S. M. Pesenti, P. Millossovich, and A. Tsanakas

Differential Sensitivity in Discontinuous Models

Differential Sensitivity in Discontinuous Models

Silvana M. Pesenti Department of Statistical Sciences, University of Toronto,
<EMAIL_ADDRESS>Pietro Millossovich Bayes Business School
(formerly Cass), City, University of London,
DEAMS, University of Trieste Andreas Tsanakas Bayes Business School (formerly
Cass), City, University of London

Differential sensitivity measures provide valuable tools for interpreting
complex computational models used in applications ranging from simulation to
algorithmic prediction. Taking the derivative of the model output in direction
of a model parameter can reveal input-output relations and the relative
importance of model parameters and input variables. Nonetheless, it is unclear
how such derivatives should be taken when the model function has
discontinuities and/or input variables are discrete. We present a general
framework for addressing such problems, considering derivatives of quantile-
based output risk measures, with respect to distortions to random input
variables (risk factors), which impact the model output through step-
functions. We prove that, subject to weak technical conditions, the
derivatives are well-defined and derive the corresponding formulas. We apply
our results to the sensitivity analysis of compound risk models and to a
numerical study of reinsurance credit risk in a multi-line insurance
portfolio.

Sensitivity analysis, importance measurement, differential sensitivity
measures, simulation, risk measures, credit risk.

##  1 Introduction

The interpretability of complex computational models is of fundamental
importance across areas of applications, with sensitivity analysis providing
tools for understanding the importance of risk factors, their interactions and
their impact on a model’s output (Saltelli et al. 2008, Borgonovo and Plischke
2016, Razavi et al. 2021, Fissler and Pesenti 2023). In recent years, the
field received renewed impetus by the widespread adoption of machine learning
and artificial intelligence models for prediction tasks, which are usually
opaque and thus require additional work to illuminate input/output
relationships. Contributions in this field range from the development of
general model-agnostic model interpretation procedures (Ribeiro et al. 2016,
Borgonovo et al. 2023), to those tailored to a class of models, such as tree
ensembles (Lundberg et al. 2018) and neural networks (Merz et al. 2022), or to
specific applications, such as image recognition (Chen et al. 2019) and credit
scoring (Chen et al. 2023). Furthermore, the interest in model interpretation
is amplified by the requirement for models’ behaviour to be fair, in the sense
that it does not generate discriminatory impacts on protected groups (Frees
and Huang 2021, Kozodoi et al. 2022, Lindholm et al. 2022) – such concerns
have generated further research at the interface of sensitivity analysis and
algorithmic fairness (Bénesse et al. 2022, Hiabu et al. 2023).

As part of sensitivity analysis, metrics are often used to assess the
importance of model inputs. A broad class of such metrics is that of
differential sensitivity measures, which rely on derivatives of (a statistical
functional of) the model output, in the direction of a perturbation of a
(random) input factor (Antoniano-Villalobos et al. 2018). Recent advances in
sensitivity analysis pertain to perturbing quantile-based risk measures of the
model output (Tsanakas and Millossovich 2016, Browne et al. 2017, Pesenti et
al. 2021, Merz et al. 2022). In that context, fundamental technical
requirements for differential sensitivity measures include differentiability
of the model function and Lipschitz continuity of the model output in the
perturbation (Broadie and Glasserman 1996, Hong 2009, Hong and Liu 2009).
These requirements are stringent, as many computational models map input
factors to outputs in a discontinuous manner; examples include credit risk
models (Chen and Glasserman 2008), financial derivatives and insurance
contracts (Albrecher et al. 2017), and tree-based predictive models (Chen and
Guestrin 2016).

In this work, we overcome such strong conditions and derive, under rather mild
assumptions, formulas for differential quantile-based sensitivity measures, in
models where the input-output relationship contains step functions. This is a
general setting, since many functions with a finite number of jump
discontinuity points can be written via a sum of step functions. We focus on
the two most common quantile-based risk measures, Value-at-Risk (VaR) and
Expected Shortfall (ES), although the expressions can be generalised for the
broader case of distortion risk measures and rank dependent expected
utilities. We consider two types of differential sensitivity measure, marginal
sensitivities and cascade sensitivities. The marginal sensitivity quantifies
an input factor’s sole effect on a model output’s risk measure (Hong 2009,
Tsanakas and Millossovich 2016). In contrast, in the cascade sensitivity
setting (Pesenti et al. 2021) a perturbation of a risk factor affects other
dependent risk factors, which in turn impact the output risk measure. To prove
the derived sensitivity formulas we use quantile differentiation and weak
convergence of generalised functions. We find that stresses propagated via
step functions naturally lead to delta functions, which in turn allow for
representation as conditional expectations. Hence, our framework allows
estimation of differential sensitivity measures by standard simulation-based
methods (Glasserman 2005, Fu et al. 2009, Koike et al. 2022).

Key to our framework is the choice of perturbation or stress on the random
input factor. In particular, the technical conditions we require pertain to
the continuity of the stressing mechanism rather than the underlying random
input factor. Consequently, our methods can also be applied to the calculation
of differential sensitivity measures with respect to discrete random inputs
for a suitably chosen stress. Sensitivity to discrete or categorical input
factors is of importance in a variety of fields, such as modelling biological
systems (Gunawan et al. 2005), chemical processes (Plyasunov and Arkin 2007),
and insurance claims (Wüthrich and Merz 2023).

The manuscript is organised as follows. Section 2 introduces the discontinuous
loss model and discusses choices of stresses on a risk factor. Following that,
expressions are derived for differential (marginal) sensitivities, with
respect to the VaR and ES risk measures. The next two sections contain
extensions within that framework. Section 3 deals with cascade sensitivities,
which reflect indirect effects via risk factors’ dependence structure. Section
4 provides differential sensitivities when the considered input random
variables are discrete, along with an application to compound distributions.
Finally, a detailed numerical study of a reinsurance credit risk portfolio is
given in Section 5. Differential sensitivity formulas for a more general model
function can be found in Appendix 7. All proofs are delegated to Appendix 8.
Finally, Appendix 9 contains additional details on the reinsurance credit risk
portfolio model used in Section 5.

##  2 Differential Sensitivity Measures

###  2.1 Portfolio Loss Model

We work on a probability space $(\Omega,\mathcal{A},{\mathbb{P}})$ and
consider a discontinuous model of the form

$L:=\sum_{j=1}^{m}g_{j}({\boldsymbol{Z}}){\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}},$ (1)

where:

  * •

The random vectors ${\boldsymbol{X}}:=(X_{1},\ldots,X_{m})$,
${\boldsymbol{Z}}:=(Z_{1},\ldots,Z_{n})$, $m,n\in{\mathbb{N}}$ are model
inputs or risk factors;

  * •

$L$ is the (univariate) random model output, which we typically interpret as a
loss;

  * •

Discontinuities emerge at those states where elements of ${\boldsymbol{X}}$
cross the thresholds $d_{1},\ldots,d_{m}\in{\mathbb{R}}$;

  * •

The functions $g_{j}\colon{\mathbb{R}}^{n}\to{\mathbb{R}}$,
$j\in{\mathcal{M}}:=\\{1,\dots,m\\}$ represent the (random) jump of the model
output at the points of discontinuity.

We assume throughout that the marginal distribution functions of $X_{j}$,
$j\in{\mathcal{M}}$ and $Z_{k}$, $k\in{\mathcal{N}}:=\\{1,\ldots,n\\}$,
denoted by $F_{j}(x):={\mathbb{P}}(X_{j}\leq x)$ and
$F_{m+k}(z):={\mathbb{P}}(Z_{k}\leq z)$, respectively, are absolutely
continuous and strictly increasing on their support and denote their
corresponding (strictly positive, a.e. on their support) densities by $f_{j}$
and $f_{m+k}$ respectively. We further denote by $F(l):={\mathbb{P}}(L\leq l)$
the distribution of the loss $L$. The functions
$g_{j}\colon{\mathbb{R}}^{n}\to{\mathbb{R}}$, $j\in{\mathcal{M}}$ are
differentiable.

A standard example of a discontinuous loss (1) is a structural model of a
credit risk portfolio (e.g. McNeil et al. 2015, Ch. 11), where $\\{X_{j}\leq
d_{j}\\}$ represents the default event of obligor $j$ and
$g_{j}({\boldsymbol{Z}})$ the corresponding loss given default. Applications
to credit risk modelling are further discussed in Example 2.6 and Section 5.
We note that the model (1) is formulated such that $g_{j}$ are functions of
${\boldsymbol{Z}}$ only. We make this assumption throughout the paper to
simplify exposition; the general case of $g_{j}$ depending on both
${\boldsymbol{Z}}$ and ${\boldsymbol{X}}$ is treated in Appendix 7.2.

The risk of a loss is assessed via a risk measure
$\rho\colon\mathcal{L}^{1}\to{\mathbb{R}}$, where $\mathcal{L}^{1}$ denotes
the set of integrable random variables on $(\Omega,\mathcal{A},{\mathbb{P}})$.
The two most widely used risk measures in practice are the _Value-at-Risk_
(VaR) and the _Expected Shortfall_ (ES). The VaR at level $\alpha\in[0,1]$ of
the portfolio loss $L$ is defined as the (left-) quantile function of $L$
evaluated at $\alpha$, that is

${\mathrm{VaR}}_{\alpha}(L):=F^{-1}(\alpha)=\inf\\{y\in{\mathbb{R}}\,|\,F(y)\geq\alpha\\}\,,$

with the convention that $\inf\emptyset=+\infty$. The Expected Shortfall at
level $\alpha\in[0,1)$ of the portfolio loss $L$ is defined by

${\mathrm{ES}}_{\alpha}(L):=\frac{1}{1-\alpha}\int_{\alpha}^{1}F^{-1}(u)\,du\,.$

While we focus on VaR and ES, the sensitivities can be generalised to other
quantile-based functionals, such as rank dependent expected utilities or
spectral risk measures (Acerbi 2002) – in the interest of concision we do not
pursue this further.

In Section 2.3 we introduce the marginal sensitivity measure, and derive
expressions in the context of the VaR/ES risk measures and the discontinuous
model (1). The sensitivity measure is defined via a partial derivative of a
risk measure in the direction of a _stressed_ version of a risk factor; hence
we first introduce ways of stressing risk factors.

###  2.2 Stressing Risk Factors

Throughout the paper, we fix the index $i$ of the risk factor with respect to
which sensitivity is calculated, such that stresses are applied to either
$X_{i}$, with $i\in{\mathcal{M}}$, or to $Z_{i}$, with $i\in{\mathcal{N}}$. We
define a stress on $X_{i}$ or $Z_{i}$ as a deformation of the risk factor
given by

$X_{i,{\varepsilon}}:=\kappa_{\varepsilon}(X_{i})\,,\quad\text{respectively},\quad
Z_{i,{\varepsilon}}:=\kappa_{\varepsilon}(Z_{i})\,,$

where $\kappa_{\varepsilon}\colon{\mathbb{R}}\to{\mathbb{R}}$ is a stress
function defined as follows.

######  Definition 2.1 (Stress Function)

A family of functions $\kappa_{\varepsilon}\colon A\to A$,
$A\subseteq{\mathbb{R}}$, where ${\varepsilon}\in[0,+\infty)$, is called a
stress function, if it satisfies the following properties:

  1. $i)$

For all ${\varepsilon}$ in a neighbourhood of 0, the function
$\kappa_{\varepsilon}(x)$ is invertible in $x\in A$, denoted by
$\kappa_{\varepsilon}^{-1}(\cdot)$;

  2. $ii)$

$\displaystyle\lim_{{\varepsilon}\searrow 0}\;\kappa_{\varepsilon}(x)=x$, for
all $x\in A$;

  3. $iii)$

$\displaystyle\lim_{{\varepsilon}\searrow 0}\;\kappa_{\varepsilon}^{-1}(x)=x$,
for all $x\in A$;

  4. $iv)$

One of the following holds:

    1. $(a)$

for all ${\varepsilon}$ in a neighbourhood of 0 and all $x\in A$, it holds
that $\kappa_{\varepsilon}(x)\geq x$; or

    2. $(b)$

for all ${\varepsilon}$ in a neighbourhood of 0 and all $x\in A$, it holds
that $\kappa_{\varepsilon}(x)\leq x$;

  5. $v)$

$\kappa_{\varepsilon}(x)$ is differentiable in ${\varepsilon}$ at
${\varepsilon}=0$, and we denote its derivative by

${\mathfrak{K}}(x):=\displaystyle\lim_{{\varepsilon}\to
0}\frac{\kappa_{\varepsilon}(x)-x}{{\varepsilon}}\,,\qquad x\in A\,;$

  6. $vi)$

$\kappa_{\varepsilon}^{-1}(x)$ is differentiable in ${\varepsilon}$ at
${\varepsilon}=0$, and we denote its derivative by

${\mathfrak{K}^{-1}}(x):=\displaystyle\lim_{{\varepsilon}\to
0}\frac{\kappa_{\varepsilon}^{-1}(x)-x}{{\varepsilon}}\,,\qquad x\in A\,.$

We further define

$c(\kappa):=\begin{cases}+1,\qquad\text{if}\quad\kappa_{\varepsilon}\text{
fulfils \ref{asm:kappa-ineq} \ref{asm:kappa-ineq-ge}}\,,\\\
-1,\qquad\text{if}\quad\kappa_{\varepsilon}\text{ fulfils \ref{asm:kappa-ineq}
\ref{asm:kappa-ineq-le}}\,.\end{cases}$ (2)

The requirements on the stress function are assumptions on its continuity.
First, if stressing $X_{i}$, we typically assume that the domain of the stress
function $A$ is equal to the support of $X_{i}$. This guarantees that $X_{i}$
and its stressed version $\kappa_{\varepsilon}(X_{i})$ have the same support.
Second, properties $i)$ to $iii)$ provide that the stressed risk factor
converges ${\mathbb{P}}$-a.s. to its unstressed form as ${\varepsilon}\searrow
0$. Property $iv)$ means that the stress, e.g. $X_{i,{\varepsilon}}$, either
approaches $X_{i}$ ${\mathbb{P}}$-a.s. from above or below. The last two
properties imply that the stress function and its inverse are differentiable,
so that the sensitivities, introduced in Sections 2.3 and 3, exist.

Different stress functions may be used, depending on the context of the
problem investigated and what type of deformation of a risk factor is
interpretable within that context. For example additive and proportional
stresses can be seen as modifications of location and scale respectively; tail
stresses may reflect risk management objectives; mixture stresses are used to
represent model uncertainty. Some stress functions and related quantities are
summarised in Table 1. The additive and proportional stresses with $\beta>0$
are such that property $iv)$ $iv)$$(a)$ is satisfied and the stress
stochastically increases the risk factor; this is easily modified by choosing
$\beta<0$. For the mixture and tail stresses both increasing ($iv)$
$iv)$$(a)$) and decreasing ($iv)$ $iv)$$(b)$) versions of the stresses are
stated. The functions ${\mathfrak{K}},~{}{\mathfrak{K}^{-1}}$ are easily
worked out; some additional detail for mixture stresses is given in Appendix
8.1. Stress functions should be designed with particular investigations in
mind; we explore this further in Example 2.6.

Table 1: Types of stress functions and related quantities. Type of stress | $\kappa_{\varepsilon}$ | ${\mathfrak{K}}$ | ${\mathfrak{K}}^{-1}$ | $c(\kappa)$
---|---|---|---|---
Additive | $x+\beta{\varepsilon}$ | $\beta$ | $-\beta$ |  sgn($\beta$)
Proportional | $x(1+\beta{\varepsilon})$ | $\beta x$ | $-\beta x$ |  sgn($\beta$)
Probability | $F_{i}^{-1}\big{(}F_{i}(x)+\beta x\big{)}$ | $\frac{\beta}{f_{i}(x)}$ | $\frac{-\beta}{f_{i}(x)}$ |  sgn($\beta$)
Mixture |  $F_{i,{\varepsilon}}^{-1}\circ F_{i}(x)$, where | $\frac{F_{i}(x)-G(x)}{f_{i}(x)}$ | $\frac{G(x)-F_{i}(x)}{f_{i}(x)}$ | sgn$\big{(}F_{i}(x)-G(x)\big{)}$
$F_{i,{\varepsilon}}(x):=(1-{\varepsilon})\,F_{i}(x)+{\varepsilon}\,G(x)$
Tail | $x+{\varepsilon}\left(x-t\right){\mathds{1}}_{\\{x\geq t\\}}$ | $(x-t)_{+}$ | $-(x-t)_{+}$ | 1
$x+{\varepsilon}\left(x-t\right){\mathds{1}}_{\\{x\leq t\\}}$ | $-(t-x)_{+}$ | $(t-x)_{+}$ | $-1$

###  2.3 Marginal Sensitivity

For a stress $Z_{i,{\varepsilon}}$ or a stress $X_{i,{\varepsilon}}$, we
denote the corresponding marginally stressed loss model by, respectively

$\displaystyle L_{{\varepsilon}}(Z_{i})$
$\displaystyle:=\sum_{j\in{\mathcal{M}}}g_{j}({\boldsymbol{Z}}_{-i},Z_{i,{\varepsilon}})\,{\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}}\,\quad\text{and}$ $\displaystyle L_{{\varepsilon}}(X_{i})$
$\displaystyle:=\sum_{\begin{subarray}{c}j\neq i\\\
j\in{\mathcal{M}}\end{subarray}}g_{j}({\boldsymbol{Z}})\,{\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}}+g_{i}({\boldsymbol{Z}})\,{\mathds{1}}_{\\{X_{i,{\varepsilon}}\leq
d_{i}\\}}\,,$

where $({\boldsymbol{Z}}_{-i},Z_{i,{\varepsilon}})$ is the vector
${\boldsymbol{Z}}$ whose $i^{\text{th}}$ component is replaced by
$Z_{i,{\varepsilon}}$. We call $L_{\varepsilon}$, denoting either
$L_{\varepsilon}(Z_{i})$ or $L_{\varepsilon}(X_{i})$, the marginally stressed
loss, since only the marginal distribution of $Z_{i}$ or $X_{i}$ is altered,
leaving all other input factors fixed. We denote by $F_{{\varepsilon}}(\cdot)$
the distribution function of $L_{\varepsilon}$ and by
$q_{{\varepsilon}}(\cdot):=F^{-1}_{\varepsilon}(\cdot)$ the quantile function
of $L_{{\varepsilon}}$ for any ${\varepsilon}\geq 0$.

For the sensitivities to exist, we require two assumptions on the stressed
loss model.

Let $0\leq\alpha\leq 1$. For all ${\varepsilon}$ in a neighbourhood of 0 the
distribution function $F_{\varepsilon}$ of the marginally stressed loss
$L_{\varepsilon}$ is continuously differentiable at
${q_{\alpha}}:=F^{-1}(\alpha)$ and let $f_{\varepsilon}(\cdot)$ be the
derivative of $F_{\varepsilon}$. For ${\varepsilon}=0$, we simply write
$F:=F_{0}$ and $f:=f_{0}$.

Let $0\leq\alpha\leq 1$. The quantile function at level $\alpha$ of the
stressed loss $L_{\varepsilon}$, $q_{\varepsilon}(\alpha)$, is differentiable
with respect to ${\varepsilon}$, that is
$\frac{\partial}{\partial{\varepsilon}}q_{\varepsilon}(\alpha)$ exists.

######  Definition 2.2 (Marginal Sensitivity)

The _marginal sensitivity_ to the risk factor $Z_{i}$ and $X_{i}$ for a risk
measure $\rho$ is defined by, respectively,

$\mathcal{S}_{Z_{i}}\,[\,\rho\,]:=\frac{\partial}{\partial{\varepsilon}}\,\rho\left(L_{\varepsilon}(Z_{i})\right)\Big{|}_{{\varepsilon}=0}\quad\text{and}\quad\mathcal{S}_{X_{i}}\,[\,\rho\,]:=\frac{\partial}{\partial{\varepsilon}}\,\rho\left(L_{\varepsilon}(X_{i})\right)\Big{|}_{{\varepsilon}=0}\,,$
(3)

whenever the derivatives exists.

