Rebalancing an investment portfolio in the presence of convex transaction costs,including market impact costs

The inclusion of transaction costs is an essential element of any realistic portfolio optimization. We extend the standard portfolio optimization problem to consider convex transaction costs incurred when rebalancing an investment portfolio. Market impact costs measure the effect on the price of a security that result from an effort to buy or sell the security, and they can constitute a large part of the total transaction costs. The loss to a portfolio from market impact costs is often modelled with a convex function that can be expressed using second-order cone constraints. The Markowitz framework of mean-variance efficiency is used. In order to properly represent the variance of the resulting portfolio, we suggest rescaling by the funds available after paying the transaction costs. This results in a fractional programming problem, which we show can be reformulated as an equivalent convex program of size comparable to the model without transaction costs. We show that an optimal solution to the convex program can always be found that does not discard assets.     

Parallel Optimization of Sparse Portfolios with AR-HMMs

In this paper we optimize mean reverting portfolios subject to cardinality constraints. First, the parameters of the corresponding Ornstein–Uhlenbeck (OU) process are estimated by auto-regressive Hidden Markov Models (AR-HMM), in order to capture the underlying characteristics of the financial time series. Portfolio optimization is then performed by maximizing the return achieved with a predefined probability instead of optimizing the predictability parameter, which provides more profitable portfolios. The selection of the optimal portfolio according to the goal function is carried out by stochastic search algorithms. The presented solutions satisfy the cardinality constraint in terms of providing a sparse portfolios which minimize the transaction costs (and, as a result, maximize the interpretability of the results). In order to use the method for high frequency trading (HFT) we utilize a massively parallel GPGPU architecture. Both the portfolio optimization and the model identification algorithms are successfully tailored to be running on GPGPU to meet the challenges of efficient software implementation and fast execution time. The performance of the new method has been extensively tested both on historical daily and intraday FOREX data and on artificially generated data series. The results demonstrate that a good average return can be achieved by the proposed trading algorithm in realistic scenarios. The speed profiling has proven that GPGPU is capable of HFT, achieving high-throughput real-time performance.     

Constrained Portfolio Selection using Particle Swarm Optimization

This paper presents a novel heuristic method for solving an extended Markowitz mean–variance portfolio selection model. The extended model includes four sets of constraints: bounds on holdings, cardinality, minimum transaction lots and sector (or market/class) capitalization constraints. The first set of constraints guarantee that the amount invested (if any) in each asset is between its predetermined upper and lower bounds. The cardinality constraint ensures that the total number of assets selected in the portfolio is equal to a predefined number. The sector capitalization constraints reflect the investors’ tendency to invest in sectors with higher market capitalization value to reduce their risk of investment.The extended model is classified as a quadratic mixed-integer programming model necessitating the use of efficient heuristics to find the solution. In this paper, we propose a heuristic based on Particle Swarm Optimization (PSO) method. The proposed approach is compared with the Genetic Algorithm (GA). The computational results show that the proposed PSO effectively outperforms GA especially in large-scale problems.     

Sparse portfolio rebalancing model based on inverse optimization

This paper considers a sparse portfolio rebalancing problem in which rebalancing portfolios with minimum number of assets are sought. This problem is motivated by the need to understand whether the initial portfolio is worthwhile to adjust or not, inducing sparsity on the selected rebalancing portfolio to reduce transaction costs (TCs), out-of-sample performance and small changes in portfolio. We propose a sparse portfolio rebalancing model by adding an l₁ penalty item into the objective function of a general portfolio rebalancing model. In this way, the model is sparse with low TCs and can decide whether and which assets to adjust based on inverse optimization. Numerical tests on four typical data sets show that the optimal adjustment given by the proposed sparse portfolio rebalancing model has the advantage of sparsity and better out-of-sample performance than the general portfolio rebalancing model.     

