# Vulnerability-CoVaR: Investigating the Crypto-market

Martin Waltz
Institute of Transportation Economics, Technische Universität Dresden,
Dresden, Saxony 01062, Germany<EMAIL_ADDRESS>
and
Abhay Kumar Singh
Department of Applied Finance, Macquarie University,
Sydney, NSW 2109, Australia<EMAIL_ADDRESS>
and
Ostap Okhrin
Institute of Transportation Economics, Technische Universität Dresden,
Dresden, Saxony 01062, Germany<EMAIL_ADDRESS>
###### Abstract

This paper proposes an important extension to Conditional Value-at-Risk
(CoVaR), the popular systemic risk measure, and investigates its properties on
the cryptocurrency market. The proposed Vulnerability-CoVaR (VCoVaR) is
defined as the Value-at-Risk (VaR) of a financial system or institution, given
that at least one other institution is equal or below its VaR. The VCoVaR
relaxes normality assumptions and is estimated via copula. While important
theoretical findings of the measure are detailed, the empirical study analyzes
how different distressing events of the cryptocurrencies impact the risk level
of each other. The results show that Litecoin displays the largest impact on
Bitcoin and that each cryptocurrency is significantly affected if an event of
joint distress among the remaining market participants occurs. The VCoVaR is
shown to capture domino effects better than other CoVaR extensions.

Keywords: copula, Conditional Value-at-Risk, cryptocurrency, systemic risk

##  1 Introduction

Various developments and crises over the last two decades, such as the
financial crisis of 2009, have demonstrated how volatile, fragile, and
interconnected the financial system and its institutions can be. This gives
rise to _systemic risk_ , which can be described as ‘the risk of the financial
system as a whole’ (Cao 2014, p. 2). The regulatory methodology focuses highly
on protecting the financial system against systemic risk events by identifying
globally systemically important financial institutions based on cross-
jurisdictional activities, size, interconnectedness, substitutability, and
complexity. These higher risk institutions are subject to higher loss
absorbency requirements, which are imposed next to general liquidity and risk-
based capital requirements (Basel Committee on Banking Supervision 2013).
However, the question of correctly quantifying systemic risk via appropriate
measures remains a crucial task and has developed into a highly researched
area. Classical univariate risk measures such as the VaR or the Expected
Shortfall are constructed to quantify the risk of an isolated institution or
asset class. Consequently, these univariate measures are unable to quantify
the impact of an institution’s distress on another institution or the whole
financial system. As a result, alternative multivariate measures which
overcome these limitations and are able to quantify the impact of the risk of
a financial institution on other institutions in a system need to be defined.

The last decade has seen a rise of a completely new, highly volatile, and
risky financial product known as Cryptocurrency (CC). The CC has recently
received increased attention in academia (Vidal-Tomás 2021; Petukhina et al.
2021). However, the economics of those financial assets are yet not well
understood, and the risks hidden in this system require thorough
investigation. The potential threats from CC have recently also been
recognized by regulating authorities, see Basel Committee on Banking
Supervision (2019). The following systemic risk discussion focuses solely on
CC as financial assets, but the methods are general and are applicable to
other inter-connected financial asset classes.

Bisias et al. (2012) and Benoit et al. (2017) provide extensive surveys of
current methodologies to quantify systemic risk. Among these several methods,
the most widely-applied market-based measure is the Conditional Value-at-Risk
(CoVaR) by Adrian and Brunnermeier (2016), which expands the approach of the
VaR to a conditional setting. The $CoVaR^{j|i}$ can be defined as a quantile
of the conditional return distribution of CC (or a system) $j$ given that the
CC $i$ is under distress, which means that if the usually stable Litecoin
(LTC) becomes risky, will this risk be transferred to Bitcoin (BTC)? Based on
that concept, Adrian and Brunnermeier (2016) define a measure called Delta-
CoVaR by taking the difference between $CoVaR^{j|i}$ with $i$ being exactly at
its VaR and with $i$ being in its median state, therefore highlighting the
strength of the effect. A list of further systemic risk measures has been
developed and analyzed by Girardi and Ergün (2013), Mainik and Schaanning
(2014), Acharya et al. (2017), and Brownlees and Engle (2017). Zhou (2010)
considers, among other measures, the Vulnerability Index (VI) that represents
the probability that the CC of interest violates its VaR under the condition
of at least one other CC violating its VaR. Several studies have expanded the
CoVaR measure to a multiple case by incorporating more than one variable in
the conditional event. Cao (2014) introduces the Multi-CoVaR (MCoVaR) with the
condition of several CCs being simultaneously in distress. Bernardi et al.
(2019) propose the System-CoVaR (SCoVaR), in which the conditional variables
are aggregated via their sum. Further extensions are detailed in Bernardi et
al. (2018), Di Bernardino et al. (2015), Bernardi et al. (2017), and
Bonaccolto et al. (2021).

The main goal of this paper is to formalise a flexible approach that allows to
capture a variety of distress events without having to specify a pre-specified
distressing situation of the given system, e.g., distress of a specific
element or group of elements. Therefore, complementary to SCoVaR and MCoVaR,
this empirical study proposes the Vulnerability-CoVaR (VCoVaR), which
translates the idea of the VI to the conditional quantile setting. The VCoVaR
is defined as the VaR of a CC (or the CC system) given there exists at least
one other CC being below or equal to its VaR. Copula-based estimation
strategies and characteristics for CoVaR and all investigated CoVaR extensions
(SCoVaR, MCoVaR, VCoVaR) are detailed and validated in a thorough simulation
study. CoVaR, MCoVaR, and VCoVaR are found to be equal in certain dependence
scenarios. Simulation-based analysis of the measures depending on the
dependence structure and intensity reveal the desirable property of the VCoVaR
of being a monotonically decreasing function of the dependence parameter for a
selected list of Archimedean copulae (AC). As an important by-product of this
research, a semi-automated univariate model selection procedure based on the
minimization of an information criterion while fulfilling the requirements on
the respective time series residuals is proposed, see Appendix B.

The paper is structured as follows: Section 2 illustrates why the VCoVaR is
particularly appropriate for the CC market and further motivates the use of
copula for estimation. Section 3 formally defines the measures and derives the
copula-based estimation. Section 4 investigates the properties of the risk
measures. Section 5 includes the simulation study, while Section 6 contains
the application study of CCs. Section 7 concludes. The `R` code to reproduce
the results from this paper will be published in a GitHub repository as soon
as the paper is accepted.

##  2 Systemic Risk in the Cryptocurrency Market

The literature identifies two highly relevant characteristic properties of the
CC market: the existence of significant spillover effects and the occurrence
of herding behaviour among CC market participants. The latter relates to the
phenomenon that investors tend to imitate each others transaction behaviour
instead of following their own information and belief basis (Hwang and Salmon
2004). The existence of spillover effects is displayed in Borri (2019), who
applies the CoVaR of Adrian and Brunnermeier (2016) based on quantile
regression to discover that CCs are highly exposed to tail-risk from other
CCs. Ji et al. (2019) use the methodology of Diebold and Yilmaz (2015) to
quantify return and volatility spillovers in the CC market. Pursuing a similar
methodological approach, Li et al. (2020) find that risk spillovers are
stronger in the direction from CCs with small market capitalization to those
with larger capitalization. Xu et al. (2021) run the TENET approach originally
developed in Härdle et al. (2016) to conclude that the market of CCs is coined
by significant effects of spillover risk and that the connectedness in the
market increased steadily over the course of time. Further spillover analysis
of the crypto-market can be found in Koutmos (2018), Luu Duc Huynh (2019), and
Katsiampa et al. (2019), while the empirical findings are greatly summarized
in the survey of Kyriazis (2019). Along with spillover effects, CCs also show
a strong behaviour of tail dependence, see Tiwari et al. (2020) and Xu et al.
(2021), which can be modelled using the copula method.

Regarding the existence of herding behaviour, a relevant contribution is Bouri
et al. (2019), who identify using the approach of Stavroyiannis and Babalos
(2017) significant herding effects whose intensity varies over time. Vidal-
Tomás et al. (2019) give evidence for herding effects during downward market
situations, based on the methodology of Chang et al. (2000) and Chiang and
Zheng (2010). They notice that the behaviour of the main CCs is crucial for
the investment decisions of traders. Ballis and Drakos (2020) and
Kallinterakis and Wang (2019) also follow the method of Chang et al. (2000)
and confirm the presence of herding effects, although detecting stronger
effects during upward market situations. Finally, Kyriazis (2020) contains a
survey about the empirical findings.

These two properties - spillover effects and herding behaviour - of the CC
market suggest that distress of a CC leads to subsequent distresses of other
CCs, and consequently, a domino effect might take place, increasing the
likelihood of a systemic risk event. Additionally, there is evidence that the
CC market can be primarily influenced by one dominant CC, for example BTC, as
stated in Smales (2020).

The VCoVaR is especially appropriate for the CC market because the measure is
tailored for quantifying tail-dependence and domino effects. For example, in
the case of extreme losses of Bitcoin (BTC) under the condition that at least
one of Ethereum (ETH), Litecoin (LTC), Monero (XMR), and Ripple (XRP) is under
distress, with the VCoVaR we capture all situations of such distress spreading
processes in the system. It is not necessary to define which CC initially was
under distress or how far the domino effect is already developed. The notion
of _at least one_ includes all possible scenarios and is hence more
appropriate in capturing domino effects than the existing alternatives CoVaR,
MCoVaR, and SCoVaR, which focus only on one pre-specified distress situation.
The use of copulae allows to model both tail dependencies and contagion risk,
with the latter being especially pronounced in this market with one dominant
CC. Consequently, the VCoVaR provides a flexible tool to depict the impact of
such systemic risk scenarios due to its natural consideration of the special
characteristics of the CC market.

##  3 Conditional Multivariate Risk Measures

###  3.1 Definitions

Before formally introducing the conditional measures, the univariate VaR
measure is reviewed. Let $X_{i,t}$ be the return of CC $i$ at time $t$. The
$VaR^{i}_{\alpha,t}$ at probability level $\alpha\in(0,1)$ is implicitly
defined as:

$P(X_{i,t}\leq VaR^{i}_{\alpha,t})=\alpha.$ (1)

If $X_{i,t}\sim F_{i,t}$, one can alternatively write
$VaR^{i}_{\alpha,t}=F_{i,t}^{-1}(\alpha)$, with $F_{i,t}^{-1}$ being the
generalized inverse of $F_{i,t}$, defined as
$F^{\leftarrow}(u)=\inf\\{x:F(x)\geq u\\}$.

Let $X_{j,t}$ be the return of CC (or the CC system) $j$ at time $t$. The
original Adrian and Brunnermeier (2016) $CoVaR^{=,j|i}_{\alpha,\beta,t}$ with
probability level $\beta$ for $j$ given $X_{i,t}$ equals its
$VaR^{i}_{\alpha,t}$ is defined as:

$P(X_{j,t}\leq
CoVaR^{=,j|i}_{\alpha,\beta,t}|X_{i,t}=VaR^{i}_{\alpha,t})=\beta,\hskip
28.45274pt\text{for}\hskip 5.69046ptj\neq i.$ (2)

The $CoVaR^{=,j|i}_{\alpha,\beta,t}$ is the quantile of the conditional return
distribution. Frequently applied probability levels in practice are
$\alpha=\beta=0.05$ or $\alpha=\beta=0.01$. We consider general cases with
$\alpha,\beta\in(0,1)$ for all measures. Girardi and Ergün (2013) modify (2)
by adding inequality to the condition:

$P(X_{j,t}\leq CoVaR^{j|i}_{\alpha,\beta,t}|X_{i,t}\leq
VaR^{i}_{\alpha,t})=\beta.$ (3)

It is argued that this definition is reasonable as it considers more extreme
distressing events of CC $i$ and gives the opportunity to apply standard
backtesting procedures, e.g., Kupiec (1995). Mainik and Schaanning (2014)
showed for selected bivariate distributions that the CoVaR in (2) is not a
monotonically increasing function of the dependence coefficient between
$(X_{j,t},X_{i,t})$, while the one in (3) is monotonically increasing. Note
that this translates into monotonically decreasing functions in our case, as
Mainik and Schaanning (2014) considered loss variables. This characteristic is
referred to as dependence consistency. More precisely, Theorem 3.6 in Mainik
and Schaanning (2014) guarantees the measure in (3) is dependence consistent
if $(X_{j,t},X_{i,t})$ follows a bivariate elliptical distribution or an
elliptical copula. Similar properties have been found for the Gumbel copula.

However, the relationships in the CC world are unlikely to be fully captured
with a bivariate distribution. It is necessary to find alternatives, including
several variables for the conditional event, to capture more complex scenarios
in which $p>1$ CCs are in distress. In the following, let
$X_{t}=(X_{1,t},\ldots,X_{p,t})^{\top}$ be the vector of returns of CCs, with
indices collected in the vector $\mathbf{i}=1,\ldots,p$ at time $t$ where $j$
is not part of these CCs. The first considered extension, the SCoVaR,
aggregates the variables in the conditional event by taking their sum and was
introduced in Bernardi et al. (2019). Building on this idea, the SCoVaR in
this paper is implicitly defined as follows:

######  Definition 3.1 (System-CoVaR).

Given the return $X_{j,t}$ of cryptocurrency/system $j$ and the returns
$X_{t}$ of cryptocurrencies $\mathbf{i}$, the SCoVaR is defined as:

$P\left\\{X_{j,t}\leq
SCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}\middle|\sum_{i=1}^{p}X_{i,t}\leq
VaR_{\alpha,t}\left(\sum_{i=1}^{p}X_{i,t}\right)\right\\}=\beta.$ (4)

Bernardi et al. (2019) impose the additional restriction that every variable
in the conditional event is below or equal its individual VaR, what leads to a
different form of (4), namely:

$P\left\\{X_{j,t}\leq
SCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}\middle|\sum_{i=1}^{p}X_{i,t}\leq
VaR_{\alpha,t}\left(\sum_{i=1}^{p}X_{i,t}\right),\forall i:X_{i,t}\leq
VaR^{i}_{\alpha,t}\right\\}=\beta.$

Building on their formulation, the authors find a generalization of the
Expected Shortfall measure, which is used to pursue a game theoretic approach
of risk allocation. However, this paper separates these naturally different
restrictions into the SCoVaR as in Definition 3.1 and the MCoVaR, which is
introduced in the following.

The MCoVaR is the second extension and was introduced in Cao (2014). This
measure covers cases when all $X_{i,t}$ are simultaneously equal or below
their $VaR^{i}_{\alpha,t}$ level. Thus, using probability levels $\alpha$ and
$\beta$, it is defined as:

######  Definition 3.2 (Multi-CoVaR).

Given the return $X_{j,t}$ of cryptocurrency/system $j$ and the returns
$X_{t}$ of cryptocurrencies $\mathbf{i}$, the MCoVaR is defined as:

$P(X_{j,t}\leq MCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}|\forall i:X_{i,t}\leq
VaR^{i}_{\alpha,t})=\beta.$ (5)

Although it is possible to consider different $\alpha$-levels for each
$X_{i,t}$ to balance individual effects, for simplicity, it is assumed that
all measures impose a common $\alpha$-level for the conditional variables. Cao
(2014) defines a measure of systemic risk contribution by taking the
difference of the MCoVaR as in (5) and the MCoVaR when the $X_{i,t}$ are at a
normal state. As for the SCoVaR, the aim of this paper is also to study the
properties and estimation of the MCoVaR given in (5).

