Extended Gini-type measures of risk and variability

In modern risk management, a large number of risk measures have been proposed in the literature. These measures are mappings from a set of random variables (financial losses) to real numbers. At first, the focus were on the variability over an expected return, as is the case for the well-known variance. After the collapses and crises in financial systems, a prominent trend associated with tail-based risk measures has emerged, especially with the most popular ones nowadays: the Value-at-Risk (VaR) and the Expected Shortfall (ES). However, this kind of risk measures does not capture the variability of a financial position, a primitive but relevant concept. In order to solution this issue, some authors have proposed and studied specific examples of risk measures.

In this sense, Fischer [8] considered combining the mean and semi-deviations. Regarding tail risk, Furman and Landsman [10] proposed a measure that weighs the mean and standard deviation in the truncated tail by VaR, while Righi and Ceretta [17] considered penalizing the ES by the dispersion of losses exceeding it. From a practical perspective, Righi and Borenstein [18] explored this concept, calling the approach as loss-deviation, for portfolio optimization. In a more general fashion, Righi [19] presents results and examples about compositions of risk and variability measures in order to ensure solid theoretical properties.

Recently, Furman et al. [11] introduced the Gini Shortfall (GS) risk measure which is coherent and satisfies co-monotonic additivity. GS is a composition between ES and tail based Gini coefficient. However, GS supposes that all individuals have the same attitude towards risk, while agents differ in the way they take personal decisions that involve risk because of discrepancies in their risk aversion. To incorporate such psychological behavior in tail risk analysis, we introduce a generalized version of the GS. This risk measure, called Extended Gini Shortfall (EGS), captures the notion of variability, satisfies the co-monotonic additivity property, and it is coherent under a necessary and sufficient condition for its loading parameter. The consideration of the decision-maker risk aversion, joined to these properties, is in consonance to what agents seek when searching for a suitable measure of risk. The approach followed in this article leads us to a new family of spectral risk measures, proposed by Acerbi [1], with an attractive weighting function.

In this sense, we discuss, in a separated manner, the properties from the variability term and our composed risk measure. Moreover, we discuss in details the role of each parameter in the mentioned weighting function. Furthermore, we expose results on analytic formulations for computation of EGS under known distribution functions. Our focus in this paper is on theoretical results, but this approach gives rise to further forthcoming investigations. In this sense, risk forecasting of our new family of risk measures is a subject that deserves a further and separate survey which will be made in a forthcoming work.

The rest of the paper is organized as follows. In Section 2, we present and discuss some preliminaries such as essential properties of measures of risk and variability, as well we elucidate the role of the signed Choquet integral. In section 3, we start with the Classical and Extended Gini functionals in order to introduce the concept and explore properties about what we call Tail Extended Gini functional and Extended Gini Shortfall. In Section 4, we give the closed-form of our risk measures class for elliptical distributions and then derive the uniform, Normal and Student-t cases. Section 5 illustrates an application of the introduced risk measures class in practice.

We first introduce some basic notation. Let $(\Omega,\mathcal{A},\mathbb{P})$ be an atomless probability space. All equations and inequalities are in the $\mathbb{P}$ almost surely sense. Let $L^{q}$, $q\in[0,\infty)$, denote the set of all random variables (rv’s from now on) in $(\Omega,\mathcal{A},\mathbb{P})$ with finite q-th moment and $L^{\infty}$ be the set of all essentially bounded rv’s. Throughout this paper, $X\in L^{0}$ is a rv modeling financial losses (profits) when it has positive (negative) values. For every $X\in L^{0},\,F_{X}$ denotes the cdf of $X$, and $U_{X}$ denote any uniform $[0,1]$ rv such that the equation $F^{-1}_{X}(U_{X})=X$ holds. The existence of such rv’s is assured in Rüschendorf ([21], Proposition 1.3). We denote $x_{p}$ as the p-quantile of $X$. Two rv’s $X$ and $Y$ are co-monotonic when $(X(\omega)-X(\omega^{\prime}))(Y(\omega)-Y(\omega^{\prime}))\geq 0\text{ for }(\omega,\omega^{\prime})\in\Omega\times\Omega\quad(\mathbb{P}\times\mathbb{P})\text{-almost surely}$. Throughout the present paper, we deal with several convex cones $\mathcal{X}$ of rv’s, of which $\mathcal{X}=L^{1}$ is of particular importance and $L^{\infty}$ is always contained in $\mathcal{X}$.

We begin by exposing definitions of both risk and variability measures. We assume throughout the paper that all functionals respect the following property, which is essential in order to obtain a functional directly from its distribution function.

The notion of signed Choquet integral plays a pivotal role thereafter. It originates from Choquet [4], in the framework of capacities, and is further characterized and studied in decision theory by Schmeidler ([22], [23]).

One of the practical and theoretical reasons for what we are particularly interested in signed Choquet integral is that we know that a suitable risk measure should be monotone as argued by Artzner et al. [2], but this issue is irrelevant for a measure of variability. In other words, signed Choquet integral is relevant as long as a measure of variability is concerned. The following theorem is enunciated with a complete proof in Furman et al. [11], it gives the characterization for co-monotonically additive and coherent measures of variability.

Next, we recall a few partial orders of variability that have been popular in economics, insurance, finance and probability theory:

In this section, we expose our main contribution, which is based on the Gini coefficient, a free-center measure of variability that was introduced by Corrado Gini as an alternative to the variance measure (e.g., Giorgi ([12], [13]) and Ceriani and Verme [3]). The Gini coefficient has been remarkably influential in numerous research areas (e.g, Yitzhaki and Schechtman [30] and the references therein). Yitzhaki [28] lists more than a dozen alternative presentations of the Gini coefficient. We now present a formal definition.

