The Current Value of the Mathematical Provision: A Financial Risk Prospect

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1 bac Poblm ad Pcv Maagm / Volum 5, Iu, 7 Th Cu Valu of h Mahmacal Povo: Facal Rk Poc Roa Cocozza *, Emla D ozo **, Mala Sbllo *** Th a add h quo of h calculao of h cu valu of h mahmacal ovo ad mould a dmc ad ochac cao, ug a o m ucu of a mad by ma of a Co-Igoll-Ro modl. I ovd a coml ad ogal ya-by-ya valuao modl fo h bu fomac, ad a clod oluo fo h cu valuao of h v, ogh wh a comhv gh o h dyamc of h v cocd o h lco of a dfd m ucu of a. Moov, h calculao of h VaR of h mahmacal ovo ocd a k mau uful o aca alo h valuao a k. Fuu ach oc coc h lco of h ochac oc ud o dcb h dyamc of h a ad h obl maagal ad gulaoy alcao of a VaR mau. Th modllg ha b ald, a a mlfcao, o a lf auy ofolo bu ca b aly lcad fo ay kd of olcy ad ay kd of ofolo v o homogou. y wod: Rk dcao, lf uac, olvcy, facal k, dmogahc k. JE Clafcao: G.. Ioduco f uac bu adoally chaacd by a coml ym of k ha ca b ally l o wo ma y of dv: acuaal ad facal; wh h cg oc, hy f o h uac comay aud o lc h gh moaly abl, ad o aly h gh dcoug oc, wh accuacy cov focag ofccy. Boh h ac ca b gadd a h am m a k dv ad valu dv, c hy ca gv o a lo o a of f h -a cd valu ov o b hgh o low ha h -o acual alzao. Wh cal fc o h mdao ofolo cog clam vu coodg a, h c ca b hacd by h cfc accoug adad ald fo h facal am; oh wod a fa valuao ym ca ubaally modfy h dclou of h coomc ul ad h olvcy aaal m ad ac. h d of Mach 4, h Iaoal ccoug Sadad Boad ud h Iaoal Facal Rog Sadad 4 Iuac Coac. Fo h f m, ovd gudac o accoug fo uac coac, ad mak h f h ISB ojc o achv h covgc of wdly vayg uac duy accoug acc aoud h wold. Mo cfcally, h IFRS m a u o chag accoug olc fo uac coac oly f, a a ul, facal am fomao ha mo lva ad o l labl, o mo labl ad o l lva. Moov, m h oduco of a accoug olcy ha volv maug dgad uac labl coly ach od o flc cu mak a ad, f h u o lc, oh cu maad aumo, hu gvg o a oal clafcao of om o all facal a a a fa valu hough of o lo. O h o had, h comomg oluo dv fom h cogo of a fa valu dclou qum, alo o comly wh a mo gal dcy cocg facal am. O h oh had, m fom h qually wdad ad d wo * Uvà d aol Fdco II, Ialy. ** Uvà d aol Fdco II, Ialy. *** Uvà d Salo, Ialy. Roa Cocozza, Emla D ozo, Mala Sbllo, 7. 7

2 Poblm ad Pcv Maagm / Volum 5, Iu, 7 cocg h lack of agm uo a dfo of fa valu a wll a of ay gudac fom h Boad o how h fa valu ha o b calculad. ccodg o h majoy of commao cludg h aoal ocao of Iuac Commo, h Iaoal ocao of Iuac Suvo, ad h Bal Comm o Bakg Suvo, h ucay may lad o fa valu dclou ha a ulabl ad coly maud amog uac. alo h mca cadmy of cua 3 claly a, mak valuao do o fo may m o h uac balac h ad h would lad o h lac o y cfc maum fo dmg uac coac ad a fa valu. Howv, uch valu would b uaoably