Twenty years of linear programming based portfolio optimization

Keywords: Survey LP computable mean-risk and mean-safety models Real features Transaction costs Exact and heuristic algorithms a b s t r a c t Markowitz formulated the portfolio optimization problem through two criteria: the expected return and the risk, as a measure of the variability of the return. The classical Markowitz model uses the variance as the risk measure and is a quadratic programming problem. Many attempts have been made to linearize the portfolio optimization problem. Several different risk measures have been proposed which are com-putationally attractive as (for discrete random variables) they give rise to linear programming (LP) problems. About twenty years ago, the mean absolute deviation (MAD) model drew a lot of attention resulting in much research and speeding up development of other LP models. Further, the LP models based on the conditional value at risk (CVaR) have a great impact on new developments in portfolio optimization during the first decade of the 21st century. The LP solvability may become relevant for real-life decisions when portfolios have to meet side constraints and take into account transaction costs or when large size instances have to be solved. In this paper we review the variety of LP solvable portfolio optimization models presented in the literature, the real features that have been modeled and the solution approaches to the resulting models, in most of the cases mixed integer linear programming (MILP) models. We also discuss the impact of the inclusion of the real features. The portfolio optimization problem considered in this paper follows the original Markowitz' formulation and is based on a single period model of investment. At the beginning of a period, an investor allocates the capital among various securities, assigning a share of the capital to each. During the investment period, the portfolio generates a random rate of return. This results in a new value of the capital (observed at the end of the period), increased or decreased with respect to the invested capital by the average portfolio return. This model has played a crucial role in stock investment and has served as basis for the development of the modern portfolio financial theory. In the original Markowitz model (Markowitz, 1952) the risk is measured by the standard deviation or variance. Several other risk measures have been later considered, creating a family of mean-risk models. Whereas the original Markowitz model is a quadratic programming problem, following Sharpe (1971a), many attempts …