DEA-Risk Efficiency and Stochastic Dominance Efficiency of Stock Indices *

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1 JEL Classfcaon: C61, D81, G11 Keywords: Daa Envelopmen Analyss, rsk measures, ndex effcency, sochasc domnance DEA-Rsk Effcency and Sochasc Domnance Effcency of Sock Indces * Marn BRANDA Charles Unversy n Prague, Faculy of Mahemacs and Physcs, Deparmen of Probably and Mahemacal Sascs (branda@karln.mff.cun.cz) Mloš KOPA Insue of Informaon Theory and Auomaon, Academy of Scences of he Czech Republc (kopa@karln.mff.cun.cz) Absrac In hs arcle, we deal wh he effcency of world sock ndces. Bascally, we compare hree approaches: mean-rsk, daa envelopmen analyss (DEA), and sochasc domnance (SD) effcency. In he DEA mehodology, effcency s defned as a weghed sum of oupus compared o a weghed sum of npus when opmal weghs are used. In DEArsk effcency, several rsk measures and funconals whch quanfy he rsk of he ndces (var, VaR, CVaR, ec.) as DEA npus are used. Mean gross reurn s consdered as he only DEA oupu. When only one rsk measure as he npu and mean gross reurn as he oupu are consdered, he DEA-rsk effcency s relaed o he mean-rsk effcency. We es he DEA-rsk effcency of 25 ndces and we analyze he sensvy of our resuls wh respec o he seleced npus. Usng sochasc domnance crera, we es parwse effcency as well as porfolo effcency, allowng full dversfcaon across asses. Whle SD parwse effcency esng s performed for frs-order sochasc domnance (FSD) as well as for second-order sochasc domnance (SSD), he SD porfolo effcency es s consdered only for he SSD case. Our numercal analyss compares he resuls usng wo sample daases: before- and durng-crss. The resuls show ha SSD porfolo effcency s he mos powerful effcency creron, ha s, classfes only one ndex as effcen, whle FSD (SSD) parwse effcency ends o be very weak. The proposed DEA-rsk effcency approach represens a compromse offerng a reasonable se of effcen ndces. 1. Inroducon In he heory of decson-makng, quesons abou how o maxmze prof and how o dversfy rsk have been around for cenures; however, boh of hese quesons ook anoher dmenson wh he work of Markowz (1952). In hs work, Markowz denfed he wo man componens of porfolo performance mean reward and rsk represened by varance and by applyng a smple paramerc opmzaon model he found he opmal rade-off beween hese wo componens. If he parameer s known one can easly fnd he opmal porfolo. If no, a leas he se of effcen porfolos (he effcen froner) can be denfed. In hs case, he porfolo s seen as effcen f here s no beer porfolo,.e., a porfolo wh a hgher mean and smaller varance. In he las 60 years, he heory of mean-rsk models has been enrched by usng oher rsk measures nsead of varance, for example, sem-varance (see Markowz, 1959), Value a Rsk (VaR), and Condonal Value a Rsk (CVaR) (see Rockafellar and Uryasev, 2000, 2002). * Acknowledgemens: The research was parly suppored by he Czech Scence Foundaon under grans P402/10/1610, P402/12/ Fnance a úvěr-czech Journal of Economcs and Fnance, 62, 2012, no. 2

