NOTE ON SIMPLE AND LOGARITHMIC RETURN

Transcription

1 Appled udes Agrbusess ad Commerce AAC Ceer-r ublshg House, Debrece DOI:.94/AAC/27/-2/6 CIENIFIC AE NOE ON IME AND OGAIHMIC EUN aa Mskolcz Uversy of Debrece, Isue of Accoug ad Face Absrac: I hs paper we descrbe ad clarfy he defos ad he usage of he smple ad logarhmc reurs for facal asses lke socks or porfolos. I ca be prove ha he dsrbuos of he smple ad logarhmc reurs are really close o each oher. Because of hs fac we vesgae he queso wheher he calculaed facal rsk depeds o he use of smple or log reurs. o show he effec of he reur-ype o he calculaos, we cosder ad compare he rskess order of socks ad porfolos. For our purposes, he emprcal sudy we use seve Hugara daly sock prces ad for he rsk calculao we focus o he followg rsk measures: sadard devao, semvarace, Value a sk ad Expeced horfall. he resuls clearly show ha he rskess order ca deped o he use of he reur ype (.e. log or smple reur). Geerally, ofe due o mssg daa or he aure of he aalyss oe has o use approxmaos. We also exame he effec of hese approxmaos o he rskess order of socks ad of porfolos. We foud dffereces he rskess order usg exac or approxmaed values. herefore, we beleve, f hs s possble, exac values sead of approxmaed oes should be used for calculaos. Addoally, s mpora ha oe uses he same ype of reur wh oe sudy ad oe has o be aware of he possble sables whe comparg reur resuls. Keywords: smple reur, logarhmc reur, rskess order, sock, porfolo (JE. Code: C8) INODUCION I he facal area of oday a mpora queso s: how oe defes ad measures he rsk of facal asses such as socks ad porfolos. Furhermore, s o eough oly o measure rsks hey also eed o be compared o help us o ake decsos o dffere facal quesos. Because of hese comparably reasos oe uses sead of he prces he reurs of a asse. I hs paper we wll be dealg wh he smple ad he logarhmc reurs. I s self-evde ad aural ha hese wo reurs are dffere from each oher. For example Hudso cosders hs relaoshp by comparg meas ad cocludes ha he mea of he logarhmc reurs s less ha he mea of he smple reur (compued o he same se of reurs) (Hudso, 2). We wll dscuss he possble correlaos ad dffereces bewee he wo reurs from a oher po of vew. For our purposes s mpora o udersad how he choce of he reur-ype effecs he rskess order of he cosdered se of asses. For example, do we cosder a sock respecvely a porfolo equally rsky (compare o he ohers) usg smple or logarhmc reurs o calculae he rsk. o aswer hs queso we wll do a emprcal sudy. he objecve of hs paper s o descrbe ad o clarfy he defos ad he usage of he smple ad logarhmc reurs. I he frs par of he heorecal backgroud we wll sae he defos of he oe- ad mul-perod smple ad log reur ad we wll descrbe he relaoshp bewee hem. hese defos wll be exeded o porfolos he secod par of he heorecal backgroud. he secod par of he sudy s he emprcal par. Frs, we would lke o cofrm pracce va usg Hugara sock daa mahemacal formulas, equaos ad resuls preseed he heorecal par. ecod, we wll aswer o our ma queso,.e. wheher usg smple or logarhmc reur could have a effec o our decso. EUN, HE HEOEICA BACKGOUND I hs heorecal par of he paper we summarze he mpora defos, expressos coeced wh smple ad logarhmc reurs ad we clarfy ad esablsh relaos bewee he wo oos. Defos of he followg chaper are based o say (say, 25) ad Calafore (Calafore, 24). AAC Vol.. Number pages IN aa Mskolcz

