On multivariate contribution measures of systemic risk with applications in cryptocurrency market

In actuarial science and finance, risks are often not isolated events but are highly correlated and capable of spreading. When financial institutions face adverse conditions, such risks can quickly propagate through complex market interconnections, creating a domino effect that escalates losses from a single institution to the entire market, potentially triggering widespread systemic risk. Systemic risk is a core concept in these fields, as it leads to market failures, large-scale financial crises, and profound, long-lasting impacts on the real economy. Typical examples of systemic risk include financial crises, market crashes, bank runs, and contagion effects spreading across multiple industries (Dičpinigaitienė & Novickytė, 2018; Zhang et al., 2023).

In this paper, we focus on the “capital” type of systemic risk model, which is a method to assess systemic risk losses and their probability of occurrence, and can be characterized by a risk measure¹¹1Systemic risk models can be categorized into the following five types: (i) early warning systems (EWS); (ii) capital; (iii) liquidity; (iv) contagion; and (v) network. For further details, the reader can refer to Silva et al. (2017) and Ellis et al. (2022).. Among widely recognized risk measures, Value-at-Risk (VaR) and Expected Shortfall (ES) have been extensively discussed in the literature (Duffie & Pan, 1997; Delbaen & Biagini, 2000; Acerbi & Tasche, 2002). However, they fail to effectively quantify systemic risk as they only consider isolated individual economic entities. Hence, there is a need for novel conditional risk (co-risk) measures to quantify and evaluate systemic risk in financial systems. The Conditional Value-at-Risk (CoVaR), introduced by Adrian & Brunnermeier (2016), measures the VaR of a particular asset under a certain level of systemic stress. Mainik & Schaanning (2014) proposed Conditional Expected Shortfall (CoES), which measures the ES of a specific asset or portfolio under a certain level of systemic stress and is defined by the average tail integral of CoVaR. Acharya et al. (2017) introduced the concept of Marginal Expected Shortfall (MES) and empirically validated its effectiveness in predicting emerging risks during the 2007-2009 financial crisis. The theoretical properties of these co-risk measures and their applications in finance and insurance can be found in Kritzman et al. (2011); Asimit & Gerrard (2016); Feinstein et al. (2017); Asimit & Li (2018); Lin et al. (2018); Duarte & Eisenbach (2021); Liu & Yang (2021); Waltz et al. (2022); Yang et al. (2024). However, these co-risk measures can only assess the interaction effect from one entity to another entity, but cannot characterize the absolute or relative spillover effects, which lays the foundation for relevant research on proposing various risk contribution measures.

The class of risk contribution measures can be divided into two types: difference-based and ratio-based contribution measures. The former is usually defined as the difference between conditional and unconditional risk measures, while the latter is defined as the ratio between the difference-based contribution measure and the benchmark unconditional risk measure. For example, Adrian & Brunnermeier (2016) introduced the well-known difference-based and ratio-based contribution measures in terms of CoVaR when the conditional systemic event is taken as VaR at some fixed level. Girardi & Ergün (2013) also defined the difference-based contribution measures (denoted as $\Delta$CoVaR and $\Delta$CoES) based on CoVaR and CoES with different types of conditional events. From the perspective of stochastic orders and dependence structures, Sordo et al. (2018) provided sufficient conditions to rank CoVaR, CoES, and their risk contribution measures $\Delta{\rm CoVaR}$ and $\Delta{\rm CoES}$. Dhaene et al. (2022) introduced conditional distortion (CoD) risk measures and the difference-based distortion risk contribution measures ($\Delta{\rm CoD}$), discussing sufficient conditions to rank different bivariate vectors with respect to these measures. Recently, Zhang (2024) introduced several types of ratio-based distortion risk contribution measures ($\Delta^{\rm R}{\rm CoD}$) and examined sufficient conditions for comparing these measures in terms of a new characterization of the convex transform order.

