Higher moments under dependence uncertainty with applications in insurance

TL;DR

This paper develops bounds for higher mixed moments under dependence uncertainty and explores their impact on actuarial measures like expected shortfall and annuities, extending financial moment analysis to actuarial science.

Contribution

It introduces analytical bounds for mixed moments with known marginals but unknown dependence, and studies their effects on key actuarial quantities.

Findings

Bounds for mixed moments are derived and dependence structures identified.

Higher-order moments influence expected shortfall and annuity premiums monotonically.

Marginal expected shortfall may remain unaffected by higher moments depending on dependence.

Abstract

Recent studies have highlighted the significance of higher-order moments - such as coskewness - in portfolio optimization within the financial domain. This paper extends that focus to the field of actuarial science by examining the impact of these moments on key actuarial applications. In the first part, we derive analytical lower and upper bounds for mixed moments of the form $\mathbb{E}(X_1X_2^d)$, where $X_i \sim F_i$ for $i=1,2$, assuming known marginal distributions but unspecified dependence structure. The results are general and applicable to arbitrary marginals and positive integer orders $d$, and we also identify the dependence structures that attain these bounds. These findings are then applied to bound centered mixed moments and explore their mathematical properties. The second part of the paper investigates the influence of higher-order centered mixed moments on key actuarial quantities, including expected shortfall (ES), marginal expected shortfall (MES), and life annuity valuation. Under a copula-based mixture model, we show that coskewness and other odd-order mixed moments exhibit a monotonic relationship with both ES and annuity premiums. However, the effect on MES is more nuanced and may remain invariant depending on the underlying dependence structure.