Forecasting Value at Risk and Expected Shortfall Using a Semiparametric. Approach Based on the Asymmetric Laplace Distribution

Transcription

1 Forecasing Value a Risk and Expeced Shorfall Using a Semiparameric Approach Based on he Asymmeric Laplace Disribuion James W. Taylor Saïd Business School Universiy of Oxford Journal of Business and Economic Saisics, forhcoming. Address for Correspondence: James W. Taylor Saïd Business School Universiy of Oxford Park End Sree Oxford OX1 1HP, UK james.aylor@sbs.ox.ac.uk

2 Forecasing Value a Risk and Expeced Shorfall Using a Semiparameric Approach Based on he Asymmeric Laplace Disribuion Absrac Value a Risk (VaR) forecass can be produced from condiional auoregressive VaR models, esimaed using quanile regression. Quanile modeling avoids a disribuional assumpion, and allows he dynamics of he quaniles o differ for each probabiliy level. However, by focusing on a quanile, hese models provide no informaion regarding Expeced Shorfall (ES), which is he expecaion of he exceedances beyond he quanile. We inroduce a mehod for predicing ES corresponding o VaR forecass produced by quanile regression models. I is well known ha quanile regression is equivalen o maximum likelihood based on an asymmeric Laplace (AL) densiy. We allow he densiy s scale o be ime-varying, and show ha i can be used o esimae condiional ES. This enables a join model of condiional VaR and ES o be esimaed by maximizing an AL log-likelihood. Alhough his esimaion framework uses an AL densiy, i does no rely on an assumpion for he reurns disribuion. We also use he AL log-likelihood for forecas evaluaion, and show ha i is sricly consisen for he join evaluaion of VaR and ES. Empirical illusraion is provided using sock index daa. Keywords: Quanile regression; CAViaR; Eliciabiliy. 1

3 1. INTRODUCTION Value a Risk (VaR) is a ail quanile of he condiional disribuion of he reurn on a porfolio. I has become he sandard measure of marke risk, and hence has been used by banks over he pas wo decades for seing regulaory capial requiremens. Alhough i is an inuiive risk measure, VaR gives no informaion regarding possible exceedances beyond he quanile. A measure addressing his, and which can be viewed as a complemen o VaR, is Expeced Shorfall (ES), which is he condiional expecaion of exceedances beyond he VaR. ES possesses a number of aracive properies (Acerbi and Tasche 2002). For example, in conras o VaR, ES is a subaddiive risk measure (Arzner e al. 1999), which means ha he measure for a porfolio canno be greaer han he sum of he measure for he consiuen pars of he porfolio. Fuure regulaory frameworks are likely o pu increased emphasis on ES (Embrechs e al. 2014). Alhough many banks already calculae ES for heir own risk measuremen purposes, esimaion is inherenly challenging, as ES is a ail risk measure. Furhermore, here is no suiable loss funcion for evaluaing ES forecass (Gneiing 2011). In his paper, we provide a new approach o ES esimaion, and a new loss funcion for joinly evaluaing VaR and ES. Forecass of ES can be produced as a by-produc of many VaR forecasing mehods. The popular nonparameric mehods, namely hisorical simulaion and kernel densiy esimaion, produce densiy forecass from which VaR and ES predicions can be obained. This is also he case for parameric approaches, which involve a model for he condiional variance, such as a GARCH model, and a disribuional assumpion. Semiparameric approaches o VaR forecasing include hose ha use exreme value heory (EVT) (see, for example, Chavez-Demoulin, Embrechs and Sardy 2014), and hose ha direcly model he condiional quanile for a chosen probabiliy level using quanile regression, such as condiional auoregressive VaR (CAViaR) modeling (see Engle and Manganelli s 2004). Direcly modeling a quanile avoids he need for a 2

4 disribuional assumpion, and allows he dynamics of he quaniles o differ for each probabiliy level. In empirical sudies of VaR forecas accuracy, CAViaR models have performed well (see, for example, Sener, Baronyan and Mengüürk 2012). However, by focusing on a paricular quanile, quanile regression models provide no apparen way of producing ES forecass. In his paper, we address his using he asymmeric Laplace (AL) densiy. Our approach uses he equivalence beween quanile regression and maximum likelihood based on an AL densiy (see Koenker and Machado 1999). In his framework, he locaion and skewness parameers of he AL densiy are he quanile and probabiliy level, respecively. The maximum likelihood esimaor for he consan scale of he AL densiy is equal o he minimized quanile regression objecive funcion divided by he sample size. Basse, Koenker and Kordas (2004) highligh he simple relaionship beween his minimized objecive funcion and he uncondiional ES. This leads us o propose ha a ime-varying scale of he AL densiy can be used o produce an esimae of he ime-varying condiional ES. This enables a join model of condiional VaR and ES o be esimaed by maximizing an AL likelihood. The approach is semiparameric because, alhough a model is specified for he VaR and ES, we do no make a disribuional assumpion for he reurns. In decision heory, a scoring funcion is he erm for a loss funcion when used o evaluae a predicion of some measure of a probabiliy disribuion, such as he mean. The measure is referred o as being eliciable if he correc forecas of he measure is he unique minimizer of he expecaion of a leas one scoring funcion (Fissler and Ziegel 2016). The exisence of such a scoring funcion enables he comparison of forecass from differen mehods, wih he bes mehod deemed o be he one wih he lowes value of he scoring funcion. I is possible ha a measure may no be eliciable on is own, bu is eliciable in combinaion wih anoher measure; for example, alhough he variance is no eliciable, he mean and variance are 3