######  Theorem 2.3 (Marginal Sensitivity VaR)

Let Assumptions 2.3 and 2.3 be fulfilled for a given $\alpha\in(0,1)$. Then,
the marginal sensitivity for ${\mathrm{VaR}}_{\alpha}$ to input factor $Z_{i}$
for a stress with stress function $\kappa_{\varepsilon}$ is

${\mathcal{S}}_{Z_{i}}\,[\,{\mathrm{VaR}}_{\alpha}\,]=\sum_{j\in{\mathcal{M}}}\left.{\mathbb{E}}\left[\,{\mathfrak{K}}(Z_{i})\partial_{i}\,g_{j}({\boldsymbol{Z}}){\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}}~{}\right|~{}L={q_{\alpha}}\,\right]\,,$

where $\partial_{i}\,g_{j}({\boldsymbol{z}}):=\frac{\partial}{\partial
z_{i}}g_{j}({\boldsymbol{z}})$ is the partial derivative in the
$i^{\text{th}}$ component. The marginal sensitivity to input factor $X_{i}$ is
given by

${\mathcal{S}}_{X_{i}}\,[\,{\mathrm{VaR}}_{\alpha}\,]=c(\kappa){\mathfrak{K}^{-1}}(d_{i})\,\frac{f_{i}(d_{i})}{f\left({q_{\alpha}}\right)}\;\;{\mathbb{E}}\left.\left[\left({\mathds{1}}_{\left\\{L\leq{q_{\alpha}}+c(\kappa)g_{i}({\boldsymbol{Z}})\,\right\\}}-{\mathds{1}}_{\\{L\leq{q_{\alpha}}\\}}\right)~{}\right|~{}X_{i}=d_{i}\right]\,.$

######  Theorem 2.4 (Marginal Sensitivity ES)

Let Assumptions 2.3 and 2.3 be fulfilled for a given $\alpha\in(0,1)$. Then,
the marginal sensitivity for ${\mathrm{ES}}_{\alpha}$ to input factor $Z_{i}$
for a stress with stress function $\kappa_{\varepsilon}$ is

${\mathcal{S}}_{Z_{i}}\,[\,{\mathrm{ES}}_{\alpha}\,]=\sum_{j\in{\mathcal{M}}}{\mathbb{E}}\left[\,{\mathfrak{K}}(Z_{i})\,\partial_{i}\,g_{j}({\boldsymbol{Z}}){\mathds{1}}_{\left\\{X_{j}\leq
d_{j}\right\\}}~{}|~{}L\geq{q_{\alpha}}\right]\,.$

The marginal sensitivity to input factor $X_{i}$ for a stress with stress
function $\kappa_{\varepsilon}$ is

${\mathcal{S}}_{X_{i}}\,[\,{\mathrm{ES}}_{\alpha}\,]=\frac{-c(\kappa){\mathfrak{K}^{-1}}(d_{i})\,f_{i}(d_{i})}{1-\alpha}\left.{\mathbb{E}}\left[\big{(}L-c(\kappa)g_{i}({\boldsymbol{Z}})\,-{q_{\alpha}}\big{)}_{+}-\big{(}L-{q_{\alpha}}\big{)}_{+}~{}\right|~{}X_{i}=d_{i}\right]\,.$

The marginal sensitivity measures to $Z_{i}$ for both the ${\mathrm{VaR}}$ and
${\mathrm{ES}}$ generalise the sensitivities derived in Hong (2009), Hong and
Liu (2009) to loss functions $L$ that are not Lipschitz continuous and to
general types of stresses. Related, Fu et al. (2009) proposes a conditional
Monte-Carlo approach to estimate quantile sensitivities. Note however, that
their key assumption is that the perturbed distribution function of
$L_{\varepsilon}$ can be written as
$F_{L_{\varepsilon}}(t)={\mathbb{E}}[G(t,{\varepsilon},Y({\varepsilon}))]$,
where $G$ is ${\mathbb{P}}$-a.s. continuous w.r.t. ${\varepsilon}$ and
$Y({\varepsilon})$ is an arbitrary random variable. This assumption does not
hold in our setting as can be seen in, e.g., Equation (15) of the Proof of
Theorem 2.3. Furthermore, one could derive the marginal sensitivities of
${\mathrm{ES}}_{\alpha}$ – as well as those of other spectral risk measures
(Acerbi 2002) – using its representation as the integral of
${\mathrm{VaR}}_{\alpha}$. Interchanging the limit and the integral, however,
requires that the sensitivities for ${\mathrm{VaR}}_{\beta}$ to exist, for all
$\beta\in[\alpha,1)$. This in particular would imply that Assumptions 2.3 and
2.3 need to hold for all $\beta\in(\alpha,1)$, which is in contrast Theorem
2.4 which requires Assumptions 2.3 and 2.3 to hold for $\alpha$ only.

We now provide an expression for the marginal sensitivity of the mean. While
this could be obtained as a special case of Expected Shortfall with
$\alpha=0$, it is simpler to derive Corollary 2.5 as a direct consequence of
Lemma 8.2 in Appendix 8.

######  Corollary 2.5 (Marginal Sensitivity Mean)

Let $\kappa_{\varepsilon}$ be a stress function, then the marginal sensitivity
for the mean $({\mathbb{E}})$ to input factor $Z_{i}$ respective $X_{i}$ are

$\displaystyle{\mathcal{S}}_{Z_{i}}\,[\,{\mathbb{E}}\,]$
$\displaystyle=\sum_{j\in{\mathcal{M}}}{\mathbb{E}}\left[\,{\mathfrak{K}}(Z_{i})\,\partial_{i}\,g_{j}({\boldsymbol{Z}}){\mathds{1}}_{\left\\{X_{j}\leq
d_{j}\right\\}}\right]\,\quad\text{and}$
$\displaystyle{\mathcal{S}}_{X_{i}}\,[\,{\mathbb{E}}\,]$
$\displaystyle={\mathfrak{K}^{-1}}(d_{i})\,f_{i}(d_{i})\left.{\mathbb{E}}\left[\,g_{i}({\boldsymbol{Z}})~{}\right|~{}X_{i}=d_{i}\right]\,.$

We conclude the section with an example of how the marginal sensitivity
measure can be applied in the context of a standard portfolio credit risk
model, with two different analysis objectives in mind.

###### Example 2.6

Consider a credit risk setting, where ${\boldsymbol{Z}}$ has the same
dimension as ${\boldsymbol{X}}$, with
$g_{j}({\boldsymbol{Z}})=Z_{j},~{}j\in\mathcal{M}$ representing the loss given
default and ${\\{X_{j}\leq d_{j}\\}}$ the default events with corresponding
probabilities $F_{j}(d_{j})$. Hence we have that

$L=\sum_{j\in\mathcal{M}}Z_{j}{\mathds{1}}_{\\{X_{j}\leq d_{j}\\}}\,.$

A first analysis pertains to the calculation of the sensitivity of the
portfolio ES with respect to the probability of the $i$-th default event. To
achieve this, we need to formulate an appropriate stress function. Consider
the probability stress from Table 1,
$\kappa_{\varepsilon}(x)=F_{i}^{-1}(F_{i}(x)-{\varepsilon}),$ leading to

$\displaystyle X_{i,{\varepsilon}}$
$\displaystyle=F_{i}^{-1}(F_{i}(X_{i})-{\varepsilon})\quad\text{and}$
$\displaystyle{\mathbb{P}}(X_{i,{\varepsilon}}\leq d_{i})$
$\displaystyle={\mathbb{P}}\left(F_{i}(X_{i})\leq
F_{i}(d_{i})+{\varepsilon}\right)=F_{i}(d_{i})+{\varepsilon}.$

Hence the chosen stress function gives an additive stress on the default
probability, such that the sensitivity
${\mathcal{S}}_{X_{i}}\,[\,{\mathrm{ES}}_{\alpha}\,]$ becomes precisely the
derivative of the portfolio risk in direction of the default probability of
the $i$-th obligor. Using $c(\kappa)=-1$ and
${\mathfrak{K}^{-1}}(x)=\frac{1}{f_{i}(x)}$; Theorem 2.4 yields:

$\displaystyle{\mathcal{S}}_{X_{i}}\,[\,{\mathrm{ES}}_{\alpha}\,]$
$\displaystyle=\frac{1}{1-\alpha}\left.{\mathbb{E}}\left[\big{(}L-({q_{\alpha}}-Z_{i})\big{)}_{+}-\big{(}L-{q_{\alpha}}\big{)}_{+}~{}\right|~{}X_{i}=d_{i}\right].$

The resulting sensitivity can thus be understood as the difference between two
expectations, each representing the excess portfolio loss over a threshold,
conditioned on the least adverse outcome of $X_{i}$ that gives a default of
the $i$-th obligor. The difference between the two terms lies in the lower
threshold used in the first term, which is reduced by the loss given default
$Z_{i}$.

Second, we consider the sensitivity to a proportional increase in the loss
given default $Z_{i}$, that is, using
$\kappa_{\varepsilon}(z)=z(1+{\varepsilon})$. Application of Theorem 2.4 then
gives us:

${\mathcal{S}}_{Z_{i}}\,[\,{\mathrm{ES}}_{\alpha}\,]={\mathbb{E}}\left[\,Z_{i}{\mathds{1}}_{\left\\{X_{i}\leq
d_{i}\right\\}}~{}|~{}L\geq{q_{\alpha}}\right]\,.$

Note that this is precisely the Euler allocation of the risk
${\mathrm{ES}}_{\alpha}(L)$ to the loss $Z_{i}{\mathds{1}}_{\left\\{X_{i}\leq
d_{i}\right\\}}$ to the $i$-th obligor (Tasche 1999).

Finally, within the same model, we turn our attention to assessment of the
relative importance of common factors that drive dependence between defaults.
The dependence of the critical variables $X_{j}$ is often modelled via factor
models (McNeil et al. 2015, Ch. 6.4, 11) and a question of interest is the
relative importance of underlying factors for portfolio risk. Consider the
following representation

$X_{j}:=\sum_{t=1}^{\tau}\beta_{j,t}W_{t}+V_{j},\quad j\in\mathcal{M},$

where $W_{t},~{}t=1,\ldots,\tau$, $\tau\in{\mathbb{N}}$, are the common
factors, and $V_{j}$ are idiosyncratic error terms. We are interested in the
sensitivity of the portfolio loss to the factor $W_{s}$. To that effect,
define:

$\displaystyle\tilde{X}_{j,{\varepsilon}}$ $\displaystyle:=\sum_{t\neq
s}\beta_{j,t}W_{t}+\beta_{j,s}(W_{s}-{\varepsilon})+V_{j}=X_{j}-\beta_{j,s}{\varepsilon}\,,$
$\displaystyle\tilde{\kappa}_{{\varepsilon}}(x)$
$\displaystyle:=x-\beta_{j,s}{\varepsilon}\,,$
$\displaystyle\tilde{L}_{\varepsilon}$
$\displaystyle:=\sum_{j\in\mathcal{M}}Z_{j}{\mathds{1}}_{\\{\tilde{\kappa}_{{\varepsilon}}(X_{j})\leq
d_{j}\\}}\,.$

The sensitivity of the portfolio risk to the factor $W_{s}$ can then be
written as

$\displaystyle\left.\frac{\partial}{\partial{\varepsilon}}{\mathrm{ES}}_{\alpha}(\tilde{L}_{\varepsilon})\right|_{{\varepsilon}=0}=\sum_{j\in\mathcal{M}}S_{X_{j}}[{\mathrm{ES}}_{\alpha}]\,,$

where the sensitivities $S_{X_{j}}[{\mathrm{ES}}_{\alpha}]$ are now calculated
with the stress functions $\tilde{\kappa}_{{\varepsilon}}$ above. Applying
again Theorem 2.4 leads to

$\displaystyle\left.\frac{\partial}{\partial{\varepsilon}}{\mathrm{ES}}_{\alpha}(\tilde{L}_{\varepsilon})\right|_{{\varepsilon}=0}=\sum_{j\in\mathcal{M}}\frac{f_{j}(d_{j})\beta_{j,s}}{1-\alpha}\left.{\mathbb{E}}\left[\big{(}L-({q_{\alpha}}-Z_{j})\big{)}_{+}-\big{(}L-{q_{\alpha}}\big{)}_{+}~{}\right|~{}X_{i}=d_{i}\right].$

Hence, intuitively, the sensitivity to the common factor $W_{s}$ is expressed
as sum of sensitivities for each obligor, weighted by the factor loadings
$\beta_{j,s}$.

##  3 Measuring Cascading Effects

The marginal sensitivity introduced in Section 2.3 quantifies the differential
impact of stressing a risk factor on the portfolio loss. Here, we provide the
first generalisation/adjustment of the framework, by considering cascade
sensitivity measures, as discussed in Pesenti et al. (2021). These sensitivity
measures quantify not only the sensitivity to an individual input $X_{i}$, but
also consider (joint) perturbation of all other risk factors $X_{j}$, $j\neq
i$, and $Z_{k}$, $k\in{\mathcal{N}}$, induced by their statistical dependence
on $X_{i}$. This is achieved by using the _inverse Rosenblatt transform_ ,
recalled next (Rosenblatt 1952, Rüschendorf and de Valk 1993).

######  Definition 3.1 (Inverse Rosenblatt Transform)

An inverse Rosenblatt transform of an $m$-dimensional random vector
${\boldsymbol{Y}}$, starting at $Y_{i}$, for fixed $i\in{\mathcal{M}}$, is
given by a function
${\boldsymbol{\Psi}}=(\Psi^{(1)},\ldots,\Psi^{(m)})^{\top}\colon\mathbb{R}^{m}\to\mathbb{R}^{m}$
and an $(m-1)$-dimensional random vector
${\boldsymbol{V}}=(V_{1},\ldots,V_{m-1})$, consisting of independent standard
uniform variables, independent of $Y_{i}$, such that

$\displaystyle{\boldsymbol{Y}}={\boldsymbol{\Psi}}\left(Y_{i},{\boldsymbol{V}}\right)=\left(\Psi^{(1)}(Y_{i},{\boldsymbol{V}}),\,\ldots,\,\Psi^{(m)}(Y_{i},{\boldsymbol{V}})\right)\,,\quad{\mathbb{P}}\text{-a.s.}\,.$

In particular, $Y_{k}=\Psi^{(k)}(Y_{i},{\boldsymbol{V}})$ ${\mathbb{P}}$-a.s.
for all $k\in{\mathcal{M}}$.

To construct the cascade sensitivity to input $X_{i}$, we proceed via the
following steps. First, we write the vector of risk factors
$({\boldsymbol{X}},{\boldsymbol{Z}})$ via its inverse Rosenblatt transform
starting from $X_{i}$. From this we obtain
$({\boldsymbol{X}},{\boldsymbol{Z}})={\boldsymbol{\Psi}}(X_{i},{\boldsymbol{V}})$
${\mathbb{P}}$-a.s., that is $X_{j}=\Psi^{(j)}(X_{i},{\boldsymbol{V}})$, for
all $j\in{\mathcal{M}}$, and $Z_{k}=\Psi^{(m+k)}(X_{i},{\boldsymbol{V}})$ for
all $k\in{\mathcal{N}}$. Second, we stress $X_{i}$ to $X_{i,{\varepsilon}}$,
such that the stressed vector of risk factors becomes
${\boldsymbol{\Psi}}(X_{i,{\varepsilon}},{\boldsymbol{V}})$. Thus, using the
inverse Rosenblatt transform, all other risk factors are perturbed according
to their dependence on $X_{i}$. Third, we apply the marginal sensitivity to
the portfolio loss as a function of the stressed input vector
${\boldsymbol{\Psi}}(X_{i,{\varepsilon}},{\boldsymbol{V}})$. When stressing
$Z_{i}$ the process is analogous. For simplicity of notation, we use
${\boldsymbol{\Psi}}$ for the inverse Rosenblatt transform regardless of
whether the transform starts at $X_{i}$ or $Z_{i}$. We further write
${\boldsymbol{\Psi}}^{({\boldsymbol{Z}})}=(\Psi^{(m+1)},\ldots,\Psi^{(m+n)})$,
so that
${\boldsymbol{Z}}={\boldsymbol{\Psi}}^{({\boldsymbol{Z}})}(X_{i},{\boldsymbol{V}})$
or
${\boldsymbol{Z}}={\boldsymbol{\Psi}}^{({\boldsymbol{Z}})}(Z_{i},{\boldsymbol{V}})$
(depending on whether the inverse Rosenblatt transform starts at $X_{i}$ or
$Z_{i}$). Specifically, following the above process, we can define the
stressed loss models for stresses $Z_{i,{\varepsilon}}$ and
$X_{i,{\varepsilon}}$ respectively by

$\displaystyle L^{{\boldsymbol{\Psi}}}_{{\varepsilon}}(Z_{i})$
$\displaystyle:=\sum_{j\in{\mathcal{M}}}g_{j}\left({\boldsymbol{\Psi}}^{({\boldsymbol{Z}})}\left(Z_{i,{\varepsilon}},{\boldsymbol{V}}\right)\right)\,{\mathds{1}}_{\\{\Psi^{(j)}(Z_{i,{\varepsilon}},{\boldsymbol{V}})\leq
d_{j}\\}}\quad\text{and}\quad$ $\displaystyle
L^{{\boldsymbol{\Psi}}}_{{\varepsilon}}(X_{i})$
$\displaystyle:=\sum_{j\in{\mathcal{M}}}g_{j}\left({\boldsymbol{\Psi}}^{({\boldsymbol{Z}})}\left(X_{i,{\varepsilon}},{\boldsymbol{V}}\right)\right)\,{\mathds{1}}_{\\{\Psi^{(j)}(X_{i,{\varepsilon}},{\boldsymbol{V}})\leq
d_{j}\\}}\,.$

With these building blocks in place, we can now define the cascade sensitivity
measure in the specific context of this paper.

######  Definition 3.2 (Cascade Sensitivity)

The _cascade sensitivity_ to the risk factor $Z_{i}$ and $X_{i}$ for a risk
measure $\rho$ is defined by, respectively,

$\mathcal{C}_{Z_{i}}\,[\,\rho\,]:=\frac{\partial}{\partial{\varepsilon}}\,\rho\big{(}\,L^{{\boldsymbol{\Psi}}}_{\varepsilon}(Z_{i})\,\big{)}\Big{|}_{{\varepsilon}=0}\,,\quad\text{and}\quad\mathcal{C}_{X_{i}}\,[\,\rho\,]:=\frac{\partial}{\partial{\varepsilon}}\,\rho\big{(}\,L^{{\boldsymbol{\Psi}}}_{\varepsilon}(X_{i})\,\big{)}\Big{|}_{{\varepsilon}=0}\,,$
(5)

whenever the derivative exists.

Note that if the cascade sensitivity is exists, it is independent of the
choice of Rosenblatt transform, see Prop. 3.6 in Pesenti et al. (2021). In
order to establish existence, in this section we make the assumption that the
inverse Rosenblatt transforms are differentiable and locally monotone in their
first argument. This means that stressing a model input leads to perturbation
of elements of ${\boldsymbol{X}}$ that makes them ${\mathbb{P}}$-a.s. greater
(or smaller) than the original input $X_{j}$.

Let $\kappa_{\varepsilon}$ be a stress function and $Y,~{}Y_{{\varepsilon}}$
be such that either $Y:=Z_{i},~{}Y_{i,{\varepsilon}}:=Z_{i,{\varepsilon}}$ or
$Y:=X_{i},~{}Y_{i,{\varepsilon}}:=X_{i,{\varepsilon}}$. Let
${\boldsymbol{\Psi}}$ be a differentiable inverse Rosenblatt transform
starting at $Y$, such that
$({\boldsymbol{X}},{\boldsymbol{Z}})={\boldsymbol{\Psi}}(Y,{\boldsymbol{V}})$.
Then, for each $j\in{\mathcal{M}}$, one of the following holds

  1. $(a)$

for all ${\varepsilon}$ in a neighbourhood of 0, it holds
$\Psi^{(j)}\left(Y_{i,{\varepsilon}},{\boldsymbol{V}}\right)\geq X_{j}$
${\mathbb{P}}$-a.s.; or

  2. $(b)$

for all ${\varepsilon}$ in a neighbourhood of 0, it holds
$\Psi^{(j)}\left(Y_{i,{\varepsilon}},{\boldsymbol{V}}\right)\leq X_{j}$
${\mathbb{P}}$-a.s.

In the case (a) we denote $c(\kappa;\;j)=1$ and in the case (b)
$c(\kappa;\;j)=-1$.

With these assumptions in place, we can now obtain explicit formulas for the
cascade sensitivity measure of Definition 3.2. In Theorems 3.3 and 3.4 below
we deal with the case of ${\mathrm{ES}}$, while formulas for ${\mathrm{VaR}}$
are given in Appendix 7.1. We observe that the cascade sensitivity to both
$X_{i}$ and $Z_{i}$ entails a decomposition, reflecting the indirect
contribution of the vector being stressed via the other inputs
$X_{j},~{}Z_{k}$.

######  Theorem 3.3 (Cascade Sensitivity ES to $X_{i}$)

Let Assumptions 2.3, 2.3 and 3 (for $Y=X_{i}$) be fulfilled for the stressed
model $L^{{\boldsymbol{\Psi}}}_{\varepsilon}(X_{i})$ and $\alpha\in(0,1)$.
Denote $\Psi_{1}^{(j)}(x,{\boldsymbol{v}}):=\frac{\partial}{\partial
x}\Psi^{(j)}(x,{\boldsymbol{v}})$. Then, the cascade sensitivity for
${\mathrm{ES}}_{\alpha}$ to input $X_{i}$ is given by

$\displaystyle{\mathcal{C}}_{X_{i}}\,[\,{\mathrm{ES}}_{\alpha}\,]$
$\displaystyle=\sum_{j\in{\mathcal{M}}}{\mathcal{C}}_{X_{i},X_{j}}+\sum_{k\in{\mathcal{N}}}{\mathcal{C}}_{X_{i},Z_{k}},$

where, for all $k\in{\mathcal{N}}$,

${\mathcal{C}}_{X_{i},Z_{k}}=\sum_{j\in{\mathcal{M}}}\left.{\mathbb{E}}\left[{\mathfrak{K}}(X_{i})\,\partial_{k}\,g_{j}({\boldsymbol{Z}})\Psi_{1}^{(m+k)}(X_{i},{\boldsymbol{V}}){\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}}~{}\right|~{}L\geq{q_{\alpha}}\right]\,,$

and for all $j\in{\mathcal{M}}$,

${\mathcal{C}}_{X_{i},X_{j}}=-\left.\frac{c(\kappa;\,j)f_{j}(d_{j})}{1-\alpha}{\mathbb{E}}\left[{\mathfrak{K}^{-1}}(X_{i})\Psi_{1}^{(j)}(X_{i},{\boldsymbol{V}})\left(\big{(}L-c(\kappa;\,j)g_{j}({\boldsymbol{Z}})\,-{q_{\alpha}}\big{)}_{+}-\big{(}L-{q_{\alpha}}\big{)}_{+}\right)\,\right|\,X_{j}=d_{j}\right]\,.$

######  Theorem 3.4 (Cascade Sensitivity ES to $Z_{i}$)

Let Assumptions 2.3, 2.3 and 3 (for $Y=Z_{i}$) be fulfilled for the stressed
model $L^{{\boldsymbol{\Psi}}}_{\varepsilon}(Z_{i})$ and $\alpha\in(0,1)$.
Then, the cascade sensitivity for ${\mathrm{ES}}_{\alpha}$ to input $Z_{i}$ is
given by

$\displaystyle{\mathcal{C}}_{Z_{i}}\,[\,{\mathrm{ES}}_{\alpha}\,]$
$\displaystyle=\sum_{j\in{\mathcal{M}}}{\mathcal{C}}_{Z_{i},X_{j}}+\sum_{k\in{\mathcal{N}}}{\mathcal{C}}_{Z_{i},Z_{k}},$

where, for all $k\in{\mathcal{N}}$,

${\mathcal{C}}_{Z_{i},Z_{k}}=\sum_{j\in{\mathcal{M}}}\left.{\mathbb{E}}\left[{\mathfrak{K}}(Z_{i})\,\partial_{k}\,g_{j}({\boldsymbol{Z}})\Psi_{1}^{(m+k)}(Z_{i},{\boldsymbol{V}}){\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}}~{}\right|~{}L\geq{q_{\alpha}}\right]\,,$

and for $j\in{\mathcal{M}}$,

${\mathcal{C}}_{Z_{i},X_{j}}=-\left.\frac{c(\kappa;\,j)f_{j}(d_{j})}{1-\alpha}{\mathbb{E}}\left[{\mathfrak{K}^{-1}}(Z_{i})\Psi_{1}^{(j)}(Z_{i},{\boldsymbol{V}})\left(\big{(}L-c(\kappa;\,j)g_{j}({\boldsymbol{Z}})\,-{q_{\alpha}}\big{)}_{+}-\big{(}L-{q_{\alpha}}\big{)}_{+}\right)\,\right|\,X_{j}=d_{j}\right]\,.$

To calculate the cascade sensitivities, we need the derivative of the inverse
Rosenblatt transform. This calculation is simplified by noting that the value
of the cascade sensitivity is independent of the specific choice of Rosenblatt
transform. Hence, when calculating, for example,
$\Psi_{1}^{(j)}(X_{i},{\boldsymbol{V}})$, we can without loss of generality
use the standard construction (Rüschendorf and de Valk 1993)
$\Psi^{(j)}(X_{i},{\boldsymbol{V}})=F_{X_{j}|X_{i}}^{-1}(V_{1}|X_{i})$ in
Theorem 3.3 – analogously if $Z_{i}$ is being stressed (Theorem 3.4). As a
result, it is sufficient to consider the derivatives of inverse Rosenblatt
transforms corresponding to the bivariate dependence structure of, e.g.,
$(X_{i},X_{j})$. If the bivariate copula between the risk factors are known,
analytical expressions for the required derivatives may be available. We refer
also to Pesenti et al. (2021), where the formulas given below for the Gaussian
and t copulas are derived.

