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

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.

Keywords

• Dependent tail value-at-risk
• dependent conditional tail variance
• FGM copula
• metaheuristic algorithms
• energy risk