2 28 aa Mskolcz Asse eur Frs, we wll defe he smple ad logarhmc reur of a asse. I addo we wll show he mos mpora equaos ad expressos coeced wh he opc. a. mple eur he Oxford dcoary defes he reur as a prof o a vesme over a perod of me, expressed as a proporo of he orgal vesme. I he ex paragraphs we express reurs a more mahemacal framework. I he case of asse reurs le us cosder a me horzo [, ]. Furhermore be he prce of a asse a me ad he prce of a asse a me. If here s o cash flow hs [, ] me erval, we speak of he oe-perod smple e reur ad we roduce he oao [, ][]. o he oeperod smple e reur of a asse ca be defed by : = =. [, ][] ( ) he correspodg oe-perod smple gross reur of a asse s gve erms of he smple reur: Gr : = + =. ( 2 ) [ ][], [, ][] aer o f we speak of he smple reur we hk of he oe-perod smple e reur. For he mul-perod case, le us dvde he erval [, ] o par wse dsjo subervals: le : = be he h me po, : = be he las me po ad le be he me pos -bewee, such ha <, =,,. Accordg o our defo we ca calculae o hese subervals he oeperod smple gross reur: Gr[, ][] = + [, ][] = ad hus he reur o he whole [ ], erval mus be he produc of he gross reurs of he subervals. hs reur s called he -perod smple gross reur: 2 [, ][] [, ][] ( ) [, ][ ] Gr = + =,, = = : Gr. ( 3 ) = = We would lke o add ha sce = ad = he -perod smple gross reur equals he oe-perod smple gross reur: [, ][ ] Gr = Gr[, ][]. = = ( 4 ) Aalogously o he oe-perod case, we defe he -perod smple e reur by usg he -perod gross reur ad subracg oe: = ( 5 ) [,][ ]:. Wh Equao ( 3 ) ad ( 5 ) a had oe ca rewre he - perod smple gross reurs by: Gr [, ][ ] = = + [, ][ ]. = ( 6 ) = b. ogarhmc eur/couously Compouded eur o udersad he logarhmc reur, smply he log reur, le us dvde he erval [, ] o equdsa ervals. I hs paragraph we use he same oao as was roduced for he mul-perod smple reur. Assume ow, ha o every [, ] suberval he reur s he same, moreover ha s he h par of some oe-perod reur o [ ],, deoed by [ ] [],.e., Equao ( 3 ) ca be wre as follows: ( ) ( ) : =. I hs case, [, ][] Gr[, ][ ] = + [, ][] = + = + = ( 7 ) [, ][] ce ad are he h ad he las me pos respecvely, Equao ( 7 ) ca be wre as follows: Gr [, ][ ] = =, = ( 8 ) ad herefore usg Equao ( 7 ) ad Equao ( 8 ) holds ha = +. e us make he legh of he [, ] [, ][] ( 9 ) subervals smaller ad smaller. hs meas ha he umber of equdsa subervals of [, ] mus grow,. Hece we have o compue lms: lm = lm +, [, ][] ( ) ad herefore follows by he defo of he expoeal fuco ha [, ][] e =. ( ) AAC Vol.. Number pages IN

3 Noe O mple Ad ogarhmc eur 29 ce we are eresed reurs, we apply he logarhm: l = [, ][]. ( 2 ) he reur Equao ( 2 ) s called he oe-perod logarhmc reur of a asse. o, we defe he oe-perod log reur as he logarhm of he oe-perod smple gross reur ad we use he oao [ ] []:, ( ) [, ][] : = l = l + [, ][]. ( 3 ) mlarly o he smple reur s case, oe defes he - perod logarhmc reur: 2 [, ][ ] : = l,, l = = =. ( 4 ) = We ca see ha hs case he -perod log reur s he sum of he oe-perod log reurs. Ad hs s oe of he reasos why oe uses he log reur raher ha he smple reur: addg umbers close o zero s o a problem, bu mulplyg umbers close o zero ca cause arhmec overflow. I addo s easer o derve he me seres properes of sums ha of producs (Daíelsso, 2). Aalogously o he smple reur s case sce ad are he h ad he las me pos respecvely: [ ][], [, ][ ] =. ( 5 ) We would lke o add, ha more geerally o every