Wavelet evolutionary network for complex-constrained portfolio rebalancing

Portfolio rebalancing problem deals with resetting the proportion of different assets in a portfolio with respect to changing market conditions. The constraints included in the portfolio rebalancing problem are basic, cardinality, bounding, class and proportional transaction cost. In this study, a new heuristic algorithm named wavelet evolutionary network (WEN) is proposed for the solution of complex-constrained portfolio rebalancing problem. Initially, the empirical covariance matrix, one of the key inputs to the problem, is estimated using the wavelet shrinkage denoising technique to obtain better optimal portfolios. Secondly, the complex cardinality constraint is eliminated using k-means cluster analysis. Finally, WEN strategy with logical procedures is employed to find the initial proportion of investment in portfolio of assets and also rebalance them after certain period. Experimental studies of WEN are undertaken on Bombay Stock Exchange, India (BSE200 index, period: July 2001–July 2006) and Tokyo Stock Exchange, Japan (Nikkei225 index, period: March 2002–March 2007) data sets. The result obtained using WEN is compared with the only existing counterpart named Hopfield evolutionary network (HEN) strategy and also verifies that WEN performs better than HEN. In addition, different performance metrics and data envelopment analysis are carried out to prove the robustness and efficiency of WEN over HEN strategy.     

An artificial bee colony algorithm with feasibility enforcement and infeasibility toleration procedures for cardinality constrained portfolio optimization

One of the most studied variant of portfolio optimization problems is with cardinality constraints that transform classical mean–variance model from a convex quadratic programming problem into a mixed integer quadratic programming problem which brings the problem to the class of NP-Complete problems. Therefore, the computational complexity is significantly increased since cardinality constraints have a direct influence on the portfolio size. In order to overcome arising computational difficulties, for solving this problem, researchers have focused on investigating efficient solution algorithms such as metaheuristic algorithms since exact techniques may be inadequate to find an optimal solution in a reasonable time and are computationally ineffective when applied to large-scale problems. In this paper, our purpose is to present an efficient solution approach based on an artificial bee colony algorithm with feasibility enforcement and infeasibility toleration procedures for solving cardinality constrained portfolio optimization problem. Computational results confirm the effectiveness of the solution methodology.     

Large scale portfolio optimization with piecewise linear transaction costs

We consider the fundamental problem of computing an optimal portfolio based on a quadratic mean-variance model for the objective function and a given polyhedral representation of the constraints. The main departure from the classical quadratic programming formulation is the inclusion in the objective function of piecewise linear, separable functions representing the transaction costs. We handle the non-smoothness in the objective function by using spline approximations. The problem is first solved approximately using a primal-dual interior-point method applied to the smoothed problem. Then, we crossover to an active set method applied to the original non-smooth problem to attain a high accuracy solution. Our numerical tests show that we can solve large scale problems efficiently and accurately.     

An optimization model of the portfolio adjusting problem with fuzzy return and a SMO algorithm

Based on possibilistic mean and variance theory, this paper deals with the portfolio adjusting problem for an existing portfolio under the assumption that the returns of risky assets are fuzzy numbers and there exist transaction costs in portfolio adjusting precess. We propose a portfolio optimization model with V-shaped transaction cost which is associated with a shift from the current portfolio to an adjusted one. A sequential minimal optimization (SMO) algorithm is developed for calculating the optimal portfolio adjusting strategy. The algorithm is based on deriving the shortened optimality conditions for the formulation and solving 2-asset sub-problems. Numerical experiments are given to illustrate the application of the proposed model and the efficiency of algorithm. The results also show clearly the influence of the transaction costs in portfolio selection.     

Optimal Portfolio Hedging with Nonlinear Derivatives and Transaction Costs

We consider the problem of dynamically hedging a fixed portfolio of assets in the presence of non-linear instruments and transaction costs, as well as constraints on feasible hedging positions. We assume an investor maximizing the expected utility of his terminal wealth over a finite holding period, and analyse the dynamic portfolio optimization problem when the trading interval is fixed. An approximate solution is obtained from a two-stage numerical procedure. The problem is first transformed into a nonlinear programming problem which utilizes simulated coefficient matrices. The nonlinear programming problem is then solved numerically using standard constrained optimization techniques.     

Growth optimal investment in discrete-time markets with proportional transaction costs

We investigate how and when to diversify capital over assets, i.e., the portfolio selection problem, from a signal processing perspective. To this end, we first construct portfolios that achieve the optimal expected growth in i.i.d. discrete-time two-asset markets under proportional transaction costs. We then extend our analysis to cover markets having more than two stocks. The market is modeled by a sequence of price relative vectors with arbitrary discrete distributions, which can also be used to approximate a wide class of continuous distributions. To achieve the optimal growth, we use threshold portfolios, where we introduce a recursive update to calculate the expected wealth. We then demonstrate that under the threshold rebalancing framework, the achievable set of portfolios elegantly form an irreducible Markov chain under mild technical conditions. We evaluate the corresponding stationary distribution of this Markov chain, which provides a natural and efficient method to calculate the cumulative expected wealth. Subsequently, the corresponding parameters are optimized yielding the growth optimal portfolio under proportional transaction costs in i.i.d. discrete-time two-asset markets. As a widely known financial problem, we also solve the optimal portfolio selection problem in discrete-time markets constructed by sampling continuous-time Brownian markets. For the case that the underlying discrete distributions of the price relative vectors are unknown, we provide a maximum likelihood estimator that is also incorporated in the optimization framework in our simulations.     

