Regulator-based risk statistics with scenario analysis

TL;DR

This paper introduces a new class of regulator-based risk statistics incorporating scenario analysis, extending existing risk measures to better focus on potential losses relevant to regulators.

Contribution

It proposes a novel risk statistic framework that integrates scenario analysis and extends prior loss-based risk measures, enhancing regulatory risk assessment methods.

Findings

New class of risk statistics derived for regulatory purposes

Extension of risk statistics with scenario analysis

Connection to existing loss-based risk measures

Abstract

As regulators pay more attentions to losses rather than gains, we are able to derive a new class of risk statistics, named regulator-based risk statistics with scenario analysis in this paper. This new class of risk statistics can be considered as a kind of risk extension of risk statistics introduced by Kou et al. \cite{11}, and also data-based versions of loss-based risk measures introduced by Cont et al. \cite{5} and Sun et al. \cite{12}.

Equations

$$\underset{Q\in {\mathbb{R}}^N,\min Q_i\ge 1-ϵ}{\min }\alpha (Q)=0\text{ for any }ϵ\in (0,1)$$

$$\rho (M)=\underset{Q\in {\mathbb{R}}^N}{\max }\{-\sum _{i=1}^N{Q}_i(X_i\wedge 0)-\alpha (Q)\}.$$

$$\rho (M)=\underset{Q\in {\mathbb{R}}^N}{\max }\{-\sum _{i=1}^N{Q}_i(X_i\wedge 0)\}.$$

$$f(M)=\underset{M^{\ast }\in {\mathbb{R}}^N}{\max }\{M^{\ast }(M)-f^{\ast }(M^{\ast })\}$$

$$f^{\ast }(M^{\ast })=\underset{M\in {\mathbb{R}}^N}{\sup }\{M^{\ast }(-M)-\rho (M)\}.$$

$$\rho (M)=f(-M)=\underset{M^{\ast }\in {\mathbb{R}}^N}{\max }\{M^{\ast }(-M)-f^{\ast }(M^{\ast })\}.$$

$$\rho (M)=\underset{Q\in {\mathbb{R}}^N}{\max }\{-\sum _{i=1}^N{Q}_i(X_i)-f^{\ast }(Q)\},$$

$$f^{\ast }(Q)=\underset{Q\in {\mathbb{R}}^N}{\sup }\{-\sum _{i=1}^N{Q}_i(X_i)-\rho (M)\}.$$

$${\alpha }_{\min }(Q):=\underset{Q\in {\mathbb{R}}^N}{\sup }\{-\sum _{i=1}^N{Q}_i(X_i)-\rho (M)\},$$

$$\rho (M)=\underset{Q\in {\mathbb{R}}^N}{\max }\{-\sum _{i=1}^N{Q}_i(X_i\wedge 0)-{\alpha }_{\min }(Q)\}.$$

$$\rho (M)\ge -\sum _{i=1}^N{Q}_i(X_i)-\alpha (Q_1,\cdots ,Q_N).$$

$$\alpha (Q_1,\cdots ,Q_N)\ge -\sum _{i=1}^N{Q}_i(X_i)-\rho (M).$$

$$\alpha (Q)$$

$$={\alpha }_{min}(Q)$$

$$a$$

$$=\underset{Q\in {\mathbb{R}}^N}{\max }\{a\sum _{i=1}^N{Q}_i-{\alpha }_{\min }(Q)\}$$

$$\displaystyle\leq\max\Big{\{}\max\limits_{Q\in\mathbb{R}^{N},\max\limits_{1\leq i\leq N}Q_{i}<1-\epsilon}\{a\sum\limits_{i=1}^{N}{Q_{i}}-\alpha_{\min}(Q)\},\max\limits_{Q\in\mathbb{R}^{N},\min\limits_{1\leq i\leq N}Q_{i}\geq 1-\epsilon}\{a\sum\limits_{i=1}^{N}{Q_{i}}-\alpha_{\min}(Q)\},$$

