A unified approach to portfolio selection in a tracking error framework with additional constraints on risk

• Restrictions of the set of βPB values giving rise to dominant portfolios with additional constraints on beta.
• Relationships between TEVfrontier with beta constraints and with other kinds of constraints, described in detail from a theoretical point of view and explained with a practical example.
• Exactly the same solution of the various optimization problems in terms of weights, that is invariance of the optimal allocation strategy whatever the constraint.
• Closed-form of the variance of tracking error under all kinds of constraint.
• TEV frontier with additional constraints on CVaR under portfolio return normality.

Most methods of performance evaluation and most allocation strategies are based on tracking error, that is the excess return of the managed portfolio with respect to the benchmark return. Analysis of the tracking error in a mean-variance framework has been performed by Roll (1992) who also investigated the impact of additional beta constraints, while Jorion (2003) considers constraints on total risk (portfolio variance). Alexander and Baptista (2010) add a constraint on the alpha of minimum tracking error variance portfolios. In other recent works, Alexander and Baptista (2008) and Palomba and Riccetti (2012) analyze the problem with Value at Risk (VaR) constraints under return normality assumption. This paper investigates the relationships between all these different approaches and provides a unified treatment. Moreover, analysis of the frontier of Conditional VaR constrained tracking error variance has been performed.