Vulnerability-CoVaR: Investigating the Crypto-market

BY KEYHUNTER 05.03.2025

Vulnerability-CoVaR: Investigating the Crypto-market

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\textsuperscript{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\textsuperscript{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 \texttt{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 \textit{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:

$$P(X_{i,t} \leq VaR^{i}_{\alpha,t}) = \alpha.$$ \hspace{1cm} (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 $F^{-1}(u) = \inf{x : F(x) \geq u}$.

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

$$P(X_{j,t} \leq CoVaR^{i,j}{\alpha,\beta,t} | X{i,t} = VaR^{i}_{\alpha,t}) = \beta, \quad \text{for } j \neq i.$$ \hspace{1cm} (2)

The $CoVaR^{i,j}_{\alpha,\beta,t}$ 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:

$$P(X_{j,t} \leq CoVaR^{i,j}{\alpha,\beta,t} | X{i,t} \leq VaR^{i}_{\alpha,t}) = \beta.$$ \hspace{1cm} (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 ( 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 ( i ), the SCoVaR is defined as:

[ P \left{ X_{j,t} \leq SCoVaR^j_{\alpha,\beta,t} \left| \sum_{i=1}^{p} X_{i,t} \leq VaR^i_{\alpha,t} \left( \sum_{i=1}^{p} X_{i,t} \right) \right. \right} = \beta. ]

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_{\alpha,\beta,t} \left| \sum_{i=1}^{p} X_{i,t} \leq VaR^i_{\alpha,t} \left( \sum_{i=1}^{p} X_{i,t} \right), \forall i : X_{i,t} \leq VaR^i_{\alpha,t} \right. \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 ( i ), the MCoVaR is defined as:

[ P(X_{j,t} \leq MCoVaR^j_{\alpha,\beta,t} | \forall i : X_{i,t} \leq VaR^i_{\alpha,t}) = \beta. ]

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 $i$, the VCoVaR is defined as:

$$P(X_{j,t} \leq VCoVaR_{i,\alpha,\beta,t} | \exists i : X_{i,t} \leq VaR_{i,\alpha,t}) = \beta.$$

(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 ( \text{CoVaR}{q,t}^{i,j} ) estimate is time-variant – the impact of ( \text{VaR}{q,t}^i ) on ( \text{CoVaR}{q,t}^{i,j} ) 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 \text{CoVaR}{\alpha,\beta,t}^{j|i}, X{i,t} \leq \text{VaR}{\alpha,t}^i)}{P(X{i,t} \leq \text{VaR}_{\alpha,t}^i)} = \beta, ]

which reduces to:

[ P(X_{j,t} \leq \text{CoVaR}{\alpha,\beta,t}^{j|i}, X{i,t} \leq \text{VaR}_{\alpha,t}^i) = \alpha \beta, ]

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

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 ( \text{VaR}_{\alpha,t}^i ).

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 ).

Step 3: Solve for ( \text{CoVaR}_{\alpha,\beta,t}^{j|i} ) the equation:

[ \int_{-\infty}^{\text{CoVaR}{\alpha,\beta,t}^{j|i}} \int{-\infty}^{\text{VaR}{\alpha,t}^i} \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 ( 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:

[ P(X_{j,t} \leq \text{CoVaR}{\alpha,\beta,t}^{j|i}, X{i,t} \leq \text{VaR}{\alpha,t}^i) = F{X_{j,t},X_{i,t}}(\text{CoVaR}{\alpha,\beta,t}^{j|i}, \text{VaR}{\alpha,t}^i) = C_{X_{j,t},X_{i,t}}{F_{X_{j,t}}(\text{CoVaR}{\alpha,\beta,t}^{j|i}), F{X_{i,t}}(\text{VaR}_{\alpha,t}^i); \theta_t}, ]

9 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 $^{ij}_{\alpha,\beta,t}$ is estimated by solving:

$$C_{X_{j,t},X_{i,t}}{F_{X_{j,t}}(\text{CoVaR}^{ij}_{\alpha,\beta,t}), \alpha; \theta_t} = \alpha \beta,$$

where $F_{X_{i,t}}(\text{VaR}^{i}{\alpha,t}) = \alpha$. Note that in the case of AC (9) can be solved analytically for CoVaR $^{ij}{\alpha,\beta,t}$, 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 $\text{VaR}{\alpha,t} = (\text{VaR}^{1}{\alpha,t}, \ldots, \text{VaR}^{p}{\alpha,t})^\top$, where $X_t \leq \text{VaR}{\alpha,t}$ holds componentwise. Furthermore, set $\alpha = (\alpha, \ldots, \alpha)^\top$ and $F_{X_t}(\text{VaR}{\alpha,t}) = {F{X_1,t}(\text{VaR}^{1}{\alpha,t}), \ldots, F{X_p,t}(\text{VaR}^{p}_{\alpha,t})}^\top$. To estimate the MCoVaR, (5) can be rewritten as:

$$\frac{P(X_{j,t} \leq \text{MCoVaR}{\alpha,\beta,t}, X_t \leq \text{VaR}{\alpha,t})}{P(X_t \leq \text{VaR}_{\alpha,t})} = \beta.$$

(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:

$$C_{X_{j,t},X_t}{F_{X_{j,t}}(\text{MCoVaR}^{ij}{\alpha,\beta,t}), F{X_t}(\text{VaR}^{\alpha}{\alpha,t}); \theta{1,t}}$$

leading to:

$$C_{X_{j,t},X_t}{F_{X_{j,t}}(\text{MCoVaR}^{ij}{\alpha,\beta,t}), \alpha; \theta{1,t}}$$

$$C_{X_t}(\alpha; \theta_{2,t}) = \beta,$$

Useful information for enthusiasts:

Contact me via Telegram: @ExploitDarlenePRO