ࡱ> y{ xw}q`bjbjqPqP .::5;%(> ^^^r%%%8&"'\rCVx' f1"111111TTTTTTT$Wh#Z^U^31133U^^11UEEE3 ^1^1TE3TEERh^^S1~' @Κ;B%= ST4V0CV.SZDxZ SZ^S1h1JE:2<v2111UUE^111CV3333rrrd %rrr%rrr^^^^^^ MODELIRANJE MJERE RIZIKA Dominika Crnjac Faculty of Electrical Engineering in Osijek, J.J. Strossmayer University of Osijek, Trpimirova bb, HR 31000 Osijek, Croatia Tel.: +385 (0)31 224 600; Fax: +385 (0)31 224 605 e-mail:  HYPERLINK "mailto:dominika.crnjac@etfos.hr" dominika.crnjac@etfos.hr Sa~etak Rizik kao fenomen je latentna koli ina novca ili protuvrijednost koja je potrebna kao garancija. Pristup potencijalnim gubicima uzrokovanim razli itim uzro nicima ~elimo na neki esencijalni na in modelirati. Potrebno je odrediti granicu ako je u fokusu interesa sigurnost. Svrha ovog rada je uputiti na relevantnu literaturu i pokazati kako teoriju mjere kao matemati ku disciplinu mo~emo ugraditi u ekonomsku teoriju i time dati alat menad~erima rizika s kojim e povezati matemati ku i ekonomsku misao. Klju ne rije i: Mjera, mjere rizika, momenti, slu ajna varijabla, vjerojatnost, aproksimacija, varijanca, odlu ivanje, funkcija rizika Uvod Statisti ki gledano problem odlu ivanja se mo~e smatrati igrom dva igra a. Jedan igra  je stvarnost, a drugi statisti ar, pri emu je stanje stvarnosti koje ozna avamo s  EMBED Equation.DSMT4  nepoznato statisti aru. Skup svih stanja stvarnosti (prostor parametara) ozna imo sa  EMBED Equation.DSMT4 . Statisti ar (operater) poduzima akciju (odluku)  EMBED Equation.DSMT4 , ako je saznao da je stanje stvarnosti  EMBED Equation.DSMT4 . Neka je A skup svih akcija ili odluka. Rezultat promatranja je slu ajna varijabla X kojoj zakon vjerojatnosti  EMBED Equation.DSMT4  zavisi od nepoznatog parametra  EMBED Equation.DSMT4 . Ako je slu ajna varijabla X uzela vrijednost  EMBED Equation.DSMT4 , operater donosi odluku  EMBED Equation.DSMT4 , pri emu je  EMBED Equation.DSMT4 funkcija odluke. Sam postupak odlu ivanja se sastoji u slijedeem: Definira se skup svih moguih vrijednosti koje  EMBED Equation.DSMT4  mo~e uzeti u promatranom problemu. Definira se skup moguih akcija ili odluka, koje se mogu donijeti. Definira se funkcija odluke  EMBED Equation.DSMT4  slu ajnog uzorka  EMBED Equation.DSMT4  U ovoj igri operater (statisti ar) e imati dobit ili gubitak, ovisno o odluci, koja e zavisiti od  EMBED Equation.DSMT4  i  EMBED Equation.DSMT4 . U svrhu kvantitativnog mjerenja uvedimo funkciju gubitka  EMBED Equation.DSMT4  kao numeri ku funkciju, koja svakoj odluci  EMBED Equation.DSMT4  iz A i svakoj vrijednosti parametra  EMBED Equation.DSMT4  iz  EMBED Equation.DSMT4  pridru~uje broj  EMBED Equation.DSMT4  koji predstavlja gubitak. Korektnom odlukom nazivamo odluku za koju je gubitak jednak nuli. Jasno, funkcija gubitka  EMBED Equation.DSMT4  je slu ajna varijabla za koju o ekivana vrijednost  EMBED Equation.DSMT4  predstavlja rizik koji se dobiva odlukom  EMBED Equation.DSMT4  kada je stanje stvarnosti  EMBED Equation.DSMT4 . Na osnovu prethodnog uvedimo funkciju rizika EMBED Equation.DSMT4  kao o ekivanu vrijednost funkcije gubitka  EMBED Equation.DSMT4 . Pojavljuju se dva slu aja: Ako je  EMBED Equation.DSMT4  neprekidna slu ajna