On Efficiency of Mean-Variance Based Portfolio Selection in DC Pension Schemes

Elena Vigna
2010 Social Science Research Network  
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 (2000) and show that it is equivalent to a target-based optimization problem, consisting in 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
more » ... selecting the individual's risk aversion coefficient into the easiest 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 comparison between an efficient portfolio and a portfolio that is optimal according to the more general criterion of maximization of 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. As a particular case, we investigate the quite popular classes of CARA and CRRA utility functions. In these cases, we prove the intuitive but not trivial results 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. Numerical investigations stress the impact of the time horizon on the extent of mean-variance inefficiency of CARA and CRRA utility functions. While at instantaneous level EU-optimality and efficiency coincide (see Merton (1971)), we find that for short durations they do not differ significantly. However, for longer durations -that are typical in pension funds -the extent of inefficiency turns out to be remarkable and should be taken into account by pension fund investment managers seeking appropriate rules for portfolio selection. Indeed, this result is a further element that supports the use of the target-based approach in DC pension schemes. . A first version of this paper was circulated under the title "Mean-variance inefficiency of CRRA and CARA utility functions for portfolio selection in defined contribution pension schemes". I am grateful to Paolo Ghirardato and Massimo Marinacci for useful advice. I especially thank Bjarne Højgaard and Luis Viceira for fruitful discussions. All remaining errors are mine. The mean-variance approach Most of the results of this section can be found in Højgaard and Vigna (2007). The model A member of a defined contribution pension scheme is faced with the problem of how to invest optimally the fund at her disposal and the future contributions to be paid in the fund. The financial market available for her portfolio allocation problem is the Black-Scholes model (see e.g. Björk (1998)). This consists of two assets, a riskless one, whose price B(t) follows the dynamics:
doi:10.2139/ssrn.1775806 fatcat:qnjlg6p36reo7bu4bns2dfc4ti