Comparison of different copula assumptions and their application in portfolio construction

Transcription

1 Comparson of dfferent copula assumptons and ther applcaton n portfolo constructon Frantšek ŠTULAJTER, ČSOB AM, Slovak Republc Abstract The paper deals wth modelng of mutual dependences among fnancal assets. Its am s to nvestgate the mpact of dfferent copula assumptons on optmal portfolos, when CVaR optmzaton s used. Strategc asset allocaton perspectve s supposed. It s demonstrated that copula functons enable us to separate the modelng of dependency features of fnancal assets from the modelng of margnal dstrbuton characterstcs, n the context of practcal portfolo constructon tasks. The dfference between portfolos constructed usng normal copula and student t copula s shown when mutual or penson fund exposed to long-only constran s assumed. The fund s consdered to nvest solely nto equty and fxed-ncome nstruments. As expected, the exclusve use of lnear correlaton coeffcents leads to underestmaton of total portfolo rsk. The superorty of student t copula portfolos ntensfes as the confdence level of CVaR rses and/or as the CVaR target ncreases. Keywords Copula functons, correlaton, CVaR, fnancal modelng, portfolo constructon JEL Classfcaton: C0, G CSOB Asset Management, Kolarska 6, Bratslava, Slovak Republc. fstulajter@csob.sk. Introducton Fnancal applcatons often deal wth the multvarate dstrbutons of random numbers. The obvous task for fnancal modelers s to descrbe the features and behavor of these multvarate dstrbutons. Statstcal propertes of fnancal data are a central pont of portfolo constructon and evaluaton problems, whch consttute the generc asset allocaton problem. Followng Meucc (2005), the quanttatve framework of solvng a generc asset allocaton problem can be summarzed to several steps: () detectng quanttes that fully descrbe behavor of asset prces: so-called market nvarants. For equtes the nvarants are the returns; for bonds the nvarants are the changes n the yeld to maturty, for vanlla dervatves t s the change n mpled volatlty; () estmatng the dstrbuton of market nvarants; () mappng the dstrbuton of market nvarants nto the dstrbuton of asset prces at a generc tme n the future; (v) defnng optmalty dependng on nvestor s profle; (v) computng the optmal allocaton, solvng portfolo selecton problem. Further, the rsk estmaton can be ncorporated. Ablty to correctly descrbe and estmate propertes of fnancal asset returns are crucally mportant to successfully solve asset allocaton problems. If the dstrbuton of market nvarants s assumed to be multvarate normal, the estmaton process and optmal portfolo optmzaton s straghtforward. However, dstrbutons of market nvarants are usually fat-taled and skewed. Not only do margnal characterstcs not conform to normalty assumpton, but the dependences are source of confuson as well. There can be found many drawbacks of normal specfcaton of the dependency structure n the non Gaussan world. Embrechts et al. (2002) dentfed and llustrated several major problems assocated wth a correlaton coeffcent, defned as Pearson s product moment: 2009 Publshed by VŠB-TU Ostrava. All rghts reserved. ER-CEREI, Volume 2: (2009). ISSN do:0.7327/cere

2 92 Ekonomcká revue Central European Revew of Economc Issues 2, 2009 The correlaton coeffcent s a measure of lnear assocaton of random varables and as such, t cannot capture non-lnear dependences. Feasble values for the correlaton depend on the margnal dstrbutons. Due to the specfcaton of correlaton as a scaled covarance, the correlaton coeffcent wll always be nfluenced by the dstrbuton of margnals. Perfect postve dependence does not mply a correlaton of one. Zero correlaton does not mply ndependence. Another problem s that the correlaton descrbes the dependency as one sngle number rather than expressng t as some functonal (e.g. the returns of fnancal assets usually posses hgher dependency n lower tals of ther multvarate dstrbuton). There are two major methods how to work wth non-normal dstrbutons n fnance, namely statc and dynamc method. Statc method assumes that there exsts a generc multvarate dstrbuton of market nvarants, whch s unobservable. The dynamc method n the smplest form works wth condtonal normalty assumpton and tres to descrbe dynamcally changng parameters of ths dstrbuton. Ths artcle focuses on statc method and copula functons as tool to descrbe the dependences among market nvarants and theoretcal concept to overcome correlaton coeffcents shortcomngs. Ptfalls of the correlaton coeffcent lead to many falures of fnancal applcatons. In rsk management or actuaral fnance, the estmates of rsk metrcs, lke Value at Rsk, possbly underestmate rsks. In the area of portfolo management, the optmal portfolos could be far less dversfed than expected. As a result, a new assessment of dependency modelng for fnance was proposed n the last decade of the 20 th century. Sklar s theorem s used to show the exstence of a specfc functon called copula, whch lnks together pure unvarate features of margnal dstrbutons nto the whole multvarate dstrbuton. The copula functon enables to ndependently model the specfcaton of margnal dstrbutons and the purely jont features. In ths context, stronger tal dependency can be easly justfed as well as skewed behavor of random numbers. A comprehensve ntroducton to copula functons can be found n Embrechts (2008) as well as a lst of copula must-reads. Copula methods n fnance attracted a lot of attenton n recent years. Thorough analyss of ths subject can be found n the publcatons of Alexander (2008) or Nelsen (2006). Fnancal applcatons of copula methods are covered n papers by Bouyé et al. (2000), Chorós et al. (2009) or the publcaton of Cherubn et al. (2004). Interestng applcaton of copula methods n hydrology s mentoned by Genest and Favre (2007). As copula tres to overcome ptfalls of correlaton coeffcents, new rsk measures were developed to assgn more emphass on extreme rsks. Downsde rsks measures lke Value at Rsk or ts extenson Expected Shortfall are the most popular ones. They focus on relaxng strong assumptons of Modern Portfolo Theory and the framework of mean-varance analyss and mean-varance optmzaton. The foundatons of portfolo selecton problem were made by Markowtz n hs paper Portfolo Selecton publshed n 952. As s shown n Fabozz et al. (2006) mean-varance approach s consstent wth: () expected utlty maxmzaton under quadratc utlty assumptons, or (2) the assumpton that securty returns are jontly normally dstrbuted. Blndly followng assumpton of jont normalty has caused falures n rsk assessment n many fnancal nsttutons; see e.g. Stulz (2008). The development of new rsk measure was most extensve at JP Morgan. The Rsk metrcs techncal documents and ntroduced Value at Rsk framework became standard n the feld. Despte ts wde use, VaR has several undesrable propertes, well documented n lterature; see e.g. Rau-Bredow (2004). The set of natural propertes that a reasonable measure of rsk should satsfy was proposed by Artzner et al. (999). Ths led to the ntroducton of a set of rsk measures that are based on estmatng the mean value of losses, whch exceed VaR value. These measures are referred to as () Mean Excess Loss or Expected Shortfall when expected losses strctly exceed VaR; () Tal VaR when expected losses weakly exceed VaR and fnally () Condtonal Value at Rsk (CVaR) gven as weghted average of VaR and Expected Shortfall; see e.g. Rockafellar and Uryasev (2000). Optmzaton framework utlzng CVaR was proposed by Rockafellar and Uryasev (2000). An attractve formulaton of ther soluton can also be found n Meucc (2006). Value at Rsk method armed wth copula dependency structure descrpton s subject to many studes. Palaro and Hotta (2006) proposed usng copula functons to model GARCH nnovatons when VaR s beng estmated. The authors show the superorty of the Joe-Clayton copula. Ba and Sun (2007) descrbe the copula theory n relaton to CVaR measures. The GARCH model wth three-dmensonal Archmedean copula s nvestgated. Integrated rsk management wth emphass on credt rsk s scrutnzed n He and Gong (2009). The copula based CVaR model for market and credt rsk s used.

