PORTFOLIO OPTIMIZATION WITH DRAWDOWN CONSTRAINTS

Transcription

1 PORTFOLIO OPTIMIZATIO WITH DRAWDOW COSTRAITS Aleei Chehlov Sanislav Urasev 2 Michael Zabaranin 2 Ris Managemen and Financial Engineering Lab Cener for Applied Opimizaion Deparmen of Indsrial and Ssems Engineering Universi of Florida Gainesville FL 326 Dae: Janar Correspondence shold be addressed o: Sanislav Urasev Absrac We propose a new one-parameer famil of ris fncions defined on porfolio rern sample-pahs which is called condiional drawdown-a-ris (CDaR These ris fncions depend on he porfolio drawdown (nderwaer crve considered in acive porfolio managemen For some vale of he olerance parameer he CDaR is defined as he mean of he wors ( 00% drawdowns The CDaR ris fncion conains he maimal drawdown and average drawdown as is limiing cases For a pariclar eample we find opimal porfolios wih consrains on he maimal drawdown average drawdown and several inermediae cases beween hese wo The CDaR famil of ris fncions originaes from he condiional vale-a-ris (CVaR measre Some recommendaions on how o selec he opimal ris measre for geing pracicall sable porfolios are provided We solved a real life porfolio allocaion problem sing he proposed ris fncions Inrodcion Opimal porfolio allocaion is a longsanding isse in boh pracical porfolio managemen and academic research on porfolio heor Varios mehods have been proposed and sdied (for a review see for eample Grinold and Kahn 999 All of hem as a saring poin assme some measre of porfolio ris From a sandpoin of a fnd manager who rades cliens' or ban s propriear capial and for whom he cliens' accons are he onl sorce of income coming in he form of managemen and incenive fees losing hese accons is eqivalen o he deah of his bsiness This is re wih no regard o wheher he emploed sraeg is long-erm valid and has ver aracive epeced Thor Asse Managemen Inc 55 Fifh Ave Sie 60 6h Floor ew Yor Y 007; a_chehlov@horcapialcom 2 Universi of Florida ISE PO Bo Weil Hall Gainesville FL ; rasev@isefled zabaran@fled

2 rern characerisics Sch fnd manager's primar concern is o eep he eising accons and o arac he new ones in order o increase his revenes A pariclar clien who was persaded ino opening an accon wih he manager hrogh reading he disclosre docmen lisening o he manager's aracive sor nowing his previos rerns ec will decide on firing he manager based mos liel on his accon's drawdown magnide and draion In pariclar i is highl ncommon for a Commodi Trading Advisor (CTA o sill hold a clien whose accon was in a drawdown even of small size for longer han 2 ears B he same oen i is nliel ha a pariclar clien will olerae a 50% drawdown in an accon wih an average- or small-ris CTA Similarl in an invesmen ban sep a propriear ssem rader will be epeced o mae mone in ear a he longes ie he canno be in a drawdown for longer han a ear Also he/she ma be sh down if a cerain maimal drawdown condiion will be breached which normall is arond 20% of his bacing eqi Addiionall he will be given a warning drawdown level a which he will be reviewed for leing him eep rnning he ssem (arond 5% Obviosl hese isses mae managed accons praciioners ver concerned abo boh he magnide and draion of heir cliens' accons drawdowns Firs we wan o menion he paper b Grossman and Zho 993 where an assmpion of lognormali of eqi saisics and se of dnamic programming heor led o an eac analical solion of a maimal drawdown problem for a one-dimensional case A sbseqen generalizaion of his wor for mliple dimensions was done in Cvianic and Karazas 995 In difference o hese wors which find ime-dependen porfolio allocaion sraegies nder some assmpions on disribions of porfolio rerns his paper defines ris and finds opimal weighs (consan over ime for a single sample-pah of porfolio rerns We will be looing o find a consan se of weighs which will saisf a cerain ris condiion over a period of ime We mae no assmpion abo he nderling probabili disribion which allows considering varie of pracical applicaions We concenrae on he porfolio eqi crves over a pariclar sample-pah (hisorical or mos probable fre sample-pah ie we define some sample-pah ris fncion raher han a ris measre on se of sample-pahs Being perfecl aware of his insfficienc we leave he isse of definiion and opimizaion of he drawdown-pe ris measres on a se of sample-pahs for fre research Also we do no invesigae he predicive power of a consan se of weighs ring o inrodce and es he new approach in a simplified version To some eend we consider a sep similar o he inde racing problem considered b Dembo and King 992 where an inde hisorical performance is replicaed b a porfolio wih consan weighs In his paper we have inrodced and sdied a one-parameer famil of ris fncions called condiional drawdown-a-dis (CDaR This ris fncion qanifies in aggregaed forma he freqenc and magnide of he porfolio drawdowns over some period of ime for a pariclar sample-pah B definiion a drawdown is he drop in he porfolio vale comparing o he maimm achieved in he pas We can define drawdown in absole or relaive (percenage erms For eample if a he presen ime he porfolio vale eqals $9M and he maimal porfolio vale in he pas was $0M we can sa ha he porfolio drawdown in absole erms eqals $M and in relaive erms eqals 0% For some vale of he confidence parameer he -CDaR is defined as he mean of he wors ( 00% drawdowns eperienced over some period of ime on a sample-pah For insance 095-CDaR (or 95%- CDaR is he average of he wors 5% drawdowns over he considered ime inerval The CDaR ris fncion conains he average drawdown and maimal drawdown as is limiing cases I aes ino accon boh he magnide and draion of he drawdowns whereas he maimal drawdown fncion concenraes on a single even maimal porfolio loss compared o is previos highes vale 2

