Robust Optimization Methods in Modern Portfolio Theory
One of the main questions that an investor faces when investing, is how to allocate resources among the variety of different assets. Markowitz portfolio optimization problem as a part of modern portfolio theory (MPT) solve this problem. The main idea of the Markowitz model is to create such portfolio management strategies that lead to the biggest expected returns with minimal possible risk. The expected returns (represent profit) and the covariance matrix (represents risk measure) of assets of the investor's portfolio are his inputs. After solving the optimization problem. its direct outputs are the weights of the distribution of funds over specific assets in order to maximize expected profit. However, needed inputs are difficult to predict which causes low credibility of the results. Therefore, we will introduce a robust way to solve this problem. We define filter matrices that make the selection of the right parameters more uncertain. After that, we solve this problem by so-called worst-case optimization, when at every point of the domain of optimization problem we deliberately choose the worst possible parameter from the relevant uncertainty set of parameters. In our case, the set of uncertainty consists of filter matrices, which change the data slightly so also expected returns and covariance matrix will change. The main goal of this paper is to examine this problem in the real time portfolio modeling when at every point in time we recalculate weights for assets and adjust portfolio composition. Finally, we discuss variations of this model when the addition of the new parameter gives a minimization function that will enable us to use the Markowitz model in solving more difficult non-linear Hamilton-Jacobi-Bellman differential equation.
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