3 F. Štulajter Comparson of dfferent copula assumptons and ther applcaton n portfolo constructon 93 The objectve of ths paper s to pont out the dfferences between two copula functon assumptons n portfolo constructon context. We consdered two dependency structure assumptons; normal copula and student t copula. The real world examples are used to emphasze the need of a copula modelng framework when estmatng optmal portfolos wth Condtonal Value at Rsk chosen as the rsk measure. Ths problem s solved from a strategc asset allocaton perspectve. Portfolos are exposed to long-only nvestment constrants and nvest solely nto equty and fxed-ncome nstruments. Monte Carlo smulatons are utlzed to estmate optmal portfolos. Impact of dfferent confdence levels and dfferent CVaR targets are nvestgated to emphasze the dfferences between optmal portfolos. The CVaR approach s chosen as a tool to estmate optmal portfolos and complement analyss of dfferent dependency structure assumptons. The paper s organzed as follows. The next secton defnes copula functons together wth the most mportant topcs: ts representaton, ellptcal and Archmedean copulas and estmaton. Frstly, the need of advanced dependency structure models s supported by emprcal observatons. In the thrd secton, the Condtonal VaR optmzaton problem s brefly ntroduced. In the fourth part we show numercal examples whch demonstrate the mportance of copula modelng n portfolo constructon. The pure lnear correlaton model (or normal copula model) s confronted wth student t copula assumpton. Dfferences among optmal portfolos estmated under these assumptons are shown n varous characterzatons of optmzaton problem. The last part concludes. In the artcle the terms copula and copula functon wll be used nterchangeably. We try to offer a valuable ntroducton to copula modelng framework n the context of portfolo management and always attempt to provde the nterested reader wth standard references. 2. Dependency patterns of fnancal data need of copulas The fat-taled character of margnal dstrbutons of market nvarants s well known and observed from most of the emprcal data. The asymmetrc dependences wthn mpled volatltes are obvous as well. However the asymmetry of market nvarants of equtes and bonds, namely of returns and changes n yelds to maturty, can be questoned. To graphcally depct the pure dependency structure between dfferent tme seres of market nvarants the gradegrade chart s created. It s the scatter plot of percentles, whch are called grades. Emprcal cumulatve dstrbuton functons are used to estmate percentles. In ths way, margnal characterstcs of data are totally fltered out. Every pont n the chart shows the combnaton of percentles of two nvarants. Combnatons of low percentles represent the market observaton of parallel negatve realzaton of market nvarants. In other words, both tme seres realze returns (n the case of equtes) from the left tal of ther margnal dstrbutons. Fgure represents the dependency structure of weekly returns of stock ndexes S&P 500 and DJ Stoxx 600 dated from 30 th December 988 to 7 th July The total number of observatons s thus 073. The sample correlaton coeffcent of these ndexes when estmated usng over 20 years of weekly data s Fgure Grade-grade chart of S&P 500 and DJ Stoxx 600 Fgure 2 shows the scatter plot of two normally dstrbuted random numbers, a and b, wth the same correlaton and the same number of observatons. The stronger tal dependence (occurrence of returns n the tals of margnal dstrbuton) of real world returns s obvous. The observatons n Fgure are more concentrated to the downward left corner of the chart whch represents stronger tal dependency than would have been predcted by multvarate normal dstrbuton wth sample correlaton coeffcent (by normal copula as wll be seen n the followng sectons). To more precsely defne the shown dfferences, the total number of observatons n the varous downward left corners was calculated as a fracton of all data ponts. Ths quantty s related to quantledependent measure of dependence ntroduced by Coles et al. (999) whch exactly defnes the probablty that one varable s extreme gven that the other s also extreme. It s also known as lower tal 2009 Publshed by VŠB-TU Ostrava. All rghts reserved. ER-CEREI, Volume 2: (2009).