For simplicity of presentation, we only provide the expressions for
$\Psi_{1}^{(j)}(X_{i},V)$, where $V$ is a suitably defined random variable
such that $X_{j}=\Psi^{(j)}(X_{i},V)$. The formulas for
$\Psi_{1}^{(j)}(Z_{i},V)$, $\Psi_{1}^{(m+k)}(X_{i},V)$, and
$\Psi_{1}^{(m+k)}(Z_{i},V)$, for $j\in{\mathcal{M}}$, $k\in{\mathcal{N}}$
follow analogously.

######  Proposition 3.5 (Bivariate Inverse Rosenblatt Transform)

Denote by $\Phi,~{}\phi$, the distribution function and density of a standard
normal variable, and by $t_{\nu},~{}s_{\nu}$ the distribution function and
density of a t-distributed random variable with $\nu$ degrees of freedom.

  1. 1.

Assume $(X_{i},X_{j})$ follows a Gaussian copula with correlation parameter
$r_{ij}$ and define $Y_{i}:=\Phi^{-1}(F_{i}(X_{i}))$ and
$Y_{j}:=\Phi^{-1}(F_{j}(X_{j}))$. Then,

$\Psi_{1}^{(j)}(X_{i},V)=r_{ij}\;\frac{f_{i}(X_{i})}{\phi\left(Y_{i}\right)}\,\frac{\phi\left(Y_{j}\right)}{f_{j}(X_{j})}\,,$

  2. 2.

Assume $(X_{i},X_{j})$ follows a t copula with correlation parameter $r_{ij}$
and $\nu$ degrees of freedom and define $Y_{i}:=t_{\nu}^{-1}(F_{i}(X_{i}))$
and $Y_{j}:=t_{\nu}^{-1}(F_{j}(X_{j}))$. Then,

$\Psi_{1}^{(j)}(X_{i},V)=\left(r_{ij}+\frac{Y_{i}\,Y_{j}-r_{ij}Y_{i}^{2}}{\nu+Y_{i}^{2}}\right)\frac{f_{i}(X_{i})}{s_{\nu}(Y_{i})}\frac{s_{\nu}(Y_{j})}{f_{j}(X_{j})}\,.$

  3. 3.

Assume $(X_{i},X_{j})$ follows a Archimedean copula with generator
$\psi\colon[0,+\infty]\to[0,1]$, i.e., the copula is given by

${\textrm{C}}(u_{1},u_{2})=\psi\left(\psi^{-1}(u_{1})+\psi^{-1}(u_{2})\right)\,,\quad
u_{1},u_{2}\in[0,1]\,,$

where $\psi^{-1}$ denotes the inverse of the generator $\psi$. Then, for
$i\neq j$

$\Psi_{1}^{(j)}(X_{i},V)=\frac{\dot{\psi}\left(\psi^{-1}(U_{j})\right)}{\dot{\psi}\left(\psi^{-1}(U_{i})\right)}\left(\frac{\dot{\psi}\left(\psi^{-1}(U_{i})+\psi^{-1}(U_{j})\right)}{\ddot{\psi}\left(\psi^{-1}(U_{i})+\psi^{-1}(U_{j})\right)}\;\frac{\ddot{\psi}\left(\psi^{-1}(U_{i})\right)}{\dot{\psi}\left(\psi^{-1}(U_{i})\right)}-1\right)\frac{f_{i}(X_{i})}{f_{j}(X_{j})}\,,$

where $U_{j}:=F_{j}(X_{j})$, $\dot{\psi}(x):=\frac{\partial}{\partial
x}\psi(x)$, and $\ddot{\psi}(x):=\frac{\partial}{\partial x}\dot{\psi}(x)$.

##  4 Sensitivity to Discrete Random Variables

In this section, we adapt the techniques developed so far, to calculate
differential sensitivities to discrete risk factors. Given the different
portfolio structure we consider here, we change notation to avoid confusion
with previous sections. We consider the loss model

$T:=h\left(W,\,{\boldsymbol{Y}}\right)\,,$ (6)

where ${\boldsymbol{Y}}:=(Y_{1},\dots,Y_{d})$, the function
$h\colon{\mathbb{R}}^{d+1}\to{\mathbb{R}}$ is differentiable, and $W$ is a
discrete random variable which sensitivity we aim to assess. Such a
sensitivity calculation presents both technical and conceptual challenges.
While $h$ is differentiable, the corresponding differential (or infinitesimal
increment) in its first argument is hard to interpret, given the discreteness
of $W$. Here we propose to calculate the differential sensitivity with respect
to a continuous variable, from which $W$ is obtained via a (discontinuous)
transformation. In other words, we exchange the problem of discreteness with
the one of non-differentiability, which we have established.

We assume that $W$ takes values $w_{1}<\cdots<w_{r}$ with ${\mathbb{P}}(W\leq
w_{k})=p_{k}$, $k=1,\ldots,r$, such that $0=:p_{0}<p_{1}<\cdots<p_{r}=1$.
Next, we rewrite the loss model (6) into a form analogous to (1). For this,
let $V\sim\mathrm{U}(0,1)$ be independent of $(W,\,{\boldsymbol{Y}})$ and
define the uniform random variable $U$ by

$U:=\tilde{F}_{W}(W;V)\,,$

where $\tilde{F}_{W}(w;\lambda):={\mathbb{P}}(W<w)+\lambda{\mathbb{P}}(W=w)$
is the generalised distributional transform of $W$ (Rüschendorf 2013). It then
follows that $U\sim\mathrm{U}(0,1)$, $U$ is comonotonic to $W$, and

$\displaystyle
W=F_{W}^{-1}(U)=\sum_{k=1}^{r}w_{k}{\mathds{1}}_{\\{p_{k-1}<U\leq
p_{k}\\}}\,,\quad{\mathbb{P}}\text{-a.s.}\,.$

Then, following some manipulations, the loss model admits the form:

$T=\sum_{k=1}^{r}h\left(w_{k},{\boldsymbol{Y}}\right){\mathds{1}}_{\\{p_{k-1}<U\leq
p_{k}\\}}=\sum_{k=1}^{r}\Delta_{k}\,h\left(W,{\boldsymbol{Y}}\right){\mathds{1}}_{\\{U\leq
p_{k}\\}}\,,$

where
$\Delta_{k}\,h\left(W,{\boldsymbol{Y}}\right):=h\left(w_{k},{\boldsymbol{Y}}\right)-h\left(w_{k+1},{\boldsymbol{Y}}\right)$,
for $k=1,\ldots r-1$, and
$\Delta_{r}h\left(W,{\boldsymbol{Y}}\right):=h(w_{r},{\boldsymbol{Y}})$.

We next stress the portfolio loss $T$ with respect to $W$ by applying a stress
function to $U$. Hence, we write the stressed model as follows:

$T_{W,{\varepsilon}}:=\sum_{k=1}^{r}\,\Delta_{k}\,h\left(W,{\boldsymbol{Y}}\right){\mathds{1}}_{\\{\kappa_{\varepsilon}(U)\leq
p_{k}\\}}\,.$ (7)

Stressing the uniform variable that generates $W$ allows for a cohesive
stress, given the comonotonicity of $(W,U)$. Next, we define the
differentiable sensitivity to $W$ via a stress on $U$ by:

$\tilde{{\mathcal{S}}}_{W}\,[\,\rho\,]:=\frac{\partial}{\partial{\varepsilon}}\rho(T_{W,{\varepsilon}})\Big{|}_{{\varepsilon}=0}\,.$

Formulas for this sensitivity are given in the following result.

######  Theorem 4.1 (Marginal Sensitivity – Discrete)

Let Assumptions 2.3 and 2.3 be fulfilled for the loss model (6) and for a
fixed $\alpha\in(0,1)$. Then the sensitivity for VaR to the discrete input $W$
is

$\tilde{{\mathcal{S}}}_{W}\,[\,{\mathrm{VaR}}_{\alpha}\,]=\frac{c(\kappa)}{f({q_{\alpha}})}\sum_{k=1}^{r}{\mathfrak{K}}^{-1}(p_{k}){\mathbb{E}}\left[\left({\mathds{1}}_{\\{T\leq{q_{\alpha}}+c(\kappa)\Delta_{k}\,h(W,{\boldsymbol{Y}})\\}}-{\mathds{1}}_{\\{T\leq{q_{\alpha}}\\}}\right)~{}|~{}W=w_{k}\right]\,,$

where, for simplicity of notation, ${q_{\alpha}}$ is the $\alpha$-quantile of
$T$ and $f$ its density. The sensitivity for ES to the discrete input $W$ is

$\tilde{{\mathcal{S}}}_{W}\,[\,{\mathrm{ES}}_{\alpha}\,]=-\frac{c(\kappa)}{1-\alpha}\sum_{k=1}^{r}{\mathfrak{K}}^{-1}(p_{k}){\mathbb{E}}\left[\Big{(}T-c(\kappa)\Delta_{k}\,h(W,{\boldsymbol{Y}})-{q_{\alpha}}\Big{)}_{+}-(T-{q_{\alpha}})_{+}~{}|~{}W=w_{k}\right]\,.$

In the next example, we apply Theorem 4.1 for the ES-sensitivity calculation
of the frequency variable in a compound loss model. Compound distributions are
canonical tools in modelling insurance claims, as well as credit and
operational risk losses, and the impact of the choice of frequency
distribution is well attested, see e.g., McNeil et al. (2015).

###### Example 4.2

We represent by $T=h(W,{\boldsymbol{Y}})$ a compound random variable.
Specifically, we set $r=d+1$ and assume that $W$ is a discrete loss frequency,
taking values in $\\{w_{1}=0,\dots,w_{d+1}=d\\}$, while the $d$ elements of
${\boldsymbol{Y}}=(Y_{1},\dots,Y_{d})$ are loss severities. The variable $W$
has distribution ${\mathbb{P}}(W\leq k-1)=p_{k},~{}k=1,\dots,d+1$.
Furthermore, we assume that $Y_{1},\dots,Y_{d}$ are i.i.d., continuously
distributed with $Y_{1}\sim F_{Y}$, and independent of $W$. The portfolio loss
is:

$T=h(W,{\boldsymbol{Y}})=\sum_{\ell=1}^{W}Y_{\ell}\,,$

with the understanding that for $W=0$ we have $T=0$. Our aim is to calculate
the sensitivity of the portfolio’s ES to the frequency variable, i.e. to
evaluate the quantity
$\tilde{{\mathcal{S}}}_{W}\,[\,{\mathrm{ES}}_{\alpha}\,]$, and to compare this
with the impact of the vector of loss severities,
$\tilde{{\mathcal{S}}}_{{\boldsymbol{Y}}}\,[\,{\mathrm{ES}}_{\alpha}\,]$,
which are defined below.

The sensitivity $\tilde{{\mathcal{S}}}_{W}\,[\,{\mathrm{ES}}_{\alpha}\,]$ is
evaluated by application of Theorem 4.1. As before let $W=F_{W}^{-1}(U)$. To
stress $U$ we need to specify a stress function
$\kappa_{\varepsilon}(u):(0,1)\to(0,1)$. Let
$\kappa_{\varepsilon}(u):=\Phi\left(\Phi^{-1}(u)+{\varepsilon}\right),$ where
$\Phi$ is the standard normal distribution. This choice is consistent with the
well-known Wang Transform (Wang 2000) in risk measure theory and satisfies the
conditions of Definition 2.1, with $c(\kappa)=1$. Then, for
$U_{{\varepsilon}}:=\Phi\left(\Phi^{-1}(U)+{\varepsilon}\right)$ we obtain,
using Theorem 4.1 and after a few manipulations not documented here, that

$\displaystyle\tilde{{\mathcal{S}}}_{W}\,[\,{\mathrm{ES}}_{\alpha}\,]=\sum_{k=1}^{d}\big{(}v(p_{k})-v(p_{k+1})\big{)}\,{\mathbb{E}}\left[\left(\sum_{\ell=1}^{k}Y_{\ell}-{q_{\alpha}}\right)_{+}\right]\,,$

where $v(p):=\frac{\phi\left(\Phi^{-1}(p)\right)}{1-\alpha}$, $p\in[0,1]$.
Hence, the sensitivity becomes a linear combination of the stop-loss terms
${\mathbb{E}}[(\sum_{\ell=1}^{k}Y_{\ell}-{q_{\alpha}})_{+}]$, with the
coefficient weights driven by the distribution of the loss frequency $W$.

We now turn our attention to stressing the loss severities ${\boldsymbol{Y}}$.
We choose to stress all elements of ${\boldsymbol{Y}}$ at the same time, using
a stress function consistent with the one used for $W$, i.e. the same
$\kappa_{\varepsilon}$. Specifically, for
$U_{\ell}:=F_{Y}(Y_{\ell}),~{}\ell=1,\dots,d$, we define the stressed
portfolio

$\displaystyle T_{{\boldsymbol{Y}},{\varepsilon}}$
$\displaystyle:=\sum_{\ell=1}^{W}F_{Y}^{-1}\left(\kappa_{\varepsilon}(U_{\ell})\right)=\sum_{k=1}^{d}{\mathds{1}}_{\\{W=k\\}}\sum_{\ell=1}^{k}F_{Y}^{-1}\left(\kappa_{\varepsilon}(U_{\ell})\right)\,$

and calculate the sensitivity

$\tilde{{\mathcal{S}}}_{{\boldsymbol{Y}}}\,[\,{\mathrm{ES}}_{\alpha}\,]:=\frac{\partial}{\partial{\varepsilon}}{\mathrm{ES}}_{\alpha}(T_{{\boldsymbol{Y}},{\varepsilon}})\Big{|}_{{\varepsilon}=0}\,.$

By the pointwise continuity of the mapping ${\varepsilon}\mapsto
T_{{\boldsymbol{Y}},{\varepsilon}}$ we can calculate
$\tilde{{\mathcal{S}}}_{{\boldsymbol{Y}}}\,[\,{\mathrm{ES}}_{\alpha}\,]$ by
standard methods (Hong and Liu 2009), yielding:

$\tilde{{\mathcal{S}}}_{{\boldsymbol{Y}}}\,[\,{\mathrm{ES}}_{\alpha}\,]:=\sum_{k=1}^{d}{\mathbb{P}}(W=k)\sum_{\ell=1}^{k}{\mathbb{E}}\left[{\mathds{1}}_{\\{T>q_{\alpha}\\}}\frac{v(U_{\ell})}{f_{Y}(Y_{\ell})}~{}|~{}W=k\right]\,.$

We now calculate the sensitivities
$\tilde{{\mathcal{S}}}_{W}\,[\,{\mathrm{ES}}_{\alpha}\,]$ and
$\tilde{{\mathcal{S}}}_{{\boldsymbol{Y}}}\,[\,{\mathrm{ES}}_{\alpha}\,]$ for a
model with the following baseline assumptions. The confidence level of the
risk measure is $\alpha=0.95$. The frequency $W$ follows a Negative Binomial
distribution with mean ${\mathbb{E}}[W]=5$ and over-dispersion
$\rm{Var}(W)/{\mathbb{E}}[W]=2.5$, truncated at the 99.9th percentile. The
severities $Y_{\ell}$ follow Gamma distributions with shape parameter
$\theta=5$, corresponding to a skewness coefficient of $0.894$. With these
choices, we find that the sensitivities, scaled by the portfolio risk, take
values
$\frac{\tilde{{\mathcal{S}}}_{W}\,[\,{\mathrm{ES}}_{\alpha}\,]}{{\mathrm{ES}}_{\alpha}(T)}=0.414$
and
$\frac{\tilde{{\mathcal{S}}}_{{\boldsymbol{Y}}}\,[\,{\mathrm{ES}}_{\alpha}\,]}{{\mathrm{ES}}_{\alpha}(T)}=0.429$.
This indicates that the compound sum $T$ is approximately equally sensitive to
the loss frequency and severity.

Figure 1: Changes in the scaled ES-sensitivity to the frequency (blue) and
severity (red) of a compound Negative Binomial-Gamma distribution. The
vertical dashed line in each plot represent baseline assumptions.

In Figure 1 we depict how the scaled sensitivities change after varying the
baseline assumptions, one at a time, regarding frequency mean, frequency over-
dispersion, the skewness of the severity distribution, and the confidence
level of the ES risk measure. In each plot the baseline assumption is
indicated by a vertical dashed line. We observe that, as the frequency mean
increases, the importance of severities dominates, given the larger overall
number of individual losses. On the other hand, when the frequency over-
dispersion increases, the importance of frequency dominates, since the
variance of the frequency distribution becomes the key risk driver.
Furthermore, as one would expected, the sensitivity of the severities
${\boldsymbol{Y}}$ increases in the skewness, which reflects a riskier loss
profile. Finally, as the confidence level increases, severities become more
important than the frequency $W$, representing a more pronounced impact on the
extreme tail of the portfolio loss.

##  5 Application to reinsurance credit risk modelling

Reinsurance credit risk modelling represents a prominent example where credit
risk exposures are non-granular and inhomogeneous. Insurers buy reinsurance
products in order to transfer some of the risk of (typically) higher than
expected claim amounts to a third party. By taking on insurers’ excess
liabilities, the reinsurance market thus operates as an industry-wide risk
pooling arrangement (Albrecher et al. 2017). Credit risk then arises from the
possibility that, in the event of high (industry) losses, reinsurers will not
be able to make good on their obligations to insurers.

Reinsurance credit risk has two features particularly relevant to our setting.
First, dependence is of primary importance. Different reinsurers’ ability to
fulfil obligations is highly dependent on each other, given the systemic
impact of overall (re)insurance market conditions and industry shocks. As a
result, reinsurers’ default indicators should also be considered dependent on
insurers’ gross (i.e. before-reinsurance) losses; hence one needs to account
for the event that reinsurers default precisely at those times when insurers
need them most. Second, reinsurance credit risk exposures are highly
inhomogeneous. Different reinsurers often reinsure different lines of business
at different levels of extreme loss. Furthermore, the credit rating of
reinsurers varies and insurers typically transfer the most extreme layers of
their gross losses to a small number of highly rated reinsurers – while this
is a rational strategy, it creates non-trivial concentration effects. The
concern with the risk of reinsurance defaults, and particularly with their
dependence, has been thoroughly reflected in actuarial modelling practice (Ter
Berg 2008, Britt and Krvavych 2009).

Here we present a numerical example of differential sensitivity analysis to
reinsurance defaults, working with an illustrative model of reinsurance credit
risk. In equation (1), we interpret the terms as follows:

  * •

$L$ is the total reinsurance credit risk loss for an insurer.

  * •

${\boldsymbol{Z}}=(Z_{1},\dots,Z_{n})$ are the gross losses of the insurer,
from its $n=12$ lines of business (LoB).

  * •

$g_{j}({\boldsymbol{Z}}),~{}j\in{\mathcal{M}}$ are the reinsurance recoveries
expected from each of $m=8$ reinsurers.

  * •

$\\{X_{j}\leq d_{j}\\}$ is the event that the $j$-th reinsurer defaults.

The 12 LoB are marginally Lognormal distributed with the same mean and
coefficient of variation (CoV) given in Table 2, and which are consistent with
the Solveny II standard formula parameters (Lloyd’s 2022). In specifying the
form of the $g_{j}$s we make the simplifying assumption that all reinsurance
contracts bought consist of reinsurance layers on the gross losses
$Z_{1},\dots,Z_{12}$.