Along these lines, we propose the VCoVaR, which is to the best of our
knowledge not existent in the current literature, although allowing for a new
perspective on systemic risk. It translates the idea of the VI of Zhou (2010)
into a conditional quantile setting. The VI was originally defined on loss
distributions and measures the probability of $X_{j,t}$ violating its VaR
given there exists at least one other CC violating its VaR. Transferring this
approach, the VCoVaR is implicitly defined as follows:

######  Definition 3.3 (Vulnerability-CoVaR).

Given the return $X_{j,t}$ of cryptocurrency/system $j$ and the returns
$X_{t}$ of cryptocurrencies $\mathbf{i}$, the VCoVaR is defined as:

$P(X_{j,t}\leq VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}|\exists i:X_{i,t}\leq
VaR^{i}_{\alpha,t})=\beta.$ (6)

This approach allows to cover a variety of distress events and naturally
generalizes the CoVaR of (3) and the MCoVaR of (5). It is straightforward to
see that the conditional event of the MCoVaR is a subset of the conditional
events of the VCoVaR. In a setting of positive dependencies, the distressing
event of the MCoVaR relates to the worst case covered in the VCoVaR, namely
all $X_{i,t}$ are below or equal to their VaR. On the other side, the VCoVaR
is able to cover situations that are less negative than the bivariate CoVaR.
Having, e.g., the return of three CCs LTC, XMR, and XRP as conditional
variables, the VCoVaR captures situations in which XMR violates its VaR while
LTC and XRP do not. This crypto market situation can be assessed more positive
than the one of the bivariate CoVaR with XMR in the conditional event as
additional positive information about LTC and XRP exist.

###  3.2 Estimation of Systemic Risk Measures

####  3.2.1 CoVaR Estimation

The original CoVaR of Adrian and Brunnermeier (2016) given in (2) was
estimated using a quantile regression approach (Koenker and Bassett Jr 1978).
Girardi and Ergün (2013) point out that - although the resulting
$CoVaR^{=,j|i}_{q,t}$ estimate is time-variant - the impact of $VaR^{i}_{q,t}$
on $CoVaR^{=,j|i}_{q,t}$ is constant, which is unlikely to be the case in
practice. In contrast, they propose to estimate their CoVaR modification based
on the bivariate distribution of $(X_{j,t},X_{i,t})$, thus rewrite (3) as:

$\frac{P(X_{j,t}\leq CoVaR^{j|i}_{\alpha,\beta,t},X_{i,t}\leq
VaR^{i}_{\alpha,t})}{P(X_{i,t}\leq VaR^{i}_{\alpha,t})}=\beta,$

which reduces to:

$P(X_{j,t}\leq CoVaR^{j|i}_{\alpha,\beta,t},X_{i,t}\leq
VaR^{i}_{\alpha,t})=\alpha\beta,$ (7)

as per definition $P(X_{i,t}\leq VaR^{i}_{\alpha,t})=\alpha$, see (1). On this
basis, the following three-step procedure was proposed for the estimation:

  1. Step 1:

Fit a suitable univariate time-series process (selected, e.g., through our
newly proposed procedure, see Section 6.1.2) to $X_{i,t}$ and estimate
$VaR^{i}_{\alpha,t}$.

  2. Step 2:

Estimate the bivariate conditional heteroscedasticity model (e.g. the DCC-
GARCH model of Engle 2002) to obtain an estimate of the time dependent
bivariate density $\hat{f_{t}}(x_{j,t},x_{i,t})$ with observations $x_{j,t}$,
$x_{i,t}$ of $X_{j,t}$, $X_{i,t}$ with $i=1,\ldots,p$.

  3. Step 3:

Solve for $CoVaR^{j|i}_{\alpha,\beta,t}$ the equation:

$\int_{-\infty}^{CoVaR^{j|i}_{\alpha,\beta,t}}\int_{-\infty}^{VaR^{i}_{\alpha,t}}\hat{f_{t}}(x_{j,t},x_{i,t})dx_{j,t}dx_{i,t}=\alpha\beta.$
(8)

This procedure might be computationally demanding as it involves numerical
evaluation of a double integral. Overcoming this issue, we base our estimation
on copulae. Copulae are multivariate distribution functions with margins being
$\mathcal{U}[0,1]$, see Joe (2014). Copulae give the opportunity to specify
the dependence structure of random variables in a flexible way, allowing to go
beyond the commonly applied multivariate Gaussian and $t$-distribution. This
is also handy because CC returns are even less normal than fiat stocks, see,
e.g. Szczygielski et al. (2020) for an extensive investigation of proper CC
return distributions.

To estimate the CoVaR as given in (7), Reboredo and Ugolini (2015) express the
bivariate distribution function $F_{X_{j,t},X_{i,t}}$ of $(X_{j,t},X_{i,t})$
as:

$\displaystyle P(X_{j,t}\leq CoVaR^{j|i}_{\alpha,\beta,t},X_{i,t}\leq
VaR^{i}_{\alpha,t})$
$\displaystyle=F_{X_{j,t},X_{i,t}}(CoVaR^{j|i}_{\alpha,\beta,t},VaR^{i}_{\alpha,t})$
$\displaystyle=C_{X_{j,t},X_{i,t}}\\{F_{X_{j,t}}(CoVaR^{j|i}_{\alpha,\beta,t}),F_{X_{i,t}}(VaR^{i}_{\alpha,t});\theta_{t}\\},$

using the Sklar (1959) theorem. $F_{X_{j,t}}$ and $F_{X_{i,t}}$ denote the
marginal distributions of $X_{j,t}$ and $X_{i,t}$, respectively.
$C_{X_{j,t},X_{i,t}}$ refers to the copula function with parameter
$\theta_{t}$. The $CoVaR^{j|i}_{\alpha,\beta,t}$ is estimated by solving:

$C_{X_{j,t},X_{i,t}}\\{F_{X_{j,t}}(CoVaR^{j|i}_{\alpha,\beta,t}),\alpha;\theta_{t}\\}=\alpha\beta,$
(9)

which uses $F_{X_{i,t}}(VaR^{i}_{\alpha,t})=\alpha$. Note that in the case of
AC (9) can be solved analytically for $CoVaR^{j|i}_{\alpha,\beta,t}$, see
Karimalis and Nomikos (2018). Another crucial advantage is that it is not
necessary to estimate the VaR of the conditional variable beforehand (Reboredo
and Ugolini 2015). To compute (9), it is sufficient to estimate the copula and
the marginal distribution of $X_{j,t}$. This estimation strategy is
transferred to the SCoVaR of (4). Although it involves information of $p$
conditional variables, it can be estimated using (9) while replacing $X_{i,t}$
with $\sum_{i=1}^{p}X_{i,t}$ for estimating the copula between $X_{j,t}$ and
$\sum_{i=1}^{p}X_{i,t}$.

####  3.2.2 MCoVaR Estimation

Set $VaR_{\alpha,t}=(VaR_{\alpha,t}^{1},\ldots,VaR_{\alpha,t}^{p})^{\top}$,
where $X_{t}\leq VaR_{\alpha,t}$ holds componentwise. Furthermore, set
$\bm{\alpha}=(\alpha,\ldots,\alpha)^{\top}$ and
$F_{X_{t}}(VaR_{\alpha,t})=\\{F_{X_{1,t}}(VaR^{1}_{\alpha,t}),\ldots,F_{X_{p,t}}(VaR^{p}_{\alpha,t})\\}^{\top}$.
To estimate the MCoVaR, (5) can be rewritten as:

$\frac{P(X_{j,t}\leq MCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t},X_{t}\leq
VaR_{\alpha,t})}{P(X_{t}\leq VaR_{\alpha,t})}=\beta.$ (10)

Similar to the procedure of Girardi and Ergün (2013), Cao (2014) computes the
individual VaR for each CC and assumes a parametric form of the
$(p+1)$-dimensional distribution for all involved variables. On this basis,
the denominator of (10) can be computed, leading to an expression with a
multiple integral with $p+1$ variables and the MCoVaR as the only unknown.
This is solved numerically, and Cao (2014) assumes a multivariate
$t$-distribution driving the overall dependency in the application.

Furthermore, (10) is given in terms of copulae by:

$\frac{C_{X_{j,t},X_{t}}\\{F_{X_{j,t}}(MCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}),F_{X_{t}}(VaR_{\alpha,t});\theta_{1,t}\\}}{C_{X_{t}}\\{F_{X_{t}}(VaR_{\alpha,t});\theta_{2,t}\\}}=\beta,$

leading to:

$\frac{C_{X_{j,t},X_{t}}\\{F_{X_{j,t}}(MCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}),\bm{\alpha};\theta_{1,t}\\}}{C_{X_{t}}(\bm{\alpha};\theta_{2,t})}=\beta,$
(11)

where $\bm{\alpha}=(\alpha,\ldots,\alpha)^{\top}\in(0,1)^{p}$.
$C_{X_{j,t},X_{t}}$ denotes the $(p+1)$-dimensional copula of
$(X_{j,t},X_{t})=(X_{j,t},X_{1,t},\ldots,X_{p,t})$ with the parameter
$\theta_{1,t}$. $C_{X_{t}}$ refers to the $p$-dimensional copula of $X_{t}$
with parameter $\theta_{2,t}$. This expression can be solved, as the MCoVaR is
the only unknown term. Consequently, it is sufficient to have an appropriate
estimate of the copulae and the marginal distribution $F_{X_{j,t}}$. One can
assume different structures for the copulae enabling different interpretations
of the gained MCoVaR, which will be detailed in Section 4.2. In practice,
$C_{X_{j,t},X_{t}}$ is estimated from the data, and the copula $C_{X_{t}}$ is
gained through marginalization, setting
$C_{X_{t}}(\bm{\alpha};\theta_{2,t})=C_{X_{j,t},X_{t}}(1,\bm{\alpha};\theta_{1,t})$,
from the grounding property (Nelsen 2006). Notice that (11) yields an analytic
solution for specific copula families. Let $\varphi_{\theta}$ be a generator
function for an AC with parameter $\theta$ and $\varphi_{\theta}^{-1}$ the
corresponding inverse. Let $C_{X_{t}}$ be some copula under the assumption,
that
$C_{X_{j,t},X_{t}}(u_{j},u_{1},\ldots,u_{p};\theta_{1,t})=\varphi_{\theta_{1,t}}^{-1}\left[\varphi_{\theta_{1,t}}\left\\{u_{j}\right\\}+\varphi_{\theta_{1,t}}\left\\{C_{X_{t}}(u_{1},\ldots,u_{p};\theta_{2,t})\right\\}\right]$
with $(u_{j},u_{1},\ldots,u_{p})^{\top}\in[0,1]^{p+1}$ is the proper copula
function. Simplification of (11) yields:

$MCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}=F^{-1}_{X_{j,t}}\left(\varphi_{\theta_{1,t}}^{-1}\left[\varphi_{\theta_{1,t}}\\{C_{X_{t}}(\bm{\alpha};\theta_{2,t})\beta\\}-\varphi_{\theta_{1,t}}\\{C_{X_{t}}(\bm{\alpha};\theta_{2,t})\\}\right]\right).$
(12)

Special cases are where $C_{X_{j,t},X_{t}}$ is an AC or a Hierarchical
Archimedean copula (HAC), see Okhrin et al. (2013). In the former case, (12)
transforms to:

$MCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}=F^{-1}_{X_{j,t}}\left\\{\varphi_{\theta_{1,t}}^{-1}\left(\varphi_{\theta_{1,t}}\left[\varphi_{\theta_{1,t}}^{-1}\left\\{p\varphi_{\theta_{1,t}}(\alpha)\right\\}\beta\right]-p\varphi_{\theta_{1,t}}(\alpha)\right)\right\\},$

where the dependence intensity is expressed via the AC parameter
$\theta_{1,t}$. Furthermore, from (11) follows that
$MCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}=CoVaR^{j|i}_{\alpha,\beta,t}$ for
$p=1$, as (11) reduces to (9).

####  3.2.3 VCoVaR Estimation

In the following, the copula-based representation of the VCoVaR of (6) is
derived. Set $\bm{1-\alpha}=(1-\alpha,\ldots,1-\alpha)^{\top}$ of length $p$.

######  Lemma 3.1.

The VCoVaR defined in (6) is equivalent to:

$\frac{F_{X_{j,t}}(VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t})-\bar{C}_{X_{t}}(\bm{1-\alpha};\theta_{2,t})+\bar{C}_{X_{j,t},X_{t}}\\{1-F_{X_{j,t}}(VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}),\mathbf{1-\bm{\alpha}};\theta_{1,t}\\}}{1-\bar{C}_{X_{t}}(\bm{1-\alpha};\theta_{2,t})}=\beta,$
(13)

where $\bar{C}_{X_{j,t},X_{t}}$ denotes the $(p+1)$-dimensional survival
copula associated with $(X_{j,t},X_{t})$ and parameter $\theta_{1,t}$ and
$\bar{C}_{X_{t}}$ denotes the $p$-dimensional survival copula of $X_{t}$,
characterized by a parameter $\theta_{2,t}$.

Proofs of all lemmata are provided in Appendix A. The VCoVaR is estimated by
solving (13). The estimation approach includes survival functions, which can -
analogously to the case of distribution functions - be decomposed using
survival copulae, see Georges et al. (2001). A crucial characteristic is that
the survival copula of a pair of random variables is the 180 degrees rotated
version of its copula, similar holds for any dimension. In consequence, there
exists a direct relationship between $C$ and $\bar{C}$. This allows estimation
of the involved survival copulae in practice as follows: first, estimate the
copula $C_{X_{j,t},X_{t}}$ from the data and rotate it to find
$\bar{C}_{X_{j,t},X_{t}}$. Second, extract $\bar{C}_{X_{t}}$ through
marginalization, setting
$\bar{C}_{X_{t}}(\bm{1-\alpha};\theta_{2,t})=\bar{C}_{X_{j,t},X_{t}}(1,\bm{1-\alpha};\theta_{1,t})$.
The essential point is again the sufficiency of having an estimate of the
(survival) copulae and the marginal distribution $F_{X_{j,t}}$. Additionally,
the considered risk measures in (9), (11), and (13) are continuous
transformations of these estimators. Therefore all asymptotic distributional
properties of the risk measures are directly determined by application of the
delta method, see Oehlert (1992).

The following Lemma shows the equivalence between the copula-based
representations of CoVaR and VCoVaR if only one conditional variable is
considered.

######  Lemma 3.2.

Given (9) and (13), it holds that
$VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}=CoVaR^{j|i}_{\alpha,\beta,t}$ if
$p=1$.

##  4 Properties of Systemic Risk Measures

###  4.1 Independence and Perfect Dependence

Let us establish a connection between the CoVaR measures introduced previously
if all variables are independent or perfectly positive dependent. The case of
perfect negative dependence is not considered, as countermonotonicity in
higher dimensions is problematic.

######  Lemma 4.1.

Let $X_{j,t},X_{1,t},\ldots,X_{p,t}$ be independent, given expressions (9),
(11), and (13), it holds that
$CoVaR^{j|i}_{\alpha,\beta,t}=MCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}=VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}=F_{X_{j,t}}^{-1}(\beta)$.

This observation is reasonable, as in the case of independence, the
conditional probabilities of (3), (5) and (6) equal the respective
unconditional probability, which results in the VaR at level $\beta$.
Transferred to the market of CCs, in the case of independence, an extreme
event for a crypto return under condition is completely irrelevant for the
crypto return $X_{j,t}$.

######  Lemma 4.2.

Let $X_{j,t},X_{1,t},\ldots,X_{p,t}$ be perfectly positive dependent, given
expressions (9), (11), and (13), it holds that:
$CoVaR^{j|i}_{\alpha,\beta,t}=MCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}=VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}=F_{X_{j,t}}^{-1}(\alpha\beta)$.

