Risk quantification for commodity ETFs: Backtesting value-at-risk and expected shortfall

Accepted Manuscript Risk quantification for commodity ETFs: Backtesting value-atrisk and expected shortfall

Esther B. Del Brio, Andrés Mora-Valencia, Javier Perote PII: DOI: Reference:

S1057-5219(17)30180-1 doi:10.1016/j.irfa.2017.11.007 FINANA 1163

To appear in:

International Review of Financial Analysis

Received date: Revised date: Accepted date:

16 August 2017 21 November 2017 28 November 2017

Please cite this article as: Esther B. Del Brio, Andrés Mora-Valencia, Javier Perote , Risk quantification for commodity ETFs: Backtesting value-at-risk and expected shortfall. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Finana(2017), doi:10.1016/j.irfa.2017.11.007

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Risk quantification for Commodity ETFs: Backtesting Value-at-Risk and Expected Shortfall

Esther B. Del Brio*, Andrés Mora-Valencia **, and Javier Perote*+

PT

* University of Salamanca (IME) Campus Miguel de Unamuno (Edif. F.E.S.), 37007 Salamanca, Spain

RI

** Universidad de los Andes, School of Management, Bogotá, Colombia

SC

Abstract

This paper calibrates risk assessment of alternative methods for modeling commodity

NU

ETFs. We implement recently proposed backtesting techniques for both value-at-risk (VaR) and expected shortfall (ES) under parametric and semi-nonparametric techniques.

MA

Our results indicate that skewed-t and Gram-Charlier distributional assumptions present the best relative performance for individual Commodity ETFs for those confidence levels recommended by Basel Accords. In view of these results, we recommend the

D

application of leptokurtic distributions and semi-nonparametric techniques to mitigate

PT E

regulation concerns about global financial stability of commodity business. Keywords: Value-at-risk; Expected shortfall; Backtesting; Gram-Charlier expansion.

AC

CE

JEL classification: C46, C58, G17

+

Corresponding author: Javier Perote. Universidad de Salamanca. Campus Miguel de Unamuno (Edif. F.ES.). 37007 Salamanca (Spain). Tel: +34-923294400 Ext. 6719. Fax: +34-923294686. E-mail: [email protected]. Other authors’s e-mails: [email protected] (E.B. del Brio), [email protected] (A. MoraValencia).

ACCEPTED MANUSCRIPT Risk quantification for Commodity ETFs: Backtesting Value-at-Risk and Expected Shortfall

1. Introduction Exchange Traded Funds (ETFs) have become a widespread and appealing investment

PT

instrument because they allow trading on a wide basket of assets (commodities included) with high liquidity, tradability and low transaction costs. Particularly, with the

RI

creation of Commodity ETFs, it is possible to invest in a security that replicates the

SC

price of a certain commodity without taking a position on the commodity itself and without storage costs, since ETFs are traded in stock markets. Diversification strategies have led asset managers to include commodity assets in their portfolios, due to their low

NU

correlation with other financial assets in relatively calm periods. Nevertheless, commodity assets generally exhibit higher volatilities than stocks1 which calls for the

MA

need of providing adequate tools for risk management. This paper gives fuel to the debate on risk management modeling by either using traditional Value-at-risk (VaR) or other approaches based on Lower Partial Moments, such as Expected Shortfall (ES).

D

The performance of these measures depends on the ETFs distribution considered and

PT E

although many parametric and non-parametric alternatives have been provided, results have been mostly unsatisfactory in highly volatile scenarios. The current paper overcomes this caveat by proposing the use of the semi-nonparametric approach based

CE

on the Gram-Charlier (GC) distribution for the first time in this particular setting, and by providing evidence on its superior risk assessment performance through novel

AC

backtesting ES techniques for Commodity ETFs. Commodity prices experienced a boom in the mid-2000s triggered by the need of metal supplies (such as iron, aluminum, copper, steel, and zinc) and energy resources from emerging markets, particularly, China. In 2010, large commodity traders such as Vitol, Glencore or Cargill had benefited from the positive returns of commodities, typically 68% a year. More recently, a sharp decline in the price of these assets took place because of the increase in supply and the slower economic growth in emerging markets. By end

1

For instance, annualized volatility for Gold and Silver ETFs are around 34%, whereas annualized volatility for MSCI Index for Developed Markets and Emerging Markets are approximately 18% and 22% from January 2007 to January 2016.

ACCEPTED MANUSCRIPT of August 2015, all main commodities prices (energy, industrial metals and agriculture) had plunged in a 10-20% range year-to-date. Slowdown of Chinese investment has mainly affected copper producers (e.g., Chile), since China demands 45% of the output. As a consequence, by the end of September 2015 Glencore’s share price had plummeted by almost a third. In 2013, Deutsche Bank and JP Morgan also suffered huge losses in based metals businesses. In addition, developed markets based on mining and energy industries like Canada and Australia have also been affected by the decline in

PT

commodity prices.

RI

However, the volatility in the commodity market is not only affected by demand and supply of consumers and producers. Derivatives and Commodity ETFs have been

SC

employed by financial market participants to invest in commodity-linked instruments and its speculation has been the reason for the rapid increase of oil prices in 2007 and

NU

part of 2008 (Hume et al., 2016). In fact, in the last decades, the presence of capital markets and financial industry with the so-called ‘financialization’ of commodity

MA

business has distorted commodity prices and contributed to the bubble formation in these markets.2

Looking back again, in the early 90s, the boom of commodity prices motivated the

D

creation of several ETFs in order to track the price of main commodities such as gold,

PT E

silver and oil. ETFs can be defined as baskets of securities that are traded on the stock exchange, which allow investors to buy commodities as trading stocks. Initially, ETFs used to replicate market indices, which are comprised of equity assets, although other

CE

types of assets such as fixed income, credit, emerging markets, or commodities have been also employed as underlying assets. The main advantages of these financial

AC

instruments rely on their low diversification cost and easier accessibility of certain asset classes, but also on their high liquidity and tradability. On the other hand, commodity assets present higher volatilities than equity assets even in “relatively” calm periods (since commodities cannot be easily stored). The high volatility of the underlying assets also produces a higher uncertainty on the Commodity ETFs market and ‘slippage’ effect (as it is known in the industry jargon), that is, the rapidly decline in ETFs price. For instance, gold ETFs fell closer to 33% in 2014, and 2

There is a debate about whether financialization distorts commodity prices. For a discussion concerning this topic see Cheng and Xiong (2014). We are grateful to an anonymous reviewer for pointing out the discussion of causes of volatility in commodity markets.

ACCEPTED MANUSCRIPT most of sales came from SPDR Gold Shares ETF (one of the analyzed data in this paper) in 2013, mainly triggered by the Federal Reserve reporting the reduction on its quantitative easing plan and the global sentiment.3 Financialization has helped to convert commodities into tradable securities (e.g. Commodity ETFs), and, consequently, has contributed to the decline in commodity prices and its high volatility (Tang and Xiong, 2012; Singleton, 2013 among others).

PT

From a regulatory point of view, the Financial Stability Board (FSB) and Bank for International Settlements (BIS) published separated reports in April 2011 to express

RI

their concerns about the potential systemic impact of complex (e.g. leverage and inverse ETFs) and synthetic (involving collateral baskets) ETFs. Possible financial stability

SC

risks are related to the complexity and relative opacity of these instruments, which hinder the risk assessment according to the regulators’ reports. Furthermore, the UN

NU

Conference on Trade and Development in 2015 has criticized the financialization of commodities highlighting its impact on speculative bubbles inflation.

MA

Since 1996, VaR has been the standard market risk measure to assess regulatory capital requirements.4 This risk measure, defined as the maximum loss given certain confidence level and time horizon (usually 99% and one or ten days, respectively), has been

D

criticized because it does not fulfill the ‘subadditivity’ property. This property states

PT E

that the total risk of a portfolio should be less than or equal to the sum of risks of the individual portfolio assets, i.e. a ‘coherent’ risk measure should be consistent with risk diversification. For this reason, other coherent risk measures have been proposed, the

CE

most widely used being the ES or conditional VaR (CVaR), defined as the conditional expectation of losses for losses that have exceeded a certain VaR threshold. As a matter

AC

of fact, the VaR’s inability to accurately capture tail risk during the global financial crisis made the Basel Committee on Banking Supervision (BCBS) reconsider the use of VaR and publish the review of trading book rules  the so-called Basel 2.5  on May 2012 with the aim to replace VaR by ES. However, ES also features disadvantages compared to VaR. By definition, ES is always higher than VaR at a given confidence level, and can be highly dependent on the Though Commodity ETFs have been used to amplify bets  especially on oil, gold and natural gas price, our study analyzes non-leveraged Commodity ETFs. Leveraged Commodity ETFs (LETFs) will be the focus in a future research. 4 The Basel Committee on Banking Supervision (BCBS) published the “Amendment to the Capital Accord to incorporate Market Risks” in 1996, which was revised and brought into effect in 1998. 3

ACCEPTED MANUSCRIPT assumed loss distribution. Though many banks have been using ES to assess economic capital, ES is very sensitive to extreme events and may result in unstable capital numbers at high confidence levels (reason why ES is commonly computed at 97.5%). In addition, ES is not as intuitive as VaR, and thus its figures are not easy to explain when reporting to board members. Furthermore, ES does not satisfy the ‘elicitability’ property  i.e the capacity of determination for optimal forecasts (Emmer et al., 2015), which

PT

generated a vivid debate on whether ES can be backtestable. On the contrary, VaR satisfies such property and additionally very well-known backtesting methods exist for this measure. Therefore, the discussion resulted in a tradeoff between coherence and

RI

elicitability, the former being preferred by regulators. However, recent studies (see

SC

Section 2) have shown that ES can be backtestable, although the proposed methods are less appealing than the straightforward backtesting alternatives for VaR.

NU

This paper contributes to the literature by providing a comparison between both risk measures (VaR and ES) through a wide variety of methodologies including parametric

MA

distributions (Gaussian, Student’s t and Skewed-t) and the semi-nonparametric approach based on expansions of GC series. In all these cases, we implement backtesting techniques to study model performance and risk assessment of Commodity

D

ETFs, designed in response to the concerns about global financial stability. So far, risk

PT E

management studies have been devoted to commodities directly or commodity derivatives. Hence, this is the first paper, to our knowledge, which analyzes risk management applied to Commodity ETFs and also the first to propose a semi-

CE

nonparametric methodology. For this purpose, we implement joint estimation maximum likelihood techniques and derive a closed form to calculate ES for the GC distribution,

AC

which eases the implementation of ES on this type of models. All in all, the contribution of this paper is three-fold. First, we propose a jointly (i.e. more efficient) estimation of GC parameters and ARMA-EGARCH process for risk calculation purposes and a straightforward formula to compute ES for GC expansions.5 Second, we apply both traditional and novel backtesting methods for quantifying risk to Commodity ETFs. Third, our results have interesting implications for practitioners and regulators, since our application implements appropriate backtesting techniques to evaluate the

5

Previous studies have implemented two-step procedure and estimated in a first step an AR(MA)GARCH process by Quasi Maximum Likelihood (QML) and the remaining density parameters in a second step.

