Assessing portfolio vulnerability to systemic risk: a vine copula and APARCH-DCC approach

Mba, Jules Clement

doi:10.1186/s40854-023-00559-2

Research
Open access
Published: 24 January 2024

Assessing portfolio vulnerability to systemic risk: a vine copula and APARCH-DCC approach

Jules Clement Mba ORCID: orcid.org/0000-0001-6462-6385¹

Financial Innovation volume 10, Article number: 20 (2024) Cite this article

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Abstract

This study evaluates the sensitivity and robustness of the systemic risk measure, Conditional Value-at-Risk (CoVaR), estimated using the vine copula and APARCH-DCC models. We compute the CoVaR for the two portfolios across five allocation strategies. The novel vine copula captures the complex dependence patterns and tail dynamics. The APARCH DCC incorporates volatility clustering, skewness, and kurtosis. The results reveal that the CoVaR estimates vary based on portfolio strategy, with higher values for the cryptocurrency portfolio. However, CoVaR appears relatively robust across strategies compared to ΔCoVaR. The cryptocurrency portfolio has a greater overall vulnerability. The findings demonstrate the value of CoVaR estimated via the vine copula and APARCH-DCC in assessing portfolio systemic risk. This advanced approach provides nuanced insights into strengthening risk management practices. Future research could explore the sensitivity of the CoVaR to different weighting schemes, such as equal versus market-weighted portfolios. Incorporating the Gram–Charlier expansion of normal density into the APARCH specification enables a nonparametric, data-driven fitting of the residual distribution. Furthermore, comparing the CoVaR to another systemic risk measure could provide further insights into its reliability as a systemic risk measure.

Introduction

Since the introduction of Bitcoin by Nakamoto (2019) in 2008, the cryptocurrency market has grown significantly, with approximately 19,800 cryptocurrencies in existence and a total of 930 billion USD in market capitalization at the time of this writing. More than 6000 cryptocurrencies are actively traded on 527 cryptocurrency exchanges, with daily volume transactions of 117 billion USD. These statistics indicate the importance of the cryptocurrency market size. Many studies have been conducted on these new types of assets in terms of their technology (Yermack 2015, 2017), volatility (Dwyer 2015; Katsiampa 2017; Scaillet et al. 2020), and price formation (Cheah and Fry 2015). Cryptocurrencies have become an important class of assets, and their speculative nature may result in large gains or losses. Therefore, it is important to adequately model the risk of the built portfolio and the marginal risk contribution of each asset in the portfolio. Value-at-Risk (VaR) is the most widely used risk measure owing to its simplicity. It has the advantage of being convex and easy to calculate and implement when solving portfolio optimization problems. It provides a single number that summarizes the risk exposure of a portfolio, can be used to set risk limits, and evaluates the effectiveness of risk management strategies (Jorion 2007; Embrechts et al. 2013; Embrechts et al. 2013). However, in 1999, this measure was criticized for the first time by Artzner et al. (1999) for lack of consistency, particularly for not being a sub-additive and therefore overestimating risk. A second criticism of VaR is its definition: a VaR of the 95% confidence level gives no idea of the magnitude of the loss if the potential loss exceeds the fixed 5% quantile. Before gradually shifting to coherent risk measures like the Conditional Value-at-Risk (CVaR), for a long time, regulators have used VaR to determine the amount of capital to be set aside by financial institutions against possible future market risk. The CVaR was introduced by Rockafellar and Uryasev (2002). This provides a measure of risk that captures the potential severity of losses beyond the VaR level by considering the shape of the loss distribution. However, it can be more computationally intensive than VaR and sensitive to the shape of the loss distribution and choice of confidence level. The CVaR has diverse applications, including long-term investment, portfolio optimization, and electricity price risk management. CVaR is a useful tool for capturing downside risks beyond VaR and can be used in various contexts to improve risk management and decision-making. However, VaR focuses on a single asset in isolation.

Since VaR fails to go beyond idiosyncratic risk to capture systemic risk by focusing on an asset in isolation, the scale of the risk facing a portfolio cannot be obtained from a portfolio’s VaR. Systemic risk in the Benoit et al. (2017) sense can be defined as the risk that spillovers to the whole portfolio when many assets are simultaneously affected by severe losses. Adrian and Brunnermeier (2016) introduce the Conditional Value-at-Risk (CoVaR) systemic risk measure, which captures the Value-at-Risk (VaR) of an asset under normal market conditions against the VaR of the portfolio conditional on the fact that a given asset is in distress. Other systemic risk measures include the Systemic Expected Shortfall (SES) introduced by Acharya et al. (2010) in terms of the Marginal Expected Shortfall (MES), the systemic risk measure (SRISK) introduced by Acharya et al. (2012), which is a variant of the systemic expected shortfall (SES). The cryptocurrency market has been subject to multiple crises, in which the prices of most cryptocurrencies plunged to nearly 40% in just a few days. Therefore, it is important for investors to assess the risk of their portfolios conditional on an asset being in distress. This indicates the assets to be closely monitored when the cryptocurrency market is in distress.

