A multicriteria optimization model of portfolio rebalancing with transaction costs in fuzzy environment

In this paper we propose multicriteria credibilistic framework for portfolio rebalancing (adjusting) problem with fuzzy parameters considering return, risk and liquidity as key financial criteria. The portfolio risk is characterized by a risk curve that represents each likely loss of the portfolio return and the corresponding chance of its occurrence rather than a single pre-set level of the loss. Furthermore, we consider an investment market scenario where, at the end of a typical time period, the investor would like to modify his existing portfolio by buying and/or selling assets in response to changing market conditions. We assume that the investor pays transaction costs based on incremental discount schemes associated with the buying and/or selling of assets, which are adjusted in the net return of the portfolio. A hybrid intelligent algorithm that integrates fuzzy simulation with a real-coded genetic algorithm is developed to solve the portfolio rebalancing (adjusting) problem. The proposed solution approach is useful particularly for the cases where fuzzy parameters of the problem are characterized by general functional forms.     

含有资本结构因子、交易成本和风险偏好的模糊最优化投资模型

建立了含有资本结构因子、交易成本和风险偏好的模糊最优化投资模型,在允许卖空条件下,给出最优投资策略及有效边界;在不允许卖空条件下,给出了确定其有效边界的算法,并分析了风险偏好、无风险利率和交易成本对有效边界的影响,最后通过示例进行了分析.     

Optimal hedging strategies for multi-period guarantees in the presence of transaction costs: A stochastic programming approach

Multi-period guarantees are often embedded in life insurance contracts. In this paper we consider the problem of hedging these multi-period guarantees in the presence of transaction costs. We derive the hedging strategies for the cheapest hedge portfolio for a multi-period guarantee that with certainty makes the insurance company able to meet the obligations from the insurance policies it has issued. We find that by imposing transaction costs, the insurance company reduces the rebalancing of the hedge portfolio. The cost of establishing the hedge portfolio also increases as the transaction cost increases. For the multi-period guarantee there is a rather large rebalancing of the hedge portfolio as we go from one period to the next. By introducing transaction costs we find the size of this rebalancing to be reduced. Transaction costs may therefore be one possible explanation for why we do not see the insurance companies performing a large rebalancing of their investment portfolio at the end of each year.     

考虑完整交易费用的组合证券投资求解

针对中国证券市场现存的各种交易费用,建立一个更加符合实际的组合证券投资模型.该模型不仅继承了股票不可拆分、不能卖空等特点,而且完全反映了交易费用的核算情况;最后给出一个遗传算法结合动态罚函数求解的投资实例,计算结果证明了该模型及其求解方法的有效性和可操作性.     

Mean-variance portfolio rebalancing with transaction costs and funding changes

Investment portfolios should be rebalanced to take account of changing market conditions and changes in funding. Standard mean-variance (MV) portfolio selection methods are not appropriate for portfolio rebalancing, as the initial portfolio, change in funding and transaction costs are not considered. A quadratic mixed integer programming portfolio rebalancing model, which takes account of these factors is developed in this paper. The transaction costs in this portfolio rebalancing model are composed of fixed charges and variable costs, including the market impact costs associated with large market trades of individual securities, where these variable transaction costs are assumed to be non-linear functions of traded value. The use of this model is demonstrated and it is shown that when initial portfolio, funding changes and transaction costs are taken into account in portfolio construction and rebalancing, MV efficient portfolios that include risk-free lending do not have the structure expected from portfolio theory.     

Multi-objective possibilistic model for portfolio selection with transaction cost

In this paper, we introduce the possibilistic mean value and variance of continuous distribution, rather than probability distributions. We propose a multi-objective Portfolio based model and added another entropy objective function to generate a well diversified asset portfolio within optimal asset allocation. For quantifying any potential return and risk, portfolio liquidity is taken into account and a multi-objective non-linear programming model for portfolio rebalancing with transaction cost is proposed. The models are illustrated with numerical examples.     

