An Analysis of Value at Risk Methods for U.S. Energy

Futures

Robert Trevor Samuel III∗

2 October 2012†

Abstract

We estimate Value-at-Risk (VaR) statistics using parametric, non-parametric, and

Extreme Value Theory (EVT) based techniques on the logarithmic price changes for

continuous futures prices of Crude Oil, Natural Gas and Heating Oil from the New

York Mercantile Exchange (NYMEX). Our results illustrate that the VaR confidence

level, α, matters along with the amount of data used (’window size’) but overall finds

poor results for all five methods of VaR tested with some positive results for specific

parameterizations of a few methods.

∗Master’s Candidate, Clemson University. Correspondence: rtsamue@clemson.edu†revised; first edition: 7 September 2012; second edition: 28 September 2012

1 Introduction

Until Markowitz (1952) people rarely considered risk when making investment or portfolio

allocation decisions. Investments were made in isolation and the portfolio allocation decision

was merely to arbitrarily specify portfolio weights for the disparate investments. Markowitz

in his seminal work demonstrated that there existed an optimal boundary when looking

at the aggregate portfolio’s expected return versus its expected risk. Any combination of

expected returns and risk that was not on this optimal boundary was sub-optimal. The

objective then became one of defining risk and solving for the optimal combination using

linear optimization technique(s).

Markowitz used the standard deviation of returns as his measure of risk but alluded to the

fact that there may be better measurements of risk. In Markowitz (1959) he recommended

the use of semi-variance which only uses deviations below the mean return. The assumption

is that long-only investors are only concerned with downside deviations from the mean. In

fact, they would favor right-skewed distributions and large deviations above the mean; and

since standard deviation does not distinguish between upside and downside deviations it

therefore may not be an accurate reflection of risk. Fishburn (1977) advocated the usage of

differing forms of risk/return utility depending on where an observation occurred within a

distribution of returns. In addition, he questioned the a priori assumption that semi-variance

is the best model and showed that there is a general class of models, the α− t models, that

are dominant and align with an investors risk/return utility as articulated within the Von

Neumann & Morgenstern framework. Nawrocki (1991) extended upon Fishburn (1977) and

investigated the performance of lower partial moment (LPM) estimators of risk. In their

study they can not say that LPM is superior to traditional covariance analysis as articulated

by Markowitz but can say that LPMs are part of the second-degree stochastic-dominance

efficient set.

During the same time period as these authors others were rephrasing the question by

asking whether it was the distribution below the mean that mattered or maybe that extreme

1

values are what matters in regards to risk. Davison & Smith (1990) provided a review of

models using Extreme Value Theory (EVT) by analyzing the limit distribution of extreme

values as first proposed by Fisher & Tippett (1928). They found that the Generalized Ex-

treme Value (GEV) distribution provided an excellent framework for analyzing the extreme

values of a distribution. Others began to use these models with financial data and found that

the GEV distribution had strong explanatory power when looking at extreme logarithmic

prices changes. Specifically related to the analysis in this paper Edwards & Netfci (1988)

and Longin (1999) looked at GEV models in regards to logarithmic price changes with com-

modity futures. Their analyses was related to counter-party risk and the optimal margin

level but they demonstrated that EVT provided an appropriate framework for looking at

extreme prices changes in the futures markets. More recently Gabaix et al (2006) found em-

pirical evidence to support power law distributions, of the same family as GEV, as defining

distributions for price returns of stocks. However, although GEV distributions may provide

strong explanatory power for extreme price changes what is needed is a general framework

for looking at risk and for that we turn to Value at Risk.

2 Value at Risk

Value at Risk (VaR) is concerned with quantifying the largest expected loss over a spec-

ified time period for a specified level of confidence. Formally, let rt = ln(pt/pt−1) be the

logarithmic change in price at time t then VaR is defined as

Pr(rt ≤ V aRt(α)) = α (1)

where the objective becomes finding some F where F−1(α) = V aRt(α). Jorion (1996)

proposed, amongst others, to simply use the sample standard deviation and the standard

2

Normal CDF, Φ. In that context VaR becomes

V aRt(α) = Φ−1(α)σ + µ (2)

where σ is the standard deviation and µ is the mean associated with rt. Jorion (1997)

addressed some of the issues of determining VaR such as the assumption of normality in

regards to financial returns data and the estimation error associated when using sample

quantile methods. In addition he cautioned against the dependence of defining risk with

a single estimator even though the financial industry was rapidly embracing VaR out of

necessity and regulation (I.e. Basel banking accords). Lastly he suggested the usage of kernel

density estimation when the financial returns data is ’suspected to be strongly nonnormal.’

