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If a random variable X follows the generalised hyperbolic distribution, we write. Its probability density function is given by. It should also be noted that the domain of the parameters must satisfy the following conditions. The mean and variance of this distribution Prause, are given by.

Also, for this distribution,. It should also be noted that different re-parameterisations of HYP exist. A particular case is given by. This parameterisation is very important in statistical analyses as it helps to determine the tail behaviour of the data. This subclass is called the VG distribution. Finally, we have the GH skew Student's t distribution, which is the last subclass of the GHD family, and it is obtained as a limiting distribution when the parameters The probability density function of this distribution is given by.

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The mean and the variance of this distribution are given by. This distribution is the only subclass of the GHD with one polynomial and one exponential tail, thus enabling them to handle heavy-tailed data well. These are summarised below. But in this case, a "bootstrapping" procedure is performed. Our main reason for employing this test is because our sample size is large and the data contains many ties i. The robust K-S test is useful in cases where the hypothesised distribution is not fully continuous or discrete.

More importantly, it also caters for data that contains ties, whereas the standard K-S test does not allow for ties in the data. The idea behind this test is to enlarge the region of acceptance hypothesis beyond that of the hypothesised distribution H x , defined over some closed interval Z c R. It should also be noted that this test is a two-sample test. The robust K-S test relies on the class of distributions defined by.

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The test statistic is given by. Selecting the optimal model i. Given that our analysis is principally based on fitting GHDs to data and comparing the fit of these distributions amongst one another for the optimal model, it is necessary for us to look at a criterion for model selection. In our case, we shall utilise the Akaike information criterion AIC. Value-at-Risk VaR is defined as a threshold value such that the probability of the market loss on a portfolio, over a given time horizon, exceeds this threshold value is equal to a pre-specified level.

It is widely used as a risk measure and utilised for assessments of extreme behaviour in financial data Jorion, More importantly, it can be used to measure a distribution's level of adequacy for tail fits, i. VaR backtesting.

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It should be noted that financial institutions are more prone to failure due to the shortage of capital resulting from underestimation of VaR. Furthermore, Beling, Overstreet and Rajaratnam have shown that, under the Basel framework, there is a negative profit impact due to the misestimation of VaR in either direction.

Hence, an adequate model for assessing the risk of a return series should not underestimate or overestimate VaR. In the analysis of maximum loss for a portfolio, the Kupiec likelihood ratio test Kupiec, is the most commonly used backtesting procedure. In this case, if the ratio of the number of violations is not significantly different from the level of significance, then the overall adequacy of the model is verified. Thus, under the null hypothesis that the ratio of the expected number of violations is a, the test statistic for the Kupiec test is given by.

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  7. This test statistic asymptotically follows a chi-square distribution with one degree of freedom. No averaging or corrections were made to the data. In the following section, we introduce the data set and its descriptive analyses. It should be noted that the data consists of the weighted average of the banks' daily rates at approximately Weights are based on the banks' foreign exchange transactions. The first thing to note about this graph is that the daily exchange rate increased from about R3.

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    But as time went on, this value began to change haphazardly upwards and downwards to about R Thus the plot shows some irregular movements characterised by upward and downward trends. This suggests that the series is non-stationary. There is a very high degree of variability which is a common characteristic of financial data.

    To transform the data to a stationary sequence, it is common practice to consider the log-return series i. The log-returns or just simply "returns" , at day t, of the series is given by. With this transformation, we obtain the time series plot and the corresponding histogram of the returns as shown below. From the graphs below, we observe that the series hovers around zero, suggesting that the series is now stationary about the mean.

    However, we see some heteroscedastic patterns and volatility clustering, which characterise financial returns Tsay, The histogram shows the leptokurtic behaviour of our log-returns as we have more returns at the centre than the tail parts, with a high peak around the mean, and fat tails. The leptokurtic behaviour is confirmed by Table 1 below, in which the kurtosis is as high as 9. This also suggests that the log-return series is not Gaussian as the kurtosis for normal distribution is 3.

    The Q-Q plot below also confirms this claim of non-normality; as normal distributed returns would imply that the returns lie in a straight line. Rather, our graph is S-shaped, which is due to the presence of fat tails, hence the need for heavy-tailed distributions such as the GHDs. A formal test concerning normality is the Jarque-Bera normality test also presented in Table 1.


    In our case, the test statistic has a value of The mean return of 0. The skewness of 0. It is evident from the ACF plot of log-returns that our data are uncorrelated all spikes are insignificant. However, the ACF plot of the squared returns shows some significant spikes, which suggests that the squared returns are autocorrelated.

    This is a common feature that characterises financial returns. These observations also confirm that the log-return series is stationary. Table 2 below summarises the results of both tests. Under the null hypothesis of log-returns having a unit root, both tests show that this hypothesis is rejected at all levels of significance.

    This is indicated by the low p -values both less than 0.

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    Hence our return series is stationary and we can proceed with further time series analyses. Employing the estimates below, we proceed to analyse the goodness-of-fit for these GHDs and compare them to the normality assumption. We begin with the comparison between the hyperbolic and normal distributions. Figure 5 presents the various graphical analyses for the hyperbolic subclass. The histogram shows that the skewness of the hyperbolic distribution makes it more appropriate for fitting the daily log-returns, relative to the normal distribution.

    This observation is also made clearer by the log density plot, which is very important in the analysis of tail behaviour. In this case, it is evident that a heavy-tailed distribution such as the hyperbolic with semi-heavy tail properties is needed, as it provides a better fit compared to the normal distribution. This is further confirmed by the Q-Q plot, where it is evidenced that the hyperbolic distribution portrays a closer description of the data, especially at the tails. The graphs in Figure 6 illustrate the goodness-of-fit for NIG.

    Secondly, the log density plot shows that the NIG distribution is more appropriate in fitting the tails, especially the upper tail of the log-returns and this is also finally, confirmed by the Q-Q plot. Thus once more, we obtain a better fit with NIG as compared to the normal distribution. In a similar way to the above, we compare the fit of the normal distribution to that of GH skew Student's t distribution. The resultant graphical analyses are presented in Figure 7. However, relative to the normal distribution, it still shows a better depiction of our returns data, especially for the lower tail.

    Finally, we compare the fit of the variance-gamma distribution to that of the normal distribution. Beyond Value at Risk provides a comprehensive guide to recent developments and existing approaches to VaR and risk management, going beyond traditional approaches to the subject and offering a new, far-reaching perspective on investment, hedging and portfolio decision-making. The key to this distinctive approach is a new decision rule - the 'Generalised Sharpe Rule', and its practical applications. Beyond Value at Risk The New Science of Risk Management Kevin Dowd Today's financial markets are experiencing a Value at Risk revolution, with risk management and measurement now the most important issues facing international finance houses.

    Not only are they desperate to avoid the fate of Orange County and Barings, they must also contend with fierce competition and market regulators who scrutinise their every move.

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    VaR theory and practice is therefore developing very rapidly. Beginning with a background to VaR in the context of recent developments in risk management, Kevin Dowd gives thorough descriptions of the various existing approaches to VaR - the variance-covariance, historical simulation, Monte Carlo and related simulation approaches, including a section on stress testing. These approaches, along with the innovative new rule described by the author, 'the Generalised Sharpe Rule', are discussed in conjunction with other factors, such as credit liquidity and operational and legal risks, giving a unique, integrated approach to enterprise-wide risk management.

    He also edited The Experience of Free Banking Convert currency.

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