Forecasting Portfolio-Value-at-Risk
with Nonparametric Lower Tail
Dependence Estimates
Karl Friedrich Siburg, Pavel Stoimenov, and
Gregor N.F. Weiß
Preprint 2013-04 April 2013
Fakulta¨t fu¨r Mathematik
Technische Universita¨t Dortmund
Vogelpothsweg 87
44227 Dortmund tu-dortmund.de/MathPreprints

Forecasting Portfolio-Value-at-Risk with
Nonparametric Lower Tail Dependence Estimates
Karl Friedrich Siburg∗
Fakulta¨t fu¨r Mathematik, Technische Universita¨t Dortmund
Pavel Stoimenov †
Fakulta¨t fu¨r Statistik, Technische Universita¨t Dortmund
Gregor N.F. Wei߇
Wirtschafts- und Sozialwissenschaftliche Fakulta¨t, Technische Universita¨t Dortmund
April 23, 2013
Abstract We propose to forecast the Value-at-Risk of bivariate portfolios using copulas which are calibrated on
the basis of nonparametric sample estimates of the coefficient of lower tail dependence. We compare our proposed
method to a conventional copula-GARCH model where the parameter of a Clayton copula is estimated via Canonical
Maximum-Likelihood. The superiority of our proposed model is exemplified by analyzing a data sample of nine
different financial portfolios. A comparison of the out-of-sample forecasting accuracy of both models confirms that
our model yields economically significantly better Value-at-Risk forecasts than the competing parametric calibration
strategy.
Keywords: Copula, tail dependence, nonparametric estimation, Value-at-Risk, Canonical
Maximum-Likelihood.
JEL Classification Numbers: C53, C58.
∗Vogelpothsweg 87, D-44227 Dortmund, Germany, telephone: +49 231 755 7211, e-mail: karl.f.siburg@math.tu-
dortmund.de.
†Address: Vogelpothsweg 78, D-44227 Dortmund, Germany, e-mail: pavel.stoimenov@gmail.com
‡Corresponding author: Otto-Hahn-Str. 6a, D-44227 Dortmund, Germany, telephone: +49 231 755 4608, telefax:
+49 231 755 5231, e-mail: gregor.weiss@tu-dortmund.de. Support by the Collaborative Research Center “Statisti-
cal Modeling of Nonlinear Dynamic Processes” (SFB 823) of the German Research Foundation (DFG) is gratefully
acknowledged.
1 Introduction
Following the events of the recent financial crisis, economists and practitioners alike have
stressed the need for a more prudent management of financial risks. There now exists a wide-
spread consensus in the empirical literature that the dependence between the returns of financial
assets is asymmetric, nonlinear and time-varying (Longin and Solnik, 2001; Ang and Chen, 2002).
Consequently, copula models, which allow for a flexible modeling of nonlinear dependence struc-
tures, have been advocated in the seminal papers by Embrechts et al. (2002) and Li (2000) as
alternatives to correlation-based models. Since then, several studies have found empirical evi-
dence for elliptical copulas yielding only suboptimal Value-at-Risk (VaR) estimates due to their
symmetric tail (in-)dependence (Cherubini et al., 2004; Fischer et al., 2009; Weiß, 2011). It is this
inadequacy to model the lower tail dependence often found in financial returns that has spurred the
search for alternative parametric copula models apart from the Gaussian and Student’s t copula.
In this paper, we propose to forecast the VaR of bivariate portfolios using copulas which are
calibrated solely on the basis of nonparametric sample estimates of the coefficient of lower tail
dependence (LTD). A first step in this direction has been made in Weiß (2011) and Ja¨schke et al.
(2012) where it is shown that choosing a model based on usual goodness-of-fit tests (see, e.g., Trede
and Savu, 2008, for a discussion of copula goodness-of-fit tests) may not provide a good model for
tail dependence. In the present work, we go a step further and show that calibrating a copula model
by fitting the lower tail dependence (LTD) to the data does indeed yield better VaR forecasts. To
fully capture the LTD in the returns on financial assets, we first use the nonparametric estimator
of tail dependence proposed by Schmidt and Stadtmu¨ller (2006) and estimate the LTD in a given
in-sample of portfolio returns. The estimates for the LTD are then transformed into corresponding
parameters of the lower tail dependent Clayton copula. In contrast to estimation procedures based
on maximizing the copula’s likelihood function, which estimate the copula parameters by using
information on the complete dependence structure, we calibrate the model by only incorporating
information on the tail dependence in the data. Our approach is motivated by the notion that the
knowledge of the tails as well as the center of a distribution will be merged by the ML-estimation
1
into one single parameter of an Archimedean copula. As a consequence, forecasting risk measures
in the lower left tail in the distribution will lose in accuracy as the tails are not adequately modeled.
After calibrating the Clayton copula with the nonparametric estimate of LTD, we forecast the VaR
of a portfolio using GARCH models for the marginals.1
We compare our proposed method to a conventional copula-GARCH model where the param-
eter of the Clayton copula is estimated via Canonical Maximum-Likelihood (CML) (Kim et al.,
2007). We examine the usefulness of our method by analyzing a data sample of nine different
portfolios consisting of different financial assets. To be precise, we model the distributions of the
returns on one electricity index, one natural gas index, two commodities as well as the stocks of
five electric utility service providers.2 Our choice of financial assets is guided by the wish to test
the forecasting accuracy of our proposed model based on a data sample that includes sufficient
tail events. Consequently, we model the distribution of securities that presumably were heavily
influenced by the financial crisis as well as the Fukushima Daiichi nuclear disaster in Japan.
The results presented in this paper show that our proposed method of calibrating a copula
model with a nonparametric estimate of the lower tail dependence performs significantly better
than the standard method of estimating the copula parameter via Canonical Maximum Likelihood
with respect to risk forecasting. We show in formal backtesting that our calibration strategy yields
significantly fewer VaR-exceedances than the parametric benchmark.
The remainder of this article is structured as follows. Section 2 introduces the nonparametric
estimator of the coefficient of lower tail dependence as well as the copula-GARCH model we use.
In Section 3 the results of the empirical study are presented and discussed. Concluding remarks
are given in Section 4.
1Thus yielding the standard copula-GARCH approach to estimating and forecasting multivariate portfolio VaR
(Hsu et al., 2012; Nikoloulopoulos et al., 2011; Weiß, 2011).
2For a related study of the dependence structures inherent in energy commodities, see Bo¨rger et al. (2009).
2
2 Methodology
2.1 Copulas and Tail Dependence Functions
The theory of copulas dates back to Sklar (1959). A copula is a function that embodies all the
information about the dependence structure between the components of a random vector. From
a probabilistic point of view, it is a multivariate distribution function with uniformly distributed
margins on the interval [0, 1]. Let us consider two random variables X, Y with continuous marginal
distribution functions F (x) := P (X ≤ x) and G(y) := P (Y ≤ y), respectively, where x and y
are real numbers. By Sklar’s theorem, there exists a unique copula C such that
P (X ≤ x, Y ≤ y) = C(F (x), G(y)). (1)
Conversely, if C is a copula and F,G are distribution functions, then the function defined via (1)
is a bivariate distribution function with margins F,G. It follows that copulas can be interpreted as
dependence functions since they separate the marginal distributions from the dependence structure.