5 joinly eliciable (Gneiing 2011). Fissler and Ziegel (2016) show ha, alhough ES is no eliciable, VaR and ES are joinly eliciable, and hey provide a se of suiable scoring funcions. We show ha he negaive of he AL log-likelihood is a member of his se, and hence we propose he use of his funcion o evaluae VaR and ES forecass. Secion 2 briefly describes quanile regression and is link o uncondiional ES. Secion 3 explains how condiional VaR and ES esimaes can be produced using maximum likelihood based on an AL densiy. Secion 4 presens candidae join models of VaR and ES. Secion 5 proposes he use of he AL log-likelihood for joinly evaluaing VaR and ES forecass. Secion 6 uses daily sock indices o illusrae he use of he models and he new evaluaion measure. 2. QUANTILE REGRESSION AND ES Quanile regression has been used in a variey of applicaions for he esimaion of he parameers in a quanile model (see Koenker 2005). I involves he minimizaion of he sum of ick loss funcions, as shown in expression (1), where y is he dependen variable, Q is he quanile wih probabiliy level, I(x) is he indicaor funcion, and n is he sample size. n 1 y Q Iy Q min (1) As he common probabiliy levels are 1% and 5% for VaR and ES esimaion, in his paper, for simpliciy, we consider only <50%. Wih he VaR being he condiional quanile Q, he condiional ES is wrien as ES Ey y Q. Alhough quanile regression focuses on he quanile for a chosen probabiliy level, and seemingly involves no esimaion of he disribuion eiher side of he quanile, Basse, Koenker and Kordas (2004) provide an ineresing link beween quanile regression and ES, by showing ha ES can be wrien as: 4

6 ES E y Ey Q Iy Q 1 (2) Basse, Koenker and Kordas sugges ha his expression can be evaluaed empirically using he sample mean y of y, and he minimized quanile regression objecive funcion, as follows: ^ ES y 1 n n 1 y Qˆ Iy Qˆ (3) This would seem o show ha an esimae of ES is a by-produc of quanile regression (Komunjer 2007). However, only an uncondiional esimae of ES is produced, as expression (3) involves averaging over he n values of he ick loss funcion. Our ineres is in condiional ES esimaion, and, given he heeroscedasiciy in daily reurns daa, such an esimae is likely o be ime-varying. Taylor (2008) uses exponenially weighed quanile regression for VaR esimaion, and essenially replaces he summaion in expression (3) wih he resuling exponenially weighed summaion o deliver a condiional ES esimae. In his paper, we use he AL disribuion o provide a more flexible framework for he condiional modeling of VaR and ES. 3. USING THE AL DISTRIBUTION TO ESTIMATE CONDITIONAL VAR AND ES Koenker and Machado (1999) poin ou ha he quanile regression minimizaion of expression (1) is equivalen o maximum likelihood based on he AL densiy of expression (4). For his densiy, is a scale parameer, and Q is he ime-varying locaion, which is he quanile of he densiy corresponding o he chosen probabiliy level. f y 1 exp y Q Iy Q (4) The likelihood framework has led o useful developmens for quanile regression, such as saisical inference via quasi-maximum likelihood (see Komunjer 2005) and Bayesian quanile 5

7 regression (see, for example, Gerlach, Chen and Chan 2011). In hese conexs, he observaions y are no assumed o follow an AL disribuion. To emphasize his, Gerlach, Chen and Chan (2011) noe ha he parameer is no esimaed, bu is a chosen fixed value, and ha i is only a quanile ha is esimaed. The AL likelihood simply provides a compuaionally convenien basis wih which o enable heir Bayesian approach o quanile regression. For he scale of he AL densiy of expression (4), he maximum likelihood esimaor is: y Qˆ Iy Q n 1 ˆ ˆ (5) n 1 This is he average of he ick loss funcion, which can be inerpreed as an uncondiional esimaor of he expecaion of his loss funcion. The uncondiional esimaor of ES, presened in expression (3), can, herefore, be rewrien in erms of he scale esimaor of expression (5): ˆ ES ^ y Our proposal is o adap his expression for condiional esimaion. Wih his aim, we inroduce a condiional AL scale, which can be viewed as he poenially ime-varying condiional expecaion of he ick loss funcion. We convey his in he following expression: E y Q Iy Q Using his, we can rewrie expression (2) so ha we express he condiional ES in erms of he condiional AL scale and he condiional mean as follows: ES (6) A model for he condiional scale can be esimaed, along wih a model for he condiional quanile Q, using maximum likelihood based on he following AL densiy: 6

8 f y 1 exp y Q Iy Q Using expression (6), we can rewrie his densiy in erms of ES, as: f y 1 ES exp y Q I y Q ES (7) In his paper, we adop he common assumpion ha he condiional mean of a series of daily reurns r is a small consan value c, which can be esimaed as he mean of he in-sample reurns. We define y o be he residual y = r c. The focus of our modeling is, herefore, a variable y wih zero mean, and so we rewrie he AL densiy of expression (7) as: f y 1 y Q Iy Q ES exp ES (8) Our proposal is o use maximum likelihood based on his AL densiy o esimae a join model for he condiional quanile and condiional ES. We do no assume ha he reurns follow an AL disribuion, because, insead of opimizing, i is seleced o be 1% or 5%, which are he probabiliy levels of ineres. If one also wished o model a ime-varying condiional mean, he AL densiy of expression (7) could be used. To generae he parameer covariance marix, one possibiliy is o draw on he work of Komunjer (2005) who invesigaes quanile model esimaion using quasi-maximum likelihood based on a family of ick-exponenial densiies, of which he AL densiy is a special case. An alernaive is o use a boosrapping procedure, and his is he approach ha we use in our empirical work. To selec beween model specificaions, he Bayesian Informaion Crierion could be calculaed using he AL likelihood (see Lee, Noh and Park 2014). In his secion, we have highlighed he link beween he scale of an AL densiy and ES; we have proposed ha condiional modeling of he scale can deliver a condiional model for ES; 7