More recently others have advocated using EVT so as to estimate F within the VaR

framework. Neftci (2000) found that using EVT yielded much better out-of-sample results

versus traditional VaR estimates when examining interest rate and foreign exchange data.

Their methodology, which is similar to what we will propose, is to count the number of

observations that exceed a VaR estimate at time t for a specified period of time. Over

the two year period of 1997-1998 they find that across all data sets that EVT-based VaR

methods have a proportion of exceedences that is closer to the stated level of VaR than

compared to standard VaR as defined in (2). Gencay & Selcuk (2004) examine EVT-based

VaR methods in conjunction with emerging markets stock market indices and found that

EVT-based methods offer better estimation for out-of-sample VaR. Specifically due to the

heavy-tailed distributions in emerging markets, because of their associated financial crises,

EVT-based methods are better at estimating VaR especially for lower levels of α.

Krehbiel & Adkins (2005) look at EVT-based methods for VaR dealing with commodity

futures on the NYMEX. They find the best success with conditional-dependence EVT meth-

ods versus Exponentially Weighted Moving Average (EWMA) and Autoregressive-General

Autoregressive Conditional Hetereoskedascity [AR(1)-GARCH(1,1)] methods for the time

3

period analyzed. However, they acknowledge that noise can adversely impact the results

and that the selection of the threshold parameter for Peaks Over Threshold (POT) EVT

methods requires more research. Iglesias (2012) studied EVT-based VaR methods with ex-

change rates and finds that they offer strong explanatory power but there exists varying

results in regards to the type of EVT method used: for some exchange rates EVT-based

methods that take into account the presence of GARCH effects in the data offer better

results.

3 Data

We look at daily logarithmic price changes in three continuous contract1 commodity futures

listed on the New York Mercantile Exchange (NYMEX): Crude Oil (CL), Natural Gas (NG)

and Heating Oil (HO)2. All data used is provided by Norgate Investor Services3 and Table

1 contains descriptive statistics for the three data series analyzed. All series are decidedly

non-normal with all series failing the Jarque-Bera test’s null hypothesis of normal skew and

kurtosis using standard confidence levels. In addition Natural Gas is the only series with

both a negative mean logarithmic return and positive skew. Figures 1, 2 and 3 display the

disparate log price changes and it is discernible the heavy-tailed nature of the series. Natural

Gas shows an increase in dispersion in the latter part of the series which is a function of the

deregulation of the markets in the United States. Other obvious periods of variability would

be the First Gulf War in 1991, global financial crisis of 2007-2009 and the ’Arab Spring’ of

2011-2012 for Crude Oil and Heating Oil.

Yang (1978) first proposed the usage of the Mean Excess (ME) function and Davison

& Smith (1990) used a ME plot to visually determine whether the data conforms to a

1A continuous contract is a construct performed by aggregating multiple time series sequencestogether but then removing gaps that occur due to the fact that commodity future contractswith differing maturities will trade at different price levels. An overview can be found at:http://www.premiumdata.net/support/futurescontinuous.php.

2It should be noted that all futures analyzed have daily price limits such that on certain unspecified daysprices may reach their daily limits which in turn truncates the data.

3http://www.premiumdata.net/

4

Generalized Pareto Distribution (GPD). Given an independent and identically distributed

data sample then the ME function is defined as

M(µ) =

∑ni=1(Xi − µ)I[Xi > µ]∑n

i=1 I[Xi > µ], µ ≥ 0 (3)

where µ is a specified threshold value. These function values can be plotted against a range

of µ to determine an appropriate threshold level and whether a series is suited for EVT

analysis (see Ghosh $ Resnick (2010) for an overview of ME plots). Figures 4, 5 and 6 show

ME plots for the respective commodity futures and were generated using the evir package

in R. In all of the plots we can clearly see a linear trend as the threshold values become

more negative which is an indication of a distribution that fits within the EVT framework.

