BACKTESTING VAR ESTIMATION UNDER GARCH AND GJR-GARCH MODELS

Transcription

1 The 7 h Ieraoal Days of Sascs ad Ecoomcs, Prague, Sepember 9-, 3 BACKTESTING VAR ESTIMATION UNDER GARCH AND GJR-GARCH MODELS Aleš Kresa Absrac The mpora ad o less eresg par of facal rsk maageme s he rsk modellg. Commoly ulzed measure of rsk (o oly by baks ad surace compaes) s Value a Rsk. Sce he facal me seres are ypcal by o-cosa volaly over me, s crucal for Value a Rsk calculao o model he sadard devao of reurs correcly. I he paper we assume (relavely smple) models based o GARCH ad GJR-GARCH models wh Sude dsrbuos of ovaos. These models are back-esed assumg he vesme o Prague sock marke dex. The perod ulzed for back-esg s from 993 ll,.e. 4,67 daly values. The evaluao s made by meas of he deeced umber of excepos,.e. he cases whch he erved losses were bgger ha esmaed Value a Rsk o a gve probably level. Also well-kow sascal ess due o Kupec ad Chrsofferse are ulzed. Accordg o resuls he assumed models are o accurae he rsk s uderesmaed, bu buchg of he excepos s o prese. Key words: back-esg, Value a Rsk, rsk maageme JEL Code: G7, G3 Iroduco Value a Rsk (heceforh VaR) s geerally acceped as a measure of rsks whch he facal suos are exposed o. Smply speakg, he Value a Rsk represes maxmum loss erved wh a gve cofdece level,.e. here wll be loss bgger ha VaR oly wh he probably level of oe mus cofdece level (for baks ad surace compaes he cofdece levels are 99% ad 99.5%). For Value a Rsk esmao we ca geerally recogze hree groups of mehods: varace-covarace mehod, (flered) hsorcal smulao ad Moe Carlo smulao. Whle s crucal for all he mehods o esmae he fuure volaly correcly, we ca dsgush also he mehods whch assume he volaly o be cosa over me ad mehods 7

2 The 7 h Ieraoal Days of Sascs ad Ecoomcs, Prague, Sepember 9-, 3 modellg varace over me. Boh ype models were esed for example by Alexader ad Sheedy (8). Maory of he rece papers publshed o rsk backesg opc s focused o he depedecy modellg (.e. he auhors assume he porfolo composed of more ha oe asse). We ca meo for sace papers publshed by Huag, Lee, Lag, ad L (9), Igaeva ad Plae () or Kresa ad Tchý (). I hese papers he accuracy of models was assessed, bu here were o gve much aeo o he legh of perod ulzed for parameers esmao. I he paper we exame he effec of he chose perod sze ulzed for parameers esmao o he backesg resuls. For he compuaoal smplcy we focus o oe asse porfolo,.e. oly oe rsk facor s assumed ad hus he depedece modellg problems are avoded. I he paper we assume (relavely smple) auoregressve models wh heeroscedascy modelled by GARCH ad GJR models wh Sude ovaos. The goal of he paper s o backes hese volaly models for dffere esmao perod szes. The backesg s performed for he vesme o he Prague sock marke dex over he perod The paper s orgazed as follows. Appled volaly models are defed he ex seco. The, Value a Rsk ad mehod of s backesg are descrbed. I he las seco, ulzed daase s descrbed ad backesg resuls are preseed. Volaly models Volaly models have become mpora ool me seres aalyss, parcularly facal applcaos. Egle (98) erved ha, alhough he fuure value of may facal me seres s upredcable, here s a cluserg volaly. He proposed auoregressve codoal heeroskedascy (ARCH) process, whch has bee laer expaded o geeralzed auoregressve codoal heeroskedascy (GARCH) model by Bollerslev (986). Furher exeso assumed hs paper s asymmerc GJR model proposed by Glose, Jagaaha, ad Rukle (993). There were also proposed oher volaly models such as IGARCH, FIGARCH, GARCH-M, EGARCH, ec. However hese are o subec of sudy hs paper. For boh models he codoal mea wll also be assumed. Thus he models of me x seres N wll be of geeral form as follows, x R x, () ~ 7

