In this article we solve the problem of maximizing the expected utility of future consumption and terminal wealth to determine the optimal pension or life-cycle fund strategy for a cohort of pension fund investors. The setup is strongly related to a DC pension plan where additionally (individual) consumption is taken into account. The consumption rate is subject to a time-varying minimum level and terminal wealth is subject to a terminal floor. Moreover, the preference between consumption and terminal wealth as well as the intertemporal coefficient of risk aversion are time-varying and therefore depend on the age of the considered pension cohort. The optimal consumption and investment policies are calculated in the case of a Black–Scholes financial market framework and hyperbolic absolute risk aversion (HARA) utility functions. We generalize Ye (American control conference, 2008) by adding an age-dependent coefficient of risk aversion and extend Steffensen (J Econ Dyn Control 35(5):659–667, 2011), Hentschel (Planning for individual retirement: optimal consumption, investment and retirement timing under different preferences and habit persistence. Ph.D. thesis, Ulm University, 2016) and Aase (Stochastics 89(1):115–141, 2017) by considering consumption in combination with terminal wealth and allowing for consumption and terminal wealth floors via an application of HARA utility functions. A case study on fitting several models to realistic, time-dependent life-cycle consumption and relative investment profiles shows that only our extended model with time-varying preference parameters provides sufficient flexibility for an adequate fit. This is of particular interest to life-cycle products for (private) pension investments or pension insurance in general.
- Age-dependent risk aversion
- HARA utility function
- Martingale method
- Optimal life-cycle consumption and investment
- Pension investments