In this paper we propose an optimization framework for multi-markets portfolio management, where a central headquarter relies upon local affiliates for the market-wise selection of investment options. Being averse to risk, the headquarter endogenously selects the maximum expected loss (conditional value at risk) for the affiliates, who respond designing portfolios and selecting management fees. In its essence, this problem constitutes a single-leader-multi-follower game resulting from the decentralized design of the investment selection. Starting from a bilevel formulation, our results build on the equivalence with the high point relaxation and provide theoretical insights about the decentralized portfolio properties and numerical solution approaches. We show that the problem is NP-Hard and propose a decomposition procedure and strong valid inequalities, capable of substantially 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 to validate and compare the proposed investment framework, using comprehensive firm-level data from 1999 to 2014 on 7256 U.S. listed enterprises. The numerical tests supports the effectiveness of the decomposition procedure, as well as the one resulting from the inclusion of strong valid inequalities, improving the LP relaxation by up to 98%.