First and Second Order Asymptotics of the Spectral Risk Measure for Portfolio Loss Under Multivariate Regular Variation
- PDF / 218,763 Bytes
- 12 Pages / 595.276 x 841.89 pts (A4) Page_size
- 44 Downloads / 166 Views
First and Second Order Asymptotics of the Spectral Risk Measure for Portfolio Loss Under Multivariate Regular Variation∗ XING Guodong · YANG Shanchao
DOI: 10.1007/s11424-020-8037-z Received: 5 February 2018 / Revised: 7 July 2019 c The Editorial Office of JSSC & Springer-Verlag GmbH Germany 2020 Abstract In the context of multivariate regular variation, the authors establish the first-order asymptotics of the spectral risk measure of portfolio loss. Furthermore, by the notion of second-order regular variation, the second-order asymptotics of the spectral risk measure of portfolio loss is also presented. In order to illustrate the derived results, a numerical example with Monte Carlo simulation is carried out. Keywords Asymptotics, multivariate regular variation, regular variation, second-order regular variation, spectral risk measure, value-at-risk.
1
Introduction
At the outset, let us introduce briefly two popular risk measures: Value-at-risk and expected shortfall. The value-at-risk (for short, VaR) for a random loss variable X is defined as VaRλ (X) := inf {x : P (X ≤ x) ≥ λ} for the confidence level 0 < λ < 1 (usually, λ > 0.9), i.e., VaRλ (X) is the λ-quantile of the loss distribution. To remedy the deficiency of VaR that it is lack of subadditivity, Artzner, et al.[1] proposed the coherent expected shortfall (for short, ES), which is the average of the XING Guodong (Corresponding author) School of Mathematics and Statistics, Hefei Normal University, Hefei 230601, China; School of Mathematics and Statistics, Yulin Normal University, Yulin 537000, China. Email: [email protected]. YANG Shanchao School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China. Email: [email protected]. ∗ This research was supported by the Important Natural Science Foundation of Colleges and Universities of Anhui Province under Grant No. KJ2020A0122 and the Scientific Research Start-up Foundation of Hefei Normal University. This paper was recommended for publication by Editor WANG Shouyang.
2
XING GUODONG · YANG SHANCHAO
worst (1 − λ)100 % of losses, i.e., the expected shortfall for the continuous loss variable X and confidence level 0 < λ < 1 is denoted by 1 1 ESλ (X) := E (X|X ≥ VaRλ (X)) = VaRu (X)du. 1−λ λ Using an ES measure implies taking an average of quantiles where tail quantiles have an equal weight and non-tail quantiles have a zero weight. Nevertheless, the fact that the ES gives all tail losses an equal weight implies that a user who uses this measure is risk neutral, and is inconsistent with risk aversion. Therefore, considering a user’s risk aversion, Acerbi[2, 3] proposed a more general risk measure, which is called spectral risk measure (for short, SRM) and defined as 1 Mφ (X) = φ(p)VaRp (X)dp, 0
where the weighted function φ(p) is an admissible risk spectrum satisfying the following properties: (i) Nonnegativity: φ(p) ≥ 0; 1 (ii) Normalization: 0 φ(p)dp = 1; (iii) Increasingness: φ (p) ≥ 0. Mφ (X) defines the class of quantile-based risk measures, and each individual risk mea
Data Loading...
