing risk, fring p capial and improving solvncy of company. - is dvlopmn can ponially diminis xpcd loss ra assmd by moor insrr, as invsors olding bonds av incniv o rdc loss for igr copons. B. invsors - s nw scriis, wi ir igr copon rrn compard o or gnral bonds, can b sn as a promising invsmn opporniy. - invsors wo ar abl o ffcivly dcras company s loss ra, av grar ponial in maximising ir rrn. - vn insranc policy oldrs wo can g procion agains moor accidns, sill av a las wo incnivs o invs in is scriy: insranc prmim discons nx im and igr copon paymns by rdcing loss ras. C. Govrnmn as an invsor alos. - Wi is approac, Govrnmn may gain grar iniiaiv o focs on rdcing moor azards, by ir imposing nw laws/rglaions or allocaing a grar bdg o rlad xpndir on, for xampl, fr driving classs or pblic informaion on saf driving. 6

7 D. Or companis or organisaions - os a ar rlad o moor bsinss and moor insranc indsry can b anor grop of invsors wo cold ponially bnfi in gra amons from is prodc.. Driv af and G Higr Copons!!!. Also gnral invsors in capial mark wo sk igr yild scriis can b ponial invsors. In any cas basic principl is driv saf and g igr copons. 7

8 4 ricing Modl for criizaion of Moor Insranc Risks - Assmpions : - Hr w assm a aggrga claims follow Compond oisson disribion. -W n driv pricing formlas for sop loss prmims a will b sd for pricing of scriisaion. -W considr wo pricing mods for scriisaion: cmlaiv loss mod and priodic loss mod. 4. Cmlaiv Loss Mod - For modlling aggrga losss, w considr losss accmlad from iss da nil mariy. - L N b a oisson procss wi consan paramr λ. W assm an aggrga loss disribion follows a discond compond oisson procss wi risk fr forc of inrs r dfind on a probabiliy spac Ω, F, : N r r xn d, dx X i, 0, R i wr i s ar mp ims of oisson procss N and X i s ar i.i.d claim siz disribion non-ngaiv wi pdf of f X x Nd,dx is coning masr for poin procss.. No a 8

9 following orm can b sown by sing sandard macin in probabiliy ory and proof for cas of infini im orizon can b fond in alsn 993. orm For ac 0, and vry masrabl s B sc a N0,, B <, and for any fncion f and gx sc a f g x N d, dx 0, B xp { } f g x N d, dx 0, B <, following olds, λ. f g x xp 0, dfx x d B W nd Forir ransform of disribion of for calclaion of sop loss prmim and i is calclad sing abov orm. Corollary L s dno probabiliy disribion fncion pdf of by x,. Forir ransform of disribion of for a givn is f xprssd as follows. ix i f ˆ, f x, dx R rs fˆ xp λ X ds, 3 0 wr fˆ X is Forir ransform of disribion of a claim siz random variabl X. 9

10 - inc w ar working in an incompl mark, r ar infini nmbrs of risk nral masrs sc a ac coic of masr yilds no arbirag pric of insranc risks. - sscr ransform is on of possibl coics of sc probabiliy masr cangs. - Also i is known a for gomric Lévy procss modl sscr ransform is acally minimal nropy maringal masr and i is also masr wic maximizing xpcd powr iliy fncion. - Risk adsmn paramr in sscr ransform plays a ky rol o xplain aid of mark o insranc risk and i can b obaind ndr maringal condiion. - pcifically, w dfin a probabiliy masr wos Radon-Nikodym drivaiv is d F, 0 <, 4 d qivalnly, f x x, f x,, providd a xiss for all 0 <. No a is nonngaiv drminisic fncion wic saisfis maringal condiion dscribd blow. Grbr and i 994 or Miyaara and Fiwara 003 for dails. 0