Table 2: Name of Lines of business (LoB) and Coefficient of variation (CoV) (Source: Lloyd’s (2022)). LOB Name | LoB | CoV
---|---|---
Direct and Proportional Motor Vehicle Liability | $Z_{1}$ | $0.1$
Direct and Proportional Other Motor | $Z_{2}$ | $0.08$
Direct and Proportional Marine, Aviation and Transportation | $Z_{3}$ | $0.15$
Direct and Proportional Fire & Other Damage to Property | $Z_{4}$ | $0.08$
Direct and Proportional General Liability | $Z_{5}$ | $0.14$
Direct and Proportional Credit & Suretyship | $Z_{6}$ | $0.19$
Direct and Proportional Legal Expenses | $Z_{7}$ | $0.083$
Direct and Proportional Assistance | $Z_{8}$ | $0.064$
Direct and Proportional Miscellaneous Financial Loss | $Z_{9}$ | $0.13$
Non-Proportional Casualty Reinsurance | $Z_{10}$ | $0.17$
Non-Proportional Marine, Aviation and Transportation Reinsurance | $Z_{11}$ | $0.17$
Non-Proportional Property Reinsurance | $Z_{12}$ | $0.17$

We assume that each of the first 6 reinsurers covers a layer from two LoBs,
with deductibles $s_{j,k}$ and limit $t_{j,k}$. Each of reinsurers 7 and 8
covers a higher layer from 6 LoBs. Specifically, we have:

$\displaystyle g_{j}({\boldsymbol{Z}})$
$\displaystyle=\sum_{k=2j-1}^{2i}\min\left\\{\left(Z_{k}-s_{j,k}\right)_{+},\;t_{j,k}\right\\}\,,\quad\text{for}\quad
j=1,\ldots,6\,,$ $\displaystyle g_{7}({\boldsymbol{Z}})$
$\displaystyle=\sum_{k=1}^{6}\min\left\\{\left(Z_{k}-s_{7,k}\right)_{+},\,t_{7,k}\right\\},\quad\text{and}\quad$
$\displaystyle g_{8}({\boldsymbol{Z}})$
$\displaystyle=\sum_{k=7}^{12}\min\left\\{\left(Z_{k}-s_{8,k}\right)_{+},\,t_{8,k}\right\\}\,.$

The deductibles and limits are such that the first six reinsurers cover losses
between the 55% and 85% quantile, whereas the last two reinsurers cover the
losses between the 85% and the 95% quantile, i.e.,

$\displaystyle s_{j,k}$
$\displaystyle=F_{Z_{k}}^{-1}(0.55)\quad\text{and}\quad
t_{j,k}=F_{Z_{k}}^{-1}(0.85)-s_{j,k}\,,\quad\text{for}\quad j=1,\ldots,6\,,$
$\displaystyle s_{j^{\prime},k}$
$\displaystyle=F_{Z_{k}}^{-1}(0.85)\quad\text{and}\quad
t_{j^{\prime},k}=F_{Z_{k}}^{-1}(0.95)-s_{j,k}\,,\quad\text{for}\quad
j^{\prime}=7,8\,.$

Finally, the default probabilities are set at 1.5% for the first 6 reinsurers
and 1% for reinsurers 7 and 8. We assume that the random vector
$({\boldsymbol{X}},{\boldsymbol{Z}})$ is dependent with a t-copula with 4
degrees of freedom, such that the correlation matrix of ${\boldsymbol{Z}}$
satisfies Solvency II assumptions (Lloyd’s 2022), while the random vector
${\boldsymbol{X}}$ has a homogeneous correlation matrix such that
$\mathrm{Corr}(X_{i},X_{j})=0.05$. The joint dependence of
$({\boldsymbol{X}},{\boldsymbol{Z}})$ is effected via a t-distribution factor
model; further details are given in Appendix 9.

Figure 2: Histogram of the insurer’s total reinsurance credit risk loss
conditional on a positive loss, i.e., $L|L>0$. Vertical lines are the
unconditional VaR and ES at level $\alpha=0.975$.

The distribution of the total credit risk loss $L$ is evaluated by Monte-Carlo
simulation. Specifically, since almost all scenarios of
$({\boldsymbol{X}},{\boldsymbol{Z}})$ result in a credit loss of zero, i.e., a
realisation $\\{L=0\\}$, we generated a dataset of size 500,000 (keeping track
of the total number of simulations), in which all realisations satisfy $L>0$.
The probability that $L>0$ is 5.044% in our dataset. Figure 2 depicts a
histogram of the insurer’s total credit risk loss $L$ conditional that a loss
occurred. We also report the unconditional VaR and ES at level $\alpha=0.975$.
The skewness and multimodality of the loss distribution, driven by the
portfolio’s lack of homogeneity, are apparent.

We apply stresses on each of the risk factors $X_{i}$ and $Z_{i}$.
Specifically, we apply left-tail stresses (see Table 1) on the risk factors
driving defaults, i.e.,
$X_{i,{\varepsilon}}:=X_{i}+{\varepsilon}\left(X_{i}-F_{X_{i}}^{-1}(0.2)\right){\mathds{1}}_{\\{X_{i}\leq
F_{X_{i}}^{-1}(0.2)\\}}$, $i=1,\ldots,8$. These stresses increase the
probability of reinsurance defaults, though in a more complex way compared to
Example 2.6. For each LoB, we consider a right-tailed stress
$Z_{i,{\varepsilon}}:=Z_{i}+{\varepsilon}\left(Z_{i}-F_{Z_{i}}^{-1}(0.8)\right){\mathds{1}}_{\\{Z_{i}\geq
F_{Z_{i}}^{-1}(0.8)\\}}$, $i=1,\ldots,12$, which increases the loss quantiles
of $Z_{i}$, beyond its 80% quantile.

We calculate the sensitivities with respect to the VaR and ES risk measures at
level $\alpha=0.975$, according to Theorems 2.3 and 2.4. To calculate the
sensitivities to $Z_{i}$s, we require estimates of expectation conditioned on
the event $\\{L=q_{\alpha}\\}$ and $\\{L\geq q_{\alpha}\\}$. For estimating
the expectation conditional on the event of zero probability
$\\{L=q_{\alpha}\\}$, we use the $\delta$-estimator (Glasserman 2005).
Specifically, for $\delta>0$ with $0<\alpha-\delta,$ and $\alpha+\delta<1$, we
approximate the sensitivity of VaR to $Z_{i}$ by

$\widehat{{\mathcal{S}}}_{Z_{i}}\,[\,{\mathrm{VaR}}_{\alpha}\,]=\frac{1}{2\delta}\sum_{j=1}^{8}{\mathbb{E}}\Big{[}\,{\mathfrak{K}}(Z_{i})\partial_{i}\,g_{j}({\boldsymbol{Z}}){\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}}{\mathds{1}}_{\\{L\in(F^{-1}(\alpha-\delta),\,F^{-1}(\alpha+\delta))\\}}\,\Big{]}\,.$

Mathematically, we replace the conditioning event $\\{L={q_{\alpha}}\\}$ by an
event of probability $2\delta$, i.e. by
$\\{L\in(F^{-1}(\alpha-\delta),\,F^{-1}(\alpha+\delta)\\}$. A value of
$\delta=0.005$ was used throughout. We use our sample of
$({\boldsymbol{X}},{\boldsymbol{Z}},L~{}|~{}L>0)$, which contains 500,000
simulated scenarios, and estimate the sensitivities using bootstrap with
replacement and a bootstrap size of 450,000. The reported sensitivities are
averaged over 100 bootstrap estimates.

For estimating the sensitivities to each $X_{i}$, a different dataset is
simulated. Specifically, for each $j=1,\ldots,8$, we generate a dataset of
size 500,000, in which all realisations of
$({\boldsymbol{X}},{\boldsymbol{Z}})$ satisfy
$X_{j}\in\big{(}F_{X_{j}}^{-1}(d_{j}-\delta),\,F_{X_{j}}^{-1}(d_{j}+\delta)\big{)}$,
for small $\delta>0$. Again, sensitivities were estimated by bootstrapping 100
times with replacement and bootstrap size 450,000. Figures 3 and 4 display
violin plots of the sensitivities to $Z_{i}$ and to $X_{i}$ for both VaR and
ES. Again a value of $\delta=0.005$ is used as a baseline; the effect of this
choice on sensitivity estimates is discussed in the sequel (Figure 6).

Figure 3: Marginal sensitivity to $Z_{i}$s of VaR (left; with $\delta=0.005$)
and ES (right) with $\alpha=0.975$.

Figure 4: Marginal sensitivity to $X_{i}$s of VaR (left; with $\delta=0.005$)
and ES (right; with $\delta=0.005$) with $\alpha=0.975$.

In Figure 3, where the sensitivities to the $Z_{i}$s are plotted, we observe
that business line 6 has a large sensitivity for VaR, the LoB with the largest
CoV, see Table 2. In the right panel we observe that for ES the sensitivities
are ordered similarly to the CoV of the business lines, but with a larger
spread compared to the case of VaR – this could be attributed to the higher
tail-sensitivity of the ES measure. Indeed, LoB 6 has the largest sensitivity,
followed by 10, 11, and 12, which all have the same, second largest, CoV.
Furthermore, LoB 2, 4, 7, and 8, which have the smallest CoVs, have small
sensitivities for both VaR and ES.

In Figure 4, we depict the sensitivities to the $X_{i}$s. A similar picture
emerges, with the sensitivities for VaR (left panel) being very close together
and for ES (right panel) being more spread-out. For ES, we observe that
reinsurer 3, which has a layer on LoBs 5 and 6, and reinsurer 6, which has a
layer on LoBs 11 ad 12, have the largest sensitivities. These LoBs have large
sensitivities for ES, as seen in Figure 3 (right panel). Thus, a default of
these reinsurers would naturally have a large impact on the ES of the total
loss. We also see that reinsurers 7 and 8 have large sensitivities for ES.
This is in line with expectations, since reinsurer 7 and 8 take on the highest
layers (between the 85% and 95% quantile) of 6 business lines each.
Nonetheless, this concentration effect is not picked up by the VaR
sensitivity, as in the left panel the sensitivities to $X_{7},X_{8}$ are
rather low. This points to the importance of selecting an appropriately tail-
sensitive risk measure.

Figure 5: Marginal sensitivities of ES and different choices of $\alpha$
between 0.955 and 0.99. Left: sensitivity to $Z_{i}$s. Right: sensitivity to
$X_{i}$s.

Figure 5 depicts the sensitivities of ES for different choices of $\alpha$,
from 0.955 to 0.99. The left panel contains the sensitivities to the LoBs
($Z_{i}$) and the right panel the sensitivities to the reinsurers ($X_{i}$).
We observe that the ordering of the risk factors is mostly consistent with
respect to changes in confidence level. An exception to this are the
sensitivities to $X_{7}$ and $X_{8}$ which increase faster (relative to
others) with $\alpha$, as seen by the line crossings on the right panel. Once
again this demonstrates the increased impact of default risk concentration at
high loss quantiles.

Figure 6: Marginal sensitivities to $X_{i}$ and their sample standard
deviations for different choices of $\delta$. Top panels: Marginal sensitivity
for VaR (left) and for ES (right) both with $\alpha=0.975$. Bottom panels:
Sample standard deviation of the sensitivity estimators for VaR (left) and ES
(right).

Finally, in Figure 6 top panels, we show the sensitivities to $X_{i}$s for VaR
(left panel) and ES (right panel) with $\alpha=0.975$, using different choices
of $\delta$ for approximating the expectation conditional on
$\\{X_{i}=d_{i}\\}$. We observe that the estimates are very stable for
different choices of $\delta$. Furthermore, in the bottom panels of Figure 6
we plot the standard deviation of the sensitivity estimators, thus choosing
$\delta=0.005$ provides a suitable bias and variance trade-off.

##  6 Conclusion

Taking derivatives of model outputs in the direction of inputs is a
foundational process for interpreting complex computational models. However,
differential sensitivity measures typically require stringent assumptions on
differentiability and Lipschitz continuity of the model function. This
severely limits the scope of current methods of differential sensitivity
analysis. We address the problem by noting that, when inputs are uncertain –
as is the case in settings ranging from Monte Carlo simulation to algorithmic
prediction – a global view can be more appropriate than a local one. For a
global assessment, differentiation is required across the entire input space;
but then, it is not the derivative of the model function as such that is of
primary interest, but rather the derivative of a statistical functional of the
output. Still, extent literature on sensitivity analysis of risk measures
typically requires differentiability of the model aggregation function.

In this paper, we overcome current limitations in the literature and derive
expressions for derivatives of quantile-based risk measures of model outputs,
in a general setting where aggregation functions contain step functions and
thus are not Lipschitz continuous. The conditions we require are rather weak
and the sensitivity measures obtained admit representations as conditional
expected values, which allows their estimation by standard methods. There are
multiple potential applications of our methodology. We demonstrate
applications in the area of credit risk modelling, but also in assessing
sensitivity with respect to discrete random inputs. While our work is
applicable in principle to discontinuous (e.g., tree-based) predictive models,
addressing the idiosyncratic challenges of such exercises remains a topic for
future work.

The authors would like to thank Wen Yuan (William) Tang for his help in
implementing the numerical examples. SP gratefully acknowledges the support of
the Canadian Statistical Sciences Institute (CANSSI) and the Natural Sciences
and Engineering Research Council of Canada (NSERC) with funding reference
numbers DGECR-2020-00333 and RGPIN-2020-04289.

##  7 Additional Sensitivity Formulas

Here we provide additional results for marginal and cascade sensitivities,
which are omitted from the main body of the text for reasons of concision. In
Section 7.1 we deal with cascade sensitivities of VaR, while in Section 7.2 we
present results for a more general model than (1).

###  7.1 Cascade Sensitivities to VaR

Here we report the cascade sensitivity formulas for VaR.

######  Theorem 7.1 (Cascade Sensitivity VaR to $X_{i}$)

Let Assumptions 2.3, 2.3 and 3 (for $Y=X_{i})$ be fulfilled for the stressed
model $L^{{\boldsymbol{\Psi}}}_{\varepsilon}(X_{i})$ and given
$\alpha\in(0,1)$. Then, the cascade sensitivity for ${\mathrm{VaR}}_{\alpha}$
to input $X_{i}$ is given by,

$\displaystyle{\mathcal{C}}_{X_{i}}\,[\,{\mathrm{VaR}}_{\alpha}\,]$
$\displaystyle=\sum_{j\in{\mathcal{M}}}{\mathcal{C}}_{X_{i},X_{j}}+\sum_{k\in{\mathcal{N}}}{\mathcal{C}}_{X_{i},Z_{k}},$

where for all $k\in{\mathcal{N}}$,

${\mathcal{C}}_{X_{i},Z_{k}}\,=\sum_{j\in{\mathcal{M}}}\left.{\mathbb{E}}\left[{\mathfrak{K}}(X_{i})\,\partial_{k}g_{j}({\boldsymbol{Z}})\,\Psi_{1}^{(m+k)}(X_{i},{\boldsymbol{V}}){\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}}~{}\right|~{}L={q_{\alpha}}\right]\,,$

and for $j\in{\mathcal{M}}$,

${\mathcal{C}}_{X_{i},X_{j}}\,=\frac{c(\kappa;\,j)f_{j}(d_{j})}{f\left({q_{\alpha}}\right)}\;\;{\mathbb{E}}\left.\left[\,{\mathfrak{K}^{-1}}(X_{i})\Psi_{1}^{(j)}(X_{i},{\boldsymbol{V}})\left({\mathds{1}}_{\\{L\leq{q_{\alpha}}+c(\kappa;\,j)g_{j}({\boldsymbol{Z}})\\}}-{\mathds{1}}_{\\{L\leq{q_{\alpha}}\\}}\right)~{}\right|~{}X_{j}=d_{j}\,\right]\,.$

######  Theorem 7.2 (Cascade Sensitivity VaR to $Z_{i}$)

Let Assumptions 2.3, 2.3 and 3 (for $Y=Z_{i})$ be fulfilled for the stressed
model $L^{{\boldsymbol{\Psi}}}_{\varepsilon}(Z_{i})$ and given
$\alpha\in(0,1)$. Then, the cascade sensitivity for ${\mathrm{VaR}}_{\alpha}$
to input $Z_{i}$ is given by,

$\displaystyle{\mathcal{C}}_{Z_{i}}\,[\,{\mathrm{VaR}}_{\alpha}\,]$
$\displaystyle=\sum_{j\in{\mathcal{M}}}{\mathcal{C}}_{Z_{i},X_{j}}+\sum_{k\in{\mathcal{N}}}{\mathcal{C}}_{Z_{i},Z_{k}},$

where for all $k\in{\mathcal{N}}$,

${\mathcal{C}}_{Z_{i},Z_{k}}\,=\sum_{j\in{\mathcal{M}}}\left.{\mathbb{E}}\left[{\mathfrak{K}}(Z_{i})\,\partial_{k}g_{j}({\boldsymbol{Z}})\,\Psi_{1}^{(m+k)}(Z_{i},{\boldsymbol{V}}){\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}}~{}\right|~{}L={q_{\alpha}}\right]\,,$

and for $j\in{\mathcal{N}}$,

${\mathcal{C}}_{Z_{i},X_{j}}\,=\frac{c(\kappa;\,j)f_{j}(d_{j})}{f\left({q_{\alpha}}\right)}\;\;{\mathbb{E}}\left.\left[\,{\mathfrak{K}^{-1}}(Z_{i})\Psi_{1}^{(j)}(Z_{i},{\boldsymbol{V}})\left({\mathds{1}}_{\\{L\leq{q_{\alpha}}+c(\kappa;\,j)g_{j}({\boldsymbol{Z}})\\}}-{\mathds{1}}_{\\{L\leq{q_{\alpha}}\\}}\right)~{}\right|~{}X_{j}=d_{j}\,\right]\,.$

###  7.2 Sensitivity to General Loss Models

In this section, we generalise the loss model (1) to include cases where the
functions $g_{j}$ depends on both ${\boldsymbol{Z}}$ and ${\boldsymbol{X}}$.
Specifically, we let

$L:=\sum_{j\in{\mathcal{M}}}g_{j}({\boldsymbol{X}},{\boldsymbol{Z}}){\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}},$ (9)

where a stress on $X_{i}$ results in the stressed loss model

$L_{\varepsilon}(X_{i}):=\sum_{\stackrel{{\scriptstyle j\neq
i}}{{j\in{\mathcal{M}}}}}g_{j}(X_{i,{\varepsilon}},{\boldsymbol{X}}_{-i},{\boldsymbol{Z}}){\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}}+g_{i}(X_{i,{\varepsilon}}{\boldsymbol{X}}_{-i},{\boldsymbol{Z}}){\mathds{1}}_{\\{X_{i,{\varepsilon}}\leq
d_{j}\\}}\,.$ (10)

We only present the sensitivities to $X_{i}$, as the sensitivities to $Z_{i}$
are not impacted by the model generalisation. We observe in the next result
that the marginal sensitivity to $X_{i}$ of the loss model (9) accounts for
both the stress in the indicator and the stress via the functions $g_{j}$,
$j\in{\mathcal{M}}$.

######  Theorem 7.3 (Marginal Sensitivity – General Loss Model)

Let Assumptions 2.3 and 2.3 be fulfilled for the loss model (9) and for fixed
$\alpha\in(0,1)$. Then, the marginal sensitivity for ${\mathrm{VaR}}$ to
$X_{i}$ for loss model (9) is

$\displaystyle{\mathcal{S}}_{X_{i}}\,[\,{\mathrm{VaR}}_{\alpha}\,]$
$\displaystyle=\sum_{j\in{\mathcal{M}}}\,\left({\mathbb{E}}\left[{\mathfrak{K}}(X_{i})\partial_{i}g_{j}({\boldsymbol{X}},{\boldsymbol{Z}}){\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}}~{}\Big{|}~{}L={q_{\alpha}}\right]\right)$
$\displaystyle\quad+c(\kappa){\mathfrak{K}^{-1}}(d_{i})\frac{f_{i}(d_{i})}{f\left({q_{\alpha}}\right)}\,{\mathbb{E}}\left[\left({\mathds{1}}_{\\{L\leq{q_{\alpha}}+c(\kappa)g_{i}({\boldsymbol{X}},{\boldsymbol{Z}})\\}}-{\mathds{1}}_{\\{L\leq{q_{\alpha}}\\}}\right)~{}\Big{|}~{}X_{i}=d_{i}\right]\,.$

The marginal sensitivity for ES to $X_{i}$ for the loss model (9) is

$\displaystyle{\mathcal{S}}_{X_{i}}\,[\,{\mathrm{ES}}_{\alpha}\,]$
$\displaystyle=\sum_{j\in{\mathcal{M}}}\,\left({\mathbb{E}}\left[{\mathfrak{K}}(X_{i})\,\partial_{i}g_{j}({\boldsymbol{X}},{\boldsymbol{Z}})\,{\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}}~{}\Big{|}~{}L\geq{q_{\alpha}}\right]\right)$
$\displaystyle\quad-\frac{c(\kappa){\mathfrak{K}^{-1}}(d_{i})f_{i}(d_{i})}{1-\alpha}\,{\mathbb{E}}\left[\left(L-c(\kappa)g_{i}({\boldsymbol{X}},{\boldsymbol{Z}})-{q_{\alpha}}\right)_{+}-(L-{q_{\alpha}})_{+}~{}\Big{|}~{}X_{i}=d_{i}\right]\,.$

######  Theorem 7.4 (Cascade Sensitivity VaR – General Loss Model)

Let Assumptions 2.3, 2.3, and 3 (for $Y=X_{i})$ be fulfilled for the stressed
model (10) and given $\alpha\in(0,1)$. Then, the cascade sensitivity for
${\mathrm{VaR}}_{\alpha}$ to input $X_{i}$ is given by

