Title:	Scenario Weights for Importance Measurement
Version:	1.0.0
Maintainer:	Silvana M. Pesenti <swimpackage@gmail.com>
Description:	An efficient sensitivity analysis for stochastic models based on Monte Carlo samples. Provides weights on simulated scenarios from a stochastic model, such that stressed random variables fulfil given probabilistic constraints (e.g. specified values for risk measures), under the new scenario weights. Scenario weights are selected by constrained minimisation of the relative entropy to the baseline model. The 'SWIM' package is based on Pesenti S.M., Millossovich P., Tsanakas A. (2019) "Reverse Sensitivity Testing: What does it take to break the model" <openaccess.city.ac.uk/id/eprint/18896/> and Pesenti S.M. (2021) "Reverse Sensitivity Analysis for Risk Modelling" https://www.ssrn.com/abstract=3878879.
Depends:	R (≥ 3.5.0)
Imports:	Rdpack (≥ 0.7), Hmisc, nleqslv, reshape2, plyr, ggplot2, stats
RdMacros:	Rdpack
License:	GPL-3
URL:	https://github.com/spesenti/SWIM, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3515274, https://utstat.toronto.edu/pesenti/?page_id=138
BugReports:	https://github.com/spesenti/SWIM/issues
Encoding:	UTF-8
LazyLoad:	true
LazyData:	true
VignetteBuilder:	knitr
RoxygenNote:	7.1.2
Suggests:	testthat, mvtnorm, spelling, Weighted.Desc.Stat, knitr, rmarkdown, bookdown, ggpubr
Language:	en-US
NeedsCompilation:	no
Packaged:	2022-01-09 14:39:55 UTC; silva
Author:	Silvana M. Pesenti [aut, cre], Alberto Bettini [aut], Pietro Millossovich [aut], Andreas Tsanakas [aut], Zhuomin Mao [ctb], Kent Wu [ctb]
Repository:	CRAN
Date/Publication:	2022-01-09 15:02:42 UTC

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

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)

SWIM: A Package for Sensitivity Analysis

Description

The SWIM package provides weights on simulated scenarios from a stochastic model, such that a stressed model component (random variable) fulfil given probabilistic constraints (e.g. specified values for risk measures), under the new scenario weights. Scenario weights are selected by constrained minimisation of the relative entropy or Wasserstein distance to the baseline model.

Details

The SWIM (Scenario Weights for Importance Measurement) package provides weights on simulated scenarios from a stochastic model, such that stressed random variables fulfil given probabilistic constraints (e.g. specified values for risk measures), under the new scenario weights. Scenario weights are selected by constrained minimisation of the relative entropy or Wasserstein distance to the baseline model.

The SWIM package is based on the reverse sensitivity framework developed by (Pesenti et al. 2019) and (Pesenti 2021).

Consider the random vector X = (X1,...,Xn). Let P represent the probability measure under which all simulated scenarios have the same probability. First, take the approach of minimizing the relative entropy. Then, for a random variable Xi, the package solves:

min D(P | Q)

subject to constraints on the distribution of Xi under Q, where D(P | Q) is the Kullback-Leibler divergence (relative entropy) between P and Q.

The approach of minimizing the Wasserstein distance of order 2 proceeds as follows: Let F be the distribution function of the random variable Xi under P, then the package solves

argmin_{G} W_{2}(G, F)

subject to constraints on G, W_{2}(G, F) is the 2-Wasserstein distance between G and F. The solution to the above minimisation problem is the distribution of Xi under Q. The current implementation of the Wasserstein approach is based on Kernel density estimation with Gaussian kernels.

For both approaches, the scenario weights are then formed via the Radon-Nikodym derivative dQ / dP. The weighting generates a model for which the joint distribution of (X1,...,Xn) is stressed.

Different elements of X can be understood as inputs or outputs of a model. For example, consider a model Y = g(Z) with input vector Z = (Z1,...,Z(n-1)). One can then identify X1 = Y and X2 = Z1,...,Xn = Z(n-1). Subsequently, the user of the SWIM package can stress the model output or any of the inputs, measuring the resulting impact on the distributions of other variables.

Stresses for Relative Entropy Minimization

Scenario weights for the following stresses are provided:

Stresses for Wasserstein Distance Minimization

Scenario weights for the following stresses are provided:

A SWIM object

A SWIM object is generated by applying a stress function subject to a relative entropy minimisation. An object of class SWIM contains a list of:

• x, a data.frame containing realisations of a random vector;

• new_weights, a list, each component corresponds to a different stress and is either a vector of scenario weights or a function, that applied to the kth column of x, generates the vectors of scenario weights;

• type: a list, each component corresponds to a different stress and specifies the type of the stress;

• specs, a list, each component corresponds to a different stress and contains a list with the specifications of what has been stressed. Specifications depend on the type of stress:

  • type = "VaR": k, the column of x on which the stress is applied to; alpha, the level of the stressed VaR; q, the stressed VaR at level alpha.

  • type = "VaR ES": k, the column of x on which the stress is applied to; alpha, the level of the stressed VaR and ES; q, the stressed VaR at level alpha.

  • type = "mean": k, the columns of x on which the stress is applied to; new_means, the stressed means.

  • type = "mean sd": k, the columns of x on which the stress is applied to; new_means, the stressed means; new_sd, the stressed standard deviations. s, the stressed ES at level alpha.

  • type = "moment": f, the list of functions, that, applied to x, constitute the moment constraints; k, the columns of x on which each function in f operates on; m, the stressed moments of f(x).

  • type = "prob": k, the column of x on which the stress is applied to; lower, the left endpoints of the intervals; upper, the right endpoints of the intervals; prob, stressed probabilities corresponding to the intervals defined through lower and upper.

  • type = "user": k, the column of x on which the stress is applied to.

A SWIMw object

A SWIMw object is generated by applying a stress function subject to a Wasserstein minimisation. The Wasserstein minimisation approach assumes that all model components, (random variables) are continuously distributed. If only the stressed model component is continuously distributed, the SWIMw stress should be converted to a SWIM object, see convert_SWIMw_to_SWIM. An object of class SWIMw contains a list of:

• x, a data.frame containing realisations of a random vector;

• new_weights: a list, each component corresponds to a different stress and is either a vector of scenario weights or a function, that applied to the kth column of x, generates the vectors of scenario weights;

• type: a list, each component corresponds to a different stress and specifies the type of the stress;

• h: a list, each component corresponds to a different stress and specifies the bandwidth;

• u: a list, each component corresponds to a different stress and is a vector containing the gridspace on [0, 1];

• lam: a list, each component corresponds to a different stress and is vector containing the lambda's of the optimized model;

• str_fY: a list, each component corresponds to a different stress and is a function defining the densities of the stressed component;

• str_FY: a list, each component corresponds to a different stress and is a function defining the distribution of the stressed component;

• str_FY_inv: a list, each component corresponds to a different stress and is a function defining the quantiles of the stressed component;

• gamma: a list, each component corresponds to a different stress and is a function defining the risk measure;

• specs: a list, each component corresponds to a different stress and contains a list with the specifications of what has been stressed. Specifications depend on the type of stress:

  • type = "RM": k, the column of x on which the stress is applied to; alpha, the level of the RM; q, the stressed RM at level alpha.

  • type = "mean sd": k, the columns of x on which the stress is applied to; new_mean, the stressed mean; new_sd, the stressed standard deviation.

  • type = "RM mean sd": k, the column of x on which the stress is applied to; alpha, the level of the stressed RM; q, the stressed RM at level alpha; new_mean, the stressed mean; new_sd, the stressed standard deviation.

