Figure 6: Funding ratio dynamics CPhl method
8 Conclusion
In this paper, we have used different risk allocation rules in order to study the welfare effect in a funded pension scheme. The plateau and exponential allocation rules are discussed, whereby benefits and/or pension contributions depend on the funding ratio of the fund. The main contribution of this paper are these non-linear allocation methods, which are an extension of the linear allocation methods described in (Cui et al.,2011).
We have optimized the risk allocation rules with respect to the age groups 25, 45 and 65, and the social optimum. We found that the hybrid allocation rules, where both the benefits and the pension contributions are a function of the funding ratio, lead to the highest certainty equivalent and are therefore preferred. In the optimization of the exponential and plateau allocation method, a funding ratio constraint of 80% was introduced in order to guarantee a pension scheme which is sustainable for future generations.
In the comparison of the three hybrid allocation methods, it is shown that the 5% quantile funding ratio remains stable for all three allocation methods. Furthermore, it followed that the plateau allocation and the exponential allocation methods are not an improvement of the linear allocation method in terms of the certainty equivalent and the distribution of the consumption levels. For the social optimum the CPhl has a certainty equivalent which is 1.1 and 1.4 percent higher as the CPhp and CPhemethods. It is argued that this difference is negligible, especially if we take the benefits of the plateau allocation methods into account.
In the plateau allocation methods, participants enjoy an income level which is more stable as the linear allocation rule found byCui et al.(2011). A stable income is important, as it makes participants better able to meet their financial obligations, and make financial decisions about the future. Furthermore, pension cuts occur less often in the plateau allocation model, which makes it a popular model from a political point of view.
In conclusion, the plateau allocation method could be seen as an important extension of the literature. It is not performing significantly worse in terms of certainty equivalents and funding ratio dynamics as previous studied models. However, this model contains important features, like a higher degree of income stability, which makes the model better suited to implement in practice, in comparison to previous studied models.
9 Suggestions for further research
This paper attempts to provide a realistic comparison of the consumption levels for different intergenerational risk allocation rules. However, in order to make the model insightful, several
simplifications have been made. Therefore, this paper could be improved by adding several extensions of the model.
First of all, we assumed a constant risk-free interest rate and a constant interest rate to value our liabilities. Changing these into stochastic models, which are based on the market performance, will give a better view of reality. The effect of stochastic interest rates are first of all that the target liabilities are no longer time indifferent and this makes the surplus of the fund dynamics dependent on both the asset dynamics and the liability dynamics. Furthermore, the asset dynamics will change as rf is not a constant anymore. This implies that more uncertainty will be present in the funding ratio dynamics, as these are now not only subject to the stock market volatility, but also to changes in the risk free rate.
Furthermore, we have used the conventional CRRA preference structure which is an additive and homogeneous utility function. This implies that the marginal utility of the consumption of this year is independent from the consumption in any other period. In practice, this assumption seems unrealistic and the Epstein-Zin preference model could solve this problem. In this model, the risk aversion coefficient (RRA) and the elasticity of intertemporal substitution (EIS) are separated (Epstein and Zin, 2013). The EIS measures the attitude of participants towards uneven consumption levels over time. The risk aversion measures the attitude of the participant towards the risk from varying consumption levels. In the CRRA preference structure, these measures are inversely connected, while the model described by Epstein and Zin(2013) allows these quantities to be independent (Heiberger,2018). The Epstein-Zin model has as advantage that we could more explicitly measure the effect of income fluctuations on the consumers welfare.
This gives a better view of reality. Furthermore, it could show the benefits of the plateau allocation rule, which allows for a relatively stable consumption level, more explicitly.
Another suggestion is to implement changing risk aversion coefficients over time. In for exampleSahm(2012), it is argued that risk aversion decreases with an improvement in macroe-conomic conditions and slightly increases with age. Implementing dynamic risk aversion values, would change the value of the certainty equivalents. Furthermore, taking the degree of income fluctuations into account in the risk aversion coefficients, gives a comparison which is more representative for reality.
Thirdly, mortality models, and therefore longevity risk or mortality risk are not taken into account in our model, while longevity risk is one of the major risks the pension fund is currently facing. Longevity risk is defined as the uncertainty in the long-term probability of survival (Cairns et al.,2006), and mortality risk is defined as the uncertainty in future mortality rates.
In our model, the maximum age a participant reaches is 85, this age is reached with certainty.
