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Dynamic Bivariate Mortality Modelling

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Abstract

The dependence structure of the life statuses plays an important role in the valuation of life insurance products involving multiple lives. Although the mortality of individuals is well studied in the literature, their dependence remains a challenging field. In this paper, the main objective is to introduce a new approach for analyzing the mortality dependence between two individuals in a couple. It is intended to describe in a dynamic framework the joint mortality of married couples in terms of marginal mortality rates. The proposed framework is general and aims to capture, by adjusting some parametric form, the desired effect such as the “broken-heart syndrome”. To this end, we use a well-suited multiplicative decomposition, which will serve as a building block for the framework to relate the dependence structure and the marginals, and we make the link with existing practice of affine mortality models. Finally, given that the framework is general, we propose some illustrative examples and show how the underlying model captures the main stylized facts of bivariate mortality dynamics.

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Notes

  1. The probability measure \(\mathbb {P}\) can be either interpreted as the historical measure or can refer to a pricing measure depending on the considered context.

  2. This can refer to a best estimate assumption on the evolution of mortality or a reference mortality Barrieu et al. (2012).

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Acknowledgements

Ying Jiao thanks Beijing International Center for Mathematical Research for visiting support and hospitality during this work. The work of Yahia Salhi has been supported by the CY Initiative of Excellence (grant “Investissements d’Avenir” ANR-16-IDEX-0008), Project “EcoDep” PSI-AAP2020-0000000013 as well as the BNP Paribas Cardif Chair “New Insurees, Next Actuaries” (NINA). The views expressed in this document are the authors owns and do not necessarily reflect those endorsed by BNP Paribas Cardif. Finally, the authors gratefully thank the anonymous referee for the constructive comments and recommendations which definitely help to improve the readability and quality of the paper.

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Appendix

Appendix

1.1 Examples of Affine Processes

1.1.1 Vasicek Process

We suppose that stochastic processes \(Y_t^i\) where \(i\in \{0,1,2\}\) follow Vasicek model whose dynamics are given by

$$\begin{aligned} dY_t^i = (b_i + c_i Y_t^i)dt + \sigma _idW_t^i, \ \ Y^i(0)= y_i, \end{aligned}$$

where \(b_i, c_i\) and \(\sigma _i\) are some constant parameters. Then, for \(i=1,2\), the Riccati ODE system Eq. (8) in Subsect. 3.2 can be rewritten as

$$\begin{aligned} {\left\{ \begin{array}{ll} {\beta }'_i(t) = -c_i\beta _i(t), &{} \text { with } \beta _i(T^*) = t_i,\\ {\gamma }'_i(t) = -\frac{1}{2}\sigma _i^2\beta _i^2(t) - b_i\beta _i(t),&{} \text { with }\gamma _i(T^*) = 0. \\ \end{array}\right. } \end{aligned}$$

Then, we can easily derive the explicit parameters as follows

$$\begin{aligned} \beta _i(t)&= t_i e^{c_i(T^*-t)} , \end{aligned}$$
(19)
$$\begin{aligned} \gamma _i(t)&= -\frac{b_it_i}{c_i}\left( 1- e^{c_i(T^*-t)}\right) -\frac{\sigma _i^2t_i^2}{4c_i}\left( 1- e^{2c_i(T^*-t)}\right) . \end{aligned}$$
(20)

For \(i=0\), we have the following ODEs

$$\begin{aligned} {\left\{ \begin{array}{ll} {\beta }'_0(t) = -c_0\beta _0(t), &{} \text {with } \beta _0(T^*) = \rho _1t_1+ \rho _2t_2,\\ {\gamma }'_0(t) = -\frac{1}{2}\sigma _0^2\beta _0^2(t) - b_0\beta _0(t), &{} \text {with } \gamma _0(T^*)= 0, \end{array}\right. } \end{aligned}$$

which similarly give arise the explicit solutions

$$\begin{aligned} \beta _0(t)&= e^{c_0(T^*-t)}(\rho _1t_1+ \rho _2t_2) , \end{aligned}$$
(21)
$$\begin{aligned} \gamma _0(t)&= -\frac{b_0(\rho _1t_1+ \rho _2t_2)}{c_0}\left( 1- e^{c_0(T^*-t)}\right) -\frac{\sigma _0^2(\rho _1t_1+ \rho _2t_2)^2}{4c_0}\left( 1- e^{2c_0(T^*-t)}\right) . \end{aligned}$$
(22)

