Robust Portfolio Models with Short-sales, Transaction Costs, and Floating Required Return

Transcription

1 Robust Portfolo Models wth Short-sales, Transacton Costs, and Floatng Requred Return ABSTRACT Our study develops feasble emprcal framework of robust portfolo models wth consderng varous parameters. Extended by the worst-case condtonal value-at-rsk (WCVaR) and relatve robust condtonal value-at-rsk (RRCVaR) models, the factors that are assocated wth practcal applcaton such as asset short-sales, the transacton costs, confdence levels ( s), and frequent tny portfolo rebalancng, are ncluded. In general the RRCVaR model slghtly outperforms the WCVaR model gven the same when requred return s fxed. The change from fxed to floatng requred returns n the robust models enhances effectveness of portfolo management. Ths s partcularly sgnfcant after the market downturn. The asset allocaton that may be assocated wth better modelng of uncertanty n future returns, but not the savng of transacton costs, contrbutes to superor proftablty. JEL Classfcatons: C61, G11 Keywords: Robust Portfolo Models; Dversfcaton Benefts; VaR; CVaR. * Correspondng author 0

2 1. Introducton The Markowtz (1952) model provdes the foundaton for modern portfolo theory but has been questoned for the feasblty of ts applcaton n the real world. Specfcally, a trval bas n the estmates of the covarance matrx, due to the return uncertanty, can lead to a sgnfcant msallocaton n the meanvarance (MV) model (). Furthermore, the MV model tends to yeld a concentraton of weghts n a small amount of assets n the optmal portfolo and yelds uncertan expected returns. These ssues lead ether to low performance of realzng the optmal portfolos or to poor out-of-sample returns. Though some asset prcng models have been suggested to reduce the error n estmatng return (Merton, 1971; Ross, 1976, Black & Ltterman, 1992; Fama & French, 1993), another possble soluton s to apply the robust optmzaton (e.g., Fabozz, Kolm, Pachamanova, and Focard, 2007; Quaranta & Zaffaron, 2008; Huang, Zhu, Fabozz, and Fukushma, 2010; Gregory, Darby-Dowman, and Mtra, 2011; Branger, Larsen, and Munk, 2013; Kapsos, Chrstofdes, and Rustem, 2014). But several crtcal questons reman unanswered n these robust portfolo models: How should one model asset short sellng, transacton costs, and preventng frequent rebalancng wth small changes n weghts to mprove the practcalty of the strateges? Does a hgh confdence level result n hgh performance? How should one, n practce, desgn the rebalancng mechansm for the robust portfolo models? What s the mpact of changng the requred return on modelng the optmal portfolos? Understandng the above questons helps nvestors decde how to model robust portfolos under dfferent crcumstances. Our study s the frst to model the above ssues n robust portfolo n managng nternatonal equtes. A maor challenge to apply robust portfolo models n fnancal ndustry s ther emprcal operatons are stll somehow gudeless. There have been many theoretcal studes on the robust portfolo models, yet t s relatvely slent n how to deal wth ther emprcal procedures and performance. Ths lmts the applcaton of the robust portfolo framework n practce. Our study ntends to respond the call of the need n the Wall Street and analyzes the performance of two robust portfolo models that are advanced from the framework of condtonal value-at-rsk (CVaR) n managng nternatonal dversfcaton. In comparson to the MV model, the value-at-rsk (VaR) model provdes an alternatve analytcal framework wth whch to manage rsk over a gven tme horzon, but the VaR can result n some unreasonable propertes n portfolo management such as the lack of subaddtvty and non-convexty. These characters may generate multple local solutons, unstable VaR rankngs, and low portfolo dversty (Rockafellar & Uryasev, 2000, 2002; Szegö, 2005). The CVaR model s an mprovement to the VaR regardng these ssues. Rockafellar and Uryasev (2000) document that the CVaR quantfes downsde rsk more precsely than conventonal MV-based models snce the CVaR models asymmetry, or fat tals, n asset return dstrbuton. Furthermore, 1

