R: Stressing Value-at-Risk and Expected Shortfall

stress_VaR_ES {SWIM}

R Documentation

Stressing Value-at-Risk and Expected Shortfall

Description

Provides weights on simulated scenarios from a baseline stochastic model, such that a stressed model component (random variable) fulfils a constraint on its Value-at-Risk (VaR) and Expected Shortfall (ES) risk measures, both evaluated at a given level. Scenario weights are selected by constrained minimisation of the relative entropy to the baseline model.

Usage

stress_VaR_ES(
  x,
  alpha,
  q_ratio = NULL,
  s_ratio = NULL,
  q = NULL,
  s = NULL,
  k = 1,
  normalise = FALSE,
  names = NULL,
  log = FALSE
)

Arguments

`x`	A vector, matrix or data frame containing realisations of random variables. Columns of `x` correspond to random variables; OR A `SWIM` object, where `x` corresponds to the underlying data of the `SWIM` object.
`alpha`	Numeric vector, the levels of the stressed VaR.
`q_ratio`	Numeric vector, the ratio of the stressed VaR to the baseline VaR. If `alpha` and `q_ratio` are vectors, they must have the same length.
`s_ratio`	Numeric, vector, the ratio of the stressed ES to the baseline ES. If `q` (`q_ratio`) and `s_ratio` are vectors, they must have the same length.
`q`	Numeric vector, the stressed VaR at level `alpha`. If `alpha` and `q` are vectors, they must have the same length.
`s`	Numeric, vector, the stressed ES at level `alpha`. If `q` and `s` are vectors, they must have the same length.
`k`	Numeric, the column of `x` that is stressed `(default = 1)`.
`normalise`	Logical. If true, values of the columns to be stressed are linearly normalised to the unit interval.
`names`	Character vector, the names of stressed models.
`log`	Boolean, the option to print weights' statistics.

Details

The VaR at level alpha of a random variable with distribution function F is defined as its left-quantile at alpha:

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

The ES at level alpha of a random variable with distribution function F is defined by:

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

The stressed VaR and ES are the risk measures of the chosen model component, subject to the calculated scenario weights. If one of alpha, q, s (q_ratio, s_ratio) is a vector, the stressed VaR's and ES's of the kth column of x, at levels alpha, are equal to q and s, respectively.

The stressed VaR specified, either via q or q_ratio, might not equal the attained empirical VaR of the model component. In this case, stress_VaR will display a message and the specs contain the achieved VaR. Further, ES is then calculated on the bases of the achieved VaR.

Normalising the data may help avoiding numerical issues when the range of values is wide.

Value

A SWIM object containing:

x, a data.frame containing the data;
new_weights, a list of functions, that applied to the kth column of x, generates the vectors of scenario weights. Each component corresponds to a different stress;
type = "VaR ES";
specs, a list, each component corresponds to a different stress and contains k, alpha, q and s.

See SWIM for details.

References

Pesenti SM, Millossovich P, Tsanakas A (2019). “Reverse sensitivity testing: What does it take to break the model?” European Journal of Operational Research, 274(2), 654–670.

Pesenti S BAMPTA (2020). “Scenario Weights for Importance Measurement (SWIM) - An R package for sensitivity analysis.” Annals of Actuarial Science 15.2 (2021): 458-483. Available at SSRN: https://www.ssrn.com/abstract=3515274.

Csiszar I (1975). “I-divergence geometry of probability distributions and minimization problems.” The Annals of Probability, 146–158.

Examples

set.seed(0)
x <- as.data.frame(cbind(
  "normal" = rnorm(1000),
  "gamma" = rgamma(1000, shape = 2)))
res1 <- stress(type = "VaR ES", x = x,
  alpha = c(0.9, 0.95), q_ratio = 1.05, s_ratio = 1.08)

## calling stress_VaR_ES directly
## stressing "gamma"
res2 <- stress_VaR_ES(x = x, alpha = 0.9,
  q_ratio = 1.03, s_ratio = c(1.05, 1.08), k = 2)
get_specs(res2)
summary(res2)

[Package SWIM version 1.0.0 Index]