Studies On Risk Measures And Entropy Optimization Models In Portfolio Selection

The main wish of investors is to obtain maximum profits for themselves. Since invest return is tightly associated with the risk, the commonsense principle is that the investors should not put all his eggs into one basket. He should diversify his portfolio, i.e. he should invest his funds in a spread of low and high risk securities in such a way that the total expected return for all his investments is maximized and at the same time the investment risk is minimized. So the research of invest risk becomes of a very important problem which is faced in financial field.The work of Markowitz in portfolio selection has been most influential for the development of modern mathematical finance and its applications in practice, where he applied variance to measure invest risk and constructed the mean-variance model. The common indexes for measuring security investment risk variance of investmentreturn and β. As the research on risk measure deepens on, it has been found thatthere are some very severe flaws which cannot be avoided when using these indexes. In order to overcome these shortcomings, a lot of research work has been done in theoretical fields. But up to the present, these problems have not been solved satisfactorily. This thesis does study these problems mentioned above exactly under this circumstance.In this dissertation, the new invest risk measurements are studied first and 4 new portfolio models are constructed on the basis of analyzing the shortcomings of the former risk indexes.In chapter 2, we developed a measure of risk based on the concept of entropy as a kind of complementarity for measuring risk with variance. On this basis, a new portfolio models using entropy and variance mixed to measure risk are proposed in this paper, and apply a unique example to illustrate the practicability of the model. And at the same time, the efficient frontier and the new model (mean-variance-entropy) is analyzedIn chapter 3, the definitions of entropy and the related characters are discussed.Different from the MV model of Markowitz, in order to diversify risk as soon as possible, we proposed a new entropy optimization portfolio model which use mean to measure return and entropy to measure risk. Maximum entropy theory is to obtain a distribution which is closest to uniform distribution. We can generalize this reasoning and choose the distribution that maximizes uncertainty subject to the given moment constraints. In this way, we make full use of all the information given to us but avoid making any assumption about any information that not available. At last we make a simple compare between the mean-variance and the mean-entropy.In chapter 4, the mean-cross entropy model is proposed on the analysis of the minimum cross entropy principle. The cross entropy is introduced to measure invest risk. Suppose the investor knew a prior distribution (expected return distribution), and an unknown distribution (practical return) is wanted to be obtained, we can apply mean-cross entropy model to construct the portfolio selection.In chapter 5, some problems about minimax risk function is discussed. On the one hand, a portfolio optimization model with a new lx measure has been proposed. Asimple scheme has been derived, which generates the efficient portfolio under the lamodel analytically. On the other hand, a new portfolio has been proposed which uses the minimum return as a measure of invest risk, which can guarantee the investor obtain the higher return and avoid the invest risk. Although the two methods are different in risk measure, the meaning of both is the same.Our methods are better than the prior method: the distribution is not restricted in normal distribution and the calculation is simple. It is the deepening of the present theoretical study on security investment risk. This will surely make a new contribution for further research on security investment risk.