2. Model formulation

Let $\tau _i$ be the death time of the breadwinner $i,$ for $i=1, 2$ . We assume that the marginal probability distribution functions of $\tau _i$ is given by

\begin{align*} F_i(t)=P(\tau _i\le t)=1-e^{-\int ^t_0\lambda _i(s)ds}, \end{align*}

where $\lambda _i$ is the force of mortality. It is assumed that the random variables $\tau _1$ and $\tau _2$ follow a joint probability distribution $F(\cdot, \cdot )$ with a density function of $f(\cdot, \cdot )$ . We use $T_1$ to denote the time of the first death of the couple, that is, $T_1=\tau _1\wedge \tau _2$ . We also use $F_{T_1}(\!\cdot\!)$ and $f_{T_1}(\!\cdot\!)$ to denote the probability distribution function and density function of $T_1$ , respectively.

In our model, the dynamics of the risk-free asset $B_t$ and risky asset $S_t$ are assumed to be

\begin{align*} \frac {dB(t)}{B(t)}&=r(t)dt,\\ \frac {dS(t)}{S(t)}&= \mu (t) dt+ \sigma _S(t) dZ_{S}(t), \end{align*}

where $r(t)$ is the risk-free interest rate, $\mu (t)$ is the appreciation rate, $\sigma _{S}(t)$ is the volatility, and $Z_{S}(t)$ is the Brownian motion with respect to risky asset price.

Based on Kung & Yang (Reference Kung and Yang2020) and Kraft & Munk (Reference Kraft and Munk2011), we assume that household can invest in real estate asset at a unit price (e.g., unit can be defined as the price per square meter). The dynamics of the unit house price are

\begin{align*} \frac {dH(t)}{H(t)}=(r(t)+\lambda _H\sigma _H(t)-\zeta )dt+\sigma _H(t)({\rho }_{HS}dZ_{S}(t)+{\rho }_HdZ_{H}(t)), \end{align*}

where $\sigma _H(t)$ is the house price volatility, $\lambda _H$ is the Sharpe ratio of the unit house price, ${\rho }_H=\sqrt {1-{\rho }^2_{HS}}$ , $\rho _{HS}$ is the constant correlation between house and stock, and $\zeta$ is the imputed rent or the cost of holding house unit, and $Z_{H}(t)$ is the Brownian motion with respect to house price. By using this setting, the household has the option to take a short position in housing assets, allowing them to access housing wealth through housing equity withdrawal using reverse mortgages or home reversion plans.

The rent of a housing unit is assumed to have a constant relationship $\upsilon$ with the house price, where $\upsilon \gt 0$ . Hence, the return of the household investing and renting out a unit house we have

\begin{align*} \frac {dH(t)+\upsilon H(t)dt}{H(t)}=\left [r(t)+\hat {\lambda }_H\sigma _{H}(t)\right ]dt+\sigma _H(t)\left [{\rho }_{HS}dZ_{S}(t)+{\rho }_{H}dZ_{H}(t)\right ], \end{align*}

where $\hat {\lambda }_H=\lambda _H+\frac {\upsilon -\zeta }{\sigma _{H}(t)}$ .

We use $\pi (t)$ to denote the proportions of the wealth invested in risky assets. $\phi _{1}(t)$ is the units of housing units owned. $\phi _{2}(t)$ is the units of housing units rented. $\phi _{3}(t)$ is the units invested in REIT at time $t$ . Housing consumption is defined as $\phi _{4}(t)=\phi _{1}(t)+\phi _{2}(t)$ and housing investment $\phi _{5}(t)=\phi _{1}(t)+\phi _{3}(t)$ .

For each breadwinner, denoted as $i$ where i can be either 1 or 2, $c_i(t)$ and $k_i(t)$ are the consumption amount and life insurance premium. Let $Y_i(t)$ be the deterministic income flow for breadwinner $i$ during period $[0, T]$ $(i=1,2)$ , where $T$ represents the time when the last survivor of the couple passes away. Hence, the wealth dynamics is

\begin{align*} dX(t) &=\{[r(t)+(\mu (t)-r(t))\pi (t)]X(t)-\phi _{4}(t)\upsilon H(t)\}dt+[\pi (t)X(t)\sigma _S(t)\\ &\quad +\phi _{5}(t)H(t)\rho _{HS}\sigma _H(t)]dZ_{S}(t)+\phi _{5}(t)H(t)\rho _{H}\sigma _H(t) dZ_{H}(t)\\ &\quad -[\mathbf {1}_{\{t\lt \tau _1\}}\left (c_{1}(t)+k_{1}(t)-Y_{1}(t)\right )-\mathbf {1}_{\{t\lt \tau _2\}}\left (c_{2}(t)+k_{2}(t)-Y_{2}(t)\right )]dt. \end{align*}

