###### Abstract

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

1 1 0 Academic Editor: Zhaojun Yang and Thanasis Stengos 13 November 2022 22
December 2022 28 December 2022 https://doi.org/ The Effect of COVID-19 on
Cryptocurrencies and the Stock Market Volatility: A Two-Stage DCC-EGARCH Model
Analysis The Effect of COVID-19 on Cryptocurrencies and the Stock Market
Volatility: A Two-Stage DCC-EGARCH Model Analysis Apostolos Ampountolas 1,2
Apostolos Ampountolas Ampountolas, Apostolos

##  1 Introduction

Though the pandemic is still active and has uncertain long-term outcomes
(Fisher et al., 2021), at the same time, the US administration has lately
announced that the pandemic is over (Stolberg, 2022), probably because we have
reached the stage of the COVID-19 outbreak known as endemic when the virus is
ubiquitous but substantially less deadly than it was in 2020. Regrettably, the
severe mortality toll from the COVID-19 epidemic is still being reported.
Similarly, we are all also concerned about how this crisis affected the
financial markets (Ashraf, 2020) and, more generally, the entire economic
environment, including consumer behavior and intentions (Watson and Popescu,
2021; Goldstein et al., 2021). From this perspective, asset management has
already become essential for organizations in managing their assets to
generate the highest returns due to today’s difficult circumstances and highly
volatile markets.

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, Mazur et al. (2021); Alqaralleh and Canepa (2021);
Akhtaruzzaman et al. (2022); Uddin et al. (2022)) also ominously predict
significant events like COVID-19 and its economic repercussions.

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 (Jorion, 1996). Moreover, Dynamic
Conditional Correlation (DCC) is thus proven to be an effective and practical
methodology for more robust market volatility and yielding conditional
variances (Tse and Tsui, 2002). As such, this study investigates the dynamic
correlation between four stock market indices and four different
cryptocurrencies using data from January 2019 to December 2020.

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. Conlon and McGee
(2020) findings show that Bitcoin does not function as a safe place when
considering the impact on an S&P500 portfolio that is balanced and includes
exposure to Bitcoin. They found that the S&P500 and Bitcoin trade is in
perfect synchronization during the period under review, increasing the
downside risk for an investor who has allocated money to Bitcoin.

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 (Favre and Galeano, 2002;
Conlon and McGee, 2020; Ali et al., 2021). Because these allow us to observe
and quantify how returns and volatility vary across markets, these approaches
are preferable to the traditional time series approach.

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.

##  2 Research Background

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, (Conlon and McGee (2020); Alqaralleh and Canepa
(2021); Goldstein et al. (2021); Mazur et al. (2021); Nguyen (2022); Uddin et
al. (2022); Yan et al. (2022)). For example, the emergence of COVID-19 has
raised essential concerns about how the interaction between cryptocurrencies
and other alternative investments has changed throughout the pandemic.
According to Conlon and McGee (2020), Bitcoin increased portfolio risk during
high uncertainty, indicating it would not be a safe haven for investments.
Similarly, Nguyen (2022) claimed that the returns on the S&P500 considerably
influenced the returns on Bitcoin at times of high uncertainty. During
COVID-19 and other volatile times, stock market shocks impacted Bitcoin’s
volatility. Uddin et al. (2022) study investigated the interconnected dynamics
of the affected Asian and worldwide financial markets in response to the
coronavirus (COVID-19) pandemic epidemic. Their results showed that the
COVID-19 outbreak had caused a significant, positive reliance among the
markets and an enhanced tendency for co-movements across the upper time
horizon. Yan et al. (2022) study examined the dynamic conditional correlations
between 10 cryptocurrencies throughout the COVID-19-affected timeframe of 2017
to 2022. According to their findings, all cryptocurrency return growth rates
increased from the pre-COVID-19 period to the COVID-19 period, and COVID-19
had a favorable impact on cryptocurrency returns. From pre-COVID-19 to
COVID-19, the average dynamic correlations between the return indices of
Bitcoin, Ethereum, and other cryptocurrencies were very high. It is obvious
that investments in cryptocurrencies are frequently seen with high volatility
and risk. In their work, Almeida et al. (2022) examined a sample of seven
cryptocurrencies, including the period of the COVID-19 pandemic employing
value-at-risk (VAR) and conditional VAR to quantify risk. The findings show
that, other than the Tether, the cryptocurrencies displayed comparable
patterns of risk and uncertainty. Moreover, Umar and Gubareva (2020) used the
wavelet method to investigate how COVID-19 has affected the volatility of
significant fiat and cryptocurrency markets. The findings for each index
combination in the study are relatively consistent and support the hypothesis
that cross-currency hedge techniques, which could be successful in a market
under normal conditions, are more likely to fail in market shocks such as the
COVID-19 outbreak.

