A Comparative Study of Mean-Variance and Mean Gini Portfolio Selection Using VaR and CVaR

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1 Journal of Fnancal Rsk Management, 5, 4, 7-8 Publshed Onlne 5 n ScRes. A Comparatve Study of Mean-Varance and Mean Gn Portfolo Selecton Usng VaR and CVaR Jamal Agouram *, Ghzlane Lakhnat Natonal School of Appled Scences (ENSA), Agadr, Morocco Emal: * jamal.agouram@edu.uz.ac.ma, g.lakhnat@uz.ac.ma Receved 4 5; accepted 5; publshed 5 5 Copyrght 5 by authors and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons Attrbuton Internatonal Lcense (CC BY). Abstract Ths paper focuses on two methods for optmum market portfolo selecton. We compare the Mean-Varance method wth the Mean-Gn method usng MADEX data from turbulent market perods n, and 3. We compare both strateges wth reference to value at-rsk (VaR) and condtonal value-at-rsk (CVaR) measures durng perods of fnancal crss. The results show that both strateges are proftable for nvestors. We consder the Mean-Gn strategy to be the more secure strategy durng perods of market nstablty. Keywords Condtonal Value-at-Rsk, Mean-Gn, Mean-Varance, Portfolo Selecton, Value-at-Rsk. Introducton Durng a fnancal crss, t s of crtcal mportance to mplement the best nvestment strategy possble to match an nvestor s preferences n terms of rsk and returns. Several studes have evaluated and compared dfferent pars of portfolo strateges n terms of ther return and rsk characterstcs. The Mean-Varance (MV) theory, suggested by Markowtz (95a, 95b), marked the startng pont of the development of modern fnance theory. The theory s based on the presumpton that dstrbuton of portfolo returns s normal and can be successfully descrbed by two moments: mean and varance. However, the applcaton of MV optmsaton s questonable because t does not consder the drecton of prce movement; optmsng the varance can prevent an nvestor from losses n the same manner as from gans. Roll (977, 978, and 979) was the frst to pont out other weaknesses of the theory. Ths evdence forced several theorsts to search for other, more approprate models to determne the best possble * Correspondng author. How to cte ths paper: Agouram, J., & Lakhnat, G. (5). A Comparatve Study of Mean-Varance and Mean Gn Portfolo Selecton Usng VaR and CVaR. Journal of Fnancal Rsk Management, 4,

