# Designing Annuities with Flexibility Opportunities in an Uncertain Mortality Scenario

## Abstract

**:**

## 1. Introduction

## 2. Model Setup

#### 2.1. Mortality/Longevity-Linked Annuity Benefits

#### 2.2. Policy Fund and Periodic Fees

#### 2.3. Individual Reserve and Components

#### 2.4. Pool Fund, Present Value of Future Benefits, Present Value of Future Profits and Business Value

#### 2.5. Setting the Periodic Fee

## 3. Results

#### 3.1. Mortality Model

#### 3.2. Benefit Arrangements

- Case (a): $0.75\xb7{b}_{0}\le {b}_{t}\le 1.25\xb7{b}_{0}$;
- Case (b): $0.9\xb7{b}_{0}\le {b}_{t}\le 1.1\xb7{b}_{0}$.

#### 3.3. Implementation and Discussion

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Andersen, Carsten, and Peter Skjodt. 2007. Pension Institutions and Annuities in Denmark. Policy Research Working Paper. Washington, DC: World Bank. [Google Scholar]
- Bacinello, Anna Rita, Pietro Millossovich, and An Chen. 2018. The impact of longevity and investment risk on a portfolio of life insurance liabilities. European Actuarial Journal 8: 257–90. [Google Scholar] [CrossRef] [Green Version]
- Bacinello, Anna Rita, Pietro Millossovich, Annamaria Olivieri, and Ermanno Pitacco. 2011. Variable annuities: A unifying valuation approach. Insurance: Mathematics and Economics 49: 285–97. [Google Scholar] [CrossRef]
- Bernhardt, Thomas, and Catherine Donnelly. 2019. Modern tontine with bequest: Innovation in pooled annuity products. Insurance: Mathematics and Economics 86: 168–88. [Google Scholar] [CrossRef] [Green Version]
- Blackburn, Craig, Katja Hanewald, Annamaria Olivieri, and Michael Sherris. 2017. Longevity risk management and shareholder value for a life annuity business. ASTIN Bulletin 47: 43–77. [Google Scholar] [CrossRef]
- Bravo, Jorge Miguel, and Najat El Mekkaoui de Freitas. 2018. Valuation of longevity-linked life annuities. Insurance: Mathematics and Economics 78: 212–29. [Google Scholar] [CrossRef]
- Cairns, Andrew J. G., David P. Blake, Amy Kessler, and Marsha Kessler. 2020. The Impact of COVID-19 on Future Higher-Age Mortality. May 19. Available online: https://ssrn.com/abstract=3606988 (accessed on 3 August 2021).
- Chen, An, and Manuel Rach. 2019. Options on tontines: An innovative way of combining tontines and annuities. Insurance: Mathematics and Economics 89: 182–92. [Google Scholar]
- Chen, An, Manuel Rach, and Thorsten Sehner. 2020. On the optimal combination of annuities and tontines. Astin Bulletin 50: 95–129. [Google Scholar] [CrossRef]
- Chen, An, Montserrat Guillen, and Manuel Rach. 2021. Fees in tontines. Insurance: Mathematics and Economics 100: 89–106. [Google Scholar]
- Chen, An, Peter Hieber, and Jakob K. Klein. 2019. Tonuity: A novel individual-oriented retirement plan. ASTIN Bulletin 49: 5–30. [Google Scholar] [CrossRef]
- Davidoff, Thomas, Jeffrey R. Brown, and Peter A. Diamond. 2005. Annuities and Individual Welfare. American Economic Review 95: 1573–90. [Google Scholar] [CrossRef] [Green Version]
- Denuit, Michel, Steven Haberman, and Arthur Renshaw. 2011. Longevity-indexed life annuities. North American Actuarial Journal 15: 97–111. [Google Scholar] [CrossRef]
- Donnelly, Catherine. 2015. Actuarial fairness and solidarity in pooled annuity funds. ASTIN Bulletin 45: 49–74. [Google Scholar] [CrossRef] [Green Version]
