This research examines the correlations between the return volatility of cryptocurrencies, global stock market indices, and the spillover effects of the COVID-19 pandemic. For this purpose, we employed a two-stage multivariate volatility exponential GARCH (EGARCH) model with an integrated dynamic conditional correlation (DCC) approach to measure the impact on the financial portfolio returns from 2019 to 2020. Moreover, we used value-at-risk (VaR) and value-at-risk measurements based on the Cornish–Fisher expansion (CFVaR). The empirical results show significant long- and short-term spillover effects. The two-stage multivariate EGARCH model's results show that the conditional volatilities of both asset portfolios surge more after positive news and respond well to previous shocks. As a result, financial assets have low unconditional volatility and the lowest risk when there are no external interruptions. Despite the financial assets' sensitivity to shocks, they exhibit some resistance to fluctuations in market confidence. The VaR performance comparison results with the assets portfolios differ. During the COVID-19 outbreak, the Dow (DJI) index reports VaR's highest loss, followed by the S&P500. Conversely, the CFVaR reports negative risk results for the entire cryptocurrency portfolio during the pandemic, except for the Ethereum (ETH).
Keywords: COVID-19 outbreak; value-at-risk (VaR); Cornish–Fisher expansion; stock market indices; cryptocurrencies return; stock return; spillover effects; volatility; EGARCH; DCC-GARCH
Though the pandemic is still active and has uncertain long-term outcomes ([
Asset allocation attempts to establish an equilibrium between risk and return by changing the ratio of each asset in a portfolio to achieve the goals, objectives, return expectations, and risk tolerance following an investment period for the investor. However, it is practical to anticipate that the topic of the COVID-19 pandemic and financial market volatility have, and will form, a significant interest in academic research in the near future. Future research topics might examine the effects of earlier occurrences resembling COVID-19, how COVID-19 may differ from those earlier events, and obtain an optimal portfolio in a highly dependent volatile financial market environment. Furthermore, along with studying the effects of previous pandemics, recent academic studies (see, [
Financial market information has become more widely available and has enhanced the relationship between market volatility and other factors. Based on price movements in other markets, investors forecast price changes. In other words, when one market's return grows, the other markets' returns may also alter simultaneously, forming a spillover effect. Therefore, Value-at-Risk (VaR) is one of the most commonly employed techniques for assessing market risk, a standardized volatility measurement tool ([
The COVID-19 pandemic has significantly impacted the financial markets worldwide. In addition, the outbreak has had a particularly detrimental effect on cryptocurrencies' potential as alternative investments. This research adds to the growing literature by examining the connections between the financial assets of each portfolio and the respective substantial volatility dynamics. As such, the study extends the literature by examining the correlations between the return-volatility of cryptocurrencies and global stock markets indices, such as the S&P500, DJI, GDAXI, and FTSE, the return-volatility spillover between cryptocurrencies, namely the Bitcoin, Ethereum, Cardano, and Ripple, and the effects of the COVID-19 pandemic on return-volatility by covering distinct peaks of the market during the pandemic. Considering the potential downside risk of investing in these two financial portfolios that are expanding simultaneously is critical. Therefore, studying the dynamics of the financial and cryptocurrencies bear markets during COVID-19 offers an unprecedented opportunity to examine. Comparing the behavior of cryptocurrencies to major stock market indices is worthwhile. [
Moreover, unlike earlier studies that used GARCH, and other mean-variance methods, this study addresses the cross-asset return and conditional volatilities using a multivariate two-stage dynamic conditional correlation model, the DCC-EGARCH model. The DCC model adopts a conditional correlational and time-varying impact to properly evaluate the dynamic correlation structure for addressing the volatilities and estimated returns. In addition, the study measures the market risk using the well-known VaR model. It employs a four-moment modified VaR based on the Cornish–Fisher (CFVaR) expansion, which in addition to mean and variance, also considers skewness and Kurtosis and is considered more accurate than the two moments VaR ([
This research analysis determines the essential positive and negative effects on the financial markets and quantifies the pandemic's impact. Moreover, the present research provides explanations of the market's volatility so that investor can diversify their investment approaches.