erval oe ca calculae he reur. I hs sudy we wll use daly asse prces ad hus daly reurs. o, he cosdered me erval wll always be oe day. herefore, he oe-perod smple ad logarhmc reur ca be wre as follows: : = [, ][] = ( 6 ) ad : = [, ][] : = l. ( 7 ) aer hs sudy we wll use Equao ( 6 ) ad Equao ( 7 ) for he calculaos ad we wll speak of he reur a me po. Noe ha oe ca easly see he relao bewee he smple ad log reur: ad = e ( 8 ) ( ) = l +. ( 9 ) I ca be deduced usg a approxmao of he logarhm ha l ( + x) x, f x s ear o zero ha f he smple reur s ear o zero s addo very comparable o he log reur (proof follows jus by subsuo of x wh he smple reur): [, ][] = l [, ][], f [, ][] =. ( 2 ) orfolo eur I hs seco we wll focus o how o calculae he smple ad he logarhmc reur of a porfolo. We use he followg oao: : he umber of asses he porfolo : refers o he asses he porfolo, =,, : he amou of moey vesed he porfolo a me,, : he amou of moey vesed asse a me : he prce of asse a me w, : relave weghs of he asse porfolo a me k : umber of asse porfolo e us cosder a porfolo whch cosss of asses. Usg he oao above s aural ha he amou of moey vesed asse a me ca be expressed by = k = w, ( 2 ),,, ad he amou of moey vesed he porfolo a me s gve by = = k,, = =. ( 22 ) Wh equao ( 2 ) ad ( 22 ) had we ca express he relave weghs a me : w, k = =. ( 23 ),, k =, hese relave weghs chage me accordg o he asse prces. I hs sudy laer o, f we speak of weghs we always hk of hese relave weghs. Noe, ha he relave weghs sum up o oe: w, =. ( 24 ) = a. mple eur of a orfolo I hs seco we wll show how o calculae he smple reur of a porfolo (deoed by ). mlarly o he smple reur of a asse we ca defe he smple reur of a porfolo a me he ga (or loss) value of he porfolo relave o he sarg value, mahemacally (Baco, 2): : =. ( 25 ) AAC Vol.. Number pages IN

4 3 aa Mskolcz Usg he fac ha he weghs sum up o oe ad he equao,, =, where,, s he smple reur of asse a me, Equao ( 22 ) ca be rewre as,, (, ), (, ), (, )( 26 ) = k = k + = + = w + = = = = ad hus we ca express he smple reur of a porfolo a me by = = w. ( 27 ),, = We ca see ha he porfolo smple reur s he sum of he weghed smple reurs of he cosues of he cosdered porfolo. b. ogarhmc eur/couously Compouded eur of a orfolo he logarhmc reur of a porfolo (deoed by ) a me ca be defed aalogously o he logarhmc reur of a asse: : = l. ( 28 ) (, ) w w w o hs case, w, = j =, (( +, ) ), =,, = j= j! = =. ( 3 ) EMIICA UDIE he daa. ( 32 ) For he emprcal calculao we wll work wh Hugara daly sock prces bewee.7.25 ad he daa was dowloaded from he Budapes ock Exchage homepage ( We focus o seve socks, amely FHB, MO, MEEKOM, O, aergy, aba ad cher ad aalyze hem he meoed me erval. he mssg values were flled by he prevous day daa. o perform he aalyss we use he mahemacal sofware. We plo he sock prces (Fgure), whch shows ha prces cao be used for comparsos. Fgure: ock prces (gray: FHB, black: MO, red: MEEKOM, gree: O, purple: aergy, lgh blue: aba, magea: cher) Moreover usg he relao bewee logarhmc ad smple reur (see Equao ( 3 ) ad Equao ( 27 ) he logarhmc reur of a porfolo ca be calculaed he followg way: w w e, = l l,, l, = + = ( 29 ) = = s he log reur of asse a me. where, Uforuaely he log reur of a porfolo does o have a smlar covee propery as was developed Equao ( 27 ) for he case of he smple reur, so cao be wre as he sum of he weghed log reurs of he cosues of he cosdered