Presenting a Model for Multiple-Step-Ahead-Forecasting of Volatility and Conditional Value at Risk in Fossil Energy Markets

Transcription

1 AU Journal of Modeling and Simulaion AU J. Model. Simul., 50()(08)83-94 DOI: 0.060/miscj Presening a Model for Muliple-Sep-Ahead-Forecasing of Volailiy and Condiional Value a Risk in Fossil Energy Markes E. Mohammadian Amiri, S. B. Ebrahimi * Faculy of Indusrial Engineering, K. N. oosi Universiy of echnology, ehran, Iran ABSRAC: Fossil energy markes have always been known as sraegic and imporan markes. hey have a significan impac on he macro economy and financial markes of he world. he naure of hese markes is accompanied by sudden shocks and volailiy in he prices. herefore, hey mus be conrolled and forecased using appropriae ools. his paper adops he Generalized Auo Regressive Condiional Heeroskedasiciy (GARCH)-ype models, Exponenial Smoohing (ES)-ype models, and classic model in order o muliple-sep-ahead forecas volailiy, Value a Risk, and Condiional Value a Risk of Bren oil and naural gas in wo differen esimaion window lenghs, respecively. o evaluae he accuracy of he aforemenioned models, eigh differen loss funcions are uilized. here are a lo of financial erms in his he noed par. So, i s comprehensible for financial person and ec. herefore, he HWES model is proposed o muliple-sep-ahead forecas funcions as a verb. Review Hisory: Received: 5 Sepember 07 Revised: 8 January 08 Acceped: 5 March 08 Available Online: 9 June 08 Keywords: Muliple-sep-ahead forecasing Volailiy Value a Risk Condiional Value a Risk ES models - Inroducion Fossil energy marke is one of he mos imporan energy sources, affecing he economy of many counries. Since he oil and gas prices are cardinal inpus in macro-economic models, volailiy of hese prices is always imporan for oil exporing and imporing counries. herefore, Bren oil and naural gas have a grea impac on variables such as economic growh and inflaion [,], energy markes [3,4], and financial markes [5-7]. he moneary variables and inernaional financial variables have been idenified as he mos effecive facors o oil prices. Also, he direc relaionship beween ime and uncerainies and sudden shocks in energy markes have been previously proven [8-0]. One of hese price shocks in he energy markes is he growh of oil price o 48 dollars per barrel in July 008 and hen he drop in oil price o 40 dollars per barrel in lae December. his kind of volailiy has caused he volailiy and price predicions of oil and gas o be of grea imporance for sudying. However, in previous sudies, i has been emphasized ha i is difficul o forecas he volailiy and he price of oil []. here are wo saes concerning forecasing ime series: a) according o available daa, imeseries models are aribued o hem; b) independen from he ime series ha is called model-free. One of he problems in he firs sae is he changes in he daa and he exreme volailiy ha increases he model s error and, as a resul, he forecass would be far from realiy. In fac, no paricular model can ever be aribued o all daa []. According o exising lieraure, here are differen models o esimae and forecas volailiy in energy markes. As for radiional economeric models, Auo-Regressive Inegraed Moving Average (ARIMA), Generalized Auoregressive Condiional Heeroskedasiciy (GARCH), Coinegraed Vecor Auo Corresponding auhor; B_ebrahimi@knu.ac.ir Regreurssive (VAR) and Arificial Neural Neworks (ANN) have popularly been used for he forecasing of Bren oil and naural gas volailiy. For example, Xiang and Zhuang [3] used he ARIMA model o forecas he monhly prices of Bren oil and crude oil in a sample period from November 0 o April 03. Farzanegan and Mrakward [4] uilized he Vecor Auo Regressive (VAR) model o examine he dynamic relaionship beween he volailiy of oil prices and macroeconomic variables in Iran such as inflaion, indusrial producion growh raes, and ne governmen expendiure. Kang e al. [5] modeled he volailiy of Bren, Dubai and Wes exas Inermediae (WI) based on he GARCH, IGARCH, CGARCH, and FIGARCH models. Sozen and Arcaklioglu [6] presened a new model of ANN o forecas consumpion of oil producs in urkey. hey designed hree differen models in which differen variables are used, and, in he end, by using error measure, hey chose an appropriae model o forecas he consumpion of oil producs in urkey. differen definiions and differen ools were suggesed o forecas he risks relaed o price shocks in energy markes. In recen years, Value a Risk has been a popular measure. in a way ha nowadays he risk merics are known as he equivalen of Value a Risk [7]. A seminal paper in his regard is ha of Cabedo and Moya [8] ha esimaed Value a Risk of daily oil price over using he Hisorical Simulaion Approach. Fan e al. [9] esimaed Value-a- Risk via GARCH-ype models based on he Generalized Error Disribuion (GED) for boh he exreme downside and upside of he daily spo WI and Bren crude oil prices from May 0, 987 o Augus, 006, Simulaneously. Su [0] esimaed he Value a Risk of he seven sock indices in developed and emerging markes by using EGARCH models wih generalized suden s disribuion and Hisorical Simulaion Approach. Owing o he fac ha Value a Risk is 83