3 the CVaR demonstrates better theoretcal propertes n rsk management than the VaR; for nstance, the CVaR s a lnear model and can reach the global optmum. Ths means CVaR model may be a better ft n analyzng a wde-range of assets than MV- and VaR-famly models. In ths paper, we consder some parameters affectng portfolo feasblty n the robust optmzaton n practce and then evaluate each portfolo s performance. Specfcally, we extend the worst-case condtonal value-at-rsk (WCVaR) and the relatve robust condtonal value-at-rsk (RRCVaR) models n nternatonal dversfcaton by consderng short-sales, transacton costs, bounds of weghts, and portfolo rebalancng mechansm. We further analyze the mpact of the settng mechansm of the requred return on portfolo performance. Specfcally, we change requred return from the conventonal fxed rate to floatng rate n robust portfolo modelng. The key to obtanng relable optmal portfolos s the technque of performng optmzaton wth uncertan returns. One of maor problems for the MV-famly and 1/N portfolo models s they requre certan return n constructng portfolos (e.g., DeMguel, Garlapp, & Uppal, 2009). The robust optmzaton fts ths context snce t controls estmaton errors n the sample from a based dstrbuton. Soyster (1973) develops a lnear optmzaton model for convex data sets. The advancements by Ben-Tal and Nemrovsk (1998, 2000) and El Ghaou, Oks, and Oustry (2003) mprove the ssue that the orgnal Soyster (1973) model can be too conservatve. Some emprcal studes, such as Fabozz, Kolm, Pachamanova, and Focard (2007), Scherer (2007), Quaranta and Zaffaron (2008), Huang et al. (2010), Gregory, et al. (2011), and Branger, et al. (2013) confrm the effectveness of applyng robust optmzaton n asset management. One of new advancements n robust portfolo models s to synthesze the VaR and/or CVaR models. Nataraan, Pachamanova, & Sm (2008, 2009) develop the asymmetry robust VaR (ARVaR) model. Ther results suggest that the propertes of the CVaR model may be better suted to portfolo and rsk management. Huang, Zhu, Fabozz, & Fukushma (2008) develop a CVaR-based robust portfolo model n whch the uncertan nformaton on the dstrbuton s assocated wth varous ncentves at the ext tme. Zhu and Fukushma (2009) further propose the WCVaR model n a stuaton n whch only partal nformaton on the underlyng probablty dstrbuton s avalable. To mprove the WCVaR model that only takes nto account the worst-case scenaros of the uncertan dstrbuton, Huang et al. (2010) extend t to the RRCVaR model that also takes nto account the best possble decson wth respect to each realzaton of the dstrbuton. Among the robust portfolo models, the propertes of the WCVaR and RRCVaR models show a good ft to the operatonal and emprcal needs n rsk management. These two models are developed from the 2

4 CVaR, whch s a convex and lnear model. Thus they yeld the global optmum. One can further mprove the models by addng other obectves and/or constrants n a lnearzed programmng dependng on the real-world scenaros. The returns and the CVaR values of these models are generated by n-the-sample hstorcal data but not applcaton of statstcal methods, such as the Bayesan or smulaton. Ths makes applyng these robust portfolo models straghtforward. Our study contrbutes to the lterature n four ways. Frst, we extend the requred return from conventonal fxed rate to floatng rate whle modelng return ambguty averson. In prevous studes such as Quaranta and Zaffaron (2008), Huang, et al. (2010), and Kapsos, et al. (2014), the requred return s fxed n the CVaR and robust portfolo models. Our study s the frst to develop changng requred return models accordng to the market dynamcs. Such a settng allows the portfolo optmzaton beng of more flexblty and provdes a possble method for mprovng rsk management models. Second, to ensure the feasblty of the strateges, our emprcal models allow short-sales n portfolo, ncorporate the transacton costs, and avod frequent but small transactons by addng weghtng bounds accordng to the market practces. Angel, Chrstophe, and Ferr (2003) ndcate that short sellng can be rsky but also provdes nvestors a chance to arbtrage. Atknson and Mokkhavesa (2004) suggest that the portfolo rebalancng frequences and ther scale wll be larger than what they should be f the transacton costs are not consdered. Yu and Lee (2011) suggest that the proporton of short sellng should be optmzed to avod hgh rsks. A reasonable desgn of portfolo short-sales ncreases the effectveness of dversfcaton across assets and hedgng over tme. Wthout consderng transacton costs, the results hnder the effectveness of the portfolo models n the fnancal ndustry. However, the above studes do not consder the hgh frequency of portfolo rebalancng and trval changes n asset allocaton. We model portfolo short-sales, varous transacton costs, and the weght lower bounds to enhance the flexblty of makng portfolo decsons. Thrd, our evaluatons are comprehensve snce the effectveness of these models s assessed from varous aspects n portfolo management. Gven the uncertanty of expected returns, the robust models are a new tool of analyzng the portfolo strateges. In ths study, the performance of models s assessed through ex ante return, Sharpe rato, ex post return, realzed market value, transacton costs, and portfolo dversty, by rebalancng portfolos. Forth, the models we suggested do not need varance-covarance matrx, thus t wll be more feasble to deal wth a wde-range assets or bg data. We develop the models based on CVaR model, whch are lnear, the results generated wll be a global optmum. 3