- Donnelly, Catherine, Montserrat Guillén, and Jens Perch Nielsen. 2013. Exchanging uncertain mortality for a cost. Insurance: Mathematics and Economics 52: 65–76. [Google Scholar] [CrossRef]
- Duffie, Darrell. 2001. Dynamic Asset Pricing Theory, 3rd ed. Princeton: Princeton University Press. [Google Scholar]
- Hanbali, Hamza, Michel Denuit, Jan Dhaene, and Julien Truffin. 2019. A dynamic equivalence principle for systematic longevity risk management. Insurance: Mathematics and Economics 86: 158–67. [Google Scholar] [CrossRef] [Green Version]
- Lee, Ronald, and Lawrence Carter. 1992. Modelling and forecasting US mortality. Journal of the American Statistical Association 87: 659–75. [Google Scholar]
- McKever, Kent. 2009. A short history of tontines. Fordham Journal of Corporate and Financial Law 15: 491–521. [Google Scholar]
- Milevsky, Moshe A. 2014. Portfolio choice and longevity risk in the late Seventeenth century: A re-examination of the first English tontine. Financial History Review 21: 225–58. [Google Scholar] [CrossRef]
- Milevsky, Moshe A. 2020. Is COVID-19 a Parallel Shock to the Term Structure of Mortality? Available online: https://moshemilevsky.com/wp-content/uploads/2020/05/MILEVSKY_20MAY2020_AMAZON.pdf (accessed on 3 August 2021).
- Milevsky, Moshe A., and Thomas S. Salisbury. 2015. Optimal retirement income tontines. Insurance: Mathematics and Economics 64: 91–105. [Google Scholar] [CrossRef] [Green Version]
- Olivieri, Annamaria, and Ermanno Pitacco. 2009. Stochastic mortality: The impact on target capital. ASTIN Bulletin 39: 541–63. [Google Scholar] [CrossRef]
- Olivieri, Annamaria, and Ermanno Pitacco. 2020a. Linking annuity benefits to the longevity experience: Alternative solutions. Annals of Actuarial Science 14: 316–37. [Google Scholar] [CrossRef]
- Olivieri, Annamaria, and Ermanno Pitacco. 2020b. Longevity-Linked Annuities: How to Preserve Value Creation against Longevity Risk. In Life Insurance in Europe. Edited by M. Borda, S. Grima and I. Kwiecień. Financial and Monetary Policy Studies. Berlin: Springer, vol. 50, pp. 103–26. [Google Scholar]
- Peijnenburg, Kim, Theo Nijman, and Bas J. M. Werker. 2016. The Annuity Puzzle Remains a Puzzle. Journal of Economic Dynamics and Control 70: 18–35. [Google Scholar] [CrossRef] [Green Version]
- Piggott, John, Emiliano A. Valdez, and Bettina Detzel. 2005. The simple analytics of a pooled annuity fund. The Journal of Risk and Insurance 72: 497–520. [Google Scholar] [CrossRef]
- Pitacco, Ermanno. 2016. Guarantee structures in life annuities: A comparative analysis. The Geneva Papers on Risk and Insurance—Issues and Practice 41: 78–97. [Google Scholar] [CrossRef]
- Qiao, Chao, and Michael Sherris. 2013. Managing systematic mortality risk with group self-pooling and annuitization schemes. The Journal of Risk and Insurance 80: 949–74. [Google Scholar] [CrossRef]
- Richter, Andreas, and Frederik Weber. 2011. Mortality-Indexed annuities. Managing longevity risk via product design. North American Actuarial Journal 15: 212–36. [Google Scholar] [CrossRef] [Green Version]
- Stamos, Michael Z. 2008. Optimal consumption and portfolio choice for pooled annuity funds. Insurance: Mathematics and Economics 43: 56–68. [Google Scholar] [CrossRef]
- Weinert, Jan Hendrik, and Helmut Gründl. 2021. The modern tontine. European Actuarial Journal 11: 49–86. [Google Scholar] [CrossRef]
- 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]