##  3 Methodology

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 Conlon and McGee (2020)
and estimate downside risk—value-at-risk (VaR), also based on Cornish–Fisher
expansion (CFVaR) for each financial portfolio in this study. Finally, the
section describes the criteria for choosing the best GARCH-type
specifications.

###  3.1 Multivariate GARCH

The GARCH family of models introduced by Bollerslev (1986) has been
extensively employed in financial modeling. The generalized autoregressive
conditional heteroscedasticity model (GARCH) extends the autoregressive
conditional heteroscedasticity (ARCH) model. The GARCH model maintains the
unconditional variance constant while enabling the conditional variance to
vary over time due to prior errors (Bollerslev, 1986). Due to the significant
modifications made to the original GARCH model, GARCH models have been used in
this context to forecast the variance of time series, not only in the
financial sector to estimate stock volatility performance but also to estimate
the volatility of cryptocurrencies or other financial instruments (Trucíos
(2019); Urquhart and Zhang (2019); Caporale and Plastun (2019); Conlon and
McGee (2020); Alqaralleh and Canepa (2021); Ampountolas (2022)), and to assess
the accuracy of forecasts. Moreover, GARCH models are employed except for
assessing financial instruments and evaluating the forecasting performance of
time series datasets in other industries (Ampountolas, 2021).

The current study will employ a multivariate GARCH model in two stages to
capture the asymmetric dependence structure (Ghalanos, 2015). In the first
stage, a univariate GARCH process is established, together with the
limitations for each series. We use the EGARCH (1, 1) model to analyze the
heteroscedasticity in the daily returns of portfolio instruments. To
appropriately represent the leverage effect, we are utilizing the well-known
EGARCH model, which typically provides better fits than the traditional GARCH
model. It also includes an additional asymmetric term. The model’s estimation
and evaluation are tasks for the second stage. Since it permits the
conditional correlation matrix to be time-dependent, ensuring that it is
positive definite and practical when modeling high-dimensional data sets, the
dynamic conditional correlation (DCC-GARCH) method is commonly used in
practice, as per Bauwens et al. (2006). Furthermore, we also discuss the
conditional variance distribution’s form parameter, which denotes excessive
kurtosis in the studied series.

####  3.1.1 EGARCH Model

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. Nelson (1991) introduced the exponential
generalized autoregressive conditional heteroscedasticity (EGARCH) model, a
rigorous parametric form, to solve the drawbacks of symmetric models, such as
the leverage effect conditional heteroscedasticity. As such, he achieved a
non-negative conclusion by using the natural logarithm of the conditional
variance. The EGARCH logarithm is described as follows:

$\log(h_{t})=\alpha_{0}+\alpha_{1}\Bigg{(}\Bigg{|}\frac{\epsilon_{t-1}}{\sqrt{h_{t-1}}}\Bigg{|}-\mathbb{E}\Bigg{|}\frac{\epsilon_{t-1}}{\sqrt{h_{t-1}}}\Bigg{|}\Bigg{)}+\xi\frac{\epsilon_{t-1}}{\sqrt{h_{t-1}}}+\gamma_{1}\log(h_{t-1})$
(1)

where
$\alpha_{1}\Bigg{(}\Bigg{|}\frac{\epsilon_{t-1}}{\sqrt{h_{t-1}}}\Bigg{|}-\mathbb{E}\Bigg{|}\frac{\epsilon_{t-1}}{\sqrt{h_{t-1}}}\Bigg{|}\Bigg{)}$
denotes the magnitude effect and $\xi\frac{\epsilon_{t-1}}{\sqrt{h_{t-1}}}$
the sign effect. Therefore, if $\xi<0$, negative innovations (negative news)
increase volatility more than positive innovations of equal size. The
volatility specification’s logarithmic adjustment suggests that this model’s
parameters are not restricted to positive values. The parameter for
conditional variance $h_{t}$ is the GARCH coefficient $\gamma_{1}$ is
typically close to 1, and the closer, the higher the conditional volatility.

