2. Background

Consider the probability space $(\Omega,\mathcal{F},\mathbb{P})$ . All equalities and inequalities are in the $\mathbb{P}$ -a.s. sense. We have that $L^0=L^0(\Omega,\mathcal{F},\mathbb{P})$ , $L^1=L^1(\Omega,\mathcal{F},\mathbb{P}),$ and $L^{\infty}=L^{\infty}(\Omega,\mathcal{F},\mathbb{P})$ are, respectively, the spaces of (equivalent classes under $\mathbb{P}$ -a.s. equality of) finite, integrable and essentially bounded real random variables in $(\Omega,\mathcal{F},\mathbb{P})$ . Let $\mathcal{P}$ be the set of all probability measures on $(\Omega,\mathcal{F})$ . We denote $E_{\mathbb{Q}}[X]=\int_{\Omega}Xd\mathbb{Q}$ , $F_{X, \mathbb{Q}}(x)=\mathbb{Q}(X\leq x)$ , and $F_{X, \mathbb{Q}}^{-1}(\alpha)=\inf\left\lbrace x\,:\,F_{X, \mathbb{Q}}(x)\geq\alpha\right\rbrace $ , respectively, the expected value, the (nondecreasing and right continuous) probability function and its left quantile for X under $\mathbb{Q}\in\mathcal{P}$ . We drop subscripts regarding probability measures when $\mathbb{Q}=\mathbb{P}$ . Furthermore, let $\mathcal{Q}\subseteq\mathcal{P}$ be the set of probability measures that are absolutely continuous about $\mathbb{P}$ with essentially bounded Radon-Nikodym derivatives $\frac{d\mathbb{Q}}{d\mathbb{P}}\in L^\infty$ . Moreover, $1_A$ is the indicator function of event A.

We begin with a brief background on the definitions and results from the risk measures literature we use alongside the paper. In this sense, we first define risk measures as functionals on $L^1$ .

Under some properties, we have a robust characterization for coherent risk measures. Such portrayal, known as dual representation, allows us to understand a risk measure as a worst-case scenario for the loss expectation.

Theorem 1. (Theorems 2.11 and 3.1 of Kaina and Rüschendorf, Reference Kaina and Rüschendorf2009). A map $\rho \,:\, L^1\to \mathbb{R}$ is a coherent risk measure if and only if it can be represented as

(2.1) \begin{equation} \rho(X)=\max\limits_{\mathbb{Q}\in\mathcal{Q}_\rho} E_\mathbb{Q}[{-}X],\:\forall\:X\in L^1, \end{equation}

where $\mathcal{Q}_\rho\subseteq\mathcal{Q}$ is nonempty, closed in total variation norm, and convex set called the dual set of $\rho$ . Moreover, $\rho$ is continuous in the $L^1$ norm.

Interesting features are present when there is Law Invariance, which is the case in most practical applications. We focus on dual representation. For the following results, we assume our probability space is atomless.

Theorem 2. (Theorem 7 of Kusuoka, Reference Kusuoka2001, Theorem 4.1 of Acerbi, Reference Acerbi2002, Theorem 7 of Frittelli and Gianin, Reference Frittelli and Gianin2005, Theorem 2.2 of Filipovi´c and Svindland, Reference Filipović and Svindland2012). $\rho \,:\, L^1\to \mathbb{R}$ is a law invariant coherent risk measure if and only if it can be represented as

(2.2) \begin{equation} \rho(X)=\sup\limits_{m\in\mathcal{M}_\rho} \int_{(0,1]}ES^p(X)dm(p),\:\forall\:X\in L^1, \end{equation}

where $\mathcal{M}_{\rho}=\left\lbrace m\in\mathcal{M}\,:\,\int_{(u,1]}\frac{1}{v}dm=F^{-1}_{\frac{d\mathbb{Q}}{d\mathbb{P}}}(1-u),\:\mathbb{Q}\in\mathcal{Q}_\rho\right\rbrace $ and $\mathcal{M}$ is the set of probabilities over $(0, 1]$ . If in addition $\rho$ is comonotone, then

(2.3) \begin{equation} \rho(X)= \int_{(0,1]}ES^p(X)dm(p)=\int_0^1VaR^p(X)\phi(p)dp,\:\forall\:X\in L^1, \end{equation}

where $m\in\mathcal{M}_\rho$ , $\phi\,:\,[0, 1]\to\mathbb{R}_+$ is a nonincreasing functional such that $\int_{0}^{1}\phi(u)du=1$ and $\int_{(u,1]}\frac{1}{v}dm=\phi(u)$ .

A recently highlighted statistical property is Elicitability, which enables comparing competing models in risk forecasting. See Ziegel (Reference Ziegel2016), Bellini and Bignozzi (Reference Bellini and Bignozzi2015), Kou and Peng (Reference Kou and Peng2016), Fissler and Ziegel (Reference Fissler and Ziegel2016, Reference Fissler and Ziegel2021) and the references therein for more details. We now adapt it to our framework.

Definition 2. A map $S\,:\,\mathbb{R}^{k+1}\to\mathbb{R}_+$ is called scoring function if $\omega\mapsto S(X(\omega),y)\in L^1$ for any $X\in L^1$ and any $y\in\mathbb{R}^k$ . A function $\rho\,:\, L^1\to\mathbb{R}^k$ is elicitable if exists a scoring function $S^\rho\,:\,\mathbb{R}^{k+1}\to\mathbb{R}_+$ such that

(2.4) \begin{equation} \rho(X)=-\mathop{\textrm{arg min}}\limits_{y\in\mathbb{R}^k}E\left[S^\rho(X,y) \right],\:\forall\:X\in L^1. \end{equation}

Remark 2. Elicitability, when confined to risk measures, can be restrictive depending on the demanded financial properties at hand. In this sense, we may end up with only one example of risk functional which satisfies the requisites. See Theorem 4.9 of Bellini and Bignozzi (Reference Bellini and Bignozzi2015) and Theorem 1 in Kou and Peng (Reference Kou and Peng2016). For instance, VaR and Expectile, when finite, are elicitable, respectively, under scores on $\mathbb{R}^2$