The target liabilities are based on this fact and therefore constant. Implementing mortality rates implies that first of all a model should be created to forecast the mortality rates. This could be the model suggested in Lee and Carter(1992), as this model has become the leading statistical mortality model in the current demographic literature (Deaton and Paxson, 2004).
For ages above 90, the Coale-Kisker extrapolation method is suggested, due to a lack of data for the older ages (Coale and Guo, 1989).
These mortality models should be taken into account in the asset dynamics. This implies that the contribution and the benefit payments should be corrected with the mortality rate, which will create more uncertain asset dynamics. Especially the long-run mortality rates are uncertain, and this makes the long-term benefit payments uncertain. A small decrease in the mortality rate, could have a large impact on the benefit payments in the future. Furthermore, the target liabilities should also be corrected with the mortality rate. This again implies that liabilities are not time indifferent anymore and become uncertain in the long-run. Implementing the mortality rates in both the assets and the liabilities of a pension fund creates a funding ratio and a funding surplus, which are dependent on both the investment returns, and the mortality rate. In our model, this would imply that the contribution and/or benefit levels are also dependent on both the investment returns and the mortality rates. In this adapted model, the funding dynamics will be different as created in our results. Therefore, it is possible that other funding ratio constraints are necessary in order to create stability over time. Further research is needed to establish these results into more detail
Related to this, in this paper it is assumed that all generations have the same size, while in reality the size of a generation is depending on the fertility rate. In 2019, approximately 14.6%
of the total Dutch population was in their 50’s, while only 10.4% of the total Dutch population was under the 10 years old.7 This implies that each age cohort is not contribution or receiving the same payments anymore. Therefore, the weight of each age cohort is different in the liability
7Source CBS
calculations. Furthermore, in the asset dynamics we can not simply assume that there are 40, equally sized age cohorts contributing to the system. The same results holds for the 20 equally sized age cohorts that are receiving benefits. The asset and liability dynamics are therefore subject to the number of participants starting at the age of 25. This implies that the population dynamics, and more important the rate of birth in the Dutch population, should be taken into account.
Another possible extension, is to take the career path of individuals into account. Currently an income level which is equal to unity is assumed for each participant. However, an income level that is based on the career path of an individual could further improve the model. As contribution levels are a percentage of income, this extension will lead to different contribution levels for each participant. Furthermore, benefit levels will be different for each career path. For participants with a steep career path, it could be beneficial to have relatively low contribution levels at the start of their career and high contribution levels at the end of their career. This will enable consumption smoothing which is welfare improving. In order to implement different career paths, an estimation of the future earnings of the participants should be made. Accord-ing to Bosworth et al. (2000), future earnings could be predicted making use of two possible approaches. The first approach is to make use of income data of previous years, whereby we search for a trend in the income levels, such that we can predict future income levels. Another possibility is the traditional method, whereby some ’stylized’ career paths are predetermined, that represent the average population. Implementing these career paths will give a model that is a better representation of reality and will further improve the model.
References
Bams, D., Schotman, P. C., and Tyagi, M. (2016). Optimal risk sharing in a collective defined contribution pension system.
Beetsma, R., Constandse, M., Cordewener, F., Romp, W., and Vos, S. (2015). The dutch pension system and the financial crisis. CESifo DICE Report, 13(2):14–19.
Bikker, J., Knaap, T., and Romp, W. (2011). Real pension rights as a control mechanism for pension fund solvency. SSRN Electronic Journal.
Bikker, J. A., Broeders, D. W., and De Dreu, J. (2007). Stock market performance and pen-sion fund investment policy: rebalancing, free float, or market timing. Discuspen-sion Paper Series/Tjalling C. Koopmans Research Institute, 7(27).
Bohn, H. et al. (2003). Intergenerational risk sharing and fiscal policy. UCSB Department of Economics Working Paper. Available at¡ http://econ. ucsb. edu/% 7Ebohn/papers/igrisk. pdf.
Bonenkamp, J., Meijdam, L., Ponds, E., and Westerhout, E. (2014). Reinventing intergenera-tional risk sharing. Netspar Panel Paper, (40).
Bosworth, B., Burtless, G., and Steuerle, E. (2000). Lifetime earnings patterns, the distribution of future social security benefits, and the impact of pension reform. Social security bulletin, 63(4):74–98.
Bovenberg, A., Koijen, R., Nijman, T., and Teulings, C. (2007). Saving and investing over the life cycle and the role of collective pension funds. De Economist (Netherlands), 155(4):347–415.
