# Reverse Sensitivity Analysis for Risk Modelling

## Abstract

**:**

`R`package

`SWIM`on

`CRAN`. The proposed reverse sensitivity analysis framework is model-free and allows for stresses on the output such as (a) the mean and variance, (b) any distortion risk measure including the Value-at-Risk and Expected-Shortfall, and (c) expected utility type constraints, thus making the reverse sensitivity analysis framework suitable for risk models.

## 1. Introduction

- (i)
- Specify a stress on the baseline distribution of the output;
- (ii)
- Derive the unique stressed distribution of the output that is closest in the Wasserstein distance and fulfils the stress;
- (iii)
- The stressed distribution induces a canonical Radon–Nikodym derivative $\frac{d{\mathbb{Q}}^{*}}{d\mathbb{P}}$; a change of measures from the baseline $\mathbb{P}$ to the stressed probability measure ${\mathbb{Q}}^{*}$;
- (iv)
- Calculate sensitivity measures that reflect an input factors’ change in distribution from the baseline to the stressed model.

## 2. Preliminaries

**Definition**

**1**

**.**The Wasserstein distance (of order 2) between two distribution functions ${F}_{1}$ and ${F}_{2}$ is defined as Villani (2008)

## 3. Deriving the Stressed Distribution

**Definition**

**2**

**.**The weighted isotonic projection ${\ell}^{{\uparrow}_{w}}$ of a function $\ell \in {\mathbb{L}}^{2}\left([0,1]\right)$ with weight function $w:[0,1]\to [0,+\infty )$, $w\in {\mathbb{L}}^{2}\left([0,1]\right)$, is its weighted projection onto the set of non-decreasing and left-continuous functions in ${\mathbb{L}}^{2}\left([0,1]\right)$. That is, the unique function satisfying

`R`package

`isotone`De Leeuw et al. (2010).

#### 3.1. Risk Measure Constraints

**Definition**

**3**

**.**Let $\gamma \in {\mathbb{L}}^{2}\left([0,1]\right)$ be a square-integrable function with $\gamma :[0,1]\to [0,+\infty )$ and ${\int}_{0}^{1}\gamma \left(u\right)\phantom{\rule{0.166667em}{0ex}}du=1$. Then the distortion risk measure ${\rho}_{\gamma}$ with distortion weight function γ is defined as

**Theorem**

**1**

**.**Let ${r}_{k}\in \mathbb{R}$, ${\rho}_{{\gamma}_{k}}$ be a distortion risk measure with weight function ${\gamma}_{k}$ and assume there exists a distribution function $\tilde{G}\in \mathcal{M}$ satisfying ${\rho}_{{\gamma}_{k}}\left(\tilde{G}\right)={r}_{k}$ for all $k\in \{1,\dots ,d\}$. Then, the optimisation problem

**Proposition**

**1**

**.**If ${\rho}_{\gamma}$ is a coherent distortion risk measure and $r\ge {\rho}_{\gamma}\left(F\right)$, then optimisation problem (3) with $d=1$ has a unique solution given by

**Example**

**1**

**.**The α-β risk measure, $0<\beta \le \alpha <1$, is defined by

#### 3.2. Integral Constraints

**Theorem**

**2**

**.**Let ${h}_{k},\phantom{\rule{0.166667em}{0ex}}{\tilde{h}}_{l}:[0,1]\to [0,\infty )$ be square-integrable functions and assume there exists a distribution function $\tilde{G}\in \mathcal{M}$ satisfying ${\int}_{0}^{1}{h}_{k}\left(u\right)\stackrel{\u02d8}{G}\left(u\right)\phantom{\rule{0.166667em}{0ex}}du\le {c}_{k}$ and ${\int}_{0}^{1}{\tilde{h}}_{l}\left(u\right){\left(\stackrel{\u02d8}{G}\left(u\right)\right)}^{2}\phantom{\rule{0.166667em}{0ex}}du\phantom{\rule{0.166667em}{0ex}}\le {\tilde{c}}_{l}$ for all $k=1,\dots ,d$, and $l=1,\dots ,\tilde{d}$. Then the optimisation problem