2 Alernavely, one can adop uly funcons for modelng an nvesor s rsk aude, especally n he approach maxmzng expeced uly. Agan, f he uly funcon s perfecly known, one can fnd he opmal decson. If ha s no he case, one can denfy a leas he se of effcen porfolos wh respec o a chosen se of uly funcons. Consderng all uly funcons, ha s, assumng only non-saaon for he nvesor s preferences, leads o he frs-order sochasc domnance (FSD) relaon (see Levy, 2006, and references heren). Addng he very common assumpon of rsk averson, one can oban he second-order sochasc domnance (SSD) approach. Usng parwse comparsons, an alernave (asse) s classfed as FSD (SSD) effcen f here s no oher alernave ha domnaes he former alernave wh respec o FSD (SSD). If nvesors may combne asses n porfolos, ess for porfolo effcency allowng full dversfcaon across he ndces can be of neres. Because of compuaonal consderaons, we lm our aenon o he SSD porfolo effcency ess developed n Pos (2003), Kuosmanen (2004), Kopa and Chovanec (2008), and Kopa (2010). These ess classfy an asse as SSD effcen f here s no porfolo creaed from he asses ha SSD domnaes. Alernavely, one can apply FSD effcency ess as n Kuosmanen (2004) and Kopa and Pos (2009), or sascal SSD effcency ess as n Scalle and Topaloglou (2010). A hrd approach o effcency s based on Daa Envelopmen Analyss (DEA) see, for example, Charnes e al. (1978) and Banker e al. (1984). For example, Basso and Funar (2001, 2003), Murh e al. (1997), Darao and Smar (2006), and Galagadera and Slvapulle (2002) appled DEA models o muual fund performance analyss. A fund s classfed as DEA effcen f he npus of he fund are accurae o s oupus. The mean (gross) reurn s usually consdered as he oupu, and oher fund characerscs, such as ransacon coss and rsk measures, serve as he npus. DEA models provde a popular ool for effcency measuremen even n oher recen applcaons, e.g. Cook and Zhu (2010), Roháčová (2011), and Průša (2012). In hs paper we apply all of he above-menoned approaches o effcency analyss of sock ndces. We consder 25 world sock ndces as he basc asses. We compare effcen ndces seleced accordng o dfferen crera: mean-varance effcency, DEA effcency, and FSD and SSD effcency usng parwse comparsons, as well as SSD porfolo effcency. An ndex ha s classfed as effcen wh respec o one of he crera can be aracve for any nvesor who uses he correspondng creron for modelng hs rsk aude. Moreover, f some ndex s effcen wh respec o more han one creron, a larger group of decson-makers s neresed n. In he deal case, we would lke o fnd a leas some ndces ha are classfed as effcen when usng all of he crera consdered. However, hs requremen proved o be very src. We emprcally examne he power of he effcency approaches consdered: he smaller an effcency se s, he more powerful s he creron consdered. We consder wo daases of weekly reurns: before-crss (Sepember 2006 Sepember 2008) and durng-crss (Sepember 2008 Sepember 2010). Whle mean-varance or FSD (SSD) effcency ess are gven precsely, he DEA effcency model can be consruced n varous ways. Conrary o Basso and Funar (2001) and Murh, Cho, and Desa (1997), we do no consder he ransacon coss conneced wh buyng or sellng he ndces, because we rely only on he mean and rsk characerscs of he ndces reurns. We choose varous rsk measures as npus, sarng from he sandard devaon up o he modern ones: VaR, CVaR, and Fnance a úvěr-czech Journal of Economcs and Fnance, 62, 2012, no

3 correspondng drawdown measures; moreover, he probably rsk measures.e., VaR, CVaR, and drawdown measures are used a several probably levels. The only oupu consdered s mean gross reurn. These DEA-rsk models classfy an ndex as effcen f he oal rsk s accurae o s mean gross reurn, where he oal rsk s descrbed by a lnear convex combnaon of he rsk measures consdered. The dea of a combnaon of rsk measures corresponds o rsk shapng (see, for example, Cheklov e al., 2005), where weghed values of a gven rsk measure a dfferen levels are consdered. The weghs are gven by he decson-maker. On he oher hand, n he DEA-rsk model, more han one rsk measure s consdered and he weghs are specfed by he opmzaon problems, as s usual n DEA models. Moreover, f only one npu s consdered, hen DEA-rsk effcency mples meanrsk effcency wh respec o he same rsk measure. For example, f he varance of an ndex s used as he npu, hen he DEA-rsk effcen ndex s always meanvarance effcen, oo. Therefore, DEA-rsk models can be seen as a generalzaon of mean-rsk models. In he models proposed n Basso and Funar (2001) and Murh e al. (1997), he weghs of he rsk measures had o be hgher han a small consan,.e., hey always nfluence fund effcency. In our DEA-rsk models, any rsk measure can be elmnaed mplcly f he opmzaon model deermnes zero wegh as opmal. Ths can gve us nformaon abou whch measures have he greaes or leas effec on he DEA effcency of world sock ndces. In he emprcal par of he paper, he DEA effcen ndces are compared o hose when usng mean-rsk and sochasc domnance crera. In addon, we sudy he sensvy of he DEA-rsk resuls o ncludng or excludng a sngle rsk measure or a whole group of measures. Moreover, he sochasc domnance analyss ncludes four dfferen effcency approaches: he wo mos popular orders (FSD and SSD) and he wo mos frequen choces of comparsons (parwse and porfolo effcency). Fnally, wo dfferen perods are consdered n he emprcal sudy. An neresng queson s how he se of effcen ndces dffers for pre-crss daa from ha for durng-crss daa usng varous effcency approaches. The remander of hs paper s srucured as follows. Secon 2 defnes he rsk measures consdered. I s followed (Secon 3) by DEA-rsk effcency formulaons usng rsk measures as npus and mean gross reurn as he oupu. Secon 4 recalls he basc deas of he FSD and SSD approaches and presens he parwse effcency ess. Secon 5 generalzes he prevous SSD resuls, allowng porfolo effcency esng wh full dversfcaon across he ndces. Secon 6 presens an emprcal applcaon comparng several ypes of effcency: mean-varance effcency, DEArsk effcency, FSD and SSD parwse effcency, and SSD porfolo effcency. Secon 7 concludes. 2. Rsk Measures In hs secon, we wll show how rsk measures can be compued based on dscreely dsrbued reurns. We consder a random vecor r ( r1r 2 rn ) of reurns of N ndces wh a dscree probably dsrbuon descrbed by T equprobable scenaros. The reurns of he ndces for he varous scenaros are gven by: 108 Fnance a úvěr-czech Journal of Economcs and Fnance, 62, 2012, no. 2

