Approximate dynamic programming via iterated Bellman inequalities

In this paper, we introduce new methods for finding functions that lower bound the value function of a stochastic control problem, using an iterated form of the Bellman inequality. Our method is based on solving linear or semidefinite programs, and produces both a bound on the optimal objective, as well as a suboptimal policy that appears to works very well. These results extend and improve bounds obtained in a previous paper using a single Bellman inequality condition. We describe the methods in a general setting and show how they can be applied in specific cases including the finite state case, constrained linear quadratic control, switched affine control, and multi‐period portfolio investment. Copyright © 2014 John Wiley & Sons, Ltd.     

Hunting High and Low: Visualising Shifting Correlations in Financial Markets

The analysis of financial assets’ correlations is fundamental to many aspects of finance theory and practice, especially modern portfolio theory and the study of risk. In order to manage investment risk, in‐depth analysis of changing correlations is needed, with both high and low correlations between financial assets (and groups thereof) important to identify. In this paper, we propose a visual analytics framework for the interactive analysis of relations and structures in dynamic, high‐dimensional correlation data. We conduct a series of interviews and review the financial correlation analysis literature to guide our design. Our solution combines concepts from multi‐dimensional scaling, weighted complete graphs and threshold networks to present interactive, animated displays which use proximity as a visual metaphor for correlation and animation stability to encode correlation stability. We devise interaction techniques coupled with context‐sensitive auxiliary views to support the analysis of subsets of correlation networks. As part of our contribution, we also present behaviour profiles to help guide future users of our approach. We evaluate our approach by checking the validity of the layouts produced, presenting a number of analysis stories, and through a user study. We observe that our solutions help unravel complex behaviours and resonate well with study participants in addressing their needs in the context of correlation analysis in finance.     

A memetic algorithm for cardinality-constrained portfolio optimization with transaction costs

A memetic approach that combines a genetic algorithm (GA) and quadratic programming is used to address the problem of optimal portfolio selection with cardinality constraints and piecewise linear transaction costs. The framework used is an extension of the standard Markowitz mean–variance model that incorporates realistic constraints, such as upper and lower bounds for investment in individual assets and/or groups of assets, and minimum trading restrictions. The inclusion of constraints that limit the number of assets in the final portfolio and piecewise linear transaction costs transforms the selection of optimal portfolios into a mixed-integer quadratic problem, which cannot be solved by standard optimization techniques. We propose to use a genetic algorithm in which the candidate portfolios are encoded using a set representation to handle the combinatorial aspect of the optimization problem. Besides specifying which assets are included in the portfolio, this representation includes attributes that encode the trading operation (sell/hold/buy) performed when the portfolio is rebalanced. The results of this hybrid method are benchmarked against a range of investment strategies (passive management, the equally weighted portfolio, the minimum variance portfolio, optimal portfolios without cardinality constraints, ignoring transaction costs or obtained with L₁ regularization) using publicly available data. The transaction costs and the cardinality constraints provide regularization mechanisms that generally improve the out-of-sample performance of the selected portfolios.     

Using investment satisfaction capability index based particle swarm optimization to construct a stock portfolio

The goal of this study is to construct an enhanced process based on the investment satisfied capability index (ISCI). The process is divided into two stages. The first stage is to apply the Process Capability Indices (PCI) for quality management so as to develop a new performance appreciation method. Investors can utilize the ISCI index to rapidly evaluate individual stock performance and then select those stocks which can lead to achieve investment satisfaction. In the second stage, a particle swarm optimization (PSO) algorithm with moving interval windows is applied to find the optimal investment allocation of the stocks in this portfolio. Based on those algorithms we can ensure investment risk control and obtain a more profitable stock investment portfolio.     

Robust multiperiod portfolio management in the presence of transaction costs

We study the viability of different robust optimization approaches to multiperiod portfolio selection. Robust optimization models treat future asset returns as uncertain coefficients in an optimization problem, and map the level of risk aversion of the investor to the level of tolerance of the total error in asset return forecasts. We suggest robust optimization formulations of the multiperiod portfolio optimization problem that are linear and computationally efficient. The linearity of the optimization problems is an advantage when complex additional requirements need to be imposed on the portfolio structure, e.g., limitations on positions in certain assets or tax constraints. We compare the performance of our robust formulations to the performance of the traditional single period mean-variance formulation frequently employed in the financial industry.     

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.     

Analytical VaR for international portfolios with common jumps

International portfolios which are composed of domestic assets and foreign assets are popular investment tools for financial institutions in highly integrated global financial markets. However, the focus of past studies had been on either domestic assets or foreign assets, but not both in the same context. They paid no attention to the studies of controlling the market risk of the international portfolios in the risk management literature. In contrast to the existing literature in portfolios, this paper considers not only domestic assets but also foreign assets, and provides an analytical value-at-risk (VaR) with common jump risk and exchange rate risk to manage market risk of international portfolios with exchange rate risk and common jumps over the subprime mortgage crisis. In general, the analytical solution can be used to accurately calculate VaRs by the backtesting criterion in terms of in-sample and out-of-sample fitting for an international portfolio with common jumps.     

On multiobjective combinatorial optimization and dynamic interim hedging of efficient portfolios

A dynamic portfolio policy is one that periodically rebalances an optimally diversified portfolio to account for time‐varying correlations. In order to sustain target‐level Sharpe performance ratios between rebalancing points, the efficient portfolio must be hedged with an optimal number of contingent claim contracts. This research presents a mixed‐integer nonlinear goal program (MINLGP) that is directed to solve the hierarchical multiple goal portfolio optimization model when the decision maker is faced with a binary hedging decision between portfolio rebalance periods. The MINLGP applied to this problem is formed by extending the separable programming foundation of a lexicographic nonlinear goal program (NLGP) to include branch‐and‐bound constraints. We establish the economic efficiency of applying this normative approach to dynamic portfolio rebalancing by comparing the risk‐adjusted performance measures of a hedged optimal portfolio to those of a naively diversified portfolio. We find that a hedged equally weighted small portfolio and a hedged efficiently diversified small portfolio perform similarly when comparing risk‐adjusted return metrics. However, when percentile risk measures are used to measure performance, the hedged optimally diversified portfolio clearly produces less expected catastrophic loss than does its nonhedged and naively diversified counterpart.     