porfolo. mlarly o he reur of a asse by usg he l ( + x) x approxmao oe ca show, ha f he smple reurs of a porfolo are close o zero he he smple reurs ad he log reurs of a porfolo are smlar o each oher: ( ) = l +. ( 3 ) Usg he assumpos ha he smple reurs are close o zero, ad he defo of he expoeal fuco oe ca everheless deduce he followg lear approxmao: Comparg smple ad logarhmc reurs We could see he asse ad he porfolo case ha f he smple reurs are close o zero he he smple ad log reurs are close o each oher. I he frs par of hs emprcal sudy we wll check hs heorecal fac pracce. he frs prce daa s from ad we cosder hem as prce daa a me =. he las oes are from ad we cosder hem as prce daa a me 267. Noe, ha he frs reurs ca be calculaed o he erval [ =, = 2] ad hey are deoed by 2, ad 2, respecvely, for all he seve socks ( =,, ). AAC Vol.. Number pages IN

5 Noe O mple Ad ogarhmc eur 3 able : Basc sascs of smple reurs: mmum, frs quarle, meda, mea, hrd quarle, maxmum FHB MO MEEKOM O aergy aba cher M. -, ,4975 -,85 -, ,8277 -,4989 -,979 s qu. -,34 -,768 -,99 -,354 -,7828 -,896 -,958 Meda Mea,4,83 -,7,272 -,8,435 -,62 3rd qu.,9742,833,8635,365,6466,8554,9654 Max.,232339,5583,23894,232639,49826,2893,94983 ource: ow calculao a. ock reurs Fgure 3. Comparg porfolo smple ad logarhmc reurs Frs we calculaed he daly smple ad logarhmc reurs of all he dvdual socks usg Equao ( 6 ) ad Equao ( 7 ). I order o show he resuls more clear we roduced wo oulers he case of he cher ad aergy socks (check he mmum values). he basc sascs are summarzed able ad able 2. From hese summares we ca clearly see ha boh cases he reurs are close o zero: he medas are zero ad he erquarle rages are relavely small. aer o hs sudy we wll use hs modfed daa. Comparg able ad able 2 oe ca say ha he dsrbuos of he smple ad logarhmc reurs are really close o each oher. b. orfolo reurs e us cosder a porfolo: We assume ha we ow a porfolo cossg of oe from all he seve socks,.e. k =, =,,7 (see oao he heorecal par). Frs we calculae he smple ad he log reurs of he porfolo usg Equao ( 27 ) ad Equao ( 28 ) respecvely. he values are summarzed box plos, see (Fgure 2). As we meoed he heorecal par, he log reur ad he smple reur should be smlar f he smple reurs are close o zero (see Equao ( 3 )). I (Fgure 2) we ca clearly see ha he case of our daa he smple ad he log reurs are close o zero excep oe ouler boh cases. hs meas ha he smple ad a log reur values are very close o each oher. hs cocluso could be cofrmed by akg a look a (Fgure 3), where he smple reur of he porfolo was ploed agas he log reur of he same porfolo. Excep oe ouler all he values are lyg o he 45 le. Fgure 2. Box plos of porfolo smple (lef) ad porfolo logarhmc reur (rgh) values Comparg rskess order From he fac, ha he dsrbuos of he sock smple ad logarhmc reurs are really close o each oher (see seco Comparg smple ad logarhmc reurs ) we may coclude ha does o deped o wheher we use smple or log reurs for he facal calculaos. We wll check hs assumpo usg dffere rsk measures ad usg he orderg mehod descrbed he roduco. We calculae four from he mos ofe ad wdely used rsk measures: he sadard devao, he semvarace, he Value a sk ad he Expeced horfall. Dealed descrpos of hs rsk measures oe ca fd for example Bugár (Bugár, 26) ad Embrechs (Embrechs, 25). I he ex sep we sae how o calculae hese rsk measures he case of a realzao of a radom varable. =,,, where r s he h reur (=,,) e r ( r r ) = ad r he average of hese reurs ( r = r ). a. adard Devao ( ) 2 r r = σ ( 33 ) ( r) AAC Vol.. Number pages IN

