# On the Shortfall of Tail-Based Entropy and Its Application to Capital Allocation

## TL;DR

This paper introduces a new risk measure called shortfall of tail-based entropy (STE) that captures both the magnitude and variability of extreme risks, and applies it to capital allocation in financial contexts.

## Contribution

The novelty lies in combining expected shortfall with tail-based entropy to create a new coherent risk measure that penalizes tail variability.

## Key findings

- STE generalizes several existing shortfall-type risk measures.
- Closed-form capital allocation formulas are derived for elliptical and extended skew-normal distributions.
- Empirical analysis on insurance data demonstrates the practicality of STE-based allocation.

## Abstract

We introduce and study the shortfall of tail-based entropy (STE), a tail-sensitive risk functional that combines expected shortfall (ES) and tail-based entropy (TE). Beyond the tail mean, STE imposes a rank-dependent penalty on tail variability, thereby capturing both the magnitude and variability of tail risk under extremes. The framework encompasses several shortfall-type measures as special cases, such as Gini shortfall, extended Gini shortfall, shortfall of cumulative residual entropy, shortfall of right-tail deviation, and shortfall of cumulative residual Tsallis entropy. We provide equivalent characterizations of STE, derive sufficient conditions for coherence, and establish monotonicity with respect to tail-variability order. As an application, we investigate STE-based capital allocation, deriving closed-form allocation formulas under elliptical and extended skew-normal distributions, along with several illustrative special cases. Finally, an empirical analysis with insurance company data illustrates the implementation and evaluates the performance of the allocation rule.