In this study, we consider the CoVaR of Adrian and Brunnermeier (2016) to analyze the vulnerability of two different portfolios consisting of ten major cryptocurrencies, a mixture of three cryptocurrencies, and seven top world indices. To evaluate the robustness of these systemic measures, we used three portfolio strategies: the global minimum variance (GMV) portfolio, the most diversified portfolio (MDP), and the market-weighted portfolio (MWP). Within the cryptocurrency market, Ji et al. (2019) use return and volatility spillovers to examine connectedness among the six largest cryptocurrencies. Borri (2019) uses the CoVaR to estimate the conditional tail risk of Bitcoin, Ethereum, Litecoin, and Ripple. These findings indicate that cryptocurrencies are highly exposed to tail risk. The CoVaR estimation is based on quantile regressions. Mba (2022) uses CoVaR to systematically classify important cryptocurrencies into a pool of 10 major cryptocurrencies. To compute CoVaR, we use the Flexible Dynamic Conditional Correlation (FDCC) by Engle (2002b) combined with the APARCH model. The idea behind the FDCC was that high- and low-capitalization cryptocurrencies do not have the same dynamics. In the FDCC model, the dynamics are constrained to be equal only among the groups of random variables. In this study, our CoVaR computation was based on vine copula-APARCH models. Unlike the FDCC model, the vine copula uses a cascade of bivariate copulas to model the dynamics among random variables. In our model, we allow the auto-selection of a suitable copula pair for a given pair of random variables. What makes our model interesting is the following. On the one hand, besides the leptokurtic property, the APARCH model introduced by Ding et al. (1993) captures other stylized facts in financial time series, such as volatility clustering and the leverage effect, as well as the long memory property. The power of the APARCH model stems from the fact that it encompasses several other generalized GARCH heteroskedasticity models. However, beyond linear correlation, the copula proposed by Sklar (1959) has shown great capability in modeling the dependence structure among random variables using a joint distribution from marginal distributions of the given random variables. In addition to elliptical copulas with multidimensional capabilities, Archimedean copulas are limited to bivariate random variables. To benefit from this large family of copulas, Bedford and Cooke (2001) introduced a vine copula, a flexible graphical model for describing multivariate copulas constructed using a cascade of bivariate copulas or pair copulas. Popular vine copulas include the regular vine (R-vine) copula with its subclasses, the canonical vine (C-vine) copula, and the drawable vine (D-vine) copula. They differ in their tree constructions–in how they associate pairwise random variables to model the dependence structure.

This study aimed to assess a portfolio’s vulnerability to its assets’ tail risk and investigate the sensitivity of CoVaR to various portfolio allocation and optimization strategies, namely, market-weighted portfolio (MWP), global minimum variance (GMV), most diversified portfolio (MDP), mean-variance differential evolution (MVDE), and mean-variance particle swarm optimization (MVPSO). Our analysis represents a significant contribution to portfolio risk management as it offers a comprehensive evaluation of portfolio vulnerability that considers the interdependence of asset returns and the impact of extreme events on portfolio performance. In particular, we contribute to the literature as follows:

(1)
Developing a novel approach for assessing portfolio vulnerability based on CoVaR that accounts for the tail risk of individual assets and their contribution to overall portfolio risk.
(2)
We conduct an extensive empirical analysis to assess the sensitivity of the CoVaR to different portfolio allocation and optimization strategies, including market-weighted portfolios, global minimum variance portfolios, most diversified portfolios, mean–variance-based differential evolution, and particle swarm optimization portfolios.
(3)
This study provides insights into the relative effectiveness of these strategies in mitigating tail risk and improving portfolio performance, as well as identifying potential trade-offs between risk reduction and return maximization.

Overall, our findings highlight the importance of incorporating CoVaR-based risk measures into portfolio management practices and suggest that optimal portfolio allocation and optimization strategies may vary depending on the level and nature of tail risk exposure. By providing a rigorous and systematic analysis of portfolio vulnerability, our study offers valuable insights for investors, asset managers, and risk professionals seeking to improve their understanding of portfolio risk and enhance their risk management strategies.

The remainder of this paper is organized as follows. Sect. "Literature review" presents a literature review, Sect. "Methodological framework" presents the methodological framework, and Sect. "Empirical results and discussion" presents the empirical results and discussion. Sect. "Sensitivity and robustness check" provides a Sensitivity and Robustness check, and Sect. "Conclusions" summarizes the findings and indicates future work directions.

Literature review

The rapid development and adoption of cryptocurrency and blockchain technologies have created new opportunities and challenges for investors and asset managers. Cryptocurrencies like Bitcoin, Ethereum, and Ripple are digital assets operating on decentralized networks. Blockchain is an underlying technology that enables secure and transparent cryptocurrency transactions. Blockchain has been hailed as a revolutionary innovation that can transform various industries and sectors, such as finance, supply chains, healthcare, and education (Xu et al. 2019). However, investing in cryptocurrencies also involves significant risks such as volatility, security breaches, regulatory uncertainty, and market manipulation. Cryptocurrencies exhibit complex and dynamic behaviors influenced by various factors such as supply and demand, network effects, technological innovations, media attention, and investor sentiment. These factors can lead to extreme price movements and spillover effects across cryptocurrencies and traditional assets (Fang et al. 2022). Therefore, investors and asset managers must understand and measure cryptocurrencies’ risks and interactions with other assets. Traditional risk measures such as value-at-risk (VaR) or standard deviation may not adequately capture the tail risk of cryptocurrencies, which is the risk of extreme losses in the lower or upper tails of the return distribution. Tail risk can severely affect portfolio performance and financial stability.

One of the most popular tail risk measures is the Conditional Value-at-Risk (CoVaR), defined as the VaR of a system conditional on an institution being under distress (Adrian and Brunnermeier 2016). The CoVaR captures the systemic risk contribution of an individual asset or institution to the overall system. CoVaR can be extended to measure the bilateral tail risk between two assets or institutions, called ΔCoVaR. ΔCoVaR measures the change in CoVaR due to the distress of another asset or institution. However, CoVaR and ΔCoVaR are not easy to estimate, as they require modeling the joint distribution of returns and accounting for various stylized facts, such as fat-tails, volatility clustering, skewness, and long memory. Moreover, CoVaR and ΔCoVaR may be sensitive to the portfolio composition and optimization strategy investors and asset managers use. Different portfolio strategies may result in different weights, diversification benefits, and risk-return trade-offs. Finding the right algorithm that provides better diversification benefits and risk-return trade-offs can be considered a multiple-criteria decision-making (MCDM) problem, and MCDM techniques can be used to select the best method for a problem at hand (Kou et al. 2014). Sebastião and Godinho (2021) examined the predictability of three major cryptocurrencies (Bitcoin, Ethereum, and Litecoin) and the profitability of trading strategies devised using machine learning techniques. Kou et al. (2019) survey existing research and methodologies for assessing and measuring financial systemic risk combined with machine learning technologies, including big data, network, and sentiment analysis.

