Index Tracking Problem Research Articles

A Three-phase Approach to an Enhanced Index-tracking Problem with Real-life Constraints

Enhanced index tracking is an emerging strategy for investing money in the stock market and is aimed at achieving outperformance over a given benchmark index while achieving a low tracking error. We consider the problem of rebalancing a portfolio for an enhanced index tracking strategy subject to various real-life constraints, including a lower bound and an upper bound on the expected tracking error. To solve this problem, we propose a three-phase approach consisting of preprocessing, optimization, and learning. In a computational experiment, we applied this approach to rebalance a given portfolio on a monthly basis over a time horizon of 10 years; the data for the S&P 500 benchmark index were provided by the investment company Principal Global Investors. Our approach generated portfolios that were provably close to optimality for all monthly rebalancing decisions. Over the entire horizon of 10 years, the portfolios devised by our approach yielded cumulative returns higher than the S&P 500 index after transaction costs with a moderate tracking error.

An analysis of a mean-variance enhanced index tracking problem with weights constraints

In this paper, the authors deal with a mean-variance enhanced index tracking (EIT) problem with weights constraints. Using a shrinkage approach, they show that constructing the constrained EIT portfolio is equivalent to constructing the unconstrained EIT portfolio. This equivalence allows to study the effect of weights constraints on the covariance matrix and on the EIT portfolio. In general, the effects of weights constraints on the EIT portfolio are different from those observed in the case of global minimum variance portfolio. Finally, the authors present a numerical asset allocation example, where the S&P 500 index is used as the market index to be tracked using a portfolio composed of ten stocks, in which the constrained EIT portfolio shows a satisfactory performance when compared to the unconstrained case.

A two-stage approach to the UCITS-constrained index-tracking problem

Undertakings for Collective Investments in Transferable Securities (UCITS) are investment funds that are regulated by the European Union. UCITS have become increasingly popular, resulting in a total corresponding amount of assets under management of € 8.5 trillion by the end of 2016. We present a two-stage approach to the problem of how to construct a portfolio of assets for a UCITS that aims to replicate the returns of a financial index subject to the constraints imposed by the UCITS regulations. In the first stage, we apply a genetic algorithm that treats subsets of the index constituents as individuals to construct a good feasible solution in a short CPU time. In this genetic algorithm, we use a new representation of subsets, which is the first to exhibit all of the following four desirable properties: feasibility, efficiency, locality, and heritability. In the second stage, we apply local branching based on a new mixed-integer quadratic programming formulation to improve the best solution obtained in the first stage. In a numerical experiment on real-world data, the approach yields very good feasible solutions in a short CPU time.

Эмпирический анализ работы алгоритмов решения задачи репликации индекса

Эмпирический анализ работы алгоритмов решения задачи репликации индекса

Index tracking and enhanced indexing using mixed conditional value-at-risk

Index tracking (IT) and enhanced indexing (EI) are two forms of investment strategies which revolve around the movements of the benchmark index. While IT aims to match the performance of the benchmark index, EI intents to outperform the same. In this paper, we seek to design portfolios for IT and EI problems using mixed conditional value at risk (MCVaR). We propose to use the two tail MCVaR (TMCVaR) measure to track the index. Optimizing TMCVaR is a linear program which minimizes the upper deviation and the downside deviation from the benchmark index and hence meets the objective of IT. On the other hand, we propose a two step procedure for EI problem. In step one, we design a discrete Markov chain model to filter a few stocks on the basis of their high probability of gain over the benchmark index. In step two, we assign optimal weights to the filtered assets through maximizing any of the two variants of the STARR ratio with MCVaR or the STARR ratio with deviation MCVaR (DMCVaR). Maximizing the STARR ratio either with MCVaR or DMCVaR is a linear program and hence tractable. We analyze the empirical performance of the proposed models over 17 world-wide indices using the rolling window approach. We consider two IT and four EI models from the literature for a comparative analysis. It is exhibited that the proposed IT model outperforms the other two IT models over several performance measures including higher correlation value with the benchmark index and lower tracking error, and the two proposed EI models outperform the other four EI models in achieving higher excess mean returns from the benchmark index.