ubjc o wd ag of judgm ad o gfca abu, ad may ovd fomao ha o a all comaabl amog coma. Th cau fo h coc h k mag como of h fa valu. Rk mag a claly a a of mak valu fo uca a ad labl, bu wh c o may uac coac, h valu cao b lably calbad o h mak. Hc, a mak-bad valuao ba fo hm would oduc lva fomao. Th am gv o a wd oubl. If h a amdm h valuao ca fo h v fom o ya o aoh bcau of cu mak yld o v of cu moaly abl, h a obl chag h valu of h v accodg o h alcao of a mo g o, a h oo, a mo flbl co. Th may u o a o fa valuao k. I h accoug cv, h oduco of a accoug olcy volvg maug dgad uac labl coly ach od o flc cu mak a ad, f h u o lc, oh cu ma ad aumo ml ha h fa valu of h mahmacal ovo oly a cu valu o a valu. Coly, h fa valu of h mahmacal ovo could b oly dfd a h valu of h dual db owad h olcyhold valuad a cu a ad, vually, a cu moaly a. I a, h makg o mak. Howv, h h cucal o. Th Iaoal ccoug Sadad Comm df h fa valu a h amou fo whch a a could b chagd, o a lably ld, bw kowldgabl, wllg a a amlgh aaco, whl h Facal ccoug Sadad Boad df h fa valu a a ma of a c dmd by mak aco. Hc, h CS Fa Valu Tak Foc dfd fa valu a h mak valu, f a ucly acv mak, o a mad mak valu, ohw. Fom h accoug d a fa valu o caly a qulbum c, bu mly a mak c. Th a add h quo of h calculao of h cu valu of h mahmacal ovo ad mould boh a dmc ad ochac cao, ug a o m ucu of a mad by ma of a Co-Igoll-Ro modl. To h d, h a a wh a coml ad ogal balac-h modllg of h uac bu, whch co: o dfy all h vaabl lva o h caal fomac of h u k ad valu dv; o moaly mau h mac of ay k faco chag; o u h ul fo a o k bad aoach a valu-a-k co cf. co ; ad, fom a mo gal o of vw, o ak cfc dco o maag ad hdg h k. cal ao dvod o h fa valuao k, dfd a h chag h coomc ul du o a chag h valuao ca adod fo uac labl, whch ov o b oooal o h valu ad h duao of h ofolo v cf. co.. Th cfc quo of h cu valu of h v ubquly u o a ochac co co 3 by ma of a cah flow aaly, wh lva v k dv a adomly ad. Fally, h aaal of h mahmacal v a cu a ocd by co 4, wh h fa valuao of h mahmacal ovo focud o h facal dv whl h acuaal como a acad by ma of a oouly ojcd moaly abl. I h co h ad wll fd h dal of h m ucu co 4., h cu valu a couou aoach wh a clod fom fo h aalycal oluo o h cu valuao of h mahmacal ovo co 4. ad a clu o h valu-a-k of h mahmacal ovo du o a chag h valuao a valuao a k. Rach ad accal mlcao a daw h la co, c h modllg ca gv o a wd ag of maagal, gulaoy, ad accoug alcao. 8