####  3.1.2 DCC-GARCH Model

Engle (2002) proposed the DCC-GARCH model, which features a dynamic
conditional correlation structure. The correlations are first estimated in the
DCC-GARCH model, then the GARCH parameters. Because $H_{t}$ represents the
conditional covariance matrix and the conditional variances must also agree,
the dynamic correlation model allows $R$ to vary over time, and its
parameterizations of $R$ exactly satisfy all specifications. The matrix
$R_{t}$ is the conditional correlation matrix, while the diagonal matrix
$D_{t}$ has time-varying parameterizations (Engle, 2002). The conditional
correlation variance of the DCC-GARCH is parameterized as follows, where
$\rho_{ij,t}$ stands for the correlation between the $i$th and $j$th return
series. Therefore, Engle (2002) is regarded as a dynamic matrix process that
may be written as

$\boldsymbol{H}_{t}=\boldsymbol{D}_{t}\boldsymbol{R}_{t}\boldsymbol{D}_{t}$
(2)
$\boldsymbol{D}_{t}=\text{diag}\Big{\\{}D_{1t},D_{2t},\dots,D_{kt}\Big{\\}}\quad\boldsymbol{R}_{t}=\Big{[}\rho_{ij,t}\Big{]}$
(3)
$\begin{split}\boldsymbol{Q}_{t}=\boldsymbol{\bar{Q}}+\alpha\left(z_{t-1}z_{t-1}^{\prime}-\boldsymbol{\bar{Q}}\right)+\beta\left(\boldsymbol{Q}_{t-1}-\boldsymbol{\bar{Q}}\right)\\\
=\boldsymbol{\bar{Q}}(1-\alpha-\beta)+\alpha
z_{t-1}z_{t-1}^{\prime}+\beta\boldsymbol{Q}_{t-1}\end{split}$ (4)
$\boldsymbol{R}_{t}=diag\left(\boldsymbol{Q}_{t}\right)^{-1/2}\boldsymbol{Q}_{t}diag\left(\boldsymbol{Q}_{t}\right)^{-1/2}$
(5)

where $\alpha$ is a positive and $\beta$ non-negative scalar parameter such
that $\alpha+\beta<1$ to ensure the stationarity and positive definiteness of
$\boldsymbol{Q_{t}}$, $\bar{Q}$ is the unconditional matrix of standardized
errors ($z_{t}$), which is entered into the equation via the covariance
targeted portion $\boldsymbol{\bar{Q}}(1-\alpha-\beta)$, and
$\boldsymbol{Q_{0}}$ is positive definite (Bauwens et al., 2006; Ghalanos,
2015), even though this method creates reliable correlation matrices, it does
not give positive definiteness; as a result, the correlation matrix
$\boldsymbol{R}$ is created by rescaling $\boldsymbol{Q_{t}}$ as in Equation
(5) (Bauwens et al., 2006).

###  3.2 Value-at-Risk (VaR)—Downside Risk

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
(Favre and Galeano, 2002). It quantifies the highest loss financial assets may
sustain at a specific time under a particular confidence level. If returns are
normally distributed, a two-moment Value-at-Risk (VaR) that considers return
and asset standard deviation may be used to calculate the downside risk. VaR
is defined as:

$VaR_{p(\alpha)}=-(\mu_{p}+Z_{a}\sigma_{p})$ (6)

where $\mu_{p}$ denotes the mean of the portfolio daily returns, and
$\sigma_{p}$ the standard deviation, respectively. Moreover, $Z_{\alpha}$
refers to the $\alpha$ quantile for a standard normal distribution. In
general, the typical values for $(1-\alpha)$ are 90%, 95%, and 99% which we
also report in this study to indicate the differences.