4 where x x 1x 2 x N along he -h scenaro. Tha s, 1 x x2 T x s he -h row of marx, represenng he ndex reurns x, 1 T, 1 N, s he -h realzaon of P he -h ndex reurn. I can be compued as 1 x 1, where P P are P 1 he prces of he -h ndex a he end of me perods and ( 1), respecvely. Then he lower semdevaon of order p for he -h ndex reurn can be compued as T x T 1 T 1 lsd( p) x x T 1 1 where x and [] mn{ 0}. Followng Rockafellar and Uryasev (2002), we wll defne Value a Rsk and Condonal Value a Rsk. Frsly, we can sor he realzaons of he -h ndex reurn n descendng order, ha s, p 1 p [1] [2] [ T ] x x x. Secondly, for (0 1), we fnd he unque ndex sasfyng 1 T T Then he Value a Rsk (VaR) of he -h ndex s defned as VaR ( ) x There exs several possble defnons of Condonal Value a Rsk see Pflug (2000) and Rockafellar and Uryasev (2000, 2002). If 11 T, hen he Condonal Value a Rsk of he -h ndex s equal o [ T ] ( ) ( ) CVaR VaR x or else can be compued as a weghed average T 1 1 [] CVaR( ) x x 1 T T 1 Condonal Value a Rsk can be also compued usng he mnmzaon formula nroduced by Pflug (2000) and Rockafellar and Uryasev (2000, 2002). We propose only he formula for he dscree dsrbuon consdered: T 1 CVaR ( ) mn y x y R y (1 ) T 1 Fnance a úvěr-czech Journal of Economcs and Fnance, 62, 2012, no

5 where [] max{ 0}. As he lef pon of he compac nerval of opmal soluons we oban he Value a Rsk a level. Denoe by c 1 T he uncompounded cumulave rae of he -h ndex reurn: c x 1 T k 1 The absolue drawdown a he end of perod s defned as k k max 1k AD c c The absolue drawdown funcon compares he curren rae of reurn wh he up-o-dae mnmum value of he reurns. When he reurn drops below he mnmum value, he drawdown funcon s he mrror mage of he reurn unl reurns o he mnmum value. Fnally, he Drawdown a Rsk (DaR) of he -h ndex can be compued as he VaR on equprobable absolue drawdowns of he -h ndex, and he Condonal Drawdown a Rsk (CDaR) of he -h ndex as he CVaR defned on equprobable absolue drawdowns of he -h ndex. Precse defnons can be found n Cheklov e al. (2003). 3. Daa Envelopmen Analyss Daa Envelopmen Analyss (DEA) was nroduced by Charnes e al. (1978) as a way o sae he effcency of a decson-makng un over all oher decsonmakng uns. Le Z 1 ZK denoe he npus and Y 1 Y L denoe he oupus of un from he N uns consdered. The DEA effcency of un I s hen evaluaed usng he opmal value of he followng program, where he weghed npus are compared wh he weghed oupus: max L l 1 li K k 1 yy w Z L l1 li K k1 yy li w Z l k 1 1 N w 0k 1 K y 0l 1 L li s Un I s hen DEA effcen f he opmal value s equal o 1, oherwse s DEA neffcen. The program can be rewren as a lnear program based on fraconal programmng reformulaon (see Charnes e al., 1978): max L yy li li l1 s 110 Fnance a úvěr-czech Journal of Economcs and Fnance, 62, 2012, no. 2

