Estimation and Asymptotic Properties of the Adjusted Standard-Deviatile for Heavy-Tailed Risks

Core Concepts

This research paper introduces the adjusted standard-deviatile (deviatile), a novel risk measure designed to address limitations of the variantile in capturing extreme risks, particularly for heavy-tailed distributions.

Abstract

• Bibliographic Information: Chen, H., Mao, T., & Yang, F. (2024). Estimation of the Adjusted Standard-deviatile for Extreme Risks. arXiv preprint arXiv:2411.07203v1.
• Research Objective: To propose and analyze a new risk measure, the adjusted standard-deviatile (deviatile), as a modification of the variantile, for improved assessment of extreme risks in heavy-tailed distributions.
• Methodology: The authors derive first- and second-order asymptotic expansions for the deviatile based on extreme value theory. They propose two estimators for the deviatile: an intermediate-level estimator based on the first-order expansion and an extreme-level estimator obtained by extrapolating the intermediate-level estimator. Asymptotic normality of both estimators is established for independent and identically distributed (i.i.d.) data and β-mixing time series. The performance of the estimators is evaluated through simulations using Pareto and Student's t-distributions, including scenarios with serial dependence modeled by a GARCH(1,1) process.
• Key Findings: The deviatile exhibits superior performance compared to the variantile in capturing extreme risks, particularly for heavy-tailed distributions. The proposed estimators demonstrate good asymptotic properties and finite-sample performance in estimating the deviatile at both intermediate and extreme levels. The second-order asymptotic expansion improves the accuracy of the deviatile estimation, especially for heavier-tailed distributions.
• Main Conclusions: The deviatile provides a more accurate and robust measure of extreme risks compared to the variantile, especially for heavy-tailed distributions commonly encountered in finance and insurance. The proposed estimation methods offer practical tools for estimating the deviatile at various levels of risk, including extreme levels beyond the observed data range.
• Significance: This research contributes to the field of risk management by introducing a new risk measure specifically designed for heavy-tailed distributions and providing reliable estimation methods. The findings have practical implications for financial institutions, insurance companies, and regulators in assessing and managing extreme risks more effectively.
• Limitations and Future Research: The study primarily focuses on univariate risks and assumes a specific class of heavy-tailed distributions. Future research could explore extensions to multivariate risks and other heavy-tailed distributions. Further investigation into the optimal selection of the threshold parameter (k) for the estimators is also warranted.

Stats

The numerical value of γ∗, the critical tail index for comparing deviatile and quantile, is approximately 0.2135.
For a Pareto(α, 1) distribution, the true value of dev0.9996 is 17.8283 when α = 3 and is 3.7609 when α = 5.
For a Student’s tα-distribution, the true value of dev0.9996 is 19.3173 when α = 3 and is 7.2585 when α = 5.
The parameters for the GARCH(1,1) process used in the simulation are: degree of freedom of the Student’s t-distribution is 6.54 and α0 = 0.0181, α1 = 0.1476, β0 = 0.8497.

Quotes

Key Insights Distilled From

Estimation of the Adjusted Standard-deviatile for Extreme Risks

by Haoyu Chen, ... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.07203.pdf

Deeper Inquiries

How does the deviatile compare to other risk measures, such as Expected Shortfall (ES) and Median Shortfall (MS), in terms of sensitivity to different tail characteristics and estimation robustness?

The deviatile, Expected Shortfall (ES), and Median Shortfall (MS) are all risk measures that aim to capture the tail risk of a distribution, but they differ in their sensitivity to different tail characteristics and estimation robustness:

Sensitivity to Tail Characteristics:

Deviatile: As a modification of the variantile, the deviatile is specifically designed to be sensitive to the right tail of the distribution, particularly for heavy-tailed distributions. This is achieved by scaling the positive and negative deviations from the expectile differently, giving more weight to the positive deviations.
Expected Shortfall (ES): ES, also known as Conditional Value-at-Risk (CVaR), represents the average of the worst (1-τ)% losses, making it more sensitive to the magnitude of extreme losses than the deviatile. ES is known to be a coherent risk measure, satisfying desirable properties like subadditivity.
Median Shortfall (MS): MS corresponds to the median of the distribution beyond the VaR at a given confidence level. It is less influenced by the extreme outliers in the tail compared to ES and deviatile.