We assume a time-additive Cobb–Douglas style utility for the consumption, $U_1(c(,\phi _{4})=\frac {(c^\beta \phi _4^{1-\beta })^{\gamma }}{\gamma }$ and the power utility for the bequest $U_2(x)=\frac {x^{\gamma }}{\gamma }$ for the terminal wealth, where $\beta$ is the relative weighting between housing, $0\lt \beta \lt 1$ , and consumption, and $\gamma$ is the relative risk aversion, $0\lt \gamma \lt 1$ .

We use $\delta$ to denote the discount factor. $\mathcal {A}_t$ is the set of all admissible strategies $U\triangleq (c_1,c_2, k_1,k_2,\pi, \phi _{4},\phi _{5})$ . For an arbitrary admissible control $u\in \mathcal {A}_t$ , the value function of this optimal problem is as follows:

\begin{align*} V(t,x,h)&=\max _{u \in \mathcal {A}_t}E\Big \{\int ^{\tau _1 \wedge T}_t w_1 e^{-\delta s}U_1(c_1(s), \phi _{4}(s))ds +\int ^{\tau _2\wedge T}_t w_2 e^{-\delta s}U_1(c_2(s),\phi _{4}(s))ds\\ &\quad +w_3 \mathbf {1}_{\{\tau _1\vee \tau _2\le T\}}e^{-\delta (\tau _1\vee \tau _2)}U_2\left (X(\tau _1 \vee \tau _2)+\sum ^2_{i=1}\frac {k_i(\tau _i, X(\tau _i))}{\theta _i(\tau _i)}\mathbf {1}_{\{\tau _i=\tau _1\vee \tau _2\}}\right )\\ &\quad +w_4 \mathbf {1}_{\{\tau _1\vee \tau _2\gt T\}}e^{-\delta T}U_2(X(T)) \Big \}, \end{align*}

where $w_i\ge 0$ , $i=1,2,3,4$ satisfies the condition of $\sum ^4_{i=1} w_i=1$ which ranks the relative importance of the utility type, $X(t)=x$ and $H(t)=h$ . For $i = 1, 2$ , the insurance benefit $\frac {k_i(\tau _i, X(\tau _i))}{\theta _i(\tau _i)}$ is assumed to be paid to the beneficiary upon the death of the insured at time $ \tau _i$ . Following Kung & Yang (Reference Kung and Yang2020), we define $l$ as the loading factor and $\theta _i=(1+l)\lambda _i$ . In this context, $k_i(\tau _i, X(\tau _i))$ represents the life insurance premium. When $k_i(\tau _i, X(\tau _i))$ is positive, it indicates behavior consistent with purchasing life insurance. Conversely, when $k_i(\tau _i, X(\tau _i))$ is negative, it reflects behavior associated with purchasing a variable life annuity. The utility derived from the bequest and terminal wealth will only be realized upon the death of the last surviving individual. To ensure the utility function remains well-defined, we follow the approach in Kung & Yang (Reference Kung and Yang2020) by introducing a loading factor that may result in a zero-insurance condition.

3. Optimal results

Initially, we assume the presence of a breaking heart effect, where the time of death, denoted as $\tau _i$ , for the bereaved individual is influenced by the passing of their partner. This phenomenon highlights how deeply connected our emotions and mental well-being are to our physical health, especially in close relationships. When someone we love passes away, it can set off a chain reaction in our bodies and minds that affects how healthy we are overall. In our analysis, we first address the optimization problem in the scenario where one of the couple passes away, recognizing that the optimization strategy in the case where both breadwinners are alive may be informed by insights gained from the former scenario.

3.1 The optimization problem after the first death

We firstly consider the optimization problem after one of the couple dies, that is, $T_1=\tau _1\lt \tau _2$ or $T_1=\tau _2\lt \tau _1$ . We simplify the notation $ V_i(t, x, h)$ for the value function to $ V_i$ .