  • type = "HARA RM": k, the column of x on which the stress is applied to; alpha, the level of the stressed RM; q, the stressed RM at level alpha; a a parameter of the HARA utility function; b, a parameter of the HARA utility function; eta a parameter of the HARA utility function; hu, the stressed HARA utility with parameters a, b, and eta.

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.

See Also

See get_data for extracting the data, x; get_weights for extracting the scenario weights, new_weights; get_weightsfun for extracting the functions generating the scenario weights; and get_specs for extracting the specifications of the stress on an object of class SWIM.

Distribution Function of a Stressed Model

Description

Provides the empirical distribution function of a stressed SWIM model component (random variable) or KDE distribution of a stressed SWIMw model component under the scenario weights.

Usage

cdf(object, xCol = 1, wCol = 1)

Arguments

Value

The empirical or KDE distribution function (a function) of the xCol component of the stressed model with weights wCol. The empirical distribution function can be evaluated at a vector.

Author(s)

Silvana M. Pesenti, Zhuomin Mao

See Also

See plot_cdf for plotting the empirical or KDE distribution function of the stressed model and quantile_stressed for sample quantiles 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)
grid <- seq(min(x$normal), max(x$normal), length.out = 5)
## stressed empirical distribution function
cdf(res1, xCol = 1, wCol = 1)(grid)
## baseline empirical distribution function
ecdf(x$normal)(grid)

Empirical Distribution Function of a Stressed Model

Description

Provides the empirical distribution of a stressed model component (random variable) under the scenario weights

Usage

cdf_stressed(object, xCol = 1, wCol = "all", grid, base = FALSE)

Arguments

Details

The cdf_stressed returns the values of the empirical distribution function of the xCol model components for weights wCol. In contrast, the cdf function returns a function, analogous to the ecdf from the base package. The function cdf_stressed is the cdf function applied to grid.

Value

A matrix containing the empirical distribution function applied to grid of the xCol components of the stressed model with weights wCol.

Author(s)

Kent Wu

See Also

See cdf for the empirical distribution function of a stressed model component and quantile_stressed for sample quantiles 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)
grid <- seq(min(x$normal), max(x$normal), length.out = 5)
## stressed empirical distribution function
cdf_stressed(res1, xCol = 1, wCol = "all", grid = grid)

Convert SWIMw to SWIM

Description

Convert SWIMw to SWIM

Usage

convert_SWIMw_to_SWIM(object)

Arguments

Details

Convert a SWIMw object into a SWIM object

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";

• specs, a list, each component corresponds to a different stress and contains k, alpha and q.

See SWIM for details.

Author(s)

Kent Wu

Examples

## Not run: 
set.seed(0)
x <- as.data.frame(cbind(
  "normal" = rnorm(1000),
  "gamma" = rgamma(1000, shape = 2)))
res1 <- stress_wass(type = "RM", x = x,
  alpha = 0.9, q_ratio = 1.05)
convert_SWIMw_to_SWIM(res1)

## End(Not run)

Correlation of a Stressed Model

Description

Provides the correlation of stressed model components (random variable) under the scenario weights.

Usage

cor_stressed(
  object,
  xCol = c(1, 2),
  wCol = "all",
  method = "Pearson",
  base = FALSE
)

Arguments

Details

cor_stressed: The correlation coefficient of stressed model components, subject to the calculated scenario weights.

Value

cor_stressed returns a list of correlation matrices corresponding to different stresses. Entries of the matrices denote correlation coefficients between stressed model components specified by xCol.

Author(s)

Kent Wu

See Also

See var_stressed and sd_stressed compute stressed variance and standard deviations under the scenario weights, respectively.

See cor for unweighted correlations between model components, while cor_stressed return correlations between stressed model components

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 correlation
cor_stressed(res1, xCol = c(1, 2), wCol = 1, base=TRUE)

Credit data set

Description

A dataset containing total aggregate losses from three sub-portfolios, generated through a binomial credit model.

Usage

credit_data

Format

A data frame with 100,000 rows and 7 variables:

L

total aggregate loss of a portfolio consisting of three homogeneous sub-portfolios L1, L2 and L3

L1

aggregate loss of sub-portfolio 1

L2

aggregate loss of sub-portfolio 2

L3

aggregate loss of sub-portfolio 3

H1

(conditional) default probability of sub-portfolio 1

H2

(conditional) default probability of sub-portfolio 2

H3

(conditional) default probability of sub-portfolio 3

Source

For a detailed case study of the credit data set using SWIM see
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.

Extracting from a Stressed Model

Description

Extracting the data (realisations of the stochastic model), the scenario weights, the functions generating the scenario weights, or the specifications of the stress from an object of class SWIM or SWIMw.

Usage

get_data(object, xCol = "all")

get_weights(object, wCol = "all")

get_weightsfun(object, wCol = "all")

get_specs(object, wCol = "all")

summary_weights(object, wCol = "all")

Arguments

Value

get_data: A data.frame containing the realisations of the stochastic model on which the object is based.

get_weights: A data.frame containing the scenario weights of the object. Columns corresponds to different stresses.

get_weightsfun: A list containing functions, which, when applied to a column of the data, generate the scenario weights of the object. The corresponding stressed columns can be obtained via get_specs.

Use get_weights if the SWIM object only contains scenario weights and not a list of functions.

get_specs: A data.frame containing specifications of the stresses with each row corresponding to a different stress. Only a selection of the specifications is returned; however, all input variables are stored in the object. See also SWIM.

summary_weights: print a list containing summary statistics of the stresses with each element being a table for a different stress. The summary statistics include minimum, maximum, standard deviation, Gini coefficient, entropy and effective sample size.

Gini coefficient uses the formula \frac{\sum_{i=1}^{n} \sum_{j=1}^{n}\left|x_{i}-x_{j}\right|}{2 n^{2} \bar{x}}.

Effective Sample Size is equal to n / (1+Var(W)), see Equation (9.13) in Owen, Art B. "Monte Carlo theory, methods and examples." (2013).

Functions

• get_data: extracting data.

• get_weights: extracting scenario weights.

• get_weightsfun: extracting weight functions.

• get_specs: extracting information of the stress.

• summary_weights: extracting summary statistics of scenario weights.

Author(s)

Silvana M. Pesenti

See Also

SWIM

Examples

## continuing example in stress_VaR
set.seed(0)
x <- as.data.frame(cbind(
     "normal" = rnorm(1000), 
     "gamma" = rgamma(1000, shape = 2)))
  res1 <- stress(type = "VaR", x = x, 
                 alpha = 0.9, q_ratio = 1.05, k = 1)

## returning the underlying data
all(get_data(res1) == x)
 ## the scenario weights
w <- get_weights(res1) 
get_weightsfun(res1)
get_specs(res1)
  
## now add a stress on the means of both variables
res1 <- stress(type = "mean", x = res1, k = 1:2, new_means = c(0.5,1.5))
get_specs(res1)
## the required moments for a stress of type "mean" are not displayed 
## the type of stress and the specs for the second stress can be 
## extracted directly from the SWIM object.
res1$type[[2]]
res1$specs[[2]]
                                             
                                                              
                                                                                                

Importance Ranking for a Stressed Model

Description

Provides the importance ranks of the components (random variables) of a stressed model for different sensitivity measures.

Usage

importance_rank(
  object,
  xCol = "all",
  wCol = "all",
  type = c("Gamma", "Wasserstein", "reverse", "all"),
  f = NULL,
  k = NULL,
  s = NULL
)

Arguments

Details

For the definition of the sensitivity measures (type), see sensitivity.

Value

A data.frame containing the importance ranks of the stressed model for different sensitivity measures. Small values correspond to large sensitivities. Different rows correspond to different random variables. The first two rows specify the stress and type of the sensitivity measure on which the ranking is calculated.

Author(s)

Silvana M. Pesenti

See Also

See sensitivity for the values of the sensitivity measures, plot_sensitivity for plotting sensitivity measures and summary for a summary statistic of a stressed model.

