The data obtained included monthly data pertaining to the average monthly exchange rate of the South African Rand (ZAR) and the Kenyan shilling against the US dollar. Also, the ratio of monthly reserves to base values (indicative of the adequacy of reserves and how much the foreign exchange reserves cover the most liquid money supply) was obtained for each country. Lastly, data for the movement of monthly nominal interest rates were obtained. The movements of these three components and their relative changes are schematically demonstrated in the results section. All three variables were computed to reflect rate relative changes. All three financial variables (rate-of-change in exchange rate, rate-of-change in interest rate and rate-of-change in reserves-to-M1) must demonstrate stationarity if used for subsequent analysis with EVT. Other than the general descriptive statistics (mean, median, variance, standard deviation, skewness and kurtosis), the stationarity according to the Jarque–Bera test statistic was determined as a first step. Another important component when applying EVT is the assumption of no-autocorrelation in the time series. The next section explains how the data analysis was applied in such instances.

The second step towards achieving an EMP index was to specify the time series dynamics of each of the components. When no time-varying volatilities were observed, the appropriate standard deviations were used for components that display features of conditional heteroscedasticity using autoregressive conditional heteroscedasticity (ARCH) and generalised autoregressive conditional heteroscedasticity (GARCH) models. The standard Box–Jenkins methodology was applied to arrive at the best fitting time series model. In doing so, the EMP series was standardised for each individual component if serial correlation was reported to achieve independence and identically distributed (iid) distribution. The results of the Box–Jenkins methodology and Ljung Box Q-statistics for the first and second moments are demonstrated in tables for the three components (see the results section). The basic statistics of exchange rate, nominal interest rate and ratio of reserves to base money are presented in a table (Table 1) in the results section (maximum, minimum, mean, standard deviation, variance, skewness and kurtosis). Possible periods of more and less volatility are identified within the total period of 19 years (SA) and 18 years (Kenya).

The EMP index was calculated according to the model independent approach as outlined by Eichengreen, Rose and Wyplosz (1994) and Eichengreen et al. (1995).

where e_t is the relative change in nominal exchange rate of ZAR against US dollar in month t; r_t is the relative change in ratio of gross forex reserves to narrow money (reserves-to-M1) in month t; i_t is the relative change in interest rate in month t; σ_e, σ_r, and σ_i are the standard deviations of e_t, r_t and i_t, respectively.

The descriptive statistics, tests for normality, skewness, kurtosis and autocorrelation, were analysed with statistical package R and RStudio (CRAN, R Core Team 2016). The website for the R project is http://www.r-project.org and the source code is http://www.gnu.org/licences/gpl.html. The R language is a dialect code of the S language, which was formulated by John Chambers at the Bell Labs in the 1970s.

In this study, extreme positive values of the EMP index as indicative of intense currency market pressure were examined. The probability of a crisis event, that is, an EMP index, that exceeds the [(1-p) × 100] percentile will indicate a stress event. Consequently, a threshold value for the identification of a stress event is provided. Where the observed value of the EMP index is more than this threshold, the index at that particular point can be considered under stress. The quantiles of these points (usually 95th or 99th percentile) are located in the tail regions of a probability density function. Financial data often display non-normal behaviour with asymmetry and kurtosis.

Generalised Pareto distribution: Peak over threshold: The largest 20% of the EMP index values were considered extreme observations (Guru & Sarma, 2013). A GPD was fitted to the ‘excesses’ over µ and applied the estimated GPD parameters to compute crisis thresholds given for the 95th and 99th percentiles. An observation of the EMP index larger than these thresholds will be indicative of a stress situation. Several tail quantiles for different probability levels p were computed to compare the results of the GPD to that of a normal distribution and the empirical (actually observed) quantiles.

The GPD parameters are estimated for the ‘exceeding’ values and the pth quantile on the tail of the distribution was estimated using the following equation proposed by McNeil and Frey (2000):

where n is the total number of observations, k is the number of observations above the threshold μ,σ^,ξ^ are maximum likelihood estimates of the GPD parameters and p is the probability level.

Hill estimator: The Hill estimator was also fitted to the ‘threshold’ values as discussed in the previous section. The Hill estimator was fitted to the tail of the distribution and the estimated Hill parameter (Xi) was used to compute crisis thresholds at the 95th and 99th percentiles. Again several tail quantiles (for different probability levels p) were computed to compare the results of the Hill estimator not only to that of the empirical quantiles but also to that of the GPD POT and normal distribution.