$\displaystyle{\mathcal{C}}_{X_{i}}\,[\,{\mathrm{VaR}}_{\alpha}\,]$
$\displaystyle=\sum_{j\in{\mathcal{M}}}{\mathcal{C}}_{X_{i},X_{j}}+\sum_{k\in{\mathcal{N}}}{\mathcal{C}}_{X_{i},Z_{k}},$

where for all $k\in{\mathcal{N}}$,

${\mathcal{C}}_{X_{i},Z_{k}}\,=\sum_{j\in{\mathcal{M}}}\left.{\mathbb{E}}\left[{\mathfrak{K}}(X_{i})\,\partial_{m+k}g_{j}({\boldsymbol{X}},{\boldsymbol{Z}})\,\Psi_{1}^{(m+k)}(X_{i},{\boldsymbol{V}}){\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}}~{}\right|~{}L={q_{\alpha}}\right]\,,$

and for $j\in{\mathcal{M}}$,

$\displaystyle{\mathcal{C}}_{X_{i},X_{j}}\,$
$\displaystyle=\left(\sum_{r=1}^{m}\,{\mathbb{E}}\left[{\mathfrak{K}}(X_{i})\partial_{j}g_{r}({\boldsymbol{X}},{\boldsymbol{Z}})\Psi_{1}^{(j)}(X_{i},{\boldsymbol{V}}){\mathds{1}}_{\\{X_{r}\leq
d_{r}\\}}~{}\Big{|}~{}L={q_{\alpha}}\right]\right)$
$\displaystyle\quad+c(\kappa;\,j)\frac{f_{j}(d_{j})}{f\left({q_{\alpha}}\right)}\,{\mathbb{E}}\left[{\mathfrak{K}^{-1}}(X_{i})\Psi_{1}^{(j)}(X_{i},{\boldsymbol{V}})\left({\mathds{1}}_{\\{L\leq{q_{\alpha}}+c(\kappa;\,j)g_{i}({\boldsymbol{X}},{\boldsymbol{Z}})\\}}-{\mathds{1}}_{\\{L\leq{q_{\alpha}}\\}}\right)~{}\Big{|}~{}X_{j}=d_{j}\right]\,.$

######  Theorem 7.5 (Cascade Sensitivity ES – General Loss model)

Let Assumptions 2.3, 2.3, and 3 (for $Y=X_{i})$ be fulfilled for the stressed
model (10) and given $\alpha\in(0,1)$. Then, the cascade sensitivity for
${\mathrm{ES}}_{\alpha}$ to input $X_{i}$ is given by

$\displaystyle{\mathcal{C}}_{X_{i}}\,[\,{\mathrm{ES}}_{\alpha}\,]$
$\displaystyle=\sum_{j\in{\mathcal{M}}}{\mathcal{C}}_{X_{i},X_{j}}+\sum_{k\in{\mathcal{N}}}{\mathcal{C}}_{X_{i},Z_{k}},$

where for all $k\in{\mathcal{N}}$,

${\mathcal{C}}_{X_{i},Z_{k}}\,=\sum_{j\in{\mathcal{M}}}\left.{\mathbb{E}}\left[{\mathfrak{K}}(X_{i})\,\partial_{m+k}g_{j}({\boldsymbol{X}},{\boldsymbol{Z}})\,\Psi_{1}^{(m+k)}(X_{i},{\boldsymbol{V}}){\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}}~{}\right|~{}L\geq{q_{\alpha}}\right]\,,$

and for $j\in{\mathcal{M}}$,

$\displaystyle{\mathcal{C}}_{X_{i},X_{j}}\,$
$\displaystyle=\sum_{r=1}^{m}\,\left({\mathbb{E}}\left[{\mathfrak{K}}(X_{i})\partial_{j}g_{r}({\boldsymbol{X}},{\boldsymbol{Z}})\Psi_{1}^{(j)}(X_{i},{\boldsymbol{V}}){\mathds{1}}_{\\{X_{r}\leq
d_{r}\\}}~{}\Big{|}~{}L\geq{q_{\alpha}}\right]\right)$
$\displaystyle\quad-c(\kappa;\,j)\frac{f_{j}(d_{j})}{1-\alpha}\,{\mathbb{E}}\left[{\mathfrak{K}^{-1}}(X_{i})\Psi_{1}^{(j)}(X_{i},{\boldsymbol{V}})\Big{(}(L-c(\kappa;\,j)g_{j}({\boldsymbol{X}},{\boldsymbol{Z}})-{q_{\alpha}})_{+}-(L-{q_{\alpha}})_{+}\Big{)}~{}\Big{|}~{}X_{j}=d_{j}\right]\,.$

##  8 Proofs

###  8.1 Proof of Mixture Stress

###### Proof 8.1

Proof of mixture stress stress properties. We prove that the mixture stress in
Table 1 is well-defined. First, the stress function and its inverse are given
by

$\kappa_{\varepsilon}(x)=F_{i,{\varepsilon}}^{-1}\left(F_{i}(x)\right)\quad\text{and}\quad\kappa^{-1}_{\varepsilon}(x)=F_{i}^{-1}\left(F_{i,{\varepsilon}}(x)\right)\,,$

where $F_{i,{\varepsilon}}=(1-{\varepsilon})F_{i}+{\varepsilon}\,G$. By
construction, it holds that $F_{i,{\varepsilon}}(x)$ is continuous and
strictly increasing in $x$ for all ${\varepsilon}\geq 0$. Furthermore,
$F_{i,{\varepsilon}}$ and $F_{i,{\varepsilon}}^{-1}$ converge pointwise, as
${\varepsilon}\searrow 0$, to $F_{i}$ and $F_{i}^{-1}$ respectively, thus the
stress function fulfils $ii)$ and $iii)$. Next, if $G(x)\leq F_{i}(x)$ for all
$x\in{\mathbb{R}}$, then
$F_{i,{\varepsilon}}(x)=F_{i}(x)+{\varepsilon}\,(G(x)-F_{i}(x))\leq F_{i}(x)$,
and therefore $F_{i,{\varepsilon}}^{-1}(u)\geq F_{i}^{-1}(u)$ for all
$u\in[0,1]$. Thus
$\kappa_{\varepsilon}(x)=F_{i,{\varepsilon}}^{-1}\left(F_{i}(x)\right)\geq x$
and the stress function fulfils $iv)$ $iv)$$(a)$. The case when the stress
function fulfils $iv)$ $iv)$$(b)$ if $G(x)\geq F_{i}(x)$ for all
$x\in{\mathbb{R}}$, follows similarly.

Now, to check property $v)$, note that

${\mathfrak{K}}(x)=\lim_{{\varepsilon}\searrow
0}\tfrac{1}{{\varepsilon}}\left(\kappa_{\varepsilon}(x)-x\right)=\lim_{{\varepsilon}\searrow
0}\tfrac{1}{{\varepsilon}}\left(F_{i,{\varepsilon}}^{-1}\big{(}F_{i}(x)\big{)}-F_{i}^{-1}\big{(}F_{i}(x)\big{)}\right)=\frac{\partial}{\partial{\varepsilon}}F_{i,{\varepsilon}}^{-1}(F_{i}(x))\Big{|}_{{\varepsilon}=0}\,.$

Further, from the relation
$\frac{\partial}{\partial{\varepsilon}}F_{i,{\varepsilon}}^{-1}(x)|_{{\varepsilon}=0}=-\frac{\frac{\partial}{\partial{\varepsilon}}F_{i,{\varepsilon}}(y)}{f_{i}(y)}|_{y=F_{i}^{-1}(x)}=\frac{F_{i}(y)-G(y)}{f_{i}(y)}|_{y=F_{i}^{-1}(x)}$,
it follows that

${\mathfrak{K}}(x)=\frac{F_{i}(x)-G(x)}{f_{i}(x)}\,.$

To obtain the expression for $vi)$, note that the stress function
$\kappa_{\varepsilon}(x)$ is differentiable in $x$, for all ${\varepsilon}$ in
a neighbourhood of 0, and $\frac{\partial}{\partial
x}\kappa_{\varepsilon}(x)|_{{\varepsilon}=0}\not=0$, then property $vi)$ is
satisfied with
${\mathfrak{K}^{-1}}(x)=\frac{-{\mathfrak{K}}(x)}{\frac{\partial}{\partial
x}\kappa_{\varepsilon}(x)|_{{\varepsilon}=0}}$, which follows immediately from
an application of the chain rule applied to the identity
$\kappa_{\varepsilon}^{-1}(\kappa_{\varepsilon}(x))=x$. Moreover,

$\displaystyle\frac{\partial}{\partial x}\kappa_{\varepsilon}(x)$
$\displaystyle=-\frac{f_{i}(x)}{\frac{\partial}{\partial
x}F_{i,{\varepsilon}}(F_{i,{\varepsilon}}^{-1}(F_{i}(x)))|_{{\varepsilon}=0}}=-1\,,$

which concludes the proof.

###  8.2 Proofs of Marginal Sensitivity: Theorems 2.3 and 2.4

For the proof of the marginal sensitivities to ${\mathrm{VaR}}$ and
${\mathrm{ES}}$, we need the following lemma concerning sequences of functions
that converge weakly to a Dirac delta function. For this, we first write the
marginally stressed portfolios as

$\displaystyle L(Z_{i,{\varepsilon}})$
$\displaystyle=L+\sum_{k=1}^{m}\Delta_{\varepsilon}g_{k}\quad\text{and}\quad$
$\displaystyle L(X_{i,{\varepsilon}})$
$\displaystyle=L+g_{i}({\boldsymbol{Z}})\,\left({\mathds{1}}_{\\{X_{i,{\varepsilon}}\leq
d_{i}\\}}-{\mathds{1}}_{\\{X_{i}\leq d_{i}\\}}\right)\,,$

where we define
$\Delta_{\varepsilon}g_{k}:=(g_{k}({\boldsymbol{Z}}_{-i},\kappa_{\varepsilon}(Z_{i}))-g_{k}({\boldsymbol{Z}})){\mathds{1}}_{\\{X_{k}\leq
d_{k}\\}}$. When the stress is clear from the context, we write
$L_{\varepsilon}=L(Z_{i,{\varepsilon}})$ and
$L_{\varepsilon}=L(X_{i,{\varepsilon}})$.

###### Lemma 8.2

For fixed $d\in{\mathbb{R}}$, define the family of functions

$\delta_{\varepsilon}(x)=\frac{\left|{\mathds{1}}_{\\{\kappa_{\varepsilon}(x)\leq
d\\}}-{\mathds{1}}_{\\{x\leq d\\}}\right|}{{\varepsilon}}\,,\quad
x\in{\mathbb{R}}\,,\quad{\varepsilon}>0.$

Then, $\delta_{\varepsilon}$ converges weakly to a scaled Dirac delta function
at $d$ for ${\varepsilon}\searrow 0$. Moreover, for any family of measurable
functions $h_{\varepsilon}\colon{\mathbb{R}}^{m+n}\to{\mathbb{R}}$ such that
$\lim_{{\varepsilon}\searrow
0}{\mathbb{E}}\left[|h_{\varepsilon}({\boldsymbol{X}},{\boldsymbol{Z}})|\right]<\infty$,
the following holds:

$\displaystyle\lim_{{\varepsilon}\searrow
0}{\mathbb{E}}\left[\delta_{\varepsilon}(X_{i})\,h_{\varepsilon}({\boldsymbol{X}},{\boldsymbol{Z}})\right]$
$\displaystyle=-c(\kappa){\mathfrak{K}^{-1}}(d)\,f_{i}(d)\;{\mathbb{E}}\left[\,h_{0}({\boldsymbol{X}},{\boldsymbol{Z}})~{}|~{}X_{i}=d\,\right]\,,$

where $c(\kappa)$ is given in (2), and
$h_{0}({\boldsymbol{x}},{\boldsymbol{z}})=\lim_{{\varepsilon}\searrow
0}h_{\varepsilon}({\boldsymbol{x}},{\boldsymbol{z}})$.

###### Proof 8.3

Proof of Lemma 8.2. First note that

$\left|{\mathds{1}}_{\\{\kappa_{\varepsilon}(x)\leq
d\\}}-{\mathds{1}}_{\\{x\leq
d\\}}\right|=-c(\kappa)\left({\mathds{1}}_{\\{\kappa_{\varepsilon}(x)\leq
d\\}}-{\mathds{1}}_{\\{x\leq d\\}}\right)\,.$ (11)

Let $\xi$ be an infinitely often differentiable function. Using the change of
variable $y=\kappa_{\varepsilon}(x)$, we obtain

$\displaystyle\int_{-\infty}^{+\infty}\xi(x)\delta_{\varepsilon}(x)\,dx$
$\displaystyle=-\frac{c(\kappa)}{{\varepsilon}}\int_{-\infty}^{+\infty}\xi(x)\left({\mathds{1}}_{\\{\kappa_{\varepsilon}(x)\leq
d\\}}-{\mathds{1}}_{\\{x\leq d\\}}\right)\,dx$
$\displaystyle=-\frac{c(\kappa)}{{\varepsilon}}\int_{-\infty}^{+\infty}\frac{\xi(z)}{\frac{\partial}{\partial
x}\kappa_{\varepsilon}(z)}\Big{|}_{z=\kappa_{\varepsilon}^{-1}(y)}{\mathds{1}}_{\\{y\leq
d\\}}\,dy-\frac{1}{{\varepsilon}}\int_{-\infty}^{d}\xi(x)\,dx\,.$

Letting $\Xi$ be a primitive of $\xi$ vanishing at $-\infty$, then

$\displaystyle\int_{-\infty}^{+\infty}\xi(x)\delta_{\varepsilon}(x)\,dx$
$\displaystyle=-\frac{c(\kappa)}{{\varepsilon}}\left(\int_{-\infty}^{d}\frac{d}{dy}\Xi(\kappa_{\varepsilon}^{-1}(y))\,dy-\Xi(d)\right)$
$\displaystyle=-\frac{c(\kappa)}{{\varepsilon}}\left(\Xi(\kappa_{\varepsilon}^{-1}(d))-\Xi(d)\right)\,.$

Taking the limit as ${\varepsilon}\rightarrow 0$, we obtain that

$\lim_{{\varepsilon}\searrow
0}\int_{-\infty}^{+\infty}\xi(x)\delta_{\varepsilon}(x)\,dx=-c(\kappa)\,\xi(d){\mathfrak{K}^{-1}}(d)\,.$

For the second part of the statement, note that

$\displaystyle\lim_{{\varepsilon}\searrow
0}\,{\mathbb{E}}\left[\delta_{\varepsilon}(X_{i})\,h_{\varepsilon}({\boldsymbol{X}},{\boldsymbol{Z}})\right]$
$\displaystyle=\lim_{{\varepsilon}\searrow
0}\,\int_{{\mathbb{R}}^{m+n}}\delta_{\varepsilon}(x_{i})\,h_{\varepsilon}({\boldsymbol{x}},{\boldsymbol{z}})\,f_{{\boldsymbol{X}},{\boldsymbol{Z}}}({\boldsymbol{x}},{\boldsymbol{z}})\,d{\boldsymbol{x}}\,d{\boldsymbol{z}}$
$\displaystyle=-c(\kappa)\,{\mathfrak{K}^{-1}}(d)\int_{{\mathbb{R}}^{m+n-1}}\,h_{\varepsilon}({\boldsymbol{x}}_{-i},d,{\boldsymbol{z}})\,f_{{\boldsymbol{X}},{\boldsymbol{Z}}}({\boldsymbol{x}}_{-i},d,{\boldsymbol{z}})\frac{f_{i}(d)}{f_{i}(d)}\,d{\boldsymbol{x}}_{-i}\,d{\boldsymbol{z}}$
$\displaystyle=-c(\kappa)\,{\mathfrak{K}^{-1}}(d)\,f_{i}(d){\mathbb{E}}\left[\,h_{0}({\boldsymbol{X}},{\boldsymbol{Z}})~{}|~{}X_{i}=d\,\right]\,.$

###### Lemma 8.4

For fixed $0<\alpha<1$ and ${\boldsymbol{z}}\in{\mathbb{R}}^{n}$, define the
sequence of functions

$\delta_{\varepsilon}(l)=\frac{{\mathds{1}}_{\left\\{l\leq{q_{\alpha}}-\sum_{k=1}^{m}\Delta_{\varepsilon}g_{k}\right\\}}-{\mathds{1}}_{\left\\{l\leq{q_{\alpha}}\right\\}}}{{\varepsilon}}\quad
l\in{\mathbb{R}}\,,\quad{\varepsilon}>0.$

where
$\Delta_{\varepsilon}g_{k}=(g_{k}({\boldsymbol{z}}_{-i},\kappa_{\varepsilon}(z_{i}))-g_{k}({\boldsymbol{z}})){\mathds{1}}_{\\{x_{k}\leq
d_{k}\\}}$, ${\boldsymbol{z}}\in{\mathbb{R}}^{n}$, and $l\geq 0$. Then,
$\delta_{\varepsilon}$ converges weakly to a scaled Dirac delta function at
${q_{\alpha}}$ for ${\varepsilon}\searrow 0$. Moreover, for any family of
measurable functions $h_{\varepsilon}\colon{\mathbb{R}}^{m+n}\to{\mathbb{R}}$
such that $\lim_{{\varepsilon}\searrow
0}{\mathbb{E}}\left[|h_{\varepsilon}({\boldsymbol{X}},{\boldsymbol{Z}})|\right]<\infty$,
the following holds:

$\lim_{{\varepsilon}\searrow
0}{\mathbb{E}}\left[\delta_{\varepsilon}(L)h_{\varepsilon}({\boldsymbol{X}},L)\right]=-f({q_{\alpha}})\sum_{k=1}^{m}\left.{\mathbb{E}}\left[{\mathfrak{K}}(Z_{i})\partial_{i}\,g_{k}({\boldsymbol{Z}}){\mathds{1}}_{\\{X_{k}\leq
d_{k}\\}}\,h_{0}({\boldsymbol{X}},L)\right|L={q_{\alpha}}\right]\,.$ (12)

###### Proof 8.5

Proof of Lemma 8.4. Let $\xi(\cdot)$ be an infinitely often differentiable
function. Applying Taylor’s Theorem of $g_{k}$ around $z_{i}$, and then using
$\kappa_{\varepsilon}(z_{i})=z_{i}+{\varepsilon}{\mathfrak{K}}(z_{i})+o({\varepsilon})$,
we obtain for all $k=1,\ldots,n$, that

$\displaystyle
g_{k}({\boldsymbol{z}}_{-i},\kappa_{\varepsilon}(z_{i}))-g_{k}({\boldsymbol{z}})$
$\displaystyle=\left(\kappa_{\varepsilon}(z_{i})-z_{i}\right)\partial_{i}\,g_{k}({\boldsymbol{z}})+o\left(\kappa_{\varepsilon}(z_{i})-z_{i}\right)$
$\displaystyle={\varepsilon}{\mathfrak{K}}(z_{i})\,\partial_{i}\,g_{k}({\boldsymbol{z}})+o\left({\varepsilon}\right)\,,$
(13)

where $\partial_{i}\,g_{k}({\boldsymbol{z}})=\tfrac{\partial}{\partial
z_{i}}g_{k}({\boldsymbol{z}})$ is the derivative in the $i^{\text{th}}$
component. Thus, we have that for all ${\boldsymbol{z}}\in{\mathbb{R}}^{n}$,
using the Mean Value Theorem for some
$l^{*}\in({q_{\alpha}},{q_{\alpha}}-\Delta_{\varepsilon}g\,]$ (or
$l^{*}\in({q_{\alpha}}-\Delta_{\varepsilon}g\,,{q_{\alpha}}]$) in the second
equation, and then (13) that

$\displaystyle\int_{-\infty}^{+\infty}\xi(l)\delta_{\varepsilon}(l)\,dl$
$\displaystyle=\frac{1}{{\varepsilon}}\int_{{q_{\alpha}}}^{{q_{\alpha}}-\sum_{k=1}^{m}\Delta_{\varepsilon}g_{k}}\xi(l)\,dl=-\frac{1}{{\varepsilon}}\sum_{k=1}^{m}\Delta_{\varepsilon}g_{k}\;\xi(l^{*})$
$\displaystyle=-\left(\sum_{k=1}^{m}\,{\mathfrak{K}}(z_{i})\,\partial_{i}\,g_{k}({\boldsymbol{z}}){\mathds{1}}_{\\{x_{k}\leq
d_{k}\\}}+o\left(1\right)\right)\xi(l^{*})\,.$

Taking the limit for ${\varepsilon}\searrow 0$, we have

$\lim_{{\varepsilon}\searrow
0}\int_{-\infty}^{+\infty}\xi(l)\delta_{\varepsilon}(l)\,dl=-{\mathfrak{K}}(z_{i})\sum_{k=1}^{m}\partial_{i}\,g_{k}({\boldsymbol{z}}){\mathds{1}}_{\\{x_{k}\leq
d_{k}\\}}\xi\left({q_{\alpha}}\right)\,.$

Equation (12) follows using a similar argument as in Lemma 8.2.

###### Proof 8.6

Proof of Theorem 2.3 (Marginal Sensitivity VaR) By Proposition 2.3 in
Embrechts and Hofert (2013) it holds for all ${\varepsilon}\geq 0$ that
$F_{{\varepsilon}}\left(q_{\varepsilon}(\alpha)\right)=\alpha$. Taking
derivative with respect to ${\varepsilon}$ and evaluating at ${\varepsilon}=0$
(note that Assumptions 2.3 and 2.3 are fulfilled), we obtain

$f\left({q_{\alpha}}\right)\;\left.\tfrac{\partial}{\partial{\varepsilon}}q_{\varepsilon}(\alpha)\right|_{{\varepsilon}=0}+\tfrac{\partial}{\partial{\varepsilon}}F_{\varepsilon}({q_{\alpha}})=0\,\quad\text{and}\quad\left.\frac{\partial}{\partial{\varepsilon}}q_{\varepsilon}(\alpha)\right|_{{\varepsilon}=0}=-\frac{1}{f\left({q_{\alpha}}\right)}\frac{\partial}{\partial{\varepsilon}}F_{\varepsilon}({q_{\alpha}})\,,$
(14)

whenever
$\frac{\partial}{\partial{\varepsilon}}F_{\varepsilon}({q_{\alpha}}(\alpha))$
exists. Next, we show that the derivative of $F_{\varepsilon}$ with respect to
${\varepsilon}$ exists.
Part 1: We first consider the case of stressing $X_{i}$ and calculate

$\displaystyle F_{\varepsilon}({q_{\alpha}})-F({q_{\alpha}})$
$\displaystyle=\;{\mathbb{P}}\left(L_{{\varepsilon}}\leq{q_{\alpha}}\right)-{\mathbb{P}}\left(L\leq{q_{\alpha}}\right)$
(15a)
$\displaystyle={\mathbb{E}}\left[{\mathds{1}}_{\left\\{L\leq{q_{\alpha}}-g_{i}({\boldsymbol{Z}})\,\left({\mathds{1}}_{\\{\kappa_{\varepsilon}(X_{i})\leq
d_{i}\\}}-{\mathds{1}}_{\\{X_{i}\leq
d_{i}\\}}\right)\right\\}}-{\mathds{1}}_{\\{L\leq{q_{\alpha}}\\}}\right]$
(15b)
$\displaystyle={\mathbb{E}}\left[\left|{\mathds{1}}_{\\{\kappa_{\varepsilon}(X_{i})\leq
d_{i}\\}}-{\mathds{1}}_{\\{X_{i}\leq
d_{i}\\}}\right|\,\left({\mathds{1}}_{\left\\{L\leq{q_{\alpha}}+c(\kappa)g_{i}({\boldsymbol{Z}})\,\right\\}}-{\mathds{1}}_{\\{L\leq{q_{\alpha}}\\}}\right)\right]\,,$
(15c)

where the last equality follows from (11). Invoking Lemma 8.2 we obtain

$\displaystyle\frac{\partial}{\partial{\varepsilon}}F_{\varepsilon}({q_{\alpha}})$
$\displaystyle=-c(\kappa){\mathfrak{K}^{-1}}(d_{i})\,f_{i}(d_{i})\;\;\left.{\mathbb{E}}\left[\,\left({\mathds{1}}_{\left\\{L\leq{q_{\alpha}}+c(\kappa)g_{i}({\boldsymbol{Z}})\,\right\\}}-{\mathds{1}}_{\\{L\leq{q_{\alpha}}\\}}\right)~{}\right|~{}X_{i}=d_{i}\right]\,.$

Combining with Equation (14) concludes the first part.

