<p>The application of truncated distributions is highly extensive and vital in practice. They are utilized in real-world operations (not merely in theory) by large insurance companies and banks. One of their applications is fitting models to loss data and estimating risk measures. In this article, an attempt has been made to transfer the concepts of tail risk measures to the family of truncated distributions and to establish a connection between these risk measures in the truncated case and other indices. Additionally, a redefinition of truncated risk measures has been presented based on risk measures in the non-truncated case and inequality indices. The proposed approach provides algebraically closed-form expressions for these risk measures, allowing us to study their properties and evaluate their connections with inequality indices. This redefinition provides a new perspective in the field of risk quantification. To enhance the empirical validity of the proposed formulas, a non-parametric bootstrap methodology with 1,000 replications is implemented to quantify parameter-estimation uncertainty. The bootstrap analysis provides confidence intervals for VaR, ES, and the shape parameter, confirming the numerical stability of the analytical formulas across truncation regimes. Comprehensive validation over truncated Pareto and truncated Exponential models shows that the closed-form expressions achieve extremely high accuracy, with average relative errors of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varvec{7.5 \times 10^{-6}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn mathvariant="bold">7.5</mn> <mo mathvariant="bold">×</mo> <msup> <mn mathvariant="bold">10</mn> <mrow> <mo mathvariant="bold">-</mo> <mn mathvariant="bold">6</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> for the Pareto and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varvec{2.8 \times 10^{-6}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn mathvariant="bold">2.8</mn> <mo mathvariant="bold">×</mo> <msup> <mn mathvariant="bold">10</mn> <mrow> <mo mathvariant="bold">-</mo> <mn mathvariant="bold">6</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> for the Exponential distribution. Additionally, the analytical approach yields substantial computational speed improvements–ranging from 17–31<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\times $</EquationSource> <EquationSource Format="MATHML"><math> <mo>×</mo> </math></EquationSource> </InlineEquation> for the Pareto model and 3–22<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\times $</EquationSource> <EquationSource Format="MATHML"><math> <mo>×</mo> </math></EquationSource> </InlineEquation> for the Exponential model–when compared with numerical integration methods. These results highlight the practicality, robustness, and computational efficiency of the proposed closed-form framework for risk-management applications.</p>

Tail Risk Measures in Truncated Distributions and their Relationship with Inequality Indices

期刊论文
发表日期：

2026年4月24日

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Roghayeh Ghorbani Gholi Abad,
Mohammad Amini,
Gholam Reza Mohtashami Borzadaran,
Zahra Behdani

摘要

The application of truncated distributions is highly extensive and vital in practice. They are utilized in real-world operations (not merely in theory) by large insurance companies and banks. One of their applications is fitting models to loss data and estimating risk measures. In this article, an attempt has been made to transfer the concepts of tail risk measures to the family of truncated distributions and to establish a connection between these risk measures in the truncated case and other indices. Additionally, a redefinition of truncated risk measures has been presented based on risk measures in the non-truncated case and inequality indices. The proposed approach provides algebraically closed-form expressions for these risk measures, allowing us to study their properties and evaluate their connections with inequality indices. This redefinition provides a new perspective in the field of risk quantification. To enhance the empirical validity of the proposed formulas, a non-parametric bootstrap methodology with 1,000 replications is implemented to quantify parameter-estimation uncertainty. The bootstrap analysis provides confidence intervals for VaR, ES, and the shape parameter, confirming the numerical stability of the analytical formulas across truncation regimes. Comprehensive validation over truncated Pareto and truncated Exponential models shows that the closed-form expressions achieve extremely high accuracy, with average relative errors of $\varvec{7.5 \times 10^{-6}}$ $7.5 \times 10^{- 6}$ for the Pareto and $\varvec{2.8 \times 10^{-6}}$ $2.8 \times 10^{- 6}$ for the Exponential distribution. Additionally, the analytical approach yields substantial computational speed improvements–ranging from 17–31 $\times $ $\times$ for the Pareto model and 3–22 $\times $ $\times$ for the Exponential model–when compared with numerical integration methods. These results highlight the practicality, robustness, and computational efficiency of the proposed closed-form framework for risk-management applications.