We have
C(u, v) = P (U ≤ u, V ≤ v) = P (X ≤ F−1(u), Y ≤ G−1(v)), (2)
for all u, v ∈ [0, 1], where F−1 and G−1 denote the (generalized) inverses of F and G, respectively.
Let ¯F (x) := 1 − F (x) = P (X > x) and ¯G(y) := 1 − G(y) = P (Y > y) denote the
corresponding survival functions of X and Y . Define a function ˆC by
ˆC(u, v) = u + v − 1 + C(1 − u, 1 − v), (3)
for all u, v ∈ [0, 1]. Then we obtain
P (X > x, Y > y) = ˆC( ¯F (x), ¯G(y)). (4)
The function ˆC is a copula itself and is called the survival copula of X and Y . In view of (4), the
3
survival copula links the joint survival function to its univariate margins in a manner completely
analogous to the one in which the copula connects the joint distribution function to its margins.
Hence, for all u, v ∈ [0, 1], we have
ˆC(u, v) = P (U > 1 − u, V > 1 − v)
= P (X > F−1(1 − u), Y > G−1(1 − v)).
(5)
For further details regarding the theory of copulas we refer the reader to Nelsen (2006).
As the focus of this paper is to characterize and measure extreme dependence, the rest of this
section is devoted to the concept of tail dependence, which concentrates on the upper and lower
quadrant tails of the joint distribution. The standard way to determine whether X and Y are tail
dependent is to look at the so-called lower and upper tail dependence coefficients, denoted by LTD
and UTD, respectively. LTD is the limit (if it exists) of the conditional probability that X is less
than or equal to the u-th quantile of F , given that Y is less than or equal to the u-th quantile of G
as u approaches 0, i.e.
LTD := lim
u→0+
P (X ≤ F−1(u) | Y ≤ G−1(u)). (6)
Similarly, UTD is the limit (if it exists) of the conditional probability that X is greater than the
u-th quantile of F , given that Y is greater than the u-th quantile of G as u approaches 1, i.e.
UTD := lim
u→1−
P (X > F−1(u) | Y > G−1(u)). (7)
From a practitioner’s point of view, lower tail dependence can be interpreted as the limiting
likelihood of two financial assets to crash together. Consequently, several studies in economics
have underlined the necessity for econometric models to account for the possibility of lower tail
dependent assets in financial portfolios (see, e.g., Poon et al., 2004). While many empirical studies
like, e.g., Christoffersen et al. (2012) use parametric models for describing tail dependent financial
4
data, it is interesting to note that the tail dependence coefficients are nonparametric and depend
only on the copula C of X and Y . In particular, we have
LTD = lim
u→0+
C(u, u)
u
(8)
and
UTD = lim
u→1−
1 − 2u+ C(u, u)
1 − u
= lim
u→0+
ˆC(u, u)
u
, (9)
where ˆC is the survival copula of X and Y defined in (3). In view of these identities, passing from
LTD to UTD is the same as passing from C to ˆC so we may, and will, concentrate on LTD and
disregard UTD for the rest of our paper.3
2.2 Nonparametric Estimation of Tail Dependence
To estimate the lower tail dependence coefficient, we will use the estimator for LTD introduced
in Schmidt and Stadtmu¨ller (2006). Let (X, Y ) be a bivariate random vector with distribution
function F and copula C, having marginal distributions functions G and H , respectively, and
(X(1), Y (1)), . . . , (X(T ), Y (T )) be an independent and identically distributed sample of (X, Y ) of
size T . If we denote the empirical distribution functions corresponding to F,G,H by FT , GT , HT ,
respectively, the empirical copula CT is given by
CT (u, v) = FT (G−1T (u), H
−1
T (v)).
In this setting, the estimator introduced in Schmidt and Stadtmu¨ller (2006) is defined as
̂LTDT :=
T
k
CT
( k
T
,
k
T
)
3In the context of modeling the profit and loss distributions of financial assets, LTD is usually of much larger
interest to investors and risk managers than UTD.
5
where k = k(T ) is some parameter satisfying k → ∞ and k/T → 0 as T → ∞. To compute the
nonparametric LTD estimate, consider the ranks R(t)T1 and R
(t)
T2 (t = 1, . . . , T ) of the observations
X(t) and Y (t) in the sample. The LTD estimate is then given by
̂LTDT =
1
k
T
∑
t=1
1
{
R(t)T1≤k and R(t)T2≤k
} (10)
where the parameter k ∈ {1, . . . , T} is chosen by the use of a plateau-finding algorithm.
2.3 Value-at-Risk Forecasting
In this subsection, we quickly review the modeling of the marginals as well as the procedure to
forecast portfolio Value-at-Risk.
Financial data are commonly found to exhibit conditional heteroscedasticity, skewness and lep-
tokurtosis in logarithmic returns. As most theoretical results for copulas including Sklar’s theorem
only hold for i.i.d. samples, financial data are usually filtered using univariate GARCH(1,1) mod-
els to yield approximately i.i.d. samples of standardized residuals.4 We follow the vast majority of
studies in the financial econometrics literature on copula models (Jondeau and Rockinger, 2006;
Fantazzini, 2009; Hafner and Reznikova, 2010) and employ standard GARCH(1,1) models with
Student’s t-distributed innovations to filter our initial data sample.5
We consider time series of daily mid prices Pt (t = 0, 1, . . . , T ) of a financial asset. The
logarithmic return Rt on the asset on day t is defined as Rt := log(Pt/Pt−1). Our main focus lies on
the modeling of the joint distribution of the return on a portfolio of 2 assets with returns (R t1, Rt2).
The marginal distributions of this random vector are modeled using GARCH(1,1) specifications
Rtj = μj + σtjZtj , (11)
σ2tj = α0j + α1jR
2
t−1,j + βjσ
2
t−1,j , j = 1, . . . , d; t = 1, . . . , T, (12)
4Garmann and Grundke (2013) show for copula goodness-of-fit tests that using GARCH filters to account for the
time-varying volatility in financial returns is indeed necessary when later estimating a copula model.
5Hansen and Lunde (2005) show that the GARCH(1,1) specification, in general, cannot be outperformed by more
complex variants of the ARCH or GARCH models.
6
where Ztj are Student’s t-distributed innovations. The vectors Zt = (Zt1, Zt2) with t = 1, . . . , T
are assumed to be distributed according to
F
Z
(z; ν
1
, ν
2
, ω) = C [F
1
(z
1
; ν
1
), F
2
(z
2
; ν
2
);ω] (13)
with parameters ν
1
, ν
2
for the innovations’ distribution and a copula with parameter ω. As the
dependence in financial market data is commonly characterized by strong lower tail dependence,
we only consider the Clayton copula as a parametric family for C.6 The parameters of the marginal
models are estimated via Maximum-Likelihood. For the parameter of the copula C, two different
estimation procedures will be used. First, the parameter of the copula will be estimated by us-
ing the standard Canonical Maximum-Likelihood (CML) method with rank-transformed pseudo-
observations (McNeil et al., 2005).7 Second, we alternatively obtain the copula parameter by es-
timating the coefficient of lower tail dependence nonparametrically with the estimator of Schmidt
and Stadtmu¨ller (2006) and converting this nonparametric estimate into the (uniquely identified)
parameter of the Clayton copula.