5 The emprcal fndngs usng the data of a wde range of nternatonal equty returns show that, n general, the RRCVaR model yelds slghtly hgher return, lower tradng costs, and hgher portfolo dversty than the WCVaR model, gven the same confdence level and under fxed-requred return settng. However, the key for effectve portfolo management les n the mechansm of the requred return. Specfcally, the ex post performances of the floatng-return robust portfolo models are hgher than the correspondng fxed-return models. Notably, the outperformance n realzed portfolo return of the floatng-return robust portfolo models over the fxed-rate models s sgnfcant after market downturn perod. The better asset allocaton generated by the RRCVaR model, but not the savng of transacton costs, attrbutes to hgher proftablty. The rest of the paper s structured as follows. Secton 2 presents the models and ther emprcal applcatons n ths study. Secton 3 descrbes how to evaluate the effectveness of the models. Secton 4 presents the data. Secton 5 reports the maor emprcal results. The analyses of the mpact of the requred return settng on the effectveness of the robust portfolo models are presented n Secton 6. Secton 7 concludes. 2. Robust Portfolo Models wth Transacton Costs and Short-sales Permtted We evaluate the economc value of the two robust optmal models n managng portfolos under dfferent confdence levels. The portfolos are rebalanced consderng stuatons n whch short-sales and upper bounds are modeled and transacton costs are ncluded. Later, n our analyss, we relax the requred return n portfolo optmzaton from fxed to floatng Worst-Case Condtonal Value-at-Rsk (WCVaR) The condtonal value-at-rsk (CVaR) model was developed to deal wth ssues n the value-at-rsk (VaR) that affect ts feasblty, such as a lack of sub-addtvty and convexty. Ths attrbute may dscourage dversfcaton. Zhu and Fukushma (2009) modfy the CVaR model by Rockafellar and Uryasev (2002) and propose the worst-case condtonal value-at-rsk (WCVaR) model that takes nto account the uncertanty n the asset return. It models portfolos gven that partal nformaton on the underlyng probablty dstrbuton s avalable. To consder mnmzng the loss of a portfolo,, the model s Mn (1) T 1 1 s.t., t 1 T t1,1 2,..., l, 3, l (2) 4

6 n t ( r wt ), t,1 2,..., T,,1 2,..., l, l 3, (3) 1 t 0, t 1,2,..., T,,1 2,..., l, l 3, (4) n 1 w 1, (5) n r w E, 1,1 2,..., l, 3, l (6) where denotes the th lkelhood dstrbuton; l s the number of the lkelhood dstrbutons; endng perod of the th lkelhood dstrbuton; and T s the t are the auxlary varables transformed from the orgnal obectve of the loss functon to a lnear functon of the th lkelhood dstrbuton. Eqs. (1) and (2) are to mnmze the maxmal worst-case CVaR. Eq. (3) s the loss value larger than the threshold of the lkelhood under dfferent scenaros. Three scenaros are assumed n the analyss. As shown n Eq. (4), the portfolo loss n each tradng day cannot be less than 0. We set the sum of the portfolo weghts as one and the portfolo return as greater than the threshold return n Eqs. (5) and (6), respectvely. To enhance feasblty, we further model short-sales and transacton costs n portfolo rebalancng. Prevous lterature has ndcated that short-sale can lower nvestment rsks and provde arbtrage opportuntes (Kwan, 1997; Angel et al., 2003). We extend Kapsos et al. (2014) to rebalance the portfolos by applyng the WCVaR model wth (1) allowng asset short sellng, (2) consderng transacton costs, (3) ncorporatng the mnmal threshold for the change of portfolo weghts. The reason of ncludng the thrd condton s frequent portfolo rebalances of tny weghts ncrease the transacton costs wth a margnal ncrease n portfolo performance. The above three mprovements strengthen the usefulness of the robust portfolo models n the real world. follows: Our advanced WCVaR model wth consderng short sellng and transacton costs s demonstrated as n n ( Mn w p l p l p s p s ) (7) T 1 1 s.t., t 1 T n t1,1 2,..., l, l 3, (8) ( r t ( wt w ) ), t 1, 2,..., T, 1, 2,..., l, l 3, (9) 1 5

7 0, t 1, 2,..., T, 1, 2,..., l, l 3, (10) t n 1 n 1 r ( w w ) E, 1, 2,..., l, l 3, (11) ( w kw p l p l p s p s ) 1, (12) w w l l, 1, 2.,. n,., (13),0 w w s s, 1, 2.,. n,., (14), v w v v 1,.01 u w u,,1 2,..., n,,,1 2,..., n, u,1 2,..., n, u, v {0, 1},,1 2,..., n, (15) (16) (17) (18) where p s are the transacton costs of buyng, sellng, short sellng, and repurchasng, respectvely; k s the ntal margn requrement for short sellng; rebalancng; w s the total proporton of securty nvested at portfolo w s the total weght of securty sold by nvestors at portfolo rebalancng; poston weght of securty pror to portfolo rebalancng; and pror to portfolo rebalancng. Wth each rebalancng, short-sellng weght of securty ; 6 w,0 s the long w,0 s the short poston weght of securty l s the buyng weght of securty ; s s the short-sellng weght of securty ; and l s the s s the repurchasng weght of securty. The bnary varables u and v are used to ndcate the long and short poston and to model the upper bounds of the weght. The obectve of the WCVaR shown n Eq. (7) s to smultaneously mnmze the obectves of the loss value, short-sellng weghts, and the transacton costs by usng the concept of the smple weghted method n multple-obectve programmng. The portfolo weght w (= w w ) s unrestrcted. Eq. (12) defnes the weght constrant ncludng the tradng cost and short sellng. The long and short poston after rebalancng are defned n Eq. (13) and (14). To ensure that the portfolo does not center on a small number of assets, we set the upper and lower bounds of the long and short poston for each securty n Eqs. (15) and (16). u and v n Eqs. (17) and (18) are used for ndcatng the poston of long and short sellng. Addng Eqs. (15) (18) n the analytcal framework enhances computatonal complexty but advances feasblty of portfolo n practce Relatve Robust Condtonal Value-at-Rsk (RRCVaR)