Optimal construction and rebalancing of index-tracking portfolios

Index funds aim to track the performance of a financial index, such as, e.g., the Standard & Poor’s 500 index. Index funds have become popular because they offer attractive risk-return profiles at low costs. The index-tracking problem considered in this paper consists of rebalancing the composition of the index fund’s tracking portfolio in response to new market information and cash deposits and withdrawals from investors such that the index fund’s tracking accuracy is maximized. In a frictionless market, maximum tracking accuracy is achieved by investing the index fund’s entire capital in a tracking portfolio that has the same normalized value development as the index. In the presence of transaction costs, which reduce the fund’s capital, one has to manage the trade-off between transaction costs and similarity in terms of normalized value developments. Existing mathematical programing formulations for the index-tracking problem do not optimize this trade-off explicitly, which may result in substantial transaction costs or tracking portfolios that differ considerably from the index in terms of normalized value development. In this paper, we present a mixed-integer linear programing formulation with a novel optimization criterion that directly considers the trade-off between transaction costs and similarity in terms of normalized value development. In an experiment based on a set of real-world problem instances, the proposed formulation achieves a considerably higher tracking accuracy than state-of-the-art formulations.

Genetic algorithm versus classical methods in sparse index tracking

The main objective in index tracking is to replicate the performance of a target index by using a small subset of its constituents. Non-convex regularization techniques, such as the $$\ell _q$$ and the log penalization, which are able to enhance portfolio sparsity by selecting a low number of active weights, recently proved to perform remarkably well in index tracking problems. The resulting non-convex optimization is NP-hard and deterministic optimization methods, such as interior point and gradient projection algorithms, may not efficiently reach the optimal solution due to the presence of multiple local optima and discontinuities in the search space. Therefore, heuristic approaches can be more helpful and easy to implement, thanks to recent hardware development. In this paper, we compare three state-of-the-art estimation techniques, i.e., the interior point, the gradient projection and the coordinate descent algorithms, to a popular heuristic method, the genetic algorithm, in index tracking optimization. We show and evaluate the performance of the four methods in a penalized framework on different simulated settings and on real-world financial data.

Algorithms for <i>l</i><SUB align="right">1-norm minimisation of index tracking error and their performance

The paper considers the index tracking problem with cardinality constraint and examines different methods for the numerical solution of the problem. Index tracking is a passive financial strategy that tries to replicate the performance of a given index or benchmark. The aim of investor is to find the weights of assets in her/his portfolio that minimise the tracking error, i.e., difference between the performance of the index and the portfolio. In this paper, we examine three different algorithms for index tracking error minimisation in l1-norm (greedy algorithm, algorithm for l1-norm minimisation with relaxation and differential evolution algorithm) and compare the empirical performance of the portfolios obtained by means of the algorithms.

Index tracking with controlled number of assets using a hybrid heuristic combining genetic algorithm and non-linear programming

In this paper, we discuss the index tracking strategy using mathematical programming. First, we use a non-linear programming formulation for the index tracking problem, considering a limited number of assets. Since the problem is difficult to be solved in reasonable time by commercial mathematical packages, we apply a hybrid solution approach, combining mathematical programming and genetic algorithm. We show the efficiency of the proposed approach comparing the results with optimal solutions, with previous developed methods, and from real-world market indexes. The computational experiments focus on Ibovespa (the most important Brazilian market index), but we also present results for consolidated markets such as S&P 100 (USA), FTSE 100 (UK) and DAX (Germany). The proposed framework shows its ability to obtain very good results (gaps from the optimal solution smaller than 5 % in 8 min of CPU time) even for a highly volatile index from a developing country.