Allen et al. (2012) propose a measure of aggregate systemic risk called CATFIN, which complements microlevel systemic risk measures by focusing on direct interbank connections and can determine the macroeconomic implications of aggregate risk-taking in the financial system. This can provide bank regulators with early warning signals to calibrate a micro-level systemic risk premium (or tax) to macroeconomic conditions. Huang et al. (2011) present a systemic risk indicator measured as a financial firm's marginal contribution to the financial sector's distress insurance premium. This systemic risk approach indicates that a bank’s contribution to systemic risk is roughly linear in its default probability but highly nonlinear concerning institution size and asset correlation. This risk is associated with the degree of interdependence among financial firms, as in Hartmann et al. (2006). In the same direction of research, we can mention the works of Billio et al. (2012), Ang and Longstaff (2013), Diebold and Yılmaz (2014), Hautsch et al. (2015), and Georgosouli and Goldby (2017). In a different direction, Brownlees and Engle (2017) introduce a systemic risk measure called SRISK, defined as the expected capital shortfall of a financial entity conditional on a prolonged market decline. It considers the size of the firm, its degree of leverage, and expected equity loss, conditional on market decline. This measure is related to Acharya et al. (2010) Systemic Expected Shortfall (SES), which measures a financial firm’s conditional capital shortfall. In the SES framework, the financial system is vulnerable because financial institutions do not factor in the negative externality costs they generate during a crisis. To compute the SRISK, Brownlees and Engle (2017) use Engle’s (2002a) GARCH-DCC model, which is widely used in financial time series analysis because of its ability to capture the stylized facts of financial data well. To measure systemic risk in the Chinese banking sector, Xu et al. (2018) used the CoVaR approach based on a DCC-MIDAS-t model. The performance of this model, which introduces the Student’s t distribution into the standard DCC-MIDAS to adequately capture fat-tailed returns, was compared to those of DCC-MIDAS-N and DCC-GARCH in the CoVaR measurement. The findings show that the DCC-MIDAS-t model outperforms conventional models in volatility prediction and CoVaR measurements. Adrian and Brunnermeier’s (2016) CoVaR links the systemic risk contribution of a financial institution in distress to an increase in the VaR of the entire financial system. The CoVaR has been widely used owing to its simplicity and effectiveness. Traditional CoVaR-based models include quantile regression, multivariate GARCH, and copulas, such as Girardi and Ergün (2013), Rösch and Scheule (2016), Karimalis and Nomikos (2018), and Xu et al. (2018). Adrian and Brunnermeier (2016) employ linear quantile regressions to obtain CoVaR estimates. However, the estimates derived from this procedure do not include the time-varying exposure to the institution’s VaR. Numerous studies indicate that the correlation between financial series is not constant over time (Engle, (2002a) Patton, (2006)). This tends to be more pronounced during downturns than during upturns, a stylized feature that should be incorporated in estimating systemic risk. Girardi and Ergün (2013) incorporate time-varying correlations into their CoVaR estimates through a three-step procedure based on a univariate GARCH-type model and the bivariate DCC model of Engle (2002a). However, their approach requires numerical integration, which can be computationally intensive and time-consuming. In addition, the specification of the marginal distribution in this procedure depends on the choice of bivariate distribution of assets ${R}_{1}$ and${R}_{2}$. In practice, as Karimalis and Nomikos (2018) point out, the distributional characteristics of ${R}_{1}$ and ${R}_{2}$ can differ substantially; hence, by restricting the marginal specification, a misspecification bias can be introduced in the CoVaR estimation. Focusing on a portfolio of large European banks, Karimalis and Nomikos (2018) use copula functions to estimate the CoVaR and measure the contribution of each bank to systemic risk. They show that the ranking of systemically important institutions and the magnitude of their corresponding CoVaRs are affected by the choice of underlying distributions modeled by a broad range of copula families (Frank, Gumbel, Clayton, and BB7). However, instead of relying only on bivariate copulas, such as Frank, Gumbel, and Clayton, to model dependence, the vine copulas introduced by Bedford and Cooke (2001) appear more appropriate in a multivariate setting. These flexible graphical models describe the multivariate copulas constructed using a cascade of bivariate copulas. Copula has been used in the computation of CoVaR (Usman et al. 2019), Ji et al. (2018a, b), Sun et al. (2020), Ji et al. (2018a, b). Copula models perform better than other models and are more suitable for dealing with nonlinear, non-stationary, and tail dependence between random variables (Patton 2012; Kayalar et al. 2017).

This study proposes a novel approach to estimate CoVaR and ΔCoVaR using Vine copula-APARCH models. The vine copula is a flexible tool that can capture complex non-linear dependencies and tail dynamics across assets. APARCH is a generalized version of GARCH that incorporates volatility clustering, skewness, leptokurtosis, and long memory into conditional variance. We apply our approach to two portfolios comprising major cryptocurrencies and world indices. We also consider five portfolio allocation and optimization strategies: market-weighted portfolio (MWP), global minimum variance (GMV) portfolio, maximum diversification portfolio (MDP), mean–variance differential evolution (MVDE), and mean–variance particle swarm optimization (MVPSO). We conduct a sensitivity analysis to assess the impact of portfolio composition and optimization strategy on CoVaR and ΔCoVaR.

Our results demonstrate that the CoVaR values are highly sensitive to portfolio strategy. The tracking lines for each asset are not parallel to the horizontal lines, indicating that different portfolio strategies can significantly impact the tail risk of individual assets and the overall portfolio. Moreover, the high CoVaR values for cryptocurrencies compared with traditional indices suggest that investing in cryptocurrencies carries a higher level of risk. Furthermore, the results suggest that CoVaR is even more sensitive to portfolio strategies than CoVaR. The GMV strategy provides the most vulnerable portfolio in terms of ΔCoVaR, emphasizing the importance of carefully selecting a portfolio strategy to mitigate tail risk.

Our findings have important implications for investors and asset managers seeking to enhance portfolio resilience and mitigate tail risk. First, our results suggest that traditional portfolio optimization strategies may not adequately mitigate tail risk in today’s rapidly changing and complex financial markets. Therefore, investors and asset managers may need to consider more sophisticated portfolio optimization techniques that explicitly account for tail risk, such as CoVaR and ΔCoVaR. Second, our findings suggest that investing in cryptocurrencies carries a higher level of risk than traditional indices do; therefore, investors should exercise caution when considering investing in cryptocurrencies. We recommend that investors conduct thorough due diligence on cryptocurrencies and consider their risk tolerance before investing. Overall, this study highlights the value of CoVaR estimated via the vine copula-APARCH in assessing portfolio systemic risk. This study gives investors and risk managers meaningful insights to enhance portfolio resilience. Our approach represents a significant methodological contribution to applying advanced econometric modeling to produce robust systemic risk metrics.