3 Poblm ad Pcv Maagm / Volum 5, Iu, 7 Fom a mhodologcal o of vw, moa o ha, fo h ak of clay, a yhc ao of h balac h, whch v a a gudl, ha b mloyd h a alhough h modl, wh uful ad aoa, ca b dd od o clud alo como uch a loadg ad mak-u mag. h am m, h whol aaly ha b ald o a lf auy ofolo bu ca b aly lcad fo ay kd of olcy ad ay kd of ofolo v o homogou.. Th balac-h chm u cod h mum P fo a mmda moay uay auy: P, wh h aaou a of obvd a h bgg of h bu ad ald fo mum g, ad df h coodg obably abl ud o c h olcy. Th fomulao ml ha h fal qulbum m of a coho of c dcal olc coad by: cp, wh h aaou oal a of u o a uchad wh w mum obvd a h d of h bu, ad h acual umb of uvvo a ag +. Th fomulao comu h ul of h ofolo: f ov ov, h bu oducd of ad, of cou, lo, f gav, whl h ull lvl h mmal qulbum ha o ay h olvcy hhold Cocozza al., ad 3. howd lwh Cocozza al., 5, h fal qulbum gv by h cly dd o h ul accud ach m buck ad hfo olvcy codoal uo h ably o k ov h dc bw h caald ul u o m < ad h valu of h dual db; ca b fomally d a h obably of cg maly ha o ay ach od h gal qulbum codo d by: - Pob cp, 3 wh h lvl of h obably a olcal quo ad h amou of caal adqua o k olvcy wh a lm, whch codd baabl, wh fc o boh caal co bo by h mda ad h k lvl facd by olcyhold. Wh h ul of h al caal vd by h u lcly cludd o h fomulao, h valu of h mdao ofolo a h d of h f ya qual o cp fom whch ca b aly fd ha h valu of h mdao ofolo dd o h u al caal, o h acual umb of coac g a h d of h f ya, o h abl adod h valuao of v a h d of h f ya +, ad o 4 Th a coa com, caal ga ad lo Pak,

4 Poblm ad Pcv Maagm / Volum 5, Iu, 7 h u o a ad h f ya a wll a o h valuao a lcd fo h mahmacal ovo a h d of h f ya. Th valu dcly flucd by boh h wo a whl vly acd by a ca h acual umb of uvvo a h d of h ya, c: cp, 5, 6. 7 Coly, h lco fo h valuao of h v a h d of h ya of a moaly abl dvg fom ha ald fo ay vou valuao boh fo cg ad vg oduc a vaao of h valu of h mdao ofolo, whch v wh c o h g of h vaao of h abl lf. a ma of fac, h lco fo aml of a mo ud abl howg h ca hgh obably of uvvg wll oduc a ca h valu of fuu ouflow ad hfo a dca h valu, whl h lco of a abl howg low obably wll gv o a ca h am valu. fa a h y of h mac cocd, ay o vfy ha fld by h z of h ofolo ud obvao ad by mlc facal dcoug oc. Fomulao 5, 6 ad 7 df a coml of k fl cocd o lva k dv ad a abl o cau h c of a chag h lva vaabl o h valu of h mdao ofolo. Th mac maud ough h moay chag ad h coodg fomulao ca b oly dfd a k dcao. Th f wo fomulao 5 ad 6 mau facal k ad ca b addd cvly a vm k ad valuao a k, c h f aoma h valu chag du o a vaao h u o a, whl h cod aoma h valu chag du o a modfcao h a ald fo h valuao of h mahmacal ovo. Th la o cf. fomulao 7 mau acuaal k, c aoma h valu chag du o a vaao h acual umb of uvvo a h d of h ya. fa a h z of h mac a cocd, h mo lva k dv ca b aly dfd h vm k ad h valuao a k, c hy a cvly fld by h valu of h oal a a h bgg of h ya, ad o of h ofolo mahmacal ovo. Smlaly, h valu of h mdao ofolo a h d of h cod ya ca b d o of h wo quval followg fomulao cp. 8 vouly, h valu of h ofolo dd o h valu, o h acual umb of coac g a h d of h cod ya [ ], o h abl adod h valuao of v a h d of h f ad cod ya + ; + ad o h f ad cod ya u o 3

