“Comparing riskiness of exchange rate volatility using the Value at Risk and Expected Shortfall methods”

This paper uses theValue at Risk (VaR) and the Expected Shortfall (ES) to compare the riskiness of the two currency exchange rate volatility, namely BitCoin against the US dollar (BTC/USD) and the South African Rand against the US dollar (ZAR/USD). The risks calculated are tail-related measures, so the Extreme Value Theory is used to capture extreme risk more accurately. The Generalized Pareto distribution (GPD) is assumed under Extreme Value Theory (EVT). The family of Generalized Autoregressive Conditional Heteroscedasticity (GARCH) models was used to model the volatility-clustering feature. The Maximum Likelihood Estimation (MLE) method was used in parameter estimation. Results obtained from the GPD are compared using two underlying distributions for the errors, namely: the Normal and the Student-t distributions. The findings show that the tail VaR on the BitCoin averaging 1.6 and 2.8 is riskier than on South Africa’s Rand that averages 1.5 and 2.3 at 95% and 99%, respectively. The same conclusion is made about tail ES, the BitCoin average of 2.3 and 3.6 is higher (riskier) than the South African Rand averages at 2.1 and 2.9 at 95% and 99%, respectively. The backtesting results confirm the model adequacy of the GARCH-GPD in the estimation of VaR and ES, since all p-values are above 0.05.


INTRODUCTION
Since its introduction in 2008, BitCoin is the number one traded cryptocurrency in the world in terms of volume. This decentralized currency can transact without the involvement of the central bank or any financial intermediaries. The transactions using BitCoin are done using a BitCoin network where transactions are authenticated on a blockchain. Due to the lack of backing from a central bank or any regulation, BitCoin users and traders are generally exposed or assumed to be at higher risk (volatility). Like all countries in the world, BitCoin trading in South Africa has gained momentum. This means that there are growing movements of people's investments between the South African Rand, which is the currency used to transact in the Republic of South Africa, and the BitCoin, and investors are thus exposed to market risk.
To measure market risk associated with any financial asset, the Basel Committee on Banking Supervision (BCBS) is responsible for developing supervisory guidelines for banks and financial trading desks. BCBS has recommended that Value at Risk (VaR) be computed and reported. VaR is a statistic that measures the riskiness of financial en-

LITERATURE REVIEW
Regardless of the popularity of VaR, its limitations are well documented by many researchers. Rockafellar and Uryasev (2002) showed that VaR is not only incoherent but also fails to precisely estimate the risk of loss when the loss distributions have'fat tails'. "This significantly discreditsthe accuracy of this risk measure" (Chen, 2018). Nonetheless, VaR remains a popular risk measure as it is very simple to calculate and understand. Artzneret al. (1999) not only showed the incoherence of VaR but also introduced the Expected Shortfall, and called it a perfect risk measure. Pflug (2000) showed that ES is a coherent risk measure, based on coherent risk measure theory.ESisarisk measure sensitive to the shape of the tail of the distribution of returns on a portfolio, unlike the more commonly used VaR.
In order for risk practitioners to fully captureextreme or'tail'risk (very big uncertainty or fluctuation in returns)as mentioned in BCBS (2019), Extreme Value Theory (EVT) has been developed. The field of EVT was pioneered by Fisher and Tippett (1928) and Pickands (1975). Fisher and Tippett (1928) obtained three asymptotic limits describing the distributions of extremes assuming variables are independent. The arguments leading to modelling extremes using the generalized Pareto distribution are attributable to Pickands (1975). EVT is the theory of measuring and modeling extreme events (large fluctuations) (tails of statistical distributions), i.e. it is well suited to financial assets with extreme returns (very large fluctuations in returns). The EVT assumes independent and identically distributed (iid) observations. This iid assumption does not always hold for financial time series data. To correct this, McNeil and Frey (2000) proposed a twostage methodology in the form of a GARCH-EVT model using five index returns in his illustrations. The first step is to capture the heteroscedasticity (non-constant variation or fluctuations) features by fitting a GARCH model. The second step is to apply the EVT to residuals extracted from a selected GARCH model using the Generalized Pareto Distribution (GPD) or the Generalized Extreme Value Distribution (GEVD). The second part of this modelling process allows one to capture or describe the large fluctuations in prices and returns. The merits of the GARCH-EVT hybrid model lie in its ability to capture conditional heteroscedasticity (changing variation) in the data through the GARCH framework, while at the same time modelling the extreme tail (large fluctuations) behavior through the EVT method. Byström (2004) applied the GARCH-EVT model to the Swedish and American stock markets and compared different EVT-AMS methods and the Peak over Threshold (POT) method, which in general perform similarly. Murenzi et al.(2015) used the hybrid GARCH models with the EVT model to estimate VaR for the Rwandese currency exchange rate. Their findings indicated that the filtered EVT model performed better than ARMA-GARCH models. Chebbi and Hedhli (2014) applied the GARCH-EVT in managing portfolios with time-varying copula in the Tunisian, American, French, and Moroccan stocks. The copula is able to quantify the interdependence betweenthe different countries' stocks. Koliai (2016) presented a GARCH-EVT model with an R-vine model to manage portfolio risks that consisted of equity, currency indices, and commodities. Chinhamu et al. (2017) investigated the performance of the Generalized Lambda Distribution (GLD), the Generalized Pareto Distribution (GPD), and the Generalized Extreme Value Distribution (GEVD) in modeling daily VaR and ES in platinum, gold, and silver price log-returns. Their findings showed that GPD and GLD generally outperform GEVD for VaR and ES estimation for negative precious metal returns.
This study seeks to apply this hybrid model (GARCH-GPD) in the calculation of the VaR and ES using data from exchange rates BTC/USD and ZAR/USD and to compare their riskiness.

