Methodological Problems in Solvency Assessment of an Insurance Company 1

Transcription

1 Invsmn Managmn and Financial Innovaions, /4 95 Mhodological Poblms in Solvncy Assssmn of an Insuanc Comany 1 Rosa Cocozza, Emilia Di Lonzo 3, Mailna Sibillo 4 Absac Th cn wid dvlomn and changs in insuanc maks highlighd h ncssiy o ma ou h solvncy analysis in a mo coml famwok. Th aoach w sn in h a coms u wih an ingad analysis of h isk ofil of an insuanc businss, aking ino accoun h acual Euoan dicivs abou solvncy assssmn. Th aim of h a is o consuc a mhodology a o incooa oly h ffc of h isk soucs in calculaing mahmaical ovisions lad o a ofolio of insuanc olicis. Jl classificaion: G, G8, G13 Ky wods: Lif insuanc, financial isk, dmogahic isk, caial adquacy, svs, condiional andom ocsss. 1. Lif insuanc businss: a isk managmn aoach An insuanc comany is solvn if i is abl o fulfil is obligaions und all conacs und all asonably fosabl cicumsancs (IAIS ). Nvhlss, in od o com o a acicabl dfiniion, i is ncssay o mak cla und which siuaion h aoianss of h asss o cov claims is o b considd. Th qusion, fing o h valuaion uoss, dos no hav a comhnsiv answ, sinc i dnds on many issus. Indd, i is lvan whh h comany is dmd as a closd oaion o a going concn, hus including only win businss (un off basis) o also fuu nw businss (going concn basis). Addiionally, i dnds on h aim of h valuaion ha is h m financial ogss of h comany o is abiliy o m claims and oh obligaions und all bu h mos m cicumsancs. Th fis oic dfins h lvan isk facos: on a going concn basis also flucuaions aound h cd valu of h nw businss will b akn ino accoun (hus allowing fo ning wihin diffn ools), whil on a unoff basis only flucuaions wihin h singl ool will b considd (hus scluding claing). Th scond on dfins h amliud of h valuaion, which in obabilisic ms would cosond o h confidnc lvl choic (fom inmdia lvls o m vns). Thfo, solvncy valuaion is a ocss, whos logical aadigm should squnially consis of h main ss: lvan isk cogniion, isk masumn and dfiniion of caial quimns o absob onial losss. In gnal, h main isk fo a fim is ha vnus ov o b unabl o cov nss and, gad h valuaion also h shaholds, o muna adqualy caial invsd. This is namly businss isk. This vy boad dfiniion dos mbac all isks and ovid fo masuing hm hough h vaianc of h cd ofi ov a im buck (isk hoizon, accouning iod c.) o also ov h ni duaion of h businss. Pmiums and claims scivly a yical insuanc vnus and nss: hfo businss isk sms fom h onial inqualiy among hs lmns, wih h fuh difficuly ha vnus hav o b simad bfo nss bcaus of h invd cycl. Thfo, all h facos ha can induc h inqualiy and in fac o a loss a lvan and dfin h whol isk subsysm. This aoach dos imly ha 1 This sach was aially suod by Ialian MIUR (Pojc: PRIN Modi sumni l analisi la gsion di ischi ni soi dll assicuazioni di son vidnzial. Alhough h a is h sul of a common sudy, h fis scion is win by R. Cocozza and h s wo a by E. Di Lonzo and M. Sibillo. Ph.D., Associa ofsso of financial inmdiais a h Univsiy of Nals, Ialy. 3 Pofsso of financial mahmaics a h Univsiy of Nals, Ialy. 4 Pofsso of financial mahmaics a h Univsiy of Salno, Ialy.

