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On the valuation of life insurance policies for dependent coupled lives

Kira Henshaw, Cedric H. A. Koffi, Olivier Menoukeu Pamen, Raghid Zeineddine

Abstract

In this paper, we investigate a complex variation of the standard joint life annuity policy by introducing three distinct contingent benefits for the surviving member(s) of a couple, along with a contingent benefit for their beneficiaries if both members pass away. Our objective is to price this innovative insurance policy and analyse its sensitivity to key model parameters, particularly those related to the joint mortality framework. We employ the $QP$-rule (described in Section \ref{secgenset}), which combines the real-world probability measure $P$ for mortality risk with risk-neutral valuation under $Q$ for financial market risks. The model enables explicit pricing expressions, computed using efficient numerical methods. Our results highlight the interdependent risks faced by couples, such as broken-heart syndrome, providing valuable insights for insurers and policyholders regarding the pricing influences of these factors.

On the valuation of life insurance policies for dependent coupled lives

Abstract

In this paper, we investigate a complex variation of the standard joint life annuity policy by introducing three distinct contingent benefits for the surviving member(s) of a couple, along with a contingent benefit for their beneficiaries if both members pass away. Our objective is to price this innovative insurance policy and analyse its sensitivity to key model parameters, particularly those related to the joint mortality framework. We employ the -rule (described in Section \ref{secgenset}), which combines the real-world probability measure for mortality risk with risk-neutral valuation under for financial market risks. The model enables explicit pricing expressions, computed using efficient numerical methods. Our results highlight the interdependent risks faced by couples, such as broken-heart syndrome, providing valuable insights for insurers and policyholders regarding the pricing influences of these factors.

Paper Structure

This paper contains 21 sections, 6 theorems, 171 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The general setup
  3. Preliminaries
  4. Financial and Bond markets
  5. The $QP$-rule
  6. Mortality and surrender models
  7. Mortality model
  8. Surrender model
  9. Pricing the insurance policy
  10. Price of the guaranteed minimum accumulation benefit
  11. Price of the surrender benefit
  12. Price of the death benefit
  13. Proofs
  14. Proof of Theorem \ref{['Guaranteed benefit']}
  15. Proof of Theorem \ref{['surrender']}
  16. ...and 6 more sections

Key Result

Proposition 1

There exist a unique probability measure, denoted $Q\odot P$, on $(\Omega, \mathcal{H})$, such that $Q\odot P=Q$ on $\mathcal{F}$ and for all $A\in \mathcal{H}$ it holds that $Q\odot P(A|\mathcal{F})=P(A|\mathcal{F})$. This measure $Q\odot P$ satisfies for any random variable $X\geq 0$:

Figures (6)

  • Figure 6.1: $M(u,T), T = 3\text{ years }(K=2)$. Left panel : real part of $M(u,T)$. Right panel: imaginary part of $M(u,T)$. Parameters: table \ref{['table: Params_VA_contract']}.
  • Figure 6.2: $N(v,T), T = 3\text{ years }(K=2)$. Left panel : real part of $N(v,T)$. Right panel: imaginary part of $N(v,T)$. Parameters: table \ref{['table: Params_VA_contract']}.
  • Figure 6.3: Sensitivity analysis : the parameters $(\beta,C)$ Top panels: left-hand-side—GMAB; right-hand-side: SB. Bottom panels: left-hand-side—DB; right-hand-side—VA. Maturity: T = 10 years.
  • Figure 6.4: Sensitivity analysis : the parameters $(\beta,\delta)$ Top panels: left-hand-side—GMAB; right-hand-side: SB. Bottom panels: left-hand-side—DB; right-hand-side—VA. Maturity: T = 10 years.
  • Figure 6.5: Sensitivity analysis : the mortality parameter $(\epsilon_{x1},\epsilon_{x2})$ Maturity: T = 10 years.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Proposition 1
  • Theorem 1: jevtic2017joint, Theorem 2.1.
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • proof