Part 2: Next, we consider the case of stressing $Z_{i}$. For this, it holds
that

$F_{\varepsilon}({q_{\alpha}})-F({q_{\alpha}})={\mathbb{E}}\left[\left({\mathds{1}}_{\left\\{L\leq{q_{\alpha}}-\sum_{k=1}^{m}\Delta_{\varepsilon}g_{k}\,\right\\}}-{\mathds{1}}_{\\{L\leq{q_{\alpha}}\\}}\right)\right]\,.$

Applying Lemma 8.4 and Equation (14) conclude the proof.

###### Proof 8.7

Proof of Theorem 2.4 (Marginal Sensitivity ES). We first calculate the
sensitivity to $X_{i}$, and in a second step to $Z_{i}$.

Part 1: To calculate the sensitivity to $X_{i}$, we observe that

$\displaystyle\frac{{\mathrm{ES}}_{\alpha}(L_{\varepsilon})-{\mathrm{ES}}_{\alpha}(L)}{{\varepsilon}}$
$\displaystyle=\frac{1}{{\varepsilon}(1-\alpha)}{\mathbb{E}}\left[\left(L_{\varepsilon}-q_{\varepsilon}\right)_{+}-\left(L-{q_{\alpha}}\right)_{+}\right]+\frac{q_{\varepsilon}-{q_{\alpha}}}{{\varepsilon}}$
$\displaystyle=\underbrace{\frac{1}{{\varepsilon}(1-\alpha)}{\mathbb{E}}\left[\left(L_{\varepsilon}-q_{\varepsilon}\right)_{+}-\left(L_{\varepsilon}-{q_{\alpha}}\right)_{+}\right]}_{:=A({\varepsilon})}+\underbrace{{\mathbb{E}}\left[\left(L_{\varepsilon}-{q_{\alpha}}\right)_{+}-\left(L-{q_{\alpha}}\right)_{+}\right]}_{:=B({\varepsilon})}+\underbrace{\frac{q_{\varepsilon}-{q_{\alpha}}}{{\varepsilon}}}_{:=C({\varepsilon})}\,.$
(16)

To calculate the expectation in $A({\varepsilon})$, we use integration by
parts in the third equation, and interpret
$\int_{b}^{a}h(x)\,dx=-\int_{a}^{b}h(x)\,dx$, if $a<b$.

$\displaystyle A({\varepsilon})\,{\varepsilon}(1-\alpha)$
$\displaystyle=\int_{q_{\varepsilon}}^{+\infty}(y-q_{\varepsilon})\,dF_{\varepsilon}(y)-\int_{{q_{\alpha}}}^{+\infty}(y-{q_{\alpha}})\,dF_{\varepsilon}(y)$
$\displaystyle=\int_{q_{\varepsilon}}^{{q_{\alpha}}}y\,dF_{\varepsilon}(y)-q_{\varepsilon}\left(1-\alpha\right)+{q_{\alpha}}\left(1-F_{\varepsilon}({q_{\alpha}})\right)$
$\displaystyle={q_{\alpha}}F_{\varepsilon}({q_{\alpha}})-q_{\varepsilon}\,\alpha-\int_{q_{\varepsilon}}^{{q_{\alpha}}}F_{\varepsilon}(y)\,dy-
q_{\varepsilon}\left(1-\alpha\right)+{q_{\alpha}}\left(1-F_{\varepsilon}({q_{\alpha}})\right)$
$\displaystyle=({q_{\alpha}}-q_{\varepsilon})-\int_{q_{\varepsilon}}^{{q_{\alpha}}}F_{\varepsilon}(y)\,dy\,.$

Next, we collect parts $A({\varepsilon})$ and $C({\varepsilon})$, and use the
Mean Value Theorem, that is there exists a
$q^{*}\in(q_{\varepsilon},{q_{\alpha}}]$ (or
$q^{*}\in({q_{\alpha}},q_{\varepsilon}]$, if ${q_{\alpha}}<q_{\varepsilon}$)
such that
$\int_{q_{\varepsilon}}^{{q_{\alpha}}}F_{\varepsilon}(y)\,dy=({q_{\alpha}}-q_{\varepsilon})\,F_{\varepsilon}(q^{*})$.
Thus, we obtain

$\displaystyle A({\varepsilon})+C({\varepsilon})$
$\displaystyle=\frac{1}{{\varepsilon}(1-\alpha)}\big{(}({q_{\alpha}}-q_{\varepsilon})\,(1-F_{\varepsilon}(q^{*}))+\frac{q_{\varepsilon}-{q_{\alpha}}}{{\varepsilon}}$
$\displaystyle=\frac{(q_{\varepsilon}-{q_{\alpha}})}{{\varepsilon}}\,\left(1-\frac{1-F_{\varepsilon}(q^{*})}{1-\alpha}\right).$

Talking the limit for ${\varepsilon}\searrow 0$, and noting that the
derivative of the quantile function with respect to ${\varepsilon}$ exists by
Theorem 2.3, we obtain

$\lim_{{\varepsilon}\searrow 0}A({\varepsilon})+C({\varepsilon})=0\,.$

Next, for part $B({\varepsilon})$ we obtain using Equation (11)

$\displaystyle B({\varepsilon})$
$\displaystyle=\frac{1}{{\varepsilon}(1-\alpha)}{\mathbb{E}}\left[\left(L+g_{i}({\boldsymbol{Z}})\,\left({\mathds{1}}_{\\{X_{i,{\varepsilon}}\leq
d_{i}\\}}-{\mathds{1}}_{\\{X_{i}\leq
d_{i}\\}}\right)-{q_{\alpha}}\right)_{+}-\left(L-{q_{\alpha}}\right)_{+}\right]$
$\displaystyle=\frac{1}{{\varepsilon}(1-\alpha)}{\mathbb{E}}\left[\Big{|}{\mathds{1}}_{\\{X_{i,{\varepsilon}}\leq
d_{i}\\}}-{\mathds{1}}_{\\{X_{i}\leq
d_{i}\\}}\Big{|}\Big{(}\left(L-c(\kappa)g_{i}({\boldsymbol{Z}})\,-{q_{\alpha}}\right)_{+}-\left(L-{q_{\alpha}}\right)_{+}\Big{)}\right]\,.$

Applying Lemma 8.2, we obtain

$\lim_{{\varepsilon}\searrow
0}B({\varepsilon})=\frac{-c(\kappa){\mathfrak{K}^{-1}}(d_{i})\,f_{i}(d_{i})}{1-\alpha}{\mathbb{E}}\left[\left(L-c(\kappa)g_{i}({\boldsymbol{Z}})\,-{q_{\alpha}}\right)_{+}-\left(L-{q_{\alpha}}\right)_{+}~{}|~{}X_{i}=d_{i}\right]\,.$

Part 2: For the sensitivity to $Z_{i}$, we write similarly to part 1,

$\frac{{\mathrm{ES}}_{\alpha}(L_{\varepsilon})-{\mathrm{ES}}_{\alpha}(L)}{{\varepsilon}}=A({\varepsilon})+B({\varepsilon})+C({\varepsilon})\,,$

where $A({\varepsilon})$ and $C({\varepsilon})$ are the same as in (16), while
$B({\varepsilon})$ is

$\displaystyle B({\varepsilon})$
$\displaystyle=\frac{1}{{\varepsilon}(1-\alpha)}\,{\mathbb{E}}\left[\left(L+\sum_{k=1}^{m}\Delta_{\varepsilon}g_{k}\,-{q_{\alpha}}\right)_{+}-\left(L-{q_{\alpha}}\right)_{+}\right]$
(17a)
$\displaystyle=\frac{1}{{\varepsilon}(1-\alpha)}\,{\mathbb{E}}\left[\left(L-{q_{\alpha}}\right)\left({\mathds{1}}_{\\{L\leq{q_{\alpha}}\\}}-{\mathds{1}}_{\\{L\leq{q_{\alpha}}-\sum_{k=1}^{m}\Delta_{\varepsilon}g_{k}\\}}\right)+\sum_{k=1}^{m}\Delta_{\varepsilon}g_{k}{\mathds{1}}_{\\{L\geq{q_{\alpha}}-\sum_{k=1}^{m}\Delta_{\varepsilon}g_{k}\\}}\right]\,,$
(17b)

where in the last equality we used that
${\mathds{1}}_{\\{L>{q_{\alpha}}\\}}=1-{\mathds{1}}_{\\{L\leq{q_{\alpha}}\\}}$.
Note that the argument that $A({\varepsilon})+C({\varepsilon})$ converges to 0
for ${\varepsilon}\searrow 0$ only depends on the fact that $F_{\varepsilon}$
converges to $F$ for ${\varepsilon}\searrow 0$. Thus, also here, it holds that
$\lim_{{\varepsilon}\searrow 0}A({\varepsilon})+C({\varepsilon})=0$. To
calculate the limit of $B({\varepsilon})$, we apply Lemma 8.4 to the first
term, which turns out to be equal to zero. For the second term, note that
$\frac{1}{{\varepsilon}}\Delta_{\varepsilon}g_{k}$ converges to
${\mathfrak{K}}(Z_{i})\partial_{i}\,g_{k}({\boldsymbol{Z}}){\mathds{1}}_{\\{X_{k}\leq
d_{k}\\}}$ ${\mathbb{P}}$-a.s. for ${\varepsilon}\searrow 0$, see also
Equation (13). Thus,

$\displaystyle\lim_{{\varepsilon}\searrow 0}B({\varepsilon})$
$\displaystyle=\frac{1}{1-\alpha}\sum_{k=1}^{m}{\mathbb{E}}\left[{\mathfrak{K}}(Z_{i})\partial_{i}\,g_{k}({\boldsymbol{Z}}){\mathds{1}}_{\\{X_{k}\leq
d_{k}\\}}{\mathds{1}}_{\\{L\geq{q_{\alpha}}\\}}\right]$
$\displaystyle=\sum_{k=1}^{m}{\mathbb{E}}\left[{\mathfrak{K}}(Z_{i})\partial_{i}\,g_{k}({\boldsymbol{Z}}){\mathds{1}}_{\\{X_{k}\leq
d_{k}\\}}~{}|~{}L\geq{q_{\alpha}}\right]\,.$

###  8.3 Proof of Cascade Sensitivity: Theorems 3.3, 3.4, 7.1, and 7.2.

For the proofs of the cascade sensitivities to ${\mathrm{VaR}}$ and
${\mathrm{ES}}$, we need the following lemmas concerning sequences of
functions that converge weakly to Dirac delta functions. For this, we first
provide a representation of the stressed loss, when stressing $X_{i}$.

For a stress function $\kappa_{\varepsilon}$ and a Rosenblatt transform
$\Psi$, we define for all $j\in{\mathcal{M}}$ and fixed $\mathbf{v}$,

$a_{{\varepsilon},j}(x):=|{\mathds{1}}_{\\{\eta_{{\varepsilon},j}(x)\leq
d_{j}\\}}-{\mathds{1}}_{\\{x\leq d_{j}\\}}|\,,$

where
$\eta_{{\varepsilon},j}(x):=\Psi^{(j)}\left(\kappa_{\varepsilon}\left(\Psi^{(j),-1}(x,\mathbf{v})\right),\mathbf{v}\right)$
and $\Psi^{(j),-1}$ denotes the inverse in the first component of
$\Psi^{(j)}$. Further, we let

$A_{{\varepsilon},j}:=a_{{\varepsilon},j}(X_{j})\,,$

where it is implicit that $\mathbf{v}$ is replaced by ${\boldsymbol{V}}$. Note
that $X_{j}=\Psi^{(j)}(X_{i},{\boldsymbol{V}})$ ${\mathbb{P}}$-a.s., and
therefore

$\Psi^{(j)}\left(X_{i,{\varepsilon}},{\boldsymbol{V}}\right)=\Psi^{(j)}\left(\kappa_{\varepsilon}(X_{i}),{\boldsymbol{V}}\right)=\Psi^{(j)}\left(\kappa_{\varepsilon}\left(\Psi^{(j),-1}(X_{j},{\boldsymbol{V}})\right),{\boldsymbol{V}}\right)=\eta_{{\varepsilon},j}(X_{j})\quad{\mathbb{P}}\text{-a.s}.\,.$

######  Lemma 8.8 (Stressed Portfolio Loss)

For a stress $X_{i,{\varepsilon}}$, the stressed portfolio admits
representation

$L^{\boldsymbol{\Psi}}(X_{i,{\varepsilon}})=L+\sum_{k=1}^{m}\tilde{\Delta}_{\varepsilon}g_{k}-\sum_{j=1}^{m}c(\kappa;\,j)g_{j}({\boldsymbol{Z}})\,A_{{\varepsilon},j}\,,$

where
$\tilde{\Delta}_{\varepsilon}\,g_{k}=\left(g_{k}\left(\Psi^{({\boldsymbol{Z}})}(X_{i,{\varepsilon}},{\boldsymbol{V}})\right)-g_{k}({\boldsymbol{Z}})\right){\mathds{1}}_{\\{\Psi^{(k)}(X_{i,{\varepsilon}},{\boldsymbol{V}})\leq
d_{k}\\}}$.

###### Proof 8.9

Proof of Lemma 8.8. We obtain

$\displaystyle L^{\boldsymbol{\Psi}}(X_{i,{\varepsilon}})$
$\displaystyle=L+\sum_{k=1}^{m}\tilde{\Delta}_{\varepsilon}g_{k}+\sum_{j=1}^{m}g_{j}({\boldsymbol{Z}})\,\left({\mathds{1}}_{\\{\eta_{{\varepsilon},j}(X_{j})\leq
d_{j}\\}}-{\mathds{1}}_{\\{X_{j}\leq d_{j}\\}}\right)$
$\displaystyle=L+\sum_{k=1}^{m}\tilde{\Delta}_{\varepsilon}g_{k}-\sum_{j=1}^{m}c(\kappa;\,j)g_{j}({\boldsymbol{Z}})\,A_{{\varepsilon},j}\,,$

since by Assumption 3 it holds that
${\mathds{1}}_{\\{\eta_{{\varepsilon},j}(x)\leq
d_{j}\\}}-{\mathds{1}}_{\\{x\leq
d_{j}\\}}=-c(\kappa;\,j)a_{{\varepsilon},j}(x)$ for all $j\in{\mathcal{M}}$.

###### Lemma 8.10

Let ${\mathcal{K}}\subset{\mathcal{M}}$ and its complement
${\mathcal{K}}^{\complement}={\mathcal{M}}/{\mathcal{K}}$ and define the
sequence of functions

$\delta^{\mathcal{K}}_{\varepsilon}({\boldsymbol{x}})=\frac{1}{{\varepsilon}}\prod_{k\in{\mathcal{K}}}a_{{\varepsilon},k}(x_{k})\prod_{l\in{\mathcal{K}}^{\complement}}a_{{\varepsilon},l}^{\complement}(x_{l}),\,\qquad{\varepsilon}>0,$

where $a_{{\varepsilon},k}^{\complement}(x)=1-a_{{\varepsilon},k}(x)$.

Then, for all functions
$h_{\varepsilon}\colon{\mathbb{R}}^{m+n}\to{\mathbb{R}}$ such that
$\lim_{{\varepsilon}\searrow
0}{\mathbb{E}}\left[|h_{\varepsilon}({\boldsymbol{X}},{\boldsymbol{Z}})|\right]<\infty$,
the following holds:

  1. i)

if ${\mathcal{K}}$ contains one element, ${\mathcal{K}}=\\{k\\}$, then

$\lim_{{\varepsilon}\searrow
0}\,{\mathbb{E}}\left[\delta_{\varepsilon}^{\mathcal{K}}({\boldsymbol{X}})\,h_{\varepsilon}({\boldsymbol{X}},{\boldsymbol{Z}})\right]=-c(\kappa;\,k)\,f_{k}(d_{k})\;{\mathbb{E}}\left.\left[\,{\mathfrak{K}^{-1}}\left(X_{i}\right)\,\Psi_{1}^{(k)}(X_{i},{\boldsymbol{V}})\,h_{0}({\boldsymbol{X}},{\boldsymbol{Z}})~{}\right|~{}X_{k}=d_{k}\right]\,.$

  2. ii)

if ${\mathcal{K}}$ contains two or more elements, then

$\lim_{{\varepsilon}\searrow
0}\,{\mathbb{E}}\left[\delta_{\varepsilon}^{\mathcal{K}}({\boldsymbol{X}})\,h_{\varepsilon}({\boldsymbol{X}},{\boldsymbol{Z}})\right]=0\,.$

###### Proof 8.11

Proof of Lemma 8.10. First, let ${\mathcal{K}}=\\{k\\}$ and note that
$\lim_{{\varepsilon}\searrow
0}a_{{\varepsilon},j}^{\complement}(x)=\lim_{{\varepsilon}\searrow
0}1-a_{{\varepsilon},j}(x)=1$, for all $j\in{\mathcal{M}}$ and
$x\in{\mathbb{R}}$. Next, we calculate the inverse of
$\eta_{{\varepsilon},k}(x)$ in $x$, which is given by

$\eta_{{\varepsilon},k}^{-1}(x)=\Psi^{(k)}\left(\kappa_{\varepsilon}^{-1}\left(\Psi^{(k),-1}(x,\mathbf{v})\right),\mathbf{v}\right)=\Psi^{(k)}\left(\kappa_{\varepsilon}^{-1}(x),\mathbf{v}\right).$

Its derivative is, noting that $\eta_{0,k}(x)=\eta^{-1}_{0,k}(x)=x$,

$\lim_{{\varepsilon}\searrow
0}\tfrac{1}{{\varepsilon}}\left(\eta_{{\varepsilon},k}^{-1}(x)-x\right)=\Psi^{(k)}_{1}\left(x,\mathbf{v}\right){\mathfrak{K}^{-1}}(x)\,.$

Using similar arguments as in the proof of Lemma 8.2, replacing
$\kappa_{\varepsilon}^{-1}$ with $\eta_{{\varepsilon},k}^{-1}$, we obtain that

$\lim_{{\varepsilon}\searrow
0}{\mathbb{E}}[\delta_{\varepsilon}^{k}({\boldsymbol{X}})h_{\varepsilon}({\boldsymbol{X}},{\boldsymbol{Z}})]=-c(\kappa;\,k)f_{i}(d_{k})\;{\mathbb{E}}\left[\,{\mathfrak{K}^{-1}}(X_{i})\,\Psi^{(k)}_{1}\left(X_{i},{\boldsymbol{V}}\right)h_{0}({\boldsymbol{X}},{\boldsymbol{Z}})~{}|~{}X_{i}=d_{k}\,\right]\,.$

Next, assume that ${\mathcal{K}}=\\{k,j\\}$ contains two indices and let
$\xi\colon{\mathbb{R}}^{2}\to{\mathbb{R}}$ be an infinitely often
differentiable function. Then, using (11) and the following change of variable
$y_{j}=\eta_{{\varepsilon},j}(x_{j})$ in the first equation

$\displaystyle\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\xi(x_{j},x_{k})\delta_{\varepsilon}^{\mathcal{K}}(x_{j},x_{k})\,dx_{j}dx_{k}$
$\displaystyle=-c(\kappa)\frac{1}{{\varepsilon}}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\xi(x_{j},x_{k})\left({\mathds{1}}_{\\{\eta_{{\varepsilon},j}}(x_{j})\leq
d_{j}\\}-{\mathds{1}}_{\\{x_{j}\leq
d_{j}\\}}\right)\,dx_{j}\,a_{{\varepsilon},k}(x_{k})\prod_{l\neq
j,k}a_{{\varepsilon},l}^{\complement}(x_{l})\,dx_{k}$
$\displaystyle=-c(\kappa)\int_{-\infty}^{+\infty}\left(\frac{1}{{\varepsilon}}\left(\int_{-\infty}^{+\infty}\frac{\xi(\eta_{{\varepsilon},j}^{-1}(y_{j}),x_{k})}{\eta_{{\varepsilon},j}^{\prime}(\eta_{{\varepsilon},j}^{-1}(y_{j}))}{\mathds{1}}_{\\{y_{j}\leq
d\\}}\,dy_{j}-\int_{-\infty}^{d_{j}}\xi(x_{j},x_{k})\,dx_{j}\right)\right)a_{{\varepsilon},k}(x_{k})\prod_{l\neq
j,k}a_{{\varepsilon},l}^{\complement}(x_{l})\,dx_{k}\,.$

Define the function $\Xi(x,y)$, such that $\frac{d}{dx}\Xi(x,y)=\xi(x,y)$, so
that

$\frac{1}{{\varepsilon}}\int_{-\infty}^{+\infty}\frac{\xi(\eta_{{\varepsilon},j}^{-1}(y_{j}),x_{k})}{\eta_{{\varepsilon},j}^{\prime}(\eta_{{\varepsilon},j}^{-1}(y_{j}))}{\mathds{1}}_{\\{y_{j}\leq
d\\}}\,dy_{j}-\int_{-\infty}^{d}\xi(x_{j},x_{k})\,dx_{j}=\frac{1}{{\varepsilon}}\left(\Xi(\eta_{{\varepsilon},j}^{-1}(d,x_{k}))-\Xi(d,x_{k})\right)\,.$
(18)

The limit of (18) for ${\varepsilon}\searrow 0$ exists, moreover
$a_{{\varepsilon},k}(x)$ converges to 1, for ${\varepsilon}\searrow 0$, while
$a_{{\varepsilon},l}^{\complement}(x)$, $l\neq\\{j,k\\}$, converge to 0 for
${\varepsilon}\searrow 0$. Thus, we obtain that
$\delta^{\mathcal{K}}_{\varepsilon}(\cdot)$ converges weakly to 0, for
${\varepsilon}\searrow 0$.

