The SWIM package provides weights on simulated scenarios from a stochastic model, such that stressed model components (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.
You can install the SWIM package from GitHub with:
The SWIM package provides sensitivity analysis tools for stressing model components (random variables). Implemented stresses are:
| R functions | Stress |
|---|---|
| stress_VaR | VaR risk measure, a quantile |
| stress_VaR_ES | VaR and ES risk measures |
| stress_mean | means |
| stress_mean_sd | means and standard deviations |
| stress_mean_moment | moments, functions of moments |
| stress_prob | probabilities of intervals |
| stress_user | user defined scenario weights |
Implemented functions allow to graphically display the change in the probability distributions under different stresses and the baseline model as well as calculating sensitivity measures.
Consider a portfolio Y = X1 + X2 + X3 + X4 + X5, where (X1, X2, X3, X4, X5) are correlated normally distributed with equal mean and different standard deviations. We stress the VaR (quantile) of the portfolio loss Y at levels 0.75 and 0.9 with an increase of 10%.
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)Summary statistics of the baseline and the stressed model can be obtained via the summary method.
summary(rev.stress, base = TRUE)
#> $base
#> Y X1 X2 X3
#> mean 500.184291828 100.135056213 100.01338838 99.932103379
#> sd 232.130670252 69.794259775 44.93395000 49.830325367
#> skewness 0.001027402 -0.005813524 -0.00128250 0.006633601
#> ex kurtosis -0.042151894 -0.026753965 -0.02114883 -0.011999996
#> 1st Qu. 342.560075159 53.236729959 69.70386870 66.231008832
#> Median 500.448531178 100.164130552 100.05638434 100.111395770
#> 3rd Qu. 657.218752825 147.329019975 130.30594874 133.511595023
#> X4 X5
#> mean 99.978360151 100.125383710
#> sd 59.851583021 74.759264648
#> skewness -0.006786036 0.008161327
#> ex kurtosis -0.021171370 -0.001472384
#> 1st Qu. 59.451872727 49.718118990
#> Median 100.233169690 99.936799228
#> 3rd Qu. 140.330428942 150.802559421
#>
#> $`stress 1`
#> Y X1 X2 X3
#> mean 534.02818733 108.2030353 104.85033601 105.38314950
#> sd 245.47309988 72.3243323 46.35055592 51.48309275
#> skewness -0.05821136 -0.0371887 -0.02747930 -0.01815501
#> ex kurtosis -0.28745974 -0.1305017 -0.09925808 -0.09586772
#> 1st Qu. 361.72627897 58.8089753 73.37887844 70.21298089
#> Median 532.14703274 108.5880504 105.08297474 105.65481777
#> 3rd Qu. 722.93817377 158.2370810 136.70730826 140.65967942
#> X4 X5
#> mean 106.69950599 108.89216051
#> sd 61.90358992 77.59609460
#> skewness -0.03554931 -0.02271947
#> ex kurtosis -0.11514632 -0.11299972
#> 1st Qu. 64.41052149 55.96741500
#> Median 107.32886639 109.01903412
#> 3rd Qu. 149.33842136 162.57599633
#>
#> $`stress 2`
#> Y X1 X2 X3
#> mean 524.20279709 105.86690581 103.44133916 103.81593690
#> sd 249.61597039 73.14092875 46.81249165 52.01382241
#> skewness 0.09284579 0.04378723 0.04119682 0.05173337
#> ex kurtosis -0.18695800 -0.09144983 -0.06441099 -0.06210128
#> 1st Qu. 352.14364373 55.98554812 71.59853268 68.34345109
#> Median 516.02838613 104.72071617 102.95358654 103.26233084
#> 3rd Qu. 687.57987615 155.32653750 134.95169086 138.82267713
#> X4 X5
#> mean 104.74531784 106.33329737
#> sd 62.57010854 78.45638368
#> skewness 0.04131518 0.06092755
#> ex kurtosis -0.07551079 -0.06676053
#> 1st Qu. 61.99518576 52.93072435
#> Median 104.11525415 104.94047561
#> 3rd Qu. 146.79280929 159.23900025Visual display of the change of empirical distribution functions of the portfolio loss Y from the baseline to the two stressed models.
Sensitivity measures allow to assess the importance of the input components. Implemented sensitivity measures are the Kolmogorov distance, the Wasserstein distance and Gamma. Gamma, the Reverse Sensitivity Measure, defined for model component Xi, i = 1, …, 5, and scenario weights w by
Gamma = ( E(Xi * w) - E(Xi) ) / c,
where c is a normalisation constant such that |Gamma| <= 1, see https://doi.org/10.1016/j.ejor.2018.10.003. Loosely speaking, the Reverse Sensitivity Measure is the normalised difference between the first moment of the stressed and the baseline distributions of Xi.
sensitivity(rev.stress, type = "all")
#> stress type Y X1 X2 X3
#> 1 stress 1 Gamma 1.0000000000 0.7949823659 0.7395474096 0.7513647749
#> 2 stress 2 Gamma 1.0000000000 0.7933125049 0.7379627871 0.7497716932
#> 3 stress 1 Kolmogorov 0.0007369138 0.0007369138 0.0007369138 0.0007369138
#> 4 stress 2 Kolmogorov 0.0004847135 0.0004847135 0.0004847135 0.0004847135
#> 5 stress 1 Wasserstein 0.2509137852 0.0753582665 0.0486278071 0.0540026075
#> 6 stress 2 Wasserstein 0.1383234997 0.0412356303 0.0270125296 0.0297424588
#> X4 X5
#> 1 0.7723759014 0.8052671614
#> 2 0.7696561928 0.7983822590
#> 3 0.0007369138 0.0007369138
#> 4 0.0004847135 0.0004847135
#> 5 0.0645056776 0.0811074417
#> 6 0.0354541720 0.0447881192
plot_sensitivity(rev.stress, xCol = 2:6, type = "Gamma") 
Sensitivity to all sub-portfolios, (X1 + X2), (X1 + X3), … (X4, X5):
importance_rank(rev.stress, xCol = NULL, wCol = 1, type = "Gamma", f = rep(list(function(x)x[1] + x[2]), 10),
k = list(c(2,3), c(2,4), c(2,5), c(2,6), c(3,4), c(3,5), c(3,6), c(4,5), c(4,6), c(5,6)))
#> stress type f1 f2 f3 f4 f5 f6 f7 f8 f9 f10
#> 1 stress 1 Gamma 7 6 3 1 10 9 5 8 4 2Ranking the input components according to the chosen sensitivity measure, in this example using Gamma.
importance_rank(rev.stress, xCol = 2:6, type = "Gamma")
#> stress type X1 X2 X3 X4 X5
#> 1 stress 1 Gamma 2 5 4 3 1
#> 2 stress 2 Gamma 2 5 4 3 1Visual display of the change of empirical distribution functions and density from the baseline to the two stressed models of X5, the portfolio component with the largest sensitivity. Stressing the portfolio loss Y, results in a distribution function of X5 that has a heavier tail.
