Measuring Value-at-Risk and Expected Shortfall of crude oil portfolio using extreme value theory and vine copula

Accepted Manuscript Measuring value-at-risk and expected shortfall of crude oil portfolio using extreme value theory and vine copula Wenhua Yu, Kun Yang, Yu Wei, Likun Lei

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Received date : 20 April 2017 Revised date : 22 July 2017 Please cite this article as: W. Yu, K. Yang, Y. Wei, L. Lei, Measuring value-at-risk and expected shortfall of crude oil portfolio using extreme value theory and vine copula, Physica A (2017), http://dx.doi.org/10.1016/j.physa.2017.08.064 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.

*Highlights (for review)

Highlights

(1) VaR and ES of a portfolio consists of various oil assets are calculated. (2) Marginal distribution of oil assets are modeled by GARCH-type models and extreme value theory. (3) Complicated dependences among oil assets are modeled by various R-vine copulas.

Measuring Value-at-Risk and Expected Shortfall of crude oil portfolio using extreme value theory and vine copula ∗

Wenhua Yuaa , Kun Yanga , Wei Yub,c , Likun Leic a b c

Commercial College, Chengdu University of Technology, Chengdu, China

School of Finance, Yunnan University of Finance and Economics, Kunming, China

School of Economics and Management, Southwest Jiaotong University, Chengdu,China

∗

Corresponding author

E-mail: [email protected] Tel: +86-18682551075 Corresponding Address: 237 Longquan Road, Kunming, Yunnan, China

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Measuring Value-at-Risk and Expected Shortfall of crude oil portfolio using extreme value theory and vine copula

Abstract Volatilities of crude oil price have important impacts on the steady and sustainable development of world real economy. Thus it is of great academic and practical significance to model and measure the volatility and risk of crude oil markets accurately. This paper aims to measure the Value-at-Risk (VaR) and Expected Shortfall (ES) of a portfolio consists of four crude oil assets by using GARCH-type models, extreme value theory (EVT) and vine copulas. The backtesting results show that the combination of GARCH-type-EVT models and vine copula methods can produce accurate risk measures of the oil portfolio. Mixed R-vine copula is more flexible and superior to other vine copulas. Different GARCH-type models, which can depict the long-memory and/or leverage effect of oil price volatilities, however offer similar marginal distributions of the oil returns.

Keywords: Crude oil markets; Extreme value theory; Mixed R-vine copula; Value-at-Risk; Expected shortfall; Monte Carlo simulation

1. Introduction Since June 2014, the international crude oil price continues to decline. The West

1

Texas light crude oil (WTI) price falls by nearly 75%. As important raw material, crude oil has been widely used in various industry productions and thus the sharp fluctuations in oil prices will have a profound impact on the development of the real economy. In addition, crude oil is also playing an important role as financial asset, and the huge fluctuations in oil price will furthermore cause the large volatility in financial markets. Therefore modeling volatility and measuring risk of crude oil markets are of great importance for investors and policy makers. On the other hand, with the increasing uncertainty of global political and economic situations, the dependence among different crude oil markets is becoming more and more complex. Therefore investors may want to allocate their investment among different crude oil assets as a portfolio to diversify the unsystematic oil market risks. Before doing investment diversification, one need understand clearly the dependence of these oil assets, and thus can accurately measure the market risk of oil portfolio. After the Basel Accord II, Value-at-Risk (VaR) has become the standard method of market risk measurement [1]. VaR answers the following question: “What loss level is such that it will only be exceeded p × 100% of the time in the next K trading days?”. However, the VaR measurement does not always meet the requirement of subadditivity. If the subadditivity is not satisfied, portfolio investment cannot effectively disperse risk. Besides, VaR cannot be used for depicting the extreme risk profile beyond itself, while extreme risk often requires more attentions. On the other hand, Artzner et al. [2] propose the Expected Shortfall (ES) method to meet the requirements of subadditivity and "coherent risk measure", which is more consistent

2

with the economic meaning of market risk. This paper aims to measure the VaR and ES of a crude oil assets portfolio accurately. Specifically, first we model the volatilities of four major oil markets, i.e., WTI, Brent, Dubai and Cinta, using GARCH-type models. Second the conditional returns of individual oil assets are fitted by distributions based on extreme value theory. Third we employ a range of vine copulas to describe the complicated dependences among these oil assets. Then a Monte Carlo simulation method is employed to calculate the VaR and ES of the oil portfolio. Finally an unconditional coverage method and a bootstrap test are used to backtest the accuracy of the proposed models. Compared with the risk assessment of single asset, the risk measurement of portfolio needs to make clear the interdependence among portfolio assets first. In the existing literature, Behmiri et al. [3] employ a Granger causality test to study the linkage among the crude oil price, the crude oil consumption, the US dollar exchange rate and the economic growth. It is found that crude oil price is the Granger cause of crude oil consumption and economic growth. Chen et al. [4] investigate the impacts of oil price shocks on the bilateral exchange rates of the U.S. dollar against currencies using Vector Autoregression (VAR) method, and point out oil price shocks can explain about 10%-20% of long-term variations in exchange rates. Mensi et al. [5] use VAR-BEKK-GARCH and VAR-DCC-GARCH models to study the relationship among four types of petroleum products and grain products. They find a significant volatility spillover effect between the oil market and the grain market. Singhal et al. [6]