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
(Cornish and Fisher, 1938) to calculate the quantiles of such a non-normal
distribution (Favre and Galeano, 2002). Conlon and McGee (2020), and Ali et
al. (2021) used a similar process in measuring the downside risk in
cryptocurrencies and the Dow Jones Islamic stock indices, respectively.

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_ :

$Z_{CF}=z_{c}+(z^{2}_{c}-1)\frac{S}{6}+(z^{3}_{c}-3z_{c})\frac{K}{24}-(2z^{3}_{c}-5z_{c})\frac{S^{2}}{36},$
(7)

where $Z_{c}$ as the critical value for probability (1 $-$ $\alpha$), $S$ as a
skewness parameter, and $K$ as an excess kurtosis parameter. It is important
to realize that $S$ and $K$ are parameters. As such, they can be very
different from the actual skew and excess kurtosis of the obtained
distribution following the Cornish–Fisher expansion.

$VaR=W(\mu-Z_{CF}\sigma),$ (8)

where $W$ is the amount at risk or portfolio, and $\sigma$ refers to the
yearly standard deviation.

###  3.3 Model Selection

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 Hansen and Lunde (2005), there was little evidence that alternative, more
complex models could outperform a GARCH(1,1). A GARCH(1,1) model with constant
mean could be expressed as follows: $y_{t}=\mu_{t}+e_{t}$ with $e_{t}\sim
N(0,\sigma^{2}_{t})$.

$\begin{cases}y_{t}=\mu_{t}+e_{t},\\\
e_{t}=\sqrt{h_{t}}\cdot\eta_{t},\eta_{t}\sim iid(0,1),\\\
h_{t}=\alpha_{0}+\alpha_{1}\epsilon^{2}_{t-1}+\gamma_{1}h_{t-1}\end{cases}$
(9)

The Akaike information criterion (AIC) (Akaike, 1981) was used to assess the
discrimination as follows:

$\text{AIC}=2k-2\ln L\left(\hat{\Theta}\right)\text{,}$ (10)

where $k$ represents the total number of unknown parameters, $\Theta$ is the
vector of unknown parameters, and $L(\hat{\Theta})$ their maximum likelihood
(ML) estimations; therefore, the optimal models were obtained by eliminating
the criteria. Additionally, AIC promotes more sophisticated models in general
(Gilli et al., 2019).

##  4 Data

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),Data are publicly available at https://www.cryptodatadownload.com/data/
(accessed on 14 June 2022).” for the cryptocurrencies in the EDT time zone.

Portfolio returns were calculated using the formula
$r_{t}=ln\bigg{(}\frac{P_{t}}{P_{t-1}}\bigg{)}$ and
define$\alpha_{t}=r_{t}-E_{t-1}$, where $P_{t}$ was the closing price of the
assets on time $t$. We initiated our research by determining the data
stationarity using the Dickey–Fuller (ADF) coefficient test for identifying
non-seasonal unit roots (Dickey and Fuller, 1979). Thus, as the evaluated
models needed stationary data, we computed the log returns and performed the
augmented Dickey–Fuller (ADF) test to confirm that there were no unit roots.
The unit root tests rejected the null hypothesis at 0.01 level for the
analyzed portfolio returns. We also used the KPSS stationarity test
(Kwiatkowski et al., 1992), which accepts the null hypothesis for portfolio
return rates, to confirm the findings of the ADF test (Table 1—stock market
indices, Table 2—cryptocurrencies assets, respectively).

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 (Jarque and Bera, 1987). The test statistic equation incorporating
skewness and kurtosis is:
$JB=\frac{n}{6}\left(S^{2}+\frac{1}{4}(K-3)^{2}\right)$ where n = the number
of values for the data. $S$ is the sample skewness (how much the data lean
away from the mean), and $K$ refers to the sample kurtosis (how wide the
distribution’s tails are).