2 no a coheren risk measure and is deprived of sub-addiiviy propery, i can be bounded by coheren risk measures like Condiional Value a Risk. For example, Youssef e al. [] muliple-sep-ahead forecased Value a Risk and Condiional Value a Risk of crude oil and gasoline marke via hree long-memory-garch-models, including FIGARCH, HYGARCH, FIAPARCH, and Exreme Value heory (EV). Kim and Lee [] esimaed Value a Risk and Condiional Value a Risk of sock reurns of Hyundai Moors, Randgold Resources Limied (Gold), and NASDAQ by using nonlinear regression models of 000 observaions from Ocober, 005 o July, 03. Mabrouk [3] esimaed and muliplesep-ahead forecased he volailiy and Condiional Value a Risk of seven sock indices (Dow Jones, Nasdaq 00, S&P 500, DAX30, CAC40, FSE00, and Nikkei 5) and hree exchange raes vis-a-vis he US dollar (GBP- USD, YEN- USD and Euro-USD) via hree long memory GARCH-ype models (FIGARCH, HYGARCH, and FIAPARCH). aylor [4] esimaed and muliple-sep-ahead forecased Value a Risk and Condiional Value a Risk of FSE 00, NIKKEI 5 and S&P 500 by using a semi-parameric approach based on he Asymmeric Laplace disribuion a 95% and 99% confidence levels. Degiannakis and Poamia [5] esimaed muliple-sep-ahead forecased Condiional Value a Risk of sock indices, commodiies, and exchange raes by using GARCH-ype models. In recen sudies, Exponenial Smoohing (ES) models have been used o forecas demand [6, 7], hea [8], pig prices [9], and air ransporaion [30]. Bu he muliple-sep-ahead forecasing of Value a Risk and Condiional Value a Risk has no been applied o any sudy by ES-ype models. he Hol- Winers Exponenial Smoohing model (a kind of ES models) modified he daa a he level and rend wih wo parameers ( λ,λ ). his propery has caused he aforemenioned model o be robus and compuaionally sable. hus, in his paper, volailiy, Value a Risk, and Condiional Value a Risk of fossil energy markes are forecased via Hol-Winers Exponenial Smoohing model (HWES) and oher ES-ype models, and resuls were compared o GARCH-ype models and Classic model. - Mehodology - - Framework In his sudy, we seek o obain he bes model for muliple-sepahead forecasing of volailiy, Value a Risk, and Condiional Value a Risk of fossil energy markes from February 00 o December 06. o his end, we divide he hisorical Bren oil and naural gas daa ino a raining daase (from February 00 o Augus 06) and a esing daase (from Augus 06 o December 06). In he raining daase, Value a Risk and Condiional Value a Risk are esimaed by using he GARCHype models consising of GARCH, Exponenial GARCH (EGARCH), and hreshold GARCH (GARCH) wih wo Esimaion window lenghs of 600 and 000 samples. o idenify a benchmark model (he model ha has he lowes esimaion of Value a Risk s and Condiional Value a Risk s errors), he uncondiional coverage es, condiional coverage es, and Lopez loss funcion es are uilized. hen, in he esing daase, he Value a Risk and Condiional Value a Risk are forecased based on he benchmark model one, five, and weny seps ahead via he ES-ype models consising of Simple ES (ESE), Hol-Winers ES (HWES), and Double Hol-Winers ES (DHWES). o assess he performance, he proposed models are compared wih he classic model (he mos common model for muliple-sep-ahead forecasing of Value a Risk and Condiional Value a Risk in previous sudies) via Blanco and Ihle loss funcion es and Lopez loss funcion es. In addiion, he volailiy is forecased by using GARCH-ype models and ES-ype models one and five seps ahead. hen, he aforemenioned models are ranked by he Roo-Mean-Square Error (RMSE), RMSE-LOG, Mean Absolue Error (MAE), and MAE-LOG. - - Value a Risk Value a Risk is a saisical measure of risk, and i esimaes how much a se of invesmens migh lose, given normal marke condiions, in a ime period such as a day [3]. Value a Risk can sugges ha a cerain amoun of money be kep. herefore, even if he maximum loss occurs, he invesors will be able o fulfil heir obligaions. ha is why he Value a Risk is referred o as a Capial Adequacy Raio (CAR) for financial insiuions and capial markes. Value a Risk can be described as a measure o a percenile of profi disribuion or loss disribuion