3

discuss the co-movements between Brent crude oil price and Indian stock market. The results show that the direct volatility spillover from oil market to Indian stock market is not significant. In the above literature, Granger causality test, VAR and multivariate GARCH model are commonly used in the research of multi-market relativity. However, it should be noted that these methods cannot accurately describe the non-linear dependence of financial markets [7, 8]. Furthermore the multivariate GARCH model requires variables studied following a specific multivariate distribution, which is apparent too restrictive to be fulfilled in real situations. In recent years, more and more studies introduce copula methods to describe the nonlinear dependence among financial assets [9]. But the traditional binary copula faces the problem of a “dimensional disaster” while the multivariate copula lacks accuracy and flexibility. To address the problems discussed above, Joe [10] first proposes that a multi-variable joint distribution could be decomposed into some pair-copula construction modules. Based on that, Bedford et al [11, 12] introduce the "vine" graph theory into describing the logical decomposition structure of variables and propose the Regular vine (R-vine) model. Similarly, using pair-copula method, Aas et al. [13] construct the Canonical vine (C-vine) which includes a central variable and the Drawable vine (D-vine) where variables’ concentration is weak. Existing studies have shown that vine copula model is superior to the traditional methods, such as historical simulation, variance-covariance, DCC-GARCH and multivariate copula models [14, 15]. Therefore it is widely used in the academic research and practical application of

4

financial risk management. For example, Weiß et al. [16] employ C-vine, D-vine, Mixed C-vine and Mixed D-vine models to measure VaR of multi-portfolio, and prove the flexibility of Mixed vine; Koliai [17] compares the risk measurement capability of Mixed R-vine, Mixed C-vine and Mixed D-vine on various stock markets, foreign exchange markets and commodity markets, and find that R-vine can achieve better measurement result. However, there is little literature on comparing risk measurement capability of different R-vine copulas. Furthermore no research is found to focus on measurement of both portfolio’s VaR and ES using R-vine copula method especially in the fields of crude oil assets. Furthermore the existing researches show that financial and oil price returns usually show some stylized facts such as conditional fat-tailed and skewed distribution, as well as volatility clustering and leverage effect [18]. Capturing these stylized facts can help to construct dependence structure of crude oil assets accurately and improve the effect of risk measurement. Ayusuk et al. [19] and Reboredo et al. [20] employ simple GARCH model to characterize the volatility clustering of financial series, and then use vine copula method to establish the dependency structure among multi-assets. While it is necessary to point out that the long memory and leverage effect are rarely included in multi-assets risk analysis framework. Thus we argue that FIGARCH and FIEGARCH models, which can depict long memory and leverage effect in oil volatility, may improve the risk measurement efficiency. In the dependence construction of different assets, another important question is the measure of extreme correlation. The tail risk of financial assets often represents

5

huge losses that may occur, and extreme dependence relation indicates the possibility that financial assets prices will rise or fall sharply at the same time, which requires special attention of investors. GARCH-type models are widely used to construct marginal distribution. However, it doesn’t model tail distribution directly [21] and it is necessary to assume asset distribution beforehand. Therefore extreme value theory (EVT) is usually adopted to depict extreme risk by tail modeling, which is important for risk measurement [22]. In order to accurately describe extreme dependence relation between crude oil assets, we will introduce extreme value theory to build marginal distributions. Thus, under the background of violent fluctuation in crude oil markets, this paper taking West Texas light (WTI), Brent, Dubai and Cinta crude oil as example, tries to measure the VaR and ES of a portfolio with the four oil assets. Firstly, marginal distributions of crude oil returns are constructed by the combination of several GARCH-type models (GARCH, FIGARCH and FIEGARCH) and generalized Pareto distribution (GPD) raised from extreme value theory. Secondly the complicated dependences among those crude oil assets are depicted by using three different R-vine copulas, including Mixed R-vine, R-vine all Gumbel and R-vine all Joe. Thirdly the VaR and ES of the oil assets portfolio are calculated through Monte Carlo simulations. Lastly the VaR and ES measured by different risk models are backtested by an unconditional coverage test and a bootstrap method, respectively. The remainder of this paper is organized as follows: Section 2 introduces the marginal distribution model, R-vine copula model, portfolio VaR and ES calculating

6

with Monte Carlo simulation and backtesting methods; Section 3 reports the empirical results; Section 4 concludes this paper.