4 94 Ekonomcká revue Central European Revew of Economc Issues 2, 2009 Fgure 2 Grade-grade chart of normally dstrbuted random numbers wth = dependence coeffcent. Three dfferent corners were assumed; namely 20 th 20 th percentle, 0 th 0 th percentle and 5 th 5 th percentle corner. These quanttes represent the total number of jont observatons of both varables that are lesser than x th percentles. In the case of real fnancal data the percentles are gven as emprcal estmates. For normally dstrbuted random numbers, Monte Carlo smulaton was used to calculate the number of observatons n the downward left corners assumng emprcal correlatons. Normal cumulatve densty functon was used to calculate normal percentles. As margnal characterstcs of emprcal and random numbers are fltered out, only pure dependency structure s nvestgated. Table shows the defned quanttes for dfferent equty and fxed ncome nvarants: The hypothess of stronger tal dependency cannot be rejected as the rato of emprcal tal frequency and normal copula frequency occurrences s bgger than one except n the case of the Amercan and Japanese equty ndex. The grade-grade charts of nvarant pars from Table can be found n Appendx B as well as grade-grade charts of selected dependency structures when daly data sets are used. There are presented charts representng dependency structures of dfferent equty markets as well as changes of generc 2 and 0 year US treasury bonds yelds. The same amount of fnancal data s used as n the prevous example. Appendx A shows relatve frequences of dfferent dependency structure assumptons when daly data sets are used. Appendx C contans a descrpton of data used to calculate quanttes from Table as well as a descrpton of data used n examples n Secton Defnton of copulas The dstrbuton of a multvarate random varable X can be factored nto two separate components. The margnal dstrbutons of each entry vector X represent the purely unvarate features of X. On the other hand, the purely jont component of the dstrbuton of X can be summarzed n standardzed dstrbuton, copula. The copula represents the true nterdependence structure of the random varable. Intutvely, the copula s a standardzed verson of the purely jont features of the multvarate dstrbuton, whch s obtaned by flterng out all the purely onedmensonal features, namely the margnal dstrbutons of each entry X n. In order to factor out the margnal components, t s necessary to determnstcally transform each entry X n n a new random varable U n, whose dstrbuton s the same for each entry. Snce all U n have the same dstrbuton, the unvarate features of X are removed. It s natural n fnancal modelng to consder cumulatve dstrbuton functon F X to map a generc random varable X nto a random varable U. Followng Meucc (2005) the random varable U s called grade of X and reads: U F X. () The grade of X s a determnstc transformaton of the random varable X that assumes values n the nterval [0, ]. In partcular, each margnal component X n can be standardzed by means of unform dstrbuton. The random varable U can be expressed as the vector of the grades: X Table Relatve frequences of dfferent dependency structure assumptons Emprcal tal frequency Normal copula tal frequency Invarants par 5th 5th 0th 0th 20th 20th 5th 5th 0th 0th 20th 20th S&P 500 DJ STOXX % 5.88% 2.69% 2.6% 4.93%.80% S&P 500 NASDAQ % 6.44% 4.55% 2.70% 5.77% 3.5% S&P 500 NIKKEI % 3.36% 8.86%.6% 2.9% 8.0% DJ STOXX 600 NIKKEI % 4.29% 9.89%.24% 3.25% 8.79% US 2YR TR US 0YR TR % 6.25% 3.99% 2.64% 5.59% 3.26%

5 F. Štulajter Comparson of dfferent copula assumptons and ther applcaton n portfolo constructon 95 U F X X U. (2) U N FX X N N In other words, the random varable U represents percentles of random varable X. The copula of the multvarate random varable X s the jont dstrbuton of ts grades. Sklar s theorem shows, that gven any multvarate random varable X wth contnuous margnal dstrbutons, there s a unque copula functon C such that: F,,, X x xn C FX X F. X X N N (3) Correspondng probablty densty functon of the multvarate random varable can be expressed as the product of the pdf of copula and the pdf of the margnal denstes of ts entres: f x N f x. x,, x f F x, F x (4) N C X X N N n Snce the copula s a dstrbuton, namely dstrbuton of the grades U (or percentles of the random number realzaton) t can be represented n terms of the probablty densty functon or the cumulatve dstrbuton functon, or the characterstc functon. It s proved n appendx of Meucc (2005) that the pdf of the copula reads: where f C X n Q u,..., Q u X X X N N Q u... f Q u, f u,..., u N (5) f X X X N X N N s the quantle functon (equvalent to QX n nverse cdf) of generc n-th margnal entry of X. And the cdf of the copula reads: F u,..., u F Q u,..., Q u. (6) C N X X X N N The copula of random varable X could be equvalently represented n terms of ts characterstc functon as well. There are two more concepts, central to the copula functons theory; tal dependence and bounds for dependence. The lower/upper tal dependence coeffcent represents the condtonal probablty that one random varable takes a value n ts lower/upper tal, gven that the other random varable takes a value n ts lower/upper tal. Copula s sad to have symmetrc tal dependence f ther lower and upper tal dependence coeffcents are equal. Conversely, t has asymmetrc tal dependence f ts tal dependence coeffcents dffer. There exst ndependence copulas, copulas wth perfect postve dependence and copulas wth perfect negatve dependence. Perfect postve/negatve dependency s defned as Fréchet upper/lower bound copulas. No other copula can take a value that s greater than the value of Fréchet upper bound copula and no other copula can take a value n that s less than the value of Fréchet lower bound copula. As t s hghlghted n Alexander (2008), less than perfect (postve or negatve) dependence s lnked to certan parametrc copulas. Copula captures postve or negatve dependence between the varables f t tends to one of Fréchet bounds as ts parameter values change. The Gaussan copula does not tend to Fréchet upper bound as the correlaton ncreases to and nether does t tend to Fréchet lower bound as approachng correlaton of. 2.2 Copula famles There exst many methods to derve copula functons; see e.g. Nelsen (2006). The most commonly used are the nverson method and generator functons method. The nverson method derves copula representatons from multvarate dstrbutons such as normal or student t dstrbuton. The most common examples of nverson method derved copulas are ellptcal copulas. An alternatve method for buldng copulas s based on a generator functon. These copulas are called Archmedean copulas. Ellptcal copulas A normal copula s derved usng the nverson method from the multvarate and unvarate standard normal dstrbuton functons, denoted and It s defned as: C u,, u ; u, u. 2 N (7) The cumulatve dstrbuton functon of normal copula cannot be wrtten n a smple closed form. The normal copula probablty densty functon s gven by:,, ; 2 c u exp 2 u n, (8) where denotes the correlaton matrx,, s ts determnant and,, n, where s the u quantle of the standard normal random varable X. The only one unknown parameter s, n ths case, the correlaton matrx. The normal copula densty s calculated as follows: Frstly, the grades of margnals are quantfed usng cumulatve dstrbuton functons of correspondng margnals: u F X x for = n. It s mportant to hghlght that copula modelng enables to use dfferent probablty specfcaton for every margnal. Also emprc cdf could be used. u P X, X ~ N0,,,, n Publshed by VŠB-TU Ostrava. All rghts reserved. ER-CEREI, Volume 2: (2009).