3 The CDaR ris fncion is relaed o Vale-a-Ris (VaR ris measre and o Condiional Valea-Ris (CVaR ris measre sdied in he paper b Rocafellar and Urasev 2000; 2002 B definiion wih a specified probabili level he -VaR of a porfolio is he lowes amon ζ sch ha he probabili ha he loss will no eceed ζ is greaer or eqal o in a specified ime τ (Jorion 996 whereas he -CVaR is -ail epecaion of losses (Rocafellar and Urasev 2000; 2002 For coninos disribions -ail epecaion of losses coincides wih he condiional epecaion of losses above he VaR vale ζ The CDaR ris fncion is based on CVaR concep and can be viewed as a modificaion of he CVaR o he case when he loss-fncion is defined as a drawdown for discree ime momens having eqal weighs (insead of scenario probabiliies CDaR and CVaR are concepall closel relaed percenile-based fncions The opimizaion approach developed for CVaR b Rocafellar and Urasev 2000; 2002 can be direcl eended o CDaR opimizaion A nice feare of he CVaR opimizaion approach is ha he resling problem is linear in an eended se of variables and he VaR vale ζ is calclaed aomaicall sing an ailiar variable (Rocafellar and Urasev 2000; 2002 The paper b Krohmal e al 2002 considers several eqivalen approaches for generaing rern-cvar efficien froniers; in pariclar i considers an approach which maimizes rern wih CVaR consrains In he case of coninos disribions CVaR is nown also as Mean Ecess Loss Mean Shorfall (Masser and Rosen 999 or Tail Vale-a-Ris (Arzner e al 999 A case sd on he hedging of a porfolio of opions sing he CVaR minimizaion echniqe is inclded in Rocafellar and Urasev 2000 Also he CVaR minimizaion approach was applied o credi ris managemen of a porfolio of bonds (Andersson e al 999 A case sd on opimizaion of a porfolio of socs wih CVaR consrains is considered in Krohmal e al 2002 Similar o he Marowiz mean-variance approach (Marowiz 952 in a sample-variance seing we formlae and solve he opimizaion problem wih he rern performance fncion and CDaR consrains The rern-cdar opimizaion problem is a conve opimizaion problem wih a linear performance fncion and piece-wise linear conve consrain (see definiion of convei in Rocafellar 970 This paper redces he rern-cdar opimizaion problem o a linear programming problem Linear programming allows solving opimizaion problems wih man hosands of insrmens The algorihm is fas nmericall sable and provides a solion dring one rn (wiho adsing parameers lie in geneic algorihms or neral newors Linear programming approaches are roinel sed in porfolio opimizaion wih varios crieria inclding mean absole deviaion (Konno and Yamazai 99 maimm deviaion (Yong 998 and mean regre (Dembo and King 992 A reader ineresed in oher applicaions of opimizaion echniqes in he finance area can find relevan papers in Ziemba and Mlve General Sep Denoe b fncion w ( he ncomponded porfolio rern a ime where he componens of he porfolio vecor ( 2 h m are weighs of m insrmens in he porfolio The drawdown fncion a ime is defined as he difference beween he maimm of he fncion w ( over he hisor preceding ime and he vale of his fncion a ime D( ma 0τ { w( τ } w( ( 3