11 - main goal of is sdy is associad wi calclaion of a risk adsd pric of sop loss conrac wi rnion lvl of d, in or words, d dx x f d x d, 5 - Dirc applicaion of orm 3.4 in Dfrsn, Garrido and Morals 006 givs following: d i f V d id 0, ˆ R π 6 d i i f V id, ˆ π 7 wr V rfrs o Cacy principl val ingral. - n, for ac, } {log, 0 x dx x f x. 8 inc, ˆ i f, lar wold b rdcd o rs X ds i f 0 ˆ λ { } X X r r λ 9 - scond qaliy can b obaind by sing Corollary and canging ordr of ingraion and xpcaion. Also, 0, ˆ, ˆ,, ˆ i f i f dx x f f x ix { } { } rs X rs X ds i f i f 0 ˆ ˆ xp λ 0

12 By sbsiing 9 and 0 o 6 or 7, w av a risk adsd pric formla of sop loss conrac. - Undr assmpion of no arbirag bwn insranc mark and capial mark, discond srpls procss sold b a maringal ndr a risk nral masr. pcifically w dfin accmlad srpls procss as follow, U r r r 0 G a, wr 0 is iniial srpls and G is prmim ra wic is ypically largr an λ X. - W can find an qivaln maringal masr a saisfis following, r U F s rs U s, -a.s. for any 0 s. By 9, for ac, w can sow a is solion of following qaion wic also saisfis xisnc of sscr ransform 4, { } λ r X X r G a.

13 3 xampl. ppos X follows xponnial disribion wi paramr sc a X. n followings olds: ˆ i f X } { r r λ r a λ, <. 3 By solving w can g following xplici xprssion for, / 4 G r r r λ. 4 Forir ransform of pdf of is, ˆ ib A f 5 wr r r r r r A λ λ and r B r λ arcan arcan. By sbsiing 3 and 5 ino 6, w g following formla, } { r r d λ 0 } cos {cos d d B A d π 6

14 - acal loss raio is dfind as acal aggrga loss dividd by oal prmim ovr a priod of im 0,. - annal insranc prmim G is masrd sing man val prmim principl wic is xprssd as G θ 7 wr : is aggrga claim on 0,, and θ is a loading facor. - L s dno acal cmlaiv loss ra by q, q Ga L, 8 Gs wr L r is cmlaiv losss nil im, and a and s ar prsn val and accmlad val of coninosly paying anniis wi ni of annal paymn rspcivly. - simad arg loss raio is dnod by qˆ. loss vn riggring occrs in cas wn acal loss raio xcds prdrmind arg loss raio, q > qˆ. a is q > qˆ implis a riggr L > qˆ G s and loss vn is riggrd, and q qˆ implis no loss cmlaiv xcss loss amon a im, L, can b dscribd by mliplying oal prmims and diffrnc bwn acal loss raio and arg loss raio, wic can b xprssd by following formla, L L qg ˆ s q qˆ G s, if q > qˆ. 9 0, orwis 4

15 - Now w dsign scriizaion of moor insranc loss ras and considr pricing mod. W sow dails of rancs of scriy in abl. abl. Fars of rancs in Moor Insranc criisaion rancs qiy ranc C Nos B Nos 3 A Nos 4 Raing NR BBB/BBB- A/A AAA/AAA Wig W W W 3 W 4 Noional Amon N N W N N W N 3 N W 3 N 4 N W 4 prad on s s s 3 s 4 Incoms ranc iz 0 q, q q, q 3 q, q 3 4 q, q - oal noional amon is dnod by N wic sold b drmind by C. facors a nd o b considrd for rancs wigs W or qivalnly noional amons of N ar csomrs prfrnc, mark racion and compiion in financial scor. - Howvr, is sdy xplors ranc wig only from mamaical viw poin, assming all xrnal facors ar ngligibl. - sprads on copon ras can b drmind according o ranc sizs. Or w can find ranc sizs giv sprads. Or pricing prpos is o drmin sprads or ranc sizs. 5

16 ranc loss amon L is L, {0, Min q q G s Max L q qˆ G s } L l l L 0 wr l q s 0 G, and q qˆ arg loss raio 3. - For ranc C no, as an xampl, wn acal cmlaiv loss occrs bwn l q s G and l G q s, oldr of qiy ranc ranc will los ir wol qiy and C no oldrs will los a porion of ir copons or noional amons. Figr 3. Figr 3. Acal Losss in ranc Modlling Acal loss L qˆ 0 q qˆ G s q q G s q q s 3 q 4 q Loss raio G Loss amon - L s dno discond val of cmlaiv loss of a ranc by, wi discond aacmn poin d r l and discond dacmn poin d 0 wi d r l 0 r l qˆ G a, r L. 3 arg loss raio can b diffrn according o scriy dsign, i.. ˆ q q for som >0. 6