2. Methodology 2.1. Marginal distribution modeling 2.1.1. GARCH-type models In financial econometric studies, we often assume asset return rt satisfies the following equation:

rt  t   t ,

(1)

where  t is the conditional mean of the return.  t is the residual and can be modeled as  t   t zt , in which  t2 is the conditional variance and the innovation item zt ~i.i.d . D(0,1) . At the same time, considering the fact that historical return has an influence on current return, we use the Autoregressive (AR) model with p orders of lag to describe the autocorrelation of financial asset returns. Then Eq. (1) can be expressed as: p

rt  a0  ai rt i   t .

(2)

i 1

On the basis of ARCH model, Bollerslev [23] propose a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model to describe the volatility clustering in financial returns. The GARCH (p, q) model is defined as: p

q

i 1

j 1

 t2    i t2i   j t2 j ,

(3)

where  t2 i is the conditional variance at time t  i and  t  j is the residual at time 7

t  j .  j and i are the coefficients of ARCH and GARCH items. Existing researches furthermore suggest that financial returns show a slower rate of decay after a volatility shock than a simple GARCH model indicates [24]. Some authors suggest that this long memory in volatility of financial returns may be better characterized by a fractionally integrated GARCH (FIGARCH) model. Baillie et al. [25] propose the FIGARCH (p, d, q) model as follows: 1

1

 t2   1    L   {1  1    L    L 1  L  } t2 , d

(4)

q

p

where   L   1   i L ，   L     j Lj , L is the lag operator, and d is the i

i 1

j 1

fractional order used to reflect the long memory. If d  0 , the FIGARCH model degrades into the simple GARCH model. Besides long-memory effect, leverage effect is another stylized fact of asset volatilities, which is characterized by the different magnitudes of volatility caused by good news and bad news [26]. In order to model this asymmetric leverage effect, Bollerslev et al. [27] define a Fractional Integrated Exponential Generalized Autoregressive Conditional Heteroscedasticity (FIEGARCH) model as: log( t2 )      L 

1

1  L 

d

[1   ( L)]g ( zt 1 ),

(5)

where g ( zt )   zt   [ zt  E zt ] ,  is the scale coefficient,  is the leverage parameter, and   0 states that leverage effect exists.

2.1.2 Extreme value theory and generalized Pareto distribution

The advantage of EVT method is its capability of describing separately the lower and upper tails of a distribution beyond a threshold. First, the conditional tail 8

distribution function Fu over a tail threshold u can be defined as: Fu  y   p  z  u  y | z  u  

F u  y   F u  , 0  y  z F  u, 1  F u 

(6)

where z is the conditional return filtered by the volatility estimates of some volatility models, e.g. the GARCH-type models mentioned above. y  z  u is the extreme statistic, zF is the end of this distribution. According to Pickands [28] and Balkema et al. [29], given a sufficiently large tail threshold, the distribution beyond threshold u can be modeled by the generalized Pareto distribution (GPD). Its probability density function can be expressed as: 1      1  1  y ,   0,    G ,   y       y    1  e  ,   0,

(7)

where  is the scale parameter, and   0.  is the tail parameter.   0 states that this tail distribution is a fat-tailed distribution;  =0 indicates a Gaussian distribution, an exponential distribution or a logarithmic distribution;   0 states a short tail distribution. For any z  u , if y  z  u , we can get from Eq. (6) and Eq. (7):

F  z   1  F  u   G ,  z  u   F u  .

(8)

A simulation method can be used to estimate F  u  

n  Nu , where n is the n

sample size, N u is the observations beyond tail threshold. Bring F  u  into Eq. (8), the tail estimation can be calculated as:

9

1   N    1  u 1   z  u  ,   0，    n    Fˆ  z    z u  N  1  u e  ,   0.  n 

(9)

2.2 R-vine copula model The different hierarchical decomposition structures and node connection orders will result in different types of vine copula models. Among them, R-vine method proposed by Bedford et al. [11, 12] can be constructed according to the actual dependence between various assets, which enables R-vine with more flexibility. Taking five-dimension R-vine as an example, its decomposition structure is shown in Fig. 1.

1

1,2

2,3

2

2,4

4

1,2

3

2,4

5

1,4|23

2,3

3,4|2

3,5

Tree 1

1,3|2

1,3|2

2,5|3

3,5 Tree 2

3,4|2 1,4|23

4,5|23

1,5|234

4,5|23

2,5|3 Tree 4

Tree 3

Fig. 1. One decomposition structure of five-dimension R-vine It should be noted that since the multivariate decomposition structure of R-vine is not pre-determined, Fig. 1 is only one type of the possible five-dimension R-vine decomposition structures. Generally, one n-dimension R-vine can be expressed with n-1 trees ( T ). The tree Ti has n-i+1 nodes and n-i edges, and the edges in the tree

