# Journal of Risk

**ISSN:**

1465-1211 (print)

1755-2842 (online)

**Editor-in-chief:** Farid AitSahlia

# An examination of the tail contribution to distortion risk measures

Miguel Santolino, Jaume Belles-Sampera, José María Sarabia and Montserrat Guillen

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Need to know

- The size of the risk measure estimate attributable to the tail of the distribution is analyzed.
- The tail contribution to distortion risk measures is defined and their additive properties are investigated.
- Decision makers are informed about the part of the regulatory reserve estimate attributable to the highest losses.
- Decision makers will identify the part of the diversification benefits attributable to the right tail.

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Abstract

Extreme losses are specially relevant in the finance and insurance sectors. Here, we analyze the tail behavior of risks and its influence on risk measures. Specifically, we examine the part of the risk value of a distortion risk measure (DRM) that is attributable to extreme losses. We analyze the additive properties of tail contributions to risk values when several risks are aggregated. We show that the partial contributions are subadditive if the distortion function is concave in the tail. We examine the tail behavior for quantile-based DRMs, including value-at-risk and tail value-at-risk. We conclude that such an evaluation will allow decision makers to obtain relevant information about the contribution of extreme losses to risk values and about the fraction of the diversification benefit attributable to the tails. An example is used to illustrate our results.

####
Introduction

## 1 Introduction

Risk is typically defined as the possibility of suffering future losses, where these losses are represented by a nonnegative random variable, $X$. This risk can then be quantified using risk measures, which encapsulate the risk associated with the loss distribution in a single value. In calculating the solvency requirements of financial institutions, such risk measures are applied to determine their economic capital, that is, the amount they need to hold to ensure that future liabilities are covered with an acceptable degree of confidence (McNeil et al 2005), and, in so doing, they are influenced by the occurrence of extreme losses located in the right tail of the loss distribution.

Here, our goal is to analyze the influence of extreme losses in the risk value computed using a particular risk measure: specifically, the family of distortion risk measures (DRMs) (Denneberg 1990; Wang 1995, 1996). DRMs are based on a function that distorts the probability measure applied to all subsets of a sample space. The class of DRMs includes the most frequently used quantile-based risk measures in finance and insurance, most notably value-at-risk (VaR) and tail value-at-risk (TVaR). We evaluate the importance of the tail in the risk values of DRMs. It is our belief that this should help decision makers anticipate the impact of an extreme loss on the risk estimate and, therefore, on other amounts derived from it, such as economic capital and premium loadings. Finally, we investigate the aggregation properties of the tail contributions when several risks are considered.

## 2 Literature review

Tail behavior of risks has gained much attention in the literature in recent years (Liu and Wang 2019; Belles-Sampera et al 2014; Landsman et al 2016; Cai and Li 2005; Yin and Zhu 2016). Belles-Sampera et al (2013, 2014) defined a new class of DRMs, named GlueVaR, and examined their tail aggregation properties. They pointed out that subadditivity might be a tough requirement on the determinations of premiums and regulatory capitals. Instead of subadditivity, they introduced a weaker concept of tail subadditivity and showed that this property is satisfied as long as the distortion function is concave in the common tail region. Cai et al (2017) generalized the concept of common tail region considered by Belles-Sampera et al (2014) and gave sufficient and necessary conditions for a DRM to be tail subadditive. These two theoretical papers mathematically analyzed the tail subadditivity condition in the field of nonadditive measure theory (Denneberg 1994).

The approach we adopt here, however, differs from previous works. This paper investigates how the theoretical concept of tail subadditivity can be evaluated in practice. This is carried out in two steps. First, we show how to assess the contribution of the random variable’s tail distribution on the value reported by the DRM. Closed-form expressions are given to compute the contribution of the tail to the risk value for a set of DRMs (VaR, TVaR and GlueVaR). Second, we show that if two risks are aggregated, the contributions of the tails to the DRM values satisfy the tail subadditivity property. To our knowledge, this practical analysis of the contribution of the tail to the value of these DRMs, and the evaluation of their tail aggregation properties in order to provide closed-form expressions, has not previously been performed. We argue that our approach helps to provide a deeper understanding of the theoretical concept of tail subadditivity. This is a convenient property because it implies that tail risk can be diversified.

An interesting approach to investigate the behavior of the tail distribution was developed by Liu and Wang (2019), who defined tail risk measures as those in which the risk measure value is solely determined by the distribution of the risk beyond its quantile. Liu and Wang (2019) defined the generator of a tail risk measure as a law-invariant risk measure applied to the conditional distribution of the tail and analyzed whether some properties often required by risk measures are transferred from the generator to the tail risk measure. They showed that subadditivity and convexity can be transferred when accompanied by other properties. Quantile-based DRMs, such as VaR, TVaR and GlueVaR, are such a class of tail risk measures (Liu and Wang 2019, Example E.2, p. 26). Non-quantile-based DRMs, such as dual power or the proportional-hazard-transform measures, are not included in the class of tail risk measures.

Although both quantile-based and non-quantile-based approaches focus on the tail of the distribution, our approach is different. The analysis carried out by Liu and Wang (2019) was based on the conditional distribution of the tail. Our analysis is based on the impact of the tail on the unconditional distorted distribution of the risk. We investigate how to measure the contribution of the distribution tail to the risk measure value, and how to evaluate the tail subadditivity property from a practical standpoint. As we deal with the unconditional distribution, the contribution of the tail to the value of the risk measure cannot be interpreted as a risk measure itself. We focus on DRMs, and their properties have been widely investigated from an axiomatic approach (see, for example, Balbás et al 2009; Wirch and Hardy 2002; Kou and Peng 2016). Our analysis covers all DRMs, including the dual power and the proportional-hazard-transform measures, as we show in an example.

Our practical approach has the advantage that the results are easily interpretable. Decision makers may obtain information about the contribution of extreme losses to the risk value reported by a specific risk measure so that, when risks are aggregated, the part of the diversification benefit that is attributable to the tails of the loss distributions can be identified. DRMs are often used to compute the economic capital. For instance, Basel Committee on Banking Supervision (2019) sets TVaR at a 97.5% confidence level for computing the minimum capital requirements. Decision makers may wish to know the impact of the $q$% of highest losses on the value reported by ${\mathrm{TVaR}}_{97.5\%}$, with $$, and the part of the diversification benefit that is attributable to these losses when risks are aggregated. In this paper we show how these figures can be computed.

The rest of the paper is structured as follows. In Section 3 risk measures are introduced. In Section 4 we analyze the influence of tails on DRMs, and we study tail aggregation properties, with particular attention on $q$-based risk measures. Finally, in Section 5, we present an example based on motor insurance claims to illustrate our results.

## 3 Risk quantification

### 3.1 Risk measures

Nonnegative random variables are often deemed suitable for defining losses in the context of enterprise risk quantification. Let $X$ be a nonnegative random variable with finite expectation, called a loss. The cumulative distribution function of $X$, denoted by ${F}_{X}$, is defined by ${F}_{X}(x)=P(X\le x)$, and it is often referred to as the loss distribution. The survival function is denoted by ${S}_{X}(x)=P(X>x)$. The mathematical expectation can be written as

$$?(X)={\int}_{0}^{+\mathrm{\infty}}xd{F}_{X}(x).$$ |

We follow standard notation, so that the derivative function of ${F}_{X}$, when it can be defined, is the density function ${f}_{X}$, so $\mathrm{d}{F}_{X}(x)={f}_{X}(x)\mathrm{d}x$. In general, the mathematical expectation can also be obtained from the survival distribution as

$$?(X)={\int}_{0}^{+\mathrm{\infty}}{S}_{X}(x)dx.$$ |

The inverse function of ${F}_{X}$, ${F}_{X}^{-1}$, is known as the quantile function; that is, ${F}_{X}^{-1}(\alpha )=inf\{x\mid {F}_{X}(x)\ge \alpha \}$, where $\alpha \in (0,1)$. The mathematical expectation can be obtained from the quantile function as

$?(X)$ | $={\displaystyle {\int}_{0}^{1}}{S}_{X}^{-1}(u)du$ | |||

$={\displaystyle {\int}_{0}^{1}}{F}_{X}^{-1}(1-u)du.$ | (3.1) |

A risk measure $\rho $ assigns a value to $X$. Let $\mathrm{\Gamma}$ be the set of all random variables defined for a given probability space $(\mathrm{\Omega},?,P)$. The risk measure $\rho $ is a mapping from $\mathrm{\Gamma}$ to $\mathbb{R}$, so $\rho (X)$ is a real value for each $X\in \mathrm{\Gamma}$. The goal of any risk measure is to summarize the risk associated with the loss distribution (McNeil et al 2005).