In the following as well as in our empirical study, we consider equally-weighted portfolios
consisting of two financial assets with log returns (Rt1) and (Rt2). Using an in-sample of T
observations (i.e., trading days), we then wish to forecast the Value-at-Risk and Expected Shortfall
of the portfolio for the following day T + 1 via Monte Carlo simulation. The algorithm we use
for forecasting the VaR and ES is an adapted version of the procedure laid out by Nikoloulopoulos
et al. (2011). Their algorithm, however, only considers an in-sample estimation whereas we employ
an out-of-sample version of their original simulation procedure extended by Weiß (forthcoming).
Then, the VaR and ES are forecasted as follows:
6Other parametric copula families are obviously possible and more flexible approaches like mixture copulas could
yield an even better fit of the model to the data. The focus of our paper, however, lies on an accurate modeling of the
lower tail dependence in financial market data. We therefore opted for a simple Clayton copula as our dependence
model as the coefficient of lower tail dependence is easily transformed into the parameter of the Clayton copula.
7Note that the CML estimator has been shown to yield less biased parameter estimates than, e.g., the Inference
For Margins or Full Maximum-Likelihood methods (Kim et al., 2007). It has since been accepted as the method of
choice for estimating copula parameters (Genest et al., 2009).
7
• For K = 5, 000, simulate K observations u(k)T+1,1, u
(k)
T+1,2 (k = 1, . . . , K) from the fitted
copula.
• Convert u(k)T+1,j to z
(k)
T+1,j (j = 1, 2) using the quantile function of the Student’s t-distribution.
• Transform z(k)T+1,j into the simulated return R
(k)
T+1,j = μˆj + σˆT+1,jz
(k)
T+1,j where σˆT+1,j and μˆj
(j = 1, 2) are the forecasted one-step ahead conditional volatility and mean values from the
previously fitted marginal models.
• Compute the simulated portfolio return as R(k)T+1,p = 2−1
∑
2
j=1R
(k)
T+1,j .
• Smooth the simulated log returns by kernel density estimation (Epanechnikov kernel)
(Pritsker, 2006; Alexander and Sheedy, 2008).
• Compute the VaR at the 100(1−α)% confidence level for day T +1 as the α-quantile of the
kernel density estimate.
• Update the information set with the actual portfolio return RT+1,p, reestimate all models and
forecast the portfolio return for day T + 2 and so forth.
The procedure described above is repeated for both the copula model based on the CML pa-
rameter estimate as well as the parameter estimate obtained by nonparametrically estimating the
coefficient of lower tail dependence.
The resulting time series of VaR-exceedances, i.e., the days on which the portfolio suffered
losses exceeding the VaR forecast on that day, are backtested by the use of the test of conditional
coverage (Christoffersen, 1998; Christoffersen and Pelletier, 2004). The p-values of the test are
approximated via parametric bootstrapping.8
After forecasting the VaR for each day in the out-of-sample, we additionally compute the esti-
mate for the portfolios’ sample Expected Shortfall as the mean of the simulated returns beyond the
8Additionally, the forecasting performance of the models could also be analyzed by the ranking model proposed by
S¸ener et al. (2012). As we only consider two models, however, such a ranking would yield only a marginal incremental
insight into the forecasting accuracy of the models.
8
estimated VaRs in the out-of-sample. The models’ Expected Shortfall estimates are then backtested
using the sample asymptotics derived by Wong (2008) under a standard normal null hypothesis.
3 Empirical Study
3.1 Data
Our sample includes the log returns on one electricity index (ICE UK Electricity Baseload
Index), one natural gas index (EEX EGIX NCG Index), two commodities (Crude Oil - Brent
Dated FOB and MLCX - South African Coal) and the stocks of five global electric utility service
providers (BP, E.ON, Royal Dutch Shell, RWE and Tohoku Electric Power). The data were ob-
tained from Thomson Reuters Datastream.9 The complete sample covers the period from May 12,
2008 to June 3, 2011 and includes 800 trading days. Our sample thus includes both the peak of
the recent financial crisis as well as the Fukushima Daiichi nuclear disaster in Japan (March 2011).
Time series plots of the data used in our empirical study are presented in Figure 1.
— insert Figure 1 here —
The plots in Figure 1 highlight several distinct challenges for the econometric forecasting of
risk measures for portfolios built from these assets. First, we can see from the plots that the dif-
ferent assets are characterized by extremely different levels of overall volatility as well as price
movements (see, e.g., the extreme spikes in the ICE index at the beginning of our sample as com-
pared to the relatively calm period following this volatility cluster). For all nine time series, we can
observe increased levels of volatility in the returns during the first 250 trading days in the in-sample
coinciding with the climax of the financial crisis. The Fukushima Daiichi nuclear disaster at the
end of our sample seems to have caused extreme tail events in the stock returns of Tohoku Electric
Power. At the same time, both E.ON and Shell experienced only moderate losses on their stocks.
9Ince and Porter (2006) propose several screening procedures for stock prices obtained from Thomson Reuters
Datastream. We control our sample both for stock prices below $ 1 (which could lead to erroneous log returns due to
Datastream’s practice of rounding prices) and log returns above 300% that are reversed within one month. None of
our time series suffer from these known data errors in Datastream.
9
While BP also did not experience any downside movements of its stock during the Fukushima Dai-
ichi disaster, the stock nevertheless suffered severe losses between April and July 2010 following
the Deepwater Horizon oil spill (sample days 508 to 559). Finally, the plots reveal that both the
initial in- as well as the out-of-samples for the nine assets we consider differ considerably with
respect to the returns’ volatility and tail events. While for some assets return volatility is low in
the initial in-sample and increases thereafter (see, e.g., the plot for Tohoku Electric Power), the op-
posite (i.e., lower asset return volatilities in the out-of-sample) is true for several other time series
(see, e.g., E.ON, BP and Shell).
Descriptive statistics as well as results of the Jarque-Bera test of normally distributed log re-
turns are given in Table 1.
— insert Table 1 here —
The results given in Table 1 underline our first impression of the time series plots that several
of the assets used in our study experienced extreme tail events both in our in-sample as well as
the out-of-sample. For example, the ICE UK Electricity index exhibited a minimum log return of
−69.31% in the in-sample while its minimum in the out-of-sample was just −5.35%. Conversely,
both RWE and Tohoku Electric Power had modest minimum log returns of −8.47% and −9.45%
in the in-sample but suffered extreme minimum returns of −14.04% and −23.84% in the out-
of-sample, respectively. Unreported results on the skewness and kurtosis of the data confirm the
stylized facts of skewed and fat-tailed returns in our full sample. As a consequence, the Jarque-
Bera test of normally distributed log returns can be rejected for all time series in the in-sample.
Although the data for two time series (MLCX and Royal Dutch Shell) seem to be approximately
normally distributed in the out-of-sample, we nevertheless employ a GARCH(1,1) model with
Student’s t-distributed innovations as the Jarque-Bera test is rejected for both series in the initial
in-sample. Also note that the volatility of the time series differs significantly both across the nine
assets we use as well as across time.