Examples

## example with a stress on VaR
set.seed(0)
x <- as.data.frame(cbind(
  "log-normal" = rlnorm(1000),
  "gamma" = rgamma(1000, shape = 2)))
res1 <- stress(type = "VaR", x = x,
  alpha = c(0.9, 0.95), q_ratio = 1.05)

importance_rank(res1, wCol = 1:2, type = "Gamma")
## sensitivity of log-transformed data
importance_rank(res1, wCol = 1, type = "all",
  f = list(function(x)log(x), function(x)log(x)), k = list(1, 2))

Mean of a Stressed Model

Description

Provides the mean of stressed model components (random variables) under the scenario weights.

Usage

mean_stressed(object, xCol = "all", wCol = "all", base = FALSE)

Arguments

Details

mean_stressed: Sample mean of chosen stressed model components, subject to the calculated scenario weights.

Value

A matrix containing the means of the xCol components of the stressed model with weights wCol.

Author(s)

Kent Wu

See Also

See var_stressed and sd_stressed compute stressed variance and standard deviations under the scenario weights, respectively.

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 mean
mean_stressed(res1, xCol = "all", wCol = "all", base = TRUE)

Merging Two Stressed Models

Description

This function is a method for an object of class SWIM.

Usage

## S3 method for class 'SWIM'
merge(x, y, ...)

Arguments

Details

Merges two objects of class SWIM, that are based on the same data.

Value

An object of class SWIM containing:

• x, a data.frame containing the data;

• new_weights, a list, each component corresponds to a different stress and is either a vector of scenario weights or a function, that applied to a column of x, generates the vectors of scenario weights;

• type, a list, each component corresponds to a different stress and specifies the type of the stress;

• specs, a list, each component corresponds to a different stress and contains a list with the specifications of what has been stressed.

See SWIM for details.

Author(s)

Silvana M. Pesenti

Merging Two Stressed Models

Description

This function is a method for an object of class SWIMw.

Usage

## S3 method for class 'SWIMw'
merge(x, y, ...)

Arguments

Details

Merges two objects of class SWIMw, that are based on the same data.

Value

A SWIMw object containing:

• x, a data.frame containing the data;

• h, bandwidths;

• u, vector containing the gridspace on [0, 1]

• lam, vector containing the lambda's of the optimized model

• str_fY, function defining the densities of the stressed component;

• str_FY, function defining the distribution of the stressed component;

• str_FY_inv, function defining the quantiles of the stressed component;

• gamma, function defining the risk measure;

• 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, specifies the stress type

• specs, a list, each component corresponds to a different stress and contains a list with the specifications of what has been stressed.

See SWIM for details.

Author(s)

Zhuomin Mao

Plotting the Distribution Functions of a Stressed Model

Description

Plots the empirical distribution function of a stressed SWIM model component (random variable) or KDE distribution function of a stressed SWIMw model component under the scenario weights.

Usage

plot_cdf(
  object,
  xCol = 1,
  wCol = "all",
  base = FALSE,
  n = 500,
  x_limits,
  y_limits,
  displ = TRUE
)

Arguments

Value

If displ = TRUE, a plot displaying the empirical or KDE distribution function of the stochastic model under the scenario weights.

If displ = FALSE, a data.frame for customised plotting with ggplot. The data.frame contains the columns: the column, xCol, of the data of the stressed model, stress (the stresses) and value (the values).
Denote by res the return of the function call, then ggplot can be called via:

ggplot(res, aes(x = res[ ,1], w = value))

+ stat_{ecdf}(aes(color = factor(stress)), n = n).

Note that the ggplot2 default of stat_ecdf does not take weight as an aesthetic. We use the workaround by Nicolas Woloszko, see Note below.

Note

This function is based on the ggplot stat_ecdf function. However, the stat_ecdf does not allow for specifying weights, thus the function is based on the workaround by Nicolas Woloszko, see https://github.com/NicolasWoloszko/stat_ecdf_weighted.

Author(s)

Silvana M. Pesenti, Zhuomin Mao

See Also

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

Examples

## example with a stress on VaR
set.seed(0)
x <- as.data.frame(cbind(
  "normal" = rnorm(10 ^ 5),
  "gamma" = rgamma(10 ^ 5, shape = 2)))
res1 <- stress(type = "VaR", x = x,
  alpha = c(0.75, 0.95), q_ratio = 1.15)
plot_cdf(res1, xCol = 1, wCol = 1:2, base = TRUE)
plot_cdf(res1, xCol = 1, wCol = 1:2, base = TRUE,
  x_limits = c(0, 5), y_limits = c(0.5, 1))

Plotting Histograms of a Stressed Model

Description

Plots the histogram of a stressed model component (random variable) under the scenario weights.

Usage

plot_hist(
  object,
  xCol = 1,
  wCol = "all",
  base = FALSE,
  x_limits,
  displ = TRUE,
  binwidth,
  displLines = FALSE
)

Arguments

Value

If displ = TRUE, a histogram of the stochastic model under the scenario weights.

If displ = FALSE, a data.frame for customised plotting with ggplot. The data.frame contains the columns: the column, xCol, of the data of the stressed model, stress (the stresses) and value (the values).
Denote by res the return of the function call, then ggplot can be called via:

ggplot(res, aes(x = res[ ,1], y = ..density.., weight = value)))

+ geom_{freqpoly}(binwidth, aes(color = factor(stress))).

See Also

See cdf and plot_cdf for values and plotting of the empirical distribution function of a stressed model, respectively, and quantile_stressed for sample quantiles of a stressed model.

Examples

## example with a stress on VaR
set.seed(0)
x <- data.frame("gamma" = rgamma(10^5, shape = 2))
res1 <- stress(type = "VaR", x = x,
  alpha = c(0.75, 0.95), q_ratio = 1.1)
plot_hist(res1, xCol = "gamma", wCol = 1:2, base = TRUE, binwidth = 0.4)
plot_hist(res1, xCol = "gamma", wCol = 1:2, base = TRUE, binwidth = 0.4, displLines = TRUE)

Plotting Quantile Functions of a Stressed Model

Description

Plots the empirical quantile function of a stressed SWIM model component (random variable) or KDE quantile function of a stressed SWIMw model component under the scenario weights.

Usage

plot_quantile(
  object,
  xCol = 1,
  wCol = "all",
  base = FALSE,
  n = 500,
  x_limits,
  y_limits,
  displ = TRUE
)

Arguments

Value

If displ = TRUE, a plot displaying the empirical or KDE quantile function of the stochastic model under the scenario weights.

If displ = FALSE, a data.frame for customised plotting with ggplot. The data.frame contains the following columns: grid, the grid points to plot the quantiles, stress (the stresses) and value (the quantile values).
Denote by res the return of the function call, then ggplot can be called via:

ggplot(res, aes(x = res[ ,1], y = value))

+ geom_lines(aes(color = factor(stress))).

Author(s)

Silvana M. Pesenti, Zhuomin Mao

See Also

See quantile_stressed for sample quantiles of a stressed model and plot_cdf for plotting empirical or KDE distribution functions under scenario weights.

Examples

## example with a stress on VaR
set.seed(0)
x <- as.data.frame(cbind(
  "normal" = rnorm(10 ^ 5),
  "gamma" = rgamma(10 ^ 5, shape = 2)))
res1 <- stress(type = "VaR", x = x,
  alpha = c(0.75, 0.95), q_ratio = 1.15)
plot_quantile(res1, xCol = 1, wCol = 1:2, base = TRUE)
plot_quantile(res1, xCol = 1, wCol = 1:2, base = TRUE, x_limits = c(0.8, 1), 
              y_limits = c(0, 5))

Plotting Sensitivities of a Stressed Model

Description

Plots the sensitivity measures for components (random variables) of a stochastic model under the scenario weights.