6 96 Ekonomcká revue Central European Revew of Economc Issues 2, 2009 Apply the quantle standardzed normal for = n. functon on grades: u Use the correlaton matrx to calculate the normal copula densty. Ths sequental, two-stage copula modelng approach can be used to plausbly estmate copula functons parameters as well as to create Monte Carlo smulatons where dfferent dependency structure assumptons are utlzed. Further presentaton of estmaton and smulaton topcs s covered n followng sectons. Fgure 3 shows the probablty densty functon of a bvarate normal copula wth correlaton coeffcent 0.5 as a functon of u and u 2 whch each range from 0 to. The normal copulas have symmetrc tal dependence behavor, whch s obvous from the graphcal representaton. They have zero or very weak tal dependence unless the correlaton s. When the margnals are also normally dstrbuted, there s always zero tal dependency. Obvously, ths s not approprate for modelng dependences among fnancal assets. c,, ; 2 n u u n K n 2 / 2, where t u,, t u n / 2 (0) s a vector of realzatons of student t varables and K() s a gamma functon defned as: K 2 n n 2 n. 2 () Fgure 4 shows a bvarate t copula probablty densty functon wth 4 degrees of freedom and correlaton of 0.5 also as a functon of u and u 2. Note that the peaks n the tals are symmetrc because the copula has symmetrc tal dependency but they are hgher than those of normal copula wth correlaton of 0.5 because the t copula has stronger tal dependence. Fgure 3 PDF of normal copula wth = 0.5. Source: Fgure II. 6. reproduced from Market Rsk Analyss: Practcal fnancal econometrcs, 2008, C. Alexander publshed by John Wley & Sons Ltd, wth permsson from the author. Another ellptcal copula, whch s derved mplctly from a multvarate dstrbuton functon, s student t copula. It s defned as: C u,, u ; t t u,, t u, (9) n v n t and t are multvarate and unvarate Student t dstrbuton functons wth degrees of freedom and denotes the correlaton matrx. Lke the normal copula, the student t copula cumulatve dstrbuton cannot be wrtten n a sngle closed form. The student t copula probablty densty functon s defned as: Fgure 4 PDF of Student t copula wth = 0.5 and 4 degrees of freedom Source: Fgure II. 6. reproduced from Market Rsk Analyss: Practcal fnancal econometrcs, 2008, C. Alexander publshed by John Wley & Sons Ltd., wth permsson from the author. However, the combnaton of the normal copula and student t margnals can create multvarate dstrbuton wth stronger tal dependence. Although both normal and student t copulas are ellptcal copulas wth symmetrc dstrbutons, 2 for practcal purposes, a shape of the whole dstrbuton of random varable X (the jont dstrbuton of margnals and copula) s mportant. The combnatons of ellptcal copulas and asymmetrcally dstrbuted margnals gve 2 There exsts a wde varety of Student t copulas; many of them wth asymmetrc tal dependence. See e.g. Demarta and McNel (2005).

7 F. Štulajter Comparson of dfferent copula assumptons and ther applcaton n portfolo constructon 97 rse to non-symmetrcal behavor of random varables, whch are partcularly mportant n fnancal modelng, especally when dervatves are consdered. Archmedean copulas Archmedean copulas are constructed usng the generator functons method. Gven any generator functon, t s possble to defne the correspondng Archmedean copula as: C u,, u u u. A n n (2) Its assocated densty functon s c A n n n n u,, u u u u, (3) where n s the n th dervatve of the nverse generator functon. When the generator functon u lnu the Archmedean copula becomes the ndependent copula. More generally, the generator functon can be any strctly convex and monotonc decreasng functon. Hence, there exsts a large number of dfferent Archmedean copulas. Alexander (2008) ntroduced two smple Archmedean copulas that are used n market rsk analyss thanks to ts asymmetrc tal dependence. These are Clayton and Gumbel copulas. The correspondng generator functon of Clayton copula s defned as u u, 0 Clayton, (4) and the generator functon of Gumbel copula as Gumbel u ln u,. (5) Interested readers are referred to Alexander (2008) or Nelsen (2006) for further dscusson of Archmedean copulas. 2.3 Estmaton of copulas The multvarate dstrbuton of the random number X comprses from the unvarate dstrbutons of ts margnals and the dstrbuton of the copula. In general, t s possble to estmate the copula and margnals parameters together n one step usng the so-called full Maxmum Lkelhood Estmaton; see for nstance Km et al. (2007). However, ths aggregated estmaton approach can become too complex to formulate and to effectvely solve n most cases. When estmatng the copula s the prmary objectve, the unknown margnal dstrbutons of the data enter the problem as unnecessary parameters. The frst step s usually to quantfy the grades of the margnals usng ts cumulatve dstrbuton functons. In general, the margnal modelng can be done by means of () fttng parametrc dstrbuton to each margnal, () modelng the margnals nonparametrcally usng a verson of the emprcal dstrbuton functons, () usng a hybrd of the parametrc and nonparametrc methods, or (v) make use of varous Bayesan estmators. Indeed, all the mentoned technques can also be used to estmate parameters of copula functons. Afterwards, selected margnal dstrbutons cdf s are used to calculate the grades and the copula parameters can be estmated ether by Maxmum Lkelhood Estmaton (MLE hereafter) or n some cases calbrated by the Generalzed Method of Moments lke n Demarta and McNel (2004). When the copula probablty densty functon s defned, constructng the log lkelhood functon s straghtforward. It s possble to estmate all unknown parameters n one step. Usng the MLE method s farly robust when the number of dmensons are low, but optmzaton of the lkelhood functon can become cumbersome when the quantty of estmated parameters ncrease (ths can be seen when dealng wth the ellptcal copulas where the number of unknown parameters n the correlaton matrx can grow quckly). Dependng on the copula functon representaton, MLE can be somehow changed; e.g. when estmatng student t copula, correlaton matrx and number of degrees of freedom (df hereafter) can be estmated smultaneously; or for set df, correlaton matrx s estmated. In other words, for every df, MLE s used to fnd out the correlaton matrx. Fnally, the resultng values of lkelhood functon are compared. When estmatng specal cases of copulas, e.g. student t copulas, the curse of the dmenson problem can be solved usng the Generalzed Method of Moments. The Method of Moments explots the correspondence between copulas and rank correlatons; for detaled dscusson about rank correlatons see Alexander (2008). For example, t can be shown that Kendall s, has a drect relatonshp wth a bvarate copula functon C u,u as follows: u, u d C u, u. C (6) 2 If the copula depends on one parameter then t can be calbrated usng a sample estmate of the rank correlaton. For nstance, the bvarate normal copula depends on one parameter, the correlaton coeffcent, and the above relatonshp yelds: sn. (7) 2 Ths relatonshp between the sample estmate of Kendall s tau and correlaton coeffcent holds also for other ellptcal copulas, lke student t copula. Smlar relatonshps can be also found between rank correlatons and non-ellptcal copulas n the referenced lterature. After specfyng the correlaton coeffcent usng the rank correlaton, the rest of the Publshed by VŠB-TU Ostrava. All rghts reserved. ER-CEREI, Volume 2: (2009).