6 K K k 1 k li l k1 l1 L w Z 1 w Z yy 0 1 N w 0k 1 K y 0l 1 L li In he numercal comparson of world sock ndces we wll consder several rsk measures as npus and mean gross reurn as he oupu. Ths specal case of DEA effcency can be called DEA-rsk effcency. The combnaon of mean, rsk measures, and DEA models can also be found n Lozano and Guerrez (2008). Conrary o ha approach, we consder several rsk measures jonly n one DEA model. 3.1 DEA-Rsk Effcency Le us consder K values of rsk measures 1 K as npus Z 1 ZK and he mean gross reurn 1x as oupu Y of he -h ndex. We say ha ndex I s DEA-rsk effcen f he opmal objecve value of he followng problem: max y I I s K k 1 w K 1 w k yi 0 1 N (1) k 1 w 0k 1 K y 0 I s equal o 1. In he specal case where only one ype of rsk measure (for example, CVaR) bu a dfferen levels α s consdered, DEA-rsk modelng s relaed o he rsk-shapng approach (see, for example, Rockafellar and Uryasev, 2002, or Cheklov e al., 2005). The man dfference s n he choce of weghs. Our model allows anoher wegh scheme, w, for each ndex. The choce s always done n he opmal way, whch leads o measuremen of he rsk of ndex I usng he bes combnaon of he rsk measures consdered. Example 1 In hs example we consder only one rsk measure. We wll show he basc propery of a DEA-rsk effcen ndex. Moreover, we wll prove ha he DEA-rsk effcen ndex s always mean-rsk effcen, oo. Le be he value of he rsk measure consdered and be he mean reurn of he -h ndex for 1 N. Then we say ha ndex I s mean-rsk effcen f here s no ndex such ha I and I wh a leas one src nequaly. In hs seng, he defnon of he lnear problem (1) smplfes o: Fnance a úvěr-czech Journal of Economcs and Fnance, 62, 2012, no

7 whch can be easly rewren as: max y I I s w I I 1 w y 0 1 N I I w 0 y I I 0 max y I I s yi 1 N (2) I y 0 Snce he opmal soluon of (2) s 1 mn y I I he ndex I s DEA-effcen f has a mnmal rsk-mean rao, ha s, he rao of he value of he consdered rsk measure and mean reurn s mnmal. Assume now ha ndex I s no mean-rsk effcen. Then here exss anoher ndex wh a lower rsk-mean rao. Hence he opmal objecve value of (2) s srcly smaller han 1, ha s, ndex I s no DEA-effcen. Summarzng, we have shown ha DEA-rsk effcency wh only one npu always mples mean-rsk effcency. 4. Sochasc Domnance Relaons respec o he frs-order sochasc domnance (FSD) f j Sochasc domnance s an appealng approach for comparng random varables. In our case we apply o compare random sock ndex reurns. Followng Levy (2006) and references heren, he -h ndex domnaes he j-h ndex wh Eu r Eu r for all uly funcons wh src nequaly for a leas one uly funcon. If he same propery s sasfed for all concave uly funcons wh src nequaly for a leas one such uly funcon, hen he -h ndex domnaes he j-h ndex wh respec o second-order sochasc domnance (SSD). I s clear ha an FSD relaon mples an SSD relaon. Le F ( x) denoe he cumulave probably dsrbuon funcon of he reurns of he -h ndex. The wce cumulave probably dsrbuon funcon of he reurns of he -h ndex s defned as: F (2) () F( x)dx Then he FSD and SSD relaons can be alernavely defned as follows: he -h ndex domnaes he j-h ndex wh respec o FSD f I 112 Fnance a úvěr-czech Journal of Economcs and Fnance, 62, 2012, no. 2

8 F() F () R wh src nequaly for a leas one R ; he -h ndex domnaes he j-h ndex wh respec o SSD f (2) (2) j j F () F () R wh src nequaly for a leas one R. 4.1 SD Parwse Effcency In hs secon, we frs formulae a compuaonally aracve algorhm of FSD parwse effcency esng. Then we presen a modfcaon for SSD parwse effcency. Followng Levy (2006), le v, 12 T 1 2 T denoe he ordered reurns of he -h ndex n ascendng order, ha s, v v v. Then he -h ndex domnaes he j-h ndex wh respec o frs-order sochasc domnance f and only f j v v 12 T (3) wh a leas one src nequaly. Moreover, we classfy he j-h ndex as FSD parwse neffcen f here exss some -h ndex sasfyng (3). Oherwse, he j-h ndex s FSD parwse effcen. Therefore, he algorhm for esng he FSD parwse effcency of he j-h ndex consss of wo seps. Frsly, we order he reurns n ascendng order v for all 12 N, s 12 T. Secondly, we ry o fnd some sasfyng (3). If such exss hen he j-h ndex s SSD parwse neffcen. If no, hen he j-h ndex s FSD parwse effcen. Tesng of SSD parwse effcency s performed n a smlar way o he prevous algorhm usng creron s s j 1 1 v v s 12 T nsead of (3). Moreover, we use he smple fac ha FSD parwse neffcency mples SSD parwse neffcency. Or, equvalenly, an SSD parwse effcen ndex s always FSD parwse effcen, oo. 5. Sochasc Domnance Porfolo Effcency Conrary o he prevous case, an nvesor may combne ndces no hs for a vecor of porfolo weghs, and porfolo. We wll use 1 2 N he porfolo possbles are gven by N R 1 1 0n 12 N For compuaonal reasons, we lm our aenon only o he SSD case, ha s, we defne SSD porfolo effcency wh respec o all possble porfolos ha can be creaed from he ndces. The j-h ndex s SSD porfolo neffcen f here exss porfolo such ha r domnaes rj by SSD. Oherwse, he j-h ndex s SSD porfolo effcen. Smlarly o how we defned he SSD porfolo effcency for n Fnance a úvěr-czech Journal of Economcs and Fnance, 62, 2012, no