17 Using pric formlas 6 or 7, w can calcla risk adsd pric of, d d. No a 0 d q G a, wi d qˆ G a. - L s considr copon paymns for invsors in ranc. If < d n invsors in ranc xprinc no losss and ir copons ar no affcd. If d < < d n y sold av som copon or principal rdcions. If copons. > d, n y rciv no - W wan o find a sprad 4, s, for ranc sc a xpcaion, wi rspc o a risk adsd probabiliy masr, of loss amons for ranc is qivaln o xpcaion of incoms of ranc invsors, 0 r dl 0 r d, 3 wr is incom paymn for ranc invsors a im. - In is papr, w assm a incom paymns ar drmind by lvl of loss amons and a y ar paid on discr im priods lik a nd of ac yar 5, l l L s discr vrsion of pricing formla is,. 4 s s, b w assm a i is a consan for ac 4 sprad can b varid according o im, ranc. 5 W can ads priodic copon paymns or copon ras sc as smi-annally or qarrly paid copons. 7

18 8 r L L r s L l l N, wr N is noional amon of ranc. qivalnly w av, r s d d 5 sprad, s, for ranc is calclad by, r d d s 6 For a pr-fixd sprad s for ranc, mark pric of ranc is diffrnc bwn xpcd loss and xpcd prmim paymns. r d d s V 0. 7 W noic a iniial invsmn amon N of ranc can b affcd according o lvl of losss in dsign of abov scriy. - om invsors may wan iniial invsmn a mariy. o saisfy s invsors w dsign scriy as follows. invsors may wan igr yild so w assm a copon ras av sprads ovr an inrs ras of a vry ig qaliy risk fr scriy. N r L L r r N C ~ 8

19 ~ wr C is copon for ranc invsors, wic is a random amon, a im. ~ - W assm a C is invrsly proporional o ranc loss amon and dfind by, ~ C Λ N R s 9 wr R is copon ra of a ig qaliy scriy sc as LIBOR, s is sprad for ranc assmd o b a consan for ac, and L Λ l l d d is proporion of losss in ranc wi 0 Λ. 30 bsiing 9 ino 8 and n aking xpcaion wi rspc o risk adsd probabiliy masr, w av formla for sprad of ranc, s r r Λ R Λ r. 3 o find sprad w may nd a modl for ra of a ig qaliy scriy sc as LIBOR. Wn copon ra of a ig qaliy scriy sc as LIBOR is consan, R R, sprad is xprssd by s r Λ r R. a is, sprad dcrass wn copon ra of a ig qaliy scriy incrass and i will b ngaiv if R is vry larg compard o risk fr ra r. 9

20 If w calcla copons by dfining ~ C Λ N s n copon ra s is xprssd as follows,, 3 s r Λ r. 33 inc copon ra of a ig qaliy scriy sc as LIBOR can b random, by rmoving i from formla, w don nd o worry abo rm srcr of ras. 0

21 4. riodic Loss Mod - cmlaiv loss mod discssd abov may no appal o som invsors bcas of a fw dmris. - For xampl, an iniial sar wi larg losss can sablis wr invsors will no rciv any of ir copons afr a crain im, bfor mariy da. - Also w can dsign scriy sc a no oldrs ar garand ir principal a mariy. is mans a maximm loss ac priod is copon amon only and iniially invsd noional amon is no affcd. - acal loss raio, q,, is dfind as acal aggrga loss dividd by oal prmim ovr a priod of im -, and arg loss raio, qˆ, is prdrmind by C. annal insranc prmim, G,, ovr a priod of im -, is calclad by θ L 34 G,, wr L, is aggrga claim on -,, i.. priodically basd aggrga claim amon, and θ is a loading facor. - acal loss raio on -, is q, L,. 35 G,

