Linear programming models based on Omega ratio for the Enhanced Index Tracking Problem

Abstract

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.

Stochastic Dominance and Omega Ratio: Measures to Examine Market Efficiency, Arbitrage Opportunity, and Anomaly

“…According to Caporin et al (2016), Bellini et al (2017) and the references provided therein, the Omega ratios are strongly related to expectiles, which are a type of inverse of the Omega ratio and present interesting properties as risk measures. Guastaroba et al (2016) discuss the advantages of using the Omega ratio further. Thus, the Omega ratio has been commonly used by academics and practitioners as noted by Kapsos et al (2014) and the references therein.…”

“…Additionally, it is also poor at measuring risk with asymmetric payoff profiles. The poor performance of the standard deviation will lead to poor performance of the Sharpe ratio, which establishes a relationship between the ratio of return versus volatility (Kapsos et al (2014); Guastaroba et al (2016)). A number of studies developed some theories that propose to circumvent the limitations.…”

Efficient optimization of the reward-risk ratio with polyhedral risk measures

“…Therefore, the corresponding reward-risk ratio (12) and the standard Omega ratio optimization models need a nonconvex constraint to enforce positive values of the risk measure. The latter requires introduction of additional constraints and auxiliary binary variables to deal with those critical situations where the risk measure at the denominator of the rewardrisk ratio may take null value (Guastaroba et al 2016). Moreover, the reward-risk ratio optimization for the mean below-target deviation measure results in a Mixed Integer LP model that contains T + n variables and T + 2 constraints (Mansini et al 2003) and its dimensionality cannot be reduced with any dual reformulation.…”

“…Explicit use of this constraint generate additional integer programming relations and dramatically increases the computational complexity. We experienced this phenomenon while studying the standard approach to the Omega ratio maximization (Guastaroba et al 2016), which corresponds to the reward-risk maximization for the mean below-target deviation. Generally, the reward-risk models are computationally not simpler that the primal risk reward models.…”

“…However, the risk-reward ratio optimization (15) enables easy control of the denominator positivity by simple inequality μ(x) ≥ τ + ε added to the problem constraints. In the standard reward-risk ratio optimization the corresponding inequality (x) ≥ τ + ε with convex risk measures requires complex discrete model even for LP computable risk measures (Guastaroba et al 2016). Model (15) may also be additionally regularized for the case of multiple risk-free solutions.…”