Interconnected nodes with quantile highlights and financial graph overlay.

Quantile Aggregation: How to Navigate Uncertainty in Risk and Economics

Beau Callahan in Business & Economy December 2025 • 5 min read.

"Discover analytical bounds that unify results and offer insights into complex systems."

In fields like finance, risk management, statistics, and operations research, quantile aggregation under conditions of dependence uncertainty is a long-standing challenge. Imagine needing to estimate the total risk of a portfolio where you know the individual assets' risk profiles but not how they might interact. This is where quantile aggregation comes into play, helping to find the possible values of quantiles of an aggregate variable. This aggregate could represent a total risk or the completion time of a task, making it crucial for informed decision-making.

Quantile aggregation is essentially about understanding the range of possible outcomes when combining several uncertain quantities. Instead of knowing exactly how these quantities depend on each other, we only have information about their individual distributions. More formally, the problem involves finding the highest and lowest possible quantiles of the sum of these variables, given their marginal distributions. For example, if you're summing random variables X1 through Xn, you know the distribution of each variable separately, but not how they correlate. Quantile aggregation helps determine the possible range of values for a specific quantile of the total sum.

This field draws on a rich history in probability theory, with applications spanning diverse areas. However, due to the complexity arising from unspecified dependence structures, finding analytical solutions has been notoriously difficult. In this article, we explore a recent advancement that offers a new perspective on this problem: convolution bounds. This approach not only provides new analytical bounds but also unifies existing results, offering a more comprehensive understanding of quantile aggregation.

Unlocking New Insights with Convolution Bounds

Interconnected nodes with quantile highlights and financial graph overlay.

Convolution bounds, derived from recent research on inf-convolution of quantile-based risk measures, represent a significant step forward in tackling the challenges of quantile aggregation. They offer a way to establish new analytical bounds, which unify every analytical result available in quantile aggregation and enlighten our understanding of these methods. These bounds are the best available in general, are easy to compute, and sharp in many relevant cases. The convolution bounds also allow for interpretability on the extremal dependence structure.

This method introduces a class of bounds on Range-Value-at-Risk (RVaR) based on inf-convolution formulas. RVaR encompasses key regulatory risk measures like Value-at-Risk (VaR) and Expected Shortfall (ES), providing a versatile tool for risk assessment. These convolution bounds enjoy multifaceted advantages such as being applicable to both quantile and RVaR aggregations, combining different existing sharpness results of quantile aggregation and some new cases into a unified form, leading to tractable extremal dependence structures for interpretation or approximation, and they are computationally convenient and efficient.

Versatile Applications: Applicable to both quantile and RVaR aggregations.
Unified Framework: Combines existing sharpness results and new cases.
Tractable Extremal Structures: Enables interpretation and approximation.
Computational Efficiency: Convenient and efficient computation.

Because RVaR includes the two regulatory risk measures, Value-at-Risk (VaR) and the Expected Shortfall (ES, also known as CVaR), as special cases, the results on RVaR give rise to useful bounds on quantile aggregation problems. As our main contributions, convolution bounds can provide by far the most convenient and sharpest theoretical results on quantile aggregation in a wide range of practical settings, and they can be applied to any marginal distributions, discrete, continuous, or mixed. Convolution bounds enjoy multifaceted advantages. They can be applied to both the quantile and RVaR aggregations; they combine different existing sharpness results of quantile aggregation and some new cases into a unified form; they lead to tractable extremal dependence structures for interpretation or approximation; and they are computationally convenient and efficient.

The Future of Understanding Uncertainty

The theoretical difficulty in quantile aggregation leaves ample room for future adventures and challenges. For instance, the sharpness of convolution bounds under general conditions, other than those in Theorems 1, 2, A.1 and A.2, is an open question. For the interested reader, we connect our results to the theory of joint mixability in Appendix E, where many questions remain to be open. Additional information on the dependence structure, other than the marginal distributions, can be incorporated in the quantile aggregation problem, and it usually leads to highly challenging questions; see e.g., Bernard et al. (2017a,b) and Bartl et al. (2022). In view of the broad appearance of quantile aggregation, its application domain includes many problems in economics, finance, risk management, statistics, and scheduling, in addition to the two applications discussed Section 2. We mention some applications in Appendix G, on which many relevant questions warrant thorough future investigation.

About this Article -

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This article is based on research published under:

DOI-LINK: https://doi.org/10.48550/arXiv.2007.0932,

Title: Convolution Bounds On Quantile Aggregation

Subject: q-fin.rm math.oc math.pr q-fin.mf

Authors: Jose Blanchet, Henry Lam, Yang Liu, Ruodu Wang

Published: 17-07-2020

Everything You Need To Know

What is quantile aggregation and why is it important in risk management?

Quantile aggregation is a method used to determine the range of possible outcomes when combining several uncertain quantities. In risk management, it is crucial because it helps estimate the total risk of a portfolio even when the interactions between individual assets are unknown. Instead of knowing exactly how quantities depend on each other, quantile aggregation uses only information about their individual distributions to find the possible values of quantiles of an aggregate variable like total risk or the completion time of a task, enabling informed decision-making in uncertain environments. This contrasts with knowing the exact correlations, which is often not available.

How do convolution bounds improve upon existing methods for quantile aggregation?

Convolution bounds, derived from recent research on inf-convolution of quantile-based risk measures, offer significant improvements. They provide new analytical bounds that unify every analytical result available in quantile aggregation. These bounds are the best available in general and are easy to compute. They allow for interpretability on the extremal dependence structure, which is not always possible with other methods. Compared to traditional methods, convolution bounds provide a more comprehensive understanding of quantile aggregation and are more versatile because they apply to both quantile and Range-Value-at-Risk (RVaR) aggregations, including Value-at-Risk (VaR) and Expected Shortfall (ES).

What are the key advantages of using convolution bounds for Range-Value-at-Risk (RVaR) aggregation?

Convolution bounds offer several advantages when applied to RVaR aggregation. First, they are applicable to both quantile and RVaR aggregations, providing a unified framework. Second, they combine existing sharpness results of quantile aggregation with new cases. Third, they lead to tractable extremal dependence structures, which facilitates interpretation and approximation. Finally, they are computationally convenient and efficient, which makes them practical for use in various applications. Because RVaR includes Value-at-Risk (VaR) and Expected Shortfall (ES), the results on RVaR give rise to useful bounds on quantile aggregation problems.

In what practical settings can convolution bounds be applied, and what types of marginal distributions are supported?

Convolution bounds are applicable in a wide range of practical settings, particularly in economics, finance, risk management, statistics, and scheduling. They can be applied to any marginal distributions, whether discrete, continuous, or mixed. This versatility allows them to be used in various real-world scenarios where uncertain quantities need to be aggregated. The bounds are particularly useful when dealing with Value-at-Risk (VaR) and Expected Shortfall (ES) due to their inclusion within the RVaR framework.

What are some open questions and future research directions related to quantile aggregation?

The field of quantile aggregation still has many open questions and potential directions for future research. One key area is the sharpness of convolution bounds under general conditions beyond those already established. Another direction involves incorporating additional information on the dependence structure beyond marginal distributions, which often leads to highly challenging questions. Furthermore, the broad applicability of quantile aggregation across economics, finance, risk management, statistics, and scheduling suggests that thorough investigation into specific applications is warranted. Exploring these areas could lead to even more refined and practical methods for navigating uncertainty in complex systems.