8 98 Ekonomcká revue Central European Revew of Economc Issues 2, 2009 unknown parameters (e.g. degrees of freedom) are estmated usng MLE. The Method of Moments s sutable when dealng wth hgher dmensonalty. All components of the correlaton matrx can be thus estmated usng the sample Kendall s tau. As n every MLE procedure the best ft from parametrc specfcatons can be determned by Akake nformaton crteron or Bayesan nformaton crteron. 3. Condtonal Value at Rsk optmzaton As was depcted n the above subsecton, not only margnal characterstcs of fnancal assets nvarants are fat-taled, ther dependency structure s also the source of non-normal behavor. As a consequence new optmzaton frameworks, whch utlze hgher moments and extreme rsk measures, are used to properly solve portfolo selecton problems. Probably the most well-known s the Value at Rsk (VaR hereafter) model orgnally developed by JP Morgan. VaR s related to the percentles of loss dstrbutons and measures the predcted maxmum loss at a specfed probablty level over a gven horzon and s defned as: VaR R mnr P R R, P P (8) where P denotes the probablty functon, R P represent s the portfolo s expected return over a gven horzon and s a chosen confdence level. In portfolo selecton, ratonal nvestors choose a portfolo that mnmzes VaR for targeted expected returns. The optmzaton problem can also be reversed. The nvestor maxmzes expected returns for targeted values of VaR. A more detal descrpton of the VaR framework can be found n Joron (2007). As was stated n the ntroductory secton, VaR has several undesrable propertes, whch has led to many extensons of the VaR framework. One of the most popular s Condtonal Value at Rsk (CVaR). Wth no loss n generalty, the CVaR can be defned as: CVaR R E R R VaR R. P P P P (9) CVaR s by defnton always at least as large as VaR and t s a coherent rsk measure by means of Artzner et al. (999). It can be shown that CVaR s a concave functon and, therefore has a unque mnmum. The resultng optmzaton problem takes the form subject to CVaR max w, (20) w w, w C w, C 0 where represents the vector of expected returns, w are optmal portfolo weghts, whle C 0 and C w are sets of constrans. Usng the expresson n (9) to estmate CVaR nvolves knowng the VaR functon representaton. Ths can become neffectve n largesze problems. Rockafellar and Uryasev (2000) proposed a smpler approach where CVaR s reformulated as a mnmzer of the auxlary functon F. Ther approach s partcularly sutable when the probablty densty functon s expressed by Monte Carlo smulatons and hence the portfolo selecton problem can be solved as lnear programmng problem. For a descrpton of the resultng optmzaton problem refer to Rockafellar and Uryasev (2000) or Meucc (2006). 4. Copulas n portfolo management The crucal role of copulas n modelng the behavor of portfolo, whch nvests n dervatve nstruments, s clear. The pay off of dervatves s usually non-lnear and so too are dependences between dervatves and underlyng assets. Therefore the use of copulas n that feld of study s natural. When the portfolo nvests only n stocks and bonds, and s possbly exposed to non-negatve weght constrants, the need for copula formulaton of dependence structure s not obvous. The examples from the second part ntutvely support the dea of copula modelng n regular, long-only portfolo management. Most of the presented gradegrade charts underlne the emprcal observaton from fnancal markets whch s sometmes called lack of dversfcaton. In the exceptonally negatve market shocks nearly all assets perform n the same fashon. The fnancal markets stress from the second half of 2008 s the latest case. In ths secton we compare the dfferences n optmal portfolos that are estmated usng two dfferent copula specfcatons normal copula and student t copula. The optmal portfolo problem was solved by means of mean CVaR optmzaton brefly ntroduced n a prevous secton. The ratonal behnd usng CVaR as a rsk measure s ts mplct focus on optmal portfolos tal characterstcs. Consequently, there s supposed to be a relatonshp between dfferent copula assumptons and dfferent CVaR confdence levels. If the emprcal dependency structure exhbts fat tals student t copula better descrbes these relatonshps optmal portfolos should be more dstnct for hgher confdence levels. In other words, the deeper we look nto the dstrbutonal tals, the more evdent the dfference between created portfolos should be, when normal and student t copula assumptons take place.

