Multivariate geometric expectiles

Abstract

A generalization of expectiles for d-dimensional multivariate distribution functions is introduced. The resulting geometric expectiles are unique solutions to a convex risk minimization problem and are given by d-dimensional vectors. They are well behaved under common data transformations and the corresponding sample version is shown to be a consistent estimator. We exemplify their usage as risk measures in a number of multivariate settings, highlighting the influence of varying margins and dependence structures.

Multivariate Geometric Tail- And Range-Value-at-Risk

“…In this paper, we focus on a related multivariate concept. In order to establish the notation we will use for the remainder of this article, we provide a streamlined introduction to multivariate geometric quantiles in the following paragraphs, but refer to Chaudhuri (1996), Stupfler (2015, 2017) and Herrmann et al (2018) for a more detailed discussion. We denote by x 2 = √ x x and x, y = x y the Euclidean norm and inner product of two vectors x, y ∈…”

“…It is important to notice that the univariate version of geometric VaR implied by (2.3) is based on a confidence level α ∈ [−1, 1], instead of the traditional α ∈ [0, 1]. The traditional confidence level in [0, 1] can be recovered by an appropriate linear re-indexing; see Section 6.3 and also Herrmann et al (2018) for an in-depth discussion. To clearly distinguish both concepts in the univariate setting, the standard quantiles with α ∈ [0, 1] are denoted by F −1 , while univariate geometric quantiles will be denoted by VaR.…”

“…When α i = 0, P α penalizes the difference between E[X i ] and c i , and the geometric VaR is thus the penalized version of the spatial median. Likewise, the geometric expectiles of Herrmann et al (2018) can be interpreted as penalized versions of the component-wise mean. In comparison to other generalizations of quantiles to d > 1, geometric VaR has a number of attractive properties; see Theorem 2.1.…”

“…More recently, Maume-Deschamps et al (2017) presented a multivariate extension of expectiles, where the risk decomposition is embedded in the risk measure and the multivariate version of expectiles presented assumes risk mitigation throughout the portfolio, but allows for a multivariate model which presumably has a better fit for the considered data. Based on the multivariate geometric quantiles defined in Chaudhuri (1996), Herrmann et al (2018) developed multivariate geometric expectiles. While univariate expectiles can be defined for random variables with finite first moment (see Newey and Powell, 1987), multivariate extensions are based on the existence of second moments (see Herrmann et al, 2018;Maume-Deschamps et al, 2017).…”

“…Based on the multivariate geometric quantiles defined in Chaudhuri (1996), Herrmann et al (2018) developed multivariate geometric expectiles. While univariate expectiles can be defined for random variables with finite first moment (see Newey and Powell, 1987), multivariate extensions are based on the existence of second moments (see Herrmann et al, 2018;Maume-Deschamps et al, 2017).…”