2 return/rsk relatonshp. For nstance, Markowtz (959), Fshburn (977), Bawa (977) proposed the use of the mean-lower partal moment approach, Ytzhak (98), Shalt and Ytzhak (984) proposed the use of the Mean- Gn (MG) portfolo selecton model, and Konno and Yamazak (99) proposed the use of the Mean-Absolute Devaton (MAD) approach. The restrctve character of varance as a rsk parameter, led us to choose the MG strategy as an alternatve to the MV strategy. The MG strategy uses Gn as a parameter of rsk nstead of varance. The concept of MG was proposed by Shalt and Ytzhak (984) as an alternatve method to the MV approach proposed by Markowtz (95) because t can outstrp normal assumptons of return dstrbuton and utlty functon quadratcs. Ytzhak (98) has shown that the Gn coeffcent satsfes the second degree stochastc domnance, whch makes the MG model compatble wth the theory of expected utlty. Ths study provdes a comprehensve statstcal analyss of four strateges: the MV strategy versus the MG strategy and the Mnmum Varance (Mn-V) versus the Mnmum Gn (Mn-G) strategy. Followng Agouram and Lakhnat (5), n order to verfy the relablty of these strateges, we used quasanalytc VaR, computng and only portfolo VaR, wth any addtonal nformaton pertanng to mportant characterstcs of fnancal asset returns (.e. ther volatlty, clusterng and non-normal dstrbutons). We consdered the followng aspects n forecastng VaR for each strategy. Frstly, we used the GARCH (.) model to explore the senstvty of VaR to return dstrbuton characterstcs assumng that portfolo return follows the classcal normal dstrbuton and the student-t dstrbuton. Secondly, we numercally computed VaR usng the Cornsh-Fsher expanson and the Johnson SU approxmaton to forecast portfolo VaR, to take nto account fat tals and skewness on forecastng VaR. Fnally, we utlsed CVaR to compare the two pars of strateges because the outcomes are smlar and CVaR can better capture the tal rsk. Ths rsk measure follows drectly from the value-at-rsk. Ths paper s organsed as follows: Frstly, we present the framework of the four models: MV, MG, Mn-V and Mn-G. Secondly, we provde a comprehensve explanaton of the data and methodology used, and defne the key terms. Fnally, we apply the GARCH (.) model wth the Cornsh-Fsher expanson and the Johnson SU approxmaton to forecast VaR and CVaR.. Materals.. Mean-Varance A portfolo s defned to be a lst of weghts x for assets S, =,, n, whch represent the amount of captal to be nvested n each asset. We assume that one unt of captal s avalable and requre that captal to be fully nvested. Thus we must respect the constrant that x n = =. The return of portfolo ( R p ), obtaned by n Rp = xr = ( R s the return of asset per perod). In the tradtonal Markowtz portfolo optmsaton, the objectve s to fnd a portfolo whch has mnmal varance for a gven expected return. More precsely, one seeks such that: Subject to: Mn σ n n P = xx jσj = j= ( p ) E R µ n x = = () x, n where σ j s the covarance between the returns of S, and S j and µ s the mnmal rate of return requred by an nvestor. The Mnmum Varance analyss conssts of constructng a portfolo wthout a gven expected return; the optmsaton program s presented mathematcally as follows: Subject to: Mn σ n n P = xx jσj = j= 73

3 n x = () = x, n.. MG Analyss The MG approach s consstent wth stochastc domnance for decsons about rsk and s deal for portfolo analyss for a varety of fnancal assets. The MG analyss ntroduced by Shalt and Ytzhak (984) defnes the Gn coeffcent as an ndex of varablty of a varable random. The approach used by these authors assumes that the cumulatve dstrbuton correspondng to the observaton wth rank t s tt. Specfcally, Dorfman (979) and Shalt and Ytzhak (984) retan as a measure of the Gn coeffcent: Γ p = cov ( Rp, F( Rp) ) where F( R p ) s the cumulatve dstrbuton functon of R p. n ( ) ( R F R ) x ( R F( R = )) Γ = cov, = cov, p p p p The MG mathematcal model s presented as follows: Mnmze: Γ p Subject to: ( p ) E R µ n x = = (3) x, n where Γ p s the portfolo Gn, x s the amount nvested n asset S, R s the expected return of asset S per perod, and µ s the mnmal rate of return requred by an nvestor. The Mnmum Gn as the Mnmum Varance analyss. The optmsaton program s presented mathematcally as follows: Mnmze: Subject to:.3. VaR Γ p n x = (4) = x, n Value-at-Rsk s a measure of rsk. It represents the maxmum loss of the portfolo wth a certan confdence probablty, over a certan tme horzon. Formally, f the portfolo s prce P( t ) at tme t s a random varable where S( t ) represents a vector of rsk factors at tme t, then the value-at-rsk ( VaR ) s mplctly gven by the formula: { ( ) ( ) } Prob P t + P > VaR = In the case of normal dstrbuton, the parametrc VaR s calculated by: VaR = R σ z where R : average return, σ: the standard devaton of returns and z s the quantle from a normal dstrbuton. Zangar (996), Favre and Galeano () provde a modfed VaR calculaton that takes the hgher moments of non-normal dstrbutons (skewness and kurtoss) nto account through the use of the Cornsh-Fsher expanson. z f ( ) ( 3 ) ( 5 ) z S z z K z z S = z

