A New Robust Estımator for Value at Rısk

Transcription

1 Amercan Research Journal of Busness and Management Orgnal Artcle Volume 1, Issue1, Feb-2015 A New Robust Estımator for Value at Rısk Nur Celk a1, Chan Dncer b a Bartn Unversty, Department of Statstcs,74100 Merkez, Bartn,Turkey b Cologne Busness School, Department of Internatonal Busness, 50677, Cologne, Germany Abstract: We consder a common rsk measurng method namely Value-at-Rsk (VaR). The easest and the most prevalent method of calculatng VaR s the varance-covarance method. Ths method s based on normal dstrbuton assumpton. However, there are a lot of nferences n lterature that non-normal dstrbutons are much more common than the normal dstrbuton. Because of economc growth and poltcal and fnancal ssues, there can be possble hgher or lower prces than normal ones n economc data, whch are named outler n statstc theory. In order to handle these data anomales and dstrbuton dfferences, robust estmaton and testng methods have been determned and studed for last decades. In ths study, we propose a new robust varancecovarance estmator for calculatng VaR value of a gven portfolo. Smulaton results show that the proposed estmator s more robust than the correspondng normal theory solutons. Also, a real data for dfferent economcal markets are analyzed to show the performances of the proposed estmators. Keywords: Value at Rsk, Varance-Covarance Method, Robust Statstcs, Outler I. INTRODUCTION Estmatng a loss n assets wll not only contrbute to fnancal busnesses but also to other companes and ndvduals. Busnesses, operatng globally undertake more rsk than those runnng ther busness on a natonal level. Ths addtonal threat needs to have an effcent rsk management plan or otherwse the busnesses wll face fnancal problems or even bankruptcy. Understandng ths threat nternatonal companes have started to develop countermeasures n order to asses and to defeat the rsk. Value at rsk (VaR) s a method, recommended by Basel standards, whch s appled by several fnancal busnesses to measure the rsk. It dentfes the hghest probablty of loss of value n assets or a gven portfolo n a gven tme perod. In other words, t dsplays the hghest possble loss n value held wthn a certan perod of tme and a gven confdence nterval. Researches have shown that busnesses can protect themselves from rsk f an accurate estmaton of VaR s done prevously. Ths wll also ensure a company's sustanablty. Due to ths, many partes ncludng governments, external audtors, busnesses wthn a supply chan, and labor unons have a hgh nterest n these nformaton. Calculatng VaR s wdespread not only because of ts easness to compute, also due to the acceptance by several busnesses. Furthermore, t s recommended by the Bank of Internatonal Settlement (BIS). The rsk of a portfolo s llustrated by a one dgt number, whch appears convenent to fnancal enttes. There are several methods to calculate VaR, for nstance the varance-covarance method (Jordan and Mackay, 1997), Exponentally Weghted Movng Average (EWMA) method (Hendrcks, 1996), Hstorcal Smulaton and Monte Carlo smulaton method (Holton, 1998), Extreme Values method (Longn, 2000 and Ho et. all. 2000), Kernel densty method (Butler and Schacter, 1997) Generalzed Auto Regressve Condtonal Heteroscedastcty (GARCH) method (Alexander, 1996) and Fractonally Integrated ARCH (Beltratt and Maronna, 1999). The smplest and the most common method s the varance-covarance method. Ths method s based on the assumpton of normalty. However, n lterature, there are several studes underlnng that non-normal dstrbutons are more prevalent than the normal dstrbuton n practce, see for example, Pearson (1932), Geary (1947), Huber (1981) and Tan and Tku (1999). In addton, observatons n a sample whch are too small or too large as compared to the bulk of observatons are called outlers. Snce ther presence adversely affects the effcency of most statstcal procedures (Tku and Akkaya, 2004). Therefore, nonparametrc methods and sem parametrc methods have been studed to handle these dstrbuton dfferences and data anomales n order to calculate VaR values, see Duffe and Pan (1997), Prtsker (1997) and Kuester et all. (2006). However, usng robust methods for analyzng n such stuatons are much more relable than sem parametrc and nonparametrc methods. An estmator s called robust f 1 Correspondng Author: ncelk@bartn.edu.tr 12