All conditional measures equal the VaR at level $\alpha\beta$ in the given
scenario, being directly influenced by the VaR level of the conditional
variables, which means that in the case of perfect positive dependence, it is
sufficient to consider one conditional crypto return, as additional CCs would
not generate any additional information.

###  4.2 General Positive Dependencies

We want to gain further understanding of the measures based on (9), (11), and
(13). Two major objectives are pursued. First, to detect the general behaviour
of the measures as the function of the dependence parameter for a given
copula. Second, to investigate differences between different copula families.
This is realized by solving (9), (11), and (13) for a range of copula
parameters. The marginal distribution $F_{X_{j,t}}$ is set to be standard
normal.

For the bivariate CoVaR of (9), the Gaussian, $t$, Clayton, and Gumbel copulae
are chosen, while we consider two approaches to analyze the MCoVaR of (11) and
the VCoVaR of (13). Generally, we set $p=2$. First, the copula
$C_{X_{j,t},X_{1,t},X_{2,t}}(u_{1},u_{2},u_{3})$ is assumed to be Clayton or
Gumbel, thus belonging to the AC family. We excluded the Gaussian and
$t$-copula as they require more correlation parameters in higher dimensions,
and we want to start the analysis by varying one dependence parameter at a
time. The copula of $(X_{1,t},X_{2,t})$ is attained by marginalization:
$C_{X_{1,t},X_{2,t}}(u_{2},u_{3})=C_{X_{j,t},X_{1,t},X_{2,t}}(1,u_{2},u_{3})$.
As a consequence, the measure depends only on one parameter for the Clayton
and Gumbel copula and can be visualized comparable to the bivariate CoVaR
case. This allows interpreting how the measure changes if the dependence of
$(X_{j,t},X_{1,t},X_{2,t})$ as a whole changes. Second, the copula of
$(X_{j,t},X_{1,t},X_{2,t})$ is assumed to be a HAC. The idea of a HAC is to
nest AC in a hierarchical structure to allow for a more flexible specification
of the dependence structure, as the property of AC of having one parameter
materializes in practice often as a limitation. The following structure is
imposed:
$C_{X_{j,t},X_{1,t},X_{2,t}}(u_{1},u_{2},u_{3})=C_{1}\\{u_{1},C_{2}(u_{2},u_{3})\\}.$
The bivariate copula $C_{2}$ describes the dependency inside the conditional
variables, while $C_{1}$ describes the dependency between the target variable
and the conditional variables. Notice that for the VCoVaR each copula is
rotated, as the measure is based on survival copulae.

For comparability, each copula parameter is converted into Kendall’s $\tau$
(Joe 2014). As in Section 4.1, only positive dependencies with Kendall’s
$\tau\in[0,1]$ are considered, although $\tau$ does not range over the whole
domain in some cases as numerical issues at the limits appeared. Note that
$\tau<0$ results in negative pair-dependencies, which is controversial in
dimensions $d>2$. The computations in the following are realized using the
R-packages `copula` (Kojadinovic and Yan 2010) and `HAC` (Okhrin and Ristig
2014).

####  4.2.1 CoVaR Properties

Starting with the bivariate CoVaR, Figure 1 shows the measure depending on the
selected copula and Kendall’s $\tau$. The probability levels
$\alpha=\beta=0.05$ and $\alpha=\beta=0.01$ are considered. In general, the
CoVaR decreases monotonically for all copulae if Kendall’s $\tau$ increases.
This is consistent with the findings of Mainik and Schaanning (2014). For
example, this implies the stronger LTC depends on XRP, the stronger a
distressing event of XRP will impact LTC. Furthermore, the results of Section
4.1 are special cases for those copulae where independence ($\tau=0$) and
perfect positive dependence ($\tau=1$) are attained. Given the standard normal
distribution for the margins, these are $F_{X_{j,t}}^{-1}(\beta)\approx-1.645$
and $F_{X_{j,t}}^{-1}(\alpha\beta)\approx-2.807$, respectively, for
$\alpha=\beta=0.05$. The CoVaR converges for all copulae towards these
theoretical limits, except for the $t$-copula if $\tau\rightarrow 0$. This is
reasonable as $\tau=0$ restricts the correlation of the $t$-copula to be 0,
but does not affect the degrees of freedom $\nu$. However, the $t$-copula
converges towards the Gaussian one if $\nu\rightarrow\infty$ (Eling and Toplek
2009). This is reflected in Figure 1, as with increasing $\nu$ the curve of
the $t$-copula comes closer to the one of the Gaussian copula. The curve of
the Clayton copula decreases very fast, while the curves of the Gumbel and
Gaussian copulae decrease slowly. Thus, the Clayton copula leads to more
conservative estimates of the CoVaR and should produce in an application fewer
exceedances than the other copulae.

Figure 1: Bivariate CoVaR for different copulae and standard normal margins.
The upper grey line corresponds to the independence case
$F_{X_{j,t}}^{-1}(\beta)$ and the lower one to perfect positive dependence
$F_{X_{j,t}}^{-1}(\alpha\beta)$, as derived in Section 4.1.

####  4.2.2 MCoVaR and VCoVaR Properties

Figure 2 shows the results of the first approach with three-dimensional AC.
The figure also contains the theoretical limits of Section 4.1, and both the
MCoVaR and the VCoVaR converge towards them if $\tau\rightarrow 0$ or
$\tau\rightarrow 1$. This is the case for both considered probability levels.
However, the MCoVaR is not a monotonically decreasing function of Kendall’s
$\tau$. For the Clayton copula, the MCoVaR achieves its minimum for
$\tau\approx 0.2$, while for the Gumbel copula it is around $\tau\approx
0.75$. Thus, the MCoVaR measure of (11) does not reflect the behaviour of the
bivariate CoVaR for the given copula specification. Nevertheless, it is
reasonable to detect lower values of the MCoVaR in comparison to the CoVaR, as
the conditioning event describes a worse market situation. In contrast to the
MCoVaR, the VCoVaR is a monotonically decreasing function of Kendall’s $\tau$.
Thus the measure decreases if the dependency of $(X_{j,t},X_{1,t},X_{2,t})$ as
a whole intensifies. Furthermore, the curve of the Clayton copula again
decreases faster than the one of the Gumbel copula. Transferring this to
practice, we can expect the Clayton copula to lead to more conservative
estimates for the VCoVaR in the empirical part.

Figure 2: MCoVaR and VCoVaR assuming Archimedean copulae for $C_{X_{j,t},X_{1,t},X_{2,t}}$. The upper grey lines correspond to the independence case $F_{X_{j,t}}^{-1}(\beta)$ and the lower ones to perfect positive dependency $F_{X_{j,t}}^{-1}(\alpha\beta)$, as derived in Section 4.1. |
---|---
(a) Multi-CoVaR | (b) Vulnerability-CoVaR
Figure 3: MCoVaR and VCoVaR assuming Hierarchical Archimedean copulae with
$C_{X_{j,t},X_{1,t},X_{2,t}}(u_{1},u_{2},u_{3})=C_{1}\\{u_{1},C_{2}(u_{2},u_{3})\\}$.
$\tau_{1}$ refers to $C_{1}$ and $\tau_{2}$ to $C_{2}$. $\alpha=\beta=0.05$.

The second approach using the HAC allows decomposing these characteristics
even further. Figure 3 shows the surfaces for Clayton and Gumbel generators if
a HAC is assumed. Regarding notation, the figure contains $\tau_{1}$ and
$\tau_{2}$, referring to $C_{1}$ and $C_{2}$, respectively. Note that a HAC is
required to fulfil the sufficient nesting condition in order to be a proper
copula (Okhrin et al. 2013). This condition corresponds to
$\tau_{1}\leq\tau_{2}$ for considered copulae, the area in the surface plots
violating this condition remains empty. The edges of the surfaces at
$\tau_{1}=\tau_{2}$ in Figure 3 equal the corresponding curves in Figure 2, as
the HAC is the equivalent with the AC of the first approach.

The MCoVaR decreases for both copulae if the dependency between the target
variable and the conditional variables, represented through $\tau_{1}$,
intensifies. For example, if BTC is the variable of interest and XRP and XMR
are under condition, the impact of joint distresses of XRP and XMR on BTC
increases if BTC has a larger dependency with XRP and XMR. This observation is
reasonable and justifies the application of the MCoVaR. However, this mainly
reflects the dependence consistency of the bivariate CoVaR. Given a fixed
$\tau_{2}$, the value $C_{2}(u_{2},u_{3})$ can be calculated with
$u_{2},u_{3}$ set to $\alpha$, as seen from (11). In consequence, the analysis
with varying $\tau_{1}$ becomes analogous to a bivariate CoVaR analysis with
an $\alpha$-level adjusted by the dependency between $X_{1,t}$ and $X_{2,t}$.
If on the other side $\tau_{1}$ is fixed and $\tau_{2}$ increases, the MCoVaR
is much less affected and tends to increase. Regarding our example with BTC,
XMR, and XRP, this could potentially be interpreted as follows: if the
conditional CCs XMR and XRP only have a weak dependence, the conditional event
in (5) will be unlikely. If such a joint event still happens, the market
situation will be devastating and the MCoVaR needs to be very small. If in the
other case, XRP and XMR depend highly on each other, distress of XRP suggests
distress of XMR and vice versa. The fulfilment of the condition is a
consequence of the dependency between XRP and XMR and not of the situation of
the overall market. Thus, the values of the MCoVaR are slightly higher.
However, the overall impact of $\tau_{1}$ on the MCoVaR is of primary
importance and determines the general behaviour of the measure.
Differentiating between the two considered copulae shows the Clayton copula
decreases faster in $\tau_{1}$, although there is an overlapping of the two
surfaces in Figure 3(a) for high $\tau_{1},\tau_{2}$.

The VCoVaR using the HAC structure in Figure 3(b) decreases if $\tau_{1}$
increases. This observation confirms that the VCoVaR as calculated in (13), is
a reasonable extension of the bivariate CoVaR measure. In contrast to the
MCoVaR, the VCoVaR decreases if the dependency between the conditional
variables, expressed through $\tau_{2}$, increases. This could be explained
via different conditional events. Consider the example of BTC with XRP and XMR
under condition again. If the dependency between XRP and XMR increases, the
probability that the conditional event of the VCoVaR in (6) yields a bad
scenario, namely both CCs are in distress, increases. Consequently, the VCoVaR
needs to decrease to capture these potentially worse situations. However, the
impact of $\tau_{2}$ on the VCoVaR is less pronounced than the one of
$\tau_{1}$. Finally, the Clayton copula produces again smaller VCoVaR values
than the Gumbel copula.

##  5 Simulation Study

A simulation study is performed to test whether (9), (11), and (13) can
reliably calculate the respective measure if the true copula is known, and no
temporal dependency is present. The study is performed as follows: _Step 1:_
Assume $(X_{j,t},X_{i,t})$ for the CoVaR or $(X_{j,t},X_{1,t},X_{2,t})$ for
the M- and VCoVaR follow a certain copula with dependence parameter $\tau$.
_Step 2:_ Sample $n=10\,000$ iid observations from the copula with margins
being $\mathcal{U}[0,1]$ and estimate the assumed copula via maximum
likelihood (ML). _Step 3:_ Compute the respective conditional measure through
(9), (11), and (13), and the VaR of the conditional variables as the empirical
$\alpha$-quantile. _Step 4:_ Compute the violation rate of the respective
measure using the sample equivalents of $\beta$ from the definitions in (3),
(5), and (6), respectively. _Step 5:_ Repeat Steps 2 to 4 for $N=100$ times
and calculate the average violation rate.

Table 1: Average violation rates for CoVaR, MCoVaR, and VCoVaR estimation. Measure | $\tau=0.25$ | $\tau=0.50$ | $\tau=0.75$
---|---|---|---
Clayton | Gumbel | Clayton | Gumbel | Clayton | Gumbel
| $\alpha=\beta=0.05$
CoVaR | 0.0489 | 0.0493 | 0.0499 | 0.0507 | 0.0506 | 0.0506
MCoVaR | 0.0491 | 0.0516 | 0.0501 | 0.0501 | 0.0490 | 0.0480
VCoVaR | 0.0503 | 0.0505 | 0.0507 | 0.0495 | 0.0496 | 0.0502
| $\alpha=\beta=0.01$
CoVaR | 0.0102 | 0.0082 | 0.0102 | 0.0089 | 0.0093 | 0.0085
MCoVaR | 0.0080 | 0.0062 | 0.0082 | 0.0112 | 0.0101 | 0.0117
VCoVaR | 0.0110 | 0.0099 | 0.0115 | 0.0113 | 0.0091 | 0.0089

The Gumbel and Clayton copulae are analyzed with Kendall’s
$\tau\in\\{0.25,0.50,0.75\\}$. The sample equivalents of $\beta$ in Step 4 are
calculated by considering only the simulated observations which fulfill the
conditional event. Of those observations, the number of violations of the
respective CoVaR measure is computed. The equivalent of $\beta$ is then the
ratio between the latter and the number of observations fulfilling the
conditional event, which were considered in the first stage. This is
equivalent to the procedure of Mainik and Schaanning (2014), in which the
bivariate CoVaR was analyzed. Table 1 shows that for both selected probability
levels, both copulae, and all values of Kendall’s $\tau$, the violation rates
are close to nominal level $\beta$. Concluding this simulation study, the
copula-based (9), (11), and (13) are able to reliably compute the respective
measure if the correct copula is assumed for the data-generating process.

##  6 Empirical Study

###  6.1 In-sample Estimation

####  6.1.1 Proceeding and Data Investigation

The study uses five CCs BTC, ETH, LTC, XMR, and XRP. These five CCs constitute
64.54% (https://coinmarketcap.com/, accessed: 07/12/2021) of the overall CC
market capitalization and offer relatively long time series compared to other
CCs, thus providing a sufficient database. The data contains daily closing
prices in USD stemming from the Community Network Data kindly provided by
CoinMetrics (https://coinmetrics.io/, accessed: 01/12/2021). The sample
includes $n=2\,283$ observations from 01/09/2015 to 30/11/2021 as CCs are
traded every day, including weekends. For the analysis, the prices are
transformed in log-returns. We calculate: (1) The bivariate CoVaR for all
possible combinations of the five CC; (2) The SCoVaR, MCoVaR, and VCoVaR of
each CC if the remaining four CCs are treated as conditional variables. We
fix: $\alpha=\beta=0.05$. Figure 4 shows the prices (scaled to [0,1]) and the
log-returns. Descriptive statistics, tests, and estimates of Kendall’s $\tau$
for the log-returns are given in Appendix B.

Figure 4: Standardized prices and log-returns of the cryptocurrencies.

To estimate the different measures of (3), (4), (5), and (6), the following
empirical procedure is used: _Step 1:_ Estimate the respective marginal model
for each time series separately. _Step 2:_ Perform the parametric probability
integral transformation on residuals to obtain iid $\mathcal{U}[0,1]$ data.
_Step 3:_ Estimate the copula based on the resulting pseudo-sample
observations. _Step 4:_ Solve (9), (11), or (13), respectively.

