Portfolio selection problem with nonlinear wealth equations under non-extensive statistical mechanics for time-varying SDE

This paper is dedicated to the study of continuous-time mean–variance optimal portfolio selection problem with non-linear wealth equations under non-extensive statistical mechanics for the time-varying stochastic differential equation model. Firstly, we allow the returns and variance of risky assets are time-varying functions, which can fit the financial data better. Secondly, we consider an investor with the non-linear wealth equation. In fact, the wealth equations are not linear in many cases. The investor has to pay some taxes, which leads to a non-linear wealth equation. Moreover, since the return of the stocks price may be affected by a large investors portfolio selection, the wealth equation is non-linear in this case. Thirdly, the non-linear wealth equation driven by Tsallis distribution is constructed under non-extensive statistical mechanics, which can capture the characteristics of fat tails and aiguilles of the risky asset’s return. The viscosity solution of the HJB equation for the portfolio problem is proposed by the optimal stochastic control theory and Lagrange multiplier method. Finally, the efficient portfolio strategy and efficient frontier under non-extensive statistical mechanics are obtained. Furthermore, numerical analysis and real data study are discussed to show our results.     

风险资产收益序列相关的多阶段均值-方差投资组合选择

基于多阶段均值-方差框架, 研究任意多种风险资产存在一般收益序列相关时的投资组合选择问题. 首先, 采用Lagrange 对偶原理与动态规划相结合的方法对模型进行求解, 得到多阶段均值-方差模型的有效投资策略和有效边界的解析表达式; 然后, 证明在含有无风险资产的情形下有效边界仍为均值-标准差平面上的一条射线; 最后, 应用所得结论给出一个具体的实例分析.

     

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.     

Uncertain exit time multi-period mean–variance portfolio selection with endogenous liabilities and Markov jumps

This paper considers an uncertain exit time multi-period mean–variance portfolio selection problem with endogenous liabilities in a Markov jump market, where assets and liabilities of the balance sheet are simultaneously optimized. The random returns of assets and liabilities depend on the states of the financial market. By applying the Lagrange duality method, the Khatri–Rao matrix product technique and the dynamic programming approach, the explicit expressions for the mean–variance efficient strategy and efficient frontier are derived. In addition, the optimal balance sheet structures in both cases with and without boundary constraints are studied. Moreover, some degenerate cases are discussed, and some results in the literature are recovered as degenerate cases under our setting. Furthermore, a numerical example based on real data from the Chinese stock market is provided to illustrate the results obtained in this paper, and some interesting findings are presented.     

Continuous-time mean–variance portfolio selection with value-at-risk and no-shorting constraints

An investment problem is considered with dynamic mean–variance(M-V) portfolio criterion under discontinuous prices which follow jump–diffusion processes according to the actual prices of stocks and the normality and stability of the financial market. The short-selling of stocks is prohibited in this mathematical model. Then, the corresponding stochastic Hamilton–Jacobi–Bellman(HJB) equation of the problem is presented and the solution of the stochastic HJB equation based on the theory of stochastic LQ control and viscosity solution is obtained. The efficient frontier and optimal strategies of the original dynamic M-V portfolio selection problem are also provided. And then, the effects on efficient frontier under the value-at-risk constraint are illustrated. Finally, an example illustrating the discontinuous prices based on M-V portfolio selection is presented.     

A portfolio selection model with borrowing constraint based on possibility theory

Compared with the conventional probabilistic mean-variance methodology, fuzzy number can better describe an uncertain environment with vagueness and ambiguity. In this paper, the portfolio selection model with borrowing constraint is proposed by means of possibilistic mean, possibilistic variance, and possibilistic covariance under the assumption that the returns of assets are fuzzy numbers. And a quadratic programming model with inequality constraints is presented when the returns of assets are trapezoid fuzzy numbers. Furthermore, Lemke algorithm is utilized to solve the model. Finally, a numerical example of the portfolio selection problem is given to illustrate our proposed effective means and variances. The results of the numerical example also show that the investor can make different decisions according to different requirements for the values of expected returns. And the efficient portfolio frontier of the model with borrowing constraints can be easily obtained.     

风险资本约束下保险公司的最优比例再保险-投资策略

本文研究保险公司的再保险-投资问题.假定保险公司的整体风险由风险资本(Capital-at-Risk,CaR)来度量;盈余过程由扩散模型近似表示;在任意时刻保险公司可购买比例再保险(或获取新业务)和投资无风险资产与多种风险资产;风险资产的价格由几何布朗运动驱动.保险公司的目标是在整体风险CaR受约束的条件下最大化终端财...     

