[Paper Review] Convolution Bounds on Quantile Aggregation

José Blanchet, Henry Lam+2 more|arXiv (Cornell University)|Jul 18, 2020

Risk and Portfolio Optimization4 citations

TL;DR

This paper introduces convolution bounds—a novel analytical framework for quantile aggregation under dependence uncertainty, leveraging inf-convolution of quantile-based risk measures. The method provides sharp, computationally efficient bounds for worst-case and best-case quantiles of sums of random variables with arbitrary dependence, unifying and extending prior results across diverse marginal distributions and settings.

ABSTRACT

Quantile aggregation with dependence uncertainty has a long history in probability theory with wide applications in finance, risk management, statistics, and operations research. Using a recent result on inf-convolution of quantile-based risk measures, we establish new analytical bounds for quantile aggregation which we call convolution bounds. Convolution bounds both unify every analytical result available in quantile aggregation and enlighten our understanding of these methods. These bounds are the best available in general. Moreover, convolution bounds are easy to compute, and we show that they are sharp in many relevant cases. They also allow for interpretability on the extremal dependence structure. The results directly lead to bounds on the distribution of the sum of random variables with arbitrary dependence. We discuss relevant applications in risk management and economics.

Motivation & Objective

• Address the challenge of quantile aggregation when dependence structures among random variables are unknown, aiming for robust and analytically tractable bounds.
• Unify and generalize existing analytical results on quantile aggregation under arbitrary dependence, including cases with discrete, continuous, or mixed marginals.
• Develop a framework that ensures sharpness under general conditions and enables interpretation of extremal dependence structures.
• Extend the approach to Range-Value-at-Risk (RVaR) and multiplicative risk models, broadening applicability in risk measurement.
• Provide computationally efficient bounds that are both theoretically rigorous and numerically accurate, even beyond proven sharpness regimes.

Proposed method

• Propose convolution bounds via inf-convolution of quantile-based risk measures, derived from recent duality results in risk sharing and distributional robustness.
• Use the transformation $\mu^{t-}$ to reduce the quantile aggregation problem to extremal dependence structures involving normalized distributions below the $t$-quantile.
• Establish lower bounds for the worst-case quantile using suprema over $t\Delta_n$ of weighted RVaR terms: $\sup_{\boldsymbol{\beta} \in t\Delta_n} \sum_{i=1}^n R_{1-\beta_i - \beta_0, \beta_0}(\mu_i)$.
• Demonstrate that the bounds are sharp under specific conditions, including increasing or decreasing densities below the quantile, or when total mass constraints are satisfied.
• Extend the framework to multiplicative risks by applying exponential transformations and proving analogous bounds for products of positive random variables.
• Leverage duality and convex analysis to derive bounds on the distribution of the sum of random variables under arbitrary dependence, enabling direct application in risk management.

Experimental results

Research questions

• RQ1 Can we derive a unified, analytical framework for worst-case and best-case quantile aggregation that subsumes all known analytical results under dependence uncertainty?
• RQ2 Under what conditions are the proposed convolution bounds sharp, and how do they compare to existing bounds in terms of accuracy and computational efficiency?
• RQ3 Can the convolution bounds be extended to Range-Value-at-Risk (RVaR) and multiplicative risk models, and what are the corresponding extremal dependence structures?
• RQ4 How do the bounds perform numerically in cases where theoretical sharpness cannot be proven, and what insights do they offer on extremal dependence?
• RQ5 Can the bounds be used to compute tight estimates for the distribution of the sum of random variables with arbitrary dependence, even when the total risk is not fixed?

Key findings

• The convolution bounds unify and generalize all known analytical results in quantile aggregation, subsuming prior bounds from Embrechts, Puccetti, Wang, and Rüschendorf.
• The bounds are sharp in multiple cases: when $n \leq 2$, when marginals have increasing or decreasing densities below the quantile, or when the total mass of the lower tails satisfies $\sum_{i=1}^n \mu_i[q_0^+(\mu_i), q_t^-(\mu_i)) \leq t$.
• The lower bound for the worst-case quantile is given by $\sup_{\boldsymbol{\beta} \in t\Delta_n} \sum_{i=1}^n R_{1-\beta_i - \beta_0, \beta_0}(\mu_i)$, which is tight under the stated conditions.
• For multiplicative risks, the bound $\exp\left\{\sup_{\boldsymbol{\beta} \in t\Delta_n} \sum_{i=1}^n R_{1-\beta_i - \beta_0, \beta_0}(\mu_i \circ \exp)\right\}$ is exact when $x f_i(x)$ is monotonic beyond the $t$-quantile.
• The infimum of the $t$-quantile over all dependence structures is always attainable, meaning $\inf_{\nu \in \Lambda(\boldsymbol{\mu})} q_t^-(\nu) = q_t^-(\nu_-)$ for some $\nu_- \in \Lambda(\boldsymbol{\mu})$, ensuring existence of extremal dependence.
• Numerical performance suggests high accuracy even beyond theoretical sharpness regimes, indicating strong practical utility in risk management and operations research applications.