**Proposition**

**2**

**Example**

**2**

**.**Here, we illustrate Proposition 2 with the ES risk measure and three different stresses. The top panels of Figure 2 display the baseline quantile function ${\stackrel{\u02d8}{F}}_{Y}$ and the stressed quantile function ${\stackrel{\u02d8}{G}}_{Y}^{*}$ of Y, where the baseline distribution ${F}_{Y}$ of Y is again $Lognormal(\mu ,{\sigma}^{2})$ with parameters $\mu =\frac{7}{8}$ and $\sigma =0.5$. The bottom panels display the corresponding baseline and stressed densities. The left panels correspond to a stress, where, under the stressed model, the ${ES}_{0.95}$ and the mean are kept fixed at their corresponding values under the baseline model, while the standard deviation is increased by 20%. We observe, both in the quantile and density plot, that the stressed distribution is more spread out indicating a larger variance. Furthermore, at $y\approx 5.77$ the stressed density ${g}_{Y}^{*}\left(y\right)$ drops to ensure that ${ES}_{0.95}\left({G}_{Y}^{*}\right)={ES}_{0.95}\left({F}_{Y}\right)$. This drop is due to the fact that a stress composed of a 20% increase in the standard deviation while fixing the mean (i.e., without a constraint on $ES$) results in an ES that is larger compared to the baseline’s. Indeed, under this alternative stress (without a constraint on ES) we obtain that ${ES}_{0.95}\left({G}_{Y}^{*}\right)\approx 7.70$ compared to ${ES}_{0.95}\left({F}_{Y}\right)\approx 6.87$.

#### 3.3. Value-at-Risk Constraints

**Theorem**

**3**

**.**Let $q\in \mathbb{R}$ and consider the optimisation problem

- (i)
- under constraint (a), if $q\le {VaR}_{\alpha}\left(F\right)$, then the unique solution is given by$${\stackrel{\u02d8}{G}}^{*}\left(u\right)=\stackrel{\u02d8}{F}\left(u\right)+\left(q-\stackrel{\u02d8}{F}\left(u\right)\right){\U0001d7d9}_{\left\{u\in \left({\alpha}_{F},\alpha \right]\right\}}\phantom{\rule{0.166667em}{0ex}};$$if $q>{VaR}_{\alpha}\left(F\right)$, then there does not exist a solution.
- (ii)
- under constraint (b), if $q\ge {VaR}_{\alpha}^{+}\left(F\right)$, then the unique solution is given by$${\stackrel{\u02d8}{G}}^{*}\left(u\right)=\stackrel{\u02d8}{F}\left(u\right)+\left(q-\stackrel{\u02d8}{F}\left(u\right)\right){\U0001d7d9}_{\left\{u\in \left(\alpha ,{\alpha}_{F}\right]\right\}}\phantom{\rule{0.166667em}{0ex}};$$if $q<{VaR}_{\alpha}^{+}\left(F\right)$, then there does not exist a solution.

#### 3.4. Expected Utility Constraint

**Theorem**

**4**

**.**Let $u:\mathbb{R}\to \mathbb{R}$ be a differentiable concave utility function, ${r}_{k}\in \mathbb{R}$, and ${\rho}_{{\gamma}_{k}}$ be distortion risk measures, for $k=1,\dots ,d$. Assume there exists a distribution function $\tilde{G}$ satisfying ${\int}_{\mathbb{R}}u\left(x\right)\phantom{\rule{0.166667em}{0ex}}d\tilde{G}\left(x\right)\ge c$ and ${\rho}_{{\gamma}_{k}}\left(\tilde{G}\right)={r}_{k}$ for all $k=1,\dots ,d$. Then the optimisation problem