####  6.1.2 Univariate Models

Empirical results from the literature indicate that CC returns exhibit
characteristics as volatility clustering, fat tails, and leverage effects, see
Zhang et al. (2018) and Phillip et al. (2018). To account for these dynamics
and building on the stationarity assumption, the margins are assumed to follow
an autoregressive moving-average (ARMA) model for the conditional mean
$\mu_{t}$ and a GJR-GARCH model (Glosten et al. 1993) for the conditional
variance $\sigma_{t}^{2}$. For example, when $X_{j,t}$ denotes the log-return
of a CC at time $t$, the full ARMA($p_{l}$, $q_{l}$)-GJR-GARCH(P,Q) model can
be outlined as follows (Ghalanos 2020):

$\displaystyle
X_{j,t}=\mu_{t}+\varepsilon_{t},\quad\mu_{t}=\mu+\sum_{i=1}^{p_{l}}\phi_{i}(X_{j,t-i}-\mu)+\sum_{j=1}^{q_{l}}\psi_{j}\varepsilon_{t-j},\quad\varepsilon_{t}=\sigma_{t}z_{t},$
$\displaystyle\sigma_{t}^{2}=\omega+\sum_{i=1}^{P}(\lambda_{i}+\gamma_{i}\mathbf{1}_{\\{\varepsilon_{t-i}\leq
0\\}})\varepsilon_{t-i}^{2}+\sum_{j=1}^{Q}\delta_{j}\sigma_{t-j}^{2},$ (14)

with iid $z_{t}\sim F_{z}(0,1)$ and $\mathbf{1}_{\\{\cdot\\}}$ being an
indicator function. This results in the ability of the GJR-GARCH specification
to model positive and negative shocks, represented by $\varepsilon_{t}$,
differently and accounts for the leverage effect (Ghalanos 2020). For $F_{z}$,
the skew-$t$ distribution with skewness $\zeta$ and shape $\nu$ of Fernández
and Steel (1998) is selected, as the time series exhibited a strong indication
of non-normality and skewness. For parameter constraints and additional
remarks on the model, see Glosten et al. (1993). In the context of this
application, the ARMA-GJR-GARCH model acts as a filter for temporal
dependencies inside the time series and the empirical counterparts of $z_{t}$
are extracted for further modeling. These standardized residuals $\hat{z}_{t}$
should be as serially independent as possible. In addition, it might be
unnecessary to specify the model as presented above, and a more parsimonious
version would be sufficient to capture the dynamics of the data. Considering
these facts, we use a semi-automated process which selects the best fitting
model according to an information criterion while fulfilling necessary
requirements on the residuals, for details see Appendix B.

Building on the selected univariate models, the VaR for each time series at
level $\alpha$ can be calculated parametrically as described in a forecasting
context in Kuester et al. (2006):

$\widehat{VaR}_{\alpha,t}=\hat{\mu}_{t}+\hat{\sigma}_{t}F_{z}^{-1}(\alpha|\hat{\zeta},\hat{\nu}),$
(15)

where $\hat{\mu}_{t},\hat{\sigma}_{t},\hat{\zeta},\hat{\nu}$ are the estimates
from the fitted univariate model. The generated in-sample VaR estimates are
necessary for evaluation and comparison with the conditional measures. To
validate the VaR estimates, Table 2 shows the realized equivalents of $\alpha$
in (1) and the absolute number of observations, in which the respective log-
return was equal or below the VaR estimate. The rates are close to
$\alpha=0.05$ in all cases, indicating the accuracy of the models.

| BTC | ETH | LTC | XMR | XRP | Sys:BTC | Sys:ETH | Sys:LTC | Sys:XMR | Sys:XRP
---|---|---|---|---|---|---|---|---|---|---
Rate in $\%$ | 6.00 | 5.43 | 5.35 | 4.78 | 5.04 | 4.69 | 4.86 | 4.73 | 5.17 | 5.04
Number | 137 | 124 | 122 | 109 | 115 | 107 | 111 | 108 | 118 | 115

  * •

NOTE: Sys:BTC denotes the sum of the log-returns of ETH, LTC, XMR, and XRP.
For the selected $\alpha=0.05$ we should observe 114.1 violations.

Table 2: Violations for the VaR estimates.

####  6.1.3 Copula Models

With the univariate models being estimated, the standardized residuals
$\hat{z}_{t}$ are parametrically transformed to $\mathcal{U}[0,1]$, denoted
$u_{i,t}=F_{z}(\hat{z}_{i,t}|\hat{\zeta}_{i},\hat{v}_{i})$ with
$i\in\\{1,\ldots,5\\}$ for BTC, ETH, LTC, XMR, and XRP. Based on these pseudo-
observations, the copulae are estimated. The following time-invariant copula
models are chosen for the application: Gaussian, $t$, Clayton, and Gumbel.
Time-invariance relates to having the same copula parameter at every point of
time $t$. In this case, the measures in (9), (11), and (13) become time-
variant only through the dynamic nature of the univariate model of BTC, while
the dependencies are assumed to be constant. However, it might be beneficial
to investigate time-variant copulae to capture the potential dynamics of the
dependencies, which is why we incorporate the dynamic model of Patton (2006)
and the DCC-copula approach of Jin (2010). Detailed descriptions of these
models alongside resulting parameter estimates of all copulae can be found in
Appendix B.

####  6.1.4 Estimates of the Systemic Risk Measures

Building on the estimated marginal distributions and copulae, (9), (11), and
(13) are solved for the systemic risk measures. Table 3 shows descriptive
statistics alongside violation rates, which are the sample equivalents of
$\beta$ and computed similar to Section 5. Figures 5 and 6 illustrate selected
measures when the log-return of BTC is $X_{j,t}$. Appendix B contains further
plots.

Measure |  Mean ($t$-copula) |  Sd ($t$-copula) | Violation rates for copula
---|---|---|---
Gaussian | $t$ | Clayton | Gumbel |  Patton-$t$ |  DCC-$t$
BTC-VaR | -0.0545 | 0.0271 | - | - | - | - | - | -
BTC-ETH | -0.1641 | 0.0791 | 0.0645 | 0.0484 | 0.0403 | 0.1774 | 0.0645 | 0.0645
BTC-LTC | -0.1720 | 0.0829 | 0.0574 | 0.0492 | 0.0492 | 0.1066 | 0.0492 | 0.0574
BTC-XMR | -0.1642 | 0.0792 | 0.0917 | 0.0642 | 0.0550 | 0.2477 | 0.0734 | 0.0826
BTC-XRP | -0.1609 | 0.0776 | 0.1130 | 0.0609 | 0.0522 | 0.2435 | 0.0696 | 0.0957
BTC-SCoVaR | -0.1694 | 0.0816 | 0.0654 | 0.0561 | 0.0561 | 0.1776 | 0.0654 | 0.0654
BTC-MCoVaR | -0.2819 | 0.1352 | 0.0263 | 0.0263 | 0.0263 | 0.1316 | - | 0.0263
BTC-VCoVaR | -0.1309 | 0.0633 | 0.1070 | 0.0576 | 0.0412 | 0.1358 | - | 0.0741
ETH-VaR | -0.0860 | 0.0350 | - | - | - | - | - | -
ETH-BTC | -0.2442 | 0.0972 | 0.0657 | 0.0292 | 0.0219 | 0.1387 | 0.0511 | 0.0365
ETH-LTC | -0.2472 | 0.0984 | 0.0738 | 0.0328 | 0.0246 | 0.1475 | 0.0410 | 0.0410
ETH-XMR | -0.2455 | 0.0977 | 0.0826 | 0.0367 | 0.0275 | 0.1560 | 0.0459 | 0.0459
ETH-XRP | -0.2458 | 0.0979 | 0.0870 | 0.0348 | 0.0261 | 0.1652 | 0.0609 | 0.0609
ETH-SCoVaR | -0.2533 | 0.1008 | 0.0541 | 0.0270 | 0.0270 | 0.1171 | 0.0270 | 0.0360
ETH-MCoVaR | -0.4026 | 0.1596 | 0.0227 | 0.0227 | 0.0227 | 0.0682 | - | 0.0227
ETH-VCoVaR | -0.1950 | 0.0779 | 0.0753 | 0.0460 | 0.0418 | 0.1004 | - | 0.0711
LTC-VaR | -0.0726 | 0.0359 | - | - | - | - | - | -
LTC-BTC | -0.1974 | 0.0976 | 0.0876 | 0.0657 | 0.0511 | 0.1533 | 0.0730 | 0.0730
LTC-ETH | -0.1916 | 0.0948 | 0.1290 | 0.0726 | 0.0484 | 0.1774 | 0.0806 | 0.0645
LTC-XMR | -0.1863 | 0.0921 | 0.1468 | 0.0917 | 0.0550 | 0.2202 | 0.0917 | 0.1009
LTC-XRP | -0.1916 | 0.0947 | 0.1304 | 0.0783 | 0.0522 | 0.1826 | 0.0783 | 0.0870
LTC-SCoVaR | -0.1968 | 0.0973 | 0.0926 | 0.0648 | 0.0556 | 0.1852 | 0.0741 | 0.0741
LTC-MCoVaR | -0.3122 | 0.1544 | 0.0256 | 0.0513 | 0.0513 | 0.1282 | - | 0.0256
LTC-VCoVaR | -0.1551 | 0.0767 | 0.1000 | 0.0885 | 0.0731 | 0.1231 | - | 0.0885
XMR-VaR | -0.0876 | 0.0361 | - | - | - | - | - | -
XMR-BTC | -0.2276 | 0.0925 | 0.0730 | 0.0438 | 0.0438 | 0.1387 | 0.0584 | 0.0657
XMR-ETH | -0.2286 | 0.0929 | 0.0726 | 0.0484 | 0.0484 | 0.1290 | 0.0565 | 0.0565
XMR-LTC | -0.2239 | 0.0910 | 0.0820 | 0.0574 | 0.0492 | 0.1557 | 0.0656 | 0.0656
XMR-XRP | -0.2204 | 0.0896 | 0.0870 | 0.0783 | 0.0522 | 0.1826 | 0.0783 | 0.0870
XMR-SCoVaR | -0.2324 | 0.0944 | 0.0763 | 0.0508 | 0.0508 | 0.1186 | 0.0593 | 0.0678
XMR-MCoVaR | -0.3388 | 0.1373 | 0.0488 | 0.0488 | 0.0488 | 0.1463 | - | 0.0488
XMR-VCoVaR | -0.1861 | 0.0757 | 0.0675 | 0.0397 | 0.0397 | 0.0833 | - | 0.0476
XRP-VaR | -0.0817 | 0.0552 | - | - | - | - | - | -
XRP-BTC | -0.2250 | 0.1522 | 0.1095 | 0.0730 | 0.0365 | 0.1971 | 0.0730 | 0.0657
XRP-ETH | -0.2308 | 0.1561 | 0.1129 | 0.0887 | 0.0484 | 0.1935 | 0.0887 | 0.0887
XRP-LTC | -0.2321 | 0.1570 | 0.0984 | 0.0738 | 0.0328 | 0.1885 | 0.0902 | 0.0820
XRP-XMR | -0.2211 | 0.1495 | 0.1560 | 0.0917 | 0.0459 | 0.2661 | 0.1193 | 0.1009
XRP-SCoVaR | -0.2345 | 0.1586 | 0.1043 | 0.0783 | 0.0435 | 0.1652 | 0.0870 | 0.0783
XRP-MCoVaR | -0.3644 | 0.2463 | 0.0488 | 0.0488 | 0.0244 | 0.0976 | - | 0.0488
XRP-VCoVaR | -0.1821 | 0.1231 | 0.1004 | 0.0763 | 0.0562 | 0.1285 | - | 0.0763

  * •

NOTE: The copula with the closest rate to $\beta=0.05$ for each row is marked
bold. BTC-LTC denotes the CoVaR of BTC with LTC under condition. Mean and sd
of each measure (except VaR) are given for the static $t$-copula case.

Table 3: Statistics and in-sample rates of the risk measures. Figure 5: CoVaR
of BTC with LTC under condition using a time-invariant $t$-copula. Realized
violation rate: 0.0492. Figure 6: VCoVaR, SCoVaR, and MCoVaR of BTC with time-
invariant $t$-copula.

This leads to the following findings:

  1. 1.

On average, all conditional measures are below the respective univariate VaR,
reflecting the positive dependencies in the crypto-market. Moreover, the
figures show that the conditional measures are driven by similar dynamics as
the VaR, which is a consequence of the chosen static $t$-copula and the
inverse margin operation (15).

  2. 2.

According to average bivariate CoVaR, BTC and LTC primarily affect each other.
Consequently, Figure 5 displays the CoVaR of BTC drastically below its VaR.
This finding agrees with Luu Duc Huynh (2019) and Xu et al. (2021), who
noticed BTC as a risk recipient of other CCs. Investors might keep that in
mind when driving towards BTC. ETH responds similarly to isolated distressing
events of BTC, LTC, XMR, and XRP. However, a notable impact has LTC on XRP
since it leads to the lowest average CoVaR estimates for this CC.

  3. 3.

Generally, the average SCoVaR estimate for each CC is below the bivariate
CoVaR estimates. The only exception being the pair BTC and LTC. This implies
that knowing LTC is in distress appears to be worse for BTC than knowing the
summation of ETH, LTC, XMR, and XRP is in a critical state.

  4. 4.

The MCoVaR yields the lowest average estimates for each CC, illustrating how
strongly a joint distressing event of other major CCs can impact a particular
currency. In addition, the MCoVaR has the highest variance among all
considered measures.

  5. 5.

Finally, the VCoVaR is estimated below the VaR, but larger than all
conditional measures. This means that knowing that at least one out of the
other four CC is in distress appears less critical than knowing that exactly
one is in distress without information on the other three. This reflects the
nature of the conditional event and can be interpreted in two ways. First,
from a methodological point of view, the conditional event of the VCoVaR
includes more scenarios than all other considered measures. Consequently, more
observations will fulfil it, see Figure 6. To achieve a violation rate at
level $\beta$ for this amount of observations, the VCoVaR estimate has to lie
closer to the log-return of BTC than the other conditional measures. Second,
from an economic point of view, the cases of the conditional event of the
VCoVaR when one CC is in distress have the additional information that the
other CCs are not in distress, which the bivariate CoVaR does not include. For
example, knowing LTC is in distress is worse than knowing LTC is in distress
and ETH, XMR, and XRP are not. This information advantage is decisive, as it
gives positive information about the market. This was discussed theoretically
in Section 3.1 and perfectly materializes in this application. Furthermore,
the VCoVaR exhibits the lowest variance among all other conditional measures.

  6. 6.

All measure can be reliably estimated via copulae. However, the selected
dependence model is crucial for the estimation quality. In general, the best
copulae are the $t$-copula and the Clayton one. The time-variant Patton (2006)
and DCC models can reliably be used as well, but offer only in distinct cases
advantages over the static $t$ and Clayton models. The worst performance
yields the Gumbel copula since it cannot the model the lower tail dependence
adequately. These observations are in line with the results of the simulations
in Section 4.2. However, the violation rates should be treated carefully due
to the evaluation approach, which only considers the observations fulfilling
the respective conditional event.

###  6.2 Out-of-sample Estimation

To validate the estimation performance in an out-of-sample setting, we analyze
the CCs based on a moving window approach. As in Section 6.1 holds:
$\alpha=\beta=0.05$, and we calculate the same systemic risk measures. The
window size covers $w=500$ observations, and a one-day-ahead forecasting
strategy is pursued. The univariate model is always assumed to be a GJR-
GARCH(1,1) model with $\mu=0$ and skew-$t$ innovations. The CoVaR, SCoVaR,
MCoVaR, and VCoVaR forecasts for $t+1$ are evaluated based on calculating the
realized equivalents of $\beta$ in (3), (4), (5), and (6) while using the
forecasted VaR and the log-return of $t+1$. This is motivated by Girardi and
Ergün (2013), although this work focuses directly on violation rates instead
of tests like Kupiec (1995). To perform the one-day-ahead VaR forecasting,
(15) is calculated using the forecasts of the conditional mean and conditional
variance. The procedure for the conditional measures for each window is as
follows: _Step 1:_ Fit a GJR-GARCH(1,1) model to each time series and
transform the resulting standardized residuals parametrically to
$\mathcal{U}[0,1]$. _Step 2:_ Based on these pseudo-observations, estimate the
respective copula via ML. _Step 3:_ Forecast the measure by solving (9), (11),
and (13) using the one-day-ahead forecast of the univariate model (14) of the
$X_{j,t}$ variable:

$\hat{\sigma}_{j,t+1}^{2}=\hat{\omega}_{j}+(\hat{\alpha}_{j}+\hat{\gamma}_{j}\mathbf{1}_{\\{\hat{\varepsilon}_{j,t}\leq
0\\}})\hat{\varepsilon}_{j,t}^{2}+\hat{\beta}_{j}\hat{\sigma}_{j,t}^{2}.$

Table 4: Out-of-sample rates of the risk measures.

