High-dimensional index tracking based on the adaptive elastic net

When a portfolio consists of a large number of assets, it generally incorporates too many small and illiquid positions and needs a large amount of rebalancing, which can involve large transaction costs. For financial index tracking, it is desirable to avoid such atomized, unstable portfolios, which are difficult to realize and manage. A natural way of achieving this goal is to build a tracking portfolio that is sparse with only a small number of assets in practice. The cardinality constraint approach, by directly restricting the number of assets held in the tracking portfolio, is a natural idea. However, it requires the pre-specification of the maximum number of assets selected, which is rarely practicable. Moreover, the cardinality constrained optimization problem is shown to be NP-hard. Solving such a problem will be computationally expensive, especially in high-dimensional settings. Motivated by this, this paper employs a regularization approach based on the adaptive elastic-net (Aenet) model for high-dimensional index tracking. The proposed method represents a family of convex regularization methods, which nests the traditional Lasso, adaptive Lasso (Alasso), and elastic-net (Enet) as special cases. To make the formulation more practical and general, we also take the full investment condition and turnover restrictions (or transaction costs) into account. An efficient algorithm based on coordinate descent with closed-form updates is derived to tackle the resulting optimization problem. Empirical results show that the proposed method is computationally efficient and has competitive out-of-sample performance, especially in high-dimensional settings.     

Cardinality versus q-norm constraints for index tracking

Index tracking aims at replicating a given benchmark with a smaller number of its constituents. Different quantitative models can be set up to determine the optimal index replicating portfolio. In this paper, we propose an alternative based on imposing a constraint on the q-norm (0?<?q?<?1) of the replicating portfolios’ asset weights: the q-norm constraint regularises the problem and identifies a sparse model. Both approaches are challenging from an optimization viewpoint due to either the presence of the cardinality constraint or a non-convex constraint on the q-norm. The problem can become even more complex when non-convex distance measures or other real-world constraints are considered. We employ a hybrid heuristic as a flexible tool to tackle both optimization problems. The empirical analysis of real-world financial data allows us to compare the two index tracking approaches. Moreover, we propose a strategy to determine the optimal number of constituents and the corresponding optimal portfolio asset weights.     

Index tracking through deep latent representation learning

We consider the problem of index tracking whose goal is to construct a portfolio that minimizes the tracking error between the returns of a benchmark index and the tracking portfolio. This problem carries significant importance in financial economics as the tracking portfolio represents a parsimonious index that facilitates a practical means to trade the benchmark index. For this reason, extensive studies from various optimization and machine learning-based approaches have ensued. In this paper, we solve this problem through the latest developments from deep learning. Specifically, we associate a deep latent representation of asset returns, obtained through a stacked autoencoder, with the benchmark index's return to identify the assets for inclusion in the tracking portfolio. Empirical results indicate that to improve the performance of previously proposed deep learning-based index tracking, the deep latent representation needs to be learned in a strictly hierarchical manner and the relationship between the returns of the index and the assets should be quantified by statistical measures. Various deep learning-based strategies have been tested for the stock market indices of the S&P 500, FTSE 100 and HSI, and it is shown that our proposed methodology generates the best index tracking performance.     

Non-fully invested derivative-free bond index replication

The problem we address here is the replication of a bond benchmark when only a fraction of the portfolio is invested for the replication. Our methodology is based on a minimization of the tracking error subject to a set of constraints, namely (1) the fraction invested for the replication, (2) a no-short-selling constraint, and (3) a null-active-duration constraint, the last of which may be relaxed. The constraints can also be adapted to accommodate the use of interest rate and bond futures. Our main contribution, however, is our derivative-free approach to replication, which should prove very useful for managing assets when the use of derivatives is prohibited, for instance, by certain investors. We can, thus, still benefit from replicating a traditional investment in a bond index with a fraction of the portfolio according to our risk appetite. The rest of the portfolio can be invested in alpha-portable strategies. An analysis without the use of derivatives over the period from January 1, 2008 to October 3, 2011 shows that with 70–90 % invested for the replication, the annualized ex-ante tracking error can range from 0.41 to 0.07 %. We use principal component analysis to extract the main drivers of the size of the tracking error, namely, the volatility of and the differential between the yields in the objective function’s covariance matrix of spot rates. These results highlight our contribution of a generic and intuitive yet robust approach to bond index replication.     

