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Expansion and Bounds for the Bias of Empirical Tail Value-at-Risk

Nadezhda Gribkova, Jianxi Su, Mengqi Wang

TL;DR

This work tackles the finite-sample negative bias of the empirical Tail Value-at-Risk estimator by deriving a leading-term asymptotic expansion $B_n=\ell_n+o(n^{-1})$ with $\ell_n=-\frac{p}{2nf(\xi_p)}$, clarifying how tail density at the high quantile governs bias magnitude. It also provides an explicit upper bound on the bias, incorporating a Hölder continuity parameter $\gamma$ of the quantile function to ensure finite-sample guarantees. The authors validate the framework via Pareto-based simulations and a Danish fire loss data example, showing the leading-term approximation tracks the true bias well (especially when tail density is well-estimated) and that the upper bound reliably bounds the bias, even in finite samples. Practically, these results enable bias-aware TVaR estimation and informed bias-reduction decisions without heavy resampling, with clear guidance on estimating the required constants and choosing probability levels. Overall, the paper contributes a rigorous, implementable bias-analysis toolkit for empirical TVaR in risk management settings.

Abstract

Tail Value-at-Risk (TVaR) is a widely adopted risk measure playing a critically important role in both academic research and industry practice in insurance. In data applications, TVaR is often estimated using the empirical method, owing to its simplicity and nonparametric nature. The empirical TVaR has been explicitly advocated by regulatory authorities as a standard approach for computing TVaR. However, prior literature has pointed out that the empirical TVaR estimator is negatively biased, which can lead to a systemic underestimation of risk in finite-sample applications. This paper aims to deepen the understanding of the bias of the empirical TVaR estimator in two dimensions: its magnitude as well as the key distributional and structural determinants driving the severity of the bias. To this end, we derive a leading-term approximation for the bias based on its asymptotic expansion. The closed-form expression associated with the leading-term approximation enables us to obtain analytical insights into the structural properties governing the bias of the empirical TVaR estimator. To account for the discrepancy between the leading-term approximation and the true bias, we further derive an explicit upper bound for the bias. We validate the proposed bias analysis framework via simulations and demonstrate its practical relevance using real data.

Expansion and Bounds for the Bias of Empirical Tail Value-at-Risk

TL;DR

This work tackles the finite-sample negative bias of the empirical Tail Value-at-Risk estimator by deriving a leading-term asymptotic expansion with , clarifying how tail density at the high quantile governs bias magnitude. It also provides an explicit upper bound on the bias, incorporating a Hölder continuity parameter of the quantile function to ensure finite-sample guarantees. The authors validate the framework via Pareto-based simulations and a Danish fire loss data example, showing the leading-term approximation tracks the true bias well (especially when tail density is well-estimated) and that the upper bound reliably bounds the bias, even in finite samples. Practically, these results enable bias-aware TVaR estimation and informed bias-reduction decisions without heavy resampling, with clear guidance on estimating the required constants and choosing probability levels. Overall, the paper contributes a rigorous, implementable bias-analysis toolkit for empirical TVaR in risk management settings.

Abstract

Tail Value-at-Risk (TVaR) is a widely adopted risk measure playing a critically important role in both academic research and industry practice in insurance. In data applications, TVaR is often estimated using the empirical method, owing to its simplicity and nonparametric nature. The empirical TVaR has been explicitly advocated by regulatory authorities as a standard approach for computing TVaR. However, prior literature has pointed out that the empirical TVaR estimator is negatively biased, which can lead to a systemic underestimation of risk in finite-sample applications. This paper aims to deepen the understanding of the bias of the empirical TVaR estimator in two dimensions: its magnitude as well as the key distributional and structural determinants driving the severity of the bias. To this end, we derive a leading-term approximation for the bias based on its asymptotic expansion. The closed-form expression associated with the leading-term approximation enables us to obtain analytical insights into the structural properties governing the bias of the empirical TVaR estimator. To account for the discrepancy between the leading-term approximation and the true bias, we further derive an explicit upper bound for the bias. We validate the proposed bias analysis framework via simulations and demonstrate its practical relevance using real data.
Paper Structure (9 sections, 4 theorems, 109 equations, 7 figures)

This paper contains 9 sections, 4 theorems, 109 equations, 7 figures.

Table of Contents

  1. Introduction
  2. A commentary on empirical Tail Conditional Expectation
  3. Main results
  4. Simulation analysis
  5. Leading-term approximation
  6. Negative-bias upper bound
  7. Real-data illustration
  8. Conclusions
  9. Technical proofs

Key Result

Lemma 3.1

Suppose that condition c1 holds, then we have where Furthermore,

Figures (7)

  • Figure 4.1: Comparison of the exact TVaR bias (red circle-cross markers), the theoretical leading-term approximation (blue star markers), and the estimated leading-term approximation (boxplots), with fixed parameters $p=95\%$, $\alpha=5$, and varying sample sizes $n\in \{100,\, 300,\, 500,\, 700,\, 900\}$.
  • Figure 4.2: Comparison of the exact TVaR bias (red circle-cross markers), the theoretical leading-term approximation (blue star markers), and the estimated leading-term approximation (boxplots), with fixed parameters $n=500$, $\alpha=5$, and varying probability levels $p\in \{80,\, 85,\, 90,\, 95,\, 97.5\}\%$.
  • Figure 4.3: Comparison of the exact TVaR bias (red circle-cross markers), the theoretical leading-term approximation (blue star markers), and the estimated leading-term approximation (boxplots), with fixed parameters $n=500$, $p=95\%$, and varying values of the shape parameter $\alpha\in \{3,\, 5,\, 10,\, 20,\, 30\}$.
  • Figure 4.4: Comparison of the exact TVaR bias with the theoretical negative-bias upper bound, with fixed parameters $p = 95\%$, $\alpha = 3$, and $h = 0.05$, across varying sample sizes $n \in \{100,\,300,\,500,\,700,\,900\}$ and different choices of the slack parameter $\delta \in \{0.01,\,0.05,\,0.1\}$.
  • Figure 4.5: Comparison of the exact TVaR bias (red circle-cross markers), the theoretical negative-bias upper bound (blue diamond markers), and the estimated negative-bias upper bound (boxplots), with fixed parameters $p = 95\%$, $\alpha = 3$, $\delta = 0.05$, and $h = 0.05$, across varying sample sizes $n \in \{100,\,300,\,500,\,700,\,900\}$.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Lemma 3.1
  • Remark 3.1
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Theorem 3.2
  • Remark 3.6
  • Remark 3.7
  • ...and 4 more