Two hybrid models for dependent death times of couple: a common shock approach

Abstract

We combine two recent credit risk models with the Marshall–Olkin setup to capture the dependence structure of bivariate survival functions. The main advantage of this approach is to handle fatal shock events in the dependence structure since these two credit risk models allow one to match the time of death of an individual with a catastrophe time event. We also provide a methodology for adding other sources of dependency to our approach. In such a setup, we derive the no-arbitrage prices of some common life insurance products for coupled lives. We demonstrate the performance of our method by investigating Sibuya's dependence function. Calibration is done on the data of joint life contracts from a Canadian company.

Keywords:

• Intensity-based models
• dependence structure
• fatal shock events
• joint life insurance

Acknowledgments

This paper has benefited from the valuable comments of one anonymous reviewer, whom the authors wish to thank. The remaining errors are the authors' only responsibility.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Throughout the paper x and y denote the ages of the husband and wife, respectively.

2 Here, we have considered the independent case just for simplicity, but we could also consider, in the same spirit, the dependent case for the joint survival function using a copula function.

3 In our setup, we have $\mathbb{P}({\tau }^x>s|{\mathcal{F}}_t)=\mathbb{E}[e^{-{\mathrm{\Gamma }}_s^x}|{\mathcal{F}}_t^W]\mathbb{E}[e^{-K_s}|{\mathcal{F}}_t^K]$. Analogous calculations can be done for the marginal probability $\mathbb{P}({\tau }^y>s|{\mathcal{F}}_t)$.

4 The authors wish to thank the Society of Actuaries, through the courtesy of Edward (Jed) Frees and Emiliano A. Valdez, for making available the data in this paper.

5 This functional form of the marginal survival probability comes from assuming a stochastic intensity of the form $d{\mu }_h(u)=a_h{\mu }_h(u)+{\sigma }_h\sqrt{{\mu }_h(h)}\mathrm{d}{W}_h(u)$, with $a,\sigma >0$. A sufficient condition for $S_h(u;t)$ to be a valid survival function is ${\sigma }^2<2\mathrm{dc}$. Additional details about this model can be found in Luciano et al. (2008), Luciano & Vigna (2005).

6 Luciano et al. (2008) refers to this model of association as the 4.2.20 Nelsen copula function. Originally proposed in Nelsen (2007), a detailed study can be found in Spreeuw (2006). The choice of this particular model of association is because it produces the best fit in a range of several Archimedean copulas for the data used in this paper (Luciano et al. 2008).

Additional information

Funding

Domenico De Giovanni gratefully acknowledges financial support from the PNRR project ‘Italian Ageing, Age-It’ (PE0000015 - CUP H73C22000900006) and Ministry of University and Research of Italy.