Asymptotic Analysis for Spectral Risk Measures Parameterized by Confidence Level

TL;DR

This paper analyzes the asymptotic behavior of spectral risk measures as the confidence level approaches one, revealing how tail heaviness of variables influences risk contributions and demonstrating non-monotonicity in risk differences.

Contribution

It extends asymptotic analysis of risk measures to spectral risk measures, detailing their behavior as confidence levels approach one, especially considering tail thickness of involved variables.

Findings

Risk difference behavior depends on tail heaviness of variables.

When one variable's tail is much thicker, risk difference peaks below confidence level 1.

Spectral risk measures exhibit non-monotonic risk differences with increasing confidence levels.

Abstract

We study the asymptotic behavior of the difference $\Delta \rho ^{X, Y}_\alpha := \rho _\alpha (X + Y) - \rho _\alpha (X)$ as $\alpha \rightarrow 1$, where $\rho_\alpha $ is a risk measure equipped with a confidence level parameter $0 < \alpha < 1$, and where $X$ and $Y$ are non-negative random variables whose tail probability functions are regularly varying. The case where $\rho _\alpha $ is the value-at-risk (VaR) at $\alpha $, is treated in Kato (2017). This paper investigates the case where $\rho _\alpha $ is a spectral risk measure that converges to the worst-case risk measure as $\alpha \rightarrow 1$. We give the asymptotic behavior of the difference between the marginal risk contribution and the Euler contribution of $Y$ to the portfolio $X + Y$. Similarly to Kato (2017), our results depend primarily on the relative magnitudes of the thicknesses of the tails of $X$ and $Y$. We also conducted a numerical experiment, finding that when the tail of $X$ is sufficiently thicker than that of $Y$, $\Delta \rho ^{X, Y}_\alpha $ does not increase monotonically with $\alpha$ and takes a maximum at a confidence level strictly less than $1$.