10

Ti will become the nodes in the tree Ti 1 . The construction of R-vine model mainly includes the following three steps: (1) Determining the connection order of the nodes: Since the decomposition of R-vine is complex, its estimation and simulation is more difficult than that of C-vine and D-vine. While the R-vine matrices (RVM) proposed by Dißmann [30] can well construct the R-vine decomposition structure. RVM should meet the following conditions:

 m , m

LM  i   LM  j  , 1  j  i  n,

(10)

mi ,i  LM  i  1 , i  1,, n 1,

(11)



, , mn ,i   BM  i  1  BM  n  1  Bˆ M  i  1   Bˆ M  n  1 , i  1,, n  1, k  i  1,, n  1, k ,i

k 1,i

(12)

where M {1,..., n}nn , LM (i )  {mi ,i ,..., mn ,i } denote the set of all the different entries in the i-th column, BM (i )  {(mi ,i , D) | k  i  1,..., n; D  {mk ,i ,..., mn ,i }} and

Bˆ M (i)  {(mk ,i , D) | k  i  1,..., n; D  {mi ,i }  {mk 1,i ,..., mn ,i }}. For one n-dimension R-vine, there is 2n1 RVMs that satisfy the conditions. In this paper, we will use the maximum spanning tree method proposed by Brechmann [31] to select a more suitable RVM. This method can maximize the sum of absolute values of Kendall-  in every tree. (2) Selecting the binary copula function: In order to discuss the measurement capability of different R-vine models, this paper constructs three types of R-vine models (Mixed R-vine, R-vine all Gumbel, R-vine all Joe). The R-vine model, which can select different binary copula functions at each edge, is called Mixed R-vine. And

11

the R-vine model which uses the same Gumbel copula (Joe copula) function is called R-vine all Gumbel (R-vine all Joe). According to the Akaike information criterion (AIC), this paper selects binary copula functions of Mixed R-vine in 31 categories: Gaussian copula, t copula, Frank copula, all of which are symmetrically distributed; Gumbel copula, Joe copula, BB6 copula and BB8 copula, which are sensitive to the upper tail; Clayton copula, which is sensitive to the lower tail; BB1 copula and BB7 copula, which are sensitive to both the upper and lower tails, and their rotated forms (90 degrees, 180 degrees, 270 degrees). (3) Estimating parameters of vine copula: In this paper, the maximum likelihood estimation (MLE) method is used to estimate parameters of vine copula. The specific steps could be described as follows: Step 1. Using the marginal distribution data to estimate the binary copula parameters of T1 . Step 2. Using the binary copula parameters obtained from step 1 and Eq. (13) to calculate the observations of T2 .

F  x|v  



Cxv j |v j F  x|v  j  , F  v j |v  j  F  v j |v  j 

,

(13)

where F  x|v  is the conditional distribution function, v j is one component of n-dimension vector v , v  j is the (n-1)-dimension vector except v j and Cxv j |v j is the bivariate Copula distribution function to F  x|v  j  and F  v j |v  j  . Step 3. Using the observations in step 2 to estimate the binary copula parameters of T2 . 12

Step 4. Repeating step 2 and step 3 until the parameters of Tn 1 are estimated. Based on the above analysis, we can construct a complete R-vine copula model. The density function of the R-vine copula is defined as: n

f k 1

1

k

k 1

( xk )   cmk ,k ,mi ,k |mi1,k ,...,mn ,k ( Fmk ,k |mi1,k ,..., mn ,k , Fmi ,k |mi1,k ,..., mn ,k ).

(14)

k  n 1 i  n

2.3 Portfolio VaR and ES calculating In this paper, we use the Monte Carlo method to simulate the marginal distribution of vine copula, and then calculate the portfolio VaR and ES. The steps of this method are as follows: (1) For one n-dimension portfolio, according to the vine copula model, 1000 random arrays w1 , w2 ,..., wn which are uniformly distributed between [0, 1] are generated. (2) Using Eq. (13) and Eq. (15) to calculate simulated series of the marginal distributions.

u1  w1 , u2  F 1 ( w2 | u1 ), ... un  F 1 ( wn | u1 ,..., un 1 ).

(15)

(3) Using the parameter estimation results of GPD and simulation series, the simulated standard residual series z1 , z2 ,..., zn are obtained. (4) Using the construction results of GARCH-type models, we can calculate returns of single asset in the portfolio according to xi ,t  ai ,0  ai ,1ri ,t 1   i ,t zi , and i  1, 2,..., n . n

(5) Calculating portfolio returns

 x i 1

i i ,t

13

, and then computing the VaR and ES

of this portfolio at a specific quantile q using Eq. (16) and Eq. (17). n

P ( i xi ,t  VaR t  q  |t 1 )  q,

(16)

i 1

q t

ES

 

q

VaR it

i q / M

M

,

(17)

where i is the weight of portfolio asset xi , t1 is the information set at time

t 1 , and M is the number of equal intervals between [0, q]. 2.4 Backtesting methods In this section, the unconditional coverage method proposed by Kupiec [32] is used to backtest various VaR models. While to backtest the ES measures, we employ the method proposed by McNeil et al. [33] based on a bootstrap process. The backtesting method of Kupiec [32] is based on the following statistic:



