Stochastic Regularization for the Mean-Variance Allocation Scheme

Despite being based on sound principles, the original Markowitz portfolio allocation theory cannot produce sound allocations, and restrictions or modifications need to be imposed from outside the theory in order to obtain meaningful portfolios. This is unsatisfactory, and the reasons for this failure are discussed, in particular the unavoidable small eigenvalues of the covariance. Within the original principles of risk minimization and return maximization, several modifications of the original theory are introduced. First, the strategic and tactical time horizons are separated. A base long term allocation is chosen at the strategic time horizon, while the portfolio is optimized at the tactical time horizon using information from the price histories. Second, the tactical portfolio is financed by the strategic one, and a funding operator is introduced. The corresponding optimal allocation (without constraints) has one free parameter fixing the leverage. Third, the transaction costs are taken into account. This includes the costs for the current re-allocation, but crucially the expected costs of the next reallocation. This last term is depending on the sensitivity of the allocation with respect to the covariance, and the expectation introduces another dependency on the (inverse) covariance. This new term regularizes the original minimization problem by modifying the lower part of the spectrum of the covariance, leading to meaningful portfolios. Without constraints, the final Lagrangian can be minimized analytically, with a solution that has a similar structure compared to the original Markowitz solution, but with the inverse covariance regularized by the expected transaction costs