The research is structured as follows: we start with a brief literature review in Section 2, then in Section 3, we describe the employed models, followed by defining the data collection in Section 4, analyze the findings in Section 5, and finally proceed to the study conclusion in Section 6.
The COVID-19 outbreak and the uncertainty arising from each country's administration restrictions have shown a growing number of academic literature discussing the implications of cryptocurrencies, market indices, and alternative investments, see, ([
This section describes the proposed two stages GARCH modeling framework. To simulate the time-varying volatility in the stock market indices and the cryptocurrency return series, we first employed the alternative Generalized Autoregressive Conditionally Heteroscedastic (GARCH)-type specification, the exponential GARCH (EGARCH) model. Second, we follow [
The GARCH family of models introduced by [
The current study will employ a multivariate GARCH model in two stages to capture the asymmetric dependence structure ([
The volatility of financial instruments and commodities is frequently modeled using the GARCH-class models. A maximum likelihood (ML) estimator is practically used to estimate the GARCH models' vector of unstructured parameters. The asymmetric relationship between asset returns and volatility changes, which is crucial when working with time series financial data, is ignored by the symmetric ARCH and GARCH models, which can capture volatility clustering and leptokurtosis. [
(
where
[
(
(
(
(
where
Value at Risk (VaR) is a statistical tool used in assessing the volatility of stock indexes and has gained popularity as a tool for market risk analysis ([
(
where
In periods of market instability when returns are not normally distributed, a four-moment modified VaR, which supplements skewness and excess kurtosis to the two-moment VaR, gives more robust estimates on the downside risk of a portfolio. The downside risk is then quantified using higher statistical moments when simulating potential diversification advantages across various financial assets under consideration.
Using the VaR calculation will result in skewed findings since the log returns of financial assets are frequently skewed and, as a result, not normally distributed. One workable alternative is using the Cornish–Fisher expansion ([
The following equation describes how the Cornish–Fisher expansion, employing four moments, converts a conventional Gaussian variable z into a non-Gaussian random variable Z:
(
where
(
where W is the amount at risk or portfolio, and
To investigate the dynamic correlation between equity and commodity markets, we rely on DCC- GARCH. The maximum likelihood (MLE) estimation was used to estimate the ((p = 1) and (q = 1)) GARCH models for this study. According to [
(
The Akaike information criterion (AIC) ([
(
where k represents the total number of unknown parameters,
We examine the impact of the COVID-19 pandemic taking into account two portfolios, including major financial assets' daily close prices not only in the US market but also in Europe: (a) four stock market indices, namely in the US: the S&P 500 (GPSC), Dow Jones (DJIA), and Europe, the German DAX Performance (GDAXI), and in the UK the FTSE 100 (FTSE); (b) daily historical closing prices of cryptocurrencies such as the Bitcoin (BTC-USD), Ethereum (ETH-USD), Ripple (XRP-USD), and Cardano (ADA-USD) and correlates them with the COVID-19 period. The data set observations covering the period from 1 January 2019 to 31 December 2020 (1 January 2019 0:00 to 31 December 2020 23:59 EDT), and the entries for each observation included the exchange date, time, symbol, open, high, low, close price, and volume for all trades in US dollars. The data set range spans the coronavirus outbreak period after a significant upswing and downswing in asset prices following the second COVID-19 wave. The data sets were obtained from Alpha Vantage for the stock market indices and from "CryptoDataDownload.com (accessed on 14 June 2022),[
Portfolio returns were calculated using the formula
The models are arranged in a certain order using the Akaike Information Criterion (AIC). The Jarque–Bera Test is a test to determine if a set of data values follows the normal distribution based on the data's skewness and kurtosis ([
Figure 1 presents the historical price performance for the stock market indices and the log returns during the observed period. The red lines indicate that the market declined significantly during the four days in March 2020, generating excessive losses. Similarly, Figure 1 suggests that the log returns were moderately symmetrically distributed before the COVID-19 outbreak, with some peaks during the observed period. The stock market indices had experienced growth until the end of February or the beginning of March when the first actions in response to the coronavirus pandemic were disclosed.