2 t s fully effcent under the assumed model and mantans hgh effcency under the plausble alternatves of the assumed model, see Tku and Akkaya (2004). There are only a few studes predctng VaR values n a robust way n lterature. The man focus of these studes s to estmaton of tals of proft and loss dstrbutons, see Mancn and Trojan, 2007 and Gebzloglu et. all, Addtonally, there s no prevous work of estmatng robust verson of volatlty. For ths reason, we propose a new robust method for calculatng VaR usng robust varance-covarance matrx. The rest of the paper organzes as follows, n chapter 2, we gve a bref nformaton about the standard varancecovarance method, then we defne a robust method for estmatng varance-covarance matrx and wth ths new matrx we calculate new robust VaR value for a gven asset or portfolo. In the next chapter, we smulate data from varous dstrbutons used n statstcal analyss and compare two estmators. Lastly, we apply ths estmator for dfferent real data examples. Concluson s gven at the end of the paper. II. VARIANCE-COVARIANCE METHOD Ths parametrc approach dentfes parameters, whch nfluence the value of a portfolo. It also llustrates the hghest possble loss whch occur due to fluctuatons on a certan probablty level. It s the method used by the RskMetrcs methodology and developed by JP Morgan. The method s based on the assumpton of normal dstrbuton. By assumng the normal dstrbuton VaR of portfolos can be calculated n terms of a lnear functon of standard devaton of assets. Wth the varance-covarance method prmarly the mean and the volatlty (standard devaton) have to be attaned n order to calculate VaR of portfolo, whch contans n fnancal assets. Afterwards, the weght vector (w) has to be determned. Based on ths, the mean and the standard devaton of the portfolo can be obtaned n w, 1, 2,..., N (1) p 1 And N N w w 1,2,..., N ; j 1,2,..., N (2) p j j 1 j 1 respectvely. where s the mean of th asset, w s the weght of th asset n portfolo, j s the covarance between th and jth asset and N s the number of assets n a gven portfolo. Then VaR s calculated wth the multplcaton of ths standard devaton and requred probablty, whch s constant n normal dstrbuton ( 1.96 wth the probablty %95, 2.58 wth the probablty %99.) The advantage of ths method s the easness of computng. On the other hand, the dsadvantage s the assumpton of normal dstrbuton. As llustrated n ntroducton part, usng robust methods s more relable n data analyss n the presence of outlers and dstrbuton dfferences. The varance of a sample s not robust and affected by even one outler. Therefore, VaR value based on the varance-covarance method s not robust as well. For ths reason, we determne a robust estmator of varance-covarance matrx and the method for calculatng VaR based on ths robust varance-covarance matrx. III. ORTHOGONALIZED GNANADESIKAN-KETTERNRING ESTIMATE The Orthogonalzed Gnanadeskan-Ketternrng (OGK) estmator s based on the robust covarance matrx estmate defned by Gnanadeskan and Ketternrng (1972). The defned varance-covarance matrx s symmetrc but not necessarly postve sem defnte. To overcome the problem of sem defnteness, Maronna and Zamar (2002) proposed a new estmate for mean vector and covarance matrces. The dstnctve feature of the OGK estmaton s that t combnes the use of the actual measurements wth an exstng estmate of the covarance and hence mplctly accounts for any correlaton between the sources of the measurements and for the way that these were obtaned (Sequera et all, 2011). The estmaton procedure s very fast and easy to compute. p Let x1, x2,..., xn be a dataset and (.) and (.) be robust unvarate dsperson and locaton statstcs and Let v (.,.) be a robust estmate of the covarance of two random varables. Maronna and Zamar(2002) defne a new robust varance-covarance matrx Z(X) and mean vector m(x) as follows: 13