5 Poblm ad Pcv Maagm / Volum 5, Iu, 7 3 a ; a wll a o lcd valuao a a h d of h f ad cod ya ;. lo fo h h dc ddc o h mo c a hold, c: cp, 9. Gally, fo ay mda od, h followg lao hold:,. 3 Coly wh fomulao, wh fc alo o a oal aaou a of u o a quval o h gl od a, h ulma ul quao a: cp gh o h valuao a k If w coca o h valuao a k ay o dmoa ha h dolla chag h valu of h mdao ofolo dcly oooal o h Macaulay duao of h mahmacal ovo D R c: R D R, 6

6 D R R Poblm ad Pcv Maagm / Volum 5, Iu, 7 D R R. 7 Thfo, fomulao 7 gv h oouy o a ha h valuao a k ca b cvly maud by a o of valu a k of h mahmacal ovo du o a chag h valuao a. I ha b ovd ha h valu chag dcly dd o h cu valu of h v R, o a oooaly faco od by Macaulay duao ad o a oal adv movm yld. fa a h g cocd, a ca dca h yld wll ul a valu fall, coodg o a low hgh valu of h lably. Moov, c h quy/v ao by dfo l ha o, h mac of a valuao a modfcao o h valu of h mdao ofolo dcly oooal o h lvl of h facal-uac lvag, c: R D. 8 R Thfo, aa fom h gl k faco m voluo, h z of h mac dd dcly o h ogo of h v/quy ao ad of h v duao. Sc h wo qua d o duc alogd h lgh of h olcy, a vaao h valuao a ucbl o oduc mo lva c a h bgg ha owad h d of h olcy; a h am m, c abu, h mac wll duc alogd h duco of h ac ag of h ud... k bad aoach If w do h fuu adom valu of h valu of h mdao ofolo ad k h coodg kow cu valu a ad F h dbuo fuco of who dy fuco f, h bac ucual oy of VaR h gv by Pob k VaR 9 fom whch ay o f ha h Valu-a-Rk a h cofdc lvl ha alzao of h lo amou k whch wll o b cdd wh obably. By dog h - qual dbuo of a F w hav ha Pob ad, coquly, VaR k,. Th valu of h -qual dbuo of c quoa Z E Z VaR k E ca b l o a cd a E[] ad a o ha fomulao ca b -w a. 3 If w gad h fomulao 3 a h ummao of h cd ul ad a addoal quoa cocd o h vaac of h ul lf, h valu-a-k of h valu ca b - 3

7 Poblm ad Pcv Maagm / Volum 5, Iu, 7 d a h VaR of a ofolo od o ho k faco ha a h chagg aam of h valu lf ad ca b acad by modllg h k faco ad by ug k fl o oooa h c o h valu uch a, fo aml, h ca of h v duao. Th aoach ca b ald o valua h c of a vaao h k dv wh c o boh h al g,.. cg hyoh, ad ya-by-ya vual valuao of h al hyoh. Moov, ca b dcly mloyd wh fc o maagal goal ad o uoal ag k bad caal qum ad ca gv o a fm cfc valuao f obl o oba a coml daa of colao amog k faco o a adadd valuao f h colao a uod o b - o fd valu. 3. Th cah flow aaly u cod a ofolo cog of c ud agd acly, ach of whom havg a dfd lf auy olcy wh mum ayabl a h bgg of h f ya ad bf ayabl a h bgg of ach ya af, h ca h ud alv. m of h co o ovd a cu valuao of h ofolo v a ochac vom fo h a, by ma of a gadual couco of h valuao fomula. Wh h followg oao.g. Cocozza al., 4: j = cua fuu lfm j =,,,... of h ud =,,...,c, h F, = flow of cah a m =,,,... lad o h olcy ofolo a m, h B,= bf ayabl o h ud a m, h P,= mum ayabl by h ud a m, w ca df h lo of ach olcy ad of h ofolo. 