8 The foundaton of the robust portfolo models s the worst-case analyss. Huang et al. (2010) and Lu and Chen (2014) suggest that such a settng lkely leads to conservatve nvestment decsons and nsenstvty to the changes n parameters. One of the possble remedes s to model both up-sde and down-sde scenaros n decson makng. Huang et al. (2010) advance ths theory by applyng the relatve robust CVaR (RRCVaR) model. One of the advantages of the RRCVaR s to consder the varaton n the benchmarks under dfferent scenaros. Furthermore, t takes nto account the worst and best scenaros n the realzaton dstrbuton. Specfcally, Mn T 1 1 s.t. t CVa R,,1 2,..., l, l 1,2,3, (19) 1 T and Eq. (2) (6), t1 where s the threshold of loss value under the th scenaro, CVaR s the CVaR of the th scenaro and serves as the benchmark. In Eq. (19), the benchmark s set for each scenaro and s used to calculate the mnmal CVaR gven the dstrbuton. The optmum s to seek the potental greatest relatve rsk yelded by the set of portfolo weghts that maxmze the dstance between the benchmark and the CVaR of each scenaro. If CVaR 0, the problem s reduced as the WCVaR portfolo optmzaton. If CVaR 0and l 1, the problem then s reduced as the CVaR portfolo selecton. We further mprove the Huang et al. (2010) model by addng the portfolo weghts and the transacton costs to the RRCVaR model. Specfcally, n n ( Mn w p l p l p s p s ) s.t. Eq. (9) (19). Smlar to the orgnal RRCVaR model, the portfolo takes ndvdual scenaros nto account to generate the CVaR. The optmum s to mnmze the maxmum dstance between the benchmark and the CVaR of each scenaro wth varous weght constrants and tradng costs constrants. 3. Performance Measures To evaluate the effectveness of the dversfyng strateges that are formed by the above rsk-return portfolo models, we measure the dversfcaton benefts by usng (1) portfolo expected return, (2) Sharpe rato, (3) Omega rato, (4) realzed return, and (5) realzed market value. Sharpe rato (SR) s defned as 7

9 r p r f SR, (20) p where r p s the portfolo expected return, rf s the rsk free rate, and p s the standard devaton of the portfolo. Consderng the mpact of hgher moment rsk on performance assessment and estmaton errors, we also use the Omega rato to measure portfolo performance: Ω P T 1 t1 max( 0, n 1 where T s the endng perod, T 1 w, tr, t1 rb, t1) / max( 0, rb, t1 w r t1 1 n, t, t1 ), t 1, 2,..., T, (21) w s the portfolo weght allocated to asset, r s the return of securtes, and rb, t1 s the benchmark of realzed return at perod t+1, whch s assumed to be zero n our case. We further evaluate the performance when nvestors follow the dversfcaton strateges. Prevous studes such as DeMguel, Garlapp, and Uppal (2009) ndcate that a poor estmaton of asset returns challenges the applcaton of portfolo models. Thus, evaluatng the out-of-sample performance of the optmal portfolo benefts nvestors how to select the optmal portfolo. The portfolos are marked to market when they are rebalanced every perod accordng to the optmzaton results. We calculate realzed portfolo values and realzed return by exercsng the strateges. We also evaluate the models by measurng ther transacton costs and dversty. Though rebalancng shapes an effcent portfolo by replacng assets of low performance wth proftable ones, frequent portfolo rebalancng and/or hgh amount of asset replacement means hgh transacton costs that possbly erode the market value. To evaluate the dversty, we also present the statstcs of the Herfndahl ndex of the weghts and the number of assets n the portfolo. In ths paper, we evaluate the performance of the WCVaR and the RRCVaR models by ncorporatng varous weghtng and tradng cost condtons. We later evaluate the models flexblty by changng the settng of the requred return from a fxed rate to a floatng rate. 4. Data We collect daly data from the Exchange Traded Funds (ETFs) on the Morgan Stanley Captal Internatonal (MSCI) stock ndces, the prces or ndces of varous commodtes, and the Real Estate Investment Trusts (REITs) ndex between October 10, 2001 and September 24, The portfolo strateges presented n ths research are feasble and are of hgh lqudty. The selected ETFs represent more than 90% of the world market captalzaton durng the sample perod. The other assets ncluded are regarded as the most popular alternatve nvestments. We dd not nclude fxed ncomes n the study due to 8