9 and we have suggesed ha his can be esimaed simulaneously wih a condiional quanile model using maximum likelihood based on an AL densiy. Alhough an AL densiy has previously been used wihin parameric approaches o ES esimaion (see Chen, Gerlach and Lu 2012; Komunjer 2007), we are no aware of any sudies ha have linked he condiional scale of an AL densiy o condiional ES esimaion. 4. JOINT MODELS FOR VAR AND ES Our proposal is o model VaR and ES joinly, wih parameers esimaed by maximum likelihood based on he AL densiy of expression (8). In his secion, we consider formulaions for he VaR componen of he model, and hen presen proposals for he ES componen. For he VaR componen, we simply propose a CAViaR model. Expressions (9)-(10) presen wo of he CAViaR models inroduced by Engle and Manganelli (2004). In hese models, he i are consan parameers. The asymmeric slope CAViaR model aims o capure he leverage effec, which is he endency for volailiy o be greaer following a negaive reurn han a posiive reurn of equal size. Symmeric Absolue Value: Q 0 1 y 1 2 Q 1 (9) Q (10) 0 I y 0 y I y 0 y Q Asymmeric Slope: For he ES componen, we require model formulaions ha avoid ES esimaes crossing he corresponding VaR esimaes. For < 50%, he ES esimae mus be a value below he quanile esimae. I is sraighforward o avoid crossing if we specify condiional ES o be a funcion of condiional VaR. This seems reasonable, as ES and VaR are, o some exen, likely o vary ogeher, as boh will vary wih he ime-varying volailiy. The simple formulaion for ES in expression (11) shows ES modeled as he produc of he quanile and a consan muliplicaive 8

10 facor (see Gourieroux and Liu 2012). To avoid crossing, we ensure his facor is greaer han 1 by expressing i in erms of an exponenial funcion of an unconsrained parameer 0. ES 1 exp 0 Q (11) The simpliciy of his formulaion is appealing. Furhermore, i correcly describes he relaionship beween ES and VaR for some daa generaing processes, such as a GARCH process wih a Suden disribuion. However, expression (11) is raher resricive, as he dynamics of VaR may no be he same as he dynamics of ES. An alernaive formulaion for ES is presened in expressions (12)-(13), where he difference x beween ES and he quanile is modeled using an auoregressive (AR) expression, which essenially smoohes he magniude of exceedances beyond he quanile. To ensure ha he quanile and ES esimaes do no cross, we consrain he parameers i o be non-negaive. ES Q x (12) x 0 1 Q 1 y 1 2 x x 1 1 if y 1 Q oherwise 1 (13) In our empirical sudy of Secion 6, we implemen his AR formulaion for ES, and he simpler ES formulaion of expression (11). However, a variey of oher models could cerainly be considered for condiional ES. For example, he expression for x could ake he same form as he CAViaR models, so ha lagged values of y or 2 y influence, in poenially differing ways, he dynamics of boh he quanile and he difference beween he quanile and ES. Anoher possibiliy is he use of a dynamic model wihin he muliplicaive facor of expression (11). 9

11 5. EVALUATION OF VAR AND ES FORECASTS 5.1. Exising Approaches for Evaluaing VaR and ES Forecass VaR forecas evaluaion ypically focuses on coverage ess. A quanile forecas Qˆ, for he probabiliy level, has correc uncondiional coverage if he variable Hi I y Qˆ ) ( has zero uncondiional expecaion, and correc condiional coverage if Hi has zero condiional expecaion (see Engle and Manganelli 2004). An alernaive way o evaluae quanile forecass is o use a scoring funcion. Given is use in quanile regression, a reasonable choice is he ick loss funcion (Giacomini and Komunjer 2005), and his has been ermed he quanile score. We presen his score in expression (14). A risk measure is eliciable if he correc forecas of he measure is he unique minimizer of he expecaion of a leas one scoring funcion. Such scoring funcions are called sricly consisen for he risk measure (Fissler and Ziegel 2016). VaR is an eliciable risk measure, for which he quanile score is sricly consisen. S Q y y Q Iy Q, (14) ES is no eliciable (Gneiing 2011). In he absence of a suiable scoring funcion for ES, he es of McNeil and Frey (2000) is ofen used. This focuses on he discrepancy beween he observed reurn and he ES forecas for he periods in which he reurn exceeds he VaR forecas. The sandardized discrepancies should have zero uncondiional and condiional expecaion. Due o he ypically small sample of discrepancies, a es of zero condiional expecaion is generally no performed, which implies ha he dynamic properies of he ES esimaes are no evaluaed. McNeil and Frey es for zero uncondiional mean using a boosrap es o avoid a disribuional assumpion. As his es focuses on observaions exceeding he VaR forecass, he assessmen of ES forecass is no independen of he VaR forecass. This, along wih ES no being eliciable, promps consideraion of a scoring funcion for joinly evaluaing ES and VaR forecass. 10

