  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 (2008) (2008 American Control Conference, 356-362)
by adding an age-dependent coefficient of risk aversion and extend Steffensen
(2011) (Journal of Economic Dynamics and Control, 35(5), 659-667), Hentschel
(2016) (Doctoral dissertation, Ulm University) and Aase (2017) (Stochastics,
89(1), 115-141) 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.