This study contributes to the literature by:

(1)
Including the vine copula in the CoVaR estimation allows a collection of copulas to be used in modeling the dependence structure. The model autoselects a suitable copula for each pair of random variables based on their shared characteristics.
(2)
Our study is also related to that of Brownlees and Engle (2017), who use the standard GARCHDCC model to capture the stylized facts of financial data. However, instead of the standard GARCH-DCC, we use the APARCH-DCC. What makes our model interesting is that, besides the leptokurtic property, volatility clustering, and leverage effect, the APARCH model introduced by Ding et al. (1993) captures other stylized facts in financial time series, such as the long memory property. The power of the APARCH model stems from the fact that it encompasses several other generalized GARCH heteroskedasticity models.
(3)
Unlike Karimalis and Nomikos (2018), who consider one portfolio and check the robustness of their procedure across different systemic risk measures (CoVaR and CoES), we consider one systemic risk measure (CoVaR) and check its robustness across different portfolio strategies, namely, global minimum variance, maximum diversification, and market portfolios. We believe that for the CoVaR to be a reliable systemic risk measure, it should be less sensitive to the type of portfolio strategy chosen. We apply these three portfolio strategies to two portfolios consisting of cryptocurrencies and stock indices.
(4)
We also aim to assess the impact of a shock to the cryptocurrency market on the traditional stock market and vice versa. Some previous systemic risk assessments of the cryptocurrency market include that of Ji et al. (2019), who use return and volatility spillovers to examine connectedness among the six largest cryptocurrencies. Borri (2019) uses CoVaR to estimate the conditional tail risk of Bitcoin, Ethereum, Litecoin, and Ripple and finds that these cryptocurrencies are highly exposed to tail risk.

Methodological framework

Portfolio theories

Modern portfolio theory and global minimum variance portfolio

Harry Markowitz was the pioneer of portfolio theory. In 1952, he published a seminal paper titled "Portfolio Selection" in the Journal of Finance. Markowitz (1959) introduced the concept of Modern Portfolio Theory (MPT). This theory is considered one of the foundations of modern portfolio construction and is widely used in investment management. MPT assumes that investors are risk averse and seek to maximize the expected return for a given level of risk or minimize risk for a given expected return. Despite many criticisms (Assumptions of rational investors (see, Shefrin 2002b; Shefrin 2002a; Prast 2003; Tversky and Kahneman 1981), Efficient Market Hypothesis (EMH) (see, Malkiel 2003; Sewell 2011), Single period optimization (see, Grossman 1995), it is still widely used in practice. In the Markowitz mean–variance portfolio theory, the goal is to optimally choose portfolio weighting factors, that is, one in which the portfolio achieves an acceptable baseline expected return with minimal volatility.

Consider a portfolio P consisting of n assets. Let ${r}_{i}$ be the return series for asset i and $r=\left({r}_{1},{r}_{2},\dots ,{r}_{n}\right).$ Set ${\mu }_{i}=E[{r}_{i}]$, $\mu =\left({\mu }_{1},{\mu }_{2},\dots ,{\mu }_{n}\right)$ and $\Sigma =cov(r)$ the positive semi-definite variance–covariance matrix of the portfolio’s assets. Given a set of weights $\omega =\left({\omega }_{1},{\omega }_{2},\dots ,{\omega }_{n}\right)$ associated with the portfolio,

${r}_{p}=\sum_{i=1}^{n}{r}_{i}{\omega }_{i}$, ${\mu }_{p}={\mu }^{T}\omega$ and ${\sigma }_{p}^{2}={\omega }^{T}\Sigma \omega$ are respectively the return, the mean, and the variance of the portfolio. If ${\mu }_{p}$ is the acceptable baseline expected return, then in the MPT framework, the optimization problem for the minimal variance portfolios can be stated as:

$$\left\{ {\begin{array}{*{20}l} {\mathop {\min }\limits_{\omega } \frac{1}{2}\omega^{T} \Sigma \omega } \hfill \\ {Subject\; to\; \omega^{T} \mu \ge \mu_{p} , and \omega^{T} i = 1 } \hfill \\ \end{array} } \right.$$

where ${\varvec{i}}$ is a column vector of ones.

According to this function, given a target return ${\mu }_{p}$, the weight vector for a minimal variance portfolio is given by

$${\omega }^{*}={\mu }_{p}{\omega }_{1}^{*}+{\omega }_{2}^{*}$$

(1)

where ${\omega }_{1}^{*}=\frac{1}{d}\left(c{\Sigma }^{-1}\mu -b{\Sigma }^{-1}{\varvec{i}}\right)$, ${\omega }_{2}^{*}=-\frac{1}{d}\left(b{\Sigma }^{-1}\mu -a{\Sigma }^{-1}{\varvec{i}}\right)$. The portfolio standard deviation is given by

$$\sqrt{\frac{1}{d}\left(c{\mu }_{b}^{2}-2b{\mu }_{b}+a\right)}$$

(2)

where $a={\mu }^{T}{\Sigma }^{-1}\mu$, b = ${\mu }^{T}{\Sigma }^{-1}{\varvec{i}}$, c = ${{\varvec{i}}}^{T}{\Sigma }^{-1}{\varvec{i}}$, and $d=ac-{b}^{2}$. Merton (1972) was consulted for a detailed explanation. Equation 2 describes a hyperbola for efficient mean–variance portfolios. The apex of this hyperbola is the Global Minimum Variance (GMV) portfolio with weights given by ${\omega }_{GMV}^{*}=-\frac{1}{c}{\Sigma }^{-1}{\varvec{i}}$. The GMV portfolio is the point on the efficient frontier where the portfolio has the lowest possible risk for a given expected return or the highest expected return for a given level of risk. This is important because it provides a baseline portfolio that can be used as a starting point for constructing efficient portfolios or as a standalone portfolio for investors prioritizing risk minimization over return maximization. This motivates our choice in this study to include the GMV portfolio in evaluating the sensitivity of the CoVaR measure.