6 32 aa Mskolcz able 2: Basc sascs of logarhmc reurs: mmum, frs quarle, meda, mea, hrd quarle, maxmum FHB MO MEEKOM O aergy aba cher M. -,972 -, , , , , ,35382 s qu. -,44 -,837 -,95 -,34 -,7859 -,9 -,9553 Meda Mea -,245,75 -,34 -,89 -,62,26 -,767 3 rd qu.,9695,764,8597,353,6445,858,968 Max.,2894, ,2957,396,247,9739 ource: ow calculao b. emvarace () V r ( m{ r }) 2 r, = ( 34 ) c. Value a sk - Va (a α level) ( ) ˆ Vaα r F{ r r }( α), ( 35 ),, where { x,,x } ( ) { x x} Fˆ x : = s he emprcal dsrbu- o = fuco ad { x x } s he dcaor fuco of he se x x. { } d. Expeced horfall E (a α level) E α ( r) k r =, ( 36 ) k where k [ α ] max{ m m α, m } r s he h = = < N ad eleme he creasg order of he reurs r : r r2 r. he oly rsk measure whch sasfes he expeced properes (mooocy, subaddvy, posve homogeey, cash varace/raslao varace) s he Expeced horfall. Furher dscusso o hs opc for example (Acerb, 22) ad (Arzer, 999). a. ocks Frs we cosder he sock reurs ad we calculae he sadard devao, he semvarace, he Va ad he E values. hey are show (Fgure 4). he purple bars dcae he values calculaed usg smple reurs ad he blue bars dcae he values calculaed usg log reurs. We ca see ha he case of he semvarace ad Va (a boh α =,5 ad α =, levels) he order does o deped o he ype of reur. If we use he semvarace as a rsk measure he rskes sock s he cher, followed by aergy, O, FHB, MO, aba ad MEEKOM. I he case of Va, he rskes sock s he O, followed by he socks MO, FHB, cher, aba, aergy ad MEEKOM. A α =, level he order s he followg: O, FHB, MO, aba, aergy, MEEKOM, cher. I he case of sadard devao ad E he corary was observed: he order does deped o he ype of reur. Usg he smple reur for rsk calculao he O sock has he hghes sadard devao value. he O s followed by cher, aergy, FHB, MO, aba ad MEEKOM. If we use he log reur for rsk calculao, he he aforemeoed order chages: he rskes s he cher, followed by aergy, O, FHB, MO, aba, MEEKOM. For example, he O sock wha was he rskes usg smple reurs, he case of he log reur s oly o he 3rd place. e us cosder ow he E. A he level of 5%, usg smple reurs for he calculaos we go he followg order: O, FHB, MO, aergy, cher, aba, MEEKOM; whle usg log reurs he order chages as follows: O, cher, aergy, FHB, MO, aba, MEEKOM. We ca see, ha for example he cher sock s he secod rskes sock he case of usg log reurs, bu s jus o he 5h place he case of smple reurs. A he level of % he rskess order also dffere cocerg smple or log reurs. I he case of smple reurs he rskes asse s he O, followed by aergy, cher, FHB, aba, MO ad MEEKOM. I he case of logarhmc reurs he rskes asse s he cher, followed by aergy, O, FHB, aba, MO ad MEEKOM. hs calculao shows, ha despe he fac ha he smple ad log reurs are comparable, our assumpo, ha he resul does o deped o wheher we use smple or log reur seems o be o correc. We could show, ha he oly cohere rsk measure, he Expeced horfall, gves dffere rskess orders for he same socks depedg o wheher was calculaed usg smple or log reurs. Ad hs ca lead o dffere decsos. b. orfolos I he case of porfolos, we cosder seve porfolos, each of hese porfolos coss of sx dsgushg socks (we jus leave away oe of he seve socks) ad we calculae he rsk of all hese porfolos order o geerae a rskess order. o calculae he rsk we cosder he wo mos ofe used rsk measures: he Value a sk (see Equao ( 35 )) AAC Vol.. Number pages IN

