Empirical estimator of diversification quotient

TL;DR

This paper develops and analyzes empirical estimators for the Diversification Quotient (DQ), a measure of portfolio diversification, demonstrating their convergence, robustness, and practical applicability in financial risk management.

Contribution

It introduces empirical estimators for DQ based on VaR and ES, proving their asymptotic properties and robustness, and compares them favorably to the diversification ratio.

Findings

Empirical DQ estimators converge and are asymptotically normal.

DQ estimators are more robust than the diversification ratio under various distributions.

Confidence intervals for DQ can be constructed using bootstrap methods.

Abstract

The Diversification Quotient (DQ), introduced by Han et al. (2025), is a recently proposed measure of portfolio diversification that quantifies the reduction in a portfolio's risk-level parameter attributable to diversification. Grounded in a rigorous theoretical framework, DQ effectively captures heavy tails, common shocks, and enhances efficiency in portfolio optimization. This paper further explores the convergence properties and asymptotic normality of empirical DQ estimators based on Value at Risk (VaR) and Expected Shortfall (ES), with explicit calculation of the asymptotic variance. In contrast to the diversification ratio (DR) proposed by Tasche (2007), which may exhibit diverging asymptotic variance due to its lack of location invariance, the DQ estimators demonstrate greater robustness under various distributional settings. We further verify their asymptotic properties under elliptical distributions through simulation, and construct confidence intervals for DQ estimates using AR-GARCH models with a residual-based bootstrap on real financial data. These results establish a solid statistical foundation for applying DQ in financial risk management and decision-making.