Portfolio revision under mean-variance and mean-CVaR with transaction costs

The portfolio revision process usually begins with a portfolio of assets rather than cash. As a result, some assets must be liquidated to permit investment in other assets, incurring transaction costs that should be directly integrated into the portfolio optimization problem. This paper discusses and analyzes the impact of transaction costs on the optimal portfolio under mean-variance and mean-conditional value-at-risk strategies. In addition, we present some analytical solutions and empirical evidence for some special situations to understand the impact of transaction costs on the portfolio revision process.     

Index tracking with constrained portfolios

Passive portfolio management strategies, such as index tracking, are popular in the industry, but so far little research has been done on the cardinality of such a portfolio, i.e. on how many different assets ought to be included in it. One reason for this is the computational complexity of the associated optimization problems. Traditional optimization techniques cannot deal appropriately with the discontinuities and the many local optima emerging from the introduction of explicit cardinality constraints. More recent approaches, such as heuristic methods, on the other hand, can overcome these hurdles. This paper demonstrates how one of these methods, differential evolution, can be used to solve the constrained index-tracking problem. We analyse the financial implication of cardinality constraints for a tracking portfolio using an empirical study of the Down Jones Industrial Average. We find that the index can be tracked satisfactorily with a subset of its components and, more important, that the deviation between computed actual tracking error and the theoretically achievable tracking error out of sample is negligibly affected by the portfolio's cardinality. Copyright © 2007 John Wiley & Sons, Ltd.     

Optimizing a portfolio of mean-reverting assets with transaction costs via a feedforward neural network

Optimizing a portfolio of mean-reverting assets under transaction costs and a finite horizon is severely constrained by the curse of high dimensionality. To overcome the exponential barrier, we develop an efficient, scalable algorithm by employing a feedforward neural network. A novel concept is to apply HJB equations as an advanced start for the neural network. Empirical tests with several practical examples, including a portfolio of 48 correlated pair trades over 50 time steps, show the advantages of the approach in a high-dimensional setting. We conjecture that other financial optimization problems are amenable to similar approaches.     

A model for tax advantages of portfolios with many assets

Taxable portfolios present challenges for optimization models with even a limited number of assets. Holding many assets, however, has a distinct tax advantage over holding few assets. In this paper, we develop a model that takes an extreme view of a portfolio as a continuum of assets to gain the broadest possible advantage from holding many assets. We find the optimal strategy for trading in this portfolio in the absence of transaction costs and develop bounding approximations on the optimal value. We compare the results in a simulation study to a portfolio consisting only of a market index and show that the multi-asset portfolio’s tax advantage can lead either to significant consumption or bequest increases.     

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.     

Prospect theory–based portfolio optimization: an empirical study and analysis using intelligent algorithms

The behaviourally based portfolio selection problem with investor’s loss aversion and risk aversion biases in portfolio choice under uncertainty is studied. The main results of this work are: developed heuristic approaches for the prospect theory model proposed by Kahneman and Tversky in 1979 as well as an empirical comparative analysis of this model and the index tracking model. The crucial assumption is that behavioural features of the prospect theory model provide better downside protection than traditional approaches to the portfolio selection problem. In this research the large-scale computational results for the prospect theory model have been obtained for real financial market data with up to 225 assets. Previously, as far as we are aware, only small laboratory tests (2–3 artificial assets) have been presented in the literature. In order to investigate empirically the performance of the behaviourally based model, a differential evolution algorithm and a genetic algorithm which are capable of dealing with a large universe of assets have been developed. Specific breeding and mutation, as well as normalization, have been implemented in the algorithms. A tabulated comparative analysis of the algorithms’ parameter choice is presented. The prospect theory model with the reference point being the index is compared to the index tracking model. A cardinality constraint has been implemented to the basic index tracking and the prospect theory models. The portfolio diversification benefit has been found. The aggressive behaviour in terms of returns of the prospect theory model with the reference point being the index leads to better performance of this model in a bullish market. However, it performed worse in a bearish market than the index tracking model. A tabulated comparative analysis of the performance of the two studied models is provided in this paper for in-sample and out-of-sample tests. The performance of the studied models has been tested out-of-sample in different conditions using simulation of the distribution of a growing market and simulation of the t-distribution with fat tails which characterises the dynamics of a decreasing or crisis market.     