3 -1 1. Let D dag( ( X1),..., ( X n )) and y = D x, 1,2,..., n 2. Compute the correlaton matrx C [ ],applyng v to the columns of Y,that s C 1, and C v( Y, Y ), j k. jk j k C jk 3. Compute the egenvalues j and egenvectors that C = EΛE',where Λ dag( 1,..., p ). 4. Let A = DE, and e j of C and call E the matrx whose columns are the -1 z = E'y = A x, so that x = Az and defne Z(X) = AΓA' and m(x) = A where Γ dag( var( Z1),..., var( Z p )) and ( ( Z1),..., ( Z p ))'. jj e j `s, so Maronna and Zamar(2002) take v the Gnanadeskan-Kettenrng estmator (Gnanadeskan and Kettenrng,1972) whch s, U jk YJ Yk YJ Yk, j k 4 Then the resultng estmate s called an OGK estmate. The procedure can be extended by teraton and by reweghtng algorthm. To determne (.) and (.) and the propertes of OGK matrx (consstent, sem defnte, affne-equvarant) see Maronna and Zamar, (2002). For the nfluence functon and other robustness propertes of Gnanadeskan-Kettenrng estmator also see Genton and Ma, (1999). Wth ths predefned robust estmator of varance-covarance matrx, we calculate VaR value of a portfolo wth tradtonal calculaton method. The only dfference between standard varance-covarance method and ths proposed method s the estmator of volatlty, whch contans robustness and effcency. IV. SIMULATION STUDY In ths secton, we compare varance-covarance method whch s tradtonally used wth the proposed robust method n terms of VaR values. All the smulaton results are based on [100,000/n] Monte Carlo runs. We use some models, whch dstrbuted near normal or contans outlers namely Dxon's outler model -(n-1) observaton come from normal dstrbuton and one outler (not known whch one) comes from normal dstrbuton wth hgher standard devaton value than the dstrbuton that the bulks of data come from-, Contamnaton model -(1-p)% of the observaton come from normal dstrbuton wth standard devaton S and p% of the observaton come from normal dstrbuton wth standard devaton not equal to S, where p s a proporton dffers from [0,1]- and Mxture model - (1-p)% of the observaton come from normal dstrbuton wth standard devaton S and p% of the observaton come from other dstrbutons near normal-. We use the followng sample models to represent a large number of plausble alternatves. Sample Models: Model (1): Dxon's outler model: (n-1) observatons come from N(0,0.01) but one observaton (we do not know whch one) comes from N(0,0.04) Model (2): Dxon's outler model: (n-1) observatons come from N(0,0.01) but one observaton (we do not know whch one) comes from N(0,0.1) Model (3): Contamnaton model: 0.90N(0,0.01) +0.10N(0,0.03) Model (4): Contamnaton model: 0.90N(0,0.01) +0.10N(0,0.05) Model (5): Mxture model: 0.90N(0,0.01) +0.10Student t(2) Model (6): Mxture model: 0.90N(0,0.01) +0.10Student t(7) (3) 14

4 Smulaton results are gven n Table 1. Table1 Smulated VaR values of dfferent dstrbutons Dstrbuton Var-Cov Method OGK Method Model (1) Model (2) Model (3) Model (4) Model (5) Model (6) Table 1 shows that, VaR values calculated wth OGK method are smaller than the values calculated wth tradtonal method. In addton, the standart devatons, estmated wth robust method are much more effcent than tradtonally estmated standart devatons. Ths outcome approves the expectaton snce t s wthn the nature of robust statstcs. V. APPLICATION We analyzed several dfferent real data from dfferent fnancal markets. We nvestgated Dow Jones from Unted States of Amerca, DAX from Germany, Nkke from Japan and BIST from Turkey. For obtanng a good portfolo, we took 5 dfferent assets from each stock exchange wthn the years to Two operatng n the fnancal market, one n the ftness sector, one n avaton and one n nformaton technology. It may be noted that for the applcaton part the weght matrx w s [0.2;0.2;0.2;0.2;0.2]. In other words the proporton of dfferent assets n portfolo s equal. After obtanng the data from dfferent stock exchanges, the descrptve statstcs are calculated. Then the Q-Q plots are drawn for determnng the dstrbuton. The data are transformed nto the form Pt Pt 1, where P t s the closng prce for a gven asset n certan tme t. P t 1 The descrptve statstcs of the assets of the German market can be found n Table 2. Respectvely, these descrptve statstcs of the Japanese, US Amercan and Turksh stock exchange are gven n Table 3,4 and 5. Table2. Descrptve statstcs of assets from DAX Mean Standard Devaton Skewness Kurtoss Asset Asset Asset Asset Asset Table3. Descrptve statstcs of assets from Nkke Mean Standard Devaton Skewness Kurtoss Asset Asset Asset Asset Asset Table4. Descrptve statstcs of assets from Dow Jones Mean Standard Devaton Skewness Kurtoss Asset Asset Asset Asset Asset

5 Table5. Descrptve statstcs of assets from BIST Mean Standard Devaton Skewness Kurtoss Asset Asset Asset Asset Asset In order to determne the dstrbuton of each assets, we use Q-Q plot technque. The Q-Q plots of each assets from dfferent stock exchange are gven n Fgure 1-4. As llustrated n Fgure 1, the Q-Q plots of asset 1,2,3 and 5 have some outlers, on the other hand, the dstrbuton of asset 4 s not exactly normal but near normal. In Fgure 2,3 and 4 we have also some outlers and dvergence from normal dstrbuton. Fgure1. Q-Q plots of the assets from DAX 16

6 Fgure2. Q-Q plots of the assets from Nkke 17

7 Fgure3. Q-Q plots of the assets from Dow Jones 18

8 Fgure4. Q-Q plots of the assets from BIST To conclude, we gve VaR values of dfferent portfolos from each stock exchanges n Table 6. VaR values are calculated wth %95 confdence levels, for one week and for 1000 dollars. As Table 6 shows, VaR values calculated wth the tradtonal varance-covarance method are hgher than VaR values calculated wth the robust method. 19