Table 3 illustrates the summary statistics for the stock market portfolio returns. Each stock market index group in this study has 516 daily portfolio returns. To gain a more transparent overview of the data, we separated the dataset into three periods, i.e., the entire dataset period 2019 to 2020, one year before the coronavirus outbreak (2019), and one year during the heavy coronavirus pandemic period (2020) (Table 3). We note that the mean daily returns are positive for every assessed period for each market index and significantly during the pandemic period, respectively, except for the FTSE index, which reports a negative mean during the pandemic outbreak. On one side, it was expected as the European market indices have strongly crashed due to inconsistent decisions taken by the countries' governments and the imposed restrictions for the pandemic. Thus, in Table 3, the statistics provide such observations as the standard deviation that evaluates the market volatility and how widely prices are dispersed from the average price. Therefore, we observed that the standard deviations were higher for the DJI index than any other market indices returns for the entire study period and during the pandemic, followed by the S&P500. It confirms the fact that during the COVID outbreak between 12 February and 23 March 2020, the Dow Industrial Index (DJI) lost 37% of its value in four trading days ([
Furthermore, we have performed and provided in the tables the Jarque–Bera (J&B) test for all data frequencies, a formal test for the standard series distribution. The time series across all stock market indices groups are not distributed normally, according to the Jarque–Bera (J&B) test results for normality. At the 1% level of statistical significance, these findings reject the null hypothesis of a unit root and constant variance.
Table 1 presents the Pearson correlation coefficients for the stock market indices. As such, the S&P500 index (GSPC) reports a high correlation with the Dow Jones index (DJI) (0.980), and similarly, the German DAX index (GDAXI) with the Financial Times index (FTSE) (0.869), respectively. Moreover, the S&P500 index presents a moderate correlation with the FTSE index (0.681) and the GDAXI (0.677). Table 1 also shows the Augmented Dickey–Fuller (ADF) ([
Figure 2 displays this study's daily closing prices of the four cryptocurrencies. We observe that each cryptocurrency has a distinct tendency; for instance, Bitcoin and Ethereum followed a rising trend that held steady up to the days of the market crash in February and March, when the first coronavirus response measures were revealed. Similarly, we see that Cardano exhibits a nearly identical pattern through the end of January; as a result, they might be correlated. On the other hand, Ripple initially showed a stabilized gain before following a dramatic fall pattern until the beginning of April. Finally, the charts indicate that the prices of the remaining three cryptocurrencies have been steadily rising since the second half of April, except for Ripple, where we initially see a stable pattern followed by a sharp surge and decline.
The daily log returns of the observed market price indices for all exchanges trading in the studied cryptocurrencies are also shown in Figure 2, jointly with their associated daily log returns. The charts exhibit that the log returns are moderately symmetrically distributed, with specific spikes within the analyzed period, comparable to the earlier data presented in the descriptive statistics table. Moreover, Ripple (XRP-USD) demonstrates the tenuous proof of volatility clustering, which, although symmetrically distributed, does not follow similar spillover effects as the other three cryptocurrencies.