Usage

plot_sensitivity(
  object,
  xCol = "all",
  wCol = "all",
  type = c("Gamma", "Kolmogorov", "Wasserstein", "reverse", "all"),
  f = NULL,
  k = NULL,
  s = NULL,
  displ = TRUE,
  p = 1
)

Arguments

Details

For the definition of the sensitivity measures (type), see sensitivity.

Note that the Kolmogorov distance is the same for all inputs under the same stress for a SWIM object. Thus, it should only be used to compare different stresses, not individual components.

Value

If displ = TRUE, a plot displaying the sensitivity measures of the stochastic model under the scenario weights. If displ = FALSE, a data.frame for customised plotting with ggplot. The data.frame contains the columns: stress (the stresses), type (the types of sensitivity), X_all (the random variables), value (the values of the sensitivities).
Denote by result the return of the function call, then ggplot can be called via:

ggplot(result, aes(x = X_{all}, y = value))

+ geom_{point}(aes(color = factor(stress), shape = type)).

Author(s)

Silvana M. Pesenti

See Also

See sensitivity for the values of the sensitivity measures of a stressed model and importance_rank for ranking of random variables according to their sensitivities.

Examples

## Consider the portfolio Y = X1 + X2 + X3 + X4 + X5,
## where (X1, X2, X3, X4, X5) are correlated normally
## distributed with equal mean and different standard deviations,
## see the README for further details.


set.seed(0)
SD <- c(70, 45, 50, 60, 75)
Corr <- matrix(rep(0.5, 5 ^ 2), nrow = 5) + diag(rep(1 - 0.5, 5))
if (!requireNamespace("mvtnorm", quietly = TRUE))
   stop("Package \"mvtnorm\" needed for this function
   to work. Please install it.")
x <- mvtnorm::rmvnorm(10 ^ 5,
   mean =  rep(100, 5),
   sigma = (SD %*% t(SD)) * Corr)
data <- data.frame(rowSums(x), x)
names(data) <- c("Y", "X1", "X2", "X3", "X4", "X5")
rev.stress <- stress(type = "VaR", x = data,
   alpha = c(0.75, 0.9), q_ratio = 1.1, k = 1)

sensitivity(rev.stress, type = "all")
plot_sensitivity(rev.stress, xCol = 2:6, type = "Gamma")
plot_sensitivity(rev.stress, xCol = 6, wCol = 1, type = "all")

Plotting the scenario weights of a Stressed Model

Description

Plots the scenario weights of a stressed model against a model component.

Usage

plot_weights(
  object,
  xCol = 1,
  wCol = "all",
  n,
  x_limits,
  y_limits,
  displ = TRUE
)

Arguments

Value

If displ = TRUE, a plot displaying the scenario weights of a stochastic model against a model component.

If displ = FALSE, a data.frame for customised plotting with ggplot. The data.frame contains the following columns: grid, the grid points to plot the quantiles, stress (the stresses) and value (the quantile values).
Denote by res the return of the function call, then ggplot can be called via:

ggplot(res, aes(x = res[ ,1], y = value))

+ geom_lines(aes(color = factor(stress))).

See Also

See plot_quantile for plotting sample quantiles of a stressed model and plot_cdf for plotting empirical distribution functions.

Examples

 
## example with a stress with \code{credit_data} data set:
data("credit_data")
## two stresses in VaR
model_stress <- stress_VaR(credit_data, alpha = c(0.9, 0.95), q_ratio = 1.1, k =1) 
plot_weights(model_stress, xCol = "L", wCol = 1:2)

## additional stress on VaR and ES
model_stress <- stress_VaR_ES(model_stress, alpha = 0.9, q_ratio = 1.1, s_ratio = 1.2, k =1) 
plot_weights(model_stress, xCol = "L", wCol = "all", n = 1000, x_limits = c(0, 3500), 
             y_limits = c(0, 10))
             

Sample Quantiles of a Stressed Model

Description

Provides sample quantiles for components (random variables) of a stochastic model, corresponding to distribution functions under the scenario weights.

Usage

quantile_stressed(
  object,
  probs = seq(0, 1, 0.25),
  xCol = "all",
  wCol = 1,
  type = c("quantile", "(i-1)/(n-1)", "i/(n+1)", "i/n"),
  base = FALSE
)

Arguments

Details

type defines the choice of algorithm used for calculating the estimate of the sample quantiles. "quantile" corresponds to the default interpolation used in quantile. Further options are "(i-1)/(n-1)", "i/(n+1)", "i/n" the inverse of the empirical distribution function, using, respectively, (wt - 1)/T, wt/(T+1), wt/T, where wt is the cumulative weight and T the total weight (usually total sample size). See wtd.quantile for further details on type, on which quantile_stressed is based. type is ignored for when evaluating quantiles for SWIMw objects.

Value

Returns a matrix with estimates of the distribution quantiles at the probabilities, probs, under the scenario weights wCol.

Author(s)

Silvana M. Pesenti, Zhuomin Mao

See Also

See wtd.quantile on which the function quantile_stressed is based.
See cdf for the empirical 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 sample quantiles  
quantile_stressed(res1, probs = seq(0.9, 0.99, 0.01), wCol = 2)    
    

Rename Stressed Models

Description

Rename Stressed Models

Usage

rename(object, names, k = 1)

Arguments

Details

Get a new SWIM object with desired name

Value

An renamed object of class SWIM containing:

• x, a data.frame containing the data;

• new_weights, a list, each component corresponds to a different stress and is either a vector of scenario weights or a function, that applied to a column of x, generates the vectors of scenario weights;

• type, a list, each component corresponds to a different stress and specifies the type of the stress;

• specs, a list, each component corresponds to a different stress and contains a list with the specifications of what has been stressed.

See SWIM for details.

Author(s)

Kent Wu

Examples

set.seed(0)
x <- as.data.frame(cbind(
  "normal" = rnorm(1000),
  "gamma" = rgamma(1000, shape = 2)))
res1 <- stress(type = "VaR", x = x,
  alpha = 0.9, q_ratio = 1.05)
res1 <- rename(res1, "VaR_09", 1)

Standard Deviation and Variance of a Stressed Model

Description

Provides the standard deviation and variance of stressed model components (random variables) under the scenario weights.

Usage

sd_stressed(object, xCol = "all", wCol = "all", base = FALSE)

var_stressed(object, xCol = "all", wCol = "all", base = FALSE)

Arguments

Details

sd_stressed: The standard deviation of a chosen model component, subject to the calculated scenario weights.

var_stressed: The variance of a chosen stressed model component, subject to the calculated scenario weights.

Value

sd_stressed: Return the standard deviation of the xCol component of the stressed model with weights wCol. The quantity can be evaluated at a vector.

var_stressed: Return the variance of the xCol component of the stressed model with weights wCol. The quantity can be evaluated at a vector.

Functions

• sd_stressed: Sample standard deviation of model components

• var_stressed: Sample variance of model components

Author(s)

Kent Wu

See Also

See mean_stressed for means of stressed model components, and cor_stressed for correlations between stressed model components.

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 standard deviation
sd_stressed(res1, xCol = "all", wCol = "all", base = TRUE)

## stressed variance
var_stressed(res1, xCol = "all", wCol = "all", base = TRUE)

Sensitivities of a Stressed Model

Description

Provides different sensitivity measures that compare the stressed and the baseline model.