12 5.2. A New Scoring Funcion for Joinly Evaluaing VaR and ES Forecass Fissler and Ziegel (2016) explain ha VaR and ES are joinly eliciable, even hough ES is no eliciable individually. They show ha sricly consisen scoring funcions, for joinly evaluaing VaR and ES forecass, are of he following form: S Q, ES, y Iy Q G1 Q Iy Q G1 y G ES ES Q Iy Q Q y ES ay 2 where G 1, G 2, 2 and a are funcions saisfying a number of condiions, including he properies ha G 2 = 2 ; G 1 is increasing; and 2 is increasing and convex. (The domain of 2 conains only negaive values, because we are considering <50%, which implies ha ES is negaive.) These condiions clearly allow a variey of alernaive funcions o be chosen. We consider here hree examples from he se of scoring funcions of expression (15). Our firs example is he score used in he empirical analysis of Fissler, Ziegel and Gneiing (2016). They consider he scoring funcion produced by using G 1 (x)=x and G 2 (x)=exp(x)/(1+exp(x)) in expression (15). We se a=ln(2) o give posiive values for he scoring funcion. We refer o his as he FZG score, and presen i in expression (16). 2 (15) S Q, ES, y Iy Q Q Iy Q ES 1 exp ES ES Q I y Q Q y y 2 ln 1 exp ES (16) A second example from he se of scoring funcions of expression (15) is he funcion proposed by Acerbi and Székeley (2014). We presen his in expression (17), where W is a consan parameer ha is large enough o ensure WQ <ES for <50%. (Noe ha ES <0 and Q <0.) We refer o his as he AS score. Fissler and Ziegel (2016) explain ha, if WQ <ES, he AS score is a sricly consisen scoring funcion ha can be produced by seing G 1 (x)=-(w/2)x 2, G 2 (x)=x and a=0 in expression (15). In our empirical sudy of Secion 6, we implemened he 11

13 AS score wih W=4, as his was he smalles ineger ha ensured WQ <ES for all pairs of forecass of ES and Q from all mehods considered in our sudy. S Q, ES, y ES 2 W Q 2 Q ES Iy Q ES y Q W y Q 2 (17) As a hird example of a scoring funcion of he form of expression (15), le us consider G 1 =0, G 2 (x)=-1/x, x)=-ln(-x), and a=1-ln(1-). Expression (15) hen becomes: S Q 1 y Q Iy Q, ES, y ln ES (18) ES ES As we have defined y o have zero mean, he expecaion of he final summand of expression (18) is zero. Therefore, forecass of VaR and ES ha minimize he expecaion of expression (18) also minimize he expecaion of his scoring funcion if he final summand is removed, as in expression (19). This implies ha expression (19) is also a sricly consisen scoring funcion. This funcion is he negaive of he AL log-likelihood. We refer o i as he AL log score. Averaging he score across a sample gives a join measure of VaR and ES forecas accuracy. S Q 1 y Q Iy Q, ES, y ln ES (19) ES Noe ha if a scoring funcion is sricly consisen, i can also be used as he loss funcion in model esimaion (Gneiing and Rafery 2007). This secion, herefore, provides suppor for our proposal of esimaing join VaR and ES models by maximizing he AL log-likelihood. Our proposal of using he AL log score o compare he forecas accuracy of mehods could be viewed as advanageous for mehods esimaed using he AL log-likelihood. However, a similar criicism could be made for oher popular scoring funcions, such as he quanile score, as i is no he only sricly consisen scoring funcion for quanile forecass (Gneiing 2011). Using he AL log score o evaluae VaR and ES forecass has he heoreical appeal of being a member y 12

14 of he se of scoring funcions proposed by Fissler and Ziegel (2016), and i has he inuiive appeal ha he AL likelihood is well esablished in he lieraure on quanile esimaion. 6. EMPIRICAL STUDY OF VAR AND ES FORECASTS USING STOCK INDICES We evaluaed day-ahead VaR and ES forecass for daily log reurns of he FTSE 100, NIKKEI 225 and S&P 500 sock indices. Following common convenion, we considered he 1% and 5% probabiliy levels. Each series consised of he 3500 daily log reurns ending on 16 April We used a rolling window of 2500 observaions for repeaed re-esimaion of each mehod, and evaluaed day-ahead VaR and ES forecass for he final 1000 observaions. As we saed in Secion 3, our modeling focuses on a residual erm, defined as y = r c, where r is he daily reurn and c is a consan erm, which we esimaed using he mean of he in-sample reurns VaR and ES Forecasing Mehods Hisorical Simulaion and GARCH Mehods As a simple benchmark, we produced VaR and ES forecass using hisorical simulaion wih a moving window consising of he 2500 observaions in each esimaion sample. We also considered hisorical simulaion wih moving windows of 100 and 25 observaions, as in he work of Chen e al. (2012a). A shor moving window has he poenial advanage of enabling fas adapaion in VaR and ES esimaion during periods when he marke experiences major change. We esimaed GARCH(1,1) and GJRGARCH(1,1) models using maximum likelihood based on a Suden disribuion. We produced VaR and ES forecass using hree approaches: (i) A Suden disribuion wih degrees of freedom opimized wih he model parameers. (ii) Filered hisorical simulaion, which applied hisorical simulaion o all 2500 in-sample residuals sandardized by he esimaed volailiy. 13