The most frequently used risk measures in finance and insurance are VaR and TVaR. VaR at confidence level $\alpha \in (0,1)$ is defined as ${\mathrm{VaR}}_{\alpha}(X)={F}_{X}^{-1}(\alpha )$. The mathematical expectation can then be represented as

$$?(X)={\int}_{0}^{1}{\mathrm{VaR}}_{1-u}(X)du.$$ |

The TVaR at confidence level $\alpha $ is defined as

$${\mathrm{TVaR}}_{\alpha}(X)=\frac{1}{1-\alpha}{\int}_{0}^{1-\alpha}{\mathrm{VaR}}_{1-u}(X)du.$$ |

The TVaR can be understood as the mathematical expectation beyond VaR and be expressed as ${\mathrm{TVaR}}_{\alpha}(X)=?[X\mid X>{\mathrm{VaR}}_{\alpha}(X)]$. Interpreted in this way, TVaR is sometimes known as the expected shortfall (McNeil et al 2005). Many other risk measures have been defined in the literature (see, for example, Denuit et al 2005; Furman et al 2017).

### 3.2 DRMs

DRMs were first introduced by Denneberg (1990) and further developed by Wang (1995, 1996). This class of risk measure is closely related to distortion expectation theory (Yaari 1987). A key element in defining a DRM is its associated distortion function $g$, which can be defined as a left-continuous nondecreasing function $g:[0,1]\to [0,1]$ such that $g(0)=0$ and $g(1)=1$.

Consider a nonnegative random variable $X$ and its survival function ${S}_{X}$; the function ${\rho}_{g}$ defined by

$${\rho}_{g}(X)={\int}_{0}^{+\mathrm{\infty}}g({S}_{X}(x))dx$$ | (3.2) |

is called a DRM.

A DRM can be understood as the distorted expectation of $X$. The mathematical expectation is a DRM whose distortion function is the identity function; that is, ${\rho}_{\mathrm{id}}(X)=?(X)$ (Denuit et al 2005). DRMs can be expressed in terms of the quantile function with the Lebesgue–Stieltjes integral representation.

The DRM ${\rho}_{g}$ is represented by the Lebesgue–Stieltjes integral as follows:

$${\rho}_{g}(X)={\int}_{0}^{1}{F}_{X}^{-1}(1-u)dg(u).$$ | (3.3) |

Note that $g({S}_{X}(x))={\int}_{0}^{{S}_{X}(x)}dg(u)$. Later, Fubini’s theorem is applied to change the order of integrals (Dhaene et al 2012, Theorem 6). In the case where the distortion function $g$ is right continuous, the DRM can be represented by the Lebesgue–Stieltjes integral in terms of ${F}_{X}^{-1+}(\alpha )$, where ${F}_{X}^{-1+}(\alpha )=sup\{x\mid {F}_{X}(x)\le \alpha \}$ (Dhaene et al 2012, Theorem 4; Wang et al 2018, Lemma 2.6).

The VaR and TVaR risk measures can be represented as DRMs. A flexible family of four-parameter DRMs, known as GlueVaR risk measures, was introduced by Belles-Sampera et al (2014). GlueVaR risk measures include VaR and TVaR as particular cases. The associated distortion functions of the VaR, TVaR and GlueVaR risk measures are shown in Table 1.

Risk | |
---|---|

measure | Distortion function |

VaR | $$ |

TVaR | $$ |

GlueVaR | $$ |

The equivalence of (3.2) and (3.3) is illustrated graphically for the TVaR measure in Figures 1–3. Figure 1 shows the functions involved in the computation of the ${\mathrm{TVaR}}_{\alpha}(X)$ when the risk measure is expressed as a DRM in (3.2). The value of the ${\mathrm{TVaR}}_{\alpha}(X)$ corresponds to the area under the solid-line function shown in Figure 1(c).

Figure 2 shows the functions involved in the computation of ${\mathrm{TVaR}}_{\alpha}(X)$ when the risk measure is expressed as the DRM defined in (3.3). The value of ${\mathrm{TVaR}}_{\alpha}(X)$ corresponds to the area under the solid-line function shown in Figure 2(c). Figure 3 shows the equivalence between the two forms to compute TVaR.

## 4 Tail contributions and their aggregation properties in distortion risk measures

In this section we analyze the role of the tail in distortion risk values. In other words, we study the contribution of the tail to the magnitude of a DRM.

### 4.1 Tail contribution to risk value

We refer to the Lebesque $q$-tail contribution in DRMs as a part of the risk measure value that can be associated with values located in the right tail of the loss distribution. Let us consider (3.3) and focus only on one part of the integral. We define the Lebesgue $q$-tail contribution in DRMs as follows.

###### Definition 4.1 ($q$-tail contribution).

Given a $q\in [0,1]$, the $q$-tail contribution of a DRM ${c}_{{\rho}_{g}}^{q}$ is represented by the Lebesgue–Stieltjes integral as

$${c}_{{\rho}_{g}}^{q}(X)={\int}_{0}^{q}{F}_{X}^{-1}(1-u)dg(u).$$ | (4.1) |

The information provided by (4.1) is relevant to decision makers, because it serves as an indicator of the importance of the tail on the risk value. Computing the risk value according to the DRM defined in (3.3), (4.1) provides information about the part of the risk value attributable to the $q$-right tail of the loss distribution. The tail contribution can then be interpreted as an indicator of the sensitivity of the risk measure value to the $q$-right tail of the loss distribution. In calculating the solvency requirements of financial institutions, the selection of the DRM to apply in the computation of the economic capital is often set by regulators. In this context, risk managers will be interested in evaluating the part of the economic capital estimate that is attributable to the $q$% of the highest losses of the loss distribution.

###### Proposition 4.2.

Given a random variable $X$, if the left-continuous function $g$ is not differentiable at points in a countably finite set $\mathrm{(}{u}_{\mathrm{1}}\mathrm{,}{u}_{\mathrm{2}}\mathrm{,}\mathrm{\dots}\mathrm{,}{u}_{n}\mathrm{)}$, the risk value of the DRM ${\rho}_{g}$ can be expressed as

$${\rho}_{g}(X)={\int}_{0}^{q}{F}_{X}^{-1}(1-u)dg(u)+{\int}_{q}^{1}{F}_{X}^{-1}(1-u)dg(u),$$ |

regardless of the allocation of $\mathrm{(}{u}_{\mathrm{1}}\mathrm{,}{u}_{\mathrm{2}}\mathrm{,}\mathrm{\dots}\mathrm{,}{u}_{n}\mathrm{)}$, for $q\mathrm{\in}\mathrm{[}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{]}$.

