# Optimal Time to Enter a Retirement Village

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## Abstract

**:**

## 1. Introduction

## 2. Model and Method

#### 2.1. Case 1: No Bequest and Incomplete Insurance Market

#### 2.2. Case 2: With Bequest and Complete Insurance Market

#### 2.3. Case 3: With Bequest, Complete Insurance Market and Wealth Floor

## 3. Parameter Values

## 4. Numerical Results and Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Optimal Exercise Price

## References

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^{1.}See, for example, Ross [32].^{2.}Retirees need to pay a large amount in upfront fees to live in such a community and have the right to re-sell the occupation rights of the property.^{3.}Note that $P(t)>0$ reflects the retiree (continuously) purchasing life insurance, while $P(t)<0$ indicates the retiree has entered into something akin to a (instantaneous) variable annuity contract with an insurer [38].

**Figure 1.**Expected consumption path for case 1 retirees, starting in the healthy state at age 65 with total wealth of 10Y, and truncated at the optimal case 1 stopping times. This captures the expected consumption outcomes of agents with no bequest motive. Note that these agents have no access to insurance markets, and are assumed to purchase a term certain annuity to protect their basic consumption needs—which is much more expensive than a life annuity, particularly at older ages.

**Figure 2.**Expected consumption, truncated at optimal stopping times, for case 2 and case 3 agents commencing at age 65 in the healthy state with total wealth of 10Y. The figure captures the expected consumption paths for agents with bequest motives. These agents, in contrast to case 1, have access to perfect insurance markets.

**Figure 3.**Expected wealth, truncated at optimal stopping times, for case 1, 2 and 3 agents commencing at age 65 in the healthy state with total wealth of 10Y. Recall case 1 agents have no access to insurance markets, while case 3 agents replicate an American put option to ensure their savings target is met.

**Figure 4.**Expected insurance premiums paid by case 2 and case 3 agents, truncated at optimal stopping times, for those commencing at age 65 in the healthy state with a total wealth of 10Y. Negative insurance premiums mean the agents are receiving funds from the insurers, that is, they are in receipt of an annuity.

**Figure 5.**Expected proportion of surplus wealth, $\tilde{W}$, invested in the risky assets, or ${\pi}^{*}W/\tilde{W}$, by agents starting at age 65 in the healthy state with a total wealth of 10Y. The expected paths are truncated at the optimal stopping times for case 2 and case 3, respectively. The differing behaviour of the case 3 target savers is clear.

t = 65 | τ = 109 |

${q}_{12}$ = 0.04 | ${q}_{21}$ = 0.4 |

$\alpha $ = 0.08112 | r = 0.034 |

$\rho $ = 0.034 | $\sigma $ = 0.15685 |

Y = AUD$47,736 | $\gamma $ = −0.5 |

$\upsilon $ = 0.8 | u = 1.2 |

${D}_{1}$ = 0.01 | ${D}_{2}$ = 0.02 |

h = AUD$12,000 p.a. |

**Table 2.**Expected stopping times by level of risk aversion for case 1 and case 2 agents aged 65, in the healthy state, and with total wealth of 10Y—and other parameters as given in Table 1. Increasing levels of risk aversion lead to falling stopping times. Also, the bequest motives of case 2 agents produce results much more sensitive to the level of risk aversion. Indeed, at $\gamma =-0.5$ case 2 agents abandon their conservative behaviour and embrace the risky investment environment.

Expected Stopping Time (Years) | ||
---|---|---|

Gamma | Case 1 | Case 2 |

$-0.5$ | 11.65 | 13.22 |

$-0.6$ | 11.04 | 10.53 |

$-0.7$ | 10.56 | 8.28 |

$-0.8$ | 10.04 | 6.23 |

$-0.9$ | 9.54 | 4.50 |

$-1$ | 9.13 | 2.99 |

**Table 3.**Expected stopping times by level of equity premium for case 1 and case 2 agents aged 65, in the healthy state, and with total wealth of 10Y—and other parameters as given in Table 1. Increasing the equity premium results in longer stopping times, as agents exploit the more profitable investment environment. The situation illustrated is for agents with risk aversion of $\gamma =-0.5$, where case 2 agents are less conservative than case 1 agents. With more risk averse agents, this boldness of agents with bequest motives over those without is reversed.

Expected Stopping Time (Years) | ||
---|---|---|

$\mathit{\alpha}-\mathit{r}$ | Case 1 | Case 2 |

0.02 | 5.49 | 6.38 |

0.03 | 7.76 | 9.02 |

0.04 | 10.07 | 11.60 |

0.05 | 12.27 | 13.87 |

0.06 | 14.31 | 16.06 |

**Table 4.**Expected stopping times by level of market volatility for case 1 and case 2 agents aged 65, in the healthy state, and with total wealth of 10Y—and other parameters as given in Table 1. Increasing market volatility results in shorter stopping times, as agents shy away from the riskier environment. The situation illustrated is for agents with risk aversion of $\gamma =-0.5$, where case 2 agents are less conservative than case 1 agents. With more risk averse agents, this boldness of agents with bequest motives over those without is reversed.

Expected Stopping Time (Years) | ||
---|---|---|

$\mathit{\sigma}$ | Case 1 | Case 2 |

0.12 | 14.62 | 16.45 |

0.13 | 13.67 | 15.46 |

0.14 | 13.44 | 14.51 |

0.15 | 12.10 | 13.70 |

0.16 | 11.42 | 12.99 |

**Table 5.**Expected stopping times by frailty factor u for case 1 and case 2 agents aged 65, in the healthy state, and with total wealth of 10Y—and other parameters as given in Table 1. Increasing frailty results in shorter stopping times, as less healthy agents choose the safer retirement village world sooner. The situation illustrated is for agents with risk aversion of $\gamma =-0.5$, where case 2 agents are less conservative than case 1 agents. With more risk averse agents, this boldness of agents with bequest motives over those without is reversed.

Expected Stopping Time (Years) | ||
---|---|---|

$\mathit{u}$ | Case 1 | Case 2 |

1.1 | 13.40 | 14.15 |

1.2 | 11.65 | 13.22 |

1.3 | 10.15 | 12.38 |

1.4 | 8.85 | 11.58 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, J.; Purcal, S.; Wei, J.
Optimal Time to Enter a Retirement Village. *Risks* **2017**, *5*, 20.
https://doi.org/10.3390/risks5010020

**AMA Style**

Zhang J, Purcal S, Wei J.
Optimal Time to Enter a Retirement Village. *Risks*. 2017; 5(1):20.
https://doi.org/10.3390/risks5010020

**Chicago/Turabian Style**

Zhang, Jinhui, Sachi Purcal, and Jiaqin Wei.
2017. "Optimal Time to Enter a Retirement Village" *Risks* 5, no. 1: 20.
https://doi.org/10.3390/risks5010020