4 VaR CF = R σ where S s the skewness of R and K s the excess kurtoss of R. The Johnson SU dstrbuton we use here dffers from the Cornsh-Fsher approach. It transforms a random varable z nto a standard normal varable x, and wrtng n general: z f x snh z γ = ξ + λ δ where z s a standard normal varable; ξ and λ shape parameters; γ the locaton parameter and δ scale parameter. The Johnson SU value-at-rsk s obtaned by:.4. CVaR VaR JSU snh z γ = λ ξ δ CVaR, also known as Expected Shortfall, can be defned as the expectaton of the loss when the loss exceeds VaR. Snce VaR measures the value that separates the ( )% of the dstrbuton, the am was to focus on the tal of the loss, the remanng %, of whch we know nether the dstrbuton nor the expectaton. So we defned a complementary measure to the rsk of loss, Condtonal VaR. For a random varable, X s defned by:.5. GARCH (.) Model ( ) CVaR = E X X < VaR X In ths study, volatlty was estmated by applyng a GARCH (.) model to each portfolo. Ths s a famlar model n econometrcs; see Shephard (996). If y t denotes the observed seres (n ths case, the observed daly return) on day t, assumed standardsed to mean, then the model represents y t n the form: y = σε, where form: ε are..d. ( ;) 3. Methods t t t t N random varables, and the volatlty σ t s assumed to satsfy an equaton of the σ = + y + βσ t t t Ths paper focuses on the MADEX. We propose to buld a portfolo composed only of assets from the MADEX over a perod of natonal and global fnancal crss spannng // to //4. Our dataset conssts of a daly seres of returns, whch served as a benchmark for comparng relatve proftablty of strateges MV and MG. The sx rsky assets selected are those most senstve durng the examned perod: Addoha, Atlanta, BCP, Delta Holdng, Managem and Maroc Telecom. Ths study begns wth an analyss of the characterstcs of sx selected assets that allows the constructon of a portfolo usng the MV strategy, the MG strategy, the Mn-V strategy and Mn-G strategy. Ths analyss determnes the weghts of the sx assets. Descrptve statstcs are presented n Table. The strong results for the normalty test (Jarque-Bera) for each stock, led us to reject the null hypothess of the normalty test at 99% confdence level. These results ndcate a well-known property of fnancal data seres: returns are usually not normally dstrbuted. In addton, skewness and kurtoss, other propertes of rsky assets, have been dscovered n our data seres. Snce both propertes are apparent n our data, we assume that usng the Mean- Gn strategy should provde the best portfolo due to the fact that the Gn strategy exceeds normal return dstrbuton assumptons. Based on these results, we assume that n the context of our data, MG strategy must produce better results than the MV strategy. After the applcaton of optmsaton programs of the MV strategy, the MG strategy, the Mnmum Varance strategy (Mn-V) and the Mnmum Gn strategy (Mn-G), we obtaned ther optmum portfolos n Table and Table 3 presents summary statstcs of optmal portfolos obtaned by the resoluton of optmzaton programs. 75

5 Table. Descrptve statstcs. Addoha Atlanta BCP Delta Holdng Mangem Maroc Telecom Mean Std. Dev Gn Skewness Kurtoss Jarque-Bera Probablty Observatons Table. Percentage of stocks n optmal portfolos. MV MG Mn-V Mn-G Addoha Atlanta BCP Delta Holdng Managem Maroc Telecom Table 3. Summary statstcs of optmal portfolos. MV MG Mn-V Mn-G Mean Std. Dev Gn Skewness Kurtoss Jarque-Bera Return seres for optmal portfolos are plotted n Fgure. Plots demonstrate that the return seres are extremely unstable. In order to make the comparson of the two strateges clearer, we used quas-analytc VaR and CVaR methods usng the GARCH (.) model, because the varance s not homoscedastc as the ARCH test result proves. Ths was done n order to take nto account those specfc characterstcs apparent n our data. In order to move away from a normal dstrbuton framework for the predcton of VaR and CVaR, we used the Cornsh-Fsher expanson and the Johnson SU approxmaton. Table 4 presents the dfferent tests of statonary. We accept the alternatve hypothess that the seres of returns of the four portfolos are statonary and the result of the ARCH test leads us to reject the null hypothess. Therefore, t s assumed that the resdual varance s not homoscedastc. The predcton the quas-analytc VaR and CVaR of portfolos wll be made on a GARCH (; ) model. 4. Results 4.. Estmatng Parameters The results of the goodness-of-ft tests for dfferent models ARMA/GARCH show clearly that a combnaton of AR ()-GARCH (.) and Gaussan resduals and student-t resduals are the approprate models, from a statstcal 76