ACCEPTED MANUSCRIPT performance of ES for individual assets, which may help to mitigate main regulators’ concerns about commodity financialization. The rest of the paper is organized as follows: Section 2 reviews the literature on risk measures and their applications to commodity assets, Section 3 describes the risk models and VaR and ES methodologies; Section 4 presents the empirical performance of the models for forecasting VaR and ES with a sample of six different Commodity

PT

ETFs, Section 5 discusses the main results of this paper in light of the new regulation

RI

and Section 6 concludes.

SC

2. Literature Overview

This section presents an overview of recent advances on risk measures focusing on their

NU

properties and applications to commodity markets. This review not only frames the

MA

contributions of the paper but also identifies its main research hypotheses.

2.1. Risk measures

D

VaR has been the standard risk measure for financial industry since mid-1990s and a

PT E

vast literature has been devoted to it  see, e.g. Jorion (2006). The use of ES was proposed by Artzner et al. (1997) and its properties studied in Acerbi and Tasche (2002), and Rockafellar and Uryasev (2002). As discussed in the previous section, VaR

CE

is not a coherent risk measure since it does not satisfy subadditivity, unlike ES.6 Empirical studies have shown that under extreme fluctuations VaR may underestimate

AC

risk but ES may overcome this shortcoming; however, ES strongly depends on the estimation accuracy (Yamai and Yoshiba, 2005). As a consequence, capital calculated from ES might be less stable than that of the VaR when losses exhibit fat tails. This evidence is consistent with Cont et al. (2010), who find that ES is less robust and more data-sensitive than VaR. Nevertheless, under a new notion of (qualitative) robustness, Embrechts et al. (2015) show that ES is more robust than VaR. Recently, the elicitability of risk measures has been discussed due to its importance for studying model performance in forecasting applications. Formally, a risk measure is 6

See Artzner et al. (1999) for more details about the properties that a risk measure should satisfy to be coherent.

ACCEPTED MANUSCRIPT elicitable if it minimizes a suitable expected scoring function (Bellini and Bignozzi, 2015). Gneiting (2011) shows that ES does not satisfy this property and thus the search of accurate techniques for backtesting ES becomes challenging. In this line, Acerbi and Székely (2014) provide three methods to backtest ES; they show that ES can be used for model testing but elicitability is still important to make comparisons of different models. Particularly, if elicitability is not satisfied, no consistent scoring function exists and thus it is not possible to calibrate the model performance to identify the best one.

PT

More recently, Fissler et al. (2016) have proposed a method based on conditional elicitability of ES, which is applied in this paper. Since ES is jointly elicitable with

RI

VaR, then a suitable scoring function can be employed to test ES (Fissler and Ziegel,

SC

2016; Fissler et al., 2016). Given that Commodity ETF returns exhibit heavy tails, we conjecture that backtesting ES procedures will show that skewed and heavy-tailed

NU

distributions, particularly the GC, perform better than the traditional alternatives (either Gaussian or Student’s t), after filtering returns with an adequate process which captures volatility clustering. This finding would be in line with other studies that show good

MA

performance of skewed-t distribution (Giot and Laurent, 2003; Aloui and Mabrouk, 2010) and GC distribution (Del Brio et al. 2014a, b) for VaR quantification.

D

Moreover, we are interested in measuring the impact of ES estimates compared to VaR

PT E

amounts, since it would be expected a sharp increase in regulatory capital with the new proposal (97.5%-ES). We hypothesize that, although capital requirements with all the leptokurtic distributions are higher than with the Gaussian, the replacement of 99%-

CE

VaR to 97.5%-ES is only fair for the GC distribution. This goal is treated in Section 5 “Discussion” in light with the results obtained in the paper.

AC

On the other hand, Bellini et al. (2014) and Ziegel (2016) finds that expectiles (introduced by Newey and Powell, 1987) are both coherent and elicitable risk measures. Expectiles are derived by employing asymmetric least squares and can be used to estimate VaR and ES (Taylor, 2008). This measure has the advantage of assuming a free distribution, but its concept is less intuitive than VaR and ES. As discussed above, every risk measure has its own advantages and drawbacks. Emmer et al. (2015) overview the desired properties of risk measures (coherence, elicitability, robustness, comonotonicity and additivity, among others) and their impact on capital allocation. These authors conclude that ES is a good risk measure in practice, although the discussion of the best risk measure is still an open question and depends on the

ACCEPTED MANUSCRIPT characteristics to be evaluated. For instance, in a recent work Koch-Medina and Munari (2016) warn about the use of ES for capital adequacy and portfolio risk measurement since ES does not necessarily guarantee protection of liability holders.

2.2. Risk quantification in Commodity Markets

PT

Recent studies on risk management applications to commodity markets have undertaken different approaches and have been applied to different assets (Giot and Laurent, 2003;

RI

Hung et al., 2008; Chiu et al., 2010; Aloui and Mabrouk, 2010; Youssef et al., 2015; Andriosopoulos and Nomikos, 2015; Aloui and Jammazi, 2015), although as far as we

SC

know they have never been used for modeling Commodity ETFs. All these papers, after testing a wide variety of models, point to the adequacy of different (asymmetric and

NU

long memory) GARCH models with (skewed) heavy-tailed distributions for risk assessment. Particularly, Giot and Laurent (2003) and Aloui and Mabrouk (2010) show

MA

that the APARCH (or FIAPARCH) model with skewed-t innovations outperforms other alternatives in several commodity and energy markets. Similarly, Youssef et al. (2015) support a FIAPARCH with EVT for estimating both VaR and ES  a circular bootstrap

D

being employed to backtest ES. On the other hand, Hung et al. (2008) find that GARCH

PT E

models with heavy-tailed innovations provide accurate risk measures for energy commodities and Steen et al. (2015) and Andriosopoulos and Nomikos (2015) show that quantile regression techniques and Monte Carlo simulations, respectively, outperform

CE

many other alternatives for modeling risk for most commodities and, particularly those of energy. Finally, other authors find evidence in favor Hull-White (Chiu et al., 2010) and three wavelet-based (Aloui and Jammazi, 2015) models for Brent and WTI crude

AC

oil prices.

In summary, risk assessment studies on commodity markets have proposed the use of (asymmetric and long memory) models for conditional variance and (skewed) heavy tailed distributions, but none of them have adopted a semi-nonparametric approach, despite it is deemed a very flexible measure to accurately account for salient empirical regularities of financial data (particularly leptokurtosis and wavy-thicked tails)  see Mauleón and Perote (2000) as the first application of this method in risk management. In this paper we show the simplicity and outperformance of this approach for commodity ETFs, compared to other parametric alternatives proposed in the literature.

ACCEPTED MANUSCRIPT

3. Models and Methodology It is well-known that financial returns are highly leptokurtic, negatively skewed and exhibit clustering and persistence in conditional volatility.7 Commodity ETFs are not an exception on such patterns and thus non-Gaussian models seem also appropriate for providing accurate risk measures. This paper compares the relative performance of

PT

different parametric and semi-nonparametric methods for measuring both VaR and ES. In what follows, we review these models, their implementation for computing VaR and

RI

ES (remarkably, the ES application for the GC distribution) and the tests for relative

SC

performance based on backtesting techniques.

Commodity returns present a predictable component in the conditional mean that has

NU

traditionally been modeled according to simple ARMA structures. Squared returns, however, exhibit particular dynamics (conditional heteroskedasticity, volatility

MA

clustering and long memory) that have been extensively studied since Engle (1982) and Bollerslev (1986) introduced ARCH and GARCH models. As we focus on VaR and ES performance due to the distributional hypotheses, we implement standard models in risk

D

management applications for the first and second conditional moments. Particularly, our

PT E

applications assume an ARMA(1,1) for modeling conditional mean  equation (2)  and an EGARCH(1,1) for modeling conditional variance  equation (3), the latter process being able to capture the ‘leverage effect’ or the asymmetric impact in volatility from

CE

negative and positive shocks (Nelson, 1991). Therefore, we model asset returns ( ) as in equation (1), where

are the

are the standardized errors defined

AC

conditional mean and variance, respectively, and

and

in equation (4), which we assume to be distributed according to a certain distribution H, and

is a white noise. The complete model is described below. ,

(1) ,

(2) ,

, 7

See, e.g. Cont (2001) for a description of the main stylized empirical regularities of financial data.

(3) (4)

ACCEPTED MANUSCRIPT where

and

are the intercepts of the conditional mean and variance models,

the parameters of AR(1) and MA(1) structures of the conditional mean, and

and and

the parameters associated with the size and sign effects of the EGARCH model (respectively). Different standardized (i.e. zero mean and unit variance) density specifications are considered for H.

PT

3.1. Distributional hypotheses

RI

Besides the Gaussian density, included as benchmark, and the most widely used parametric distributions to capture leptokurtosis (Student’s t) and skewness (skewed-t),

SC

we also consider the semi-nonparametric GC density that, as far as we know, has never been employed for modeling Commodity ETFs. The outstanding performance of this

NU

method is based on the asymptotic property of the GC type A series for approximating any frequency function under weak regularity conditions (Kendall and Stuart, 1977). All

MA

these probability density functions (pdf) are described below.

where

PT E

CE

(ii) Student’s t pdf:

.

(5)

D

(i) Gaussian pdf:

is the gamma function and

(6) the degrees of freedom parameter. As the number

AC

of degrees of freedom grows to infinity, the Student’s t distribution tends to the normal distribution.