Estimation Robustness:

Deviatile: The estimation of the deviatile relies on estimating both the expectile and the tail index, which can be challenging for heavy-tailed distributions. The choice of the tail index estimator (e.g., Hill estimator) and the threshold parameter can significantly impact the estimation accuracy.
Expected Shortfall (ES): ES estimation also involves tail estimation and can be sensitive to the choice of the tail index estimator and threshold. However, ES is generally considered more robust than the deviatile due to its reliance on averaging over tail losses, reducing the impact of individual extreme observations.
Median Shortfall (MS): MS is considered the most robust of the three measures as it is less sensitive to the extreme values in the tail. However, this robustness comes at the cost of being less informative about the magnitude of potential extreme losses.
In summary:

The deviatile is a right-tail-focused risk measure, particularly suitable for heavy-tailed distributions, but its estimation can be less robust than ES.
ES provides a comprehensive view of the tail risk by averaging over tail losses, offering a balance between sensitivity and robustness.
MS is the most robust but least informative about the magnitude of extreme losses.
The choice of the most appropriate risk measure depends on the specific application and the desired balance between sensitivity to tail characteristics and estimation robustness.

Could the deviatile be extended to incorporate risk aversion preferences, allowing for a more flexible and tailored risk assessment based on specific risk appetites?

Yes, the deviatile can potentially be extended to incorporate risk aversion preferences, enabling a more flexible and tailored risk assessment aligned with specific risk appetites. Here are a few potential approaches:

Weighted Deviatile: Introduce a risk aversion parameter into the deviatile formula to adjust the relative weights assigned to positive and negative deviations from the expectile. A higher risk aversion parameter would amplify the contribution of larger positive deviations, reflecting a greater aversion to extreme losses.

Utility-Based Deviatile: Instead of using the squared deviations in the deviatile calculation, incorporate a utility function that reflects the decision-maker's risk preferences. A concave utility function would penalize losses more heavily than gains, capturing risk aversion. The deviatile could then be defined as the minimum expected utility loss.

Expectile-Based Risk Aversion: Leverage the relationship between the deviatile and the expectile. Adjust the confidence level (τ) used in the expectile calculation based on the risk aversion level. A higher risk aversion would correspond to a higher τ, leading to a more conservative estimate of the deviatile.

Generalized Loss Function: Modify the loss function used in the definition of the deviatile to incorporate risk aversion. This could involve using a loss function that is more sensitive to losses than gains or introducing a risk aversion parameter into the loss function.
These extensions would allow the deviatile to be tailored to specific risk preferences, providing a more nuanced and insightful risk assessment for decision-making. Further research is needed to explore these extensions formally and develop robust estimation methods for the risk-averse deviatile.

How can the concept of the deviatile be applied to non-financial contexts, such as environmental risk assessment or public health modeling, where extreme events can have significant consequences?

The concept of the deviatile, with its focus on capturing extreme risks, holds significant potential for application in non-financial contexts where extreme events can have substantial consequences. Here are a few examples:

Environmental Risk Assessment:

Extreme Weather Events: The deviatile can be used to assess the risk associated with extreme weather events like hurricanes, floods, and droughts. By modeling the distribution of rainfall, wind speed, or other relevant variables, the deviatile can provide insights into the potential magnitude of extreme events, aiding in disaster preparedness and infrastructure planning.
Pollution Levels: In monitoring air or water pollution, the deviatile can help assess the risk of exceeding critical thresholds. By analyzing historical pollution data, the deviatile can estimate the likelihood and potential severity of extreme pollution events, informing mitigation strategies and public health advisories.

Public Health Modeling:

Disease Outbreaks: The deviatile can be applied to model the spread of infectious diseases, particularly for outbreaks with the potential for rapid transmission. By incorporating factors like infection rates and population density, the deviatile can estimate the risk of a large-scale outbreak, guiding resource allocation and intervention strategies.
Hospital Capacity Planning: The deviatile can be used to assess the risk of hospitals exceeding their capacity due to unexpected surges in patient admissions, such as during pandemics or natural disasters. By modeling patient arrival rates and lengths of stay, the deviatile can inform capacity planning and resource allocation to mitigate the risk of overwhelming healthcare systems.

Other Applications:

Earthquake Risk: Assessing the potential magnitude of earthquakes and their impact on infrastructure.
Cybersecurity: Estimating the potential damage from cyberattacks and informing cybersecurity investments.
Wildfire Risk: Modeling the potential spread and intensity of wildfires to guide prevention and response efforts.
In these applications, the deviatile's ability to capture the behavior of heavy-tailed distributions, where extreme events are more likely, makes it a valuable tool for risk assessment and decision-making. By quantifying the potential magnitude of extreme events, the deviatile can help stakeholders make informed decisions to mitigate risks and enhance resilience.