3 The 7 h Ieraoal Days of Sascs ad Ecoomcs, Prague, Sepember 9-, 3 where s ucodoal mea of he seres, ~ ~,, () are auocorrelao coeffces for lag up o R, s modelled sadard devao (volaly) ad ~ s a radom umber from Sude probably dsrbuo (heceforh ).. GARCH model The GARCH model was proposed as he exeso of ARCH model order o avod problemac parameers esmao, whe here are may of hem. The model akes he followg form, where, ad P Q, (3) are parameers eeded o be esmaed. The posve varace s assured f, ad. The model s saoary f. P Q. GJR model I was show, frsly by Black (976), ha here s usually dffere mpac of he posve ad egave shocks o he volaly. GJR model, proposed by by Glose e al. (993), akes hs o accou. I s smlar o he GARCH model (3), bu f he prevous ovao was egave (dummy varable model akes he followg form, ) he mpac o volaly s bgger (by he parameer ). The P Q, (4) Q where s a dummy varable whch equals o oe whe ovao s egave ad ull oherwse. Varables,, ad are parameers eeded o be esmaed. The posve varace s assured f,, ad ad model s P Q Q saoary f. 7

4 The 7 h Ieraoal Days of Sascs ad Ecoomcs, Prague, Sepember 9-, 3 VaR ad backesg procedure Value a Rsk (VaR) s owadays commoly acceped measure of he rsk. If we assume a radom varable X he prof from asse / porfolo wh he (u)kow dsrbuo fuco F X, VaR a a gve probably level s he maxmum loss whch wll occur cases (cofdece level), X x R : F x VaR sup X. (5) VaR s usually esmaed for oe day ahead perod ad he (f eeded) recalculaed for loger perods. Mosly ulzed values of are 5%, 5%, % ad.5%. For furher explaao of VaR cocep ad mehods ulzed for s esmao see e.g. (Joro, 6). There are hree basc approaches o VaR esmao () aalycal formula ulzg paramercal probably dsrbuo fuco, () sochasc (Moe Carlo) smulao ha esmaes he quale of a gve dsrbuo umercally, ad () (flered) hsorcal smulao ha relaes VaR esmao o he quale obaed from hsorcal ervaos. For all he models he accurae predco of reurs volaly s fudameal. I hs paper we assume ha he facal reurs ca be modelled by he AR-GARCH ad AR-GJR processes, whch were descrbed he prevous seco. Whe parameers of hese processes are esmaed, VaR ca be calculaed as follows, where ˆ ad VaR R, VaR ˆ x ˆ ˆ x q :, (6) ˆ are esmaed coeffces of codoal mea, ˆ s esmaed sadard devao by oe of he models defed prevous seco ad q s a approprae quale of he ovaos dsrbuo (Sude dsrbuo). By meas of backesg procedure he model s verfed. Ths procedure s based o he comparso of he rsk esmaed a me for me wh he rue loss erved a me. Wh he backesg procedure o a gve me seres he followg wo suaos ca arse he loss s hgher or lower ha s esmao, I f r VaR. (7) f r VaR Whle he former case s deoed by as a excepo, he laer oe s deoed by zero. If he model s accurae, ha roughly excepos (where s he legh of he daa se 73

5 The 7 h Ieraoal Days of Sascs ad Ecoomcs, Prague, Sepember 9-, 3 ulzed for backesg) should be expereced. Bgger quay of excepos meas, ha he model uderesmaes he rsk ad vce versa. For furher deals see e. g. (Hull, 6). Mosly appled sascal ess are due o Kupec (995) ad Chrsofferse (998). Kupec s es (heceforh K-es) s derved from a relave amou of excepos,.e. wheher her quay s from he sascal po of vew dffere from he assumpo. The ull hypohess s ha he erver probably of excepo occurrg s equal o he assumed. A gve lkelhood rao o he bass of probably dsrbuo wh oe degree of freedom s formulaed as follows: LR ex l ex, (8) where ex s expeced probably of excepo occurrg, s erved probably of excepo occurrg, s he umber of zeros ad s he umber of oes (excepos). By coras, order o assess wheher he excepos are dsrbued equally me,.e. whou ay depedece (auocorrelao), we should defe he me lag frs. Therefore, we replace he orgal sequece by a ew oe, where,, or are recorded. The ull hypohess s ha he probably of excepo occurrg s depede o he formao wheher he excepo has occurred also prevous day. The we have he lkelhood rao as follows (C-es): LR where PrI I ad l probably dsrbuo wh oe degree of freedom., (9). Ths es sasc has 3 Resuls Parcular models descrbed he prevous seco wll be appled o daa seres of logreurs calculaed from hsorcal seres of Prague sock marke dex (dex PX-5 ad PX, heceforh oly dex) for perod from Sepember 5, 993 ll Ocober 5,. The pu The dex PX s calculaed from March, 6. I ha dae ook over he values of he dex PX-5, whch s calculaed from Aprl 5, 994. The prevous values of dex PX 5 has bee calculaed ex-pos. 74