9 he j-h ndex, we can defne he SSD porfolo effcency of any porfolo, usng r nsead of r j. The followng SSD porfolo effcency ess are formulaed for hs more general seng. However, one could easly consder (10 0), (01 0),..., (001) o es he effcency of he ndces. 5.1 SSD Porfolo Effcency Tes The followng SSD porfolo effcency es was nroduced by Kopa and Chovanec (2008). Alernavely, one can use he SSD porfolo effcency ess developed n Pos (2003) or Kuosmanen (2004). Le k k T, kk 01 T 1. Le CVaR D ( ) max T 1 Dkn k0 D k s ( r) CVaR ( r) D kk k k k D 0 k K k Usng lnear programmng reformulaon of CVaR (see Rockafellar and Uryasev, 2002), we can compue he measure of neffcency D ( ) as follows: CVaR k1 T D ( ) max T Dknbk wk k 1 D k s 1 ( r ) b w D kk T k k 1 k k 1 T T 1 k w x b k K k w 0 kk k D 0k K k If D ( ) 0, hen s SSD porfolo neffcen and r SSD r. Oherwse, D ( ) 0 and s SSD porfolo effcen. Tesng of FSD porfolo effcency s much more compuaonally demandng han n he SSD case and herefore we were no able o apply n he followng emprcal sudy. However, formulaons of FSD porfolo effcency ess can be found n Kuosmanen (2004) and Kopa and Pos (2009). 6. Sock Index Effcency Emprcal Sudy We consder he followng 25 world fnancal (sock) ndces lsed on Yahoo Fnance: Amerca (5): MERVAL BUENOS AIRES, IBOVESPA, S&P TS Compose ndex, S&P 500 INDE RTH, IPC, 114 Fnance a úvěr-czech Journal of Economcs and Fnance, 62, 2012, no. 2

10 Table 1 Descrpve Sascs of Index Reurns (before crss) Index mean sd Skewness Kuross mn max MERVAL BUENOS AIRES IBOVESPA S&PTS Compose ndex S&P 500 INDE.RTH IPC ALL ORDINARIES SSE Compose Index HANG SENG INDE BSE SENSE Jakara Compose Index FTSE Bursa Malaysa KLCI NIKKEI NZ 50 INDE GROSS STRAITS TIMES INDE KOSPI Compose Index TSEC weghed ndex AT CAC DA AE SMSI OM Sockholm PI SMI FTSE TEL AVIV TA-100 IND Asa/Pacfc (11): ALL ORDINARIES, SSE Compose Index, HANG SENG INDE, BSE SENSE, Jakara Compose Index, FTSE Bursa Malaysa KLCI, NIKKEI 225, NZ 50 INDE GROSS, STRAITS TIMES INDE, KOSPI Compose Index, TSEC weghed ndex, Europe (8): AT, CAC 4, DA, AE, SMSI, OM Sockholm PI, SMI, FTSE 100, Mddle Eas (1): TEL AVIV TA-100 IND. In our analyss we descrbe each ndex by s weekly raes of reurn. We dvded he reurns no wo perods: before-crss (B): Sepember 11, 2006 Sepember 15, 2008, durng-crss (D): Sepember 16, 2008 Sepember 20, Of course, he exac sarng dae of he crss s no known and can be debaed a lengh. We chose Sepember 16, 2008 because all sock ndces plummeed n he week sarng ha day. The descrpve sascs of he reurns are summarzed n Tables 1 and 2. Almos all he reurns are negavely skewed. Moreover, comparng he before-crss daa wh he durng-crss daa, we fnd ha he durng-crss reurns usually have a hgher sandard devaon and kuross. Focusng on he mean and varance of he reurns, we can sar our effcency analyss by denfyng he meanvarance effcen ndces. Followng Markowz (1952, 1959) we say ha he -h Fnance a úvěr-czech Journal of Economcs and Fnance, 62, 2012, no