Arrangement | Moderate Aggregate Deviations | Major Aggregate Deviations |
---|---|---|

Fixed benefits | 0.069% | 0.242% |

Benefits linked to surv. prob., case (a) | 0.003% | 0.025% |

Benefits linked to act. value, case (a) | 0.013% | 0.033% |

Benefits linked to surv. prob., case (b) | 0.006% | 0.093% |

Benefits linked to act. value, case (b) | 0.013% | 0.019% |

**Table 2.**Benefit amount ${b}_{t}$ (expected value and 0.01- and 0.99-quantiles) for selected times. Moderate aggregate deviations.

Timet | Fixed benefits | Benefits Linked to Surv. Prob., Case (a) | Benefits Linked to Act. Value, Case (a) | ||||

Exp. value | 0.01-quant. | 0.99-quant. | Exp. value | 0.01-quant. | 0.99-quant. | ||

0 | 5.199 | 5.242 | 5.236 | ||||

5 | 5.199 | 5.242 | 5.213 | 5.272 | 5.236 | 5.069 | 5.408 |

10 | 5.199 | 5.242 | 5.168 | 5.320 | 5.236 | 5.052 | 5.426 |

15 | 5.199 | 5.242 | 5.097 | 5.398 | 5.236 | 5.037 | 5.443 |

20 | 5.199 | 5.243 | 4.984 | 5.526 | 5.236 | 5.029 | 5.454 |

25 | 5.199 | 5.246 | 4.805 | 5.741 | 5.236 | 5.037 | 5.447 |

30 | 5.199 | 5.253 | 4.527 | 6.114 | 5.236 | 5.093 | 5.388 |

Time $\mathit{t}$ | Benefits Linked to Surv. Prob., Case (b) | Benefits Linked to Act. Value, Case (b) | |||||

Exp. value | 0.01-quant. | 0.99-quant. | Exp. value | 0.01-quant. | 0.99-quant. | ||

0 | 5.240 | 5.236 | |||||

5 | 5.240 | 5.211 | 5.270 | 5.236 | 5.069 | 5.408 | |

10 | 5.240 | 5.166 | 5.318 | 5.236 | 5.052 | 5.426 | |

15 | 5.240 | 5.095 | 5.396 | 5.236 | 5.037 | 5.443 | |

20 | 5.241 | 4.982 | 5.524 | 5.236 | 5.029 | 5.454 | |

25 | 5.243 | 4.804 | 5.739 | 5.236 | 5.037 | 5.447 | |

30 | 5.244 | 4.716 | 5.764 | 5.236 | 5.093 | 5.388 |

**Table 3.**Benefit amount ${b}_{t}$ (expected value and 0.01- and 0.99-quantiles) for selected times. Major aggregate deviations.

Timet | Fixed benefits | Benefits Linked to Surv. Prob., Case (a) | Benefits Linked to Act. Value, Case (a) | ||||

Exp. value | 0.01-quant. | 0.99-quant. | Exp. value | 0.01-quant. | 0.99-quant. | ||

0 | 5.090 | 5.228 | 5.223 | ||||

5 | 5.090 | 5.228 | 5.143 | 5.326 | 5.221 | 4.717 | 5.784 |

10 | 5.090 | 5.228 | 5.011 | 5.487 | 5.222 | 4.671 | 5.846 |

15 | 5.090 | 5.231 | 4.806 | 5.757 | 5.223 | 4.629 | 5.905 |

20 | 5.090 | 5.240 | 4.494 | 6.223 | 5.224 | 4.607 | 5.946 |

25 | 5.090 | 5.253 | 4.030 | 6.534 | 5.225 | 4.635 | 5.929 |

30 | 5.090 | 5.247 | 3.921 | 6.534 | 5.226 | 4.804 | 5.736 |

Time $\mathit{t}$ | Benefits Linked to Surv. Prob., Case (b) | Benefits Linked to Act. Value, Case (b) | |||||

Exp. value | 0.01-quant. | 0.99-quant. | Exp. value | 0.01-quant. | 0.99-quant. | ||

0 | 5.184 | 5.231 | |||||

5 | 5.185 | 5.101 | 5.282 | 5.229 | 4.725 | 5.755 | |

10 | 5.185 | 4.970 | 5.442 | 5.229 | 4.708 | 5.755 | |

15 | 5.187 | 4.767 | 5.703 | 5.229 | 4.708 | 5.755 | |

20 | 5.185 | 4.666 | 5.703 | 5.229 | 4.708 | 5.755 | |

25 | 5.179 | 4.666 | 5.703 | 5.231 | 4.708 | 5.755 | |

30 | 5.175 | 4.666 | 5.703 | 5.233 | 4.812 | 5.745 |

Arrangement | Moderate Aggregate Deviations | Major Aggregate Deviations |
---|---|---|

Fixed benefits | 0.845% | 2.933% |

Benefits linked to surv. prob., case (a) | 0.038% | 0.311% |

Benefits linked to act. value, case (a) | 0.155% | 0.400% |

Benefits linked to surv. prob., case (b) | 0.076% | 1.132% |

Benefits linked to act. value, case (b) | 0.155% | 0.236% |

**Table 5.**Individual reserve ${V}_{t}$ and components ${V}_{t}^{\left[\mathrm{ben}\right]}$, ${V}_{t}^{\left[\mathrm{fee}\right]}$, at selected times t. Moderate aggregate deviations.