**Example**

**3**

**.**The Hyperbolic absolute risk aversion (HARA) utility function is defined by

#### 3.5. Smoothing of the Stressed Distribution

**Remark**

**1.**

## 4. Analysing the Stressed Model

#### 4.1. The Stressed Probability Measures

**Example**

**4**

**.**We continue Example 3 and illustrate the RN-densities $\frac{d{\mathbb{Q}}^{*}}{d\mathbb{P}}$ for the following three stresses (from the left to the right panel): a 10% decrease in ${ES}_{0.8}$ and a 10% increasing ${ES}_{0.95}$ for all three stresses, and a 0%, 1%, and 3% increase in the HARA utility, respectively.

#### 4.2. Reverse Sensitivity Measures

**Definition**

**4**

**.**For a function $s:\mathbb{R}\to \mathbb{R}$, the reverse sensitivity measure to input ${X}_{i}$ with respect to a stressed probability measure ${\mathbb{Q}}^{*}$ is defined by

**Definition**

**5.**

**Proposition**

**3**

**.**The reverse sensitivity measure possesses the following properties:

- (i)
- ${S}_{i}^{{\mathbb{Q}}^{*}}\in [-1,1]$;
- (ii)
- ${S}_{i}^{{\mathbb{Q}}^{*}}=0$ if $(s\left({X}_{i}\right),\frac{d{\mathbb{Q}}^{*}}{d\mathbb{P}})$ are independent under $\mathbb{P}$;
- (iii)
- ${S}_{i}^{{\mathbb{Q}}^{*}}=1$ if and only if $(s\left({X}_{i}\right),\frac{d{\mathbb{Q}}^{*}}{d\mathbb{P}})$ are comonotonic;
- (iv)
- ${S}_{i}^{{\mathbb{Q}}^{*}}=-1$ if and only if $(s\left({X}_{i}\right),\frac{d{\mathbb{Q}}^{*}}{d\mathbb{P}})$ are counter-comonotonic.

**Definition**

**6**

**.**For a function $s:{\mathbb{R}}^{2}\to \mathbb{R}$, the reverse sensitivity measure to inputs $({X}_{i},{X}_{j})$ with respect to a stressed probability measure ${\mathbb{Q}}^{*}$ is defined by