Table 4 reports the summary statistics for the cryptocurrency market returns. Similar to the stock market indices dataset, we have separated the statistics into three periods. For the entire period, the Bitcoin (BTC-USD) (−0.002) and the Ethereum (ETH-USD) (−0.001) present a negative mean (Panel A), however a year before the COVID-19 outbreak (Panel B), only the Bitcoin (BTC-USD) reports negative returns mean (−0.001). The remaining cryptocurrencies report positive means (Panel B). The standard deviation for Bitcoin (BTC-USD) 0.039 is the lowest among the cryptocurrencies, which indicates low volatility in each reported period, followed by Ripple (XRP-USD) 0.039 in the pre-COVID-19 pandemic period (Panel B). For the remaining cryptocurrencies, the standard deviations are 0.047, 0.055, and 0.057 for Ethereum (ETH-USD), Ripple (XRP-USD), and Cardano (ADA-USD), respectively (Panel A). The high level of volatility is characterized by comparatively high standard deviations as well as minimum and maximum rates. Additionally, the average return for all four cryptocurrencies is positive. XRP is the only cryptocurrency that displays more consistent positive returns over the evaluation period, according to Table 4. It implies that XRP has a lower systematic risk in the cryptocurrency market because its correlation with the market is more minimal than other cryptocurrencies. When the time correlation drops and a short uncorrelated Itô process is revealed, the GARCH models typically exhibit weak persistent behavior, especially over extended time horizons ([
Table 2 shows that the cryptocurrency pairs have positive and significant Pearson correlation coefficients. Additionally, a significant correlation of 0.834 is reported between Bitcoin and Ethereum. Furthermore, Cardano exhibits the lowest correlation with Bitcoin (0.681) and a moderately high correlation with Ethereum (0.724) and Ripple (0.716). Similarly, Ripple has the lowest correlation 0.554 and 0.616 with Bitcoin and Ethereum, respectively. Table 2 also presents the unit root tests for the study's cryptocurrencies.
This study employed an estimation approach in two steps for each portfolio. Constraints are established in the first stage for the univariate GARCH algorithm for each series. As such, the GARCH model with the maximum likelihood (MLE) estimator is frequently utilized since these features necessitate volatility modeling. Moreover, the exponential GARCH model (EGARCH), one of the well-known and often used model specifications for the GARCH process, offers better fits than traditional GARCH (
We employed a multivariate DCC-GARCH model in the second step to estimate pairwise models between the study portfolios. According to statistically significant estimated coefficients that are primarily near the 1% level, the volatility of an asset is commonly strongly influenced by its historical squared shocks and historical volatility, independent of the asset pair under study. We can observe evidence of considerable cross-market impacts between the variability of the returns of all assets pairs, particularly with shock and volatility spillovers, as demonstrated by the findings of the DCC-GARCH model in Table 5 and Table 6. At the 1% level, the estimates of
The multivariate DCC-GARCH parameter estimates for the most suitable EGARCH-type model are summarized in Table 5, which presents the performance of the study stock market indices. Empirical findings exhibit that all stock market indices except for the Dow Jones (DJI) index and the German DAX (GDAXI) index, which both show non-significant leverage impact parameter
Considering the previous discussion, GARCH models have been used to study the relationship between conditional variance and asset risk premium. The results show that volatility considerably rises in the wake of negative news. The parameter
To improve the estimation accuracy, in the second stage, Table 5, we employed a multivariate EGARCH DCC(
The performance of the cryptocurrency returns for the most acceptable EGARCH-type model (First Stage) is shown in Table 6, which also reports the multivariate DCC-GARCH parameter estimates (Second Stage). This empirical study revealed that cryptocurrency returns reported computed coefficients
In the Second Stage, Table 6, we used a multivariate EGARCH DCC(
