Robust Mean-Conditional Value at Risk Portfolio Optimization

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1 3 rd Coferece o Facal Matheatcs & Applcatos, 30,3 Jauary 203, Sea Uversty, Sea, Ira, pp. xxx-xxx Robust Mea-Codtoal Value at Rsk Portfolo Optato M. Salah *, F. Pr 2, F. Mehrdoust 2 Faculty of Matheatcal Sceces, Uversty of Gula, Rasht, Ira *Correspodg author: salah@gula.ac.r 2 Faculty of Matheatcal Sceces, Uversty of Gula, Rasht, Ira faraehpr@yahoo.co, fehrdoust@gula.ac.r Abstract: I the portfolo optato, the goal s to dstrbute the fxed captal o a set of vestet opportutes to axe retur whle aagg rsk. Rsk ad retur are quatt es that are used as put paraete rs for the optal allocato of the captal the suggested odels. But these quattes are ot kow at the te of the forulato ad solvg proble. hus they shou ld be estated to solve the proble whch ght lead to large error. Oe of the wdely used approaches to deal wth such a stuato, s robust optato. I ths paper we study the ea- Codtoal Value at Rsk (M-CVaR) portfolo selecto probles uder the estato rsk ea retur for both terval ad ellpsodal ucertaty sets. Equvalet forulatos of the robust couterparts are gve. At ed a exaple s gve to deostrate the pact of ucertaty. KeyWords: Portfolo Optato, Robust Optato, Value at Rsk, Codtoal Value at Rsk, Coc Optato.. Itroducto he ature of the vestet ad busess actvtes s such that to acheve retur are requred to bear the rsk. herefore, whe decdg for the vestet, a vestor has to accept a balace betwee rsk ad retur. Hece, portfolo optato has bee deostrated as a portat ploy vestet ad has led to create ay theores ad odels ths cotext [8]. Oe of the ost faous theores s optal selecto of portfolo theory troduced by Harry Markowt 952 [0]. he Markowt ea-varace (MV) odel has bee used as the stadard fraework for optal portfolo selecto probles. I ths odel, portfolo retur s represeted by the expected retur ad rsk of the portfolo s easured by the varace of the portfolo returs. he varace s a statstcal dsperso easure that gves the average of the squared dstace of the possble returs fro the expected retur. So, a asset wth retur better tha expected retur s assued to be as rsky as a asset wth retur lower tha expected retur, whereas ost vestors do't cosder rsk of the hgh retur. Hece, the varace s a adequate easure for calculatg rsk oly whe the returs of the uderlyg assets are dstrbuted syetrcally. Hece, facal sttutos ad dvduals atteto was attracted to rsk aageet ters of percetles of loss dstrbuto such as Value at Rsk (VaR). Istead of regardg the whole spreadg aroud the expected retur, VaR cosders oly the spreadg to the left of the expected retur as rsk ad represets the predcted axu loss wth a specfed cofdece level (e.g. 95%) over a certa perod of te (e.g. oe day) [4,2]. VaR s wdely used the facal dustry, but despte ths popularty, t has udesrable propertes that restrct ts use [2, 9, ]. Oe of these propertes s that VaR lacks subaddtvty (for a rsk easure f, we should have f ( x x2) f ( x ) f( x2) ). Obvously dversfcato reduces rsk, thus "he total rsk of two dfferet vestet portfolos does ot exceed the su of the dvdual