Hill (1975) offered a non-parametric approach to GEV distributions with his Hill estimator.

Define the Hill estimator as

ξ =1

k

k∑i=j

lnXj,n − lnXk,n (4)

where the data are ordered such that X1,n ≥ X2,n ≥ X3,n, . . . ,≥ Xn,n then α = 1

ξis called

the tail index statistic. Again using evir package in R we create plots of α for varying

order statistics, and their corresponding values, using the negative value for each element of

a series: this is done since by definition the Hill estimator deals with maxima and for the

purposes of our analysis we are only considering negative extremals which means we use the

negative of the returns, r′t = −rt, for our analysis. Figures 7, 8 and 9 show the respective

Hill plots for each series. In each we can clearly see that the standard error of the estimate

is a function of the order statistic selected with the confidence intervals narrowing as the

order statistic gets smaller in value.

5

Crude Oil (CL) Heating Oil (HO) Natural Gas (NG)T 7364 8211 5597Mean 0.0001 0.0002 -0.0004Median 0.0002 0.0002 -0.0001Maximum 0.089 0.1062 0.1305Minimum -0.1343 -0.213 -0.0825Std. Dev. 0.01 0.0136 0.0109Skewness -0.5822 -0.5136 0.1555Kurtosis 12.0376 12.2362 10.4824Jarque-Bera 44810 51517 25596Augmented Dickey-Fuller -17.5707 -18.5048 -17.7102

Table 1: Descriptive statistics of daily continuous futures contract logarithmic pricechanges. Time periods: Crude Oil, 4/15/1983 - 8/10/2012; Heating Oil, 11/26/1979 -

8/10/2012; and Natural Gas, 4/17/1990 - 8/10/2012.

4 VaR Models

For the purposes of our analysis we select five different implementations of Value at Risk.

The intent is to illustrate some of the implementation issues arising from estimating VaR

and to contrast the effectiveness of different types of techniques (I.e. parametric vs. non-

parametric).

4.1 Value at Risk - Hill Estimator (V aRH)

The limiting distribution of a Frechet, Weibull or Gumbel distribution is the GEV and is

represented as

Gξ(x) =

exp(−(1 + ξx)−

1ξ ) if ξ 6= 0

exp(−e−x) if ξ = 0

(5)

where where ξ = 1α

for the Frechet distribution, ξ = − 1α

for the Weibull distribution and

ξ = 0 for the Gumbel distribution. The two primary techniques for the estimation for GEV

is the block maxima approach and the peaks over threshold (POT) approach. The POT

approach uses EVT but uses extreme observations exceeding a ’high’ threshold. As such

more information is used from the entire data set versus the block maxima approach which

6

only uses the maximum from each block of a specified dimension. Specifically, if we let x0 be

the infinite end-point of a distribution then the distribution of the excesses over a threshold

µ is

Fµ(x) = P [X − µ ≤ x|X > µ] =F (x+ µ)− F (µ)

1− F (µ)(6)

for any 0 ≤ x < x0−µ. Within the POT framework the Hill estimation method of ξ becomes

ξ =1

Tu

Tu∑i=j

ln(r′

j)− ln(µ), where µ ≥ 0 (7)

where µ is an arbitrary threshold, Tu is the number of observations exceeding the threshold

and r′t = −rt. For our analysis we use the inverse of sample quantile function with a specified

value of α to find the appropriate µ. Then if we recognize that 1−Fµ(x) = TuT

then our VaR

estimate becomes

V aR(α)H = −exp(−ξ[ln(αT

Tu+ ln(µ)]) (8)

where T is the total number of observations.

4.2 Value at Risk - Historical Simulation (V aRHS)

Jorion motivated the usage of Historical Simulation method of VaR by demonstrating that

in most cases the distribution of financial data is both non-normal and unknown. In those

cases he advocated using the inverse of the sample quantile function such that VaR was

defined as

V aRHS(α) = F−1(α) (9)

where we look at the left tail of the distribution in lieu of taking the negative value of the

right tail observations.