11 Table 2 Descrpve Sascs of Index Reurns (durng crss) Index mean sd Skewness Kuross mn max MERVAL BUENOS AIRES IBOVESPA S&PTS Compose ndex S&P 500 INDE.RTH IPC ALL ORDINARIES SSE Compose Index HANG SENG INDE BSE SENSE Jakara Compose Index FTSE Bursa Malaysa KLCI NIKKEI NZ 50 INDE GROSS STRAITS TIMES INDE KOSPI Compose Index TSEC weghed ndex AT CAC DA AE SMSI OM Sockholm PI SMI FTSE TEL AVIV TA-100 IND Table 3 Mean-Varance Effcen Indces (B before crss, D durng crss) Perod B D IBOVESPA S&PTS Compose ndex S&P 500 INDE,RTH IPC Jakara Compose Index FTSE Bursa Malaysa KLCI NZ 50 INDE GROSS TEL AVIV TA-100 IND ndex s mean-varance effcen f here s no oher ndex havng reurns wh hgher or equal mean and smaller or equal varance (wh a leas one src nequaly). I s no surprse ha four ou of he fve Amercan ndces consdered are classfed as mean-varance effcen n he before-crss case. Perhaps surprsngly, all of hem become mean-varance neffcen n he durng-crss perod. Moreover, all he European ndces are mean-varance neffcen n boh perods. The ls of meanvarance effcen ndces s presened n Table 3. In he las wo columns, he meanvarance ndces are marked by. 116 Fnance a úvěr-czech Journal of Economcs and Fnance, 62, 2012, no. 2

12 Table 4 Mean-Varance Effcen and SSD Parwse Effcen Indces (B before crss, D durng crss) Perod mean-varance SSD parwse B D B D IBOVESPA S&PTS Compose ndex S&P 500 INDE,RTH IPC ALL ORDINARIES HANG SENG INDE BSE SENSE Jakara Compose Index FTSE Bursa Malaysa KLCI NZ 50 INDE GROSS TSEC weghed ndex DA TEL AVIV TA-100 IND 6.1 Effcency wh Respec o Sochasc Domnance Usng mehods nroduced n Secons 4 and 5, we analyze ndex effcency wh respec o sochasc domnance crera. We sar wh FSD parwse effcency esng, because denfes he larges number of effcen ndces. In hs case, an ndex s effcen f he expeced uly of s reurns s maxmal for a leas one uly funcon. Ths means ha he FSD parwse effcen ndex s he bes choce (ou of all he 25 ndces consdered) for a leas one decson-maker, because maxmzes hs expeced uly. Usng he algorhm descrbed n Secon 4.1, we classfy all ndces n boh perods as FSD parwse effcen. Therefore, we wan o observe how he se of effcen ndces s reduced when only rsk-averse decsonmakers are consdered, whch leads o SSD parwse effcency esng. The resuls of hese ess are compared o he mean-varance effcency n Table 4. I s well known ha mean-varance effcency does no mply sochasc domnance effcency (see Levy, 2006). However, Table 4 shows ha every meanvarance effcen ndex s SSD parwse effcen, oo. Hence, n hs sudy, SSD parwse effcency can be seen as a generalzaon of he mean-varance effcency approach. Moreover, here are only sx ndces ha are classfed dfferenly fve of hem for he before-crss daa and wo of hem for he durng-crss daa. Fnally, we allow full dversfcaon among he ndces and apply he Kopa and Chovanec (2008) es o denfy SSD porfolo effcen ndces. Ths propery s much sronger han SSD parwse effcency because a gven ndex s SSD porfolo effcen f here exss no lnear convex combnaon of he consdered ndces whch SSD-domnaes he underlyng ndex. Therefore, every SSD porfolo effcen ndex s also SSD parwse effcen. The comparson of hese wo approaches can be seen n Table 5. As we expeced, he SSD porfolo effcency classfcaon s very srong. Only he ndex wh he hghes mean s SSD porfolo effcen (n boh perods). All oher ndces are SSD porfolo neffcen, ha s, all rsk-averse decson-makers prefer some combnaon of oher ndces o he ndex. Fnance a úvěr-czech Journal of Economcs and Fnance, 62, 2012, no