Usage

sensitivity(
  object,
  xCol = "all",
  wCol = "all",
  type = c("Gamma", "Kolmogorov", "Wasserstein", "reverse", "all"),
  f = NULL,
  k = NULL,
  s = NULL,
  p = 1
)

Arguments

Details

Provides sensitivity measures that compare the stressed and the baseline model. Implemented sensitivity measures:

1. Gamma, the Reverse Sensitivity Measure, defined for a random variable Y and scenario weights w by

   Gamma = ( E(Y * w) - E(Y) ) / c,

   where c is a normalisation constant such that |Gamma| <= 1, see (Pesenti et al. 2019). Loosely speaking, the Reverse Sensitivity Measure is the normalised difference between the first moment of the stressed and the baseline distributions of Y.

2. Kolmogorov, the Kolmogorov distance, defined for distribution functions F,G by

   Kolmogorov = sup |F(x) - G(x)|.

3. Wasserstein, the Wasserstein distance of order 1, defined for two distribution functions F,G by

   Wasserstein = \int |F(x) - G(x)| dx.

4. reverse, the General Reverse Sensitivity Measure, defined for a random variable Y, scenario weights w, and a function s:R -> R by

   epsilon = ( E(s(Y) * w) - E(s(Y)) ) / c,

   where c is a normalisation constant such that |epsilon| <= 1. Gamma is a special instance of the reverse sensitivity measure when s is the identity function.

If f and k are provided, the sensitivity of the transformed data is returned.

Value

A data.frame containing the sensitivity measures of the stressed model with rows corresponding to different random variables. The first two rows specify the stress and type of the sensitivity measure.

Author(s)

Silvana M. Pesenti, Zhuomin Mao

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.

See Also

See importance_rank for ranking of random variables according to their sensitivities, plot_sensitivity for plotting sensitivity measures and summary for summary statistics of a stressed model.

Examples

## example with a stress on VaR
set.seed(0)
x <- as.data.frame(cbind(
  "log-normal" = rlnorm(1000),
  "gamma" = rgamma(1000, shape = 2)))
res1 <- stress(type = "VaR", x = x,
  alpha = c(0.9, 0.95), q_ratio = 1.05)

sensitivity(res1, wCol = 1, type = "all")
## sensitivity of log-transformed data
sensitivity(res1, wCol = 1, type = "all",
  f = list(function(x)log(x), function(x)log(x)), k = list(1,2))

## Consider the portfolio Y = X1 + X2 + X3 + X4 + X5,
## where (X1, X2, X3, X4, X5) are correlated normally
## distributed with equal mean and different standard deviations,
## see the README for further details.


## Not run: 
set.seed(0)
SD <- c(70, 45, 50, 60, 75)
Corr <- matrix(rep(0.5, 5 ^ 2), nrow = 5) + diag(rep(1 - 0.5, 5))
if (!requireNamespace("mvtnorm", quietly = TRUE))
   stop("Package \"mvtnorm\" needed for this function
   to work. Please install it.")
x <- mvtnorm::rmvnorm(10 ^ 5,
   mean =  rep(100, 5),
   sigma = (SD %*% t(SD)) * Corr)
data <- data.frame(rowSums(x), x)
names(data) <- c("Y", "X1", "X2", "X3", "X4", "X5")
rev.stress <- stress(type = "VaR", x = data,
   alpha = c(0.75, 0.9), q_ratio = 1.1, k = 1)

sensitivity(rev.stress, type = "all")
## sensitivity to sub-portfolios X1 + X2 and X3 + X4
sensitivity(rev.stress, xCol = NULL, type = "Gamma",
  f = rep(list(function(x)x[1] + x[2]), 2), k = list(c(2, 3), c(4, 5)))
plot_sensitivity(rev.stress, xCol = 2:6, type = "Gamma")
importance_rank(rev.stress, xCol = 2:6, type = "Gamma")

## End(Not run)

Stressing Random Variables

Description

Provides weights on simulated scenarios from a baseline stochastic model, such that stressed random variables fulfil given probabilistic constraints (e.g. specified values for risk measures), under the new scenario weights. Scenario weights are selected by constrained minimisation of the relative entropy to the baseline model.

Usage

stress(
  type = c("VaR", "VaR ES", "mean", "mean sd", "moment", "prob", "user"),
  x,
  ...
)

Arguments

Value

An object of class SWIM, see SWIM for details.

Author(s)

Silvana M. Pesenti

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.

See Also

Other stress functions: stress_HARA_RM_w(), stress_RM_mean_sd_w(), stress_RM_w(), stress_VaR_ES(), stress_VaR(), stress_mean_sd_w(), stress_mean_sd(), stress_mean_w(), stress_mean(), stress_moment(), stress_prob(), stress_user(), stress_wass()

Examples

set.seed(0)
x <- as.data.frame(cbind(
  "normal" = rnorm(1000), 
  "gamma" = rgamma(1000, shape = 2)))
res <- stress(type = "VaR", x = x, 
  alpha = 0.9, q_ratio = 1.05)
summary(res)   

Stressing Risk Measure and HARA Utility

Description

Provides weights on simulated scenarios from a baseline stochastic model, such that a stressed model component (random variable) fulfills a constraint on its HARA utility defined by a, b and eta parameter and risk measure defined by a gamma function and evaluated at a given level alpha. Scenario weights are selected by constrained minimisation of the Wasserstein distance to the baseline model.

Usage

stress_HARA_RM_w(
  x,
  alpha = 0.8,
  a,
  b,
  eta,
  q_ratio = NULL,
  q = NULL,
  hu_ratio = NULL,
  hu = NULL,
  k = 1,
  h = 1,
  gamma = NULL,
  names = NULL,
  log = FALSE,
  method = "Nelder-Mead"
)

Arguments

Details

This function implements stresses on distortion risk measures. Distortion risk measures are defined by a square-integrable function gamma where

\int_0^1 gamma(u) du = 1.

The distortion risk measure for some gamma and distribution G is calculated as:

\rho_{gamma}(G) = \int_0^1 \breve(G)(u) gamma(u) du.

Expected Shortfall (ES) is an example of a distortion risk measure. 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 HARA Utility is defined by

u(x) = \frac{1-eta}{eta}(\frac{ax}{1 - eta} + b)^eta

.

Value

A SWIMw object containing:

• x, a data.frame containing the data;

• h, h is a multiple of the Silverman’s rule;

• u, vector containing the gridspace on [0, 1];

• lam, vector containing the lambda's of the optimized model;

• str_fY, function defining the densities of the stressed component;

• str_FY, function defining the distribution of the stressed component;

• str_FY_inv, function defining the quantiles of the stressed component;

• gamma, function defining the risk measure;

• 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 = "HARA RM";

• specs, a list, each component corresponds to a different stress and contains k, alpha, a, b, eta, q, and hu.

See SWIM for details.

Author(s)

Zhuomin Mao

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.

Pesenti SM (2021). “Reverse Sensitivity Analysis for Risk Modelling.” Available at SSRN 3878879.

See Also

Other stress functions: stress_RM_mean_sd_w(), stress_RM_w(), stress_VaR_ES(), stress_VaR(), stress_mean_sd_w(), stress_mean_sd(), stress_mean_w(), stress_mean(), stress_moment(), stress_prob(), stress_user(), stress_wass(), stress()

Examples

## Not run: 
set.seed(0)
x <- as.data.frame(cbind(
  "normal" = rnorm(1000),
  "gamma" = rgamma(1000, shape = 2)))
res1 <- stress_wass(type = "HARA RM", x = x, a=1, b=5, eta=0.5, alpha=0.95,
 q_ratio=1.05, hu_ratio=1.05, k=1)
  summary(res1)

## calling stress_RM_w directly
## stressing "gamma"
res2 <- stress_HARA_RM_w(x = x, a=1, b=5, eta=0.5, alpha=0.95,
 q_ratio=1.05, hu_ratio=1.05, k=2)
summary(res2)

## End(Not run)

Stressing Risk Measure, Mean and Standard Deviation

Description

Provides weights on simulated scenarios from a baseline stochastic model, such that a stressed model component (random variable) fulfils a constraint on its mean, standard deviation, and risk measure defined by a gamma function and evaluated at a given level alpha. Scenario weights are selected by constrained minimisation of the Wasserstein distance to the baseline model.

