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
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
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, 997. 9
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
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 4. 5.. 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
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
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
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 9 3.4. 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
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, 734-735: 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
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 3 4 5 6 7 8 9 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 3 4 5 6 7 8 9 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
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 3 4 5 6 7 8 9 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
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
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://www.ab.og/doc/d5/d5-cl9.df. 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, 9-7. 3. 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, 57-7. 6. 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, 8-86. h://www.labcooma.ua./maaalo 8. COCOZZ, R., DI OREZO, E. ad SIBIO M. 4b. Mhodologcal oblm olvcy am of a uac comay. Ivm Maagm ad Facal Iovao, Iu, 95-. 9. 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://www.mom4.com. COPPO, M., DI OREZO, E. ad SIBIO, M. : Rk Souc a f uy Pofolo: Dcomoo ad Maum Tool. Joual of cuaal Pacc, 8,. 43-6.. 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,,. 9-4.. COX, J.C., IGERSO, J.E. ad ROSS, S.. 985: hoy of h m ucu of h a. Ecoomca 53,. 385-47. 3. DEESTR, G. ad PRER, G. 995. Covaac Equval Dcao Of Th CIRMod Pocdg of h V FIRColloquum, Rom, I, 73-747. 4. DE FEICE,M. ad MORICOI, M. 4. Mak bad ool fo maagg h lf uac comay. h://www.mah.hz.ch/fac/f DFM.df 39
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://www.a.ucl.ac.b/samo/ocdsbllo.df 6. FEES, E.W. 99. Sochac lf cogc wh olvcy codao. Ta of h Socy of cua, XII, 9-9. 7. 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. 973. Ioduco o Pobably ad Sochac Poc. Pc-Hall, w Jy. 3. PRER, G. 997. Sochac aaly of h aco bw vm ad uac k. oh mca cuaal Joual, 55-84. 4. VRDERHOOF I. ad TM, E.I., Ed.. Th fa valu of Iuac Bu. luw cadmc Publh, Boo. 4