10 the focus of the study s on managng rsky assets. Table 1 shows the summary statstcs of the sample stocks, ncludng ther standard devaton (SD), Sharpe rato (SR), skewness, and kurtoss durng the sample perod. Table 1. Investment Assets Asset Symbol SD SR Skewness Kurtoss Shares MSCI Australa Index EWA Shares MSCI Austra Index EWO Shares MSCI Belgum Index EWK Shares MSCI Brazl Index EWZ Shares MSCI Canada Index EWC Shares MSCI France Index EWQ Shares MSCI Germany Index EWG Shares MSCI Hong Kong Index EWH Shares MSCI Italy Index EWI Shares MSCI Japan Index EWJ Shares MSCI Malaysa Index EWM Shares MSCI Mexco Index EWW Shares MSCI Netherlands Index EWN Shares MSCI Sngapore Index EWS Shares MSCI South Korea Index EWY Shares MSCI Span Index EWP Shares MSCI Sweden Index EWD Shares MSCI Swtzerland Index EWL Shares MSCI Tawan Index EWT Shares MSCI Unted Kngdom Index EWU SPDR S&P 500 SPY Goldman Sachs Commodty Prce Index Commodty S&P Global 1200 Energy Index Energy Gold Bullon Prce-New York (US$/Ounce) Gold S&P Dversfed Metals & Mnng Index Metals Slver Bullon Prce-New York (US$/Ounce) Slver Real Estate Investment Trust REIT To generate the nputs for the robust portfolos under dfferent scenaros, the data n each sample perod s splt nto varous subsamples. For our study, we dvde 180 daly observatons nto three equal subsamples to form the optmal portfolo to generate the CVaR values. The portfolos are then rebalanced every 20 tradng days by rollng the sample to the next 180 daly data. We fnd smlar outcome from the results of dfferent rebalancng frequences. The weghts of each portfolo are obtaned by usng updated asset returns n each perod and assumng the ntal nvestments to be $1 mllon. For the computaton of market value when short-sales s allowed, the margn needs to be pad before the asset s short sold. The budget for the nvestment n the next perod thus depends on the market value at the end of the prevous perod. We desgn a portfolo rebalancng mechansm wth short-sales, tradng costs, and the bounds of asset weghts. The tradng costs vary from broker to broker and from asset to asset. In ths study, we nclude the fees that are generally accepted n the U.S. market and set all transacton costs (p 1, p 2, p 3, and p 4 ) at 25 9

11 bass ponts of the tradng value and the tradng margn (k) at 100%. The requred return s 0.01% for varous robust models for the fxed-return analyss. The benchmark of each scenaro s set up by the mnmal CVaR that s computed under the current state. The uncertanty n return then s captured by dfferent scenaros. The maor dfference between the WCVaR and RRCVaR models s that the latter consders the benchmark that s set up by portfolos of each scenaro before computng the mnmum CVaR. Snce the RRCVaR model consders both the worst scenaro and the best possble decson, t demonstrates superorty n theoretcal foundaton than the WCVaR model. Each portfolo uses the weght of ndvdual scenaro to generate the CVaR at the same tme. We obtan the asset allocaton that mnmzes the maxmal dstance between the benchmark and the CVaR of each scenaro. The RRCVaR model s expected to yeld portfolo strategy of hgher effectveness than the WCVaR model does. 5. Emprcal Results of Fxed-Requred-Rate Models 5.1. Portfolo Performance Table 2 reports the summary statstcs of the over-tme portfolo performance of the WCVaR and RRCVaR models wth fxed requred return under dfferent confdence levels ( s) from both ex ante and ex post vews. We consder allowng short sellng and transacton costs n the models. We rebalance each of the portfolos every 20 tradng days by usng the data of the prevous 180 tradng days, whch are used to generate the CVaR values n three even subsamples and to capture the return uncertanty durng the perod. For ths study, each portfolo s rebalanced 132 tmes over the perod. Panel A presents the statstcs of the monthly expected return, and Panel B reports the Sharpe rato and the Omega rato. In Panel C, we compute the realzed return by usng the weghts to rebalance the portfolos. We also report the dstrbuton of the benefts over the sample perod. Panel A shows that the expected return s low, around 1% for the two models under dfferent scenaros, due to the fact that the goal of robust portfolo models s to mnmze the ex ante loss but not to maxmze the return. Though the RRCVaR model demonstrates slghtly hgher expected returns than the correspondng WCVaR model, the selecton of the confdence level () seems not to sgnfcantly affect the expected return across the same model. Snce the RRCVaR model tends to yeld lower volatlty, ts Sharpe rato s hgher than that of the correspondng WCVaR model. The results n Panels A and B show that the portfolo flexblty suggested by the RRCVaR model slghtly mproves the ex ante expected return and the mean-varance effcency. 10

12 Table 2. Performance The table reports the summary of the over-tme performance of the worst-case condtonal value-at-rsk (WCVaR) model and the relatve robust condtonal value-at-rsk (RRCVaR) model under dfferent confdence levels ( s). A. Expected Return Model WCVaR RRCVaR CVaR = 50% 75% 90% 95% 99% 50% 75% 90% 95% 99% Mean St. Dev Max Mn Dstrbuton (%) E( r)< < E( r) < E( r) < E( r) B. Sharpe Rato and Omega Rato Model WCVaR RRCVaR CVaR = 50% 75% 90% 95% 99% 50% 75% 90% 95% 99% Mean St. Dev Max Mn Dstrbuton (%) SR < <SR <SR <SR WCVaR RRCVaR 50% 75% 90% 95% 99% 50% 75% 90% 95% 99% Omega C. Realzed Return Model WCVaR RRCVaR CVaR = 50% 75% 90% 95% 99% 50% 75% 90% 95% 99% Mean St. Dev Skew Kurtoss SR Dstrbuton (%) R( r) < R( r) < R( r) <R( r) One of mportant ssues n asset management s whether realzng these models generate proft. Panel C shows that exercsng both the WCVaR and the RRCVaR generate a proft from 6.3% to 8.8% per 11