Most of the above-mentioned works only consider a single risk as the systemic risk event, which runs counter to the reality in the financial market that there might be multiple risks collapsing simultaneously. This scenario hinders the usage of the aforementioned systemic risk measures, which calls for the definitions of multivariate systemic risk measures. In fact, multivariate risks are attracting increasing attention from many researchers such as Lee & Prékopa (2013), Sun et al. (2018), and Ling (2019). Let ${\bm{X}}=(X_{1},\cdots,X_{n})$ represent a portfolio of risks. Here, we assume $X_{2},\cdots,X_{n}$ represent systemic risk, capturing the volatility and uncertainty of the entire system or market, rather than just the impact of any individual asset or event. The definition of multivariate marginal mean excess (in short, MMME) is initially introduced by Das & Fasen-Hartmann (2018) and the asymptotic behavior is studied under suitable conditions within the framework of multivariate regular variation, hidden regular variation and asymptotic tail independence. The multivariate CoVaR (in short, MCoVaR) and multivariate CoES (in short, MCoES) are firstly formally defined in Ortega-Jiménez et al. (2021) to quantify the risk of $X_{2},\cdots,X_{n}$ spilled over to $X_{1}$. Besides, the difference-based contribution risk measures are also introduced. Utilizing MCoVaR, MCoES and MMME, and their associated difference-based contribution measures (with unconditional risk measure as the benchmark), Ortega-Jiménez et al. (2021) investigated sufficient conditions for implementing comparisons between two different sets of multivariate risk vectors.

In this paper, we introduce two types of multivariate risk contribution ratios to reassess the concepts of MCoVaR, MCoES, and MMME. Our research motivation stems from the fact that when we are more concerned with relative contributions rather than absolute contributions of systemic risks, the effectiveness of those contribution measures studied in Ortega-Jiménez et al. (2021) is limited. Therefore, this paper introduces the multivariate risk contribution ratio measures to address this issue, where the first type of benchmark measure uses unconditional risk values such as VaR or ES, which are univariate risk measures, and the second type of benchmark measure uses a multivariate joint risk measure based on the median of systemic events, a method commonly used in financial markets. Based on the newly proposed contribution measures, we theoretically analyze sufficient conditions for comparing two distinct multivariate risk portfolios. In particular, for two risk vectors $(X_{1},\dots,X_{n})$ and $(Y_{1},\dots,Y_{n})$, the consistency of co-risk measures is examined under different stochastic orders and dependence assumptions. Further, we compute the values of these new risk measures using real-world dataset in the cryptocurrency market, comparing with existing related risk measures, and analyzing the market interactions.

The remaining sections of this paper are structured as follows. Section 2 reviews some basic concepts, including univariate and multivariate stochastic orders, copula functions, and some well-known (conditional) risk measures. Section 3 introduces several new definitions of multivariate systemic risk contribution ratio measures based on MCoVaR, MCoES and MMME, and establishes sufficient conditions to compare two different risk portfolios under these new measures. Section 4 analyzes the risk co-movement effect in the cryptocurrency market by computing and comparing these proposed contribution risk measures. Section 5 concludes the paper. All proofs and supplementary definitions are provided in the appendix.

Throughout this paper, let $\bm{X}=(X_{1},\dots,X_{n})$ and $\bm{Y}=(Y_{1},\dots,Y_{n})$ represent two $n$-dimensional random vectors with joint distribution functions (d.f.) denoted by $F$ and $G$, and joint density functions $f({\bm{x}})$ and $g({\bm{x}})$, respectively, which will be abbreviated to “$\bm{X}\sim F$” and “$\bm{Y}\sim G$”. Their marginal distributions are denoted by $F_{1},\dots,F_{n}$ and $G_{1},\dots,G_{n}$, which are continuous and have finite expectations. Additionally, their joint survival functions are denoted by $\overline{F}$ and $\overline{G}$, that is, $\overline{F}({\bm{x}})$$=\mathbb{P}({\bm{X}}>{\bm{x}})$ and $\overline{G}({\bm{x}})=\mathbb{P}({\bm{Y}}>{\bm{x}})$ for ${\bm{x}}\in\mathbb{R}^{n}$. Let ${\bm{x}}=(x_{1},\dots,x_{n})$ and ${\bm{y}}=(y_{1},\dots,y_{n})$ be two real-valued vectors in $\mathbb{R}^{n}$, we denote ${\bm{x}}\vee{\bm{y}}=\left({\rm min}\{x_{1},y_{1}\},\dots,{\rm min}\{x_{n},y_{n}\}\right)$ and ${\bm{x}}\wedge{\bm{y}}=\left({\rm max}\{x_{1},y_{1}\},\dots,{\rm max}\{x_{n},y_{n}\}\right)$.