$$\displaystyle\quad\quad\max\limits_{Q\in\mathbb{R}^{N},\min\limits_{1\leq i\leq N}Q_{i}\leq 1-\epsilon,\max\limits_{1\leq i\leq N}Q_{i}\geq 1-\epsilon}\{a\sum\limits_{i=1}^{N}{Q_{i}}-\alpha_{\min}(Q)\}\Big{\}}$$

$$\displaystyle\leq\max\Big{\{}Na(1-\epsilon),\max\limits_{Q\in\mathbb{R}^{N},\min\limits_{1\leq i\leq N}Q_{i}\geq 1-\epsilon}\{a\sum\limits_{i=1}^{N}{Q_{i}}-\alpha_{\min}(Q)\},$$

$$\displaystyle\quad\quad\max\limits_{Q\in\mathbb{R}^{N},\min\limits_{1\leq i\leq N}Q_{i}\leq 1-\epsilon,\max\limits_{1\leq i\leq N}Q_{i}\geq 1-\epsilon}\{a\sum\limits_{i=1}^{N}{Q_{i}}-\alpha_{\min}(Q)\}\Big{\}}$$

$$\displaystyle\leq\max\Big{\{}Na(1-\epsilon),a-\min\limits_{Q\in\mathbb{R}^{N},\min\limits_{1\leq i\leq N}Q_{i}\geq 1-\epsilon}\alpha_{\min}(Q),$$

$$card(i\in \{i:Q_i(1)<1-ϵ\})a(1-ϵ)+card(i\in \{i:Q_i(1)\ge 1-ϵ\})a-$$

$$\displaystyle\quad\quad\min\limits_{Q\in\mathbb{R}^{N},\min\limits_{1\leq i\leq N}Q_{i}<1-\epsilon,\max\limits_{1\leq i\leq N}Q_{i}>1-\epsilon}\alpha_{\min}(Q)\Big{\}},$$

$$\sum _{i=1}^N{Q}_i(X_i^1\wedge 0)\le \sum _{i=1}^N{Q}_i(X_i^2\wedge 0),$$

$${\alpha }_{\min }(Q)$$

$$=\underset{M\in {\mathbb{R}}^N}{\sup }\{-\sum _{i=1}^N{Q}_i(-\lambda X_i)-\rho (\lambda M)\}$$

$$=\lambda \underset{M\in {\mathbb{R}}^N}{\sup }\{-\sum _{i=1}^N{Q}_i(-X_i)-\rho (M)\}$$

$$=\lambda {\alpha }_{\min }(Q)$$

$$\rho \left(M-a1\right)=\rho (M)+a.$$

$$\rho \left(M-a1\right)=\bar{\rho }\left(M-a1\right)=\bar{\rho }(M)+a=\rho (M)+a.$$

$$\bar{\rho}(M)=\rho\Big{(}M-a_{M}1\Big{)}-a_{M}$$

$$\bar{\rho }\left(M-a1\right)$$

$$=\rho \left(M-a_{M}1\right)-a_M+a$$

$$=\bar{\rho }(M)+a.$$

$$\bar{\rho }(M_1)$$

$$\ge \rho (M_2-a_{M_2}1)-a_{M_2}$$

$$=\bar{\rho }(M_2),$$

$$\bar{\rho }\left(t{M}_1+(1-t)M_2\right)$$

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1 Introduction

Research on risk is a hot topic in both quantitative and theoretical research, and risk models have attracted considerable attention. The quantitative calculation of risk involves two problems: choosing an appropriate risk model and allocating the risk to individual institutions. This has led to further research on risk statistics.

In their seminal paper, Artzner et al. [3][4] first introduced the class of coherent risk measures, by proposing four basic properties to be satisfied by every sound financial risk measure. Further, Föllmer and Schied [10] and, independently, Frittelli and Rosazza Gianin [9] introduced the broader class, named convex risk measure, by dropping one of the coherency axioms. Later, Hlatshwayo et al. [11] studied the risk measures with bank failure, Li [13] studied the systemic risk.

