Some Mathematical Models Of Asset Portfolio Selection

There are already pretty rich achievement in the theory studying and it's application of portfolio selection during the past 60 years' development. But the application of portfolio theory has also had various concrete condition with the sum perfects development of financial market. In this paper we introduce several kinds of main theory , model and method of the portfolio selection according to the way of the portfolio selection's development. And the focus of the introduction is to incorporate fuzzy set theory into a semi-absolute deviation, multi-objective portfolio selection model.That so-called portfolio selection is to assign the wealth being to different assets and we can ensure the avails by dispersing the risk. The earliest studying of modern portfolio selection theory started from pioneering research work of Markowiz in 1952. Markowiz used the variance to depict the risk of the stock's avails and formulated the mean-variance analysis method of portfolio selection. In fact this model is a pair of target decision-making problems taking account into expected return (mean value) and risk (variance). The (EV) model is formulated as:where (μ_i,σ_ij) is given by x_i,. We also pay attention to a fact that it is really not enough althoughthe anticipated return and risk are the most essential elements of the investors' thinkingin developing process of portfolio selection. So the multi-objective portfolio selection models also develop quickly.In 1991 Konno and Yamazaki used the expected absolute deviation to depict the risk and gave a portfolio selection linear program model, which is often called mean-absolute deviation model and it developed into semi-absolute deviation later.where R_j represents the rate of return and x_j is the amount of money invested in security j while r_ji is the realization of the random variable R_j during the period t and is obtainable through historical data.In 1965 the USA auto control expert Zadeh presented the concept of membership function.For every mappingμ_A : Uâ†'[0,1],μâ†'μ_A(u), it ascertains a fuzzy subset A of U. Theμ_A is the membership function of A. And the value ofμ_A(u) at u is the membership degree of A to u. Zadeh used membership function to describe the difference of objective things. And it's the first time for people to apply the mathematics method to describe fuzzy concept. Fuzzy set theory is a forceful implement handling non-random uncertainty and it has advantage on describing the uncertainty of people's knowledge and behavior. We believe it to be a effective implement in the finance studying. Probably it can play a great important role in the behavior finance as the probability theory to the classical modern finance theory. Portfolio selection using fuzzy mathematical programming have been being a new studying direction during the past few years.In this paper the focus of our introduction is to incorporate fuzzy set theory into a semi-absolute deviation portfolio selection model(FP) for investors' taking account into five criteria: short term return, long term return, dividend, risk and liquidity. Additionally, in the model they consider constrains like the minimal and maximal fraction of the capital that can be invested in a single asset and also the number of assets in the portfolio to avoid impractical solutions that contain many assets with a very small percentage of the portfolio, a situation that occurs frequently in the classical mean-variance model. And they use a logistic function, i. e. , a non-linear S-shape membership function to express vague aspiration levels of an investor. The S-shape membership function is given bywhere 0 <α<∞is a fuzzy parameter which measures the degree of vagueness. We also introduce the two-phase approach to find efficient solutions and introduce the example analyze of the fuzzy programming model using the history data of 20 kinds of assets of India country stock trading market. Finally, We bring forward some new study direction of asset portfolio optimization with a view of our country reality. The first is to develop asset portfolio optimization using fuzzy mathematical programming combined with the behavior finance just as the focus of our introduction. The second is to carry out the study of asset being in debt. Another study direction is to develop the study of early warning of financial risks which occur suddenly. It's a common view that we should apply the optimization theory and method into finance decision-making practice with the fact that the course of our country's financial reform and economy unify in the whole world is being accelerated. But the finance is also confronted with risk and challenge during its developing. So the theory study and method application of portfolio optimization still remains to be gone deep continuously.