From the nine assets, we build nine bivariate portfolios for which we forecast the portfolio
Value-at-Risk on each day of the out-of-sample containing 300 trading days. The portfolios we
10
consider are built as follows: ICE UK Electricity Baseload and EEX EGIX NCG Index (portfolio
1), Crude Oil Brent and MLCX (portfolio 2), Crude Oil Brent and E.ON (portfolio 3), Crude Oil
Brent and Shell (portfolio 4), Crude Oil Brent and BP (portfolio 5), RWE and E.ON (portfolio
6), RWE and Shell (portfolio 7), RWE and BP (portfolio 8) and Shell and Tohoku Electric Power
(portfolio 9). Both assets enter each portfolio with equal weights to control for a possibly biasing
influence the specific choice of weights might have on the forecasting accuracy of our models.
3.2 Results
We start the discussion of the results of our empirical study by first comparing the parametric
and nonparametric estimates of the coefficient of lower tail dependence. The parametric and non-
parametric estimates of the LTD for all nine bivariate portfolios in the out-of-sample are presented
in Figure 2.
— insert Figure 2 here —
Across all nine portfolios, we can observe a general trend of the parametric model to yield
lower LTD estimates than the nonparametric estimator and thus possibly underestimate the true
level of lower tail dependence between the two portfolio constituents. With the exception of port-
folios five and six (Crude Oil Brent/BP and RWE/E.ON) and few forecasting days for portfolios
four and eight (Crude Oil Brent/Shell and RWE/BP), the parametric estimates of the lower tail
dependence are always lower than the corresponding nonparametric estimates. For portfolios one
and three (ICE/EEX EGIX NCG and Crude Oil Brent/E.ON), both the time series of the paramet-
ric and nonparametric LTD estimates seem to comove with the difference between the two staying
constant for any given day in the out-of-sample. In contrast, estimates for portfolios seven and
eight (RWE/Shell and RWE/BP) differed only slightly at the beginning of the out-of-sample but
dispersed considerably afterwards. For portfolio nine (Shell/Tohoku), the parametric model yields
LTD estimates that are constantly zero due to an overall bad fit of the parametric Clayton copula
model. Only for portfolio six (RWE/E.ON) does the nonparametric LTD estimator yield lower
11
LTD estimates than the parametric model via Canonical Maximum Likelihood estimation. The
plots given in Figure 2 thus confirm our hypothesis that the LTD estimates extracted from the para-
metric models fitted via CML and the nonparametric estimates differ considerably. It thus remains
to be seen whether these differences in the way the LTD in a portfolio is estimated has a significant
influence on the accuracy of Value-at-Risk forecasts for the portfolio.
To test our hypothesis that copula models calibrated by the use of nonparametric estimates of
lower tail dependence can significantly improve the accuracy of VaR forecasts in comparison to a
parametric model calibrated via Maximum Likelihood, we use both model approaches to forecast
the VaR and ES for the nine portfolios described above for each day in our out-of-sample consisting
of 300 trading days. Following the general procedure laid out in Section 2.3, we compare the VaR
and ES estimates from two different model setups.
The first model we consider is a Clayton copula model with GARCH(1,1) marginals (t-
distributed innovations) and the parameter of the Clayton copula being estimated via Canonical
Maximum Likelihood (parametric model). The second model we consider equals the first one with
the exception that the parameter of the Clayton copula is estimated via the transformed nonpara-
metric estimate of the coefficient of lower tail dependence (nonparametric model). Both models
are estimated based on samples of size 500. We use both fitted models to forecast the portfolio
returns, Value-at-Risk as well as Expected Shortfall for the following trading day. The estimation
sample is then updated with the next day in the out-of-sample and all models are reestimated using
the updated sample. The procedure is repeated for all 300 trading days in the out-of-sample and the
resulting VaR and ES estimates are backtested using the formal tests of Christoffersen and Pelletier
(2004) and Wong (2008). The confidence level for the VaR forecasting is 95%. For each day in the
out-of-sample, the VaR and ES forecasts are computed based on 5, 000 simulated daily portfolio
returns. The resulting VaR forecasts from the two models as well as the realized portfolio returns
for all nine portfolios are shown in Figure 3.
— insert Figure 3 here —
12
The plots given in Figure 3 provide us with a first hint at the superiority of our proposed
model based on nonparametric LTD estimates. With the exception of portfolio one (ICE/EEX
EGIX), the model based on the CML parameter estimates seems to underestimate risk for all
remaining eight portfolios compared to the approach based on nonparametric LTD estimates. The
reason for the considerably bad fit of both models for portfolio one is easily seen from the time
series plot of the ICE index in Figure 1 which shows that the initial in-samples of the portfolio
are characterized by extreme volatility in the log returns that vanishes completely after about 70
trading days. Consequently, the VaR forecasts computed from in-samples containing these highly
volatile times are too conservative.10 For the remaining eight portfolios, both models seem to yield
adequate VaR forecasts with the VaR forecasts based on the nonparametric LTD estimates being
slightly smaller than the corresponding forecasts from the model estimated via CML.
To formally test the accuracy of the VaR and ES forecasts of both models, we performe both
the test of conditional coverage (CC test) by Christoffersen and Pelletier (2004) and the Expected
Shortfall (ES) backtest proposed by Wong (2008) on the two models’ results. The results for the
tests are presented in Table 2. The table presents the number of VaR-exceedances, the p-value of
the test of conditional coverage as well as the test decision for the ES backtest for all nine portfolios
and both models (parametric and nonparametric) used for forecasting the portfolio VaR and ES.
The expected number of VaR-exceedances for each portfolio is 15.
— insert Table 2 here —
The results in Table 2 confirm our finding that the VaR forecasts based on CML parameter es-
timates significantly underestimate portfolio risk. The number of VaR-exceedances is significantly
higher for the parametric model than for the forecasts computed by using the nonparametric LTD
estimates. Only for portfolio one (ICE/EEX EGIX) does the nonparametric modeling of the lower
tail dependence not improve the accuracy of the VaR forecasts. However, this result is most likely
10Note that both models are capable of adapting quickly to extreme portfolio returns as can be seen from the plot
for portfolio nine (Shell/Tohoku). Although both models are not able to anticipate the extreme negative return around
trading day 240 (i.e., the Fukushima Daiichi nuclear disaster), VaR forecasts immediately drop down to about −13%
just five trading days after the extreme shock.
13
due to the highly irregular time series behaviour of the ICE index (extremely high volatility in the
beginning compared to virtually no price movements in the later parts of the sample). In contrast,
the parametric model yielded 19 VaR-exceedances more than expected for portfolios three and
four. While this result alone is indicative of an inaccurate forecasting of portfolio risk, we cannot
decide per se whether the number of VaR-exceedances is still acceptable or not. To this end, the
p-values for the test of conditional coverage reveal that for four portfolios (three, four, six and
seven), the VaR forecasts of the model using the CML parameter estimates are rejected while the
model based on the nonparametric LTD estimates yields acceptable VaR forecasts. For portfolios
two, five, eight and nine, the parametric model yields considerably more VaR-exceedances which,
however, cannot be rejected by the formal backtest. In summary, our proposed method of calibrat-
ing a copula model with a nonparametric estimate of the lower tail dependence with the purpose
of forecasting a quantile-based risk measure in the lower tail of a portfolio’s distribution performs
significantly better than the standard method of estimating the copula parameter via Canonical
Maximum Likelihood. The backtesting results underline the finding that the differences between
the two models’ VaR forecasts are highly significant.