13 Table 5 SSD Porfolo Effcen and SSD Parwse Effcen Indces (B before crss, D durng crss) Perod mean-varance SSD parwse B D B IBOVESPA S&PTS Compose ndex S&P 500 INDE,RTH IPC ALL ORDINARIES HANG SENG INDE BSE SENSE Jakara Compose Index FTSE Bursa Malaysa KLCI NZ 50 INDE GROSS TSEC weghed ndex DA TEL AVIV TA-100 IND Table 6 DEA-rsk (B before crss, D durng crss) Perod B D S&PTS Compose ndex S&P 500 INDE,RTH ALL ORDINARIES FTSE Bursa Malaysa KLCI NZ 50 INDE GROSS FTSE DEA-Rsk Effcency In boh perods, we compue he rsk measures (a gven levels) ha are used as he npus o he DEA-rsk model: Inpus (28): sd; lsd for powers 1,2,3; VaR, CVaR, DaR and CDaR a levels 0.75, 0.9, 0.95, 0.99, Oupu: mean gross reurn. In general, one can choose arbrary rsk measures a arbrary levels as he npus of a DEA-rsk model. The only lmaon s ha all npus of all ndces mus be non-negave. We employ he measures ha have ended o be he mos popular n recen years. On he oher hand, he oupu of he DEA-rsk model s precsely specfed. Snce he oupus of all ndces mus be non-negave, he mean reurn canno generally be used. Therefore, we modfed o he mean gross reurn. The resuls of DEA-effcency esng can be found n Table 6. Smlar o he case of mean-varance effcency, only afew ndces are classfed as effcen and he crss almos compleely changed he se of effcen ndces. However, conrary o he mean-varance case, an ndex exss ha s DEAeffcen n boh perods. As a by-produc of DEA-effcency esng, we can analyze he opmal weghs of he npus. Specfcally, we can dsngush beween zero and non-zero weghs. For a gven ndex, omng he npu causes no harm f he opmal wegh 118 Fnance a úvěr-czech Journal of Economcs and Fnance, 62, 2012, no. 2

14 Table 7 Number of Indces wh Posve Wegh of Seleced Rsk Measure n DEA-Rsk Model VaR Level Before crss Durng crss CVaR Level Before crss Durng crss DaR Level Before crss Durng crss CDaR Level Before crss Durng crss lsd Power Before crss Durng crss sd of an npu s zero; ha s, he npu has no relevan mpac on DEA-rsk effcency. In Table 7, we presen he number of ndces havng posve opmal weghs of parcular rsk measures n he DEA-rsk model. As we can see, Value a Rsk a all consdered levels plays he crucal role n our DEA-rsk analyss. Parcularly n he before-crss perod, VaR a level 0.99 s an mporan measure for 15 ndces. In Secon 6.1 we found ha frs-order sochasc domnance parwse effcency esng denfed he larges se of effcen ndces. Snce he FSD creron can be expressed usng VaR (see Ogryczak and Ruszczynsk, 2002), he mporan role of VaR s n accordance wh our FSD esng resuls. On he oher hand, omng he sandard devaon or CDaR measures from DEA-rsk esng causes no harm for nearly all he ndces. 6.3 Sensvy of DEA-Rsk Effcency We also suded he sensvy of our resuls wh respec o he npus. We sar wh sably analyss wh respec o seleced ypes of rsk measures. Tables 8 and 9 presen he effcen ndces when only one ype of rsk measure (a all consdered levels) s consdered. In boh perods, we fnd ha Value a Rsk denfes he larges se of effcen ndces. If anoher ype of rsk measure s consdered, hen always only one ndex s DEA-rsk effcen. Smlarly, we can es DEA-effcency f one ype of rsk measure s omed. The resuls are summarzed n Tables 10 and 11. Agan, we can see he crucal role of VaR n boh perods. If s kep n he model wh reduced npus, he resuls obaned are he same as hose n he full Fnance a úvěr-czech Journal of Economcs and Fnance, 62, 2012, no

15 Table 8 Sensvy of Resuls wh One Type of Rsk Measure (before crss) Only wh VaR CVaR DaR CDaR lsd and sd S&PTS Compose ndex S&P 500 INDE,RTH NZ 50 INDE GROSS FTSE 100 Table 9 Sensvy of Resuls wh One Type of Rsk Measure (durng crss) Only wh VaR CVaR DaR CDaR lsd and sd FTSE Bursa Malaysa KLCI NZ 50 INDE GROSS Table 10 Sensvy of Resuls wh One Type of Rsk Measure Omed (before crss) Whou VaR CVaR DaR CDaR lsd and sd S&PTS Compose ndex S&P 500 INDE,RTH ALL ORDINARIES NZ 50 INDE GROSS FTSE 100 Table 11 Sensvy of Resuls wh One Type of Rsk Measure Omed (durng crss) Whou VaR CVaR DaR CDaR lsd and sd FTSE Bursa Malaysa KLCI NZ 50 INDE GROSS model (see Table 6). On he oher hand, f VaR s no consdered as an npu, he number of effcen ndces decreases. The same can be concluded for DaR, bu only n he case of he before-crss daa. Smlar o he prevous analyss, we wll examne he sensvy of he resul wh respec o he levels consdered. In Tables 12, 13, 14, and 15 we can see resuls for models n whch only rsk measures a parcular levels are eher kep as npus or dropped. Droppng a level hardly changed he se of effcen ndces a all. Fnally, we esed he sensvy of he resuls o droppng exacly one npu. If we drop almos any npu, we ge he same resuls as n he case of he full model (Table 6). The only excepons are DaR075, whch causes he All Ordnares ndex o become neffcen n he before-crss perod, and VaR09, whch causes he NZ 50 Index Gross o be denfed as neffcen based on durng-crss daa. 120 Fnance a úvěr-czech Journal of Economcs and Fnance, 62, 2012, no. 2