###### Proof.

Given a random variable $X$, the risk value of the DRM ${\rho}_{g}$ is computed as (3.3). When $g$ is a left-continuous function, the following equivalence holds:

$${\int}_{0}^{1}{F}_{X}^{-1}(1-u)dg(u)={\int}_{0}^{1}{F}_{X}^{-1}(u)d\overline{g}(u),$$ |

where $\overline{g}(u)=1-g(1-u)$ is a right-continuous function (Dhaene et al 2012). For a right-continuous increasing function $\overline{g}$ that is differentiable on $\mathbb{R}$ except at points in a countably finite set $({u}_{1},{u}_{2},\mathrm{\dots},{u}_{n})$, the Lebesgue–Stieltjes integral can be computed as follows:

${\int}_{0}^{1}}{F}_{X}^{-1}(u)d\overline{g}(u)$ | $={\displaystyle {\int}_{0}^{{u}_{1}}}{F}_{X}^{-1}(u){\overline{g}}^{\prime}(u)du+{\displaystyle {\int}_{{u}_{1}}^{{u}_{2}}}{F}_{X}^{-1}(u){\overline{g}}^{\prime}(u)du+\mathrm{\cdots}$ | ||

$\mathrm{\hspace{1em}\hspace{1em}}+{\displaystyle {\int}_{{u}_{n}}^{1}}{F}_{X}^{-1}(u){\overline{g}}^{\prime}(u)du+{\displaystyle \sum _{i=1}^{n}}{F}_{X}^{-1}({u}_{i})\mathrm{\Delta}\overline{g}({u}_{i}),$ |

where ${\overline{g}}^{\prime}$ is the derivative of $\overline{g}$ and $\mathrm{\Delta}\overline{g}({u}_{i})=\overline{g}({u}_{i})-\overline{g}(u_{i}{}^{-})$. For any given $q\in [0,1]$, it is easy to check that the risk measure ${\rho}_{g}$ can be represented by the Lebesgue–Stieltjes integral as

$${\int}_{0}^{1-q}{F}_{X}^{-1}(u)d\overline{g}(u)+{\int}_{1-q}^{1}{F}_{X}^{-1}(u)d\overline{g}(u),$$ |

or, equivalently, as

$${\int}_{q}^{1}{F}_{X}^{-1}(1-u)dg(u)+{\int}_{0}^{q}{F}_{X}^{-1}(1-u)dg(u).$$ |

∎

###### Remark 4.3.

The first term on the right-hand side in Proposition 4.2 can be interpreted as the part attributable to the $q$-right tail in the risk value returned by the risk measure ${\rho}_{g}$. In case of a jump of $\overline{g}$ in $(1-q)$, $(1-q)$ contributes

$${F}_{X}^{-1}(1-q)(\overline{g}(1-q)-\overline{g}({(1-q)}^{-}))$$ |

to ${\int}_{0}^{1-q}{F}_{X}^{-1}(u)d\overline{g}(u)$ (or, equivalently, to ${\int}_{q}^{1}{F}_{X}^{-1}(1-u)dg(u)$) and

$${F}_{X}^{-1}(1-q)(\overline{g}({(1-q)}^{+})-\overline{g}(1-q))$$ |

to ${\int}_{1-q}^{1}{F}_{X}^{-1}(u)d\overline{g}(u)$ (or, equivalently, to ${\int}_{0}^{q}{F}_{X}^{-1}(1-u)dg(u)$). The function $\overline{g}(u)$ is right continuous, so the difference $(\overline{g}({(1-q)}^{+})-\overline{g}(1-q))$ is $0$.

###### Example 4.4.

Let $X$ be a uniform random variable between 0 and 1. Assume that the proportional-hazard-transform distortion function, ${g}_{\mathrm{ph}}(u)={u}^{r}$, is applied, where $0\le r\le 1$. The value of the risk measure is equal to

$${\rho}_{{g}_{\mathrm{ph}}}={\int}_{0}^{1}{(1-u)}^{r}du=\frac{1}{(r+1)}.$$ |

The $q$-tail contribution to the risk measure ${c}_{{\rho}_{{g}_{\mathrm{ph}}}}^{q}(X)$ is equal to the following:

$${c}_{{\rho}_{{g}_{\mathrm{ph}}}}^{q}(X)={\int}_{0}^{q}{F}_{X}^{-1}(1-u)dg(u)={\int}_{0}^{q}(1-u)r{u}^{r-1}du={q}^{r}-\frac{r}{(r+1)}{q}^{r+1}.$$ |

### 4.2 Tail importance in quantile-based risk measures

We analyze the Lebesgue $q$-tail contribution for the three DRMs shown in Table 1.

#### 4.2.1 VaR

We fix $q$ in $[0,1-\alpha ]$. In the case of VaR, the distortion function ${\psi}_{\alpha}$ is 0 in $[0,q]$. So, it holds that

$${c}_{{\mathrm{VaR}}_{\alpha}}^{q}(X)={\int}_{0}^{q}{F}_{X}^{-1}(1-u)d{\psi}_{\alpha}(u)=0.$$ |

Here, it can be seen that (4.1) reflects the importance of the tail on the risk value. The VaR${}_{\alpha}$ value does not take into account the $q$-tail of the random variable if $q\le 1-\alpha $, so the impact of the $q$-right tail on the risk value is null.

Let us analyze the case $q>1-\alpha $. Now, the $q$-tail contribution defined in (4.1) is exactly ${\mathrm{VaR}}_{\alpha}$. Note that ${\psi}_{\alpha}$ is a step function. If the $q$-tail includes the point value at which ${\psi}_{\alpha}$ has the step, then the solution of the Lebesgue–Stieltjes integral is ${F}_{X}^{-1}(\alpha )$. That is, the $q$-tail contribution is the total value of ${\mathrm{VaR}}_{\alpha}$. So, only the value located in the $\alpha $-quantile (step point) matters to determine the value of the risk measure.

An alternative interpretation when considering the ${\mathrm{VaR}}_{\alpha}$ risk measure is that $q$-tail contributions provide information about the extremes beyond a certain threshold in the right tail of interest (the $1-\alpha $ right tail). In larger tails, that is, when $q>1-\alpha $, the $q$-tail contribution is equal to the value of the risk measure, ${\mathrm{VaR}}_{\alpha}$.

#### 4.2.2 TVaR

In the case of the TVaR, the distortion function ${\gamma}_{\alpha}$ takes the value $u/(1-\alpha )$ for all $u$, when $q\in [0,1-\alpha ]$. Therefore, the $q$-tail contribution to the risk value of ${\mathrm{TVaR}}_{\alpha}$ is computed as

$${c}_{{\mathrm{TVaR}}_{\alpha}}^{q}(X)={\int}_{0}^{q}{F}_{X}^{-1}(1-u)d{\gamma}_{\alpha}(u)=\frac{1}{1-\alpha}{\int}_{0}^{q}{F}_{X}^{-1}(1-u)du.$$ |

Now, we substitute

$$\frac{1}{1-\alpha}=\frac{q}{1-\alpha}\frac{1}{q},$$ |

so the tail contribution can be expressed as

$${c}_{{\mathrm{TVaR}}_{\alpha}}^{q}(X)=\frac{q}{1-\alpha}{\mathrm{TVaR}}_{1-q}(X).$$ |

As in the previous case, when $q>1-\alpha $, the Lebesgue $q$-tail contribution in the risk value of ${\mathrm{TVaR}}_{\alpha}$ is then

$$\frac{1}{1-\alpha}{\int}_{0}^{1-\alpha}{F}_{X}^{-1}(1-u)du.$$ |

Note that this is the definition of TVaR. This means that when $q\ge 1-\alpha $ the Lebesgue $q$-tail contribution to the risk value of ${\mathrm{TVaR}}_{\alpha}$ represents the complete risk value. Let us recall that ${\mathrm{TVaR}}_{\alpha}$ can be understood as the mathematical expectation beyond ${\mathrm{VaR}}_{\alpha}$. So, the $q$-tail contribution explains the whole risk value when the $q$-tail includes the $(1-\alpha )$ tail. Similarly to ${\mathrm{VaR}}_{\alpha}$, it can be interpreted as if our definition of $q$-tail contribution has an embedded threshold right tail of interest when considering the ${\text{TVaR}}_{\alpha}$ measure.

#### 4.2.3 GlueVaR

Finally, the Lebesgue $q$-tail contribution to GlueVaR measures is analyzed. As shown in Table 1, the distortion function for GlueVaR has an additional confidence level $\beta $. The shape of the GlueVaR distortion function is determined by the distorted survival probabilities ${h}_{1}$ and ${h}_{2}$ at levels $1-\beta $ and $1-\alpha $, respectively. Parameters ${h}_{1}$ and ${h}_{2}$ are the heights of the distortion function.