Moreover, we assessed the stock market indices portfolio's historical returns for the entire dataset sample ranging from 1 January 2019 to 31 December 2020. We employ a VaR model, and a four-moment modified VaR model using the Cornish–Fischer (CFVaR) expansion for three confidence levels, i.e., 90%, 95%, and 99%, following a common approach of measuring the downside risk of a portfolio ([
During the COVID-19 outbreak period (January–December 2020), the Dow (DJI) index reports VaR's highest loss of 7.3%, followed by the S&P500 (6.6%), GDAXI with 5.4%, and the FTSE with 4.6%, respectively, similar to the entire period dataset. Next, we assess the period with the CFVaR approach (see Table 7). Our results show a similar loss direction, with Dow (DJI) reporting the highest loss of 8.9%, followed by the S&P500 (8.1%) index. The highest difference between the VaR and the CFVaR estimation is 2.4% in the GDAXI index, and the average difference is 1.9%. The results are not unexpected. A more significant CFVaR estimate than the typical VaR will be provided if the returns are not normally distributed, which is the case in this study. The results align with the study by [
Table 8 presents the confidence level of the historical returns of the cryptocurrencies market portfolio. Accordingly, at the 90% confidence level for the value-at-risk (VaR) during the entire period, we encounter the worst loss with Cardano (ADA) of 6.1%, followed by Ethereum (ETH) with a loss of 5.2%, which is consistent with the correlation between them. Bitcoin reports less estimated risk (4.2%) than the remaining cryptocurrencies, followed by Ripple (XRP) (4.4%). This trend continues, with BTC reporting a smaller loss at 99% confidence, at 9.1%, followed by ETH with 11.4%. For the same period, the CFVaR generated contradictory results. At the confidence level of 90%, all cryptocurrencies except Ethereum (ETH) yielded negative results, i.e., BTC (−1.0%), ADA (−4.0%), and XRP (−9.2%). It would be implied by the negative CFVaR that there is a significant likelihood of a profit for the portfolio. Ethereum (ETH) is the only cryptocurrency to report positive values at every confidence level. Because of the correlation between BTC and ETH, we would expect something similar here. As opposed to that, ETH reports the lowest loss of 9.6% at the 99% confidence level, followed by BTC. The XRP and the ADA report the highest loss of 31.9%, and 25.6%, respectively. A reason might be that both cryptocurrencies show a robust correlation.
During the COVID-19 outbreak period (January–December 2020), the VaR results in Table 8 show that at 90% confidence level, BTC generated the lowest loss (4.1%) followed by XRP (4.9%). At a 99% confidence level, ADA reported the worst loss (14.9%) followed by XRP (14.7%) loss. BTC reported the lowest loss (8.4%), which is significantly lower than the ADA. Similar to the entire period of the study, as well as during the COVID-19 pandemic, there is a robust correlation between ADA and XRP. In Table 8, the CFVaR during the pandemic reports negative results for the entire cryptocurrencies portfolio except the Ethereum (ETH) for confidence levels of 90% and 95%. For the 99% confidence level, ADA and XRP show similar risk at 21.1%, followed by BTC with 13.9% loss. Ethereum (ETH) reports less risky results at a 9.7% loss. The maximum difference between the cryptocurrencies loss is 11.4%, ADA and XRP to ETH at the 99% confidence level. These results contradict the risk reported in this study for the pre-pandemic period (January–December 2019), where Bitcoin (BTC) showed a higher risk value noting at the 99% confidence level of 11.9% risk, higher than the remaining cryptocurrencies. During the pandemic, BTC's trading behavior was more stable. The results confirm the study by [
Figure 3 provides the downside risk of both portfolios' during the entire period of the study. It is an evaluation of how much funds could be lost due to a security's ability to lose value in the event that market conditions change. It is employed to comprehend the worst-case event of asset investment. In Figure 3a, we encounter a sharp drawdown that recovers relatively quickly. In Figure 3b, the cryptocurrencies portfolio experiences a longer modest drawdown that lasts for a long time, sometimes less hurting than a sharp one.