2 96 Invsmn Managmn and Financial Innovaions, /4 h caabiliy of coving nss by mans of miums dnds on h dsign of lvan cos divs. Fo lif businss, h main uzzls a moaliy and ins as 1, which could imac of h singl iod fomanc (incom), by h modificaion hy oduc on lvan coss and vnus. Thfo, concnaing on h u mium comonns and foming an analysis on a un-off basis, h isk imac valuaion can sa off on h bakdown of h incom comonns ov a singl accouning iod. Pofi ( ) can b oulind as h algbaic sum of: ovisions a h bginning of h ya (R -1 ), and miums (P ), invsmn incom (R -1 +P )( -1), aymns o olicyholds (S ) and ovisions a h nd of h ya (R ), ha is R 1 P S R. (1) Fo an immdia moay (n) uniay annuiy, quaion (1) can b scifid as 3 N N , 1 N 1 1 () wh N is h acual numb of suvivos a ag + and is h cd singl sv fo ach ising olicy a h nd of iod. In his sciv accud incom can b sn as a funcion of h foc of h oal a of un fo h iod (-1,) and of h acual numb of suvivos N -1 and of h obabiliy +-1, and N , (3) N 1 1, (4) 1 N 1. (5) Th fis divaiv (quaion 3) givs h fil of h insuanc isk, ha is a isk indicao fo h vn of an acual numb of suvivos divs fom h cd on. Th indicao is always osiiv bcaus a high numb of suvivos in h cding iod focs h insu o incas h cosonding ovision, giving is o a caialisd valu lag han cd nss. This bnfi, alhough always osiiv, shows a dynamics conncd o h ag () and h valuaion im () as shown by Figu 1. 1 Fo aml, fo moay annuiis, h minimal quilibium is dicly dndn on h singl iod diffnials bwn h ingal of h insananous oal un on asss uchasd wih win mium and ha of h oiginal ins a alid in mium aing and h logaihm of h acual numb of suvivos and h cd numb of suvivos (Cocozza al., 3b). Fo h sak of claiy, miums a assumd o b and a h bginning of h accouning iod and aymns o olicyholds a assumd o b fomd immdialy bfo h nd of h iod. 3 Th aim of h a is o oulin a valuaion mhodology: w hav dcidd o wok all h amls hough an annuiy bu any oh kind of olicy could sv ou uos.

3 Invsmn Managmn and Financial Innovaions, /4 97 Fig. 1. Dynamics conncd o h ag and h valuaion im Th scond divaiv (quaion 4) givs h longviy isk indicao, ha is a oy fo h vn ha h is a shock in h moaliy funcion. Th ngaiv sign accouns fo h ns incas bcaus of a lag numb of suvivos fo h cun iod. Th imac of h longviy isk is ooional o h siz of h ofolio and o h sv valu, hus incasing no only fo young olicyholds and/o fo valuaion im clos o h issu (as shown by Figu ) bu also fo lag siz ofolios. 1+ SIM 1991 n= =9% longviy isk fil ag a issu () im () Fig.. Th ooion of longviy isk imac h h siz of h ofolio Th hid divaiv (quaion 5) givs h invsmn isk, sinc i givs h mulili of h a shock. Th osiiv sign of h indicao accouns fo h dic ooionaliy bwn h a shock and h incom ov iod ; naually h high h valu of h sv is, h song h imac aas o b (Figu 3).