(iii) Skewed-t pdf (Fernández and Steel, 1998):

(7)

where

is the shape parameter, which incorporates the skewness, and

Student’s t pdf in equation (6). When Student’s t distribution.

is the

, the skewed-t distribution collapses to the

ACCEPTED MANUSCRIPT (iv) GC Type A pdf: (8) where

denotes the standard normal pdf in equation (5),

a vector of parameters such that

and

is

is the Hermite polynomial (HP) of

order s, which is defined in terms of the sth order derivative of

as

PT

(9)

RI

In particular, the first four HP are:

. These polynomials form an orthonormal basis, thus satisfying

NU

SC

the orthogonality property,

,

(10)

which is the basis of the characterization of GC series as a pdf (i.e. the GC density integrates to one) and its parameters in terms of density moments. For instance, even

variance,

MA

non-central moments depend on the even

parameters (e.g.

accounts for the

for the excess kurtosis and the rest of the even parameters capture higher-

D

order moments) and skewness is represented by the odd parameters. A well-known

PT E

shortcoming of this expansion is that it is not necessarily positive for all the values of the parameters, which calls for the implementation of positive transformations (Gallant and Nychka, 1987; León et al. 2009) or positivity constraints (Jondeau and Rockinger,

CE

2001). However, we use the original GC Type A expansion in equation (8), which is simpler and more useful for risk measure applications.

AC

Most of the empirical studies that implement GC density in finance  mainly for option valuation, e.g. Jarrow and Rudd (1982)  truncate the expansion in n = 4 and employ only two terms of the expansion,

and

. These models, as well as the other

parametric distributions, were jointly estimated with the ARMA(1,1)-EGARCH(1,1) models  equations (2) and (4)  by maximum likelihood (ML) techniques.

3.2. Conditional VaR measures The estimated VaR with a probability  is computed as a linear transformation of the estimated conditional α-quantile,

, of the assumed standardized distributions in

ACCEPTED MANUSCRIPT the above section. That is, a quantile is calculated as where

,

denotes the cumulative distribution function (cdf). Different quantiles are

obtained for each model (Student’s t, skewed-t and GC distributions) at each time , except for the normal distribution whose -quantile is a fixed value and depends only on the probability . For our applications, we employ

and

,

according to Basel II and the Fundamental Review of Trading Book (FRTB), respectively (i.e. we estimate VaR in the left tail of the returns distribution). Therefore,

PT

the predicted VaR for the variable R at the time horizon t+1 and with probability  is

RI

given in equation (11).

,

where

and

(11)

SC



are the predictions for the mean and standard deviation

NU

conditioned on the available information at time t, t, obtained from the ARMAEGARCH model described in equations (2) and (3). This measure amounts for the

MA

maximum expected loss of the variable in t+1 obtained with a probability . The α-quantiles can be easily estimated by numerical integration and, for the GC satisfying

D

distribution, may be obtained as those

PT E

, (12)

as a direct application of equation (12), where

is the

CE

standard normal pdf.

denotes the GC cdf and

AC

3.3. Conditional ES measures The ES for a given probability  is the expected return of the variable on the worst  per cent of the cases. In terms of the standardized distribution of

it can be calculated

as (13) Thus, the expected shortfall for the variable R at the time horizon t+1 and with probability  is obtained through the following transformation 

,

(14)

ACCEPTED MANUSCRIPT where

and

are the predictions for the mean and standard deviation

conditioned on t, and

. The ES estimates are usually

computed by numerical integration, but in most cases the use of closed forms can simplify the estimation procedures. Particularly, closed forms of the cases studied in Section 3.1 are displayed below.8

is the pdf of standard normal  equation (5)  and

RI

where

PT

(i) Gaussian ES:

NU

(ii) Student’s t ES:

is the pdf of standard Student’s t  equation (6),

quantile and

MA

where

(16) its corresponding -

the degrees of freedom parameter.

PT E

D

(iii) GC Type A ES:

where

is its corresponding

SC

-quantile.

(15)

is the pdf of standard normal  equation (5)  and

(17) is the -quantile of

, is similar to the expression for Gaussian-ES, but the argument

AC

The first term

CE

the GC Type A distribution equation (8). The proof is provided in the Appendix A.

is different, since it depends on the quantile function of the GC distribution, while the second term depends on combinations of Hermite polynomials weighted by the GC parameters (or moments). It is worth mentioning that we refer (1−)100%-VaR and (1−)100%-ES for risk measures calculations in our applications.

3.4. Backtesting methods

8

The ES under the Skewed-t in the empirical application of next section was obtained by numerical integration.

ACCEPTED MANUSCRIPT 3.4.1. Tests for VaR a. Bernoulli coverage test The VaR backtesting is usually based on the assumption that the number of exceptions is generated by an iid Bernoulli process. The violation or exception is 1 if the actual loss is greater than the predicted VaR, otherwise it is 0. The null hypothesis of Bernoulli Coverage test indicates that the VaR model is correct according to the exceptions ratio.

PT

A coverage test may also be performed by the likelihood ratio (LR), which



(18)

SC



RI

asymptotically follows a chi-square distribution with one degree of freedom,

where  is the proportion of violations to be tested  1% and 2.5% in next section’s and

are the number of ones (violations) and zeros (no

NU

empirical application.

violations) in the backtesting period of size . The p-value is then calculated as:

MA

where

(19)

is the cdf of a chi-square random variable with one degree of freedom.

D

b. Relative Comparison for VaR

PT E

The relative performance of a risk measure is based on a scoring function. This function is denoted as

, where

is the forecast provided by a certain model and

the

realized observation. The scoring function can be viewed as a prediction error function

CE

that must be minimized. A scoring function consistent for VaR is (Gneiting, 2011; Fissler and Ziegel, 2016)

AC

where

,

is a strictly increasing function,

represents the observed return and

(20) the

estimated VaR. This scoring function can be used to rank VaR forecasts and to implement Diebold and Mariano’s (1995) test for relative VaR model performance. The Diebold-Mariano’s test statistic (DM) is calculated as (21) where

denotes the mean of

which is

ACCEPTED MANUSCRIPT (22) where

and

denote the scores obtained from VaR models

respectively, and

and

,

is the heteroskedasticity and autocorrelation consistent (HAC)

standard error (e.g., Newey-West estimator). The null hypothesis is

, meaning

that both models present equal predictive accuracy and the composite alternative is . Since the limiting distribution of DM statistic

PT

and

under null hypothesis is standard normal, the null hypothesis is rejected when for large

RI

at 5% significance level, indicating the outperformance of model

SC

negative (positive) values.

NU

3.4.2. Tests for ES a. t-test

is the actual loss,

(23)

is the estimated ES given in equation (14),

is

PT E

where

D

MA

A first test for ES is based on the violation residuals which is calculated as

the conditional mean of the model  equation (2). The indicator function takes value 1 when the actual loss has exceeded the estimated VaR, and

CE

0 otherwise. Then, the null hypothesis of zero mean violation residuals may be tested by

where

AC

a simple t-test on this variable (McNeil et al., 2015),

is the sample mean of the violation residuals of size , and

(24) denotes its

standard deviation. b. Relative Comparison for ES Fissler et al. (2016) show that VaR and ES are jointly elicitable and therefore a possible scoring function for both arguments is given by

ACCEPTED MANUSCRIPT

(25)

,

where

represents the observed return,

In addition,

,

and

and

the estimated VaR and ES respectively.

are continuous and differentiable functions9, see Fissler

and Ziegel (2016) for more details. In this paper, we employ

and

PT

. For such scoring function, the relative performance of the models can be

RI

assessed by implementing DM test in equation (21).

SC

4. Risk quantification for individual commodity ETFs 4.1. Data

NU

The sample comprises nine years of daily prices from January 2007 to January 2016 for Gold, Silver, Oil, Agriculture, Energy and Broad Commodity ETFs. All data were obtained from Bloomberg; further details on the data are provided in Appendix B.10

MA

Table 1 displays the descriptive statistics for continuously compounded returns of these series, defined as Rt = 100log(Pt/Pt-1), where Pt represents ETFs prices at time t.

D

[INSERT TABLE 1]

PT E

Descriptive statistics confirm that Commodity ETFs feature the same regularities as most financial data: Location statistics (daily mean and median) are around zero reflecting market efficiency; they exhibit negative skewness (i.e. more probability mass

CE

around positive returns, but also more extreme values at the left tail of negative returns); high volatility and kurtosis (Oil ETF featuring the highest volatility followed by Silver

AC

ETF, the latter having the highest excess kurtosis). Figures 1 and 2 display prices and returns, respectively, featuring the instability and volatility clustering of the data. [INSERT FIGURE 1] [INSERT FIGURE 2] 4.2. Model Performance for Commodity ETFs This section presents the parameter estimation of the ARMA(1,1)-EGARCH(1,1) model under the alternative distributional hypotheses in Section 3.1 and the model comparison 9

and are strictly increasing functions. The selected assets, available on Bloomberg, are the main ETFs per segment according to the Assets Under Management (AUM) ranking found in ETF.com website as of January 7, 2016. 10

ACCEPTED MANUSCRIPT according to the tests mentioned in Section 3.2. We examine backtesting for both VaR (at 99% and 97.5% confidence level) and ES (at 97.5% confidence level), and discuss the relative model assessment. Recall that we are estimating VaR and ES in the left tail of the return distributions at probability , but we refer the risk measures to (1−)100%-VaR and (1−)100%-ES.

PT

4.2.1. Model estimates

RI

Table 2 presents the ML estimates11 of the parameters of the ARMA(1,1)– EGARCH(1,1) model under four distributional hypotheses: Gaussian (Panel A),

SC

Student’s t (Panel B), Skewed-t (Panel C) and GC (Panel D). For the first three models, the rugarch package is employed (Ghalanos, 2015). It is noteworthy that, unlike Del

NU

Brio and Perote (2012), where parameters are estimated in two steps, we estimate ARMA-EGARCH and GC innovation parameters jointly by ML. The loglikelihood

MA

function is given by (after dropping out some constants)

and

CE

PT E

D

where

(26)

,

(27) (28)

see Appendix C. The parameters are estimated by see

AC

employing the R command nlminb, which uses a quasi-Newton algorithm

Appendix D for the R Code. P-values for testing the significance of every parameter are given in parentheses. The estimates of the AR ( ) and MA () structures are significant for most cases of Commodity ETF series and the values of the EGARCH(1,1) parameters are also statistically significant. The estimates for the shape (degrees of freedom), , and skew parameter (ξ) of the Student’s t distributions are significant and reveal the leptokurtic and (negatively) skewed nature of the data, which is confirmed by

11

Only the estimates for the first in-sample window of the backtesting procedure are displayed. However, the dynamics of the fitted parameters along the whole backtesting period is illustrated in the Appendix E.

ACCEPTED MANUSCRIPT the estimates of the parameters of the GC densities:

(skewness coefficient) and

(excess kurtosis). [INSERT TABLE 2] The skewness parameters for Gold and Agricultural ETFs are close to one, when filtering the returns by an ARMA(1,1)-EGARCH(1,1) with innovations skewed-t distributed. This fact indicates that the distribution for these variables is close to the

PT

Student’s t (as revealed by the similar loglikelihood and information criteria values in Table 3, Panels A and B), and thus the performance would be expected to be the same

RI

for both distributions, when calculating risk measures for Gold and Agriculture ETFs.