After $T_1$ , the alive bread winner $i$ has the value function of this optimal problem is as follows:

\begin{align*} V_i&=\max E\Big \{\int ^{\tau _i \wedge T}_t w_i e^{-\delta (s-t)}U_1(c_i(s), \phi _{4}(s))ds \\ &\quad +w_3 \mathbf {1}_{\{\tau _i\le T\}}e^{-\delta (\tau _i-t)}U_2\left (X_i(\tau _i)+\frac {k_i(\tau _i, X_i(\tau _i))}{\theta _i(\tau _i)}\right )\\ &\quad +w_4 \mathbf {1}_{\{\tau _i\gt T\}}e^{-\delta (T-t)}U_2(X_i(T)) \Big \}, \end{align*}

where $X_i(\!\cdot\!)$ is given by

\begin{align*} dX_i(t)&=\{[r(t)+(\mu (t)-r(t))\pi _{i}(t)]X_i(t)-\phi _{ 4_i}(t)\upsilon H(t)\}dt+[\pi _{i}(t)X_i(t)\sigma _S(t)\\ &\quad +\phi _{5_i}(t)H(t)\rho _{HS}\sigma _H(t)]dZ_{S}(t)+\phi _{5_i}(t)H(t)\rho _{H}\sigma _H(t) dZ_{H}(t)-\left (c_{i}(t)+k_{i}(t)-Y_{i}(t)\right )dt. \end{align*}

The proof of the following proposition can be found in Appendix A.

Proposition 3.1. The value function is given by

\begin{equation*} V_i=\frac {1}{\gamma }g_i(t,h)^{1-\gamma }(x+b_i(t))^{\gamma }, \end{equation*}

where $ b_i(t)= \int ^{\omega }_{t}e^{-\int ^s_t(r(u)+\lambda _i(u))du}Y_i(s)ds$ represents the human capital for breadwinner $ i$ . The optimal strategies are given by

(3.1) \begin{align} c^*_i(t)&=\frac {\beta }{(1-\beta )g_i(t,h)}\upsilon \eta _i h^k (x+b_i(t)),\nonumber \\ k^*_i(t,x)&=\theta _i(t)\left [\left (\frac {\gamma }{w_3}\right )^{\frac {1}{\gamma -1}}g_i(t,h)^{-1}(x+b_i(t))-x\right ],\nonumber \\ \pi _i^*(t)&=\frac {-(\mu (t)-r(t))}{(1-\rho ^2_{HS})\sigma ^2_S(t)x(\gamma -1)}(x+b_i(t)),\nonumber \\ \phi ^*_{4_i}(t)&=\frac {\eta _i h^{q-1}}{g_i(t,h)}(x+b_i(t)),\nonumber \\ \phi ^*_{5_i}(t)&=\left [\frac {\rho _{HS}(\mu (t)-r(t))}{(1-\rho ^2_{HS})\sigma _S(t)\sigma _H(t)h(\gamma -1)}+\frac {g_{ih}(t,h)}{g_i(t,h)}\right ](x+b_i(t)). \end{align}

where

\begin{align*} g_i(t,h)&=\alpha _1h^k \int ^T_te^{\alpha _2(s)}ds+\alpha _3h^{\gamma (\beta +k-1)}\int ^T_t e^{\alpha _4(s)}ds+\alpha _5,\\ \alpha _1&=\frac {\eta _i \upsilon }{(1-\beta )(\gamma -1)},\end{align*}

\begin{align*} \alpha _2(s)&=\frac {d_1(s-T)}{(1-\gamma )\alpha _1},\\\alpha _3&=-\frac {w_1}{1-\gamma }\left (\frac {\beta }{1-\beta }\right )^{\beta \gamma }\eta ^{\gamma }_i\upsilon ^{\beta \gamma },\\ \alpha _4(s)&=\frac {d_2(s-T)}{(1-\gamma )\alpha _3},\\ \alpha _5&=w_4^{\frac {1}{1-\gamma }}+\left (\frac {\gamma -1}{d_3}w_3^{\frac {1}{1-\gamma }}+w_4^{\frac {1}{1-\gamma }}\right )(e^{\frac {d_1(T-t)}{\gamma -1}}-1),\\ d_1&=\alpha _1d_3+\alpha _1(\gamma -1)(\gamma -\zeta +\lambda _H\sigma _H(t))k-\frac {1}{2}\alpha _1(\gamma -1)\sigma ^2_H(t)k(k-1),\\ d_2&=\alpha _3d_3+\alpha _3(\gamma -1)(\gamma -\zeta +\lambda _H\sigma _H(t))\gamma (\beta +k-1)\\ &\quad -\frac {1}{2}\alpha _3(\gamma -1)\sigma ^2_H(t)\gamma (\beta +k-1)[\gamma (\beta +k-1)-1]\\ d_3&=\delta -(\gamma +1)\theta _i(t)-\gamma r-\frac {1}{2}\frac {\gamma (\mu -r)^2}{\sigma ^2_S(\rho ^2_{HS}-1)(\gamma -1)}\\ q&=\frac {-\gamma +\beta \gamma }{1-\gamma },\\ \eta _i&=(w_i\beta )^{\frac {1}{1-\gamma }}\left (\frac {\beta \upsilon }{1-\beta }\right )^{k-1}. \end{align*}