(2.5) \begin{align} S^{VaR^\alpha}(x,y)=\alpha(x - y)^+ + (1 - \alpha)(x - y)^- \end{align}

and

(2.6) \begin{align} S^{\text{Expectile}^\alpha}(x,y)=\alpha((x-y)^+)^2+(1-\alpha)((x-y)^-)^2, \end{align}

while ES and SDR are not. However, under joint elicitability of $(VaR^\alpha(X), ES^\alpha(X))$ , we can have a useful score for ES, on $\mathbb{R}^3$ , as

(2.7) \begin{align} S^{ES^\alpha}(x,y,z)=y\left(1_{x<y}-\alpha \right) - x1_{x<y} + e^{z} \left(z -y + \frac{1_{x<y}}{\alpha}\left( y - x \right) \right) - e^{z} + 1-\log\!\left(1- \alpha\right). \end{align}

3. Range-based risk measures

We are now in conditions to define the main functional in our approach. The goal is to consider a risk measure parameterized by a tail level $\alpha\in[0, 1]$ , as is the case of those we presented as examples, and to derive a range-based formulation.

Definition 3. A collection of risk measures $\{\rho^s\,:\, L^1\to\mathbb{R}\cup{\{-\infty,\infty\}},\:s\in[0, 1]\}$ defines a tail level functional if $s\mapsto \rho^s(X)$ is nonincreasing and integrable in $[\alpha,\beta]$ for any $X\in L^1$ . Its range-based risk measure is defined as

(3.1) \begin{equation} R_\rho(X)\,:\!=\, R_\rho^{\alpha,\beta}(X)=\begin{cases} \frac{1}{\beta-\alpha}\int_{\alpha}^{\beta}\rho^s(X)ds,\:\text{if}\:\alpha<\beta,\\[6pt] \rho^\alpha(X),\:\text{if}\:\alpha=\beta, \end{cases} \end{equation}

where $\:0\leq \alpha\leq \beta\leq 1$ .

We now explore some results of our approach. We focus on the case $\beta>\alpha$ as otherwise claims are trivially obtained. We begin with the preservation of properties since it is crucial for using $R_\rho$ in financial applications.

We now derive in the next result the dual set for range-based risk measures.

Proposition 2. Let $X\mapsto \rho^s(X)$ be a coherent risk measure with fixed $\mathbb{Q}^\prime\in\mathcal{Q}_{\rho^s}$ for $\lambda$ -almost all $s\in[\alpha,\beta]$ . Then $R_\rho$ also is finite and coherent and its dual set $\mathcal{Q}_{R_\rho}$ is given by the closure in total variation norm of

(3.4) \begin{equation}\left\lbrace \mathbb{Q}\in\mathcal{Q}\,:\, \mathbb{Q}(A)=\frac{1}{\beta-\alpha}\int_{[\alpha,\beta]}\mathbb{Q}^s(A)d\lambda(s)\:\forall\:A\in\mathcal{F},\mathbb{Q}^s\in\mathcal{Q}_{\rho^s}\:\text{for}\:\lambda\:\text{-almost all}\:s\in[\alpha,\beta]\right\rbrace.\end{equation}

Moreover, $\mathbb{Q}^{*}\in\mathcal{Q}_{R_\rho}$ is optimal for $R_\rho(X)$ , that is, $\mathbb{Q}^{*}=\mathop{\textrm{arg\,max}}\limits_{\mathbb{Q}\in\mathcal{Q}_{R_\rho}}E_\mathbb{Q}[{-}X]$ , if and only if $\mathbb{Q}^s$ is optimal for $\rho^s(X)$ for $\lambda$ -almost all $s\in[\alpha,\beta]$ .

Proof. Coherence and finiteness of $R_\rho$ are straightforward. We then focus on the dual representation. By Theorem 1, $\mathcal{Q}_{R_\rho}$ is nonempty. We claim that the proposed dual set is composed by probability measures. Let $\mathbb{Q}(A)=\frac{1}{\beta-\alpha}\int_{[\alpha,\beta]}\mathbb{Q}^s(A)d\lambda(s)$ for any $A\in\mathcal{F}$ , where $\mathbb{Q}^s\in\mathcal{Q}_{\rho^s}$ for $\lambda$ -almost all $s\in[\alpha,\beta]$ . Note that such $\mathbb{Q}$ exists since some $\mathbb{Q}^\prime\in\mathcal{Q}_{\rho^s}$ for $\lambda$ -almost all $s\in[\alpha,\beta]$ . Further, by definition of $\mathbb{Q}$ we have that $s\mapsto\mathbb{Q}^s(A)$ is measurable for any $A\in\mathcal{F}$ . It is direct that both $\mathbb{Q}(\emptyset)=0$ and $\mathbb{Q}(\Omega)=1$ . For countably additivity, let $\{A_n\}_{n\in\mathbb{N}}$ be a collection of mutually disjoint sets. Then, since both $s\mapsto\mathbb{Q}^s(A)$ and $s\mapsto\sum_{n=1}^\infty \mathbb{Q}^s(A_n)$ are bounded we have by Monotone Convergence Theorem that

\begin{equation*}\mathbb{Q}(\!\cup_{n=1}^\infty A_n)=\frac{1}{\beta-\alpha}\int_{[\alpha,\beta]}\sum_{n=1}^{\infty}\mathbb{Q}^s(A_n)d\lambda(s)=\sum_{n=1}^{\infty}\left( \frac{1}{\beta-\alpha}\int_{[\alpha,\beta]}\mathbb{Q}^s(A_n)d\lambda(s)\right) =\sum_{n=1}^{\infty}\mathbb{Q}(A_n).\end{equation*}

