M. Salah, F. Mehrdoust, F. Pr Uversty of Gula, Rasht, Ira CVaR Robust Mea-CVaR Portfolo Optmzato Abstract: Oe of the most mportat problems faced by every vestor s asset allocato. A vestor durg makg vestmet decsos has to search for equlbrum betwee rsk ad returs. Rsk ad retur are ucerta parameters the suggested portfolo optmzato models ad should be estmated to solve the problem. he estmato mght lead to large error the fal decso. Oe of the wdely used ad effectve approaches for optmzato wth data ucertaty s robust optmzato. I ths paper, we preset a ew robust portfolo optmzato techque for mea-cvar portfolo selecto problem uder the estmato rsk mea retur. We addtoally use CVaR as rsk measure, to measure the estmato rsk mea retur. Moreover, to solve the model effcetly, we use the smoothg techque of Alexader et al. []. We compare the performace of the CVaR robust mea-cvar model wth robust mea-cvar models usg terval ad ellpsodal ucertaty sets. It s observed that the CVaR robust mea-cvar portfolos are more dversfed. Moreover, we study the mpact of the value of cofdece level o the coservatsm level of a portfolo ad also o the value of the maxmum expected retur of the portfolo. Keywords: Portfolo Optmzato; Robust Optmzato; Value at Rsk; Codtoal Value at Rsk; Smoothg Itroducto Portfolo optmzato s oe of the best kow approaches facal portfolo selecto. he earlest techque to solve the portfolo selecto problem was developed by Harry Markowtz the 952. I ths method that s called mea-varace (MV portfolo optmzato model, the portfolo retur s measured by the expected retur of the portfolo ad the assocated rsk s measured by the varace of portfolo returs [6]. Varace as the rsk measure has ts weakesses. Cotrollg the varace does ot oly lead to low devato from the expected retur o the dowsde, but also o the upsde [7]. Hece, alteratve rsk measures have bee suggested to replace the varace such as Value at Rsk ( VaR that maage ad cotrol rsk terms of percetles of loss dstrbuto. Istead of regardg the both upsde ad dowsde of the expected retur, VaR cosders oly the dowsde of the expected retur as rsk ad represets the predcted maxmum loss wth a specfed cofdece level (e.g. 95% over a certa perod of tme (e.g. oe day [5, 6, 5].
VaR s a popular rsk measure. However, VaR may have drawbacks ad udesrable propertes that lmt ts use [2,, 4]. Such as lack of subaddtvty,.e., VaR of two dfferet vestmet portfolos may be greater tha the sum of the dvdual VaR s. Also, VaR s ocovex ad osmooth ad has multple local mmum, whle we seek the global mmum [6, 0, 4]. So alteratve rsk measures was troduced such as Codtoal Value at Rsk ( CVaR - the codtoal expected value of loss, uder the codto that t exceeds the value at rsk [5]. VaRmples that what s the maxmum loss that we realze? but CVaR asks: How do we expect to cur losses whe stuato s udesrable?. Numercal expermets show that mmum CVaR ofte lead to optmal solutos close to the mmum VaR, because VaR ever exceeds CVaR [5]. CVaR has better propertes tha VaR. CVaR optmzato s a covex optmzato problem ad thus t s easy to optmze [6]. It s demostrated that lear programmg techques ca be used for optmzato of CVaR rsk measure [0, 5]. he rest of the paper s arraged as follows. I Secto 2, we state the mea- CVaR portfolo selecto problem. he because of the evtable estmato error of the mea retur of the assets, we preset robust optmzato by CVaR Secto 3. o solve the model effcetly, we use the smoothg techque of Alexader et al. []. Fally, Secto 4, we compare the performace of the CVaR robust mea-cvar model wth robust mea-cvar models usg terval ad ellpsodal ucertaty sets o a example. We have observed that the CVaR robust mea-cvar portfolos are more dversfed ad they are sestve to tal data used to geerate each set of samples. Moreover, we demostrate the value of cofdece level affects o the coservatsm level, dversfcato ad also o the value of the maxmum expected retur of the resultg portfolos. 