for any given ime horizon and confidence level of α. Value a Risk follows he following equaion [3]: VaR where: α α ( α x )( X ) = q ( ), () [ X : P( X > α ] q ( x) = inf x). () Value a Risk can also be formulaed as follows [33]: Pr( V + V VaR + ) α or c + c Pr( V V VaR + ) α, where V and V + are he values of he porfolio a he presen ime and V+ is he value of he porfolio a he fuure ime, respecively. However, Value a Risk is no a coheren risk measure because i is deprived of sub-addiiviy propery. Subaddiiviy propery suggess ha if a porfolio is composed of several sub-porfolios, hen he risk of he porfolio will no be greaer han he sum of he risks of he sub-porfolios. Subaddiiviy propery is shown as follows [34]: X, Y, X + Y V, X Y P( X + Y ) P( X ) + P( Y ) herefore, Condiional Value a Risk is used insead of Value a Risk in recen sudies Condiional Value a Risk If X is a coninuous random variable, hen condiional Value a Risk is defined as follows: xdf ( x ) x p CV ar p( X ) = E ( X X > x p) = F( x ) And, if he funcion is a discree disribuion, hen Condiional Value a Risk is calculaed as follows: n n n n CVaR = α X i, ni X X w n I i, (6) w n i= n i = w Also, if he funcion is a coninuous disribuion, hen Condiional Value a Risk is formulaed as follows [35]: P 84

3 CVaR α = ( E( X X qα ( X ) qα (Pr[ X qα ( X )] α ) (7) α - 4- Esimaing and forecasing mehodology Auoregressive Condiional Heeroscedasiciy (ARCH) Auoregressive Condiional Heeroscedasiciy was inroduced by Engel [36] as one of he nonlinear models for financial ime series. ARCH models assume ha he volailiy is ime-dependen. his propery helps models o mainain he dynamics. ARCH model is shown as follows: σ α 0 α = +, (8) whereσ is he variance of he forecas a ime, is he error (reurn residuals) a ime, and α 0,α denoe consan coefficien and ARCH coefficien, respecively. Also, he ARCH model (q) can be formulaed as follows: q σ α α = +. (9) 0 i i i = Generalized Auo Regressive Condiional Heeroskedasiciy (GARCH) he GARCH was presened as a generalized ARCH model by Bollerslev [37]. he mos common version of he model is GARCH (, ). his model can be wrien as follows: σ = ω+ α + β σ (0) whereσ is he variance of forecass a ime, σ is he variance of forecass a ime, and ω, α, β are consan coefficien, ARCH coefficien, and GARCH coefficien, respecively. Also, he GARCH (p, q) model can be formulaed as follows: q p σ = ω + α + σ i i j j i= j= β () GARCH model is used o esimae he parameers of he maximum likelihood model: z σ z r E ( r ) σ σ =, = = ω+ α + β () he above fracion has he sandard normal disribuion and he denominaor of he fracion is calculaed by using he maximum likelihood model: max L = max ϕ (, ) max exp z µ σ = = = π Each parameer ha maximizes L can maximize ln L as well. herefore, we maximize he logarihmic likelihood funcion as follows, max ln L = max ln ϕ (, ) max ln exp z µ σ = = = π z = max ln, = π max ln L = max ln ϕ (, ) max ln exp z µ σ = = = π z = max ln. = π z z z hreshold Generalized Auo Regressive Condiional Heeroskedasiciy (GARCH) GARCH model was inroduced by Zakoian [38]. he GARCH model can be defined as follows: σ = ω + α + γ I + β σ where I follows he following equaion: < (6), if 0 I = (7) 0, if 0 In his model, he good news, i.e. 0, and bad news, i.e. < 0, have differen effecs on he condiional variance. he good news has he α effec and he bad news has he α + γ effec. If γ > 0 hen, we can conclude ha here is a leverage effec. On he oher hand, if γ 0 hen, he effec of news is asymmeric. Also, he GARCH (p, q) model is formulaed as follows [39]: q p σ = + α + I + σ. (8) ω γ β i i j j i= j= Exponenial Generalized Auo Regressive Condiional Heeroskedasiciy (EGARCH) EGARCH model was inroduced by Nelson [40]. he aforemenioned model follows he following equaion: σ σ σ ω α γ β log( ) = log( ) (9) his equaion suggess ha he leverage effec is exponenial. Also, he non-negaive predicions of he condiional variance in his equaion are guaraneed. he original version of EGARCH model can be wrien as follows: σ log( σ ) = ω+ α + γ + βlog( ) π σ σ σ (0) Simple Exponenial Smoohing (SES) model he Simple Exponenial Smoohing model is based on a recursive formula. he forecas for each new observaion is updaed and he newer informaion gains more weigh han older informaion [4]. he forecased value of each year in his model is equal o he sum of he oal forecased amoun of he previous year; In addiion o difference beween he acual amoun of he same year, and he forecased amoun of he previous year [4]. Y Y ( λ) Y + λ ( λ) Y + = λ Y + λ + ( λ) [ λ Y + λ ( λ) Y + ] = λ Y + () + Y + = λ Y + ( λ) Y Y + = λ Y + Y λ Y Y + = Y + λ ( Y Y where Y + is he forecased amoun a ime +, Y is he acual amoun a ime and λ is he smoohing coefficien Hol-Winers Exponenial Smoohing model (HWES) Whenever here is an increasing or decreasing rend, he resuls obained from he Simple Exponenial Smoohing model are lower or higher han he acual value, respecively. o solve his problem, a rend parameer is added o he Simple Exponenial Smoohing model, which is referred o ) 85

