Given a fitted model with fitted conditional means and conditional standard deviations
or given point forecasts of such series based on a fitted model, the risk measures
VaR and ES can be computed (at arbitrary confidence levels) following the conditional
loss distribution defined through the estimated / forecasted conditional mean value,
the estimated / forecasted conditional standard deviation value, and the assumed
conditional distribution (including potential estimates of distribution parameters).
Let \(\hat{\mu}_t\) be the estimated / forecasted conditional mean and \(\hat \sigma_t\) be the
estimated / forecasted conditional standard deviation at some time point \(t\). Furthermore,
define \(\text{VaR}_{\eta,\alpha}\) and \(\text{ES}_{\eta,\alpha}\) be
the time-invariant VaR and ES, respectively, of some identically but independently
distributed random variables \(\eta_t\) with mean zero and variance one. Given that
the relationship \(r_t = \mu_t + \sigma_t\eta_t\), where \(\mu_t\) and \(\sigma_t\)
are the true conditional mean and conditional standard deviation at time \(t\), is assumed
for some return series \(\{r_t\}\), the estimated / forecasted conditional VaR and ES of \(r_t\) are simply
$$\widehat{\text{VaR}}_{r,\alpha}(t) = \hat{\mu}_t + \hat{\sigma}_t \text{VaR}_{\eta,\alpha} \hspace{3mm} \text{ and } \hspace{3mm} \widehat{\text{ES}}_{r,\alpha}(t) = \hat{\mu}_t + \hat{\sigma}_t \text{ES}_{\eta,\alpha}.$$
This definition holds, when losses and therefore also
\(\text{VaR}_{\eta,\alpha}(t)\) and \(\text{ES}_{\eta,\alpha}(t)\) (for common \(\alpha\) such as
\(\alpha = 0.975\) or \(\alpha = 0.99\)) are considered
to be negative in sign.
Define
$$\text{VaR}_{\eta,\alpha} = f_{\eta}^{-1}(1-\alpha) \hspace{3mm} \text{ and } \hspace{3mm} \text{ES}_{\eta,\alpha} = (1-\alpha)^{-1}\int_{\alpha}^{1} \text{VaR}_{\eta, x} dx,$$
which also need to be estimated for some distributions, if a distribution parameter needed to be estimated. \(f\) in the previous formula
is the cumulative distribution function of the random variables \(\eta_t\). Therefore,
\(f^{-1}_{\eta}(1-\alpha)\) returns the quantile of the innovation distribution at level
\(1-\alpha\).
In some cases, when rolling
one-step forecasts of the conditional standard deviations and the conditional means
were obtained following a nonparametric approach, for example through
neural networks or similar approaches, VaR and ES are not directly to be calculated
because distribution assumptions have not been made. If an object that
is a fitted distribution to the model's standardized in-sample residuals is provided,
and if also test observations as well as forecasted conditional standard deviations
and conditional means for the test time points are passed to the method, VaR
and ES will be computed using the fitted distribution in object. Note
that object must be of class "fEGarch_distr_est". A natural
selection of object is the output of find_dist, which returns
the best fitted model among a normal distribution, a \(t\)-distribution, a
generalized error distribution, an average Laplace distribution, and their skewed
variants, following either BIC (the default) or AIC. It is recommended to then
set fix_mean = 0 and fix_sdev = 1 in the call to
find_dist to reflect the known property that the residuals are assumed
to be estimated from innovations with mean zero and variance one.