16 Table 12 Sensvy of Resuls wh Only One Level (before crss) Index Wh level S&PTS Compose ndex S&P 500 INDE,RTH NZ 50 INDE GROSS FTSE 100 Table 13 Sensvy of Resuls wh Only One Level (durng crss) Index Wh level FTSE Bursa Malaysa KLCI NZ 50 INDE GROSS Table 14 Sensvy of Resuls whou Only One Level (before crss) Index Whou level S&PTS Compose ndex S&P 500 INDE,RTH ALL ORDINARIES NZ 50 INDE GROSS FTSE 100 Table 15 Sensvy of Resuls whu Only One Level (durng crss) Index Whou level FTSE Bursa Malaysa KLCI NZ 50 INDE GROSS 6.4 Comparson of Dfferen Effcency Approaches In hs secon we compare all four effcency approaches consdered n hs paper: mean-varance effcency, DEA-rsk effcency wh all npus, SSD parwse effcency, SSD porfolo effcency. We do no nclude resuls of FSD parwse effcency esng, because he FSD creron s oo weak o classfy any ndex as neffcen. The comparson s presened n Table 16. Unforunaely, no ndex s classfed as effcen usng all four mehods. Snce he SSD porfolo effcency ess denfed only one effcen ndex n boh perods, we lm our aenon o he oher hree approaches. In he before-crss case, we can fnd hree ndces (S&PTS Comp. ndex; S&P 500 INDE, RTH; NZ 50 INDE GROSS) ha are mean-varance effcen and DEA-rsk effcen as well as SSD parwse effcen, and 14 ndces are classfed as neffcen n all hree cases. Usng Fnance a úvěr-czech Journal of Economcs and Fnance, 62, 2012, no

17 Table 16 All Four Types of Effcency (B before crss, D durng crss) SSD Mean-varance DEA-rsk Perod Parwse Porfolo B D B D B D B D IBOVESPA S&PTS Comp. ndex S&P 500 INDE,RTH IPC ALL ORDINARIES HANG SENG INDE BSE SENSE Jakara Compose Index FTSE Bursa Malaysa NZ 50 INDE GROSS TSEC weghed ndex DA FTSE 100 TEL AVIV TA-100 IND durng-crss daa, only one ndex (FTSE Bursa Malaysa) s mean-varance effcen and DEA-rsk effcen as well as SSD parwse effcen, and 21 ndces are classfed as neffcen n all hree cases. Fnally, we compare he effcency classfcaons of he before-crss daa wh hose of he durng-crss daa. Perhaps surprsngly, here are only hree ndces (BSE SENSE; Jakara Compose Index; NZ 50 INDE GROSS) ha are classfed as effcen n boh perods usng a leas one approach. 7. Conclusons In hs paper we analyzed he effcency of 25 world sock ndces usng hree dfferen approaches: mean-rsk effcency, sochasc domnance effcency, and DEA-rsk effcency. We denfed effcen porfolos wh respec o mean-varance, DEA-rsk effcency, FSD parwse, SSD parwse, and SSD porfolo crera. We consdered wo perods: before-crss and durng-crss. We sared wh he classcal mean-varance mehod. We found ha four ou of he fve Amercan ndces consdered were classfed as mean-varance effcen n he before-crss perod, bu none were so n he durng-crss perod. Followng, for example, Basso and Funar (2001) and Murh e al. (1997), we proposed a new DEA-rsk effcency es based on he classcal DEA model of Charnes e al. (1978), where several rsk measures and funconals whch quanfy he rsk are used as npus, and mean gross reurn s used as he oupu. Usng he sochasc domnance approach, we found he FSD parwse es oo weak because classfed all ndces as effcen (n boh perods). Moreover, he SSD porfolo effcency es ended o be oo srong only one ndex was SSD porfolo effcen (n each perod). Therefore, we focused manly on comparng he mean-varance, DEA-rsk, and SSD parwse resuls. We found four ndces ha were classfed as effcen usng all hree mehods: S&PTS Comp., S&P 500 INDE, RTH, NZ 50 INDE GROSS (before he crss), and FTSE Bursa Malaysa (durng he crss). Comparng he number of effcen ndces, we 122 Fnance a úvěr-czech Journal of Economcs and Fnance, 62, 2012, no. 2

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