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.     

Polyhedral coherent risk measures and investment portfolio optimization

The paper studies a class of polyhedral coherent risk measures for risk-return portfolio optimization problems under partial uncertainty, with unknown scenario probabilities estimated by some polyhedron. Such portfolio problems are reduced to linear programming problems. As an example, continuous problems of optimal investment allocation under risk of catastrophic floods are described. The study was supported from the Ukrainian Science and Technology Center, Project G3127. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 120–133, March–April 2008.     

Resolving uncertainty and creating value from the exercise of e‐commerce investment options

This study examines the effect of the resolution of uncertainty on real options exercise decisions with respect to three e‐commerce investment options: scale‐up, stage and joint investment, and the relationship between exercise of these options and firm performance. The results of a study of 172 public e‐commerce investment announcements show that resolution of external (exogenous) and internal (endogenous) uncertainty has a significant effect on option exercise decisions. However, the results also imply that simply waiting without investment in active learning does not create significant value from real options. The key differentiator is how a firm resolves endogenous uncertainty as this endows it with the ability to successfully undertake the information technology investment and exploit the economic opportunity implied by the resolution of exogenous uncertainty. Furthermore, our results imply that different options should be used to manage situations involving certainty of loss on one hand and severity of loss on the other hand. Thus, it is important for firms to make the right choices when using options‐based investing to manage risk. We suggest that, perhaps, managers need to maintain a portfolio of options to manage the two dimensions of risk simultaneously.     

Selection of a dynamic supply portfolio in make-to-order environment withrisks

The problem of a multi-period supplier selection and order allocation in make-to-order environment in the presence of supply chain disruption and delay risks is considered. Given a set of customer orders for finished products, the decision maker needs to decide from which supplier and when to purchase product-specific parts required for each customer order to meet customer requested due date at a low cost and to mitigate the impact of supply chain risks. The selection of suppliers and the allocation of orders over time is based on price and quality of purchased parts and reliability of supplies. For selection of dynamic supply portfolio a mixed integer programming approach is proposed to incorporate risk that uses conditional value-at-risk via scenario analysis. In the scenario analysis, the low-probability and high-impact supply disruptions are combined with the high probability and low impact supply delays. The proposed approach is capable of optimizing the dynamic supply portfolio by calculating value-at-risk of cost per part and minimizing expected worst-case cost per part simultaneously. Numerical examples are presented and some computational results are reported.     

An extended ASLD trading system to enhance portfolio management

An adaptive supervised learning decision (ASLD) trading system has been presented by Xu and Cheung (1997) to optimize the expected returns of investment without considering risks. In this paper, we propose an extension of the ASLD system (EASLD), which combines the ASLD with a portfolio optimization scheme to take a balance between the expected returns and risks. This new system not only keeps the learning adaptability of the ASLD, but also dynamically controls the risk in pursuit of great profits by diversifying the capital to a time-varying portfolio of N assets. Consequently, it is shown that: 1) the EASLD system gives the investment risk much smaller than the ASLD one; and 2) more returns are gained through the EASLD system in comparison with the two individual portfolio optimization schemes that statically determine the portfolio weights without adaptive learning. We have justified these two issues by the experiments.     

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.     

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。     

The trading on the mutual funds by gene expression programming with Sortino ratio

The aim of this paper is to combine several techniques together to provide one systematic method for guiding the investment in mutual funds. Many researches focus on the prediction of a single asset time series, or focus on portfolio management to diversify the investment risk, but they do not generate explicit trading rules. Only a few researches combine these two concepts together, but they adjust trading rules manually. Our method combines the techniques for generating observable and profitable trading rules, managing portfolio and allocating capital. First, the buying timing and selling timing are decided by the trading rules generated by gene expression programming. The trading rules are suitable for the constantly changing market. Second, the funds with higher Sortino ratios are selected into the portfolio. Third, there are two models for capital allocation, one allocates the capital equally (EQ) and the other allocates the capital with the mean variance (MV) model. Also, we perform superior predictive ability test to ensure that our method can earn positive returns without data snooping. To evaluate the return performance of our method, we simulate the investment on mutual funds from January 1999 to September 2012. The training duration is from 1999/1/1 to 2003/12/31, while the testing duration is from 2004/1/1 to 2012/9/11. The best annualized return of our method with EQ and MV capital allocation models are 12.08% and 12.85%, respectively. The latter also lowers the investment risk. To compare with the method proposed by Tsai et al., we also perform testing from January 2004 to December 2008. The experimental results show that our method can earn annualized return 9.07% and 11.27%, which are better than the annualized return 6.89% of Tsai et al.     

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.     

The Impact of Short-Sale Constraints on Asset Allocation Strategies via the Backward Markov Chain Approximation Method

This paper considers an asset allocation strategy over a finite period under investment uncertainty and short-sale constraints as a continuous-time stochastic control problem. Investment uncertainty is characterised by a stochastic interest rate and inflation risk. If there are no short-sale constraints, the optimal asset allocation strategy can be obtained analytically. We consider several kinds of short-sale constraints and employ the backward Markov chain approximation method to explore the impact of short-sale constraints on asset allocation decisions. Our results show that the short-sale constraints do indeed have a significant impact on these decisions.     

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.