LRuc  2ln 1  q 

T N





q N  2ln 1  N / T 

T N

N /T 

N

,

(18)

where q is the VaR quantile, T is the number of measurement, N is the number of measurement failures. By comparing the actual failure rate with the quantile level, we can find out the model’s measurement accuracy. The null hypothesis of this test is

N / T  q , and then the statistic LRuc ~  2 1 . To backtest the ES measurement accuracy, McNeil et al. [33] propose a bootstrap process as this: Step 1. Using Eq. (19) to construct a series of exceedance residuals yt, where xt is the real returns beyond the VaR measurement.

yt 

xt  EStq

t

.

(19)

Step 2. Assuming the exceedance residuals yt have m observations and

14

calculating the initial sample lt as: lt  yt  y , t  1, 2, , m.

(20)

Step 3. Calculating t  l   l / std , where l is the sample mean and std is the standard deviation of the sample. Step 4. Generating m random numbers uniformly distributed in the scope of

1, 2,, m ,

and identifying the corresponding sample points in lt to form a new

bootstrap sample. And then by repeating 1000 trials, we can obtain 1000 new bootstrap samples. Step 5. The t  l  calculated from initial sample is denoted as t0  l  , and we can

t l  , t l  ,, t l  . The p value of the

denote the 1000 bootstrap samples as backtesting is the proportion of

1

2

t l  , t l  ,, t l  1

2

1000

1000

beyond t0  l  .

For both the unconditional coverage test of Kupiec [32] and the bootstrap test of McNeil et al. [33], the null hypotheses of the two methods are the risk model can produce accurate risk measurement. If the p-values are larger than some commonly-used significant levels, such as 0.1 or 0.05, we cannot reject the null hypotheses and conclude that the corresponding risk models are acceptable. Furthermore, the larger the p value is, the more accurate the risk model will be.

3 Empirical results 3.1 Data and descriptive statistics West Texas Light crude oil (WTI), North Sea Brent crude oil (Brent) and Dubai crude oil (Dubai) are the three most important crude oil markets, and their prices are

15

often regarded as the benchmark of the international crude oil prices. Besides, crude oil price in Cinta is also regarded as an important price benchmark in Far East area. Therefore we select WTI, Brent, Dubai and Cinta crude oil as the representatives of international crude oil market. The data used in this study are spot oil prices and are collected from US Energy Information Administration. The data sample is from 20 June 2014 to 30 September 2016, during which a sharp decrease of oil price from over 100 dollars to about 30 dollars per barrel is observed. After removing the unmatched data among the four oil prices, 544 daily price observations are selected. The logarithmic oil return is calculated as:

rt  ln  pt   ln  pt 1  ,

(21)

where pt is the daily oil spot price. The descriptive statistics for crude oil returns are shown in Table 1. Table 1 Descriptive statistics for crude oil returns Ljune-Box Q *** *** *** WTI -0.0015 0.0297 4.3580 0.1566 43.7848 10.485 41.034** Brent -0.0016 0.0007 4.9458*** 0.3069*** 94.1887*** 3.666*** 32.395* *** *** *** *** Dubai -0.0017 0.0008 6.1643 0.3227 235.9612 16.217 33.381* *** ** *** *** Cinta -0.0017 0.0009 5.8019 0.4997 200.2222 10.561 30.561* Notes: ***, ** and * indicate significance at 1%, 5% and 10% levels, respectively. Mean

Std.

Kurtosis

Skewness

Jarque-Bera

ARCH-LM

ADF -24.408*** -21.434*** -24.100*** -24.434***

Table 1 indicates that the oil returns are skewed and fat-tailed distributed in most cases. The Jarque-Bera tests show that all the oil returns reject the null hypothesis of normal distribution at the 1% significance level. The ARCH-LM tests reveal that there is significant volatility clustering in the oil price. Besides, the Ljune-Box Q statistics show that the crude oil returns are auto-correlated. Furthermore the ADF unit root tests suggest that all of the oil returns are stationary, indicating that we may model them by econometrics analysis without further transformation. 16

3.2 Constructing marginal distribution model In this section, we use AR(1)-GARCH(1,1), AR(1)-FIGARCH(1,1) and AR(1)-FIEGARCH(1,1) models to account for the stylized facts existing in oil returns. The estimation method used in GARCH-type modeling is maximum likelihood estimation method and the estimation results are summarized in Table 2. Table 2 Estimation results of different GARCH-type models a1

ω

α (φ)

β

d

γ

θ

-0.0022**

-0.0563

0.0000

0.1220**

0.8578***

—

—

—

(0.0010) -0.0022**

(0.0452) 0.0543

(0.1994) 0.0000

(0.0486) 0.0961***

(0.0592) 0.9288***

— —

— —

— —

(0.0009) -0.0026***

(0.0373) -0.0451

(0.0378) 0.0000

(0.0281) 0.1648***

(0.0254) 0.8512***

— —

— —

— —

(0.0007) -0.0029***

(0.0430) -0.0598

(0.0455) 0.0000

(0.0413) 0.1635***

(0.0326) 0.8494***

— —

— —

— —

(0.0008)