Algorithm for constructing the efficient frontier of an investment portfolio

An algorithm for calculating the limiting (for large time values) efficient frontier of an investment portfolio with its assets given by stochastic differential equations is considered. The model also takes into account the influence of macroeconomic factors. What makes this algorithm special is that it uses only the simplest operations of linear algebra.     

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.     

Sensitivity analysis on modified variable returns to scale model in Data Envelopment Analysis using facet analysis

One of the concerns in Data Envelopment Analysis (DEA) is the sensitivity and stability analysis of specific Decision Making Unit (DMU), which is under evaluation. In economical point of view, the stability region in input–output space for maintaining the efficiency score of efficient DMU is important. In this paper, a new sensitivity analysis approach based on Banker, Charnes and Cooper (BCC) model which is modified by facet analysis, is developed. An extended stability region is determined especially for DMUs that are placed on intersection of efficient and weak efficient frontier. The results are shown by numerical examples.     

Possibilistic mean-variance models and efficient frontiers for portfolio selection problem

In this paper, it is assumed that the rates of return on assets can be expressed by possibility distributions rather than probability distributions. We propose two kinds of portfolio selection models based on lower and upper possibilistic means and possibilistic variances, respectively, and introduce the notions of lower and upper possibilistic efficient portfolios. We also present an algorithm which can derive the explicit expression of the possibilistic efficient frontier for the possibilistic mean-variance portfolio selection problem dealing with lower bounds on asset holdings.     

Robust optimization framework for cardinality constrained portfolio problem

One of the primary concerns on any asset allocation problem is to maintain a limited number of assets from the market. The problem becomes more complicated when the return of all risky assets are subject to uncertainty. In this paper, we propose a new portfolio modeling approach with uncertain data and it is also analyzed using different robust optimization techniques. The proposed formulations are solved using genetic algorithm. The implementation of the proposed method is examined on variety of well known benchmark data sets.     

Comparative issues in large-scale mean-variance efficient frontier computation

One of the functions of a portfolio management system is to return quickly an efficient frontier. However, in the large-scale problems (1000 to 3000 securities) that are beginning to appear with greater frequency, the task of computing the mean-variance efficient frontier, even when all constraints are linear, can range from the significant to the prohibitive. For ease of reference, we call mean-variance problems with all linear constraints Markowitz problems. With little on the time to compute a Markowitz-problem efficient frontier in the literature, we conduct experiments that involve varying problem sizes, methods employed, and optimizers used to present an overall picture of the situation and establish benchmarks in the large-scale arena. One of the conclusions of the experiments is the superiority of the class of techniques that would fall under the title of parametric quadratic programming.     

Tax impact on multi-stage mean-variance portfolio allocation

We investigate the sensitivity to tax change of multi‐stage portfolio allocation over a discrete time investment horizon. Special taxation rules within wrappers grouped a number of risky assets are integrated with multi‐stage linear or quadratic stochastic programming in the mean‐variance framework. The uncertainty on the returns of assets is specified as a scenario tree generated by a simulation‐based approach. We adjust different values on capital gains tax under different asset bounds and risk levels. The tax impact in the yearly reallocation of the investments for a typical case with an annual fixed withdrawal that utilizes completely the option of taper relief is also explored. Our computational results show that taxes, combined with other effects such as risk and investment upper bounds, have a significant performance impact on portfolio allocation as well as diversification over wrappers. Yet, investment strategies can be made robust to changes in taxation.     

Prediction of bankruptcy Iranian corporations through artificial neural network and Probit-based analyses

In this study, multilayer perceptron (MLP) of artificial neural networks is utilized to build a new model for bankruptcy prediction. A precise MLP-based relationship is obtained to classify samples of 136 bankrupt and non-bankrupt Iranian corporations using their financial ratios. A Probit analysis is performed to benchmark the MLP model. Ratios of sales to current assets ratio, operational income to sales, quick assets to total assets, and total liability to total assets are used as the effective predictive financial ratios. A comparative study is further conducted on the classification accuracy of the MLP, Probit, and other existing models. The proposed MLP model has a significantly better performance than the Probit and other models found in the bankruptcy prediction literature.     

