One-step-ahead Value at Risk and Expected Shortfall

Calculates the one-day Value at Risk (VaR) and Expected Shortfall (ES) of the returns from the predictive distribution of y_t given y_1, ..., y_{t-1}, for every observation and for the next day after the data.

Usage

var_es(object, level = c(0.95, 0.99), grid_size = 500)

Arguments

object: A stochvolTMB object returned from estimate_parameters.
level: Vector of confidence levels between 0.5 and 1.
grid_size: Number of grid points for the log-volatility.

Value

A data.table with one row for each time point and level, and the columns time (1 to nobs + 1), level, y (the observed return, NA for the forecast), var, es and exceedance (y < -var).

Details

The predictive distribution is calculated exactly with the same filter on a grid for the log-volatility as residuals.stochvolTMB. It is a mixture over h_t of the conditional distribution of the returns. For a level $\alpha$, for example 0.99, let $q_t$ be the $1 - \alpha$ quantile of the predictive distribution. Then $$VaR_t = -q_t, \qquad ES_t = -E[y_t \mid y_t \le q_t, y_1, \ldots, y_{t-1}].$$ Both are reported as positive numbers when they are losses, on the scale of the returns. A return is an exceedance if $y_t < -VaR_t$.

The parameters are fixed at their estimates from the full data, and the uncertainty of the parameters is not included. The values for the observed data are therefore in-sample, and a backtest with backtest_var on them is slightly optimistic. The last row (time = nobs + 1) is the forecast for the next day. For more than one day, use forecast_var_es.

Examples

# \donttest{
opt <- estimate_parameters(spy$log_return, model = "skew_t_leverage", silent = TRUE)
risk <- var_es(opt, level = c(0.95, 0.99))
# Forecast for the next day
risk[is.na(y)]
#>     time level     y        var         es exceedance
#>    <int> <num> <num>      <num>      <num>     <lgcl>
#> 1:  3523  0.95    NA 0.03181108 0.04560319         NA
#> 2:  3523  0.99    NA 0.05359064 0.06893175         NA
backtest_var(risk)
#>    level     n expected observed        rate      lr_uc      p_uc    lr_ind
#>    <num> <int>    <num>    <int>       <num>      <num>     <num>     <num>
#> 1:  0.95  3522   176.10      191 0.054230551 1.29303116 0.2554901 1.3599331
#> 2:  0.99  3522    35.22       34 0.009653606 0.04318348 0.8353802 0.9112984
#>        p_ind     lr_cc      p_cc  es_ratio
#>        <num>     <num>     <num>     <num>
#> 1: 0.2435491 2.6529642 0.2654093 0.9890637
#> 2: 0.3397702 0.9544819 0.6204930 1.0067746
# }