Forecasting Portfolio Risk Estimation by Using Garch And Var Methods

Transcription

1 ISSN -697 (Paper) ISSN -847 (Onlne) Vol 3, No., 0 Forecastng Portfolo Rsk Estmaton by Usng Garch And Var Methods. Noor Azlnna Azzan, Faculty of Technology, Unverst Malaysa Pahang, Lebuhraya Tun Razak, 6300 Kuantan, Pahang, Malaysa. Tel: Lee Cha Kuang, Faculty of Technology, Unverst Malaysa Pahang, Lebuhraya Tun Razak, 6300 Kuantan, Pahang, Malaysa. 3. Zeenat Ahmed, Insttute of Mathematcal Sccences, Unverst of Malaya, Malaysa. *Correspondng author: Emal: azlnna@ump.edu.my Abstract Rsk management or rsk predctng are closely related wth the market volatlty whch affect the return of portfolo estmaton. Portfolo managers around the world concerned wth rsk estmaton because portfolo rsk management s part of ther decson-makng process. Accordng to Hull (006), VaR s wdely used by fund managers to provde a sngle number summarzng the total rsk n a portfolo of fnancal assets. Motvates from ths, we conducted an analyss to compare the effectveness of VaR analyss and GARCH method n forecastng rsk estmaton. Rsk manager can used the best methods n reducng ther customers rsk volatlty and rank the rsk level. Keywords: Forecastng, Value at Rsk, GARCH, Portfolo estmaton, Rsk.. Introducton Value at rsk (VaR) s wdely used by banks, securtes frms, commodty merchants, energy merchants, and other tradng organzaton. Such frms could track ther portfolo market rsk by usng hstorcal volatlty as a rsk metrc. VaR has become a very popular measure of market rsk. VaR s the loss on the portfolo that wll not be exceeded wth a specfed probablty over a specfed tme horzon. VaR s an extremely powerful rsk measure, because looks at downsde rsk, that s well suted for asymmetrcal dstrbuton, and because n prncple t can calculated assumng any knd of dstrbuton of portfolo returns. VaR s wdely used for controllng traders, for determnng captal requrements and for dsclosure to external subjects, both nvestors and regulators. (Raffaele.Z &Massmlano.P, 000) Adaptng VaR measures for asset managers (rather than traders) nvolves fndng a proper way to model future scenaros, preservng the multvarate propertes of asset returns, when tme horzon s relatvely long. Accordng to Raffaele.Z &Massmlano.P, (000) the VaR concept has been further extended to the portfolo value at rsk (PVaR) measure used to evaluate the maxmum potental loss of a portfolo wth a gven probablty over a specfed perod (Manganell & Engle, 00). Accordngly, our paper explores the queston of whether VaR analyss s better than GARCH model n forecastng rsk. We wll compare dfferent VaR analyss methods such as hstorcal smulaton method and normal dstrbuton. They are several mportances; frst, practtoners are redscoverng the mportance of portfolo rsk management as part of ther decson-makng process. Second, Levy and Levy (004) show that ths model can be used for makng portfolo selecton decsons and thrd accordng to Hull (006, p. 435) notes, VaR s wdely used by fund managers to provde a sngle number summarzng the total rsk n a portfolo of fnancal assets. Fnally, the economc losses arsng from gnorng estmaton rsk can be partcularly large (see, e.g., Best and Grauer (99), Chopra and Zemba (993), and Chan, Karcesk, and Lakonshok (999)).. Methodology Descrpton of the data The data set conssts of daly stock ndces between 000 and 009 for the followng market: a) Malaysa Kuala Lumpur composte Index (KLCI). b) Inda: Bombay Stock Exchange (BSE). c) Japan: Nkke Stock Average 5. 6