Pricing American options under multi-state regime switching with an efficient L- stable method

An efficient second-order method based on exponential time differencing approach for solving American options under multi-state regime switching is developed and analysed for stability and convergence. The method is seen to be strongly stable (L-stable) in each regime. The implicit predictor–corrector nature of the method makes it highly efficient in solving nonlinear systems of partial differential equations arising from multi-state regime switching model. Stability and convergence of the method are examined. The impact of regime switching on option prices for different jump rates and volatility is illustrated. A general framework for multi-state regime switching in multi-asset American option has been provided. Numerical experiments are performed on one and two assets to demonstrate the performance of the method with convex as well as non-convex payoffs. The method is compared with some of the existing methods available in the literature and is found to be reliable, accurate and efficient.     

Directors' digital fiduciary duties

We explore the potential legal and technical risks inherent in attempts to implement digital protection. Specifically, we consider how liability might arise for those who have fiduciary responsibility for sensitive information assets, including the emerging trend toward imposing liability where digital protections are severely deficient or digital security has been breached. We also focus on obstacles created by disparities in the knowledge and expertise of professionals who are responsible for corporate assets and who risk legal liability if their efforts are insufficient or ineffective. We address these disparities to bridge the gap between executive personnel responsible for corporate governance and technical personnel responsible for corporate digital security     

Bidding games with several risky assets

This paper considers multistep bidding models where several types of risky assets (shares) are traded between two agents that have different information on the liquidation prices of traded assets. These random prices depend on “a state of nature” determined by the initial chance move according to a probability distribution that is known to both players. Player 1 (insider) is aware of the state of nature, but Player 2 is not. Player 2 knows that Player 1 is an insider. The bids may take any integer values. The n-step model is reduced to a zero-sum repeated game with incomplete information of Player 2. It is shown that, if the liquidation prices of shares have finite variances, then the value sequence of n-step games is bounded. This property makes it reasonable to consider the bidding of unlimited duration. The solutions of the corresponding infinite-duration games are constructed. By analogy with the case of two risky assets (see [9]), the optimal strategy of Player 1 induces a random walk of the transaction prices. The symmetry of this random walk is broken at the final steps of the game.     

A novel nonlinear programming approach for estimating CAPM beta of an asset using fuzzy regression

The evaluation of risky assets is one of the major research tasks in the finance theory. There are several Capital Asset Pricing Models (CAPM) in the literature; the most popular one of those is the Sharpe–Lintner–Black mean–variance CAPM. According to this model, the typical measure of systematic risk is the beta coefficient. The beta coefficient can be evaluated by means of least squares method (LSM), Robust Regression Techniques (RRT), or similar approaches. However, the statistical assumptions of LSM might be invalid in the existence of extreme observations in data set. In order to decrease influence on the beta coefficient of extreme observations, most analyst apply to RRT’s. However, either RRT’s remove the extreme observations from the data set, or decrease their influences on the beta coefficient. Whereas the omitted observations might be valuable for investors since they carry substantial information about the state of nature. In other words, there is a clash between statistical and financial theory. In this study, to overcome this incompatibility, and to take into account the extreme observations carried worthy information, a novel fuzzy regression approach is proposed. The proposed approach is based on both possibility concepts and central tendency in the estimation of beta coefficient. In application section of this paper, the beta coefficients of three assets traded in Istanbul Stock Exchange (ISE) are estimated by the proposed fuzzy approach and the traditional techniques, and then the results of analysis are compared, and discussed.     

A portfolio optimization model with three objectives and discrete variables

We formulate the portfolio selection as a tri-objective optimization problem so as to find tradeoffs between risk, return and the number of securities in the portfolio. Furthermore, quantity and class constraints are introduced into the model in order to limit the proportion of the portfolio invested in assets with common characteristics and to avoid very small holdings. Since the proposed portfolio selection model involves mixed integer decision variables and multiple objectives finding the exact efficient frontier may be very hard. Nevertheless, finding a good approximation of the efficient surface which provides the investor with a diverse set of portfolios capturing all possible tradeoffs between the objectives within limited computational time is usually acceptable. We experiment with the current state of the art evolutionary multiobjective optimization techniques, namely the Non-dominated Sorting Genetic Algorithm II (NSGA-II), Pareto Envelope-based Selection Algorithm (PESA) and Strength Pareto Evolutionary Algorithm 2 (SPEA2), for solving the mixed-integer multiobjective optimization problem and provide a performance comparison among them using metrics proposed by the community.