Considering the volatility of any financial market portfolio involves the evaluation of the model's characteristics. This research focuses on evaluating the volatility dynamics of financial asset returns on four global stock market indices and four leading cryptocurrencies during the COVID-19 outbreak utilizing a two-stage multivariate DCC-GARCH Student-t distribution model. During the pandemic, the stock market indices collapsed in March 2020, leading to one of the most significant stock market crashes showing a decline of close to 40%. For example, the Dow Industrial Index (DJI) lost 37% of its value in four trading days ([
Investors should be mindful while making investment decisions in financial assets. We observed that the assets portfolio's current conditional variance was significantly affected by its historical volatility and shocks. Our findings indicate that the COVID-19 pandemic has improved the underlying assets' broad correlation. This work contributes by initially extending and verifying earlier GARCH models using EGARCH terms, an extension of the GARCH-family model to predict and anticipate volatility. Moreover, the research incorporated pairwise models between the assets portfolios estimated using the multivariate DCC-EGARCH model to produce more robust estimations. The DCC-EGARCH model also offers the benefit of including the analysis of dynamic beta values.
We quantified the relative risk using two widely used metrics of downside risk, portfolio value-at-risk (VaR) and value-at-risk (CFVaR), based on the Cornish–Fisher expansion, an approach suited for incorporating higher-order distributional properties associated with drastic price changes. According to the accuracy estimation, the VaR performance comparison results with the assets portfolios differ. For example, during the COVID-19 outbreak, the Dow (DJI) index reports VaR's highest loss, followed by the S&P500. Similarly, a close loss direction is informed by assessing the stock market using the CFVaR approach, with Dow (DJI) showing the highest loss, followed again by the S&P500 index. Conversely, during the pandemic, the CFVaR reports negative risk results for the entire cryptocurrency portfolio except the Ethereum (ETH) for confidence levels of 90% and 95%. As such, investment managers should choose GARCH-type models with a long memory to estimate the VaR of the portfolios, considering the high volatility dynamics observed in all financial assets.
Future research topics might examine the effects of earlier occurrences resembling COVID-19, how COVID-19 may differ from those earlier events, and obtain an optimal portfolio in a highly dependent volatile financial market environment. In addition, investors and portfolio managers actively investing in financial assets will find our observations of considerable interest. Overall, our findings offer information to regulators and investors on risk management and optimal asset allocation. Investors can use optimal portfolios to create portfolios that decrease risk exposure during and after a crisis. However, if authorities wish to prevent negative repercussions from infectious shocks, they must closely monitor changes in the financial assets and follow up with caution.
Graph: Figure 1 Stock market indices performance and returns period January 2019–December 2020.
Graph: Figure 2 Cryptocurrencies price-performance and returns period January 2019–December 2020.
Graph: Figure 3 Financial Market's drawdown during January 2019–December 2020. (a) Stock market returns drawdown. (b) Crypto market returns drawdown.
Table 1 Correlation and Unit Root Test—Stock market indices.
DJI FTSE GDAXI GSPC B KPSS DJI 1 −6.4190 * 0.06104 FTSE 0.704 1 −7.1827 * 0.07928 GDAXI 0.693 0.869 1 −7.0149 * 0.05182 GSPC 0.980 0.681 0.677 1 −6.0856 * 0.06295
Table 2 Correlation and Unit Root Test—Cryptocurrencies.
BTC ETH XRP ADA B KPSS BTC 1 −8.6000 * 0.20587 ETH 0.834 1 −8.6093 * 0.16746 XRP 0.554 0.616 1 −7.7385 * 0.04243 ADA 0.681 0.724 0.716 1 −8.6460 * 0.11464
Table 3 Descriptive statistics—Stock market indices returns January 2019–December 2020.