6 Mean-Varance Mean-Gn Mnmum Varance Mnmum Gn Fgure. Evoluton of four portfolos. 3 Table 4. Unt root tests of the seres of returns. Tests MV MG Mn-V Mn-G Test crtcal value: 5% level ADF KPSS ERS ARCH test pont of vew, for portfolos n tmes of market turbulence. Table 5 and Table 6 present parameters of GARCH (.) Model wth Normal Dstrbuton and Student-t Dstrbuton. 4.. Back-Testng VaR Estmates We evaluated the accuracy of the proposed VaR estmates over a 5 day perod usng the now standard coverage tests of Chrstoffersen (998). We combned the GARCH (.) model wth the approxmaton method. The Cornsh-Fsher expanson, and the Johnson SU approxmaton derve the VaR estmates for each portfolo where = % ; 5% and %. In fnance lterature tells us that there are two fundamental test procedures used to compare the performances of VaR: Uncondtonal and Condtonal. We make use of Kupec s (995) Test to evaluate GARCH specfcatons for uncondtonal coverage, and the Chrstoffersen Test to embrace both uncondtonal coverage and the ndependence of volatons. The Kupec Test and Chrstoffersen Test results for the portfolos are reported n Tables

7 Table 5. Estmatng parameters of GARCH (.) model wth normal dstrbuton. MV MG Mn-V Mn-G Probablty β Table 6. Estmatng parameters of GARCH (.) model wth student-t dstrbuton. MV MG Mn-V Mn-G Probablty β Table 7. Uncondtonal coverage and condtonal coverage of VaR GARCH (.) model wth normal dstrbuton. % 5% Coverage test MV MG MV MG MV MG N Rate LRnd NA NA N Rate LRnd N Rate % LRuc NA LRnd LRcc NA Fnally, we utlsed CVaR to compare the two pars of strateges because the outcomes are smlar and CVaR can better capture the tal rsk. Results are reported n Tables Dscusson Ths paper dscusses and compares analytcal results obtaned wth MV, MG, the Mn-V and the Mn-G strateges on the Moroccan fnancal market (MADEX) durng perods of market nstablty, and demonstrates, emprcally, that quas-analytc GARCH VaR forecasts can be accurately constructed usng analytc formulae for hgher moments of aggregated GARCH returns by usng Cornsh-Fsher expanson and the Johnson SU dstrbuton. Results show that the composton of two pars of strateges: MV versus MG and Mn-versus Mn-G strategy are smlar but not dentcal for that we have very close results n the VaR whch s not convncng to compare the two strateges. The judgment s conclusve when consderng the CVaR because the MG strategy has lower 78

8 Table 8. Uncondtonal coverage and CONDITIONAL coverage of VaR GARCH (.) model wth student-t dstrbuton. Coverage test MV MG MV MG MV MG N Rate.8.4 % LRnd NA NA NA NA N Rate % LRnd N Rate % LRuc LRnd LRcc Table 9. Uncondtonal coverage and condtonal coverage of VaR GARCH (.) model wth normal dstrbuton. Coverage test Mn-V Mn-G Mn-V Mn-G Mn-V Mn-G N Rate.4.8 % LRnd NA NA NA NA N Rate % LRuc NA NA NA NA LRnd LRcc NA NA NA NA N Rate % LRuc LRnd LRcc

