Paper: arXiv 2508.12606

Authors: Hamza Hanbali, Jan Dhaene, Daniel Linders

Abstract

This paper considers the difference of stop-loss payoffs where the underlying is a difference of two random variables. The goal is to study whether the comonotonic and countermonotonic modifications of those two random variables can be used to construct upper and lower bounds for the expected payoff, despite the fact that the payoff function is neither convex nor concave. The answer to the central question of the paper requires studying the crossing points of the cdf of the original difference with the cdfs of its comonotonic and countermonotonic transforms. The analysis is supplemented with a numerical study of longevity trend bonds, using different mortality models and population data. The numerical study reveals that for these mortality-linked securities the three pairs of cdfs generally have unique pairwise crossing points. Under symmetric copulas, all crossing points can reasonably be approximated by the difference of the marginal medians, but this approximation is not necessarily valid for asymmetric copulas. Nevertheless, extreme dependence structures can give rise to bounds if the layers of the bond are selected to hedge tail risk. Further, the dependence uncertainty spread can be low if the layers are selected to hedge median risk, and, subject to a trade-off, to hedge tail risk as well.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical framework addressing dependence bounds for non-convex payoff functions via stochastic orders and CDF crossing points. It balances theoretical proof development with an extensive empirical evaluation using multiple mortality models across 21 country datasets.

Research Flowchart

  flowchart TD
    A["Research Question: Can comonotonic/countermonotonic modifications bound stop-loss payoffs on R.V. differences?"] --> B["Methodology: CDF Crossing-Point Analysis & Copula Theory"]
    C["Inputs: Mortality Models, Population Data, Longevity Trend Bonds"] --> D["Computational Process: Evaluate Payoffs & CDF Crossings (Symmetric vs. Asymmetric Copulas)"]
    B --> D
    D --> E["Findings: Unique crossing points exist; bounds depend on layer selection (tail vs. median risk)"]

Cited References (85)

