This paper implements different approaches used to compute the one-day Value-at-Risk (VaR) forecast for a portfolio of four currency exchange rates. The concepts and techniques of the conventional methods considered in the study are first reviewed. These approaches have shortcomings and therefore fail to capture the stylized characteristics of financial time series returns such as; non-normality, the phenomenon of volatility clustering and the fat tails exhibited by the return distribution. The GARCH models and its extensions have been widely used in financial econometrics to model the conditional volatility dynamics of financial returns. The paper utilizes a conditional extreme value theory (EVT) based model that combines the GJR-GARCH model that takes into account the asymmetric shocks in time-varying volatility observed in financial return series and EVT focuses on modeling the tail distribution to estimate extreme currency tail risk. The relative out-of-sample forecasting performance of the conditional-EVT model compared to the conventional models in estimating extreme risk is evaluated using the dynamic backtesting procedures. Comparing each of the methods based on the backtesting results, the conditional EVT-based model overwhelmingly outperforms all the conventional models. The overall results demonstrate that the conditional EVT-based model provides more accurate out-of-sample VaR forecasts in estimating the currency tail risk and captures the stylized facts of financial returns.

In this paper the generalized autoregressive conditional heteroscedastic models are applied in modeling exchange rate volatility of the USD/KES exchange rate using daily observations over the period starting 3 rd January 2003 to 31 st December 2015. The paper applies both symmetric and asymmetric models that capture most of the stylized facts about exchange rate returns such as volatility clustering and leverage effect. The performance of the symmetric GARCH (1, 1) and GARCH-M models as well as the asymmetric EGARCH (1, 1), GJR-GARCH (1, 1) and APARCH (1, 1) models with different residual distributions are applied to data. The most adequate models for estimating volatility of the exchange rates are the asymmetric APARCH model, GJR-GARCH model and EGARCH model with Student's t-distribution.

GARCH models have been commonly used to capture volatility dynamics in financial time series. A key assumption utilized is that the series is stationary as this allows for model identifiability. This however violates the volatility clustering property exhibited by financial returns series. Existing methods attribute this phenomenon to parameter change. However, the assumption of fixed model order is too restrictive for long time series. This paper proposes a change-point estimator based on Manhattan distance. The estimator is applicable to GARCH model order change-point detection. Procedures are based on the sample autocorrelation function of squared series. The asymptotic consistency of the estimator is proven theoretically.

The limit theory of a change-point process which is based on the Manhattan distance of the sample autocorrelation function with applications to GARCH processes is examined. The general theory of the sample autocovariance and sample autocorrelation functions of a stationary GARCH process forms the basis of this study. Specifically the point processes theory is utilized to obtain their weak convergence limit at different lags. This is further extended to the change-point process. The limits are found to be generally random as a result of the infinite variance.

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