6 The 7 h Ieraoal Days of Sascs ad Ecoomcs, Prague, Sepember 9-, 3 daa was dowloaded from he webserver The sze of he seres s 4,67 daly reurs, from whch he frs 5 ervaos are lef for al esmao of parameers ad 4,377 ervaos are ulzed for backesg procedure. The evoluo of he log-reurs s depced Fg.. Fg. The evoluo of reurs of dex from ll 5... Reur 8% 3% 8% 3% -% -7% -% -7% 9/993 6/996 /999 / 8/4 5/7 / Dae Source: daa from ow elaborao I s appare ha here are clusers wh hgher volaly of reurs. The hghes volaly s he secod half of he year 8 las facal crss. Also oher perods possess creased volaly of reurs summer 6, secod half of he year 998, he ed of 993, ec. Basc descrpve characerscs of reurs are depced Tab.. Tab. : Sasc descrpo of erved reurs Characerscs Value Mea.3% Meda.8% Sadard devao.59% Skewess.3 Kuross Reurs auocorrelao.9 Auocorrelao of he square of he reurs.495 Source: ow calculao 75

7 The 7 h Ieraoal Days of Sascs ad Ecoomcs, Prague, Sepember 9-, 3 The followg resuls from he characerscs are appare: () alhough he meda s close o he mea, due o ozero skewess we ca coclude ha he probably dsrbuo of he reurs s skewed, () also hgh kuross suggess he presece of heavy als so ha he ormal dsrbuo cao be ulzed for modellg purposes, () he auocorrelao of he mea s small, bu prese ad (v) he auocorrelao of he square of he reurs s relavely hgh hs sugges ha he use of codoal volaly model s ecessary. For he parameers esmao of whole perod process oly he frs auoregressve coeffce ad oly frs order coeffces for volaly equao were foud sascally sgfca for boh assumed models. Thus we furher assume oly -- models. I he Tab. we provde he summary of parameers esmaed from he whole daase as well as he values of log-lkelhood fuco (LLF), Akake formao crera (AIC) ad Bayesa formao crera (BIC). As ca be see from he able, LLF (ad hus also AIC ad BIC) s slghly hgher for he GJR model, bu esmaed parameers are smlar for boh models (parameer s relavely small). Tab. : Esmaed parameers of parcular models from he whole perod Model AR()-GARCH(,)- AR()-GJR(,)- Esmaed parameers e e LLF,45.4,4.9 AIC -,87 -,83 BIC -,797 -,7978 Source: ow calculao Akake formao crera s compued as k LLF ad Bayesa formao crera (BIC) s compued as log k LLF esmaed ad LLF s he value of log-lkelhood fuco., where s he umber of ervaos, k s he quay of parameers o be 76

8 The 7 h Ieraoal Days of Sascs ad Ecoomcs, Prague, Sepember 9-, 3 The models (each wh R, P, Q ) were backesed wh dffere perods ulzed for parameers esmao ad he rao of erved quay of excepos o he expeced quay were recorded. The resuls are depced o Fg.. The raos should be deally equal o oe or a leas close o oe (as we wa he quay of erved excepos o be approxmaely equal o he expeced quay). As ca be see, he red of he rao s he same for boh models (ad also for all he probably levels). The rao of erved quay of excepos o he expeced quay s decreasg wh he creasg sze of perod ulzed for parameer esmao. The explaao of hs s probably followg. Wh he crease of he perod legh he esmaed parameer s creasg o average (as wh he loger perod he exreme values are more lkely o be erved als are heaver), whch creases he esmaed Value a Rsk. From he fgure we ca clearly dsgush he dfferece bewee wo groups of raos. Raos for probably levels 5% ad 5% are aroud. For he shor esmao perods (lower ha 7 days) hey are above he rsk s uderesmaed; for loger esmao perods (above 7 days) hey are below he rsk s overesmaed. Raos for probably levels % ad.5% are above for all he assumed esmao perods. Thus, he rsk s always uderesmaed. For he bes rsk esmao, he perod of legh 4 days should be ulzed for parameers esmao. Fg. The raos of erved ad expeced quaes of excepos for dffere perods ulzed for parameers esmao (from days ll 5 days wh he sep of days) ad dffere models (GARCH- lef, GJR- rgh) Source: ow calculao 5% 5% %,5% 77

