Construction of Risk-Averse Enhanced Index Funds

Abstract

We propose a partial replication strategy to construct risk-averse enhanced index funds. Our model takes into account the parameter estimation risk by defining the asset returns and the return covariance terms as random variables. The variance of the index fund return is forced to be below a low-risk threshold with a large probability, thereby limiting the market risk exposure of the investors and the moral hazard associated with the wage structure of fund managers. The resulting stochastic integer problem is reformulated through the derivation of a deterministic equivalent for the risk constraint and the use of a block decomposition technique. We develop an exact outer approximation method based on the relaxation of some binary restrictions and the reformulation of the cardinality constraint. The method provides a hierarchical organization of the computations with expanding sets of integer-restricted variables and outperforms the Bonmin and the Cplex 12.1 solvers. The method can solve very large (up to 1000 securities) instances, converges fast, scales well, and is general enough to be applicable to problems with buy-in threshold constraints. Cross-validation tests show that the constructed funds track closely and are consistently less risky than the benchmark on the out-of-sample period.

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

“…The problem is formulated as a stochastic model which aims at maximizing the probabilistic excess return of the portfolio compared to the benchmark while ensuring that the relative risk, given by the downside absolute deviation, does not exceed a chosen maximum level. A stochastic mixed integer nonlinear model for the EITP where asset returns and the return covariance terms are treated as random variables is proposed in Lejeune and Samatlı-Paç [20]. Two models for the EITP that aim at selecting a portfolio whose return distribution dominates the distribution of the benchmark with respect to the second-order stochastic dominance paradigm are introduced in Roman et al [31].…”

“…Active management strategies often involve frequent trading to rebalance the portfolio composition in an attempt to beat the benchmark (cf. Lejeune and Samatlı-Paç [20]), thus generating high transaction costs which diminish the fund return.…”

A constrained cluster-based approach for tracking the S&P 500 index

“…Stoyan and Kwon (2010) consider a two-stage stochastic mixed integer programming model with several discrete choice constraints such as buy-in thresholds, cardinality constraints, as well as round lots to track the Toronto Stock Exchange (TSX). Lejeune and Samatlõ-Paç, 2013 consider a chance constrained stochastic programming formulation used for the risk averse indexing problem with cardinality constraints and develop an associated outer approximation method. Cornuejols and Tutuncu (2007) consider an index tracking model which maximizes similarity between selected assets and the assets of the target index.…”

A heuristic framework for the bi-objective enhanced index tracking problem

“…Lejeune [36] formulates the enhanced index tracking problem as a stochastic game theoretic model where the probabilistic excess return of the portfolio over the benchmark is maximized while ensuring that the risk, measured by the downside absolute deviation, does not exceed a chosen maximum level. Lejeune and Samatlı-Paç [37] introduce a stochastic mixed integer non-linear model for the enhanced index tracking problem where stock returns and the return covariance terms are treated as random variables. Two enhanced index tracking formulations based on the second-order stochastic dominance are presented in Roman et al [51].…”

“…A common active management strategy consists in underweighting or overweighting stocks compared with the benchmark based on the manager beliefs (e.g., see Li et al [38]). Therefore, fund managers adopting an active management strategy frequently rebalance their portfolio composition in an attempt to beat the benchmark [37], but incurring expensive transaction and management costs. Additionally, several researchers highlighted that the majority of actively managed funds do not outperform the benchmark in the long term (see, among others, Gruber [23]).…”