4 We consider hree ris fncions defined on a sample-pah of porfolio rerns: (i maimm drawdown (MaDD (ii average drawdown (AvDD and (iii condiional drawdown-a-ris (CDaR The CDaR ris fncion is acall a famil of ris fncions parameerized b he parameer I conains as a special case he maimm drawdown and average drawdown ris fncions The maimm drawdown on he inerval [ 0 T ] is calclaed b maimizing he drawdown fncion D ( ie The average drawdown is eqal o { D( } M ( ma (2 0T T A ( D( d (3 T 0 Le denoe he nmber of ime sb-periods in he ime inerval [ 0 T ] and [0] denoe he confidence level In a case when ( is an ineger nmber (ie we are able o precisel con ( 00% of he drawdowns he CDaR is defined as he mean of he wors ( 00% drawdowns For insance if 0 CDaR eqals he average drawdown over all sb-periods and if 0 95 hen CDaR is he average of he wors 5% drawdowns Le ζ ( denoe a hreshold sch ha eacl ( 00% of drawdowns eceed his hreshold (i is spposed ha ( is an ineger nmber In his case CDaR wih a confidence level is he average of ( 00% of he drawdowns ( ( ( D d T Ω where Ω { [ 0 T] : D( ζ ( } However when ( is no ineger which is a general case CDaR is epressed as a liner combinaion of ζ ( and drawdowns sricl eceeding ζ ( similar as i was done for CVaR b Rocafellar and Urasev 2002 The following opimizaion formla presens CDaR in a general case T + ( min ζ + [ D( ζ ] d (4 ζ ( T 0 where [ g ] + ma {0 g} If ( is ineger hen opimal vale of ζ in (4 coincides wih ζ ( oherwise opimal ζ is greaer han or eqal o ζ ( When ends o CDaR ends o he maimm drawdown ie ( M( and when 0 CDaR coincides wih he average drawdown ie ( A( 0 In porfolio opimizaion problems depending on or ris preference we can impose consrains on he maimm drawdown given b (2 on average drawdown given b (3 M( νc 4

5 on CDaR given b (4 or combine several consrains ogeher A ( ν 2C ( ν 3C M ( ν C A( ν C ( ν C (5 2 3 where he consan C represens he available capial and coefficiens ν ν 2 and ν 3 define he capial porions allowed o be los Usall ν ν ν (6 Le be an insrmen inde m Sppose ha for a pariclar sample-pah rerns ( r ( r (2 r ( m for he porfolio insrmens are available Denoe b ( r ( he ncomponded cmlaive rern of he h insrmen p o he ime momen Le he vecor ( ( ( 2( l m ( be a se of ncomponded cmlaive rerns for m porfolio insrmens a ime momen The cmlaive porfolio rern eqals w( ( ( m i i i The average annalized rern R ( over he period [ 0 T ] which is a linear fncion of is defined as follows R( w( ( (7 where d is he nmber of ears in he ime inerval [ 0 T ] and C is he available capial We impose he so-called echnological consrains on he vecor given b bo ineqaliies { : ma i m } X i (8 min for some consan vales of min and ma However oher feasible consrains can be imposed In he ne secion we will consider he problem of maimizing rern R ( sbec o consrains on varios ris fncions and echnological consrains (8 on he porfolio posiions 3 Porfolio Opimizaion wih Drawdown Consrains The maimizaion of he average rern wih consrains on maimm drawdown (see (9 average drawdown (see (0 or CDaR (see ( can be formlaed as he following mahemaical programming problems: ma X s R( M( νc (9 5

6 6 ( ( ma 2C s R X ν A (0 ( ( ma 3C s R X ν ( Similar o Rocafellar and Urasev 2000; 2002 problems (9 (0 and ( can be redced o linear programming problems sing ailiar variables 4 Discree Formlaion: Redcion o Linear Programming Problems Denoe he vecor of cmlaive insrmen rerns p o he ime momen b ( (3 hen he drawdown fncion is defined b he epression D } ma { ( (4 and he average annalized rern fncion is given b he inner prodc R ( (5 Problems (9 (0 and ( are formlaed accordingl The opimizaion problem wih a consrain on maimm drawdown is } } ma {ma{ ma C s X ν which is redced o he following linear programming (LP problem 0 ma ma min 0 m i C s i ν (6 where are ailiar variables The opimizaion problem wih a consrain on average drawdown can be wrien as follows