2 Mea-Codtoal Value at Rsk S Cosder assets,..., S, 2, wth radom returs. Suppose S deote the expected retur of asset x ad also cosder as the proporto of holdg the th asset. We ca represet the expected retur of the resultg portfolo x as follows: E[ x] x... x x. ( Also, we wll assume that the set of feasble portfolos s a oempty polyhedral set ad represet that as x Ax b, Cx d where A s a m matrx, b s a m -dmesoal vector, C s a p matrx ad d s a p -dmesoal vector [6]. I partcular, oe of the costrats the set s x =. f ( x, y Let deote the loss fucto whe we choose the portfolo x from a set of feasble portfolos ad y s the realzato of the radom evets (the vector of the asset returs of assets. We cosder f ( x, the portfolo retur loss, y, the egatve of the portfolo retur that s a covex (lear fucto of the portfolo varables x :
f ( x, y y x [ y x... y x ]. (2 We assume that the radom vector y has a probablty desty fucto deoted by py (. For a fxed decso vector x, the cumulatve dstrbuto fucto of the loss assocated wth that vector s computed as follows: he, for a gve cofdece level, the Also, we defe the CVaR assocated wth portfolo x as: f ( x, y ( x, p( y dy. (3 VaR assocated wth portfolo x s represeted as VaR ( x m x,. (4 CVaR ( x f ( x, y p( y dy. (5 ( f ( x, y VaR ( x CVaR ( x VaR ( x heorem -2. We always have:, that meas CVaR of a portfolo s always at least as bg as ts VaR. Cosequetly, portfolos wth small CVaR also have small VaR. However, geeral mmzg CVaR ad VaR are ot equvalet. Proof: See [6]. Sce the defto of CVaR mples the VaR fucto clearly, t s dffcult to work wth ad optmze ths fucto. Istead, the followg smpler auxlary fucto s cosdered: ad or a where max a, 0 F ( x, ( f ( x, y p( y dy, (6 ( f ( x, y F ( x, ( f ( x, y p( y dy, (7 (. hs fucto, cosdered as a fucto of, has the followg mportat propertes that makes t useful for the computato of VaR ad CVaR [6]:. F s a covex fucto of. 2. VaR s a mmzer over of F. 3. he mmum value over of the fucto F s CVaR. As a cosequece of the lsted propertes, we mmedately deduce that, order to mmze over x F ( x,, we eed to mmze the fucto wth respect to x ad smultaeously: CVaR ( x m CVaR ( x m F ( x,. (8 x x,
Cosequetly, we ca optmze CVaR drectly, wthout eedg to compute VaR frst. Sce we assumed that the loss fucto f ( x, y s the covex (lear fucto of the portfolo varables x, so F ( x, s also a covex (lear fucto of x. I ths case, provded the feasble portfolo set s also covex, the optmzato problems equato (8 are covex optmzato problems that ca be solved usg well kow optmzato techques for such problems. Istead of usg the desty fucto py ( of the radom evets formulato (7 that t s ofte mpossble or udesrable to compute t, we ca use a umber of scearos the ames of,..., F ( x,. I ths case, we cosder the followg approxmato to the fucto : y for F ( x, ( f ( x, y. (9 ( Now, the problem m CVaR ( x x, we replace F ( x, wth F ( x, : m ( f ( x, y. (0 x, ( z o solve ths optmzato problem, we troduce artfcal varables so, we add the costrats z 0 z (, ad f x y to the problem [5]: to replace ( f ( x, y. o do m xz,, ( s. t. z 0,,..., ( z f ( x, y,,..., x. It should be oted that rsk maagers ofte try to optmze rsk measure whle expected retur s more tha a threshold value. I ths case, we ca represet mea- CVaR model as follows: or m xz,, ( z s. t. x R, (2 z 0,,..., z f ( x, y,,..., x, z
