Measuring Multivariate Negative Dependence in Higher Dimensions Using Copula-based Approach

Determination of aggregate risk as a function of dependence amongst random variables is a vital step in modelling any insurance or financial portfolio.The standard independence assumption simplifies calculations but has the downside of overstating or understating aggregate risk.Compared to comonotonicity, negative dependence has received little attention due to difficulty in extending its bivariate results into multivariate cases despite its natural risk minimization properties and potential to create internal hedging.In this paper, we investigate properties of counter-monotonicity (Fréchet-Hoeffding lower bound) in higher dimensions (d ≥ 3) by restricting characteristics of individual random variables X i .Using Archimedean copulas, we introduce a measure of multivariate negative dependence derived from d-dimensional hypervolumes of independent and counter-monotonic distribution functions.The choice of integrals facilitates natural boundaries of the measure within (0, 1), enabling smooth derivation of its properties.In addition, we establish a moment-based version for raw statistics and a numerical example illustrating an application using an arbitrary seven-dimensional portfolio of health insurance risks.Analysts can apply the findings of this study within decentralized insurance or finance characterized by fewer risks that do not obey the Central Limit Theorem.The measure facilitates accurate measurement of portfolio countermovements and selects appropriate hedging strategies for dependent portfolios.