# An Indexation Mechanism for Retirement Age: Analysis of the Gender Gap

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Motivation

#### Increasing Retirement Age and Pension Systems Sustainability

## 2. The Indexation Mechanism for Retirement Age

^{(M)}). In other terms the EPPD

^{(M)}is a fixed number that represents the expected number of years during which pension payments are due for cohort C*. A similar index is the basis for the analysis pursued in Bisetti and Favero (2014). According to the longevity trend, the real life expectancy at age ${x}_{0}^{*}$ will change for future generations. Then, given the benchmark, for a fixed mortality model M and for each of the selected cohort C, we determine the age ${x}_{0}^{*}+j$ at which life expectancy does not exceed the EPPD

^{(M)}for the first time. To this aim, we evaluate ${e}_{{x}_{0}^{*}+j,C}^{\left(M\right)}$ for increasing age span j = 1, 2…. and we index the retirement age ${x}_{0}^{*}$ by shifting it onwards to reach the EPPD

^{(M)}. We define the lag for the cohort C under the model M as follows:

^{(M)}, if model M is applied for mortality forecasts. In other words, it will be obliged to pay a constant monthly payment for an expected number of years not exceeding the EPPD

^{(M)}. For the illustrative purposes of the study, here we consider benchmark cohort ${C}^{*}$, retirement age ${x}_{0}^{*}$ and corresponding EPPD

^{(M)}but it should be noted that for real applications these values could be updated in a dynamic way, for instance being re-evaluated on the basis of subsequent cohorts.

## 3. Modelling Mortality

#### 3.1. Generalised Age-Period-Cohort Stochastic Mortality Models (GAPC)

- the Lee-Carter model (LC)
- the cohort extension of the Lee-Carter model given by Renshaw and Haberman (RH)
- the reduced Plat model (Plat)

