On Optimization of Copula-Based Extended Tail Value-at-Risk and its Application in Energy Risk

Bony Parulian Josaphat,Khreshna Syuhada,Moch Fandi Ansori

doi:10.1109/access.2021.3106715

Bony Parulian Josaphat, Khreshna Syuhada + Show 1 more

Open Access

https://doi.org/10.1109/access.2021.3106715

Copy DOI

Abstract

In this paper, we study a novel risk measure, which is a copula-based extension of tail value-at-risk (TVaR). This measure is called dependent tail value-at-risk (DTVaR), which is a generalization of TVaR. Moreover, we describe a second conditional tail moment of the tail distribution with the center being the DTVaR itself, which is called the dependent conditional tail variance (DCTV). Both DTVaR and DCTV contain two contraction parameters, which make them much more flexible than some of the more familiar measures of risk, such as TVaR and conditional tail variance (CTV). We derive analytical formulas of the DTVaR and DCTV for exponential risk associated with another risk where their dependence structure is represented by Farlie-Gumbel-Morgenstern (FGM) copula. This paper proposes an optimization method for DTVaR by applying two metaheuristic algorithms: spiral optimization (SpO) and particle swarm optimization (PSO). Furthermore, we perform SpO and PSO by utilizing DCTV and CTV to estimate two contraction parameters that maximize DTVaR. This work presents an application of DTVaR optimization in predicting the DTVaR of energy risk of New York Harbor (NYH) gasoline associated with energy risk of West Texas Intermediate (WTI) crude oil. We find that the values of the objective function using both algorithms converge to zero, which implies that the SpO and PSO algorithms are very suitable for application to DTVaR optimization. However, according to the values of the objective function, we find that the PSO algorithm is more suitable than the SpO algorithm in optimizing DTVaR.

Highlights

The idea is that, regardless of the usefulness and calculates the mean losses of a target risk by providing59 desired properties, tail value-at-risk (TVaR) only records the mean losses on the condition that the target risk is bounded by two values60 the tail and ignores the variability and makes of VaR and considering the dependence on another risk61 sense to include a second central moment or variance of the associated with the target risk
[6] The dependent tail VaR (DTVaR) has a (1)47 larger risk than or equal to MTVaR and a lower risk than or equal to copula tail value-at-risk (CTVaR). where SN is the aggregate risk, Y is another risk associated with SN, or SN depends on Y, α1 = α + (1 − α)a+1 and δ1 = δ + (1 − δ)d+1
The dependent conditional tail variance (DCTV) of SN associated with Y is given by d that maximize DTVaR by applying a constraint, where the corresponding kDCTV must be equal to CTV for some multiplier k ∈ R+

Summary

INTRODUCTION

Presence of two contraction parameters a, d > 0 They proved that CTVaR is not less than TVaR when the Jadhav et al [4] did not optimize MTVaR and they target and associated risks are positive quadrant-dependent. set the values of a when calculating MTVaR. The dependent conditional tail variance of SN associated with Y is given by DCTV((δα,,da))(SN |Y ; C) = E SN − DTVaR((δα,,da))(SN |Y ; C) 2. The dependent conditional tail variance (DCTV) of SN associated with Y is given by d that maximize DTVaR by applying a constraint, where the corresponding kDCTV must be equal to CTV for some multiplier k ∈ R+. We apply the Lagrange multiplier method in [20] by defining a Lagrangian function, The following property holds for the variability measure of DCTV. We illustrate DCTV using the two risks joined by FGM the objective function as follows: copula via Example VI. in the Appendix

DTVAR OPTIMIZATION USING DCTV AND CTV

SPIRAL OPTIMIZATION METHOD

PARTICLE SWARM OPTIMIZATION METHOD

Objective function J Objective function J

CONCLUSIONS