varijabla, tada je  EMBED Equation.DSMT4  Ako je  EMBED Equation.DSMT4  prekidna slu ajna varijabla, tada je  EMBED Equation.DSMT4  Nadalje, dvije funkcije odluke  EMBED Equation.DSMT4  i  EMBED Equation.DSMT4  mo~emo usporediti pomou odgovarajuih funkcija rizika  EMBED Equation.DSMT4  i  EMBED Equation.DSMT4 . Bolja je ona funkcija odluke pri kojoj funkcija rizika uzima manju vrijednost. Da bi doali do izbora funkcije odluke, prirodno je koristiti maksimalne vrijednosti funkcije rizika. Bayesovski pristup Bayesovski pristup je jednostavan i time interesantan s motriata upotrebljivosti u teoriji odlu ivanja. Stanje stvarnosti smatramo slu ajnom varijablom  EMBED Equation.DSMT4  sa zakonom vjerojatnosti  EMBED Equation.DSMT4  ato je rezultat osobnog uvjerenja operatera (statisti ara) o stanju stvarnosti. Budui da je  EMBED Equation.DSMT4  slu ajna varijabla biti e i funkcija rizika  EMBED Equation.DSMT4  slu ajna varijabla, pri emu je zdru~eni zakon vjerojatnosti  EMBED Equation.DSMT4 . U ovom slu aju o ekivana vrijednost funkcije rizika u odnosu na zakon vjerojatnosti a priori parametra  EMBED Equation.DSMT4  definira novu funkciju (Bayesov rizik)  EMBED Equation.DSMT4 . Prema 1) i 2) imamo respektivno I)  EMBED Equation.DSMT4  i II)  EMBED Equation.DSMT4 , zavisno o tome da li je neprekidna ili prekidna slu ajna varijabla. U ovom radu emo ovaj pristup implementirati u teoriju rizika s posebnim osvrtom na mjerenje rizika, ime emo optimizirati strategije sigurnosti. Naravno, funkcija sigurnosti ovisi o viae varijabli i njihovim ograni enjima i problem rizika je lociran na kona ni vremenski interval  EMBED Equation.DSMT4 . Interesantnu definiciju rizika je dao  EMBED Equation.DSMT4  koja je bez pretpostavke o potpunosti tr~iata, ali definira rizik kao buduu neprihvatljivu vrijednost u intervalu  EMBED Equation.DSMT4 . Pretpostavimo da je D skup prihvatljivih situacija, X slu ajna varijabla promatranih situacija, te da je  EMBED Equation.DSMT4  referentni povratni instrument. Rizik e se mjeriti kao minimalni dodatni kapital C, koji treba investirati u projekt da bi vrijednost novo nastale situacije C i +X bila prihvatljiva. Mjeru rizika mo~emo shvatiti kao preslikavanje  EMBED Equation.DSMT4   EMBED Equation.DSMT4 , pri emu prihvaamo sve zakonitosti iz matemati ke teorije mjere kao ato su nenegativnost, subaditivnost, translacija itd. Budui da je rije  o aproksimaciji mo~emo koristiti varijancu, matemati ki pojam esto upotrebljavan u statistici:  EMBED Equation.DSMT4  ili  EMBED Equation.DSMT4 , gdje je E matemati ko o ekivanje. Jasno, ovako definirane mjere nisu subaditivne. Nije jednostavno nai distribuciju vjerojatnosti, pa se susreemo s poteakoom opisa kroz familiju  EMBED Equation.DSMT4  moguih situacija.Uzimamo mjeru rizika kao o ekivani gubitak iz najnepo~eljnije situacije  EMBED Equation.DSMT4 , najmanje gornje ograni enje. Navedimo neke primjere mjera prethodno spomenutog tipa: Funkcija prosje nog viaka  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Retardirana mjera  EMBED Equation.DSMT4 , gdje se uvodi konkavna funkcija  EMBED Equation.DSMT4  sa svojstvima  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Opa