مدل تحمل خطر و پرتفوی تنظیم کننده مسائل همراه با هزینه های مبادله بر اساس زمان قطعی

Due to changes of situation in financial markets and investors’ preferences towards risk, an existing portfolio may not be efficient after a period of time. In this paper, we propose a possibilistic risk tolerance model for the portfolio adjusting problem based on possibility moments theory. A Sequential Minimal Optimization (SMO)-type decomposition method is developed for finding exact optimal portfolio policy without extra matrix storage. We present a simple method to estimate the possibility distributions for the returns of assets. A numerical example is provided to illustrate the effectiveness of the proposed models and approaches.

In conventional portfolio analysis, a financial asset is usually characterized as a random variable with a probability distribution over its returns (Markowitz, 1959 and Markowitz, 1987). The key principle of the mean–variance methodology for portfolio selection is that the investment return is measured by the mean and the investment risk is measured by the variance (or standard deviation). The mean–variance models are usually described mathematically in two ways: maximizing return when a level of risk is given and minimizing risk when a level of return is given. To formulate the models of portfolio selection, it is necessary to estimate the probability distribution, strictly speaking, a mean vector and a covariance matrix. It means that all expected returns, variances, and covariances of risky assets can be accurately estimated by an investor. However, the returns of risky assets are in an uncertain economic environment and vary from time to time. The future states of returns and risks of risky assets cannot be predicted accurately. So it is impossible for investors to get the precise probability distribution. Based on the idea of approximations, the mean–variance model of portfolio selection is extended (Liu, 2004 and Samuelson, 1970). Though probability theory is one of the main tools used for analyzing uncertainty in finance, it cannot describe uncertainty completely since there are some other uncertain factors that differ from the random ones found in financial markets. Non-probabilistic factors affect the financial markets such that the return of risky asset is fuzzy uncertain. Recently, a number of researchers investigated the fuzzy portfolio selection problem. Watada (1997) and León et al. (2002) presented two approaches for portfolio selection using fuzzy decision theory. Inuiguchi and Tanino (2000)introduced a possibilistic programming approach to the portfolio selection problem based on the minimax regret criterion. Lai et al. (2002) and Giove and Funari (2006) constructed interval programming models of portfolio selection. Zhang and Nie (2004), Zhang et al. (2006) and Zhang and Wang (2008) studied the admissible efficient portfolio selection problems under the assumption that the expected return and risk of asset have admissible errors to reflect the uncertainty in real investment actions, the admissible portfolio selection models are extensions of the conventional mean–variance models. Liu et al. (2006) proposed a linear belief function (LBF) approach to evaluate portfolio performance. Carlsson et al. (2002) introduced a possibilistic approach to selecting portfolios with highest utility score under the assumption that the returns of assets are trapezoidal fuzzy numbers. On the basis of the (crisp) possibilistic mean and possibilistic variance introduced by Carlsson and Fullér (2001), Zhang et al. (2009) dealt with the portfolio selection problem when the returns of assets obey LR-type possibility distributions and there exist limits on holdings. Zhang and Xiao (2009) proposed the weighted lower and upper possibilistic portfolio selection models with tolerated risk level, where the return rates of risky assets are trapezoidal fuzzy numbers. Zhang et al. (2007) proposed an algorithm which can derive the explicit expression of the possibilistic efficient frontier based on the return rates of risky assets with general possibility distributions. Based on Carlsson et al. (2002), Zhang et al. (2009) discussed the portfolio selection problem for bounded assets with the maximum possibilistic mean–variance utility. Li and Xu (2007) proposed a new portfolio selection model in a hybrid uncertain environment under the assumption that the returns of securities are fuzzy random variables. The above-mentioned portfolio selection models only considered how to select the optimal portfolio at the beginning of investment period. Due to changes of situation in financial markets and the investors’ preferences towards risk, an existing portfolio may not be efficient after a period of time. Thus, the investors would consider to adjust the existing portfolio by buying or selling assets in response to changing conditions in financial markets. Therefore, the portfolio adjusting problem for an existing portfolio is an important issue for researchers and investors. Furthermore, this adjusting will incur a certain amount of transaction cost. Arnott and Wanger (1990) and Yoshimoto (1996) found that ignoring transaction costs would result in an inefficient portfolio. The purpose of this paper is to discuss the portfolio adjusting problem for an existing portfolio under the assumptions that the uncertain returns of assets in financial markets are fuzzy numbers. The contribution is in the following two aspects. This paper presents a risk tolerance model with transaction costs for adjusting an existing portfolio, which uses possibilistic moments of fuzzy number provided by Saeidifar and Pasha (2009). Meanwhile, this paper also proposes a Sequential Minimal Optimization(SMO)-type decomposition method for finding exact optimal portfolio policy without extra matrix storage. The rest of this paper is organized as follows. We recall the notions of possibilistic moments of fuzzy numbers in Section 2. We propose the possibilistic risk tolerance model for the portfolio adjusting problem based on possibilistic moments and transform the problem into several special cases in Section 3. We design a SMO-type decomposition method specially for the portfolio adjusting problem in Section 4. We present a simple method to estimate the possibility distributions and give a numerical example to illustrate the proposed model and algorithm in Section 5. Finally, some concluding remarks are given in Section 6.

Inthispaper,wediscusstheportfolioadjustingproblemwith transaction costs based on possibilistic moments under the as- sumptionthatthereturnsoftheassetsarefuzzynumbers.Analo- goustoMarkowitz'smeanvariancemethodology,weusethefirst possibilistic moment about zero of a portfolio as the investment return, and use the second possibilistic moment about the pos- sibilistic mean value of the portfolio as the investment risk. We propose a possibilistic risk tolerance model of the portfolio ad- justingproblem.Astherisktoleranceparametervariesfrom0to C1 , the efficient frontier can be traced out. A Sequential Mini- malOptimization(SMO)-typedecompositionmethodisdeveloped tofindtheexactoptimalportfoliopolicy,whichcansolvetheprob- lemswithoutextrastorageandspecialoptimizationsoftwarefor investors.Meanwhile,webringupasimpleestimationmethodfor possibilitydistributionsoffuzzyreturnsofassets.Thenumerical resultsalsoillustratetheeffectivenessoftheproposedmodelsand methods.