2 3 rd Coferece o Facal Matheatcs & Applcatos, 30,3 Jauary 203, Sea Uversty, Sea, Ira, pp. xxx-xxx rsks". Aother drawback of VaR s that t s a o-covex fucto []. hus, optg VaR ths aer ght lead to local opters, whle we look for the global u. hese drawbacks of VaR led to the developet of alteratve rsk easures. Oe well-kow odfcato of VaR s obtaed by coputg the expected loss gve that the loss exceeds VaR. hs quatty s called Codtoal Value-at-Rsk (CVaR). VaR ples that "what s the axu loss that we reale?" but CVaR asks: "How do we expect to cur losses whe stuato s udesrable?" CVaR also cosders oly oe tal of the loss dstrbuto, whch correspods to hgh losses, ad does ot accout for the opposte tal represetg hgh returs. hs s a preferece over Markowt approach that as was etoed t cosders both of the tal of the loss dstrbuto. Nuercal experets show that u CVaR ofte lead to optal solutos close to the u VaR, because VaR ever exceeds CVaR. hus, portfolos wth low CVaR should also have a low VaR. I addto, whe retur (loss) dstrbuto s oral, VaR ad CVaR are equal aely they result sae optal portfolos. However for skewed retur dstrbutos, optal VaR ad CVaR portfolos ay be qute dfferet. I addto, the tal of the loss dstrbuto hgher losses of VaR ay be log, because usg VaR, the hgher losses of VaR are ot cotrolled. But CVaR cotrols the hgher losses of VaR, thereupo t s ore coservatve tha VaR. Also CVaR optato s a covex prograg proble ad thus t s easy to opte [5]. I ths paper, frst we troduce ea- CVaR odel secto 2. he due to the uavodable estato error of the expected retur of the assets, Secto 3 we utle robust optato to deal wth t for terval ad ellpsodal ucertaty sets. he orgal odel ad the robust odels are copared o a exaple showg soe useful feature of the latter oes. 2. Mea-Codtoal Value at Rsk Cosder assets S,..., S, 2, wth rado returs. Suppose ad retur ad the stadard devato of the retur of asset S. Moreover j deote the expected j, deote the letfor correlato coeffcet of the returs of assets S ad S j, = [,..., ] ad Q ( j ) be the syetrc covarace atrx. Now f we deote by x as the proporto of holdg the asset, oe ca represet the expected retur ad the varace of the resultg portfolo x as follows: E [ x ] x... x x, Var ( x ) x x x Qx,, j j j j where. Also, we wll assue that the set of feasble portfolos s a oepty polyhedral set = x Ax b, Cx d where A s a atrx, b s a -desoal ad represet t as vector, C s a p atrx ad d s a p -desoal vector. I partcular, oe of the costrats the set s x =. Let f ( x, y ) deote the loss fucto whe we choose the portfolo x fro a set of feasble portfolos ad y s the realato of the rado evets. We cosder the portfolo retur loss, f ( x, y ), the egatve of the portfolo retur that s a covex (lear) fucto of the portfolo varables x : f ( x, y) - y x -[ yx... yx]. () We assue that the rado vector y has a probablty desty fucto deoted by py ( ). For a fxed decso vector x, the cuulatve dstrbuto fucto of the loss assocated wth that vector s coputed as follows:

3 3 rd Coferece o Facal Matheatcs & Applcatos, 30,3 Jauary 203, Sea Uversty, Sea, Ira, pp. xxx-xxx ( x, ) p( y) dy. (2) f ( x, y) he, for a gve cofdece level, the -VaR assocated wth portfolo x s defed by VaR ( x) x,. (3) Moreover, we defe the -CVaR assocated wth portfolo x as follows: CVaR ( x) f ( x, y) p( y) dy. (4) ( ) f ( x, y) VaR ( x) heore -2. We always have: CVaR( x) VaR( x), that eas CVaR of a portfolo s always at least as bg as ts VaR. Cosequetly, portfolos wth sall CVaR also have sall VaR. However, geeral g CVaR ad VaR are ot equvalet. Proof: See [5]. Sce the defto of CVaR ples the VaR fucto clearly, t s dffcult to work wth ad opte ths fucto. Istead, the followg spler auxlary fucto s cosdered [5]: F ( x, ) ( (, ) ) ( ) (5) ( ) f x y p y dy f ( x, y) ad or F ( x, ) ( (, ) ) ( ) (6) ( ) f x y p y dy where a ax a,0 coputg VaR ad CVaR [5]:. F s a covex fucto of. 2. VaR s a er over of F.. As a fucto of, F has the followg u propertes that are useful 3. he u value over of the fucto F s CVaR. Usg these propertes, we have CVaR ( x) F ( x, ). (7) x x, hus, we ca opte CVaR drectly, wthout eedg to copute VaR frst. Sce we cosder the loss fucto f ( x, y ) s a covex (lear) fucto of the portfolo varables x, the F ( x, ) s also a covex (lear) fucto of x. hus f the feasble portfolo set s also covex, the optato probles (7) are covex optato probles that are effcetly solvable. Istead of usg the desty fucto py ( ) of the rado evets forulato (6) whch s ofte possble or udesrable to copute, we ca use a uber of scearos y,...,. hus we cosder the followg approxato to the fucto F ( x, ) : F ( x, ) ( f ( x, y ) ) (8) ( ) Now (7) becoes ( f ( x, y ) ). (9) x, ( ) hs tself ca be reforulated as follows:

4 3 rd Coferece o Facal Matheatcs & Applcatos, 30,3 Jauary 203, Sea Uversty, Sea, Ira, pp. xxx-xxx x,, ( ) s.t. 0, =,..., (0) x. whch s a covex optato proble for covex fucto f. f ( x, y ), =,..., It should be oted that rsk aagers ofte try to opte rsk easure whle expected retur s ore tha a threshold value. I ths case, the ea-cvar odel s as follows: x,, ( ) s.t. x R, () 0, =,..., f ( x, y ), =,..., x. 3. Robust ea-cvar odel he expected retur of the assets as put paraeter for the ea-cvar proble s ot kow, thus t should be estated order to solve the proble. But estatg t ght lead to large error. he effect of the estato error ca be observed Fgure () that shows the resultg effcet froters of the s ea-cvar proble wth true ad estated paraeters for a 8-asset exaple take fro [3]. Fgure : rue effcet froter ad estated effcet froter o deal wth the estato error, several ethods such as robust optato have bee suggested [6, 7]. Robust optato refers to the odelg of optato probles wth data ucertaty. Ucertaty the paraeters s descrbed through ucertaty sets that cota all (or ost) possble