4 98 Invsmn Managmn and Financial Innovaions, /4 SIM 1991 n = =9% invsmn isk fil im () ag a issu () Fig. 3. Th valu of h sv and h imac aio I can b asily shown ha h imac of financial and longviy isk is fa lag han insuanc isk sinc hi lvanc is fild hough h sv valu conncd o h ofolio siz. This imlis ha hi imac is dicly ooional o h acual numb of suvivos N -1 and, in a sns, o h numb of issud olicis c. In conclusion, h vaiaion vlociy of h ofi is masud by mans of h gadin, N 1, 1 sinc N. (6) 1, 1, Thfo, h oal vaiabiliy of h ofi ov h singl im buck can b dividd ino h main h comonns, hus giving h oouniy o saaly idnify h imac of a scific isk faco, also considing h im voluion of h indicaos.. Th mahmaical snaion As alady oind ou in h vious scions, in iskinss valuaion concning h liabiliy comonns, h join ffc of dmogahic and financial facos lays a fundamnal ol. In his sns i is ncssay o cocly valua h imac of such facos and hi inacions on h mahmaical sv. On h basis of h balanc quaion, w can agu ha flucuaions of h a of un a aiculaly lvan fo h asss, whil on h liabiliis h inacions bwn dmogahic and financial comonns hav a mo and mo makd ffc. As a fis s in solving h oblm, w focus on h iskinss analysis concning h mahmaical sv, fo which w inoduc a siml and a suiabl masu, a o quanify h syngy oducd by h andom flucuaion of ins and moaliy (suvival). L us consid a ofolio of c idnical n-ya moay lif annuiy-immdia, ach olicy bing of 1 uni ayabl a h nd of ach ya whil h lif agd () suvivs. L us dno h andom vaiabl sning h fuu lifim of h i-h insud (fo ach i) by T i ()

5 Invsmn Managmn and Financial Innovaions, /4 99 and h cua fuu lifim of () by K i (), h is h numb of coml fuu yas livd by () in,n-. Th osciv loss a im, fo h i-h olicy, is dfind as h diffnc L (i) a im bwn h sn valu of fuu bnfi aymns and h sn valu of fuu mium aymns, assuming ha T i ()>: L (i) K i j ds s, (7) wh (s) is h foc of ins. Th n mium sv a im is dfind as h condiional caion of L (i), givn ha T i ()>. L us dno by L h osciv loss fo h ni ofolio, accoding o h following noaion: j1 n sds L, (8) N 1 wh N () is h numb of suvivos a im. Sinc in ou analysis L is affcd by wo isk soucs, (s) and K i (), h gssion funcion E L(s) (Fs, 1998) ovids an avag of L ov all valus of (s). In his way w consid a condiional man valu wih sc o h flucuaions of h insud s lifim ino h sidual olicy duaion,n. In his famwok vae L(s) sns a masu of h volailiy of L aising fom h uncain bhaviou of h ins a. By mans of his masu w quanify h financial isk, aking ino accoun also h snc, vn hough avagd, of h dmogahic comonn. W obsv ha: va n va E L s E N 1 sds s vac n 1 sds (9) c n n 1 s1 s cov u du s u du,. W can also obsv ha h sidual isk comonn is quanifid by h obvious diffnc bcaus of h uncainy of lif duaions va L. (1) va E L s

6 1 Invsmn Managmn and Financial Innovaions, /4 3. How o quanify h uncainy imac L us assum ha h foc of ins in h iod und considaion is govnd by a dminisic low, (), dducibl on h basis of h cun lvan as, cocd by a sochasic ocss, X(), a o summaiz h andomnss affcing h a voluion in im and o cau all is ossibl dviaions fom h dminisic an. So, i is naual o s (Di Lonzo al., 1999) X. (11) In ou aml X() is an Onsin-Uhlnbck ocss, wih aams > and >, and iniial osiion X()=, involvd by h following sochasic diffnial quaion dx X d dw, (1) wh W() is a sandad Win ocss. On h basis of sochasic calculus incils and af som lins of algba, i is ossibl o obain h following covaianc and vaianc funcions of h ocss cov 3 va h h X X s ds, X s ds minh, k k h k hk 1 1 h s ds h h hk Xsds: ; h. Fo a siml snaion of h fomulas fo h vaianc and h covaianc funcions of h valuaion sochasic faco in fomula (9), w os: h k cov Xs ds, Xs dsh,k and va h Xs ds (h). Considing, in ou hyohss, ha h valuaion faco is log-nomally disibud, w can wi: cov h k X sds X s ds, 1 h k h, k 1.

