Mean-variance dynamic optimality for DC pension schemes

In this paper we deal with the mean-variance portfolio selection for a defined contribution (DC) pension fund. Since this problem is time-inconsistent, a number of papers have proposed to tackle it through either a Nash equilibrium approach or a precommitment strategy. Here, we adopt the dynamically optimal approach introduced by Pedersen and Peskir (Math Financ Econ 11:137-160, 2017), and we compare the dynamically optimal strategy with the precommitment one. While it is well known that the precommitment strategy is the solution to a target-based problem, we show that the same holds for the dynamically optimal strategy. In particular, 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 applications show that (i) the median of the risky asset's share is lower for the precommitment than the dynamically optimal strategy; (ii) the amount of money invested in the precommitment risky portfolio is highly more volatile than in the dynamically optimal case; (iii) the variance of wealth is lower with the precommitment strategy than with the dynamically optimal one; (iv) under scenarios of extreme market returns (either good or bad), the dynamically optimal strategy allows a more effective reaction because of the continuous adjustment of the final target.