# Risk concentration under second order regular variation

## TL;DR

This paper investigates the asymptotic behavior of risk concentration measures in portfolios with heavy-tailed risks under second order regular variation, providing convergence rates and relationships between multivariate and univariate cases.

## Contribution

It derives the asymptotic rate of convergence for risk concentration measures under second order regular variation and explores the link between multivariate and univariate second order regular variation.

## Key findings

- Established the asymptotic convergence rate of risk concentration measures.
- Demonstrated the relationship between multivariate and univariate second order regular variation.
- Provided illustrative examples for the theoretical results.

## Abstract

Measures of risk concentration and their asymptotic behavior for portfolios with heavy-tailed risk factors is of interest in risk management. Second order regular variation is a structural assumption often imposed on such risk factors to study their convergence rates. In this paper, we provide the asymptotic rate of convergence of the measure of risk concentration for a portfolio of heavy-tailed risk factors, when the portfolio admits the so-called second order regular variation property. Moreover, we explore the relationship between multivariate second order regular variation for a vector (e.g., risk factors) and the second order regular variation property for the sum of its components (e.g., the portfolio of risk factors). Results are illustrated with a variety of examples.