Let us consider the following notation:

$$\begin{array}{cc}\hfill {\omega}_{1}& ={h}_{1}-\frac{({h}_{2}-{h}_{1})(1-\beta )}{\beta -\alpha},\hfill \\ \\ \hfill {\omega}_{2}& =\frac{{h}_{2}-{h}_{1}}{\beta -\alpha}(1-\alpha ).\hfill \end{array}\}$$ |

Belles-Sampera et al (2014) showed that

$${\mathrm{GlueVaR}}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}(X)={\omega}_{1}{\mathrm{TVaR}}_{\beta}(X)+{\omega}_{2}{\mathrm{TVaR}}_{\alpha}(X)+{\omega}_{3}{\mathrm{VaR}}_{\alpha}(X),$$ | (4.2) |

where ${\omega}_{3}=1-{\omega}_{1}-{\omega}_{2}$. We analyze the $q$-tail contribution to GlueVaR measures when $$. The following proposition is shown.

###### Proposition 4.5.

The $q$-tail contribution of ${\mathrm{GlueVaR}}_{\beta \mathrm{,}\alpha}^{{h}_{\mathrm{1}}\mathrm{,}{h}_{\mathrm{2}}}$ when $$ is equal to

$${c}_{{\mathrm{GlueVaR}}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}}^{q}(X)={\omega}_{1}{\mathrm{TVaR}}_{\beta}(X)+{\omega}_{2}\frac{q}{1-\alpha}{\mathrm{TVaR}}_{1-q}(X).$$ |

Note that, when $q=1-\alpha $, we have

$${c}_{{\mathrm{GlueVaR}}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}}^{q}(X)={\omega}_{1}{\mathrm{TVaR}}_{\beta}(X)+{\omega}_{2}{\mathrm{TVaR}}_{\alpha}(X).$$ |

###### Proof.

The $q$-tail contribution of ${\mathrm{GlueVaR}}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}$ when $$ can be expressed as

${c}_{{\mathrm{GlueVaR}}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}}^{q}(X)$ | $={\displaystyle {\int}_{0}^{q}}{F}_{X}^{-1}(1-u)d{\kappa}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}(u)$ | ||

$={\displaystyle \frac{{h}_{1}}{\beta -\alpha}}{\displaystyle {\int}_{0}^{1-\beta}}{F}_{X}^{-1}(1-u)du+{\displaystyle \frac{{h}_{2}-{h}_{1}}{\beta -\alpha}}{\displaystyle {\int}_{1-\beta}^{q}}{F}_{X}^{-1}(1-u)du,$ |

where

$\frac{{h}_{2}-{h}_{1}}{\beta -\alpha}}{\displaystyle {\int}_{1-\beta}^{q}}{F}_{X}^{-1}(1-u)du$ | ||

$\mathrm{\hspace{1em}\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}}={\displaystyle \frac{{h}_{2}-{h}_{1}}{\beta -\alpha}}({\displaystyle {\int}_{0}^{q}}{F}_{X}^{-1}(1-u)\mathrm{d}u-{\displaystyle {\int}_{0}^{1-\beta}}{F}_{X}^{-1}(1-u)\mathrm{d}u).$ |

Therefore,

${c}_{{\mathrm{GlueVaR}}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}}^{q}(X)$ | $=\left({h}_{1}-{\displaystyle \frac{({h}_{2}-{h}_{1})(1-\beta )}{\beta -\alpha}}\right){\displaystyle \frac{1}{1-\beta}}{\displaystyle {\int}_{0}^{1-\beta}}{F}_{X}^{-1}(1-u)du$ | ||

$\mathrm{\hspace{1em}\hspace{1em}}+{\displaystyle \frac{({h}_{2}-{h}_{1})(1-\alpha )}{\beta -\alpha}}{\displaystyle \frac{q}{1-\alpha}}{\displaystyle \frac{1}{q}}{\displaystyle {\int}_{0}^{q}}{F}_{X}^{-1}(1-u)du.$ |

This can be expressed as follows:

${c}_{{\mathrm{GlueVaR}}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}}^{q}(X)$ | $=\left({h}_{1}-{\displaystyle \frac{({h}_{2}-{h}_{1})(1-\beta )}{\beta -\alpha}}\right){\mathrm{TVaR}}_{\beta}(X)$ | ||

$\mathrm{\hspace{1em}\hspace{1em}}+\left({\displaystyle \frac{({h}_{2}-{h}_{1})(1-\alpha )}{\beta -\alpha}}\right){\displaystyle \frac{q}{1-\alpha}}{\mathrm{TVaR}}_{1-q}(X).$ |

Following (4.2.3), this is equivalent to

$${c}_{{\mathrm{GlueVaR}}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}}^{q}(X)={\omega}_{1}{\mathrm{TVaR}}_{\beta}(X)+{\omega}_{2}\frac{q}{1-\alpha}{\mathrm{TVaR}}_{1-q}(X).$$ |

∎

To conclude, the $q$-tail contribution of ${\mathrm{GlueVaR}}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}$ can easily be derived when $q\le 1-\beta $ and $q>1-\alpha $. If $q\le 1-\beta $, then

$${c}_{{\mathrm{GlueVaR}}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}}^{q}(X)={h}_{1}\frac{q}{1-\beta}{\mathrm{TVaR}}_{1-q}(X).$$ |

As in previous cases, if $q>1-\alpha $, then the $q$-tail contribution of the risk measure is equal to the risk measure value.

### 4.3 Tail subadditivity

DRMs satisfy a set of properties, including positive homogeneity, translation invariance, comonotonic additivity and monotonicity (Balbás et al 2009). These DRM properties are desirable in many contexts of risk quantification.

When aggregating risks, risk measures are often required to satisfy the subadditivity property, because the risk of the sum can be bounded by the sum of risks.

###### Definition 4.6.

Let $X,Y$ be nonnegative random variables. A DRM ${\rho}_{g}$ is subadditive for $X$, $Y$ if

$${\int}_{0}^{+\mathrm{\infty}}g({S}_{Z}(z))dz\le {\int}_{0}^{+\mathrm{\infty}}g({S}_{X}(x))dx+{\int}_{0}^{+\mathrm{\infty}}g({S}_{Y}(y))dy,$$ |

where $Z=X+Y$.

In other words, a risk measure is subadditive if ${\rho}_{g}(X+Y)\le {\rho}_{g}(X)+{\rho}_{g}(Y)$. Subadditivity distinguishes between VaR and TVaR measures, since this property is only satisfied by the latter. Embrechts et al (2015) showed several ways to demonstrate that the TVaR risk measure is subadditive. Based on (4.2), the GlueVaR measure satisfies the subadditivity property if and only if a null weight is given to VaR and TVaRs are not negatively weighted.

The subadditivity property of DRMs is guaranteed when the distortion function $g$ is concave in $[0,1]$ (Denneberg 1994; Wang and Dhaene 1998; Wirch and Hardy 2002). Based on (3.3), the subadditivity property of a risk measure ${\rho}_{g}$ can be expressed as

$${\int}_{0}^{1}{F}_{Z}^{-1}(1-u)dg(u)\le {\int}_{0}^{1}{F}_{X}^{-1}(1-u)dg(u)+{\int}_{0}^{1}{F}_{Y}^{-1}(1-u)dg(u),$$ |

where $Z=X+Y$.