METHODOLOGY
This section describes the three steps that will be taken to fit the model. Secondly, extreme value theory (EVT) is then used to model the tail behavior of the data. The VaR q (X) is then calculated using the Generalized Pareto Distribution (GPD) tail estimation procedure.
Finally, the VaR of the asset is computed using the following formula: where is µ t+1 is the forecasts from the mean equation and σ t+1 is estimated from the volatility prediction model. VaR q (X) and ES q (X) are the VaR and ES of the standardized residuals.

ARMA -GARCH
GARCH models allow one to explain the varying, up and down movements (volatility), in asset prices,e.g. Bitcoin prices, and ZAR/USD exchange rates. The GARCH family volatility models to be considered are GARCH (1,1), EGARCH(1,1), GJRGARCH(1,1), and APARCH(1,1). All four volatility models are to be fitted to the exchange rates and their residuals are used in modelling the tail behavior of the series.
The GARCH(p, q) volatility model is mathematically defined as: where α i and β j are respectively ARCH and GARCH terms, and w is a constant; σ t 2 and σ t-1 2 are respectively the fitted conditional volatility from the model and its previous value. ε t 2 are the squared error terms in the model. The simplest model, the random walk with GARCH(1,1) variance/volatility model, is of the form: Exchange rates are unpredictable, and it makes sense to assume a random walk model.

Exponential GARCH (EGARCH) model
The EGARCH model allows efficient capturing of volatility clustering and asymmetric effects. An EGARCH(1, 1) volatility model can be expressed as: where w, α 1 , β 1 are model coefficients. γ is the leverage effect.

GJR-GARCH model
The GJR-GARCH volatility model was proposed by Glosten et al. (1993). This model takes into account the asymmetry property of financial data in obtaining the conditionals.

EVT -The Generalized Pareto Distribution (GPD)
The peak over threshold (POT) approach in fitting the Generalized Pareto Distribution (GPD) is used to model the standardized residuals from the selected GARCH family model. Balkema and deHaan (1974) and Pickands (1975) showed that for threshold u that is large enough, the POT approach leads to the use of the GPD. The GPD is defined as follows: where x> 0 when ξ ≥ 0 and 0 ≤x≤ -β/ξ. When ξ< 0 and β> 0, G ξ,β (x) is a GPD with the shape parameter or tail index ξ, a scale parameter β and a threshold u The value of ξ shows how heavy the tail is, with a bigger value indicating a heavy tail.

Parameter estimation of GPD
Let u be a sufficiently high threshold, assuming n observations y such that y iu ≥ 0, the subsample {y 1u, ..., y n -u} has an underlying distribution of a GPD, where y iu ≥ 0 for ξ ≥ 0, 0 ≤ y i -u ≤ -β/ξ for ξ< 0, then the logarithm of the probability density function of y iu is: Then the log-likelihood L(ξ, β| y i -u) for the model is the logarithm of the joint density of the n observations, i.e.
The parameters (ξ, β) are obtained by maximizing the log-likelihood function of the subsample under a suitable threshold u.