To ensure the non-negativity of $ X_i(\tau _i) + \frac {k_i(\tau _i, X_i(\tau _i))}{\theta _i(\tau _i)}$ , we need to verify that the expression for $ g_i(t, h)$ is non-negative. Given that $ w_i \gt 0$ , $ \upsilon \gt 0$ , and $ 0 \lt \beta \lt 1$ , it follows that $ \eta _i \gt 0$ . Given $ \gamma \lt 1$ , we can ensure that $ \alpha _1 \gt 0$ and $ \alpha _3 \gt 0$ and $ X_i(\tau _i) + \frac {k_i(\tau _i, X_i(\tau _i))}{\theta _i(\tau _i)}$ can be non-negative.

For this case, we build upon the optimization framework proposed by Wei et al. (Reference Wei, Cheng, Jin and Wang2020). Our approach initiates by considering the optimization problem after the first death and subsequently addresses the optimization problem before the first death. Compared to their work, our model further includes housing consumption and investment, thereby exploring the area of housing assets. By incorporating housing consumption and insurance demand, our model addresses two scenarios: after the first death, where the surviving breadwinner cannot purchase life insurance, a variable annuity, or real estate assets, which is considered an extension of the traditional Richard’s model (Richard, Reference Richard1975). Our extended model integrates housing elements such as consumptiona and investment component, distinguishing it from other extensions of Richard’s model (e.g., Pliska & Ye, Reference Pliska and Ye2007; Zhang et al., Reference Zhang, Purcal and Wei2021 and Chen et al., Reference Chen, Luo and Yao2024), which initially did not consider these aspects. This incorporation broadens the scope of the model and allows for a more comprehensive analysis of the interplay between housing decisions and life insurance.

3.2 The optimization problem before the first death

We now consider the case that both breadwinners are alive. For each breadwinner, the optimal strategy has been discussed in Section 3.1 when $t\gt T_1$ . The optimal strategies are written as $\bar {c}_i(t,x_t)$ , $\bar {k}_i(t,x_t)$ , $\bar {\pi }(t,x_t)$ , $\bar {\phi }_{4_i}(t,x_t)$ and $\bar {\phi }_{5_i}(t,x_t)$ when both breadwinners are alive.

We simplify the notation $ V_i(t, x, h)$ for the value function to $ V_i$ . The proof of the following dynamic equation is stated in Appendix B.

(3.2) \begin{align} V&=\frac {1}{1-F_{T_1}(t)}\max _{u \in {\mathscr{A}}_t }E\left \{ \int ^T_t\left [{\int ^{\infty }_z f(s,z)ds}\right ]V_1\left (z, X(z)+\frac {\bar {k}_2(z, X(z))}{\theta _2(z)},H(z)\right )dz\right .\nonumber \\ &\left .\quad +\int ^T_t\left [{\int ^{\infty }_s f(s,z)dz}\right ]V_2\left (s, X(s)+\frac {\bar {k}_1(s, X(s))}{\theta _1(s)},H(z)\right )ds\right .\nonumber \\ &\left .\quad +\int ^T_t \left [{1-F_{T_1}(s)} \right ] e^{-\delta (s-t)} \left [w_1 U_1(\bar {c}_1(s,X_1(s)))+ w_2 U_1(\bar {c}_2(s,X_2(s)))\right ] ds\right .\nonumber \\ &\left .\quad + {w_4e^{-\delta (T-t)}U_2(X(T))}\int ^{\infty }_T\int ^{\infty }_T f(s,z)dsdz\right \}, \end{align}