7 Noe O mple Ad ogarhmc eur 33 Fgure 4. adard Devao, emvarace, Va ad E values calculaed usg smple (purple bars) ad logarhmc (blue bars) reurs ource: ow calculao ad he Expeced horfall (see Equao ( 36 )) a α =,5 ad α =, levels. he resuls are show able 3. he umbers he able show he rskess order of he porfolo calculaed by usg Va ad E case of smple reurs respecve log reurs a wo dffere alpha levels. Oe ca observe, ha Va s sable o boh levels, meag he order does o deped o he choce of reur. I he case of he E a α =, level smlarly o he Va he wo orders are he same. o he corary for α =,5 level: he frs ad he secod porfolo swched posos. Ad our decso ca be flueced by hs dffere rskess order. We ca clearly see from he resuls, ha o oly he ype of he reur, or he chose rsk measure bu also he level of alpha (gve he rsk measure) has a decsve effec o he order, ad hece o he decso. For example a α =,5 level he E measures orfolo7 s oe of he rskes porfolo. Bu, coras, a α =, level, orfolo7 s he leas rsky porfolo from hese seve porfolos. able 3. skess order of porfolos usg smple ad logarhmc reurs Va E alpha,5,,5, reur smple log smple log smple log smple log orfolo orfolo orfolo orfolo orfolo orfolo orfolo AAC Vol.. Number pages IN

8 34 aa Mskolcz Usg approxmaos I leraure oe ca regularly see ha he relave weghs are subsued by /, where s he umber of asses he porfolo. Oe reaso for hs could be ha oe was ha he weghs are cosa me, because he relave weghs are chagg me sce hey are calculaed from he prces (see Equao ( 23 )). Aoher argume for usg approxmao s ha pracce somemes oe does o kow he asse prces (for example, a smulao resul gves oly reurs). I he absece of he prces oe cao calculae he relave weghs ad he absece of he relave weghs s o possble o calculae he porfolo reur. I hs las par of our sudy we would lke o show usg our orderg mehod he effec of a approxmao o he rskess order. We wll cosder aga he porfolos whch were cosruced seco Comparg rskess order, orfolos. o calculae he smple reur of a porfolo we wll use equao ( 27 ), ad we wll approxmae he weghs here s oe ouler. Up o hs he po clouds are sll dsrbued alog he 45 le. We may coclude from hs, ha usg hese approxmaos we ca ge smlar resul ha usg o approxmaed, exac reur values. We would lke o aswer he followg quesos. Frs we wll check wheher he rskess order chages f we use approxmaed smple or approxmaed logarhmc reurs. ecod, we wll compare hese orders he case of approxmaed ad exac smple ad logarhmc reurs. o calculae he rsk we wll use aga he Va ad he E rsk measures a wo dffere α =, 5 ad α =, levels. Fgure 5. Approxmaos of porfolo smple ad logarhmc reurs wh /: w, =,, ad such ha we cosder a equally weghed porfolo. hs approxmao urs o exac equao f we cosder a porfolo whch cosss of same umber of all he asses ( k = kj, j,, j =,, ) ad he asse prces are he same. o, we wll use he followg approxmao for he porfolo smple reur: ˆ =,. ( 37 ) = I he case of log reurs we wll cosder wo dffere approxmaos. For he frs oe we use Equao ( 28 ) ad he same assumpo as before: we assume ha he weghs are / ( w =, =,, ). herefore: ˆ = l, e. ( 38 ) = For he secod approxmao we wll use Equao ( 3 ), whch s already a approxmao ad as a furher assumpo we cosder he weghs equal o / ( w =, =,, ), smlarly o he prevous oes. o, he secod approxmao