CAPM and APT

In practice, one would like to better understand the risk-return trade-off because we want to maximize returns while minimizing risk. One way to achieve this is to solve a quadratic programming

$$\left\{ {\begin{array}{*{20}l} {\mathop {\min }\limits_{\omega } \frac{1}{2}\omega^{T} {\Sigma }\omega - \lambda \omega^{T} \mu } \hfill \\ {Subject \; to \;\omega^{T} i = 1 } \hfill \\ \end{array} } \right.$$

where i is a column of ones.

The MPT analysis assumes that asset f is added with a risk-free return, ${r}_{f}.$ Let ${\omega }_{0}$ be the weight to be assigned to the asset f. The Markowitz quadratic program can be written as follows:

$$\left\{ {\begin{array}{*{20}l} {\mathop {\min }\limits_{\omega } \frac{1}{2}\omega^{T} {\Sigma }\omega } \hfill \\ {Subject \;to\; r_{f} \omega_{0} \omega^{T} \mu \ge \mu_{b} , and \omega_{0} + \omega^{T} i = 1 } \hfill \\ \end{array} } \right.$$

Let ${r}_{M}={r}_{f}\left(1-{{\varvec{i}}}^{T}{\Sigma }^{-1}\left(\mu -{r}_{f}{\varvec{i}}\right)\right)+r{\Sigma }^{-1}\left(\mu -{r}_{f}{\varvec{i}}\right)$ be the market portfolio and let $r={\omega }_{0}{r}_{f}+{\mu }^{T}\omega$ be any efficient portfolio with ${\omega }_{0}=1-\alpha {{\varvec{i}}}^{T}{\Sigma }^{-1}\left(\mu -{r}_{f}{\varvec{i}}\right)$ and $\omega =\alpha {\Sigma }^{-1}\left(\mu -{r}_{f}{\varvec{i}}\right)$ for some value $\alpha$. Then the expected return ${\mu }_{r}$ of the portfolio $r$ satisfies the equation

$${\mu }_{r}={r}_{f}+\frac{{\mu }_{M}-{r}_{f}}{{\sigma }_{M}}{\sigma }_{r}$$

where ${\mu }_{r}=E\left[r\right], {\sigma }_{r}^{2}=variance\left(r\right), {\mu }_{M}={r}_{f}+{\left(\mu -{r}_{f}{\varvec{i}}\right)}^{T}{\Sigma }^{-1}\left(\mu -{r}_{f}{\varvec{i}}\right)$ and ${\sigma }_{M}^{2}={\left(\mu -{r}_{f}{\varvec{i}}\right)}^{T}{\Sigma }^{-1}\left(\mu -{r}_{f}{\varvec{i}}\right)$. This line describes the efficient frontier and is called the capital market line. The slope of this line $\frac{{\mu }_{M}-{r}_{f}}{{\sigma }_{M}}$ is called the price of risk. Consider a portfolio that combines ${r}_{M}$ and asset i. The capital

market line (CML) is tangent to the mean-standard deviation curve for this portfolio at portfolio ${r}_{M}$, so that

$${\mu }_{i}={r}_{f}+{\beta }_{i}\left({\mu }_{M}-{r}_{f}\right)$$

where ${\mu }_{i}$ is the expected return of asset i, ${\beta }_{i}=\frac{{\sigma }_{iM}}{{\sigma }_{M}^{2}}$, and ${\sigma }_{iM}$ is the covariance of the return on asset i

and market portfolio ${r}_{M}$. This is the Capital Asset Pricing Model (CAPM). Sharpe (1964) first introduced this concept. While MPT provides the rules for making investment decisions and a systematic approach to determining a set of efficient portfolios and selecting optimal portfolios to evaluate financial assets, the CAPM and the Arbitrage Pricing Theory (APT) models assume that all investors respect these rules for making investment decisions. The CAPM and APT contribute significantly to the understanding of the linear relationship between return, risk, and asset valuation in the capital market. APT, developed by Ross in the 1970s, provides a multifactor model for security pricing based on the concept of arbitrage. APT assumes that securities are priced based on multiple factors, not just a single market risk factor, and that these factors interact to determine the expected return on the security.

$${r}_{i}={r}_{f}+{\lambda }_{1}{\beta }_{i1}+{\lambda }_{2}{\beta }_{i2}+\dots +{\lambda }_{k}{\beta }_{ik}$$

where ${r}_{i}$ is the expected return of the asset i; ${r}_{f}$ is the risk-free rate; ${\lambda }_{j}\left(j=\mathrm{1,2},\dots ,k\right)$ is the factor risk premium related with the j-th factor and ${\beta }_{ij}$ is the return sensitivity of the asset i to the value of the factor $j, j=\mathrm{1,2},\dots ,k$. The key shortcoming of the APT model is that it does not specify the systemic risk factors. Many attempts have been made to evaluate these factors through factor analysis (Roll and Ross 1980; Dhrymes et al. 1984), the specification of macroeconomic factors (Faruque 2011; Zhu 2012; Jamaludin et al. 2017), and the specification of microeconomic factors (Tudor 2010; Uwubanmwenand and Obayagbona 2012; Idris and Bala 2015).

These portfolio theories suggest that diversification is a key element in portfolio construction. The MPT is built on the concept of diversification and argues that investors can maximize the expected returns for a given level of risk or minimize the risk for a given expected return by diversifying their investments across different asset classes. The CAPM is an extension of the MPT and incorporates diversification as a key factor in determining the expected returns for individual assets within a portfolio. Diversification comprises at least two dimensions. The first addresses the underlying common characteristics of diverse assets. The second one seeks to measure the degree of diversification regarding underlying characteristics. The variance–covariance matrix of returns is often used to assess the riskiness of assets and the dependencies between them. However, diversification is defined regarding the overall return dependencies.

Choueifaty and Coignard (2008) and Choueifaty et al. (2011) address portfolios’ theoretical and empirical properties when diversification is used as a criterion. They establish a measure to assess the degree of diversification of a long-only portfolio.

Most diversified portfolio

The diversification ratio (DR) is given by Eq. (3)

$${DR}_{\omega \epsilon\Omega }=\frac{1}{\sqrt{\rho +CR-\rho CR}}$$

(3)

where $\rho$ and $CR$ denote the volatility-weighted average correlation and the volatility-weighted concentration ratio, respectively.