**Remark**

**2.**

## 5. Application to a Spatial Model

## 6. Concluding Remarks

`R`package

`SWIM`which is available on

`CRAN`.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proofs

**Proof**

**of**

**Theorem**

**1.**

**Proof**

**of**

**Proposition**

**1.**

**Proof**

**of**

**Theorem**

**2.**

**Proof**

**of**

**Proposition**

**2.**

**Proof**

**of**

**Theorem**

**3.**

**Proof**

**of**

**Theorem**

**4.**

**Proof**

**of**

**Proposition**

**3.**

- (i)
- We first define for a random variable Z with $\mathbb{P}$-distribution ${F}_{Z}$ the random variable ${U}_{Z}:={F}_{Z}\left(Z\right)$. Then, ${U}_{Z}$ and Z are comonotonic and ${U}_{Z}$ has a uniform distribution under $\mathbb{P}$. Next, recall that for any random variables ${Y}_{1},{Y}_{2}$ it holds that Rüschendorf (1983)$$\mathbb{E}\left[{Y}_{1}\phantom{\rule{0.166667em}{0ex}}{F}_{{Y}_{2}}^{-1}\left(1-{U}_{{Y}_{1}}\right)\right]\le \mathbb{E}\left[{Y}_{1}\phantom{\rule{0.166667em}{0ex}}{Y}_{2}\right]\le \mathbb{E}\left[{Y}_{1}\phantom{\rule{0.166667em}{0ex}}{F}_{{Y}_{2}}^{-1}\left({U}_{{Y}_{1}}\right)\right].$$Thus, we can rewrite the maximum in the normalising constant of the reverse sensitivity measure as follows$$\underset{\mathbb{Q}\in \mathcal{Q}}{max}\phantom{\rule{0.166667em}{0ex}}{\mathbb{E}}^{\mathbb{Q}}\left[s\left(X\right)\right]=\underset{Z\stackrel{\mathbb{P}}{=}\frac{d{\mathbb{Q}}^{*}}{d\mathbb{P}}}{max}\mathbb{E}\left[s\left(X\right)\phantom{\rule{0.166667em}{0ex}}Z\right]\phantom{\rule{0.166667em}{0ex}}=\mathbb{E}\left[s\left(X\right)\phantom{\rule{0.166667em}{0ex}}{F}_{\frac{d{\mathbb{Q}}^{*}}{d\mathbb{P}}}^{-1}\left({U}_{s\left(X\right)}\right)\right]\phantom{\rule{0.166667em}{0ex}}$$$$\underset{\mathbb{Q}\in \mathcal{Q}}{min}{\mathbb{E}}^{\mathbb{Q}}\left[s\left(X\right)\right]=\underset{Z\stackrel{\mathbb{P}}{=}\frac{d{\mathbb{Q}}^{*}}{d\mathbb{P}}}{min}\mathbb{E}\left[s\left(X\right)\phantom{\rule{0.166667em}{0ex}}Z\right]\phantom{\rule{0.166667em}{0ex}}=\mathbb{E}\left[s\left(X\right)\phantom{\rule{0.166667em}{0ex}}{F}_{\frac{d{\mathbb{Q}}^{*}}{d\mathbb{P}}}^{-1}\left(1-{U}_{s\left(X\right)}\right)\right]\phantom{\rule{0.166667em}{0ex}}.$$The reverse sensitivity for the case ${\mathbb{E}}^{{\mathbb{Q}}^{*}}\left[s\left({X}_{i}\right)\right]\ge \mathbb{E}\left[s\left({X}_{i}\right)\right]$ then becomes$${S}_{i}^{{\mathbb{Q}}^{*}}=\frac{\mathbb{E}\left[s\left({X}_{i}\right)\frac{d{\mathbb{Q}}^{*}}{d\mathbb{P}}\right]-\mathbb{E}\left[s\left({X}_{i}\right)\right]}{\mathbb{E}\left[s\left(X\right)\phantom{\rule{0.166667em}{0ex}}{F}_{\frac{d{\mathbb{Q}}^{*}}{d\mathbb{P}}}^{-1}\left({U}_{s\left(X\right)}\right)\right]-\mathbb{E}\left[s\left({X}_{i}\right)\right]}\phantom{\rule{0.166667em}{0ex}},$$$${S}_{i}^{{\mathbb{Q}}^{*}}=-\frac{\mathbb{E}\left[s\left({X}_{i}\right)\frac{d{\mathbb{Q}}^{*}}{d\mathbb{P}}\right]-\mathbb{E}\left[s\left({X}_{i}\right)\right]}{\mathbb{E}\left[s\left(X\right)\phantom{\rule{0.166667em}{0ex}}{F}_{\frac{d{\mathbb{Q}}^{*}}{d\mathbb{P}}}^{-1}\left(1-{U}_{s\left(X\right)}\right)\right]-\mathbb{E}\left[s\left({X}_{i}\right)\right]}\phantom{\rule{0.166667em}{0ex}},$$
- (ii)
- Assume that $s\left({X}_{i}\right)$ and $\frac{d{\mathbb{Q}}^{*}}{d\mathbb{P}}$ are independent under $\mathbb{P}$, then$$\mathbb{E}\left[s\left({X}_{i}\right){\textstyle \frac{d{\mathbb{Q}}^{*}}{d\mathbb{P}}}\right]=\mathbb{E}\left[s\left({X}_{i}\right)\right]\mathbb{E}\left[{\textstyle \frac{d{\mathbb{Q}}^{*}}{d\mathbb{P}}}\right]=\mathbb{E}\left[s\left({X}_{i}\right)\right]\phantom{\rule{0.166667em}{0ex}},$$
- (iii)
- From property $\left(i\right)$ we observe that $s\left({X}_{i}\right)$ and $\frac{d{\mathbb{Q}}^{*}}{d\mathbb{P}}$ are comonotonic, if and only if, ${S}_{i}^{{\mathbb{Q}}^{*}}=1$ since in this case the right inequality in Equation (A4) becomes equality.
- (iv)
- From property $\left(i\right)$ we observe that $s\left({X}_{i}\right)$ and $\frac{d{\mathbb{Q}}^{*}}{d\mathbb{P}}$ are counter-comonotonic, if and only if, then ${S}_{i}^{{\mathbb{Q}}^{*}}=1$ as in this case left inequality in Equation (A4) becomes equality.