3.. dmc cao a f, w a fom a dmc cao, codg, fo ach olcy comg ofolo a m ad fo a gv valu of j, h followg chm fo h cah flow: F, f j, j, f j wh F P B, f f,,. Th lo k k,,,... cocd o h -h olcy h um of h dcoud cah flow cf. F, 99: jk jk j, k j, k, k k, j v k 5 whch, accodg o h uual oao, k moay u a m. 3.. Ig h adom h fuu lfm Rlacg h fuu lfm h v h valu a m k of o j by a adom vaabl, h ojcd cah flow ul: 33

8 Poblm ad Pcv Maagm / Volum 5, Iu, 7 F, wh obably, 6 wh obably havg dcad by h obably fo a olcy a m o b ll ofolo a m. I k ach olcy cu h cd adom lo, wh c o h fuu lfm, gv by h o: k F, v k. 7 k k k 3.3. Ig h adom h umb of olc ofolo Idcag by k h umb of uvvo a m k blogg o h al gou of c ud a m, w ca w h o of h lo k of h ofolo g a h m, o h ba of h fomao h u ow a m : k k k, 8 j Suog h ' muually dd ad dcally dbud adom vaabl, h adom lo a m k of h ofolo g a ha m gv by h um of h lo fd o ach olcy: k k k Ig h adom h a ud fo h valuao ow w d wha ad abov o a ochac a cao. Th dcoug faco dcbd by a ochac oc v h h modllg h voluo m of h a ad h cao had by all h olc ofolo. Two mo aumo hold: h adom vaabl k a dd ad dcally dbud gv h qucv h h ad h j adv h a muually dd. Wh a mo gal fomal chm, l u do by,, P,,, P cvly, wo obably ac, wh h -algba coag h facal v ad h - algba coag h uvval v. Thfo h obably ac,, P, gad by h cdg wo, h -algba coa boh h fomao abou moaly ad facal hoy, by ma of h flao k wh k k k, wh k ad k. I h followg w wll aum a fcol mak wh couou adg, o co o boowg o ho-al, zo-bod ad ock boh fly dvbl. So w ak h cao quao 7 ad 8 ud h k-ual obably mau, followg a caocal ocdu wll ad h facal lau: E k k k k F Ev k,. 3 k k 34

9 Poblm ad Pcv Maagm / Volum 5, Iu, 7 Th w oba h cd valu k of h lo fd o h ofolo: E k k k k, k k k. 3 k E k k c F E v k Th oblm of h hyoh of mak coml h dmogahc famwok wll kow lau; om a h couco of a oou obably mau, od o guaa h aoa o of h c fuco cf. D Flc al., 4 fo a bf uvy o h ubjc. a ma of fac, by vu of h c chaacc of h v, w cod aoabl o h cu valuao by ma of h cao famd wh h b dco of h dmogahc cao. I h od of da, w oba o of h v mak valu, whch a dducd coly wh h modl, o ha w ca ay hy a makd o modl. 4. lcao 4.. Backgoud hyoh obl alcao gv by h calculu of 6 ad 7 fo h ca of a mmda moay = ya uay auy fo a mal olcyhold = 4, by ma of a m ucu of a ad of a ojcd moaly abl. Th m ucu wa bad o a Co- Igoll-Ro qua oo modl accodg o a ml dcao Cha al., 99; Dla al., 995, whch h couou cd a dfd by h ochac dal quao d k d db, 3 wh k ad a ov coa, h log m ma, B a Bowa moo ad h hfd a qual o h gl obvao of h hocal ma. Th oluo of 3 gv by k k ku u db 33 u fo whch cd valu, covaac ad vaac a cvly Dla al., 995, : k E 34 va cov k k k k k k k k k. 36 k k, fa a daa a cocd, a dv fom Bak of Ialy ocal ac ad co of auald a of Govm 3-moh T-Bll a covg h od fom Jauay 996 ul Jauay 4; moaly a hav b dvd fom h RG48 Iala Mal abl. Th followg ml dcao 35 a a 37 mad a how by Dla al. 995, 74 gav h ul how by Tabl. Tabl 35