The data are sorted from the largest to the smallest (X1, n > … Xn, n). The formula as proposed by Hill (1975) is given below:

where K represents the cut-off between the tail and central observations so that Xj,  n can be considered extreme observations.

Alternatively, this formula can also be structured in a different way that is more user-friendly (Kang & Song 2017).

where n represents the largest observations (cut-off between tail and central observations).

The 100pth quantile using the Hill estimator is computed by the following formula (Kang & Song 2017):

where n is the tail observations, N is the entire sample and Xi is the Hill estimator.

Once the stress events are identified, an analysis of the stress events can be performed in the context of prevailing conditions in these countries around the stress periods. Furthermore, the contribution of each of the three components of the EMP index can be assessed during the stress periods.

Using the exchange rate, interest rate and reserves-to-M1 data from South Africa, the EMP index can be calculated.

However, before the respective standard deviations can be applied in the formula, we need to use the standardised standard deviations (for the variables in the EMP index that displayed conditional heteroscedasticity) to ensure that the assumption of iid holds. As such, the EMP index requires the standardisation of the time-varying volatilities. Consequently, the time series dynamics are displayed for each constituent of the series (Tables 2–4).

The e_t series demonstrated autocorrelation with significant Ljung–Box statistics for the mean series, revealing a moving average (MA)[1] process (Table 1).

The i_t series displayed autocorrelation with significant Ljung–Box statistics signifying an ARMA process. Moreover, the squared series also demonstrated serial correlation suggesting a GARCH effect (Table 1). The time series method of Box and Jenkins was applied and an appropriate model for the rate of changes in interest rate was found to be an AR(1)-GARCH(1,1) process. The information for this model and the diagnostic examination is presented in Table 3.

Lastly, the r_t series showed significant Ljung–Box statistics for the squared series (heteroscedasticity) demonstrating a GARCH [1,1] process. The process is depicted in Table 4.

The standardised standard deviations of the respective ARMA and/or GARCH processes for each factor of the EMP index are used for the calculation of the EMP index.

The calculation of the EMP index was performed in Microsoft Excel (2013) using the standardised standard deviations with no heteroscedasticity. Using these estimated volatilities and the EMP formula listed above, the EMP index series from 1999 to 2017 was calculated. The descriptive statistics for the series is depicted in Table 5 and the movement of the EMP index series is schematically demonstrated in Figure 4. The series is shown to be non-normally distributed with a positive skew. Other studies performed on the analysis of EMP similarly demonstrated non-Gaussian distributions (Guru & Sarma 2013; Heinz & Rusinova 2015; Pozo & Amuedo-Dorantes 2003). The time series analysis reported in Table 4 demonstrates that the EMP series is stationary, with no autocorrelation for the series or squared series.

As demonstrated in Figure 4, the EMP index reveals more or less constant volatility during the entire period. Several spikes of depreciation (positive value of the EMP index) can be seen.

As we have studied the EMP index in South Africa from 1999 to 2017, we now need to demonstrate the identification of a stress period. These are periods of extreme positive values in the EMP index. Periods of sustained stress over two or more periods are indicative of a currency crisis. The application of EVT will be used to demonstrate these periods of extreme pressure and/or currency crisis.

The study by Guru and Sarma (2013) chose a threshold value of the largest 20%. In our study, we decided to use a threshold value of 15% of the data to be extreme observations as our South African data set is less volatile than that presented by Guru and Sarma (2013). The tail threshold (largest 15% of values) chosen is 7.00. This equates to the largest 35 values. From Figure 5, it is evident that the POT estimate levels off and becomes stable after 35 observations.

Using this threshold value, we proceeded to fit a GPD to the values exceeding 7.00. The maximum likelihood estimation method was utilised. The package fExtremes (Wuertz 2013) was used for the analysis. Consequently, the estimated parameters were used to calculate crisis thresholds at the 95th and 99th percentiles. The formula used for the POT method is as follows:

where n = 216

k = 35 (number of observations above the threshold µ)

σ^,ξ^: maximum likelihood estimates of the GPD parameters (see Table 6).

Usually, when using the POT approach, one would implement the probability stress threshold fixed at 0.05 (95th percentile) or at 0.01 (99th percentile). Using the Equation 6, the entire tail region can be calculated using different p-quantiles. Table 6 depicts the parameters estimated by the GPD.

The shape parameter was shown to be 0.400, which is a positive number and indicative of the heavy-tailed and Frechet distribution. The results in Table 5 also revealed non-normality (Jarque–Bera statistic) and excess kurtosis. The histogram of the EMP index confirms the positive skew and kurtosis (Figure 6).