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**Figure 1.**

**Top**panels: Baseline quantile function ${\stackrel{\u02d8}{F}}_{Y}$ (blue dashed) compared to the stressed quantile function ${\stackrel{\u02d8}{G}}_{Y}^{*}$ (red solid) for a 10% increase on the $\alpha $-$\beta $ risk measure with $\beta =0.1$, $\alpha =0.9$, and various values of p. The green line $\ell (\xb7)$ is the function, whose isotonic projection equals ${\stackrel{\u02d8}{G}}_{Y}(\xb7)$.

**Bottom**panels: corresponding baseline ${f}_{Y}$ and stressed ${g}_{Y}^{*}$ densities.

**Figure 2.**

**Top**: Baseline quantile function ${\stackrel{\u02d8}{F}}_{Y}$ compared to the stressed quantile function ${\stackrel{\u02d8}{G}}_{Y}^{*}$.

**Bottom**: corresponding baseline ${f}_{Y}$ and stressed ${g}_{Y}^{*}$ densities.

**Left**: ${\mathrm{ES}}_{0.95}$ and the mean being fixed and a 20% increase in the standard deviation.

**Middle**: 10% increase in ${\mathrm{ES}}_{0.95}$, 10% decrease in the mean, and fixed standard deviation.

**Right**: 10% increase in ${\mathrm{ES}}_{0.95}$, 10% increase in the mean, and 10% decrease in standard deviation. Note that in the middle and right panel the green lines are equal to the red lines.

**Figure 3.**

**Top**panels: Baseline quantile function ${\stackrel{\u02d8}{F}}_{Y}$ compared to the stressed quantile function ${\stackrel{\u02d8}{G}}_{Y}^{*}$, for a 10% decrease in ${\mathrm{ES}}_{0.8}$, and a 10% increase in ${\mathrm{ES}}_{0.95}$, and, from left to right, a 0%, 1%, and 3% increase in the HARA utility, respectively. The function $\ell (\xb7)$ (solid green) is the function whose isotonic projection equals ${\stackrel{\u02d8}{G}}^{*}(\xb7)$.

**Bottom**panels: Corresponding baseline ${f}_{Y}$ and stressed ${g}_{Y}^{*}$ densities.

**Figure 4.**RN-densities for following stresses: a 10% decrease in both ${\mathrm{ES}}_{0.8}$ and ${\mathrm{ES}}_{0.95}$, and an increase in the HARA utility. The change in HARA utility is 0%, 1%, and 3%, respectively, from left to right.

**Figure 5.**Reverse sensitivity measures with $s\left(x\right)=x$, $s\left(x\right)={\U0001d7d9}_{\{x>{\stackrel{\u02d8}{F}}_{i}\left(0.8\right)\}}$, and $s\left(x\right)={\U0001d7d9}_{\{x>{\stackrel{\u02d8}{F}}_{i}\left(0.95\right)\}}$ (left to right), for two different stresses on the output Y. First stress (salmon) is keeping the HARA utility and $\mathrm{ES}{\left(Y\right)}_{0.8}$ fixed and increasing the $\mathrm{ES}{\left(Y\right)}_{0.85}$ by 1%. Second stress (violet) is an increase of 1% in HARA utility, 1% $\mathrm{ES}{\left(Y\right)}_{0.8}$, and 3% in $\mathrm{ES}{\left(Y\right)}_{0.85}$.

