Linear Models for Portfolio Optimization

Nowadays, Quadratic Programming (QP) models, like Markowitz model, are not hard to solve, thanks to technological and algorithmic progress. Nevertheless, Linear Programming (LP) models remain much more attractive from a computational point of view for several reasons. In order to guarantee that a portfolio takes advantage from diversification, no risk or safety measures can be a linear function of the weights of the assets. Is it possible to have linear models for portfolio optimization? How can we measure the risk or safety in order to have a linear model? In this chapter, we show how it is possible to achieve a linear form of the overall optimization problem for several different risk measures through the concept of scenarios for the rates of return. The variance is the classical statistical quantity used to measure the dispersion of a random variable from its mean. However, there are several other ways to measure dispersion. We introduce the mean absolute deviation (MAD), the Gini’s mean difference (GMD) as basic LP computable risk measures and the worst realization (Minimax) and the Conditional Value-at-Risk (CVaR) as basic LP computable safety measures. We show how from each risk measure it is possible to build its safety measure and vice versa. Ratio measures and further enhanced risk measures and shortfall risk measures based on the concept of risk as failure to achieve a defined target are also discussed.