The following four figures demonstrate the diagnostic plots for the fitted distribution (Figures 7–10). The observations are fitted well for the tail and excess of the underlying distribution (Figures 7 and 8). The figures demonstrate a good fit with the POT method (using maximum likelihood estimation), with the use of the top 15% values. In Figure 9, the scatterplot of the residuals is random and spread, with the red line nearly parallel to the X-axis indicating a good fit. The residuals in the Quantile-Quantile (QQ)-plot are also fitted close to the 45° line in the next figure (Figure 10).

The estimated 95th percentile and 99th percentile for the EMP index are 12.13 and 24.48, respectively. These are indicative of the crisis threshold or stress periods at 0.05 and 0.01 probabilities, respectively. Exchange market pressure values greater than these thresholds would indicate a period of stress or crisis at the corresponding probability levels. Table 7 demonstrates that the EVT quantiles fit the empirical quantiles better than the normal distribution quantiles as reported by Guru and Sarma (2013). The normal distribution crisis-based thresholds are very low and, therefore, many false alarms would be raised. From Table 7, it is clear that the model is accurate for values up to the 95th percentile, and that the accuracy of the estimated quantiles for the 96th to 99th percentile decreases. However, the model still provides a better fit when compared to the normal distribution.

In the current study, an EMP index greater than 12.13 would be indicative of currency stress at p = 0.05 (GPD POT). As such, eight periods of stress are highlighted. At p = 0.01 (values greater than 24.48), four periods of extreme stress or crisis are highlighted between 1999 and 2017 (GPD POT). The months affected are December 2001, March 2004, June 2006, June 2007, October 2008, February 2014, December 2015 and February 2016. If currency crisis is defined as periods of two consecutive months of stress, no such periods are found. The four periods of extreme pressure with p = 0.01 are December 2001, June 2006, October 2008 and February 2014. If the current study had used normally distributed thresholds at p = 0.05, 92 exceedances would have been raised, causing many false positive warnings. If the Hill estimate was used (right hand column), the same periods of time would have been selected at p = 0.05, but at p = 0.01 only one period of extreme pressure is flagged.

In the next section, the stress period (p = 0.05) is reflected in Table 8 with the individual factors of the EMP index. The bold cells show the individual components mostly responsible for the high EMP values during the respective periods. In December 2001, June 2006, October 2008 and December 2015, the rates of change in exchange rates were the most influential values on the EMP index. In June 2007, February 2014 and February 2016, the rates of change in interest rates were the most powerful values for driving the EMP index into an extreme positive index. Lastly, in March 2004, the rate-of-change in reserves-to-M1 was the most prominent component to influence the EMP index. Although a near currency crisis was elicited in January and February 2014, values of the EMP index returned to normal in March 2014. Also, fortunately the value for the EMP index for January 2014 was not indicative of severe currency pressure (p = 0.01).

Similar studies showing the results for periods of stress or crisis are also reported in Asian studies (Guru & Sarma 2013; Pozo & Amuedo-Dorantes 2003). The study by Guru and Sarma (2013) also reported the influence of the individual components in periods of extreme pressure.

Quantiles calculated according to the Hill estimate are also provided in Table 7. Figure 11 concurs with the earlier decision to use a threshold value of 7.00 (i.e. K = 35). A parallel line to the X-axis can be drawn up to k = 35. The largest 35 values were sorted from the largest to the smallest (rank ordered), and Equation 4 (Hill 1975; Kang & Song 2017) was applied to determine the Hill estimate (Xi [Hill]).

The Hill estimate calculated is 0.54172.

The 100pth quantile using the Hill estimator was also computed according to Equation 5 (Kang & Song 2017). The analysis was also run in R (package ReIns).

From Table 7, it is clear that the GPD POT method outperforms the Hill estimate (compared to the empirical quantiles) at higher quantiles (p = 0.001 to p = 0.10) where the risk of extreme crises becomes more formidable. However, there are instances where the Hill estimate provides a better index compared to the GPD POT method and especially so at percentiles of 0.80–0.89.

Using the three components of the EMP index for Kenya, the exchange market pressure for monthly data can be calculated (Equation 1).

As stated earlier, the respective standard deviations can only be applied when each variable is independent and identically distributed. Consequently, EMP index requires the standardisation of the time-varying volatilities (Guru & Sarma 2013). None of the three variables for the Kenyan data displayed time-varying volatility for the mean and series (no heteroscedasticity).

The EMP index series was calculated from August 2000 to February 2017. The descriptive statistics for the series (Table 10) and a schematic presentation of the EMP index (Figure 15) are provided. The results of the series show that it is not normally distributed with a positive skew. A large value for kurtosis is reported. The time series analysis depicted in Table 10 reveals that the series is stationary with no serial correlation for the mean or squared series.