Usage

stress_RM_mean_sd_w(
  x,
  alpha = 0.8,
  new_means,
  new_sd,
  q_ratio = NULL,
  q = NULL,
  k = 1,
  h = 1,
  gamma = NULL,
  names = NULL,
  log = FALSE,
  method = "Nelder-Mead",
  ...
)

Arguments

Details

This function implements stresses on distortion risk measures. Distortion risk measures are defined by a square-integrable function gamma where

\int_0^1 gamma(u) du = 1.

The distortion risk measure for some gamma and distribution G is calculated as:

\rho_{gamma}(G) = \int_0^1 \breve(G)(u) gamma(u) du.

Expected Shortfall (ES) is an example of a distortion risk measure. 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.

Value

A SWIMw object containing:

• x, a data.frame containing the data;

• h, h is a multiple of the Silverman’s rule;

• u, vector containing the gridspace on [0, 1];

• lam, vector containing the lambda's of the optimized model;

• str_fY, function defining the densities of the stressed component;

• str_FY, function defining the distribution of the stressed component;

• str_FY_inv, function defining the quantiles of the stressed component;

• gamma, function defining the risk measure;

• 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 = "RM mean sd";

• specs, a list, each component corresponds to a different stress and contains k, alpha, q, new_means, and new_sd.

See SWIM for details.

Author(s)

Zhuomin Mao

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.

Pesenti SM (2021). “Reverse Sensitivity Analysis for Risk Modelling.” Available at SSRN 3878879.

See Also

Other stress functions: stress_HARA_RM_w(), stress_RM_w(), stress_VaR_ES(), stress_VaR(), stress_mean_sd_w(), stress_mean_sd(), stress_mean_w(), stress_mean(), stress_moment(), stress_prob(), stress_user(), stress_wass(), stress()

Examples

## Not run: 
set.seed(0)
x <- as.data.frame(cbind(
  "normal" = rnorm(1000),
  "gamma" = rgamma(1000, shape = 2)))
res1 <- stress_wass(type = "RM mean sd", x = x,
  alpha = 0.9, q_ratio = 1.05, new_means=1, new_sd=0.9)
  summary(res1)

## calling stress_RM_w directly
## stressing "gamma"
res2 <- stress_RM_mean_sd_w(x = x, alpha = 0.9,
  q_ratio = 1.05, new_means=2.2, new_sd=1.5, k = 2)
summary(res2)

## End(Not run)

Stressing Risk Measure

Description

Provides weights on simulated scenarios from a baseline stochastic model, such that a stressed model component (random variable) fulfils a constraint on its risk measure defined by a gamma function. The default risk measure is the Expected Shortfall at level alpha.

Usage

stress_RM_w(
  x,
  alpha = 0.8,
  q_ratio = NULL,
  q = NULL,
  k = 1,
  h = 1,
  gamma = NULL,
  names = NULL,
  log = FALSE,
  method = "Nelder-Mead"
)

Arguments

Details

This function implements stresses on distortion risk measures. Distortion risk measures are defined by a square-integrable function gamma where

\int_0^1 gamma(u) du = 1.

The distortion risk measure for some gamma and distribution G is calculated as:

\rho_{gamma}(G) = \int_0^1 \breve(G)(u) gamma(u) du.

Expected Shortfall (ES) is an example of a distortion risk measure. 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.

Value

A SWIMw object containing:

• x, a data.frame containing the data;

• h, h is a multiple of the Silverman’s rule;

• u, vector containing the gridspace on [0, 1];

• lam, vector containing the lambda's of the optimized model;

• str_fY, function defining the densities of the stressed component;

• str_FY, function defining the distribution of the stressed component;

• str_FY_inv, function defining the quantiles of the stressed component;

• gamma, function defining the risk measure;

• 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 = "RM";

• specs, a list, each component corresponds to a different stress and contains k, alpha, and q.

See SWIM for details.

Author(s)

Zhuomin Mao

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.

Pesenti SM (2021). “Reverse Sensitivity Analysis for Risk Modelling.” Available at SSRN 3878879.

See Also

Other stress functions: stress_HARA_RM_w(), stress_RM_mean_sd_w(), stress_VaR_ES(), stress_VaR(), stress_mean_sd_w(), stress_mean_sd(), stress_mean_w(), stress_mean(), stress_moment(), stress_prob(), stress_user(), stress_wass(), stress()

Examples

## Not run: 
set.seed(0)
x <- as.data.frame(cbind(
  "normal" = rnorm(1000),
  "gamma" = rgamma(1000, shape = 2)))
res1 <- stress_wass(type = "RM", x = x,
  alpha = 0.9, q_ratio = 1.05)
  summary(res1)

## calling stress_RM_w directly
## stressing "gamma"
res2 <- stress_RM_w(x = x, alpha = 0.9,
  q_ratio = 1.05, k = 2)
summary(res2)

# dual power distortion with beta = 3
# gamma = beta * u^{beta - 1}, beta > 0 

gamma <- function(u){
  .res <- 3 * u^2
  return(.res)
}
res3 <- stress_wass(type = "RM", x = x, 
  gamma = gamma, q_ratio = 1.05)
summary(res3)

## End(Not run)
 

Stressing Value-at-Risk

Description

Provides weights on simulated scenarios from a baseline stochastic model, such that a stressed model component (random variable) fulfils a constraint on its quantile at a given level, also known as Value-at-Risk (VaR). Scenario weights are selected by constrained minimisation of the relative entropy to the baseline model.

Usage

stress_VaR(
  x,
  alpha,
  q_ratio = NULL,
  q = NULL,
  k = 1,
  names = NULL,
  log = FALSE
)

Arguments

Details

The stressed VaR is the quantile of the chosen model component, subject to the calculated scenario weights. 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).

If one of alpha, q (q_ratio) is a vector, the stressed VaR's of the kth column of x, at levels alpha, are equal to q.

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.

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";

• specs, a list, each component corresponds to a different stress and contains k, alpha and q.

See SWIM for details.

Author(s)

Silvana M. Pesenti

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.

See Also

Other stress functions: stress_HARA_RM_w(), stress_RM_mean_sd_w(), stress_RM_w(), stress_VaR_ES(), stress_mean_sd_w(), stress_mean_sd(), stress_mean_w(), stress_mean(), stress_moment(), stress_prob(), stress_user(), stress_wass(), stress()

Examples

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

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

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

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.

See Also

Other stress functions: stress_HARA_RM_w(), stress_RM_mean_sd_w(), stress_RM_w(), stress_VaR(), stress_mean_sd_w(), stress_mean_sd(), stress_mean_w(), stress_mean(), stress_moment(), stress_prob(), stress_user(), stress_wass(), stress()

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)

Stressing Means

Description

Provides weights on simulated scenarios from a baseline stochastic model, such that stressed model components (random variables) fulfil the mean constraints. Scenario weights are selected by constrained minimisation of the relative entropy to the baseline model.

Usage

stress_mean(x, k, new_means, normalise = TRUE, names = NULL, log = FALSE, ...)

Arguments

Details

The function stress_mean is a wrapper for the function stress_moment. See stress_moment for details on the additional arguments to ... and the underlying algorithm.

Value

A SWIM object containing:

• x, a data.frame containing the data;

• new_weights, a list, each component corresponds to a different stress and is a vector of scenario weights;

• type = "mean";

• specs, a list, each component corresponds to a different stress and contains k and new_means.

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.

See Also

See stress_mean_sd for stressing means and standard deviations jointly, and stress_moment for moment constraints.