Hence, $\mathbb{Q}$ is a probability measure. By continuity of probability measures, limit points are also probability measures. Further, it is clear that $\mathcal{Q}_{R_\rho}$ is closed and convex. Notice that $\mathcal{Q}_{\rho^{s}}\subseteq\mathcal{Q}_{\rho^{r}}$ if and only if $s\leq r$ . Moreover, we have that

\begin{align*} E_{\mathbb{Q}}[{-}X]&=\int_{0}^{\infty}\frac{1}{\beta-\alpha}\int_{[\alpha,\beta]}(1-\mathbb{Q}^s(X\leq x))d\lambda(s) dx+\int_{-\infty}^{0}\frac{1}{\beta-\alpha}\int_{[\alpha,\beta]}\mathbb{Q}^s(X\leq x)d\lambda(s) dx\\[5pt] &=\frac{1}{\beta-\alpha}\int_{[\alpha,\beta]}\left( \int_{0}^{\infty}(1-\mathbb{Q}^s(X\leq x))dx+\int_{-\infty}^{0}\mathbb{Q}^s(X\leq x)dx\right) d\lambda(s)\\[5pt] &=\frac{1}{\beta-\alpha}\int_{[\alpha,\beta]}E_{\mathbb{Q}^s}[{-}X]d\lambda(s). \end{align*}

The measurability of $s\mapsto\rho^s(X)$ for any $X\in L^1$ it is attained from the definition of $\{\rho^s,\:s\in[0, 1]\}$ . This also implies that the maps $s\mapsto E_{\mathbb{Q}^s_{*}}[{-}X]=\max_{\mathbb{Q}\in\mathcal{Q}_{\rho^s}}E_\mathbb{Q}[{-}X]$ are measurable any $X\in L^1$ . From Hölder inequality, we have $E_{\mathbb{Q}^s_{*}}[|-X|]<\infty$ . Thus, $R_\rho(X)=\frac{1}{\beta-\alpha}\int_{[\alpha,\beta]}E_{\mathbb{Q}^s_{*}}[{-}X]d\lambda(s)\geq E_\mathbb{Q}[{-}X]$ for any $\mathbb{Q}\in\mathcal{Q}_{R_\rho}$ . By taking supremum (which is not affected by closure operation), we get $R_\rho(X)\geq\sup_{\mathbb{Q}\in\mathcal{Q}_{R_\rho}}E_\mathbb{Q}[{-}X]$ . For the converse, consider for each $n\in\mathbb{N}$ the partition $P^n$ of $[\alpha,\beta]$ as $P^n=\left\lbrace t^k_n=\alpha+\frac{k(\beta-\alpha)}{n},\:k=0,\dots,n\right\rbrace $ . Define for each $n\in\mathbb{N}$ the set $\mathcal{Q}^n$ as and

\begin{equation*}\left\lbrace \mathbb{Q}\in\mathcal{Q}\,:\, \mathbb{Q}(A)=\frac{1}{\beta-\alpha}\sum_{k=0}^{n-1}\mathbb{Q}^{t^n_{k}}(A)(t^n_{k+1}-t^n_k)\:\forall\:A\in\mathcal{F},\mathbb{Q}^{t^n_{k}}\in\mathcal{Q}_{\rho^{t^n_{k}}}\:\forall\:k=0,\dots,n-1\right\rbrace\!.\end{equation*}

It is clear that $\mathcal{Q}^n\subseteq \mathcal{Q}_{R_\rho}$ . Define for each $n\in\mathbb{N}$ the map

\begin{equation*}R^n_\rho(X)=\sup\limits_{\mathbb{Q}\in\mathcal{Q}^n}E_\mathbb{Q}[{-}X]=\dfrac{1}{\beta-\alpha}\sum_{k=0}^{n-1}\rho^{t^n_{k+1}}(X)(t^n_{k+1}-t^n_k),\:X\in L^1.\end{equation*}

We then have that $R^n_\rho(X)\to R_\rho(X)$ for any $X\in L^1$ . Thus, we get for any $X\in L^1$ that

\begin{equation*}R_\rho(X)\geq\sup_{\mathbb{Q}\in\mathcal{Q}_{R_\rho}}E_\mathbb{Q}[{-}X]\geq \lim\limits_{n\to\infty}\sup_{\mathbb{Q}\in\mathcal{Q}^n}E_\mathbb{Q}[{-}X]=\lim\limits_{n\to\infty} R^n_\rho(X)=R_\rho(X).\end{equation*}

Hence, from Theorem 1, $R_\rho(X)=\max_{\mathbb{Q}\in\mathcal{Q}_{R_\rho}}E_\mathbb{Q}[{-}X]$ . Moreover, $\mathbb{Q}^{*}\in\mathcal{Q}_{R_\rho}$ is the argmax if and only if $R_\rho(X)- E_{\mathbb{Q}^{*}}[{-}X]=\frac{1}{\beta-\alpha}\int_\alpha^\beta\left(\rho^s(X)-E_{\mathbb{Q}^s}[{-}X] \right)d\lambda(s)=0$ . Since $R_\rho(X)\geq E_{\mathbb{Q}^{*}}[{-}X]$ , we have that this is equivalent to $\rho^s(X)=E_{\mathbb{Q}^s}[{-}X]$ for $\lambda$ -almost all $s\in[\alpha,\beta]$ .

We now expose a result for the representation of $R_\rho$ under Law Invariance and Comonotonic Additivity. For such a claim, we assume that our probability space is atomless.

Proposition 3. If $X\mapsto \rho^s(X)$ is a law invariant coherent risk measure for $\lambda$ -almost all $s\in[\alpha,\beta]$ , then the representation is

(3.5) \begin{equation} R_\rho(X)=\sup\limits_{m\in cl\left(\mathcal{M}_{R_\rho}\right)} \int_{(0,1]}ES^p(X)dm(p),\:\forall\:X\in L^1, \end{equation}

with $\mathcal{M}_{R_\rho}=\left\lbrace m\in\mathcal{M}\,:\, m=\frac{1}{\beta-\alpha}\int_{[\alpha,\beta]}m^sd\lambda(s),\:m^s\in\mathcal{M}_{\rho^s}\:\forall\:s\in[\alpha,\beta]\right\rbrace$ and the closure taken under the total variation norm. If, in addition, we have Comonotonic Additivity, then

(3.6) \begin{equation} R_\rho(X)= \int_{(0,1]}ES^p(X)dm(p)=\int_{0}^{1}VaR^p(X)\phi(p)dp,\:\forall\:X\in L^1, \end{equation}

where $m\in cl(\mathcal{M}_{R_\rho})$ and $\phi(u)=\frac{1}{\beta-\alpha}\int_{[\alpha,\beta]}\phi^s(u)d\lambda(s)$ .

