PELVaR: Probability equal level representation of Value at Risk through the notion of Flexible Expected Shortfall

Abstract: This paper proposes a novel perspective on the relationship between Value at Risk (VaR) and Expected Shortfall (ES) by employing the mixing framework of Flexible Expected Shortfall (FES) to construct coherent representations of VaR. The methodology enables a reinterpretation of VaR within a coherent risk measure framework, thereby addressing well-known limitations of VaR, including non-subadditivity and insensitivity to tail risk. A central feature of the framework is the flexibility parameter inherent in FES, which captures salient distributional properties of the underlying risk profile. This parameter is formalized as the $\theta$-index, a normalized measure designed to reflect tail heaviness. Theoretical properties of the $\theta$-index are examined, and its relevance to risk assessment is established. Furthermore, risk capital allocation is analyzed using the Euler principle, facilitating consistent and meaningful marginal attribution. The practical implications of the approach are illustrated through appropriate simulation studies and an empirical analysis based on an insurance loss dataset with pronounced heavy-tailed characteristics.