A VaR Black-Litterman Model for the Construction of Absolute Return Fund-of-Funds

This research was motivated by our work with the private investment group of an international bank. The objective is to construct fund-of-funds (FoF) that follow an absolute return strategy and meet the requirements imposed by the Value-at-Risk (VaR) market risk measure. We propose the VaR-Black Litterman model which accounts for the VaR and trading (diversification, buy-in threshold, liquidity, currency) requirements. The model takes the form of a probabilistic integer, non-convex optimization problem. We develop a solution method to handle the computational tractability issues of this problem. We first derive a deterministic reformulation of the probabilistic problem, which, depending on the information on the probability distribution of the FoF return, is the equivalent or a close approximation of the original problem. We then show that the continuous relaxation of the reformulated problem is a nonlinear and convex optimization problem for a wide range of probability distributions. Finally, we use a specialized nonlinear branch-and-bound algorithm which implements the new portfolio return branching rule to construct the optimal FoF. The practical relevance of the model and solution method is shown by their use by the private investment group of a financial institution for the construction of four FoFs that are now traded worldwide. The computational study attests that the proposed algorithmic technique is very efficient, outperforming, in terms of both speed and robustness, three state-of-the-art alternative solution methods and solvers.