Retirement Lifecycle Model

This page demonstrates the retirement lifecycle model described in the research paper Optimal consumption, investment and housing with means-tested public pension in retirement. The model starts at retirement and solves for optimal control given total wealth at time of retirement. The user can chose which Age Pension policy to apply with Age Pension rates as of 20th March 2017, depending on when the accounts was opened.

TThe model starts at the current age $t_0$ of the retiree, with the option to allocate part of total wealth $\mathsf{W}$ into housing $H$ and leave the rest as liquid wealth $W_{t_0}$. The objective is to maximize utility throughout retirement by controlling the proportion drawdown $\alpha_t$ from liquid wealth, and the proportion $\delta_t $ of liquid wealth that is invested into a risky asset, for each annual time period $t = t_0,t_0+1,...,T$. The consumption for each period is then given by $C_t = \alpha_t W_t + P_t$, where $P_t$ is the amount Age Pension received. Between each time period the remaining wealth grows based on the allocation to a risky asset and a risk free asset. The risky asset follows the i.i.d real log returns $Z_{t+1} \sim \mathcal{N}(\mu, \sigma)$ where $\mu$ is the real rate of return and $\sigma$ is the standard deviation. The evolution of wealth is defined as \[ W_{t+1} = \left[ W_t - \alpha_t W_t \right] \left[\delta_t e^{Z_{t+1}} + (1-\delta_t) e^{r} \right], \] where $r$ is the deterministic real risk free return.

The retiree receives utility from consumption $U_C(\cdot)$ and housing $U_H(\cdot)$ if alive, and bequest $U_B(\cdot)$ if death occured during $(t-1,t]$. Parameters are subject to a sequential family status where couples use subscript $d=\mathrm{C}$ and singles $d=\mathrm{S}$. The utility from consumption is defined as \[ U_C (C_t) = \frac{1}{\psi^{t-t_0} \gamma_d} \left(\frac{C_t - \overline{c}_d}{\zeta_d} \right)^{\gamma_d}, \] where $\gamma_d$ is the risk aversion, $\overline{c}_d$ is the consumption floor, $\zeta_d$ a normalization factor between singles and couples, and $\psi$ is a parameter for decreasing consumption with age. The housing function is defined as \[ U_H (H) = \frac{1}{\gamma_\mathrm{H}} \left(\frac{\lambda_d H}{\zeta_d} \right)^{\gamma_\mathrm{H}}, \] where $\gamma_\mathrm{H}$ is the risk aversion for housing, $\lambda_d$ is the housing preference and $H$ is the market value of the house at the time of purchase. The bequest function is defined as \[ U_B(W_t) = \left(\frac{\theta}{1-\theta}\right)^{1-\gamma_\mathrm{S}} \frac{\left(\frac{\theta}{1-\theta} a+W_t\right)^{\gamma_\mathrm{S}}}{\gamma_\mathrm{S}}, \] where $a$ is the threshold for luxury bequest and $\theta$ is the preference of bequest over consumption. The discounting of utility from time $t$ to $t'$ is denoted as $\beta_{t,t'}$.

The reward for each period is given by a reward function that is subject to family state $G_t = \{\Delta, 0, 1, 2\}$, where $\Delta$ corresponds to already deceased, $0$ corresponds to death during $(t-1, t]$ and $1$ and $2$ correspond to a single or couple household respectively. The reward function is defined as \begin{equation} \label{ext_RewardFunction} R_{t}(W_t,G_t,H) = \left\{ \begin{array}{ll} U_C(C_t,t) + U_H(H), & \mbox{if $G_t = 1,2$},\\ U_B(W_t), & \mbox{if $G_t = 0$},\\ 0, & \mbox{if $G_t = \Delta.$}\end{array} \right. \end{equation} The terminal ($t=T$) reward is defined as \begin{equation} \label{ext_TerminalRewardFunction} \widetilde{R}(W_T, G_T) = \left\{ \begin{array}{ll} U_B(W_T), & \mbox{if $G_T \geq 0,$}\\ 0, & \mbox{if $G_T = \Delta.$}\end{array} \right. \end{equation}

Finally, the objective function to maximize is defined as \begin{equation} \label{eq:FinalValueFunction} \underset{H}{\max} \left[ \underset{\boldsymbol{\pi}}{\sup} \: \mathbb{E}^{\boldsymbol{\pi}}_{t_0} \left[\beta_{t_0,T} \widetilde{R}(W_T,G_T) + \sum_{t={t_0}}^{T-1} \beta_{t_0,t} R_{t}(W_t,G_t,H_T) \, \Big| \, W_{t_0}, G_{t_0} \right] \right], \end{equation} where $\mathbb{E}^{\boldsymbol{\pi}}_{t_0}[\cdot]$ is the expectation under policy $\boldsymbol{\pi} = (\alpha_t, \delta_t)^{T-1}_{t_0}$ of control variables with respect to the state variables $W_t$ and $G_t$ for $t=t_0+1, ..., T$, conditional on the state variables at time $t=t_0$. The mortality risk is based on unisex survival probabilities for 2009-2011, taken from the Australian Bureau of Statistics.

For a detailed description of the model, please refer to the research paper Optimal consumption, investment and housing with means-tested public pension in retirement.

Note that the model is solved via backward induction using a numerical dynamic programming, hence the accuracy of the solution depends on the number of grid steps used. A larger number of grid steps results in a more accurate result, but increases computational time. Hover the input box for further explanations of the inputs.

Family status
Include housing
Enforce minimum withdrawals
Pension Policy
Total wealth ($)	$ \mathsf{W} = $
Age (years)	$ t_0 = $
Terminal time (age)	$ T = $
Real risky return p.a (decimal)	$ \mu = $	Please enter a value in range [-1, 1]
Risky volatility (decimal)	$ \sigma = $	Please enter a value in range [0, 1]
Real risk-free return p.a (decimal)	$ r = $	Please enter a value in range [-1, 1]
Risk aversion (single)	$ \gamma_\mathrm{S} = $	Please enter a negative value
Risk aversion (couple)	$ \gamma_\mathrm{C} = $	Please enter a negative value
Risk aversion (house)	$ \gamma_\mathrm{H} = $	Please enter a negative value
Consumption floor single ($)	$ \bar{c}_\mathrm{S} = $
Consumption floor couple ($)	$ \bar{c}_\mathrm{C} = $
Luxury bequest threshold($)	$ a = $
Bequest preference	$ \theta = $	Please enter a value in range (0, 1)
Health proxy	$ \psi = $	Please enter a value in range [1, 3]
Housing preference	$ \lambda = $	Please enter a value in range (0, 1]
Discount factor	$ \beta = $	Please enter a value in range (0, 1]
Wealth grid steps		Please enter a value in range [15, 40]
House grid steps		Please enter a value in range [10, 40]
Control grid steps		Please enter a value in range [15, 40]

Important! This model is for online demonstration purposes only. Calculations can take a long time if the number of grid steps are high. The time-out for calculations has been set to 1 hour, after which the webpage will return an error. We suggest you start with low values for the grid steps and then increase them if better accuracy is needed.