9 Therefore, robust method s not affected by outlers and dstrbuton dvergences. The outcome of ths method s more relable and more accurate. Table6. VaR values of dfferent stock exchanges Stock Exchange Var-Cov Method OGK Method Germany Japan USA Turkey VI. CONCLUSION Tradtonally, varance-covarance method are used n the context of Value at Rsk calculaton. The method s appled wth the normal dstrbuton assumpton. However, effcences of the standard devaton estmator are low when the normalty assumpton s not satsfed. Also, the sample standard devaton estmator s nonrobust when the dstrbuton s not normal and outlers n a sample are exstng. In ths paper, we defne a new method of calculatng VaR value wth robust varance-covarance matrx, snce the robust methods have been used to handle these data anomales and departures from normalty. Smulaton studes and real data analyss underlne that the VaR values calculated wth robust method s more relable and more accurate than those calculated wth varance-covarance method. Addtonally, varance-covarance matrces estmated wth robust method are more effcent and more robust than the matrces estmated wth normal theory. REFERENCES [1] Alexander, C. (1996), Volatlty and Correlaton: Measurement, models and applcatons, Rsk Management and Analyss, John Wley and Sons. [2] Beltratt, A. and Maronna, C. (1999), Computng value at rsk wth hgh frequency data, Journal of Emprcal Fnance, 6, [3] Butler, J.S. and Schachter, B. (1997), Estmatng Value-at-rsk wth a precson measure by combnng Kernel estmaton wth hstorcal smulaton, Revew of Dervatve Research, 1, [4] Duffe, D. and Pan, J. (1997), An overvew of Value at rsk, Journal of Dervatves, 4, [5] Geary, R.C. (1947), Testng for normalty, Bometrka, 34, [6] Gebzloglu, O. Senoglu, B. and Kantar, Y.M. (2011), Comparson of certan value-at-rsk estmaton methods for the two-parameter Webull loss dstrbuton, Journal of Computatonal and Appled Mathematcs, 235, [7] Genton, M.G. and Ma, Y. (1999), Robustness propertes of dsperson estmators Statstcs and Probablty Letters, 44, [8] Gnanadeskan, R. and Kettenrng, J.R. (1972), Robust estmates, resduals and outler detecton wth multresponse data., Bometrcs, 28-1, [9] Hendrcks, D. (1996), Evaluaton of Value-at-Rsk models usng hstorcal data, FRBNY Economc Polcy Rewew, 2, [10] Ho, L.C., Burrdge, P., Caddle, J. and Theobald, M. (2000), Value-at-Rsk applyng the extreme value approach to Asan markets n the recent fnancal turmol, Pacfc-Basn Fnance Journal, 8, [11] Holton, A.G. (1998), Smulatng Value-at-Rsk, Rsk, 11-5, [12] Huber, P.J., (1981), Robust Statstcs, Jonh Wley, New York [13] Jordan, J.V. and Mackay, R.J. (1997), Assesng Value at Rsk for equty portfolos: Implementng alternatve technques, Dervatves Handbook Rsk Management and Control, John Wley and Sons. [14] Kuester, K., Mttnk, S. and Paolella, M.S. (2006), Value at Rsk npredcton: A comparson of alternatve strateges, Journal of Fnancal Econometrcs, 4, [15] Longn, F.M. (2000), From value at rsk to stress testng: The extreme value approach, Journal of Bankng and Fnance, 24, [16] Mancn, L. and Trojan, F. (2010), Robust Value at Rsk Predcton, Journal of Fnancal Econometrcs, 9, [17] Maronna, R.A. and Zamar, R.H. (2002), Robust estmates of locaton and dsperson for hgh-dmensonal datasets, Technometrcs, 44-4, [18] Pearson, E.S. (1932), The analyss of varance n cases of nonnormal varaton, Bometrka, 23,

10 [19] Prtsker, M. (1997), Towards, assesng the magntude of Value at Rsk errors due to the errors n the correlaton matrx, Fnancal Engneerng News, 2, [20] Sequera, J., Tsourdos, A., and Lazarus, S.B. (2011), Robust Covarance Estmaton for Data Fuson From Multple Sensors, Instrumentaton and Measurement, IEEE Transactons on, 12, [21] Tan, W.Y. and Tku, M.L. Samplng dstrbutons n terms of Laguerre Polynomals wth applcatons, New Age Internatonal (formerly, Wley Eastern), New Delh, [22] Tku, M.L. and Akkaya, A.D. Robust Estmaton and Hypothess Testng, New Age Internatonal (P) Lmted, Publshers (2004), New Delh, 337pp. 21