The diversification ratio measures the degree of portfolio diversification. The higher the DR, the more diversified the portfolio. Portfolio solutions characterized by highly concentrated allocations or highly correlated asset returns qualify as poorly diversified. The most diversified portfolio (MDP) is obtained by maximizing DR:

$${P}_{MDP}=\mathrm{arg}\underset{\mathit{\omega \epsilon }\Omega }{\mathrm{max}}DR$$

(4)

The diversification ratio is maximized by minimizing ${\omega }^{T}C\omega$, where $C$ denotes the correlation matrix of the initial asset returns. Hence, the objective function coincides with that of a GMV portfolio. However, instead of using the variance–covariance matrix, a correlation matrix is employed. In the GMV approach, asset volatilities enter directly into the quadratic form of the objective function to be minimized, whereas the impact of asset volatilities is smaller for an MDP.

This study assesses the sensitivity and robustness of the CoVaR measure for portfolio strategies. In other words, is it a reliable measure independent of the portfolio strategy considered? To achieve this, we consider five portfolio strategies: global minimum variance (GMV) portfolio, most diversified portfolio (MDP), mean–variance differential evolution (MVDE), mean–variance particle swarm optimization (MVPSO), and market-weighted portfolio (MWP). In a market-weighted portfolio is the one in, weights are determined by the market share of each asset.

APARCH model

Consider a return series ${r}_{t}={\mu }_{t}+{a}_{t}$, where ${\mu }_{t}$ is the conditional expected return and ${a}_{t}=\sigma {z}_{t}$ is a zero-mean white noise, where ${z}_{t}\sim D(\mathrm{0,1})$, and $D$ is the skew Student’s t distribution. (Ding et al. 1993) found that ${|{a}_{t}|}^{s}$ often displays a strong and persistent autocorrelation for various values of s illustrating a long memory property. We say that ${a}_{t}\sim APARCH(p,q)$ if

$${\sigma }_{t}^{\delta }=\omega +\sum_{i=1}^{p}{\alpha }_{i}{\left(\left|{a}_{t-i}\right|-{\gamma }_{i}{a}_{t-i}\right)}^{\delta }+\sum_{j=1}^{q}{\beta }_{j}{\sigma }_{t-j}^{\delta }$$

(5)

where $\omega >0$, ${\alpha }_{i}\ge 0, {\beta }_{j}\ge 0, \delta >0$ and $-1<$ ${\gamma }_{i}<1$. The parameter ${\gamma }_{i}$ captures the leverage effect. In this study, we use $\left(p,q\right)=\left(\mathrm{1,1}\right)$ since it is usually the option that best fits the financial time series. Brooks (2002) proves that using a GARCH class model with one lag order is sufficient to describe volatility clustering in asset returns. The innovations in this model are assumed to follow a skewed Student’s t-distribution with a density function:

$$d\left( {x;\eta ,\lambda } \right) = \left\{ {\begin{array}{*{20}c} {bc\left( {1 + \frac{1}{\eta - 2}\left( {\frac{bx + a}{{1 - \lambda }}} \right)^{2} } \right)^{{ - \frac{\eta + 1}{2}}} , if x > - \frac{a}{b}} \\ {bc\left( {1 + \frac{1}{\eta - 2}\left( {\frac{bx + a}{{1 + \lambda }}} \right)^{2} } \right)^{{ - \frac{\eta + 1}{2}}} , if x \le - \frac{a}{b}} \\ \end{array} } \right.$$

where $a=4\lambda c\frac{\eta -2}{\eta +1}$, $b=1+3{\lambda }^{2}-{a}^{2}$, $c=\frac{\Gamma \left(\frac{\eta +1}{2}\right)}{\sqrt{\pi \left(\eta -2\right)\Gamma \left(\frac{\eta }{2}\right)}}$ and $\Gamma$ is the gamma function.

Vine copula

Although the DCC-GARCH model allows for a time-varying conditional correlation, it fails to reproduce the non-linear dependence that may exist between variables and does not provide information about tail dependence. Tail dependence corresponds to the possibility of joint events, such as a low or high occurrence of extreme events. To achieve this, an alternative approach based on copula functions was adopted. The main advantage of copulas is that they separate the dependence structure from the marginals without making any assumptions about the distribution. Using several copula functions, Nguyen and Bhatti (2012) provide evidence of left-tail dependence in Vietnam but not China. In the case of six CEE countries (Bulgaria, Czech Republic, Hungary, Poland, Romania, and Slovenia), Aloui et al. (2013) also find left-tail dependence. Copula functions are statistical tools for modeling the dependence structure by coupling multiple marginal distributions to represent a joint distribution function. This was introduced by Sklar (1959) in the following theorem:

Theorem 3.1

Assume $F=\left({F}_{1},\dots ,{F}_{n}\right)$ is an n-dimensional joint distribution function with the marginal distribution function ${F}_{i} \left(i=1,\dots ,n\right)$. Then there exists a copula $C$ such that for all ${\varvec{x}}=\left({x}_{1},\dots ,{x}_{n}\right) \epsilon { I}^{n}$,

$$F\left(x\right)=C\left({F}_{1}\left({x}_{1}\right),\dots ,{F}_{n}\left({x}_{n}\right)\right)$$

If ${F}_{1},\dots ,{F}_{n}$ continue, then $C$ is unique. Otherwise, $C$ is not unique to ${I}^{n}.$ In addition, if ${F}_{1},\dots ,{F}_{n}$ are distribution functions on I and if $C$ is a copula, then the function $F\left(x\right)=C\left({F}_{1}\left({x}_{1}\right),\dots ,{F}_{n}\left({x}_{n}\right)\right)$ is a joint distribution function on ${I}^{n}$. The canonical representation of the copula density function is

$$c\left({u}_{1},\dots ,{u}_{n}\right)=\frac{{\partial }^{n}C\left({u}_{1},\dots ,{u}_{n}\right)}{\partial {u}_{1}\dots \partial {u}_{n}}$$

(7)

To obtain the density of the n-dimensional distribution $F$, the following relationship is used:

$$f\left({x}_{1},\dots ,{x}_{n}\right)=c\left({F}_{1}\left({x}_{1}\right),\dots ,{F}_{n}\left({x}_{n}\right)\right)\prod_{i=1}^{n}{f}_{i}\left({x}_{i}\right)$$

(7)

where ${f}_{i}$ is the density of the marginal distribution ${F}_{i}$.