Subadditivity is an appealing property for decision makers. Suppose a risk manager holds a pair of risks $X$ and $Y$. A subadditive risk measure considers that the risk of holding $X$ and $Y$ together is lower than the risk of holding $X$ and $Y$ individually. Hence, subadditivity means that diversification benefits are reflected in the risk measure. Subadditivity in the whole domain is a strong condition. When dealing with fat-tail losses (ie, low-frequency and large loss events), risk managers are especially interested in the tail region. Fat right tails have been extensively studied in insurance and finance (Wang 1998; Embrechts et al 2009a, b; Degen et al 2010; Nam et al 2011; Chen et al 2012). Belles-Sampera et al (2014) introduced a weaker concept of tail subadditivity and proved mathematically that this property is satisfied if the distortion function is concave in the common tail region. Cai et al (2017) gave sufficient and necessary conditions for a DRM to be tail subadditive. These two papers did not analyze how tail subadditivity could be evaluated for frequently used DRMs. In our paper we introduce the definition of $q$-tail subadditivity for a pair of risks in terms of the tail contribution to the risk measure value, as follows.

###### Definition 4.7 ($q$-tail subadditivity).

Given $q\in [0,1]$ and nonnegative random variables $X$, $Y$, a distortion risk measure ${\rho}_{g}$ is $q$-tail subadditive if

$${\int}_{0}^{q}{F}_{Z}^{-1}(1-u)dg(u)\le {\int}_{0}^{q}{F}_{X}^{-1}(1-u)dg(u)+{\int}_{0}^{q}{F}_{Y}^{-1}(1-u)dg(u),$$ |

where $Z=X+Y$ or, equivalently,

$${c}_{{\rho}_{g}}^{q}(X+Y)\le {c}_{{\rho}_{g}}^{q}(X)+{c}_{{\rho}_{g}}^{q}(Y).$$ |

The $q$-tail subadditivity property is useful for decision makers to know the aggregate behavior of the loss distributions of a pair of risks in tails. If a subadditive risk measure is fixed as the regulatory risk measure in the computation of the economic capital, when risks $X$ and $Y$ are aggregated, decision makers may identify the portion of the diversification benefits attributable to the $q$-tails of the loss distributions of both risks. If the regulatory risk measure is not subadditive but it is $q$-tail subadditive, diversification benefits of the $q\%$ of the extreme losses of both risks are captured by the risk measure even if the total diversification benefit cannot be guaranteed to be positive.

###### Proposition 4.8.

Under the corresponding $q$-tail subadditivity property conditions, the $q$-tail subadditivity property is satisfied when the distortion function $g$ is concave in $\mathrm{[}\mathrm{0}\mathrm{,}q\mathrm{]}$.

The proof of Proposition 4.8 is provided in the online appendix.

###### Example 4.9 (Continuation).

Consider $X$ and $Y$ as two independent uniform random variables between 0 and 1. The density function of $Z=X+Y$ is

$$ |

The proportional-hazard-transform distortion function is applied. Assume that $r=0.5$, and that the value of the risk measure for $Z$ is equal to

${\rho}_{{g}_{\mathrm{ph}}}(Z)$ | $={\displaystyle {\int}_{0}^{1}}{\left(1-{\displaystyle \frac{{u}^{2}}{2}}\right)}^{0.5}du+{\displaystyle {\int}_{1}^{2}}{\left({\displaystyle \frac{{u}^{2}}{2}}-2u+2\right)}^{0.5}du$ | ||

$=\sqrt{{\displaystyle \frac{1}{2}}}\left(1+\mathrm{arcsin}\left(\sqrt{{\displaystyle \frac{1}{2}}}\right)\right).$ |

From (4.4) we know that for $X$ and $Y$ the risk value is equal to ${\rho}_{{g}_{\mathrm{ph}}}(X)={\rho}_{{g}_{\mathrm{ph}}}(Y)=1/1.5$ and that the $q$-tail contribution to the risk measure value is equal to

$${c}_{{\rho}_{{g}_{\mathrm{ph}}}}^{q}(X)={c}_{{\rho}_{{g}_{\mathrm{ph}}}}^{q}(Y)={q}^{0.5}-\frac{0.5}{1.5}{q}^{1.5}.$$ |

Subadditivity is satisfied since ${\rho}_{{g}_{\mathrm{ph}}}(X)+{\rho}_{{g}_{\mathrm{ph}}}(Y)-{\rho}_{{g}_{\mathrm{ph}}}(Z)=0.07$. If $q\le 0.5$, the tail contribution to the risk measure, ${c}_{{\rho}_{{g}_{\mathrm{ph}}}}^{q}(Z)$, is equal to

$${c}_{{\rho}_{{g}_{\mathrm{ph}}}}^{q}(Z)={\int}_{0}^{q}(2-{2}^{0.5}u)0.5{u}^{0.5-1}du=2({q}^{0.5})-0.5\sqrt{2}q.$$ |

Then, tail subadditivity is satisfied since

$${c}_{{\rho}_{{g}_{\mathrm{ph}}}}^{q}(X)+{c}_{{\rho}_{{g}_{\mathrm{ph}}}}^{q}(Y)-{c}_{{\rho}_{{g}_{\mathrm{ph}}}}^{q}(Z)=0.5\sqrt{2}q-\frac{{q}^{1.5}}{1.5}$$ |

is positive for any $0\le q\le 0.5$.

### 4.4 Tail subadditivity for quantile-based DRMs

We analyze the $q$-tail subadditivity property for the three risk measures shown in Table 1.

#### 4.4.1 VaR

In the case of VaR, the distortion function ${\psi}_{\alpha}$ is concave in $[0,1-\alpha ]$. Given a $q$ in $[0,1-\alpha ]$, it holds that, for nonnegative random variables $X$, $Y$, we can write

$${\int}_{0}^{q}{F}_{Z}^{-1}(1-u)d{\psi}_{\alpha}(u)\le {\int}_{0}^{q}{F}_{X}^{-1}(1-u)d{\psi}_{\alpha}(u)+{\int}_{0}^{q}{F}_{Y}^{-1}(1-u)d{\psi}_{\alpha}(u),$$ |

where $Z=X+Y$.

The ${\mathrm{VaR}}_{\alpha}$ measure satisfies $q$-tail subadditivity when $q\le 1-\alpha $. Note that the integrals are equal to zero on both sides of the inequality.

#### 4.4.2 TVaR

The distortion function of TVaR, $\gamma $, is concave in the whole interval $[0,1]$. Therefore, from Proposition 4.8, for nonnegative random variables $X$, $Y$, it holds that

$${\int}_{0}^{q}{F}_{Z}^{-1}(1-u)d{\gamma}_{\alpha}(u)\le {\int}_{0}^{q}{F}_{X}^{-1}(1-u)d{\gamma}_{\alpha}(u)+{\int}_{0}^{q}{F}_{Y}^{-1}(1-u)d{\gamma}_{\alpha}(u),$$ |

where $Z=X+Y$ for any $q$ in $[0,1]$. In other words, TVaR has $q$-tail subadditivity for any $q$ in $[0,1]$.

Given $q\in [0,1-\alpha ]$,

$\frac{1}{1-\alpha}}{\displaystyle {\int}_{0}^{q}}{F}_{Z}^{-1}(1-u)du\le {\displaystyle \frac{1}{1-\alpha}}{\displaystyle {\int}_{0}^{q}}{F}_{X}^{-1}(1-u)du+{\displaystyle \frac{1}{1-\alpha}}{\displaystyle {\int}_{0}^{q}}{F}_{Y}^{-1}(1-u)du$ |

is equivalent to

$$\frac{q}{1-\alpha}{\mathrm{TVaR}}_{1-q}(Z)\le \frac{q}{1-\alpha}{\mathrm{TVaR}}_{1-q}(X)+\frac{q}{1-\alpha}{\mathrm{TVaR}}_{1-q}(Y),$$ |

where $q/(1-\alpha )$ is nonnegative. Therefore, the inequality holds since ${\mathrm{TVaR}}_{1-q}(Z)\le {\mathrm{TVaR}}_{1-q}(X)+{\mathrm{TVaR}}_{1-q}(Y)$.

When $q$ $\in [1-\alpha ,1]$, we have ${\mathrm{TVaR}}_{1-q}(X)={\mathrm{TVaR}}_{\alpha}(X)$.

