# Life Insurance and Annuity Demand under Hyperbolic Discounting

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## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. Naïve Case

#### 2.2. Sophisticated Case

## 3. Results

#### 3.1. Parameter Values

#### 3.2. Hyperbolic Discounting Results

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Barro, Robert J. 1999. Ramsey meets Laibson in the neoclassical growth model. The Quarterly Journal of Economics 114: 1125–52. [Google Scholar] [CrossRef]
- Bernheim, B. Douglas. 1991. How strong are bequest motives? Evidence based on estimates of the demand for life insurance and annuities. Journal of Political Economy 99: 899–927. [Google Scholar] [CrossRef]
- Butler, Monika, Kim Peijnenburg, and Stefan Staubli. 2017. How much do means-tested benefits reduce the demand for annuities? Journal of Pension Economics & Finance 16: 419–49. [Google Scholar]
- Carroll, Christopher D. 1997. Buffer-stock saving and the life cycle/permanent income hypothesis. The Quarterly Journal of Economics 112: 1–55. [Google Scholar] [CrossRef]
- Finkelstein, Amy, and James Poterba. 2004. Adverse selection in insurance markets: Policyholder evidence from the UK annuity market. Journal of Political Economy 112: 183–208. [Google Scholar] [CrossRef]
- Friedman, Benjamin M., and Mark J. Warshawsky. 1990. The cost of annuities: Implications for saving behavior and bequests. Quarterly Journal of Economics 105: 135–54. [Google Scholar] [CrossRef]
- Gourinchas, Pierre-Olivier, and Jonathan A. Parker. 2002. Consumption over the life cycle. Econometrica 70: 47–89. [Google Scholar] [CrossRef]
- Laibson, David. 1997. Golden eggs and hyperbolic discounting. The Quarterly Journal of Economics 112: 443–78. [Google Scholar] [CrossRef]
- Lockwood, Lee M. 2012. Bequest motives and the annuity puzzle. Review of Economic Dynamics 15: 226–43. [Google Scholar] [CrossRef] [PubMed]
- Marín-Solano, Jesús, and Jorge Navas. 2010. Consumption and portfolio rules for time-inconsistent investors. European Journal of Operational Research 201: 860–72. [Google Scholar] [CrossRef]
- Merton, Robert C. 1969. Lifetime portfolio selection under uncertainty: The continuous-time case. The Review of Economics and Statistics 51: 247–57. [Google Scholar] [CrossRef]
- Ministry of Health, Labour and Welfare. 2010. Abridged Life Tables for Japan 2010; Tokyo: Ministry of Health, Labour and Welfare.
- Ministry of Health, Labour and Welfare. 2015. Final Report of Monthly Labour Survey—June 2015; Tokyo: Ministry of Health, Labour and Welfare.
- Oppers, Erik, Ken Chikada, Frank Eich, Patrick Imam, John Kiff, Michael Kisser, Mauricio Soto, and Tao Sun. 2012. The Financial Impact of Longevity Risk. Washington: International Monetary Fund. [Google Scholar]
- Pliska, Stanley R., and Jinchun Ye. 2007. Optimal life insurance purchase and consumption/investment under uncertain lifetime. Journal of Banking & Finance 31: 1307–19. [Google Scholar]
- Pollak, Robert A. 1968. Consistent planning. The Review of Economic Studies 35: 201–8. [Google Scholar] [CrossRef]
- Purcal, Sachi, and John Piggott. 2008. Explaining low annuity demand: An optimal portfolio application to Japan. Journal of Risk and Insurance 75: 493–16. [Google Scholar] [CrossRef]
- Richard, Scott F. 1975. Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model. Journal of Financial Economics 2: 187–203. [Google Scholar] [CrossRef]
- Strotz, Robert Henry. 1955. Myopia and inconsistency in dynamic utility maximization. The Review of Economic Studies 23: 165–80. [Google Scholar] [CrossRef]
- Tang, Siqi. 2009. Life Insurance and Annuity Demand under Hyperbolic Discounting. BCom Honours thesis, School of Actuarial Studies, Faculty of Commerce, University of New South Wales, Sydney, Australia. [Google Scholar]
- World Health Organization. 2015. World Report on Ageing and Health. Geneva: World Health Organization. [Google Scholar]
- Yaari, Menahem E. 1965. Uncertain lifetime, life insurance, and the theory of the consumer. The Review of Economic Studies 32: 137–50. [Google Scholar] [CrossRef]
- Zhang, Jinhui, Sachi Purcal, and Jiaqin Wei. 2017. Optimal time to enter a retirement village. Risks 5: 20. [Google Scholar] [CrossRef]
- Zhang, Jinhui, Sachi Purcal, and Jiaqin Wei. 2018. Optimal Life Insurance and Annuity Demand under Hyperbolic Discounting When Bequests Are Luxury Goods. Working paper, Department of Actuarial Studies and Business Analytics, Macquarie University, Sydney, Australia. [Google Scholar]

1 | Adverse selection occurs when agents with a higher-than-average risk profile purchase insurance products at a premium set for the average consumer. It is thus predicated on an information asymmetry between insurer and insured. |

2 | The function $m{(t)}^{1-\gamma}$ does not necessarily discount the bequest motive. In fact, it can be any function that impacts the bequest motive directly. |

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6 | Numerical solutions can be found in Zhang et al. (2018). |

7 | Daily data of the Japanese Nikkei 225 index from 2005 to 2014 were obtained from FactSet and further analysed (https://company-security.apps.factset.com/price-history/180461). |

8 | |

9 | Gourinchas and Parker (2002), pp. 67–69. |

t = 30 $\alpha =0.0418$ $\rho =0.0022$ Y = 5,102,412 Yen $\xi =0.5$ | $\tau =110$ r = 0.00364 $\sigma =0.25$ $\gamma =-0.5$ $\zeta =0.25$ |

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**MDPI and ACS Style**

Tang, S.; Purcal, S.; Zhang, J.
Life Insurance and Annuity Demand under Hyperbolic Discounting. *Risks* **2018**, *6*, 43.
https://doi.org/10.3390/risks6020043

**AMA Style**

Tang S, Purcal S, Zhang J.
Life Insurance and Annuity Demand under Hyperbolic Discounting. *Risks*. 2018; 6(2):43.
https://doi.org/10.3390/risks6020043

**Chicago/Turabian Style**

Tang, Siqi, Sachi Purcal, and Jinhui Zhang.
2018. "Life Insurance and Annuity Demand under Hyperbolic Discounting" *Risks* 6, no. 2: 43.
https://doi.org/10.3390/risks6020043