9 F. Štulajter Comparson of dfferent copula assumptons and ther applcaton n portfolo constructon 99 We consdered an nvestment entty that was assumed to be a long-only mutual or a penson fund whch nvests n 3 bond market ndexes and 6 stock ndexes. The bond nvestments were represented by Boxx 3 ndexes and cover US Treasures, the Euro Government Bond Market and the Euro A Corporate Bond Market. Consdered stock nvestments were: S&P 500, DJ Stoxx 600, CECE Traded Eur Index, Nkke 225, NASDAQ 00 and Hang Seng Chna Enterprses. All calculatons were done usng weekly data from 5 th January 999 to 24 th July 2009, whch add up to 550 weekly observatons of return data. The detaled statstcal descrpton of data s presented n appendx C. In the case of bond nvestments we adopted a more elementary approach than s usually presented n lterature. Bond market nvarants were not used drectly but returns of total return ndexes were used. The proposed smplfcaton s legtmated by two reasons: frstly, there can be found nvestment vehcles that offer nvestors the chance of exposure to fxed-ncome ndexes drectly va exchange traded funds or mutual funds. Ther total returns could be assumed as market nvarants. Secondly and more mportantly, portfolo constructon problem perspectve enables us to demand lower precson as many more errors arse from stochastc estmates. The global asset allocaton s focused on estmatng optmal rsk exposures for gven expected returns whereas nvestment vehcle selectons are subject to tactcal asset allocaton. As s ponted out by Joron (2007), the rsk management and dervatve valuaton approaches have much methodology n common. However, valuaton methods requre more precson as accurate asset prces are needed for tradng purposes. We suggest that the portfolo constructon problem, specfcally global asset allocaton, enables us to use less precson as well and focus on overall exposures. Frstly, we solated margnal dstrbutons usng emprcal cumulatve dstrbuton functons F E (x). They were further smoothed by means of the Gaussan kernel functon to obtan more spaced observatons as n Meucc (2006). Thus the skewness and/or the kurtoss of margnal dstrbutons were preserved. The parameters to be estmated remaned correlaton matrx n the case of normal copula and correlaton matrx and degrees of freedom n the case of student t copula. The estmaton of normal copula correlaton matrx s straghtforward, as s ts MLE sample estmator. Recall that the copula operates on grades or percentles of underlyng dstrbutons. The estmaton of normal copula and smulaton from ts dstrbuton can be vsualzed as n Fgure 5. X F U X E copula x estmaton ˆ smulaton Fgure 5 Copula modelng framework Y F U Y copula E x The X (550 x 9) matrx conssts of weekly observatons of nne consdered nvestments. X was transformed to equally dstrbuted grades matrx U usng emprcal cdf functons. Standard normal nverse functon was further used to create X copula matrx from whch correlaton matrx was estmated. The parameters of student t copula were estmated followng the same steps. Obvously we used student t nverse functon. Several values of df were chosen and for every assumed number, we estmated correlaton matrx. Havng set number of df and correlaton matrx, the log lkelhood functon was calculated. However, the local maxmum was not found. Frstly, the values of the log lkelhood ncreased quckly when the number of df was small, but after reachng the level of 9 ts values became not elastc to addtonal changes n number of df. Usng ths number of df represented approxmately 95% of the total mprovement n the log lkelhood functon. For further analyss we consdered 9 as the estmated number of df. After specfyng the parameters of normal and student t copula, we generated Monte Carlo smulatons from correspondng copula dstrbuton usng algorthms proposed n Alexander (2008) and vsualzed n the rght hand sde of Fgure 5. Y copula (0 000 x 9) matrx represents Monte Carlo smulatons whch were frstly transformed to equally dstrbuted grades matrx U usng normal or student t cdf functons. The nverse emprcal cumulatve densty functons were fnally used to calculate weekly returns, panel Y. To get rd of unstable expected returns estmate, smulated sample averages of Y were adjusted to mpled returns that were nduced by emprcal covarance matrx; assumed market portfolo market weghts w and assumed nvestors rsk averson coeffcent. The mpled returns vector IMPL s defned as: IMPL E market w. (2) The concepts of reverse optmzaton and mpled returns were developed by Sharpe (974) and are wdely used n Black Ltterman models. Standard penson fund benchmark portfolo was assumed as market w. Rsk averson coeffcent was set to or Publshed by VŠB-TU Ostrava. All rghts reserved. ER-CEREI, Volume 2: (2009).

10 200 Ekonomcká revue Central European Revew of Economc Issues 2, 2009 Obvously, separate smulaton panels were created for both copula assumptons. Smulated panels Y were used to estmate optmal portfolos by solvng (20). We estmated values of CVaR as a sample average loss that exceeded correspondng VaR. We defned VaR as emprcal quantle of smulated portfolo returns. Frstly, a portfolo wth mnmum attanable CVaR was estmated. Afterwards, we reversed the optmzaton settng and estmated optmal portfolos by maxmzng expected returns for targeted CVaR. The CVaR targets equalled to mnmum CVaR subsequently ncreased by 0.25% up to the optmal portfolo equty allocaton reached unty (ths examnaton crteron wll be ntroduced shortly). We estmated optmal portfolos wth three dfferent CVaR confdence levels: %, 3% and 5%. As an examnaton crteron we adopted the dfference n the optmal portfolos total weghts of equty nvestments for the same CVaR target. Snce the optmal portfolos were estmated from dfferent smulated panels Y, t was not possble to use formal statstcal tests to evaluate ther correspondence. Followng emprcal observatons we can conclude that equty nvestments usually suffer from heavy tals and strong lower tal dependence. So student t copula s supposed to capture these effects and should lead to more conservatve optmal portfolos. Fgure 6 shows dfference curves for chosen confdence levels. They represent equty allocaton dfferences between normal and student t copula. ncreased by 0.25% and portfolos were optmzed accordng to these values. Full nvestments nto equtes under student t copula assumpton were reached wth CVaR target set to 6.25% ( = 5%), 7.5% ( = 3%) and 0.25% ( = %). Normal copula leads to full equty allocatons sgnfcantly sooner. The 5% confdence level dfference curve s the flattest. The equty allocatons do not dffer by more than 2.5%. However, the more rsky profle of optmal portfolos estmated under normal copula assumpton became more apparent when ncreasng CVaR confdence levels. As expected, growng CVaR confdence level ntensfes conceptual dfferences between copula assumptons. Ths result has strong consequences as a rather hgh confdence level of % s requred for nternal rsk models by the Basel Supervsory Commttee. So t s hghly recommended to nvestgate the possblty of usng dfferent dependency structures when assessng captal adequacy requrements. The % confdence level dfference curve gves another nsght as well. The largest dfferences are reached when the most aggressve CVaR targets are selected. The same explanaton as n the case of confdence level changes s vald. Hgher CVaR targets focus on deeper dstrbutonal tals where dfferences between normal and student t copula assumptons are more strkng. On the other hand, hgh CVaR targets are usually only used by rsk seekng mutual funds. Dfferent copula assumptons resulted n dfferent ndvdual postons as well. Table 2 shows estmated portfolos weghts for several optmzaton targets and confdence levels. These dfferences can easly lead to the msunderstandng of proper rsk exposures and neffectve hedgng scheme. Compostons of other estmated optmal portfolos are avalable upon request. 5. Concluson Fgure 6 Derenceurves As Fgure 6 shows, optmal portfolos estmated under normal copula assumpton often leads to hgher allocaton nto equty nvestments. We pont out that allocaton dfferences are solely the result of dependency structure assumptons as the margnal characterstcs are treated equally. The mnmum attanable CVaR targets were.04% n the case of 5% confdence level,.5%.6% n the case of 3% confdence level and.36%.40% n the last case. As was stated above, CVaR targets were further The copula functons can help to realstcally descrbe the true dependences among random numbers. Moreover, they propose a practcal soluton to the lack of dversfcaton problem. The term copula modelng framework was used throughout the artcle and ts basc concepts were descrbed. The most mportant feature s the ablty to model non-lnear dependences among market nvarants. Addtonally, t enables us to separately model margnal characterstcs and dependency structures. Dfferent statstcal and econometrcal tools can be used for both parts. There exst many dfferent classes of copulas. For modelng fat-taled dependences, the ellptcal class of copulas s usually suffcent. When the asymmetrc dependences are present, the