Measure | Violation rates for copula
---|---
Gaussian | $t$ | Clayton | Gumbel
BTC-ETH | 0.1237 | 0.0825 | 0.0928 | 0.1959
BTC-LTC | 0.0583 | 0.0485 | 0.0485 | 0.1165
BTC-XMR | 0.1023 | 0.0568 | 0.0568 | 0.1932
BTC-XRP | 0.1466 | 0.0862 | 0.0345 | 0.2328
BTC-SCoVaR | 0.1099 | 0.0549 | 0.0549 | 0.1758
BTC-MCoVaR | 0.0513 | 0.0513 | 0.0513 | 0.1026
BTC-VCoVaR | 0.0909 | 0.0642 | 0.0481 | 0.1390
ETH-BTC | 0.1078 | 0.0784 | 0.0686 | 0.1667
ETH-LTC | 0.0583 | 0.0583 | 0.0485 | 0.1068
ETH-XMR | 0.1023 | 0.0795 | 0.0455 | 0.1705
ETH-XRP | 0.0948 | 0.0690 | 0.0431 | 0.1552
ETH-SCoVaR | 0.0968 | 0.0753 | 0.0430 | 0.1290
ETH-MCoVaR | 0.0233 | 0.0233 | 0.0233 | 0.0698
ETH-VCoVaR | 0.0952 | 0.0635 | 0.0423 | 0.1270
LTC-BTC | 0.0588 | 0.0490 | 0.0392 | 0.1176
LTC-ETH | 0.0722 | 0.0515 | 0.0515 | 0.1031
LTC-XMR | 0.1023 | 0.0682 | 0.0568 | 0.1477
LTC-XRP | 0.0776 | 0.0431 | 0.0345 | 0.1121
LTC-SCoVaR | 0.0814 | 0.0581 | 0.0465 | 0.1279
LTC-MCoVaR | 0.0250 | 0.0750 | 0.0750 | 0.1250
LTC-VCoVaR | 0.0681 | 0.0576 | 0.0419 | 0.1099
XMR-BTC | 0.1078 | 0.0784 | 0.0490 | 0.1667
XMR-ETH | 0.0825 | 0.0825 | 0.0412 | 0.1340
XMR-LTC | 0.0874 | 0.0777 | 0.0485 | 0.1359
XMR-XRP | 0.0948 | 0.0690 | 0.0431 | 0.1466
XMR-SCoVaR | 0.0968 | 0.0860 | 0.0430 | 0.1290
XMR-MCoVaR | 0.0732 | 0.0732 | 0.0732 | 0.1220
XMR-VCoVaR | 0.0737 | 0.0474 | 0.0474 | 0.1105
XRP-BTC | 0.1667 | 0.0980 | 0.0882 | 0.2941
XRP-ETH | 0.1134 | 0.0825 | 0.0825 | 0.1959
XRP-LTC | 0.1165 | 0.0680 | 0.0583 | 0.1845
XRP-XMR | 0.1250 | 0.0795 | 0.0795 | 0.2386
XRP-SCoVaR | 0.0769 | 0.0769 | 0.0769 | 0.1429
XRP-MCoVaR | 0.0732 | 0.0244 | 0.0244 | 0.0732
XRP-VCoVaR | 0.1561 | 0.1040 | 0.0636 | 0.1676

NOTE: The copula with the closest rate to $\beta=0.05$ for each row is marked
bold. BTC-LTC denotes the CoVaR of BTC with LTC under condition.

The resulting VaR violation rates for the CC are with 0.0572 for BTC, 0.0544
for ETH, 0.0578 for LTC, 0.0494 for XMR, and 0.0651 for XRP all close to
$\alpha=0.05$. Further, the rates for the systems are similarly accurate:
0.0511 for Sys:BTC, 0.0522 for Sys:ETH, 0.0483 for Sys:LTC, 0.0522 for
Sys:XMR, and 0.0511 for Sys:XRP. Table 4 shows the rates for the conditional
measures while using the time-invariant Gaussian, $t$, Clayton, and Gumbel
copulae. Especially for the Clayton model, the CoVaR, SCoVaR, MCoVaR, and
VCoVaR forecasts are close to $\beta=0.05$, although the $t$-copula is also a
valid choice in most cases. Furthermore, the performance of the copulae
relative to each other is as in the in-sample scenario.

##  7 Conclusion

Quantifying systemic risk in the financial system is a key focus for
regulators and risk management practitioners. Especially the rapidly-growing
and highly volatile market of CC has attracted the attention of regulating
authorities and researchers due to its potential impact on the status of the
global financial system. An important methodological contribution is the
systemic risk measure CoVaR, introduced by Adrian and Brunnermeier (2016). Cao
(2014) proposed an extension called Multi-CoVaR, while Bernardi et al. (2019)
introduced a measure named System-CoVaR. Complementing these extensions, a new
measure, the Vulnerability-CoVaR is proposed. The VCoVaR offers the improved
ability to capture domino effects and is advantageous in a system where there
exists at least one other CC which is facing an extreme risk scenario. Due to
evidence of spillover effects, tail-dependence, and herding behaviour, such
domino effects are an existing threat and source of systemic risk in the CC
market.

The simulation-based analysis of dependence consistency displays the property
of the VCoVaR to be a monotonically decreasing function of the dependence
parameter for selected AC. The empirical analysis on the CC market showed that
LTC displays the largest impact on BTC out of the selected currencies, and LTC
further affects XRP relatively strong. Generally, each CC is significantly
affected if an event of joint distress of the remaining currencies occurs,
which can be considered symptomatic for systemic risk events in the CC market.
Interestingly, a situation of at least on CC being in distress appears less
critical than a specification of one concrete CC in the conditional event.
This observation reflects exactly the nature of the conditional event of the
VCoVaR. Regarding the estimation quality, significant differences between the
considered copulae are detected. However, for the $t$ and Clayton copulae, the
conditional measures are reliably estimated, as the in-sample and out-of-
sample violation rates are approximately in line with the selected probability
level.

Future work on the topic of CoVaR extensions could extend the analysis of
dependence consistency to broader families of copulae and higher dimensional
scenarios. Especially in the case of the newly proposed VCoVaR, theoretical
analysis and simulations apart from the considered Archimedean copula models
could be conducted. In addition, considered CoVaR extensions could be applied
to different asset markets with varied sample sizes, to validate the
estimation quality using more observations. It could also be analysed whether
the VCoVaR could be reasonably extended to a difference-based measure
comparable to the Delta-CoVaR of Adrian and Brunnermeier (2016).

ACKNOWLEDGEMENTS

The authors would like to thank Arsen Palestini for his fruitful answer
regarding the System-CoVaR and Jianlin Zhang for writing an excellent Master’s
thesis, which inspired this work. The authors report there are no competing
interests to declare.

## References

  * Acharya et al. (2017) Acharya, V. V., Pedersen, L. H., Philippon, T., and Richardson, M. Measuring systemic risk. The Review of Financial Studies, 2017, 30, 2–47.
  * Adrian and Brunnermeier (2016) Adrian, T. and Brunnermeier, M. K. CoVaR. The American Economic Review, 2016, 106, 1705–1741.
  * Akaike (1974) Akaike, H. A new look at the statistical model identification. IEEE Transactions on Automatic Control, 1974, 19, 716–723.
  * Ballis and Drakos (2020) Ballis, A. and Drakos, K. Testing for herding in the cryptocurrency market. Finance Research Letters, 2020, 33, 101210.
  * Basel Committee on Banking Supervision (2013) Basel Committee on Banking Supervision, Global Systemically Important Banks: Updated Assessment Methodology and the Higher Loss Absorbency Requirement. Technical report, Bank for International Settlements, Basel, 2013.
  * Basel Committee on Banking Supervision (2019) —, Designing a prudential treatment for crypto-assets. Discussion paper, Bank for International Settlements, Basel, 2019.
  * Benoit et al. (2017) Benoit, S., Colliard, J.-E., Hurlin, C., and Pérignon, C. Where the risks lie: A survey on systemic risk. Review of Finance, 2017, 21, 109–152.
  * Bernardi et al. (2019) Bernardi, M., Cerqueti, R., and Palestini, A. Allocation of risk capital in a cost cooperative game induced by a modified expected shortfall. Journal of the Operational Research Society, 2019.
  * Bernardi et al. (2018) Bernardi, M., Durante, F., Jaworski, P., Petrella, L., and Salvadori, G. Conditional risk based on multivariate hazard scenarios. Stochastic Environmental Research and Risk Assessment, 2018, 32, 203–211.
  * Bernardi et al. (2017) Bernardi, M., Maruotti, A., and Petrella, L. Multiple risk measures for multivariate dynamic heavy–tailed models. Journal of Empirical Finance, 2017, 43, 1–32.
  * Bisias et al. (2012) Bisias, D., Flood, M., Lo, A. W., and Valavanis, S. A survey of systemic risk analytics. Annual Review of Financial Economics, 2012, 4, 255–296.
  * Bonaccolto et al. (2021) Bonaccolto, G., Borri, N., and Consiglio, A. Breakup and default risks in the great lockdown. Journal of Banking & Finance, 2021, 106308.
  * Borri (2019) Borri, N. Conditional tail-risk in cryptocurrency markets. Journal of Empirical Finance, 2019, 50, 1–19.
  * Bouri et al. (2019) Bouri, E., Gupta, R., and Roubaud, D. Herding behaviour in cryptocurrencies. Finance Research Letters, 2019, 29, 216–221.
  * Brownlees and Engle (2017) Brownlees, C. and Engle, R. F. SRISK: A conditional capital shortfall measure of systemic risk. The Review of Financial Studies, 2017, 30, 48–79.
  * Cao (2014) Cao, Z., Multi-CoVaR and Shapley value: A systemic risk measure. Working paper, Banque de France DSF-SMF, 2014.
  * Chang et al. (2000) Chang, E. C., Cheng, J. W., and Khorana, A. An examination of herd behavior in equity markets: An international perspective. Journal of Banking & Finance, 2000, 24, 1651–1679.
  * Chiang and Zheng (2010) Chiang, T. C. and Zheng, D. An empirical analysis of herd behavior in global stock markets. Journal of Banking & Finance, 2010, 34, 1911–1921.
  * Di Bernardino et al. (2015) Di Bernardino, E., Fernández-Ponce, J. M., Palacios-Rodríguez, F., and Rodríguez-Griñolo, M. R. On multivariate extensions of the conditional Value-at-Risk measure. Insurance: Mathematics and Economics, 2015, 61, 1–16.
  * Diebold and Yilmaz (2015) Diebold, F. X. and Yilmaz, K. Trans-Atlantic equity volatility connectedness: US and European financial institutions, 2004–2014. Journal of Financial Econometrics, 2015, 14, 81–127.
  * Eling and Toplek (2009) Eling, M. and Toplek, D. Modeling and management of nonlinear dependencies-copulas in dynamic financial analysis. Journal of Risk and Insurance, 2009, 76, 651–681.
  * Engle (2002) Engle, R. F. Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business & Economic Statistics, 2002, 20, 339–350.
  * Engle and Ng (1993) Engle, R. F. and Ng, V. K. Measuring and testing the impact of news on volatility. The Journal of Finance, 1993, 48, 1749–1778.
  * Fernández and Steel (1998) Fernández, C. and Steel, M. F. On bayesian modeling of fat tails and skewness. Journal of the American Statistical Association, 1998, 93, 359–371.
  * Fisher and Gallagher (2012) Fisher, T. J. and Gallagher, C. M. New weighted portmanteau statistics for time series goodness of fit testing. Journal of the American Statistical Association, 2012, 107, 777–787.
  * Georges et al. (2001) Georges, P., Lamy, A.-G., Nicolas, E., Quibel, G., and Roncalli, T., Multivariate survival modelling: A unified approach with copulas. Working Paper, DOI: 10.2139/ssrn.1032559 2001.
  * Ghalanos (2019) Ghalanos, A. 2019, rmgarch: Multivariate GARCH models, R package version 1.3-7, URL: https://cran.r-project.org/package=rmgarch, accessed: 23/12/20.
  * Ghalanos (2020) — 2020, rugarch: Univariate GARCH models, R package version 1.4-4, URL: https://cran.r-project.org/package=rugarch, accessed: 13/12/21.
  * Girardi and Ergün (2013) Girardi, G. and Ergün, A. T. Systemic risk measurement: Multivariate GARCH estimation of CoVaR. Journal of Banking & Finance, 2013, 37, 3169–3180.
  * Glosten et al. (1993) Glosten, L. R., Jagannathan, R., and Runkle, D. E. On the relation between the expected value and the volatility of the nominal excess return on stocks. The Journal of Finance, 1993, 48, 1779–1801.
  * Härdle et al. (2016) Härdle, W. K., Wang, W., and Yu, L. TENET: Tail-Event driven NETwork risk. Journal of Econometrics, 2016, 192, 499–513.
  * Hwang and Salmon (2004) Hwang, S. and Salmon, M. Market stress and herding. Journal of Empirical Finance, 2004, 11, 585–616.
  * Ji et al. (2019) Ji, Q., Bouri, E., Lau, C. K. M., and Roubaud, D. Dynamic connectedness and integration in cryptocurrency markets. International Review of Financial Analysis, 2019, 63, 257–272.
  * Jin (2010) Jin, X., Large portfolio risk management with dynamic copulas. Technical report, McGill University, 2010.
  * Joe (2014) Joe, H., Dependence Modeling with Copulas, 2014, Boca Raton, FL: CRC press.
  * Kallinterakis and Wang (2019) Kallinterakis, V. and Wang, Y. Do investors herd in cryptocurrencies–and why? Research in International Business and Finance, 2019, 50, 240–245.
  * Karimalis and Nomikos (2018) Karimalis, E. N. and Nomikos, N. K. Measuring systemic risk in the European banking sector: A copula CoVaR approach. The European Journal of Finance, 2018, 24, 944–975.
  * Katsiampa et al. (2019) Katsiampa, P., Corbet, S., and Lucey, B. Volatility spillover effects in leading cryptocurrencies: A BEKK-MGARCH analysis. Finance Research Letters, 2019, 29, 68–74.
  * Koenker and Bassett Jr (1978) Koenker, R. and Bassett Jr, G. Regression quantiles. Econometrica, 1978, 46, 33–50.
  * Kojadinovic and Yan (2010) Kojadinovic, I. and Yan, J. Modeling multivariate distributions with continuous margins using the copula R package. Journal of Statistical Software, 2010, 34, 1–20.
  * Koutmos (2018) Koutmos, D. Return and volatility spillovers among cryptocurrencies. Economics Letters, 2018, 173, 122–127.
  * Kuester et al. (2006) Kuester, K., Mittnik, S., and Paolella, M. S. Value-at-risk prediction: A comparison of alternative strategies. Journal of Financial Econometrics, 2006, 4, 53–89.
  * Kupiec (1995) Kupiec, P. Techniques for verifying the accuracy of risk measurement models. The Journal of Derivatives, 1995, 3, 73–84.
  * Kwiatkowski et al. (1992) Kwiatkowski, D., Phillips, P. C., Schmidt, P., and Shin, Y. Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics, 1992, 54, 159–178.
  * Kyriazis (2019) Kyriazis, N. A. A survey on empirical findings about spillovers in cryptocurrency markets. Journal of Risk and Financial Management, 2019, 12, 170.
  * Kyriazis (2020) —Herding behaviour in digital currency markets: An integrated survey and empirical estimation. Heliyon, 2020, 6, e04752.
  * Li and Mak (1994) Li, W. K. and Mak, T. K. On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity. Journal of Time Series Analysis, 1994, 15, 627–636.
  * Li et al. (2020) Li, Z., Wang, Y., and Huang, Z. Risk Connectedness Heterogeneity in the Cryptocurrency Markets. Frontiers in Physics, 2020, 8, 243.
  * Ljung and Box (1978) Ljung, G. M. and Box, G. E. On a measure of lack of fit in time series models. Biometrika, 1978, 65, 297–303.
  * Luu Duc Huynh (2019) Luu Duc Huynh, T. Spillover risks on cryptocurrency markets: A look from VAR-SVAR Granger causality and student’s $t$ copulas. Journal of Risk and Financial Management, 2019, 12, 52.
  * Lux and Papapantoleon (2017) Lux, T. and Papapantoleon, A. Improved Fréchet–Hoeffding bounds on $d$-copulas and applications in model-free finance. The Annals of Applied Probability, 2017, 27, 3633–3671.
  * Mainik and Schaanning (2014) Mainik, G. and Schaanning, E. On dependence consistency of CoVaR and some other systemic risk measures. Statistics & Risk Modeling, 2014, 31, 49–77.
  * Manner and Reznikova (2012) Manner, H. and Reznikova, O. A survey on time-varying copulas: Specification, simulations, and application. Econometric Reviews, 2012, 31, 654–687.
  * McLeod and Li (1983) McLeod, A. I. and Li, W. K. Diagnostic checking ARMA time series models using squared-residual autocorrelations. Journal of Time Series Analysis, 1983, 4, 269–273.
  * Nelsen (2006) Nelsen, R. B., An Introduction to Copulas, 2006, New York: Springer Science & Business Media.
  * Oehlert (1992) Oehlert, G. W. A note on the delta method. The American Statistician, 1992, 46, 27–29.
  * Okhrin et al. (2013) Okhrin, O., Okhrin, Y., and Schmid, W. On the structure and estimation of hierarchical Archimedean copulas. Journal of Econometrics, 2013, 173, 189–204.
  * Okhrin and Ristig (2014) Okhrin, O. and Ristig, A. Hierarchical Archimedean Copulae: The HAC Package. Journal of Statistical Software, 2014, 58, 1–20.
  * Patton (2006) Patton, A. J. Modelling asymmetric exchange rate dependence. International Economic Review, 2006, 47, 527–556.
  * Petukhina et al. (2021) Petukhina, A., Trimborn, S., Härdle, W. K., and Elendner, H. Investing with cryptocurrencies–evaluating their potential for portfolio allocation strategies. Quantitative Finance, 2021, 1–29.
  * Phillip et al. (2018) Phillip, A., Chan, J. S., and Peiris, S. A new look at cryptocurrencies. Economics Letters, 2018, 163, 6–9.
  * Phillips and Perron (1988) Phillips, P. C. and Perron, P. Testing for a unit root in time series regression. Biometrika, 1988, 75, 335–346.
  * Reboredo and Ugolini (2015) Reboredo, J. C. and Ugolini, A. Systemic risk in European sovereign debt markets: A CoVaR-copula approach. Journal of International Money and Finance, 2015, 51, 214–244.
  * Said and Dickey (1984) Said, S. E. and Dickey, D. A. Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika, 1984, 71, 599–607.
  * Sklar (1959) Sklar, A. Fonctions de Répartition à n Dimension et Leurs Marges. Publications de l’Institut de Statistique de l’Université de Paris, 1959, 8, 299–231.
  * Smales (2020) Smales, L. A. One Cryptocurrency to explain them all? Understanding the importance of Bitcoin in Cryptocurrency returns. Economic Papers. Special Section: Economics of Blockchain Technology, 2020, 39, 118–132.
  * Stavroyiannis and Babalos (2017) Stavroyiannis, S. and Babalos, V. Herding, faith-based investments and the global financial crisis: Empirical evidence from static and dynamic models. Journal of Behavioral Finance, 2017, 18, 478–489.
  * Szczygielski et al. (2020) Szczygielski, J. J., Karathanasopoulos, A., and Zaremba, A. One shape fits all? A comprehensive examination of cryptocurrency return distributions. Applied Economics Letters, 2020, 27, 1567–1573.
  * Tiwari et al. (2020) Tiwari, A. K., Adewuyi, A. O., Albulescu, C. T., and Wohar, M. E. Empirical evidence of extreme dependence and contagion risk between main cryptocurrencies. The North American Journal of Economics and Finance, 2020, 51, 101083.
  * Tsay (2010) Tsay, R. S., Analysis of Financial Time Series, 2010, New Jersey: John Wiley & Sons.
  * Vidal-Tomás (2021) Vidal-Tomás, D. An investigation of cryptocurrency data: The market that never sleeps. Quantitative Finance, 2021, 1–18.
  * Vidal-Tomás et al. (2019) Vidal-Tomás, D., Ibáñez, A. M., and Farinós, J. E. Herding in the cryptocurrency market: CSSD and CSAD approaches. Finance Research Letters, 2019, 30, 181–186.
  * Xu et al. (2021) Xu, Q., Zhang, Y., and Zhang, Z. Tail-risk spillovers in cryptocurrency markets. Finance Research Letters, 2021, 38, 101453.
  * Zhang et al. (2018) Zhang, W., Wang, P., Li, X., and Shen, D. Some stylized facts of the cryptocurrency market. Applied Economics, 2018, 50, 5950–5965.
  * Zhou (2010) Zhou, C. Are banks too big to fail? Measuring systemic importance of financial institutions. International Journal of Central Banking, 2010, 6, 205–250.