for he porfolo log reur ca be expressed he followg way: ˆ =. ( 39 ) =, I (Fgure 5) we ca see how far away he approxmaed values are from he exac values. he frs plo shows he approxmaed values calculaed usg Equao ( 37 ), he mddle oe shows he approxmaed values evaluaed usg Equaos ( 38 ) ad he hrd plo shows he approxmaed values calculaed usg Equao ( 39 ). I all he hree cases he resuls are show (Fgure 6)., 2,, 7 dcae he seve dffere porfolos. he purple bars sad for he rsk calculaed usg he approxmaed smple reur daa (see Equao ( 37 )), he orage ad blue oes for he rsk calculaed usg he approxmaed log reur daa (see Equao ( 38 ) ad Equao ( 39 ) respecvely). I he lef colum of he fgure we ca see he Value a sk values ad he rgh colum we ca see he E values. mlarly o he prevous cases he Va seems o be more sable, sce he order does o deped o wheher we use smple or log reur. A α =, 5 level he rskes porfolo s 5 followed by 7, 3, 6,, 2 ad 4. Ad a α =, level we calculaed he followg order: 7, 3, 5, 2, 6,, ad 4. hese orders are he same usg approxmaed smple or oe of he log reur daa. If we ake a look a he E values we ca see ha here he order of he porfolos chages depedg o he ype of used approxmao mehod. A α =,5 level we go he same order as he case of he approxmaed smple reur ad he frs approxmao of he log reur (see Equao ( 38 )), amely: 5, 3, 7,6,, 2, 4. Bu hs s dffere from he order whch we ge f we use he secod approxmao of he log reur (see Equao ( 39 )), ha s: 3, 5, 6,, 2, 7, 4. hese resuls are also show he las hree colums of able 4 AAC Vol.. Number pages IN

9 Noe O mple Ad ogarhmc eur 35 ad able 5, where ad deoe he smple ad logarhmc reurs respecvely, whle ˆ deoes he approxmaed smple reurs, ad ˆ (see Equao ( 38 )) ad ˆ 2 (see Equao ( 39 )) he approxmaed log reur. Fgure 6. Va ad E values for seve porfolos usg approxmaed daa ource: ow calculao able 5. Order of he porfolos usg he rsk measure E o he level of alpha=5% ad alpha= %. alpha,5, E Fally le us exame wheher he ype of daa used has a effec o he order,.e. wheher approxmaed or exac daa. I able 4 we ca see he resuls usg rsk measure Va ad able 5 we ca see he resuls usg rsk measure E. he resul clearly shows a oally dffere rskess order o all he cases. For example a 5% level he Va raked orfolo2 o he secod place usg o approxmaed daa ad s o he sxh place whe measured wh approxmaed daa. I s smlar he case of E: orfolo2 s he secod or frs place (depedg o he ype of he reur) usg o approxmaed daa bu coras he porfolo s o he ffh or sxh place usg approxmaed daa. Or orfolo5 s o oe had he less rskes porfolo f we calculae E a alpha=,5 level from he exac smple or logarhmc reur, bu o he oher had s he rskes porfolo f we calculae he E a alpha=,5 level from approxmaed daa. mlar resuls have bee foud o he level of alpha=,. For example he case of Va orfolo5 s o he sxh place f he value calculaed from o approxmaed daa ad o he hrd place f he E s calculaed from approxmaed reurs. he E s less sable. Depedg o he ype of reur or wheher we use approxmao he order ca vary srogly, see for example orfolo4 or orfolo5. reur able 4. Order of he porfolos usg he rsk measure Va o he level of alpha=5% ad alpha= %. Va alpha,5, reur ˆ ˆ ˆ 2 ˆ ˆ ˆ 2 orfolo orfolo Va alpha,5, orfolo orfolo orfolo orfolo orfolo orfolo orfolo2 orfolo3 orfolo4 orfolo5 orfolo6 orfolo7 ˆ ˆ ˆ 2 ˆ ˆ ˆ ource: ow calculao UMMAY AND CONCUION I hs sudy our goal was o