ES_stressed {SWIM}

Value-at-Risk and Expected Shortfall of a Stressed Model

Description

Provides the Value-at-Risk (VaR) and the Expected Shortfall (ES) for components (random variables) of a stochastic model.

Usage

ES_stressed(
  object,
  alpha = 0.95,
  xCol = "all",
  wCol = 1,
  base = FALSE,
  gamma = NULL
)

VaR_stressed(object, alpha = 0.95, xCol = "all", wCol = 1, base = FALSE)

Arguments

object

A SWIM or SWIMw object.

alpha

Numeric vector, the levels of the stressed VaR and ES (default = 0.95).

xCol

Numeric or character vector, (names of) the columns of the underlying data of the object (default = "all").

wCol

Numeric, the column of the scenario weights of the object (default = 1).

base

Logical, if TRUE, statistics under the baseline are also returned (default = "FALSE").

gamma

Function that defines the gamma of the risk measure. If null, the Expected Shortfall (ES) will be used.

Details

ES_stressed: The ES of a stressed model is the ES of a chosen stressed model component, subject to the calculated scenario weights. The ES at level alpha of a stressed model component is given by:

ES_{alpha} = 1 / (1 - alpha) * \int_{alpha}^1 VaR_u^W d u,

where VaR_u^W is the VaR of the stressed model component, defined below.

VaR_stressed: The VaR of a model is the VaR (quantile) of a chosen stressed model component, subject to the calculated scenario weights. The VaR at level alpha of a stressed model component with stressed distribution function F^W is defined as its left-quantile at alpha:

VaR_{alpha}^W = F^{W,-1}(alpha).

The function VaR_stressed provides the empirical quantile, whereas the function quantile_stressed calculates quantiles of model components with different interpolations.

Value

ES_stressed: Returns a matrix with the empirical or KDE ES's at level alpha of model components specified in xCol, under the scenario weights wCol.

VaR_stressed: Returns a matrix with the empirical or KDE VaR's at level alpha of model components specified in xCol, under the scenario weights wCol.

Functions

• ES_stressed: Expected Shortfall of a stressed model

• VaR_stressed: Value-at-Risk of a stressed model.

Author(s)

Silvana M. Pesenti, Zhuomin Mao

See Also

See quantile_stressed for quantiles other than the empirical quantiles and cdf for the empirical or KDE distribution function of a stressed model.

Examples

## example with a stress on VaR
set.seed(0)
x <- as.data.frame(cbind(
  "normal" = rnorm(1000),
  "gamma" = rgamma(1000, shape = 2)))
res1 <- stress(type = "VaR", x = x,
  alpha = c(0.9, 0.95), q_ratio = 1.05)
## stressed ES
quantile_stressed(res1, probs = seq(0.9, 0.99, 0.01),
                    xCol = 1, wCol = 2, type = "i/n")
quantile(x[, 1],  probs = seq(0.9, 0.99, 0.01), type = 1)
VaR_stressed(res1, alpha = seq(0.9, 0.99, 0.01), xCol = 1,
                    wCol = 2, base = TRUE)

## the ES of both model components under stress one
ES_stressed(res1, alpha = seq(0.9, 0.99, 0.01), xCol = "all",
                    wCol = 1)
## the ES of both model components under stress two
ES_stressed(res1, alpha = seq(0.9, 0.99, 0.01), xCol = "all",
                    wCol = 2)