where

\begin{align*} dX(t)&=[r(t)+(\mu (t)-r(t))\bar {\pi }(t)]X(t)dt+\sum ^2_{i=1}[\!-\!\bar {\phi }_{ 4_i}(t)\upsilon H(t)-\left (\bar {c}_{i}(t)+\bar {k}_{i}(t)-Y_{i}(t)\right )]dt\\ &\quad +\bar {\pi }(t)X(t)\sigma _S(t)dZ_{S}(t)+\sum ^2_{i=1}[\bar {\phi }_{5_i}(t)H(t)\rho _{HS}\sigma _H(t)]dZ_{S}(t)\\ &\quad +\sum ^2_{i=1}\bar {\phi }_{5_i}(t)H(t)\rho _{H}\sigma _H(t) dZ_{H}(t). \end{align*}

The proof of the following proposition can be found in Appendix C.

Proposition 3.2. We write $\tilde {V}= (1-F_{T_1}(t))V$ . Rewriting this, We assume the value function follows the ansatz:

\begin{equation*} V=\frac {g(t,h)^{1-\gamma }}{\gamma [1-F_{T_1(t)}]}(x+b(t))^{\gamma } \end{equation*}

and

\begin{equation*} \tilde {V}=\frac {g(t,h)^{1-\gamma }}{\gamma }(x+b(t))^{\gamma }. \end{equation*}

The optimal strategies are given by

(3.3) \begin{align} \bar {c}^*_i(t)&=\frac {\beta }{(1-\beta )g(t,h)}\upsilon \eta _i h^k (x+b(t)),\nonumber \\ \bar {\pi }^*(t)&=\frac {-(\mu (t)-r(t))}{(1-\rho ^2_{HS})\sigma ^2_S(t)x(\gamma -1)}(x+b(t)),\nonumber \\ \bar {\phi }^*_{4_i}(t)&=\frac {\eta _i h^{k-1}}{g(t,h)}(x+b(t)),\nonumber \\ \bar {\phi }^*_{5}(t)&=\left [\frac {\rho _{HS}(\mu (t)-r(t))}{(1-\rho ^2_{HS})\sigma _S(t)\sigma _H(t)h(\gamma -1)}+\frac {g_{h}(t,h)}{g(t,h)}\right ](x+b(t)). \end{align}

where

\begin{align*} g(t,h)&=\left (w^{\frac {1}{1-\gamma }}_4+\frac {\tilde {d}_2}{\frac {\gamma }{1-\gamma }\tilde {d}_1}\right )e^{\frac {\gamma }{1-\gamma }\tilde {d}_1(T-t)}-\frac {\tilde {d}_2}{\frac {\gamma }{1-\gamma }\tilde {d}_1},\\ k&=\frac {-\gamma +\beta \gamma }{1-\gamma },\end{align*}

\begin{align*} \eta _i&=(w_i\beta )^{\frac {1}{1-\gamma }}\left (\frac {\beta \upsilon }{1-\beta }\right )^{k-1},\\b(t)&= \sum ^2_{i=1}\int ^{\omega }_{t}e^{-\int ^s_t(r(u)+\theta _i(u))du}Y_i(s),\\ \tilde {d}_1&=r(t)+\frac {(\mu (t)-r(t))^2}{(1-\rho ^2_{HS}) \sigma ^2_S(t) (\gamma -1)}+\frac {1}{2}\frac {(\mu (t)-r(t))^2}{(1-\rho ^2_{HS})^2 \sigma ^2_S(t) (\gamma -1)}+\frac {\tilde {P}_1}{\gamma }+\frac {\tilde {P}_3}{x},\\ \tilde {d}_2&=(1-F_{T_1}(t))^{-\frac {1}{\gamma -1}}\left [w_1\frac { \eta _1^{\gamma }}{\gamma }+w_2\frac { \eta _2^{\gamma }}{\gamma }\right ]\left (\frac {\beta }{1-\beta }\right )^{\beta \gamma }\upsilon ^{\beta \gamma } h^{\gamma (k+\beta -1)},\\ &-(1-F_{T_1}(t))^{-\frac {1}{\gamma -1}}\frac {(\eta _1+\eta _2)\upsilon h^k}{1-\beta }+\tilde {P}_2\gamma ^{-\frac {\gamma }{\gamma -1}}x^{\frac {1}{\gamma -1}},\\\ \tilde {P}_1&= -\left (\delta +\int ^{\infty }_t f(s,t)ds+\int ^{\infty }_t f(t,z)dz\right ),\\ \tilde {P}_2&=\theta _1(x+b_2)+\theta _2(x+b_1),\\ \tilde {P}_3&=\left (1-\frac {1}{\gamma }\right )\left (\frac {\theta _1}{\int ^{\infty }_tf(t,z)dz}\right )^{\frac {\gamma }{\gamma -1}}g_2(t,h)\int ^{\infty }_tf(t,z)dz\\ &\quad +\left (1-\frac {1}{\gamma }\right )\left (\frac {\theta _2}{\int ^{\infty }_tf(s,t)ds}\right )^{\frac {\gamma }{\gamma -1}}g_1(t,h)\int ^{\infty }_tf(s,t)ds. \end{align*}