4 as Hol-Winers Exponenial Smoohing model [43]. Y + = λ Y + ( λ )( Y + F ), :Level equaion () F ( Y Y ) + ( ) F + = + λ λ, :rend equaion Y + = Y + hf, : Forecasing equaion h where F + is smoohing index a ime + ; Y + is he forecased value based on Simple Exponenial Smoohing a ime + ; Y is he acual amoun a ime. he parameers λ and λ are he smoohing coefficiens in he level and rend, respecively, and h is he number of seps in forecasing Double Hol-Winers Exponenial Smoohing model (DHWES) DHWES model can be considered as a special case of Hol- Winers Exponenial Smoohing model where λ is equal o λ [44]. + = Y + ( λ Y λ )( Y + F ), :Level equaion (5) F ( Y Y ) + ( ) F + = + λ λ, : rend equaion (6) Y + = Y + hf. : Forecasing equaion (7) h Family of Exponenial Smoohing models requires o deermine he smoohing coefficien. If he smoohing coefficien is close o zero, hen, i obains more weigh o he recen evens. By increasing he weigh of recen evens, he number of days decreases in he volailiy forecasing. On he oher hand, if he smoohing coefficien is near o one, hen i is less sensiive o he recen evens, making forecasing more sable (no necessarily more accurae). herefore, he smoohing coefficien is beween 0 and and he opimal value is obained from he following equaion [45]: λ op = ˆ (8) n arg min (Y Y ). = Classic model he classic model is he mos common model for muliplesep-ahead forecasing of Value a Risk and Condiional Value a Risk in previous sudies. he following equaions are used o muliple-sep-ahead forecasing of Value a Risk and Condiional Value a Risk via classic model [ 46]: VaR CVaR = VaR (9) day day = CVaR (30) day day For example, in mos banks he covered ime period is one day; on he oher hand, he Basle Commiee requires en days. his means ha Value a Risk mus be accumulaed. If risks are no correlaed over ime, hen, aggregaion is simple, summarized by heir sum. In his case, moving from a oneday o a en-day Value a Risk is calculaed as follows: VaR 0 day VaR day 0 =, where VaR denoes Value a Risk Esimaion and forecas evaluaion Esimaion and forecas evaluaion of volailiy o evaluae he forecasing performance of volailiy models, four loss funcions were used [47]: RMSE = N RMSE _ LOG = MAE = N + N ( = + + N = + MAE _ LOG = N δ δ ), N + N ( = + Log( δ ) Log( δ ), (33) δ δ, (34) + N = + Log( δ ) Log( δ ) (35) where δ denoes he volailiy forecas obained using a GARCH-ype model or ES-ype models a ime ; δ is he acual volailiy, and N is he number of ime horizon Evaluaion of VaR and CvaR esimaion a) Uncondiional coverage es his es was presened by Kupiec [48] which is based on he rae of failure. If he amoun of he acual loss is larger han he VaR, hen i is known as a failure. If he probabiliy of each failure is consan, hen, he oal number of failures follows a binomial disribuion B( ν, a) in which ν and α are he number of samples and coverage level, respecively. he saisical hypohesis esing is as follows: H 0 : α = α (36) H : α α where α is he raio of a number of failures o he oal forecasing. he saisical likelihood of his es is as follows: LRucc ν0 ν ν 0 α ( α), (37) = Ln ν0 ν ν 0 α ( α) where LR ucc has he chi-square disribuion wih one degree of freedom. If he raio of he failure probabiliy is higher han his, he null hypohesis is rejeced, and i canno be acceped ha he model forecased he VaR correcly; hence he model is invalid. Oherwise, he accuracy of he forecased VaR is confirmed. b) Condiional coverage es Chrisoffersen [49] presened condiional coverage es based on firs-order Markov chain. o implemen he condiional coverage es, a ransiion marix is formed as follows: π00 π = 0 π0 π, (38) where ij Pr I = j I = i and calculaed as follows: ν 0 ν π0 = π = π00 = π0 π0 = π (39) ν0 + ν00 ν0 + ν where ν denoes he number of imes ha he sae j π is equal o [ ] ij 86