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

Note: * Significant at the 0.05 level; ADF: augmented Dickey–Fuller
statistics; KPSS: KPSS test statistics using residuals from regressions.

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

Note: * Significant at the 0.05 level; ADF: augmented Dickey–Fuller
statistics; KPSS: KPSS test statistics using residuals from regressions.

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.

Figure 1: Stock market indices performance and returns period January
2019–December 2020.

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 (Frazier, 2021). The market returns
indicate a significant frequency of negative returns showing that the
distribution has a left-tail skewness in addition to the Sharpe ratio.

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 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
---|---|---|---|---|---|---|---|---|---|---
Panel A: (1 January 2019–31 December 2020) |  |  |  |  |  |
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 ***
Panel B: (1 January–31 December 2019) |  |  |  |  |  |
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 ***
Panel C: (1 January–31 December 2020) |  |  |  |  |  |
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 ***

Note: *** Significant at the 0.001 level; SR: Sharpe Ratio; J&B: Jacque–Bera
test statistics.

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)
(Dickey and Fuller, 1979) and the KPSS (Kwiatkowski et al., 1992) tests, which
could indicate whether the data series are stationary. The KPSS test is
frequently employed in empirical studies to examine trend stationarity. It
serves as an addition to the traditional ADF unit root test when reviewing the
characteristics of time series data.

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.

Figure 2: Cryptocurrencies price-performance and returns period January
2019–December 2020.

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 (Carbone et al., 2004). As a
result, investors looking to build a market portfolio might be more interested
in it. Lastly, the Jarque–Bera (J&B) test (Jarque and Bera, 1987) reveals that
the price returns of all cryptocurrencies generally deviate from normality
(Table 4).

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.

Table 4: Descriptive Statistics—Cryptocurrencies returns January 2019–December
2020.

| Mean | Std | Min | Max | Kurt | Skew | Q0.25 | Q0.75 | J&B Test
---|---|---|---|---|---|---|---|---|---
Panel A: (1 January 2019–31 December 2020) |  |  |  |  |
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 ***
Panel B: (1 January–31 December 2019) |  |  |  |  |
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 ***
Panel C: (1 January–31 December 2020) |  |  |  |  |
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 ***

Note: *** Significant at the 0.001 level. J&B: Jacque–Bera test statistics.

##  5 Results

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 (1,1) models (Lee and
Hansen, 1994). Hence, the leverage impact is captured by an additional
asymmetric term in the EGARCH model. As a result of comparing the stock market
and cryptocurrencies portfolios, our objective supports employing multivariate
GARCH rather than univariate 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 Tables 5 and 6. At the 1% level, the estimates of $\alpha_{1}$, and
$\beta_{1}$ are statistically significant. The portfolio pairs in the paired
models exhibit bidirectional transmission and volatility relationships. The
results indicate that lagged shocks and volatility strongly and significantly
impact both the own volatility spillovers ($\beta_{1}$) and the conditional
volatility that is currently existing in both assets portfolio ($\alpha_{1}$).

Table 5: Results from multivariate EGARCH (DCC) model—Stock Market Indices.

| $\omega$ | $\alpha_{1}$ | $\beta_{1}$ | $\gamma_{1}$ | Skew | Shape | $Log(L)$ | 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

Note: GARCH models were estimated with the Student’s $t$ distribution.
Asymptotic standard errors are given in parentheses. Mean model—ARFIMA: DJI(1,
0, 2), FTSE(0, 0, 0), GDAXI(2, 0, 0), GSPC(4, 0, 1), respectively. We create
GARCH estimates for the four stock market indices portfolios in the first
phase. In the second stage, GARCH estimates of the joint series are evaluated
while considering the conditional correlation among the four groups. **
Significance at the 5% level, *** at the 1% level.