9 The 7 h Ieraoal Days of Sascs ad Ecoomcs, Prague, Sepember 9-, 3 I he Tab. 3 ad Tab. 4 he p-values of he defed sascal ess are summarzed for esmao perod sze of 7 ad 4 days. Accordg o he p-values we ca reec he buchg of excepos (all he p-values of C-es are hgher ha 5%). However, he accuracy of he model s problemac. For probably levels 5% ad 5% he models are accurae oly whe parameers are esmaed from las 7 days. For he probably levels % ad.5% (he levels mpora for facal suos ad her regulaory auhores) he GJR model ca be reeced as accurae (all he p-values are lower ha 5%) ad he GARCH model s accurae for 4 days perod ulzed for parameers esmao. Tab. 3: P-values of ess for ulzed perod of parameers esmao of 7 days model GJR- GARCH- probably level 5% 5% %,5% 5% 5% %,5% P-value of K es 95,% 7,%,%,% 6,9% 55,%,%,% P-value of C es 94,7% 44,7% 9,% 8,4% 8,6% 6,8% 8,9% 6,8% Source: ow calculao Tab. 4: P-values of ess for ulzed perod of parameers esmao of 4 days model GJR- GARCH- probably level 5% 5% %,5% 5% 5% %,5% P-value of K es,3% 6,% 4,7%,4%,%,%,4%,% P-value of C es 66,% 6,3% 55,% 9,7% 49,% 7,5% 35,9% 6,8% Source: ow calculao Cocluso The developme of he accurae model for he rsk esmao s esseal for all he facal suos. Challegg par of each mehod s volaly modellg ad predco. I hs paper we have show ha he backesg resuls are also flueced by he sze of he perod ulzed for parameer esmao. For he vesme o Prague sock marke dex we have show ha for he backesg resuls of descrbed models he buchg ca be reeced,.e. models reac o he volaly crease saly. Bu hese models are o accurae ad more sophscaed models should be assumed. We also foud ou ha wh he creasg sze of he perod for parameers esmao he quay of excepos s decreasg. 78

10 The 7 h Ieraoal Days of Sascs ad Ecoomcs, Prague, Sepember 9-, 3 Ackowledgme The research was suppored by he Europea Socal Fud uder he Opporuy for youg researchers proec (CZ..7/.3./3.6) as well as by GAČR (Czech Scece Foudao Graová Ageura České Republky) uder he proec o. GP3-83P. The access o compug ad sorage facles owed by pares ad proecs corbug o he Naoal Grd Ifrasrucure MeaCerum, provded uder he programme "Proecs of Large Ifrasrucure for Research, Developme, ad Iovaos" (LM5) s hghly apprecaed. Refereces Alexader, C., & Sheedy, E. (8). Developg a sress esg framework based o marke rsk models. Joural of Bakg ad Face, 3(), -36. do:.6/.bakf.7..4 Black, F. (976). The prcg of Commody Coracs. Joural of Facal Ecoomcs, 3(- ), do:.6/34-45x(76)94-6 Bollerslev, T. (986). Geeralzed auoregressve codoal heeroskedascy. Joural of Ecoomercs, 3(3), do:.6/34-476(86)963- Egle, R. F. (98). Auo-regressve codoal heeroscedascy wh esmaes of he varace of Ued Kgdom flao. Ecoomerca, 5(4), do:.37/9773 Glose, L. R., Jagaaha, R., & Rukle, D. E. (993). O he relao bewee he expeced value ad he volaly o he omal excess reurs o socks. Joural of Face, 48(5), do:./ b58.x Huag, J. J., Lee, K. J., Lag, H., & L, W.F. (9). Esmag value a rsk of porfolo by codoal copula-garch mehod. Isurace: Mahemacs ad Ecoomcs, 45(3), do:.6/.smaheco Hull, J. (6). Rsk Maageme ad Facal Isuos. Upper Saddle Rver: Prece Hall. Chrsofferse, P. F. (998). Evaluag erval forecass. Ieraoal Ecoomc Revew, 39(4), do:.37/5734 Igaeva, K., & Plae, E. (). Modellg Co-movemes ad Tal Depedecy he Ieraoal Sock Marke va Copulae. Asa-Pacfc Facal Markes, 7(3) do:.7/s

11 The 7 h Ieraoal Days of Sascs ad Ecoomcs, Prague, Sepember 9-, 3 Joro, P. (6). Value a Rsk: The New Bechmark for Maagg Facal Rsk. (3rd ed.). New York: McGraw-Hll. Kresa, A., & Tchý, T. (). Ieraoal Equy Porfolo Rsk Modelg: The case of NIG model ad ordary copula fucos. Face a úvěr Czech Joural of Ecoomcs ad Face, 6(), 4-5. Rereved from hp://oural.fsv.cu.cz/mag/arcle/show/d/44 Kupec, P. (995). Techques for verfyg he accuracy of rsk measureme models. Joural of Dervaves, 3(), do:.395/od Coac Aleš Kresa VŠB Techcal Uversy of Osrava, Faculy of Ecoomcs Sokolská řída 33, Osrava, Czech Republc ales.kresa@vsb.cz 7