#### 4.4.3 GlueVaR

The $q$-tail subadditivity of GlueVaR measures is now analyzed. Figure 5 shows two examples of distortion functions of GlueVaR measures.

Let us consider ${\kappa}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}$ in Figure 5(a). The distortion function has a discontinuity at $1-\alpha $. Note that

$${\kappa}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}(1-\alpha )={h}_{2},$$ |

where $$ and

$$\underset{u\to {(1-\alpha )}^{+}}{lim}{\kappa}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}(u)=1.$$ |

This distortion function is concave in the interval $[0,1-\alpha ]$. However, the great flexibility of GlueVaR measures allows us to define a distortion function ${\kappa}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}$ that is convex in the interval $[0,1-\alpha ]$. This is the case with the distortion function in Figure 5(b).

We fix $q=1-\alpha $ so that for nonnegative random variables $X$, $Y$ it holds that

${\int}_{0}^{1-\alpha}}{F}_{Z}^{-1}(1-u)d{\kappa}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}(u)\le {\displaystyle {\int}_{0}^{1-\alpha}}{F}_{X}^{-1}(1-u)d{\kappa}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}(u)+{\displaystyle {\int}_{0}^{1-\alpha}}{F}_{Y}^{-1}(1-u)d{\kappa}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}(u),$ |

where $Z=X+Y$, if the distortion function ${\kappa}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}$ is concave in $[0,1-\alpha ]$.

Our starting point is

$${\int}_{0}^{1-\alpha}{F}_{X}^{-1}(1-u)d{\kappa}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}(u)={\omega}_{1}{\mathrm{TVaR}}_{\beta}(X)+{\omega}_{2}{\mathrm{TVaR}}_{\alpha}(X).$$ |

Therefore, $q$-tail subadditivity is ensured if ${\omega}_{1}$ and ${\omega}_{2}$ are positive. The weights ${\omega}_{1}$ and ${\omega}_{2}$ are positive if and only if the distortion function $\kappa $ is concave in $[0,1-\alpha ]$. Indeed, ${\omega}_{1}$ is positive if and only if ${h}_{1}/(1-\beta )\ge ({h}_{2}-{h}_{1})/(\beta -\alpha )$. The distortion function $\kappa $ is concave in $[0,1]$ if ${\omega}_{1}$ is positive and ${h}_{2}=1$.

When $q$ is in $(1-\beta ,1-\alpha )$, it can be shown that $q$-tail subadditivity is satisfied if $g$ is concave in $[0,q]$ (see Proposition 4.5). Finally, $q$-tail subadditivity when $q\le 1-\beta $ is directly derived.

## 5 Example

To illustrate the quantification of the tail contribution, we use the insurance data described in Belles-Sampera et al (2017). The data comprise three types of claim: property damage (${X}_{1}$), bodily injuries (${X}_{2}$) and medical expenses (${X}_{3}$). The sample contains $n=350$ observations of the cost of individual claims expressed in thousands of euros. The cost of bodily injuries contains compensation for bodily injuries. It is relatively low compared with that of property damage, because it does not include that part already covered by public health insurance.

In Table 2 a set of quantile-based risk measures, including three different measures of GlueVaR, are shown. These GlueVaR measures reflect different risk attitudes. ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{11/30,2/3}$ corresponds to a balanced attitude between ${\mathrm{TVaR}}_{99.5\%}$, ${\mathrm{TVaR}}_{95\%}$ and VaR${}_{95\%}$. Indeed, ${h}_{1}=11/30$ and ${h}_{2}=2/3$ correspond to ${\omega}_{1}={\omega}_{2}={\omega}_{3}=1/3$. ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{0,1}$ has associated weights ${\omega}_{1}=-1/9$, ${\omega}_{2}=10/9$ and ${\omega}_{3}=0$. It corresponds to an extreme scenario in which the lowest feasible ${\omega}_{1}$ is allocated to ${\mathrm{TVaR}}_{99.5\%}$ and the highest ${\omega}_{2}$ to ${\mathrm{TVaR}}_{95\%}$. A zero weight is allocated to ${\mathrm{VaR}}_{95\%}$, ${\omega}_{3}=0$. Finally, ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{1/20,1/8}$ assigns a high weight to VaR${}_{95\%}$ (${\omega}_{3}=21/24$) and low weights to ${\mathrm{TVaR}}_{99.5\%}$ and ${\mathrm{TVaR}}_{95\%}$ (${\omega}_{1}=1/24$ and ${\omega}_{2}=1/12$). Risk measures are estimated based on the empirical survival function.

A | B | C | D | Difference* | |
---|---|---|---|---|---|

VaR${}_{\text{95\%}}$ | 2.5 | 0.6 | 1.1 | 5.9 | $-$1.7 |

${\mathrm{TVaR}}_{\text{95\%}}$ | 12.5 | 8.0 | 1.3 | 19.7 | 2.1 |

GlueVaR${}_{\text{99.5\%,95\%}}^{\text{11/30,2/3}}$ | 18.6 | 16.9 | 1.4 | 35.6 | 1.3 |

GlueVaR${}_{\text{99.5\%,95\%}}^{\text{1/20,1/8}}$ | 4.9 | 2.9 | 1.1 | 10.2 | $-$1.3 |

GlueVaR${}_{\text{99.5\%,95\%}}^{\text{0,1}}$ | 9.4 | 4.2 | 1.2 | 12.9 | 1.9 |

Table 2 shows that ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{11/30,2/3}$ returns higher risk values than the other two selected GlueVaR measures. This result can be generalized to any random variable because the associated distortion function of ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{11/30,2/3}$ is greater than the other two distortion functions in the whole domain. It is also observed in Table 2 that ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{1/20,1/8}\le $ ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{0,1}$. Note that this outcome is a feature of this data set and cannot be generalized. We now analyze the subadditivity property. The only risk measure with a concave distortion function is TVaR. The other risk measures shown in Table 2 do not have concave distortion functions in $[0,1]$. The last column of Table 2 records the difference between the sum of risk values and the risk value of the sum. A negative value indicates that the risk value of the sum is higher, so the subadditivity is not satisfied, ie, there is no benefit of diversification. VaR${}_{95\%}$ and ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{1/20,1/8}$ fail to be subadditive for ${X}_{1}$, ${X}_{2}$ and ${X}_{3}$ since $$ and $$, respectively. The fact that risk values are subadditive for ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{11/30,2/3}$ and ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{0,1}$ is a characteristic attributable to this data set but cannot be generalized for all contexts.

(a) $q=\text{5\%}$ | |||||
---|---|---|---|---|---|

A | B | C | D | Difference* | |

VaR${}_{95\%}$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

${\mathrm{TVaR}}_{\text{95\%}}$ | 12.5 | 8.0 | 1.3 | 19.7 | 2.1 |

GlueVaR${}_{\text{99.5\%,95\%}}^{\text{11/30,2/3}}$ | 17.8 | 16.7 | 1.0 | 33.6 | 1.9 |

GlueVaR${}_{\text{99.5\%,95\%}}^{\text{1/20,1/8}}$ | 2.7 | 2.4 | 0.1 | 5.0 | 0.2 |

GlueVaR${}_{\text{99.5\%,95\%}}^{\text{0,1}}$ | 9.4 | 4.2 | 1.2 | 12.9 | 1.9 |

(b) $q=\text{0.5\%}$ | |||||

A | B | C | D | Difference* | |

VaR${}_{\text{95\%}}$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

${\mathrm{TVaR}}_{\text{95\%}}$ | 4.1 | 4.2 | 0.2 | 8.1 | 0.4 |

GlueVaR${}_{\text{99.5\%,95\%}}^{\text{11/30,2/3}}$ | 15.0 | 15.4 | 0.7 | 29.7 | 1.3 |

GlueVaR${}_{\text{99.5\%,95\%}}^{\text{1/20,1/8}}$ | 2.0 | 2.1 | 0.1 | 4.1 | 0.2 |

GlueVaR${}_{\text{99.5\%,95\%}}^{\text{0,1}}$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

The $q$-tail contribution to the risk values and tail subadditivity is analyzed for $q=5\%$ and $q=0.5\%$ in Table 3. We focus on column D, which shows the $5\%$- and $0.5\%$-tail contributions to the aggregate risk value, ${\rho}_{g}({X}_{1}+{X}_{2}+{X}_{3})$. The tail contribution indicates the sensitivity of the risk measure value to the right $q$-tails of the loss distribution. Both contributions are equal to 0 when the VaR${}_{95\%}$ is analyzed. The contribution of the 5% of the most extreme losses to VaR${}_{95\%}$ is null. The same occurs when the contribution of the 0.5% of the most extreme losses is considered; see Table 3(b). In other words, the information provided by the tail contribution to VaR${}_{95\%}$ is that the 5% and 0.5% right tails of the loss distribution do not have an impact on the risk value.