7

4.3 Value at Risk (V aRP)

Jorion (1996) along with others laid the groundwork for the traditional implementation of

VaR which assumes a normal distribution such that our estimate of VaR is

V aRP (α) = Φ−1(α)σ + µ (10)

where Φ is the standard Normal CDF and where we use the sample standard deviation, s,

and sample mean, x, as our estimates for the distribution parameters.

4.4 Value at Risk - Kernel Density (V aRK)

We can expand upon the strengths of nonparametric VaR by looking at alternative methods

of estimating F . Li & Racine (2007) provide an excellent overview of kernel density estima-

tion techniques of F . In particular, let h be a bandwidth size that determines the amount of

smoothness then we are interested in estimating the leave-one-out kernel density estimator

F−i(x) =

∑nj 6=iG(x−Xj

h)

(n− 1)(11)

where G(x) =∫ x−∞ k(v)dv is the CDF for a specified kernel k(· ). For our analysis we select

the Epanechnikov kernel which we define as

k(u) =3

4(1− u2)I[|u| ≤ 1] (12)

which then leads to our estimate of VaR as

V aRK(α) = F−1−i (α) (13)

where we use the quantile function within the BMS package of R for our estimate of the

inverse CDF given our estimate of f−i(x), the PDF associated with F−i(x).

8

4.5 Value at Risk - Cornish-Fisher (V aRCF)

Early in the development of VaR it was noted the restrictive and implausible assumptions of

normality in regards to financial data. Mandelbrot (1963) noted that cotton futures prices

did not exhibit normality but a heavy-tail behavior in regards to their price changes and

Fallon (1996) provided a derivation for VaR called ’Modified VaR’ that uses an expansion

series as proposed by Cornish & Fisher (1938) which in turn uses the first four moments

of a distribution. In that case, the Cornish-Fisher expansion is an expansion of the normal

density function with a inverse CDF of

F−1CF (α) = φ−1(1− α){1 +g13!

[(φ−1(1− α))2 − 1] +K

4![(φ−1(1− α))3 − 3φ−1(1− α)]} (14)

where g1 is the sample skewness and K is the sample kurtosis. Given this expansion of the

normal density function our VaR becomes

V aRCF (α) = F−1CF (α)σ + µ (15)

where again we use the sample estimates for σ and µ respectively.

5 Evaluation of Models

As VaR is a risk statistic, we are concerned with its efficiency in helping investors avoid

excessive losses which we define as large negative returns. In that case we perform backtesting

analysis of the VaR estimates for negative returns borrowing upon the work by Krehbiel &

Adkins (2005) and Gencay & Selcuk (2004) on the three commodity futures listed in Table

1. In that case we define a window size, denoted as m, and perform a ’rolling’ analysis by

calculating all five VaR statistics for the period from t − (m + 1) through t − 1 repeating

that process until we reach T , the terminal observation. For each time t we compare the

V aRt−1 statistic against the observed rt so as to calculate the number of exceptions a which

9

is defined as

aj =T∑

t=m+1

I[rt < V aRj,t−1] (16)

where j is an index representing each of the five VaR methods tested. Since a follows a

binomial distribution our test statistic becomes

zj =aj − α(T −m)√α(1− α)(T −m)

(17)

where α is a specified confidence level for VaR and is the same for all j for every window size

m analyzed. For window sizes Krehbiel & Adkins (2005) looked at m = 1000 and Gencay &

Selcuk (2004) looked at m = {500, 1000, 1500}. However we want to test whether the window

size has any influence on the performance of the disparate VaR models and therefore we test

the following window size ranges: m ∈ [950, 1050], m ∈ [450, 550] and m ∈ [60, 120]. For our

VaR confidence levels we look at α = {0.01, 0.05} as as these are the widely used levels within

published research. In regards to V aRCF , Cavenaile & Lejeune (2012) show that α = 0.05

leads inconsistent investor preferences for a risk statistic but we continue to evaluate due

to its prevalence in the literature. Lastly, in regards to the V aRH method we select as our

quantile level 0.1 which represents 10% of the data for each rolling window. This level is

strictly arbitrary and would warrant further research so as to determine whether this level

has any impact on the performance of the V aRH model.