SC

In addition, the degrees of freedom parameter ( ) for Oil ETF is relatively high (12.28) when filtering the returns by an ARMA(1,1)-EGARCH(1,1) with innovations Student’s

NU

t distributed. The higher the value of this shape parameter the less leptokurtic (i.e. closer to the Gaussian pdf) the distribution is. This feature was also clear from the descriptive statistics (see Table 1), where Oil ETF presents the minimum excess kurtosis (2.47). It

MA

is worth noting that, after filtering Oil ETF returns by an ARMA(1,1)-EGARCH(1,1) model with skewed-t distribution for the innovations, the estimation for degrees of freedom is still relatively high (13) and skewness parameter close to one. The diagnostic

D

tests for serial correlation (Ljung Box statistic) on the ARMA(1,1)-EGARCH(1,1)

PT E

model (levels and squared) standardized residuals are displayed in Table 3, as well as the accuracy criteria (AIC, BIC and HQC) under the different distributional hypotheses.

CE

[INSERT TABLE 3]

Accuracy criteria show that ARMA-EGARCH-Gaussian is outperformed by the other

AC

models for all series. Though the hypothesis of absence of serial correlation is rejected for squared standardized residuals in metal (Gold and Silver) ETFs, the ARMAEGARCH model seems to be adequate for Commodity ETF returns, since their most important parameters to capture volatility persistence and leverage effects are significant. Appendix E show the dynamics of the parameter estimates for the whole backtesting period and all distributions.

4.2.2. Model performance for VaR and ES a. Tests for 99%-VaR

ACCEPTED MANUSCRIPT According to Bernoulli Coverage test for VaR  see equation (18)  presented in Table 4, ARMA-EGARCH-skewed-t and ARMA-EGARCH-GC perform adequately for all Commodity ETFs, but the ARMA-EGARCH-GC fails in Agriculture ETF. ARMAEGARCH-Gaussian and ARMA-EGARCH-t tend to underestimate risk, whereas ARMA-EGARCH-skewed-t seems to overestimate it. As seen in Table 4, ARMAEGARCH-Gaussian and ARMA-EGARCH-t only perform reasonably well for

PT

Agriculture ETF returns according to the coverage test. Details about the relative performance tests of VaR at 99% (pairwise DM test) can be found in Appendix F. Results for relative performance shows the outperformance of non-Gaussian models

RI

with the exception of Oil ETF (as commented before, Oil ETF features the less

SC

leptokurtic pattern of all series), but ARMA-EGARCH-t performs poorly in most cases (all cases except for Silver and Oil ETFs). ARMA-EGARCH-GC is significantly better

NU

than ARMA-EGARCH-skewed-t for Silver and Oil ETFs, whereas ARMA-EGARCHskewed-t performs more accurately than ARMA-EGARCH-GC for Energy ETF.

MA

[INSERT TABLE 4]

D

b. Tests for 97.5%-VaR and 97.5%-ES

PT E

Table 5 displays the results of the Bernoulli Coverage test for VaR at 97.5%. They confirm the good performance of ARMA-EGARCH-skewed-t and ARMA-EGARCHGC models. However, the ARMA-EGARCH-Gaussian for Silver, Agriculture and

CE

Broad ETFs are not rejected at 5% confidence. As expected, Gaussian model performs better when the confidence level of the VaR is not so high (e.g. 97.5%). The 97.5%-

AC

VaR relative performance (see results in Appendix G) supports the ARMA-EGARCHGC and ARMA-EGARCH-skewed-t as the best models. ARMA-EGARCH-Gaussian (ARMA-EGARCH-t) only performs well for one (two) case(s). [INSERT TABLE 5] Table 6 presents the results of the t-test  equations (23) and (24)  for model performance in terms of ES, the null hypothesis being zero mean of violation residuals. Hence, a model is good enough if there is no evidence to reject null hypothesis. The results of t-test for ES at 97.5% show that ARMA-EGARCH-skewed-t and ARMAEGARCH-GC model are the best (these models are never rejected) followed by

ACCEPTED MANUSCRIPT ARMA-EGARCH-t (rejected in 4 occasions). ARMA-GARCH-Gaussian is always rejected when risk is measured by ES at 97.5%. [INSERT TABLE 6] Relative performance (DM pairwise) test for ES is displayed in Appendix H. In this case, the best model is ARMA-EGARCH-t followed by ARMA-EGARCH-GC and ARMA-EGARCH-skewed-t. Gaussian model works better than skewed-t model for Oil

PT

and Silver ETFs respectively. Once more, this is not an unexpected result, since excess kurtosis in the Oil ETF is the lowest. However, it is outperformed by the rest of the

RI

models in two cases (Gold and Broad ETFs).

SC

In summary, ARMA-EGARCH-GC outperforms the other models for most series and according to Bernoulli test, t-test and relative performance tests for VaR and ES

NU

estimations. Nevertheless, ARMA-EGARCH-skewed-t and ARMA-EGARCH-t seem to be the best models according to coverage test for 99%-VaR and relative performance

MA

for 97.5%-ES, respectively. ARMA-EGARCH-Gaussian performs very poorly according to both coverage and relative performance test for all series with the

D

exception of the Oil ETFs series (the less leptokurtic one).

PT E

5. Discussion

ETFs have provided investors and portfolio managers with access to a variety of

CE

alternative strategies and investments that guarantee relevant advantages in terms of liquidity transparency and cost effectiveness. During financial crises, the main concern

AC

for financial institutions is dealing with high volatility. Asset managers have been considering the inclusion of commodities in their portfolios, as a solution for fluctuations in implied market volatility, since commodities are negatively correlated to other financial assets in “relatively” calm periods, and also for market-neutral strategy reasons. One of the best ways to include commodities in portfolios is through ETFs due to the abovementioned advantages. However, correlation of equity and commodities has increased with financialization of commodities (Basak and Pavlova, 2016) and during crisis periods the volatility of Commodity ETFs has become particularly extreme. Since the global financial crisis, regulators and policy makers have been concerned about financial stability and an important instrument to achieve this goal has been

ACCEPTED MANUSCRIPT setting capital requirements adequacy. In this line, the Basel Committee published a consultative document (BCBS, 2013) on the FRTB. One of the main changes proposed in the FTRB is replacing 99%-VaR with a 97.5%-ES in the internal model’s approach for financial institutions. With the new rules, regulators estimate that banks would have to increase capital buffer around 40% in average to mitigate potential market risk losses in their trading books. Nevertheless, few studies are devoted to examine the impact for capital requirements with the new regulatory proposal. For instance, Kellner and Rösch

PT

(2016) find that ES is more sensitive towards regulatory arbitrage and model risk, especially under parameter misspecification. As Commodity ETFs represent an

RI

increasing proportion in portfolios, their risk exposure also impacts capital

SC

requirements. In this section, we illustrate how capital requirements on positions related to the Commoditiy ETFs studied in the previous section would be affected by the new

NU

regulatory environment. Table 7 presents the average ratio of 97.5%-ES to 99%-VaR for the backtesting of the six Commodity ETFs analyzed.

MA

[INSERT TABLE 7] As expected, the ratio 97.5%-ES/99%-VaR for the Gaussian distribution is 1, which was the reason why Basel Committee established the 97.5% confidence level for ES.

D

Nevertheless, under the more reasonable assumption of leptokurtic distributions for

PT E

daily returns replacing 99%-VaR with 97.5%-ES would result in average increments of capital requirements between 14% and 42% under Student’s t and 16% under skewed-t. In other words, to obtain similar capital requirements with ES and VaR under Student’s

CE

t distributional assumption, ES should be computed at a lower confidence level than 97.5% (if VaR is computed at 99%). Nevertheless, the ratio 97.5%-ES/99%-VaR for

AC

GC is also close to one as in the Gaussian case, since GC distribution is an expansion in terms of derivatives of the Gaussian pdf, see equations (8) and (9). This means that replacing 99%-VaR by 97.5%-ES is equivalent in terms of capital requirements under GC distribution as was in the Gaussian case, for which the rule was initially proposed. However, due to the fact that GC allows for skewness and kurtosis, capital requirements based on GC-ES are much higher than Gaussian-ES, and more accurate to represent the true risk according to their superior backtesting performance (see also Del Brio et al., 2014a,b for similar results on other assets). In view of our results, one relevant conclusion of this investigation is that we highly recommend the application of leptokurtic distributions and semi-nonparametric

ACCEPTED MANUSCRIPT techniques to mitigate regulation concerns about global financial stability and financialization of commodity business. Furthermore, for the GC distribution replacing 99%-VaR with 97.5%-ES does not suppose additional capital requirements than those due to the consideration of the thick tails, unlike when using other leptokurtic distributions for which the proper confidence levels employed for ES should be

PT

reconsidered.

6. Conclusions

RI

The Basel Committee’s FRTB proposes to replace Value-at-Risk (VaR) at 99% by

SC

Expected Shortfall (ES) at 97.5% to obtain more accurate market risk measures. This proposal has initiated a controversial debate in the academy and financial industry about

NU

the appropriateness of such measure, mainly due to its troublesome backtesting implementation. However, recent studies have shown that, under certain conditions, it is

MA

possible to implement performance tests in terms of ES jointly with VaR. This paper presents an application of these methods to several Commodity ETFs in order to shed light on the risk assessment of different techniques, which cover parametric (Gaussian,

D

Student’s t and skewed-t), and semi-nonparametric (GC). Particularly, as far as we

PT E

know, the ES assessment of the latter is being tested for the first time. Coverage tests for backtesting 97.5%-VaR show that ARMA-EGARCH-skewed-t and ARMA-EGARCH-GC perform satisfactorily for Commodity ETFs. The latter and

CE

ARMA-EGARCH-t, however, are the best models in terms of relative performance for 97.5%-VaR and 97.5%-ES, which is the new confidence level proposed by Basel

AC

Accords. Furthermore, measures for 99%-VaR and 97.5%-ES seem to be ‘equivalent’ for GC, but not for Student’s t or skewed-t. Therefore, our study recommends the implementation of semi-nonparametric methods based on the GC expansion, for which we provide a straightforward closed form for ES. Its application to Commodity ETFs, as stated above, is found to be essential to handle with high systematic risk in financial markets during financial crises and extreme shocks. An accurate assessment of risk under these techniques may enable ETFs, and more specifically Commodity EFTs, to control for implied market volatility where other financial assets and econometric techniques have failed to do it.