5 happens afer i. Finally, he es saisic is calculaed by he following equaion: ν ν 00 0 ν0 ν ( π0 ) π0 ( π ) π LRcc = Ln. ν0 ν ν0 α ( α ) (40) he LRcc saisic has a chi-square disribuion wih one degree of freedom, and when he raio probabiliy of failure is higher han his, he null hypohesis is rejeced. Oherwise, i will obain a passing mark. c) Lopez loss funcion es he lopez loss funcion es assumes each loss higher han VaR as a failure and assigns one o ha number. Oherwise, his funcion adops a zero. he lopez loss funcion is defined as follows [50]: if L > VaR C =. 0 if L < VaR d) Blanco and Ihle loss funcion es his loss funcion is similar o Lopez loss funcion. If he loss is higher han he VaR, hen i is assumed as a failure and he funcion is as follows: ( L VaR ), i.e. VaR he Blanco and Ihle loss funcion es are defined as follows [5]: ( L VaR ) if L > VaR C = VaR if L < VaR 0 he final score of Lopez loss funcion es and Blanco and Ihle loss funcion es are calculaed by equaion (43), where C is equaion or equaion, P is confidence level, and n is he number of observaions, Fig.. Daily Bren oil and naural gas prices (from February 00 o December 06) n QPS = ( C P ). n i = (43) 3- Resuls and discussion 3- - Resuls In his paper, as menioned earlier, he daa used is he daily Bren oil and naural gas logarihmic reurns from February 00 o December 06. hey follow he equaion P R = Log( ), P where R is reurn a ime ; P and P are prices a ime and, respecively. he diagrams of prices and logarihmic reurns of Bren oil and naural gas are shown in Figs. and, respecively. According o Fig., i can be noiced ha he daily Bren oil and naural gas logarihmic reurns have an exreme volailiy which can be regarded as oulier daa. Ignoring he oulier daa reduces he accuracy of forecasing models. Also, a summary of saisics of he variables is presened in able. Fig.. Daily Bren oil and naural gas logarihmic reurns (from February 00 o December 06) able. Saisics of Bren oil and naural oil. Bren oil Naural gas Mean Maximum Minimum Sd. Dev Skewness Kurosis

6 able shows ha he daily logarihmic reurns of Bren oil and naural gas have an asymmeric disribuion wih posiive and negaive skewness coefficien, respecively. On he oher hand, kurosis coefficien of boh indices is higher han hree whereas he kurosis coefficiens of he normal disribuion is approximaely equal o hree. his implies ha boh indices have kurosis coefficiens ha are bigger han kurosis coefficiens of he normal disribuion. Finally, he high Jarque-Bera coefficiens in Figs. 3 and 4 show ha aforemenioned indices are far apar from a normal disribuion (he Jarque-Bera coefficien of he normal disribuion is equal o zero). hus, based on he skewness, kurosis, and Jarque-Bera coefficiens, i can be concluded ha he daa of Bren oil and naural gas follows suden s disribuion. For his reason, he esimaing and muliple-sepahead forecasing are assumed wih suden s disribuion. able. Resuls of uni roo ess. Variable ADF PP Level (Consan and rend) Bren oil (0.0) (0.0) Naural gas (0.0) (0.0) (Consan, no rend) Bren oil (0.0) (0.0) Naural gas (0.0) (0.0) Figures in brackes are probabiliy values. able presens he resuls of uni roo ess based on Augmened Dickey Fuller (ADF) es and Phillips Perron (PP) es. In ADF and PP ess, he null hypohesis implies he ime-series has a uni roo agains he alernaive of saionariy. Resuls of uni roo ess show ha he Bren oil and naural gas series are saionary. Fig. 3. Normaliy es resuls of Bren oil 3- - Esimaion and forecasing resuls of volailiy models able 3 presens he esimaion parameers of he volailiy models for Bren oil naural gas reurns. In accordance wih prior discussions, i is necessary o measure consan coefficien ( ω ), ARCH coefficien ( α ), GARCH coefficien ( β ), level coefficien ( λ ), and rend coefficien ( λ ) for he esimaing and he muliple-sep-ahead forecasing of volailiy. hey are as follows: Fig. 4. Normaliy es resuls of naural gas able 3. Esimaion parameers of models for Bren oil and naural gas reurns. Parameer GARCH(,) EGARCH(,) GARCH(,) SES DHWES HWES ω α β * ** * ** 0.460* ** * ** * ** * ** γ * ** λ λ * ** * 0.99** * ** * ** * ** * ** * ** * ** * ** Esimaes marked wih an aserisk ( ) and ( ) are hose of Bren oil and naural gas reurns, respecively. 88

7 Parameer GARCH(,) able 4. Evaluaion of one-sep-ahead volailiy forecas of Bren oil and naural gas reurn: GARCH as a benchmark Benchmark RMSE RMSE_LOG MAE MAE_LOG EGARCH(,) () () () () GARCH(,) () () () () SES DHWES () () () () HWES () () () () able 5. Evaluaion for he five-sep-ahead volailiy forecas of Bren oil and naural gas reurn: GARCH as a benchmark Parameer RMSE RMSE_LOG MAE MAE_LOG GARCH(,) Benchmark EGARCH(,) GARCH(,) SES () () () () () () () () DHWES HWES () () () () () () () () ables 4 and 5 show he evaluaion resuls of one- and five-sep-ahead volailiy forecas of Bren oil and naural gas reurn. In he one-sep-ahead forecasing, HWES and GARCH models have an accepable forecasing performance for volailiy esimaion. Also, in five-sepahead forecasing, HWES and SES models have an accurae forecasing. Overally, he HWES model has he leas predicion volailiy error compared o oher models across all forecasing horizons and subsamples used Esimaing and forecasing resuls of VaR and CVaR o forecas Value a Risk and Condiional Value a Risk, GARCH models were combined wih HWES model (he model ha has he leas forecasing volailiy error compared parameer in he equaion of HWES model is esimaed by he mos accurae esimaion beween GARCH models (benchmark model). o idenify he benchmark model (on which he Value a Risk and Condiional Value a Risk forecased via HWES model are based), he uncondiional coverage es, condiional coverage es, and Lopez loss funcion es are uilized. he resuls of he aforemenioned ess are as follows: o oher models). herefore, he Y 89