In the case of ${\mathrm{TVaR}}_{95\%}$, losses located in the 5% right tail of the distribution contribute the whole amount to the risk value, 19.7. As argued above, this is expected because the 5% tail is precisely the threshold right tail embedded in our $q$-tail contribution definition for ${\mathrm{TVaR}}_{95\%}$. However, if we only consider the $0.5\%$ right tail of the loss distribution, the tail contribution is $\text{\u20ac}8100/\text{\u20ac}\mathrm{19\hspace{0.17em}700}$. That is, 0.5% of the highest aggregate losses contribute 41% (8.1/19.7) of the total risk value. This information is important in risk management. Mandatory reserves are often computed according to a risk measure set by regulators. Let us consider that regulatory reserves must be computed based on ${\mathrm{TVaR}}_{95\%}$. Risk managers will know that 41% of the regulatory reserve value is due to the 0.5% right tail of the loss distribution. The same interpretation can be made for individual risks. If risks ${X}_{1}$, ${X}_{2}$ and ${X}_{3}$ are individually analyzed, losses located at the 0.5% right tails of their loss distributions represent 33% (4.1/12.5), 52% (4.2/8.0) and 15% (0.2/1.3) of the ${\mathrm{TVaR}}_{95\%}$ risk values, respectively. The tail contribution for the rest of risk measures can be interpreted in the same fashion.

Tail-subadditivity is now investigated. Nonnegative values in the last column of Table 3 reflect the part of the diversification benefit captured by the risk measures when the 5% and the 0.5% of the most extreme losses of all risks are considered. When looking at subadditivity in the tail, the associated distortion functions of VaR${}_{95\%}$, ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{11/30,2/3}$ and ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{1/20,1/8}$ are concave in [0,0.05]. Since

$$\frac{{h}_{1}}{0.005}\ge \frac{({h}_{2}-{h}_{1})}{(0.995-0.95)},$$ |

the concavity of the distortion functions of ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{11/30,2/3}$ and ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{1/20,1/8}$ in [0,0.05] holds. The distortion functions of these three risk measures have a discontinuity at the point 0.05; that is,

${\psi}_{0.05}(0.05)$ | $=0\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\psi}_{0.05}({0.05}^{+})=1$ | $\text{for}{\mathrm{VaR}}_{95\%},$ | |||

${\kappa}_{99.5\%,95\%}^{11/30,2/3}(0.05)$ | $=\frac{2}{3}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\kappa}_{99.5\%,95\%}^{11/30,2/3}({0.05}^{+})=1$ | $\text{for}{\mathrm{GlueVaR}}_{99.5\%,95\%}^{11/30,2/3},$ | |||

${\kappa}_{99.5\%,95\%}^{1/20,1/8}(0.05)$ | $=\frac{1}{8}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}{\kappa}_{99.5\%,95\%}^{1/20,1/8}({0.05}^{+})=1$ | $\text{for}{\mathrm{GlueVaR}}_{99.5\%,95\%}^{1/20,1/8}.$ |

Finally, the associated distortion function of ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{0,1}$ is convex in the interval $[0,0.05]$. In this case, the concavity of the distortion function is only satisfied in the interval [0,0.005], where the distortion function ${\kappa}_{99.5\%,95\%}^{0,1}$ is equal to 0. So, only the 0.5%-tail subadditivity of ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{0,1}$ can be guaranteed.

If we look at ${\mathrm{TVaR}}_{95\%}$, the proportion of the benefit of diversification associated with 0.5% of the most extreme losses of all risks is 19% of €2.1 million. In the case of ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{11/30,2/3}$, a total diversification benefit of €1.3 million is considered when risks are aggregated (Table 2). The diversification benefit raises to €1.9 million when only the 5% right tail of the loss distribution is considered (Table 3). The net diversification benefit in the region (0.05,1] is then equal to $-$€0.6 million. If the focus is on 0.5% of the highest losses, the diversification benefit is again €1.3 million. The information given to decision makers is that a diversification benefit of €1.9 million is computed by the risk measure in the region in which the distortion function is concave [0,0.05], and the main part of this diversification benefit is due to 0.5% of the highest losses: 1.3 (out of 1.9). This means that a relatively low weight is given to the likelihood of the 0.5% highest losses of all three individual risks occurring at the same time.

An interesting result is observed in the case of ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{1/20,1/8}$. The total diversification benefit is negative, $-\text{\u20ac}1.3$ million. However, the diversification benefit associated with 5% and the 0.5% of the highest losses is €0.2 million. The risk measure takes into account that there are benefits of diversification in the 5% right tail of the loss distribution and these benefits are concentrated in the most extreme adverse scenarios (0.5% of the highest losses). To conclude, there are no diversification benefits computed by VaR and ${\mathrm{GlueVaR}}_{99.5\%,95\%}^{0,1}$ in the regions where tail subadditivity is guaranteed (5% and 0.5% respectively).

## 6 Conclusion

In this paper, we report a method for analyzing the influence of the tail in calculations of DRMs. By concentrating on the tail, we define the $q$-tail contribution as being the size of the risk measure estimate that is attributable to the tail of the distribution. As such, the $q$-tail contribution represents the weight of the tail in the risk measure. For the ${\mathrm{VaR}}_{\alpha}$, ${\mathrm{TVaR}}_{\alpha}$ and ${\mathrm{GlueVaR}}_{\beta ,\alpha}^{{h}_{1},{h}_{2}}$ measures, these weights are below 100% if the $q$-tail is smaller than some embedded threshold right tail (the $(1-\alpha )$-tail for each of these three quantile-based risk measures).

The tail contribution is valuable information for decision makers. It reports the impact of the $q\%$ extreme losses on the risk measure value. The selection of risk measures, and risk-aversion coefficients, to compute economic reserves is often set by regulators. The $q$-tail contribution will inform decision makers about the part of the regulatory reserve estimate that is attributable to the $q\%$ of the highest losses.

Under straightforward conditions, we have proven that tail subadditivity holds and that $q$-tail contributions satisfy the subadditivity property. So, if a subadditive DRM is fixed in the computation of the aggregate regulatory reserve, decision makers will identify those diversification benefits that are attributable to the $q\%$ of the highest losses. If the regulatory risk measure is not subadditive but it is $q$-tail subadditive, diversification benefits due to the $q$-right tail of the loss distributions will be captured by the risk measure. In other words, even in the case where the total diversification benefit could be negative, the risk measure allows us to contemplate diversification benefits for extreme losses (in the $q$-right-tail region), and decision makers are provided with an instrument to quantify the size of this diversification benefit in the tail.

In our example based on motor insurance claims, an examination of the risk of the severity of claims along three dimensions (property damage, bodily injury and additional medical expenses) allows us to conclude that the weight of the tail of each dimension of the final risk estimate can be assessed. Moreover, since subadditivity holds for that part of the domain, we can identify the role of each type of cost in the final risk. For example, the contribution of additional medical expenses to risk is almost negligible compared with the contribution of the claim costs of property damage and bodily injury. This holds for all the DRMs analyzed in this paper. However, because not all these DRMs satisfy subadditivity in general, the overall diversification of risk could not be analyzed.

## Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.