Table 6: Results from multivariate EGARCH (DCC) model—Cryptocurrencies.

| $\omega$ | $\alpha_{1}$ | $\beta_{1}$ | $\gamma_{1}$ | Skew | Shape | $Log(L)$ | 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

Note: GARCH models were estimated with the Student’s $t$ distribution.
Asymptotic standard errors are given in parentheses. Mean model—ARFIMA: BTC(0,
0, 1), ETH(0, 0, 2), XRP(1, 0, 0), ADA(0, 0, 2), respectively. We create GARCH
estimates for the four cryptocurrencies in the first phase. In the second
stage, GARCH estimates of the joint series are evaluated while considering the
conditional correlation among the four groups. * Significance at 10% level, **
at the 5% level, *** at the 1% level.

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 $\gamma_{1}$, have
computed coefficients $\omega$, $\alpha_{1}$, $\beta_{1}$, $\gamma_{1}$, and
shape that are statistically significant. In addition, the Dow Jones (DJI)
index reports a non-significant $\omega$ $-$0.20355, for the other three
indices $\omega$ is significant to 0.001 level, for FTSE is $-$0.10745, GDAXI
is $-$0.05417, and for S&P500 is $-$0.26431, respectively. The findings show
that lagged shocks and volatility significantly and positively impact the
current conditional volatility. The values of the permanent parameters
$\beta_{1}$ range from low 0.971682 for the S&P500 (GSPC) index to high
0.994266 for the German DAX (GDAXI) index are close to one, positive, and
significant. The German DAX index (GDAXI) has a leverage effect $\gamma_{1}$
parameter value of low 0.03079, while the S&P500 index (GSPC) reports a value
of 0.26354 high. Additionally, adjustments to previous shock parameters
($\alpha_{1}$) are negative ($-$0.15858 for the DJI, $-$0.14098 for the FTSE,
$-$0.18539 for the GDAXI, and $-$0.13035 for the S&P 500), respectively.
Finally, all of the parameters for the leverage effect are positive, and the
excess kurtosis parameter $shape$ yield values ranging from low 3.66400 for
the German DAX index (GDAXI) to high 6.07973 for the S&P500 index (GSPC),
following the same pattern as the leverage effect $\gamma_{1}$ 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 $\gamma_{1}$ is typically negative for the stock market, according
to a study by Nelson (1991) discussing the conditional heteroskedasticity in
asset returns. All stock indices in this analysis had significant positive
$\gamma_{1}$ values, which shows that the own-shock spillover has no effect on
current volatility and that the daily fluctuations of these indexes rise after
good news. The only exceptions are the DJI and the GDAXI indices, both
positive but not statistically significant. It demonstrates that pandemic
crashes may have significantly impacted these indices.

To improve the estimation accuracy, in the second stage, Table 5, we employed
a multivariate EGARCH DCC(1, 1) model on a student’s t-distribution to
parameterize the distributions following the first stage with the EGARCH model
and to improve the accuracy of the estimation (Engle and Sheppard, 2001). The
41 parameters estimated by the DCC model are all statistically significant at
the 1% level. The presented joint estimates for $\alpha_{1}$ and $\beta_{1}$
are, respectively, 0.05458 and 0.88763. The multivariate joint shape is
similarly substantial and moderately low, and the multivariate parameters are
significantly positive (6.13999). This moderate to strong positive correlation
between the stock market indices may affect their combined $\alpha_{1}$,
leading to favorable reactions to early shocks.

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
$\omega$, $\alpha_{1}$, $\beta_{1}$, $\gamma_{1}$, and $shape$ that are
statistically significant except for Cardano (ADA) that report a non-
significant $\alpha$ $-$0.049195. All cryptocurrency report significant
$\omega$ to various statistically significant levels. For example, BTC is
$-$0.02862 ($\alpha$ = 0.001), ETH is $-$0.333921 ($\alpha$ = 0.05), XRP is
$-$0.085255 ($\alpha$ = 0.05), and ADA is $-$0.896513 ($\alpha$ = 0.01). The
results demonstrate that lagged shocks and volatility have an extensive and
favorable effect on the current conditional volatility. The values of the
permanent parameters $\beta_{1}$ range from low 0.846718 for the Cardano (ADA)
to high 0.994565 for the Bitcoin (BTC) and are close to one, positive, and
significant ($\alpha$ = 0.001). Moreover, Ethereum (ETH) has a leverage effect
$\gamma_{1}$ parameter value of 0.164941 low, while Ripple (XRP) provides a
value of 0.274431 high. The adjustments to previous shock parameters
($\alpha_{1}$) are negative as expected with values as $-$0.089649 for BTC,
$-$0.083583 for ETH, $-$0.114611 for the XRP, and $-$0.049195 for ADA,
respectively. The excess kurtosis parameter $shape$ product values range from
2.332726 low for the BTC to 3.545309 high for ADA.