**Figure 6.**Contour plots of the bivariate copulae of $({L}_{5},{L}_{10})$ (

**top**panels) and $({L}_{9},{L}_{10})$ (

**bottom**panels) under different models. The left contour plots correspond to the baseline model and the right panels to the stress ${\mathbb{Q}}_{2}^{*}$ (solid lines) with the baseline contours reported using partially transparent lines. Red points are simulated realisations.

**Table 1.**Summary of the stresses applied to the portfolio loss Y represented in relative increases of the stressed model from the baseline model.

HARA Utility | ${\mathbf{ES}}_{0.8}\left(\mathit{Y}\right)$ | ${\mathbf{ES}}_{0.95}\left(\mathit{Y}\right)$ | |
---|---|---|---|

Stress 1: ${\mathbb{Q}}_{1}^{*}$ | 0% | 0% | 1% |

Stress 2: ${\mathbb{Q}}_{2}^{*}$ | 1% | 1% | 3% |

**Table 2.**Comparison of different sensitivity measures: First two columns correspond to the reverse sensitivity measures with $s\left(x\right)={\U0001d7d9}_{\{x>\stackrel{\u02d8}{F}\left(0.95\right)\}}$ and stressed models ${\mathbb{Q}}_{1}^{*}$, and ${\mathbb{Q}}_{2}^{*}$, respectively. The last three columns are the delta measure under $\mathbb{P}$, ${\mathbb{Q}}_{1}^{*}$, and ${\mathbb{Q}}_{2}^{*}$, respectively.

${\mathit{S}}_{\mathit{i}}^{{\mathbb{Q}}_{1}^{*}}$ | ${\mathit{S}}_{\mathit{i}}^{{\mathbb{Q}}_{2}^{*}}$ | ${\mathit{\xi}}^{\mathbb{P}}$ | ${\mathit{\xi}}^{{\mathbb{Q}}_{1}^{*}}$ | ${\mathit{\xi}}^{{\mathbb{Q}}_{2}^{*}}$ | |
---|---|---|---|---|---|

${L}_{1}$ | 0.45 | 0.68 | 0.38 | 0.38 | 0.38 |

${L}_{2}$ | 0.47 | 0.62 | 0.29 | 0.29 | 0.29 |

${L}_{3}$ | 0.51 | 0.57 | 0.30 | 0.30 | 0.29 |

${L}_{4}$ | 0.52 | 0.63 | 0.30 | 0.30 | 0.29 |

${L}_{5}$ | 0.34 | 0.58 | 0.33 | 0.34 | 0.33 |

${L}_{6}$ | 0.41 | 0.62 | 0.34 | 0.34 | 0.32 |

${L}_{7}$ | 0.54 | 0.72 | 0.40 | 0.40 | 0.38 |

${L}_{8}$ | 0.60 | 0.69 | 0.38 | 0.39 | 0.39 |

${L}_{9}$ | 0.24 | 0.66 | 0.40 | 0.40 | 0.38 |

${L}_{10}$ | 0.41 | 0.73 | 0.39 | 0.38 | 0.37 |

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**MDPI and ACS Style**

Pesenti, S.M.
Reverse Sensitivity Analysis for Risk Modelling. *Risks* **2022**, *10*, 141.
https://doi.org/10.3390/risks10070141

**AMA Style**

Pesenti SM.
Reverse Sensitivity Analysis for Risk Modelling. *Risks*. 2022; 10(7):141.
https://doi.org/10.3390/risks10070141

**Chicago/Turabian Style**

Pesenti, Silvana M.
2022. "Reverse Sensitivity Analysis for Risk Modelling" *Risks* 10, no. 7: 141.
https://doi.org/10.3390/risks10070141