10 Poblm ad Pcv Maagm / Volum 5, Iu, 7 CIR aam ad coodg m ucu of a volaly coff. dc ca df coff. dc ca log m dffuo ag valu ma al valu coff.,7% 4,5% 97,4%,5%,63%,53% Tm Ecd a,7%,48% 3,3% 3,43% 3,73% 3,94% 4,% 4,% 4,9% 4,35% 4,4% Sadad dvao,%,3%,3%,37%,4%,43%,44%,46%,47%,47%,48% 4.. Cu valu of ojcd cah-flow a couou aoach Cha al. 99, w ado h am aam boh h dc ad h couou oc, obag h mao fo k og k ad ug h quao a. W calcula h cd valu of h cu valu of h v. Ou aml of alcao f o h ca of a mmda moay uay auy fo a ud agd 4. O h ba of quao 3 w g h cu valu of h v a h bgg of ach of h ya coug h olcy duao. I Tabl w o h ul ad coma hm wh h coodg valu calculad a h coacual aual a.4. Th cu valu of h v a l ha h coacual mum. Th fluc of h facal k du o h valuao a ca b obvd h bhavou of h dc bw h cu valu mum ad h coacual o, alway gav bu dcag wh h fc m ca. Cu valu of h mahmacal ovo Tabl Evaluao m Fd valu 4% 8,6 7,39 6,69 5,97 5, 4,43 3,6,77,88,96, Cu valu CIR-dc 8,4 7,3 6,59 5,87 5,4 4,37 3,57,74,87,96, Cu valu CIR-couou Dffc 4%- CIR dc. Dffc 4%- CIR co. 7,6 7, 6,39 5,73 5,3 4,3 3,53,7,85,95,,,9,,,8,6,4,3,,,,45,37,3,4,8,3,9,5,3,, 4.3. Rk dcao fo ojcd cah-flow a dc aoach Th calculao gv u h oouy o valua o oly a yhc k dcao uch a h duao bu alo h VaR, bad o h cah flow mag, a how by Tabl 3. Sg aa fom h acuaal k como, a f od aomao of h vaao du o a chag h valuao a ca b obad hough h cah flow mag of h ofolo v. Th mahmacal ovo ca b a a ofolo of zo couo bod who omal valu h ca-quval of h codoal aym wh mauy qual h ach gl c- 36

11 Poblm ad Pcv Maagm / Volum 5, Iu, 7 cal mag a ad wh ofolo cobuo qual o w. Th coodg duao fomulao 6 ca b calculad a D R w, wh w. 38 VaR of h mahmacal ovo a u m Tabl 3 Tm/m Toal Ca quval RG48,999,998,997,9959,9946,993,996,9899,988,9863 Cu valu CIR dc Pofolo wgh Modfd duao Idvdual m chag Idvdual m VaR99% 8,43,9749,94,9,86,898,785,749,77,6733,643 %,%,69%,%,7%,9% 9,7% 9,4% 8,79% 8,37% 7,98%,9758,94,94 3,856 4,84 5,7638 6,77 7,676 8,645 9,5787,,57,95,33,68,,8,53,74,93,48,5,33,,39,39,464,53,587,638,68 Va Ro 99% Valu 8,4 Udvfd,48 Pofolo,788 Colao -, % 4,98% 3,47% -,5% I h cv, h valu of h v, a a ofolo, dd o h aag of m ad o a vco of k faco f f, f,..., f mad u by h a o ach gl od of h m ucu. Th f od aomao gv by R f ; R f ; R f, 39 T w w f R f wh μ R h chag valu ulg fom h aag of m, ad h o-calld aggga dla-faco. Oc w call ha h vco of chag h udlyg faco follow a mulvaa omal wh kow ma ad covaac ma cf. co 4. fomulao 34, 35 ad 36, h f dvav of v valu wh c o bod-quval c ca b calculad fom h dla of h dvdual um ha o ay hough h duao como of h v. Mo cfcally, a u m h f adddum of h fomulao 39 ull whl h cod ca b d fo cho lvl of cofdc lkd o h aam a VaR R T T R R R ww VaRC VaR, 4 Th k faco vco mad u acly of lm, qual o h umb of a coug h ofolo, c ach of hm od o a cfc a. 37