clarfy he oo of smple ad logarhmc reur ad o show he dffereces ad he coecos bewee hem. I he heorecal par we saed he defos of he oe- ad he mul-perod smple ad logarhmc reurs. Equaos - preseed he sock case show, ha he logarhmc reur has a advaage agas he smple reur, amely ha he mul-perod logarhmc reur ca be calculaed as a sum of he oe-perod logarhmc reurs, whle he mul-perod smple reur s he produc of he oe-perod smple reurs, whch ca lead o compuaoal problems for values close o zero. I he case of a porfolo s mpora o hghlgh, ha he porfolo weghs deped o he prce of socks he porfolo. o hey chage me. I he case of a equally weghed porfolo oe has o balace regularly he porfolo. I s also mpora o oe, ha he smple reur of a porfolo s he sum of he weghed smple reurs of he cosues of he cosdered porfolo. I coras, he logarhmc AAC Vol.. Number pages IN

10 36 aa Mskolcz reur of a porfolo ca oly be approxmaed by he sum of he weghed logarhmc reurs of he cosues of he cosdered porfolo. I addo we could see, ha f he smple reur values are close o zero, he he dsrbuo of he smple ad logarhmc reurs are very ear o each oher. hs rases he queso wheher he used reur-ype (.e. smple or log reur) has a effec o he calculaos ad hus o he resuls. I he emprcal par of our sudy we waed o aswer hs queso. We were eresed wheher he used reurype he calculaos resuls a dffere rskess order. Frs we compared he order he case of he socks. We foud ha whle he case of semvarace ad Va he order does o deped o he ype of reur, he case of sadard devao ad E does. Afer he socks we cosdered porfolos. x dffere porfolos were compared ad ordered accordg o her rsks. he resul of our calculao shows, ha he Va does o deped o he use reur-ype, bu he case of he E we go dffere orders he wo cases. Furhermore, we vesgaed wha s he effec o hs order f we use approxmaed reur values for example we cosdered equal weghs, whch s commo pracce - sead of he exac values. We have foud every case dffere rskess orders, somemes eve serous dffereces. herefore, we beleve, f hs s possble, exac values sead of approxmaed oes should be used for calculaos. I summary, eve hough he wo reur-ype values are very smlar, s o ecessary ha he rskess orders are he same. I s mpora ha oe uses he same ype of reur wh oe sudy ad oe has o be aware of he possble sables whe comparg reur resuls. EFEENCE Acerb, C., ad asche, D. Expeced shorfall: a aural cohere alerave o value a rsk. Ecoomc oe, 22, 3, 2, Acerb, C., ad asche, D. O he coherece of expeced shorfall. Joural of Bakg & Face 22, 26, 7, Arzer,., Delbae, F., Eber, J.-M., ad Heah, D. Cohere measures of rsk. Mahemacal face 999, 9, 3, Baco, C.. (2): raccal porfolo performace measureme ad arbuo, volume 568. Joh Wley ad os. Bugár, Gy és Uzsok, M. Befekeések kockázaáak mérése. aszka zemle 26, 84, 9. Calafore, G. C., ad El Ghaou,. Opmzao models. Cambrdge uversy press, 24. Daíelsso, J. (2): Facal rsk forecasg: he heory ad pracce of forecasg marke rsk wh mplemeao ad Maab, volume 588. Joh Wley & os. Embrechs,., Frey,., ad McNel, A. Quaave rsk maageme. rceo eres Face, rceo, 25. Hudso, ; Gregorou, A. Calculag ad comparg secury reurs s harder ha you hk: A comparso bewee logarhmc ad smple reurs. Ieraoal evew of Facal Aalyss, Elsever, 2, 38(C), pages:5-62. Oxford Dcoary, hps://e.oxforddcoares.com/defo/ rae_of_reur say,.. (25): Aalyss of facal me seres, vol Joh Wley & os. AAC Vol.. Number pages IN