We will propose a branch and bound algorithm for calculating a globally optimal solution of a portfolio construction/rebalancing problem under concave transaction costs and minimal transaction unit constraints. We will employ the absolute deviation of the rate of return of the portfolio as the measure of risk and solve linear programming subproblems by introducing (piecewise) linear underestimating function for concave transaction cost functions. It will be shown by a series of numerical experiments that the algorithm can solve the problem of practical size in an efficient manner.

We will propese a branch and bound algoriLhm for solving a portfolio optimization model under nonconvex transaction costs, It is well knewn that the llnit transaction cost is larger when the amollnt of

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Index tracking is a very common and popular approach in portfolio management. When there is neither (nonconvex) transaction costs nor minimal transaction unit constraints, the problem can be formulated as a convex least square problem, so that it can be solved by standard methods. However, when the transaction cost is nonconvex and not negligible, or if there is a minimal unit constraint on the amount of transaction, the problem becomes a nonconvex minimization problem with discrete variables. In this paper, we will propose a branch and bound algorithm for solving this class of problems and show that it can solve an index tracking problem of practical size in a reasonable amount of computation time.

No abstract