2 ISSN -697 (Paper) ISSN -847 (Onlne) Vol 3, No., 0 d) Sngapore: Strats Tmes Index. 3. Data Analyss and Dscussons 3. Dstrbutons of Returns The followng tables dsplay the results for normalty test for the data tested. Table : Normalty test results Return Malaysa Sngapore Inda Japan Test Stat p.value Dst. under Null: ch-square wth degrees of freedom Based on table, the null of normalty s rejected usng ths test snce P value s less than 0.05 and t s sgnfcant. Table : Descrptve Statstcs for daly returns Return Mean Std Dev Skewness Kurtoss MALAYSIA JAPAN INDIA SINGAPORE From the above table, we can see that daly return of market ndexes have hgh Kurtoss for daly seres. Ths means that the daly returns are not normally dstrbuted, and the mean of daly return seres s very close to zero. Daly returns for Malaysa has low standard devatons compare to other market ndexes and Inda has the hghest standard devaton so t wll be more rsky. When the data s not normal, uncondtonal volatlty s not realstc. Condtonal volatlty s emprcally observed and probably s the culprt behnd fat-taled asset returns. 3. Estmaton of ARCH/GARCH Models ARCH models assume the varance of the current error term or nnovaton to be a functon of the actual szes of the prevous tme perods' error terms: often the varance s related to the squares of the prevous nnovatons. To Test ARCH Effects we used the Lagrange multpler (LM) prncple can be appled. Consder the null hypothess of no ARCH errors versus the alternatve hypothess that the condtonal error varance s gven by an ARCH (q) process. The test approach proposed n Engle [98] s to regress the squared resduals on a constant and q lagged values of the squared resduals. From the results of ths auxlary regresson, a test statstc s calculated as: (N-q) R There s evdence to reject the null hypothess f the test statstc exceeds the crtcal value from a ch-square dstrbuton wth q degrees of freedom. Null Hypothess H0 :no ARCH effects Table 3: Test for ARCH Effects for ndex return: Lagrange Multpler (LM) Test Index Malaysa Sngapore Inda Japan Test Stat p.value

3 ISSN -697 (Paper) ISSN -847 (Onlne) Vol 3, No., 0 The above table 3, stated that the value of test Statstcs for the four returns are very bg f we compare t wth statstcal table for wth 33 degrees of freedom, so F s sgnfcant, so reject H0.There are ARCH effects. To avod ths problem we model all the market tested daly return usng GARCH model. The followng tables 4(a),(b),(c) and (d) are the results for GARCH model for all the market tested. Table 4(a): Results of GARCH model for daly return (Malaysa) Model coeffcent Std.Error t value Pr(> t ) φ β AIC = Table 4, provdes some descrptve statstcs of KLCI daly return. The sample sze data are 086 observatons. Our results show that GARCH(,) model s the most sgnfcant compare to other GARCH model wth hgher order rank and ths s prove by the lowest AIC = So our GARCH (,) model s the followng. a t= tε t = + + t 0 a t β t = t a t 0.7 t where εt s a sequence of ndependent and dentcally dstrbuted (d) random varables wth mean zero and varance, 0 > 0, and 0 for > 0. Table 4(b): Results of GARCH model for Sngapore daly return Model coeffcent Std.Error t value Pr(> t ) φ β AIC = For AIC value we choose the model wth the smallest AIC value, from table 4(b) above the model has the smallest AIC value, whch show that there s GARCH (, ) effects then a t = t = + + t 0 a t β t ε t = t a t 0.7 t 4 64