Proof. Law invariance implies $\mathbb{P}\in\mathcal{Q}_{\rho^s}$ for $\lambda$ -almost all $s\in[\alpha,\beta]$ . By Theorems 1 and 2, we have that for any $m\in\mathcal{M}_{R_\rho}$ , there is $\mathbb{Q}^\prime\in\mathcal{Q}_{R_\rho}$ such that

\begin{align*}\int_{(0,1]}ES^\alpha(X)dm&=\sup\left\lbrace E_\mathbb{Q}[{-}X]\,:\, \frac{d\mathbb{Q}}{d\mathbb{P}}\sim \frac{d\mathbb{Q}^\prime}{d\mathbb{P}},\int_{(u,1]}\frac{1}{v}dm=F^{-1}_{\frac{d\mathbb{Q}^\prime}{d\mathbb{P}}}(1-u)\right\rbrace\\&=E_{\mathbb{Q}^\prime}[{-}X],\:\forall\:X\in L^1.\end{align*}

Thus, the result for (3.5) follows similarly steps of those for Proposition 2 by considering the maps $s\mapsto\int_{(0,1]}ES^p(X)dm^s(s)=E_{\mathbb{Q}^s}[{-}X]=\max_{\mathbb{Q}\in\mathcal{Q}_{\rho^s}}E_\mathbb{Q}[{-}X]$ . For (3.6), the claim is a consequence of Theorem 2. The measurability of $s\mapsto\phi^s(u)$ for any $u\in[0, 1]$ can be found in Remark 4.15 of Righi (Reference Righi2023).

In the context of tail risk, one typically has that smaller values for $\alpha$ lead to larger losses. Thus, studying the role of significance levels is relevant, and the following result explores such features.

Proof. In the following, we fix for once $0\leq\alpha\leq\beta\leq 1$ and $X\in L^1$ . For (i), for any $s\in[\alpha,\beta]$ we have that $\rho^\alpha(X)\geq \rho^s(X)\geq \rho^\beta(X)$ . Thus, by monotonicity of integral we have that

\begin{equation*}\rho^\alpha(X)=\dfrac{1}{\beta-\alpha}\int_{\alpha}^\beta\rho^\alpha(X)ds\geq R_\rho^{\alpha,\beta}(X)\geq\dfrac{1}{\beta-\alpha}\int_{\alpha}^\beta\rho^\beta(X)ds\geq\rho^\beta(X).\end{equation*}

Concerning (ii), let $(\alpha_1,\beta_1)\geq(\alpha_2,\beta_2)$ . Then we have that $R_\rho^{\alpha_1,\beta_1}(X)\leq R_\rho^{\alpha_2,\beta_1}(X)\leq R_\rho^{\alpha_2,\beta_2}(X)$ . Item (iii) is direct the Hadamard-type inequality.

Regarding (iv), we prove for $\alpha\mapsto R_\rho^{\alpha,\beta}(X)$ . We have that the map $\alpha\mapsto\rho^\alpha(X)=R_\rho^{\alpha,\alpha}$ is monotone. We then get that

\begin{align*} \dfrac{\partial R_\rho^{\alpha,\beta}(X)}{\partial\alpha}&=\dfrac{1}{(\beta-\alpha)^2}\int_{\alpha}^\beta\rho^s(X)ds-\dfrac{1}{\beta-\alpha}\rho^\alpha(X)\\[5pt] &=\dfrac{1}{\beta-\alpha}\left(R_\rho^{\alpha,\beta}(X)-\rho^\alpha(X)\right)\leq 0. \end{align*}

Since the map has a nonnegative derivative, it is nonincreasing. For concavity, we look for the second derivative. We have that

\begin{align*} \dfrac{\partial^2 R_\rho^{\alpha,\beta}(X)}{\partial\alpha^2}&=\dfrac{1}{(\beta-\alpha)^2}\left(R_\rho^{\alpha,\beta}(X)-\rho^\alpha(X) \right)+\dfrac{1}{\beta-\alpha}\dfrac{\partial R_\rho^{\alpha,\beta}(X)}{\partial\alpha}\leq 0. \end{align*}

Nonnegativity thus implies the map is concave. Together with monotonicity we have, it is continuous. For $\beta\mapsto R_\rho^{\alpha,\beta}(X)$ the deduction is quite similar. We have that

\begin{align*} \dfrac{\partial R_\rho^{\alpha,\beta}(X)}{\partial\beta}&=-\dfrac{1}{(\beta-\alpha)^2}\int_{\alpha}^\beta\rho^s(X)ds+\dfrac{1}{\beta-\alpha}\rho^\beta(X)=\dfrac{1}{\beta-\alpha}\left(\rho^\beta(X)-R_\rho^{\alpha,\beta}(X)\right)\leq 0,\\[5pt] \dfrac{\partial^2 R_\rho^{\alpha,\beta}(X)}{\partial\beta^2}&=-\dfrac{1}{(\beta-\alpha)^2}\left(\rho^\beta(X)-R_\rho^{\alpha,\beta}(X) \right)-\dfrac{1}{\beta-\alpha}\dfrac{\partial R_\rho^{\alpha,\beta}(X)}{\partial\beta}\geq 0. \end{align*}

Finally, continuity in the product Euclidean metric is obtained from the counterpart property in real line for both the first and second arguments.

Despite the continuity of probability tail levels, we can expect an asymmetric variation rate at extreme tails. This pattern can represent, for instance, more sensibility to greater losses than to smaller ones reflecting risk aversion, that is, $s\mapsto\rho^s(X)$ be convex and continuous. This is the case, for instance of $\rho^s=\frac{1}{s}E\left[e^{-sX}\right]$ for $s\in(0,1]$ and $\rho^0(X)=-E[X]$ , which is linked to the Entropic risk measure; see Föllmer and Schied (Reference Föllmer and Schied2016) for details. The following result addresses this feature.