## Appendix A: Proofs

###### Proof of Lemma 3.1.

Transform (6) to:

$\frac{P\\{X_{j,t}\leq VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t},(\exists
i:X_{i,t}\leq VaR^{i}_{\alpha,t})\\}}{P(\exists i:X_{i,t}\leq
VaR^{i}_{\alpha,t})}=\beta.$ (A.1)

Based on: $P(\exists i:X_{i,t}\leq VaR^{i}_{\alpha,t})=1-P(\forall
i:X_{i,t}>VaR^{i}_{\alpha,t})$, and:

$\displaystyle P(X_{j,t}\leq VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t})$
$\displaystyle=P\\{X_{j,t}\leq VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t},(\exists
i:X_{i,t}\leq VaR^{i}_{\alpha,t})\\}$ $\displaystyle+P\\{X_{j,t}\leq
VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t},(\forall
i:X_{i,t}>VaR^{i}_{\alpha,t})\\},$

(A.1) is rewritten:

$\frac{P(X_{j,t}\leq VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t})-P\\{X_{j,t}\leq
VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t},(\forall
i:X_{i,t}>VaR^{i}_{\alpha,t})\\}}{1-P(\forall
i:X_{i,t}>VaR^{i}_{\alpha,t})}=\beta.$ (A.2)

Recognizing:

$\displaystyle P(\forall i:X_{i,t}>VaR^{i}_{\alpha,t})$
$\displaystyle=P\\{X_{j,t}\leq VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t},(\forall
i:X_{i,t}>VaR^{i}_{\alpha,t})\\}$
$\displaystyle+P\\{X_{j,t}>VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t},(\forall
i:X_{i,t}>VaR^{i}_{\alpha,t})\\},$

allows to transform (A.2) to:

$\frac{P(X_{j,t}\leq VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t})-P(\forall
i:X_{i,t}>VaR^{i}_{\alpha,t})+P\\{X_{j,t}>VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t},(\forall
i:X_{i,t}>VaR^{i}_{\alpha,t})\\}}{1-P(\forall
i:X_{i,t}>VaR^{i}_{\alpha,t})}=\beta.$

Noticing this expression consists of survival functions, Lemma 3.1 follows. ∎

###### Proof of Lemma 3.2.

(13) becomes:

$\frac{F_{X_{j,t}}(VCoVaR^{j|i=1}_{\alpha,\beta,t})-(1-\alpha)+\bar{C}_{X_{j,t},X_{1,t}}\\{1-F_{X_{j,t}}(VCoVaR^{j|i=1}_{\alpha,\beta,t}),1-\alpha;\theta_{1,t}\\}}{1-(1-\alpha)}=\beta.$
(A.3)

Noticing $\bar{C}(u_{1},u_{2})=u_{1}+u_{2}-1+C(1-u_{1},1-u_{2})$, rewrite:

$\displaystyle\bar{C}_{X_{j,t},X_{1,t}}\\{1-F_{X_{j,t}}(VCoVaR^{j|i=1}_{\alpha,\beta,t}),1-\alpha;\theta_{1,t}\\}$
$\displaystyle=1-F_{X_{j,t}}(VCoVaR^{j|i=1}_{\alpha,\beta,t})+1-\alpha-1$
$\displaystyle+C_{X_{j,t},X_{1,t}}\\{F_{X_{j,t}}(VCoVaR^{j|i=1}_{\alpha,\beta,t}),\alpha;\theta_{1,t}\\}.$

Simplifying and inserting into (A.3) yields:
$C_{X_{j,t},X_{1,t}}\\{F_{X_{j,t}}(VCoVaR^{j|i=1}_{\alpha,\beta,t}),\alpha;\theta_{1,t}\\}=\alpha\beta$,
equaling the result of the bivariate CoVaR in (9). Thus,
$VCoVaR^{j|i=1}_{\alpha,\beta,t}=CoVaR^{j|i}_{\alpha,\beta,t}$. ∎

###### Proof of Lemma 4.1.

The CoVaR of (9) becomes:
$F_{X_{j,t}}(CoVaR^{j|i}_{\alpha,\beta,t})\alpha=\alpha\beta$, which reduces
for $\alpha,\beta\in(0,1)$ to:
$CoVaR^{j|i}_{\alpha,\beta,t}=F_{X_{j,t}}^{-1}(\beta)$. $F_{X_{j,t}}^{-1}$
denotes the inverse marginal distribution of $X_{j,t}$. The MCoVaR of (11)
becomes:

$\frac{F_{X_{j,t}}(MCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t})\alpha^{p}}{\alpha^{p}}=\beta,$

which simplifies to:
$MCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}=F_{X_{j,t}}^{-1}(\beta)$. Finally, the
VCoVaR in (13) becomes:

$\frac{F_{X_{j,t}}(VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t})-(1-\alpha)^{p}+\\{1-F_{X_{j,t}}(VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t})\\}(1-\alpha)^{p}}{1-(1-\alpha)^{p}}=\beta.$

Simplification yields:

$\frac{F_{X_{j,t}}(VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t})\\{1-(1-\alpha)^{p}\\}}{1-(1-\alpha)^{p}}=\beta,$

leading to: $VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}=F_{X_{j,t}}^{-1}(\beta)$.
∎

###### Proof of Lemma 4.2.

The CoVaR of (9) becomes:
$\min\\{F_{X_{j,t}}(CoVaR^{j|i}_{\alpha,\beta,t}),\alpha\\}=\alpha\beta,$ as
the copula equals the upper Fréchet-Hoeffding bound. Solving leads to:
$CoVaR^{j|i}_{\alpha,\beta,t}=F_{X_{j,t}}^{-1}(\alpha\beta)$, based on
$\alpha,\beta\in(0,1)$. Similarly for the MCoVaR, (11) transforms in:

$\frac{\min\\{F_{X_{j,t}}(MCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}),\bm{\alpha}\\}}{\min(\bm{\alpha})}=\beta,$

which is solved as:
$MCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}=F_{X_{j,t}}^{-1}(\alpha\beta)$. For
the VCoVaR, (13) becomes:

$\frac{F_{X_{j,t}}(VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t})-\min(\bm{1-\alpha})+\min\\{1-F_{X_{j,t}}(VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}),\bm{1-\alpha}\\}}{1-\min(\bm{1-\alpha})}=\beta,$
(A.4)

which applies the upper Fréchet-Hoeffding bound to survival copulae, see Lux
and Papapantoleon (2017). Notice that
$F_{X_{j,t}}(VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t})\geq\alpha$ implies
$\beta=1$. This follows directly as (A.4) would reduce in this case to:

$F_{X_{j,t}}(VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t})-(1-\alpha)+\\{1-F_{X_{j,t}}(VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t})\\}=\alpha\beta,$

which is simplified to: $\beta=1$. As $\alpha,\beta\in(0,1)$ is imposed, it
must hold: $F_{X_{j,t}}(VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t})<\alpha$. In
this scenario, (A.4) can be transformed to:

$\frac{F_{X_{j,t}}(VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t})-(1-\alpha)+(1-\alpha)}{\alpha}=\beta,$

which yields:
$VCoVaR^{j|\mathbf{i}}_{\alpha,\beta,t}=F_{X_{j,t}}^{-1}(\alpha\beta)$. ∎

## Appendix B: Supplements to the Empirical Study

###  B.1 Data Description

Tables B.1, B.2 provides descriptive statistics and tests of the log-returns
of the empirical study for the CCs and system variables. For example,
’Sys:BTC’ denotes the system for BTC, and is the equally weighted sum of the
log-returns of ETH, LTC, XMR, and XRP. ADF refers to the test of Said and
Dickey (1984), PP to Phillips and Perron (1988), and KPSS to Kwiatkowski et
al. (1992). An asterisk (*) indicates rejection of the null hypothesis at
level $\alpha^{*}=0.05$.

| BTC | ETH | LTC | XMR | XRP
---|---|---|---|---|---
Min | -0.4706 | -0.5656 | -0.4588 | -0.4922 | -0.6365
Mean | 0.0024 | 0.0036 | 0.0019 | 0.0027 | 0.0021
Median | 0.0024 | 0.0013 | -0.0005 | 0.0018 | -0.0015
Max | 0.2241 | 0.3006 | 0.5568 | 0.5963 | 1.0087
Sd | 0.0399 | 0.0615 | 0.0567 | 0.0620 | 0.0705
Kurtosis | 11.2569 | 6.7074 | 12.4750 | 11.2825 | 30.2736
Skewness | -0.7495 | -0.3117 | 0.7160 | 0.5130 | 1.9136
$p$-value: Jarque-Bera | 0.0000* | 0.0000* | 0.0000* | 0.0000* | 0.0000*
$p$-value: Ljung-Box (8 lags) | 0.0950 | 0.0046* | 0.0004* | 0.0000* | 0.0001*
Statistics: ADF | -15.7728* | -15.7839* | -16.0902* | -14.7867* | -13.2823*
Statistics: PP | -49.6547* | -49.9640* | -48.7192* | -51.2537* | -49.6844*
Statistics: KPSS | 0.1178 | 0.2188 | 0.1194 | 0.3638 | 0.1115

Table B.1: Descriptive statistics and tests of the log-returns of the CC.

| Sys:BTC | Sys:ETH | Sys:LTC | Sys:XMR | Sys:XRP
---|---|---|---|---|---
Min | -1.9056 | -1.8106 | -1.9174 | -1.8918 | -1.9793
Mean | 0.0103 | 0.0092 | 0.0109 | 0.0100 | 0.0106
Median | 0.0122 | 0.0090 | 0.0133 | 0.0111 | 0.0141
Max | 1.3151 | 1.0483 | 0.9200 | 1.2294 | 0.9068
Sd | 0.1950 | 0.1816 | 0.1812 | 0.1818 | 0.1807
Kurtosis | 8.5834 | 8.8517 | 8.8186 | 9.2215 | 9.7159
Skewness | -0.6068 | -0.6277 | -0.7920 | -0.5603 | -0.9170
$p$-value: Jarque-Bera | 0.0000* | 0.0000* | 0.0000* | 0.0000* | 0.0000*
$p$-value: Ljung-Box (8 lags) | 0.0000* | 0.0000* | 0.0000* | 0.0001* | 0.0000*
Statistics: ADF | -14.8125* | -14.8316* | -14.6531* | -15.0416* | -15.6975*
Statistics: PP | -50.5398* | -50.8404* | -51.1797* | -50.3953* | -50.6902*
Statistics: KPSS | 0.2609 | 0.2261 | 0.2738 | 0.1905 | 0.2673

Table B.2: Descriptive statistics and tests of the log-returns of the systems.