ACCEPTED MANUSCRIPT This study presents the basic methodology for backtesting ES with semi-nonparametric distributions. Nevertheless, it is focused on the univariate case, whilst for portfolio managers would be also interesting the extension to the multivariate case. Furthermore, risk assessment comparisons have been made only with VaR, disregarding other alternative risk measures. Consequently, future research will study alternative seminonparametric methods for portfolio assessment  e.g. the moments expansion density

PT

in Ñíguez and Perote (2016, 2017)  and the applications of new risk measures such as median shortfall. This measure has been proven to be elicitable and to satisfy a wide set

RI

of economic axioms (Kou and Peng, 2014), which might help to overcome the sensitivity of ES to extreme events and incoherence of VaR. Spectral risk measures

SC

introduced by Acerbi (2002) appear to be other interesting avenues for research, as well as calculating confidence intervals of these risk measures on a specific model, as urged

NU

by the European Banking Authority.

Other interesting line of research would be the testing of the relative performance of the

MA

methods proposed by Fissler and Ziegel (2016) and Acerbi and Székely (2014), the analysis of interdependence in commodity market caused by financialization and the application of a new measure called “contagion Expected Shortfall” proposed by Suh

D

(2015). From the point of view of risk managers would also be interesting the study of

PT E

the role of commodities (and commodity ETFs) in risk diversification, since the negative correlation between commodities and stocks has recently become strongly

CE

positive.

AC

Acknowledgments

This work was supported by the Spanish Ministry of Economics and Competitiveness [ECO2013-44483-P and ECO2016-75631-P]; FAPA-Uniandes [PR.3.2016.2807]; and Junta de Castilla y León [SA072U16]. We gratefully acknowledge these institutions for their funding. We also thank the Editor Brian M. Lucey, two anonymous referees and participants of the Conference on the Energy & Commodity Finance (Paris, 2016), Conference Mathematics of Energy Markets (Vienna, 2016), INFINITI 2017 Conference (Valencia 2017) and Econometrics and Financial Data Science (Reading, 2017) for their comments and suggestions.

ACCEPTED MANUSCRIPT References Acerbi, C. 2002. Spectral measures of risk: A coherent representation of subjective risk aversion. Journal of Banking and Finance 26, 1505-1518. Acerbi, C. and Székely, B. 2014. Back-testing expected shortfall. Risk magazine 27, 76–81. Acerbi, C., Tasche, D. 2002. On the coherence of expected shortfall. Journal of Banking and Finance 26, 1487-1583.

PT

Aloui, C., Jammazi, R. 2015. Dependence and risk assessment for oil prices and exchange rate portfolios: A wavelet based approach. Physica A 436, 62-86.

RI

Aloui, C., Mabrouk, S. 2010. Value-at-risk estimations of energy commodities via long memory, asymmetry and fat-tailed GARCH models. Energy Policy 38, 2326-2339.

SC

Andriosopoulos, K., Nomikos, N. 2015. Risk management in the energy markets and Value-at-Risk modelling: a hybrid approach. The European Journal of Finance 21 (7), 548-574.

NU

Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. 1997. Thinking coherently. Risk 10 (11), 68-71.

MA

Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. 1999. Coherent measures of risk. Mathematical Finance 9 (3), 203-228. Basak, S. Pavlova, A. 2016. A Model of financialization of commodities. The Journal of Finance 71 (4), 1511-1555.

PT E

D

BCBS (Basel Committee on Banking Supervision). 2013. Fundamental Review of the Trading Book: A Revised Market Risk Framework, Bank for International Settlements, Basel.

CE

Bellini, F., Bignozzi, V. 2015. On elicitable risk measures. Quantitative Finance 15 (5), 725-733. Bellini, F., Klar, B., Müller, A., Rosazza Gianin, E. 2014. Generalized quantiles as risk meausres. Insurance: Mathematics and Economics 54, 41-48.

AC

Bollerslev, T., 1986. Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 31, 307-327. Cheng, I-H., Xiong, W. 2014. Financialization of Commodity Markets. Annual Review of Financial Economics 6 (1), 419-441. Chiu, Y.-C., Chuang, I.-Y., Lai, J.-Y. 2010. The performance of composite forecast models of value-at-risk in the energy market. Energy Economics 32, 423-431. Cont, R., 2001. Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues. Quantitative Finance 1, 223-236. Cont, R., Deguest, R., Scandolo, G. 2010. Robustness and sensitivity analysis of risk measurement procedures. Quantitative Finance 10, 593-606.

ACCEPTED MANUSCRIPT Del Brio, E. B., Perote J., 2012. Gram-Charlier densities: Maximum likelihood versus the Method of Moments. Insurance: Mathematics and Economics 51, 531–537. Del Brio E.B., Níguez, T.M., Perote, J. 2011. Multivariate semi–nonparametric distributions with dynamic conditional correlations. International Journal of Forecasting 27, 347–364. Del Brio, E.B., Mora-Valencia, A., Perote, J. (2014a). VaR performace during the subprime and sovereign debt crises: An application to emerging markets. Emerging Markets Review 20, 23–41.

PT

Del Brio, E.B., Mora-Valencia, A., Perote, J. (2014b). Semi–nonparametric VaR forecasts for hedge funds during the recent crisis. Physica A 401, 330–343.

SC

RI

Diebold, F.X., Mariano, R.S. 1995. Comparing predictive accuracy. Journal of Business and Economic Statistics 13, 253-263. Embrechts, P., Wang, B., Wang, R. 2015. Aggregation-robustness and model uncertainty of regulatory risk measures. Finance and Stochastics 19 (4), 763-790.

NU

Emmer, S., Kratz, M., Tasche, D. 2015. What is the best risk measure in practice? A comparison of standard measures. Journal of Risk 18 (2), 31-60.

MA

Engle, R.F., 1982. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom. Econometrica, 50, 987-1007. Fernández, C., Steel M.F.J., 1998. On bayesian modeling of fat tails and skewness. Journal of the American Statistical Association 93, 359–371.

PT E

D

Fissler, T., Ziegel, J.F. 2016. Higher order elicitability and Osband’s principle. Annals of Stastics 44 (4), 1680-1707. Fissler, T., Ziegel, J.F., Gneiting, T. 2016. Expected Shortfall is jointly elicitable with value-at-risk: Implications for backtesting. Risk 29 (1) 58-61.

CE

Gallant, A. R., Nychka, D. W., 1987. Semi-Nonparametric maximum likelihood estimation. Econometrica 55, 363-390. Ghalanos, A. 2015. rugarch: Univariate GARCH models. R package version 1.3-6.

AC

Giot, P., Laurent, S. 2003. Market risk in commodity markets: a VaR approach. Energy Economics 25, 433-457. Gneiting, T. 2011. Making and evaluating point forecasts. Journal of the American Statistical Association 106, 746-762. Hume, N., Sanderson, H. Sheppard, D. 2016, April 28. China futures gripped as speculators rush in. The Financial Times. Retrieved from http://www.ft.com. Hung, J.-C., Lee, M.-C., Liu, H.-C. 2008. Estimation of value-at-risk for energy commodities via fat-tailed GARCH models. Energy Economics 30, 1173-1191. Jarrow, R., Rudd A., 1982. Approximate option valuation for arbitrary stochastic processes. Journal of Financial Economics 10, 347–369. Johnson, N.L., Kotz, S. 1970. Continuous univariate distributions, Vol. 1 and 2. Houghton Mifflin Co., Boston.

ACCEPTED MANUSCRIPT Jondeau, E., Rockinger, M. 2001. Gram-Charlier densities. Journal of Economic Dynamics and Control 25, 1457-1483. Jorion, P., 2006. Value at Risk: The New Benchmark for Managing Financial Risk, Third Edition, McGraw-Hill. Kellner, R., Rösch, D. 2016. Quantifying market risk with value-at-risk or expected shortfall? – Consequences for capital requirements and model risk. Journal of Economic Dynamics & Control 68, 45–63.

PT

Kendall, M., Stuart, A., 1977. The Advanced Theory of Statistics. Griffin&Co, London.

RI

Koch-Medina, P., Munari, C. 2016. Unexpected shortfalls of Expected Shortfall: Extreme default profiles and regulatory arbitrage. Journal of Banking and Finance 62, 141-151.

SC

Kou, S., Peng, X. 2014. On the measurement of economic tail risk. Preprint arXiv: 1401.4787.

NU

León, A., Mencía, J., Sentana, E., 2009. Parametric properties of semi-nonparametric distributions. Journal of Business and Economic Statistics 27, 176-192.

MA

Mauleón, I., Perote, J. 2000. Testing densities with financial data: An empirical comparison of the Edgeworth-Sargan density to the Student’s t. European Journal of Finance 6, 225-239. McNeil, A.J., Frey, R., Embrechts, P. 2015. Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press (Revised Edition).

D

Nelson, D.B. 1991. Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59, 347—370.

PT E

Newey, W.K., Powell, J.L. 1987. Asymmetric least squares estimation and testing. Econometrica 55, 819-847.

CE

Ñíguez, T.M., Perote, J., 2012. Forecasting heavy-tailed densities with positive Edgeworth and Gram-Charlier expansions. Oxford Bulletin of Economics and Statistics 74, 600-627.

AC

Ñíguez, T.M., Perote, J., 2016. Multivariate moments expansion density: Application of the dynamic equicorrelation model. Journal of Banking and Finance 72, S216-S232. Ñíguez, T.M., Perote, J., 2017. Moments expansion densities for quantifying financial risk. North American Journal of Economics and Finance 42, 53-69. Rockafellar, R.T., Uryasev, S. 2002. Conditional value-at-risk for general loss distributions. Journal of Banking and Finance 26 (7) 1443-1471. Singleton, K.J. 2013. Investor flows and the 2008 boom/bust in oil prices. Management Science 60 (2), 300-318. Steen, M., Westgaard, S., Gjolberg, O. 2015. Commodity value-at-risk modeling: comparing RiskMetrics, historic simulation and quantile regression. Journal of Risk Model Validation 9 (2), 49-78. Suh, S. 2015. Measuring sovereign risk contagion in the Eurozone. International Review of Economics and Finance 35, 45–65.

