3 Scope of the SWIM package

3.1 Stressing a model

We briefly introduce key concepts, using slightly more technical language compared to Section 2. A model consists of a random vector of model components \(\mathbf X = (X_1,\dots,X_d)\) and a probability measure; we denote the probability measure of a baseline model by \(P\) and that of a stressed model by \(P^W\), where \(W= \frac{dP^W}{dP}\), satisfying \(E(W)=1\) and \(W\geq 0\), is a Radon-Nikodym derivative. In a Monte Carlo simulation context, the probability space is discrete with \(n\) states \(\Omega=\{\omega_1,\dots,\omega_n\}\), each of which corresponds to a simulated scenario. To reconcile this formulation with the notation of Section 2, we denote, for \(i=1, \dots, n,~j=1,\dots, d\), the realisations \(X_j(\omega_i):= x_{ji}\) and \(W(\omega_i):=w_i\); the latter are the scenario weights. Under the baseline model, each scenario has the same probability \(P(\omega_i)=1/n\), while under a stressed model it is \(P^W(\omega_i)=W(\omega_i)/n=w_i/n\).

The stressed model thus arises from a change of measure from \(P\) to \(P^W\), which entails the application of scenario weights \(w_1,\dots, w_n\) on individual simulations. SWIM calculates scenario weights such that model components fulfil specific stresses, while the distortion to the baseline model is as small as possible when measured by the Kullback-Leibler divergence (or relative entropy). Mathematically, a stressed model is derived by solving

\[\begin{equation} \min_{ W } ~E(W \log (W)), \quad \text{subject to constraints on } \mathbf X \text{ under } P^W. \tag{3.1} \end{equation}\] In what follows, we denote by a superscript \(W\) quantities of interest under the stressed model, such as \(F^W, ~ E^W\) for the probability distribution and expectation under the stressed model, respectively. We refer to Pesenti, Millossovich, and Tsanakas (2019) and references therein for further mathematical details and derivations of solutions to (3.1).

Table 3.1 provides a collection of all implemented types of stresses in the SWIM package. The precise constraints of (3.1) are explained below.

Table 3.1: Implemented types of stresses in SWIM.

R function	Stress	type	Reference
stress	wrapper for the stress_type functions		Sec. 3.1.1
stress_user	user defined scenario weights	user	Sec. 3.1.5
stress_prob	probabilities of disjoint intervals	prob	Eq. (3.2)
stress_mean	means	mean	Eq. (3.3)
stress_mean_sd	means and standard deviations	mean sd	Eq. (3.4)
stress_moment	moments (of functions)	moment	Eq. (3.5)
stress_VaR	VaR risk measure (quantile)	VaR	Eq. (3.6)
stress_VaR_ES	VaR and ES risk measures	VaR ES	Eq. (3.7)

The solutions to the optimisations (3.2) and (3.6) are worked out fully analytically (Pesenti, Millossovich, and Tsanakas 2019), whereas problems (3.3), (3.4), (3.5) and (3.7) require some root-finding. Specifically, problems (3.3), (3.4) and (3.5) rely on the package nleqslv, whereas (3.7) uses the uniroot function.

3.1.1 The stress function and the SWIM object

The stress function is a wrapper for the stress_type functions, where stress(type = "type", ) and stress_type are equivalent. The stress function solves optimisation (3.1) for constraints specified through type and returns a SWIM object, that is, a list including the elements shown in Table 3.2:

Table 3.2: The SWIM object, returned by any stress function.

x	realisations of the model
new_weights	scenario weights
type	type of stress
specs	details about the stress

The data frame containing the realisations of the baseline model, x in the above table, can be extracted from a SWIM object using get_data. Similarly, get_weights and get_weightsfun provide the scenario weights, respectively the functions that, when applied to x, generate the scenario weights. The details of the applied stress can be obtained using get_specs.