11 F. Štulajter Comparson of dfferent copula assumptons and ther applcaton n portfolo constructon 20 Archmedean copulas are a better choce. The potental problem s the estmaton of exact parametrc specfcaton of copula functon when the number of dmensons s large. The portfolo selecton problem defned n terms of Condtonal VaR optmzaton enables us to fully utlze copula modelng framework whle the meetng requrements of coherent rsk measure. Despte ts lnear representaton, the effcently soluton to the CVaR problem s rather complex. The presented examples showed better propertes of student t copula than normal n the feld of portfolo management. Optmal portfolos estmated under normal copula assumpton often lead to hgher allocaton nto equty nvestments that usually suffer from heavy tals and strong lower tal dependence. The extent of equty allocaton overweght proved to be a functon of CVaR targets and CVaR confdence levels. Both hgher confdence levels and targets ntensfed the dfferences between assumed copula specfcatons. Our results recommend focusng on advanced dependency structure models n varous felds of fnancal modelng. References ALEXANDER, C. (2008). Market Rsk Analyss: Practcal Fnancal Econometrcs. Hoboken NJ: Wley. ARTZNER, P., DELBAEN, F., EBER, J.M., HEATH, D. (999). Coherent Measures of Rsk. The Journal of Mathematcal Fnance 9: BAI, M., SUN, L. (2007). Applcaton of Copula and Copula-CVaR n the Multvarate Portfolo Optmzaton. In: Chen, B., Paterson, M., Zhang, G. (eds.): Combnatorcs, Algorthms, Probablstc and Expermental Methodologes, Berln/ Hedelberg: Sprnger BOUYÉ, E., DURRLEMAN, V., NIKEGHBALI, A., RIBOULET, G., RONCALLI, T. (2000). Copulas for fnance: A Readng Gude and Some Applcatons. Groupe de Recherche Operatonnelle, Credt Lyonnas. Workng Paper. OLES, S., HEFFERNAN, J., TAWN, J. (999). Dependence Measures for Extreme Value Analyses. Extremes 2: DEMARTA, S., McNEIL, A.J. (2004). The t Copula and Related Copulas. Internatonal Statstcal Revew 73: EMBRECHTS, P., McNEIL, A.J., STRAUMANN, D. (2002). Correlaton and dependence n rsk management: Propertes and ptfalls. In.: Dempster, M. (eds.): Rsk Management: Value at Rsk and Beyond, Cambrdge: Cambrdge Unversty Press. Table 2 Composton of selected optmal portfolos Copula Student Normal Student Normal Student Normal Student Normal Student Normal Student Normal % 3% 5% % 3% 5% CVaR.5% 4.0% QWA Index 74.2% 66.0% 49.3% 39.0% 47.5% 27.% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% ITRROV Index 2.7% 7.7% 27.0% 32.2% 6.% 33.9% 43.4% 24.7% 2.7% 29.8% 2.% 28.7% QX5N Index 0.0% 0.% 2.7% 6.5%.6% 3.2% 4.5% 29.4% 2.8% 0.5% 2.5% 2.9% S&P % 4.57% 5.42% 7.4% 7.48% 9.7% 0.49% 4.% 6.54% 2.75% 22.85% 27.43% DJ STOXX % 4.78%.56% 8.97%.8% 8.83% 5.84% 4.02% 8.90% 5.96% 8.93% 6.82% CECE-EUR 0.00% 2.08% 0.54%.26%.29% 2.% 3.40% 5.63% 5.63% 8.37% 8.3% 9.07% NIKKEI %.89% 2.27% 3.00% 3.4% 3.37% 2.9% 5.69% 6.70% 6.58% 9.% 7.45% NASDAQ 00 HSCEI Index equty allocaton 0.07%.% 0.00% 0.00% 0.4% 0.00% 2.20% 0.02% 3.0% 0.00% 0.95% 0.00% 0.40%.67%.5%.88%.5%.72% 7.25% 6.44% 5.69% 6.98% 6.4% 7.6% 3.% 6.% 20.9% 22.3% 24.7% 25.7% 42.% 45.9% 56.5% 59.6% 66.4% 68.4% 2009 Publshed by VŠB-TU Ostrava. All rghts reserved. ER-CEREI, Volume 2: (2009).