Bovenberg, L. and Nijman, T. (2018). New dutch pension contracts and lessons for other countries. Journal of Pension Economics and Finance, 18:331 – 346.
Broadbent, J., Palumbo, M., and Woodman, E. (2006). The shift from defined benefit to defined contribution pension plans–implications for asset allocation and risk management. Reserve Bank of Australia, Board of Governors of the Federal Reserve System and Bank of Canada, pages 1–54.
Cairns, A. J., Blake, D., and Dowd, K. (2006). A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration. Journal of Risk and Insurance, 73(4):687–718.
Coale, A. and Guo, G. (1989). Revised regional model life tables at very low levels of mortality.
Population index, pages 613–643.
Cui, J., De Jong, F., and Ponds, E. (2011). Intergenerational risk sharing within funded pension schemes. Journal of Pension Economics & Finance, 10(1):1–29.
Deaton, A. S. and Paxson, C. (2004). Mortality, income, and income inequality over time in britain and the united states. In Perspectives on the Economics of Aging, pages 247–286.
University of Chicago Press.
Dijsselbloem, J., De Waegenaere, A., van Ewijk, C., van der Horst, A., Knoef, M., and Steenbeek, O. (2019). Advies commissie parameters.
Epstein, L. G. and Zin, S. E. (2013). Substitution, risk aversion and the temporal behavior of consumption and asset returns: A theoretical framework. In Handbook of the fundamentals of financial decision making: Part i, pages 207–239. World Scientific.
Goecke, O. (2013). Pension saving schemes with return smoothing mechanism. Insurance:
Mathematics and Economics, 53(3):678–689.
Gollier, C. (2008). Intergenerational risk-sharing and risk-taking of a pension fund. Journal of Public Economics, 92(5-6):1463–1485.
Heiberger, C. (2018). Asset Prices, Epstein Zin Utility, and Endogenous Economic Disasters.
doctoralthesis, Universit¨at Augsburg.
Kemna, A. A., Ponds, E. H., and Steenbeek, O. W. (2011). Pension funds in the netherlands.
Journal of Investment Consulting, 12(1):28–34.
Kortleve, N. and Ponds, E. (2009). Dutch pension funds in underfunding: Solving generational dilemmas. SSRN Electronic Journal.
Lee, R. D. and Carter, L. R. (1992). Modeling and forecasting us mortality. Journal of the American statistical association, 87(419):659–671.
Pennacchi, G. and Rastad, M. (2011). Portfolio allocation for public pension funds. Journal of Pension Economics & Finance, 10(2):221–245.
Ponds, E. H. and Van Riel, B. (2009). Sharing risk: the netherlands’ new approach to pensions.
Journal of Pension Economics & Finance, 8(1):91.
Queisser, M. and Whitehouse, E. (2007). Neutral or fair?: Actuarial concepts and pension-system design.
Rauh, J. D. (2009). Risk shifting versus risk management: Investment policy in corporate pension plans. The Review of Financial Studies, 22(7):2687–2733.
Romp, W. and Beetsma, R. (2020). Sustainability of pension systems with voluntary participa-tion. Insurance: Mathematics and Economics, 93:125–140.
Sahm, C. R. (2012). How much does risk tolerance change? The quarterly journal of finance, 2(04):1250020.
Siegmann, A. (2007). Optimal investment policies for defined benefit pension funds. Journal of pension economics & finance, 6(1):1–20.
Teulings, C. N. and De Vries, C. G. (2006). Generational accounting, solidarity and pension losses. de Economist, 154(1):63–83.
van der Lecq, S., Dellaert, B., Swinkels, L., and Alserda, G. (2016). Pension risk preferences: A personalized elicitation method and its impact on asset allocation.
10 Appendix
10.1 Simulation consumption exponential allocation methods
(a) Confidence interval for the consumption level of workers for the DBceallocation method
(b) Confidence interval for the consumption level of retirees for the CPbe allocation method
(c) Confidence interval for the consumption level of workers for the CPhe allocation method
(d) Confidence interval for the consumption level of retirees for the CPhe allocation method
Figure 7: Consumption dynamics for DBec CPbe and CPhe method
(a) DBce method contribution dynamics (b) CPbemethod benefit dynamics
(c) CPhemethod contribution dynamics (d) CPhemethod benefit dynamics Figure 8: Contribution and benefit dynamics for exponential methods in year 30 of the simulation