On efficiency of mean--variance based portfolio selection in defined contribution pension schemes

We consider the portfolio selection problem in the accumulation phase of a defined contribution (DC) pension scheme. We solve the mean--variance portfolio selection problem using the embedding technique pioneered by Zhou and Li [ Appl. Math. Optimiz ., 2000, 42 , 19--33] and show that it is equivalent to a target-based optimization problem, consisting of the minimization of a quadratic loss function. We support the use of the target-based approach in DC pension funds for three reasons. Firstly, it transforms the difficult problem of selecting the individual's risk-aversion coefficient into the easier task of choosing an appropriate target. Secondly, it is intuitive, flexible and adaptable to the member's needs and preferences. Thirdly, it produces final portfolios that are efficient in the mean--variance setting. We address the issue of the comparison between an efficient portfolio and a portfolio that is optimal according to the more general criterion of maximization of the expected utility (EU). The two natural notions of Variance Inefficiency and Mean Inefficiency are introduced, which measure the distance of an optimal inefficient portfolio from an efficient one, focusing on their variance and on their expected value, respectively. We illustrate the general procedure for finding the mean--variance inefficiency for the HARA class of utility functions, and then focus on the popular classes of CARA and CRRA utility functions. In these cases we prove the intuitive but non-trivial result that the mean--variance inefficiency decreases with the risk aversion of the individual and increases with the time horizon and the Sharpe ratio of the risky asset. As a by-product we prove that the CARA and CRRA optimal portfolios are not mean--variance efficient. Numerical investigations stress the impact of the time horizon on extent of the mean--variance inefficiency of CARA and CRRA utility functions. While at the instantaneous level EU optimality and efficiency coincide (see Merton [ J. Econ. Theory , 1971, 3 , 373--413]), we find that, for short durations, they do not differ significantly. However, for longer durations, which are typical for pension funds, the extent of the inefficiency turns out to be remarkable and should be taken into account by pension fund investment managers seeking appropriate rules for portfolio selection. Indeed, we see this result as a fourth reason for supporting the use of the target-based approach in DC pension schemes.