Fixed Benefits | ||||

Time $\mathit{t}$ | Age $\mathit{x}+\mathit{t}$ | ${\mathit{V}}_{\mathit{t}}$ | $\frac{{\mathit{V}}_{\mathit{t}}^{\left[\mathrm{ben}\right]}}{{\mathit{V}}_{\mathit{t}}}$ | $\frac{{\mathit{V}}_{\mathit{t}}^{\left[\mathrm{fee}\right]}}{{\mathit{V}}_{\mathit{t}}}$ |

0 | 65 | 100.000 | 99.155% | 0.845% |

5 | 70 | 80.512 | 99.286% | 0.714% |

10 | 75 | 62.970 | 99.410% | 0.590% |

15 | 80 | 47.576 | 99.525% | 0.475% |

20 | 85 | 34.378 | 99.632% | 0.368% |

25 | 90 | 23.095 | 99.736% | 0.264% |

30 | 95 | 12.387 | 99.847% | 0.153% |

Benefits Linked to Surv. Prob., Case (a) | ||||

Time $\mathit{t}$ | Age $\mathit{x}+\mathit{t}$ | ${\mathit{V}}_{\mathit{t}}$ | $\frac{{\mathit{V}}_{\mathit{t}}^{\left[\mathrm{ben}\right]}}{{\mathit{V}}_{\mathit{t}}}$ | $\frac{{\mathit{V}}_{\mathit{t}}^{\left[\mathrm{fee}\right]}}{{\mathit{V}}_{\mathit{t}}}$ |

0 | 65 | 100.000 | 99.962% | 0.038% |

5 | 70 | 80.614 | 99.968% | 0.032% |

10 | 75 | 63.126 | 99.974% | 0.026% |

15 | 80 | 47.750 | 99.979% | 0.021% |

20 | 85 | 34.545 | 99.984% | 0.016% |

25 | 90 | 23.242 | 99.988% | 0.012% |

30 | 95 | 12.496 | 99.993% | 0.007% |

Benefits Linked to Act. Value, Case (a) | ||||

Time $\mathit{t}$ | Age $\mathit{x}+\mathit{t}$ | ${\mathit{V}}_{\mathit{t}}$ | $\frac{{\mathit{V}}_{\mathit{t}}^{\left[\mathrm{ben}\right]}}{{\mathit{V}}_{\mathit{t}}}$ | $\frac{{\mathit{V}}_{\mathit{t}}^{\left[\mathrm{fee}\right]}}{{\mathit{V}}_{\mathit{t}}}$ |

0 | 65 | 100.000 | 99.845% | 0.155% |

5 | 70 | 80.597 | 99.869% | 0.131% |

10 | 75 | 63.102 | 99.892% | 0.108% |

15 | 80 | 47.721 | 99.913% | 0.087% |

20 | 85 | 34.515 | 99.933% | 0.067% |

25 | 90 | 23.207 | 99.952% | 0.048% |

30 | 95 | 12.458 | 99.972% | 0.028% |

Arrangement | Moderate Aggregate Deviations | Major Aggregate Deviations | ||
---|---|---|---|---|

${\mathrm{PVFP}}_{0}^{\left[\mathrm{pool}\right]}$ | $\frac{{\mathrm{BV}}_{0}^{\left[\mathrm{pool}\right]}}{{\mathrm{PVFP}}_{0}^{\left[\mathrm{pool}\right]}}$ | ${\mathrm{PVFP}}_{0}^{\left[\mathrm{pool}\right]}$ | $\frac{{\mathrm{BV}}_{0}^{\left[\mathrm{pool}\right]}}{{\mathrm{PVFP}}_{0}^{\left[\mathrm{pool}\right]}}$ | |

Fixed benefits | 0.820 | 25.160% | 2.713 | 29.071% |

Benefits linked to surv. prob., case (a) | 0.034 | 52.960% | 0.341 | 28.146% |

Benefits linked to act. value, case (a) | 0.151 | 42.137% | 0.387 | 23.347% |

Benefits linked to surv. prob., case (b) | 0.076 | 17.690% | 1.140 | 24.734% |

Benefits linked to act. value, case (b) | 0.151 | 42.137% | 0.246 | 9.468% |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Olivieri, A.
Designing Annuities with Flexibility Opportunities in an Uncertain Mortality Scenario. *Risks* **2021**, *9*, 189.
https://doi.org/10.3390/risks9110189

**AMA Style**

Olivieri A.
Designing Annuities with Flexibility Opportunities in an Uncertain Mortality Scenario. *Risks*. 2021; 9(11):189.
https://doi.org/10.3390/risks9110189

**Chicago/Turabian Style**

Olivieri, Annamaria.
2021. "Designing Annuities with Flexibility Opportunities in an Uncertain Mortality Scenario" *Risks* 9, no. 11: 189.
https://doi.org/10.3390/risks9110189