7 7 } (ma{ ma 2 C s X ν or eqivalenl 0 ( ma ma min 0 2 m i C s i ν (7 Following he approach for Condiional Vale-a-Ris (CVaR see Rocafellar and Urasev 2002 i can be proved ha he opimizaion problem wih a consrain on CDaR ma be formlaed as follows ] } [ma{ ma 3 ( C s X ν ζ ζ + + which is redced o he LP 0 0 ma ma min 0 3 ( m i z z C z s i z + ζ ν ζ ζ (8 An imporan feare of formlaion (8 is ha i does no involve he hreshold fncion ( ζ A opimal solion of problem (8 variables and ζ give an opimal porfolio and corresponding vale of he hreshold fncion

8 5 Calclaion Resls For nmerical eperimens we have considered he eqi crves generaed b a fres echnical rading ssem in m 32 differen mares (a wide range of maor liqid mares: crrencies crrenc crosses US reasries boh shor- and long-erm foreign long-erm reasries inernaional eqi indices and meals The lis of mare icer smbols provided in he resls below is mnemonic and corresponds o he widel sed daa provider FreSorce The individal eqi crves covered a ime span of //988 hrogh 9//999 The eqi crves were based on $20M bacing eqi in a margin accon and were ncomponded ie he ris was aen based alwas on he original $20M and did no accon for he mone made or los dring a rading sraeg As for he echnological consrains (8 we chose min 0 2 and ma 0 8 This choice was dicaed b he need o have he reslan margin-o-eqi raio in he accon wihin admissible bonds which are specific for a pariclar porfolio In his fres rading sep hese consrains are analogos o he fll-invesed condiion from classical Sharpe-Marowiz heor The define bonds on he leverage of he sraeg and mae an efficien fronier o be concave If all posiions are eqal o he lower bond 02 hen he sm of he posiions eqals and he minimal leverage eqals 64 However if all posiions are eqal o he pper bond 08 hen he sm of he posiions eqals and he maimal leverage eqals 256 The opimal allocaion of weighs pics boh he opimal leverage and proporions beween insrmens The absence of echnological consrains (8 leads o he infinie leverage of hese pes of sraegies and as a resl an efficien fronier wold be a sraigh line passing hrogh (00 Anoher sble isse is he sabili of he opimal porfolios in he case when he consrains are "oo la" I is a maer of empirical evidence ha he more la he consrains are he beer porfolio eqi crve o can ge hrogh opimal miing and he less sable hese resls wold be wih respec o wal-forward analsis The aforemenioned se of consrains was empiricall fond o be boh leading o sfficienl sable porfolios and allowing enogh mi of he individal eqi crves The problem hen is o find a se of weighs ( 2 l m sch ha i solves he minimizaion problems (6 (7 or (8 These linear programming problems have been solved b he CPLEX solver (inps are prepared wih C++ programming langage An alernaive verificaion of he solions was obained via solving hese opimizaion problems b a geneic algorihm implemened in VB6 discssion of which is beond he presen scope Le s denoe (6 (7 and (8 b MaDD AvDD and -CDaR problems respecivel We have solved hese opimizaion problems for cases of ( and As we have noed before cases of ( 0 and ( correspond o MaDD and AvDD problems accordingl Tables and 2 presen he lis of mares and corresponding ses of opimal weighs for MaDD and AvDD problems Table 3 presens he weighs for he CDaR case wih he confidence level ( 005 In hese ables he solion achieving maimal reward/ris raio is boldfaced oe ha he smalles vale of ris is chosen in sch a wa ha he solions of he opimizaion problem sill eis This means ha each problem does no have a solion beond he pper and 8