3.1.2 Stressing disjoint probability intervals

Stressing probabilities of disjoint intervals allows defining stresses by altering the probabilities of events pertaining to a model component. The scenario weights are calculated via stress_prob, or equivalently stress(type = "prob", ), and the disjoint intervals are specified through the lower and upper arguments, the endpoints of the intervals. Specifically,

stress_prob solves (3.1) with the constraints \[\begin{equation} P^W(X_j \in B_k) = \alpha_k, ~k = 1, \ldots, K, \tag{3.2} \end{equation}\] for disjoint intervals \(B_1, \ldots, B_K\) with \(P(X_j \in B_k) >0\) for all \(k = 1, \ldots, K\), and \(\alpha_1, \ldots, \alpha_K > 0\) such that \(\alpha_1 + \ldots + \alpha_K \leq 1\) and a model component \(X_j\).

3.1.3 Stressing moments

The functions stress_mean, stress_mean_sd and stress_moment provide stressed models with moment constraints. The function stress_mean returns a stressed model that fulfils constraints on the first moment of model components. Specifically,

stress_mean solves (3.1) with the constraints \[\begin{equation} E^W(X_j) = m_j, ~j \in J, \tag{3.3} \end{equation}\] for \(m_j, ~ j \in J\), where \(J\) is a subset of \(\{1, \ldots, d\}\).

The arguments \(m_j\) are specified in the stress_mean function through the argument new_means. The stress_mean_sd function allows to stress simultaneously the mean and the standard deviation of model components. Specifically,

stress_mean_sd solves (3.1) with the constraints \[\begin{equation} E^W(X_j) = m_j \text{ and Var}^W(X_j) = s_j^2 , ~j \in J, \tag{3.4} \end{equation}\] for \(m_j, s_j, ~ j \in J\), where \(J\) is a subset of \(\{1, \ldots, d\}\).

The arguments \(m_j, s_j\) are defined in the stress_mean_sd function by the arguments new_means and new_sd respectively. The functions stress_mean and stress_mean_sd are special cases of the general stress_moment function, which allows for stressed models with constraints on functions of the (joint) moments of model components. Specifically

For \(k = 1, \ldots, K\), \(J_k\) subsets of \(\{1, \ldots, d\}\) and functions \(f_k \colon \mathbb{R}^{|J_k|} \to \mathbb{R}\), stress_moment solves (3.1) with the constraints \[\begin{equation} E^W(f_k(\mathbf X_{J_k}) ) = m_k, ~k = 1, \ldots, K, \tag{3.5} \end{equation}\] for \(m_k, ~k=1, \dots,K\) and \(\mathbf X_{J_k}\) the subvector of model components with indices in \(J_k\).

Note that stress_moment not only allows to define constraints on higher moments of model components, but also to construct constraints that apply to multiple model components simultaneously. For example, the stress \(E^W(X_h X_l) =m_k\) is achieved by setting \(f_k(x_h, x_l) = x_h x_l\) in (3.5) above. The functions stress_mean, stress_mean_sd and stress_moment can be applied to multiple model components and are the only stress functions that have scenario weights calculated via numerical optimisation, using the nleqslv package. Thus, depending on the choice of constraints, existence or uniqueness of a stressed model is not guaranteed.

3.1.4 Stressing risk measures

The functions stress_VaR and stress_VaR_ES provide stressed models, under which a model component fulfils a stress on the risk measures Value-at-Risk (\(\text{VaR}\)) and/or Expected Shortfall (\(\text{ES}\)). The \(\text{VaR}\) at level \(0 < \alpha < 1\) of a random variable \(Z\) with distribution \(F\) is defined as its left-inverse evaluated at \(\alpha\), that is \[\text{VaR}_\alpha(Z) = F^{-1}(\alpha) = \inf\{ y \in \mathbb{R} ~|~F(y) \geq \alpha\}.\] The \(\text{ES}\) at level \(0 < \alpha < 1\) of a random variable \(Z\) is given by \[\text{ES}_\alpha(Z) =\frac{1}{1-\alpha}\int_\alpha^1 \text{VaR}_u(Z) \mathrm{d}u.\] The details of the constraints that stress_VaR and stress_VaR_ES solve, are as follows:

