Classical decomposition of Markowitz portfolio selection

In this study, we enhance Markowitz portfolio selection with graph theory for the analysis of two portfolios composed of either EU or US assets. Using a threshold-based decomposition of their respective covariance matrices, we perturb the level of risk in each portfolio and build the corresponding sets of graphs. We show that the "superimposition" of all graphs in a set allows for the (re)construction of the efficient frontiers. We also identify a relationship between the Sharpe ratio (SR) of a given portfolio and the topology of the corresponding network of assets. More specifically, we suggest SR = f(topology) ≈ f(ECC/BC), where ECC is the eccentricity and BC is the betweenness centrality averaged over all nodes in the network. At each threshold, the structural analysis of the correlated networks provides unique insights into the relationships between assets, agencies, risks, returns and cash flows. We observe that the best threshold or best graph representation corresponds to the portfolio with the highest Sharpe ratio. We also show that simulated annealing performs better than a gradient-based solver.