On Efficient Optimization of the LP Computable Risk Measures for Portfolio Selection

The portfolio optimization problem is modeled as a mean-risk bicriteria optimization problem where the expected return is maximized and some (scalar) risk measure is minimized. In the original Markowitz model the risk is measured by the variance while several polyhedral risk measures have been introduced leading to Linear Programming (LP) computable portfolio optimization models in the case of discrete random variables represented by their realizations under specified scenarios. Recently, the second order quantile risk measures have been introduced and become popular in finance and banking. The simplest such measure, now commonly called the Conditional Value at Risk (CVaR) or Tail VaR, represents the mean shortfall at a specified confidence level. The corresponding portfolio optimization models can be solved with general purpose LP solvers. However, in the case of more advanced simulation models employed for scenario generation one may get several thousands of scenarios. This may lead to the LP model with huge number of variables and constraints thus decreasing the computational efficiency of the model. Actually, the number of constraints (matrix rows) is proportional to the number of scenarios. while the number of variables (matrix columns) is proportional to the total of the number of scenarios and the number of instruments. We show that the computational efficiency can be then dramatically improved with an alternative model taking advantages of the LP duality. In the introduced model the number of structural constraints (matrix rows) is proportional to the number of instruments thus not affecting seriously the simplex method efficiency by the number of scenarios. Moreover, similar reformulation can be applied to more complex quantile risk measures like the Gini’s mean difference and the tail Gini’s measures as well as to the mean absolute deviation.