Diverging from the framework presented by Kung & Yang (Reference Kung and Yang2020), we introduce a novel insurance component that extends Richard’s foundational model (Richard, Reference Richard1975). While Kung & Yang (Reference Kung and Yang2020) focus on integrating housing and life insurance decisions within a continuous time setting, our approach enhances this integration by specifically addressing the optimal consumption and investment strategies for households with two breadwinners following the first death. It is important to note that although $ \bar {k}_i^*(t,x)$ can be negative, we must ensure that the expression $ \frac {\bar {k}_i^*(t,x)}{\theta _i(t)} + x$ remains positive. Following the approach of Kung & Yang (Reference Kung and Yang2020), we apply a loading factor to reduce insurance demand. As demonstrated in Section 4, by choosing specific value of $l=0.1$ , we can ensure $ \frac {k_i^*(t,x)}{\theta _i(t)} + x$ is non-negative.

The optimal solution after the first death represents a significant advancement over Richard’s original model, as it now accounts for the housing components crucial for realistic financial planning and decision-making. By structuring the problem to capture the dynamic decision-making process of both individuals before the first death, our model accurately captures the dynamics of life insurance, housing, and financial decision-making over the life cycle of couples with correlated lifetimes, providing a more realistic and practical framework for optimizing consumption and investment strategies. Compared to Wei et al. (Reference Wei, Cheng, Jin and Wang2020), our model includes housing elements for consumption and investment, broadening the study of optimal strategies. By incorporating this additional insurance component and integrating housing elements, our model provides a more robust framework for understanding and optimizing the consumption and investment behaviors of individuals facing the dual challenges of housing and life insurance decisions in a continuous time context.

4. Numerical results and discussion

In our numerical demonstration, two cases are studied. For the optimisation problem after the first death (case 1), there is only one alive breadwinner. This person can purchase life insurance, a variable annuity, and real estate assets. For the second case (the optimisation problem before the first death), both breadwinners are alive. The couple can purchase life insurance, a variable annuity, and real estate assets. Calibration parameters in the mortality model are shown in Table 1.

Table 1. Calibrated parameters in the mortality model

The marginal probability distribution functions for male and female are assumed to be expressed by

\begin{align*} \begin{split} F_1(t)=1-\exp \{\exp\!(\!-\!m_1/\sigma _1)[1-\exp\!(t/\sigma _1)]\},\\ F_2(t)=1-\exp \{\exp\!(\!-\!m_2/\sigma _2)[1-\exp\!(t/\sigma _2)]\}. \end{split} \end{align*}

These two-parameter Gompertz distributions imply the following density functions of $\tau _1$ and $\tau _2$ :

\begin{align*} \begin{split} f_1(t)=\frac {1}{\sigma _1}\exp \left \{\exp\!(\!-\!m_1/\sigma _1)[1-\exp\!(t/\sigma _1)]-\frac {m_1}{\sigma _1}+\frac {t}{\sigma _1}\right \},\\ f_2(t)=\frac {1}{\sigma _2}\exp \left \{\exp\!(\!-\!m_2/\sigma _2)[1-\exp\!(t/\sigma _2)]-\frac {m_2}{\sigma _2}+\frac {t}{\sigma _2}\right \}. \end{split} \end{align*}