Given this analytical framework, we are interested in testing the hypothesis

H0 : α = α0

Ha : α 6= α0

(18)

where α0 = {0.01, 0.05} are the VaR confidence levels we intend to test and where αj =aj

T−m

10

is our test statistic. Then let us define a p-value as

p− value(z) =

2− 2Φ(|z|) if Ha : α 6= αj

1− Φ(|z|) else

(19)

where Φ is the standard Normal CDF. We then use the p-value to reject or fail-to-reject our

null hypothesis as stated in (18) by comparing its value to oft-used hypothesis confidence

levels {0.01, 0.05, 0.1}. Tables 2, 3 and 4 list summary statistics of p-values for specified

ranges of our window size, m, for Crude Oil (CL), Heating Oil (HO) and Natural Gas (NG)

respectively. We see that for all three commodity futures that both the α level used and the

window size have a significant impact on p-values for all five VaR models. Namely we see

that α = 0.05 is the best VaR confidence level for all three commodities and that smaller

window sizes are better as well. In contrast, Gencay & Selcuk (2004) found that for GEV-

based VaR that the performance was superior for lower levels of α which is not supported

by our results. In addition, Krehbiel & Adkins (2005) found different results with similar

VaR methods but their time period only went through 2003 and did not include some of the

recent extreme variability found in energy futures prices due to geopolitical and regulatory

changes.

6 Conclusions and Future Work

Overall we find that most of our p-values reject the hypothesis stated in (18) for most

VaR methods, using oft-used hypothesis test confidence levels, but that V aRK and V aRCF

perform better when using shorter window sizes and with α = 0.05. However our research

does illustrate that the specified α level matters in regards to performance, but also as

importantly, that the window size m has an impact on the results. It may be that given some

of the heterogeneity in our data we should test methods that utilize a GARCH framework

as they will offer a flexibility deemed necessary by some the change in variance exhibited by

11

our data. In addition due to ’limit’ price movements, which in turn truncate the data, there

may be unusual behavior at the tails of the return distribution(s). Lastly, we only look at

left-tail VaR but investors may be interested in VaR estimation issues for short positions (I.e.

right-tail VaR). Both Krehbiel & Adkins (2005) and Gencay & Selcuk (2004) found differing

results for either tail for the VaR methods tested. This would warrant further analysis to

see whether the results we see for left-tail VaR holds for right-tail VaR as well.

References

[1] Cavenaile, L. and Lejeune, T. (2012) ’A Note on the use of Modified Value-at-Risk’,

Journal of Alternative Investments 14(4), 79-83

[2] Cornish, E. and Fisher, R. (1938) ’Moments and Cumulants in the Specification of

Distributions’, Review of the International Statistical Institute 5(4), 307-320

[3] Davison, A. and Smith, R. (1990) ’Models for Exceedances over High Thresholds’,

Journal of the Royal Statistical Society: Series B 52(3), 393-442

[4] Edwards, F and Neftci, S (1988) ’Extreme Price Movements and Margin Levels in

Futures Markets’, Journal of Futures Markets 8(6), 639-655

[5] Fallon, W. (1996) ’Calculating Value-at-Risk’, Working Paper, The Wharton School

[6] Fishburn, P. (1977) ’Mean-Risk Analysis with Risk Associated with Below-Target Re-

turns’, The American Economic Review 67(2), 116-126

[7] Fisher, R and Tippett, L (1928) ’Limiting forms of the frequency distributions of the

largest or smallest member of a sample’, Proceeding of the Cambridge Philosophical

Society 24, 180-190

[8] Gabaix, X., Gopikrishnan, P., Plerou, V., and Stanley, H. (2006) ’Institutional Investors

and Stock Market Volatility’, Quarterly Journal of Economics May 2006

12

[9] Gencay, R. and Selcuk, F. (2004) ’Extreme value theory and Value-at-Risk: Relative

performance in emerging markets’ International Journal of Forecasting 20, 287-303

