Right-truncated Archimedean and related copulas

TL;DR

This paper derives analytical formulas for right-truncated copulas, especially Archimedean types, enabling easier analysis of tail dependence and sampling, with applications in insurance and risk management.

Contribution

It introduces a general formula for right-truncated copulas, characterizes right-truncated Archimedean copulas as tilted versions, and derives properties and limits relevant for practical applications.

Findings

Right-truncated Archimedean copulas are analytically tractable.

Limiting Clayton copula obtained as truncation points approach zero.

Characterization of right-truncated Archimax and nested Archimedean copulas.

Abstract

The copulas of random vectors with standard uniform univariate margins truncated from the right are considered and a general formula for such right-truncated conditional copulas is derived. This formula is analytical for copulas that can be inverted analytically as functions of each single argument. This is the case, for example, for Archimedean and related copulas. The resulting right-truncated Archimedean copulas are not only analytically tractable but can also be characterized as tilted Archimedean copulas. This finding allows one, for example, to more easily derive analytical properties such as the coefficients of tail dependence or sampling procedures of right-truncated Archimedean copulas. As another result, one can easily obtain a limiting Clayton copula for a general vector of truncation points converging to zero; this is an important property for (re)insurance and a fact already known in the special case of equal truncation points, but harder to prove without aforementioned characterization. Furthermore, right-truncated Archimax copulas with logistic stable tail dependence functions are characterized as tilted outer power Archimedean copulas and an analytical form of right-truncated nested Archimedean copulas is also derived.