Enhanced index tracking with CVaR-based ratio measures

Abstract

The enhanced index tracking problem (EITP) calls for the determination of an optimal portfolio of assets with the bi-objective of maximizing the excess return of the portfolio above a benchmark and minimizing the tracking error. The EITP is capturing a growing attention among academics, both for its practical relevance and for the scientific challenges that its study, as a multi-objective problem, poses. Several optimization models have been proposed in the literature, where the tracking error is measured in terms of standard deviation or in linear form using, for instance, the mean absolute deviation. More recently, reward-risk optimization measures, like the Omega ratio, have been adopted for the EITP. On the other side, shortfall or quantile risk measures have nowadays gained an established popularity in a variety of financial applications. In this paper, we propose a class of bi-criteria optimization models for the EITP, where risk is measured using the weighted multiple conditional value-at-risk (WCVaR). The WCVaR is defined as a weighted combination of multiple CVaR measures, and thus allows a more detailed risk aversion modeling compared to the use of a single CVaR measure. The application of the WCVaR to the EITP is analyzed, both theoretically and empirically. Through extensive computational experiments, the performance of the optimal portfolios selected by means of the proposed optimization models is compared, both in-sample and, more importantly, out-of-sample, to the one of the portfolios obtained using another recent optimization model taken from the literature. KeywordsEnhanced index tracking · Quantile risk measures · Conditional value-at-risk · Mean-risk models · Risk-reward ratios · Risk-averse optimization · Stochastic dominance · Linear programming Annals of Operations ResearchSortino ratio (see Sortino and Price 1994) are widely used to evaluate, compare and rank different investment strategies. To the best of our knowledge, Meade and Beasley (2011) are the first ones attempting to use a reward-risk ratio in the context of enhanced indexation. The authors introduce a non-linear optimization model, based on the maximization of a modified Sortino ratio, and solve it by means of a genetic algorithm. However, the non-linearity of this model may represent an undesirable limitation to its use in financial practice, especially when portfolios have to meet several side constraints (such as cardinality constraints or buy-in thresholds) or when large-scale instances have to be solved since, in most cases, the inclusion of these features requires the introduction of binary and integer variables (see the survey by Mansini et al. 2014). Based on this observation, Guastaroba et al. (2016) introduce two mathematical formulations for the EITP based on the Omega ratio. The Omega ratio is a performance measure introduced by Keating and Shadwick (2002) which, broadly speaking, can be defined as the ratio between expected value of the profits, defined as the portfolio returns over a predetermined target τ , and e...

Enhanced indexation via chance constraints

“…While the former measure represents the weighted sum of the left tails (worst scenarios) and right tails (best scenarios), the latter is the weighted sum of the CVaR calculated for different confidence levels. The mixed CVaR has been also used by Guastaroba et al (2017) in the framework of risk-reward ratio. The contribution is similar to the model proposed by the same authors in Guastaroba et al (2016) where they consider the maximization of the Omega Ratio in the standard and the extended forms.…”