Proof. First, we get that any $s\in [\alpha-\epsilon,\beta+\epsilon]$ , with $\epsilon\in[0,\min\{\alpha,(1-\beta)\}]$ , lies in the unit interval. From convexity of $s\mapsto\rho^s(X)$ , we have that $\epsilon\mapsto R_\rho^{\alpha-\epsilon,\beta+\epsilon}(X)$ is differentiable. We then have that

\begin{align*}\dfrac{\partial R_\rho^{\alpha-\epsilon,\beta+\epsilon}(X)}{\partial\epsilon}&=-\dfrac{2}{(\beta-\alpha+2\epsilon)^2}\int_{\alpha-\epsilon}^{\beta+\epsilon}\rho^s(X)ds+\dfrac{1}{\beta-\alpha+2\epsilon}\left(\rho^{\beta+\epsilon}(X)+\rho^{\alpha-\epsilon}(X) \right) \\&=\dfrac{1}{\beta-\alpha+2\epsilon}\left(\rho^{\beta+\epsilon}(X)+\rho^{\alpha-\epsilon}(X)-2 R_\rho^{\alpha-\epsilon,\beta+\epsilon}(X)\right)\geq 0.\end{align*}

The last inequality comes from item (iii) in Proposition 4. Hence, we have that the map $\epsilon\mapsto R_\rho^{\alpha-\epsilon,\beta+\epsilon}(X)$ is nondecreasing.

A question that naturally arises in the context of $\rho^\alpha\geq R_\rho^{\alpha,\beta}\geq \rho^\alpha$ is if exists an equivalent probability tail level $s\in[\alpha,\beta]$ such that $\rho^s=R_\rho^{\alpha,\beta}$ . This feature is studied in Li and Wang (Reference Li and Wang2023), which introduce the probability equivalent level of VaR-ES (PELVE) as the ratio of the ES confidence level to that of VaR, which yields an equivalent risk value. We now define a functional to deal with this task in our proposed context.

Definition 4. The probability equivalent level to $R_\rho$ is a functional $\Pi^{\alpha,\beta}\,:\!=\,\Pi\,:\, L^1\to [0, 1]$ defined as

(3.7) \begin{equation}\Pi^{\alpha,\beta}(X)=\inf\left\lbrace x\in[\alpha,\beta]\,:\,\rho^x(X)\leq R_\rho^{\alpha,\beta}(X)\right\rbrace .\end{equation}

The next Proposition explores the properties of the probability equivalent-level functional we have defined. More specifically, we explore the existence and uniqueness of the satisfying value, an alternative representation, monotonicity with respect to the probability levels, invariance, and quasi-concavity for comonotone pairs.

Regarding statistical properties, it does not have to necessarily exist a score $S^{R_\rho}$ such that $R_\rho$ is elicitable at all. For the particular case of RVaR, as pointed out in Fissler and Ziegel (Reference Fissler and Ziegel2021), there is a scoring function over $\mathbb{R}^4$ that makes it jointly elicitable with $(VaR^\alpha,VaR^\beta,RVaR^{\alpha,\beta})$ . Such scoring function is as

(3.8) \begin{align}S^{RVaR^{\alpha,\beta}}(x,y,z,w) &= y\left(1_{x<y} - \alpha\right)- x1_{x<y} + z\left(1_{x<z} - \beta\right)-x1_{x<z}\nonumber\\& \quad + (\beta - \alpha) \tanh\!\left(\left(\beta - \alpha\right)w \right)\left[ w + \frac{1}{\beta - \alpha}\left(S^{VaR^\beta}(x,z)-S^{VaR^\alpha}(x,y)\right)\right] \nonumber\\&\quad - \log\!\left( \cosh\!\left((\alpha - \beta)w\right)\right)+ 1 - \log\!\left(1 - \alpha\right).\end{align}

Nonetheless, in the case $\{\rho^s,\:s\in[0, 1]\}$ are elicitable, we may consider a range criteria for comparison of forecasting for the resulting $R_\rho$ . We now define such concept.

Definition 5. Let $\rho^s$ be elicitable under $S^{\rho^s}\,:\,\mathbb{R}^{k+1}\to\mathbb{R}_+$ for $\lambda$ -almost all $s\in[\alpha,\beta]$ , and $(s,\omega)\mapsto S^{\rho^s}(X(\omega),y)$ integrable, in the product measure space, for any $X\in L^1$ and any $y\in \mathbb{R}^{k}$ . In this case its range-based score is a map $S^{R_\rho}\,:\, \mathbb{R}^{k+1}\to\mathbb{R}_+$ defined as

(3.9) \begin{equation}S^{R_\rho}(x)=\frac{1}{\beta-\alpha}\int_{\alpha}^\beta S^{\rho^s}(x)ds.\end{equation}

Remark 6. We have that the scores for VaR, ES, and Expectile are contemplated by such definition. In fact, they are all continuous and bounded in the $\alpha$ parameter. For SDR, one can consider as an approximation the score for $ES^\alpha(Y)$ , where $Y=X-SD^\alpha(X)$ . Furthermore, note that

\begin{equation*}\mathop{\textrm{arg min}}\limits_{y\in\mathbb{R}^k}\frac{1}{\beta-\alpha}\int_{[\alpha,\beta]}E\left[S^{\rho^s}(X,y)\right]ds=\mathop{\textrm{arg min}}\limits_{y\in\mathbb{R}^k}E\left[S^{R_\rho}(X,y)\right].\end{equation*}

However, such quantity does not have to coincide with

\begin{equation*}\frac{1}{\beta-\alpha}\int_{[\alpha,\beta]}\mathop{\textrm{arg min}}\limits_{y\in\mathbb{R}^k}E[S^{\rho^s}(X,y)]ds=R_\rho(X).\end{equation*}

This is exactly why elicitability does not have to be inherited.

4. Numerical example

This section presents two numerical examples to explain the concepts and theoretical results of the proposed approach from a practical point of view. In our experiments, we utilize as tail risk measures ( $\rho^{\alpha}$ ) the VaR, ES, Expectile, and SDR, and the range-based risk measure generated from these functionals. The number of Monte Carlo replications was set at 1,000. We use this value because it provides satisfactory results when comparing risk forecasting models in simulation studies (Escanciano and Olmo, Reference Escanciano and Olmo2011; Yi et al., Reference Yi, Feng and Huang2014). For window estimation (n), we employ 250 and 1000 observations. Both n are common in the risk forecasting literature (Kuester et al., Reference Kuester, Mittnik and Paolella2006; Yi et al., Reference Yi, Feng and Huang2014; Righi and Ceretta, Reference Righi and Ceretta2016), and 250 is the minimum sample size recommended by the Basel Committee to determine daily risk forecasts of banks and other Authorized Deposit-taking Institutions (ADIs) (Basel Committee on Banking Supervision, 2013).