The Gini functional supposes that all individuals have the same attitude towards risk. Nonetheless, the concept can be extended into a family of measures of variability differing from each other in the decision-maker’s degree of risk aversion, which is reflected in this paper by the parameter $r$. The basic definition of the Extended Gini coefficient is based on the covariance term. We refer to Yitzhaki [27], Shalit and Yitzhaki [24], Yitzhaki and Schechtman [29], and Yitzhaki and Schechtman [30] for an overview of the Extended Gini properties. We now expose it in a formal sense.

We now explore the characterization of the Extended Gini coefficient as a signed Choquet integral. In this sense, note that from equation (2.5) in Theorem 2.9, if one sets $g_{r}(u)=-r(1-u)^{{}^{r-1}}$ for $r>1$ and $u\in[0,1]$, we run into $\nu(X)=-r\,Cov[X,(1-F_{X}(X))^{r-1}]$. With that in mind, we now state and prove the formal result.

Based on the exposed content, we now turn the focus to an adaptation to the tails of the distribution function. In this sense, we now introduce the Tail Extended Gini (TEG) functional, as well formally prove its properties and Choquet integral representation.

Despite the fact that TEG is not a coherent measure of variability, we will show now that a linear combination of the Expected Shortfall with the Tail Extended Gini gives rise to a coherent risk measure, the Extended Gini Shortfall, that quantifies both the magnitude and the variability of tail risks. We now define such a combination.

In order to be a reasonable tool for risk management, the properties of coherent risk measures are desired. However, as mentioned in the previous section, TEG is not sub-additive, and as a measure of variability is not monotone. However, when $\lambda$ is zero, then EGS obviously inherits all the properties of the ES which is coherent, but when $\lambda$ is sufficiently large, then the TEG starts to dominate ES, and thus coherence of EGS cannot be expected. Intuitively, as suggested by Furman et al. [11], there might be a threshold that delineates the value of $\lambda$ for which EGS is coherent. We now verify it in a formal way.

We now provide a result and interpretation about how this weighting function $\phi$ behaves in relation to changes (partial derivatives) of each variable (parameter) among $u$, $r$, $p$ and $\lambda$. It is valid to point out that, by the spectral representation, results can be directly understood as the effect each parameter has over values for EGS.

Since $p$ reflects the prudence level, which is usually close to $1$ in practice, then $r_{0}$ in the proof of item (i) is in general close to 1. Thus, for practical matters, the superior limit for $\lambda$ is non-decreasing in most relevant values of $r$. The most complex parameter sensitivities are regarding the prudence level $p$ and the generalization term $r$, respectively in items (iv) and (v), because both $\frac{\partial\phi}{\partial p}$ and $\frac{\partial\phi}{\partial r}$ are expressions that can assume both positive and negative values since some terms in numerator posses distinct signs without a domination. This can be explained due to the fact that $\phi$ is a weighting function and changes in $p$ and $r$ alter how much ‘mass’ is put to any probability level $u$. Because $\int_{0}^{1}\phi_{r,p}^{\lambda}(u)du=1$, it is necessary that the increase on $\phi(u)$ for some values of $u$ be compensated by some decrease in others.

Regarding prudence level, item (iv) in the previous proposition emphasizes that $\phi$ non-decreases in $p$ for larger values of $u$. This is in consonance with practical intuition because more weight is put to extreme probabilities. Moreover, this is corroborated when we verify the variations of $\phi$ regarding $u$ and $p$, where we obtain the non-negative expression

When we consider the sensibility of variations regarding $u$ and $r$, in a case we get

there is divergence about sign – first term inside brackets is non-negative while the second is non-positive, corroborating with the previous argument. Nonetheless, $\phi$ is non-decreasing in $r$, a situation where $r$ behaves more like a risk aversion coefficient, when a non-decreasing function of $p$, $(1-p)^{(1-p)}$ is greater than a threshold that is an expression depending on both $u$ and $r$. Repeating the argument of practical choices for the prudence level, $p$ will be close to $1$ and $\phi$ will be increasing in $r$ for more values of $u$. In this sense, $r$ must be understood as a generalization parameter for a family of EGS risk measures rather than a single linear risk aversion coefficient.

In this section we provide analytical formulations to compute the proposed Extended Gini Shortfall to known and very used distribution functions. In that sense, a location-scale family is a family of probability distributions parameterized by a location parameter and a non-negative scale parameter. Suppose that $Z$ is a fixed rv taking values in $\mathbb{R}$. For $\alpha\in\mathbb{R}$ and $\beta\in(0,\infty)$, let $X=\alpha+\beta Z$. The two-parameter family of distributions associated with $X$ is called the location-scale family associated with the given distribution of $Z$; $\alpha$ is called the location parameter and $\beta$ the scale parameter. The standard form of any distribution is the form whose location and scale parameters are $0$ and $1$, respectively. In this section, we restrain our attention into standardized rv’s. In general, when $X=\alpha+\beta Z$ for $\alpha\in\mathbb{R}$ and $\beta\in(0,\infty)$, we have, directly from their properties, both $ES_{p}(X)=\alpha+\beta ES_{p}(Z)$ and $TEGini_{r,p}(X)=\beta TEGini_{r,p}(Z)$.