13 annum. In general, the RRCVaR performs better n terms of raw return and rsk-adusted return; partcularly, when values are 50% and 75%. Translatng to the endng market value, the dfference of value between WCVaR and RRCVaR s about 13%. For nstance, when =50%, the endng market value of the WCVaR s $2,133,176 whle the RRCVaR s $2,473,049; for =75%, the WCVaR s $2,107,383 whle the RRCVaR s $2,273,881. But there s no sgnfcant dfference n the other hgher confdence levels. Due to the low predctablty of the out-of-sample return, the dstrbutons of realzed performance are volatle over the sample perod. Unlke the expected return, the mean of realzed return s hgh but s tme-varyng, wth about from 35% to 40% of sample perod of negatve value Transacton Costs There are many possble explanatons how the robust portfolo models contrbute to hgher portfolo market value, ncludng a better modelng uncertanty n future returns and the low transacton cost. Table 3 shows the statstcs of the transacton costs n terms of dollar value and the percentage of portfolo value. We also report the dstrbuton of the annualzed percentage of transacton costs over the portfolo market value. When the confdence levels are lower,.e., 50% and 75%, the transacton costs, both n terms of dollar value and the percentage of portfolo value, are lower than hgh s. The percentages of hgher tradng cost rato ncrease when the confdence level ncreases for both robust portfolo models. There s no sgnfcant and unform dfference n tradng costs between the WCVaR and RRCVaR models. Table 3. Transacton Costs The table reports the summary of the over-tme transacton costs of the worst-case condtonal value-at-rsk (WCVaR) model and the relatve robust condtonal value-at-rsk (RRCVaR) model under dfferent confdence levels ( s). Model WCVaR RRCVaR CVaR = 50% 75% 90% 95% 99% 50% 75% 90% 95% 99% $ Cost Mean 1,319 1,461 1,724 2,124 2,197 1,455 1,603 1,754 2,009 2,282 St. Dev 1,604 1,654 1,599 1,750 1,774 1,776 1,723 1,520 1,656 1,532 Max 7,379 7,607 7,046 7,857 8,727 8,604 7,831 8,114 8,335 7,144 Mn %Cost to Portfolo Value Mean SD Max Mn Dstrbuton (%) 0<Cost 0.5% %<Cost 1% %<Cost

14 5.3. Portfolo Dversty We report the statstcs of the Herfndahl (H) ndex of portfolo weghts and the number of assets n the portfolo and ther dstrbutons n Table 4. Shown n Panel A, the weght concentraton s lowest for the RRCVaR when s 75% but s not very dfferent n other scenaros. The means of the Herfndahl ndex of portfolo weghts, rangng between 0.32 and 0.35, suggest that the portfolo on average conssts of about equvalent 3 assets over the sample perod. The dstrbutons of the Herfndahl ndex show the portfolo concentraton, over all, s not nsgnfcant for the models. Table 4. Portfolo Dversty The table reports the summary of the over-tme measures of portfolo dversty of the worst-case condtonal value-at-rsk (WCVaR) model and the relatve robust condtonal value-at-rsk (RRCVaR) model under dfferent confdence levels ( s). A. Herfndahl Index of Portfolo Weghts Model WCVaR RRCVaR CVaR = 50% 75% 90% 95% 99% 50% 75% 90% 95% 99% Mean St. Dev Max Mn Dstrbuton (%) 0 <H <H <H <H B. Number of Assets Model WCVaR RRCVaR CVaR = 50% 75% 90% 95% 99% 50% 75% 90% 95% 99% Mean St. Dev Max Mn Dstrbuton (%) 0<N N N N Though both portfolo models show smlar weght concentraton accordng to the Herfndahl ndex, Panel B suggests that the RRCVaR portfolo contans more assets than the WCVaR model. On average, each RRCVaR model has a hgher number of assets (N) ncluded n the portfolo, from 5.8 to 7.1, than ts 13

