Analysis and solution of cardinality constrained quadratic portfolio optimization problem using eigen portfolios

Abstract

The aim of this study is to simplify the classic Markowitz quadratic portfoliooptimization model by transforming it into an equivalent but simpler optimizationmodel so that when additional realistic constraints, such as cardinality and minimum trading constraints, are added to the model, the transformed model can be solvedwith widely available MIP solvers. The columns of the matrix that defines the lineartransformation of variables are the eigen vectors of the sample covariance matrix foundin the objective function of the Markowitz classic model. The classic model and the transformed model with and without realistic addi-tional constraints are compared. During the comparison, two directions are followed.The first one is to compare both models in terms of their solution times required bythe exact optimization techniques. Second direction followed is to approximate the separable objective function of the transformed model with piecewise linear functionsand to compare the classic model that has a quadratic objective function with thetransformed model that has a linear objective function in terms of the approximationerror and their solutions times. To sum up, due to the simple structure of the transformed model and its separableobjective function, cardinality constrained quadratic optimization model can be solvedthrough solvers that are widely used by practitioners such as Excel Solver.