(0.0443)

(0.0906)

(0.0518)

(0.0444)

—

—

—

a0

LL

AR(1)-GARCH(1,1) WTI Brent Dubai Cinta

1183.34 1270.79 1233.69 1190.78

AR(1)-FIGARCH(1,1) WTI Brent Dubai Cinta

-0.0022**

-0.0583

0.0000

0.1139

0.7213

0.7302

—

—

(0.0010) -0.0021***

(0.0457) 0.0505

(0.3320) 0.0000

(0.4197) -0.5457***

(0.6591) 0.9978***

(1.0738) 1.6018***

— —

— —

(0.0008) -0.0025***

(0.0367) -0.0263

(0.0209) 0.0000*

(0.1299) -0.3709**

(0.0020) 0.9539***

(0.1086) 1.4042***

— —

— —

(0.0008) -0.0029***

(0.0440) -0.0495

(0.0262) 0.0000

(0.1887) -0.0352

(0.0446) 0.5395

(0.2323) 0.6966**

— —

— —

(0.0009)

(0.0449)

(0.2585)

(0.1459)

(0.3538)

(0.2896)

—

—

1182.83 1275.62 1236.14 1190.43

AR(1)-FIEGARCH(1,1) WTI Brent Dubai Cinta

-0.0025**

-0.0580

-4.7347***

1.1864

0.5605***

-0.1946***

0.0092

0.2783**

(0.0011) -0.0027***

(0.0544) 0.0696*

(744.57) -5.0389***

(0.7824) 4.3640

(0.1724) 0.7959***

(0.0366) -0.2642***

(0.0409) -0.0309

(0.1241) 0.1032

(0.0010) -0.0029***

(0.0411) -0.0361

(2752.8) 0.0000

(5.0609) 2.0098

(0.0673) 0.7929***

(0.0524) -0.1024***

(0.0374) -0.0239

(0.1056) 0.1986*

(0.0007) -0.0030***

(0.0442) -0.0475

(58.390) -4.4495***

(1.4884) 1.4271*

(0.0474) 0.7383***

(0.0132) -0.2212***

(0.0222) -0.0228

(0.1130) 0.2722**

(0.0009)

(0.0442)

(731.06)

(0.7521)

(0.1112)

(0.0376)

(0.0335)

(0.1209)

1174.84 1268.73 1224.78 1187.40

Notes: LL is the log-likelihood value of the estimation. The standard errors are reported in the second row for each series. ***, ** and * indicate significance at 1%, 5% and 10% levels, respectively.

Table 2 shows that most of the coefficients of conditional variance are significant, implying that the GARCH-type models adopted here fit the volatilities of oil price well. Furthermore to model the skewed and fat-tailed conditional returns, the GPD are 17

used here. Before GPD estimation, the conditional returns (standardized residual of the GARCH-type models) are tested by BDS test for the null hypothesis of i.i.d. The results are shown in Table 3. Table 3 Results of BDS test for standardized residuals of GARCH-type models Models WTI Brent Dubai Cinta -0.248 0.178 -0.441 0.005 AR(1)-GARCH(1,1) [0.804] [0.859] [0.659] [0.996] -0.320 -1.063 -0.941 -0.365 AR(1)-FIGARCH(1,1) [0.749] [0.288] [0.347] [0.715] 1.114 -0.896 -0.634 -0.768 AR(1)-FIEGARCH(1,1) [0.265] [0.370] [0.526] [0.443] Notes: The p values of the BDS test are reported in square brackets.

Table 3 shows that all the p values of BDS test are larger than 0.1, indicating that the conditional returns are i.i.d. and therefore can be modeled by GPD. Since too large or too small tail thresholds will affect the accuracy or unbiasedness of GPD estimation, this paper choose 10% tail observations as the extreme values according to the method of DuMouchel [34]. The tail thresholds and GPD estimation results are reported in Table 4. Table 4 Tail threshold and GPD estimation results Lower tail Upper tail

L AR(1)-GARCH(1,1)-EVT WTI -1.2371 Brent -1.1184 Dubai -1.2277 Cinta -1.1888 AR(1)-FIGARCH(1,1)-EVT WTI -1.2219 Brent -0.9872 Dubai -1.2518 Cinta -1.2300 AR(1)-FIEGARCH(1,1)-EVT WTI -1.2557 Brent -1.2788 Dubai -1.2047 Cinta -1.1858

