1 Simulate (-log-return) data $(X_t)$ from an ARMA-GARCH process | 2 Fit an ARMA-GARCH model to the (simulated) data | 3 Calculate the VaR time series | 4 Backtest VaR estimates | 5 Predict VaR based on fitted model | 6 Simulate future trajectories of $(X_t)$ and compute corresponding VaRs | 7 Plot

Introduction | Estimating Risk Measures for $X\sim NVM_1(\mu, \sigma, F_W)$ | Value-at-risk | Expected Shortfall

1 Introduction | 2 Evaluating the distribution function | 2.1 Exponential mixture distribution | 2.2 Three-point mixture distribution | 2.3 The wrappers pNorm() and pStudent() | 2.4 The effect of algorithm-specific parameters | 3 Evaluating the density function | 3.1 3-point mixture | 3.2 Inverse-gamma mixture | 4 (Quasi-)random number generation | 5 Parameter estimation

1 Geometric VaR and expectile for two sets of confidence levels | 2 Bootstrapped geometric expectiles | 3 Comparison of geometric VaR and expectile for a given direction

1 Homogeneous case | 1.1 Checks for method = "dual" | 1.2 Checks for method = "Wang"/"Wang.Par" | 1.2.1 Check of auxiliary functions with numerical integration (for $\theta = 2$) | 1.2.2 Check of $h(c)$ without numerical integration (for a range of $\theta$) | 1.3 Compute best/worst $\mathrm{VaR}_\alpha$ (via "Wang.Par") | 1.4 Comparison between various methods for computing worst value-at-risk | 2 Inhomogeneous case | 2.1 A motivation for (column) rearrangements | 2.2 Run-time comparison (straightforward vs efficient implementation) | 2.3 How rearrange() acts on specific matrices | 2.4 Convergence | 2.5 A real data application | 2.6 Worst VaR copula samples