15 (iii) The mehod of McNeil and Frey (2000), which applies he peaks-over-hreshold (POT) EVT mehod o he sandardized residuals. Sandard CAViaR wih Simple ES Mehods We fied he wo CAViaR models of expressions (9)-(10), wih he parameers esimaed using quanile regression, as in he work of Engle and Manganelli (2004). We firs sampled 10 4 candidae parameer vecors from uniform disribuions wih lower and upper bounds based on iniial experimenaion. Of hese vecors, he hree giving he lowes quanile regression objecive funcion were used, in urn, as he iniial vecor in a quasi-newon algorihm. The resuling vecor, wih lowes objecive funcion, was chosen as he final parameer vecor. When esimaing he parameers for he second moving window of 2500 observaions, and for all subsequen moving windows, we included, as an addiional candidae, he parameer vecor ha had been opimized for he previous window of observaions. Afer producing CAViaR model quanile forecass, we used he following wo approaches o forecas ES: (i) As suggesed by Manganelli and Engle (2004), we performed leas squares regression, wih dependen variable se as he vecor of observaions ha exceeded he quanile esimaes, and regressor se as he vecor of quanile esimaes. Forecass from his model were used as ES predicions. In our resuls ables, we refer o his as QR for VaR: ES = muliple of VaR, where QR emphasizes he use of quanile regression o esimae he CAViaR model. (ii) We produced ES forecass by summing he quanile forecas and he average in-sample quanile exceedance. In our ables, we refer o his as QR for VaR: ES = mean exceedance. 14

16 CAViaR wih EVT Manganelli and Engle (2004) adap he EVT-based mehod of McNeil and Frey (2000). A CAViaR model is firs esimaed for a ail quanile ha is no as exreme as he VaR of ineres. We followed Manganelli and Engle by esimaing he 7.5% quanile. POT EVT is hen applied o exceedances beyond his quanile, afer sandardizing he exceedances by he corresponding quanile esimaes. The fied EVT disribuion is hen used o obain he 1% and 5% quanile and ES esimaes of he reurns. The CAViaR model, herefore, only provides he EVT hreshold for he mehod. In our resuls ables, we refer o he mehod as QR for 7.5% wih EVT. Join Models for VaR and ES Esimaed using AL Densiy We implemened our proposed approach of Secions 3 and 4, which involves maximum likelihood based on he AL densiy of expression (8). We considered four join models for VaR and ES, which each involved one of he wo CAViaR formulaions of expressions (9)-(10), and one of he wo ES formulaions of expressions (11)-(13). In our resuls ables, we refer o he ES formulaion of expression (11) as AL: ES = muliple of VaR, and he ES formulaion of expressions (12)-(13) as AL: ES = AR model, where AL is used o emphasize ha he models have been esimaed wih maximum likelihood based on an AL densiy. The likelihood maximizaion followed a similar opimizaion procedure o he one ha we described for CAViaR models, wih wo noable differences. Firs, he quanile regression objecive funcion was replaced by he negaive of he AL log-likelihood. Second, for he candidae parameer vecors, we se he CAViaR parameers o be he values opimized separaely using quanile regression, while he ES model parameers were randomly sampled. We did his o assis he opimizaion when he AR model of expressions (12)-(13) was used for he ES, due o he relaively large number of parameers involved. For his model, we used

17 candidae parameer vecors, while, for he simpler ES formulaion of expression (11), we found ha 10 3 was sufficien. In using, as saring values, CAViaR model parameers, esimaed separaely using quanile regression, we have followed he approach employed by Whie, Kim and Manganelli (2015) for heir muli-equaion models. For he 5% probabiliy level, expressions (20)-(22) presen a join model wih asymmeric slope CAViaR formulaion for VaR, and he AR formulaion for ES. The parameers were esimaed using he firs moving window of 2500 S&P 500 reurns. The expressions also presen parameer sandard errors in parenheses below each parameer. The parameer covariance marix was esimaed using boosrapping, as described in he supplemenary maerial o his paper. In he quanile model of expression (20), here is asymmery in he response o he size of he previous period s reurn, and he AR parameer is relaively close o 1, which is ypical of CAViaR, as well as GARCH, models. The AR parameer is also quie high in he model for x in expression (22). Q ( ) (0.052) y 0 y Iy I y Q 1 (0.046) (0.022) (20) ES Q x (21) Q ( ) (0.076) x x 1 1 y x (0.224) 1 if y 1 Q 1 (22) oherwise For he pos-sample period, Figure 1 shows he S&P 500 reurns and he 5% VaR and ES forecass from he model of expressions (20)-(22), implemened wih parameer re-esimaion. The plo also shows he difference beween VaR and ES forecass, which is represened by x in he model. As shown in expression (22), x responds o exceedances beyond he VaR. We 16

18 highligh hese exceedances in Figure 1. The figure shows x varying across he pos-sample period, wih a clear response o he increased volailiy around period % 2.5% 0.0% % -5.0% S&P 500 VaR forecas ES forecas ES forecas - VaR forecas -7.5% Exceedance beyond VaR forecas Figure 1. For he S&P 500 and 5% probabiliy level, forecass from asymmeric slope CAViaR wih AR model for ES, joinly esimaed by maximizing he AL likelihood. The simpler formulaion for ES is used in he join model of expressions (23)-(24), which was also esimaed for he 5% probabiliy level using he firs moving window of 2500 S&P 500 reurns. The parameers of expression (23) are quie similar o hose of expression (20). Q ( ) (0.014) y 0 y Iy I y Q 1 (0.029) (0.034) (23) ES (0.054) 1 exp 1.11 Q (24) Figure 2 relaes o he model of expressions (23)-(24), implemened wih repeaed reesimaion of parameers. In addiion o he variables ploed in Figure 1, Figure 2 presens he reesimaed muliplicaive facor (1+exp( 0 )) of expression (11) and (24). Alhough his facor is 17

