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

Area of Science:

• Quantitative Finance
• Risk Management
• Actuarial Science

Background:

• Traditional risk measures like expected shortfall (ES) may not fully capture tail risk variability.
• Tail-based entropy (TE) offers insights into the distribution of extreme losses.
• A combined approach is needed for a comprehensive assessment of tail risk.

Purpose of the Study:

• Introduce and analyze the shortfall of tail-based entropy (STE), a novel tail-sensitive risk functional.
• Investigate STE's properties, including its relationship to existing shortfall measures and coherence.
• Apply STE to capital allocation problems and evaluate its empirical performance.

Main Methods:

• Define STE as a combination of expected shortfall (ES) and tail-based entropy (TE).
• Derive theoretical properties of STE, including characterizations, coherence conditions, and monotonicity.
• Develop closed-form capital allocation formulas based on STE for elliptical and extended skew-normal distributions.
• Conduct an empirical analysis using insurance company data.

Main Results:

• STE is shown to capture both the magnitude and variability of tail risk.
• The framework unifies several existing shortfall measures.
• Sufficient conditions for STE's coherence and monotonicity are established.
• Closed-form capital allocation formulas are derived for specific distributions.
• Empirical analysis demonstrates STE's practical implementation and performance.

Conclusions:

• STE provides a robust and comprehensive measure of tail risk.
• The derived capital allocation rules offer practical tools for financial institutions.
• STE enhances risk management strategies by considering both loss magnitude and variability.