Conditional VaR
If F is an extreme distribution above some threshold u, then the F u (x) = G ξ,β (x), where 0 ≤ x<x Fu and ξ∈ℝ and β> 0, if x≥u then: Given F̅ (u), F̅ (x) is the formula for survival tail probabilities, it is inverse, gives the highest quantile of the distribution which represent the Value at Risk and is given by: and the Expected Shortfall is given as: . 11

Backtesting
To assess model adequacy and effectiveness in the computation of VaR, two backtesting methodologies are used. The Kupiec unconditional coverage test (Kupiec, 1995) and Christoffersen conditional coverage test (Christoffersen, 1998 (15) where Φ ij is the number of returns in state i who have been in state j previously (state 1 indicates that the VaR estimate is violated and state 0 indicates that it is not) and π i is the probability of having an exception that is conditional on state i the previous day. This statistic follows a chi-square distribution with two degrees of freedom.

RESULTS
Quantitative exchange rates data was collected and modelled so as to achieve the set objectives. Data was obtained from the finance sector website (Investing.com). The currencies considered are the South African Rand (ZAR) against the US dollar (USD) and the BitCoin (BTC) against the US dollar. The data was analyzed in an R-programming environment. The daily exchange rates considered were from 1/1/2015 to 30 /06/2021. The log returns were calculated and used to do the modelling.
The formula used is: whereP t and P t-1 are today and yesterday's closing values of daily prices (exchange rates), respectively.
In Figures 1 and 2, the time series plots reveal several trends in the mean and variance, confirming non-stationarity of the exchange rates prices. The log returns are stationary, around the zero-mean, although volatility is non-constant and clustered, indicating heteroscedasticity, and is common with financial data. Isolated extreme returns are visible and are caused by shocks in the financial markets.  Table 1 gives the descriptive statistics of the two exchange rates.
A positive mean for BTC/USD indicates a small increasing trend over time, whereas the opposite is true with a negative mean for the ZAR/USD, indicating a slight decreasing trend over time for the return series.
The Jarque-Bera test rejects the null hypothesis of Normality at the 5% level of significance, suggesting that extreme value theory distributions could be useful in capturing any heavy tails.
Source: Authors' own work.  The p-value of the Ljung-Box test for ZAR/USD returns confirms non-rejection of the null hypothesis of no autocorrelation. The fitting of a statistical distribution usually assumes homoscedasticity and no autocorrelation. However, autocorrelation is confirmed for the BTC/USD sincep-value = 0.0006249 < 0.05. The two step approach will therefore be used to help deal with the autocorrelation.
The null hypothesis of no ARCH effects is rejected at the 5% level of significance using the ARCH LM test, suggesting the use of GARCH family models should be considered when analyzing the above-mentioned return series The unit root test and stationary tests show that, at the 5% level of significance, the null hypothesis of a unit root is rejected, and it can be concluded that both exchange rate return series are stationary. The KPSS test results shows that all returns are stationary, since all p-values are greater than 0.05, therefore the null hypothesis of stationarity is accepted

GARCH-EVT models
Based on literature and the financial characteristics presented above, using GARCH-EVT model could lead to a better measure of risk in the tail. As the return series are non-Normal and have a heteroscedasticity feature, as suggested in other research work, using EVT and GARCH model to capture some features of the data is necessary.
All the four GARCH(1,1) models were estimated with both Normal distributed errors and Student's t-distributed errors. Their residuals were extracted, standardized and used to fit the Generalized Pareto distribution models. The aim is to fit a statistical distribution (the GPD) to the extreme residuals and use the estimated parameters and formulas to calculate VaR and ES.
To estimate the GPD model, a thresholdu must be selected. This threshold determines the number of observations above the threshold, N u . As the rule of thumb also suggests that it is ideal to choose the threshold that gives about 100 observations for fitting the Pareto distribution when the data set is large enough (McNeil & Frey, 2000).
The mean excess plots determine a suitable threshold, which is necessary for fitting the GPDmodel.
The choice of a threshold should be depicted by linear increases in the mean excess plot. Figure 3 presents the mean excess function of BTC/USD returns. By observing the meanexcess function in Figure 3, a threshold of between 0.8 and 1.5 seems to be a reasonable choice. The 90th percentile was selected. It provided a reasonable choice as it yielded enough data points for analyses and it falls within the above range.
The parameters of the GPD were estimated using the Maximum Likelihood method and are presented in Table 3.
n and t represent the assumption of Normal and Students't-distributed errors, respectively.Most of the ξ are positive, suggesting the presence of heavy tails, except for the APARCH-GPD-N and eGARCH-GPD-N models.
After observing the mean excess function for ZAR/USD in Figure 4, a threshold of between 0.8 and 1.8 seems to be a reasonable choice for ZAR/USD returns. Again, the threshold at 90th percentile was selected. This is a reasonable choice as it yields enough data points for analyses and it falls within the above range. The pa-Source: Authors' own work.  rameters of the GPD were estimated using the Maximum Likelihood method and are presented in Table 4.
Again, n and t represent the assumption of Normal and Student's-distributed errors.
All the ξ are positive suggesting the presence of heavy tails. The ξ for the ZAR/USD are bigger and positive, suggesting that it may be a riskier asset; however, the standard errors of the estimates of ξ are bigger as well. The VaR and ES will give a better picture of the risk.

