“Value-at-risk (VAR) estimation and backtesting during COVID-19: Empirical analysis based on BRICS and US stock markets”

Value-at-risk (VaR) is the most common and widely used risk measure that enter- prises, particularly major banking corporations and investment bank firms employ in their risk mitigation processes. The purpose of this study is to investigate the value- at-risk (VaR) estimation models and their predictive performance by applying a series of backtesting methods on BRICS (Brazil, Russia, India, China, South Africa) and US stock market indices. The study employs three different VaR estimation models, namely normal (N), historical (HS), exponential weighted moving average (EMWA) procedures, and eight backtesting models. The empirical analysis is conducted during three different periods: overall period (2006–2021), global financial crisis (GFC) period (2008–2009), and COVID-19 period (2020–2021). The results show that the EMWA model performs better compared to N and HS estimation models for all the six stock market indices during overall and crisis sample periods. The results found that VaR models perform poorly during crisis periods like GFC and COVID-19 compared to the overall sample period. Furthermore, the study result shows that the predictive accuracy of VaR methods is weak during the COVID-19 era when compared to the GFC period. The goal of this paper is to assess and backtest market risk during times of crisis. First, the study computes three popular VaR estimation techniques (namely, historical simulation model (HS), normal distribution method (N), and exponential weighted moving average (EWMA) model) and compares their predictive performance using eight different backtesting approaches (namely, Traffic Light (TL), Binomial (Bin), proportion of failure (POF), time till first failure (TUFF), conditional coverage independence (CCI), conditional coverage (CC), time between failures independence (TBFI), and time between failures (TBF) tests). Second, the study aims to investigate how well the VaR estimation models perform during crisis periods like the global financial crisis (2008–2009) and COVID-19 crisis (2020–2021) compared to the


INTRODUCTION
Financial markets have become increasingly global and sophisticated in the current stage of the world economy. The stock market has never been more pessimistic than it has been in response to the COVID-19 outbreak. The COVID-19 disaster has had grave undesirable consequences on a global scale, hurting multiple economies and deteriorating their conditions, perhaps leading to a catastrophic recession akin to that seen during the Global Recession of 2008-2009. The goal of this paper is to assess and backtest market risk during times of crisis. First, the study computes three popular VaR estimation techniques (namely, historical simulation model (HS), normal distribution method (N), and exponential weighted moving average (EWMA) model) and compares their predictive performance using eight different backtesting approaches (namely, Traffic Light (TL), Binomial (Bin), proportion of failure (POF), time till first failure (TUFF), conditional coverage independence (CCI), conditional coverage (CC), time between failures independence (TBFI), and time between failures (TBF) tests). Second, the study aims to investigate how well the VaR estimation models perform during crisis periods like the global financial crisis (2008)(2009)) and COVID-19 crisis (2020-2021) compared to the

LITERATURE REVIEW
The value-at-risk (VaR) has been endorsed widely in measuring market risks (Jorion, 2011). Multiple research has been undertaken on VaR in emerging Southeast Asian countries (Cheong et al., 2011), the European Union nations (Iglesias, 2015), the Latin American countries (Ozun & Cifter, 2007), Nordic markets (Jobayed, 2017), the South African market (Mabitsela et  There is a dearth of research on VaR estimation and predictive performance based on BRICS countries, which are the world's five most attractive and strongest emerging markets (Mukta & Muneer, 2020a). Aside from economic growth, the BRICS stock indices provide greater and more reliable market proceeds (Jiang et  Burdorf and van Vuuren (2018) examined stock portfolios from the banking and retail sectors in developed (UK) and emerging (South Africa) markets, finding that industries and periods affected risk measure accuracy, but not economies.
The present study fills the gap and contributes to the existing literature by analyzing the three popular VaR estimation models along with a comprehensive eight backtesting methods to understand the predictive performance of BRICS nations along with developed US stock market indices. Degiannakis et al. (2012) looked at the performance of three different VaR models to come up with estimates that might be used to measure and anticipate market risk. During times of financial crisis, they discovered evidence that generally acknowledged methodologies provide solid VaR estimates and projections. Miletic and Miletic (2015) study the performance of value-at-risk (VaR) models in selected Central and Eastern European (CEE) developing capital markets during the global financial crisis. In comparison to GARCH-type models with normal distribution, historical simulations, and RiskMetrics approaches, backtesting analysis for the crisis shows that GARCH-type models with t-distribution of residuals produce higher VaR estimates. At a 95 percent confidence level, Su et al. (2010) found that the Historical Simulation VaR estimate model significantly outperforms the GARCH (1,1) -AR(1) model. For the nations of the United States, the United Kingdom, France, Germany, Italy, Japan, China, Spain, and Portugal, Ramalho (2020) estimate VaR using Historical Simulation, GARCH(1,1), and Dynamic EVT-POT with a time horizon of January 1, 2007, to August 31, 2020. It was discovered that as the number of deaths grew throughout the COVID-19 era, so did the volatility in these markets; the bulk of exceedances occur during crisis moments rather than normal ones. Using conditional extreme value theory, Omari