Pareto‐archived evolutionary wavelet network for financial constrained portfolio optimization

The multi‐objective portfolio optimization problem is too complex to find direct solutions by traditional methods when constraints reflecting investor's preferences and/or market frictions are included in the mathematical model and hence heuristic approaches are sought for their solution. In this paper we propose the solution of a multi‐criterion (bi‐objective) portfolio optimization problem of minimizing risk and maximizing expected return of the portfolio which includes basic, bounding, cardinality, class and short sales constraints using a Pareto‐archived evolutionary wavelet network (PEWN) solution strategy. Initially, the empirical covariance matrix is denoised by employing a wavelet shrinkage denoising technique. Second, the cardinality constraint is eliminated by the application of k‐means cluster analysis. Finally, a PEWN heuristic strategy with weight standardization procedures is employed to obtain Pareto‐optimal solutions satisfying all the constraints. The closeness and diversity of Pareto‐optimal solutions obtained using PEWN is evaluated using different measures and the results are compared with existing only solution strategies (evolution‐based wavelet Hopfield neural network and evolution‐based Hopfield neural network) to prove its dominance. Eventually, data envelopment analysis is also used to test the efficiency of the non‐dominated solutions obtained using PEWN. Experimental results are demonstrated on the Bombay Stock Exchange, India (BSE200 index: period July 2001–July 2006), and the Tokyo Stock Exchange, Japan (Nikkei225 index: period March 2002–March 2007), data sets. Copyright © 2010 John Wiley & Sons, Ltd.     

Index tracking with utility enhanced weighting

Passive index investing involves investing in a fund that replicates a market index. Enhanced indexation uses the returns of an index as a reference point and aims at outperforming this index. The motivation behind enhanced indexing is that the indices and portfolios available to academics and practitioners for asset pricing and benchmarking are generally inefficient and, thus, susceptible to enhancement. In this paper we propose a novel technique based on the concept of cumulative utility area ratios and the Analytic Hierarchy Process (AHP) to construct enhanced indices from the DJIA and S&P500. Four main conclusions are forthcoming. First, the technique, called the utility enhanced tracking technique (UETT), is computationally parsimonious and applicable for all return distributions. Second, if desired, cardinality constraints are simple and computationally parsimonious. Third, the technique requires only infrequent rebalancing, monthly at the most. Finally, the UETT portfolios generate consistently higher out-of-sample utility profiles and after-cost returns for the fully enhanced portfolios as well as for the enhanced portfolios adjusted for cardinality constraints. These results are robust to varying market conditions and a range of utility functions.     

A methodology for index tracking based on time-series clustering

With the increased acceptance of capital market efficiency, there has been a significant increase in the money managed on an indexed basis. Several methodologies are available to replicate the target index. In this paper, we discuss the problems of (1) defining suitable performance objectives and tracking error that scale properly over the entire management period and (2) implementing an optimal investment strategy when full replication of an index is not deemed suitable. We then argue that clustering might be a viable methodology for building parsimonious tracking portfolios. With suitably defined distances between the time series of asset prices, clustering ‘discovers’ the correlation and cointegration structure of an index. Sampling the clusters with appropriate heuristics and optimization techniques, an optimal tracking portfolio can be constructed. One advantage of this approach is that it eschews the difficulties and computational burden of density forecasts and full optimization.     

Portfolio frontiers with restrictions to tracking error volatility and value at risk

Asset managers are often given the task of restricting their activity by keeping both the value at risk (VaR) and the tracking error volatility (TEV) under control. However, these constraints may be impossible to satisfy simultaneously because VaR is independent of the benchmark portfolio. The management of these restrictions is likely to affect portfolio performance and produces a wide variety of scenarios in the risk-return space. The aim of this paper is to analyse various interactions between portfolio frontiers when risk managers impose joint restrictions upon TEV and VaR. Specifically, we provide analytical solutions for all the intersections and we propose simple numerical methods when such solutions are not available. Finally, we introduce a new portfolio frontier.     

Constructing Asset Pricing Models With Specific Factor Loadings

We demonstrate how one can build pricing formulae in which factors other than beta may be viewed as determinants of asset returns. This is important conceptually as it demonstrates how the additional factors can compensate for a market portfolio proxy that is mis‐specified, and also shows how such a pricing model can be specified ex ante. The procedure is implemented by first selecting an ‘orthogonal’ portfolio which falls on the mean‐variance efficient frontier computed from the empirical average returns, variances and covariances on the equity securities of a large sample of firms. One then determines the inefficient index portfolio which leads to a vector of betas that when multiplied by the average return on the orthogonal portfolio, and which when subtracted from the vector of average returns for the firms comprising the sample, yields an error vector that is equal to the vector of numerical values for the variables that are to form the basis of the asset pricing formula. There will then be a perfect linear relationship between the vector of average returns for the firms comprising the sample, the vector of betas based on the inefficient index portfolio and such other factors that are deemed to be important in the asset pricing process. We illustrate computational procedures using a numerical example based on the quality of information contained in published corporate financial statements.     