Table B.3 shows estimates of Kendall’s $\tau$ between the log-returns. BTC and
LTC display the strongest dependency between the five CCs, while $\tau$
between the systems is by design relatively high.

Table B.3: Estimates of Kendall’s $\tau$ between log-returns.

| BTC | ETH | LTC | XMR | XRP | Sys:BTC | Sys:ETH | Sys:LTC | Sys:XMR | Sys:XRP
---|---|---|---|---|---|---|---|---|---|---
BTC | 1.0000 | 0.4218 | 0.5558 | 0.4278 | 0.3805 | 0.5070 | 0.5978 | 0.5570 | 0.5892 | 0.6049
ETH |  | 1.0000 | 0.4558 | 0.4206 | 0.4166 | 0.6428 | 0.5013 | 0.6469 | 0.6617 | 0.6560
LTC |  |  | 1.0000 | 0.4228 | 0.4597 | 0.6262 | 0.6641 | 0.5467 | 0.6698 | 0.6543
XMR |  |  |  | 1.0000 | 0.3737 | 0.6241 | 0.6416 | 0.6391 | 0.4794 | 0.6460
XRP |  |  |  |  | 1.0000 | 0.6120 | 0.6170 | 0.6061 | 0.6273 | 0.4655
Sys:BTC |  |  |  |  |  | 1.0000 | 0.8298 | 0.8858 | 0.8283 | 0.8286
Sys:ETH |  |  |  |  |  |  | 1.0000 | 0.8073 | 0.7718 | 0.7788
Sys:LTC |  |  |  |  |  |  |  | 1.0000 | 0.7997 | 0.8157
Sys:XMR |  |  |  |  |  |  |  |  | 1.0000 | 0.7704
Sys:XRP |  |  |  |  |  |  |  |  |  | 1.0000

###  B.2 Univariate Model Selection

####  B.2.1 Procedure

We propose to use the following procedure to select the univariate models,
while we perform the ML estimation in the `R`-package `rugarch` (Ghalanos
2020).

  1. Step 1:

Calculate all possible ARMA($p_{l},q_{l}$) models with $p_{l},q_{l}<6$. Of
those models, consider the ones non-rejecting the Ljung and Box (1978) (LB)
$H_{0}$ hypothesis of having no serial correlation in the residuals.

  2. Step 2:

From the models left after Step 1, select the one with the lowest Akaike
information criterion (AIC, Akaike 1974). Test the model for ARCH effects in
the residuals using the test of McLeod and Li (1983).

  3. Step 3:

If the $H_{0}$ hypothesis of having no ARCH effects of the model of Step 2 is
non-rejected, return the model of Step 2. If $H_{0}$ is rejected, calculate
all possible ARMA($p_{l},q_{l}$)-GARCH(P,Q) models with $p_{l},q_{l},P<6$ and
$Q<2$. Of those models, consider the ones non-rejecting the LB test and the
weighted version of the test of Li and Mak (1994) proposed in Fisher and
Gallagher (2012) (WLM) for correct ARCH model specification, applied to the
standardized residuals.

  4. Step 4:

From the models left after Step 3, select the one with the lowest AIC. Test
the model for asymmetries/leverage effect using the Sign Bias tests of Engle
and Ng (1993).

  5. Step 5:

If the $H_{0}$ hypotheses of having no asymmetries are non-rejected, return
the model of Step 4. If an $H_{0}$ is rejected, calculate all
ARMA($p_{l},q_{l}$)-GJR-GARCH(P,Q) models with $p_{l},q_{l},P<6$ and $Q<2$. Of
those models, consider the ones non-rejecting the LB, the WLM and the Sign
Bias tests on the standardized residuals.

  6. Step 6:

From the models of Step 5, return the one with the lowest AIC.

Following Tsay (2010), the degrees of freedom of the chi-square distribution
of the test statistic of the LB test is reduced by $(p_{l}+q_{l})$ to account
for the parametrization of the ARMA model. The same holds for the test of
McLeod and Li (1983) of Step 2 in the selection procedure, which tests the
null hypothesis of having no ARCH effects by applying the LB statistic to the
squared residuals (Tsay 2010). Common use in practice is to pursue the same
procedure for the squared standardized residuals of an ARMA-GARCH model, see,
e.g., Reboredo and Ugolini (2015). However, Li and Mak (1994) proposed an
improved test statistic to test the null hypothesis of having no ARCH effects
in GARCH-type models. In the given application, the modification of Fisher and
Gallagher (2012) is used while pursuing a correction of the respective degrees
of freedom following the implementation of the `rugarch` package. The Sign
Bias tests of Engle and Ng (1993) regress the squared standardized residuals
$\hat{z}_{t}^{2}$ on the lagged estimated residuals $\hat{\varepsilon}_{t-1}$
as follows:

$\hat{z}_{t}^{2}=c_{0}+c_{1}\mathbf{1}_{\\{\hat{\varepsilon}_{t-1}<0\\}}+c_{2}\mathbf{1}_{\\{\hat{\varepsilon}_{t-1}<0\\}}\hat{\varepsilon}_{t-1}+c_{3}\mathbf{1}_{\\{\hat{\varepsilon}_{t-1}\geq
0\\}}\hat{\varepsilon}_{t-1}+u_{t},$

where $\mathbf{1}_{\\{\cdot\\}}$ is the indicator function, to examine the
hypotheses $H_{0}:c_{i}=0$ with $i\in\\{1,2,3\\}$ and for the _Joint effect_
$H_{0}:c_{1}=c_{2}=c_{3}=0$. The test of $c_{1}$ is referred to as _Sign Bias_
, the one of $c_{2}$ as _Negative Sign Bias_ , and the one of $c_{3}$ as
_Positive Sign Bias_ (Ghalanos 2020). In case of rejection of a hypothesis,
there are significant asymmetries in the data, and the model is adjusted.

Please note that this procedure results in finding the best fitting model (in
the sense of AIC) while maintaining parsimony and fulfilling the conditions on
the standardized residuals. All models were estimated twice - with $\mu$
estimated from the data and with $\mu=0$ \- to capture possible improvements.
We remark that this procedure is statistically thoroughly debatable, as a test
is regarded as fulfilled if its $H_{0}$ hypothesis is not rejected. However,
such approaches are heavily pursued in practical time series testing, which is
the reason we specified this kind of selection procedure. The resulting models
of this procedure have not been used in the case of the systems, which are
necessary for the SCoVaR. Since each system is the sum of four CC, the
dynamics become extremely complex and the found models yielded large VaR
violations. More research has to be done to analyze the dynamics in such
systems and we stick to a standard GARCH(1,1) specification for these cases.

####  B.2.2 Results

The results of the univariate model selection are given in Tables B.4 and B.5.
Standard errors are given in parentheses, and an asterisk (*) denotes
significance at level $\alpha^{*}=0.05$. The AIC of a model with log-
likelihood $LL$ and number of parameters $\tilde{k}$ is given as:
$AIC=2\tilde{k}-2LL$ (Akaike 1974). LB is the test of Ljung and Box (1978),
while LB2 performs the test on the squared standardized residuals. WLM is the
weighted version of the test in Li and Mak (1994) proposed by Fisher and
Gallagher (2012). The Sign Bias (SB) tests relate to Engle and Ng (1993).

All time series exhibit relevant ARCH effects, and the pure
ARMA($p_{l},q_{l}$) specification was in all cases not sufficient to pass the
considered tests. Moreover, a symmetric GARCH(P,Q) specification seems to be
adequate in most cases. An exception is BTC, with significant asymmetric
effects causing the necessity of considering GJR-GARCH(P,Q) models for the
variance equation. However, in the final model, the $\gamma$-estimates were
not significant. Regarding $F_{z}$, the estimates of $\zeta$ and $\nu$ are
significant in all cases, confirming the relevance of a non-normal
distribution. Tables B.4 and B.5 further show the AIC and relevant tests of
the standardized residuals of the selected models. For comparison, the LB test
on the squared standardized residuals is included, which is in line with the
WLM test in most cases. In the following, the plots of the autocorrelation
functions (ACF) and partial autocorrelation functions (PACF) of the
standardized residuals of the selected models are displayed.

| BTC | ETH | LTC | XMR | XRP
---|---|---|---|---|---
Selected Model |  ARMA(2,2)- GJR-ARCH(5) | ARCH(3) | GARCH(1,1) | GARCH(1,1) | GARCH(1,1)
$\hat{\mu}$ | 0.0024* (0.0012) | 0.0028* (0.0011) |  | 0.0019 (0.0010) |
$\hat{\phi}_{1}$ | -0.0043* (0.0036) |  |  |  |
$\hat{\phi}_{2}$ | 0.9846* (0.0035) |  |  |  |
$\hat{\psi}_{1}$ | 0.0162* (0.0003) |  |  |  |
$\hat{\psi}_{2}$ | -0.9759* (0.0000) |  |  |  |
$\hat{\omega}$ | 0.0005* (0.0001) | 0.0020* (0.0003) | 0.0000* (0.0000) | 0.0002* (0.0001) | 0.0002* (0.0000)
$\hat{\lambda}_{1}$ | 0.1524* (0.0610) | 0.3246* (0.0720) | 0.0890* (0.0109) | 0.1669* (0.0302) | 0.2295* (0.0360)
$\hat{\lambda}_{2}$ | 0.0771* (0.0452) | 0.2498* (0.0605) |  |  |
$\hat{\lambda}_{3}$ | 0.1236* (0.0501) | 0.2063* (0.0590) |  |  |
$\hat{\lambda}_{4}$ | 0.3485* (0.0835) |  |  |  |
$\hat{\lambda}_{5}$ | 0.2230* (0.0724) |  |  |  |
$\hat{\gamma}_{1}$ | 0.1687* (0.0935) |  |  |  |
$\hat{\gamma}_{2}$ | 0.0711* (0.0674) |  |  |  |
$\hat{\gamma}_{3}$ | 0.0442* (0.0747) |  |  |  |
$\hat{\gamma}_{4}$ | -0.0400* (0.1031) |  |  |  |
$\hat{\gamma}_{5}$ | -0.0935* (0.0815) |  |  |  |
$\hat{\delta}_{1}$ |  |  | 0.9100* (0.0115) | 0.8045* (0.0291) | 0.7695* (0.0342)
$\hat{\zeta}$ | 0.979* (0.0232) | 1.0426* (0.0268) | 1.0426* (0.0226) | 1.0141* (0.0294) | 1.0694* (0.0227)
$\hat{v}$ | 3.0091* (0.1624) | 3.0941* (0.2385) | 3.4214* (0.1823) | 3.6145* (0.2881) | 2.9867* (0.1567)
AIC | -9092.8599 | -6975.8293 | -7839.1305 | -7031.3708 | -7519.3082
$p$-val.: LB | 0.0524 | 0.0635 | 0.3947 | 0.0656 | 0.1635
$p$-val.: LB2 | 0.0202* | 0.1608 | 0.9881 | 0.3693 | 0.9070
$p$-val.: WLM | 0.5244 | 0.2693 | 0.9923 | 0.4216 | 0.9117
$p$-val.: SB | 0.5536 | 0.2434 | 0.7885 | 0.7513 | 0.0916
$p$-val.: Neg. SB | 0.4161 | 0.4363 | 0.5272 | 0.1465 | 0.9266
$p$-val.: Pos. SB | 0.1552 | 0.8325 | 0.4973 | 0.6267 | 0.5489
$p$-val.: Joint SB | 0.4202 | 0.5247 | 0.8075 | 0.4924 | 0.1243

Table B.4: Parameter estimates, AIC, and tests of the univariate models.

| Sys:BTC | Sys:ETH | Sys:LTC | Sys:XMR | Sys:XRP
---|---|---|---|---|---
Selected Model | GARCH(1,1) | GARCH(1,1) | GARCH(1,1) | GARCH(1,1) | GARCH(1,1)
$\hat{\mu}$ | 0.0052* (0.0034) | 0.0052* (0.0030) | 0.0071* (0.0032) | 0.0055* (0.0028) | 0.0088* (0.0033)
$\hat{\omega}$ | 0.0018* (0.0005) | 0.0010* (0.0003) | 0.0018* (0.0005) | 0.0008* (0.0003) | 0.0015* (0.0004)
$\hat{\lambda}_{1}$ | 0.1649* (0.0322) | 0.1452* (0.0266) | 0.1683* (0.0337) | 0.1521* (0.0247) | 0.1367* (0.0264)
$\hat{\delta}_{1}$ | 0.8223* (0.0299) | 0.8538* (0.0257) | 0.8152* (0.0318) | 0.8469* (0.0248) | 0.8435* (0.0260)
$\hat{\zeta}$ | 0.9446* (0.0267) | 0.9538* (0.0265) | 0.9404* (0.0260) | 0.9517* (0.0255) | 0.9363* (0.0265)
$\hat{v}$ | 3.5653* (0.2926) | 3.3996* (0.2448) | 3.4711* (0.2838) | 3.5796* (0.2523) | 3.6289* (0.3025)
AIC | -1667.4989 | -2109.0873 | -1972.6166 | -2114.0524 | -1955.2041
$p$-val.: LB | 0.0197* | 0.0625 | 0.0085* | 0.0290* | 0.0184*
$p$-val.: LB2 | 0.4058 | 0.6909 | 0.4429 | 0.5588 | 0.3737
$p$-val.: WLM | 0.5206 | 0.8245 | 0.4270 | 0.6103 | 0.3781
$p$-val.: SB | 0.7703 | 0.7930 | 0.1400 | 0.9673 | 0.7721
$p$-val.: Neg. SB | 0.5120 | 0.4868 | 0.2593 | 0.8352 | 0.7134
$p$-val.: Pos. SB | 0.4394 | 0.5198 | 0.6966 | 0.3526 | 0.1605
$p$-val.: Joint SB | 0.7278 | 0.7927 | 0.2627 | 0.7767 | 0.3126

Table B.5: Parameter estimates, AIC, and tests of the univariate models for
the systems. Figure B.1: ACF/PACF plots of standardized residuals for BTC.
Figure B.2: ACF/PACF plots of standardized residuals for ETH. Figure B.3:
ACF/PACF plots of standardized residuals for LTC. Figure B.4: ACF/PACF plots
of standardized residuals for XMR. Figure B.5: ACF/PACF plots of standardized
residuals for XRP. Figure B.6: ACF/PACF plots of standardized residuals for
Sys:BTC. Figure B.7: ACF/PACF plots of standardized residuals for Sys:ETH.
Figure B.8: ACF/PACF plots of standardized residuals for Sys:LTC. Figure B.9:
ACF/PACF plots of standardized residuals for Sys:XMR. Figure B.10: ACF/PACF
plots of standardized residuals for Sys:XRP.