However, as pointed out by Cont et al. [5], these axioms fail to take into account some key features encountered in the practice of risk management. In fact, sometimes, when measuring the risk, it is only relevant to consider the losses, not the gains. For this reason, we are able to derive the risk based on losses, not gains.

Next, from the statistical point of view by Kou et al. [12], the behavior of a random variable can be characterized by its samples. At the same time, one can also incorporate scenario analysis into this framework. Therefore, a natural question is how about the discuss of regulator-based risk with scenario analysis.

In the present paper, we are able to derive convex and coherent regulator-based risk statistics with scenario analysis, and dual representation results for them. Finally, the relationship between regulator-based risk statistics and the convex risk statistics introsuced by Tian and Suo [16] also be given to illustrate the regulator-based risk statistics.

It is worth mentioning that the issue of risk measures with scenario analysis have already been studied by Delbaen [7]. It have also been extensively studied in the last decade. For example, see Kou et al. [12], Ahmed et al. [1], Assa and Morales [2], Tian and Jiang [15], Tian and Suo [16], and the references therein. From this point of view, the present paper can also be considered as a kind of risk extension of risk statistics.

The remainder of this paper is organized as follows. In Sect. 2, we briefly introduce some preliminaries. The main results of regulator-based risk statistics be stated in Sect. 3, and their proofs be postponed to Sect. 4. Finally, in Sect. 5, we are able to derive the relationship between regulator-based risk statistics and the convex risk statistics introsuced by Tian and Suo [16],.

2 Preliminaries

In this section, we briefly introduce the preliminaries that are used throughout this paper. Let $N\geq 1$ be a fixed positive integer. Denote $\mathscr{X}$ by a set of random losses, and $\mathscr{X}^{N}$ by the product space $\mathscr{X}_{1}\times\cdots\times\mathscr{X}_{N}$, where $\mathscr{X}_{i}=\mathscr{X}$ for $1\leq i\leq N$. Any element of $\mathscr{X}^{N}$ is said to be a portfolio of random losses. In practice, the behavior of the $N$-dimensional random vector ${\bf M}=(X_{1},\cdots,X_{N})$ under different scenarios is represented by different sets of data observed or generated under those scenarios because specifying accurate models for ${\bf M}$ is usually very difficult. Some detailed notations can be found in Kou et al. [12]. Here, we suppose that there always exist $m$ scenarios. Specifically, suppose that the behavior of ${\bf M}$ is represented by a collection of data $M=(X_{1},\cdots,X_{N})\in\mathbf{R}^{N}$ which can be a data set based on historical observations, hypothetical samples simulated according to a model, or a mixture of observations and simulated samples.

For any $M_{1}=(X^{1}_{1},\cdots,X^{1}_{N})$, $M_{2}=(X^{2}_{1},\cdots,X^{2}_{N})\in\mathbb{R}^{N}$, $M_{1}\leq M_{2}$ means $X^{1}_{i}\leq X^{2}_{i}$ for any $i=1,2,\cdots,N$. And for any $M=(X_{1},\cdots,X_{N})\in\mathbf{R}^{N}$, let M\wedge 0:=\big{(}\min\{X_{1},0\},\cdots,\min\{X_{N},0\}\big{)}. Given $a\in\mathbf{R}$, denote a1:=\big{(}a,\cdots,a\big{)}.

3 Regulator-based risk statistics

In this section, we state the main result of regulator-based risk statistics with scenario analysis. Firstly, we derive the properties related to regulator-based risk statistics.