9 lower bonds of he ris range covered (he whole efficien fronier is shown The ris and rae of rern are epressed in percenage erms wih respec o he original accon size ie $20M The efficien froniers for he reward-madd and reward-avdd problems are shown in Figres and 2 respecivel We do no show efficien froniers for reward-cdar problem in separae graphs (ecep for MaDD and AvDD However in Figre 3 we show he reward-madd graphs for porfolios opimal wih ( and CDaR consrains As epeced he case wih ( 0 CDaR corresponding o MaDD has a concave efficien fronier dominaing oher graphs The reward is no maimal for each level of MaDD when we solved he opimizaion problems wih ( and CDaR consrains Viewed from he reference poin of MaDD problem ( < solions are niforml worse However none of hese solions are rl beer or worse han ohers from a mahemaical sandpoin Each of hem provides he opimal solion in is own sense Some hoghs on qali of solions are discssed from a pracical sandpoin below Similar o Figre 3 Figre 4 depics he reward- AvDD graphs for porfolios opimal wih ( and CDaR consrains The case wih ( CDaR corresponding o AvDD has a concave efficien fronier dominaing oher graphs As in classical porfolio heor we are ineresed in a porfolio wih a maimal reward/ris raio ie he porfolio corresponding o he poin of conac of he sraigh line coming hrogh (00 and he efficien fronier For he ris defined in erms of problems (6 (7 and (8 we will call he reward/ris raios as MaDDRaio AvDDRaio and CDaRRaio which b definiion are: MaDDRaio R( M( AvDDRaio R( A( and CDaRRaio R( ( The chars of MaDDRaio and AvDDRaio qaniies are shown in Figres 5 and 6 for he same cases of ( as in Figres 3 and 4 We have solved opimizaion problem (8 for cases of ( and Le s noe ha he case of ( 0 05 (see Table 3 considering minimizaion of he wors 5% par of he nderwaer crve prodces a se of weighs significanl differen from he ( 0 case (MaDD problem and ( 0 05 CDaR case conains several ens of evens over which he averaging was performed We consider ha opimizaion wih ( 005 or 0 consrains prodces a more robs porfolio han he opimizaion wih MaDD or AvDD consrains CDaR solion aes ino accon man significan drawdowns comparing o MaDD considering onl he larges drawdown Also CDaR solion is no impaced b man small drawdowns lie a solion in AvDD case We have also verified or resls via solving he relaed nonlinear opimizaion problems corresponding o problems (6-8 These problems have opimized he drawdown raios corresponding o he same se of consrains Verificaion was done sing a geneic opimizaion algorihm We were saisfied o find ha his procedre has prodced he same ses of weighs for he opimal solions 9

10 6 Conclsions We have inrodced a new CDaR ris fncion which is qie sefl in pracical porfolio managemen This fncion is based on CVaR ris measre and has he MaDD and AvDD ris fncions as is limiing cases We have sdied reward/ris raios implied b hese ris fncions namel MaDDRaio AvDDRaio and CDaRRaio We have shown ha he porfolio allocaion problem wih CDaR MaDD and AvDD ris fncions can be efficienl solved We have formlaed and solved a porfolio allocaion problem for a real-life eample Implemened in a managed accons' environmen hese developmens will allow a rading or ris manager o allocae ris according o his/her personal assessmen of magnide and draion of ereme drawdowns of a porfolio eqi We hin ha solions prodced b he MaDD approach ma conain a significan amon of saisical error since i is based on a single observaion of maimal loss The CDaR famil of ris fncions allows a ris manager o conrol he wors ( 00% drawdowns De o he saisical averaging of drawdowns a beer predicion of ris in he fre and a more sable porfolio can be obained Or sdies indicae ha CDaR conrol wih an appropriae level (eg 095 ie opimizing over he 5% of he wors drawdowns prodces a more sable weigh allocaion compared o one prodced b MaDD conrol A deailed sd of his isse calls for a separae pblicaion 0

11 Table Lis of mares and corresponding ses of opimal weighs for he MaDD problem The solion achieving maimal reward/ris raio is boldfaced Ris % Reward % Reward/Ris OPTIMAL PORTFOLIO COFIGURATIO AAO AD AXB BD BP CD CP DGB DX ED EU FV FXADJY FXBPJY FXEUBP FXEUJY FXEUSF FXZUS FXUSSG FXUSSK GC JY LBT LFT LGL LML M SF SI SJB SI TY

12 Table 2 Lis of mares and corresponding ses of opimal weighs for he AvDD problem The solion achieving maimal reward/ris raio is boldfaced Ris % Reward % Reward/ris OPTIMAL PORTFOLIO COFIGURATIO AAO AD AXB BD BP CD CP DGB DX ED EU FV FXADJY FXBPJY FXEUBP FXEUJY FXEUSF FXZUS FXUSSG FXUSSK GC JY LBT LFT LGL LML M SF SI SJB SI TY