Portfolio optimization for heavy-tailed assets: Extreme Risk Index vs. Markowitz

Using daily returns of the S&P 500 stocks from 2001 to 2011, we perform a backtesting study of the portfolio optimization strategy based on the Extreme Risk Index (ERI). This method uses multivariate extreme value theory to minimize the probability of large portfolio losses. With more than 400 stocks to choose from, our study seems to be the first application of extreme value techniques in portfolio management on a large scale. The primary aim of our investigation is the potential of ERI in practice. The performance of this strategy is benchmarked against the minimum variance portfolio and the equally weighted portfolio. These fundamental strategies are important benchmarks for large-scale applications. Our comparison includes annualized portfolio returns, maximal drawdowns, transaction costs, portfolio concentration, and asset diversity in the portfolio. In addition to that we study the impact of an alternative tail index estimator. Our results show that the ERI strategy significantly outperforms both the minimum-variance portfolio and the equally weighted portfolio on assets with heavy tails.     

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.

     

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.     

Robust hedging with proportional transaction costs

A duality for robust hedging with proportional transaction costs of path-dependent European options is obtained in a discrete-time financial market with one risky asset. The investor’s portfolio consists of a dynamically traded stock and a static position in vanilla options, which can be exercised at maturity. Trading of both options and stock is subject to proportional transaction costs. The main theorem is a duality between hedging and a Monge–Kantorovich-type optimization problem. In this dual transport problem, the optimization is over all probability measures that satisfy an approximate martingale condition related to consistent price systems, in addition to an approximate marginal constraint.     

Model Predictive Control for Optimal Pairs Trading Portfolio with Gross Exposure and Transaction Cost Constraints

Model predictive control (MPC) is a flexible yet tractable technique in control engineering that recently has gained much attention in the area of finance, particularly for its application to portfolio optimization. In this paper, we extend the MPC with linear feedback setting in Yamada and Primbs (in: Proceedings of the IEEE conference on decision and control, pp 5705–5710, 2012) by incorporating the following two important and practical issues: The first issue is gross exposure (GE), which is the total value of long and short positions invested in risky assets (or stocks) as a proportion of the wealth possessed by a hedge fund. This quantity measures the leverage of a hedge fund, and the fund manager may limit the amount of leverage by imposing an upper bound, i.e., a GE constraint. The second issue is related to transaction costs, where the MPC algorithm may require frequent trades of many stocks leading to large transaction costs in practice. Here we assume that the transaction cost is proportional to the change in the amount of money (i.e., the change of absolute values of long or short positions) invested in each stock. We formulate the MPC strategy based on a conditional mean-variance problem which we show reduces to a convex quadratic problem, even with gross exposure and proportional transaction cost constraints. Based on numerical experiments using Japanese stock data, we demonstrate that the incorporation of the transaction cost constraint improves the empirical performance of the wealth in terms of Sharpe ratio, which may be improved further by adding the GE constraint.     

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.     

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.     

Life-cycle portfolio choice with liquid and illiquid financial assets

Traditionally, quantitative models that have studied households׳ portfolio choices have focused exclusively on the different risk properties of alternative financial assets. We introduce differences in liquidity across assets in the standard life-cycle model of portfolio choice. More precisely, in our model, stocks are subject to transaction costs, as considered in recent macroliterature. We show that when these costs are calibrated to match the observed infrequency of households׳ trading, the model is able to generate patterns of portfolio stock allocation over age and wealth that are constant or moderately increasing, thus more in line with the existing empirical evidence.     