Portfolio rebalancing model with transaction costs based on fuzzy decision theory

The fuzzy set is one of the powerful tools used to describe an uncertain environment. As well as quantifying any potential return and risk, portfolio liquidity is taken into account and a linear programming model for portfolio rebalancing with transaction costs is proposed. The level of return that an investor might aspire to, the risk and the liquidity of portfolio are vague in an uncertain financial environment. Considering them as fuzzy numbers, we propose a portfolio rebalancing model with transaction costs based on fuzzy decision theory. An example is given to illustrate the behavior of the proposed model using real data from the Shanghai Stock Exchange.     

Portfolio selection in discrete time with transaction costs and power utility function: a perturbation analysis

In this article, we study a multi-period portfolio selection model in which a generic class of probability distributions is assumed for the returns of the risky asset. An investor with a power utility function rebalances a portfolio comprising a risk-free and risky asset at the beginning of each time period in order to maximize expected utility of terminal wealth. Trading the risky asset incurs a cost that is proportional to the value of the transaction. At each time period, the optimal investment strategy involves buying or selling the risky asset to reach the boundaries of a certain no-transaction region. In the limit of small transaction costs, dynamic programming and perturbation analysis are applied to obtain explicit approximations to the optimal boundaries and optimal value function of the portfolio at each stage of a multi-period investment process of any length.     

Portfolio-optimization models for small investors

Since 2010, the client base of online-trading service providers has grown significantly. Such companies enable small investors to access the stock market at advantageous rates. Because small investors buy and sell stocks in moderate amounts, they should consider fixed transaction costs, integral transaction units, and dividends when selecting their portfolio. In this paper, we consider the small investor’s problem of investing capital in stocks in a way that maximizes the expected portfolio return and guarantees that the portfolio risk does not exceed a prescribed risk level. Portfolio-optimization models known from the literature are in general designed for institutional investors and do not consider the specific constraints of small investors. We therefore extend four well-known portfolio-optimization models to make them applicable for small investors. We consider one nonlinear model that uses variance as a risk measure and three linear models that use the mean absolute deviation from the portfolio return, the maximum loss, and the conditional value-at-risk as risk measures. We extend all models to consider piecewise-constant transaction costs, integral transaction units, and dividends. In an out-of-sample experiment based on Swiss stock-market data and the cost structure of the online-trading service provider Swissquote, we apply both the basic models and the extended models; the former represent the perspective of an institutional investor, and the latter the perspective of a small investor. The basic models compute portfolios that yield on average a slightly higher return than the portfolios computed with the extended models. However, all generated portfolios yield on average a higher return than the Swiss performance index. There are considerable differences between the four risk measures with respect to the mean realized portfolio return and the standard deviation of the realized portfolio return.     

Equal Risk Bounding is better than Risk Parity for portfolio selection

Risk Parity (RP), also called equally weighted risk contribution, is a recent approach to risk diversification for portfolio selection. RP is based on the principle that the fractions of the capital invested in each asset should be chosen so as to make the total risk contributions of all assets equal among them. We show here that the Risk Parity approach is theoretically dominated by an alternative similar approach that does not actually require equally weighted risk contribution of all assets but only an equal upper bound on all such risks. This alternative approach, called Equal Risk Bounding (ERB), requires the solution of a nonconvex quadratically constrained optimization problem. The ERB approach, while starting from different requirements, turns out to be strictly linked to the RP approach. Indeed, when short selling is allowed, we prove that an ERB portfolio is actually an RP portfolio with minimum variance. When short selling is not allowed, there is a unique RP portfolio and it contains all assets in the market. In this case, the ERB approach might lead to the RP portfolio or it might lead to portfolios with smaller variance that do not contain all assets, and where the risk contributions of each asset included in the portfolio is strictly smaller than in the RP portfolio. We define a new riskiness index for assets that allows to identify those assets that are more likely to be excluded from the ERB portfolio. With these tools we then provide an exact method for small size nonconvex ERB models and a very efficient and accurate heuristic for larger problems of this type. In the case of a common constant pairwise correlation among all assets, a closed form solution to the ERB model is obtained and used to perform a parametric analysis when varying the level of correlation. The practical advantages of the ERB approach over the RP strategy are illustrated with some numerical examples. Computational experience on real-world and on simulated data confirms accuracy and efficiency of our heuristic approach to the ERB model also in comparison with some state-of-the-art local and global optimization codes.     