Other stress functions: stress_HARA_RM_w(), stress_RM_mean_sd_w(), stress_RM_w(), stress_VaR_ES(), stress_VaR(), stress_mean_sd_w(), stress_mean_sd(), stress_mean_w(), stress_moment(), stress_prob(), stress_user(), stress_wass(), stress()

Examples

set.seed(0)
x <- data.frame(cbind(
  "normal" = rnorm(1000),
  "gamma" = rgamma(1000, shape = 2),
  "beta" = rbeta(1000, shape1 = 2, shape2 = 2)))
## stressing means
res1 <- stress(type = "mean", x = x, k = 1:3,
  new_means = c(1, 1, 0.75))
summary(res1)
res1$specs
## calling stress_mean directly
res2 <- stress_mean(x = x, k = 1:3,
  new_means = c(1, 1, 0.75))
summary(res2)

## See also examples in stress_moment and stress_mean_sd.

Stressing Mean and Standard Deviation

Description

Provides weights on simulated scenarios from a baseline stochastic model, such that stressed model components (random variables) fulfil the mean and standard deviation constraints. Scenario weights are selected by constrained minimisation of the relative entropy to the baseline model.

Usage

stress_mean_sd(
  x,
  k,
  new_means,
  new_sd,
  normalise = TRUE,
  names = NULL,
  log = FALSE,
  ...
)

Arguments

Details

The function stress_mean_sd is a wrapper for the function stress_moment. See stress_moment for details on the additional arguments to ... and the underlying algorithm.

For stressing means only, see stress_mean, for stressing higher moments and functions of moments, see stress_moment.

Value

A SWIM object containing:

• x, a data.frame containing the data;

• new_weights, a list, each component corresponds to a different stress and is a vector of scenario weights;

• type = "mean";

• specs, a list, each component corresponds to a different stress and contains k, new_means and new_sd.

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.

See Also

Other stress functions: stress_HARA_RM_w(), stress_RM_mean_sd_w(), stress_RM_w(), stress_VaR_ES(), stress_VaR(), stress_mean_sd_w(), stress_mean_w(), stress_mean(), stress_moment(), stress_prob(), stress_user(), stress_wass(), stress()

Examples

set.seed(0)
x <- data.frame(cbind(
  "normal" = rnorm(1000),
  "gamma" = rgamma(1000, shape = 2),
  "beta" = rbeta(1000, shape1 = 2, shape2 = 2)))
## stressing mean and sd of column 1
res1 <- stress(type = "mean sd", x = x, k = 1, new_means = 0.1,
  new_sd = 1.1, method = "Newton",
  control = list(maxit = 1000, ftol = 1E-15))
summary(res1)
## calling stress_mean_sd directly
res2 <- stress_mean_sd(x = x, k = 1, new_means = 0.1,
  new_sd = 1.1, method = "Newton",
  control = list(maxit = 1000, ftol = 1E-15))

## See also examples in stress_moment.

Stressing Mean and Standard Deviation

Description

Provides weights on simulated scenarios from a baseline stochastic model, such that a stressed model component (random variable) fulfils a constraint on its mean and standard deviation. Scenario weights are selected by constrained minimisation of the Wasserstein distance to the baseline model.

Usage

stress_mean_sd_w(
  x,
  new_means,
  new_sd,
  k = 1,
  h = 1,
  names = NULL,
  log = FALSE,
  method = "Nelder-Mead",
  ...
)

Arguments

Value

A SWIMw object containing:

• x, a data.frame containing the data;

• h, h is a multiple of the Silverman’s rule;

• u, vector containing the gridspace on [0, 1];

• lam, vector containing the lambda's of the optimized model;

• str_fY, function defining the densities of the stressed component;

• str_FY, function defining the distribution of the stressed component;

• str_FY_inv, function defining the quantiles of the stressed component;

• gamma, function defining the risk measure;

• 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 = "mean sd";

• specs, a list, each component corresponds to a different stress and contains k, new_means and new_sd.

See SWIM for details.

Author(s)

Zhuomin Mao

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.

Pesenti SM (2021). “Reverse Sensitivity Analysis for Risk Modelling.” Available at SSRN 3878879.

See Also

Other stress functions: stress_HARA_RM_w(), stress_RM_mean_sd_w(), stress_RM_w(), stress_VaR_ES(), stress_VaR(), stress_mean_sd(), stress_mean_w(), stress_mean(), stress_moment(), stress_prob(), stress_user(), stress_wass(), stress()

Examples

## Not run: 
set.seed(0)
x <- as.data.frame(cbind(
  "normal" = rnorm(1000),
  "gamma" = rgamma(1000, shape = 2)))
res1 <- stress_wass(type = "mean sd", x = x, k = 1,
  new_means=1, new_sd=0.9)
  summary(res1)

## calling stress_RM_w directly
## stressing "gamma"
res2 <- stress_mean_sd_w(x = x, 
  new_means=2.2, new_sd=1.5, k = 2)
summary(res2)

## End(Not run)

Stressing Mean

Description

Provides weights on simulated scenarios from a baseline stochastic model, such that a stressed model component (random variable) fulfils a constraint on its mean. Scenario weights are selected by constrained minimisation of the Wasserstein distance to the baseline model.

Usage

stress_mean_w(
  x,
  new_means,
  k = 1,
  h = 1,
  names = NULL,
  log = FALSE,
  method = "Nelder-Mead",
  ...
)

Arguments

Value

A SWIMw object containing:

• x, a data.frame containing the data;

• h, h is a multiple of the Silverman’s rule;

• u, vector containing the gridspace on [0, 1];

• lam, vector containing the lambda's of the optimized model;

• str_fY, function defining the densities of the stressed component;

• str_FY, function defining the distribution of the stressed component;

• str_FY_inv, function defining the quantiles of the stressed component;

• gamma, function defining the risk measure;

• 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 = "mean";

• specs, a list, each component corresponds to a different stress and contains k and new_means.

See SWIM for details.

Author(s)

Zhuomin Mao

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.

Pesenti SM (2021). “Reverse Sensitivity Analysis for Risk Modelling.” Available at SSRN 3878879.

See Also

Other stress functions: stress_HARA_RM_w(), stress_RM_mean_sd_w(), stress_RM_w(), stress_VaR_ES(), stress_VaR(), stress_mean_sd_w(), stress_mean_sd(), stress_mean(), stress_moment(), stress_prob(), stress_user(), stress_wass(), stress()

Examples

## Not run: 
set.seed(0)
x <- as.data.frame(cbind(
  "normal" = rnorm(1000),
  "gamma" = rgamma(1000, shape = 2)))
res1 <- stress_wass(type = "mean", x = x, k = 1,
  new_means=1)
  summary(res1)

## calling stress_RM_w directly
## stressing "gamma"
res2 <- stress_mean_w(x = x, 
  new_means=2.2, k = 2)
summary(res2)

## End(Not run)

Stressing Moments

Description

Provides weights on simulated scenarios from a baseline stochastic model, such that stressed model components (random variables) fulfill the moment constraints. Scenario weights are selected by constrained minimisation of the relative entropy to the baseline model.

Usage

stress_moment(
  x,
  f,
  k,
  m,
  normalise = TRUE,
  show = FALSE,
  names = NULL,
  log = FALSE,
  ...
)

Arguments

Details

The moment constraints are given by E^Q( f(x) ) = m, where E^Q denotes the expectation under the stressed model. stress_moment solves the subsequent set of equations with respect to theta, using nleqslv from package nleqslv:

E^Q( f(x) ) = E( f(x) * exp(theta * f(x)) ) = m.