15 correspondng WCVaR model. The dstrbutons show that about 19% to 25% of portfolos constructed by three or less assets. 6. Does Floatng Requred Return Improve Modelng Effectveness? The assumpton of a fxed requred return n the prevous models leads to an nterestng queston: Does allowng the requred return to be floatng mprove the portfolo effectveness? What s the mpact of the changng return on portfolo dversty? Though the prevous emprcal fndngs confrm the benefts of robust portfolo models, t also ndcates an occasonal lack of portfolo dversty. For nvestors, t makes sense to modfy models to mprove realzed proftablty and to decrease tradng costs. We revse the above models by settng floatng requred return n the robust portfolos and by performng asset rebalancng strateges. The floatng requred return s computed by takng average of the mnmums of asset returns of all scenaros. We update the requred return perod by perod when the portfolos are rebalanced. Table 5 reports the portfolo performance of the floatng requred return models (the WCVaR_F model and the RRCVaR_F model, respectvely). Snce the volatlty sgnfcantly reduces when the requred return s floatng, the mean-varance effcency mproves though overall the expected return slghtly decreases. In general, the n-the-sample performance that s measured by the frst two moments s not necessarly better than the results of the fxed requred rate presented n Table 2. However, the hgher values suggest uncertanty n return s better managed n the floatng return models. The dfferng results of the realzed market value from those of ex ante performance show the usefulness of the robust portfolo models n fnancal ndustry. The flexblty brought by the floatng requred rate ncreases the realzed return of all models, n terms of both raw return and rsk-adusted return. The average annual realzed returns are between 10% and 12.4% for the robust portfolos wth floatng return. The percentage of perods of postve realzed return for the floatng return models s larger than that for the fxed return models. Ths suggests that fashonng requred return accordng to market dynamcs mprove the ex post performance of robust portfolo models. 14

16 Table 5. Performance wth Floatng Requred Return The table reports the summary of the over-tme performance of the worst-case condtonal value-at-rsk wth floatng requred return (WCVaR_F) model and the relatve robust condtonal value-at-rsk wth floatng requred return (RRCVaR_F) model under dfferent confdence levels ( s). A. Expected Return Model WCVaR_F RRCVaR_F CVaR = 50% 75% 90% 95% 99% 50% 75% 90% 95% 99% Mean St. Dev Max Mn Dstrbuton (%) E( r)< < E( r) < E( r) < E( r) B. Sharpe Rato and Omega Rato Model WCVaR_F RRCVaR_F CVaR = 50% 75% 90% 95% 99% 50% 75% 90% 95% 99% SR Mean St. Dev Max Mn Dstrbuton (%) SR < <SR <SR <SR WCVaR_F RRCVaR_F CVaR = 50% 75% 90% 95% 99% 50% 75% 90% 95% 99% Omega C. Realzed Return Model WCVaR_F RRCVaR_F CVaR = 50% 75% 90% 95% 99% 50% 75% 90% 95% 99% Mean St. Dev Skew Kurtoss SR Dstrbuton (%) R( r) < R( r) < R( r) <R( r)

17 Fgure 1 shows the comparson between the fxed return models and ther correspondng floatng return models (WCVaR vs. WCVaR_F and RRCVaR vs. RRCVaR_F) under dfferent confdence levels. The hypothetcal ntal nvestng value of each portfolo s $1 mllon. The endng portfolo market values for the two models are demonstrated n each graph. The market value of the floatng return models s consstently hgher than that of the fxed return models over the entre sample perod except n one case. For the RRCVaR model when a=50%, the endng market value of the floatng return model s hgher than the fxed return model though the market values durng the nvestment perod vary sgnfcantly. Interestngly, n most cases, the outperformance of the floatng rate s trval before the second quarter of 2009, whch s the trough defned by the Natonal Bureau of Economc Research (NBER). The devaton of the portfolo market value wdens over tme and can reach more than 40% of the market value. Ths seems to suggest that the varable-return robust portfolo models are more effectve when market s expected to depart from a downturn perod. Fgure 1. Portfolo Market Value: Constant vs. Floatng Requred Return The fgures show the over-tme portfolo market value of the worst-case condtonal value-at-rsk and the relatve robust condtonal value-at-rsk models wth constant requred return (WCVaR and RRCVaR) and floatng requred return (WCVaR_F and RRCVaR_F) under dfferent confdence levels ( s). A. WCVaR and WCVaR_F when=50% B. RRCVaR and RRCVaR_F when=50% WCVaR WCVaR_F 2,133,176 3,191,670 RRCVaR RRCVaR_F 2,473,049 3,096,885 16

18 C. WCVaR and WCVaR_F when=75% D. RRCVaR and RRCVaR_F when=75% WCVaR WCVaR_F 2,107,383 2,778,498 E.WCVaR and WCVaR_F when=90% RRCVaR RRCVaR_F 2,273,881 3,068,054 F. RRCVaR and RRCVaR_F when=90% WCVaR WCVaR_F 1,946,777 2,873,872 G. WCVaR and WCVaR_F when=95% RRCVaR RRCVaR_F 1,916,634 3,393,296 H. RRCVaR and RRCVaR_F when=95% WCVaR WCVaR_F 1,924,224 3,136,149 RRCVaR RRCVaR_F 2,058,372 3,578,925 17