#### 3.2. Fitting Models and Model Risk

## 4. Forecasting

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Albarrán, Lozano Irene, Juan Miguel Marín Díazaraque, Pablo J. Alonso, and Andrés Gustavo Benchimol. 2016. Model Uncertainty Approach in Mortality Projection with Model Assembling Methodologies. WS 23434. Madrid: Universidad Carlos III de Madrid. [Google Scholar]
- Alonso-García, Jennifer, and Pierre Devolder. 2017. Liquidity and Solvency in Pay-as-You-Go Defined Contribution Pension Schemes: A Continuous OLG Sustainability Framework. Available online: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3000778 (accessed on 2 January 2019).
- Alonso-García, Jennifer, María del Carmen Boado-Penas, and Pierre Devolder. 2018. Adequacy, fairness and sustainability of pay-as-you-go-pension-systems: Defined benefit versus defined contribution. The European Journal of Finance 24: 1100–22. [Google Scholar] [CrossRef]
- Berkeley Human Mortality Database. 2014. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available online: https://www.mortality.org/ or https://www.humanmortality.de (accessed on 25 November 2016).
- Bisetti, Emilio, and Carlo A. Favero. 2014. Measuring the impact of longevity risk on pension Systems: the case of Italy. North American Actuarial Journal 18: 87–103. [Google Scholar] [CrossRef]
- Booth, Heather, and Leonie Tickle. 2008. Mortality Modelling and Forecasting: A Review of Methods. Annals of Actuarial Science 3: 3–43. [Google Scholar] [CrossRef]
- Booth, Heather, Rob Hyndman, Leonie Tickle, and Piet de Jong. 2006. Lee-Carter mortality forecasting: A multi-country comparison of variants and extensions. Demographic Research 15: 289–310. [Google Scholar] [CrossRef]
- Brouhns, Natacha, Michel Denuit, and Jeroen K. Vermunt. 2002. A Poisson log-bilinear regression approach to the construction of projected lifetables. Insurance: Mathematics and Economics 31: 373–93. [Google Scholar] [CrossRef]
- Brouhns, Natacha, Michel Denuit, and Ingrid Van Keilegom. 2005. Bootstrapping the Poisson log-bilinear model for mortality forecasting. Scandinavian Actuarial Journal 3: 212–24. [Google Scholar] [CrossRef]
- Buckland, Stephen Terrence, Kenneth P. Burnham, and Nicole H. Augustin. 1997. Model Selection: An Integral Part of Inference. Biometrics 53: 603–18. [Google Scholar] [CrossRef]
- Burnham, Kenneth P., and David R. Anderson. 2003. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd ed. New York: Springer. [Google Scholar]
- Cairns, Andrew J. G., David Blake, and Kevin Dowd. 2006. A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration. Journal of Risk and Insurance 73: 687–718. [Google Scholar] [CrossRef]
- Chlon-Dominczak, Agnieszka, Daniele Franco, and Edward Palmer. 2012. The First Wave of NDC Reforms: The Experiences of Italy, Latvia, Poland, and Sweden. In Nonfinancial Defined Contribution Pension Schemes in a Changing Pension World. Washington, DC: The World Bank, vol. 1. [Google Scholar]
- Coppola, Mariarosaria, Maria Russolillo, and Rosaria Simone. 2018a. Flexible Retirement Scheme for the Italian Mortality experience. In Demography and Health Issues. Cham: Springer, pp. 325–36. [Google Scholar]
- Coppola, Mariarosaria, Maria Russolillo, and Rosaria Simone. 2018b. Risk and Uncertainty for Flexible Retirement Schemes. In Mathematical and Statistical Methods for Actuarial Sciences and Finance. Cham: Springer. [Google Scholar]
- Denuit, Michel, Steven Haberman, and Arthur Renshaw. 2015. Longevity-contingent deferred life annuities. Journal of Pension Economics and Finance 14: 315–27. [Google Scholar] [CrossRef]
- DWP (Department of Work and Pension, UK). 2013. The Core Principle Underpinning Future State Pension Age Rises. Available online: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/263660/spa-background-note-051213_tpf_final.pdf (accessed on 27 July 2018).
- Ehiemua, Solomon. 2014. Gender longevity: Male/female disparity. International Journal of Academic Research and Reflection 2: 32–38. [Google Scholar]
- Finnish Centre for Pensions. 2018. Retirement ages in Member States. Available online: https://www.etk.fi/en/the-pension-system/international-comparison/retiremet-ages/ (accessed on 20 February 2019).
- Godínez-Olivares, Humberto, María del Carmen Boado-Penas, and Athanasios A. Pantelous. 2016a. How to finance pensions: Optimal strategies for pay-as-you-go pension systems. Journal of Forecast 35: 13–33. [Google Scholar] [CrossRef]
- Godínez-Olivares, Humberto, María del Carmen Boado-Penas, and Steven Haberman. 2016b. Optimal strategies for pay-as-you-go pension finance: A sustainability framework. Insurance: Mathematics and Economics 69: 117–26. [Google Scholar] [CrossRef]
- Hyndman, Rob J., and Yeasmin Khandakar. 2008. Automatic time series forecasting: The forecast package for R. Journal of Statistical Software 26. [Google Scholar] [CrossRef]
- Lee, Ronald D., and Lawrence R. Carter. 1992. Modelling and Forecasting U.S. Mortality. Journal of the American Statistical Association 87: 659–71. [Google Scholar]
- Pattinson, John, Klim McPherson, Steven Haberman, and Colin Blakemore. 2012. Life Expectancy: Past and future variations by gender in England & Wales. Available online: https://www.longevitypanel.co.uk/_files/life-expectancy-by-gender.pdf (accessed on 25 July 2018).
- Plat, Richard. 2009. On stochastic mortality modeling. Insurance: Mathematics and Economics 45: 393–404. [Google Scholar]
- Renshaw, Arthur E., and Steven Haberman. 2006. A cohort-based extension to the Lee-Carter model for mortality reduction factors. Insurance: Mathematics and Economics 38: 556–70. [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]
- Sartor, Nicola, Laurence J. Kotlikoff, and Willi Leibfritz. 1999. Generational Accounts for Italy. Edited by Alan J. Auerbach, Laurence J. Kotlikoff and Willi Leibfritz. In Generational Accounting Around the World. Chicago: The University of Chicago Press. [Google Scholar]
- Spedicato, Giorgio Alfredo. 2013. The lifecontingencies package: Performing financial and actuarial mathematics calculations in R. Journal of Statistical Software 55. [Google Scholar] [CrossRef]
- Sundberg, Louise, Neda Agahi, Johan Fritzell, and Stefan Fors. 2018. Why is the gender gap in life expectancy decreasing? The impact of age- and cause-specific mortality in Sweden 1997–2014. International Journal of Public Health 63: 673–81. [Google Scholar] [CrossRef]
- Tabeau, Ewa. 2001. A Review of Demographic Forecasting Models for Mortality. Edited by Ewa Tabeau, Anneke van den Berg Jeths and Christopher Heathcote. In Forecasting in Developed Countries: From Description to Explanation. Dordrecht: Kluwer Academic Publishers. [Google Scholar]
- Villegas, Andrés M., Vladimir K. Kaishev, and Pietro Millossovich. 2018. StMoMo: An R Package for Stochastic Mortality Modelling. Journal of Statistical Software 83: 1466. [Google Scholar] [CrossRef]
- Wong-Fupuy, Carlos, and Steven Haberman. 2004. Projecting mortality trends: Recent developments in the United Kingdom and the United States. North American Actuarial Journal 8: 56–83. [Google Scholar] [CrossRef]

Model | Men | Women | ||
---|---|---|---|---|

AIC | AIC Weights | AIC | AIC Weights | |

LC | 41,266.93 | 0.274 | 32894.02 | 0.306 |

RH | 26,516.93 | 0.362 | 26349.78 | 0.346 |

PLAT | 26,412.91 | 0.364 | 26220.45 | 0.348 |

© 2019 by the authors. 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Coppola, M.; Russolillo, M.; Simone, R.
An Indexation Mechanism for Retirement Age: Analysis of the Gender Gap. *Risks* **2019**, *7*, 21.
https://doi.org/10.3390/risks7010021

**AMA Style**

Coppola M, Russolillo M, Simone R.
An Indexation Mechanism for Retirement Age: Analysis of the Gender Gap. *Risks*. 2019; 7(1):21.
https://doi.org/10.3390/risks7010021

**Chicago/Turabian Style**

Coppola, Mariarosaria, Maria Russolillo, and Rosaria Simone.
2019. "An Indexation Mechanism for Retirement Age: Analysis of the Gender Gap" *Risks* 7, no. 1: 21.
https://doi.org/10.3390/risks7010021