The cases when ${\mathcal{K}}$ contains more than two indices follow
analogous.

###### Lemma 8.12

Define the sequence of functions

$\delta_{\varepsilon}(l)=\frac{{\mathds{1}}_{\left\\{l\leq{q_{\alpha}}-\sum_{k=1}^{m}\tilde{\Delta}_{\varepsilon}g_{k}\right\\}}-{\mathds{1}}_{\left\\{l\leq{q_{\alpha}}\right\\}}}{{\varepsilon}}\,,$

where
$\tilde{\Delta}_{\varepsilon}g_{k}=\left(g_{k}\left(\Psi^{({\boldsymbol{Z}})}(\kappa_{\varepsilon}(x_{i}),{\boldsymbol{v}})\right)-g_{k}({\boldsymbol{z}})\right){\mathds{1}}_{\\{\Psi^{(k)}(\kappa_{\varepsilon}(x_{i}),{\boldsymbol{v}})\leq
d_{k}\\}}$, ${\boldsymbol{z}}\in{\mathbb{R}}^{n}$, $x_{i}\in{\mathbb{R}}$, and
$l\geq 0$. Then, $\delta_{\varepsilon}$ converges weakly to a scaled Dirac
delta function at ${q_{\alpha}}$ for ${\varepsilon}\searrow 0$. Moreover, for
any function $h_{\varepsilon}\colon{\mathbb{R}}^{m+n}\to{\mathbb{R}}$ such
that $\lim_{{\varepsilon}\searrow
0}{\mathbb{E}}\left[|h_{\varepsilon}({\boldsymbol{X}},{\boldsymbol{Z}})|\right]<\infty$,
the following holds:

$\lim_{{\varepsilon}\searrow
0}{\mathbb{E}}\left[\delta_{\varepsilon}(L)h_{\varepsilon}({\boldsymbol{X}},L)\right]=-f({q_{\alpha}})\sum_{k=1}^{m}\sum_{l=1}^{n}\left.{\mathbb{E}}\left[{\mathfrak{K}}(X_{i})\partial_{l}\,g_{k}({\boldsymbol{Z}})\Psi_{1}^{(m+l)}(X_{i},{\boldsymbol{V}}){\mathds{1}}_{\\{X_{k}\leq
d_{k}\\}}\,h_{0}({\boldsymbol{X}},L)\right|L={q_{\alpha}}\right]\,.$

###### Proof 8.13

Proof of Lemma 8.12. This proof follows along the lines of the proof of Lemma
8.4. Note that $z_{l}=\Psi^{(m+l)}(x_{i},{\boldsymbol{v}})$, and that the
Taylor approximation of
$g_{k}\left(\Psi^{({\boldsymbol{z}})}(\kappa_{\varepsilon}(x_{i}),{\boldsymbol{v}})\right)$
around ${\varepsilon}=0$, becomes, using first an approximation of $g_{k}$
around ${\boldsymbol{z}}$, then of ${\boldsymbol{\Psi}}^{(m+l)}$ around
$x_{i}$, for all $l=1,\ldots,n$, and finally for $\kappa_{\varepsilon}$ around
${\varepsilon}=0$

$\displaystyle
g_{k}\left(\Psi^{({\boldsymbol{Z}})}(\kappa_{\varepsilon}(x_{i}),{\boldsymbol{v}})\right)-g_{k}({\boldsymbol{z}})$
$\displaystyle=\sum_{l=1}^{n}\partial_{l}\,g_{k}({\boldsymbol{z}})\left(\Psi^{(m+l)}(\kappa_{\varepsilon}(x_{i}),{\boldsymbol{v}})-z_{l}\right)+o\left(\Psi^{(m+l)}(\kappa_{\varepsilon}(x_{i}),{\boldsymbol{v}})-z_{l}\right)$
$\displaystyle=\sum_{l=1}^{n}\partial_{l}\,g_{k}({\boldsymbol{z}})\,\Psi_{1}^{(m+l)}(x_{i},{\boldsymbol{v}})\,(\kappa_{\varepsilon}(x_{i})-x_{i})+o\left(\kappa_{\varepsilon}(x_{i})\right)$
$\displaystyle={\varepsilon}\sum_{l=1}^{n}\partial_{l}\,g_{k}({\boldsymbol{z}})\,\Psi_{1}^{(m+l)}(x_{i},{\boldsymbol{v}})\,{\mathfrak{K}}(x_{i})+o\left({\varepsilon}\right)\,.$

The reminder of the proof follows analogous steps to those in the proof of
Lemma 8.4.

###### Proof 8.14

Proof of Theorem 7.1 (Cascade Sensitivity VaR to $X_{i}$). Analogous to the
proof of Theorem 2.3, we use Equation (14) and, thus, we only need to
calculate
$\frac{\partial}{\partial{\varepsilon}}F_{\varepsilon}({q_{\alpha}})|_{{\varepsilon}=0}$.
Using Lemma 8.8, we obtain

$\displaystyle F_{\varepsilon}({q_{\alpha}})-F({q_{\alpha}})$
$\displaystyle={\mathbb{E}}\left[{\mathds{1}}_{\left\\{L\leq{q_{\alpha}}-\sum_{k=1}^{m}\tilde{\Delta}_{\varepsilon}g_{k}+\sum_{j=1}^{m}c(\kappa;\,j)g_{j}({\boldsymbol{Z}})A_{{\varepsilon},j}\right\\}}-{\mathds{1}}_{\left\\{L\leq{q_{\alpha}}\right\\}}\right]\,,$

where we recall that
$A_{{\varepsilon},j}=|{\mathds{1}}_{\\{\eta_{{\varepsilon},j}(X_{j})\leq
d_{j}\\}}-{\mathds{1}}_{\\{X_{j}\leq d_{j}\\}}|$ and denote its complement by
$A_{{\varepsilon},j}^{\complement}$, i.e.,
$A_{{\varepsilon},j}^{\complement}=1-A_{{\varepsilon},j}$. Next, as
$A_{{\varepsilon},j}$ are indicators, we can rewrite the expectation and split
it into multiple sums, as follow: The first expectation corresponding to all
$A_{{\varepsilon},j}^{\complement}$ (19a), and then we sum over all possible
combinations of $A_{{\varepsilon},j}$ and $A_{{\varepsilon},k}^{\complement}$.

$\displaystyle F_{\varepsilon}({q_{\alpha}})-F({q_{\alpha}})$
$\displaystyle={\mathbb{E}}\left[\prod_{i=1}^{m}A_{{\varepsilon},i}^{\complement}\left({\mathds{1}}_{\left\\{L\leq{q_{\alpha}}-\sum_{k=1}^{m}\tilde{\Delta}_{\varepsilon}g_{k}\right\\}}-{\mathds{1}}_{\left\\{L\leq{q_{\alpha}}\right\\}}\right)\right]$
(19a) $\displaystyle\quad+\sum_{k=1}^{m}\;\sum_{\stackrel{{\scriptstyle
i_{1},\ldots,i_{k}=1}}{{i_{1}<\cdots<i_{k}}}}^{m}{\mathbb{E}}\left[\prod_{j=1}^{k}A_{{\varepsilon},i_{j}}\prod_{\stackrel{{\scriptstyle
l=1}}{{l\not\in\\{i_{1},\ldots,i_{k}\\}}}}^{m}A_{{\varepsilon},l}^{\complement}\right.$
$\displaystyle\quad\times\left.\left({\mathds{1}}_{\left\\{L\leq{q_{\alpha}}-\sum_{r=1}^{m}\tilde{\Delta}_{\varepsilon}g_{r}+\sum_{j=1}^{k}c(\kappa;\,j)g_{i_{j}}({\boldsymbol{Z}})\right\\}}-{\mathds{1}}_{\left\\{L\leq{q_{\alpha}}\right\\}}\right)\right]\,.$

For the first expectation above (Equation (19a)), we apply Lemma 8.12 and that
$\lim_{{\varepsilon}\searrow 0}A_{{\varepsilon},k}^{\complement}=1$ for all
$k=1,\ldots,m$. For the other terms, we apply Lemma 8.10. Specifically, we
observe that only the summands that contains exactly one $A_{{\varepsilon},k}$
do not converge to 0. Thus, we obtain the limit, noting that for all
$k=1,\ldots,m$, $\tilde{\Delta}_{\varepsilon}\,g_{k}$ converges to 0, for
${\varepsilon}\searrow 0$,

$\displaystyle\lim_{{\varepsilon}\searrow 0}$
$\displaystyle\frac{F_{\varepsilon}({q_{\alpha}})-F({q_{\alpha}})}{{\varepsilon}}$
$\displaystyle=-\sum_{j=1}^{m}\sum_{l=1}^{n}\left.f({q_{\alpha}}){\mathbb{E}}\left[{\mathfrak{K}}(X_{i})\partial_{l}\,g_{j}({\boldsymbol{Z}})\Psi_{1}^{(m+l)}(X_{i},{\boldsymbol{V}}){\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}}\right|L={q_{\alpha}}\right]$
$\displaystyle\quad-\sum_{j=1}^{m}c(\kappa;\,j)f_{j}(d_{j}){\mathbb{E}}\left[{\mathfrak{K}^{-1}}(X_{i})\Psi_{1}^{(j)}(X_{i},{\boldsymbol{V}})\right.\left.\left({\mathds{1}}_{\left\\{L\leq{q_{\alpha}}+c(\kappa;\,j)g_{j}({\boldsymbol{Z}})\right\\}}-{\mathds{1}}_{\left\\{L\leq{q_{\alpha}}\right\\}}\right)~{}|~{}X_{j}=d_{j}\right]\,.$

Combining with Equation (14) concludes the proof.

###### Proof 8.15

Proof of Theorem 3.3 (Cascade Sensitivity ES to $X_{i}$). We write analogous
to the proof of Theorem 2.4

$\lim_{{\varepsilon}\searrow
0}\tfrac{1}{{\varepsilon}}\left({\mathrm{ES}}_{\alpha}(L_{\varepsilon})-{\mathrm{ES}}_{\alpha}(L)\right)=\lim_{{\varepsilon}\searrow
0}A({\varepsilon})+B({\varepsilon})+C({\varepsilon})=\lim_{{\varepsilon}\searrow
0}B({\varepsilon})\,.$

For part $B({\varepsilon})$, we proceed similar to the proof of Theorem 7.1
and write, using the notation from the proof of Theorem 7.1 and Lemma 8.8

$\displaystyle B({\varepsilon})(1-\alpha){\varepsilon}$
$\displaystyle={\mathbb{E}}\left[\left(L+\sum_{r=1}^{m}\tilde{\Delta}_{\varepsilon}g_{r}-\sum_{k=1}^{m}c(\kappa;\,k)g_{k}({\boldsymbol{Z}})\,A_{{\varepsilon},k}-{q_{\alpha}}\right)_{+}-\left(L-{q_{\alpha}}\right)_{+}\right]$
$\displaystyle={\mathbb{E}}\left[\prod_{i=1}^{m}A_{{\varepsilon},i}^{\complement}\left(\big{(}L+\sum_{r=1}^{m}\tilde{\Delta}_{\varepsilon}g_{r}-{q_{\alpha}}\big{)}_{+}-\left(L-{q_{\alpha}}\right)_{+}\right)\right]$
(20a) $\displaystyle\quad+\sum_{k=1}^{m}\sum_{\stackrel{{\scriptstyle
i_{1},\ldots,i_{k}=1}}{{i_{1}<\cdots<i_{k}}}}^{m}{\mathbb{E}}\Big{[}\prod_{j=1}^{k}A_{{\varepsilon},i_{j}}\prod_{\stackrel{{\scriptstyle
l=1}}{{l\not\in\\{i_{1},\ldots,i_{k}\\}}}}^{m}A_{{\varepsilon},l}^{\complement}$
$\displaystyle\quad\times\left.\left(\big{(}L+\sum_{r=1}^{m}\tilde{\Delta}_{\varepsilon}g_{r}-\sum_{j=1}^{k}c(\kappa;\,j)g_{i_{j}}({\boldsymbol{Z}})\,A_{{\varepsilon},i_{j}}-{q_{\alpha}}\big{)}_{+}-\big{(}L-{q_{\alpha}}\big{)}_{+}\right)\right]\,.$
(20b)

To calculate the limit of the expectation in Equation (20a), we rewrite
similar to (17)

$\displaystyle\big{(}L+\sum_{r=1}^{m}\tilde{\Delta}_{\varepsilon}g_{r}-{q_{\alpha}}\big{)}_{+}-\left(L-{q_{\alpha}}\right)_{+}$
$\displaystyle=\big{(}L-{q_{\alpha}}\big{)}_{+}\left({\mathds{1}}_{\\{L\leq{q_{\alpha}}\\}}-{\mathds{1}}_{\\{L\leq{q_{\alpha}}-\sum_{k=1}^{m}\tilde{\Delta}_{\varepsilon}g_{k}\\}}\right)$
(21a)
$\displaystyle\quad+\sum_{r=1}^{m}\tilde{\Delta}_{\varepsilon}g_{r}{\mathds{1}}_{\\{L\geq{q_{\alpha}}-\sum_{k=1}^{m}\tilde{\Delta}_{\varepsilon}g_{k}\\}}\,.$
(21b)

For the term (21a) we apply Lemma 8.12, noting that
$A_{{\varepsilon},k}^{\complement}$ converges to 1, for all $k=1,\ldots,m$, as
${\varepsilon}\searrow 0$. For the term (21b), we note that for all
$k=1,\ldots,m$, it holds ${\mathbb{P}}$-a.s. (see the Proof of Lemma 8.12)
that

$\lim_{{\varepsilon}\searrow
0}\frac{\tilde{\Delta}_{\varepsilon}g_{k}}{{\varepsilon}}=\sum_{l=1}^{n}\partial_{l}\,g_{k}({\boldsymbol{Z}})\Psi_{1}^{(m+l)}(X_{i},{\boldsymbol{V}}){\mathfrak{K}}(X_{i}){\mathds{1}}_{\\{X_{k}\leq
d_{k}\\}}\,.$

For all the other summands in Equation (20b) we apply Lemma 8.10. Collecting,
we obtain that

$\displaystyle(1-\alpha)\;\lim_{{\varepsilon}\searrow 0}B({\varepsilon})=$
$\displaystyle\sum_{k=1}^{m}\sum_{l=1}^{n}\left.f({q_{\alpha}}){\mathbb{E}}\left[{\mathfrak{K}}(X_{i})\partial_{l}\,g_{k}({\boldsymbol{Z}})\Psi_{1}^{(m+l)}(X_{i},{\boldsymbol{V}}){\mathds{1}}_{\\{X_{k}\leq
d_{k}\\}}\,\big{(}L-{q_{\alpha}}\big{)}_{+}\right|L={q_{\alpha}}\right]$ (22a)
$\displaystyle+\sum_{j=1}^{m}\sum_{l=1}^{n}{\mathbb{E}}\left[\partial_{l}\,g_{j}({\boldsymbol{Z}})\Psi_{1}^{(m+l)}(X_{i},{\boldsymbol{V}}){\mathfrak{K}}(X_{i}){\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}}{\mathds{1}}_{\\{L\geq{q_{\alpha}}\\}}\right]$
$\displaystyle-\sum_{j=1}^{m}c(\kappa;\,j)f_{j}(d_{j}){\mathbb{E}}\left[{\mathfrak{K}^{-1}}(X_{i})\Psi_{1}^{(j)}(X_{i},{\boldsymbol{V}})\left(\big{(}L-c(\kappa;\,j)g_{j}({\boldsymbol{Z}})\,-{q_{\alpha}}\big{)}_{+}-\big{(}L-{q_{\alpha}}\big{)}_{+}\right)~{}|~{}X_{j}=d_{j}\right]\,.$

Due to the conditioning event, (22a) is equal to 0.

###### Proof 8.16

Proof of Theorems 7.2 and 3.4 (Cascade Sensitivities to $Z_{i}$). The proof
follows by noting that the stressed portfolio for a stress on $Z_{i}$, admits
an analogous representation as when stressing $X_{i}$, with the only
difference that the inverse Rosenblatt transform starts at $Z_{i}$ instead of
$X_{i}$, see Equation (4) .

###### Proof 8.17

Proof of Proposition 3.5. The first two cases follow from Proposition 4.2 in
Pesenti et al. (2021). Assume $(X_{i},X_{j})$ follow a Archimedean copula. By
independence of the cascade sensitivity to the choice of Rosenblatt transform,
we have can choose (Rüschendorf and de Valk 1993)

$\Psi^{(j)}_{1}(X_{i},{\boldsymbol{V}})=F_{j|i}^{-1}(V_{j}|X_{i})\quad{\mathbb{P}}\text{-a.s.}\,,$

where a.s. $V_{j}=F_{X_{j}\,|\,X_{i}}(X_{j}\,|\,X_{i})$ and
$F_{X_{j}\,|\,X_{i}}(x\,|\,y)={\mathbb{P}}\left(X_{j}\leq
x\,|\,X_{i}=y\right)$ denotes the conditional distribution of $X_{j}$ given
$X_{i}=y$. We observe that $\Psi^{(j)}_{1}(X_{i},{\boldsymbol{V}})$ only
depends on $V_{j}$ and we may write $\Psi^{(j)}_{1}(X_{i},V_{j})$ instead of
$\Psi^{(j)}_{1}(X_{i},{\boldsymbol{V}})$.

We use Sklar’s theorem to write the conditional distribution and quantile
functions as

$F_{j|i}(x_{j}\,|\,x_{i})={\textrm{C}}_{j|i}(F_{j}(x_{j})\,|\,F_{i}(x_{i}))\quad\text{and
}\quad
F^{-1}_{j|i}(v\,|\,x_{i})=F_{j}^{-1}\left({\textrm{C}}_{j|i}^{-1}(v\,|\,F_{i}(x_{i}))\right)\,.$
(23)

Taking derivative of the conditional quantile function with respect to the
conditioning variable

$\Psi_{1}^{(j)}(y,v)=\frac{\partial}{\partial
y}F^{-1}_{j|i}(v\,|\,y)=\frac{f_{i}(y)}{f_{j}\left(F^{-1}_{j|i}(v\,|\,y)\right)}\frac{\partial}{\partial
z}{\textrm{C}}_{j|i}^{-1}(v\,|\,z)\Big{|}_{z=F_{i}(y)}\,.$

By definition of $V_{j}$, it holds ${\mathbb{P}}$-a.s. that
$F_{j|i}^{-1}(V_{j}\,|X_{i})=X_{j}$, thus

$\Psi_{1}^{(j)}(X_{i},V_{j})=\frac{f_{i}(X_{i})}{f_{j}\left(X_{j}\right)}\frac{\partial}{\partial
z}{\textrm{C}}_{j|i}^{-1}(V_{j}\,|\,z)\Big{|}_{z=F_{i}(X_{i})}\,.$ (24)

Next, we calculate the derivative (with respect to the conditioning argument)
of the inverse of an conditional Archimedean copula with generator $\psi$. The
conditional Archimedean copula and its inverse are given by (Cambou et al.
2017)

$\displaystyle{\textrm{C}}_{j|i}(x\,|\,y)$
$\displaystyle=\frac{\dot{\psi}\left(\psi^{-1}(y)+\psi^{-1}(x)\right)}{\dot{\psi}\left(\psi^{-1}(y)\right)}\quad\text{and}$
(25a) $\displaystyle{\textrm{C}}_{j|i}^{-1}(v\,|\,y)$
$\displaystyle=\psi\left[(\dot{\psi})^{-1}\left\\{v\,\dot{\psi}\left(\psi^{-1}(y)\right)\right\\}-\psi^{-1}(y)\right]\,,$

where $\dot{\psi}(x)=\frac{d}{dx}\psi(x)$. Taking derivative

$\displaystyle\frac{\partial}{\partial y}\,{\textrm{C}}_{j|i}^{-1}(v\,|\,y)$
$\displaystyle=\dot{\psi}\left[(\dot{\psi})^{-1}\left\\{v\,\dot{\psi}\left(\psi^{-1}(y)\right)\right\\}-\psi^{-1}(y)\right]\frac{1}{\dot{\psi}\left(\psi^{-1}(y)\right)}\,\left\\{\frac{v\,\ddot{\psi}\left(\psi^{-1}(y)\right)}{\overset{..}{\psi}\left((\dot{\psi})^{-1}\\{v\,\dot{\psi}\left(\psi^{-1}(y)\right)\\}\right)}-1\right\\}\,.$

Next, we use the definition of $V_{j}$ and Equations (23) and (25a) to write

$\displaystyle
V_{j}={\textrm{C}}_{j|i}(F_{j}(X_{j})\,|\,F_{i}(X_{i}))=\frac{\dot{\psi}\left(\psi^{-1}(U_{i})+\psi^{-1}(U_{j})\right)}{\dot{\psi}\left(\psi^{-1}(U_{i})\right)}\,,$

where $U_{j}=F_{j}(X_{j})$ and $U_{i}=F_{i}(X_{i})$. Using the above, we
obtain

$\left.\frac{\partial}{\partial
y}\,{\textrm{C}}_{j|i}^{-1}(V_{j}\,|\,y)\right|_{y=U_{i}}=\frac{\dot{\psi}\left(\psi^{-1}(U_{j})\right)}{\dot{\psi}\left(\psi^{-1}(U_{i})\right)}\left\\{\frac{\ddot{\psi}\left(\psi^{-1}(U_{i})\right)}{\ddot{\psi}\left(\psi^{-1}(U_{i})+\psi^{-1}(U_{j})\right)}\frac{\dot{\psi}\left(\psi^{-1}(U_{i})+\psi^{-1}(U_{j})\right)}{\dot{\psi}\left(\psi^{-1}(U_{i})\right)}-1\right\\}\,.$

Combining with Equation (24) concludes the proof.