19 I. WCVaR and WCVaR_F when=99% J. RRCVaR and RRCVaR_F when=99% WCVaR WCVaR_F 1,969,058 3,012,160 RRCVaR RRCVaR_F 1,902,072 2,922,485 The mproved proftablty of the floatng requred return models may be credted by (1) lower transacton costs, and/or (2) better asset allocaton strateges. Table 6 shows that most of the hgher market values yelded by the varable-return models can be attrbuted to better asset allocaton strateges. For both the WCVaR and the RRCVaR models under varous scenaros, the ncrease n endng market value by swtchng from the fxed requred rate to the floatng rate can be more than $1.5 mllon. For nstance, for the RRCVaR when=95%, the market values are $2.06 mllon (fxed-rate) and $3.58 mllon (floatng-rate), respectvely. However, the reducton n transacton costs, totalng from $96,000 to $140,000 n varous models durng the entre nvestment horzon, merely represents a trval fracton of the ncrease n portfolo market value. A reasonable explanaton s that the flexblty n model settng may be more effectve n capturng the uncertanty n future returns. Table 7 shows that the changng requred return mechansm ncreases the portfolo dversty. In Panel A, all statstcs of the Herfndahl ndex are lower than the correspondng numbers of fxed requred return results n Table 4. In Panel A, the descrptve statstcs of the dstrbuton show that the concentraton n portfolo weghts s dropped. For nstance, the means and maxmums of Herfndahl ndex for all models are smaller than ther correspondng fxed-rate models. Panel B shows that the number of assets ncluded n the portfolo ncreases when the requred return changes accordng to the market condton. The percentage of perods n whch nclude four or more assets ncreases, partcularly for the RRCVaR model. Smlar to the results of the fxed requred return, the RRCVaR portfolos are more dversfed than the WCVaR portfolos. 18

20 Table 6. Transacton Costs wth Floatng Requred Return The table reports the summary of the over-tme transacton costs of the worst-case condtonal value-at-rsk wth floatng requred return (WCVaR_F) model and the relatve robust condtonal value-at-rsk wth floatng requred return (RRCVaR_F) model. We report the statstcs of the transacton costs as dollar value and as annualzed percentage over the portfolo market value wth the dstrbuton of the annualzed percentage of transacton costs over the portfolo market value. Model WCVaR_F RRCVaR_F CVaR = 50% 75% 90% 95% 99% 50% 75% 90% 95% 99% $ Cost Mean ,093 1, ,112 1,541 St. Dev ,175 1, ,151 Max 2,494 3,517 5,093 6,251 8,229 2,494 3,361 5,410 5,588 5,038 Mn %Cost to Portfolo Value Mean St. Dev Max Mn Dstrbuton (%) 0<Cost 0.5% %<Cost 1% %<Cost Table 7. Portfolo Dversty wth Floatng Requred Return The table reports the summary of the over-tme measures of portfolo dversty of the worst-case condtonal value-at-rsk wth floatng requred return (WCVaR_F) model and the relatve robust condtonal value-at-rsk wth floatng requred return (RRCVaR_F) model. We consder allowng short sellng and transacton costs n the models. A. Herfndahl Index of Portfolo Weghts Model WCVaR_F RRCVaR_F CVaR = 50% 75% 90% 95% 99% 50% 75% 90% 95% 99% Mean St. Dev Max Mn Dstrbuton (%) 0 <H <H <H <H B. Number of Assets Model WCVaR_F RRCVaR_F CVaR = 50% 75% 90% 95% 99% 50% 75% 90% 95% 99% Mean St. Dev Max Mn Dstrbuton (%) 0<N N N N

21 We emprcally test the effectveness of varous robust portfolo models to deal wth the ssue of asset return uncertanty. We compare ther ex ante return, volatlty, ex post performance, realzed market value, tradng costs, and portfolo dversty. The results show that the portfolo effectveness of the WCVaR and RRCVaR models wth varous confdence levels yeld smlar results. Changng the requred return from a conventonal fxed rate to floatng rate mproves portfolo realzed performance. The ncrease n realzed proftablty comes from the better modelng of uncertanty n future return by the floatng-rate models. 7. Concluson Prevous studes have proposed robust portfolo models to deal wth ambguty and uncertanty n return but pay lttle attenton to ther operatonal frameworks. In ths paper, we thoroughly evaluate the performance of the worst-case condtonal value-at-rsk (WCVaR) and relatve robust condtonal valueat-rsk (RRCVaR) models n managng nternatonal dversfcaton. Dfferng from the prevous studes, we model short-sales, the transacton costs, and the bounds of portfolo weghts to ensure feasblty of strateges. Specfcally, our model ncludes the mnmal threshold for the change of portfolo weghts to avod frequent rebalancng. We fnd n general that the RRCVaR model yelds slghtly hgher return, lower tradng costs, and hgher portfolo dversty than the WCVaR model gven the same confdence level when requred return s fxed. The performances of the floatng-return robust portfolo models are hgher than the correspondng fxed-return models. Notably, the outperformance of the floatng-return models over the fxed-rate models s sgnfcant durng the perod when the market recovers from the downturn. The better asset allocaton but not the savng of transacton costs attrbutes to the superor proftablty n the floatng-rate models. We contrbute to the current lterature by extendng the robust portfolo models and evaluatng ther benefts under varous scenaros whle consderng the factors affectng the feasblty of strateges, such as short-sales, the tradng costs, asset weghts, and portfolo rebalancng. Our study syntheszes the maor concepts and mod operand of the prevous research and maxmzes the feasblty n managng nternatonal portfolos. The emprcal results show the superorty of the floatng-return robust models n comparson to the correspondng fxed-return models n portfolo and rsk management. Ths paper develops feasble methodologes applyng the WCVaR and RRCVaR models n global portfolo management. 20

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