Mean-variance optimization using forward-looking return estimates

Despite its theoretical appeal, Markowitz mean-variance portfolio optimization is plagued by practical issues. It is especially difficult to obtain reliable estimates of a stock’s expected return. Recent research has therefore focused on minimum volatility portfolio optimization, which implicitly assumes that expected returns for all assets are equal. We argue that investors are better off using the implied cost of capital based on analysts’ earnings forecasts as a forward-looking return estimate. Correcting for predictable analyst forecast errors, we demonstrate that mean-variance optimized portfolios based on these estimates outperform on both an absolute and a risk-adjusted basis the minimum volatility portfolio as well as naive benchmarks, such as the value-weighted and equally-weighted market portfolio. The results continue to hold when extending the sample to international markets, using different methods for estimating the forward-looking return, including transaction costs, and using different optimization constraints.

     

Do Socially Responsible Index Investors Incur an Opportunity Cost?

We construct index‐tracking portfolios using integer programming and then compare the tracking errors and performances of portfolios formed from an unrestricted and socially screened stock universe. We find that one can construct a portfolio of socially responsible stocks that deliver market performance. Thus, the exclusion of a set of stocks from consideration does not exhaust the existence of efficient index‐tracking portfolios, especially when the exclusionary screen is for nonfinancial reasons. Our results are robust to various specifications in constructing the portfolio, for example, number of stocks included in the portfolio and weighting schemes, and robust to alternative tracking error measurement; we show that the difference induced from conducting socially responsible screen is never statistically significant.     

Factor based index tracking

Stock index tracking requires to build a portfolio of stocks (a replica) whose behavior is as close as possible to that of a given stock index. Typically, much fewer stocks should appear in the replica than in the index, and there should be no low frequency or integrated (persistent) components in the tracking error. The latter property is not satisfied by many commonly used methods for index tracking. These are based on the in-sample minimization of a loss function, but do not take into account the dynamic properties of the index components. Moreover, most existing methods do not take into account the known structure of the index weight system. In this paper we represent the index components with a dynamic factor model. In this model the price of each stock in the index is driven by a set of common and idiosyncratic factors. Factors can be either integrated or stationary. We develop a procedure that, in a first step, builds a replica that is driven by the same persistent factors as the index. This procedure is grounded in recent results which suggest the application of principal component analysis for factor estimation even for integrated processes. In a second step, it is also possible to refine the replica so that it minimizes a specific loss function, as in the traditional approach. In both steps the replica weights depend on the existing information on the index weights system. An extended set of Monte Carlo simulations and an application to the most widely used index in the European stock market, the EuroStoxx50 index, provide substantial support for our approach.     

Portfolio optimization with stochastic dominance constraints

We consider the problem of constructing a portfolio of finitely many assets whose return rates are described by a discrete joint distribution. We propose a new portfolio optimization model involving stochastic dominance constraints on the portfolio return rate. We develop optimality and duality theory for these models. We construct equivalent optimization models with utility functions. Numerical illustration is provided.     

Collective mental accounting: an integrated behavioural portfolio selection model for multiple mental accounts

We propose a behavioural portfolio selection model called collective mental accounting (CMA), which integrates all mental sub-portfolios (mental accounts) in one mathematical model. Moreover, this study contributes to the literature of behavioural portfolio selection in three further ways: first, the CMA model can determine the proportions of wealth allocated to each mental sub-portfolio with and without input from the investor. Second, unlike other mental accounting models (MA), in CMA it is possible to define constraints on total asset holdings such as short-selling, and cardinality constraints. Third, in order to make CMA more tractable and mathematically elegant, we obtain a semi-definite programming representation of the model. We also present a numerical example to investigate the effects of short-selling constraints as well as to compare the portfolio recommendations, utility functions, feasibility, and optimality of the CMA and MA models. The results reveal that although both models’ solutions are mean-variance efficient, CMA outperforms MA in terms of behavioural efficient frontier and utility functions.     

Deep hedging

We present a framework for hedging a portfolio of derivatives in the presence of market frictions such as transaction costs, liquidity constraints or risk limits using modern deep reinforcement machine learning methods. We discuss how standard reinforcement learning methods can be applied to non-linear reward structures, i.e. in our case convex risk measures. As a general contribution to the use of deep learning for stochastic processes, we also show in Section 4 that the set of constrained trading strategies used by our algorithm is large enough to ε-approximate any optimal solution. Our algorithm can be implemented efficiently even in high-dimensional situations using modern machine learning tools. Its structure does not depend on specific market dynamics, and generalizes across hedging instruments including the use of liquid derivatives. Its computational performance is largely invariant in the size of the portfolio as it depends mainly on the number of hedging instruments available. We illustrate our approach by an experiment on the S&P500 index and by showing the effect on hedging under transaction costs in a synthetic market driven by the Heston model, where we outperform the standard ‘complete-market’ solution.     