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

The index tracking problem is the problem of determining a portfolio of assets whose performance replicates, as closely as possible, that of a financial market index chosen as benchmark. In the enhanced index tracking problem the portfolio is expected to outperform the benchmark with minimal additional risk. In this paper, we study the bi-objective enhanced index tracking problem where two competing objectives, i.e., the expected excess return of the portfolio over the benchmark and the tracking error, are taken into consideration. A bi-objective Mixed Integer Linear Programming formulation for the problem is proposed. Computational results on a set of benchmark instances are given, along with a detailed out-of-sample analysis of the performance of the optimal portfolios selected by the proposed model. Then, a heuristic procedure is designed to build an approximation of the set of Pareto optimal solutions. We test the proposed procedure on a reference set of Pareto optimal solutions. Computational results show that the procedure is significantly faster than the exact computation and provides an extremely accurate approximation.

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.

Invasive weed optimization for solving index tracking problems

Index tracking belongs to one of the most important types of problems in portfolio management. In contrast to classical (active) portfolio management and optimization, passive portfolio management usually seeks to replicate a given index for a financial market. Due to transaction costs or legal or other practical limitation on the tradability of the respective assets, such indexes are often not fully replicated by a respective portfolio. Instead, one seeks to use a subset of the index assets (or other types of assets) to obtain a portfolio most similar to the index. This index tracking problem can be formulated as an optimization problem with respect to the minimization of the tracking error. In this article, we explore possibilities to solve the index tracking problem with invasive weed optimization (IWO), a rather new population-based metaheuristics algorithm. The complexity of this real-life problem and especially its solution space and restrictions require substantial adaptation of the original IWO algorithm. We explore different possibilities to adapt IWO to the considered type of problem. The adapted IWO method is tested using MSCI USA Value data, and systematic studies to find suitable parameter values are conducted. Although the method basically works well, the obtained results do not fully reach our intended benchmark. Reasons for that and possibilities for further improvement of the methodology are discussed.

The Impact of Different Economic Scenarios Towards Portfolio Selection in Enhanced Index Tracking Problem

The Impact of Different Economic Scenarios Towards Portfolio Selection in Enhanced Index Tracking Problem

An Empirical Comparison of Different Optimization Models in Enhanced Index Tracking Problem

An Empirical Comparison of Different Optimization Models in Enhanced Index Tracking Problem

Index Tracking com Controle do Número de Ativos

Index tracking é uma estratégia de investimento passiva, com objetivo de formar portfólios para reprodução do desempenho de um índice do mercado. Esse artigo apresenta uma proposta de modelagem de otimização de index tracking com controle do número de ativos na carteira, correspondente à restrição de custos de transação. O modelo é aplicado ao Ibovespa entre Jan/2009 e Jul/2012. Foram formadas carteiras sem limite de ativos e com limite de 40, 30 e 20 ativos, com rebalanceamento em 20, 40 e 60 períodos. Os resultados evidenciam a composição de carteiras estatisticamente significativas para acompanhar a rentabilidade do índice, especialmente para rebalanceamento de 60 períodos, com custos de transação mais baixos devido ao maior intervalo de rebalanceamento. Verificou-se também que alterações de parâmetros do solver Cplex não influenciaram os resultados, especialmente para tempo de solução. Assim, concluímos sobre a necessidade de aplicar métodos heurísticos para formação de carteiras com menos ativos.

Performance replication of the Spot Energy Index with optimal equity portfolio selection: Evidence from the UK, US and Brazilian markets

This paper reproduces the performance of a geometric average Spot Energy Index by investing only in a subset of stocks from the Dow Jones Composite Average, the FTSE 100 and Bovespa Composite indexes, and in two pools that include only energy-sector stocks from the US and the UK respectively. Daily data are used and the index-tracking problem for passive investment is addressed with two evolutionary algorithms – the differential evolution algorithm and the genetic algorithm. The performance of the suggested investment strategy is tested under three different scenarios: buy-and-hold, quarterly and monthly rebalancing, accounting for transaction costs where necessary.

