Asymptotics of higher-order conditional tail moments for convolution-equivalently distributed losses

TL;DR

The paper studies the asymptotics of higher-order conditional tail moments $\rho_k(x,y;\beta,\zeta)$ for two portfolios with dependent losses following convolution-equivalent distributions. It proves two robustness-focused theorems: (i) in the light-tailed convolution-equivalent setting, $\rho_k(x,y;\beta,\zeta)$ scales as $x^{\beta}$ (or $x^{\beta}/n$ when $\zeta=0$ and $\beta\in\mathbb{N}_+$); and (ii) in the heavy-tailed regularly varying setting, $\rho_k(x,y;\beta,\zeta)$ scales as $\frac{\alpha}{\alpha-\beta}x^{\beta}$ (or the $x^{\beta}/n$ variant when $\zeta=0$ under mild conditions). Crucially, the asymptotics are shown to be independent of the dependence strength parameter $\theta$, with the joint dependence modeled by a FGM copula. The authors validate the theory via numerical simulations in both regimes, illustrating accurate approximations for large thresholds and providing practical insights for MES/ES-type risk measures under dependence.

Abstract

This paper investigates the asymptotic behavior of higher-order conditional tail moments, which quantify the contribution of individual losses in the event of systemic collapse. The study is conducted within a framework comprising two investment portfolios experiencing dependent losses that follow convolution-equivalent distributions. The main results are encapsulated in two theorems: one addressing light-tailed losses with convolution-equivalent distributions and the other focusing on heavy-tailed losses with regularly varying distributions. Both results reveal that the asymptotic behavior remains robust regardless of the strength of dependence. Additionally, numerical simulations are performed under specific scenarios to validate the theoretical results.

Figures (6)

• Figure 1: Comparison between the asymptotic values for $\rho_1(x,x;\beta, \zeta)$ and its empirical values with $\theta=0.5$, $\zeta=0$, $\nu=1$, and $\mu=1$.
• Figure 2: Comparison between the asymptotic values for $\rho_1(x,x;\beta, \zeta)$ and its empirical values with $\beta=2$, $\zeta=0$, $\nu=1$, and $\mu=1$.
• Figure 3: Comparison between the asymptotic values for $\rho_1(x,x;\beta, \zeta)$ and its empirical values with $\beta=2$, $\theta=0.5$, $\zeta=0$, and $\gamma=\frac{1}{2}$.
• Figure 4: Comparison between the asymptotic values for $\rho_1(x,x;\beta, \zeta)$ and its empirical values with $\theta=0.5$, $\zeta=\frac{1}{3}$, and $\alpha=2.8$.
• Figure 5: Comparison between the asymptotic values for $\rho_1(x,x;\beta, \zeta)$ and its empirical values with $\beta=2$, $\zeta=\frac{1}{3}$, and $\alpha=2.8$.

Theorems & Definitions (11)

• Theorem 2.1
• Theorem 2.2
• Remark 2.1
• Remark 2.2
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4
• Remark 4.1
• Remark 4.2