19 consan for a given esimaion sample, Figure 2 shows ha i varies a lile over ime, moivaing he possible use of more complex formulaions for he ES. In Figure 2, he difference beween he VaR and ES forecass is generally smaller and more variable han in Figure 1. Informally, one migh ake he view ha, in Figure 1, he ES forecass look oo exreme in comparison wih he VaR exceedances. In he nex secion, we formally evaluae he VaR and ES forecass. 5.0% % % % -5.0% -7.5% S&P 500 VaR forecas ES forecas ES forecas - VaR forecas Exceedance beyond VaR forecas ES muliplicaive facor (secondary y-axis) Figure 2. For he S&P 500 and 5% probabiliy level, forecass from asymmeric slope CAViaR wih ES modeled as muliple of VaR, joinly esimaed by maximizing he AL likelihood Evaluaion of Pos-Sample VaR and ES Forecass For he 1000 pos-sample periods, we evaluaed he uncondiional coverage of he VaR predicions using a es based on he binomial disribuion o examine wheher he percenage of observaions falling below he corresponding quanile esimaes is significanly differen from he VaR probabiliy level. We refer o he proporion as he hi percenage, and presen he resuls in Table 1 for boh he 1% and 5% probabiliy levels. To save space, we do no repor he resuls for hisorical simulaion based on a moving window of 25 observaions, as his was 18

20 comforably ouperformed by he use of 100 observaions in he moving window. Table 1 shows reasonable resuls for all mehods, excep perhaps he hisorical simulaion mehods. Table 1. VaR uncondiional coverage hi percenages. Hisorical simulaion 1% 5% No. sig. FTSE NIKKEI S&P a 5% FTSE NIKKEI S&P 2500 observaions 0.2* 0.3* * observaions 2.1* 2.1* 2.1* GARCH Suden * 1 6.6* Filered hisorical simulaion * EVT GJRGARCH Suden 1.8* * Filered hisorical simulaion * EVT Symmeric absolue value CAViaR QR for VaR: ES = muliple of VaR * QR for VaR: ES = mean exceedance * QR for 7.5% wih EVT AL: ES = AR model * AL: ES = muliple of VaR * Asymmeric slope CAViaR QR for VaR: ES = muliple of VaR * QR for VaR: ES = mean exceedance * QR for 7.5% wih EVT AL: ES = AR model AL: ES = muliple of VaR * Noes. Bold indicaes bes mehod in each column. Significance a 5% level indicaed by *. No. sig. a 5% We esed for condiional coverage using Engle and Manganelli s (2004) dynamic quanile es. We included four lags in he es s regression o give a es saisic ha, under he null hypohesis of correc coverage, is disribued 2 (6). Table 2 provides he p-values for he es. The resuls for he simplisic hisorical simulaion mehods are poor. For he GARCH 19

21 models, here was benefi in using he asymmeric model for he 1% quanile. The resuls are reasonable for he models ha involve CAViaR quanile formulaions, wih no clear superioriy of one of hese models over anoher. Table 2. VaR condiional coverage dynamic quanile es p-values. Hisorical simulaion 1% 5% FTSE NIKKEI No. sig. S&P a 5% FTSE NIKKEI S&P 2500 observaions observaions GARCH Suden Filered hisorical simulaion EVT GJRGARCH Suden Filered hisorical simulaion EVT Symmeric absolue value CAViaR QR for VaR: ES = muliple of VaR QR for VaR: ES = mean exceedance QR for 7.5% wih EVT AL: ES = AR model AL: ES = muliple of VaR Asymmeric slope CAViaR QR for VaR: ES = muliple of VaR QR for VaR: ES = mean exceedance QR for 7.5% wih EVT AL: ES = AR model AL: ES = muliple of VaR Noes. Bold indicaes bes mehod in each column. No. sig. a 5% In addiion o he coverage ess, we evaluaed he VaR forecass using he quanile score of expression (14). For each mehod, we calculaed he raio of he score o ha of he hisorical simulaion mehod involving 2500 observaions, hen subraced his raio from one, and muliplied he resul by 100. We erm his he quanile skill score, and presen he resuls in 20

22 Table 3. I is essenially he quanile model pseudo R 2 presened by Koenker and Machedo (1999). Higher values indicae superior accuracy. To summarize performance across he hree series for each probabiliy level, we calculaed he geomeric mean of he raios of he score for each mehod o he score for he hisorical simulaion reference mehod, hen subraced his from one, and muliplied he resul by 100. The resuling values are presened in Table 3 in he columns eniled Geo. Mean. For boh he GARCH and CAViaR-based mehods, we see ha he asymmeric versions were more accurae. The CAViaR-based mehods compare well wih he GARCH models. Alhough our main moivaion for joinly modeling VaR and ES is o improve ES esimaion, i is ineresing o see ha he bes quanile score resuls, overall, are in he final row of Table 3, which corresponds o one of he new join models. We implemened Diebold-Mariano ess o compare he quanile score for pairs of mehods. We draw on he asympoic resuls of Giacomini and Whie (2006) o jusify our use of he Diebold-Mariano es wihou he need for a correcion for parameer esimaion error. This seems reasonable, as we are using moving windows of 2500 observaions for esimaion and a pos-sample period of 1000 observaions. (For insigh ino he condiions under which he asympoic resuls of Giacomini and Whie apply, see Clark and McCracken 2012.) Due o he inheren variabiliy in he quanile score, he es s sandard errors were high, and his resuled in few cases of saisical significance. In Table 3, he symbol * indicaes ha he quanile score for he mehod in ha row was significanly worse (a he 5% significance level) han ha of he mehod in he final row of he able, which corresponds o he model esimaed using he AL likelihood, and ES modeled as a muliple of VaR. The symbol indicaes ha he mehod in ha row was significanly worse han ha of he GJRGARCH model wih EVT, which is one of he more compeiive mehods in Table 3. The symbol * occurs many more imes han he symbol, providing suppor for he new mehod in he final row of he able. 21