12 38 Poblm ad Pcv Maagm / Volum 5, Iu, 7 wh VaR h -vco of dvdual udvfd VaR, VaR T ao, ad C h coml ma of u colao. Fomulao 4 how ha h valu-a-k of h mahmacal ovo dd o udlyg mv faco.. h vaac-covaac ma ad lva cal faco.. h wgh mbodd w ad ovall v z R. v VaR calculao a u m gv by Tabl 3, whch h VaR o co how a a 99% cofdc lvl h mamum ca h v valu w a lkly o m f, mmdaly af h u of h olcy, h a a hock fo boh mak ad accoug ao. 5. Cocluo Th a addd h quo of h calculao of h cu valu of h mahmacal ovo a dmc ad ochac cao, ug a o m ucu of a ad ovdg a ogal modl of ya-by-ya valuao of lva k faco. a f lvl, ha b howd ha h valuao of lva k faco ca b ufully fomd hough a coml vy aaly, whch ca b l a umb of gm flcg h umb of lva k faco. ca b acad fom calculao od by Tabl, h cu valu cly dd o h valuao aam. fa a h a cocd, h u of a m ucu g a log m ma lghly hgh 4,5% h h coodg fd a 4% ul a ymacally low valu of h v, boh h couou ad h dc aoach. Th valuao h ochac couou famwok co 3 uo h ul obad h oh co, by ma of clod fomulao. Th couou chm o ovd a uay vw of h ma k como acg h cu valuao of h uac bu, accodg o h c uggo fom ISB. I h od of da, ag v, bcau of h cfc au, cao b wh a fc mak. So h lably chag vual ad h mahmacal modl amd o a cg ym accodg o a coh ao of h uac alm. Th ml h choc of modl whch b h voluo m of ad moaly a, maly focug o h dmogahc fc ym. I fac h o do o h fudamal chaacc qud fo a k ual valuao ladg o mak valu; h aoabl o cod a aoa oy of oly dfd mak valu, fg o valuao co wh h modl. a cod lvl, h alcao o h ca of h valuao a k ha how ha obl o mau h mac alo of a chag h valuao ca ad o aca cly h fomac modfcao. a fom h umcal dc, h a fuh quo: h u of a g m ucu ml h cluo of fuu of ag fom oll-ov ag, alhough h mag a dfly uca boh fo h alzao of h a ad fo h alcao of oll-ov ag. Th ml ha h alcao of a g m ucu may gv o a ud-valuao of h v. Coly, h u of a fallg m ucu may gv o a ov-valuao of h v lf, c bad o dcag a. Gv h oblm, h obly of calcula a v VaR ca ovd fo a gh o valuao dyamc. I h cv, h kowldg of a mamum lklhood valu chag ca ovd fo a bchmak of h valuao lf. Fo aml, wh fc o h umcal alcao, could b codd a mamum vaao of h 3,47% o, alavly, h coodg amou could b aa a cal ovo o o fuu ualzd of mag. Oh mlcao, og fuu ach oc, coc h lco of h ochac oc ud o dcb h dyamc of h a ad h obl maagal ad gulaoy