In the Second Stage, Table 6, we used a multivariate EGARCH DCC(1, 1) model on
a student’s t-distribution to parameterize the distributions after the first
stage using the EGARCH model and to increase the accuracy of the estimation
(Engle and Sheppard, 2001). The order of the DCC model estimated 37
parameters, all of which are statistically significant. The reported joint
estimates for $\alpha_{1}$ and $\beta_{1}$ are 0.024248 and 0.936152,
respectively. The multivariate parameters are significantly positive 4.000001,
$\alpha$ = 0.001, and the multivariate joint shape is similarly significant
and comparatively small. This moderately positive correlation may affect the
aggregate $\alpha_{1}$ of the cryptocurrencies, resulting in positive
responses to initial shocks.

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 (Conlon and McGee, 2020; Ali et al., 2021). The findings in Table 7
show that for Value-at-Risk (VaR) at a 95% confidence interval, the most
significant loss is a 2.4% loss (DJI, S&P500); however, all stock market
indices show a similar comparative loss, with tiny differences. At the 99%
confidence level, DJI offers the highest risk, followed again by the S&P500
and the GDAXI, with risk high at 5.7%, 4.8%, and 4.3%, respectively. The
higher VaR is a result of the lower confidence level. A higher confidence
level will result in a more significant loss percentage, and a decreased
confidence level might lead to a lower rate of failure; this is reasonable and
assists in understanding how the VaR operates. Such results were expected if
we consider that the Dow (DJI) index crashed during March 2020 financial
market black days (Frazier, 2021). However, the financial markets unexpectedly
recovered most of their value within a month to a surge that pushed the stock
indices back to the pre-COVID period. For the same period, CFVaR at a 99%
confidence level reports the worst loss of 9.6% (DJI), followed by the S&P500
index at a high of 8.5% with the portfolio, and the FTSE index reports (6.9%).
As such, Table 7 shows that all results are higher than the VaR results. The
Cornish–Fisher VaR (CFVaR) will provide a higher loss estimate than the
typical VaR when the returns are negatively skewed (Table 3), which is
exhibited as such in this study.

Table 7: Value at Risk (VaR) and Cornish–Fisher expansion (CFVaR)
estimations—Stock market indices.

|  | (January 2019–December 2020) | (January–December 2019) | (January–December 2020)
---|---|---|---|---
Indices | _p_ | 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

Note: The Value-at-Risk (VaR) and the four-moment modified Value-at-Risk
(CFVaR) using the Cornish–Fisher expansion are reported in this table.

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 Mazur et al. (2021) found that extremely asymmetric volatility is
present in underperforming stocks and negatively correlates with stock market
returns. In this sense, businesses responded in various ways during the
COVID-19 outbreak income shock.

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 Yan et al.
(2022), which found that COVID-19 positively affected the returns of
cryptocurrencies, and the varying correlations were robust.

Table 8: Value at Risk (VaR) and Cornish–Fisher expansion (CFVaR)
estimations—Cryptocurrencies.

| Period: | (January 2019-December 2020) | (January–December 2019) | (January–December 2020)
---|---|---|---|---
Crypto | _p_ | 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

Note: The Value-at-Risk (VaR) and the four-moment modified Value-at-Risk
(CFVaR) using the Cornish–Fisher expansion are reported in this table.

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.

a

b

Figure 3: Financial Market’s drawdown during January 2019–December 2020.(a)
Stock market returns drawdown. (b) Crypto market returns drawdown.

##  6 Conclusions

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 (Frazier, 2021) due to the pandemic and public
administration decisions.

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.

This research received no external funding.

Data are publicly available.

The authors declare no conflict of interest.

[custom] References

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