ACCEPTED MANUSCRIPT Tang, K., Xiong, W. 2012. Index investment and financialization of commodities. Financial Analysts Journal 68 (6), 54-74. Taylor, J.W. 2008. Estimating value at risk and expected shortfall using expectiles. Journal of Financial Econometrics 6 (2), 231-252. Yamai, Y., Yoshiba, T. 2005. Value-at-risk versus expected shortfall: A practical perspective. Journal of Banking and Finance 29, 997-1015. Youssef, M., Belkacem, L., Mokni, K. 2015. Value-at-Risk estimation of energy commodities: A long-memory GARCH-EVT approach. Energy Economics 51, 99-110.

AC

CE

PT E

D

MA

NU

SC

RI

PT

Ziegel, J.F. 2016. Coherence and elicitability. Mathematical Finance 26 (4), 901-918.

ACCEPTED MANUSCRIPT Table 1. Descriptive statistics of Commodity ETFs Gold 0.0244 0.0433 1.2642 (19.99) -9.1905 10.6974 -0.2567 6.0464

Silver 0.0045 0.0882 2.1821 (34.50) -19.8349 13.4733 -0.9751 7.6419

Oil -0.0731 -0.0171 2.1963 (34.73) -11.2996 9.1691 -0.1966 2.4764

Agriculture -0.0099 0.0000 1.2680 (20.05) -8.9990 7.6675 -0.2513 5.4450

Energy -0.0406 0.0000 1.7523 (27.71) -9.02391 8.8686 -0.1912 2.6507

Broad -0.0279 0.0000 1.3127 (20.75) -6.9341 6.6484 -0.3048 2.6222

PT

Mean Median Standard deviation Min Max Skewness Excess Kurtosis

RI

Annualized volatility in parentheses, estimated as daily standard deviation multiplied by the square root of 250 (i.e. assuming 250 trading days in a year).

 

    

AC



0.042 (0.022) 0.449 (0.000) -0.489 (0.000) 0.003 (0.115) 0.009 (0.373) 0.992 (0.000) 0.107 (0.000) 4.960 (0.000)

NU

MA

-0.013 (0.766) 0.044 (0.153) -0.014 (0.669) 0.051 (0.000) -0.016 (0.164) 0.971 (0.000) 0.203 (0.000)

CE



0.023 (0.000) 0.952 (0.000) -0.965 (0.000) 0.014 (0.000) -0.010 (0.236) 0.981 (0.000) 0.145 (0.00)

Oil Agriculture Panel A: Gaussian -0.038 -0.032 (0.248) (0.057) -0.725 0.490 (0.000) (0.000) 0.685 -0.538 (0.000) (0.000) 0.013 0.002 (0.000) (0.113) -0.051 0.007 (0.000) (0.344) 0.991 0.997 (0.000) (0.000) 0.101 0.114 (0.000) (0.000) Panel B: Student’s t -0.019 -0.028 (0.574) (0.024) -0.715 0.402 (0.000) (0.000) 0.672 -0.457 (0.000) (0.000) 0.008 -0.001 (0.000) (0.671) -0.047 0.000 (0.000) (0.958) 0.993 0.997 (0.000) (0.000) 0.101 0.107 (0.000) (0.000) 12.285 8.292 (0.000) (0.000)

D



Silver

PT E



Gold

SC

Table 2. Estimates of ARMA(1,1)-EGARCH(1,1) models

0.038 (0.242) -0.337 (0.000) 0.322 (0.000) 0.012 (0.000) 0.011 (0.287) 0.992 (0.000) 0.117 (0.000) 4.310 (0.000)

Energy

Broad

-0.029 (0.252) -0.858 (0.000) 0.819 (0.000) 0.008 (0.000) -0.046 (0.000) 0.993 (0.000) 0.098 (0.000)

-0.037 (0.068) -0.880 (0.000) 0.858 (0.000) 0.004 (0.008) -0.023 (0.002) 0.994 (0.000) 0.103 (0.000)

-0.004 (0.857) -0.837 (0.000) 0.795 (0.000) 0.003 (0.091) -0.042 (0.000) 0.995 (0.000) 0.101 (0.000) 9.371 (0.000)

-0.020 (0.301) -0.855 (0.000) 0.827 (0.000) 0.000 (0.870) -0.025 (0.004) 0.996 (0.000) 0.102 (0.000) 9.521 (0.000)

ACCEPTED MANUSCRIPT

Skew-ξ

   

AC

-0.025 (0.312) -0.825 (0.000) 0.781 (0.000) 0.004 (0.038) -0.043 (0.000) 0.995 (0.000) 0.104 (0.000) 9.894 (0.000) 0.897 (0.000)

-0.031 (0.033) -0.814 (0.000) 0.782 (0.000) 0.006 (0.690) -0.025 (0.003) 0.996 (0.000) 0.103 (0.000) 9.902 (0.000) 0.919 (0.000)

-0.030 (0.534) -0.858 (0.000) 0.818 (0.000) 0.009 (0.003) -0.049 (0.000) 0.993 (0.000) 0.028 (0.000) -0.034 (0.002) 0.116 (0.000)

-0.087 (0.017) -0.880 (0.000) 0.855 (0.000) 0.017 (0.000) -0.057 (0.000) 0.980 (0.000) 0.033 (0.000) -0.026 (0.024) 0.228 (0.000)

PT

CE



0.003 (0.000) 0.840 (0.000) -0.853 (0.000) 0.018 (0.000) -0.005 (0.621) 0.981 (0.000) 0.055 (0.000) -0.035 (0.004) 0.166 (0.000)

RI



SC



NU



MA



D



PT E



0.023 (0.262) 0.443 (0.000) -0.485 (0.000) 0.003 (0.129) 0.009 (0.386) 0.992 (0.000) 0.107 (0.000) 5.069 (0.000) 0.939 (0.000)

Panel C: Skewed Student’s t -0.034 -0.035 -0.032 (0.245) (0.270) (0.057) -0.277 -0.701 0.405 (0.000) (0.000) (0.000) 0.257 0.655 -0.463 (0.000) (0.000) (0.000) 0.011 0.009 -0.000 (0.000) (0.000) (0.704) 0.010 -0.047 0.000 (0.320) (0.000) (0.971) 0.992 0.993 0.997 (0.000) (0.000) (0.000) 0.118 0.103 0.109 (0.000) (0.000) (0.000) 4.341 13.008 8.299 (0.000) (0.000) (0.000) 0.894 0.918 0.970 (0.000) (0.000) (0.000) Panel D: GC -0.003 -0.038 -0.031 (0.925) (0.493) (0.052) 0.034 -0.726 0.491 (0.000) (0.000) (0.022) -0.024 0.684 -0.527 (0.000) (0.000) (0.010) 0.063 0.011 0.035 (0.000) (0.002) (0.000) -0.003 -0.056 -0.059 (0.848) (0.000) (0.004) 0.965 0.992 0.970 (0.000) (0.000) (0.000) 0.056 0.020 0.046 (0.000) (0.000) (0.000) -0.042 -0.022 -0.032 (0.000) (0.034) (0.008) 0.244 0.111 0.408 (0.000) (0.000) (0.000)

, and  are the parameters of the ARMA(1,1) model; , , γ and  are the parameters of the EGARCH(1,1) model;  and ξ are the parameters of the (skewed) Student’s t; d3 and d4 are the parameters of the GC model. Pvalues for the t-test in parentheses.

ACCEPTED MANUSCRIPT Table 3. Diagnostics of ARMA(1,1)-EGARCH(1,1)

LBSSR

LL AIC BIC HQC LBSR LBSSR

-3455.90 3.051 3.074 3.060 2.185 (0.139) 33.70 (0.000) -1389.89 1.232

Broad

-4146.57 3.658 3.675 3.664 0.079 (0.778) 1.006 (0.316)

-3497.14 3.086 3.103 3.092 0.030 (0.862) 0.303 (0.582)

-4117.24 3.633 3.653 3.640 0.194 (0.659) 0.930 (0.335)

-3468.68 3.062 3.082 3.069 0.006 (0.938) 0.222 (0.637)

-4111.05 3.628 3.651 3.637 0.340 (0.560) 0.753 (0.385)

-3464.98 3.059 3.082 3.068 0.070 (0.791) 0.257 (0.612)

-2045.83 1.809

-1434.25 1.271

PT

Energy

RI

SC

NU

LL AIC BIC HQC LBSR

MA

LBSSR

-3458.40 3.053 3.073 3.060 1.967 (0.161) 33.83 (0.000)

PT E

LL AIC BIC HQC LBSR

CE

LBSSR

-3548.78 3.131 3.149 3.138 0.051 (0.821) 7.829 (0.005)

1.255

2.314

2.287

1.157

1.832

1.294

1.240

2.300

2.273

1.142

1.818

1.279

2.965 (0.085) 28.751 (0.000)

0.310 (0.577) 47.300 (0.150)

0.166 (0.683) 3.900 (0.048)

0.794 (0.373) 39.519 (0.271)

3.228 (0.0724) 83.646 (0.413)

AC

LL AIC BIC HQC LBSR

Silver Oil Agriculture Panel A: Gaussian -4738.13 -4648.64 -3323.15 4.179 4.100 2.933 4.197 4.118 2.950 4.185 4.106 2.939 0.009 0.194 0.696 (0.923) (0.659) (0.404) 10.31 1.679 0.485 (0.001) (0.195) (0.486) Panel B: Student’s t -4647.04 -4632.68 -3287.33 4.099 4.087 2.902 4.120 4.107 2.922 4.107 4.094 2.909 3.595 0.286 1.461 (0.058) (0.593) (0.227) 30.40 1.758 0.720 (0.000) (0.185) (0.396) Panel C: Skewed Student’s t -4638.37 -4628.84 -3286.86 4.093 4.084 2.903 4.115 4.107 2.925 4.101 4.093 2.911 4.077 0.461 1.750 (0.043) (0.497) (0.186) 30.18 1.552 0.624 (0.000) (0.213) (0.430) Panel D: GC -2593.33 -2562.65 -1278.65 2.292 2.265 1.134

D

Gold

0.255 (0.613) 21.690 (0.000)

LL: Loglikelihood, AIC: Akaike information criterion, BIC: Bayesian (Schwartz) information criterion, HQC: Hannan-Quinn information criterion, LBSR: Ljung Box Standardized Residuals, LBSSR: Ljung Box Squared Standardized Residuals (1 lag). P-values for testing H0: no serial correlation in parentheses.