###  B.3 Copula Models and Parameter Estimates

This section provides details on the time-variant copula models used in the
application. A crucial contribution in the field of dynamic copulae is Patton
(2006), in which the copula parameter itself is assumed to follow an
autoregressive scheme. Building on this idea, our work incorporates a time-
variant $t$-copula. The number of degrees of freedom $\nu$ of the copula is
assumed to be constant, while the copula parameter $\theta_{t}$ has the
following dynamics:

$\displaystyle\theta_{t}=\Lambda\\{\omega_{\theta}+\beta_{\theta}\theta_{t-1}+c_{\theta}\frac{1}{10}\sum_{l=1}^{10}t^{-1}_{\nu}(u_{i_{1},t-l})t^{-1}_{\nu}(u_{i_{2},t-l})\\},$

for $i_{1},i_{2}\in\\{1,2,3,4,5\\},i_{1}\neq i_{2}$, with
$\Lambda(x)=\tanh(\frac{x}{2})$ to maintain $\theta_{t}\in[-1,1]$.
$\omega_{\theta}$, $\beta_{\theta}$ and $c_{\theta}$ denote the parameters to
be estimated, and $t_{\nu}^{-1}$ is the inverse univariate $t$-distribution
with $\nu$ degrees of freedom. Following Manner and Reznikova (2012), the
model is limited to the bivariate case and will therefore be applied only to
the bivariate CoVaR and the SCoVaR. The ML estimation routine depends on the
initial values. For a second time-variant copula model, the DCC-copula
approach proposed in Jin (2010) is used. The model applies the idea of the
DCC-model of Engle (2002) to the copula correlation matrix $R_{t}$ of the
$t$-copula:

$\displaystyle
Q_{t}=(1-a-b)\bar{Q}+a{z^{*}_{t-1}}{{z^{*}_{t-1}}}^{\top}+b{Q_{t-1}},$
$\displaystyle R_{t}=diag{({Q_{t}})^{-1/2}}{Q_{t}}diag{({Q_{t}})^{-1/2}},$

s.t. $a\geq 0$, $b\geq 0$ and $a+b<1$. $Q_{t}$ denotes an auxiliary process
driving the dynamics of the copula correlation. If $R_{t}$ is five-
dimensional, $z^{*}_{t}$ is the vector of transformed pseudo-observations
$\\{\tilde{t}_{\nu}^{-1}(u_{1,t}),\tilde{t}_{\nu}^{-1}(u_{2,t}),\tilde{t}_{\nu}^{-1}(u_{3,t}),\tilde{t}_{\nu}^{-1}(u_{4,t}),\tilde{t}_{\nu}^{-1}(u_{5,t})\\}^{\top}$
while ${\tilde{t}_{\nu}}$ is the standardized univariate $t$-distribution with
unit variance and shape parameter $\nu$. $\bar{Q}$ denotes the unconditional
matrix of $z^{*}_{t}$ and the degrees of freedom $\nu$ of the $t$-copula are
assumed constant. The implementation of the model is realized using the
`R`-package `rmgarch` (Ghalanos 2019), which performs a two-stage ML
estimation.

Tables B.6, B.7 shows the parameter estimates of the copulae for the bivariate
case, while Table B.8 depicts the multivariate estimation results. $\theta$
denotes the copula correlation for the Gaussian and $t$-copulae in the
bivariate case, which expands to the correlation matrix in the five-
dimensional scenario. The tables additionally include the AIC values for the
estimates based on the full log-likelihood. The information criterion states
that the best copula is generally the $t$-copula, which further improves when
considering the dynamic Patton and DCC models. This indicates that the
dependencies of the CCs were not constant and changed over the years, as the
dynamic dependence models provide a better fit. The Gumbel copula is
considered the least appropriate one, while the Gaussian copula ranks between
the $t$ and the Clayton one.

Table B.6: Estimates of the time-invariant bivariate copulae.

CC Pairs | Copula
---|---
Gaussian | $t$ | Clayton | Gumbel
$\theta$ | AIC | $\theta$ | $\nu$ | AIC | $\theta$ | AIC | $\theta$ | AIC
BTC-ETH | 0.559 | -16953.907 | 0.604 | 2.689 | -17264.039 | 1.174 | -17087.318 | 1.592 | -16914.514
BTC-LTC | 0.720 | -18716.545 | 0.747 | 3.827 | -18951.558 | 1.934 | -18872.348 | 1.930 | -18496.948
BTC-XMR | 0.581 | -17090.670 | 0.606 | 3.330 | -17306.663 | 1.251 | -17258.100 | 1.592 | -16976.563
BTC-XRP | 0.520 | -17379.916 | 0.557 | 3.271 | -17593.135 | 1.057 | -17553.168 | 1.491 | -17279.776
BTC-Sys:BTC | 0.667 | -12139.275 | 0.691 | 2.612 | -12450.922 | 1.639 | -12287.770 | 1.816 | -12052.933
ETH-BTC | 0.559 | -16953.907 | 0.604 | 2.689 | -17264.039 | 1.174 | -17087.318 | 1.592 | -16914.514
ETH-LTC | 0.582 | -15816.171 | 0.638 | 3.144 | -16108.938 | 1.288 | -15973.963 | 1.629 | -15743.510
ETH-XMR | 0.574 | -14917.515 | 0.618 | 3.023 | -15192.795 | 1.232 | -15081.750 | 1.623 | -14871.455
ETH-XRP | 0.564 | -15393.672 | 0.622 | 2.809 | -15696.008 | 1.221 | -15564.679 | 1.604 | -15337.560
ETH-Sys:ETH | 0.655 | -10370.179 | 0.707 | 2.447 | -10776.117 | 1.624 | -10549.724 | 1.832 | -10342.260
LTC-BTC | 0.720 | -18716.545 | 0.747 | 3.827 | -18951.558 | 1.934 | -18872.348 | 1.930 | -18496.948
LTC-ETH | 0.582 | -15816.171 | 0.638 | 3.144 | -16108.938 | 1.288 | -15973.963 | 1.629 | -15743.510
LTC-XMR | 0.555 | -15761.594 | 0.588 | 4.035 | -15923.228 | 1.211 | -15963.043 | 1.526 | -15602.025
LTC-XRP | 0.588 | -16410.185 | 0.637 | 3.321 | -16683.239 | 1.342 | -16606.216 | 1.619 | -16293.759
LTC-Sys:LTC | 0.690 | -11355.003 | 0.728 | 2.655 | -11705.841 | 1.850 | -11564.733 | 1.884 | -11241.507
XMR-BTC | 0.581 | -17090.670 | 0.606 | 3.330 | -17306.663 | 1.251 | -17258.100 | 1.592 | -16976.563
XMR-ETH | 0.574 | -14917.515 | 0.618 | 3.023 | -15192.795 | 1.232 | -15081.750 | 1.623 | -14871.455
XMR-LTC | 0.555 | -15761.594 | 0.588 | 4.035 | -15923.228 | 1.211 | -15963.043 | 1.526 | -15602.025
XMR-XRP | 0.516 | -15277.939 | 0.552 | 3.751 | -15450.299 | 1.071 | -15475.696 | 1.477 | -15158.206
XMR-Sys:XMR | 0.637 | -10350.501 | 0.671 | 3.045 | -10628.356 | 1.538 | -10556.765 | 1.732 | -10246.908
XRP-BTC | 0.520 | -17379.916 | 0.557 | 3.271 | -17593.135 | 1.057 | -17553.168 | 1.491 | -17279.776
XRP-ETH | 0.564 | -15393.672 | 0.622 | 2.809 | -15696.008 | 1.221 | -15564.679 | 1.604 | -15337.560
XRP-LTC | 0.588 | -16410.185 | 0.637 | 3.321 | -16683.239 | 1.342 | -16606.216 | 1.619 | -16293.759
XRP-XMR | 0.516 | -15277.939 | 0.552 | 3.751 | -15450.299 | 1.071 | -15475.696 | 1.477 | -15158.206
XRP-Sys:XRP | 0.623 | -10621.955 | 0.666 | 2.627 | -10946.718 | 1.471 | -10811.682 | 1.719 | -10548.650

Table B.7: Estimates of the time-variant bivariate copulae.

CC Pairs | Copula
---|---
Patton-$t$ | DCC-$t$
$\omega_{\theta}$ | $\beta_{\theta}$ | $c_{\theta}$ | $\nu$ | AIC | a | b | $\nu$ | AIC
BTC-ETH | 0.208 | 1.443 | 0.525 | 5.882 | -17582.578 | 0.085 | 0.910 | 5.612 | -17887.668
BTC-LTC | 1.090 | 0.622 | 0.344 | 5.767 | -19008.341 | 0.080 | 0.883 | 6.665 | -19134.243
BTC-XMR | 0.545 | 0.745 | 0.572 | 6.220 | -17468.442 | 0.071 | 0.919 | 5.201 | -17663.209
BTC-XRP | 0.345 | 0.980 | 0.527 | 5.678 | -17804.362 | 0.076 | 0.916 | 5.443 | -17972.693
BTC-Sys:BTC | 0.699 | 0.709 | 0.608 | 5.572 | -12656.446 | 0.088 | 0.902 | 5.578 | -12940.102
ETH-BTC | 0.208 | 1.443 | 0.525 | 5.882 | -17582.578 | 0.085 | 0.910 | 5.612 | -17887.668
ETH-LTC | 0.468 | 0.979 | 0.613 | 7.518 | -16358.821 | 0.037 | 0.959 | 7.637 | -16666.221
ETH-XMR | 0.683 | 0.755 | 0.427 | 5.598 | -15271.656 | 0.039 | 0.958 | 4.448 | -15473.335
ETH-XRP | 0.242 | 1.477 | 0.477 | 6.108 | -15919.341 | 0.051 | 0.948 | 5.785 | -16247.803
ETH-Sys:ETH | 0.748 | 0.798 | 0.615 | 6.218 | -10940.029 | 0.060 | 0.937 | 5.454 | -11405.215
LTC-BTC | 1.075 | 0.647 | 0.341 | 5.767 | -19008.544 | 0.080 | 0.883 | 6.665 | -19134.243
LTC-ETH | 0.468 | 0.979 | 0.613 | 7.518 | -16358.825 | 0.037 | 0.959 | 7.637 | -16666.221
LTC-XMR | 0.462 | 0.905 | 0.484 | 6.394 | -16084.549 | 0.062 | 0.929 | 6.501 | -16243.662
LTC-XRP | 0.468 | 1.102 | 0.409 | 6.287 | -16796.757 | 0.051 | 0.944 | 5.186 | -17045.160
LTC-Sys:LTC | 0.818 | 0.770 | 0.506 | 5.911 | -11847.605 | 0.070 | 0.922 | 5.347 | -12170.773
XMR-BTC | 0.545 | 0.745 | 0.572 | 6.220 | -17468.442 | 0.071 | 0.919 | 5.201 | -17663.209
XMR-ETH | 0.683 | 0.755 | 0.427 | 5.598 | -15271.656 | 0.039 | 0.958 | 4.448 | -15473.335
XMR-LTC | 0.462 | 0.905 | 0.484 | 6.394 | -16084.549 | 0.062 | 0.929 | 6.501 | -16243.662
XMR-XRP | 0.475 | 0.785 | 0.491 | 5.757 | -15580.298 | 0.044 | 0.952 | 5.779 | -15747.537
XMR-Sys:XMR | 0.825 | 0.662 | 0.435 | 5.508 | -10713.380 | 0.053 | 0.938 | 4.537 | -10897.825
XRP-BTC | 0.345 | 0.980 | 0.527 | 5.678 | -17804.362 | 0.076 | 0.916 | 5.443 | -17972.693
XRP-ETH | 0.309 | 1.336 | 0.502 | 6.013 | -15914.526 | 0.051 | 0.948 | 5.785 | -16247.803
XRP-LTC | 0.468 | 1.102 | 0.409 | 6.287 | -16796.757 | 0.051 | 0.944 | 5.186 | -17045.160
XRP-XMR | 0.475 | 0.785 | 0.491 | 5.757 | -15580.298 | 0.044 | 0.952 | 5.779 | -15747.537
XRP-Sys:XRP | 0.392 | 1.253 | 0.478 | 5.036 | -11150.509 | 0.063 | 0.934 | 4.466 | -11445.793

Table B.8: Estimates of the multivariate copulae for BTC-ETH-LTC-XMR-XRP. Copula | Characteristic | Estimate/Value
---|---|---
Gaussian | $\theta$ | $\left[\begin{array}[]{ccccc}1.000&0.554&0.721&0.577&0.517\\\ &1.000&0.582&0.568&0.559\\\ &&1.000&0.555&0.590\\\ &&&1.000&0.511\\\ &&&&1.000\\\ \end{array}\right]$
AIC | -44087.260
$t$ | $\theta$ | $\left[\begin{array}[]{ccccc}1.000&0.639&0.746&0.633&0.593\\\ &1.000&0.659&0.621&0.629\\\ &&1.000&0.609&0.655\\\ &&&1.000&0.569\\\ &&&&1.000\\\ \end{array}\right]$
$\nu$ | 4.804
AIC | -45201.470
Clayton | $\theta$ | 1.077
AIC | -44174.270
Gumbel | $\theta$ | 1.616
AIC | -43531.790
DCC-$t$ | a | 0.036
b | 0.959
$\nu$ | 8.029
AIC | -46720.380

###  B.4 Plots of Systemic Risk Measures

Further illustrations of the estimated systemic risk measures are displayed
below.

Figure B.11: CoVaR of BTC-ETH using a time-invariant $t$-copula. Figure B.12:
CoVaR of BTC-XMR using a time-invariant $t$-copula. Figure B.13: CoVaR of BTC-
XRP using a time-invariant $t$-copula. Figure B.14: CoVaR of ETH-BTC using a
time-invariant $t$-copula. Figure B.15: CoVaR of ETH-LTC using a time-
invariant $t$-copula. Figure B.16: CoVaR of ETH-XMR using a time-invariant
$t$-copula. Figure B.17: CoVaR of ETH-XRP using a time-invariant $t$-copula.
Figure B.18: SCoVaR, MCoVaR, and VCoVaR of ETH using a time-invariant
$t$-copula. Figure B.19: CoVaR of LTC-BTC using a time-invariant $t$-copula.
Figure B.20: CoVaR of LTC-ETH using a time-invariant $t$-copula. Figure B.21:
CoVaR of LTC-XMR using a time-invariant $t$-copula. Figure B.22: CoVaR of LTC-
XRP using a time-invariant $t$-copula. Figure B.23: SCoVaR, MCoVaR, and VCoVaR
of LTC using a time-invariant $t$-copula. Figure B.24: CoVaR of XMR-BTC using
a time-invariant $t$-copula. Figure B.25: CoVaR of XMR-ETH using a time-
invariant $t$-copula. Figure B.26: CoVaR of XMR-LTC using a time-invariant
$t$-copula. Figure B.27: CoVaR of XMR-XRP using a time-invariant $t$-copula.
Figure B.28: SCoVaR, MCoVaR, and VCoVaR of XMR using a time-invariant
$t$-copula. Figure B.29: CoVaR of XRP-BTC using a time-invariant $t$-copula.
Figure B.30: CoVaR of XRP-ETH using a time-invariant $t$-copula. Figure B.31:
CoVaR of XRP-LTC using a time-invariant $t$-copula. Figure B.32: CoVaR of XRP-
XMR using a time-invariant $t$-copula. Figure B.33: SCoVaR, MCoVaR, and VCoVaR
of XRP using a time-invariant $t$-copula.