4 ISSN -697 (Paper) ISSN -847 (Onlne) Vol 3, No., 0 Where ε t s a sequence of ndependent and dentcally dstrbuted (d) random varables wth mean zero and varance, 0 > 0, and 0 for > 0. Table 4(c): Results of GARCH model for Inda daly return from Model value Std.Error t value Pr(> t ) φ β AIC = For AIC value we choose the model wth the smallest AIC value, from table 4(c) above the model has the smallest AIC value, whch show that there s GARCH (, ) effects, then a t = tε t t = 0+ at +β t = 0 + t a t t Table 4(d): Results of GARCH model for Japan Daly return Model Coeffcent Std.Error t value Pr(> t ) θ β AIC = For AIC value we choose the model wth the smallest AIC value, from table 4(d) above the model has the smallest AIC value, whch show that there s GARCH (, ) then a t = tε t t = 0+ at +β t = t a t 0.7 t Based on all table 4 (a), (b), (c) and (d) all market can be modeled by GARCH (, ). Ths means volatlty s a functon of lagged squared returns and lagged varances of one day. The coeffcent of the ARCH effect () s statstcally sgnfcant at % sgnfcance level. Ths ndcates that news about volatlty from the prevous perods has an explanatory power on current volatlty. Smlarly, the coeffcent of the lagged condtonal varance (β) s sgnfcantly dfferent from zero, ndcatng volatlty clusterng n all markets return seres. The sum of ( + β) coeffcents s unty, suggestng that shocks to the condtonal varance are hghly persstent. Ths mples that wde changes n returns tend to be followed by wde changes and mld changes tend to be followed by mld Changes. A 65

5 ISSN -697 (Paper) ISSN -847 (Onlne) Vol 3, No., 0 major economc mplcaton of ths fndng for nvestors s that stock returns volatlty occurs n cluster and as t s predctable. From Table 4(a) (b), (c) and (d), we also notce that asymmetry (gamma) coeffcent s postve. The sgn of the gamma reflects that a negatve shock nduce a larger ncrease n volatlty greater than the postve shocks. It also mples that the dstrbuton of the varance of all market returns s left skewed, mplyng greater chances of negatve returns than postve. The postve asymmetrc coeffcent s ndcatve of leverage effects evdence n Ngera stock returns. 3.3 Value at Rsk (VaR) Ths secton summarzes the steps for calculatng Value-at-Rsk (VaR) for a portfolo of equty assets usng S-PLUS 7.0 and S+FnMetrcs.0. VaR s computed usng emprcal quantles, and the normal dstrbuton. Some basc concepts of asset returns and portfolos, and defnes the market rsk concepts value-at-rsk (VaR) and expected tal loss (ETL) (whch s also called expected shortfall (ES)) Asset Returns The portfolo conssts of =,..., N equty assets. Let Pt denote the prce of asset at tme t. The one-perod smple return on asset between tmes t and t s Pt Pt = R t P t 3.3. Value-at-Rsk Defned Consder a one perod nvestment n an asset wth smple return R. Let $ W 0 denote the ntal dollar amount nvested. The value of the nvestment after one perod n terms of the smple return s $ W = $ 0(+ R) VaR Based on Smple Returns For (0,), let q R denote the 00% quntles of the probablty dstrbuton of the smple return R. Usually, q R s a low quartle such that = 0.0 or = As a result, qr s typcally a negatve number. The 00% dollar Value-at-Rsk ($VaR ) s $VaR = $W 0 q R In words, $ represents the dollar loss that could occur wth probablty. By conventon, t s reported as a postve number (hence the mnus sgn). The VaR as a percentage of the ntal portfolo value s smply the (negatve) low quartle of the smple return dstrbuton: VaR $VaR = $W = q Expected Tal Loss Defned 0 R The 00% expected tal loss (ETL), n terms of the log return, s defned as ETL = E[r r < ] In words, the ETL s the expected (negatve) return condtonal on the return beng less than the 00% percentage VaR. If the ntal nvestment s $, then the dollar ETL s $ET L = $ ET L 66 W