Then we easily have the joint (conditional) survival probability Eq. (9). We should, therefore, substitute \(\gamma _i(0)\) and \(\beta _i(0)\) by the above corresponding forms in Eqs. (19), (20), (21) and (22).

1.1.2 Cox-Ingersoll-Ross Process

We suppose that processes \(Y_t^i\) where \(i\in \{0,1,2\}\) follow Cox-Ingersoll-Ross (CIR) model and are described as solution of the following SDE

$$\begin{aligned} dY_t^i = a_i(b_i - Y_t^i)dt + \sigma _i\sqrt{Y_t^i}dW_t^i, \ \ Y^i(0)= y_i, \ \ \end{aligned}$$

where \(b_i, c_i\) and \(\sigma _i\) are some constant parameters. Then, for \(i=1,2\), the Riccati ODEs system Eq. (8) in Subsect. 3.2 are given as follows

$$\begin{aligned} {\left\{ \begin{array}{ll} {\beta }'_i(t) = a_i\beta _i(t) -\frac{1}{2}\sigma _i^2\beta _i(t)^2, &{} \text {with } \beta _i(T^*) = t_i,\\ {\gamma }'_i(t) = -a_ib_i\beta _i(t), &{} \text {with } \gamma _i(T^*) = 0. \\ \end{array}\right. } \end{aligned}$$

Then we can easily have

$$\begin{aligned} \beta _i(t)&= \frac{2a_i}{\sigma _i^2}\frac{1}{ 1- \left( 1-\frac{2a_i}{\sigma _i^2t_i}\right) e^{a_i(T^*-t)}}, \end{aligned}$$
(23)
$$\begin{aligned} \gamma _i(t)&= -\frac{2a_ib_i}{\sigma _i^2} \log \left( 1- \sigma _i^2t_i \frac{1- e^{-a_i(T^*-t)}}{2a_i} \right) .\end{aligned}$$
(24)

For \(i=0\), the Riccati ODE system Eq. (8) can be rewritten as

$$\begin{aligned} {\left\{ \begin{array}{ll} {\beta }'_0(t) = a_0\beta _0(t) -\frac{1}{2}\sigma _0^2\beta _0(t)^2, &{} \text {with } \beta _0(T^*) = \rho _1t_1+ \rho _2t_2,\\ {\gamma }'_0(t) = -a_0b_0\beta _0(t), &{} \text {with } \gamma _0(T^*) = 0. \end{array}\right. } \end{aligned}$$

Then we have

$$\begin{aligned} \beta _0(t)&= \frac{2a_0}{\sigma _0^2}\frac{1}{ 1- \left( 1-\frac{2a_0}{\sigma _0^2(\rho _1t_1+ \rho _2t_2)}\right) e^{a_0(T^*-t)}}, \end{aligned}$$
(25)
$$\begin{aligned} \gamma _0(t)&= -\frac{2a_0b_0}{\sigma _0^2}\log \left( 1- \frac{1- e^{-a_0(T^*-t)}}{\frac{2a_0}{\sigma _0^2(\rho _1t_1+ \rho _2t_2)}} \right) . \end{aligned}$$
(26)

Similarly as for the Vasicek case, we can also derive the joint (conditional) survival probability Eq. (9) using the explicit form of \(\gamma _i(0)\) and \(\beta _i(0)\) in Eqs. (23), (24), (25) and (26).

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Jiao, Y., Salhi, Y. & Wang, S. Dynamic Bivariate Mortality Modelling. Methodol Comput Appl Probab 24, 917–938 (2022). https://doi.org/10.1007/s11009-022-09955-0

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