A new multi-period investment strategies method based on evolutionary algorithms

This work introduces a new algorithmic trading method based on evolutionary algorithms and portfolio theory. The limitations of traditional portfolio theory are overcome using a multi-period definition of the problem. The model allows the inclusion of dynamic restrictions like transaction costs, portfolio unbalance, and inflation. A Monte Carlo method is proposed to handle these types of restrictions. The investment strategies method is introduced to make trading decisions based on the investor’s preference and the current state of the market. Preference is determined using heuristics instead of theoretical utility functions. The method was tested using real data from the Mexican market. The method was compared against buy-and-holds and single-period portfolios for metrics like the maximum loss, expected return, risk, the Sharpe’s ratio, and others. The results indicate investment strategies perform trading with less risk than other methods. Single-period methods attained the lowest performance in the experiments due to their high transaction costs. The conclusion was investment decisions that are improved when information providing from many different sources is considered. Also, profitable decisions are the result of a careful balance between action (transaction) and inaction (buy-and-hold).

     

Optimal portfolio selection of assets with transaction costs and no short sales

In this paper we study the optimal portfolio selection problem for assets. A double-objective programming model is first formulated for selecting optimal portfolios of asserts with transaction costs and taxes, where short sales and borrowings are not allowed. Some properties of efficient portfolios and the efficient frontier to the model are then derived. Based on these results, an interactive method that requires only paired preference comparison from the investor is established for solving the optimal portfolio selection problem. A numerical example is also presented to illustrate this method.     

Portfolio selection using neural networks

In this paper we apply a heuristic method based on artificial neural networks (NN) in order to trace out the efficient frontier associated to the portfolio selection problem. We consider a generalization of the standard Markowitz mean-variance model which includes cardinality and bounding constraints. These constraints ensure the investment in a given number of different assets and limit the amount of capital to be invested in each asset. We present some experimental results obtained with the NN heuristic and we compare them to those obtained with three previous heuristic methods. The portfolio selection problem is an instance from the family of quadratic programming problems when the standard Markowitz mean-variance model is considered. But if this model is generalized to include cardinality and bounding constraints, then the portfolio selection problem becomes a mixed quadratic and integer programming problem. When considering the latter model, there is not any exact algorithm able to solve the portfolio selection problem in an efficient way. The use of heuristic algorithms in this case is imperative. In the past some heuristic methods based mainly on evolutionary algorithms, tabu search and simulated annealing have been developed. The purpose of this paper is to consider a particular neural network (NN) model, the Hopfield network, which has been used to solve some other optimisation problems and apply it here to the portfolio selection problem, comparing the new results to those obtained with previous heuristic algorithms.     

基数约束投资组合问题的一种混合元启发式算法求解

针对资产数目和投资资金比例受约束的投资组合选择这一NP难问题,基于混沌搜索、粒子群优化和引力搜索算法提出了一种新的混合元启发式搜索算法。该算法能很好地平衡开发能力和勘探能力,有效抑制了算法早熟收敛现象。标准测试函数的测试结果表明混合算法与标准的粒子群优化和引力搜索算法相比具有更好的寻优效率;实证分析进一步对混合算法与遗传算法及粒子群优化算法在求解这类投资组合选择问题的性能进行了比较。数值结果表明,混合算法在搜索具有高预期回报的非支配投资组合方面表现更好,取得了更为满意的结果。     

International portfolio optimisation with integrated currency overlay costs and constraints

International financial portfolios can be exposed to substantial risk from variations of the exchange rates between the countries in which they hold investments. Nonetheless, foreign exchange can both generate extra return as well as loss to a portfolio, hence rather than just being avoided, there are potential advantages to well-managed international portfolios. This paper introduces an optimisation model that manages currency exposure of a portfolio through a combination of foreign exchange forward contracts, thereby creating a “currency overlay” on top of asset allocation. Crucially, the hedging and transaction costs associated with holding forward contracts are taken into account in the portfolio risk and return calculations. This novel extension of previous overlay models improves the accuracy of the risk and return calculations of portfolios. Consequently, more accurate investment decisions are obtained through optimal asset allocation and hedging positions. Our experimental results show that inclusion of such costs significantly changes the optimal decisions. Furthermore, effects of constraints related to currency hedging are examined. It is shown that tighter constraints weaken the benefit of a currency overlay and that forward positions vary significantly across return targets. A larger currency overlay is advantageous at low and high return targets, whereas small overlay positions are observed at medium return targets. The resulting system can hence enhance intelligent expert decision support for financial managers.     