m x ( xz,, ( s. t. z 0,,..., (3 z f ( x, y,,..., x, where the frst costrat of problem (2 dcates that the expected retur s o less tha the target value R ad 0 used problem (3 s rsk averso parameter that adapts the balace betwee CVaR ( x expected retur ad. It s mportat to ote that there s a equvalece betwee R ad so that the problems (2 ad (3 geerate the same effcet froters. Sce f ( x, y s lear x, all the z (, expressos f x y represet lear costrats ad therefore the problem s a lear programmg problem that ca be effcetly solved usg the smplex or teror pot methods. 3 CVaR robust mea- CVaR model Oe of the ucerta parameters for mea- CVaR model s ad usg estmatos for ths parameter leads to a estmato rsk portfolo selecto. I partcular, small dffereces the estmatos of ca create large chages the composto of a optmal portfolo. Oe way to reduce the sestvty of mea- CVaR model to the parameter estmatos s usg robust optmzato to determe the optmal portfolo uder the worst case scearo the ucertaty set of the expected retur. o ths ed, we represeted robust mea- CVaR models wth terval ad ellpsodal ucertaty sets the prevous studes that have bee demostrated formulatos (4 ad (5 respectvely. m xz,, ( L s. t. ( x R, (4 m xz,, z 0,,..., z f ( x, y,,..., x, ( s. t. M x x R, (5 z z 0,,..., z f ( x, y,,..., x, z z where L s a gve vector ad M s a -dmesoal matrx. Now, we preset CVaR robust mea- CVaR portfolo optmzato problem that estmato rsk mea retur s measured by CVaR. he CVaR robust mea- CVaR model specfes a optmal portfolo based
o the tal of the mea loss dstrbuto ad the adjustmet of the cofdece level wth regard to the preferece of the vestor correspods to the adjustmet of the coservatve level wth cosderg the ucertaty of the mea retur [8]. I ths model, CVaR s used to measure the rsk of the portfolo retur as before. I addto, whe usg the mea- CVaR model, we cosder the ucertaty of the expected retur that ca be cosdered as estmato rsk ad use CVaR to measure estmato rsk. CVaR wth ths perspectve s deoted as CVaR (We use y CVaR to deote the CVaR rsk measure dscussed Secto 2 order to dfferetate y t from CVaR F ( x, ad also we use F ( x, to deote ts assocated. hus, cosderg the problem (3, a CVaR robust mea- CVaR portfolo wll be determed as the soluto of the followg optmzato problem: m CVaR ( x ( z xz,, ( s. t. z 0,,..., (6 z f ( x, y,,..., x. For a portfolo of assets, we assume s the radom vector of the expected returs of the assets wth a probablty desty fucto p(. o determe the mea loss of the portfolo, we defe mea loss fucto, f( x,, as follows [8]: So, for cofdece level CVaR ( x, ca be defed as follows: f ( x, x [ x... x ]. (7 CVaR ( x m( E([ f ( x, ]. (8 Accordg to the defto of CVaR ad CVaR robust mea- CVaR model, we fd that CVaR wll crease as the value of creases. hs correspods to takg more pessmsm o the estmato rsk the model ad to optmze the portfolo uder worse cases of the mea loss. hus, the resultg CVaR robust portfolo s more coservatve. Coversely, coservatsm of the portfolo s reduced as the value of decreases [8]. I secto 4, we wll llustrate the mpact of the value of o the coservatsm level of a portfolo ad also o the value of the maxmum expected retur of the portfolo. As before, we ca cosder a auxlary fucto to smplfy the computatos: ad use the followg approxmato to the fucto F ( x, ( f ( x, p( d, (9 ( F ( x, :
F ( x, ( x, (20 ( m m Where,..., m are a collecto of m depedet samples for based o ts desty fucto We ca show that [5]: p(. So, wth troducg artfcal varables v m CVaR ( x m F ( x,. (2 x to replace ( f( x, F x, ( x, ad addg the v 0 costrats v (, ad f x to the problem to do so, the CVaR robust mea- CVaR portfolo optmzato problem becomes m v ( ( m ( x, z, v,, s. t. z 0,,..., z f ( x, y,,..., m v 0,,..., m v f ( x,,,..., m x. hs problem has O( m varables ad O( m costrats that m s the umber of -samples, s the umber of assets ad s the umber of y -scearos. Whe the umber of y -scearos ad -samples crease, the approxmatos s