Definition 3.1**.**

*A function $\rho:\mathbb{R}^{N}\rightarrow[0,+\infty)$ is said to be a convex regulator-based risk statistic if it satisfies the following properties,

(A.1)Normalization: for cash losses: for any $a\geq 0$, $\rho(-(a1))=a;$

(A.2)Monotonicity: for any $M_{1},M_{2}\in\mathbb{R}^{N}$, $M_{1}\leq M_{2}$ implies $\rho(M_{1})\geq\rho(M_{2});$

(A.3)Loss-dependence : for any $M\in\mathbb{R}^{N}$, $\rho(M)=\rho(M\wedge 0)$.

(A.4)Convexity: for any $M_{1},M_{2}\in\mathbb{R}^{N}$ and $0<\lambda<1$, $\rho(\lambda M_{1}+(1-\lambda)M_{2})\leq\lambda\rho(M_{1})+(1-\lambda)\rho(M_{2}).$

Moveover, a convex regulator-based risk statistic $\rho$ is said to be a coherent regulator-based risk statistic if it still satisfies

(A.5)Positive homogeneity: for any $\alpha\geq 0$ and $M\in\mathbb{R}^{N}$, $\rho(\alpha M)=\alpha\rho(M).$*

Next, we derive the dual representations of regulator-based risk statistics and the proofs were leaved in next section.

Theorem 3.1**.**

$\rho:\mathbb{R}^{N}\rightarrow[0,+\infty)$* is a convex regulator-based risk statistic in the case of that there exists a convex function $\alpha:\mathbb{R}^{N}\rightarrow[0,+\infty]$, which is satisfied*

[TABLE]

such that

[TABLE]

The function $\alpha$ for which (3.2) holds can be choose as $\alpha_{\min}(Q):=\sup\limits_{M\in\mathbb{R}^{N}}\{-\sum\limits_{i=1}^{N}{Q_{i}}(X_{i})-\rho(M)\}$ for any $Q\in\mathbb{R}^{N}$. Moreover, $\alpha_{\min}$ is the minimal penalty function in the sense that for any penalty function $\alpha$ representing $\rho$ satisfies $\alpha(Q_{1},\cdots,Q_{N})\geq\alpha_{\min}(Q_{1},\cdots,Q_{N})$ for all $(Q_{1},\cdots,Q_{N})\in\mathbb{R}^{N}$.

Theorem 3.2**.**

$\rho:\mathbb{R}^{N}\rightarrow[0,+\infty)$* is a coherent regulator-based risk statistic in the case of that for any $M\in\mathbb{R}^{N}$,*

[TABLE]

Remark 3.1**.**

The dual representation result in Theorem 3.1 depends only on the negative part of $M$ due to the loss-dependence property (A.3). In Theorem 3.2, let $N=1$, then representation result is reduced to the one-dimensional case which coincides with the representation results of Cont et al. [5].

4 Proofs of main results

In this section, we are able to derive the proof of main results in Sect. 3.

Proof of Theorem 3.1. Let $f(X)=\rho(-X)$, then $f$ is an increasing convex function. According to [Cheridto and Li [6], Th4.2], we have

[TABLE]

where

[TABLE]

Hence

[TABLE]

Hence

[TABLE]

where

[TABLE]

Define $\alpha_{\min}:\mathbb{R}^{N}\rightarrow[0,+\infty]$ by

[TABLE]

and using loss-dependence property of $\rho$, we have

[TABLE]

Now, let $\alpha$ be any penalty function for $\rho$. Then, for any $(Q_{1},\cdots,Q_{N})\in\mathbb{R}^{N}$ and $M=(X_{1},\cdots,X_{N}),$

[TABLE]

Hence,

[TABLE]

Taking supremum over $\mathbb{R}^{N}$ for $M=(X_{1},\cdots,X_{N})$ in give rise to

[TABLE]

Next, we check that $\rho$ represented in (3.2) is a convex regulator-based risk statistic. Obviously, $\rho$ is a convex function and satisfies (A.3). Hence, we need only to show that $\rho$ satisfies (A.1) and (A.2). To this end, for any $a\geq 0$ and $1<\epsilon<1$,