Ortega-Jiménez et al. (2021) introduced two definitions of difference-based multivariate risk contribution measures corresponding to MCoVaR and MCoES, where the benchmark risk measure does not involve systemic risk. However, when regulators in financial markets focus on the relative spillover effects of systemic risk, the effectiveness of these measures becomes limited. To assess relative risk, the relative spillover effect of risk can be measured by dividing the multivariate risk contribution of an entity by its benchmark. One way to evaluate the risk contribution ratio of $X_{2},\ldots,X_{n}$ to $X_{1}$ is to compare the conditional risk measure of $X_{1}$ (MCoVaR) with its unconditional risk value (VaR). Another method is to replace the unconditional risk value $\mathrm{VaR}$ with the conditional VaR of $X_{1}$ when $X_{2},\ldots,X_{n}$ are under benchmark conditions, where the benchmark state is typically defined by the median (Sordo et al., 2018).

We introduce the definition of a risk contribution ratio measure leveraging MCoVaR as follows.

Similarly, the multivariate risk contribution measures for MCoES are defined as follow.

Correspondingly, the ratio-based contribution $\rm MMME$ with unconditional mean excess as the benchmark measure is defined as follows.

These multivariate risk contribution ratio measures are new compared with the ones introduced in Ortega-Jiménez et al. (2021). Next, we shall establish sufficient conditions for comparing these newly proposed measures for multivariate risk vectors. For two $n$-dimensional portfolio of risks $\bm{X}$ and $\bm{Y}$, this section established sufficient conditions for comparing the ratio-based risk contribution measures in terms of MCoVaR, MCoES, and MMME. The next result compares $\Delta^{\rm R}{\rm MCoVaR}_{\bm{p}}[X_{1}|X_{2},\dots,X_{n}]$ and $\Delta^{\rm R}{\rm MCoVaR}_{\bm{p}}[Y_{1}|Y_{2},\dots,Y_{n}]$ under some appropriate conditions imposed on marginal risks $X_{1}$ and $Y_{1}$ and the dependence structure.

Recall that an increasing function $h:[0,1]\rightarrow[0,1]$ is said to be a distortion function if it satisfies $h(0)=0$ and $h(1)=1$. The following lemma is needed for proving the comparison results under other types of ratio-based contribution measures.

In the following theorem, sufficient conditions for comparison between $\Delta^{\rm R}{\rm MCoES}_{\bm{p}}[X_{1}|X_{2},\dots,X_{n}]$ and $\Delta^{\rm R}{\rm MCoES}_{\bm{p}}[Y_{1}|Y_{2},\dots,Y_{n}]$ are provided in terms of the expected proportional shortfall order.