Conversely, the formal ES backtest does not reject any of the models’ ES forecasts with the
exception of the nonparametric model’s forecasts for portfolio five (Crude Oil Brent/BP). It thus
seems that the ES forecasts from both models do not differ considerably. The results on the ES
backtests have to be interpreted with care, however, as the sample size for the test was regularly
smaller than ten.11
4 Conclusion
This paper proposes to forecast the VaR of bivariate portfolios by the use of copula models
which are calibrated with nonparametric estimates of the coefficient of lower tail dependence. Our
approach is motivated by the notion that copula models that are estimated based on information on
11Backtesting forecasts of Expected Shortfall is known to suffer from small sample biases as noted, e.g., by Wong
(2008).
14
the complete underlying distribution could significantly underestimate tail risk. Using the Canoni-
cal Maximum-Likelihood estimator as a benchmark, we show that the ML-estimated model yields
significantly smaller estimates of the coefficient of lower tail dependence than our proposed model.
These results hint at the possibility that a Clayton copula estimated via Maximum-Likelihood could
systematically underestimate the lower tail dependence inherent in a given data sample.
The results of our empirical study strongly corroborate this finding as we find our proposed
model to perform significantly better in VaR backtesting than the ML-estimated benchmark. Our
proposed model produces fewer VaR-exceedances in out-of-sample backtesting than the bench-
mark for all nine portfolios we consider. For four out of these nine bivariate portfolios, we find
that the differences in the number of VaR-exceedances is statistically significant as evidenced by
the formal test of conditional coverage.
It could be argued that this paper’s focus on bivariate copulas is a serious limitation of our
work’s pratical relevance. Although our proposed model significantly improves bivariate portfolio
VaR forecasts, portfolios of financial assets are often of much higher dimension. We nevertheless
believe our findings from the analysis of bivariate copulas to be highly relevant. Recent work by
Aas et al. (2009) and Dissmann et al. (2013) emphasizes that so-called pair-copula constructions
are extremely well suited for modeling financial returns in high dimensions. As these hierarchical
models are built from bivariate copulas as buildings blocks, our findings should be directly applica-
ble to high-dimensional pair-copula models. The extension of our work to the field of pair-copula
modeling thus appears to be a promising avenue for further research.
Acknowledgements
Support by the Collaborative Research Center “Statistical Modeling of Nonlinear Dynamic
Processes” (SFB 823) of the German Research Foundation (DFG) is gratefully acknowledged.
15
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18
Figures and Tables
19
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l D
ut
ch
 S
he
ll
Pa
ra
m
e
tri
c 
LT
D
N
on
pa
ra
m
. 
LT
D
0
50
10
0
15
0
20
0
25
0
30
0
0.30.40.50.6
Fo
re
ca
st
in
g 
da
y 
t
Lower Tail Dependence coef.
Pa
ra
m
et
ric
 v
s.
 N
on
pa
ra
m
et
ric
 L
TD
 e
st
im
at
es
:
RW
E 
an
d 
BP
Pa
ra
m
e
tri
c 
LT
D
N
on
pa
ra
m
. 
LT
D
0
50
10
0
15
0
20
0
25
0
30
0
0.000.050.100.150.200.25
Fo
re
ca
st
in
g 
da
y 
t
Lower Tail Dependence coef.
Pa
ra
m
et
ric
 v
s.
 N
on
pa
ra
m
et
ric
 L
TD
 e
st
im
at
es
:
R
oy
al
 D
ut
ch
 S
he
ll 
an
d 
To
ho
ku
 E
le
ct
ric
 P
o
w
er
Pa
ra
m
e
tri
c 
LT
D
N
on
pa
ra
m
. 
LT
D
Fi
gu
re
2:
C
on
tin
ue
d.
23
0
50
10
0
15
0
20
0
25
0
30
0
−0.5−0.4−0.3−0.2−0.10.0
Fo
re
ca
st
in
g 
da
y 
t
log return / Value−at−Risk in %
A
ct
ua
l p
or
tfo
lio
 re
tu
rn
s 
an
d 
Va
R
 fo
re
ca
st
s:
IC
E 
UK
 E
le
ct
ric
ity
 B
as
el
oa
d 
an
d 
EE
X 
EG
IX
 N
CG
 In
de
x
Ac
tu
al
 re
tu
rn
s
Va
R
 C
M
L
Va
R
 N
on
pa
ra
m
.L
TD
0
50
10
0
15
0
20
0
25
0
30
0
−0.04−0.03−0.02−0.010.000.010.020.03
Fo
re
ca
st
in
g 
da
y 
t
log return / Value−at−Risk in %
A
ct
ua
l p
or
tfo
lio
 re
tu
rn
s 
an
d 
Va
R
 fo
re
ca
st
s:
Cr
ud
e 
O
il 
Br
en
t D
at
ed
 F
O
B 
an
d 
M
LC
X 
− 
So
ut
h 
Af
ric
an
 C
oa
l
Ac
tu
al
 re
tu
rn
s
Va
R
 C
M
L
Va
R
 N
on
pa
ra
m
.L
TD
0
50
10
0
15
0
20
0
25
0
30
0
−0.04−0.020.000.020.04
Fo
re
ca
st
in
g 
da
y 
t
log return / Value−at−Risk in %
A
ct
ua
l p
or
tfo
lio
 re
tu
rn
s 
an
d 
Va
R
 fo
re
ca
st
s:
Cr
ud
e 
O
il 
Br
en
t D
at
ed
 F
O
B 
an
d 
E.
O
N
Ac
tu
al
 re
tu
rn
s
Va
R
 C
M
L
Va
R
 N
on
pa
ra
m
.L
TD
0
50
10
0
15
0
20
0
25
0
30
0
−0.04−0.020.000.020.04
Fo
re
ca
st
in
g 
da
y 
t
log return / Value−at−Risk in %
A
ct
ua
l p
or
tfo
lio
 re
tu
rn
s 
an
d 
Va
R
 fo
re
ca
st
s:
Cr
ud
e 
O
il 
Br
en
t D
at
ed
 F
O
B 
an
d 
Ro
ya
l D
ut
ch
 S
he
ll
Ac
tu
al
 re
tu
rn
s
Va
R
 C
M
L
Va
R
 N
on
pa
ra
m
.L
TD
Fi
gu
re
3:
C
om
pa
ri
so
n
o
fr
ea
liz
ed
po
rt
fo
lio
lo
g
re
tu
rn
s
a
n
d
Va
lu
e-
at
-R
isk
fo
re
ca
st
s.