L

L

U

U

U

0.1016 0.2040 0.3224 0.1262

0.5866 0.4340 0.3507 0.5052

1.2749 1.0176 1.2741 1.2215

-0.0818 -0.1204 -0.1699 0.0260

0.5366 0.6143 0.6252 0.5870

0.1474 0.2308 0.1488 0.0690

0.5540 0.3379 0.4330 0.5833

1.2613 0.8978 1.2889 1.2564

-0.0460 -0.0648 -0.0667 -0.1073

0.4815 0.5586 0.5447 0.7952

0.0752 0.1653 0.0122 0.1639

0.5626 0.4809 0.5710 0.4446

1.3054 1.2195 1.3018 1.2394

-0.2528 0.0066 -0.0687 0.0079

0.6806 0.6231 0.6589 0.7002

Table 4 shows that all the lower tail parameters  L are larger than 0 and most of 18

the upper tail parameters U are smaller than 0, indicating that the lower tails are significantly fat-tailed while the upper tails are not. These results may due to the large oil price drop during the sample horizon. Besides, since the construction of copula models requires that the marginal distributions are i.i.d with zero mean and unit variance, BDS and Kolmogorov-Smirnov (K-S) tests are performed for each marginal distribution and the results are shown in Table 5. The null hypotheses of both BDS and K-S tests are the distributions tested are i.i.d. Table 5 indicates that all the marginal distributions are i.i.d. and can be used for further establishment of vine copulas. Statistic BDS test

K-S test

Table 5 Results of BDS and K-S tests for marginal distribution Risk model WTI Brent Dubai AR(1)-GARCH(1,1)-EVT -0.204 -1.435 -0.404 [0.838] [0.151] [0.686] AR(1)-FIGARCH(1,1)-EVT -0.412 -1.903 -1.139 [0.681] [0.057] [0.255] AR(1)-FIEGARCH(1,1)-EVT -0.241 -1.371 -0.952 [0.810] [0.170] [0.341] AR(1)-GARCH(1,1)-EVT 0.526 0.656 0.521 [0.945] [0.783] [0.949] AR(1)-FIGARCH(1,1)-EVT 0.515 0.751 0.475 [0.953] [0.626] [0.978] AR(1)-FIEGARCH(1,1)-EVT 0.526 0.806 0.541 [0.945] [0.534] [0.931]

Cinta -0.026 [0.979] -0.524 [0.600] -2.050 [0.040] 0.546 [0.926] 0.580 [0.890] 0.608 [0.854]

Notes: The p values of the BDS and K-S tests are reported in square brackets.

3.3 Constructing R-vine copulas With the marginal distributions obtained from combination of GARCH-type models and GPD based on EVT, three types of R-vine copula models (Mixed R-vine, R-vine all Gumbel and R-vine all Joe) are estimated by maximum likelihood method. For clarity, we indicate different crude oil assets as: WTI (1), Brent (2), Dubai (3), Cinta (4), and the estimation results of Mixed R-vine copula which takes 19

AR(1)-GARCH(1,1)-EVT as marginal distribution are shown in Table 6. Other estimation results are available upon request. Table 6 Estimation results of Mixed R-vine copula with AR(1)-GARCH(1,1)-EVT

[1,] [2,] [3,] [4,]

[,1] 1 4 3 2

R-vine matrices [,2] [,3] 2 4 3

3 4

Binary copula selected [,1] [,2] [,3]

[,4] [1,] [2,] [3,] [4,]

4

SJ RC2 T

RC SBB7

[,4]

T

The estimation results of R-vine matrices (RVM) are reported in the left panel of Table 6. From the RVM, we can see that the four-dimension R-vine has three trees ( T1 ,

T2 , T3 ) in total. T1 has three edges ( C12 , C23 , C34 ), and each edge represents the correlation between two assets. T2 has two edges ( C13|2 , C 24|3 ), and each edge denotes the conditional correlation with a conditional market. T3 has one edge ( C14|23 ) which represents the conditional correlation with two assets. The right panel of Table 6 is the copula selection matrix and each copula function describes the dependence of the corresponding position in RVM. SJ is the survival Joe copula function. RC2 is the rotated Clayton copula by 270 degrees. RC is the rotated Clayton copula by 90 degrees. T is the t copula function and SBB7 is the survival BB7 function. As Mixed R-vine copula can select different binary copulas for each edge, compared with R-vine all Gumbel and R-vine all Joe copulas, it is more flexibility and able to describe various conditional dependences among oil assets.