The figure of the EMP index illustrates relatively constant volatility during the studied period. Several spikes of depreciation (positive value of the EMP index) are shown in early 2003 and mid-2014. These are discussed in greater detail in the next section. Extreme value theory was applied to demonstrate possible periods of extreme pressure and/or currency crisis.

For the Kenyan data, the researcher used a threshold value of 20% similar to the study by Guru and Sarma (2013). The data set contains 199 observations. Twenty per cent of 199 equates to 40 values. Consequently, the threshold value for the Kenyan data set is 6.50. From Figure 16, it is clear that the POT estimate levels off and becomes stable after 40 observations. Subsequently, a GPD to the values exceeding 6.50 can be fitted (Equation 6). The maximum likelihood estimation method is utilised.

Table 11 depicts the parameters estimated by the GPD.

The shape parameter is 0.166, which is a positive number and indicative of a Frechet (heavy-tailed distribution). The results in Table 10 also reveal non-normality and excess kurtosis. The histogram reiterates a positive skew and is leptokurtic (Figure 17).

Figures 18–21 demonstrate the diagnostic plots for the GPD (POT) fitted distribution for Kenya. The observations exceeding the threshold value of 6.50 are fitted well for the tail and excess of the underlying distribution (Figures 18 and 19). The scatterplot of the residuals (Figure 20) is random and spread, with the red line nearly parallel to the X-axis indicating a good fit. The residuals in the QQ-plot are also fitted close to the 45° line in Figure 21.

The estimated 95th percentile and 99th percentile for the EMP index are 16.17 and 30.53. Exchange market pressure values greater than these percentile-based thresholds would indicate a period of stress or crisis at the corresponding probability levels. Table 12 demonstrates that the EVT quantiles fit the empirical quantiles better than the normal distribution quantiles. The 95th and 9th percentile values are provided in bold in Table 12. The Hill estimated quantiles are discussed in the next section.

In this study, an EMP index greater than 16.17 would be indicative of currency stress at p = 0.05 (GPD POT). As such, 10 periods of stress are identified. At p = 0.01 (values greater than 30.53), two periods of extreme stress are identified between 2000 and 2017 (GPD POT). The months affected are March and May 2005, December 2007, April 2009, August 2009, November 2011, June 2012, June and December 2014 and October 2015. No period of currency crisis was defined, which is periods of two consecutive months of stress. The two periods of extreme pressure (p = 0.01) are March 2005 and June 2014. If the current study had employed normally distributed thresholds (p = 0.05), 82 exceedances would have been raised, causing many unnecessarily flagged periods. The same periods are flagged when the Hill estimate is used for the 95th percentile. However, at the 99th percentile, no time periods demonstrate pressure situations from 2000 to 2017.

Because the EMP index and extreme periods of stress have been identified, it is possible to explore in more detail what economic activities transpired at this time. Table 13 is representative of the individual factors of the EMP index. The bold cells display the individual components mostly responsible for the high EMP values during these 10 periods. During March 2005, the rates of change in interest rates were the most influential value on the EMP index. However, during June 2014, both the rate-of-change in exchange rate and the rate-of-change interest rate were responsible for driving the EMP index into an extreme positive index. Fortunately, during July 2014, all components returned to stable values free of extreme pressure. Regarding values exceeding p = 0.05, seven of the remaining eight pressure situations, the rate-of-change in exchange rate was mostly responsible. Moreover, during November 2011 and October 2015, the rate-of-change in interest rate was also responsible. Guru and Sarma (2013) also identified individual components as the most likely contributors of extreme pressure. In their study, the rates-in-change in exchange rate and interest rate were individually, and at different periods, responsible for all incidents of pressure.

Quantiles computed according to the Hill estimate are presented in Table 12. Figure 22 confirms the earlier decision to use a threshold value of 6.50 (i.e. K = 40). A parallel line to the X-axis can be drawn up to k = 40. The largest 40 values were sorted from the largest to the smallest (rank ordered) and Equation 3 was applied to determine the Hill estimate (Xi [Hill]).

The Hill estimate calculated is 0.68015.

The 100pth quantile using the Hill estimator was also calculated according to Equation 4.

When revisiting Table 12, it is certain that the GPD POT method provides a better estimate of the empirical quantiles compared to the Hill estimate. This is shown across all quantiles (0.80–0.999). There are only five quantiles where the Hill estimate provides a better index compared to the GPD POT method.