Mean Std Min Max Kurt Skew Q0.25 Q0.75 SR J&B Test DJI 0.00067 0.01710 −0.12927 0.11365 16.16261 −0.61453 −0.00415 0.00657 −0.05334 5532.72 *** FTSE 0.00002 0.01392 −0.10874 0.09053 12.78455 −0.95778 −0.00533 0.00624 −0.06849 3518.59 *** GDAXI 0.00063 0.01580 −0.12239 0.10976 13.42626 −0.69903 −0.00464 0.00733 −0.05790 3836.29 *** GSPC 0.00091 0.01613 −0.11984 0.09383 14.63959 −0.69860 −0.00358 0.00724 −0.05559 4553.70 *** DJI 0.00081 0.00776 −0.03046 0.03292 3.39944 −0.60943 −0.00264 0.00514 −0.16717 133.07 *** FTSE 0.00047 0.00733 −0.03231 0.02253 2.22334 −0.39732 −0.00420 0.00525 −0.17988 56.51 *** GDAXI 0.00091 0.00871 −0.03107 0.03370 2.09444 −0.30200 −0.00331 0.00577 −0.14812 48.02 *** GSPC 0.00101 0.00778 −0.02978 0.03434 3.40344 −0.57982 −0.00227 0.00576 −0.16505 131.86 *** DJI 0.00053 0.02288 −0.12927 0.11365 8.93517 −0.47372 −0.00588 0.00898 −0.02303 834.02 *** FTSE −0.00043 0.01825 −0.10874 0.09053 7.55239 −0.76400 −0.00759 0.00970 −0.03221 613.35 *** GDAXI 0.00035 0.02056 −0.12239 0.10976 8.34480 −0.57867 −0.00685 0.00993 −0.02621 733.12 *** GSPC 0.00081 0.02144 −0.11984 0.09383 8.28656 −0.55716 −0.00565 0.00906 −0.02377 722.05 ***
Table 4 Descriptive Statistics—Cryptocurrencies returns January 2019–December 2020.
Mean Std Min Max Kurt Skew Q0.25 Q0.75 J&B Test Bitcoin −0.002 0.039 −0.227 0.444 25.364 1.914 −0.018 0.012 19,759.237 *** Ethereum −0.001 0.047 −0.183 0.352 6.541 1.010 −0.024 0.018 1406.006 *** Ripple 0.002 0.055 −0.287 0.714 48.688 3.474 −0.018 0.019 72,660.308 *** Cardano 0.000 0.057 −0.181 0.704 32.830 2.711 −0.028 0.027 33,257.031 *** Bitcoin −0.001 0.039 −0.227 0.154 5.735 −0.160 −0.016 0.015 485.316 *** Ethereum 0.001 0.047 −0.183 0.208 3.979 0.597 −0.021 0.020 253.799 *** Ripple 0.002 0.039 −0.206 0.146 4.101 −0.143 −0.014 0.019 248.119 *** Cardano 0.002 0.047 −0.156 0.244 2.988 0.456 −0.023 0.027 143.205 *** Bitcoin −0.003 0.040 −0.159 0.444 43.801 3.839 −0.019 0.010 29,336.785 *** Ethereum −0.004 0.048 −0.158 0.352 9.196 1.412 −0.030 0.016 1370.999 *** Ripple 0.001 0.067 −0.287 0.714 42.206 3.806 −0.022 0.019 27,287.361 *** Cardano −0.003 0.065 −0.181 0.704 37.407 3.472 −0.035 0.026 21,473.441 ***
Table 5 Results from multivariate EGARCH (DCC) model—Stock Market Indices.
Skew Shape AIC First stage DJI −0.20355 (0.215422) −0.15858 *** (0.042943) 0.97786 *** (0.023375) 0.24883 (0.221394) 0.86471 *** (0.068518) 5.43556 ** (2.795454) 1656.919 −6.383 FTSE −0.10745 *** (0.00262) −0.14098 *** (0.028023) 0.988111 *** (0.000171) 0.08099 ** (0.037483) 0.86005 *** (0.054435) 5.56222 *** (1.331363) 1641.112 −6.334 GDAXI −0.05417 *** (0.004147) −0.18539 *** (0.028109) 0.994266 *** (0.000005) 0.03079 (0.022275) 0.86107 *** (0.046866) 3.66400 *** (0.650932) 1584.797 −6.108 GSPC −0.26431 *** (0.068048) −0.13035 *** (0.035984) 0.971682 *** (0.007229) 0.26354 *** (0.064915) 0.77979 *** (0.055043) 6.07973 *** (1.761195) 1685.150 −6.485 Second Stage Joint 0.05458 *** (0.010974) 0.88763 *** (0.025251) 6.13999 *** (0.555841) 7507.884 −28.941
Table 6 Results from multivariate EGARCH (DCC) model—Cryptocurrencies.