Next, we provide an example to illustrate the findings in Theorems 3.4 and 3.6. It is known that if a random vector $\bm{X}$ satisfies ${\rm MTP}_{2}$, it then implies that $\hat{\bm{X}}_{i}\uparrow_{\rm SI}X_{i}$ holds for all $i\in\{1,\dots,n\}$. An example of a copula satisfying ${\rm MTP}_{2}$ is provided below. As per Müller & Scarsini (2005), the Archimedean copula $C_{\psi}$ is ${\rm MTP}_{2}$ if and only if $(-1)^{n}\psi^{(n)}$ is log-convex, where $\psi^{(n)}$ denotes the $n$-th derivative. Let $\Psi^{(n)}:={\rm ln}\left((-1)^{n}\psi^{(n)}\right)$. For the Gumbel copula, $\psi(u)=(-{\rm ln}~u)^{\theta}$, with $n=3$, we have

Fixing $\theta=2$, it follows that

Furthermore, taking the second derivative with respect to $u$ yields that

which implies that $C_{2}(u_{1},u_{2},u_{3})$ is ${\rm MTP}_{2}$. Based on this observation, the following two examples can be established.

The next result compares $\Delta^{\rm R}\mathrm{MMME}_{\bm{p}_{[-1]}}[X_{1}|X_{2},\dots,X_{n}]$ and $\Delta^{\rm R}\mathrm{MMME}_{\bm{p}_{[-1]}}[Y_{1}|Y_{2},\dots,Y_{n}]$ under some appropriate conditions imposed on the dependence structure when $\bm{X}$ and $\bm{Y}$ have the same marginals.

In Theorem 3.9, we assumed that both vectors have the same marginal distributions. Under this assumption, the comparison of $\Delta^{\rm R}{\rm MMME}$ can be equivalent to the comparison of MMME, with conditions similar to Corollary 2 in Ortega-Jiménez et al. (2021). Therefore, the two measures exhibit consistent ordering under the copula structure, a property we refer to as dependence consistency.

The following example is provided to show the validity of Theorem 3.9.

Under an unconditional risk measure as benchmark, different copulas can be used to measure the risk contributions of two multivariate risk vectors, as the benchmark is unaffected by the copula. This is the reason why we can consider different copulas in the comparison results developed in Theorems 3.4, 3.6 and 3.9. However, when a median-type co-risk measure as the benchmark, which is also affected by the copula, using different copulas can make the comparison of risk contribution measures challenging. Therefore, to ensure the comparability of median-type risk contribution measures, the same copula will be adopted in the following discussions.

The next result establishes sufficient conditions for comparison between $\Delta^{\rm R-med}{\rm MCoVaR}_{\bm{p}}[X_{1}|X_{2},\dots,X_{n}]$ and $\Delta^{\rm R-med}{\rm MCoVaR}_{\bm{p}}[Y_{1}|Y_{2},\dots,Y_{n}]$ when $X_{1}$ and $Y_{1}$ are ranked by the star order.

It is worth addressing that the condition $X_{1}\uparrow_{\rm RTI}(X_{2},\dots,X_{n})$ indicates that the conditional probability $\mathbb{P}(X_{1}>x_{1}\mid X_{2}>x_{2},\dots,X_{n}>x_{n})$ increases with $x_{j}$, focusing on how $X_{1}$ behaves given $(X_{2},\dots,X_{n})$. Such condition can be implied from $X_{1}\uparrow_{\rm SI}(X_{2},\dots,X_{n})$, which is very different from $(X_{2},\dots,X_{n})\uparrow_{\rm SI}X_{1}$. In fact, the later condition $(X_{2},\dots,X_{n})\uparrow_{\rm SI}X_{1}$ means that the conditional distribution of $(X_{2},\dots,X_{n}|X_{1}=x_{1})$ is stochastically increasing in $x_{1}$, showing how the random vector $(X_{2},\dots,X_{n})$ shifts as $X_{1}$ changes. These conditions describe different stochastic relationships and do not imply any interdeducible relationship.

In the following theorem, we establish sufficient conditions for comparing $\Delta^{\rm R-med}{\rm MCoES}_{\bm{p}}[X_{1}|X_{2},\dots,X_{n}]$ and $\Delta^{\rm R-med}{\rm MCoES}_{\bm{p}}[Y_{1}|Y_{2},\dots,Y_{n}]$ when $X_{1}$ and $Y_{1}$ are ranked by the expected proportional shortfall order.