8 Parameer GARCH(,) able 6. Backesing resuls of Value a Risk (VaR) esimaion: window lengh of 600 samples (0.037) Uncondiional Coverage es accep (0.093) accep (0.497) Condiional Coverage es accep (0.4043) accep Lopez loss funcion es () EGARCH(,).038 (0.340) accep (0.4337) accep (0.5398) accep (0.085) accep 0.03 () () GARCH(,) (0.0683) accep (0.093) accep (0.4560) accep (0.4043) accep 0.06 () () Parameer GARCH(,) able 7. Backesing resuls of Value a Risk (VaR) esimaion: esimaion window lengh of 000 samples (0.064) Uncondiional Coverage es accep.5996 (0.0007) rejec (0.4688) Condiional Coverage es accep (0.349) accep Lopez loss funcion es () () EGARCH(,) (0.046) accep.5996 (0.0007) rejec (0.5398) accep (0.349) accep () () GARCH(,) (0.064) accep (0.0047) rejec (0.4688) accep (0.404) accep () () Parameer GARCH(,) EGARCH(,) GARCH(,) able 8. Backesing resuls of Condiional Value a Risk (CVaR) esimaion: window lengh of 600 samples (0.6675) (0.4048) 0.50 (0.663) Uncondiional Coverage es accep accep accep.65 (0.988).754 (0.736) 0.5 (0.4703) accep accep accep (0.5996) 0.0 (0.769) (0.694) Condiional Coverage es accep accep accep (0.5) (0.507) (0.5694) accep accep accep Lopez loss funcion es () () 0.04 () 0.04 () 0.03 () able 9. Backesing resuls of esimaion Condiional Value a Risk (CVaR) esimaion: window lengh of 000 samples. Parameer GARCH(,) EGARCH(,) GARCH(,) (0.5905) (0.344) (0.8706) Uncondiional Coverage es accep accep accep (0.369).4950 (0.4).5490 (0.33) accep accep accep 0.97 (0.6570) (0.785) (0.6978) Condiional Coverage es accep accep accep (0.67) (0.5405) (0.578) accep accep accep Lopez loss funcion es () () () 0.05 () 90

9 Based on he previous discussions presened in his sudy, he LRucc and LRcc saisics have a chi-square disribuion wih one degree of freedom (6.63). When he probabiliy raio of failure is higher han hose, he null hypohesis is rejeced; oherwise, i will obain a passing mark in he uncondiional coverage es and condiional coverage es, respecively. he esimaion models of Value a Risk and Condiional Value a Risk have been approved in all ables excep able 8 for esimaion Value a Risk of Bren oil naural gas wih an esimaion window lengh of 600 samples. According o Lopez loss funcion es, he EGARCH and GARCH models are he bes models o esimae he Value a Risk and Condiional Value a Risk in boh wo esimaion window lenghs of 600 and 000 samples for Bren oil and naural gas markes, respecively. herefore, he EGARCH and GARCH models are he benchmark models for esimaion of Value a Risk and Condiional Value a Risk. Fig. 5. Condiional Value a Risk of Bren oil reurns wih an esimaion window lengh of 600 samples Fig. 6. Condiional Value a Risk of naural gas reurns wih an esimaion window lengh of 600 samples Fig. 7. Condiional Value a Risk of Bren oil reurns wih an esimaion window lengh of 000 samples Fig. 8. Condiional Value a Risk of naural gas reurns wih an esimaion window lengh of 000 samples 9

10 able 0. Backesing resuls of one-, five- and weny-sep-ahead forecasing Value a Risk (VaR) Parameer Lopez loss funcion es One-sep-ahead Five-sep-ahead weny-sep-ahead Blanco & Ihle loss funcion es Lopez loss funcion es Blanco & Ihle loss funcion es Lopez loss funcion es Blanco & Ihle loss funcion es Classic model () 0.038() () () () () () () () () () () HWES model 0.03 () () () () () () () () 0.00 () () () () able. Backesing resuls of one-, five- and weny-sep-ahead forecasing Condiional Value a Risk (CVaR) Parameer Lopez loss funcion es One-sep-ahead Five-sep-ahead weny-sep-ahead Blanco & Ihle loss funcion es Lopez loss funcion es Blanco & Ihle loss funcion es Lopez loss funcion es Blanco & Ihle loss funcion es Classic model () () 0.03 () () () () () () () () () () HWES model () () () () () () () () 0.00() () () () According o all he figures, i can be saed ha he number of failures is almos less han or equal o he limi allowed for esimaion of Condiional Value a Risk wih esimaion window lengh of 600 and 000 samples. In addiion, here is no evidence for a cluser