Th
e
fig
ur
es
sh
ow
pl
ot
s
o
ft
he
re
al
iz
ed
lo
g
re
tu
rn
s
o
n
th
e
n
in
e
po
rtf
ol
io
sw
e
an
al
yz
e
in
th
is
st
ud
y
as
w
el
la
s
co
rr
es
po
nd
in
g
Va
lu
e-
at
-R
isk
(V
aR
)f
or
ec
as
ts
co
m
pu
te
d
by
th
e
u
se
o
fp
ar
am
et
ric
an
d
n
o
n
pa
ra
m
et
ric
es
tim
at
es
o
ft
he
co
ef
fic
ie
nt
o
fl
ow
er
ta
il
de
pe
nd
en
ce
(L
TD
).T
he
re
al
iz
ed
po
rtf
ol
io
re
tu
rn
s
ar
e
sh
ow
n
in
so
lid
bl
ac
k
lin
es
,t
he
Va
R
fo
re
ca
sts
es
tim
at
ed
by
th
eu
se
o
fp
ar
am
et
ric
es
tim
at
es
o
ft
he
LT
D
ar
e
sh
ow
n
in
da
sh
ed
re
d
lin
es
an
d
th
eV
aR
fo
re
ca
sts
co
m
pu
te
d
fro
m
n
o
n
pa
ra
m
et
ric
es
tim
at
es
o
ft
he
LT
D
ar
e
sh
ow
n
in
do
tte
d
gr
ee
n
lin
es
.E
ac
h
o
u
t-
of
-s
am
pl
e
co
n
sis
ts
o
f3
00
tr
ad
in
g
da
ys
in
th
e
pe
rio
d
04
/1
0/
20
10
to
06
/0
3/
20
11
.B
ot
h
th
e
pa
ra
m
et
ric
co
pu
la
m
o
de
ls
an
d
th
e
n
o
n
pa
ra
m
et
ric
es
tim
at
es
ar
e
co
m
pu
te
d
ba
se
d
o
n
ro
lli
ng
w
in
do
w
s
co
n
ta
in
in
g
50
0
tr
ad
in
g
da
ys
.A
fte
re
ac
h
es
tim
at
io
n,
th
e
in
-s
am
pl
e
ar
e
sh
ift
ed
o
n
e
da
y
ah
ea
d
to
fo
re
ca
st
th
e
n
ex
td
ay
’s
ta
il
de
pe
nd
en
ce
co
ef
fic
ie
nt
sa
n
d
co
rr
es
po
nd
in
g
po
rtf
ol
io
re
tu
rn
s.
24
0
50
10
0
15
0
20
0
25
0
30
0
−0.06−0.04−0.020.000.020.04
Fo
re
ca
st
in
g 
da
y 
t
log return / Value−at−Risk in %
A
ct
ua
l p
or
tfo
lio
 re
tu
rn
s 
an
d 
Va
R
 fo
re
ca
st
s:
Cr
ud
e 
O
il 
Br
en
t D
at
ed
 F
O
B 
an
d 
BP
Ac
tu
al
 re
tu
rn
s
Va
R
 C
M
L
Va
R
 N
on
pa
ra
m
.L
TD
0
50
10
0
15
0
20
0
25
0
30
0
−0.06−0.04−0.020.000.02
Fo
re
ca
st
in
g 
da
y 
t
log return / Value−at−Risk in %
A
ct
ua
l p
or
tfo
lio
 re
tu
rn
s 
an
d 
Va
R
 fo
re
ca
st
s:
RW
E 
an
d 
E.
O
N
Ac
tu
al
 re
tu
rn
s
Va
R
 C
M
L
Va
R
 N
on
pa
ra
m
.L
TD
0
50
10
0
15
0
20
0
25
0
30
0
−0.03−0.02−0.010.000.010.020.03
Fo
re
ca
st
in
g 
da
y 
t
log return / Value−at−Risk in %
A
ct
ua
l p
or
tfo
lio
 re
tu
rn
s 
an
d 
Va
R
 fo
re
ca
st
s:
RW
E 
an
d 
Ro
ya
l D
ut
ch
 S
he
ll A
ct
ua
l r
et
ur
n
s
Va
R
 C
M
L
Va
R
 N
on
pa
ra
m
.L
TD
0
50
10
0
15
0
20
0
25
0
30
0
−0.06−0.04−0.020.000.020.04
Fo
re
ca
st
in
g 
da
y 
t
log return / Value−at−Risk in %
A
ct
ua
l p
or
tfo
lio
 re
tu
rn
s 
an
d 
Va
R
 fo
re
ca
st
s:
RW
E 
an
d 
BP
Ac
tu
al
 re
tu
rn
s
Va
R
 C
M
L
Va
R
 N
on
pa
ra
m
.L
TD
0
50
10
0
15
0
20
0
25
0
30
0
−0.15−0.10−0.050.000.05
Fo
re
ca
st
in
g 
da
y 
t
log return / Value−at−Risk in %
A
ct
ua
l p
or
tfo
lio
 re
tu
rn
s 
an
d 
Va
R
 fo
re
ca
st
s:
R
oy
al
 D
ut
ch
 S
he
ll 
an
d 
To
ho
ku
 E
le
ct
ric
 P
o
w
er
Ac
tu
al
 re
tu
rn
s
Va
R
 C
M
L
Va
R
 N
on
pa
ra
m
.L
TD
Fi
gu
re
3:
C
on
tin
ue
d.
25
n
m
in
m
a
x
m
ea
n
st
.d
ev
.
5%
-q
ua
nt
ile
Ja
rq
ue
-B
er
a
Pa
n
el
(a)
:F
ul
ls
a
m
pl
e:
05
/1
2/
20
08
-0
6/
03
/2
01
1
IC
E
U
K
El
ec
tri
ci
ty
B
as
el
oa
d
In
de
x
80
0
-
0.
69
31
0.
28
72
-
2.
12
7e
-0
4
0.
03
91
-
0.
01
11
0.
00
0
EE
X
EG
IX
N
CG
In
de
x
80
0
-
0.
12
73
0.
27
96
-
1.
75
4e
-0
4
0.
03
02
-
0.
04
15
0.
00
0
Cr
ud
e
O
il
-
B
re
nt
D
at
ed
FO
B
80
0
-
0.
09
54
0.
07
48
3.
56
4e
-0
5
0.
02
04
-
0.
03
46
0.
00
0
M
LC
X
-
So
ut
h
A
fri
ca
n
Co
al
80
0
-
0.
12
31
0.
13
69
-
7.
24
0e
-0
5
0.
02
60
-
0.
04
00
0.
00
0
E.
O
N
80
0
-
0.
11
56
0.
14
26
-
8.
94
9e
-0
4
0.
01
89
-
0.
02
66
0.
00
0
R
oy
al
D
ut
ch
Sh
el
l
80
0
-
0.
09
81
0.
13
21
6.
73
9e
-0
5
0.
02
09
-
0.
03
13
0.
00
0
B
P
80
0
-
0.
10
75
0.
15
89
-
1.
05
9e
-0
3
0.
02
23
-
0.
03
42
0.
00
0
RW
E
80
0
-
0.
14
04
0.
10
58
-
3.
97
1e
-0
4
0.
02
25
-
0.
03
41
0.
00
0
To
ho
ku
El
ec
tri
c
Po
w
er
80
0
-
0.
23
84
0.
15
66
-
1.
20
6e
-0
3
0.
02
04
-
0.
02
40
0.
00
0
Pa
n
el
(b)
:I
ni
tia
li
n-
sa
m
pl
e:
05
/1
2/
20
08
-0
4/
09
/2
01
0
IC
E
U
K
El
ec
tri
ci
ty
B
as
el
oa
d
In
de
x
50
0
-
0.
69
31
0.
28
72
-
1.
14
3e
-0
3
0.
04
89
-
0.
01
81
0.
00
0
EE
X
EG
IX
N
CG
In
de
x
50
0
-
0.
12
73
0.
27
96
-
1.
42
7e
-0
3
0.
03
44
-
0.
04
74
0.
00
0
Cr
ud
e
O
il
-
B
re
nt
D
at
ed
FO
B
50
0
-
0.