VaR estimation using normal distribution
This method presumes that the returns are regularly distributed when using the normal distribution approach. The normal distribution approach has the virtue of being simple and is also known as parametric VaR.

VaR estimation using historical simulation (H) method
The H approach, unlike the normal distribution method, is nonparametric. It makes no assumptions about the distribution of asset returns. The historical simulation calculates risk by presuming that the allocation of profits and losses in the prior period of returns will be used as the allocation of profits and losses in the subsequent period of returns. The VaR "today" is calculated by taking pth-quantile of previous results. The profile of the historical simulation curve is piecewise constant. This one is since quantiles need not alter for many days until important incidents happen. As a result, the H technique reacts slowly to increases in volatility.

VaR estimation using the exponential weighted moving average (EWMA) method
Prior algorithms assume that all previous returns have equal weight. EWMA approach allocates weights that are not equal, especially weights that decrease exponentially. The most recent returns have larger weights since they have a greater effect on "today's" return compared to returns from further back in time. The EWMA variance across an estimating window of size W E is calculated by: 1 Basel Committee on Banking Supervision (1996).
where c is a normalizing constant: The decay factor is commonly used in practice at 0.94 (Danielsson, 2012).

Binomial test (Bin)
The binomial test (Jorion, 2011) compares the experiential amount of exemptions to the anticipated amount of exemptions. The outcome of the test is: where x, N denotes the number of failures, and amount of observations respectively; p = 1 -VaR level is a likelihood of detecting a failure if VaR is right.

Traffic Light Test (TL)
For a certain list of exemptions, x, the traffic light test as described in the Basel committee report 1 can calculate the cumulative likelihood up to x. That is, F(x|N, p). It is also known as the three zones test and denoted as: • Green: F(x|N, p) ≤ 0.95. If one has a small number of failures in the VaR model, they will be placed in the green zone.
• Yellow: 0.95 ≤ F(x|N, p) ≤ 0.9999. Even though the failures are high, the violation count is not exceedingly high and hence they fall in the yellow zone.
• Red: 0.9999 < F(x|N, p). A proper VaR model is unlikely to provide too many exceptions. So, in case of too many failures, it falls in the red zone.  Kupiec (1995) developed a proportion of failures (POF) test that accommodates a binomial distribution and employs a probability ratio to determine if the likelihood of exceptions is matched with the likelihood p given by the VaR level of confidence. The VaR model is rejected if the data advocates that the chance of exceptions is larger than p. The POF test statistic is:

Kupiec's POF and TUFF tests
where x, N denote the number of failures, and a number of observations respectively; p = 1 -VaR level is the likelihood of detecting a failure if the VaR model is right. Kupiec (1995) presented an alternative test called the time until first failure (TUFF) test which is likewise based on a likelihood ratio test but has a geometric distribution as the underlying distribution.
( ) Both statistics have an asymptotic distribution as a chi-square variable with one degree of freedom. If the likelihood ratio reaches a crucial value determined by the test confidence level, the VaR model fails the test. Christoffersen (1998) suggested a measure to determine if the likelihood of detecting an exemption at a given time is affected by whether an exemption happened. In Christoffersen's interval forecast technique, the test statistic for independence is provided by:  (6) where n00: the number of failure-free periods followed by a failure-free period, n10: the number of periods with failures followed by a failure-free period, n01: the number of failure-free periods followed by a failed period, n11: the number of failed periods followed by a failed period.