Skew Shape AIC First Stage BTC −0.02862 *** (0.007336) −0.089649 *** (0.020531) 0.994565 *** (0.00263) 0.165754 ** (0.056144) 1.01418 *** (0.038105) 2.332726 *** (0.065807) 1514.983 −4.123 ETH −0.333921 ** (0.129402) −0.083583 * (0.033099) 0.943549 *** (0.021325) 0.164941 ** (0.059533) 0.988406 *** (0.046471) 2.906735 *** (0.335029) 1295.614 −3.520 XRP −0.085255 ** (0.041778) −0.114611 ** (0.041343) 0.984135 *** (0.007442) 0.274431 *** (0.087779) 1.019635 *** (0.040798) 2.340565 *** (0.200588) 1397.988 −3.803 ADA −0.896513 * (0.505929) −0.049195 (0.040657) 0.846718 *** (0.086361) 0.172597 * (0.070766) 1.039076 *** (0.050585) 3.545309 *** (0.511266) 1186.803 −3.222 Second Stage Joint 0.024248 ** (0.010787) 0.936152 *** (0.025525) 4.000001 *** (0.170452) 6516.831 −17.729
Table 7 Value at Risk (VaR) and Cornish–Fisher expansion (CFVaR) estimations—Stock market indices.
Indices (January 2019–December 2020) (January–December 2019) (January–December 2020) VaR CFVaR VaR CFVaR VaR CFVaR DJI 90% 0.013 0.007 0.007 0.010 0.020 0.021 95% 0.024 0.026 0.012 0.013 0.032 0.037 99% 0.057 0.096 0.026 0.020 0.073 0.089 FTSE 90% 0.013 0.010 0.008 0.010 0.018 0.020 95% 0.022 0.024 0.010 0.012 0.033 0.033 99% 0.040 0.069 0.022 0.017 0.046 0.068 GDAXI 90% 0.014 0.009 0.009 0.011 0.020 0.020 95% 0.023 0.025 0.016 0.014 0.037 0.034 99% 0.043 0.079 0.024 0.019 0.054 0.078 GSPC 90% 0.012 0.008 0.007 0.009 0.018 0.020 95% 0.024 0.025 0.012 0.013 0.034 0.035 99% 0.048 0.085 0.025 0.020 0.066 0.081
Table 8 Value at Risk (VaR) and Cornish–Fisher expansion (CFVaR) estimations—Cryptocurrencies.
Crypto Period: (January 2019-December 2020) (January–December 2019) (January–December 2020) VaR CFVaR VaR CFVaR VaR CFVaR BTC 90% 0.042 −0.010 0.042 0.044 0.041 −0.043 95% 0.055 0.025 0.057 0.064 0.054 −0.018 99% 0.091 0.189 0.094 0.119 0.084 0.139 ETH 90% 0.052 0.048 0.045 0.053 0.055 0.042 95% 0.071 0.061 0.076 0.066 0.070 0.055 99% 0.114 0.096 0.110 0.090 0.113 0.097 XRP 90% 0.044 −0.092 0.039 0.045 0.049 −0.072 95% 0.066 −0.028 0.060 0.062 0.076 −0.034 99% 0.119 0.319 0.106 0.100 0.147 0.211 ADA 90% 0.061 −0.040 0.057 0.057 0.068 −0.050 95% 0.084 0.008 0.077 0.069 0.093 −0.013 99% 0.142 0.256 0.102 0.087 0.149 0.211
Data are publicly available.
The authors declare no conflict of interest.
By Apostolos Ampountolas
Reported by Author