09
54
0.
07
48
-
5.
96
5e
-0
4
0.
02
42
-
0.
04
43
0.
00
0
M
LC
X
-
So
ut
h
A
fri
ca
n
Co
al
50
0
-
0.
12
31
0.
13
69
-
7.
37
1e
-0
4
0.
03
03
-
0.
04
74
0.
00
0
E.
O
N
50
0
-
0.
11
56
0.
14
26
-
2.
62
0e
-0
4
0.
02
16
-
0.
02
95
0.
00
0
R
oy
al
D
ut
ch
Sh
el
l
50
0
-
0.
09
81
0.
13
21
-
1.
35
5e
-0
4
0.
02
43
-
0.
04
12
0.
00
0
B
P
50
0
-
0.
10
75
0.
15
89
-
8.
23
7e
-0
4
0.
02
62
-
0.
03
74
0.
00
0
RW
E
50
0
-
0.
08
47
0.
10
58
6.
85
5e
-0
5
0.
02
27
-
0.
03
69
0.
00
0
To
ho
ku
El
ec
tri
c
Po
w
er
50
0
-
0.
09
45
0.
08
26
-
3.
71
0e
-0
4
0.
01
68
-
0.
02
44
0.
00
0
Pa
n
el
(c)
:O
ut
-o
f-s
am
ple
-
04
/1
0/
20
10
-0
6/
03
/2
01
1
IC
E
U
K
El
ec
tri
ci
ty
B
as
el
oa
d
In
de
x
30
0
-
0.
05
35
0.
07
72
1.
33
8e
-0
3
0.
00
96
-
0.
00
46
0.
00
0
EE
X
EG
IX
N
CG
In
de
x
30
0
-
0.
06
05
0.
10
18
1.
91
0e
-0
3
0.
02
14
-
0.
02
95
0.
00
0
Cr
ud
e
O
il
-
B
re
nt
D
at
ed
FO
B
30
0
-
0.
03
53
0.
03
93
1.
08
9e
-0
3
0.
01
15
-
0.
02
15
0.
00
0
M
LC
X
-
So
ut
h
A
fri
ca
n
Co
al
30
0
-
0.
05
87
0.
04
44
1.
03
5e
-0
3
0.
01
64
-
0.
02
90
0.
10
9
E.
O
N
30
0
-
0.
07
58
0.
02
88
-
1.
95
0e
-0
3
0.
01
31
-
0.
02
18
0.
00
0
R
oy
al
D
ut
ch
Sh
el
l
30
0
-
0.
04
83
0.
04
14
4.
05
5e
-0
4
0.
01
35
-
0.
02
30
0.
03
3
B
P
30
0
-
0.
07
11
0.
03
05
-
1.
45
2e
-0
3
0.
01
35
-
0.
02
32
0.
00
0
RW
E
30
0
-
0.
14
04
0.
08
94
-
1.
17
3e
-0
3
0.
02
21
-
0.
03
00
0.
00
0
To
ho
ku
El
ec
tri
c
Po
w
er
30
0
-
0.
23
84
0.
15
66
-
2.
59
8e
-0
3
0.
02
53
-
0.
02
27
0.
00
0
Ta
bl
e
1:
D
es
cr
ip
tiv
e
st
at
ist
ic
s.
Th
e
ta
bl
e
pr
es
en
ts
de
sc
rip
tiv
e
st
at
ist
ic
s
o
n
th
e
lo
g
re
tu
rn
s
o
n
th
e
el
ec
tri
ci
ty
in
de
x
(IC
E
U
K
El
ec
tri
ci
ty
B
as
el
oa
d
In
de
x
an
d
Po
w
er
n
ex
tE
le
c.
B
as
el
oa
d),
o
n
e
n
at
ur
al
ga
si
nd
ex
(E
EX
EG
IX
N
CG
In
de
x
),t
w
o
co
m
m
o
di
tie
s(
Cr
ud
eO
il
-
B
re
nt
D
at
ed
FO
B
an
d
M
LC
X
-
So
ut
h
A
fri
ca
n
Co
al
)a
n
d
th
e
st
oc
ks
o
ffi
v
e
gl
ob
al
el
ec
tri
c
u
til
ity
se
rv
ic
e
pr
ov
id
er
s(
E.
ON
,R
W
E,
B
P,
R
o
ya
lD
ut
ch
Sh
el
la
n
d
To
ho
ku
El
ec
tri
c
Po
w
er
).
Th
e
de
sc
rip
tiv
e
st
at
ist
ic
s
as
w
el
la
s
th
e
p-
va
lu
es
o
ft
he
Ja
rq
ue
-B
er
a
te
st
o
fn
o
rm
al
ly
di
str
ib
u
te
d
lo
g
re
tu
rn
s
ar
e
gi
v
en
fo
rt
he
fu
ll
sa
m
pl
e
in
pa
ne
l(
a),
th
e
in
iti
al
in
-s
am
pl
e
in
pa
ne
l(
b)
an
d
th
e
o
u
t-
of
-s
am
pl
e
in
pa
ne
l(
c).
26
Pa
ra
m
et
ric
m
o
de
l
N
on
pa
ra
m
et
ric
m
o
de
l
Va
R
-e
xc
ee
da
nc
es
C
C
te
st
ES
ba
ck
te
st
Va
R
-e
xc
ee
da
nc
es
C
C
te
st
ES
ba
ck
te
st
Po
rt
fol
io
1
-
IC
E
U
K
/E
EX
EG
IX
4
0.
00
4
n
o
tr
ej.
4
0.
00
0
n
o
tr
ej.
2
-
Cr
ud
e
O
il
B
re
nt
/M
LC
X
17
0.
36
6
n
o
tr
ej.
9
0.
37
4
n
o
tr
ej.
3
-
Cr
ud
e
O
il
B
re
nt
/E
.O
N
34
0.
00
0
n
o
tr
ej.
17
0.
35
8
n
o
tr
ej.
4
-
Cr
ud
e
O
il
B
re
nt
/S
he
ll
34
0.
00
0
n
o
tr
ej.
19
0.
76
5
n
o
tr
ej.
5
-
Cr
ud
e
O
il
B
re
nt
/B
P
28
0.
35
6
n
o
tr
ej.
13
0.
98
5
re
j.
6
-
RW
E
/E
.O
N
28
0.
00
8
n
o
tr
ej.
22
0.
73
9
n
o
tr
ej.
7
-
RW
E
/S
he
ll
29
0.
00
3
n
o
tr
ej.
22
0.
74
6
n
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tr
ej.
8
-
RW
E
/B
P
23
0.
73
3
n
o
tr
ej.
14
1.
00
0
n
o
tr
ej.
9
-
Sh
el
l/
To
ho
ku
El
ec
tr.
Pw
r.
25
0.
36
8
n
o
tr
ej.
13
0.
98
3
n
o
tr
ej.
Ta
bl
e
2:
Va
R
-e
xc
ee
da
nc
es
a
n
d
ba
ck
te
st
s.