[10] Ghosh, S. and Resnick, R. (2010) ’A discussion on mean excess plots’ Stochastic Pro-

cesses and their Applications 120, 1492-1517

[11] Hill, B. (1975) ’A Simple General Approach to Inference About the Tail of a Distribu-

tion’ The Annals of Statistics 3(5), 1163-1174

[12] Iglesias, E. (2012) ’An analysis of extreme movements of exchange rates of the main

currencies traded in the Foreign Exchange market’, Applied Economics 44, 4631-4637

[13] Jorion, P. (1996) ’Risk2: Measuring the Risk in Value-At-Risk’, Financial Analysts

Journal 52(6), 47-56

[14] Jorion, P. (1997) Value at Risk: The New Benchmark for Controlling Derivatives Risk

[15] Krehbiel, T. and Adkins, L. (2005) ’Price Risk in the NYMEX Energy Complex: An

Extreme Value Approach’ Journal of Futures Markets 25(4), 309-337

[16] Li, Q. and Racine, J. (2007) Nonparametric Econometrics

[17] Longin, F (1999) ’Optimal Margin Level in Futures Markets: Extreme Price Movements’

Journal of Futures Markets 19(2), 127-152

[18] Mandelbrot, B. (1963) ’The Variation of Certain Speculative Prices’, Journal of Business

36, 394-419

[19] Markowitz, H. (1952) ’Portfolio Selection’, Journal of Finance 7(1), 77-91

[20] Markowitz, H. (1959) Portfolio Selection: Efficient Diversification of Investments

[21] Nawrocki, D. (1991) ’Optimal algorithms and lower partial moment: ex post results’,

Applied Economics 23, 465-470

13

[22] Neftci, S. (2000) ’Value at risk calculations, extreme events, and tail estimators’, Journal

of Derivatives 7(3), 23-37

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112-116

14

Figure 1: Daily logarithmic price changes for Crude Oil (CL)

Figure 2: Daily logarithmic price changes for Heating Oil (HO)

Figure 3: Daily logarithmic price changes for Natural Gas (NG)

15

Figure 4: Mean Excess Plot for Crude Oil (CL)

Figure 5: Mean Excess Plot for Heating Oil (HO)

Figure 6: Mean Excess Plot for Natural Gas (NG)

16

Figure 7: Hill Plot for Crude Oil (CL)

Figure 8: Hill Plot for Heating Oil (HO)

Figure 9: Hill Plot for Natural Gas (NG)

17

Win

dow

size

s:m∈

[950,1

050]

VaRH

VaRHS

VaRP

VaRK

VaRCF

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

Min

imum

2.48

E-1

38.

59E

-08

6.31

E-1

31.

93E

-08

01.

09E

-05

3.05

E-0

91.

35E

-05

00.

0081

Max

imum

6.65

E-0

91.

70E

-05

1.45

E-1

08.

94E

-06

07.

23E

-04

4.61

E-0

75.

45E

-04

00.

0491

Mea

n1.

14E

-09

2.65

E-0

63.

49E

-11

1.73

E-0

60

1.46

E-0

41.

39E

-07

1.40

E-0

40

0.02

15

Win

dow

size

s:m∈

[450,5

50]

VaRH

VaRHS

VaRP

VaRK

VaRCF

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

Min

imum

6.25

E-1

02.

40E

-06

1.25

E-1

31.

61E

-06

00.

0005

3.43

E-0

70.

0035

00.

0230

Max

imum

5.20

E-0

70.

0003

6.06

E-1

07.

74E

-05

00.

0016

3.20

E-0

50.

0169

00.

0988

Mea

n6.

16E

-08

4.61

E-0

56.

48E

-11

1.70

E-0

50

0.00

081.

19E

-05

0.00

800

0.05

00

Win

dow

size

s:m∈

[60,

120]

VaRH

VaRHS

VaRP

VaRK

VaRCF

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

Min

imum

01.

99E

-11

08.

49E

-11

08.

99E

-05

0.01

880.

2029

00.

1217

Max

imum

1.04

E-0

50.

0018

01.

42E

-07

1.39

E-1

00.

0271

0.98

120.

9979

00.

7696

Mea

n7.