23 Table 3. VaR evaluaed using quanile skill score. Hisorical simulaion 1% 5% FTSE NIKKEI S&P Geo. Mean FTSE NIKKEI S&P Geo. Mean 100 observaions 10.5 * -0.9 * 3.4 * * -0.5* -1.1 * -1.4 GARCH Suden * Filered hisorical simulaion 24.0* * * EVT 23.5* * * GJRGARCH Suden 25.6* Filered hisorical simulaion 26.0* EVT 24.9* Symmeric absolue value CAViaR QR for VaR: ES = muliple of VaR * QR for VaR: ES = mean exceedance * QR for 7.5% wih EVT AL: ES = AR model * AL: ES = muliple of VaR * Asymmeric slope CAViaR QR for VaR: ES = muliple of VaR QR for VaR: ES = mean exceedance QR for 7.5% wih EVT 26.6* AL: ES = AR model * AL: ES = muliple of VaR Noes. Higher values are beer. Bold indicaes bes mehod in each column. indicaes score is significanly worse han GJRGARCH-EVT a 5% level using Diebold-Mariano es. * indicaes score is significanly worse han AL: ES = muliple of VaR a 5% level using Diebold-Mariano es. To evaluae he ES forecass, we firs used McNeil and Frey s (2000) boosrap es, which we discussed in Secion 5.1. We sandardized by dividing each discrepancy by he corresponding VaR esimae. The resuls, which are shown in Table 4, provide lile insigh ino he relaive performance of he mehods. This moivaes he use of an addiional approach o evaluaing ES forecas accuracy. 22

24 Table 4. ES evaluaed using p-values for boosrap es for zero mean VaR exceedances. Hisorical simulaion 1% 5% FTSE NIKKEI No. sig. S&P a 5% FTSE NIKKEI S&P 2500 observaions observaions GARCH Suden Filered hisorical simulaion EVT GJRGARCH Suden Filered hisorical simulaion EVT Symmeric absolue value CAViaR QR for VaR: ES = muliple of VaR QR for VaR: ES = mean exceedance QR for 7.5% wih EVT AL: ES = AR model AL: ES = muliple of VaR Asymmeric slope CAViaR QR for VaR: ES = muliple of VaR QR for VaR: ES = mean exceedance QR for 7.5% wih EVT AL: ES = AR model AL: ES = muliple of VaR Noes. Bold indicaes bes mehod in each column. No. sig. a 5% In Secion 5.2, we inroduced he AL log score of expression (19) for joinly evaluaing VaR and ES forecass. In Table 5, we show he AL log skill score, which we calculaed as he raio of a mehod s AL log score o ha of hisorical simulaion using 2500 observaions, hen subraced one from his raio, and muliplied he resul by 100. Higher values are preferable for his skill score, which can be viewed as a pseudo R 2 for joinly evaluaing VaR and ES predicions. As wih he quanile skill score, we summarize across he hree series using he geomeric mean. The able shows ha, overall, he bes AL log skill score resuls are in he final 23

25 row, which corresponds o he join model esimaed by maximizing he AL likelihood, and wih ES modeled as a muliple of VaR. The asymmeric slope CAViaR model wih EVT and he GJRGARCH model wih EVT also performed well. As wih he quanile score, we implemened Diebold-Mariano ess o compare he AL log score for pairs of mehods. In Table 5, he symbols * and have he same inerpreaions as in Table 3. In Table 5, he symbol * occurs approximaely wice as many imes as he symbol, providing suppor for he new mehod in he final row of he able. Table 5. VaR and ES evaluaed using AL log skill score. Hisorical simulaion 1% 5% FTSE NIKKEI S&P Geo. Mean FTSE NIKKEI S&P Geo. Mean 100 observaions 6.4 * -3.5 * -0.4 * * -0.8 * 0.4 * -0.2 GARCH Suden * * Filered hisorical simulaion * * EVT * GJRGARCH Suden * Filered hisorical simulaion 16.6* EVT 16.3* Symmeric absolue value CAViaR QR for VaR: ES = muliple of VaR * QR for VaR: ES = mean exceedance * QR for 7.5% wih EVT * AL: ES = AR model * AL: ES = muliple of VaR * Asymmeric slope CAViaR QR for VaR: ES = muliple of VaR QR for VaR: ES = mean exceedance * QR for 7.5% wih EVT 17.2* AL: ES = AR model AL: ES = muliple of VaR Noes. Higher values are beer. Bold indicaes bes mehod in each column. indicaes score is significanly worse han GJRGARCH-EVT a 5% level using Diebold-Mariano es. * indicaes score is significanly worse han AL: ES = muliple of VaR a 5% level using Diebold-Mariano es. 24

26 Table 6. VaR and ES evaluaed using FZG skill score. Hisorical simulaion 1% 5% FTSE NIKKEI S&P Geo. Mean FTSE NIKKEI S&P Geo. Mean 100 observaions 10.6 * -0.9 * 3.5 * * -0.5* -1.1 * -1.4 GARCH Suden * Filered hisorical simulaion 24.1* * * EVT 23.6* * * GJRGARCH Suden 25.8* Filered hisorical simulaion 26.1* EVT 25.0* Symmeric absolue value CAViaR QR for VaR: ES = muliple of VaR * QR for VaR: ES = mean exceedance * QR for 7.5% wih EVT AL: ES = AR model * AL: ES = muliple of VaR * Asymmeric slope CAViaR QR for VaR: ES = muliple of VaR QR for VaR: ES = mean exceedance QR for 7.5% wih EVT 26.8* AL: ES = AR model * AL: ES = muliple of VaR Noes. Higher values are beer. Bold indicaes bes mehod in each column. indicaes score is significanly worse han GJRGARCH-EVT a 5% level using Diebold-Mariano es. * indicaes score is significanly worse han AL: ES = muliple of VaR a 5% level using Diebold-Mariano es. Our use of he AL log score o evaluae VaR and ES forecass could perhaps be viewed as giving an unfair advanage o mehods esimaed using he AL log-likelihood. We, herefore, also evaluae VaR and ES using he FZG and AS scores, which we described in Secion 5.2, and presened in expressions (16) and (17), respecively. Tables 6 and 7 presen he skill score values corresponding o hese wo scores, wih benchmark mehod again chosen as hisorical simulaion using 2500 observaions. Higher values are again preferable for he skill scores. The resuls of Tables 6 and 7 are broadly consisen wih hose for he AL log score in Table 5, wih he 25