6 ISSN -697 (Paper) ISSN -847 (Onlne) Vol 3, No., Hstorcal Smulaton A dfferent approach for VaR assessment s called Hstorcal Smulaton (HS). Ths technque s nonparametrc and does not requre dstrbutonal assumptons. Ths s because HS uses essentally only the emprcal dstrbuton of the portfolo returns. Hstorcal smulaton s one of the popular ways of estmatng VaR. It nvolves usng past data n a very drect way as gude to mght happen n the future. Ths data conssts of the daly movements n all market varables over the perod of tme. The frst step n ths method s to dentfy the market varables affectng the portfolo. Then collect data on the movements n these market varables over the perod of tme. The frst smulaton tral assumes that the percentage changes n each market varable are the same as those on the frst day covered by the data, the second smulaton tral assumes that the percentage changes n the portfolo value, P s calculated for each probablty dstrbuton P. Ths defnes a probablty dstrbuton for daly change n the value of portfolo. Defne ν as the value of a market varable on day I and suppose that today s day m. The I th scenaro assumes that the value of the market varables tomorrow wll be ν Hstorcal smulaton (HS) smply refers to the emprcal dstrbuton of the observed returns. As a result, the 00% VaR based on HS s just the 00% emprcal quartle of the return dstrbuton. (same dea s n Hull. J. C. 006) Normal Dstrbuton Assume the N vector of log-returns r has a multvarate normal dstrbuton wth mean vector μ and covarance matrx Σ, r N(μ, Σ) where μ has elements ( =,..., N) and Σ has elements j, ) The 00% quartle r of the normal µ dstrbuton for r s m ν ν (, j =,..., N ). For an ndvdual asset, N( Where s the z 00% quartle of the standard normal dstrbuton. The dstrbuton of gven that s q = q µ + q z truncated normal. The mean of ths dstrbuton s the normal ETL. Greene (004) shows that r φ( E [ r / r q ] = µ + z) Φ( z) VaR ) / φ(z) s the standard normal PDF and (Z) where z = (µ Φ s the standard normal CDF. Gven a random sample of sze T of observed returns on N assets from the multvarate normal dstrbuton, the mean vector μ and covarance matrx Σ may be estmated usng the sample statstcs T ˆµ = T rt, ˆ Σ= T ( rt ˆ)( µ rt µ ) t= T t= The normal quartle may then be estmated usng the plug-n method z qˆ = ˆ µ + ˆ q where s the th element, and ofµˆ ˆ s the square root of the th dagonal element of. µ Smlarly, the estmate of normal ETL s φ( Eˆ[ r/ r q ] = ˆ µ + zˆ ) ˆ Φ( zˆ ) where = )/ (µˆ ˆ, and VaR ˆ = ˆ µ + ˆ q q z z 67

7 ISSN -697 (Paper) ISSN -847 (Onlne) Vol 3, No., 0 Standard errors for these estmates may be convenently computed usng the bootstrap.(same dea s n Erc,Z. 005) VaR.0 for Malaysa, Sngapore, Inda, Japan based on hstorcal smulaton and normal dstrbuton: Malaysa Sngapore Inda Japan Hstorcal smulaton Normal dstrbuton Wth % probablty the loss s about.7%, 3%, 4.8% and 3.% or hgher for (Malaysa, Sngapore, Inda, Japan) respectvely, based on hstorcal smulaton method. Wth % probablty the loss s about %,.4%, 3.6% and.8% or hgher for (Malaysa, Sngapore, Inda, Japan) respectvely, based on normal dstrbuton method. Compare the above results, we found that for hstorcal smulaton method the % probablty loss s hgher than the normal dstrbuton method. We can also make a concluson that the hghest most rsky market s Inda, follow by Japan, Sngapore and Malaysa, ths consstent wth our data descrpton statstc n chapter 3, where the standard devaton for Inda s the hghest compare to other market. 4. Concluson Our results for Garch (,) model and VaR model for all the market tested showed that VaR has better n predcton the rsk because VaR gves the percentage and rank of rsk level. The man objectve of ths study s to detect and forecast the rsk movement and volatlty of the Kuala Lumpur Composte Index (KLCI) data and other Asan markets lke Sngapore, Inda and Japan from 000 to 009. We also compared dfferent VaR analyss method such as hstorcal smulaton method and normal dstrbuton method n portfolo rsk estmaton. Besdes that we compare the two VaR methods wth GARCH model. We dscover that wth % probablty the loss s about.7%, 3%, 4.8% and 3.% or hgher for KLCI, Sngapore, Inda, Japan respectvely based on hstorcal smulaton method. Wth % probablty the loss s about %,.4%, 3.6% and.8% or hgher for KLCI, Sngapore, Inda and Japan respectvely, based on normal dstrbuton method. Compare the above results, we found that for hstorcal smulaton method the % probablty loss s hgher than the normal dstrbuton method. Whereas the GARCH method can only forecast by usng the lag value wthout able to rank the rsk level. We concluded that the hghest most rsky market s Inda, follow by Japan, Sngapore and Malaysa, ths consstent wth our data descrpton statstc where the standard devaton for Inda s the hghest compare to other market. References Al Janb, M. A. M. (008). Integratng lqudty rsk factor nto a parametrc value at rsk method. Insttutonal Investor, Vol 3, (3), Andrey, Y. R. (005). Methodologcal Issues and Some Illustratons of Applyng Dynamc Value-at-Rsk Model n Portfolo Management. Workng paper, socal scence research networkng database. Best, Mchael J., and Robert R. Grauer, (99), On the senstvty of mean-varance-effcent portfolos to changes n asset means: Some analytcal and computatonal results, Revew of Fnancal Studes 4,