[TABLE]

which implies $\alpha_{\min}$ satisfies (3.1). Now, let $M_{1}:=(X^{1}_{1},\cdots,X^{1}_{N}),M_{2}:=(X^{2}_{1},\cdots,X^{2}_{N})$. Then, the relation $M_{1}\leq M_{2}$ implies $X^{1}_{i}\wedge 0\leq X^{2}_{i}\wedge 0$ for any $1\leq i\leq N$. Hence for any $Q:=(Q_{1},\cdots,Q_{N})\in\mathbb{R}^{N}$, we have

[TABLE]

which implies $\rho(M_{1})\leq\rho(M_{2}).$ This completes the proof of Theorem 3.1.∎

Proof of Theorem 3.2. If $\rho$ is a coherent regulator-based risk statistic, then from the proof of Theorem 3.1 and the positive homogeneity of $\rho$, for any $Q\in\mathbb{R}^{N}$ and $\lambda>0$, we have

[TABLE]

Hence, $\alpha_{\min}$ can take only the values [math] and $+\infty$. This completes the proof of Theorem 3.2.∎

5 Regulator-based version of convex risk statistics

In this section, we derive a new version of regulator-based risk statistics. It is worth noting that this version can be related to convex risk statistics introsuced by Tian and Suo [16],.

For any convex risk statistic $\bar{\rho}$ on $\mathbb{R}^{N}$ defined in Tian and Suo [16],, we can define a new risk statistic $\rho$ by $\rho(M):=\bar{\rho}(M\wedge 0)$ for any $M\in\mathbb{R}^{N}$. Obviously, $\rho$ is a convex regulator-based risk statistic defined in Sect. 3. We call $\rho$ the regulator-based version of $\bar{\rho}$.

We can prove that a convex regulator-based risk statistic $\rho$ is the regulator-based version of some convex risk statistic in the case of that it satisfies

Cash-loss additivity: for any $M\in\mathbb{R}^{N}$ and $a\in\mathbb{R}$ where $M\leq 0$, $a\geq 0$,

[TABLE]

On the one hand, if $\rho(M)=\bar{\rho}(M\wedge 0)$ for certain convex risk statistic $\bar{\rho}$ on $\mathbb{R}^{N}$, then for any $M\in\mathbb{R}^{N}$, $M\leq 0$ and $a\geq 0$,

[TABLE]

where the second equality is due to the cash-additivity property of $\bar{\rho}$.

Let us now suppose that a convex regulator-based risk statistic $\rho$ satisfies the cash-loss additivity property. Define

[TABLE]

for any $M:=(X_{1},\cdots,X_{N})\in\mathbb{R}^{N}$ where $a_{M}$ is any upper-bound of each $X_{i}$. Using the cash-loss additivity property for $\rho$, we know that $\bar{\rho}$ is well-defined. Next, we need to claim that $\bar{\rho}$ is a convex risk statistic where $\rho(M)=\bar{\rho}(M\wedge 0)$. To this end, for any $M:=(X_{1},\cdots,X_{N})\in\mathbb{R}^{N}$ and $a\in\mathbb{R}$,

[TABLE]

Next, let $M_{1}:=(X_{1}^{1},\cdots,X_{N}^{1}),M_{2}:=(X_{1}^{2},\cdots,X_{N}^{2})\in\mathbb{R}^{N}$ where $M_{1}\leq M_{2}$. Taking $a_{M_{1}}$, $a_{M_{2}}$ to be the upper-bound of each $X^{1}_{i}$ and $X^{2}_{i}$. Then,

[TABLE]

which yields $\bar{\rho}$ is monotonous. Finally, for any $M_{1},M_{2}\in\mathbb{R}^{N}$ and $0\leq t\leq 1$,

[TABLE]

which implies $\bar{\rho}$ is convex.