Initially, we present a brief illustration of the use of range-based risk measures for calculating the risk premium in an insurance setup. For the data generating process, we consider Weibull distribution because it is common in actuarial and financial risk management problems (Gebizlioglu et al., Reference Gebizlioglu, Şenoğlu and Kantar2011; Ahmad et al., Reference Ahmad, Mahmoudi and Hamedani2022). For simplicity, we omit the cumulative distribution function and probability density function of the Weibull distribution, but both functions can be seen in the research of Gebizlioglu et al. (Reference Gebizlioglu, Şenoğlu and Kantar2011). The Weibull distribution has two parameters, the scale and the shape parameter. Based on the numerical example of Gebizlioglu et al. (Reference Gebizlioglu, Şenoğlu and Kantar2011), who evaluated the performance of different estimators for Weibull distribution and VaR estimation, we consider the scale parameter equal to 1 and the shape parameters equal to 0.5, 1.5, and 3.0. We choose 99%, 97.5%, and 95% as confidence intervals since they are typical values in the insurance literature (Tsai et al., Reference Tsai, Wang and Tzeng2010; Ahmad et al., Reference Ahmad, Mahmoudi and Hamedani2022). In the numerical insurance problem, we are interested in the upper tail. For risk estimation, we consider all combinations between the referred confidence levels, that is, 97.50% and 99.00%, 95.00% and 97.50%, and 95.00% and 99.00%. For a better description of the parameterizations of each scenario, see the Online Supplementary Material.

For risk premium quantification, we consider the simulated data defined in some discrete probability space $\Omega = (w_1, \cdots, w_n)$ , as $X(w_t) = X_t, t = 1,\cdots,n$ , where n represents the number of observations. Thus, we have $\mathbb{P}(X = X_t) = \mathbb{P}(w_t) = \frac{1}{n}$ , which results in the empirical distribution and expectation given by

\begin{align*}F_X(x) = \frac{1}{n}\sum^n_{t=1}1_{X_t \leq x}, \; \; \; E[X] = \frac{1}{n}\sum^n_{t=1}X_i.\end{align*}

Based on the empirical distribution of the data, the estimation method we consider is the historical simulation (HS). This method is a nonparametric approach that makes no assumptions about the data distribution. Furthermore, HS is a common risk estimation approach (Kuester et al., Reference Kuester, Mittnik and Paolella2006).

During the simulation process, in each Monte Carlo replica, we compute the risk premium using a sample size n for each $\rho^{\alpha}$ and range-based measure generated from $\rho^{\alpha}$ . Our intention with this illustration is to show the behavior of the insurance risk premium using our approach in relation to traditional options in the literature. For this reason, our analysis will be based on the risk premium’s mean value and standard deviation in an insurance setup. These results are exposed in Table 1.

Table 1.

the average and standard deviation of the risk premiums obtained in Monte Carlo experiments.

Note: SN $^{*}$ refers to scenarios. Scenarios 1–9 consider $n = 1000$ , while Scenarios 10–18 consider $n = 250$ . All scenarios consider scale parameters equal to 1. For scenarios 1–3 and 10–12, the shape parameter is equal to 0.5; for scenarios 4–6 and 13–15, the shape parameter is equal to 1.5; and for the other analyzed scenarios, the value is equal to 3. Scenarios 2, 5, 8, 11, 14, and 17 use $\alpha = 95.00\%$ and $\beta= 97.50\%$ ; scenarios 3, 6, 9, 12, 15, and 18 consider $\alpha = 95.00\%$ and $\beta = 99.00\%$ ; and scenarios 1, 4, 7, 10, 13, and 16 use $\alpha = 97.50\%$ and $\beta = 99.00\%$ . This table describes the average and standard deviation values of risk premiums. The results are based on 1000 Monte Carlo replications. For estimation, we consider the historical simulation.

Our results indicate that risk premium increases for scenarios with lower values for the shape parameter (Weibull distribution). Distinct shape parameters result in marked effects on the behavior of the data distribution. Smaller shape values generate simulated distributions with more extreme observations, which explains the higher risk premium values in these scenarios. We verified that the difference is more accentuated when we take into consideration shape = 0.5. As expected, we identified that the average value of the risk premium of the range measures is between the value determined by $\rho^{\alpha}$ and $\rho^{\beta}$ . By illustration, for Scenario 4, the risk premiums quantified by VaR $^{\alpha}$ , VaR $^{\beta}$ , and R $_{VaR^{\alpha}}$ are, respectively, 2.373, 2.733, and 2.541. We also observe that the risk premium obtained via $\rho^{\beta}$ is greater than $\rho^{\alpha}$ . The measure $\rho^{\beta}$ considers a more extreme tail associated with a higher insurance premium. Thus, we have the premium of $\rho^{99\%} > \rho^{97.5\%} > \rho^{95\%}$ . Another interesting finding is that the interval measures maintain the behavior of the functionals used to generate them. In this sense, it can be mentioned that the risk premium determined by the ES for the same confidence interval is greater than that of the VaR. Following this behavior, we realize that the premium obtained by R $_{ES^{\alpha}}$ is greater than the premium quantified using R $_{VaR^{\alpha}}$ . Notably, this is valid when we use the same values of $\alpha$ and $\beta$ for both measures. Concerning R $_{SDR^{\alpha}}$ because it has a penalty coefficient (deviation term), it was expected that his premium would be higher compared to R $_{ES^{\alpha}}$ and consequently to R $_{VaR^{\alpha}}$ . The lowest risk premium computed by R $_{\text{Expectile}^{\alpha}}$ is consistent with the behavior of Expectile values concerning VaR and ES premiums (Bellini and Di Bernardino, Reference Bellini and Di Bernardino2017).

