Paper: arXiv 2508.16600

Authors: Carole Bernard, Jinghui Chen, Steven Vanduffel

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents strong theoretical proofs and analytical bounds for higher-order mixed moments under dependence uncertainty, extending existing copula literature into actuarial science. While it includes calibration using demographic data from Statistics Canada, it primarily functions as a theoretical and mathematical framework rather than an empirical backtest-heavy study.

Research Flowchart

  flowchart TD
    A["Research Goal: Impact of higher-order mixed moments (e.g., coskewness) under dependence uncertainty in actuarial science"] --> B["Inputs: Known Marginal Distributions F1, F2 & Unspecified Dependence Structure"]
    
    B --> C["Derive Analytical Bounds for Mixed Moments E(X1 * X2^d) & Identify Extremal Copulas"]
    
    C --> D["Apply Bounds to Centered Mixed Moments & Evaluate Copula-Based Mixture Model"]
    
    D --> E["Actuarial Applications: Expected Shortfall (ES), Marginal ES (MES), & Life Annuity Valuation"]
    
    E --> F["Outcomes: Odd-order moments monotonically affect ES & Annuities; MES effects are nuanced/invariant"]