7 Invsmn Managmn and Financial Innovaions, /4 11 Now w a abl o comu h imac of h isk soucs on a lif annuiy ofolio. As an mlificaion w imlmn ou modl in h cas of a ofolio consising of c=1 olicis a issu, ach on lad o a lif agd, wih duaion n=. Wih gad o h insananous ins a, w fi a consan dminisic comonn =.6, and h aams =.11, =.5 fo h sochasic on. In figus 4 and 5 w can obsv h bhaviou of h wo isk comonns as funcions of h ag and h sv valuaion im (and, imlicily, of h olicy sidual duaion). Fig. 4.Th Financial Risk (=,,7; =1,,15) Fig. 5.Th Insuanc Risk (=,,7; =1,,15) Th financial isk, as funcion of, dcass whn is fid; h sam bhaviou aas whn h ols of and a inchangd. Th insuanc isk, fo vy fid valu of, incass wih unil a cain valuaion im; h young h insud is, h long h iod of h incasing bhaviou is. This hnomnon dnds on h sv amoun osd o h dmogahic isk, aking ino accoun h influnc of h insuds ag and h sidual duaion of h ofolio Whn is fid, h dmogahic isk gnally incass wih, bu fo ga valus of i dcass whn h ag bcoms ga, as i is cla asoning as in h vious cas.

8 1 Invsmn Managmn and Financial Innovaions, /4 4. Conclusions Th alicaion of isk faco analysis has givn h oouniy o build u a mhodology s abl o valua h imac of diffn isk divs, boh in a dminisic and in a sochasic cons. Thfo, i could b of valuabl alicaion fo boh nal and innal conols. I has bn shown ha, in boh h cons, h is a significan and consisn voluion along im of main isk indicaos, hus suggsing h oouniy o scal h caial quimn accoding o h siz and im voluion of h businss. Finally, i is ncssay o undlin ha h nsion of h analysis o oh kind of olicis could giv is o vy diffn sul as fa h lvanc and h im voluion of h isk indicaos a concnd. Rfncs 1. Babbl D.F., F.J. Fabozzi. Invsmn managmn fo insus. - Nw Yok: Wily, Babbl, D.F., J. Gold, C.B. Mill. Fai valu of liabiliis: h financial conomics sciv//noh Amican Acuaial Jounal. N 6 (1) Cocozza, R. La gsion dl ischio di asso di inss nll assicuazioni dl amo via. Padova: CEDAM,. 4. Cocozza, R., E. Di Lonzo. Solvncy of lif insuanc comanis: mhodological issus//woking a 3a. 5. Cocozza, R., E. Di Lonzo. Risk ofils of lif insuanc businss: a combind aoach//vi Ialian-Sanish Confnc on Financial Mahmaics (Tis), 3b. 6. Di Lonzo, E., M. Sibillo, G. Tssio. A Sochasic Modl fo Financial Evaluaion: Alicaions o Acuaial Conacs//Alid Sochasic Modls in Businss and Indusy N Fs, E. Rlaiv Imoanc of Risk Soucs in Insuanc Sysms//Noh Amican Acuaial Jounal N () IAIS Solvncy & Acuaial Issus Subcommi. On Solvncy, Solvncy Assssmns and Acuaial Issus.. IAIS, Mach. 9. IAIS Solvncy & Acuaial Issus Subcommi. Pincils on caial adquacy and solvncy.. IAIS, Januay. 1. IASB Sing Commi. Insuanc Issu a IASB, Dcmb 11. Knigh, K. Mahmaical Saisics - Boca Raon, Floida: Chaman and Hall/CRC,. 1. Pak, G. Sochasic analysis of h inacion bwn invsmn and insuanc isks//noh Amican acuaial jounal N 1 () Vandhoof, I.T., E.I. Alman (ds). Th fai valu of insuanc liabiliis - Boson. Dodch, London: Kluw Acadmic Publishs