9 Table. Uncondtonal coverage and condtonal coverage of VaR GARCH (.) model wth student-t dstrbuton. % 5% % Coverage test Mn-V Mn-G Mn-V Mn-G Mn-V Mn-G N Rate.4.8 LRnd NA NA NA NA N Rate LRuc NA NA NA LRnd LRcc NA NA NA N Rate LRuc LRnd LRcc Table. CVaR GARCH (.) model wth normal dstrbuton. MV MG MV MG MV MG % ** NA NA ** 5% ** ** ** % ** ** ** ** Represent the lower CVaR of both strateges. Table. CVaR (.) model wth student-t dstrbuton. MV MG MV MG MV MG % NA NA NA NA % ** ** ** % ** ** ** ** Represent the lower CVaR of both strateges. Table 3. CVaR GARCH (.) model wth normal dstrbuton. Mn-V Mn-G Mn-V Mn-G Mn-V Mn-G % NA NA NA NA % ** **.56.4 ** %.48 ** ** ** ** Represent the lower CVaR of both strateges. 8

10 Table 4. CVaR (.) Model wth student-t dstrbuton. Mn-V Mn-G Mn-V Mn-G Mn-V Mn-G % NA NA NA NA ** 5% ** **.56.4 ** % ** ** ** ** Represent the lower CVaR of both strateges. CVaR to that of the MV strategy. Ths s also true of the Mn-G strategy n relaton to the Mn-V strategy. In vew of these results, we conclude that the MG strategy outperforms the MV strategy n our real-world examples taken from the Moroccan Fnancal Market. Ths s due to the characterstcs of the fnancal assets that do not follow a normal dstrbuton and the unstable nature of the varance n tme. There was great accuracy for all sgnfcance levels (%, 5% and %), when we consdered for GARCH VaR forecastng. Our results are even more remarkable when we consder that the analyss s entrely out-ofsample and that the testng perod (-4) covers several prolonged perods of excessvely turbulent fnancal market actvty. References Agouram, J., & Lakhnat, G. (5). Mean-Gn Portfolo Selecton: Forecastng VaR Usng GARCH Models n Moroccan Fnancal Market. Journal of Economcs and Internatonal Fnance, 7, Chrstoffersen, P. F. (998). Evaluatng Interval Forecasts. Internatonal Economc Revew, 39, Dorfman, R. (979). A Formula for the Gn Coeffcent. Revew of Economcs and Statstcs, 6, Favre, L., & Galeano, J. A. (). Mean-Modfed Value-at-Rsk Optmzaton wth Hedge Funds. Journal of Alternatve Investments, 5, -. Kupec, P. (995). Technque for Verfyng the Accuracy of Rsk Measurement Models. Journal of Dervatves,, Markowtz, H. (95a). Porfolo Selecton. Journal of Fnance, 7, Markowtz, H. (95b). The Utlty of Wealth. The Journal of Poltcal Economy (Cowles Foundaton Paper 57), LX (), Markowtz, H. (959). Portfolo Selecton: Effcent Dversfcaton of Investments. New York: Wley. Shalt, H., & Ytzhak, S. (984) Mean-Gn, Portfolo Theory, and the Prcng of Rsky Asset. Journal of Fnance, 39, Shephard, N. (996). Statstcal Aspects of ARCH and Stochastc Volatlty. Monographs on Statstcs and Appled Probablty, 65, -68. Ytzhak, S. (98). Stochastc Domnance, Mean-Varance, and Gn s Mean Dfference. Amercan Economc Revew,, Zangar, P. (996). A VaR Methodology for Portfolos That Include Optons. RskMetrcs Montor, JP Mogran-Reuters, Frst Quarter, 4-. 8