In our second example, we perform an extensive numerical risk prediction study. We consider the AR(p)-GARCH(q, s) model as a data generating process (DGP).Footnote 2 This model can be defined as

(4.1) \begin{align}& X_t = \phi_0 + \sum_{i=1}^{p}\phi_i X_{t-i} + \epsilon_t, \nonumber \\& \epsilon_t = \sigma_tz_t, \qquad z_t \sim i.i.d. \;F(z_{t};\,\boldsymbol{\theta}), \nonumber \\& \sigma_t^2 = a_0 + \sum_{j = 1}^{q}a_{j}\epsilon^2_{t-j} + \sum_{k = 1}^{s}b_{k}\sigma^2_{t-k},\end{align}

where $X_t$ is the return for period t, $\phi_i$ , for $i = 1, \cdots, p$ , being p term autoregressive order, are parameters of autoregressive model, $\epsilon_t$ is the error term, $z_t$ is a white noise process with distribution $F(z_{t};\,\boldsymbol{\theta})$ , where $\boldsymbol{\theta}$ is a vector of parameters of distribution of $z_t$ , including zero mean and unit variance in addition to additional parameters that vary as the distribution. $\sigma_t^2$ is the conditional variance, and $a_j$ , for $j = 1,\cdots, q$ , as well as $b_k$ , for $k = 1, \cdots, s$ , are parameters of the GARCH model ( $\omega >0$ , $a_j\geq 0$ , $b_k\geq 0$ ), and q and s are its order, respectively. For more details regarding GARCH models, we suggest Francq and Zakoian (Reference Francq and Zakoian2019).

We use as model a Student’s t-AR(1)-GARCH(1, 1) because it takes into account common stylized facts of financial data, which include volatility clusters and heavy tails, and it is employed by other studies in risk measures forecasting (So and Philip, Reference So and Philip2006; Angelidis et al., Reference Angelidis, Benos and Degiannakis2007; Ardia and Hoogerheide, Reference Ardia and Hoogerheide2014). Parameter values similar to those used by us are also considered by Escanciano and Olmo (Reference Escanciano and Olmo2011), Righi and Ceretta (Reference Righi and Ceretta2016), and Müller and Righi (Reference Müller and Righi2018). We use the degree of freedom parameter ( $\eta$ ) equal to 8 to simulate a series with heavy tails, as this feature is frequent in financial data. We consider $\eta = 800$ to represent a normal distribution since when $\eta \to \infty$ the t-distribution approaches normal. For this decision, we follow Christoffersen and Gonçalves (Reference Christoffersen and Gonçalves2005), which resort Student’s t-distribution with $\eta = 500$ to simulate a normal distribution.

As significance values, we employ 1%, 2.5%, and 5%. 1% and 5% are the most frequent values to forecast risk measures (Kuester et al., Reference Kuester, Mittnik and Paolella2006; Escanciano and Olmo, Reference Escanciano and Olmo2011; Müller and Righi, Reference Müller and Righi2018), and 1% and 2.5% are the levels recommended for VaR and ES forecasting, respectively, by the Basel Committee on Banking Supervision (2013). The level pairs used in each experiment to forecast range-based risk measures are named $\alpha$ and $\beta$ . The VaR, ES, SDR, and Expectile are predicted considering both levels in each scenario. Our chosen parameterizations are described in detail in the Online Supplementary Material.

In each Monte Carlo replication, we generate the returns distribution considering each of the 12 analyzed scenarios. Each simulated sample has $n + 1$ observations. We use n observations for estimation and the observation regarding the $n+1$ position to evaluate risk predictions. In every replication, we determine the real risk value corresponding to the $n+1$ position.Footnote 3 Descriptive statistics of the real risk and the out-of-sample are available under request. As expected, for scenarios generated with lower $\eta$ , that is, $\eta = 8 $ , the average value of the real risk forecasts is higher. Smaller values of $\eta$ imply heavier tails, that is, a higher probability of extreme values than the normal distribution. In contrast, higher values of $\eta$ make the t-distribution close to the normal distribution with mean 0 and standard deviation 1.

To forecast the mean ( $\mu_{t+1}$ ) and conditional standard deviation ( $\sigma_{t+1}$ ), we use an AR(1)-GARCH(1,1) model,Footnote 4 which is defined in Equation (4.1) for $p = q = s = 1$ . In the estimation, we assume that $z_t$ follows normal ( ${\text{norm}}$ ), skewed normal ( ${\text{snorm}}$ ), Student-t ( ${\text{std}}$ ), skewed Student-t ( ${\text{sstd}}$ ), generalized error ( ${\text{ged}}$ ), skewed generalized error ( ${\text{sged}}$ ), and Johnson SU ( ${\text{jsu}}$ ) distribution.Footnote 5 We use the quasi-maximum likelihood (QML) method to estimate model parameters. According to the results of Garcia-Jorcano and Novales (Reference Garcia-Jorcano and Novales2021), the risk forecast performance is associated with the probability distribution of the innovations, and model selection plays a secondary role. For this reason, we focus on using the AR(1)-GARCH(1,1) model with different probability distributions instead of considering different volatility models.

Table 2.

This table describes the average and standard deviation of the risk forecasts obtained in Monte Carlo experiments, considering Scenarios 1–6. The results are multiplied by 100.

Note: SN $^{*}$ refers to scenarios. This table describes the average and standard deviation values of risk and range-based forecasts. The results are based on 1000 Monte Carlo replications considering scenarios 1–6, as detailed in the Online Supplementary Material. Scenarios 1–3 consider $\phi_1 = 0.50, a_0 = 4.00E-06, a = 0.10, b = 0.85, \eta= 8.00$ , while scenarios 4–6 differ only in the value of $\nu$ , considering it equal to 800. For scenarios 1 and 4, we use $\alpha = 1.00\%$ and $\beta = 2.50\%$ . In the scenarios 2 and 5, we use $\alpha = 2.50\%$ and $\beta = 5.00\%$ , while for the scenarios 3 and 6, we consider $\alpha = 1.00\%$ and $\beta = 5.00\%$ . Scenarios 1–6 use a $n=1000$ . For risk estimation, we consider an AR(1)-GARCH(1,1) model, where $z_t$ follows normal ( ${\text{norm}}$ ), skewed normal ( ${\text{snorm}}$ ), Student-t ( ${\text{std}}$ ), skewed Student-t ( ${\text{sstd}}$ ), generalized error ( ${\text{ged}}$ ), skewed generalized error ( ${\text{sged}}$ ), or Johnson SU ( ${\text{jsu}}$ ) distributions.