In risk management, one is interested in capturing tail dependence and describing the behavior of random variables during extreme events. We distinguish between symmetric and asymmetric copula based on how they model tail dependence. The pairwise upper and lower tail coefficients, denoted respectively by ${\lambda }_{U}$ and ${\lambda }_{L}$, are given by the following equations:

$${\lambda }_{U}=\underset{u\to 1}{\mathrm{lim}}\frac{1-2u+C\left(u,u\right)}{1-u}$$

(8)

$${\lambda }_{L}=\underset{u\to 1}{\mathrm{lim}}\frac{C\left(u,u\right)}{u}$$

(9)

This copula is not limited to 2 dimensions. This approach could be extended to arbitrarily large dimensions. However, the disadvantage is its practicality. At higher dimensions, the copula becomes rigid and tends to lose several useful properties. Therefore, the vine copula described by Bedford and Cooke (2001, 2002) addressed this high-dimensional probabilistic modeling issue. Instead of using a multidimensional copula directly, it first decomposes the probability density into conditional probabilities and then decomposes the conditional probabilities into bivariate copulas.

For example, let ${Y}_{1}$, ${Y}_{2}$ and ${Y}_{3}$ be three random variables with distribution functions ${G}_{1}$, ${G}_{2}$ and ${G}_{3}$ respectively. The joint density can be decomposed as:

$$g\left({y}_{1},{y}_{2},{y}_{3}\right)={g}_{3|12}\left({y}_{3}|{y}_{1},{y}_{2}\right){g}_{2|1}\left({y}_{2}|{y}_{1}\right){f}_{1}\left({y}_{1}\right)$$

where ${g}_{2|1}\left({y}_{2}|{y}_{1}\right)={c}_{12}\left({G}_{1}\left({y}_{1}\right),{G}_{2}\left({y}_{2}\right)\right){g}_{2}\left({y}_{2}\right)$,

$${g}_{3|12}\left({y}_{3}|{y}_{1},{y}_{2}\right)={c}_{\mathrm{13,2}}\left({G}_{1|2}\left({y}_{1}|{y}_{2}\right),{G}_{3|2}\left({y}_{3}|{y}_{2}\right)\right){g}_{3|2}\left({y}_{3}|{y}_{2}\right)$$

${g}_{3|2}\left({y}_{3}|{y}_{2}\right)={c}_{23}\left({G}_{2}\left({y}_{2}\right),{G}_{3}\left({y}_{3}\right)\right){g}_{3}\left({y}_{3}\right)$,

with $G\left(y|{\varvec{v}}\right)=\frac{\partial {C}_{y,{v}_{j},{{\varvec{v}}}_{-j}}\left(G\left(y|{{\varvec{v}}}_{-j}\right),G\left({v}_{j}|{{\varvec{v}}}_{-j}\right)\right)}{\partial G\left({v}_{j}|{{\varvec{v}}}_{-j}\right)}$ for every ${v}_{j}$ of the vector ${\varvec{v}}$ with ${{\varvec{v}}}_{-j}=\boldsymbol{ }{\varvec{v}}-\left\{{v}_{j}\right\}$ in the general case.

Aas et al. (2009) described the statistical inference techniques for two classes of vine copulas, namely, canonical vine (C-vine) and drawable vine (D-vine), which are the most commonly used in practice for dependence modeling. Owing to its star-like structure, the C-vine is useful when a key variable governs the interactions among random variables. This appears suitable in the cryptocurrency market context, where the top cryptocurrencies, Bitcoin and Ethereum, largely govern behavioral patterns in many other cryptocurrencies.

Co-risk measure: CoVaR

Let $r=\left({r}_{1},\dots ,{r}_{d}\right)$ be a vector of the asset returns in the portfolio. Let ${r}_{p}=\sum_{i=1}^{d}{\omega }_{i}{r}_{i}$ be the portfolio return, where ${\omega }_{i}$ is the weight of asset i. We define ${CoVaR}_{q}^{p|i}$ with confidence level $q$ as the $VaR$ of the portfolio conditional on asset i in a state of distress (i.e., at ${VaR}_{q}^{i}$). It is the q-quantile of the conditional probability distribution

$$P\left({r}_{p}\le {CoVaR}_{q}^{p|i}|{r}_{i}={VaR}_{q}^{i}\right)=q$$

(10)

$CoVaR$ is calculated as a quantile-based measure of systemic risk. It estimates the potential losses of a portfolio, given the severe loss experienced by asset i which pushes asset i to the lower quantile of its distribution. The calculation of CoVaR involves two steps. First, the Value-at-Risk (VaR) of the at-risk asset is calculated at a specified confidence level (e.g., 95%). VaR measures the maximum loss an asset is expected to incur within a specified time horizon at a given confidence level. Second, the portfolio’s CoVaR is calculated as its expected losses beyond its VaR conditional on asset i being in a low quantile (VaR). We measure the vulnerability of the portfolio to tail-risk in asset i with ${\Delta CoVaR}_{q}$ which is given by

$${\Delta CoVaR}_{q}^{p|i}=\left({CoVaR}_{q}^{p|i}|{r}_{i}={VaR}_{q}^{i}\right)-\left({CoVaR}_{q}^{p|i}|{r}_{i}={VaR}_{0.5}^{i}\right)$$

(11)

This is the difference between the CoVaR of the portfolio conditional on the distress of a particular asset i and the median state (i.e., q = 0.5), that is, during normal market conditions. Therefore, the larger (in absolute value) the ΔCoVaR, the higher the vulnerability of the portfolio to contagion from tail-risk events of the asset i.

To compute the CoVaR, the below steps are followed:

(1)
Compute daily returns data.
(2)
Fit the multivariate DCC-APARCH to the data.
(3)
Extract the standardized residuals.
(4)
Estimate the coefficients of the marginal distribution (skewed student distribution) for the C-vine copula simulation.
(5)
Simulate new data from C-vine copula.
(6)
Apply the inverse transform of the skewed Student’s t-distribution to the new data to recover the returns structure.
(7)
Compute variance–covariance matrix.
(8)
Compute the ${VaR}_{i}$ of each asset i in the portfolio.
(9)
Compute the ${CoVaR}^{p|i}$ of the portfolio given that asset i is at its ${VaR}_{i}$.