## Acknowledgements

The authors are grateful for the comments by the anonymous referees and editors. The authors acknowledge support received from the Spanish Ministry of Science and Innovation under grant PID2019-105986GB-C21 and from the Riskcenter research group members at the University of Barcelona. Miguel Santolino acknowledges the support from the Catalan government under grant 2020-PANDE-00074. Montserrat Guillen thanks ICREA Academia and the Consortium for Data Analytics in Risk at the University of California, Berkeley, for their generous hospitality.

## References

- Balbás, A., Garrido, J., and Mayoral, S. (2009). Properties of distortion risk measures. Methodology and Computing in Applied Probability 11(3), 385–399 (https://doi.org/10.1007/s11009-008-9089-z).
- Basel Committee on Banking Supervision (2019). Minimum capital requirements for market risk. Report, January, Bank for International Settlements.
- Belles-Sampera, J., Guillen, M., and Santolino, M. (2013). Generalizing some usual risk measures in financial and insurance applications. In Modeling and Simulation in Engineering, Economics and Management, Fernández-Izquierdo, M., MuÃ±oz-Torres, M., and León, R. (eds), pp. 75–82. Lecture Notes in Business Information Processing, Volume 145. Springer.
- Belles-Sampera, J., Guillen, M., and Santolino, M. (2014). Beyond value-at-risk: GlueVaR distortion risk measures. Risk Analysis 34(1), 121–134 (https://doi.org/10.1111/risa.12080).
- Belles-Sampera, J., Guillen, M., and Santolino, M. (2017). Risk Quantification and Allocation Methods for Practitioners. Amsterdam University Press.
- Cai, J., and Li, H. (2005). Conditional tail expectations for multivariate phase type distributions. Journal of Applied Probability 42, 810–825 (https://doi.org/10.1239/jap/1127322029).
- Cai, J., Wang, Y., and Mao, T. (2017). Tail subadditivity of distortion risk measures and multivariate tail distortion risk measures. Insurance: Mathematics and Economics 75, 105–116 (https://doi.org/10.1016/j.insmatheco.2017.05.004).
- Chen, D., Mao, T., Pan, X., and Hu, T. (2012). Extreme value behavior of aggregate dependent risks. Insurance: Mathematics and Economics 50(1), 99–108 (https://doi.org/10.1016/j.insmatheco.2011.10.008).
- Degen, M., Lambrigger, D. D., and Segers, J. (2010). Risk concentration and diversification: second-order properties. Insurance: Mathematics and Economics 46(3), 541–546 (https://doi.org/10.1016/j.insmatheco.2010.01.011).
- Denneberg, D. (1990). Premium calculation: why standard deviation should be replaced by absolute deviation. ASTIN Bulletin 20(2), 181–190 (https://doi.org/10.2143/AST.20.2.2005441).
- Denneberg, D. (1994). Non-Additive Measure and Integral. Kluwer Academic, Dordrecht.
- Denuit, M., Dhaene, J., Goovaerts, M., and Kaas, R. (2005). Actuarial Theory for Dependent Risks: Measures, Orders and Models. Wiley.
- Dhaene, J., Kukush, A., Linders, D., and Tang, Q. (2012). Remarks on quantiles and distortion risk measures. European Actuarial Journal 2(2), 319–328 (https://doi.org/10.1007/s13385-012-0058-0).
- Embrechts, P., Lambrigger, D. D., and Wuethrich, M. V. (2009a). Multivariate extremes and the aggregation of dependent risks: examples and counter-examples. Extremes 12(2), 107–127 (https://doi.org/10.1007/s10687-008-0071-5).
- Embrechts, P., Neslehova, J., and Wuethrich, M. V. (2009b). Additivity properties for Value-at-Risk under Archimedean dependence and heavy-tailedness. Insurance: Mathematics and Economics 44(2), 164–169 (https://doi.org/10.1016/j.insmatheco.2008.08.001).
- Embrechts, P., Liu, H., and Wang, R. (2015). Seven proofs for the subadditivity of expected shortfall. Dependence Modeling 30(1), 126–140 (https://doi.org/10.1515/demo-2015-0009).
- Furman, E., Wang, R., and Zitikis, R. (2017). Gini-type measures of risk and variability: Gini shortfall, capital ]allocations, and heavy-tailed risks. Journal of Banking and Finance 83, 70–84 (https://doi.org/10.1016/j.jbankfin.2017.06.013).
- Goovaerts, M., Linders, D., Van Weert, K., and Tank, F. (2012). On the interplay between distortion, mean value and Haezendonck–Goovaerts risk measures. Insurance: Mathematics and Economics 51(1), 10–18 (https://doi.org/10.1016/j.insmatheco.2012.02.012).
- Kou, S., and Peng, X. (2016). On the measurement of economic tail risk. Operations Research 64(5), 1056–1072 (https://doi.org/10.1287/opre.2016.1539).
- Landsman, Z., Makov, U., and Shushi, T. (2016). Multivariate tail conditional expectation for elliptical distributions. Insurance: Mathematics and Economics 70, 216–223 (https://doi.org/10.1016/j.insmatheco.2016.05.017).
- Liu, F., and Wang, R. (2019). A theory for measures of tail risk. Mathematics of Operations Research. Working Paper, Social Science Research Network (https://doi.org/10.2139/ssrn.2841909).
- McNeil, A. J., Frey, R., and Embrechts, P. (2005). Quantitative Risk Management, rev. edn. Princeton Series in Finance. Princeton University Press.
- Nam, H. S., Tang, Q., and Yang, F. (2011). Characterization of upper comonotonicity via tail convex order. Insurance: Mathematics and Economics 48(3), 368–373 (https://doi.org/10.1016/j.insmatheco.2011.01.003).
- Tsanakas, A., and Desli, E. (2005). Measurement and pricing of risk in insurance markets. Risk Analysis 25(6), 1653–1668 (https://doi.org/10.1111/j.1539-6924.2005.00684.x).
- Tsanakas, A., and Millossovich, P. (2016). Sensitivity analysis using risk measures. Risk Analysis 36(1), 30–48 (https://doi.org/10.1111/risa.12434).
- Wang, R., Wei, Y., and Willmot, G. (2018). Characterization, robustness and aggregation of signed Choquet integrals. Working Paper, Social Science Research Network (https://doi.org/10.2139/ssrn.2956962).
- Wang, S. S. (1995). Insurance pricing and increased limits ratemaking by proportional hazard transforms. Insurance: Mathematics and Economics 17(1), 43–54 (https://doi.org/10.1016/0167-6687(95)00010-P).
- Wang, S. S. (1996). Premium calculation by transforming the layer premium density. ASTIN Bulletin 26(1), 71–92 (https://doi.org/10.2143/AST.26.1.563234).
- Wang, S. S. (1998). An actuarial index of the right-tail risk. North American Actuarial Journal 2(2), 88–101 (https://doi.org/10.1080/10920277.1998.10595760).
- Wang, S. S., and Dhaene, J. (1998). Comonotonicity, correlation order and premium principles. Insurance: Mathematics and Economics 22(3), 235–242 (https://doi.org/10.1016/S0167-6687(97)00040-1).
- Wirch, J. L., and Hardy, M. R. (2002) Distortion risk measures: coherence and stochastic dominance. In Proceedings of the International Congress on Insurance: Mathematics and Economics, Lisbon, pp. 15–17. URL: https://bit.ly/2XNPnrO.
- Yaari, M. E. (1987). The dual theory of choice under risk. Econometrica 55(1), 95–115 (https://doi.org/10.2307/1911158).
- Yin, C., and Zhu, D. (2016). New class of distortion risk measures and their tail asymptotics with emphasis on VaR. Preprint (arXiv:1503.08586v2 [q-fin.RM]). URL: https://arxiv.org/abs/1503.08586v2.
- Zhu, L., and Li, H. (2012). Tail distortion risk and its asymptotic analysis. Insurance: Mathematics and Economics 51(1), 115–121 (10.1016/j.insmatheco.2012.03.010).

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