For \(0< \alpha <1\) and \(q, s\) such that \(q < s\), stress_VaR solves (3.1) with the constraint \[\begin{equation} \text{VaR}_{\alpha }^W(X_j) = q; \tag{3.6} \end{equation}\] and stress_VaR_ES solves (3.1) with the constraints \[\begin{equation} \text{VaR}_{\alpha }^W(X_j) = q \text{ and ES}_{\alpha }^W(X_j) = s.\tag{3.7} \end{equation}\]

Note that, since SWIM works with discrete distributions, the exact required VaR may not be achievable. In that case, stress_VaR will return scenario weights inducing the largest quantile in the dataset smaller or equal to the required VaR (i.e. \(q\)); this guarantees that \(P^W(X_j\leq q)=\alpha\).

3.1.5 User defined scenario weights

The option type = "user" allows to generate a SWIM object with scenario weights defined by a user. The scenario weights can be provided directly via the new_weights argument or through a list of functions, new_weightsfun, that applied to the data x generates the scenario weights.

3.2 Analysis of stressed models

Table 3.3 provides a complete list of all implemented R functions in SWIM for analysing stressed models, which are described below in detail.

Table 3.3: Implemented R function in SWIM for analysing stressed models.

R function	analysis of stressed models
summary	summary statistics
cdf	cumulative distribution function
quantile_stressed	quantile function
VaR_stressed	VaR
ES_stressed	ES
sensitivity	sensitivity measures
importance_rank	importance ranks
plot_cdf	plots cumulative distributions functions
plot_hist	plots histograms
plot_sensitivity	plots sensitivity measures

3.2.1 Distributional comparison

The SWIM package contains functions to compare the distribution of model components under different (stressed) models. The function summary is a method for an object of class SWIM and provides summary statistics of the baseline and stressed models. If the SWIM object contains more than one set of scenario weights, each corresponding to one stressed model, the summary function returns for each set of scenario weights a list, containing the elements shown in Table 3.4.

Table 3.4: The output of the summary function applied to a SWIM object.

mean	sample mean
sd	sample standard deviation
skewness	sample skewness
ex kurtosis	sample excess kurtosis
1st Qu.	\(25%\) quantile
Median	median, \(50%\) quantile
3rd Qu.	\(75%\) quantile

The empirical distribution function of model components under a stressed model² can be calculated using the cdf function of the SWIM package, applied to a SWIM object. To calculate sample quantiles of stressed model components, the function quantile_stressed can be used. The function VaR_stressed and ES_stressed provide the stressed VaR and ES of model components, which is of particular interest for stressed models resulting from constraints on risk measures, see Section 3.1.4. (While quantile_stressed works very similarly to the base R function quantile, VaR_stressed provides better capabilities for comparing different models and model components.)

Implemented visualisation of distribution functions are plot_cdf, for plotting empirical distribution functions, and plot_hist, for plotting histograms of model components under different (stressed) models.

3.2.2 Sensitivity measures

Comparison of baseline and stressed models and how model components change under different models, is typically done via sensitivity measures. The SWIM packages contains the sensitivity function, which calculates sensitivity measures of stressed models and model components. The implemented sensitivity measures, summarised in the table below, are the Wasserstein, Kolmogorov and the Gamma sensitivity measures, see Pesenti, Millossovich, and Tsanakas (2016) Pesenti, Millossovich, and Tsanakas (2019) Emmer, Kratz, and Tasche (2015).

Table 3.5: Definition of the sensitivity measures implemented in SWIM.

Metric	Definition
Wasserstein	\(\int | F^W_X (x) - F_X(x)| dx\)
Kolmogorov	\(\sup_x |F^W_X (x) - F_X(x)|\)
Gamma	\(\frac{E^W(X) - E(X)}{c}\), for a normalisation \(c\)

The Gamma sensitivity measure is normalised such that it takes values between -1 and 1, with higher positive (negative) values corresponding to a larger positive (negative) impact of the stress on the particular model component. The sensitivity measures can be plotted using plot_sensitivity. The function importance_rank returns the effective rank of model components according to the chosen sensitivity measure.

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2. Note that functions implementing the empirical cdf or the quantile, ecdf and quantile, will not return the empirical distribution function or the quantile function under a stressed model.↩