In this case, the mortality rate of each wage earner is given by

\begin{align*} \begin{split} \lambda _1(t)=\frac {1}{\sigma _1}e^{\frac {t-m_1}{\sigma _1}},\\ \lambda _2(t)=\frac {1}{\sigma _2}e^{\frac {t-m_2}{\sigma _2}}. \end{split} \end{align*}

We use the following Gumbel–Hougaard copula to capture the dependence structures between the mortalities of a couple:

\begin{equation*} C(s,t)=e^{-\left [(-\ln s)^{\alpha }+(-\ln t)^{\alpha }\right ]^{-1/\alpha }},\ \ \alpha \geq 1. \end{equation*}

Under this model, the joint probability distribution of $(\tau _1, \tau _2)$ is given by

\begin{equation*} F(t,t)=e^{-\left [(-\ln F_M(t))^{\alpha }+(-\ln F_F(t))^{\alpha }\right ]^{\frac {1}{\alpha }}}. \end{equation*}

The distribution function of $T_1$ is

\begin{equation*} F_{T_1}(t)=F_1(t)+F_2(t)-F(t,t). \end{equation*}

Also, we have the following results:

\begin{align*} \begin{split} f_1(t) =\int _t^\infty f(t,z)d z+\frac {f_1(t)F(t,t)}{F_1(t)}\left [1+\left (\frac {\ln F_2(t)}{\ln F_1(t)}\right )^{\alpha }\right ]^{\frac {1-\alpha }{\alpha }},\\ f_2(t)=\int _t^\infty f(s,t)d s+\frac {f_2(t)F(t,t)}{F_2(t)}\left [1+\left (\frac {\ln F_1(t)}{\ln F_2(t)}\right )^{\alpha }\right ]^{\frac {1-\alpha }{\alpha }}. \end{split} \end{align*}

We use $\alpha _0,$ $\alpha _5$ and $\alpha _{10}$ to denote the value of $\alpha$ in the copula model when the age difference between male and female is $0,$ $5$ and $10,$ respectively.

Table 2. Values of parameters in the numerical experiments

Figure 1 Effect of $t$ on the optimal investment strategies.

Figure 2 Effect of $t$ on the optimal consumption strategies.

Figure 3 Effect of $t$ on the optimal housing consumption strategies.

Figure 4 Effect of $t$ on the optimal insurance strategies.

In the numerical examples, we assume that the age of male is $75$ , and the age of female is $70$ . The age difference is 5, which makes $\alpha =1.67.$ Also, it is supposed that $T_1=\tau _2,$ which implies that the wife dies before the husband, and so for the case of after the first death we focus on the strategies of the male. Most of the values of the model parameters shown in Table 2 are borrowed from Kung & Yang (Reference Kung and Yang2020). In Fig. 1, we show the effects of time $t$ on the optimal investment strategies. As we can see, the household will not adjust the optimal investment decisions as time varies. The optimal investment strategy is a fixed negative constant which is independent of wealth level $x$ and time $t.$ This is consistent with the expressions for $\pi _i^*$ in (3.5) and $\bar \pi ^*$ in (3.14). This is because we assume that $\mu (t),$ $r(t)$ and $\sigma _S(t)$ are constants for simplicity, and also we do not consider income of the household in the numerical demonstration.

Fig. 2 displays the effects of time on the optimal consumption strategies. It can be observed that before the first death, the consumption strategies of male and female are both decreasing functions of time $t$ . Comparatively speaking, the consumption policy for the case of after the first death is more sensitive to the change of time. Usually, it is a fact that the consumption strategy increases w.r.t. time. This is due to the fact that consumption strategy is an increasing function of labor income and usually the labor income increases as time goes by before the retirement time (see, for example, Kung & Yang, Reference Kung and Yang2020 and Wei et al., Reference Wei, Cheng, Jin and Wang2020). However, this kind of result is not obtained in our article because we assume the household has no labor income in this section to illustrate the theoretical results. The household will reduce the consumption rates to keep the wealth for future use in our case. Similar arguments can be used to analyze the effects of time $t$ on the housing consumption strategy which has been shown in Fig. 3.

In Fig. 4, we examine the effects of $t$ on the optimal insurance strategy. The household usually spends more on life insurance as their ages increase to hedge against its mortality risk and labor income risk. But in our case, the life insurance strategy declines as the human capital is not considered and purchasing life insurance makes little sense. Also, a negative life insurance strategy after the first death means the household receives payment from the insurance company. To ensure that $ \frac {k_i^*(t,x)}{\theta _i(t)} + x$ remains non-negative, we set the loading factor to $ l = 0.1$ . This adjustment helps maintain positivity in the expression by moderating the impact of $ k_i^*(t,x)$ when it is negative, effectively reducing the insurance demand. By following this approach, we ensure that the utility function $ U_2(\!\cdot\!)$ remains well-defined and applicable within the value function.

