riskmeasures function

Risk measures for normal variance mixtures

Estimation of value-at-risk and expected shortfall for univariate normal variance mixtures

VaR_nvmix(level, qmix, loc = 0, scale = 1, control = list(), verbose = TRUE, ...)
ES_nvmix(level, qmix, loc = 0, scale = 1, control = list(), verbose = TRUE, ...)

Arguments

• level: $n$-vector of confidence levels.

• qmix: see pnvmix().

• loc: numeric location, see also pnvmix()

• scale: numeric scale, see also pnvmix()

• control: list specifying algorithm specific parameters; see get_set_param().

• verbose: logical

  indicating whether a warning is given if the required precision has not been reached.

• ...: additional arguments (for example, parameters) passed to the underlying mixing distribution when qmix is a character string or function, see also pnvmix()

Returns

VaR_nvmix() and ES_nvmix() return a numeric $n$-vector with the computed risk measures and in case of ES_nvmix() corresponding attributes "abs. error" and "rel. error"(error estimates of the RQMC estimator) and "numiter" (number of iterations).

Details

VaR_nvmix calls qnvmix().

The function ES_nvmix() estimates the expected shortfall using a randomized quasi Monte Carlo procedure by sampling from the mixing variable specified via qmix and and using the identity $\int_k^{\infty} x\phi(x)dx=\phi(k)$ where $\phi(x)$ denotes the density of a standard normal distribution. Algorithm specific paramaters (such as tolerances) can be conveniently passed via the control argument, see get_set_param() for more details.

Author(s)

Erik Hintz, Marius Hofert and Christiane Lemieux

References

Hintz, E., Hofert, M. and Lemieux, C. (2021), Normal variance mixtures: Distribution, density and parameter estimation. Computational Statistics and Data Analysis 157C, 107175.

Hintz, E., Hofert, M. and Lemieux, C. (2022), Multivariate Normal Variance Mixtures in : The Package nvmix. Journal of Statistical Software, tools:::Rd_expr_doi("10.18637/jss.v102.i02") .

See Also

dnvmix(), pnvmix(), qnvmix(), rnvmix(), get_set_param()

Examples

## Example for inverse-gamma mixture (resulting in a t distribution) for
## which the expected shortfall admits a closed formula
set.seed(42) # reproducibility
level <- seq(from = 0.9, to = 0.95, by = 0.01)
df <- 4
## If 'qmix' is provided as string, ES_nvmix() uses the closed formula
ES1 <- ES_nvmix(level, qmix = "inverse.gamma", df = df)
## If 'qmix' is provided as function, the expected shortfall is estimated
ES2 <- ES_nvmix(level, qmix = function(u, df) 1/qgamma(1-u, shape = df/2, rate = df/2),
       df = df)
stopifnot(all.equal(ES1, ES2, tol = 1e-2, check.attributes = FALSE))