在线投资组合选择的半指数梯度策略及实证分析

针对传统投资组合策略的高频资产配置调整产生高额交易成本从而导致最终收益不佳这一问题，提出基于机器学习与在线学习理论的半指数梯度投资组合（SEG）策略。该策略对投资期进行划分，通过控制投资期内的交易量来降低交易成本。首先，基于仅在每段分割的初始期调整投资组合而其余时间不进行交易这一投资方式来建立SEG策略模型，并结合收益损失构造目标函数；其次，利用因子图算法求解投资组合迭代更新的闭式解，并证明该策略累积资产收益的损失上界，从理论上保证算法的收益性能。在纽约交易所等多个数据集上进行的仿真实验表明，该策略在交易成本存在时仍然能够保持较高的收益，证实了该策略对于交易成本的不敏感性。     

Optimal portfolio trading subject to stochastic dominance constraints under second‐order autoregressive price dynamics

This paper studies the optimal portfolio trading problem under the generalized second‐order autoregressive execution price model. The problem of minimizing expected execution cost under the proposed price model is formulated as a quadratic programming (QP) problem. For a risk‐averse trader, problem formulation under the second‐order stochastic dominance constraints results in a quadratically constrained QP problem. Under some conditions on the execution price model, it is proved that the portfolio trading problems for risk‐neutral and risk‐averse traders become convex programming problems, which have many theoretical and computational advantages over the general class of optimization problems. Extensive numerical illustrations are provided, which render the practical significance of the proposed execution price model and the portfolio trading problems.     

Possibilistic Approaches to Portfolio Selection Problem with General Transaction Costs and a CLPSO Algorithm

The structure of portfolio selection depends essentially on the form of transaction cost. In this paper, we deal with the portfolio selection problems with general transaction costs under the assumption that the returns of assets obey LR-type possibility distributions. For any type of transaction costs, we employ a comprehensive learning particle swarm optimizer algorithm to obtain the optimal portfolio. Furthermore, we offer numerical experiments of different forms of transaction costs to illustrate the effectiveness of the proposed model and approach.     

Evolutionary multi-objective portfolio optimization in practical context

This paper addresses evolutionary multi-objective portfolio optimization in the practical context by incorporating realistic constraints into the problem model and preference criterion into the optimization search process. The former is essential to enhance the realism of the classical mean-variance model proposed by Harry Markowitz, since portfolio managers often face a number of realistic constraints arising from business and industry regulations, while the latter reflects the fact that portfolio managers are ultimately interested in specific regions or points along the efficient frontier during the actual execution of their investment orders. For the former, this paper proposes an order-based representation that can be easily extended to handle various realistic constraints like floor and ceiling constraints and cardinality constraint. An experimental study, based on benchmark problems obtained from the OR-library, demonstrates its capability to attain a better approximation of the efficient frontier in terms of proximity and diversity with respect to other conventional representations. The experimental results also illustrated its viability and practicality in handling the various realistic constraints. A simple strategy to incorporate preferences into the multi-objective optimization process is highlighted and the experimental study demonstrates its capability in driving the evolutionary search towards specific regions of the efficient frontier.     

Portfolio optimization problems in different risk measures using genetic algorithm

This paper introduces a heuristic approach to portfolio optimization problems in different risk measures by employing genetic algorithm (GA) and compares its performance to mean–variance model in cardinality constrained efficient frontier. To achieve this objective, we collected three different risk measures based upon mean–variance by Markowitz; semi-variance, mean absolute deviation and variance with skewness. We show that these portfolio optimization problems can now be solved by genetic algorithm if mean–variance, semi-variance, mean absolute deviation and variance with skewness are used as the measures of risk. The robustness of our heuristic method is verified by three data sets collected from main financial markets. The empirical results also show that the investors should include only one third of total assets into the portfolio which outperforms than those contained more assets.