Data description

The data span from November 10, 2017, to April 07, 2022, from Yahoo Finance (https://finance.yahoo.com/), the weighted average prices from all the market exchanges. The rationale for using a weighted average is that markets with higher volumes generally have higher liquidity and are less prone to price fluctuations. The analysis is based on two portfolios, each with ten assets.

Portfolio 1 consists of ten major cryptocurrencies, namely, Bitcoin (BTC), Ethereum (ETH), Ripple (XRP), Cardano (ADA), Chainlink (LINK), Litecoin (LTC), Bitcoin Cash (BCH), Stellar (XLM), Binance Coin (BNB) and Dogecoin (DOGE). BTC was the first cryptocurrency created in 2009 and was still the largest by market capitalization. It was designed as a decentralized alternative to traditional currencies with a fixed supply of 21 million coins. ETH, created in 2015, is the second-largest cryptocurrency by market capitalization. It differs from Bitcoin in that it allows the creation of decentralized applications (dApps) on its blockchain using smart contracts. The XRP is a cryptocurrency designed for global payments and remittances. It is used by financial institutions and is known for its fast transaction speed and low fees. The ADA, created in 2017, is a third-generation blockchain platform that aims to be more secure, sustainable, and scalable than before its introduction. LINK is a decentralized oracle network that provides off-chain data to smart contracts in the blockchain. It was created in 2017 and aims to bridge the gap between blockchain technology and real-world applications. LTC is a cryptocurrency created in 2011 that is similar to Bitcoin but has faster transaction speeds and lower fees. It is designed as a more efficient alternative to Bitcoin for everyday transactions. BCH is a fork of Bitcoin created in 2017. It was designed to increase the block size limit of Bitcoin to allow for more transactions per block and lower fees. XLM is a cryptocurrency created in 2014 and has similar goals as XRP in revolutionizing cross-border payments; however, it has different target audiences, governance models, consensus mechanisms, transaction speeds and costs, and cryptocurrency use cases. The BNB, created in 2017, is a native token of the Binance Exchange, one of the world’s largest cryptocurrency exchanges. It is used to pay trading fees in exchange. It can also be used for other services, including payment for goods and services, investment in other cryptocurrencies, and participation in the Binance Launchpad platform for token sales. DOGE, a cryptocurrency created in 2013, was originally designed as a lightweight and fun alternative to BTC. Its transactions are generally processed much faster and at lower fees than BTC and many other cryptocurrencies, making it a popular choice for micropayments and tipping on social media platforms. To create a cryptocurrency portfolio, one might consider including a mix of assets with different characteristics and risk profiles, such as large-cap cryptocurrencies (BTC, ETH, BNB, XRP, and ADA), mid-cap cryptocurrencies (DOGE and LTC), and smaller-cap cryptocurrencies with higher growth potentials (LINK, BCH, and XLM).

Portfolio 2 consists of seven world indices and three cryptocurrencies: the NYSE COMPOSITE (NYA), NASDAQ Composite (IXIC), S&P 500 (GSPC), Euronext 100 Index (N100), FTSE 100 (FTSE), CAC 40 (FCHI), Dow Jones Industrial Average (DJI), Bitcoin (BTC), Ethereum (ETH) and Litecoin (LTC). NYA is an index that tracks the performance of all stocks listed on the New York Stock Exchange. It includes more than 2000 companies and covers a wide range of industries. The IXIC tracks the performance of all stocks listed on the NASDAQ stock exchange. These include technology, biotech, and other growth-oriented companies. The GSPC tracks the performance of the 500 largest publicly traded companies in the United States. It includes companies from various industries, such as technology, healthcare, and finance. The N100 tracks the performance of the 100 largest companies listed on the Euronext stock exchange, which operates in Amsterdam, Brussels, Dublin, Lisbon, Oslo, and Paris. The FTSE tracks the performance of the 100 largest companies listed on the London Stock Exchange. It includes companies from various industries, such as oil and gas, mining, and financial services. The FCHI tracks the performance of the 40 largest companies listed on the Euronext Paris Stock Exchange. It includes companies from various industries, such as healthcare, technology, and consumer goods. The DJI tracks the performance of 30 large publicly traded companies in the United States. It includes companies from various industries, such as technology, finance, and consumer goods. These indices track the performance of large and influential companies in their respective regions and industries. They can provide investors with a way to gauge the overall health of the stock market and economic conditions. To form Portfolio 2, we added to these seven indices BTC and ETH indices, the two largest cryptocurrencies in terms of market capitalization, and LTC, which is similar to BTC but offers faster transaction speeds and lower fees. The aim is to assess the level of impact that cryptocurrencies might have on the traditional stock market when in distress. However, we assess which assets are more vulnerable in both portfolios. Because many of the selected cryptocurrencies were created only in 2017, we collected data on November 10, 2017.

Table 1 presents the descriptive statistics of the assets in Portfolio 1. Except for BCH, which had a negative mean return, the remaining nine cryptocurrencies had positive mean returns. This finding suggests that investment in these cryptos during the study period will likely generate an overall positive return. The lowest percentage change in investment value over the specified period was significantly high in absolute value, with LINK having the highest at 61.5%. This indicates the worst-case scenario and potential downside risk of these cryptocurrencies. The highest percentage changes over the specified period were for XRP, ADA, and DOGE at 61%, 86%, and 152%, respectively. This indicates the best-case scenario and upside potential of these cryptocurrencies. The two leading cryptocurrencies, BTC and ETH, have the lowest maximum returns. Should this be an indication that these two cryptocurrencies are slowly reaching maturity?

Table 1 Descriptive statistics for assets in Portfolio 1

Assessing portfolio vulnerability to systemic risk: a vine copula and APARCH-DCC approach

Abstract

Introduction

Literature review

Methodological framework

Portfolio theories

Modern portfolio theory and global minimum variance portfolio

CAPM and APT

Most diversified portfolio

APARCH model

Vine copula

Theorem 3.1

Co-risk measure: CoVaR

Data description

Empirical results and discussion

Portfolio 1 & 2 under MWP, GMV, MDP, MVDE and MVPSO

Sensitivity and robustness check

Comparing CoVaR and ΔCoVaR from Portfolio 1

Comparing CoVaR and ΔCoVaR from Portfolio 2

Conclusions

Abbreviations

References

Funding

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