Value at Risk and Expected Shortfall
The Value at Risk and Expected Shortfall values are calculated at 95% and 99% levels for both exchange rates understudy. Tables 5 and 6 show the estimated values of each VaR and ES.
At both the 95% and 99% levels of significance, daily VaR data suggests that Bit Coin is riskier than the South African rand, as the VaR statistic is higher. The higher VaR statistics mean that one loses more at a specified level of significance.
Source: Authors' own work.

Back testing results
for Value at Risk The estimated VaR are back tested using the Kupiec unconditional coverage test and Christoffersen conditional coverage. The p-values of each test are presented in Table 7.
Based on the p-values from both Kupiec likelihood ratio test and Christoffersen's test likelihood ratio test, the fitted GARCH-GPD models are well suited to the returns series understudy, since the observedp-values are greater than 0.05. Hence, the null hypothesis of model adequacy is accepted.

DISCUSSION
Four GARCH models are applied to the two data sets understudy, namely: the BTC/USD and the ZAR/USD. All four models were considered under two commonly used error distributions, that is normally and Student's t distributed residuals. The residuals were extracted and used to fit Generalized Pareto Distribution, and the estimated parameters were used to estimate risk statistics.
Under the BTC/USD time series, the extreme value index (ξ) seems to be inconclusive as to whether the tails are heavy or short. The GARCH models with Normally distributed errors would mean the tails are short, while the Student's t distributed errors suggest that the tails are heavier. In the case of the ZAR/USD currency series, the EVI suggests that the tails are heavy, hence the use of the EVT model is deemed more appropriate.
The computed values of both VaR and Conditional VaR indicate that there is a higher amount expected to be lost in the BTC/USD than in the ZAR/ USD at both the 95% and 99% confidence levels. This leads to the conclusion that the BTC is riskier than the ZAR. This could be due to the fact that the BTC is not backed by any central bank and is unregulated hence making it highly risky to keep as a savings tool. This information is useful to  South African and global investors who need to understand how much risk they take when converting their savings or investments to BitCoin instead of the South African currency, the Rand (ZAR).
The backtest results gave p-values that are way above 0.05. The high p-values indicate that the hybrid model used in the study isgoodand fitsthe currency data set used. The Kupiec test suggests that the GARCH(1,1)-GPD with Normally distributed errors is the best fitting model for BTC/USD, while eGARCH(1,1)-GPD with Normally distributed errors is the best-fitted model for ZAR/USD, as they both have higher p-values at both 95 % and 99% significance levels.
Christoffersen's test likelihood ratio test enables one to ascertain whether the output model violations are independent.A violation is when the actual loss exceeds the VaR estimate. In all the models, the p-values are greater than 0.05, leading to a conclusion that indeed the violations are independent of each other.

CONCLUSION
The purpose of this study is to use VaR and ES to compare the riskiness of the daily returns of the BitCoin (BTC) and South African Rand (ZAR), both currenciesbeingagainst the US Dollar. Both VaR and ES conclude that BTC is riskier than ZAR. This could be of great help to forex market risk managers in South Africa, particularly in choosing whether to keep their savings in the local currency or consider the cryptocurrency, BitCoin.
The hybrid model did capture fat tails and improved the computation of the VaR and ES as shown by the positive EVI (ξ) and backtesting procedures. The high p-values above 0.5 suggest that the GARCH (1,1)-GPD is a very good fit for the two currencies.
These results do not imply that GARCH (1,1)-GPD will always give good fits for every currency data set.
For further research, out-of-sample backtests and comparisons with generalized POT models are recommended, such as DIPOT (Dynamic Intensity Peaks Over Threshold) and PORT (Peak Over Random Threshold).