13 Table 3 Lis of mares and corresponding ses of opimal weighs for he CDaR problem wih ( 005 The solion achieving maimal reward/ris raio is boldfaced Ris % Reward % ζ % Reward/Ris OPTIMAL PORTFOLIO COFIGURATIO AAO AD AXB BD BP CD CP DGB DX ED EU FV FXADJY FXBPJY FXEUBP FXEUJY FXEUSF FXZUS FXUSSG FXUSSK GC JY LBT LFT LGL LML M SF SI SJB SI TY

14 Figre Efficien fronier for he MaDD problem (rae of rern verss MaDD Efficien Fronier: MaDD R( MaDD M( Figre 2 Efficien fronier for he AvDD problem (rae of rern verss AvDD Efficien Fronier: AvDD R( AvDD A( 4

15 Figre 3 Reward-MaDD graphs for opimal porfolios wih ( and CDaR consrains (rae of rern verss MaDD The fronier is efficien onl for he case wih ( 0 CDaR consrains which corresponds o he MaDD ris measre R( Reward-MaDD % CDaR 5% CDaR 40% CDaR 00% CDaR MaDD M( Figre 4 Reward-AvDD graphs for opimal porfolios wih ( and CDaR consrains (rae of rern verss AvDD The fronier is efficien onl for he case wih ( CDaR consrains which corresponds o he AvDD ris measre R( Reward-AvDD % CDaR 5% CDaR 40% CDaR 00% CDaR AvDD A( 5

16 Figre 5 MaDDRaio graphs for opimal porfolios wih ( and CDaR consrains (MaDDRaio verss MaDD The maimm MaDDRaio is achieved in he case wih ( 0 CDaR consrains which corresponds o he MaDD ris measre MaDD Raio % CDaR 5% CDaR 40% CDaR 00% CDaR M( Figre 6 AvDDRaio graphs for opimal porfolios wih ( and CDaR consrains (AvDDRaio verss AvDD The maimm AvDDRaio is achieved in he case wih ( CDaR consrains which corresponds o he AvDD ris measre AvDD Raio A( 0% CDaR 5% CDaR 40% CDaR 00% CDaR 6

17 Bibliograph Andersson F Masser H Rosen D and S Urasev 200 Credi Ris Opimizaion wih Condiional Vale-A-Ris Crierion Mahemaical Programming Series B 89 pp Arzner P F Delbaen J M Eber and D Heah 999 Coheren Measres of Ris Mahemaical Finance 9 pp Cvianic J and I Karazas 995 On Porfolio Opimizaion Under Drawdown Consrains IMA Lecre oes in Mahemaics & Applicaions 65 pp 7788 Dembo R S and A J King 992 Tracing Models and he Opimal Regre Disribion in Asse Allocaion Applied Sochasic Models and Daa Analsis 8 pp 557 Jorion P 996 Vale a Ris: A ew Benchmar for Measring Derivaives Ris (Irwin Professional Pblisher Grinold R C and R Kahn 999 Acive Porfolio Managemen (ew Yor: McGraw- Hill Grossman S J and Z Zho 993 Opimal Invesmen Sraegies for Conrolling Drawdowns Mahemaical Finance 3(3 pp Konno H and H Yamazai 99 Mean Absole Deviaion Porfolio Opimizaion Model and Is Applicaion o Too Soc Mare Managemen Science 37 pp 5953 Krohmal P Palmqis J and S Urasev 2002 Porfolio Opimizaion wih Condiional Vale-A-Ris Obecive and Consrains The Jornal of Ris 4(2 Marowiz H M 952 Porfolio Selecion Jornal of Finance 7( pp 779 Masser H and D Rosen 999 Beond VaR: From Measring Ris o Managing Ris ALGO Research Qarerl (2 pp 520 Rocafellar R T 970 Conve Analsis (Princeon Universi Press Rocafellar R T and S Urasev 2000 Opimizaion of Condiional Vale-a-Ris The Jornal of Ris 2 pp 242 Rocafellar R T and S Urasev 2002 Condiional Vale-a-Ris for General Loss Disribions Jornal of Baning and Finance 26 pp Yong M R 998 A Minima Porfolio Selecion Rle wih Linear Programming Solion Managemen Science 44(5 pp Ziemba W T and J M Mlve 998 Worldwide Asse and Liabili Modeling (Cambridge Universi Press 7