Th
e
ta
bl
e
pr
es
en
ts
th
e
n
u
m
be
ro
fd
ai
ly
re
al
iz
ed
po
rtf
ol
io
re
tu
rn
s
ex
ce
ed
in
g
th
e
re
sp
ec
tiv
e
da
ily
Va
lu
e-
at
-R
isk
(V
aR
)f
or
ec
as
ts,
p-
va
lu
es
fo
r
th
e
te
st
o
f
co
n
di
tio
na
lc
o
v
er
ag
e
(C
C
te
st
)b
y
Ch
ris
to
ffe
rs
en
an
d
Pe
lle
tie
r
(20
04
)a
n
d
th
e
Ex
pe
ct
ed
Sh
or
tfa
ll
(E
S)
ba
ck
te
st
pr
op
os
ed
by
W
o
n
g
(20
08
).
Fo
r
ea
ch
po
rtf
ol
io
,t
he
ba
ck
te
sts
ar
e
pe
rfo
rm
ed
o
n
th
e
o
u
t-
of
-s
am
pl
e
co
n
sis
tin
g
o
f3
00
tr
ad
in
g
da
ys
.
Th
e
co
n
fid
en
ce
le
v
el
fo
r
th
e
Va
R
fo
re
ca
sti
ng
is
9
5
%
.
Th
e
ex
pe
ct
ed
n
u
m
be
ro
f
Va
R
-e
x
ce
ed
an
ce
s
fo
r
ea
ch
po
rtf
ol
io
is
1
5
.
Th
e
p-
va
lu
es
o
f
th
e
te
st
o
f
co
n
di
tio
na
lc
o
v
er
ag
e
ar
e
co
m
pu
te
d
v
ia
pa
ra
m
et
ric
bo
ot
str
ap
pi
ng
.T
he
se
co
n
d
to
fo
ur
th
co
lu
m
n
pr
es
en
tt
he
ba
ck
te
sti
ng
re
su
lts
fo
rt
he
fo
re
ca
sts
fro
m
th
e
co
pu
la
m
o
de
lc
al
ib
ra
te
d
v
ia
Ca
no
ni
ca
lM
ax
im
um
-L
ik
el
ih
oo
d
(pa
ram
etr
ic
m
o
de
l)
w
hi
le
th
e
fif
th
to
se
v
en
th
co
lu
m
n
pr
es
en
tc
o
rr
es
po
nd
in
g
re
su
lts
fo
rt
he
co
pu
la
m
o
de
lc
al
ib
ra
te
d
by
th
e
u
se
o
ft
he
n
o
n
pa
ra
m
et
ric
es
tim
at
es
o
ft
he
co
ef
fic
ie
nt
o
fl
ow
er
ta
il
de
pe
nd
en
ce
(no
np
ara
me
tri
cm
o
de
l).
27
Preprints ab 2011/02
2013-04 Karl Friedrich Siburg, Pavel Stoimenov, and Gregor N.F. Weiß
Forecasting Portfolio-Value-at-Risk with Nonparametric Lower Tail Dependence Estimates
2013-03 Martin Heida
ON THERMODYNAMICS OF FLUID INTERFACES
2013-02 Martin Heida
EXISTENCE OF SOLUTIONS FOR TWO TYPES OF GENERALIZED
VERSIONS OF THE CAHN-HILLIARD EQUATION
2013-01 T. Dohnal, A. Lamacz, B. Schweizer
Dispersive effective equations for waves in heterogeneous media on large time scales
2012-19 Martin Heida
On gradient flows of nonconvex functionals in Hilbert spaces with
Riemannian metric and application to Cahn-Hilliard equations
2012-18 R.V. Kohn, J. Lu, B. Schweizer, and M.I. Weinstein
A variational perspective on cloaking by anomalous localized resonance
2012-17 Margit Ro¨sler and Michael Voit
Olshanski spherical functions for infinite dimensional motion groups of fixed rank
2012-16 Selim Esedog¯lu, Andreas Ra¨tz, Matthias Ro¨ger
Colliding Interfaces in Old and New Diffuse-interface Approximations of Willmore-flow
2012-15 Patrick Henning, Mario Ohlberger and Ben Schweizer
An adaptive multiscale finite element method
2012-14 Andreas Knauf, Frank Schulz, Karl Friedrich Siburg
Positive topological entropy for multi-bump magnetic fields
2012-13 Margit Ro¨sler, Tom Koornwinder, and Michael Voit
Limit transition between hypergeometric functions of type BC and Type A
2012-12 Alexander Schnurr
Generalization of the Blumenthal-Getoor Index to the Class of Homogeneous
Diffusions with Jumps and some Applications
2012-11 Wilfried Hazod
Remarks on pseudo stable laws on contractible groups
2012-10 Waldemar Grundmann
Limit theorems for radial random walks on Euclidean spaces of high dimensions
2012-09 Martin Heida
A two-scale model of two-phase flow in porous media ranging from porespace
to the macro scale
2012-08 Martin Heida
On the derivation of thermodynamically consistent boundary
conditions for the Cahn-Hilliard-Navier-Stokes system
2012-07 Michael Voit
Uniform oscillatory behavior of spherical functions of GLn/Un at the identity
and a central limit theorem
2012-06 Agnes Lamacz and Ben Schweizer
Effective Maxwell equations in a geometry with flat rings of arbitrary shape
2012-05 Frank Klinker and Gu¨nter Skoruppa
Ein optimiertes Gla¨ttungsverfahren motiviert durch eine technische Fragestellung
2012-04 Patrick Henning, Mario Ohlberger, and Ben Schweizer
Homogenization of the degenerate two-phase flow equations
2012-03 Andreas Ra¨tz
A new diffuse-interface model for step flow in epitaxial growth
2012-02 Andreas Ra¨tz and Ben Schweizer
Hysteresis models and gravity fingering in porous media
2012-01 Wilfried Hazod
Intrinsic topologies on H-contraction groups with applications to semistability
2011-14 Guy Bouchitte´ and Ben Schweizer
Plasmonic waves allow perfect transmission through sub-wavelength metallic gratings
2011-13 Waldemar Grundmann
Moment functions and Central Limit Theorem for Jacobi hypergroups on [0,∞[
2011-12 J. Koch, A. Ra¨tz, and B. Schweizer
Two-phase flow equations with a dynamic capillary pressure
2011-11 Michael Voit
Central limit theorems for hyperbolic spaces and Jacobi processes on [0,∞[
2011-10 Ben Schweizer
The Richards equation with hysteresis and degenerate capillary pressure
2011-09 Andreas Ra¨tz and Matthias Ro¨ger
Turing instabilities in a mathematical model for signaling networks
2011-08 Matthias Ro¨ger and Reiner Scha¨tzle
Control of the isoperimetric deficit by the Willmore deficit
2011-07 Frank Klinker
Generalized duality for k-forms
2011-06 Sebastian Aland, Andreas Ra¨tz, Matthias Ro¨ger, and Axel Voigt
Buckling instability of viral capsides - a continuum approach
2011-05 Wilfried Hazod
The concentration function problem for locally compact groups revisited: Non-dissipating
space-time random walks, τ -decomposable laws and their continuous time analogues
2011-04 Wilfried Hazod, Katrin Kosfeld
Multiple decomposability of probabilities on contractible locally compact groups
2011-03 Alexandra Monzner* and Frol Zapolsky†
A comparison of symplectic homogenization and Calabi quasi-states
2011-02 Stefan Ja¨schke, Karl Friedrich Siburg and Pavel A. Stoimenov
Modelling dependence of extreme events in energy markets using tail copulas