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Robust risk measures: an averaging approach

Marcelo Righi, Rodrigo Targino

Abstract

We develop an averaging approach to robust risk measurement under payoff uncertainty. Instead of taking a worst-case value over an uncertainty neighborhood, we weight nearby payoffs more heavily under a chosen metric and average the baseline risk measure. We prove continuity in the neighborhood radius and provide a stable large-radius behavior. In Banach lattices, the approach leads to a convex risk measure and under separability of the space, a dual representation through a penalty term based on an inf-convolution taken over a Gelfand integral constraint. We also relate our veraging to aggregation at the distribution and quantile levels of payoffs, obtaining dominance and coincidence results. Numerical illustrations are conducted to verify calibration and sensitivity.

Robust risk measures: an averaging approach

Abstract

We develop an averaging approach to robust risk measurement under payoff uncertainty. Instead of taking a worst-case value over an uncertainty neighborhood, we weight nearby payoffs more heavily under a chosen metric and average the baseline risk measure. We prove continuity in the neighborhood radius and provide a stable large-radius behavior. In Banach lattices, the approach leads to a convex risk measure and under separability of the space, a dual representation through a penalty term based on an inf-convolution taken over a Gelfand integral constraint. We also relate our veraging to aggregation at the distribution and quantile levels of payoffs, obtaining dominance and coincidence results. Numerical illustrations are conducted to verify calibration and sensitivity.

Paper Structure

This paper contains 5 sections, 12 theorems, 173 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Asymptotics in r
  3. rhoMuR as a risk measure
  4. Aggregate then evaluate
  5. The Gaussian measure and numerical examples

Key Result

Lemma 1

Let $A\in\mathcal{B}$. Then the map is continuous in $(C(\mathbb{R}_+),\|\cdot\|_\infty)\times(ca(\bar{B}(X,r)),\|\cdot\|_{TV})$ at $(\varphi,\gamma)$ such that $\int_{A} \varphi(d(X, Z))d\gamma(Z) > 0$ and $\rho$ is integrable in $A$ with respect to $\gamma$.

Figures (2)

  • Figure 1: Dominance chain for the averaging measure under the Normal-Gamma base measure with $\mu_X=0$, $\sigma_X=1$, $a=0.95$, $\lambda=2$, $\alpha_{\mathrm{NG}}=25$, $k=4$, $N=10^6$. Five quantities are plotted as functions of the Wasserstein radius $r$: baseline ES $\rho(X)$ (red dashed), averaging measure $\rho_{\mu,r}(X)$ (solid black), quantile-aggregated ES $\rho(X_{\bar{\mu},r})$ (dot-dashed blue, numerically coinciding with $\rho_{\mu,r}(X)$ in agreement with \ref{['thm:avg']}), distribution-aggregated ES $\rho(X_{\mu,r})$ (dotted green), and worst-case ES $\rho^{WC}(X)$ (dashed orange, closed form \ref{['eq:WC_closed_form']}).
  • Figure 2: Sensitivity of the averaging measure $\rho_{\mu,r}(X)$ to tuning parameters ($\mu_X=0$, $\sigma_X=1$, $a=0.95$, $N=10^6$). Panel (a): varying prior concentration with $\lambda=2$ fixed; concentrated prior ($\alpha_{\mathrm{NG}}=25$, $k=4$, solid black) versus diffuse prior ($\alpha_{\mathrm{NG}}=5$, $k=1$, dashed blue). Horizontal dotted lines mark the respective large-$r$ limits $\bar{\rho}(X)$ of \ref{['thm:r']}. Panel (b): varying kernel decay $\lambda\in\{0.5,2,8\}$ with the concentrated prior fixed. Baseline ES $\rho(X)$ shown as red dashed line in both panels.

Theorems & Definitions (44)

  • Definition 1
  • Remark 1
  • Remark 2
  • Example 1: Discrete perturbation measures
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Remark 3
  • Example 2: Gaussian measure
  • ...and 34 more