Stock index tracking by Pareto efficient genetic algorithm

This paper proposes a heuristic searching approach on construction of a tracking portfolio, which is able to get the average market return and can even outperform some hedge funds that are managed actively. The tracking portfolio is expected to replicate the performance of a benchmark index return with a part of its component stocks while reducing the cost of transaction by limiting the number of rebalancing and unnecessary investment on less influential component stocks. The mathematical model being proposed is based on a hybrid genetic algorithm with a self-adaptive evolving mechanism. In order to enhance the model efficiency, we optimize the original genetic algorithm by applying Pareto efficiency as utility measure and goal programming for the inevitable conflicts of multiple objectives/interests. The proposed approach provides a comprehensive solution to index tracking problem by considering as many practical issues as possible. The constructed portfolio has a satisfactory performance on experiments based on CSI300, FTSE100 and HSI data. The proposed formulation of index tracking is therefore believed to be a good alternative to many current techniques.

Portfolio optimization and index tracking for the shipping stock and freight markets using evolutionary algorithms

This paper reproduces the performance of an international market capitalization shipping stock index and two physical shipping indexes by investing only in US stock portfolios. The index-tracking problem is addressed using the differential evolution algorithm and the genetic algorithm. Portfolios are constructed by a subset of stocks picked from the shipping or the Dow Jones Composite Average indexes. To test the performance of the heuristics, three different trading scenarios are examined: annually, quarterly and monthly rebalancing, accounting for transaction costs where necessary. Competing portfolios are also assessed through predictive ability tests. Overall, the proposed investment strategies carry less risk compared to the tracked benchmark indexes while providing investors the opportunity to efficiently replicate the performance of both the stock and physical shipping indexes in the most cost-effective way.

Exact and heuristic approaches for the index tracking problem with UCITS constraints

Index tracking aims at determining an optimal portfolio that replicates the performance of an index or benchmark by investing in a smaller number of constituents or assets. The tracking portfolio should be cheap to maintain and update, i.e., invest in a smaller number of constituents than the index, have low turnover and low transaction costs, and should avoid large positions in few assets, as required by the European Union Directive UCITS (Undertaking for Collective Investments in Transferable Securities) rules. The UCITS rules make the problem hard to be satisfactorily modeled and solved to optimality: no exact methods but only heuristics have been proposed so far. The aim of this paper is twofold. First, we present the first Mixed Integer Quadratic Programming (MIQP) formulation for the constrained index tracking problem with the UCITS rules compliance. This allows us to obtain exact solutions for small- and medium-size problems based on real-world datasets. Second, we compare these solutions with the ones provided by the state-of-art heuristic Differential Evolution and Combinatorial Search for Index Tracking (DECS-IT), obtaining information about the heuristic performance and its reliability for the solution of large-size problems that cannot be solved with the exact approach. Empirical results show that DECS-IT is indeed appropriate to tackle the index tracking problem in such cases. Furthermore, we propose a method that combines the good characteristics of the exact and of the heuristic approaches.

Tracking a benchmark index – using a spreadsheet‐based decision support system as the driver

AbstractThis paper presents a two‐phase quantitative approach for enhanced index investing based on the mean‐variance model and the goal programming method. In the first stage, we use the mean‐variance theory to select better performing stocks for an investment pool. Then, in the second stage, we use a goal programming method to weight the selected stocks by balancing both the tracking error and the rate of return. In addition to the theoretical formulation, we construct a spreadsheet‐based decision support system (DSS) based on the transaction data to help resolve the index tracking problem. The paper contributes to the literature in two ways. For academics, we present original discussions on combining an interdisciplinary mean‐variance model and a goal programming method. Unlike the conventional approach used for enhanced index investing that requires a fund manager to actively buy and sell stocks to improve returns, our approach is based on historical data and deduces subjective judgments. Meanwhile, for practitioners, we present an original discussion on using a DSS to support index investing. The results of an empirical survey of the Taiwan stock market are also presented.