5 3 rd Coferece o Facal Matheatcs & Applcatos, 30,3 Jauary 203, Sea Uversty, Sea, Ira, pp. xxx-xxx values that ca be realed by the ucerta paraeters. Ucertaty over the paraeters s dscussable several approaches. wo of the ost coo approaches are terval ad ellpsodal ucertates. I ths paper, we wll use both of the ad preset equvalet odels of robust ea-cvar odel. Frst let us cosder ea-cvar odel wth the terval ucertaty over expected retur, aely x,, ( ) where ad also equvalet to s.t. x R, (2) 0, =,..., x, f ( x, y ), =,..., S, L U S, L U ad are two gve vectors. Sce x R should hold for all S, thus (2) s x,, ( ) L s.t. ( ) x R, (3) 0, =,..., f ( x, y ), =,..., x. Obvously, f the fucto f s lear, the the proble s a lear prograg proble, whch s effcetly solvable usg teror pot ethods. However, f the ucertaty set s ellpsodal, aely S Mu, u, where M s a -desoal atrx, the to establsh x R the proble (), we should have: x R (( +Mu) x) R ( ) x + (Mu) x R. Cosequetly t s eough to have ( ) x - M x R or M x ( ) x R where s a secod order coc costrat. hus, the robust ellpsodal ucertaty set s as follows: x,, u ( ) u ea-cvar proble () uder s.t. M x ( ) x R, (4) 0, =,..., f ( x, y ), =,..., x.

6 3 rd Coferece o Facal Matheatcs & Applcatos, 30,3 Jauary 203, Sea Uversty, Sea, Ira, pp. xxx-xxx Obvously, f the fucto f s lear, (4) s a secod order coc prograg proble (SOCP) that s effcetly solvable usg teror pot ethods [3]. 4- Nuercal Results I ths secto, the perforace of the robust ea-cvar odels usg terval ad ellpsodal ucertaty sets wth ea-cvar odel are copared usg actual data. he dataset used here s avalable returs for eght assets related to 50 cosecutve oths that expected retur ad covarace atrx of the retur of assets have bee gve tables () ad (2). he ea-cvar effcet froter ad the average of 00 froters obtaed fro two odels are gve Fgure (2). We observe that the effcet froter for the terval ucertaty set les above the effcet froter obtaed fro the ellpsodal ucertaty set. hs fgure also dcates that resultg robust portfolos ca be too coservatve ad axu expected retur obtaed fro the s sgfcatly less tha ea-cvar effcet froter. hus these ethods ca be approprate for coservatve vestors to prevet losses due to tal paraeter ucertates, but t s ot a approprate choce for the vestors who are ore tolerat to estato rsk ad wsh to seek hgher returs. However, the coservats of the robust portfolos obtaed usg terval ucertaty set ca be adjusted by decreasg the ucertaty terval. But t s portat to ote that the worst saples are elated the ew ucertaty sets ad the robustess of the odel s reduced copared wth the robust froter obtaed fro orgal terval ucertaty set Fgure (3), the froters obtaed based o elatg 2.5% ad 5% of the worst saples Fgures (4) ad (5) becoe loger ad acheve hgher axu expected returs. It also shows that ths odel ca be very sestve to the choce of the boudary L. herefore, the ucertaty set ust be carefully chose. 0.0 S S2 S3 S4 S5 S6 S7 S able : Mea retur vector 0.0 S S2 S3 S4 S5 S6 S7 S8 S S S S S S S S able 2: Covarace atrx Q

7 3 rd Coferece o Facal Matheatcs & Applcatos, 30,3 Jauary 203, Sea Uversty, Sea, Ira, pp. xxx-xxx Fgure 2: Robust ea-cvar froters usg terval ad ellpsodal ucertaty sets Fgure 3: Robust ea-cvar froters usg terval ucertaty set wth elatg 0% of the saples

8 3 rd Coferece o Facal Matheatcs & Applcatos, 30,3 Jauary 203, Sea Uversty, Sea, Ira, pp. xxx-xxx Fgure 4: Robust ea-cvar froters usg terval ucertaty set wth elatg 2.5% of the saples Fgure 5: Robust ea-cvar froters usg terval ucertaty set wth elatg 5% of the saples Refereces: [] Alexader S, Colea.F, ad L Y. Mg VaR ad CVaR for a Portfolo of Dervatves. Joural of Bakg ad Face, 2006; 30(2): [2] Arter P, Delbae F, Eber J ad Heath D. hkg coheretly, Rsk, 997; 0:68 7. [3] Boyd S, Grat M. Cvx users, gude for cvx verso.2, [4] Cho W.N. Robust Portfolo Optato Usg Codtoal Value At Rsk, Iperal College Lodo, Departet of Coputg, Fal Report, [5] Coruejols G, utucu R, Optato Methods Face, Cabrdge Uversty press, 2006.

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