  1. rsconnect: Deployment Interface for R Markdown Documents and Shiny Applications (2019)
  2. shiny: Web Application Framework for R (2021)
  3. On consistency of the omega ratio with stochastic dominance rules (2018)
  4. tikzDevice: R Graphics Output in LaTeX Format (2020) — link
  5. Copula Methods in Finance (2004)
  6. Reducing model risk via positive and negative dependence assumptions ( 2015)
  7. Analyzing mortality bond indexes via hierarchical forecast reconciliation (2019)
  8. The birth of the life market (2008)
and 77 more
  1. Hedging longevity risk when interest rates are uncertain (2011)
  2. Robust hedging of longevity risk (2013)
  3. The role of longevity bonds in optimal portfolios (2007)
  4. A note on generalized inverses (2013)
  5. Comonotonic bounds on the survival probabilities in the L ee- C arter model for mortality projection (2007)
  6. Estimates for the distribution function of a sum of two random variables when the marginal distributions are fixed (1981)
  7. Approximating a sum of random variables with a lognormal (2007)
  8. An International Comparison of Recent Trends in the Population Mortality (1998)
  9. Do actuaries believe in longevity deceleration? (2017)
  10. Longevity Risk and Private Pensions (2007)
  11. Longevity Risk in Portfofolios of Pension Annuities (2008)
  12. Longevity Risk and Capital Markets (2006)
  13. The simple analytics of a pooled annuity fund (2005)
  14. Equitable retirement income tontines: mixing cohorts without discrimination (2016)
  15. Survivor Funds (2016)
  16. Longevity-indexed life annuities (2011)
  17. Tonuity: A Novel Individual-Oriented Retirement Plan (2017)
  18. Systematic risk in long term insurance business: the need for a dynamic approach (2018)
  19. The New Life Market (2013)
  20. Longevity risk and capital markets: The 2015-16 update (2017)
  21. Modelling longevity bonds: A nalysing the S wiss R e K ortis bond (2015)
  22. Multi-population mortality models: A factor copula approach (2015) — link
  23. Keeping some skin in the game: how to start a capital market in longevity risk transfers (2014)
  24. Natural hedging of life and annuity mortality risks (2007)
  25. Modelling and forecasting the time series of U S mortality (1992)
  26. Multi-population mortality models: fitting, forecasting and comparisons (2017)
  27. Mortality density forecasts: An analysis of six stochastic mortality models (2011)
  28. A gravity model of mortality rates for two related populations (2011)
  29. Living with mortality: longevity bonds and other mortality-linked securities (2006)
  30. Pricing standardized mortality securitizations: A two-population model with transitory jump effects (2013)
  31. Identifiability, cointegration and the gravity model (2017) — link
  32. Modeling Period Effects in Multi-Population Mortality models: Applications to Solvency II (2014)
  33. Multi-Population Mortality Modelling with L ' e vy Processes (2016)
  34. Producing the Dutch and Belgian mortality projections: a stochastic multi-population standard (2017)
  35. Modeling multi-country mortality dependence and its application in pricing survivor index swaps - A dynamic copula approach (2015)
  36. Mortality dependence and longevity bond pricing: A dynamic factor copula mortality model with the G A S structure (2017) — link
  37. A discussion of parameter and model uncertainty in insurance (2000)
  38. Sharp distribution free lower bounds for spread options and the corresponding optimal subreplicating portfolios (2009)
  39. Distribution free bounds for spread options and market implied antimonotonicity gap (2008)
  40. Extensions of O hlin’s L emma with applications to optimal reinsurance structures (1993)
  41. Stochastic Orders (2007)
  42. Longevity hedging 101: a framework for longevity basis risk analysis and hedge effectiveness (2011)
  43. Measuring Basis Risk in Longevity Hedges (2011)
  44. Prices of state-contingent claims implicit in option prices (1978)
  45. Comonotonic bounds on the survival probabilities in the L ee- C arter model for mortality projections (2007)
  46. Simple characterizations of comonotonicity and countermonotonicity by extremal correlations. (2003)
  47. Static Super-Replicating Strategies for a Class of Exotic Options (2008)
  48. An overview of comonotonicity and its applications in finance and insurance (2011)
  49. Index Options: A Model Free Approach (2012)
  50. Static arbitrage bounds on basket option prices (2006)
  51. On Optimal Super-Hedging and Sub-Hedging strategies (2013)
  52. Pricing and Hedging Spread Options (2003)
  53. The concept of comonotonicity in actuarial science and finance : T heory (2002a)
  54. The concept of comonotonicity in actuarial science and finance : A pplication (2002b)
  55. Convex majorization with an application to the length of critical paths (1979)
  56. Comonotonicity and maximal stop-loss premiums (2000)
  57. Static-arbitrage upper bounds for the prices of basket options (2005)
  58. Solution of a statistical optimization problem by rearrangement methods (1983)
  59. A P D E approach for bounds of basket derivatives (2018)
  60. Actuarial Theory for Dependent Risks (2005)
  61. Dependency of risks and stop-loss order (1996)
  62. Inequalities for distributions with given marginals (1980)
  63. Longevity bonds: F inancial engineering, valuation and hedging (2006)
  64. On the pricing of longevity-linked securities (2010)
  65. On an optimization problem related to static super-replicating strategies (2015)
  66. Single- and cross-generation natural hedging of longevity and financial risk (2017)
  67. Model-independent price bounds for catastrophic mortality bonds (2021)
  68. A comparative study of two-population models for the assessment of basis risk in longevity hedges (2017)
  69. copula: Multivariate Dependence with Copulas (2017) — link
  70. Coherent mortality forecasts for a group of populations: an extension of the L ee- C arter model (2005)
  71. A common age effect model for the mortality of multiple populations (2015)
  72. Modelling longevity dynamics for pensions and annuity business (2009)
  73. On the difference of random variables (in progress)
  74. Age specific copula- AR - GARCH mortality models (2015)
  75. Still living with mortality: the longevity risk transfer market after one decade (2018)
  76. Strategy for increasing the global capacity for longevity risk transfer: developing transactions that attract capital markets investors (2014)
  77. R: A Language and Environment for Statistical Computing (2020)

0 of 85 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.