A Continuous Time Financial Market with a Poisson Process as Transaction Timer

We study the influence of transactions on investors' portfolio values under the assumption that the investors' transaction times are determined by Poisson point processes, whose intensity measures can naturally be interpreted as transaction frequencies. We give lower and upper bounds on the expectations of portfolio values in terms of transaction intensities, and prove that there exist a sequence of portfolio fractions and a transaction frequency which maximize the expectation of the portfolio value for a finite horizon. We also give bounds on transaction frequencies for preventing the investors from losing money. Then the optimal transaction strategies for finite and infinite time horizons and the asymptotic effects of making transactions are discussed based on the concept of a benefit function of transactions. Finally, we investigate the influence of transactions on financial markets, with the market mean rates of return and volatilities being connected with the transaction frequency. We show that the market becomes unprofitable in a finite time if an overwhelming amount of transactions is involved and the market is suitable for some limited transactions when its trade capacity does not increase beyond any limit at a relatively high speed. Our models and simulations illustrate how the investors' collective action affects the financial market.     

基于CVaR约束的指数组合优化模型及实证分析

随着指数衍生产品日益受到重视,指数化投资组合常被投资者或机构所采用,而用有限的资金按指数构成比例进行投资显然是不现实的,所以指数的最优误差追踪就显得更加重要。本文将追踪误差定义为证券投资组合收益率与所追踪的指数基准收益率之差,并在分析CvaR(ConditionalValue at Risk)的基础上,在无交易费用和有交易费用的情况下,建立了基于CVaR约束的追踪误差最小化的指数组合优化模型,对指数进行复制,并通过实证分析,得出了基于CVaR约束的追踪误差最小时的样本期内及样本期外的最优投资策略,验证了CVaR约束控制风险的有效性。     

Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints

We will propose a branch and bound algorithm for calculating a globally optimal solution of a portfolio construction/rebalancing problem under concave transaction costs and minimal transaction unit constraints. We will employ the absolute deviation of the rate of return of the portfolio as the measure of risk and solve linear programming subproblems by introducing (piecewise) linear underestimating function for concave transaction cost functions. It will be shown by a series of numerical experiments that the algorithm can solve the problem of practical size in an efficient manner. Received: July 15, 1999 / Accepted: October 1, 2000?Published online December 15, 2000     

A multi-period bank portfolio management program

We describe a bank portfolio management program based on the complete Markowitz model, which explicitly treats risk due to unanticipated fluctuations in interest rate. Our program takes into account both inter-temporal and intra-temporal covariance. The major result of this approach is that, for the same expected return, our model yields a portfolio with significantly smaller risk than that determined by an index model. For the same risk level, our method yields a portfolio with higher expected yield. The model employs a rolling planning horizon, with time periods in the planning horizon of arbitrary length. A novelty in the model is that it permits inter-temporal transactions in the portfolio's securities by generating dummy securities to represent every possible transaction over the planning horizon. The output from the model consists of a list of portfolio strategies showing the expected after-tax return and the 1% worst case yield for each strategy. We also present an illustrative example, using real data from a large Pennsylvania bank, and compare the results from our model to the simpler variance-only and index models. The principles upon which the model is based are sufficiently general to allow the program to be expanded into a general asset-liability balance sheet management program.     

Minimal concave cost rebalance of a portfolio to the efficient frontier

One usually constructs a portfolio on the efficient frontier, but it may not be efficient after, say three months since the efficient frontier will shift as the elapse of time. We then have to rebalance the portfolio if the deviation is no longer acceptable. The method to be proposed in this paper is to find a portfolio on the new efficient frontier such that the total transaction cost required for this rebalancing is minimal. This problem results in a nonconvex minimization problem, if we use mean-variance model. In this paper we will formulate this problem by using absolute deviation as the measure of risk and solve the resulting linearly constrained concave minimization problem by a branch and bound algorithm successfully applied to portfolio optimization problem under concave transaction costs. It will be demonstrated that this method is efficient and that it leads to a significant reduction of transaction costs. Key words.portfolio optimization – rebalance – mean-absolute deviation model – concave cost minimization – optimization over the efficient set – global optimizationMathematics Subject Classification (1991):20E28, 20G40, 20C20     

Portfolio adjusting optimization with added assets and transaction costs based on credibility measures