teorija rizika je dobro obraena  EMBED Equation.DSMT4  uz obilato koriatenje stohasti ke teorije i teorije korisnosti. Rizik s obzirom na situaciju  EMBED Equation.DSMT4  je definiran kao mjera  EMBED Equation.DSMT4 , gdje je  EMBED Equation.DSMT4 , a domena je ograni ena sa f, a F funkcija distribucije. Model korisnosti prosje nog rizika postoji ako i samo ako egzistira realna funkcija H tako da je distribucija F po~eljnija od G, tj. ako i samo ako je  EMBED Equation.DSMT4  . Dinami ki pristup mjerama rizika Prethodno smo vidjeli kako se mjeri rizik u jednom periodu. Meutim, esto osiguravajui ugovori, investicije, itd. zahtijevaju du~i vremenski period, pa se vrijeme mora jasno uva~iti u definiciji rizika i modeliranju mjere rizika. Neka je  EMBED Equation.DSMT4  kona ni vremenski interval i neka je C kapital sa dospijeem t. U slu aju potpunog tr~iata pri emu nema arbitra~e, a postoji zainteresirani subjekt koji ne mo~e odmah, u vrijeme t=0 ulo~iti cijeli iznos, pa je  EMBED Equation.DSMT4 , ato garantira izvjesnu zaatitu.  EMBED Equation.DSMT4 *+Xi~   + , - .  <$bdvTxqqmqmmmimehyohhp= hp=hp=hMZhMZ5mHsHhd)hMZ0J5mHsHjhXhMZ5UjhMZ5Uhd)hMZ5mHsHhMZmHsHhd)hMZmHsH h`KhMZh!hMZB*nHphtHhMZhhMZ5 hMZ5 hp=5hp=hp=5'+X- . /   > dfjtv8x`xgdp=gdp=$a$gdMZ$a$gdMZ$a$gdp=Z68LN|~"siX!j8^H hyoCJPJUVaJjh0JUjhyohyoEHU!j]H hyoCJPJUVaJhjKhyohyoEHU!j]H hyoCJPJUVaJjhyohyoEHU!ji]H hyoCJPJUVaJhp=jhyohyoEHU!j!]H hyoCJPJUVaJhyojhyoU"8\`r Z \ #l#n#p#r#x#z#####%"%*%F%J%R%z&|&&&&&&&''淪榡wjY!jofH hfoCJPJUVaJjghfohfoEHU!jMfH hfoCJPJUVaJjhfoUh hfo5hfohfo5 h5hfojEdhL6hL6EHU!j-eH hL6CJPJUVaJj`hL6hL6EHU!jeH hL6CJPJUVaJhL6jhL6Uj1^hL6hL6EHUP$%%% %"%H%J%&'V)*&+(+~++T,V,|-.h0j01223455gdfox`xgdfo'''''( ( ((j(l((((())L)N)P)R)$*&*T*V*X*Z******淪晌{n]PjyhfohfoEHU!jhH hfoCJPJUVaJjwhfohfoEHU!jnhH hfoCJPJUVaJjBshfohfoEHU!jgH hfoCJPJUVaJjphfohfoEHU!j7gH hfoCJPJUVaJjgmhfohfoEHU!jfH hfoCJPJUVaJhfojhfoUj>jhfohfoEHU****4+6+d+f+h+j+++++++V,n,........//@/B/D/F/H/.0}l_}[hjjhhEHU!jXH hCJPJUVaJjhUhtj[hthtEHU!j{H htCJPJUVaJhjIhthtEHU!jiH htCJPJUVaJjN}hthtEHU!jiH htCJPJUVaJjhtUhfohtjhfoU!.000^0`0b0d0f0h00000<1>1l1n1p1r1122H3J3x3z3|3~333333333:4ͰpcjKhhEHU!jH hCJPJUVaJjNhhEHU!jlH hCJPJUVaJhtjhth0EHU!jA{H h0CJPJUVaJjh0Uhh0htjhthtEHU!j{H htCJPJUVaJhjhU#:4<4444555555555 6 66666667777777n8p88888鲥yudWjh.*'h.*'EHU!joH h.*'CJPJUVaJhMZjKh.*'h.*'EHU!jIoH h.*'CJPJUVaJh0jhMZ0JUjnh.*'h.*'EHU!jnH h.*'CJPJUVaJj֗h.*'h.*'EHU!jxnH h.*'CJPJUVaJjh.*'Uh.*'h' hjh0JU"5X66l8n88T9V99:H;J;;<z=>>?@@@@@@Bgd -$a$gd -gdl^gdl & Fgdlx`xgd.*'gdfo888T999999999::::@:B:p:r:t:v:::::::::;;@;qdjhlhlEHU!jcqH hlCJPJUVaJjhl0JUj`hlhlEHU!jqH hlCJPJUVaJjhlhlEHU!jpH hlCJPJUVaJj#hlhEHU!j3H hCJPJUVaJjhlUhljht0JUh.*' @;B;D;F;l;;;;;;;;"<$<R<T<V<X<==D=F=H=J=========,>բՄwfYjhlhlEHU!jrH hlCJPJUVaJjzhlhlEHU!jrH hlCJPJUVaJjFhlhEHU!jSH hCJPJUVaJjThlhlEHU!jqH hlCJPJUVaJhgZhljhlUjhlhlEHU!jqH hlCJPJUVaJ ,>.>0>2>>>>>?H@J@x@z@|@~@@@@@@@@AABBBBBB`CbCEûêǖ|x|p|_Rp|x|j1h -h -EHU!j|H h -CJPJUVaJjh -Uh0h - h -5h -h -5hfo hfo5 h.