ACCEPTED MANUSCRIPT Table 4. Backtesting 99%-VaR for Commodity ETFs returns

Agriculture 24 (0.154)

Energy 38 (0.000)

Broad 33 (0.001)

18 (0.945) 21 (0.445)

29 (0.013) 17 (0.864)

29 (0.013) 15 (0.506)

5 (0.000)

24 (0.154)

17 (0.864)

PT

Expected number of exceptions = 18 Backtesting period: December 2008 – January 2016 1771 days Gold Silver Oil ARMA-EGARCH43 (0.000) 38 (0.000) 35 (0.000) Gaussian ARMA-EGARCH-t 28 (0.023) 27 (0.039) 27 (0.039) ARMA-EGARCH25 (0.101) 15 (0.506) 15 (0.506) skewed-t ARMA-EGARCH-GC 21 (0.445) 21 (0.445) 16 (0.678) P-values for Bernoulli Coverage test in parentheses.

Agriculture 49 (0.479)

Energy 58 (0.046)

Broad 53 (0.197)

45 (0.912) 40 (0.508)

56 (0.086) 27 (0.005)

53 (0.197) 33 (0.072)

22 (0.000)

46 (0.794)

30 (0.021)

MA

NU

SC

Expected number of exceptions = 44 Backtesting period: December 2008 – January 2016 1771 days Gold Silver Oil ARMA-EGARCH62 (0.011) 57 (0.063) 62 (0.011) Gaussian ARMA-EGARCH-t 62 (0.011) 56 (0.086) 59 (0.033) ARMA-EGARCH42 (0.727) 34 (0.103) 39 (0.412) skewed-t ARMA-EGARCH-GC 42 (0.727) 41 (0.614) 44 (0.967) P-values for the Bernoulli Coverage test in parentheses.

RI

Table 5. Backtesting 97.5%-VaR for Commodity ETFs returns

Table 6. T-test for 97.5%-ES for Commodity ETFs returns

AC

CE

PT E

D

Mean of violation residual is zero Backtesting period: December 2008 – January 2016 1771 days Gold Silver ARMA-EGARCH4.443 (0.000) 3.455 (0.001) Gaussian ARMA-EGARCH-t -4.586 (0.000) -4.014 (0.000) ARMA-EGARCH1.756 (0.087) 1.206 (0.237) skewed-t ARMA-EGARCH-GC 1.855 (0.071) 1.856 (0.071) P-values for the t-test test in parentheses.

Oil 3.004 (0.004)

Agriculture 2.203 (0.032)

Energy 3.539 (0.001)

Broad 3.410 (0.001)

-1.292 (0.201) 0.342 (0.734)

-3.177 (0.003) 0.951 (0.347)

-1.785 (0.079) 1.561 (0.130)

-1.209 (0.232) 1.541 (0.133)

1.179 (0.245)

-0.856 (0.401)

3.027 (0.004)

1.288 (0.208)

Table 7. Average ratio 97.5%-ES to 99%-VaR

Backtesting period: December 2008 – January 2016 1771 days Gold Silver Oil

Gaussian Student’s t Skewed-t GC

1.005 (0.000) 1.427 (0.294) 1.160 (0.104) 0.994 (0.004)

1.005 (0.000) 1.495 (0.340) 1.160 (0.103) 0.994 (0.022)

Standard deviation in parentheses.

1.005 (0.000) 1.141 (0.227) 1.160 (0.103) 1.000 (0.004)

Agriculture 1.005 (0.000) 1.154 (0.060) 1.160 (0.104) 1.000 (0.006)

Energy 1.005 (0.000) 1.160 (0.104) 1.161 (0.104) 0.999 (0.004)

Broad 1.005 (0.000) 1.157 (0.102) 1.161 (0.104) 0.999 (0.005)

ACCEPTED MANUSCRIPT

NU

SC

RI

PT

Figure 1. Commodity ETFs prices

AC

CE

PT E

D

MA

Figure 2. Commodity ETFs returns

ACCEPTED MANUSCRIPT Appendix A. ES computation for GC ES for the GC Type A distribution can be obtained as the conditional expectation of variable

, that is conditioned on the fact that this variable had exceeded a given . Therefore,

PT E

D

MA

NU

SC

RI

PT

quantile

By direct application of Lemma 1 and Lemma 2 in Ñíguez and Perote (2012) and taking .

CE

into account that

Then, ES is equal to the latter expression divided by the probability of the conditioned

AC

event, that is,

Consequently, for the GC expansion expanded to the eighth term and with unique (odd) skewness parameter and denoting

,

as the

ACCEPTED MANUSCRIPT

AC

CE

PT E

D

MA

NU

SC

RI

PT

.

ACCEPTED MANUSCRIPT Appendix B. Data description Ticker ETF Commodity

Description

GLD

SPDR Gold Shares is an investment fund incorporated in the USA. The investment objective of the Trust is for the Shares to reflect the performance of the price of gold bullion, less the Trust’s expenses. The Trust holds gold and is expected from time to time to issue Baskets in exchange for deposits of gold and to distribute gold in connection with redemptions of Baskets. The reference benchmark is the (London Bullion Market Association) LBMA Gold Price PM.

Silver

SLV

iShares Silver Trust is a trust formed to invest in silver. The assets of the trust consist primarily of silver held by the custodian on behalf of the trust. The objective of the trust is for the shares to reflect the price of silver owned by the trust, less the trust’s expenses and liabilities. The reference benchmark is the LBMA Silver Price (August 2014).

Oil

USO

United States Oil Fund LP is a Delaware limited partnership incorporated in the USA. The Fund’s objective is to have changes in percentage terms of its unit's net asset value reflect the changes of the price of WTI Crude Oil delivered to Cushing, Oklahoma, as measured by changes in percentage terms of the price of the WTI Crude Oil futures contract on the NYMEX. The reference benchmark is the Benchmark Oil Futures Contract (price of a specified short-term futures contract on light, sweet crude oil).

Agriculture

DBA

PT E

D

MA

NU

SC

RI

PT

Gold

Energy

Broad

AC

CE

PowerShares DB Agriculture Fund is an exchange-traded fund incorporated in the USA. The Fund's objective is to reflect the performance of the Deutsche Bank Index Quant Group’s (DBIQ) Diversified Agriculture Index Excess Return plus the interest income from the Fund’s holdings on primarily US Treasury securities and money market income less Fund’s expenses.

DBE

PowerShares DB Energy Fund is an exchange-traded fund incorporated in the USA. The Fund’s objective is to track the DBIQ Optimum Yield Energy Index Excess Return plus the interest income from the Fund’s holdings of primarily US Treasury securities and money market income less the Fund's expenses.

DBC

PowerShares DB Commodity Index Tracking Fund is an investment fund incorporated in the USA. The Fund’s objective is to reflect the performance of the DBIQ Optimum Yield Diversified Commodity Index Excess plus the interest income

ACCEPTED MANUSCRIPT from the Fund’s holdings of primarily US Treasury securities and money market income less the Fund’s expenses. The Fund invests in commodities such as Light, Sweet Crude Oil, Heating Oil, Aluminum, Gold, Corn and Wheat.

AC

CE

PT E

D

MA

NU

SC

RI

PT

Source: Bloomberg LP., spdrs.com, ishares.com, uscfinvestments.com, invesco.com

ACCEPTED MANUSCRIPT Appendix C. Expected value of

with pdf

, then

SC

stands for standard normal density.

MA

NU

where

RI

PT

Let

for EGARCH-GC

D

The integrals in bold comprises the expected value of the absolute value of random ) which is equal to

PT E

variable of normal distribution (with 0 mean and variance

AC

CE

(see Johnson and Kotz, 1970 for more details).

Evaluating the definite integrals with

, and

,

the first and third integral are equal to 0, the second integral is equal to 1, while the fourth integral is equal to −1. Then,

Therefore

ACCEPTED MANUSCRIPT

Finally,

CE

PT E

D

MA

NU

SC

RI

PT

follows a standard GC distribution.

AC

where

ACCEPTED MANUSCRIPT Appendix D. R Code for parameter estimation of ARMA-EGARCH with GC innovations

SC NU MA

#initial values T <- length(dato) u <- rep(0,T) z <- rep(0,T) xpred <- rep(0,T) h <- rep(0,T) u[1] <- dato[1] - mean(dato) h[1] <- u[1]^2 z[1] <- u[1]/sqrt(h[1]) xpred[1] <- dato[1] LL <- 0 sigma <- 0

RI

PT

armaegarchgc4 <- function(dato, phi0_0,phi1_0,phi2_0,omega0,alfa0,beta0,gama0) { loglik <- function(param){ phi0 <- param[1] phi1 <- param[2] phi2 <- param[3] omega <- param[4] alfa <- param[5] beta <- param[6] gama <- param[7] d3 <- param[8] d4 <- param[9]

PT E

D

eps <-1e-16 for (t in 2:T){ xpred[t] = phi0 + phi1*dato[t-1] + phi2*u[t-1] u[t] = dato[t] - xpred[t] h[t] = exp( omega + alfa*z[t-1] + gama*( abs(z[t-1]) - 0.7978846*(1-d4) beta*log(h[t-1]) )

) +

AC

} -LL }

CE

z[t] = u[t]/sqrt(h[t]) sigma <- sqrt(h[t]) LL <- LL - 0.5*log(h[t]+eps) -0.5*(u[t]^2)/h[t] + log(1 + d3*((u[t]/sigma)^33*(u[t]/sigma)) + d4*((u[t]/sigma)^4-6*(u[t]/sigma)^2+3)+eps)

eps <-1e-16

resulta3 <- nlminb(objective=loglik, start=c(phi0_0,phi1_0,phi2_0,omega0,alfa0,beta0,gama0,eps,eps), lower=c(-Inf,1,-1,eps,eps,eps,-Inf,-Inf,-Inf),upper=c(Inf,1,1,Inf,Inf,Inf,Inf,Inf,Inf)) resum<-summary(resulta3)

#Parameter estimates phi0<-as.numeric(resulta3$par[1]) phi1<-as.numeric(resulta3$par[2]) phi2<- as.numeric(resulta3$par[3]) omega<-as.numeric(resulta3$par[4]) alfa<-as.numeric(resulta3$par[5]) beta<-as.numeric(resulta3$par[6])

#mu #ar1 #ma1

ACCEPTED MANUSCRIPT gama<-as.numeric(resulta3$par[7]) d3<-as.numeric(resulta3$par[8]) d4<-as.numeric(resulta3$par[9]) LogLik <- as.numeric(-resulta3$objective)

AIC <- -2*LogLik + 2*9 pars <- cbind(phi0,phi1,phi2,omega,alfa,beta,gama,d3,d4) crits <- cbind(AIC, LogLik)

AC

CE

PT E

D

MA

NU

SC

RI

PT

partscr <- cbind(pars,crits) print(partscr) }

ACCEPTED MANUSCRIPT Appendix E. Parameter dynamics for all distributions E.1. Student’s-t shape (degrees of freedom) parameter Agriculture ETF

