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Convolution Bounds on Quantile Aggregation

Jose Blanchet, Henry Lam, Yang Liu, Ruodu Wang

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.

Convolution Bounds on Quantile Aggregation

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.

Paper Structure

This paper contains 35 sections, 24 theorems, 117 equations, 4 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Motivating examples
  3. Robust risk management
  4. The O-ring model
  5. Notation and preliminaries
  6. Convolution bounds on RVaR aggregation
  7. Convolution bounds on quantile aggregation
  8. Convolution bounds
  9. Technical discussions
  10. Quantile aggregation at levels $0$ and $1$
  11. Approximation of the extremal dependence
  12. Extremal dependence structures: Monotone densities
  13. The general case and an approximation
  14. Dual formulation
  15. Numerical illustration
  16. ...and 20 more sections

Key Result

Theorem 1

Let $\boldsymbol \mu=(\mu_1,\dots,\mu_n)\in \mathcal{M}^n$. For any $t, s$ with $0 \leqslant t < t+s \leqslant 1$, Moreover, eq:prime1 holds as an equality in the following cases:

Figures (4)

  • Figure 1: Quantile functions for the sum. Left panel: decreasing densities ($n = 3$, quantile functions are $\frac{6}{5} r(t)$, $\frac{4}{5} r(t)$ and $\frac{4}{5} r(t)$, where $r(t) = -\log(\varepsilon + (1-\varepsilon) (1-t)), t \in [0, 1]$) and $\varepsilon =0.0001$); Right panel: increasing densities ($n = 3$, quantile functions are $-\frac{6}{5} r(1-t)$, $-\frac{4}{5} r(1-t)$ and $-\frac{4}{5} r(1-t)$, $t \in [0, 1]$.). The events $A_1, \dots, A_n, B$ are described in Theorem \ref{['th:qa-5']}.
  • Figure 2: Bounds for $\sup_{\nu \in \Lambda_3(\mu)} R^+_{0.9-s, s}(\nu)$. Left panel: $\mu =$ Pareto$(1,1/2)$ with a decreasing density $\frac{1}{2}x^{-3/2}, ~ x \in [1, \infty)$. Right panel: $\mu$ has an increasing density $\frac{5}{9}(101-x)^{-\frac{3}{2}}, \; x\in[1,100]$.
  • Figure 3: Bounds for $\sup_{\nu \in \Lambda_3(\mu) }q_t^+ (\nu)$. Left panel: $\mu =$ Pareto$(1,1/2)$ with a decreasing density $\frac{1}{2}x^{-3/2}, ~ x \in [1, \infty)$. Right panel: $\mu$ has an increasing density $\frac{5}{9}(101-x)^{-3/2}, \; x\in[1,100]$. In the left panel, "reduced dual bound", "reduced convolution bound", "convolution bound" and "RA" have the same curve, and for better visibility the "RA" curve is not plotted. In the right panel, "RA" and "convolution bound" have the same curve and "reduced dual bound" and "reduced convolution bound" have the same curve.
  • Figure 4: Performance of extremal dependence structures with settings in Table \ref{['table:subs1']}. In each panel, we plot the function of $h_i$ and the values of quantile aggregation provided by the convolution bound, the suboptimum $\boldsymbol{\beta}$ and the suboptimum $\boldsymbol{\gamma}$.

Theorems & Definitions (52)

  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Proposition 2: Reduced convolution bounds
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6: Convolution bounds at levels $0$ and $1$
  • ...and 42 more