Enhanced index tracking with CVaR-based ratio measures

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Twenty years of linear programming based portfolio optimization

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 computationally 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

Linear Programming Models based on Omega Ratio for the Enhanced Index Tracking Problem

Modern performance measures differ from the classical ones since they assess the performance against a benchmark and usually account for asymmetry in return distributions. The Omega ratio is one of these measures. Until recently, limited research has addressed the optimization of the Omega ratio since it has been thought to be computationally intractable. The Enhanced Index Tracking Problem (EITP) is the problem of selecting a portfolio of securities able to outperform a market index while bearing a limited additional risk. In this paper, we propose two novel mathematical formulations for the EITP based on the Omega ratio. The first formulation applies a standard definition of the Omega ratio where it is computed with respect to a given value, whereas the second formulation considers the Omega ratio with respect to a random target. We show how each formulation, nonlinear in nature, can be transformed into a Linear Programming model. We further extend the models to include real features, such as a cardinality constraint and buy-in thresholds on the investments, obtaining Mixed Integer Linear Programming problems. Computational results conducted on a large set of benchmark instances show that the portfolios selected by the model assuming a standard definition of the Omega ratio are consistently outperformed, in terms of out-of-sample performance, by those obtained solving the model that considers a random target. Furthermore, in most of the instances the portfolios optimized with the latter model mimic very closely the behavior of the benchmark over the out-of-sample period, while yielding, sometimes, significantly larger returns

Bridging k-sum and CVaR optimization in MILP

Mixed-Integer Linear Programming models often optimize the sum, or average, of different outcomes. In a deterministic setting, each outcome may be associated with an agent, for example a customer, an employee, or a time period. In a stochastic setting, each outcome may be associated with a discrete scenario. The average approach optimizes the overall efficiency of the solution, but neglects the possible unfair distribution of outcomes among agents or the risk of very bad scenarios. In this paper, we exploit the analogies of the two settings to derive a common optimization paradigm bridging the gap between k - sum optimization in the deterministic setting and Conditional Value-at-Risk optimization in the stochastic setting. We show that the proposed paradigm satisfies properties that make it an attractive criterion. To illustrate the proposed paradigm, we apply it to the multidimensional knapsack problem and the p - median/ p -center problem

Dual Stochastic Dominance and Related Mean-Risk Models

We consider the problem of constructing mean--risk models which are consistent with the second degree stochastic dominance relation. By exploiting duality relations of convex analysis we develop the quantile model of stochastic dominance for general distributions. This allows us to show that several models using quantiles and tail characteristics of the distribution are in harmony with the stochastic dominance relation. We also provide stochastic linear programming formulations of these models

On Stochastic Dominance and Mean-Semideviation Models

We analyze relations between two methods frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean--risk approaches. New necessary conditions for stochastic dominance are developed. These conditions compare values of a certain functional, which contains two components: the expected value of a random outcome and a risk term represented by the central semideviation of the corresponding degree. If the weight of the semideviation in the composite objective does not exceed the weight of the expected value, maximization of such a functional yields solutions which are efficient in terms of stochastic dominance. The results are illustrated graphically