# VaR and ES Computation Following Fitted Models or Forecasts Provides easy access to value-at-risk (VaR) and expected shortfall (ES) computation for available models in this package. VaR and ES can either be computed based on (a) fitted conditional means and conditional standard deviations for a training period, or following (b) point forecasts (either multistep or rolling) of the conditional means and conditional standard deviations. ```r measure_risk(object, measure = c("VaR", "ES"), level = c(0.975, 0.99), ...) ## S4 method for signature 'fEGarch_fit' measure_risk(object, measure = c("VaR", "ES"), level = c(0.975, 0.99), ...) ## S4 method for signature 'fEGarch_forecast' measure_risk(object, measure = c("VaR", "ES"), level = c(0.975, 0.99), ...) ## S4 method for signature 'fEGarch_distr_est' measure_risk( object, measure = c("VaR", "ES"), level = c(0.975, 0.99), test_obs, sigt, cmeans, ... ) ``` ## Arguments - `object`: either an object of class `"fEGarch_fit"` returned by the fitting / estimation functions of this package like returned by for example `fEGarch` among others, an object of class `"fEGarch_forecast"` as returned by `predict,fEGarch_fit-method` or `predict_roll,fEGarch_fit-method`, or an object of class `"fEGarch_distr_est"` returned by the distribution fitting functions of this package like returned by for example `find_dist` among others. - `measure`: a character vector with element `"VaR"`, `"ES"` or both; indicates, what risk measures should be computed; by default, both VaR and ES are computed. - `level`: a numeric vector of arbitrary length indicating the confidence levels to compute the VaR and the ES at; by default, the levels `0.975` and `0.99`, i.e. 97.5 percent and 99 percent, are considered. - `...`: currently without use. - `test_obs`: a series of test observations (only required when `object` is of class `"fEGarch_distr_est"`). - `sigt`: a series of forecasted conditional standard deviations for the same time points as `test_obs` (only required when `object` is of class `"fEGarch_distr_est"`). - `cmeans`: a series of forecasted conditional means for the same time points as `test_obs` (only required when `object` is of class `"fEGarch_distr_est"`). ## Returns The S4 methods all return an object of class `"fEGarch_risk"` with elements `measures`, `observations` and `model`. `observations` is the observed series at the time points, for which the risk measures are calculated. `measures` is a list with elements `VaR` and `ES`, distinguishing between computed VaR and ES values. These elements again are list with named elements representing the various computed series. `model` is the fitted model object. ## Details 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. ## Examples ```r # In-sample window.zoo <- get("window.zoo", envir = asNamespace("zoo")) rt <- window.zoo(SP500, end = "2002-12-31") model <- fEGarch(egarch_spec(), rt) risk <- measure_risk(model, measure = c("VaR", "ES"), level = c(0.95, 0.975, 0.99)) risk # Out-of-sample rolling point forecasts window.zoo <- get("window.zoo", envir = asNamespace("zoo")) rt <- window.zoo(SP500, end = "2002-12-31") model2 <- fEGarch(egarch_spec(), rt, n_test = 250) fcast <- predict_roll(model2) risk2 <- measure_risk(fcast, measure = c("VaR", "ES"), level = c(0.95, 0.975, 0.99)) risk2 # Use some model to obtain rolling point forecasts of # the conditional mean and the conditional standard deviation for # some test period; in practice, this will not be from a GARCH-type # model, because it is parametric and includes a distribution assumption, # but instead from some nonparametric model window.zoo <- get("window.zoo", envir = asNamespace("zoo")) rt <- window.zoo(SP500, end = "2005-12-31") model <- fEGarch(egarch_spec(), rt, n_test = 250) fc <- model %>% predict_roll() test_obs <- model@test_obs # Test observations sigt <- fc@sigt # Conditional volatility forecasts cmeans <- fc@cmeans # Conditional mean forecasts resids <- model@etat # In-sample standardized residuals # Given 'test_obs', 'sigt', 'cmeans' and 'resids', we can now # compute the VaR and ES forecasts for the test period dist <- find_dist(resids, fix_mean = 0, fix_sdev = 1) dist risk <- dist %>% measure_risk(test_obs = test_obs, sigt = sigt, cmeans = cmeans) plot(risk, which = 0.975) ```

measure_risk function