Paper: arXiv 2510.01971

Authors: Takaaki Koike

Abstract

Dependence among multiple lifetimes is a key factor for pricing and evaluating the risk of joint life insurance products. The dependence structure can be exposed to model uncertainty when available data and information are limited. We address robust pricing and risk evaluation of joint life insurance products against dependence uncertainty between two lifetimes. We first show that, for some class of standard contracts, the risk evaluation based on a distortion risk measure is monotone with respect to the concordance order of the underlying copula. Based on this monotonicity, we then study the most conservative and anti-conservative risk evaluations for this class of contracts. We prove that the bounds for the mean, Value-at-Risk and Expected Shortfall are computed by combinations of linear programs when the uncertainty set is defined by a norm-ball centered around a reference copula. Our numerical analysis reveals that the sensitivity of the risk evaluation against the choice of the copula differs depending on the risk measure and the type of the contract, and our proposed bounds can improve the existing bounds based on the available information.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a sophisticated theoretical framework establishing monotonicity properties for distortion risk measures under copula dependence uncertainty and formulating risk bounds via linear programming. While mathematically rigorous with complete derivations, the empirical side is limited to numerical demonstrations rather than extensive real-world actuarial backtesting.

Research Flowchart

  flowchart TD
    A[Goal: Robust Risk Evaluation of Joint Life Insurance under Dependence Uncertainty] --> B[Inputs: Reference Copula, Norm-Ball Uncertainty Set & Life Contracts]
    B --> C[Theoretical Analysis: Prove Monotonicity of Distortion Risk Measures w.r.t. Concordance Order]
    C --> D[Optimization Framework: Formulate Bounds via Combinations of Linear Programs]
    D --> E[Numerical Analysis: Compute Mean, VaR, & Expected Shortfall Bounds]
    E --> F[Key Findings: Bounds Improve Existing Baselines & Sensitivity Varies by Contract/Risk Measure]

Cited References (48)

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and 40 more
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  35. we compare the following bounds of ϱ h ( C ) for h = h E , h VaR α VaR and h ES α ES
  36. iii) Improved FH bounds with given values on a region : ϱ h ( B S ′ ,Q ) and ϱ h ( A S ′ ,Q ) in
  37. ii) Fr´echet-Hoeffding (FH) bounds : ϱ h ( W ) and ϱ h ( M )
  38. (iv) Improved FH bounds with a given value of Kendall’s tau : we also compute the bounds on the set of copulas with Kendall’s tau given by τ = 0 . 49;
  39. ii) Second-to-die annuity (S2DA) : ( t X , t Y ) = ( 65 , 62 ) and L = K ∨ k = 1 a v k , a = 1 . 169, which is increasing in K ∨
  40. (i) ε -ball bounds : ϱ h ( C S ,ε ( C ref )) and ϱ h ( C S ,ε ( C ref ))

0 of 48 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.