###  8.4 Proof of Sensitivity to General Loss Model: Theorems 7.3, 7.4, and
7.5

###### Proof 8.18

Proof of Theorem 7.3 (Marginal Sensitivity - General Loss Model). The stressed
loss model has representation (using Equation (11))

$L_{\varepsilon}(X_{i})=L+\sum_{j=1}^{m}\Delta_{\varepsilon}g_{j}-c(\kappa)g_{i}\left(X_{i,{\varepsilon}},{\boldsymbol{X}}_{-i},{\boldsymbol{Z}}\right)A_{\varepsilon}\,,$

where
$\Delta_{\varepsilon}g_{j}:=\left(g_{j}(X_{i,{\varepsilon}},{\boldsymbol{X}}_{-i},{\boldsymbol{Z}})-g_{j}({\boldsymbol{X}},{\boldsymbol{Z}})\right){\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}}$ and $A_{\varepsilon}=|{\mathds{1}}_{\\{X_{i,{\varepsilon}}\leq
d_{i}\\}}-{\mathds{1}}_{\\{X_{i}\leq d_{i}\\}}|$. To prove the case for VaR,
note that

$\displaystyle F_{\varepsilon}({q_{\alpha}})-F({q_{\alpha}})$
$\displaystyle={\mathbb{E}}\left[A_{\varepsilon}^{\complement}\left({\mathds{1}}_{\\{L+\sum_{j=1}^{m}\Delta_{\varepsilon}g_{j}\leq{q_{\alpha}}\\}}-{\mathds{1}}_{\\{L\leq{q_{\alpha}}\\}}\right)\right]$
(26a)
$\displaystyle\quad+{\mathbb{E}}\left[A_{\varepsilon}\left({\mathds{1}}_{\\{L+\sum_{j=1}^{m}\Delta_{\varepsilon}g_{j}-c(\kappa)g_{i}(X_{i,{\varepsilon}},{\boldsymbol{X}}_{-i},{\boldsymbol{Z}})\leq{q_{\alpha}}\\}}-{\mathds{1}}_{\\{L\leq{q_{\alpha}}\\}}\right)\right]\,,$
(26b)

Applying Lemma 8.4 to (26a), noting that $\lim_{{\varepsilon}\searrow
0}A_{\varepsilon}^{\complement}=1$, and Lemma 8.2 to (26b), noting that
$\lim_{{\varepsilon}\searrow 0}\sum_{j=1}^{m}\Delta_{\varepsilon}g_{j}=0$
concludes the proof for VaR.

To prove the case of ES, we have from the proof of Theorem 2.4 that

$\displaystyle\lim_{{\varepsilon}\searrow
0}\frac{{\mathrm{ES}}_{\alpha}(L_{\varepsilon})-{\mathrm{ES}}_{\alpha}(L)}{{\varepsilon}}$
$\displaystyle=\lim_{{\varepsilon}\searrow
0}\frac{1}{{\varepsilon}(1-\alpha)}\,{\mathbb{E}}[(L_{\varepsilon}-{q_{\alpha}})_{+}-(L-{q_{\alpha}})_{+}]$
$\displaystyle=\lim_{{\varepsilon}\searrow
0}\frac{1}{{\varepsilon}(1-\alpha)}\,\Bigg{\\{}{\mathbb{E}}\Big{[}A_{\varepsilon}^{\complement}\underbrace{\left((L+\sum_{j=1}^{m}\Delta_{\varepsilon}g_{j}-{q_{\alpha}})_{+}-(L-{q_{\alpha}})_{+}\right)}_{=B_{\varepsilon}}\Big{]}$
$\displaystyle\quad+{\mathbb{E}}\left[A_{\varepsilon}\left(\Big{(}L+\sum_{j=1}^{m}\Delta_{\varepsilon}g_{j}-c(\kappa)g_{i}(X_{i,{\varepsilon}},{\boldsymbol{X}}_{-i},{\boldsymbol{Z}})-{q_{\alpha}}\Big{)}_{+}-(L-{q_{\alpha}})_{+}\right)\right]\Bigg{\\}}\,.$
(27a)

Next, we see that

$B_{\varepsilon}=(L-{q_{\alpha}})_{+}\left({\mathds{1}}_{\\{L\leq{q_{\alpha}}\\}}-{\mathds{1}}_{\\{L\leq{q_{\alpha}}-\sum_{j=1}^{m}\Delta_{\varepsilon}g_{j}\\}}\right)+\sum_{j=1}^{m}\Delta_{\varepsilon}g_{j}{\mathds{1}}_{\\{L\geq{q_{\alpha}}-\sum_{j=1}^{m}\Delta_{\varepsilon}g_{i}\\}}\,.$

Using similar arguments as in the proof of Theorem 3.3 (in particular applying
Lemma 8.12), we observe that

$\lim_{{\varepsilon}\searrow
0}\frac{1}{{\varepsilon}(1-\alpha)}{\mathbb{E}}[A_{\varepsilon}^{\complement}\;B_{\varepsilon}]=\sum_{j=1}^{m}{\mathbb{E}}\left[\partial_{i}g_{j}({\boldsymbol{X}},{\boldsymbol{Z}}){\mathfrak{K}}(X_{i}){\mathds{1}}_{\\{X_{j}\leq
d_{j}\\}}~{}|~{}L\geq{q_{\alpha}}\right]\,.$

Applying Lemma 8.2 to (27a) concludes the proof.

The proofs of Theorems 7.4, and 7.5 follow along the lines of the proofs of
Theorems 7.1 and 3.3.

###  8.5 Proof of Sensitivity to Discrete Random Variable: Theorem 4.1.

###### Proof 8.19

Proof of Theorem 4.1 (Marginal Sensitivity - Discrete). The stressed loss can
be written as

$T_{W,{\varepsilon}}=T-c(\kappa)\sum_{k=1}^{r}\Delta_{k}\,h(W,{\boldsymbol{Y}})A_{{\varepsilon},k}\,,$

where $A_{{\varepsilon},k}=|{\mathds{1}}_{\\{\kappa_{\varepsilon}(U)\leq
p_{k}\\}}-{\mathds{1}}_{\\{U\leq p_{k}\\}}|$ for $k\in\\{1,\ldots,r\\}$. To
prove the formula for VaR, we calculate similarly to the proof of Theorem 7.1

$\displaystyle{\mathbb{P}}\left(T_{W,{\varepsilon}}\leq{q_{\alpha}}\right)$
$\displaystyle-{\mathbb{P}}\left(T\leq{q_{\alpha}}\right)$
$\displaystyle={\mathbb{E}}\left[{\mathds{1}}_{\\{T\leq{q_{\alpha}}+c(\kappa)\sum_{k=1}^{r}\Delta_{k}\,h(W,{\boldsymbol{Y}})A_{{\varepsilon},k}\\}}-{\mathds{1}}_{\\{T\leq{q_{\alpha}}\\}}\right]$
$\displaystyle=\sum_{k=1}^{r}\sum_{\stackrel{{\scriptstyle
i_{1},\ldots,i_{k}=1}}{{i_{1}<\cdots<i_{k}}}}^{r}{\mathbb{E}}\left[\prod_{j=1}^{k}A_{{\varepsilon},i_{j}}\prod_{\stackrel{{\scriptstyle
l=1}}{{l\not\in\\{i_{1},\ldots
i_{k}\\}}}}^{r}A_{{\varepsilon},l}^{\complement}\left({\mathds{1}}_{\\{T\leq{q_{\alpha}}+c(\kappa)\sum_{j=1}^{k}\Delta_{j}\,h(W,{\boldsymbol{Y}})\\}}-{\mathds{1}}_{\\{T\leq{q_{\alpha}}\\}}\right)\right]\,.$

Applying Lemma 8.10 we obtain

$\displaystyle\frac{\partial}{\partial{\varepsilon}}F_{\varepsilon}({q_{\alpha}})$
$\displaystyle=-\sum_{k=1}^{r}c(\kappa){\mathbb{E}}\left[{\mathfrak{K}}^{-1}(U)\left({\mathds{1}}_{\\{T\leq{q_{\alpha}}+c(\kappa)\Delta_{k}\,h(W,{\boldsymbol{Y}})\\}}-{\mathds{1}}_{\\{T\leq{q_{\alpha}}\\}}\right)~{}|~{}U=p_{k}\right]$
$\displaystyle=-c(\kappa)\sum_{k=1}^{r}{\mathfrak{K}}^{-1}(p_{k}){\mathbb{E}}\left[\left({\mathds{1}}_{\\{T\leq{q_{\alpha}}+c(\kappa)\Delta_{k}\,h(W,{\boldsymbol{Y}})\\}}-{\mathds{1}}_{\\{T\leq{q_{\alpha}}\\}}\right)~{}|~{}W=w_{k}\right]\,.$

Combining with Equation (14) concludes the proof for VaR.

Second, we prove the case ES. From the proof of Theorem 2.4, we have

$\displaystyle{\mathrm{ES}}_{\alpha}(T_{W,{\varepsilon}})-{\mathrm{ES}}_{\alpha}(T)$
$\displaystyle=\tfrac{1}{(1-\alpha)}\,{\mathbb{E}}[(T_{W,{\varepsilon}}-{q_{\alpha}})_{+}-(T-{q_{\alpha}})_{+}]$
$\displaystyle=\tfrac{1}{(1-\alpha)}\,{\mathbb{E}}\left[\Big{(}T-c(\kappa)\sum_{k=1}^{r}\Delta_{k}\,h(W,{\boldsymbol{Y}})A_{{\varepsilon},k}-{q_{\alpha}}\Big{)}_{+}-(T-{q_{\alpha}})_{+}\right]$
$\displaystyle=\tfrac{1}{(1-\alpha)}\,\sum_{k=1}^{r}\sum_{\stackrel{{\scriptstyle
i_{1},\ldots,i_{k}=1}}{{i_{1}<\cdots<i_{k}}}}^{m}{\mathbb{E}}\left[\prod_{j=1}^{k}A_{{\varepsilon},i_{j}}\prod_{\stackrel{{\scriptstyle
l=1}}{{l\not\in\\{i_{1},\ldots
i_{k}\\}}}}^{m}A_{{\varepsilon},l}^{\complement}\right.$
$\displaystyle\quad\times\left.\left(\Big{(}T-c(\kappa)\sum_{j=1}^{k}\Delta_{i_{j}}h(W,{\boldsymbol{Y}})-{q_{\alpha}}\Big{)}_{+}-(T-{q_{\alpha}})_{+}\right)\right]\,.$

Next, we apply Lemma 8.10 and obtain

$\displaystyle\lim_{{\varepsilon}\searrow
0}\frac{1}{{\varepsilon}}\left({\mathrm{ES}}_{\alpha}(L_{\varepsilon})-{\mathrm{ES}}_{\alpha}(L)\right)$
$\displaystyle=-\frac{c(\kappa)}{1-\alpha}\sum_{k=1}^{r}{\mathfrak{K}}^{-1}(p_{k}){\mathbb{E}}\left[\Big{(}T-c(\kappa)\Delta_{k}h(W,{\boldsymbol{Y}})-{q_{\alpha}}\Big{)}_{+}-(T-{q_{\alpha}})_{+}~{}\Big{|}~{}U=p_{k}\right]\,,$

which concludes the proof.

##  9 Model specification for the numerical example of Section 5

The insurance company has $n=12$ lines of business (LoB) each following a
LogNormal distribution, $Z_{k}\sim\mathcal{LN}(\mu_{k},\sigma_{k}^{2})$,
$k\in{\mathcal{N}}$. The parameters $\mu_{k}$ and $\sigma_{k}$ are such that
${\mathbb{E}}[Z_{i}]=e^{\mu_{i}+\frac{1}{2}\sigma_{i}^{2}}=100$ (reflecting
the business volume) and
$\mathrm{CoV}_{i}=\sqrt{\text{var}(Z_{i})}/{\mathbb{E}}[Z_{i}]=\sqrt{e^{\sigma_{i}^{2}}-1}$,
where $\mathrm{CoV}$ denotes the coefficients of variation. The considered CoV
and lines of business are reported in Table 2 and the correlation matrix
$\mathbf{R}$ in Table 3. These figures are taken from the Solvency II Standard
Formula parameters (Lloyd’s 2022).

Table 3: Correlation matrix $\boldsymbol{R}$ of $Z$ (source: Lloyd’s (2022)). | $Z_{1}$ | $Z_{2}$ | $Z_{3}$ | $Z_{4}$ | $Z_{5}$ | $Z_{6}$ | $Z_{7}$ | $Z_{8}$ | $Z_{9}$ | $Z_{10}$ | $Z_{11}$ | $Z_{12}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$Z_{1}$ | $1$ | $0.5$ | $0.5$ | $0.25$ | $0.5$ | $0.25$ | $0.5$ | $0.25$ | $0.5$ | $0.25$ | $0.25$ | $0.25$
$Z_{2}$ | $0.5$ | $1$ | $0.25$ | $0.25$ | $0.25$ | $0.25$ | $0.5$ | $0.5$ | $0.5$ | $0.25$ | $0.25$ | $0.25$
$Z_{3}$ | $0.5$ | $0.25$ | $1$ | $0.25$ | $0.25$ | $0.25$ | $0.25$ | $0.5$ | $0.5$ | $0.25$ | $0.5$ | $0.25$
$Z_{4}$ | $0.25$ | $0.25$ | $0.25$ | $1$ | $0.25$ | $0.25$ | $0.25$ | $0.5$ | $0.5$ | $0.25$ | $0.5$ | $0.5$
$Z_{5}$ | $0.5$ | $0.25$ | $0.25$ | $0.25$ | $1$ | $0.5$ | $0.5$ | $0.25$ | $0.5$ | $0.5$ | $0.25$ | $0.25$
$Z_{6}$ | $0.25$ | $0.25$ | $0.25$ | $0.25$ | $0.5$ | $1$ | $0.5$ | $0.25$ | $0.5$ | $0.5$ | $0.25$ | $0.25$
$Z_{7}$ | $0.5$ | $0.5$ | $0.25$ | $0.25$ | $0.5$ | $0.5$ | $1$ | $0.25$ | $0.5$ | $0.5$ | $0.25$ | $0.25$
$Z_{8}$ | $0.25$ | $0.5$ | $0.5$ | $0.5$ | $0.25$ | $0.25$ | $0.25$ | $1$ | $0.5$ | $0.25$ | $0.25$ | $0.5$
$Z_{9}$ | $0.5$ | $0.5$ | $0.5$ | $0.5$ | $0.5$ | $0.5$ | $0.5$ | $0.5$ | $1$ | $0.25$ | $0.5$ | $0.25$
$Z_{10}$ | $0.25$ | $0.25$ | $0.25$ | $0.25$ | $0.5$ | $0.5$ | $0.5$ | $0.25$ | $0.25$ | $1$ | $0.25$ | $0.25$
$Z_{11}$ | $0.25$ | $0.25$ | $0.5$ | $0.5$ | $0.25$ | $0.25$ | $0.25$ | $0.25$ | $0.5$ | $0.25$ | $1$ | $0.25$
$Z_{12}$ | $0.25$ | $0.25$ | $0.25$ | $0.5$ | $0.25$ | $0.25$ | $0.25$ | $0.5$ | $0.25$ | $0.25$ | $0.25$ | $1$

We assume that the reinsurers’ critical variables follow a standardised
student $t$ distributions with $\nu=4$ degrees of freedom, i.e. $X_{i}\sim
t(4)$, for all $i=1,\ldots,8$. (Note that the choice of marginal distribution
for $X_{i}$ is irrelevant since we consider only the event $\\{X_{i}\leq
d_{i}\\}$.) The default probabilities ${\mathbb{P}}(X_{i}\leq d_{i})=q_{i}$
are set to $q_{i}=0.015$, $i=1,\ldots,6$ and $q_{i}=0.01$, $i=7,8$.

We assume that $({\boldsymbol{X}},{\boldsymbol{Z}})$ has a multivariate $t$
copula with $\nu=4$ degrees of freedom and correlation parameter matrix
$\mathbf{\Sigma}=\\{\sigma_{i,j}\\}_{i,j=1,\dots,m+n}$ (McNeil et al. 2015,
Sec. 7.3). The elements of $\mathbf{\Sigma}$ comprise the pairwise Pearson
correlations of multivariate $t_{\nu}$-distributed random vector arising from
monotone transformations of elements of $({\boldsymbol{X}},{\boldsymbol{Z}})$,
so that each has $t_{\nu}$ marginals. As the (multivariate) margins of
multivariate $t_{\nu}$ distributions are again multivariate $t_{\nu}$, we
start by specifying the correlation parameters of the vectors
${\boldsymbol{X}}$ and ${\boldsymbol{Z}}$ separately and then consider the
dependence across the two vectors’ elements. First, ${\boldsymbol{X}}$ has
standardised $t_{\nu}$ margins and hence follows a multivariate $t_{\nu}$
distribution. Furthermore, we assume that the dependence structure is
homogeneous, such that $\sigma_{i,j}={\rm Corr}(X_{i},X_{j})=\lambda>0$, for
all $i\neq j,~{}i,j\in{\mathcal{M}}$. Second, ${\boldsymbol{Z}}$ has a
multivariate $t_{\nu}$ copula with correlation parameter matrix
$\mathbf{R}=\\{r_{k,l}\\}_{k,l\in{\mathcal{N}}}$, given in Table 2, thus
$\sigma_{m+k,m+l}:=r_{k,l},~{}k,l\in{\mathcal{N}}$. Third, to specify the
elements of $\mathbf{\Sigma}$ characterising the dependence of
$(X_{j},Z_{k})$, i.e.
$\sigma_{j,m+k},~{}j\in{\mathcal{M}},~{}k\in{\mathcal{N}}$, we build a
dependence model that links gross losses to reinsurance defaults using a
single factor model, reflecting the homogeneity in the dependence of
${\boldsymbol{X}}$. The common factor is a function of the gross losses and
acts as a proxy for industry effects.

By the representation of multivariate $t$ distributions as normal mixtures
(McNeil et al. 2015, Sec. 6.2), we can represent each $Z_{k}$ as

$Z_{k}=F_{Z_{k}}^{-1}\left(t_{\nu}\big{(}\sqrt{W}\tilde{Z}_{k}\big{)}\right),\quad
k\in{\mathcal{N}},$

where $W\sim{\rm InvGamma}(\nu/2,\nu/2-1)$, such that ${\mathbb{E}}[W]=1$, and
$(\tilde{Z}_{1},\dots,\tilde{Z}_{n})$ are multivariate standard normal, with
correlation matrix $\mathbf{R}$. Then the random variables
$(\sqrt{W}\tilde{Z}_{1},\dots,\sqrt{W}\tilde{Z}_{n})$ are multivariate
$t_{\nu}$ distributed, with correlation matrix $\mathbf{R}$ and margins
standardised to have unit variance. (Note that this is slightly different to
the standard $t_{\nu}$ construction, which has margins with variance
$\nu/(\nu-2)$. This choice, which does not affect the dependence model, is
made to simplify moment calculations.) Define:

$\beta:=var\Big{(}\sum_{k=1}^{n}\tilde{Z}_{k}\Big{)}=\sum_{k,l\in{\mathcal{N}}}r_{k,l}\,,\quad\text{and}\quad\Psi:=\frac{1}{\sqrt{\beta}}\sum_{k\in{\mathcal{N}}}\tilde{Z}_{k}\sim{\rm
N}(0,1)\,.$

Then the factor model becomes:

$\displaystyle X_{j}$
$\displaystyle=\sqrt{W}\left(\sqrt{\lambda}\Psi+\sqrt{1-\lambda}\Theta_{j}\right),\quad
j\in{\mathcal{M}},$

where $\Theta_{1},\dots,\Theta_{m}$ are i.i.d. standard normal variables,
independent of $(\tilde{Z}_{1},\dots,\tilde{Z}_{n},W)$. It follows easily that
${\mathbb{E}}[X_{i}]=0,~{}var(X_{i})=1$ and ${\rm Corr}(X_{i},X_{j})=\lambda$
is fulfilled. Furthermore, the cross-correlation values:

$\displaystyle\sigma_{i,m+k}$ $\displaystyle={\rm
Corr}(X_{i},t_{\nu}^{-1}\left(F_{Z_{k}}(Z_{k})\right)={\rm
Corr}\big{(}\sqrt{W}\left(\sqrt{\lambda}\Psi+\sqrt{1-\lambda}\Theta_{i}\right)~{},~{}\sqrt{W}\tilde{Z}_{k}\big{)}$
$\displaystyle=\sqrt{\lambda}~{}{\rm
Corr}\big{(}\Psi~{},~{}\tilde{Z}_{k}\big{)}=\sqrt{\tfrac{\lambda}{\beta}}~{}{\rm
Corr}\Big{(}\sum_{l\in{\mathcal{N}}}\tilde{Z}_{l}~{},~{}\tilde{Z}_{k}\Big{)}=\sqrt{\tfrac{\lambda}{\beta}}\sum_{l\in{\mathcal{N}}}r_{k,l}\,.$

This completes the dependence model specification.

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