Nonlinear portfolio selection using approximate parametric Value-at-Risk

As the skewed return distribution is a prominent feature in nonlinear portfolio selection problems which involve derivative assets with nonlinear payoff structures, Value-at-Risk (VaR) is particularly suitable to serve as a risk measure in nonlinear portfolio selection. Unfortunately, the nonlinear portfolio selection formulation using VaR risk measure is in general a computationally intractable optimization problem. We investigate in this paper nonlinear portfolio selection models using approximate parametric Value-at-Risk. More specifically, we use first-order and second-order approximations of VaR for constructing portfolio selection models, and show that the portfolio selection models based on Delta-only, Delta–Gamma-normal and worst-case Delta–Gamma VaR approximations can be reformulated as second-order cone programs, which are polynomially solvable using interior-point methods. Our simulation and empirical results suggest that the model using Delta–Gamma-normal VaR approximation performs the best in terms of a balance between approximation accuracy and computational efficiency.     

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.     

In search of a theory of debt management

The complete market approach to government debt management argues that a portfolio of non-contingent bonds at different maturities should be chosen so that fluctuations in market value offset changes in expected future deficits. However, this approach recommends huge fluctuations in positions, enormous changes in portfolios for minor changes in maturities and no presumption it is always optimal to issue long and invest short term in a wide array of model specifications. These extreme, volatile and unstable features are undesirable for two reasons. Firstly fragility of portfolios to small changes in assumptions means that it is often better to follow a balanced budget rather than issue the optimal debt portfolio under some possibly misspecified model. Secondly for even miniscule transaction costs, governments prefer a balanced budget rather than the large positions complete markets recommends. The complete market recommendations conflict with a number of features we believe are integral to bond market incompleteness, e.g. transaction costs, liquidity effects, robustness, etc. and which need to be explicitly incorporated into the portfolio problem.     

Constant Rebalanced Portfolio Optimization Under Nonlinear Transaction Costs

We study the constant rebalancing strategy for multi-period portfolio optimization via conditional value-at-risk (CVaR) when there are nonlinear transaction costs. This problem is difficult to solve because of its nonconvexity. The nonlinear transaction costs and CVaR constraints make things worse; state-of-the-art nonlinear programming (NLP) solvers have trouble in reaching even locally optimal solutions. As a practical solution, we develop a local search algorithm in which linear approximation problems and nonlinear equations are iteratively solved. Computational results are presented, showing that the algorithm attains a good solution in a practical time. It is better than the revised version of an existing global optimization. We also assess the performance of the constant rebalancing strategy in comparison with the buy-and-hold strategy.     

Particle swarm optimization approach to portfolio construction

Particle swarm optimization (PSO) is an artificial intelligence technique that can be used to find approximate solutions to extremely difficult or impossible numeric optimization problems. Recently, PSO algorithms have been widely used in solving complex financial optimization problems. This paper presents a PSO approach to solve a portfolio construction problem, since this methodology is a population-based heuristic algorithm that is suitable for solving high-dimensional constrained optimization problems. The computational results show that PSO algorithms have advantages in optimizing the Sortino ratio, especially in speed, when the size of the portfolio is large.     

Optimal mean-reversion strategy in the presence of bid-ask spread and delays in capital allocations

A portfolio optimization problem for an investor who trades T-bills and a mean-reverting stock in the presence of proportional and convex transaction costs is considered. The proportional transaction cost represents a bid-ask spread, while the convex transaction cost is used to model delays in capital allocations. I utilize the historical bid-ask spread in US stock market and assume that the stock reverts on yearly basis, while an investor follows monthly changes in the stock price. It is found that proportional transaction cost has a relatively weak effect on the expected return and the Sharpe ratio of the investor's portfolio. Meantime, the presence of delays in capital allocations has a dramatic impact on the expected return and the Sharpe ratio of the investor's portfolio. I also find the robust optimal strategy in the presence of model uncertainty and show that the latter increases the effective risk aversion of the investor and makes her view the stock as more risky.