8 ISSN -697 (Paper) ISSN -847 (Onlne) Vol 3, No., 0 Chan, Lous K.C., Jason Karcesk, and Josef Lakonshok, (999), On portfolo optmsaton: Forecastng covarances and choosng the rsk model, Revew of Fnancal Studes, Chopra, Vjay K. and Wllam T. Zemba, (993), The effect of errors n means, varances and covarances on optmal portfolo choce, Journal of Portfolo Management 9, 6-. Hull. J. C. (006). Optons, futures, and other dervatves. Pearson Prentce Hall. 6 th ed. Levy, Ham, and Harry M. Markowtz, 979, Approxmatng expected utlty by a functon of mean and varance, Amercan Economc Revew 69, Ln. P. C. and Ko, P.-C (008). Portfolo value- at rsk forecastng wth GA-based extreme value theory. Expert Systems wth Applcatons. Massmlano Pallotta, Raffaele zent (00) Rsk Analyss for Asset Managers: Hstorcal Smulaton, the Bootstrap Approach and Value at Rsk Calculaton, socal scence research networkng database. Pangrtzoglou, Nkolaos, and George Skadopoulos, 004, A new approach to modelng the 34 dynamcs of mpled dstrbutons: Theory and evdence from the S&P 500 optons, Journal of Bankng and Fnance 8, Porte, N. (007). Revenue volatlty and fscal rsks. Emergng markets fnance and trade, Vol.43,(6), 6-4. Smth D. R. and Pergnon C. (007). Whch value-at. rsk method works best for bank tradng revenues? Workng paper, socal scence research networkng database. 69

9 Ths academc artcle was publshed by The Internatonal Insttute for Scence, Technology and Educaton (IISTE). The IISTE s a poneer n the Open Access Publshng servce based n the U.S. and Europe. The am of the nsttute s Acceleratng Global Knowledge Sharng. More nformaton about the publsher can be found n the IISTE s homepage: CALL FOR PAPERS The IISTE s currently hostng more than 30 peer-revewed academc journals and collaboratng wth academc nsttutons around the world. There s no deadlne for submsson. Prospectve authors of IISTE journals can fnd the submsson nstructon on the followng page: The IISTE edtoral team promses to the revew and publsh all the qualfed submssons n a fast manner. All the journals artcles are avalable onlne to the readers all over the world wthout fnancal, legal, or techncal barrers other than those nseparable from ganng access to the nternet tself. Prnted verson of the journals s also avalable upon request from readers and authors. IISTE Knowledge Sharng Partners EBSCO, Index Coperncus, Ulrch's Perodcals Drectory, JournalTOCS, PKP Open Archves Harvester, Belefeld Academc Search Engne, Elektronsche Zetschrftenbblothek EZB, Open J-Gate, OCLC WorldCat, Unverse Dgtal Lbrary, NewJour, Google Scholar