27 GJRGARCH mehods performing relaively well, and he bes resuls overall produced by he join model esimaed using he AL likelihood, and wih ES modeled as a muliple of VaR. Table 7. VaR and ES evaluaed using AS skill score. Hisorical simulaion 1% 5% FTSE NIKKEI S&P Geo. Mean FTSE NIKKEI S&P Geo. Mean 100 observaions 20.0 * * * -2.2* -2.2 * -2.0 GARCH Suden * Filered hisorical simulaion 40.3* * * EVT 38.9* * GJRGARCH Suden 43.5* Filered hisorical simulaion 43.3* * EVT 40.5* Symmeric absolue value CAViaR QR for VaR: ES = muliple of VaR 42.2* * QR for VaR: ES = mean exceedance 42.2* * QR for 7.5% wih EVT 42.3* * AL: ES = AR model 42.1* * AL: ES = muliple of VaR 41.9* * Asymmeric slope CAViaR QR for VaR: ES = muliple of VaR QR for VaR: ES = mean exceedance * QR for 7.5% wih EVT 43.9* AL: ES = AR model * AL: ES = muliple of VaR Noes. Higher values are beer. Bold indicaes bes mehod in each column. indicaes score is significanly worse han GJRGARCH-EVT a 5% level using Diebold-Mariano es. * indicaes score is significanly worse han AL: ES = muliple of VaR a 5% level using Diebold-Mariano es. 26

28 Table 8. VaR and ES evaluaed using AL log skill score. Mehod parameers esimaed wihou he period of he Global Financial Crisis. Hisorical simulaion 1% 5% FTSE NIKKEI S&P Geo. Mean FTSE NIKKEI S&P Geo. Mean 100 observaions 6.4 * -3.5 * -0.4 * * -0.8 * 0.4 * -0.2 GARCH Suden Filered hisorical simulaion * EVT * GJRGARCH Suden * Filered hisorical simulaion EVT Symmeric absolue value CAViaR QR for VaR: ES = muliple of VaR * QR for VaR: ES = mean exceedance * QR for 7.5% wih EVT * AL: ES = AR model * AL: ES = muliple of VaR * Asymmeric slope CAViaR QR for VaR: ES = muliple of VaR * 4.7 QR for VaR: ES = mean exceedance * QR for 7.5% wih EVT AL: ES = AR model * AL: ES = muliple of VaR Noes. Higher values are beer. Bold indicaes bes mehod in each column. indicaes score is significanly worse han GJRGARCH-EVT a 5% level using Diebold-Mariano es. * indicaes score is significanly worse han AL: ES = muliple of VaR a 5% level using Diebold-Mariano es Influence of he Global Financial Crisis on VaR and ES Forecas Accuracy A consequence of our daase of 3500 periods ending on 16 April 2013 was ha all esimaion samples conained he period covering he heigh of he global financial crisis. To ry o assess he influence of he crisis period on our resuls, we repeaed our empirical sudy wih observaions 2251 o 2500 omied from he objecive funcions used for parameer esimaion for he various mehods. These omied observaions covered he approximaely 1-year periods 27

29 saring on 6 May 2008, 11 March 2008 and 29 April 2008, for he FTSE 100, NIKKEI 225 and S&P 500, respecively. Table 8 repors he resuling AL log skill scores. Comparing his able wih our original resuls for his measure in Table 5, we see ha, in general, he accuracy of he mehods worsened when he crisis period was omied from he esimaion sample. This was mos noiceable for he CAViaR-based mehods when esimaing he 1% quanile of he S&P 500. I is ineresing o noe ha he GJRGARCH model wih EVT was leas affeced by he removal of he crisis period from he parameer esimaion. 7. SUMMARY Using quanile regression o esimae VaR models has he appeal ha i allows he quanile dynamics o differ for differen probabiliy levels. However, i leaves open he quesion of how o esimae ES. To address his, we have proposed ha esimaion is performed by maximum likelihood based on an AL densiy. The locaion of he densiy is he quanile, and he scale is a simple funcion of ES. This esimaion framework avoids a disribuional assumpion, and enables join modeling of he ime-varying condiional VaR and ES. Esimaing VaR and ES in one sep has heoreical appeal in erms of efficiency, as well as being convenien from a pracical perspecive. In addiion o is use for esimaion, we have proposed ha he AL likelihood be used o evaluae pos-sample VaR and ES forecass. The work of Fissler and Ziegel (2016) has enabled us o provide heoreical suppor for his, and hence for he use of an AL likelihood o esimae join models for VaR and ES. Using sock index daa, we evaluaed he forecass from join models of VaR and ES esimaed in his way. The resuls were promising, wih benchmark mehods no able o ouperform he model consising of an asymmeric slope CAViaR formulaion for VaR, and ES expressed simply as a consan muliple of VaR. This model, esimaed using he AL likelihood, provides a simple exension of he CAViaR approach 28

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