13 Poblm ad Pcv Maagm / Volum 5, Iu, 7 alcao of a VaR mau. Wh fc o h f o, h lco of a CIR modl m o b aculaly lva, c h valuao a by dfo a o-gav a. Oh oc, uch a h O-Uhlbck collco fo aml, may b mo aoa fo h vgao of h vm k cf. co, c hy mbody v gav u. Wh fc o h cod ac, h maagal ad gulaoy ag could b dfd hough VaR mau fo boh olvcy ad agc dco. kowldgm Th a wa aally uod by h Iala MIUR Rach Pojc of Rlva aoal I: Valuao lf uac ad o fud: acuaal, coomc ad facal mhodolog Rach U: Rk ad fomac dcao of lf uac ofolo: mlmao of quaav modl fo h aaal of olvcy. lhough h a h ul of a jo udy of h auho, co,, 4., 4.3 a du o R. Cocozza, whl co 3, 4. a du o E. D ozo ad M. Sbllo. Rfc. MERIC CDEMY OF CTURIES 3: Comm o Eou Daf 5 Iuac Coac. h:// CH,.C., ROYI,.G., OGSTFF, F.. ad SDERS,.B. 99: Emcal Comao of lav Modl of h Sho-Tm I Ra, Th Joual of Fac 47, COCOZZ, R. : a go dl cho d ao d ll m d acuazo ulla va. CEDM: Padova, I. 4. COCOZZ, R. ad DI OREZO, E. : Solvcy of lf uac coma: mhodologcal u. Tchcal Ro 5. COCOZZ, R. ad DI OREZO, E. 3: Rk ofl of lf uac bu: a combd aoach Pocdg of h 6h Sah-Iala Mg o Facal Mahmac, Damo d Mamaca lcaa Buo d F, T, I, COCOZZ, R. ad DI OREZO, E. 5: Solvcy of lf uac coma: mhodologcal u. Joual of cuaal Pacc, fohcomg. 7. COCOZZ, R., DI OREZO, E. ad SIBIO M. 4a. Rk Pofl of f Iuac Bu: quaav aaly a maagal cv dl covgo Mod Mamac Sac l al d Da cuav Faza MF4 CUS, Salo, I, h:// 8. COCOZZ, R., DI OREZO, E. ad SIBIO M. 4b. Mhodologcal oblm olvcy am of a uac comay. Ivm Maagm ad Facal Iovao, Iu, COCOZZ, R., DI OREZO, E. ad SIBIO M. 4c. f uac k dcao: a balac h aoach Pocdg of 8h Iaoal Cog o Iuac: Mahmac & Ecoomc, Rom, I, h:// COPPO, M., DI OREZO, E. ad SIBIO, M. : Rk Souc a f uy Pofolo: Dcomoo ad Maum Tool. Joual of cuaal Pacc, 8, COPPO, M., DI OREZO, E. ad SIBIO, M. : Fuh Rmak o Rk Souc Maug h ca of a f uy Pofolo. Joual of cuaal Pacc,, COX, J.C., IGERSO, J.E. ad ROSS, S.. 985: hoy of h m ucu of h a. Ecoomca 53, DEESTR, G. ad PRER, G Covaac Equval Dcao Of Th CIRMod Pocdg of h V FIRColloquum, Rom, I, DE FEICE,M. ad MORICOI, M. 4. Mak bad ool fo maagg h lf uac comay. h:// DFM.df 39

14 Poblm ad Pcv Maagm / Volum 5, Iu, 7 5. DI OREZO, E. ad SIBIO, M.,. Th ogvy Rk: Maum ad lcao Pcv Poc. of h d Cofc cuaal Scc ad Fac o Samo, alova, h:// 6. FEES, E.W. 99. Sochac lf cogc wh olvcy codao. Ta of h Socy of cua, XII, HIRS, C.J., BESHM, D.J., BRYSO,.M., GEORGE, C.M., HRE, D. J.P., SMITH, D.. ad THOMPSO, S.. Fa valuao of labl. Iu of cua ad Faculy of cua, ovmb. 8. IIS SOVECY & CTURI ISSUES SUBCOMMITTEE. O olvcy, olvcy am ad acuaal u. IIS, Mach. 9. IIS. Gloay of m.. IIS SOVECY & CTURI ISSUES SUBCOMMITTE. Pcl o caal adquacy ad olvcy. IIS, Jauay.. ITERTIO CCOUTIG STDRDS BORD 3. Eou Daf 5 Iuac Coac. Iaoal ccoug Sadad Boad Comm Foudao, odo.. MES, J.. ad SGE,.P Ioduco o Pobably ad Sochac Poc. Pc-Hall, w Jy. 3. PRER, G Sochac aaly of h aco bw vm ad uac k. oh mca cuaal Joual, VRDERHOOF I. ad TM, E.I., Ed.. Th fa valu of Iuac Bu. luw cadmc Publh, Boo. 4