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Risk measures such as value-at-risk (VaR) and expected shortfall (ES) may require the calculation of quantile functions from quantile regression models. In a parametric set-up, we propose to regress directly on the quantiles of a distribution and demonstrate a method through the conditional autoregressive range model which has increasing popularity in recent years. Two flexible distribution families: the generalised beta type two on positive support and the generalised-t on real support (which requires log transformation) are adopted for the range data. Then the models are extended to allow the volatility dynamic and compared in terms of goodness-of-fit. The models are implemented using the module fmincon in Matlab under the classical likelihood approach and applied to analyse the intra-day high-low price ranges from the All Ordinaries index for the Australian stock market. Quantiles and upper-tail conditional expectations evaluated via VaR and ES respectively are forecast using the proposed models.

Normal mean-variance mixture distributions are widely applied to simplify a model's implementation and improve their computational efficiency under the Maximum Likelihood (ML) approach. Especially for distributions with normal mean-variance mixtures representation such as the multivariate skewed variance gamma (MSVG) distribution, it utilises the expectation-conditional-maximisation (ECM) algorithm to iteratively obtain the ML estimates. To facilitate application to financial time series, the mean is further extended to include autoregressive terms. Techniques are proposed to deal with the unbounded density for small shape parameter and to speed up the convergence. Simulation studies are conducted to demonstrate the applicability of this model and examine estimation properties. Finally, the MSVG model is applied to analyse the returns of five daily closing price market indices and standard errors for the estimated parameters are computed using Louis's method.

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