22 - loss vn riggring occrs in cas wn acal loss raio xcds prdrmind arg loss raio: q, > qˆ implis a L, G, > qˆ q, and loss vn is riggrd. qˆ implis no loss riggr. xcss loss amon, L,, on -, can b dscribd by mliplying annal prmim and diffrnc bwn acal loss raio and arg loss raio, L, q q G if q > qˆ, ˆ,, 0, orwis. 36 W wan o find sprads or ranc sizs basd on priodic losss. wr ranc loss amon L is,, L,,, {0, Min q q G Max L, q qˆ G, } l q l L l G, L 37,,, 0, wi l, qˆ. G,, Assm a noional amon of ranc, N, is invsd ino a ig qaliy risk fr scriy. noional amon of ranc sold b qal o smmaion of discond copons pls discond fac val noional amon of ranc. N r r C N 38 C ~ r N r, 39

23 wr C is copon rcivd from ig qaliy scriy a im, and ~ C is copon for ranc invsors, wic is a random amon, a im. W assm a copon paymns for ranc paid in inrval -, dpnd on ranc loss amon and dfind by, l, l, L, C N R s l l ~,, Λ N R s 40, wr R is copon ra of ig qaliy scriy, s is sprad for ranc assmd o b a consan for ac, and L Λ,, l, l, is proporion of losss in ranc wi 0 Λ., From 40, w can noic a maximm copon amon is N R s and minimm copon amon is 0. bsiing 40 ino 39 prodcs N r r Λ, N R s N. 4 Now, i is possibl o drmin sprad of ranc, s r * r Λ, R Λ, r. 4 No a sprad s is acal sprad ovr copon ra R sc as LIBOR of a ig qaliy risk fr scriy, wic can b 3

24 ngaiv. And w may nd rm srcr of R. o w may dfin copons by ~ C N s, 43 Λ, n copon ra s is xprssd by, s r Λ, r. 44 4

25 5 Nmrical xampls 6 Hr w sow nmrical xampls ndr spcifid assmpions on disribion of discond loss and is paramrs. rsls may vary on cangs of assmpions. Modl assmpion : Compond oisson / xponnial W assm oisson paramr λ40, X 5, scriy loading θ0., coninosly componding risk fr inrs ra r0.045, and mariy 4. n consan prmim ra G θ following figr sows volion of discond loss disribion ovr im and compars wo dnsiis f x, and f x,. 7 Figr 4. volion of Loss Disribion ovr im Nmrical ingraion is implmnd by Mamaica 6.0. and w s R for simlaion. 7 Obaind by 00,000 simlaion in R. 5

26 - W can s a disribions of discond loss ndr sscr ransform av avy ails on bo sid wic mans mark p mor wig on xrm vals. - following abl conains smmary saisics of loss raion disribion, q / Ga, by wic w will dfin loss raio riggr and ranc sizs. abl 3. mmary aisics of Loss Raion Disribion Min s q. Mdian Man 3 rd q. Max. 3 4 Undr Undr Undr Undr Undr Undr Undr Undr No a man of q ndr risk nral masr is s o b. 8 8 rrors in abl ar d o simlaions. 6

27 - I sms o b rasonabl o considr slling loss risks in bwn mdian o 3 rd qaril and following abl givs dails on ordr saisics for givn prcnils. abl 4. Ordr aisics for givn rcnils 45%-il 50%-il 55%-il 60%-il 75%-il 3 4 Undr Undr Undr Undr Undr Undr Undr Undr *** **** * * ** * W considr for rancs wr rancs ar dfind by prcnils of loss raio disribion a im 4 for bo ndr masr and. abl 5. prads ndr Cmlaiv Loss Mod basd on formla 6 rancs 3 4 * p, p 0.933, , , , s ** q, q , , , ,.30 s