of failures occurrence. Afer deermining he benchmark models, he Value a Risk and he Condiional Value a Risk are forecased via HWES model. o evaluae he forecasing performance of he HWES model, his model was compared o he classic model (he mos common model for he muliple-sep-ahead forecasing of Value a Risk and Condiional Value a Risk in previous sudies) via Lopez loss funcion es and Blanco and Ihle loss funcion es. he resuls of he aforemenioned ess are as given in able Conclusions here are several models for forecasing volailiy, Value a Risk, and Condiional Value a Risk. his paper analyzed he forecasing performance of wo classes of volailiy models, namely GARCH-ype models and ES-ype models and wo classes of Value a Risk and Condiional Value a Risk models, namely ES-ype models and Classic model via seven differen loss funcions. he Hol-Winers Exponenial Smoohing model modified he daa a he level and rend wih wo parameers and i is a well-known adapive model used o model ime series characerized by rend [5]. he muliplesep-ahead forecasing of Value a Risk and he Condiional Value a Risk by ES-ype models have no been used in any sudy erswhile. According o ables 4 and 5 HWES model (a kind of ES models) has he leas forecasing volailiy error compared o oher models. his model is proposed o forecas he volailiy of fossil energy markes. So, when HWES model is adoped, i leads o invesigaing of he level and rend, simulaneously, and filers he sudden changes by smoohing coefficiens. Consequenly, i provides a robus and compuaionally sable forecasing. Also, according o Lopez loss funcion es scores and Blanco and Ihle loss funcion es scores in ables 0 and, i can be concluded ha he HWES model has a beer forecasing performance han he classic model in one-, five-, and weny-sep-ahead forecasing of Value a Risk and Condiional Value a Risk. Overally, he Hol-Winers Exponenial Smoohing model provides a robus forecasing for volailiy, Value a Risk, and Condiional Value a Risk, ha fis he coninuous Bren oil and naural gas price movemens and provides an efficien risk quanificaion across all forecasing horizons. 9

11 References [] B.-N. Huang, M.-J. Hwang, H.-P. Peng, he asymmery of he impac of oil price shocks on economic aciviies: an applicaion of he mulivariae hreshold model, Energy Economics, 7 (005) [] S. Lardic, V. Mignon, he impac of oil prices on GDP in European counries: An empirical invesigaion based on asymmeric coinegraion, Energy policy, 34(8) (006) [3] R. Bhar, S. Hamori, Causaliy in variance and he ype of raders in crude oil fuures, Energy Economics, 7 (005) [4] B.. Ewing, S.M. Hammoudeh, M.A. hompson, Examining asymmeric behavior in US peroleum fuures and spo prices, he Energy Journal, (006) 9-3. [5] R.S. Pindyck, he dynamics of commodiy spo and fuures markes: a primer, he energy journal, (00) -9. [6] B.. Ewing, M.A. hompson, Dynamic cyclical comovemens of oil prices wih indusrial producion, consumer prices, unemploymen, and sock prices, Energy Policy, 35() (007) [7] P. Sadorsky, he macroeconomic deerminans of echnology sock price volailiy, Review of Financial Economics, () (003) [8] J.J. Roemberg, M. Woodford, Imperfec compeiion and he effecs of energy price increases on economic aciviy, Naional Bureau of Economic Research, 996. [9] M.G. Finn, Perfec compeiion and he effecs of energy price increases on economic aciviy, Journal of Money, Credi and banking, (000) [0] I.-M. Kim, P. Loungani, he role of energy in real business cycle models, Journal of Moneary Economics, 9() (99) [] L. Yu, Y. Zhao, L. ang, A compressed sensing based AI learning paradigm for crude oil price forecasing, Energy Economics, 46 (04) [] M. Kovářík, L. Sarga, P. Klímek, Usage of conrol chars for ime series analysis in financial managemen, Journal of Business Economics and Managemen, 6() (05) [3] Y. Xiang, X.H. Zhuang, Applicaion of ARIMA model in shor-erm predicion of inernaional crude oil price, in: Advanced Maerials Research, rans ech Publ, 03, pp [4] M.R. Farzanegan, G. Markward, he effecs of oil price shocks on he Iranian economy, Energy Economics, 3() (009) [5] S.H. Kang, S.-M. Kang, S.-M. Yoon, Forecasing volailiy of crude oil markes, Energy Economics, 3() (009) 9-5. [6] A. Sözen, E. Arcaklioglu, Predicion of ne energy consumpion based on economic indicaors (GNP and GDP) in urkey, Energy policy, 35(0) (007) [7] A. Glyn, G. Holon, Value a risk, heory and pracice, Academic, Amserdam Google Scholar, (003). [8] J.D. Cabedo, I. Moya, Esimaing oil price Value a Risk using he hisorical simulaion approach, Energy Economics, 5 (003) [9] Y. Fan, Y.-J. Zhang, H.-. sai, Y.