*'hg'jthg'h'fjEHU!j߲H h'fjCJPJUVaJjh'fjUh'fjhg'jhl0JUhljhlUjhlhlEHU!jsH hlCJPJUVaJ BtCD|ErdsttuFvdwwx~yxz2{J{}b`ށ< gd#2gd -EE2E4E6E8EbE|E~EEEEEr,s.s\s^s`sbssssstt t ttu.u0u󩥔viX!j!H h#2CJPJUVaJjh#2h#2EHU!jH h#2CJPJUVaJj=h#2h#2EHU!jH h#2CJPJUVaJh#2jh#2UUj$h -h -EHU!jm}H h -CJPJUVaJh'fjj@h -h'fjEHU!j H h'fjCJPJUVaJh -jh -U je cijena ne rizi ne investicije na tr~iatu, E o ekivanje grani nog rizika, C je rizik kojeg treba izmjeriti. Vrlo interesantan prijedlog su dali  EMBED Equation.DSMT4  Ako model ovisi o distribuciji vjerojatnosti, strategije  EMBED Equation.DSMT4 trebaju se odabrati s ciljem da se umanje o ekivane vjerojatnosti neto-gubitaka;  EMBED Equation.DSMT4  , pri emu je  EMBED Equation.DSMT4  po etni kapital, a  EMBED Equation.DSMT4  skup prihvatljivih situacija (strategija). Ova mjera je suglasna uz uvjet da su investicijska strategija  EMBED Equation.DSMT4  i obveza  EMBED Equation.DSMT4  proporcionalne. Ovaj stohasti ki problem je rijeaen pod odreenim pretpostavkama. Neizvjesnost je determinirana optimizacijom preko familije  EMBED Equation.DSMT4  strategija. U svrhu mogunosti kontroliranja rizika, odreujemo interval moguih mjera sa ograni enjima  EMBED Equation.DSMT4 , koje odgovara najnepovoljnijoj strategiji (donja mjera) i  EMBED Equation.DSMT4 , gornja mjera najgore strategije. Poznavajui zakone matemati ke analize mo~emo interval  EMBED Equation.DSMT4  podijeliti na podintervale  EMBED Equation.DSMT4 , gdje je  EMBED Equation.DSMT4  iz skupa prirodnih brojeva i uva~iti da je  EMBED Equation.DSMT4  promjena vrijednosti portfelja na intervalu  EMBED Equation.DSMT4 . Dakle, vremenski interval  EMBED Equation.DSMT4  ne ini viae cijelinu nego je podijeljen na podintervale, pa svaki podinterval promatramo posebno kao particiju  EMBED Equation.DSMT4   EMBED Equation.DSMT4 , pri emu je  EMBED Equation.DSMT4  povratni instrument. Ako je  EMBED Equation.DSMT4  osigurava investitoru ubla~avanje gubitka. Ako je  EMBED Equation.DSMT4  daje mogunost povla enja s ra una, a da se mo~e podnijeti o ekivani gubitak. Nije teako uo iti da je pomou realnih nizova  EMBED Equation.DSMT4  dan interval mjere rizika,  EMBED Equation.DSMT4   EMBED Equation.DSMT4  za koji vrijedi sve ato je dano slijedeim aneksima. Appendix A Neka je  EMBED Equation.DSMT4  realni niz i S skup svih njegovih to aka nagomilavanja (S mo~e biti i prazan skup). Gornjim limesom (limes superior, gornja to ka nagomilavanja) niza  EMBED Equation.DSMT4  naziva se element  EMBED Equation.DSMT4  skupa  EMBED Equation.DSMT4  koji se odreuje na slijedei na in:  EMBED Equation.DSMT4  Analogno se definira donji limes (limes inferior, donja to ka nagomilavanja) niza  EMBED Equation.DSMT4 . Gornji limes niza  EMBED Equation.DSMT4  ozna ava se simbolom:  EMBED Equation.DSMT4 , a donji limes istog niza simbolom:  EMBED Equation.DSMT4 . Gornji i donji limes su jednozna no odreeni (kao elementi skupa  EMBED Equation.DSMT4 ) za svaki realni niz i svaki od njih je to ka nagomilavanja niza u slu aju kad je kona an. Stoga, ako je  EMBED Equation.DSMT4 , imamo  EMBED Equation.DSMT4 , a ako je  EMBED Equation.DSMT4 , tada je  EMBED Equation.DSMT4 . esto se u slu aju kada je realni niz  EMBED Equation.DSMT4  neograni en s gornje, odnosno s donje strane, ka~e da je simbol  EMBED Equation.DSMT4 , odnosno  EMBED Equation.DSMT4 , njegova to ka nagomilavanja. Ako se ovo usvoji, tada se, na prirodan na in na skup  