gettg closer to the exact values. But the computatoal cost sgfcatly creases ad thus makes the method effcet. Istead of ths method, we ca more effcetly determe the CVaR robust mea- CVaR portfolos usg the smoothg method suggested by Alexader []. Alexader preseted the followg fucto to approxmate F ( x, : m z (22 ˆ F ( x, ( x, (23 ( m where ( a s defed as follows: For a gve resoluto parameter 0, pecewse lear fucto follows: ( a a a, 2 a ( a a a, (24 4 2 4 0 ow.. s cotuous dfferetable, ad approxmates the max( a, 0 []. We ca also use ths fucto to approxmate y F ( x, as
Usg smoothg method, the CVaR robust mea- CVaR model s as follows: ˆ y F ( x, ( f ( x, y. (25 ( m ( x x,, ( m ( ( y x ( m s. t. x. (26 I ths paper, we assume 0.005 for both smoothg fuctos. Formulato (26 has ( varables ad ( costrats. hus, the umber of varables ad costrats do ot chage as the sze of - samples ( ad y -scearos ( crease. he effcecy of the smoothg approach s show the ext secto. 4 Numercal results I ths secto, frst we wll compare the performace of the CVaR robust mea- CVaR model wth robust mea- CVaR models usg terval ad ellpsodal ucertaty sets by actual data. he, we wll compare tme requred to compute the CVaR robust portfolos usg problems (22 ad (26. he dataset used here s avalable returs for eght assets that expected retur ad covarace matrx of the retur of assets have bee gve tables ( ad (2 [9]. I addto, the computatos are based o 0,000 -samples geerated from the Mote Carlo re-samplg (RS method troduced [3] ad 96 y -scearos obtaed va computer smulato. It should be further oted that the computato s performed MALAB verso 7.2, ad ra o a Core 5 CPU 2.40 GHz Laptop wth 4 GB of RAM. Problems are solved usg CVX [3] ad fucto fmco Optmzato oolbox of MALAB. able. mea retur vector 0.0 S S2 S3 S4 S5 S6 S7 S8.060 0.4746 0.4756 0.4734 0.4742-0.0500-0.20 0.0360 able 2: covarace matrx Q 0.0 S S2 S3 S4 S5 S6 S7 S8 S 0.0980 S2 0.0659 0.549 S3 0.074 0.09 0.2738 S4 0.005 0.0058-0.0062 0.0097 S5 0.0058 0.0379-0.06 0.0082 0.046 S6-0.0236-0.0260 0.0083-0.025-0.035 0.269 S7-0.064 0.0079 0.0059-0.0003 0.0076-0.0080 0.0925 S8 0.0004-0.0248 0.0077-0.0026-0.0304 0.059-0.0095 0.0245
4. Sestvty to tal data o show the sestvty of the CVaR robust portfolo to tal data, we repeat RS samplg techque 00 tmes. Each of fgures (, (2 ad (3 dsplay 00 CVaR robust actual froters (actual froters are obtaed by applyg the true parameters o the portfolo weghts derved from ther estmated values [4] for 99%, 90%, 75% respectvely. As ca be see from fgures, the CVaR robust mea- CVaR actual froters chage wth tal data used to geerate samples. Also, ths chages crease as the cofdece level decreases. hus, we ca regard as a estmato rsk averso parameter. Wth these qualtes, a vestor who s more averse to estmato rsk wll choose a larger. O the other had, a vestor who s more tolerat to estmato rsk may choose a smaller. Fgure. 00 CVaR robust mea- CVaR actual froters ( 99% Fgure 2. 00 CVaR robust mea- CVaR 90% actual froters (
Fgure 3. 00 CVaR robust mea- CVaR actual froters ( 75% 4.2 Portfolo dversfcato As we kow, dversfcato decreases rsk [6]. Portfolo dversfcato dcates dstrbutg vestmet amog assets the portfolo. We llustrate the followg that compared wth the robust mea- CVaR portfolos wth terval ad ellpsodal ucertaty sets, the CVaR robust mea- CVaR portfolos are more dversfed. I addto, the dversfcato of the CVaR robust mea- CVaR portfolos decreases as the cofdece level decreases. o do so, we compute the CVaR robust mea- CVaR portfolos (for 99%, 90%, 75% ad robust mea- CVaR portfolos wth terval ad ellpsodal ucertaty sets for the 8-asset example. he composto graphs of the resultg optmal portfolos are preseted fgures (4, (5, (6, (7 ad (8. Cosderg these fgures, whe the expected