证券投资组合的风险度量与熵优化模型研究
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摘要
证券投资的最根本目的在于获取利益。但在投资活动中，收益总是伴随着风险．通常，收益越高，风险越大；风险越小，收益越低。为了分散风险，许多投资者将许多种证券组合在一起进行投资，即所谓的投资组合，以期获得最大收益，这就使得投资风险的研究成为金融界面临的重大课题之一。
Markowitz以证券收益率的方差作为组合证券风险的度量，开辟了金融定量分析的时代，在度量风险的基础的上建立了组合投资决策模型，该模型在理论和实际应用中都有重要意义。证券投资风险常用的度量方式主要是投资收益率的方差或β值，但是随着研究的深入，人们发现常用的风险度量指标存在不可回避的重大缺陷，为了克服现有理论的不足，理论界进行了广泛的研究，但是到目前为止，还没有一种广泛有效的度量风险的方法。本文正是在此背景下对上述问题展开研究的。
在本篇论文中，作者首先以证券投资的风险为对象，研究了不同的风险度量模型：分析了各种风险度量方法的不足。鉴于目前的风险度量方法都存在不同程度的缺陷，本文主要提出了4种新的风险度量方法，并在此基础上建立了相关的投资组合优化模型。
第二章从证券投资组合理论的风险度量方面着手，基于熵可以作为不确定性度量指标的内涵，把熵作为方差度量风险的一种有效的补偿，用熵和方差来共同度量风险，从而提出了一个新的风险度量模型：均值-方差-熵模型，并以实例作了说明。用熵和方差共同来度量风险，实际上是从一种最保守的角度出发，从而使厌恶风险的投资者从事投资的时候获得较大的收益，面临较少的风险，这在一定程度上弥补了仅仅用方差度量风险的缺陷。本章里还介绍了有效边界以及有效投资组合的内容，并从几何意义上对新模型进行了有关分析。
第三章从风险的度量出发，引进了一种新的度量方法-熵。在简单介绍了最大熵优化原理以及这个问题的对偶规划的基础上，建立了证券投资组合的均值-熵优化模型，这种方法不必计算协方差矩阵，避免了用方差度量风险计算的复杂性；并且均值-熵模型不依赖于收益率的概率分布是某种特定的分布(如正态分布)，最后将该方法与均值-方差方法进行了对比。
大连理工大学博士学位论文
第四章从最小叉嫡原理与最大嫡原理互补性出发建立了均值一叉嫡优化模型。假设
投资者已经知道一个某个时期的收益率分布，而且期望实际的收益率分布在满足所有约
束的条件下尽量靠近这个先验的概率分布，我们将最小叉嫡原理引进投资组合选择中，
应用叉嫡作为度量投资风险的指标，建立了均值一叉嫡优化模型。
第五章中采用了极大极小函数的分析方法，从投资者的角度出发，运用不同的风险
度量准则建立了两个模型，即:最大风险最小化模型和最小收益最大化模型，并对模型
进行了分析，跟以往的投资组合模型进行了比较，经过比较可以看出，极大极小风险函
数的投资组合模型是把投资者的最大损失降到最低当作目标，从最坏情形中取最优，可
达到较高的收益率，从而保证了投资者的投资收益。同时，极大极小风险函数根据有代
表性的数据进行计算，因此并不要求收益率服从正态分布。对于最大风险最小化模型，
是采用了几风险度量函数来度量整个投资组合的风险，而最小收益最大化模型是把最低
收益当作度量风险的指标，本章对前者给出了投资组合选择准则的基础上给出了优化问
题的解析解，对后者分别从收益率服从标准分布与不服从标准分布两种情况跟均值一方
差模型进行了对比。
本文提出的证券投资风险的度量方法比以往的投资度量方法具有更大的优越性，不
仅不必计算协方差矩阵，而且建立的模型不局限于投资收益服从对称的概率分布(如正
态分布)，所以更能反映证券投资风险的本质属性，它是现有投资风险度量方法研究的
继续，将帮助人们更深刻的认识证券投资风险的本质，并且进一步提高风险度量的准确
性。本文的研究成果对证券投资具有一定的指导意义。
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 investment
return and β. As the research on risk measure deepens on, it has been found that
there 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 analyzed
In 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. A
simple scheme has been derived, which generates the efficient portfolio under the la
model 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.