28 W considr for rancs wr rancs ar dfind by prcnils of loss raio disribion a im 4 for bo ndr masr and. abl 6. prads ndr Cmlaiv Loss Mod basd on formla 3 rancs 3 4 * ** p, p 0.933, , , , s q, q , , , ,.30 s W assm yr-libor ra rancs ar dfind by prcnils of loss raio disribion a im for bo ndr masr and. abl 7. prads ndr riodic Loss Mod basd on formla 4 rancs 3 4 *** **** p, p 0.898, , , ,.060 s q, q , , , ,.340 s

29 6 Conclsion - Moor insranc scriizaion is grond braking. I is an alrnaiv cannl for moor insrrs, wo onc wr limid o only radiional rinsranc mods o manag risk. Alog moor insranc companis, sc as AXA, av rcognisd ponial of scriizaion as a risk managmn ool, a rol can b frr nancd rog innovaiv priva placmns. - From wo possibl pricing mods discssd, i is mor dsirabl o modl loss raio priodically as opposd o sing cmlaiv mod. Bo of s mods ar analyically corrc, og wn bond is considrd from invsors poin of viw, priodic mod is bs. is is bcas wn a sbsanial loss is incrrd in any givn priod bond oldr will ir forfi or sll ir conrac in following priod. As a consqnc of is, r is a possibiliy of scriizaion srcr collapsing. Moor insranc scriizaion is a vry nw dvlopmn and is crrnly in is xprimnal pas. A prsn im, r is an incompl mark for s nw scriis d o rariy of is ransacion. Hnc, r ar concrns abo liqidiy of s scriis wic can b improvd by mlipl ransacions, rfor broadning invsor bas. Alog, if is rnd of sing scriizaion as a mod of moor insranc risk ransfr conins o mov forward in fr, s problms sold improv. Driv saf and g mor copons!!! 9

30 Rfrncs AXA Financial rocion 005. AXA Lancs Firs criizaion of a Moor Insranc orfolio, rss Rlas, AXA Financial rocion, Novmbr 3, 005. Availabl a Blm, C. and Ovrbck, L rcrd Crdi orfolio Analysis, Basks and CDOs, Capman Hall/CRC. Covns, F., Wow, V., and Goovars, M On Nmrical valaion of op-loss rmim, Asin Bllin Vol. 0: Dlban, F., and Hazndonck, J Classical risk ory in an conomic nvironmn, Insranc, Mamaics and conomics Vol. 6: Dfrsn, D., Garrido, J., and Morals, M Forir Invrsion Formlas in Opion ricing and Insranc, cnical Rpor No. 06/06 Dcmbr 006. Concordia Univrsiy. Fabozzi, F., and Goodman, L. 00. Invsing in Collaralizd Db Obligaions. Frank J. Fabozzi and Assoc., nnsylvania. Fingr, C. 00. Closd-form ricing of ynic CDO s and Applicaions, ICBI Risk Managmn. Gnva. Dcmbr 00. Frsfilds Brckas Dringr 006. Insranc and Rinsranc Nws. April 006. Availabl a p:// Grbr, H. and i, Opion ricing by sscr ransforms, ransacions of ociy of Acaris Vol Hll, J., and Wi, A Valaion of a CDO and an n o Dfal CD wio Mon Carlo imlaion, Jornal of Drivaivs Vol.. No : 8-3. Jang, J., and Krvavyc, Y Arbirag-fr prmim calclaion for xrm losss sing so nois procss and sscr ransform, Insranc, Mamaics and conomics Vol. 35: 97-. Jozf D My Insranc and Capial Marks, Gnva aprs Vol. 3: Lin., Y. and Cox.,.H criizaion of Moraliy Risks in Lif Anniis, Jornal of Risk and Insranc, Vol. 7, No.: 7-5. Miyaara, Y., and Fiwara, minimal nropy maringal masrs for gomric Lévy procsss, Financ and ocas. Vol. 7: Nilsn,. and alsn, J On Disribion of a Randomly Discond Compond oisson rocss, ocasic rocsss and ir Applicaions Vol alsn, J Risk ory in a socasic conomic nvironmn, ocasic rocsss and ir Applicaions Vol

31 owrs rrin, illingas AXA s Moor Insranc Book criizaion, owrs rrin, Janary 006. Availabl a p:// /UpdaAXA.pdf sions?? ank yo vry mc!!! 3