3.4 Risk measurement and backtesting Based on the dependence structure of crude oil markets measured by R-vine copulas, the portfolio VaR and ES are calculated by Monte Carlo simulation discussed

20

in Section 2.3. The data sample investigated experienced a sharp decrease of oil price from over 100 dollars to about 30 dollars per barrel, and the oil price then began to increase steadily with little likelihood of another drastic drop. Thus we anticipate that the main market risk for oil investors in the near future is the short-position risk (upper-tail risk). For this reason, we backtest the VaR and ES of the oil portfolio at two upper-tail quantile, i.e., 0.99 and 0.975 levels. Table 7 reports the p values of backtesting for the oil portfolio with equal weighted assets, and the larger the p values the better risk measuring accuracy is obtained. Table 7 Backtesting results of different risk models VaR Risk model 99.0% 97.5% 99.0% Panel A: GARCH-GPD Mixed R-vine 0.5139 0.3609 0.6900 R-vine all Gumbel 0.5139 0.4685 0.0430 R-vine all Joe 0.2983 0.4685 0.0725 Panel B: FIGARCH-GPD Mixed R-vine 0.5203 0.3060 0.3600 R-vine all Gumbel 0.8056 0.0987 0.0175 R-vine all Joe 0.1581 0.2434 0.0220 Panel C: FIEGARCH-GPD Mixed R-vine 0.8056 0.0987 0.9435 R-vine all Gumbel 0.8056 0.0987 0.3290 R-vine all Joe 0.2983 0.3609 0.0845

ES 97.5% 0.6660 0.0595 0.0130 0.5170 0.0205 0.0125 0.2995 0.0420 0.1055

Firstly, Table 7 shows that most p values of the backtesting are larger than 5%, indicating that the models based on combination of GARCH-type models, EVT and vine copulas can measure the risk of crude oil portfolio accurately. Secondly, panels A, B and C of Table 7 report the backtesting results for risk models based on three marginal distributions, GARCH-GPD, FIGARCH-GPD and FIEGARCH-GPD, respectively. Panel A indicates that Mixed R-vine copula archives three out of four best performances among different R-vine copulas. Similarly in Panels B and C of 21

Table 7, Mixed R-vine copulas are also found to be superior to the other two copulas in three out of four cases. This finding strongly suggests that Mixed R-vine copula is very flexible and can be used as a powerful tool in describing the complex dependence structures of crude oil assets. Lastly, Table 7 indicates that in each panel of A, B or C, there are about 4 to 5 p values less than 10%, implying that there is no significant difference of measurement accuracy across different marginal distribution models, i.e., GARCH-GDP, FIGARCH-GDP and FIEGARCH-GDP. This result may indicate that that the key factor determining the risk measurement is the dependence modeling of oil assets instead of the marginal distribution. It should also be pointed out that the portfolio with equal weighted assets is just a special case of portfolio investment. In order to ensure the reliability of our conclusions, we assume another new setting of asset weights for WTI, Brent, Dubai and Cinta oil to be 0.3, 0.3, 0.25 and 0.15, respectively. The backtesting results under the new weights are shown in Table 8. Table 8 presents quite similar results to those reported in Table 7. In many cases, VaR and ES measurements based on Mixed R-vine copula are more accurate than other copulas, and the three different marginal distributions performs no obvious differences under VaR and ES measurements with 0.99 and 0.975 quantile. Table 8 Backtesting results of different risk models under new asset weight VaR ES Risk model 99.0% 97.5% 99.0% Panel A: GARCH-GPD Mixed R-vine 0.5139 0.1571 0.9070 R-vine all Gumbel 0.5139 0.1829 0.4260 R-vine all Joe 0.5139 0.1829 0.2990 Panel B: FIGARCH-GPD Mixed R-vine 0.8542 0.0987 0.9045 R-vine all Gumbel 0.8056 0.0987 0.0160 R-vine all Joe 0.5139 0.2434 0.0105 22

97.5% 0.8575 0.0595 0.0245 0.2665 0.0410 0.1615

Panel C: FIEGARCH-GPD Mixed R-vine R-vine all Gumbel R-vine all Joe

0.8056 0.8542 0.1581

0.4685 0.3060 0.6637

0.9535 0.4835 0.5675

0.7840 0.3375 0.0295

4 Conclusions In this paper, we try to measure the VaR and ES of a portfolio consist of four crude oil assets by using the combination of GARCH-type-EVT models and vine copula methods. We find that the models adopted can depict the portfolio risk accurately, implying the relevance of introduction of extreme value distribution and vine copula technique into modeling and measuring the market risk of oil assets. Furthermore the backtesting results show that Mixed R-vine copula is superior to R-vine all Gumbel and R-vine all Joe copulas in modeling the complicated dependence structures of oil prices. Different GARCH-type models, which can depict the long-memory and/or leverage effect of volatilities, however offer similar marginal distributions of the oil returns and thus the key determinant for accurate measurement of oil portfolio risk is the proper modeling of dependence of various oil assets. Further robustness test with unequal asset weights in the portfolio proves the main conclusions of this study.

Acknowledgments The authors are also grateful for the financial support from the National Natural Science Foundation of China (71371157, 71671145), the humanities and social science fund of ministry of education (15YJA790031 and 16YJA790062), and the

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young scholar fund of science & technology department of Sichuan province (2015JQO010), fund of high-level social science research team of Sichuan province, Fundamental research funds for the central universities (26816WCX02), and National Training Programs of Innovation and Entrepreneurship for Undergraduates (Nos. 201610616035).

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