Multiobjective portfolio optimization with non-convex policy constraints: Evidence from the Eurostoxx 50

Our purpose in this paper is to depart from the intrinsic pathology of the typical mean–variance formalism, due to both the restriction of its assumptions and difficulty of implementation. We manage to co-assess a set of sophisticated real-world non-convex investment policy limitations, such as cardinality constraints, buy-in thresholds, transaction costs, particular normative rules, etc., within the frame of complex scenarios, which demand for simultaneous optimization of multiple investment objectives. In such a case, the portfolio selection process reflects a mixed-integer multiobjective portfolio optimization problem. On this basis, we meticulously develop all the corresponding modeling procedures and then solve the underlying problem by use of a new, fast and very effective algorithm. The value of the suggested framework is integrated with the introduction of two novel concepts in the field of multiobjective portfolio optimization, i.e. the security impact plane and the barycentric portfolio. The first represents a measure of each security's impact in the efficient surface of Pareto optimal portfolios. The second serves as the vehicle for implementing a balanced strategy of iterative portfolio tuning. Moreover, a couple of some very informative graphs provide thorough visualization of all empirical testing results. The validity of the attempt is verified through an illustrative application on the Eurostoxx 50. The results obtained are characterized as very encouraging, since a sufficient number of efficient or Pareto optimal portfolios produced by the model, appear to possess superior out-of-sample returns with respect to the underlying benchmark.     

A dynamic model of active portfolio management with benchmark orientation

This paper studies optimal dynamic portfolios for investors concerned with the performance of their portfolios relative to a benchmark. Assuming that asset returns follow a multi-linear factor model similar to the structure of Ross (1976) [Ross, S., 1976. The arbitrage theory of the capital asset pricing model. Journal of Economic Theory, 13, 342–360] and that portfolio managers adopt a mean tracking error analysis similar to that of Roll (1992) [Roll, R., 1992. A mean/variance analysis of tracking error. Journal of Portfolio Management, 18, 13–22], we develop a dynamic model of active portfolio management maximizing risk adjusted excess return over a selected benchmark. Unlike the case of constant proportional portfolios for standard utility maximization, our optimal portfolio policy is state dependent, being a function of time to investment horizon, the return on the benchmark portfolio, and the return on the investment portfolio. We define a dynamic performance measure which relates portfolio’s return to its risk sensitivity. Abnormal returns at each point in time are quantified as the difference between the realized and the model-fitted returns. Risk sensitivity is estimated through a dynamic matching that minimizes the total fitted error of portfolio returns. For illustration, we analyze eight representative mutual funds in the U.S. market and show how this model can be used in practice.     

Comparative issues between linear and non-linear risk measures for non-convex portfolio optimization: evidence from the S&P 500

Abstract

This article focuses on inferring critical comparative conclusions as far as the application of both linear and non-linear risk measures in non-convex portfolio optimization problems. We seek to co-assess a set of sophisticated real-world non-convex investment policy limitations, such as cardinality constraints, buy-in thresholds, transaction costs, particular normative rules, etc. within the frame of four popular portfolio selection cases: (a) the mean-variance model, (b) the mean-semi variance model, (c) the mean-MAD (mean-absolute deviation) model and (d) the mean-semi MAD model. In such circumstances, the portfolio selection process reflects to a mixed-integer bi-objective (or in general multiobjective) mathematical programme. We precisely develop all corresponding modelling procedures and then solve the underlying problem by use of a novel generalized algorithm, which was exclusively introduced to cope with the above-mentioned singularities. The validity of the attempt is verified through empirical testing on the S&P 500 universe of securities. The technical conclusions obtained not only confirm certain findings of the particular limited existing theory but also shed light on computational issues and running times. Moreover, the results derived are characterized as encouraging enough, since a sufficient number of efficient or Pareto optimal portfolios produced by the models appear to possess superior out-of-sample returns with respect to the benchmark.     

Optimal portfolios under worst-case scenarios

In standard portfolio theories such as Mean–Variance optimization, expected utility theory, rank dependent utility heory, Yaari’s dual theory and cumulative prospect theory, the worst outcomes for optimal strategies occur when the market declines (e.g. during crises), which is at odds with the needs of many investors. Hence, we depart from the traditional settings and study optimal strategies for investors who impose additional constraints on their final wealth in the states corresponding to a stressed financial market. We provide a framework that maintains the stylized features of the SP/A theory while dealing with the goal of security in a more flexible way. Preferences become state-dependent, and we assess the impact of these preferences on trading decisions. We construct optimal strategies explicitly and show how they outperform traditional diversified strategies under worst-case scenarios.