In response to changeful financial markets and investor’s capital, we discuss a portfolio adjusting problem with additional risk assets and a riskless asset based on credibility theory. We propose two credibilistic mean–variance portfolio adjusting models with general fuzzy returns, which take lending, borrowing, transaction cost, additional risk assets and capital into consideration in portfolio adjusting process. We present crisp forms of the models when the returns of risk assets are some deterministic fuzzy variables such as trapezoidal, triangular and interval types. We also employ a quadratic programming solution algorithm for obtaining optimal adjusting strategy. The comparisons of numeral results from different models illustrate the efficiency of the proposed models and the algorithm.     

Simultaneous pursuit of out-of-sample performance and sparsity in index tracking portfolios

Index tracking is a passive investment strategy in which a fund (e.g., an ETF: exchange traded fund) manager purchases a set of assets to mimic a market index. The tracking error, i.e., the difference between the performances of the index and the portfolio, may be minimized by buying all the assets contained in the index. However, this strategy results in a considerable transaction cost and, accordingly, decreases the return of the constructed portfolio. On the other hand, a portfolio with a small cardinality may result in poor out-of-sample performance. Of interest is, thus, constructing a portfolio with good out-of-sample performance, while keeping the number of assets invested in small (i.e., sparse). In this paper, we develop a tracking portfolio model that addresses the above conflicting requirements by using a combination of L0- and L2-norms. The L2-norm regularizes the overdetermined system to impose smoothness (and hence has better out-of-sample performance), and it shrinks the solution to an equally-weighted dense portfolio. On the other hand, the L0-norm imposes a cardinality constraint that achieves sparsity (and hence a lower transaction cost). We propose a heuristic method for estimating portfolio weights, which combines a greedy search with an analytical formula embedded in it. We demonstrate that the resulting sparse portfolio has good tracking and generalization performance on historic data of weekly and monthly returns on the Nikkei 225 index and its constituent companies.     

具有不确定交易价格的指数跟踪资产组合再平衡问题

本文考虑具有某种不确定性的交易费用及预算约束的指数跟踪资产组合的再平衡问题。在已有的模型中,预算约束中使用的交易价格通常是一个确定值。而在再平衡过程中,股票的实际交易价格是不确定的。本文使用有限状态的离散时间马尔柯夫链模型处理交易价格的不确定性,并基于情景分析方法建立了具有不确定预算关系式的再平衡模型,然后使用股票市场的实际样本数据进行了数值实验,模拟结果说明本文的模型是可行的。     

考虑保守卖空与财务困境的背景风险投资组合模型

如何合理地考虑投资者所面临的背景风险及现实市场限制来进行有效地投资决策是人们所广泛关注的重要实际管理决策问题。本文研究投资者同时面临加性和乘性两类背景风险的前提下具有保守卖空与财务困境的投资组合选择问题。假定投资者寻求使得投资收益最大、投资风险最小及证券主体财务困境最小的最优投资组合策略,进而提出考虑保守卖空与财务困境的背景风险投资组合模型。然后,利用具有精英策略的非支配排序遗传算法对模型进行求解。最后,通过实例来阐述模型的实用性。研究结果表明:考虑保守卖空能为投资者提供更大的收益;两类背景风险的变化均导致有效前沿面的变化。     

Genetic algorithms for portfolio selection problems with minimum transaction lots

Conventionally, portfolio selection problems are solved with quadratic or linear programming models. However, the solutions obtained by these methods are in real numbers and difficult to implement because each asset usually has its minimum transaction lot. Methods considering minimum transaction lots were developed based on some linear portfolio optimization models. However, no study has ever investigated the minimum transaction lot problem in portfolio optimization based on Markowitz’ model, which is probably the most well-known and widely used. Based on Markowitz’ model, this study presents three possible models for portfolio selection problems with minimum transaction lots, and devises corresponding genetic algorithms to obtain the solutions. The results of the empirical study show that the portfolios obtained using the proposed algorithms are very close to the efficient frontier, indicating that the proposed method can obtain near optimal and also practically feasible solutions to the portfolio selection problem in an acceptable short time. One model that is based on a fuzzy multi-objective decision-making approach is highly recommended because of its adaptability and simplicity.