12 202 Ekonomcká revue Central European Revew of Economc Issues 2, 2009 EMBRECHTS, P. (2008). Copulas: A personal vew. Journal of Rsk and Insurance 76: GENEST, C., FAVRE, A.C. (2007). Everythng you always wanted to know about copula modelng but were afrad to ask. Journal of Hydrologc Engneerng 2: FABOZZI, F.J., FOCARDI, S.M., KOLM, P.N. (2006). Fnancal Modelng of the Equty Market: From CAPM to Contegraton. Hoboken NJ: Wley. HE, X., GONG, P. (2009). Measurng the coupled rsks: A copula-based CVaR model. Journal of computatonal and appled mathematcs 223: CHERUBINI, U., LUCIANO, E., VECCHIATO W. (2004). Copula methods n fnance. Hoboken NJ: Wley. CHORÓS, B., HÄRDLE, W., OKHRIN, O. (2009). CDO Prcng wth Copulae. SFB 649 Dscusson Paper, , Humbolt Unverstät zu Berln. JORION, P. (2007). Value-at-Rsk: The New Benchmark for Managng Fnancal Rsk, 3 rd edton. New York: McGraw-Hll. KIM, G., SILVAPULLE, M.J., SILVAPULLE, P. (2007). Comparson of semparametrc and parametrc methods for estmatng copulas. Computatonal Statstcs & Data analyss 5: MEUCCI, A. (2005). Rsk and Asset Allocaton. New York: Sprnger. MEUCCI, A. (2006). Beyond Black Ltterman n Practce. Rsk Magazne 9: 4 9. NELSEN, R.B. (2006). An ntroducton to Copula, 2 nd edton. New York: Sprnger. PALARO, H.P., HOTTA, L.K. (2006). Usng Condtonal Copula to Estmate Value at Rsk. Journal of Data Scence 4: RAU-BREDOW, H. (2004). Value-at-Rsk, Expected Shortfall and Margnal Rsk Contrbuton. In: Szego, G. (eds.): Rsk Measures for the 2 st Centrury, Hoboken NJ: Wley. ROCKAFELLAR, R., URYASEV, S. (2000). Optmzaton of Condtonal Value-At-Rsk. The Journal of Rsk 2: 2 4. SHARPE, W.F. (974). Imputng expected returns from portfolo composton. Journal of Fnancal and Quanttatve Analyss June 974: STULZ, R. (2008). Rsk Management Falures: What are They and When Dot They Happen?. Fsher College of Busness Workng Paper Seres. October Appendx A Table A Relatve frequences of dfferent dependency structure assumptons usng daly data Emprcal tal frequency Normal copula tal frequency Invarants par 5th 5th 0th 0th 20th 20th 5th 5th 0th 0th 20th 20th S&P 500 DJ STOXX % 5.0% 9.94%.40% 3.44% 9.6% S&P 500 NASDAQ % 8.40% 6.95% 3.7% 7.88% 7.02% S&P 500 NIKKEI %.85% 5.32%.24% 3.27% 8.80% DJ STOXX 600 NIKKEI % 3.47% 8.09%.05% 2.72% 7.8% US 2YR TR US 0YR TR % 6.39% 4.02% 2.52% 5.3% 2.8% GE 0YR TR US 0YR TR % 4.6% 0.94%.24% 3.22% 8.74% CECEEUR DJ STOXX % 6.24% 2.87% 2.06% 4.72%.46% S&P 500 US 0YR TR % 4.47% 8.94% 0.89% 2.52% 7.36%

13 F. Štulajter Comparson of dfferent copula assumptons and ther applcaton n portfolo constructon 203 Appendx B In ths appendx dfferent grade-grade charts of dfferent fnancal data seres are shown. Weekly as well as daly data are used. The relatve frequences of weekly data seres s ncluded n secton 2 and that of daly data n Appendx A. Fgure B Grade-grade chart of S&P 500 and NASDAQ 00 weekly data Fgure B2 Grade-grade chart of US 2YR and US 0 YR Total Return treasury bond yelds weekly data Fgure B3 Grade-grade chart of Nkke 225 and DJ Stoxx 600 weekly data Fgure B4 Grade-grade chart of S&P 500 and NASDAQ 00 daly data Fgure B5 Grade-grade chart of DJ Stoxx 600 and CECEEUR Index daly data Fgure B6 Grade-grade chart of S&P 500 and US 0 YR Total Return treasury bond yelds daly data US 0YR TR 0,75 0,5 0, ,25 0,5 0,75 US 2YR TR Fgure B7 Grade-grade chart of US 2YR and US 0 YR Total Return treasury bond yelds daly data 2009 Publshed by VŠB-TU Ostrava. All rghts reserved. ER-CEREI, Volume 2: (2009).

14 204 Ekonomcká revue Central European Revew of Economc Issues 2, 2009 Appendx C In ths appendx descrpton of dfferent date sets s presented. Table C shows statstcs of 6 fnancal tme seres whch represent 4 equty market ndces and 2 fxed-ncome market nvarants. The samplng frequency s weekly for all seres and ncludes tme wndow from 6 th January 989 to 7 th July 2009, totalng to 072 observatons. Table C Descrpton of data used to calculate quanttes from Table S&P 500 DJ STOXX 600 NASDAQ 00 NIKKEI 225 US 2YR TR US 0YR TR μ 0.% 0.08% 0.5% 0.% 0.76 bps. 0.5 bps. μ p.a. 5.92% 4.30% 7.76% 5.66% s 2.36% 2.48% 3.27% 3.0% 4.22 bps. 3.4 bps. s p.a. 7.00% 7.86% 23.59% 22.33% bps bps. skew kurt N mn 20.0% 24.30% 29.20% 27.90% bps bps. max.40% 2.40% 7.40%.40% bps bps. VaR 5% 3.54% 3.67% 5.07% 4.98% bps bps. CVaR 5% 5.62% 5.90% 7.93% 7.8% 3.87 bps bps. Table C2 depcts statstcs of fnancal data seres used n secton 4. As n prevous table, data represent both fxed-ncome and equty market seres sampled weekly. Instead of drectly usng fxed-ncome nvarants, total return ndces were used as proxy for bond markets. The tme wndow represent perod from 5 th January 999 to 24 th July Table C2 Descrpton of data used n examples from secton 4 QWA Index ITRROV Index QX5N Index S&P 500 DJ STOXX 600 CECEEUR NIKKEI 225 NASDAQ 00 HSCEI Index μ 0.09% 0.0% 0.07% 0.05% 0.05% 0.% 0.05% 0.03% 0.35% μ p.a. 4.47% 5.36% 3.38% 2.50% 2.82% 5.56% 2.8%.66% 8.25% σ 0.5% 0.66% 0.55% 2.79% 2.90% 3.99% 3.22% 4.03% 5.5% σ p.a. 3.7% 4.76% 3.98% 20.2% 20.89% 28.77% 23.22% 29.03% 37.3% skew kurt N mn.80% 3.00% 4.40% 20.0% 24.30% 26.60% 27.90% 29.20% 23.40% max.70% 2.40%.40%.40% 2.40% 22.90%.40% 7.40% 8.00% VaR 5% 0.79%.0% 0.83% 4.44% 4.79% 6.3% 5.24% 6.28% 8.47% CVaR 5%.09%.46%.32% 6.85% 7.2% 9.70% 7.58% 9.68%.47% * QWA Index Boxx Soveregns Eurozone, ITRROV Index Boxx $ Treasures Total Return Index, QX5N Index Boxx Corporates A, HSCEI Index HANG SENG CHINA ENT INDX