Diversification Quotients: Quantifying Diversification via Risk Measures

56 Pages Posted: 7 Jul 2022 Last revised: 10 Mar 2024

See all articles by Xia Han

Xia Han

Nankai University - School of Mathematical Sciences and LPMC

Liyuan Lin

University of Waterloo - Department of Statistics and Actuarial Science

Ruodu Wang

University of Waterloo - Department of Statistics and Actuarial Science

Date Written: July 18, 2022

Abstract

We establish the first axiomatic theory for diversification indices using six intuitive axioms: non-negativity, location invariance, scale invariance, rationality, normalization, and continuity. The unique class of indices satisfying these axioms, called the diversification quotients (DQs), are defined based on a parametric family of risk measures. A further axiom of portfolio convexity pins down DQ based on coherent risk measures. DQ has many attractive properties, and it can address several theoretical and practical limitations of existing indices. In particular, for the popular risk measures Value-at-Risk and Expected Shortfall, the corresponding DQ admits simple formulas and it is efficient to optimize in portfolio selection. Moreover, it can properly capture tail heaviness and common shocks, which are neglected by traditional diversification indices. When illustrated with financial data, DQ is intuitive to interpret, and its performance is competitive against other diversification indices.

Keywords: Expected Shortfall, diversification quotient, diversification benefit, portfolios, quasi-convexity

Suggested Citation

Han, Xia and Lin, Liyuan and Wang, Ruodu, Diversification Quotients: Quantifying Diversification via Risk Measures (July 18, 2022). Available at SSRN: https://ssrn.com/abstract=4149069 or http://dx.doi.org/10.2139/ssrn.4149069

Xia Han

Nankai University - School of Mathematical Sciences and LPMC ( email )

No.94 Weijin Road, Nankai District
Tianjin, 300071
China

Liyuan Lin (Contact Author)

University of Waterloo - Department of Statistics and Actuarial Science ( email )

200 University Avenue West
Waterloo, Ontario N2L 3G1
Canada

Ruodu Wang

University of Waterloo - Department of Statistics and Actuarial Science ( email )

Waterloo, Ontario N2L 3G1
Canada

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