Mean-variance dynamic optimality for DC pension schemes

It is well known that the mean-variance portfolio selection is a time-inconsistent optimization problem. In the current literature, this time inconsistency is often tackled with either a game theoretical approach (Basak and Chabakauri, 2010, and Bjork and Murgoci, 2010) or a so-called precommitment approach (Zhou and Li, 2000). The framework of a defined contribution (DC) pension scheme, which we deal with in this work, makes no exception, with a number of papers computing either the Nash equilibrium or the precommitment strategy in the presence of a variety of financial markets. Here, we solve a mean-variance portfolio selection problem for a DC pension fund through the dynamically optimal approach introduced by Pedersen and Peskir (2017), and we compare the dynamically optimal strategy with the precommitment one. We show that both strategies are the solution to target-based problems. The precommitment strategy has a constant target, while the dynamically optimal strategy has a time-varying target whose expectation coincides with the constant target of the previous case. We also show that the expected wealth is the same under the two approaches. Numerical simulations show that, with respect to the precommitment strategy, the dynamically optimal strategy provides: (i) a larger variance of wealth, (ii) a less volatile asset allocation, and (iii) a larger effectiveness in reacting against most unfavorable and persistent market conditions.