EMBED Equation.DSMT4  proairen poredak skupa  EMBED Equation.DSMT4 , gornji i donji limes mogu bez ograni enja definirati na slijedei na in:  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4 , a takoer i ovako:  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4 . Appendix B 1. Neka su  EMBED Equation.DSMT4  i  EMBED Equation.DSMT4  ( EMBED Equation.DSMT4 ) realni nizovi. Ako je za  EMBED Equation.DSMT4  dovoljno veliko  EMBED Equation.DSMT4  EMBED Equation.DSMT4  EMBED Equation.DSMT4 , tada je  EMBED Equation.DSMT4  i  EMBED Equation.DSMT4  Neka su  EMBED Equation.DSMT4  i  EMBED Equation.DSMT4  dva realna niza. Uz isklju enje slu ajeva kad se javljaju izrazi koji nemaju smisla, va~e slijedee nejednakosti:  EMBED Equation.DSMT4  Ako je joa  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  za dovoljno veliko  EMBED Equation.DSMT4 , tada, uz isto ograni enje kao prije navedeno, va~e nejednakosti  EMBED Equation.DSMT4  Za svaki realni niz  EMBED Equation.DSMT4  je  EMBED Equation.DSMT4 , Ako je  EMBED Equation.DSMT4 , tada je  EMBED Equation.DSMT4 , Gdje se, pored ve usvojene konvencije  EMBED Equation.DSMT4 , uzima da je  EMBED Equation.DSMT4 . Neka su  EMBED Equation.DSMT4  i  EMBED Equation.DSMT4  dva realna niza. Ako  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Umjesto zaklju ka Ovim radom je djelomi no dan matemati ki instrumentarij menad~erima koji mogu povezati matemati ku i ekonomsku misao. Eksplicitno je dana ocjena (interval) mjere rizika. Literatura P.Artzner-F.Delbaen-J.M.Eber-D.Heath, Coherent risk measures, Mathematical Finance, 1999, vol.9, pp. 203-228 E.Castagnoli, Qualche riflessione sull'utilit attesa, Proceed. 13th Italian Conference of the Association for Mathematics applied to Economic and Social Sciences, 1989., pp.993-995 J.Cvitanik-I.Karatzas, On dynamic measures of risik, Finance and Stochastics, 1999., vol. 3, pp.451-482 C.Fishburn, Mean-risk analysis with risk associated with below-target returns, The American Economic Review, 1977., vol.67, pp.116-126 5. P.R. Halmos, Measure Theory, Van Nostrand, Princeton, New Jersey, 1963. 6. Karlin S., Mathematical Methods and Theory in Games, Pergamon Press, London-Paris, 1959. 7. M.Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, 1985., University Press, Warszawa 8. J. von Neumann, O. Morgenstern: Theory of Games and Economic Behavior, Princenton University Press, 1953. 9. H.L. Royden, Real Analysis, Macmillan, New York, 1968. 10. Sarapa, N., Teorija vjerojatnosti, `kolska knjiga, Zagreb, (1988) 11. Stoller S.D., Operations Research: Process and Strategy, University of California Press, Berkley and Los Angeles, 1964. 12. J.L.Wirch-M.R.Hardy, A synthesis of risk measures for capital adequacy, Insurance: Mathematics and Economics, 1999., vol.25, pp. 337-347 13. Williams J.D., The Compleat stratgyst, being a primer on the theory of games of strategy, Mc Graw-Hill Book Co., New York, 1966.  Sarapa, N., Teorija vjerojatnosti, `kolska knjiga, Zagreb, (1988)  P.R. Halmos, Measure Theory, Van Nostrand, Princeton, New Jersey, 1963.  H.L. Royden, Real Analysis, Macmillan, New York, 1968.  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