In general, the risk forecasts are obtained in the following way:

(4.2) \begin{align} \rho^{\alpha}_{t+1} = -\mu_{t+1} + \sigma_{t+1}\rho^{\alpha}\left(z_t+1\right), \end{align}

where $\rho^{\alpha}$ can be VaR $^{\alpha}$ , $\text{ES}^{\alpha}$ , $\text{SD}^{\alpha}$ , $\text{SDR}^{\alpha}$ , and $\text{Expectile}^{\alpha}$ (see Example 1) and their range-based counterparts (see Equation (3.1)). The Equation (4.2) is also valid when $\rho^{\alpha}$ refers to $\text{RVaR}^{\alpha}$ . For $\text{SDR}^{\alpha}$ , we use $k = 1$ , as performed by Righi and Borenstein (Reference Righi and Borenstein2018) when comparing risk measures for portfolio optimization. In each replicate, we compute a 1-step-ahead forecast for each risk measure.Footnote 6

According to the resultsFootnote 7 of Table 2, the average value of the range-based risk measures is between the average values of the tail risk measures for significance levels $\alpha$ and $\beta$ , that is, $\rho^{\alpha}(X) \geq R_{\rho}(X) \geq \rho^{\beta}(X)$ . By a illustration, see the Scenario 1 and normal distribution, the average values of VaR $^{\alpha}$ and VaR $^{\beta}$ are 1.934 and 1.629, respectively, while for R $_{\text{VaR}^{\alpha}}$ is 1.762. Thus, we have VaR $^{\alpha} \geq \text{R}_{\text{VaR}^{\alpha}} \geq $ VaR $^{\beta}$ . We can conclude that R $_{\rho}$ gives greater protection than $\rho^{\beta}$ and is more conservative than $\rho^{\alpha}$ . For visual analysis of this inequality, we present Figure 1, which illustrates the left tail of the sample generated using Scenario 5Footnote 8 with $n =10^5$ . This figure has four illustrations, one for each range measure and tail risk measure used to generate it with $\alpha = 2.5\%$ and $\beta = 5\%$ . In each illustration, we also include the ES $^{2.5\%}$ value.Footnote 9 The risk values are with the sign adjusted.Footnote 10 We perceive that this inequality is maintained in most cases for deviation values.

Figure 1.

Tail and range risk measures with $\alpha = 2.5\%$ and $\beta = 5\%$ for sample generated considering a AR(1)-GARCH(1,1) model with $\phi_1 = 0.50, a_0 = 4.00E-06, a = 0.10, b = 0.85, \eta= 800.00$ and $n =10^5$ . The risk values are with the sign adjusted.

We verify as expected that lower significance levels imply higher average risk values. We inform a pair of significance levels ( $\alpha$ and $\beta$ ) to compute range-based risk measures. For level pairs with lower values, the average risk value is higher. So, we have that higher risk values for $\alpha = 1\%$ and $\beta = 2.5\%$ , followed by $\alpha = 1\%$ and $\beta = 5\%$ . On the other hand, the lowest risk values are found for $\alpha = 2.5\%$ and $\beta = 5\%$ . From a risk management point of view, lower significance levels result in higher levels of security, which implies higher risk estimates.

For scenarios in which there is a change only in the value of degrees of freedom, we find that the average risk forecasts tend to be higher for a smaller $\eta$ . Lower degrees of freedom result in more extreme values than when considering an $\eta = 800$ . The measures considered in this study are applied under the left tail. Thus, for the series with more extreme observations, it is expected that the value of the risk measure will be higher than a distribution with light-tailed. By way of illustration, for Scenarios 2 ( $\eta = 8$ ) and 5 ( $\eta = 800$ ) and R $_{\text{VaR}^{\alpha}}$ forecasts considering normal distribution, we have an average risk forecast equal to 1.525 and 1.494, respectively. We also observe that the risk forecasts of scenarios with a smaller estimation window but with the same GARCH parameters and significance level have a greater standard deviation (in absolute value). This result can be justified by the fact that smaller window estimations tend to present bias and variability in the GARCH model estimates (Hwang and Valls Pereira, Reference Hwang and Valls Pereira2006; Fantazzini, Reference Fantazzini2009). As the sample size increases, this problem becomes insignificant. Hwang and Valls Pereira (Reference Hwang and Valls Pereira2006) point out that for the estimation of the GARCH model, it is recommended to use a sample of at least 500 observations.

Expectile and R $_{\text{Expectile}^{\alpha}}$ result in the lowest risk estimates when compared with other risk measures. For a visual description, we suggest reviewing Figure 1. According to Bellini and Di Bernardino (Reference Bellini and Di Bernardino2017), for a normally distributed X, the Expectile is closely comparable to the VaR $^{1\%}$ and ES $^{2.5\%}$ when we consider a significance level equal to $0.145\%$ (Expectile $^{0.145\%}$ ). Moreover, the forecasts of the Expectile and R $_{\text{Expectile}^{\alpha}}$ have less variability in absolute terms. However, this result is not maintained when assessing the relative standard deviation (RSD), that is, $ \text{RSD} \,:\!=\, \frac{\text{Standard Deviation}}{\text{Average value}}$ . The R $_{\text{Expectile}}$ is based on Expectile, which unlike quantile measures, such as VaR, is obtained by minimizing the asymmetrically weighted quadratic loss function (Newey and Powell, Reference Newey and Powell1987). Due to the squared error loss function used for its estimation (see Example 1, (iii)), Expectile is more sensitive to the tails distributions (Xie et al., Reference Xie, Zhou and Wan2014). Thus, changes in the left tail have a more significant impact on the results of the Expectile and the measure based on it compared to the other measures used, which naturally leads to greater relative variability.

In summary, our results show the similarity between the range and tail measures used to generate them. We numerically confirm our theoretical results and the representations. The mean value of the range measures is between the values of $\rho^{\alpha}$ to $\rho^{\beta}$ , and R $_{\text{VaR}}$ coincide with the RVaR. In the Online Supplementary Material, we further evaluate the risk predictions obtained from the numerical example. Also, we present an illustration of our approach to capital determination.