retur value creases from left to rght, the allocated assets the portfolos wth mmum expected retur are replaced by a composto of other assets, getly. Observg the rght-most ed of each graph, we ca coclude the composto of the assets of the portfolo acheved from CVaR robust mea- CVaR model wth 99% s more dversfed tha that acheved from other models. I fgures (9, (0 ad (, the CVaR robust mea- CVaR actual froters for dfferet values of are compared wth robust mea- CVaR actual froters wth terval ad ellpsodal ucertaty sets ad mea- CVaR true effcet froter. Sce portfolos o the robust mea- CVaR actual froters wth terval ad ellpsodal ucertaty sets are less dversfed, they should accept more rsk for a gve level of expected retur ad also acheve a lower maxmum expected retur. Cosequetly, the fgures, ther actual froters are more rght ad lower tha the other froters ad ths s oe of the dsadvatages of the low dversfcato the portfolo. Seeg these froters, we also deduce that the maxmum expected retur ad the assocated retur rsk crease as the cofdece level decreases. But ths case, the varatos o the compostos of the resultg maxmum-retur portfolos mght be large, ad so the exact soluto wll ot always acheve. Istead, the maxmum expected retur of 99% the portfolo s low for ad the varatos wll be low. So, the probablty of the havg poor performace of the portfolo wll be reduced whe there s a bg estmato rsk of. hus, resultg robust portfolos wll be too coservatve. Cosequetly, a vestor who s more rsk averse to
estmato rsk selects a larger ad obtas a more dversfed portfolo. hs justfes that t s reasoable to regard as a estmato rsk averso parameter. Fgure 4. Compostos of robust mea- CVaR portfolo weghts wth terval ucertaty set Fgure 5. Compostos of robust mea- CVaR portfolo weghts wth ellpsodal ucertaty set Fgure 6. Compostos of CVaR robust mea- CVaR 99% portfolo weghts ( Fgure 7. Compostos of CVaR robust mea- CVaR 90% portfolo weghts (
Fgure 8. Compostos of CVaR robust mea- CVaR portfolo weghts ( 75% Fgure 9. robust mea- CVaR wth terval ad ellpsodal ucertaty sets ad CVaR robust ( 99% actual froters Fgure 0. Robust mea- CVaR wth terval ad ellpsodal ucertaty sets ad CVaR robust ( 90% actual froters
Fgure. Robust mea- CVaR wth terval ad ellpsodal ucertaty sets ad CVaR robust ( 75% actual froters 4.3 Comparso of effcecy of two approaches for computg CVaR robust portfolos I secto 3, we troduced two formulatos (22 ad (26 to compute CVaR robust portfolos. Now, we show that computg CVaR robust portfolos va the smoothg approach (problem (26 s more effcet. o do so, we compare the tme requred to solve problems (22 ad (26 wth dfferet umber of assets ad dfferet umber of -samples ad y -scearos. he results have bee gve tables (3 ad (4. able 3. tme requred to compute maxmum-retur ( 0 portfolos for problem (22 ( 99%, 0.005 scearos( samples(m 8 assets 50 assets 48 assets 500 5,000 3.0 5.7 23.05,000 0,000 4.85 9.95 37.2 3,000 25,000 3.86 45.32 85.49 able 4. tme requred to compute maxmum-retur ( 0 portfolos for problem (26 ( 99%, 0.005 scearos( samples(m 8 assets 50 assets 48 assets 500 5,000.3 3.08 5.67,000 0,000.28 4.67 26. 3,000 25,000.69 2.22 79.3 As we see, the tme requred to compute CVaR robust portfolos va two approaches dffer sgfcatly whe the sample sze ad the umber of assets crease. For example, the tme requred to solve the CVaR robust mea- CVaR problem by two approaches for problem wth 8 assets ad 5000 samples ad 500 scearos dffer slghtly. But, whe the umber of assets s more tha 50, the umber of scearos s more tha 500 ad the sample sze s more tha 5000, dffereces become sgfcat. Problem wth 48 assets, 3000 y -scearos ad 25000 -samples s solved less tha 80 secod usg smoothg
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