-M. Wei, Esimaing Value a Risk of crude oil price and is spillover effec using he GED-GARCH approach, Energy Economics, 30(6) (008) [0] Su, J. B., Value-a-risk esimaes of he sock indices in developed and emerging markes including he spillover effecs of currency marke, Economic Modelling, 46: 04-4, 05. [] M. Youssef, L. Belkacem, K. Mokni, Value-a-Risk esimaion of energy commodiies: A long-memory GARCH EV approach, Energy Economics, 5 (05) [] M. Kim, S. Lee, Nonlinear expecile regression wih applicaion o value-a-risk and expeced shorfall esimaion, Compuaional Saisics & Daa Analysis, 94 (06) -9. [3] S. Mabrouk, Forecasing daily condiional volailiy and h-sep-ahead shor and long Value-a-Risk accuracy: Evidence from financial daa, he Journal of Finance and Daa Science, () (06) [4] J.W. aylor, Forecasing value a risk and expeced shorfall using a semiparameric approach based on he asymmeric Laplace disribuion, Journal of Business & Economic Saisics, (07) -3. [5] S. Degiannakis, A. Poamia, Muliple-days-ahead value-a-risk and expeced shorfall forecasing for sock indices, commodiies and exchange raes: Iner-day versus inra-day daa, Inernaional Review of Financial Analysis, 49 (07) [6] G. Sudheer, A. Suseelaha, Shor erm load forecasing using wavele ransform combined wih Hol Winers and weighed neares neighbor models, Inernaional Journal of Elecrical Power & Energy Sysems, 64 (05) [7] L.F. raar, B. Mojškerc, A. oman, Demand forecasing wih four-parameer exponenial smoohing, Inernaional Journal of Producion Economics, 8 (06) [8] L.F. raar, E. Srmčnik, he comparison of Hol Winers mehod and Muliple regression mehod: A case sudy, Energy, 09 (06) [9] L. Wu, S. Liu, Y. Yang, Grey double exponenial smoohing model and is applicaion on pig price forecasing in China, Applied Sof Compuing, 39 (06) 7-3. [30].M. Danas, F.L.C. Oliveira, H.M.V. Repolho, Air ransporaion demand forecas hrough Bagging Hol Winers mehods, Journal of Air ranspor Managemen, 59 (07) 6-3. [3] P. Bes, Implemening value a risk, John Wiley & Sons, 000. [3] G.N. Gregoriou, he VaR implemenaion handbook, McGraw Hill Professional, 009. [33] J. Danielsson, Financial Risk Forecasing: he heory and Pracice of Forecasing Marke Risk, wih Implemenaion in R and Malab (Wiley Finance Series), John Wiley & Sons Incorporaed, 0. 93

12 [34] C. Acerbi, D. asche, Expeced shorfall: a naural coheren alernaive o value a risk, Economic noes, 3() (00) [35] H.H. Panjer, Operaional risk: modeling analyics, John Wiley & Sons, 006. [36] R.F. Engle, Auoregressive condiional heeroscedasiciy wih esimaes of he variance of Unied Kingdom inflaion, Economerica: Journal of he Economeric Sociey, (98) [37]. Bollerslev, Generalized auoregressive condiional heeroskedasiciy, Journal of economerics, 3 (986) [38] J.-M. Zakoian, hreshold heeroskedasic models, Journal of Economic Dynamics and conrol, 8 (994) [39] A.A. Chrisie, he sochasic behavior of common sock variances: Value, leverage and ineres rae effecs, Journal of financial Economics, 0 (98) [40] D.B. Nelson, Condiional heeroskedasiciy in asse reurns: A new approach, Economerica: Journal of he Economeric Sociey, (99) [4] C.C. Hol, Forecasing seasonals and rends by exponenially weighed moving averages, Inernaional journal of forecasing, 0() (959) 5-0. [4] S. Makridakis, S.C. Wheelwrigh, F. Hyndman, Mehods and applicaions, in, NY, 998. [43] P.R. Winers, Forecasing sales by exponenially weighed moving averages, Managemen science, 6 (960) [44] R. Hyndman, A.B. Koehler, J.K. Ord, R.D. Snyder, Forecasing wih exponenial smoohing: he sae space approach, Springer Science & Business Media, 008. [45] C. Croux, S. Gelper, K. Mahieu, Robus conrol chars for ime series daa, Exper Sysems wih Applicaions, 38() (0) [46] C.S. apiero, Risk and financial managemen: mahemaical and compuaional mehods, John Wiley & Sons, 004. [47] Z. Shen, M. Rier, Forecasing volailiy of wind power producion, Applied energy, 76 (06) [48] P. Kupiec, echniques for verifying he accuracy of risk measuremen models, (995). [49] P.F. Chrisoffersen, Evaluaing inerval forecass, Inernaional economic review, (998) [50] J.A. Lopez, Mehods for evaluaing value-a-risk esimaes, Economic review, (999) 3-7. [5] C. Blanco, G. Ihle, How good is your VaR? Using backesing o assess sysem performance, Financial Engineering News, (8) (999) -. [5] M. Szmi, A. Szmi, Usage of modified Hol-Winers mehod in he anomaly deecion of nework raffic: Case sudies, Journal of Compuer Neworks and Communicaions, 0 (0). Please cie his aricle using: E. Mohammadian Amiri, S. B. Ebrahimi, Presening a Model for Muliple-Sep-Ahead-Forecasing of Volailiy and Condiional Value a Risk in Fossil Energy Markes, AU J. Model. Simul. Eng., 50() (08) DOI: 0.060/miscj