Conditional coverage mixed test
Asymptotically, this statistic is distributed as a chisquare with one degree of freedom. This statistic may be used with the frequency POF test to create a conditional coverage (CC) mixed test: .
As a chi-square variable with two degrees of freedom, this test is asymptotically distributed.

Time between failures (TBF) or mixed Kupiec's test
Haas (2001) In this statistic, p = 1 -VaR level and n i is the number of days between failures i -1 and i (or until the first exception for i = 1). Asymptotically, this statistic is distributed as a chi-square variable with x degrees of freedom, where x is the rate of failure.
As a chi-square variable with x + 1 degrees of freedom, this test is asymptotically distributed.

DATA
The study uses daily data of stock market indices of BRICS nations and the US index for the peri-

RESULTS AND DISCUSSION
This section first estimates the VaR using three different models like normal (N), historical simulation (HS), and exponential weighted moving average  Figure 2 displays the plots of returns of the indices and VaR estimation methods at a 95% confidence level for the GFC period from 2008 to 2009. Figure  3 shows the plots of returns of the indices and VaR estimation methods at a 95% confidence level for the COVID-19 period from 2020 to 2021. Similar findings are observed in both the GFC and COVID-19 periods, that the EMWA model performs better compared to HS and N models.
All three Figures 1, 2, and 3 exhibit the degree to which the three VaR estimation models N, HS, and EWMA are successful to evaluate the index returns related to the actual index returns. It is observed that actual index returns in all three periods exhibit stylized facts like volatility cluster- ing. In Figure 1   tially more effective in predicting risk exposures for all three periods under study. On the other hand, N and HS estimation methods are less fruitful in projecting the market risks and their correlation of predictive accuracy is seen to be relatively near to one across all periods. By looking at Figures 2 and 3, it is observed that during extreme event periods like GFC and COVID-19, both the models N and HS underestimate the risks whenever the moment of recession uncovers, and overestimate the risks whenever the market starts to stabilize again.
It is known that computing VaR is critical for enterprises and institutional investors to make wise financial decisions but it is also vital to guar-   antee that the calculation is computed with the least estimation error and in the most accurate manner. Therefore, the estimation models in this study are evaluated through eight different backtesting methods to understand their predictive performance.
In Tables 2 and 3, the headings TL, Bin, POF,  TUFF, CC, CCI, TBF, and TBFI denote   spectively. The following criteria are used to assess the outcomes. In a TL test, G stands for the green zone, Y stands for the yellow zone, and Rd stands for the red zone. Under the prospective backtesting approach, A signifies acceptance and R signifies rejection of the VaR model for all other tests. The VaR backtesting is applied to all the three VaR estimation models at both the 95% and 99% levels of confidence.
The first column of Table 2    the models N and HS underestimate the risks whenever the moment of recession uncovers, and overestimate the risks whenever the market starts to stabilize again. Amongst the three estimation methods considered in this study, the EWMA method performs better. Furthermore, it is observed that VaR estimation models have poor predictive accuracy, especially during the COVID-19 period compared to the global financial crisis.

VaR at a 99% level of confidence
Overall, the study rates the VaR models as EWMA > historical > normal based on their predictive performance. It is obvious that dynamic VaR models such as EWMA outperform static methods such as historical and conventional simulation approaches. During a crisis, however, the predictive performance of the VaR models fails catastrophically. This raises a crucial concern regarding the VaR models' usefulness during extreme event times.
As for recommendations for future study, it will be interesting to test parametric GARCH family models alongside alternative semi-parametric approaches such as CAViaR and EVT methods, which may bring novel views on this line of research. Because VaR has distinct limitations as a risk measure, it would be thought provoking to examine market risks using anticipated shortfall (ES) models in addition to VaR models in future studies.