PT

05/01/2007 1/10/2008 1/06/2010 1/02/2012 1/10/2013

SC

05/01/2007 1/10/2008 1/06/2010 1/02/2012 1/10/2013

RI

5

5

10

15

20

10 15 20 25 30 35 40

Gold ETF

date

date

Energy ETF

D

PT E

05/01/2007 1/10/2008 1/06/2010 1/02/2012 1/10/2013

40 20

5

MA

10

60

15

NU

80

20

100

Silver ETF

05/01/2007 1/10/2008 1/06/2010 1/02/2012 1/10/2013

date

date

05/01/2007 1/10/2008 1/06/2010 1/02/2012 1/10/2013

40

60

80

100

Broad ETF

20

CE AC

0

20

40

60

80

100

Oil ETF

05/01/2007 1/10/2008 1/06/2010 1/02/2012 1/10/2013

ACCEPTED MANUSCRIPT E.2. Skewed-t shape (degrees of freedom) parameter Agriculture ETF

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

PT RI

5

5

10

15

10 15 20 25 30 35 40

Gold ETF

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

date

50

30

40

NU

25

10 0

MA D

PT E

20

30

20 15 10 5

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

date

date

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

20

30

40

50

60

Broad ETF

10

CE AC

10

20

30

40

50

60

Oil ETF

0

date

Energy ETF

60

SC

Silver ETF

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

ACCEPTED MANUSCRIPT

Agriculture ETF 1.2

Gold ETF

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

date

date

Energy ETF

PT E CE AC

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

date

Broad ETF 0.80 0.85 0.90 0.95 1.00

date

Oil ETF

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

D

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

0.8

MA

1.0

NU

1.2

SC

1.4

0.75 0.80 0.85 0.90 0.95 1.00

Silver ETF

RI

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

PT

0.8

0.80

0.9

0.90

1.0

1.1

1.00

1.10

E.3. Skewed-t skew parameter

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

ACCEPTED MANUSCRIPT E.4. Gram-Charlier d3 parameter Agriculture ETF

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

date

SC 0.10 0.15 0.00 0.05

MA D

PT E

-0.10

0.05 -0.05 -0.15

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

date

date

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

-0.05

0.00

0.05

Broad ETF

-0.10

CE AC

-0.05

0.00

0.05

Oil ETF

-0.10

date

Energy ETF

NU

0.15

Silver ETF

PT RI

-0.15

-0.05

-0.10 -0.05 0.00

0.05

0.05

0.10

0.15

Gold ETF

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

ACCEPTED MANUSCRIPT E.5. Gram-Charlier d4 parameter Agriculture ETF

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

date

0.15

SC 0.10 0.05

MA D

PT E

0.00

0.10 0.05 0.00

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

date

0.04

0.08

Broad ETF

0.00

CE AC

0.04

0.08

0.12

Oil ETF

0.12

date

0.00

date

Energy ETF

NU

0.15

Silver ETF

PT

0.04 0.00

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

RI

0.05 0.00

0.10

0.08

0.15

Gold ETF

05/01/2007 9/10/2008 1/07/2010 5/03/2012 6/11/2013

ACCEPTED MANUSCRIPT Appendix F. Pairwise Diebold Mariano test for 99%-VaR Panel F.1. Gold ETF returns Model A →

ARMAEGARCH-t

ARMA-EGARCHskewed-t

ARMAEGARCH-GC

Model B ↓ ARMA-EGARCH-normal

-2.346 (0.009)

ARMA-EGARCH-t

-1.507 (0.145)

-1.912 (0.056)

0.089 (0.535)

-1.161 (0.123) -0.931 (0.176)

PT

ARMA-EGARCH- skewed-t

Panel F.2. Silver ETF returns -1.600 (0.055)

4.304 (0.000)

RI

ARMA-EGARCH-normal

4.151 (0.000)

ARMA-EGARCH- skewed-t

SC

ARMA-EGARCH-t

-0.303 (0.381) 0.071 (0.528) -2.647 (0.004)

Panel F.3. Oil ETF returns 1.311 (0.095)

3.828 (0.000)

NU

ARMA-EGARCH-normal ARMA-EGARCH-t

-1.311 (0.095)

ARMA-EGARCH- skewed-t

-2.129 (0.017) -1.311 (0.095) -4.363 (0.000)

MA

Panel F.4. Broad ETF returns ARMA-EGARCH-normal

-2.996 (0.001)

ARMA-EGARCH- skewed-t

-1.682 (0.092)

-2.218 (0.013)

-0.509 (0.305) 1.564 (0.108)

D

ARMA-EGARCH-t

-2.532 (0.006)

PT E

Panel F.5. Agriculture ETF returns

ARMA-EGARCH-normal ARMA-EGARCH-t

-1.482 (0.069)

-2.367 (0.009)

-2.212 (0.013)

-0.700 (0.242)

-1.130 (0.129)

CE

ARMA-EGARCH- skewed-t

Panel F.6. Energy ETF returns

ARMA-EGARCH-normal ARMA-EGARCH-t

AC

-0.944 (0.173)

ARMA-EGARCH- skewed-t

-1.826 (0.068)

-1.825 (0.068)

1.438 (0.075)

1.697 (0.090)

2.620 (0.004) 2.620 (0.004)

The table shows the Diebold-Mariano statistics for different models. Bold figures indicate Model A is preferred than Model B, figures in red indicate the opposite. Otherwise both models present the same performance. P-values in parentheses.

ACCEPTED MANUSCRIPT Appendix G. Pairwise Diebold Mariano test for 97.5%-VaR Panel D.1. Gold ETF returns Model A →

ARMAEGARCH-t

ARMA-EGARCHskewed-t

ARMAEGARCH-GC

Model B ↓ ARMA-EGARCH-normal

-0.268 (0.394)

ARMA-EGARCH-t

-1.170 (0.121)

-2.064 (0.019)

-1.342 (0.089)

1.566 (0.059) -0.691 (0.245)

PT

ARMA-EGARCH- skewed-t

Panel D.2. Silver ETF returns 0.633 (0.737)

5.172 (0.000)

RI

ARMA-EGARCH-normal

4.871 (0.000)

ARMA-EGARCH- skewed-t

SC

ARMA-EGARCH-t

-0.2285 (0.409) -0.356 (0.361) -3.488 (0.000)

Panel D.3. Oil ETF returns 1.309 (0.095)

4.997 (0.000)

NU

ARMA-EGARCH-normal ARMA-EGARCH-t

-1.309 (0.095)

ARMA-EGARCH- skewed-t

-2.685 (0.004) -1.309 (0.095) -5.619 (0.000)

MA

Panel D.4. Broad ETF returns ARMA-EGARCH-normal

-2.645 (0.004)

ARMA-EGARCH- skewed-t

-1.777 (0.038)

-3.244 (0.001)

-1.035 (0.150) 1.990 (0.023)

D

ARMA-EGARCH-t

-3.239 (0.001)

PT E

Panel D.5. Agriculture ETF returns

ARMA-EGARCH-normal ARMA-EGARCH-t

-1.133 (0.129)

-3.508 (0.000)

-2.604 (0.005)

-2.752 (0.003)

-1.788 (0.074)

CE

ARMA-EGARCH- skewed-t

Panel D.6. Energy ETF returns

ARMA-EGARCH-normal ARMA-EGARCH-t

AC

0.305 (0.620)

ARMA-EGARCH- skewed-t

-1.555 (0.060)

-1.555 (0.060)

1.068 (0.143)

1.738 (0.082)

1.866 (0.062) 1.866 (0.062)

The table shows the Diebold-Mariano statistics for different models. Bold figures indicate Model A is preferred than Model B, figures in red indicate the opposite. Otherwise both models present the same performance. P-values in parentheses.

ACCEPTED MANUSCRIPT Appendix H. Pairwise Diebold Mariano test for 97.5%-ES Panel E.1. Gold ETF returns Model A →

ARMAEGARCH-t

ARMA-EGARCHskewed-t

ARMAEGARCH-GC

Model B ↓ ARMA-EGARCH-normal

-3.092 (0.002)

ARMA-EGARCH-t

-2.759 (0.006)

-2.853 (0.004)

0.446 (0.655)

0.358 (0.719) -0.143 (0.885)

PT

ARMA-EGARCH- skewed-t

Panel E.2. Silver ETF returns -3.071 (0.002)

3.314 (0.001)

RI

ARMA-EGARCH-normal

3.389 (0.001)

ARMA-EGARCH- skewed-t

SC

ARMA-EGARCH-t

0.977 (0.328) 1.025 (0.305) 0.376 (0.706)

Panel E.3. Oil ETF returns 1.309 (0.190)

2.895 (0.004)

NU

ARMA-EGARCH-normal ARMA-EGARCH-t

-1.309 (0.190)

ARMA-EGARCH- skewed-t

-2.502 (0.012) -1.309 (0.190) -4.083 (0.000)

MA

Panel E.4. Broad ETF returns ARMA-EGARCH-normal

-2.188 (0.029)

ARMA-EGARCH- skewed-t

-2.205 (0.027)

-2.706 (0.007)

-1.922 (0.054) 1.917 (0.055)

D

ARMA-EGARCH-t

-2.339 (0.019)

PT E

Panel E.5. Agriculture ETF returns

ARMA-EGARCH-normal ARMA-EGARCH-t

-0.954 (0.340)

-2.608 (0.009)

-2.785 (0.005)

-1.400 (0.161)

-1.439 (0.150)

CE

ARMA-EGARCH- skewed-t

Panel E.6. Energy ETF returns

ARMA-EGARCH-normal ARMA-EGARCH-t

AC

0.877 (0.380)

ARMA-EGARCH- skewed-t

-2.099 (0.035)

-2.099 (0.035)

1.535 (0.124)

0.638 (0.523)

2.026 (0.042) 2.026 (0.042)

The table shows the Diebold-Mariano statistics for different models. Bold figures indicate Model A is preferred than Model B, figures in red indicate the opposite. Otherwise both models present the same performance. P-values in parentheses.

ACCEPTED MANUSCRIPT Highlights

Commodity ETFs present high volatilities and accurate risk measures must be provided Gram-Charlier (GC) distribution provides accurate risk measures for commodity ETFs Recent Basel Accords propose to replace 99%-VaR by 97.5%-Expected Shortfall (ES) Backtesting ES under GC innovations is possible and we provide a closed-form for it

AC

CE

PT E

D

MA

NU

SC

RI

PT

Replacing 99%-VaR by 97.5%-ES is fair for the GC but not for the (skewed) Student’s t