Portfolio frontiers with restrictions to tracking error volatility and value at risk

Managers are often given the task of restricting their activity by maintaining both the value at risk (VaR) and the tracking error volatility (TEV) under control. Sometimes, these constraints can not be simultaneously satisfied because the VaR is independent from the benchmark portfolio. The possibility of managing these restrictions surely affects the portfolio performances and produces a wide variety of scenarios in the risk-return space. This paper aims to analyse various interactions between portfolio frontiers when restrictions upon TEV and VaR are jointly imposed. Specifically, the analytic solutions for the intersections among the "Mean-Variance Frontier" (Markowitz, 1959), "Mean-TEV Frontier" (Roll, 1992), "Constrained TEV Frontier (Jorion, 2003) and "Constrained Mean-TEV Frontier" (Alexander and Baptista, 2008) are provided; moreover, short numerical methods are proposed when those solutions are not available. All the results are obtained under the classical hypothesis of normally distributed expected returns which translates into an optimization in the mean-variance space. Short sales are also allowed. However, a new portfolio frontier is introduced: this is the ``Fixed VaR-TEV Frontier'' (FVTF) which has two important properties: on the one hand, it allows managers to satisfy TEV and VaR restrictions at the same time and, on the other hand, it contains portfolios less risky than those lying on the Constrained Mean-TEV Frontier. In order to illustrate the theoretical results of the paper, an empirical example is also provided: in practice, it consists of an asset allocation problem among ten asset classes taking TEV and VaR constraints into consideration. The available data are given by the quarterly returns of the 50 stocks composing the DJ Eurostoxx 50 index over the period which ranges from the first quarter of 2003 to the fourth quarter 2010. Finally, all the program  files used to carry out the empirical analysis are freely available at http://utenti.dea.univpm.it/palomba/TEV-VaR.html