39E

-07

0.00

020

2.42

E-0

84.

99E

-12

0.00

350.

5158

0.59

350

0.35

38

Tab

le2:

p-v

alues

for

asso

ciat

edV

aRα

leve

lsfo

rsp

ecifi

edw

indow

size

s,m

,fo

rC

rude

Oil

(CL

)re

late

dtoHa

:α6=α0.

18

Win

dow

size

s:m∈

[950,1

050]

VaRH

VaRHS

VaRP

VaRK

VaRCF

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

Min

imum

0.00

050.

1098

0.00

050.

0957

6.04

E-1

40.

5656

0.01

100.

6618

00.

0425

Max

imum

0.00

420.

2384

0.00

260.

2175

6.97

E-1

20.

9957

0.07

740.

9568

00.

1196

Mea

n0.

0021

0.15

940.

0012

0.13

821.

78E

-12

0.82

770.

0425

0.80

290

0.06

58

Win

dow

size

s:m∈

[450,5

50]

VaRH

VaRHS

VaRP

VaRK

VaRCF

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

Min

imum

5.28

E-0

50.

0639

3.30

E-0

70.

0583

00.

7078

0.00

470.

4842

00.

0402

Max

imum

0.00

110.

3815

0.00

050.

1737

01

0.06

020.

9979

00.

2309

Mea

n0.

0003

0.15

684.

58E

-05

0.09

350

0.86

770.

0178

0.81

900

0.10

35

Win

dow

size

s:m∈

[60,

120]

VaRH

VaRHS

VaRP

VaRK

VaRCF

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

Min

imum

01.

75E

-10

02.

31E

-10

00.

0136

0.00

010.

4115

00.

5857

Max

imum

8.47

E-0

60.

0130

03.

51E

-05

00.

3380

0.12

080.

9067

01

Mea

n2.

69E

-07

0.00

160

4.58

E-0

60

0.14

780.

0147

0.61

340

0.82

66

Tab

le3:

p-v

alues

for

asso

ciat

edV

aRα

leve

lsfo

rsp

ecifi

edw

indow

size

s,m

,fo

rH

eati

ng

Oil

(HO

)re

late

dtoHa

:α6=α0.

19

Win

dow

size

s:m∈

[950,1

050]

VaRH

VaRHS

VaRP

VaRK

VaRCF

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

Min

imum

00

00

00

1.55

E-1

50

07.

24E

-12

Max

imum

00

2.22

E-1

60

01.

95E

-14

1.88

E-1

20

03.

72E

-08

Mea

n0

02.

86E

-17

00

2.00

E-1

52.

25E

-13

00

7.12

E-0

9

Win

dow

size

s:m∈

[450,5

50]

VaRH

VaRHS

VaRP

VaRK

VaRCF

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

Min

imum

2.70

E-1

21.

40E

-14

1.87

E-1

19.

33E

-15

05.

12E

-05

6.46

E-0

74.

42E

-10

00.

0279

Max

imum

2.31

E-0

73.

41E

-09

5.55

E-0

94.

94E

-11

00.

0008

0.00

022.

36E

-06

00.

1448

Mea

n2.

20E

-08

1.50

E-1

01.

21E

-09

3.73

E-1

20

0.00

043.

73E

-05

2.67

E-0

70

0.06

34

Win

dow

size

s:m∈

[60,

120]

VaRH

VaRHS

VaRP

VaRK

VaRCF

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

α=

0.01

α=

0.05

Min

imum

02.

22E

-16

06.

66E

-16

00.

0004

1.79

E-0

50.

0408

00.

0644

Max

imum

2.61

E-0

81.

92E

-05

04.

86E

-09

5.67

E-1

00.

0075

0.00

990.

9238

00.

4898

Mea

n7.

08E

-10

1.43

E-0

60

4.18

E-1

02.

56E

-11

0.00

280.

0009

0.29

900

0.18

80

Tab

le4:

p-v

alues

for

asso

ciat

edV

aRα

leve

lsfo

rsp

ecifi

edw

indow

size

s,m

,fo

rN

atura

lG

as(N

G)

rela

ted

toHa

:α6=α0.

20