Markowitz以证券收益率的方差作为组合证券风险的度量，开辟了金融定量分析的时代，在度量风险的基础的上建立了组合投资决策模型，该模型在理论和实际应用中都有重要意义。证券投资风险常用的度量方式主要是投资收益率的方差或β值，但是随着研究的深入，人们发现常用的风险度量指标存在不可回避的重大缺陷，为了克服现有理论的不足，理论界进行了广泛的研究，但是到目前为止，还没有一种广泛有效的度量风险的方法。本文正是在此背景下对上述问题展开研究的。
在本篇论文中，作者首先以证券投资的风险为对象，研究了不同的风险度量模型：分析了各种风险度量方法的不足。鉴于目前的风险度量方法都存在不同程度的缺陷，本文主要提出了4种新的风险度量方法，并在此基础上建立了相关的投资组合优化模型。
第二章从证券投资组合理论的风险度量方面着手，基于熵可以作为不确定性度量指标的内涵，把熵作为方差度量风险的一种有效的补偿，用熵和方差来共同度量风险，从而提出了一个新的风险度量模型：均值-方差-熵模型，并以实例作了说明。用熵和方差共同来度量风险，实际上是从一种最保守的角度出发，从而使厌恶风险的投资者从事投资的时候获得较大的收益，面临较少的风险，这在一定程度上弥补了仅仅用方差度量风险的缺陷。本章里还介绍了有效边界以及有效投资组合的内容，并从几何意义上对新模型进行了有关分析。
第三章从风险的度量出发，引进了一种新的度量方法-熵。在简单介绍了最大熵优化原理以及这个问题的对偶规划的基础上，建立了证券投资组合的均值-熵优化模型，这种方法不必计算协方差矩阵，避免了用方差度量风险计算的复杂性；并且均值-熵模型不依赖于收益率的概率分布是某种特定的分布(如正态分布)，最后将该方法与均值-方差方法进行了对比。
大连理工大学博士学位论文
第四章从最小叉嫡原理与最大嫡原理互补性出发建立了均值一叉嫡优化模型。假设
投资者已经知道一个某个时期的收益率分布，而且期望实际的收益率分布在满足所有约
束的条件下尽量靠近这个先验的概率分布，我们将最小叉嫡原理引进投资组合选择中，
应用叉嫡作为度量投资风险的指标，建立了均值一叉嫡优化模型。
第五章中采用了极大极小函数的分析方法，从投资者的角度出发，运用不同的风险
度量准则建立了两个模型，即:最大风险最小化模型和最小收益最大化模型，并对模型
进行了分析，跟以往的投资组合模型进行了比较，经过比较可以看出，极大极小风险函
数的投资组合模型是把投资者的最大损失降到最低当作目标，从最坏情形中取最优，可
达到较高的收益率，从而保证了投资者的投资收益。同时，极大极小风险函数根据有代
表性的数据进行计算，因此并不要求收益率服从正态分布。对于最大风险最小化模型，
是采用了几风险度量函数来度量整个投资组合的风险，而最小收益最大化模型是把最低
收益当作度量风险的指标，本章对前者给出了投资组合选择准则的基础上给出了优化问
题的解析解，对后者分别从收益率服从标准分布与不服从标准分布两种情况跟均值一方
差模型进行了对比。
本文提出的证券投资风险的度量方法比以往的投资度量方法具有更大的优越性，不
仅不必计算协方差矩阵，而且建立的模型不局限于投资收益服从对称的概率分布(如正
态分布)，所以更能反映证券投资风险的本质属性，它是现有投资风险度量方法研究的
继续，将帮助人们更深刻的认识证券投资风险的本质，并且进一步提高风险度量的准确
性。本文的研究成果对证券投资具有一定的指导意义。
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 investment
return and β. As the research on risk measure deepens on, it has been found that
there 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 analyzed
In 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. A
simple scheme has been derived, which generates the efficient portfolio under the la
model 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.
引文
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