There is no guarantee that the set of equations has a solution, or that the solution is unique. SWIM will return a warning if the termination code provided by nleqslv is different from 1 (convergence has been achieved). It is recommended to check the result of the call to nleqslv using the "show" argument. The user is referred to the nleqslv documentation for further details.

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, each component corresponds to a different stress and is a vector of scenario weights;

• type = "moment";

• specs, a list, each component corresponds to a different stress and contains f, k and m.

See SWIM for details.

The function call will print a message containing the termination code returned by the call to nleqslv and a table with the required and achieved moment, and the absolute and relative error.

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.

See Also

See stress_mean for stressing means and stress_mean_sd for stressing mean and standard deviation jointly.

Other stress functions: stress_HARA_RM_w(), stress_RM_mean_sd_w(), stress_RM_w(), stress_VaR_ES(), stress_VaR(), stress_mean_sd_w(), stress_mean_sd(), stress_mean_w(), stress_mean(), stress_prob(), stress_user(), stress_wass(), stress()

Examples

set.seed(0)
x <- data.frame(cbind(
  "normal" = rnorm(1000),
  "gamma" = rgamma(1000, shape = 2),
  "beta" = rbeta(1000, shape1 = 2, shape2 = 2)))

## stressing covariance of columns 1, 2 while leaving the means unchanged
res1 <- stress_moment(x = x,
  f = list(function(x)x, function(x)x, function(x)x[1] * x[2]),
  k = list(1, 2, c(1, 2)), m = c(0, 2, 0.5),
  method = "Newton", control = list(maxit = 1000, ftol = 1E-10))
## means under the stressed model
summary(res1)
apply(x, 2, stats::weighted.mean, w = get_weights(res1))
## covariance of columns 1,2 under the stressed model
stats::weighted.mean(x[, 1] * x[, 2], w = get_weights(res1))

## stressing jointly the tail probabilities of columns 1, 3
res2 <- stress_moment(x = x,
  f = list(function(x)(x > 1.5), function(x)(x > 0.9)),
  k = list(1, 3), m = c(0.9, 0.9))
summary(res2)
## probabilities under the stressed model
mean((x[, 1] > 1.5) * get_weights(res2))
mean((x[, 3] > 0.9) * get_weights(res2))

Stressing Intervals

Description

Provides weights on simulated scenarios from a baseline stochastic model, such that a stressed model component (random variable) fulfils constraints on probability of disjoint intervals. Scenario weights are selected by constrained minimisation of the relative entropy to the baseline model.

Usage

stress_prob(x, prob, lower = NULL, upper, k = 1, names = NULL, log = FALSE)

Arguments

Details

The intervals are treated as half open intervals, that is the lower endpoint are not included, whereas the upper endpoint are included. If upper = NULL, the intervals are consecutive and prob cumulative.
The intervals defined through lower and upper must be disjoint.

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, generate the vectors of scenario weights. Each component corresponds to a different stress;

• type = "prob";

• specs, a list, each component corresponds to a different stress and contains k, lower, upper and prob.

See SWIM for details.

Author(s)

Silvana M. Pesenti

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.

See Also

Other stress functions: stress_HARA_RM_w(), stress_RM_mean_sd_w(), stress_RM_w(), stress_VaR_ES(), stress_VaR(), stress_mean_sd_w(), stress_mean_sd(), stress_mean_w(), stress_mean(), stress_moment(), stress_user(), stress_wass(), stress()

Examples

set.seed(0)
x <- rnorm(1000)
## consecutive intervals
res1 <- stress(type = "prob", x = x, prob = 0.008, upper = -2.4)
# probability under the stressed model
cdf(res1, xCol = 1)(-2.4)

## calling stress_prob directly
## multiple intervals
res2 <- stress_prob(x = x, prob = c(0.008, 0.06), 
  lower = c(-3, -2), upper = c(-2.4, -1.6))
# probability under the stressed model
cdf(res2, xCol = 1)(c(-2.4, -1.6)) - cdf(res2, xCol = 1)(c(-3, -2))

User Defined Stress

Description

Returns a SWIM object with scenario weights defined by the user.

Usage

stress_user(
  x,
  new_weights = NULL,
  new_weightsfun = NULL,
  k = 1,
  names = NULL,
  log = FALSE
)

Arguments

Value

A SWIM object containing:

• x, a data.frame containing the data;

• new_weights, a list, each component corresponds to a different stress and is either a vector of scenario weights (if new_weights is provided) or (if new_weightsfun is provided) a function, that applied to the kth column of x, generates the vectors of scenario weights;

• type = "user";

• specs, a list, each component corresponds to a different stress and contains k.

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.

See Also

Other stress functions: stress_HARA_RM_w(), stress_RM_mean_sd_w(), stress_RM_w(), stress_VaR_ES(), stress_VaR(), stress_mean_sd_w(), stress_mean_sd(), stress_mean_w(), stress_mean(), stress_moment(), stress_prob(), stress_wass(), stress()

Examples

set.seed(0)
x <- as.data.frame(cbind(
  "normal" = rnorm(1000),
  "gamma" = rgamma(1000, shape = 2)))
res1 <- stress(type = "user", x = x, new_weightsfun = function(x)x ^ 2, k = 1)
## plot user defined weights against the first column of x.
plot(x$normal, get_weights(res1), pch = ".")

Stressing Random Variables Using Wasserstein Distance

Description

Provides weights on simulated scenarios from a baseline stochastic model, such that stressed random variables fulfill given probabilistic constraints (e.g. specified values for risk measures), under the new scenario weights. Scenario weights are selected by constrained minimisation of the Wasserstein Distance to the baseline model.

Usage

stress_wass(type = c("RM", "mean sd", "RM mean sd", "HARA RM"), x, ...)

Arguments

Value

An object of class SWIMw, see SWIM for details.

Author(s)

Zhuomin Mao

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.

See Also

Other stress functions: stress_HARA_RM_w(), stress_RM_mean_sd_w(), stress_RM_w(), stress_VaR_ES(), stress_VaR(), stress_mean_sd_w(), stress_mean_sd(), stress_mean_w(), stress_mean(), stress_moment(), stress_prob(), stress_user(), stress()

Examples

## Not run: 
set.seed(0)
x <- as.data.frame(cbind(
  "normal" = rnorm(1000), 
  "gamma" = rgamma(1000, shape = 2)))
res <- stress_wass(type = "RM", x = x, 
  alpha = 0.9, q_ratio = 1.05)
summary(res)   

## End(Not run)

Summarising Stressed Models

Description

This function is a methods for an object of class SWIM or SWIMw. Provides summary statistics of the stochastic model, stressed using the scenario weights.

Usage

## S3 method for class 'SWIM'
summary(object, ..., xCol = "all", wCol = "all", base = FALSE)

## S3 method for class 'SWIMw'
summary(object, ..., xCol = "all", wCol = "all", base = FALSE)

Arguments

Value

summary.SWIM returns a list with components corresponding to different stresses. Components contain a summary statistic of each column of the data of the SWIM object:

Functions

• summary.SWIM: Summarising Stressed Models

• summary.SWIMw: Summarising Stressed Models

Author(s)

Silvana M. Pesenti

Zhuomin Mao

See Also

summary, SWIM

Examples

     
## Example with the Relative Entropy
## continuing example in stress_VaR 
set.seed(0)
x <- as.data.frame(cbind(
  "normal" = rnorm(1000), 
  "gamma" = rgamma(1000, shape = 2)))
  
res1 <- stress(type = "VaR", x = x, 
  alpha = 0.9, q_ratio = 1.05)
summary(res1, xCol = "normal", base = TRUE) 


     
## Example with the Wasserstein distance 
## Not run: 
resW <- stress_wass(type = "RM", x = x, 
alpha = 0.9, q_ratio = 1.05)
summary(resW, xCol = "normal", base = TRUE) 

## End(Not run)