Fig. 5 demonstrates how time $t$ impacts optimal housing investment strategies for different scenarios. Considering that the pandemic has introduced significant uncertainty and volatility into the financial markets, which increases the demand for safer investments and leads to a decrease in stock prices and potentially lower expected returns, we assume a smaller value for the expected return of risky asset in this section. This is why the the values of the optimal investment strategy for risky asset in Fig. 1 are negative. On the other hand, housing and risk-free asset may appear relatively more attractive compared to stocks, which explains why the optimal housing investment strategies in Fig. 5 are positive. Finally, Fig. 6 shows the effects of wealth on the value functions for two cases. Not surprisingly, we find that the value functions increase w.r.t. the level of wealth.

In what follows, we show the effects of risk aversion parameter $\gamma$ on the optimal control policies. In each of the following figures, we vary the value of $\gamma$ from $0.01$ to $0.5.$ Fig. 7 shows the impact of $\gamma$ on the optimal investment decisions in risky asset. As we can observe, if the wage earner is more risk averse, they tend to sell more risky assets to mitigate financial risk, which makes intuitive sense.

Figure 5 Effect of $t$ on the optimal housing investment strategies.

Figure 6 Effect of $x$ on the value functions.

In Figs. 8 and 9, we illustrate how the optimal consumption strategies and optimal housing consumption strategies vary w.r.t. $\gamma$ , respectively. It can be seen that a more risk-averse wage earner consumes less. This stems from the tendency of risk-averse individuals to prefer more certain outcomes, often sacrificing potential higher utility from consumption to avoid potential financial instability or uncertainty. In this sense, our results are consistent with those in Chen et al. (Reference Chen, Luo and Yao2024).

From Fig. 10, we can see that optimal housing investment strategies increase as $\gamma$ increases and the difference between before and after the first death is insignificant. In both cases, housing is considered as a relatively stable and less volatile asset compared to stocks, and hence more risk-averse breadwinner tends to invest more in housing.

Fig. 11 displays how $\gamma$ influences the optimal life insurance purchasing strategies. In the literature, different patterns of life insurance demand w.r.t. $\gamma$ have been reveled, see, for example, Kwak & Lim (Reference Kwak and Lim2014), Han & Hung (Reference Han and Hung2017) and Chen et al. (Reference Chen, Luo and Yao2024). In our model setup, the results show the following economic implications. First, the impact of $\gamma$ for the case of after the first death is not significant. Second, for the other two cases, the optimal insurance strategies increase and then decrease w.r.t. $\gamma$ . When $\gamma$ is small, the wage earner who is more risk averse tends to purchase more life insurance to hedge against the potential losses from mortality risk and protect their family. However, if $\gamma$ is relatively larger, housing investment may be considered as a more attractive and effective tool than life insurance for ensuring an adequate legacy and future consumption for dependents after the death of breadwinner, which results in a reduction in life insurance demand.

Figure 7 Effects of $\gamma$ on the optimal investment strategies.

Figure 8 Effects of $\gamma$ on the optimal consumption strategies.

Figure 9 Effects of $\gamma$ on the optimal housing consumption strategies.

Figure 10 Effects of $\gamma$ on the optimal housing investment strategies.

Figure 11 Effects of $\gamma$ on the optimal insurance strategies.

The numerical examples consider a couple with a 5-year age difference. Mortality dependency significantly influences consumption, investment, and insurance decisions. Before the first death, mortality dependency is reflected in a decreasing consumption pattern. Conversely, after the first death (without mortality dependency), consumption displays an increasing trend over time. The effect of mortality dependency is further evident in insurance purchase and housing investment. Mortality dependency prompts a demand for insurance, leading to a gradual decrease over time. In scenarios without mortality dependency (after the first death), the theoretical demand shifts towards annuities, as illustrated in our example. Moreover, mortality dependency influences housing investment, resulting in a rising trend in housing consumption. These findings highlight the intricate interplay between mortality dependency and diverse financial decisions within the specified scenarios.