Multi-market portfolio optimization with conditional value at risk

The delegated portfolio management has been at the core of financial debates, leading to a growing research effort to provide modeling and solution approaches. This class of problems focuses on investors relying upon decentralized affiliates for the specialized selection of investment options. In this paper we propose a novel optimization framework for multi-market portfolio management, where a central headquarter delegates the market-wise portfolio selection to specialized affiliates. Being averse to risk, the headquarter endogenously sets the maximum expected loss (in the form of conditional value at risk) for the affiliates, who respond designing portfolios and retaining portions of the expected investment returns as management fees. In its essence, this problem constitutes a single-leader-multi-follower game, resulting from the decentralized investment design. Starting from a bilevel formulation, our results build on the equivalence with the high point relaxation to provide theoretical insights and numerical solution approaches. We show that the problem is NP-Hard and propose a decomposition procedure and strong valid inequalities, capable of boosting the efficiency of the computational solution, when instances become large. In the same line, optimality bounds exploiting overlooked properties of the conditional value at risk are deduced, to provide almost exact solutions with few seconds of computation. Building on this theoretical development, we conduct computational tests using comprehensive firm-level data from 1999 to 2014 on 7256 U.S. listed enterprises. These tests support the effectiveness of the decomposition procedure, as well as the one of the strong valid inequalities, improving the LP relaxation by up to 99.18%.